chapter 1 number toolbox

Chapter 1 Number Toolbox

Chapter 2 Introduction to Algebra Chapter 2 Introduction to Algebra

Chapter 3 Decimals

Ch t 4 N b Th d F tiChapter 4 Number Theory and Fractions

Chapter 5 Fraction Operations

Chapter 6 Collect and Display Data

Chapter 7 Plane Geometry

Chapter 8 Ratio, Proportion, and Percent

Chapter 9 Integers

Chapter 10 Perimeter, Area, and Volume

Chapter 11 Probabilityp y

Chapter 12 Functions and Coordinate Geometry

1-1 Comparing and Ordering Whole Numbers

1-2 Estimating with Whole Numbers1-2 Estimating with Whole Numbers

1-3 Exponents

1 4 O d f O ti1-4 Order of Operations

1-5 Mental Math

1-6 Choose the Method of Computation

1-7 Find a Pattern

2-1 Variables and Expressions

2-2 Translate Between Words and Math2-2 Translate Between Words and Math

2-3 Equations and Their Solutions

2 4 S l i Additi E ti2-4 Solving Addition Equations

2-5 Solving Subtraction Equations

2-6 Solving Multiplication Equations

2-7 Solving Division Equations

3-1 Representing, Comparing, and Ordering Decimals

3-2 Estimating Decimals3-2 Estimating Decimals

3-3 Adding and Subtracting Decimals

3 4 D i l d M t i M t3-4 Decimals and Metric Measurement

3-5 Scientific Notation

3-6 Multiplying Decimals

3-7 Dividing Decimals by Whole Numbers

3-8 Dividing by Decimals

3-9 Interpret the Quotient

3-10 Solving Decimal Equations

4-1 Divisibility

4-2 Factors and Prime Factorization4-2 Factors and Prime Factorization

4-3 Greatest Common Factor

4 4 D i l d F ti4-4 Decimals and Fractions

4-5 Equivalent Fractions

4-6 Comparing and Ordering Fractions

4-7 Mixed Number and Improper Fractions

4-8 Adding and Subtracting with Like Denominators

4-9 Multiplying Fractions by Whole Numbers

5-1 Multiplying Fractions

5-2 Multiplying Mixed Numbers5-2 Multiplying Mixed Numbers

5-3 Dividing Fractions and Mixed Numbers

5 4 S l i F ti E ti M lti li ti d Di i i5-4 Solving Fraction Equations: Multiplication and Division

5-5 Least Common Multiple

5-6 Estimating Fraction Sums and Differences

5-7 Adding and Subtracting with Unlike Denominators

5-8 Adding and Subtracting Mixed Numbers

5-9 Renaming to Subtract Mixed Numbers

5-10 Solving Fraction Equations: Addition and Subtraction

6-1 Make a Table

6-2 Range Mean Median and Mode6-2 Range, Mean, Median, and Mode

6-3 Additional Data and Outliers

6 4 B G h6-4 Bar Graphs

6-5 Frequency Tables and Histograms

6-6 Ordered Pairs

6-7 Line Graphs

6-8 Misleading Graphs

6-9 Stem-and-Leaf Plots

7-1 Points, Lines, and Planes

7-2 Angles7-2 Angles

7-3 Angle Relationships

7 4 Cl if i Li7-4 Classifying Lines

7-5 Triangles

7-6 Quadrilaterals

7-7 Polygons

7-8 Geometric Patterns

7-9 Congruence

7-10 Transformations

7-11 Symmetryy y

7-12 Tessellations

8-1 Ratios and Rates

8-2 Proportions8-2 Proportions

8-3 Proportions and Customary Measurement

8 4 Si il Fi8-4 Similar Figures

8-5 Indirect Measurement

8-6 Scale Drawings and Maps

8-7 Percents

8-8 Percents, Decimals, and Fractions

8-9 Percent Problems

8-10 Using Percents

9-1 Understanding Integers

9-2 Comparing and Ordering Integers9-2 Comparing and Ordering Integers

9-3 The Coordinate Plane

9 4 Addi I t 9-4 Adding Integers

9-5 Subtracting Integers

9-6 Multiplying Integers

9-7 Dividing Integers

9-8 Solving Integer Equations

10-1 Finding Perimeter

10-2 Estimating and Finding Area10-2 Estimating and Finding Area

10-3 Break into Simpler Parts

10 4 C i P i t d A10-4 Comparing Perimeter and Area

10-5 Circles

10-6 Solid Figures

10-7 Surface Area

10-8 Finding Volume

10-9 Volume of Cylinders

11-1 Introduction to Probability

11-2 Experimental Probability11-2 Experimental Probability

11-3 Theoretical Probability

11 4 M k O i d Li t11-4 Make an Organized List

11-5 Compound Events

11-6 Making Predictions

12-1 Tables and Functions

12-2 Graphing Functions12-2 Graphing Functions

12-3 Graphing Translations

12 4 G hi R fl ti12-4 Graphing Reflections

12-5 Graphing Rotations

12-6 Stretching and Shrinking


Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day


Course 1


Warm UpWarm UpCompare. Use <, >, or =.

1 8 9 2 27 14< >1. 8 9 2. 27 143. 56 23 4. 10 155. 11 12 6. 37 16

< >> << >5. 11 12 6. 37 16< >

Course 1



Subtract your age from your age multiplied by 100 Divide the result by multiplied by 100. Divide the result by 11, and then divide the quotient by 9. What number do you get?What number do you get?

The answer will be the student’s age.

Course 1


Learn to compare and order whole pnumbers using place value or a number line.

Course 1


Course 1

1-1 Comparing and Ordering Whole NumbersAdditi l E l 1 U i Pl V l t Additional Example 1: Using Place Value to

Compare Whole Numbers

Belize’s 2000 population was 249 183 people Belize s 2000 population was 249,183 people. Iceland’s 2000 population was 276,365 people. Which country had more people?

249,183 BelizeStart at the left and compare digits in the same place value position Look for the first

276,365 Icelandposition. Look for the first place where the values are different.

40 thousand is less than 70 thousand.

249,183 is less than 276,365.

Course 1

Iceland had more people.


T Thi E l 1Try This: Example 1

In 2000, the population of San Diego, California was 1 223 400 people In 2000 the population was 1,223,400 people. In 2000, the population of Dallas, Texas was 1,188,580 people. Which city had more people?

Start at the left and compare 1,223,400 San Diego

1 188 580 Dallas

Start at the left and compare digits in the same place value position. Look for the first

1,188,580 Dallas

200 th d i t th 100 th d

place where the values are different.

200 thousand is greater than 100 thousand.

1,223,400 is greater than 1,188,580.

Course 1

San Diego had more people.


To order numbers, you can compare them , y pusing place value and then write them in order from least to greatest. You can also

h h b b lgraph the numbers on a number line. As you read the numbers from left to right, they will be ordered from least to greatestthey will be ordered from least to greatest.

Course 1


Remember!< means

“is less than.”3 < 5 120 < 504

> means “is greater than.”

17 > 9 212 > 83

Course 1

1-1 Comparing and Ordering Whole NumbersAdditional E ample 2 Using a N mber Line Additional Example 2: Using a Number Line

to Order Whole NumbersOrder the numbers from least to greatest:

Graph the numbers on a number line:The number 675 is between 600 and 700

675; 1,044; 497

The number 675 is between 600 and 700.The number 1,044 is between 1,000 and 1,100.The number 497 is between 400 and 500The number 497 is between 400 and 500.

400 600 800 1,000

497 675 1,044

The numbers are ordered when you read the number line from left to right.The numbers in order from least to greatest are

Course 1

The numbers in order from least to greatest are 497, 675, and 1,044.


Try This: Example 2

Order the numbers from least to greatest: 732 923 502Graph the numbers on a number line:The number 732 is between 700 and 800

732; 923; 502

The number 732 is between 700 and 800.The number 923 is between 900 and 1,000.The number 502 is between 500 and 600The number 502 is between 500 and 600.

400 600 800 1,000

502 732 923

The numbers are ordered when you read the number line from left to right.The numbers in order from least to greatest are

Course 1

The numbers in order from least to greatest are 502, 732, and 923.

Insert Lesson Title Here1-1 Comparing and Ordering Whole Numbers

Lesson QuizCompare. Write < or >.

1. 47,328 47,238

2. 933,826 933,520 >

>

, ,

3. Write the numbers in order from least to greatest: 726 847 221 221 726 847greatest: 726, 847, 221.

4. The are of Panama is 78,200 square kil d h f Li h i i 65 200

221, 726, 847

kilometers, and the area of Lithuania is 65,200 square kilometers. Which country is smaller?Lith iLithuania

Course 1

1-2 Estimating with Whole Numbers

Warm UpWarm Up




Course 1


Warm UpWarm UpFind each sum.

1 3 214 + 5 4901. 3,214 + 5,4902. 9,225 + 8,652

8,70417,877

3. 3,210 + 1,2004. 8,774 + 2,156

4,41010,9300,930

Course 1



Continue the number pattern below. Explain the pattern you foundExplain the pattern you found.3, 6, 10, 15, ___, ___

21, 28; One possible pattern is to increase the difference between consecutive terms b th th diff b t by one more than the difference between preceding consecutive terms.

Course 1


Learn to estimate with whole numbers.

Course 1

Insert Lesson Title Here1-2 Estimating with Whole Numbers

Vocabularycompatible numberunderestimateoverestimate

Course 1


Sometimes in math you do not need an t I t d exact answer. Instead, you can use an

estimate. Estimates are close to the exact answer but are usually easier and faster to answer but are usually easier and faster to find.

When estimating, you can round the numbers in the problem to compatible numbers Compatible numbers are close numbers. Compatible numbers are close to the numbers in the problem, and they can help you do math mentally.

Course 1

can help you do math mentally.


Remember!When rounding, look at the digit to the right of the place to which you are

Remember!

right of the place to which you are rounding.

If that digit is 5 or greater round up• If that digit is 5 or greater, round up.

• If that digit is less than 5, round down.

Course 1


Additional Example 1A: Estimating a Sum or Difference by Rounding

E i h b di h l Estimate the sum by rounding to the place value indicated.

A 12 345 + 62 167;A. 12,345 + 62,167;ten thousands

10 000 Round 12 345 down10,000 Round 12,345 down.Round 62,167 down.+ 60,000__________

70 000

The sum is about 70,000.

70,000

Course 1

,


Additional Example 1B: Estimating a Sum or Difference by Rounding

E i h diff b di h Estimate the difference by rounding to the place value indicated.

B 4 983 2 447;B. 4,983 – 2,447;thousands

5 000 Round 4 983 up5,000 Round 4,983 up.Round 2,447 down.– 2,000__________

3 000

The difference is about 3,000.

3,000

Course 1

,


Try This: Example 1A

Estimate the sum by rounding to the place Estimate the sum by rounding to the place value indicated.

A. 13,235 + 41,139;, , ;

ten thousands

10 000 Round 13 235 down10,000 Round 13,235 down.Round 41,139 down.+ 40,000__________

50 000

The sum is about 50,000.

50,000

Course 1

,


Try This: Example 1B

Estimate the difference by rounding to the y gplace value indicated.

B. 5,723 – 1,393;

thousands

6 000 Round 5 723 up6,000 Round 5,723 up.Round 1,393 down.– 1,000__________

5 000

The difference is about 5,000.

5,000

Course 1

,


An estimate that is less than the exact answer is an underestimate.

An estimate that is greater than the exact answer is an overestimate.

Course 1


Remember!The area of a rectangle is found by multiplying the length by the width.

Remember!

g y

A = l w

l

w

Course 1

Insert Lesson Title Here1-2 Estimating with Whole NumbersAdditi l E l 2 E ti ti P d t b Additional Example 2: Estimating a Product by

Rounding

The sixth grade class wants to paint a wall The sixth-grade class wants to paint a wall with white paint. The wall is a rectangle 9 feet tall and 18 feet wide. One quart of paint will cover 100 square feet. How many quarts of paint should the students buy?

9 18 9 20 Overestimate the area of the wall9 18 9 20 Overestimate the area of the wall.The actual area is less than 180 square feet.

9 20 = 180square feet.

If one quart of paint will cover 100 square feet, then 2 quarts will cover 200 square feet. The

Course 1

q qstudents should buy 2 quarts of paint.


Try This: Example 2

The seventh-grade class wants to paint a wall g pwith blue paint. The wall is a rectangle 9 feet tall and 29 feet wide. One quart of paint will cover 100 square feet How many quarts of cover 100 square feet. How many quarts of paint should the students buy?

9 29 9 30 Overestimate the area of the wall9 29 9 30 Overestimate the area of the wall.The actual area is less than 270 square feet.

9 30 = 270square feet.

If one quart of paint will cover 100 square feet, then 3 quarts will cover 300 square feet. The

Course 1

q qstudents should buy 3 quarts of paint.

Insert Lesson Title Here1-2 Estimating with Whole NumbersAdditi l E l 3 E ti ti Q ti t Additional Example 3: Estimating a Quotient

Using Compatible Numbers

Mr Dehmel will drive 243 miles to the fair at Mr. Dehmel will drive 243 miles to the fair at 65 mi/h. About how long will his trip take?

240 and 60 are compatible 243 ÷ 65 240 ÷ 60

240 and 60 are compatible numbers. Underestimatethe speed.

Because he underestimated the speed, the actual time will be less than 4 hours

240 ÷ 60 = 4will be less than 4 hours..

The trip will take about 4 hours.

Course 1



Try This: Example 3

Mrs. Blair will drive 103 miles to the airport at p55 mi/h. About how long will her trip take?

100 and 50 are compatible 103 ÷ 55 100 ÷ 50

100 and 50 are compatible numbers. Underestimatethe speed.

Because she underestimated the speed, the actual time will be less than 2 hours

100 ÷ 50 = 2will be less than 2 hours..


Course 1


L Q i

Insert Lesson Title Here1-2 Estimating with Whole NumbersLesson Quiz

Estimate each sum or difference by rounding to the place value indicated.to the place value indicated.

1. 7,420 + 3,527; thousands 11,000

2. 47,821 + 19,925; ten thousands

3. 8,254 – 5,703; thousands

70,000

2,000, , ;

4. 66,845 – 24,782; ten thousands

,000

50,000

5. One quart of paint covers an area of 100 square feet. How many quarts are needed to paint a wall 8 f t t ll d 19 f t id ?feet tall and 19 feet wide?

Course 1

2

1-3 Exponents

Warm UpWarm Up




Course 1

1-3 Exponents

Warm UpWarm UpMultiply.

1 3 3 31. 3 3 32. 4 4 4

2764

3. 2 2 2 24. 5 5 5 5

16625625

Course 1

1-3 Exponents


Replace the letters a, b, and c with the numbers 3 4 and 5 to make a true numbers 3, 4, and 5 to make a true statement.2a + 2a = bc 2a + 2a = bc

25+ 25 = 43

Course 1

1-3 Exponents

Learn to represent numbers by using p y gexponents.

Course 1

Insert Lesson Title Here1-3 Exponents

Vocabularyexponentbaseexponential form

Course 1

1-3 Exponents

An exponent tells how many times a number called the base is used as a factor.

A number is in exponential formwhen it is written with a base and when it is written with a base and an exponent.

7733BaseExponent

= 7 7 7= 343

Course 1

77 = 7 7 7= 343

1-3 Exponents

Exponential Form 101

Read “10 to the 1st power”Multiply 10Value 10

Course 1

1-3 Exponents


Read“10 squared” or “10 to the 2nd power”

M l i l 10 10Multiply 10 10Value 100

Course 1

1-3 Exponents


Read“10 cubed” or “10 to the 3rd power”

M l i l 10 10 10Multiply 10 10 10Value 1,000

Course 1

1-3 Exponents


Read “10 to the 4th power”Multiply 10 10 10 10Value 10,000

Course 1


Additional Example 1: Writing Numbers in Exponential Form

Write each expression in exponential form.

A. 5 5 5 5

5 is a factor 4 times.54

B. 3 3 3 3 3

35 3 is a factor 5 times.35

Course 1


Try This: Example 1

Write each expression in exponential form.Write each expression in exponential form.

A. 7 7 7

7 is a factor 3 times73 7 is a factor 3 times.73

B. 6 6 6 6 6 6

66 6 is a factor 6 times.66

Course 1


Additional Example 2: Finding the Value of Numbers in Exponential Form

Find each value.

A. 26

26 = 2 2 2 2 2 2= 64

B. 45

45 4 4 4 4 4 45 = 4 4 4 4 4 = 1,024

Course 1


Try This: Example 2

Find each value.

A. 34

34 = 3 3 3 3 = 81

B. 25

25 2 2 2 2 2 25 = 2 2 2 2 2 = 32

Course 1

1-3 Exponents

A phone tree is used to contact families at P l’ h l Th t ll 4 f ili

Additional Example 3: Problem Solving Application

Paul’s school. The secretary calls 4 families. Then each family calls 4 other families, and so on. How many families will notified during the y gfourth round of calls?

11 Understand the Problem11 Understand the ProblemThe answer will be the number of families called in the 4th round.

List the important information:

• The secretary calls 4 families.

Course 1

The secretary calls 4 families.

• Each family calls 4 families.

1-3 Exponents

You can draw a diagram to see how many

22 Make a Plan

g ycalls are in each round.

SecretarySecretary

1st round – 4 calls1st round 4 calls

2nd round–16 calls

Course 1

1-3 Exponents

Solve33Notice that in each round, the number of calls is a power of 4is a power of 4.

1st round: 4 calls = 4 = 41

2nd round: 16 calls = 4 x 4 = 42

So during the 4th round, there will be 44 calls. 44 4 4 4 4 25644 = 4 4 4 4 = 256

During the 4th round of calls, 256 families will have been notifiedhave been notified.

Look Back44Drawing a diagram helps you see how to use

Course 1

Drawing a diagram helps you see how to use exponents to solve the problem.

1-3 ExponentsTr This E ample 3Try This: Example 3

A phone tree is used to contact families at Paul’s school The secretary calls 3 families Paul s school. The secretary calls 3 families. Then each family calls 3 other families, and so on. How many families will notified during

f fthe fourth round of calls?

11 Understand the Problem

The answer will be the number of families called in the 3rd round.


• The secretary calls 3 families.

Course 1

The secretary calls 3 families.

• Each family calls 3 families.

1-3 Exponents

You can draw a diagram to see how many

22 Make a Plan

g ycalls are in each round.

SecretarySecretary

1st round – 3 calls1st round 3 calls

2nd round–9 calls

Course 1

1-3 Exponents

Solve33Notice that in each round, the number of calls is a power of 3is a power of 3.

1st round: 3 calls = 3 = 31

2nd round: 9 calls = 3 x 3 = 32

So during the 4th round, there will be 34 calls. 34 3 3 3 3 8134 = 3 3 3 3 = 81

During the 4th round of calls, 81 families will have been notifiedhave been notified.

Look Back44Drawing a diagram helps you see how to use

Course 1

Drawing a diagram helps you see how to use exponents to solve the problem.

L Q i

Insert Lesson Title Here1-3 ExponentsLesson Quiz

Write each expression in exponential form.

1. 12 12 12

2. 9 9 9 9 9 9 9 97

123

2. 9 9 9 9 9 9 9

Find each value.

9

3. 202 4. 64

5. In a phone tree, each of three people will call

400 1,296

p , p pthree people, and then each of those will call three more. If there are five levels of the tree, how many people will be called?

Course 1

243

1-4 Order of Operations

Warm UpWarm Up




Course 1


Warm UpWarm UpPerform the operations in order from left to right.

1. 8 + 4 – 22 9 3 + 1

10282. 9 3 + 1

3. 7 – 3 + 54 20 ÷ 4 + 6

289114. 20 ÷ 4 + 6 11

Course 1



0 1 2 3 4 5 6 7 8 9 = 1P t th i t l i i Put the appropriate plus or minus signs between the numbers so that the total equals 1equals 1.

0 + 1 – 23 + 45 + 67 – 89 = 1

Course 1


Learn to use the order of operations.p

Course 1

Insert Lesson Title Here1-4 Order of Operations

Vocabularynumerical expressionsevaluateorder of operations

Course 1


A numerical expression is a mathematical phase that includes only numbers and operation phase that includes only numbers and operation symbols.

Numerical Expressions 4 + 8 ÷ 2 6 371 – 203 + 2 5,006 19

When you evaluate a numerical expression, you find its value.

Course 1


When an expression has more than one operation, you must know which operation to do first. To make sure that everyone gets the same answer, we use the order of operations.

ORDER OF OPERATIONS1. Perform operations in parentheses.1. Perform operations in parentheses.2. Find the values of numbers with exponents.3. Multiply or divide from left to right as ordered in the problem.4. Add or subtract from left to right as ordered in the problem

Course 1

in the problem.


The first letters of these words Helpful Hint

can help you remember the order of operations.

Pl P thPlease Parentheses

Excuse Exponents

My Multiply

Dear Divide

Aunt Add

Sally Subtract

Course 1

y


Additional Example 1A: Using the Order of Operations

Evaluate each expression.

A. 15 – 10 ÷ 2 There are no parentheses or exponents.

15 – 5 Divide.

10 Subtract.10 Subtract.

Course 1

1-4 Order of OperationsAdditi l E l 1B U i th O d f Additional Example 1B: Using the Order of

Operations

B. 8 + 14 ÷ 2 + 6 33There are no parentheses. Find the value of the number with exponents.8 + 14 ÷ 2 + 6 27Divide.8 + 7 + 6 27

Multiply.8 + 7 + 162

Add15 + 162

177

Add.15 + 162

Add.

Course 1


Additional Example 1C: Using the Order of Operations

C. 3 (4 + 7) + 22Perform operations within parentheses 3 11 22 parentheses.

3 11 + 4

3 11 + 22

Find the value of the number with exponents

Multiply.33 + 4

number with exponents.

37 Add.

Course 1



Evaluate each expressionEvaluate each expression.

There are no parentheses or A. 12 – 6 ÷ 2 There are no parentheses or exponents.

12 – 3 Divide.

9 Subtract.9 Subtract.

Course 1



B. 4 + 14 ÷ 2 + 4 23 There are no parentheses. Find the value of the 4 10 2 4 8 Find the value of the number with exponents.

4 + 10 ÷ 2 + 4 8

Divide.4 + 5 + 4 8

Multiply.4 + 5 + 32

Add9 + 32

41

Add.9 + 32

Add.

Course 1


Try This: Example 1C

2C. 2 (3 + 6) + 42

Perform operations within parentheses 2 9 42 parentheses.

2 9 + 16

2 9 + 42

Find the value of the number with exponents

Multiply.18 + 16

number with exponents.

34 Add.

Course 1


Additional Example 2: Consumer Application

Mr. Kellett bought 6 used CDs for $4 each and 5 used CDs for $3 each. Evaluate the following expression to find the amount Mr. Kellett spent on CDs.Kellett spent on CDs.

6 4 + 5 3

24 + 15

39

Mr. Kellett spent $39 on CDs.

Course 1


Try This: Example 2

Ms. Nivia bought 4 new CDs for $8 each and 6 used CDs for $4 each. Evaluate the following expression to find the amount Ms. Nivia spent on CDs.on CDs.

4 8 + 6 432 + 2432 + 24

56

Ms. Nivia spent $56 on CDs.

Course 1

L Q i

Insert Lesson Title Here1-4 Order of OperationsLesson Quiz

Evaluate each expression.1 15 + 4 2 231. 15 + 4 2

2. (12 – 5)2 – 10 39

23

( )

3. 3 + 9 2 – 5 16

4. 43 – 30 ÷ 2 49

5. Chaz bought 4 football cards for $2 each and 8 baseball cards for $3 each. Evaluate the expression to find the amount Chaz spent on p pcards: 4 2 + 8 3.

Course 1

$32

1-5 Mental Math

Warm UpWarm Up




Course 1

1-5 Mental Math

Warm UpWarm UpFind each sum or product.

1 17 + 151. 17 + 152. 29 + 39

3268

3. 8(24)4. 7(12)

19284

5. 3(91)6. 6(15)

827390( )

Course 1

90

1-5 Mental Math

Problem of the Day

Determine the secret number from the following clues:• The number is a multiple of 5.• It is divisible by 3.• It is less than 200.• Its tens digit equals the sum of its g qother two digits.165

Course 1

1-5 Mental Math

Learn to use number properties to p pcompute mentally.

Course 1

Insert Lesson Title Here1-5 Mental Math

VocabularyCommutative PropertyAssociative Propertyp yDistributive Property

Course 1

1-5 Mental Math

l h “d hMental math means “doing math in your head.”

Course 1

1-5 Mental Math

COMMUTATIVE PROPERTY (Ordering)Words NumbersWords Numbers

You can add or multiply numbers in

18 + 9 = 9 + 18multiply numbers in any order. 15 2 = 2 15

Course 1

1-5 Mental Math

ASSOCIATIVE PROPERTY (Grouping)ASSOCIATIVE PROPERTY (Grouping)Words Numbers

Wh l When you are only adding or only multiplying, you (17 + 2) + 9 = 17 + (2 + 9) multiplying, you can group any of the numbers

h

( ) ( )(12 2) 4 = 12 (2 4)

together.

Course 1

1-5 Mental MathAdditi l E l 1A U i P ti t Additional Example 1A: Using Properties to

Add and Multiply Whole Numbers

A Evaluate 17 + 5 + 3 + 15A. Evaluate 17 + 5 + 3 + 15.

17 + 5 + 3 + 15 Look for sums that are multiples of 10multiples of 10

Use the Commutative Property. 17 + 3 + 5 + 15

(17 + 3) + (5 + 15)Use the Associative Property to make groups of compatible numbers

40 Use mental math to add.

20 + 20 of compatible numbers.

Course 1

1-5 Mental MathAdditional Example 1B: Using Properties to Additional Example 1B: Using Properties to

Add and Multiply Whole Numbers

B E l t 4 13 5B. Evaluate 4 13 5.

4 13 5 Look for products that are multiples of 10multiples of 10

Use the Commutative Property.13 4 5

13 (4 5)

3 20

Use the Associative Property to group compatible numbers.

260

Use mental math to multiply.13 20

Course 1

1-5 Mental Math


A. Evaluate 12 + 5 + 8 + 5.A. Evaluate 12 + 5 + 8 + 5.

12 + 5 + 8 + 5 Look for sums that are multiples of 10multiples of 10

Use the Commutative Property.12 + 8 + 5 + 5

(12 + 8) + (5 + 5)Use the Associative Property to make groups of compatible numbers

30 Use mental math to add.

20 + 10 numbers.

Course 1

1-5 Mental Math


B. Evaluate 8 3 5.

8 3 5 Look for products that are multiples of 10multiples of 10

Use the Commutative Property.3 8 5

3 (8 5) Use the Associative Property to group compatible numbers.

120

Use mental math to multiply.3 40

Course 1

1-5 Mental Math

DISTRIBUTIVE PROPERTYWords NumbersWords Numbers

When you multiply a number times a sum 6 (10 + 4) = 6 14number times a sum, you can• find the sum first

6 (10 + 4) = 6 14= 84

6 (10 + 4) = (6 10) + (6 4)and then multiply, or• multiply by each number in the sum

( ) ( ) ( )= 60 + 24= 84

number in the sum and then add.

Course 1

1-5 Mental Math

When you multiply two numbers, you can y p y , y“break apart” one of the numbers into a sum and then use the Distributive Property.

Helpful HintBreak the greater factor into a sum that contains a multiple of 10 and a one-digit number. You can add

Helpful Hint

and multiply these numbers mentally.

Course 1

1-5 Mental MathAdditional Example 2A: Using the Distributive Additional Example 2A: Using the Distributive

Property to Multiply

Use the Distributive Property to find the productUse the Distributive Property to find the product.

A. Evaluate 6 35.

6 35 = 6 (30 + 5) “Break apart” 35 into 30 + 5.

Use the Distributive Property.= (6 30) + (6 5) Use the Distributive Property. (6 30) + (6 5)

= 180 + 30 Use mental math to multiply.= 210 Use mental math to add.

Course 1

1-5 Mental MathAdditi l E l 2B U i th Di t ib ti Additional Example 2B: Using the Distributive

Property to Multiply

Use the Distributive Property to find the Use the Distributive Property to find the product.

B Evaluate 9 x 87B. Evaluate 9 x 87.

9 87 = 9 (80 + 7) “Break apart” 87 into 80 + 7.

h i ib i (9 80) (9 7) Use the Distributive Property.= (9 80) + (9 7)

= 720 + 63 Use mental math to multiply.

= 783 Use mental math to add.

Course 1

1-5 Mental Math


Use the Distributive Property to find the p yproduct.

A. Evaluate 4 x 27.

4 27 = 4 (20 + 7) “Break apart” 27 into 20 + 7.




Course 1

1-5 Mental Math


Use the Distributive Property to find the p yproduct.

B. Evaluate 6 43.

6 43 = 6 (40 + 3) “Break apart” 43 into 40 + 3.




Course 1

Insert Lesson Title Here1-5 Mental Math

Lesson QuizEvaluate.

1. 18 + 24 + 2 + 6 2. 10 5 3

3. 13 + 42 + 7 + 8

15050

703. 13 + 42 + 7 + 8

Use the Distributive Property to find each product

70

product.

4. 8 12 5. 6 1596 90

6. Angie wants to buy 3 new video games. How much will she need to save if each game costs g$27?

Course 1

$81


Warm UpWarm Up




Course 1


Warm UpWarm UpUse mental math to find each fraction of 80.

1. 2.40 8110__1

2

3. 4.2010

1614

15

Course 1



About 23% of 600 firefighters are not on duty on any particular day Steven on duty on any particular day. Steven estimates that about 200 firefighters have the day off. Is his estimate too have the day off. Is his estimate too high or too low? Explain.

too high; 25% of 600 is 150too high; 25% of 600 is 150

Course 1


Learn to choose an appropriate method of pp pcomputation and justify your choice.

Course 1


Additional Example 1

Find the sum: 4 + 3 + 2 + 10 + 8 + 2 + 5 + 1.

There are probably too many numbers to add mentally, but the numbers are small. You can y,use paper and pencil.35

Course 1


Try This: Example 1

Find the sum: 3 + 7 + 2 + 18 + 7 + 1 + 3 + 1.

It might be hard to keep track of all of these numbers if you tried to add mentally. But the y ynumbers themselves are small. You can use paper and pencil.4242

Course 1



Find the difference: 4,562 – 397.,

397 is close to a multiple of 100.

You can use mental mathYou can use mental math.

(4,562 + 3) – (397 + 3)Think: Add 3 to 397 to make 400. Add 3 to 4 562 to

4,565 – 400

4,165

to 4,562 to compensate.

,

Course 1


Try This: Example 2

Find the difference: 3,442 – 298.,

298 is close to a multiple of 100.

You can use mental math Think: Add 2 to 298 You can use mental math.

(3,442 + 2) – (298 + 2)

Think: Add 2 to 298 to make 300. Add 2 to 3,442 to

3,444 – 300

3,144

compensate.

,

Course 1



Find the quotient: 9,288 ÷ 24.q ,

The numbers are not compatible, so mental math is not a good idea. You could use paper g p pand pencil, but long division requires several steps. Using a calculator will probably be fasterfaster.

9,288 ÷ 24 = 387

Course 1


Try This: Example 3

Find the quotient: 9,248 ÷ 32.q ,

The numbers are not compatible, so mental math is not a good idea. You could use paper and pencil, but long division requires several steps. Using a calculator will probably be faster.faster.

9,248 ÷ 32 = 289

Course 1

L Q i

Insert Lesson Title Here1-6 Choose the Method of Computation

Lesson Quiz

Evaluate the expression and state the method f t ti dof computation you used.

1. 17 + 6 + 24 + 35 + 3 + 5 90; mental math

2. 63 197

3 It takes Jupiter approximately 4 344 days to

12,411; paper and pencil

3. It takes Jupiter approximately 4,344 days to complete one revolution around the Sun. It takes Earth 365 days to revolve around the Sun How Earth 365 days to revolve around the Sun. How many more days does it take Jupiter to revolve around the Sun than Earth? 3,979 days3,979 days

Course 1

1-7 Find a Pattern

Warm UpWarm Up




Course 1

1-7 Find a Pattern

Warm UpWarm UpDetermine what could come next.

1 3 4 5 6 71. 3, 4, 5, 6, ___2. 10, 9, 8, 7, 6, ___3. 1, 3, 5, 7,

7593. 1, 3, 5, 7, ___

4. 2, 4, 6, 8, ___5. 5, 10, 15, 20, ___

91025

Course 1

1-7 Find a Pattern

Problem of the Day

How can you place the numbers 1 through 6 in the circles so that the

l h d lsums along each side are equal?

612

6

534

Course 1

1-7 Find a Pattern

Learn to find patterns and to recognize, p gdescribe, and extend patterns in sequences.

Course 1

Insert Lesson Title Here1-7 Find a Pattern

Vocabularyperfect squaresequenceqterm

Course 1

1-7 Find a Pattern

Whole numbers raised to the second power are called perfect squares. This is because they can be represented by objects arranged in the shape be represented by objects arranged in the shape of a square.

1 4 9 16

Course 1

1-7 Find a Pattern

The perfect squares can be written as a sequence. A sequence is an ordered set of numbers. Each number in the sequence is called a term. In a sequence, there is often a pattern between one term and the next.term and the next.

Course 1

1-7 Find a Pattern

1 4 9 161 t 2 d 3 d 4th

Sequence1st term 2nd term 3rd term 4th term Term

+3 +5 +73 5 7

You can use this pattern to find the fifth and sixth terms in the sequence. To get the fifth term, add 9. te s t e seque ce o get t e t te , add 9To get the sixth term, add 11.

1, 4, 9, 16, , , . . . 1, 4, 9, 16, , , . . .

16 + 9 = 25 25 + 11 = 36

Course 1

So the next two perfect squares are 25 and 36.

1-7 Find a Pattern

Look for a relationship between the 1st term and Helpful HintLook for a relationship between the 1st term and the 2nd term. Check if this relationship works between the 2nd term and the 3rd term, and so on.

Course 1

1-7 Find a PatternAdditional Example 1A: Extending Sequences

with Addition and Subtraction

Id tif tt i h d Identify a pattern in each sequence and name the next three terms.

A 48 42 36 30 A. 48, 42, 36, 30, , , , . . .

–6 –6 –6 –6 –6 –6

A pattern is to subtract 6 from each term to get the next term.30 – 6 = 24 24 – 6 = 18 18 – 6 = 12

So 24, 18, and 12 will be the next three terms.

Course 1

, ,

1-7 Find a PatternAdditional Example 1B: Extending Sequences

with Addition and Subtraction

Id tif tt i h d Identify a pattern in each sequence and name the next three terms.

B 24 34 31 41 38 48 B. 24, 34, 31, 41, 38, 48, , , , . . .

+10 –3 +10 –3 +10 –3 +10 –3

A pattern is to add 10 to one term and subtract 3 from the next.48 – 3 = 45 45 + 10 = 55 55 – 3 = 52


Course 1

, ,

1-7 Find a PatternTry This: Example 1A

Identify a pattern in each sequence and name Identify a pattern in each sequence and name the next three terms.

A 39 34 29 24 A. 39, 34, 29, 24, , , , . . .

–5 –5 –5 –5 –5 –5

A pattern is to subtract 5 from each term to get the next term.24 – 5 = 19 19 – 5 = 14 14 – 5 = 9


Course 1

, ,

1-7 Find a Pattern


Identify a pattern in each sequence and name Identify a pattern in each sequence and name the next three terms.

B 12 23 17 28 22 33 B. 12, 23, 17, 28, 22, 33, , , , . . .

+11 –6 +11 –6 +11 –6 +11 –6

A pattern is to add 11 to one term and subtract 6 from the next.33 – 6 = 27 27 + 11 = 38 38 – 6 = 32


Course 1

, ,

1-7 Find a PatternAdditional Example 2A: Completing

Sequences with Multiplication and Division

Id tif tt i h d Identify a pattern in each sequence and name the missing terms.

A 2 6 18 162 A. 2, 6, 18, , 162, ,…

3 3 3 3 3

A pattern is to multiply each term by 3.

18 3 54 162 3 48618 3 = 54 162 3 = 486

So 54 and 486 are the missing terms.

Course 1

g

1-7 Find a PatternAdditi l E l 2B C l ti S Additional Example 2B: Completing Sequences

with Multiplication and Division

Identify a pattern in each sequence and name Identify a pattern in each sequence and name the missing terms.

A 12 6 24 12 48 96 48 96 A. 12, 6, 24, 12, 48, , 96, 48, , 96, . . .

÷2 4 ÷2 4 ÷2 4 ÷2 4 ÷2 A pattern is to divide one term by 2 and multiply the next by 4.

48 ÷ 2 = 24 48 4 = 192


Course 1

g

1-7 Find a PatternTry This: Example 2A

Identify a pattern in each sequence and name th i i tthe missing terms.

A 6 12 24 96 A. 6, 12, 24, , 96, , . . .

2 2 2 2 2

A pattern is to multiply each term by 2.

24 2 = 48 96 2 = 192


Course 1

g

1-7 Find a PatternTry This: Example 2B

Identify a pattern in each sequence and name th i i tthe missing terms.

A 8 2 16 4 32 64 16 A. 8, 2, 16, 4, 32, , 64, 16, ,…

÷4 8 ÷4 8 ÷4 8 ÷4 8

A pattern is to divide one term by 4 and multiply the next by 8.

32 ÷ 4 = 8 16 8 = 128


Course 1

g

Insert Lesson Title Here1-7 Find a Pattern

Lesson Quiz

Identify a pattern in each sequence, and name the next three terms.

1. 12, 24, 36, 48, , , , … add 12; 60, 72, 84

2. 75, 71, 67, 63, , , ,…

Id tif tt i h d

subtract 4; 59, 55, 51

; , ,

Identify a pattern in each sequence, and name the missing terms.

3. 1000, 500, , 125,…

4. 100, 50, 200, , 400, ,…

divide by 2; 250

divide by 2 then , , , , , ,multiply by 4; 100, 200

Course 1

Course 1

Warm Up

Lesson Presentation

Problem of the Day


Course 1

Warm Up Simplify. 1. 4 + 7 × 3 − 1 2. 87 − 15 ÷ 5 3. 6(9 + 2) + 7 4. 35 ÷ 7 × 5

24 84 73 25


Course 1

Problem of the Day

How can the digits 1 through 5 be arranged in the boxes to make the greatest product?

× 4 3 1

5 2


Course 1

Learn to identify and evaluate expressions.


Course 1

Vocabulary variable constant algebraic expression


Course 1

A variable is a letter or symbol that represents a quantity that can change.

A constant is a quantity that does not change.


Course 1

An algebraic expression contains one or more variables and may contain operation symbols. p × 7 is an algebraic expression.

Algebraic Expressions NOT Algebraic Expressions

150 + y 85 ÷ 5

35 × w + z 10 + 3 × 5

To evaluate an algebraic expression, substitute a number for the variable and then find the value.


Course 1

Evaluate the expression to find the missing values in the table.

Additional Example 1A: Evaluating Algebraic Expressions

Substitute for y in 5 × y. y 5 × y

16

27

35

80 y = 16; 5 × 16 = 80

135

175

27 135

35 175

y = 27; 5 × =

y = 35; 5 × =


Course 1

The missing values are 135 and 175.


Additional Example 1B: Evaluating Algebraic Expressions

Substitute for z in z ÷ 5 + 4. z z ÷ 5 + 4

20

45

60

8 z = 20; 20 ÷ 5 + 4 = 8

z = 45; __ ÷ 5 + 4 = __

z = 60; __ ÷ 5 + 4 = __

13

16

45 13

60 16


Course 1




Substitute for x in x ÷ 9. x x ÷ 9

18

36

54

2 x = 18; 18 ÷ 9 = 2

4

6

4

54 6 x = 36; ÷ 9 = x = 54; ÷ 9 =

36


Course 1




Substitute for z in 8 × z + 2. z 8 × z + 2

7

9

11

58 z = 7; 8 × 7 + 2 = 58

z = 9; 8 × __+ 2 = __

z = 11; 8 × __ + 2 = __ 74

90

9 74

11 90


Course 1


Multiplication and division can be written without using the symbols × and ÷.

Instead of . . . You can write . . .

x × 3 x • 3

35 ÷ y

x(3)

3x

35 y

When you are multiplying a number times a variable, the number is written first. Write “3x” not “x3.” Read 3x as “three x.”

Writing Math


Course 1

Find an expression for the table.

Additional Example 2A: Finding an Expression

39 ÷ 13 = 3

n

39

52

65

3

4

5 65 ÷ 13 = 5

52 ÷ 13 = 4

n ÷ 13

An expression is n ÷ 13, or . n 13


Course 1


Additional Example 2B: Finding an Expression

7 • 4 = 28

p

4

6

8

28

42

56 7 • 8 = 56

7 • 6 = 42

7 • p

An expression is 7 • p, or 7p.


Course 1


21 ÷ 3 = 7

u

21

42

63

7

14

21 63 ÷ 3 = 21

42 ÷ 3 = 14

u ÷ 3

An expression is u ÷ 3, or . u 3



Course 1



4 • 5 = 20

n

5

8

11

20

32

44 4 • 11 = 44

4 • 8 = 32

4 • n

An expression is 4 • n, or 4n.


Course 1

1. Evaluate each expression to find the missing values in the tables.

Lesson Quiz: Part 1

x 10 7 5

x 1 3 5

7 21 35

x – 5

2. Find an expression for the table.

5 2

0

7x


Course 1

Lesson Quiz: Part 2

3. Evaluate 8x for x = 5.

4. Evaluate 4x – 1 for x = 12.

40

47


Course 1

Course 1

2-2 Translating Between Words and Math

Course 1

Warm Up

Lesson Presentation

Problem of the Day

16 36

19

4

Course 1


Warm Up Evaluate each expression for x = 9.

1. 7 + x 2. 4x 3. 2x + 1 4. 36

x

Problem of the Day

Draw a square around the numbers of 4 adjacent days on the calendar for this month. Add all the numbers in the square and subtract 4 times the first number. What number do you get? 16

Solving Subtraction Equations

Course 1


Learn to translate between words and math.

Course 1


In word problems, you may need to translate words to math.

Course 1


Put together or combine

Find how much more or less

Put together groups of equal parts

Separate into equal groups

Add

Subtract

Multiply

Divide

Action Operation

Lake Superior is the largest lake in North America. Let a stand for the area in square miles of Lake Superior. Lake Erie has an area of 9,910 square miles. Write an expression to show how much larger Lake Superior is than Lake Erie.

Additional Example 1A: Social Studies Applications

Course 1


To find how much larger, subtract the area of Lake Erie from the area of Lake Superior.

a – 9,910 Lake Superior is a – 9,910 square miles larger than Lake Erie.

Additional Example 1B

Course 1


Let p represent the number of colored pencils in a box. If there are 26 boxes on the shelf, write an algebraic expression to represent the total number of pencils on the shelf.

To put together 26 equal groups of p, multiply 26 times p.

26p

There are 26p pencils on the shelf.

The Nile River is the world’s longest river. Let n stand for the length in miles of the Nile. The Paraná River is 3,030 miles long. Write an expression to show how much longer the Nile is than the Paraná.


Course 1


To find how much longer, subtract the length of the Paraná from the length of the Nile.

n – 3,030

The Nile is n – 3,030 miles longer than the Paraná.


Course 1


Let p represent the number of paper clips in a box. If there are 125 boxes in a case, write an algebraic expression to represent the total number of paper clips in a case.

To put together 125 equal groups of p, multiply 125 times p.

125p

There are 125p paper clips in a case.

Course 1


37 + 28

• 28 added to 37 • 37 plus 28 • the sum of 37 and 28 • 28 more than 37

• 28 added to x • x plus 28 • the sum of x and 28 •28 more than x

x + 28

+ Operation

Numerical Expression

Words

Algebraic Expression

Words

Course 1


• 12 subtracted from k • 12 minus k • the difference of k and 12 • take away 12 from k

k — 12

Operation


Words


Words

− 90 — 12

• 12 subtracted from 90 • 90 minus 12 • the difference of 90 and 12 • 12 less than 90 • take away 12 from 90

Course 1


• 8 times w • w multiplied by 8 • the product of 8 and w • 8 groups of of w

8 • w or (8)(w) or 8w

Operation


Words


Words

8 × 48 or 8 • 48

• 8 times 48 • 48 multiplied by 8 • the product of 8 and 48

× or (8)(48) or

8(48) or (8)48

Course 1


• n divided by 3 • the quotient of n and 3

n ÷ 3 or

Operation


Words


Words

37 ÷ 3 or

• 327 divided by 3 • the quotient of 327 and 3

÷ 327 3

n 3

Write each phrase as a numerical or algebraic expression.

Additional Example 2: Translating Words into Math

Course 1


A. 987 minus 12

987 − 12

B. x times 45

45 • x or 45x

Write each phrase as a numerical or algebraic expression.

Try This: Example 2

Course 1


A. 42 less than 79

79 − 42

B. y divided by 22

y ÷ 22 or y 22

Write two phrases for each expression.

Additional Example 3: Translating Math into Words

Course 1


A. 16 + b

• 16 added to b

B. (75)(32)

• 75 times 32

• b more than 16

• the product of 75 and 32

Write two phrases for each expression.

Try This: Example 3

Course 1


A. 17 – 14

• 14 subtracted from 17

• 16 divided by b

• 17 minus 14

• the quotient of 16 and b

B. 16 b

Write an expression 1. Let x represent the number of minutes Kristen works out in one week. Write an expression for the number of minutes she works out in 4 weeks.

Lesson Quiz

Write each phrase as a numerical or algebraic expression 2. 7 less than x 3. The product of 12 and w

4x

x − 7 12w

x more than 17 or x added to 17 n divided by 12 or the quotient of n and 12

Course 1


Write a phrase for each expression 4. 17 + x 5. n ÷ 12

Course 1


Course 1

Warm Up

Lesson Presentation

Problem of the Day

29 16 9

16

Warm Up Evaluate each expression for x = 8.

1. 3x + 5 2. x + 8 3. 2x – 7 4. 8x ÷ 4 5. 7x – 1 6. x – 3

Course 1


55 5

Problem of the Day

Complete the magic square so that every row, column, and diagonal add up to the same total.


10 7

4

9 2

8 6

12 5

Course 1


Learn to determine whether a number is a solution of an equation.

Course 1


equation solution

Course 1


Vocabulary

An equation is a mathematical statement that two quantities are equal. You can think of a correct equation as a balanced scale.

Course 1


3 + 2 5

Equations may contain variables. If a value for a variable makes an equation true, that value is a solution of the equation.

Course 1


12 + 15 27

s + 15 = 27

s = 12 s = 10 10 + 15

27

s = 12 is a solution because 12 + 15 = 27.

s = 10 is not a solution because 10 + 15 ≠ 27.

Determine whether the given value of the variable is a solution.

Additional Example 1A: Determining Solutions of Equations

b — 447 = 1,203 for b = 1,650

Because 1,203 = 1,203, 1,650 is a solution to b — 447 = 1,203.

Course 1


b — 447 = 1,203 Substitute 1,650 for b.

1,203 = 1,203 ?

1,650 — 447 = 1,203 ?

1,203 1,203

Subtract.


Additional Example 1B: Determining Solutions of Equations

27x = 1,485 for x = 54

Because 1,458 ≠ 1,485, 54 is not a solution to 27x = 1,485.

Course 1


27x = 1,485 Substitute 54 for x.

1,458 = 1,485 ?

27 • 54 = 1,485 ?

1,458 1,485

Multiply.

Determine whether the given value of the variable is a solution. u + 56 = 139 for u = 73

Because 129 ≠ 139, 73 is not a solution to u + 56 = 139.

Course 1


u + 56 = 139 Substitute 73 for u.

129 = 139 ?

73 + 56 = 139 ?


129 139

Add.


45 ÷ g = 3 for g = 15

Because 3 = 3, 15 is a solution to 45 ÷ g = 3.

Course 1


45 ÷ g = 3 Substitute 15 for g.

3 = 3 ?

45 ÷ 15 = 3 ?


3 3

Divide.

Paulo says that his yard is 19 yards long. Jamie says that Paulo’s yard is 664 inches long. Determine if these two measurements are equal.


Because 684 ≠ 664, 19 yards are not equal to 664 inches.

Course 1


Substitute 19 for y

684 = 664 ?

Multiply.

36 • yd = in.

36 • 19 = 664 ? 36 • y = 664

Anna says that the table is 7 feet long. John says that the table is 84 inches long. Determine if these two measurements are equal.

Because 84 = 84, 7 feet is equal to 84 inches.

Course 1


12f = 84 Substitute 7 for f.

84 = 84 ?

12 • 7 = 84 ?

Try This: Example 2

12 • ft. = in.

Multiply.

Determine whether the given value of the variable is a solution. 1. 85 = 13x for x = 5 2. w + 38 = 210 for w = 172 3. 8y = 88 for y = 11 4. 16 = w ÷ 6 for w = 98

Lesson Quiz

no yes

yes no

no

5. The local pizza shop charged Kylee $172 for 21 medium pizzas. The price of a medium pizza is $8. Determine if Kylee paid the correct amount of money. (Hint: $8 • pizzas = total cost.)

Course 1


Course 1

2-4 Solving Addition Equations

Course 1

Warm Up

Lesson Presentation

Problem of the Day

yes no yes yes

Warm Up Determine whether each value is a solution. 1. 86 + x = 102 for x = 16 2. 18 + x = 26 for x = 4 3. x + 46 = 214 for x = 168 4. 9 + x = 35 for x = 26

Course 1


Problem of the Day

After Renee used 40 m of string for her kite and gave 5 m to her sister for her wagon, she had 8 m of string left. How much string did she have to start with?


53 m

Course 1


Learn to solve whole-number addition equations.

Course 1


The equation h + 14 = 82 can be represented as a balanced scale.

h + 14 82

Course 1


To find the value of h, you need h by itself on one side of the scale.

h ?

To get h by itself, first take away 14 from the left side of the scale. Now the scale is unbalanced.

h 68 To rebalance the scale, take away 14 from the other side.

h + 14 82

Taking away 14 from both sides of the scale is the same as subtracting 14 from both sides of the equation.

h + 14 = 82 –14

Course 1


–14 h = 68

Subtraction is the inverse, or opposite, of addition. If an equation contains addition, solve it by subtracting from both sides to “undo” the addition.

Solve the equation. Check your answer.

Additional Example 1A: Solving Addition Equations

x + 87 = 152

Check x + 87 = 152

x + 87 = 152 87 is added to x.

x = 65

Course 1


– 87 – 87 Subtract 87 from both sides to undo the addition.

65 + 87 = 152 ?

152 = 152 ?

Substitute 65 for x in the equation.

65 is the solution.


Additional Example 1B: Solving Addition Equations

72 = 18 + y

Check 72 = 18 + y

72 = 18 + y 18 is added to y.

54 = y

Course 1


–18 –18 Subtract 18 from both sides to undo the addition.

72 = 18 + 54 ?

72 = 72 ?

Substitute 54 for y in the equation.

54 is the solution.



u + 43 = 78

Check u + 43 = 78

u + 43 = 78 43 is added to u.

u = 35

Course 1


– 43 – 43 Subtract 43 from both sides to undo the addition.

35 + 43 = 78 ?

78 = 78 ?

Substitute 35 for u in the equation.

35 is the solution.



68 = 24 + g

Check 68 = 24 + g

68 = 24 + g 24 is added to g.

44 = g

Course 1


–24 –24 Subtract 24 from both sides to undo the addition.

68 = 24 + 44 ?

68 = 68 ?

Substitute 44 for g in the equation.

44 is the solution.

Johnstown, Cooperstown, and Springfield are located in that order in a straight line along a highway. It is 12 miles from Johnstown to Cooperstown and 95 miles from Johnstown to Springfield. How far is it from Cooperstown to Springfield?

Additional Example 2: Social Studies Application

It is 83 miles from Cooperstown to Springfield.

95 = 12 + d

Course 1


distance between Johnstown and Springfield

distance between Johnstown and Cooperstown

distance between Cooperstown and Springfield

= +

95 = 12 + d –12 –12 83 = d

12 is added to d.

Subtract 12 from both sides to undo the addition.

Patterson, Jacobsville, and East Valley are located in that order in a straight line along a highway. It is 17 miles from Patterson to Jacobsville and 35 miles from Patterson to East Valley. How far is it from Jacobsville to East Valley?

Try This: Example 2

The distance between Jacobsville and East Valley is 18 miles.

35 = 17 + d

Course 1


distance between Patterson and East Valley

distance between Patterson and Jacobsville

distance between Jacobsville and East Valley

= +

35 = 17 + d –17 –17 18 = d

17 is added to d.

Subtract 17 from both sides to undo the addition.

Solve each equation. 1. x + 15 = 72 2. 81 = x + 24 3. x + 22 = 67 4. 93 = x + 14

Lesson Quiz

x = 57 x = 57 x = 45

x = 79

56 inches 5. Kaitlin is 2 inches taller than Reba. Reba is 54 inches tall. How tall is Kaitlin?

Course 1


Course 1


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Simplify.

1. x + 7 = 22 2. 18 + x = 105 3. 16 = x + 9 4. 23 = x + 4

x = 15 x = 87

x = 7 x = 19


Course 1

Problem of the Day

Bruce has 25 CD’s remaining after giving 14 to John, 17 to Mary, and 25 to Sue. How many CD’s did he begin with? 81

Solving Subtraction Equations 2-5 Solving Subtraction Equations

Course 1

Learn to solve whole-number subtraction equations.


Course 1

When John F. Kennedy became president of the United States, he was 43 years old, which was 8 years younger than Abraham Lincoln was when Lincoln became President. How old was Lincoln when he became president?


Course 1

Abraham Lincoln’s age

John F. Kennedy’s age – = 8

Let a represent Abraham Lincoln’s age.

a – = 8 43

a – 8 = 43 + 8 + 8

a = 51 Abraham Lincoln was 51 years old when he became president.


Course 1

y – 23 = 39 23 is subtracted from y. + 23

y = 62 Add 23 to both sides to undo the subtraction.

Check y – 23 = 39 Substitute 62 for y in the equation.

Solve y – 23 = 39. Check your answer.

62 is the solution. 39 = 39 ?

Additional Example 1A: Solving Subtraction Equations

62 – 23 = 39 ?

+ 23


Course 1

78 = s – 15 15 is subtracted from s. + 15

93 = s Add 15 to both sides to undo the subtraction.

Check 78 = s – 15 Substitute 93 for s in the equation.

Solve 78 = s – 15. Check your answer.


Additional Example 1B: Solving Subtraction Equations

78 = 93 – 15 ?

+ 15


Course 1

z – 3 = 12 3 is subtracted from z. + 3

z = 15 Add 3 to both sides to undo the subtraction.

Check z – 3 = 12 Substitute 15 for z in the equation.

Solve z – 3 = 12. Check your answer.


Additional Example 1C: Solving Subtraction Equations

15 – 3 = 12 ?

+ 3


Course 1

a – 4 = 7 4 is subtracted from a. + 4

a = 11 Add 4 to both sides to undo the subtraction.

Check a – 4 = 7 Substitute 11 for a in the equation.

Solve a – 4 = 7. Check your answer.


11 – 4 = 7 ?

+ 4



Course 1

57 = c – 13 13 is subtracted from c. + 13

70 = c Add 13 to both sides to undo the subtraction.

Check 57 = c – 13 Substitute 70 for c in the equation.

Solve 57 = c – 13. Check your answer.


57 = 70 – 13 ?

+ 13



Course 1

g – 62 = 14 62 is subtracted from g. + 62

g = 76 Add 62 to both sides to undo the subtraction.

Check g – 62 = 14 Substitute 76 for g in the equation.

Solve g – 62 = 14. Check your answer.


76 – 62 = 14 ?

+ 62



Course 1

1. x – 9 = 21

2. 14 = x – 3

3. x – 7 = 11

4. 16 = x – 14

5. x – 9 = 11

Lesson Quiz

6. Susan is taller than James. The difference in their height is 4 inches. James is 62 inches tall. How tall is Susan?

x = 30

x = 17

x = 18

x = 30

x = 20

66 inches


Course 1

Course 1


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Divide. 1. 2.

3.

4.

5.

6.

6

13

10

4

72 12 65 5 60 6 40 10 130 5 91 7

26

13

Course 1


Problem of the Day

Katie’s little brother is building a tower with blocks. He adds 6 blocks and then removes 4 blocks. If he adds 8 more blocks, the tower will have twice as many blocks as when he started. How many blocks did he start with? 10

Course 1


Learn to solve whole-number multiplication equations.

Course 1


Division is the inverse of multiplication. To solve an equation that contains multiplication, use division to “undo” the multiplication.

Course 1


4m = 32 4m = 32 4 4 m = 8

Remember! 4m means “4 × m.”

5p = 75 p is multiplied by 5.

5 p = 15

Divide both sides by 5 to undo the multiplication.

Check 5p = 75 Substitute 15 for p in the equation.

Solve 5p = 75. Check your answer.


Additional Example 1A: Solving Multiplication Equations

5(15)= 75 ?

5

Course 1


5p = 75

16 = 8r r is multiplied by 8.

8 2 = r


Check 16 = 8r Substitute 2 for r in the equation.

Solve 16 = 8r. Check your answer.


Additional Example 1B: Solving Multiplication Equations

16 = 8(2) ?

8

Course 1


16 = 8r

8a = 72 a is multiplied by 8.

8 x = 9


Check 8a = 72 Substitute 9 for a in the equation.

Solve 8a = 72. Check your answer.



8(9)= 72 ?

8

Course 1


8a = 72

18 = 3w w is multiplied by 3.

3 6 = w


Check 18 = 3w Substitute 6 for w in the equation.

Solve 18 = 3w. Check your answer.



18 = 3(6) ?

3

Course 1


18 = 3w

Course 1


The area of a rectangle is 56 square inches. Its length is 7 inches. What is its width?

Additional Example 2: Problem Solving

Course 1


Additional Example 2 Continued


The answer will be the width of the rectangle in inches.


• The area of the rectangle is 56 square inches.

• The length of the rectangle is 7 inches.

Draw a diagram to represent this information.

56

7

w

Course 1



2 Make a Plan

You can write and solve an equation using the formula for area. To find the area of a rectangle, multiply its length by its width.

w

l

A = lw 56 = 7w

Course 1



56 = 7w 56 = 7w

Solve 3

7 7 8 = w

So the width of the rectangle is 8 inches.

w is multiplied by 7. Divide both sides by 7 to undo the multiplication.

Course 1



Look Back

Arrange 56 identical squares in a rectangle. The length is 7, so line up the squares in rows of 7. You can make 8 rows of 7, so the width of the rectangle is 8.

4

Try This: Example 2

Course 1


The area of a rectangle is 48 square inches. Its width is 8 inches. What is its length?

Course 1


Try This: Example 2


The answer will be the length of the rectangle in inches.


• The area of the rectangle is 48 square inches.

• The width of the rectangle is 8 inches.

Draw a diagram to represent this information.

48

l

8

Course 1


Try This: Example 2 Continued

2 Make a Plan

You can write and solve an equation using the formula for area. To find the area of a rectangle, multiply its length by its width.

w

l

A = lw 48 = l8

Course 1



48 = l 8 48 = l 8

Solve 3

8 8 6 = l

So the length of the rectangle is 6 inches.

l is multiplied by 8. Divide both sides by 8 to undo the multiplication.

Course 1



Look Back

Arrange 48 identical squares in a rectangle. The width is 8, so line up the squares in columns of 8. You can make 6 columns of 8, so the length of the rectangle is 6.

4

1. 10y = 300

2. 2y = 82

3. 63 = 9y

4. 78 = 13x

Lesson Quiz

5. The area of a board game is 468 square inches. Its width is 18 inches. What is the length?

y = 30

y = 41

y = 7

x = 6

26 inches

Course 1


Course 1


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Solve.

1. 5x = 20

2. 7n = 84

3. 18 = y – 4

4. 21 = n + 3

x = 4

n = 12

y = 22

n = 18

Course 1


Course 1

Problem of the Day

If 11 is 4 less than 3 times a number, what is the number? 5


Course 1

Learn to solve whole-number division equations.


Course 1

Multiplication is the inverse of division. When an equation contains division, use multiplication to “undo” the division.


Course 1

x = 35

Multiply both sides by 7 to undo the division.




Additional Example 1A: Solving Division Equations


Course 1

x 7 = 5

x is divided by 7. x 7 = 5

7 • x 7 = 7 • 5

Check x 7 = 5

35 7 = 5 ?

78 = p


Substitute 78 for p in the equation.



Additional Example 1B: Solving Division Equations


Course 1

p 6 13 =

6 • p 6

6 • 13 =

p is divided by 6. p 6 13 =

Check p 6 13 =

? 78 6 13 =

x = 18







Course 1

x 2 = 9

x is divided by 2. x 2 = 9

2 • x 2 = 2 • 9

Check x 2 = 9

18 2 = 9 ?

288 = p


Substitute 288 for p in the equation.





Course 1

p 4 72 =

4 • p 4

4 • 72 =

p is divided by 4. p 4 72 =

Check p 4 72 =

? 288 4 72 =

An aspen tree is one-third the height of a pine tree at Elk Meadow Park.



Course 1

height of aspen = height of pine 3

The aspen tree is 14 feet tall. How tall is the pine tree? Let h represent the height of the pine tree.

42 = h


3 • h 3

3 • 14 =

Substitute 14 for height of aspen. h is divided by 3.

h 3 14 =

The pine tree is 42 feet tall.

Try This: Example 2


Course 1

Let w represent her father’s weight.

180 = w


2 • w 2

2 • 95 =

Substitute 95 for Jamie’s weight. w is divided by 2.

w 2 95 =

Jamie’s father weighs 180 pounds.

Jamie weighs one-half as much as her father.

Jamie’s weight = father’s weight 2

Jamie weighs 95 pounds. How many pounds does her father weigh?

1. = 7

2. 8 =

3. = 11

4. = 7

Lesson Quiz

5. The area of Sherry’s flower garden is one-fourth the area of her vegetable garden. The area of the flower garden is 17 square feet. Let x represent the area of her vegetable garden. Find the area of her vegetable garden?

x = 70

x = 32

x = 99

x = 105

68 square feet


Course 1

x 4

x 9

Solve each equation. Check your answers. x 10

x 15


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Order the numbers from least to greatest.

1. 242, 156, 224, 165 2. 941, 148, 914, 814, 721 3. 345, 376, 354, 397

156, 165, 224, 242

148, 721, 814, 914, 941

345, 354, 376, 397

Course 1


Problem of the Day

Lupe is taller than Reba and shorter than Miguel. Tory is shorter than Lupe but taller than Reba. List the four brothers and sisters in order from tallest to shortest.

Miguel, Lupe, Tory, Reba

Course 1


Learn to write, compare, and order decimals using place value and number lines.

Course 1


Additional Example 1A & B: Reading and Writing Decimals

Write each decimal in standard form, expanded form, and words.

A. 1.07

B. 0.03 + 0.006 + 0.0009

1 + 0.07 Expanded form: one and seven hundredths Word form:

0.0369 Standard form: three hundred sixty-nine ten-thousandths Word form:

Course 1


Additional Example C: Reading and Writing Decimals

Write the decimal in standard form and expanded form.

C. fourteen and eight hundredths 14.08 Standard form:

10 + 4 + 0.08 Expanded form:

Course 1


Try This: Example 1A & 1B

Write each decimal in standard form, expanded form, and words.

A. 1.12

B. 0.6 + 0.008 + 0.0007

1 + 0.12 Expanded form: one and twelve hundredths Word form:

0.6087 Standard form: six thousand eighty-seven ten-thousandths

Word form:

Course 1



Write each decimal in standard form and expanded form.

C. eleven and two hundredths

11.02 Standard form:

10 + 1 + 0.02 Expanded form:

Course 1


Additional Example 2: Earth Science Application

The star Wolf 359 has an apparent magnitude of 13.5. Suppose another star has an apparent magnitude of 13.05. Which star has the smaller magnitude?

13.50 Line up the decimal points.

Start from the left and compare the digits. 13.05 Look for the first place where the digits are

different. 0 is less than 5. 13.05 < 13.50 The star that has an apparent magnitude of 13.05 has the smaller magnitude.

Course 1


Try This: Example 2

Tina reported on a star for her science project that has a magnitude of 11.3. Maven reported on another star that has a magnitude of 11.03. Which star has the smaller magnitude?

11.30 Line up the decimal points.

Start from the left and compare the digits. 11.03 Look for the first place where the digits are

different. 0 is less than 3. 11.03 < 11.30 The star that has a magnitude of 11.03 has the smaller magnitude.

Course 1


Additional Example 3: Comparing and Ordering Decimals

Order the decimals from least to greatest.

16.67, 16.6, 16.07

16.67

16.60

Compare two of the numbers at a time.

Write 16.6 as “16.60.” 16.60 < 16.67 16.67

16.07 Start at the left and compare the digits. 16.07 < 16.67

Course 1


16.60

16.07 Look for the first place where the digits are different. 16.07 < 16.60


Graph the numbers on a number line.

16.67 16.07 16.6

Course 1


16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17

The numbers are ordered when you read the number line from left to right. The numbers in order from least to greatest are 16.07, 16.6, and 16.67.

Try This: Example 3

Order the decimals from least to greatest.

12.42, 12.4, 12.02

12.42

12.40

Compare two of the numbers at a time.

Write 12.4 as “12.40.” 12.40 < 12.42

12.42

12.02 Start at the left and compare the digits. 12.02 < 12.42

Course 1


12.40

12.02 Look for the first place where the digits are different. 12.02 < 12.40


Graph the numbers on a number line.

Course 1


12.42 12.02 12.4

12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13

The numbers are ordered when you read the number line from left to right. The numbers in order from least to greatest are 12.02, 12.4, and 12.42.

Lesson Quiz: Part 1 Write each in standard form, expanded form and words.

1. 8.0342

2. 18 + 0.3 + 0.006

3. eight and twelve hundredths

18.306; eighteen and three hundred six thousandths

8 + 0.03 + 0.004 + 0.0002; eight and three hundred forty-two ten thousandths

Insert Lesson Title Here

8.12; 8 + 0.1 + 0.02

Course 1


Lesson Quiz: Part 2

4. It takes Pluto 246.7 years to orbit the Sun, and it takes Neptune 164.8 years. Which planet takes longer to orbit the Sun?

5. Order the decimals from least to greatest: 16.35, 16.3, 16.5.


Pluto

16.3, 16.35, 16.5

Course 1


3-2 Estimating Decimals

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Order the decimals from least to greatest.

1. 18.74, 18.7, 18.47 2. 9.06, 9.66, 9.6, 9.076 Write each in words. 3. 3.072 4. 6.1258

18.47, 18.7, 18.74

9.06, 9.076, 9.6, 9.66

three and seventy-two thousandths

Course 1


six and one thousand two hundred fifty-eight ten-thousandths

Problem of the Day

Calculate your age in months.

Possible answer: 11 yr 8 mo = 140 mo

Course 1


Learn to estimate decimal sums, differences, products, and quotients.

Course 1


Vocabulary clustering front-end estimation


Course 1


Course 1


When numbers are about the same value, you can use clustering to estimate. Clustering means rounding the numbers to the same value.

Course 1

3-2 Estimating Decimals Additional Example 1: Health Application

Nancy wants to cycle, ice skate, and water ski for 30 minutes each. About how many calories will she burn in all? (Cycling = 165.5 calories, ice skating = 177.5 calories, and water skiing = 171.5 calories) 165.5 170 The addends cluster around 170. 177.5 170 To estimate the total number of

calories, round each addend to 170.

171.5 + 170

Add. 510

Nancy burns about 510 calories

Course 1

3-2 Estimating Decimals Try This: Example 1

Abner wants to run, roller blade, and snow ski for 60 minutes each. About how many calories will he burn in all? (Running = 185.5 calories, roller blading = 189.5 calories, and snow skiing = 191.5 calories) 185.5 190 The addends cluster around 190. 189.5 190 To estimate the total number of

calories, round each addend to 190. 191.5 + 190

Add. 570

Abner burns about 570 calories

Course 1


When rounding, look at the digit to the right of the place to which you are rounding.

• If it is 5 or greater, round up.

• If it is less than 5, round down.

Remember!

Course 1

3-2 Estimating Decimals Additional Example 2: Rounding Decimals to

Estimate Sums and Differences

Estimate by rounding to the indicated place value.

A. 7.13 + 4.68; ones

B. 9.705 – 0.2683; tenths

7.13 + 4.68 Round to the nearest whole number. 7 + 5 = 12 The sum is about 12.

9.705 9.7 Round to the tenths. Align.

9.4 Subtract. – 0.2683 –0.3

Course 1


Try This: Example 2


A. 6.09 + 3.72; ones

B. 8.898 – 0.4619; tenths

6.09 + 3.72 Round to the nearest whole number. 6 + 4 = 10 The sum is about 10.

8.898 8.9 Round to the tenths. Align.

8.4 Subtract. –0.4619 –0.5

Course 1


Compatible numbers are close to the numbers that are in the problem and are helpful when you are solving the problem mentally.

Remember!

Course 1


Additional Example 3: Using Compatible Numbers to Estimate Products and Quotients

Estimate each product or quotient.

A. 33.83 × 1.98

B. 72.77 ÷ 26.14

35 × 2 = 70 35 and 2 are compatible.

75 ÷ 25 = 3 75 and 25 are compatible.

So, 72.77 ÷ 26.14 is about 3.

So 33.83 × 1.98 is about 70.

Course 1


Try This: Example 3

Estimate each product or quotient.

A. 22.12 × 4.98

B. 62.31 ÷ 18.52

20 × 5 = 100 20 and 5 are compatible.

60 ÷ 20 = 3 60 and 20 are compatible.

So, 62.31 ÷ 18.52 is about 3.

So 22.12 × 4.98 is about 100.

Course 1


You can also use front-end estimation to estimate with decimals. Front-end estimation means to use only the whole-number part of the decimal.

Course 1


Additional Example 4: Using Front-End Estimation

Estimate a range for the sum.

7.86 + 36.97 + 5.40

Use front-end estimation.

7.86 7 Add the whole numbers only. 36.97 36 The whole-number values of the

decimals are less than the actual numbers, so the answer is an underestimate.

5.40 + 5 at least 48

The exact answer of 7.86 + 36.97 + 5.40 is 48 or greater.

Course 1



You can estimate a range for the sum by adjusting the decimal part of the numbers. Round the decimals to 0, 0.5, or 1.

0.86 1.00 Add the decimal part of the numbers.

0.97 1.00 Add the whole-number estimate and the adjusted estimate. 0.40 + 0.50

2.50

48.00 + 2.50 = 50.50

The adjusted decimals are greater than the actual decimal, so 50.50 is an overestimate.

The estimated range for the sum is from 48.00 to 50.50.

Course 1


Try This: Example 4

Estimate a range for the sum.

8.92 + 47.88 + 3.41

Use front-end estimation.

8.92 8 Add the whole numbers only. 47.88 47 The whole-number values of the

decimals are less than the actual numbers, so the answer is an underestimate.

3.41 + 3 at least 58

The exact answer of 8.92 + 47.88 + 3.41 is 58 or greater.

Course 1


Try This: Example 4

You can estimate a range for the sum by adjusting the decimal part of the numbers. Round the decimals to 0, 0.5, or 1.

0.92 1.00 Add the decimal part of the numbers.

0.88 1.00 Add the whole-number estimate and the adjusted estimate. 0.41 +0.50

2.50

58.00 + 2.50 = 60.50

The adjusted decimals are greater than the actual decimal, so 60.50 is an overestimate.

The estimated range for the sum is from 58.00 to 60.50.

Lesson Quiz: Part 1


3

4.5


Course 1


1. 3.07442 + 1.352; tenths

2. 7.305 – 4.12689; nearest whole number

Lesson Quiz: Part 2 Estimate each product or quotient.

3

14


80

Course 1


3. 6.75 × 1.82

4. 10.5 ÷ 3.42

5. The snowfall in December, January, and February was 18.26 cm, 29.36 cm, and 32.87 cm, respectively. About how many total centimeters of snow fell during the three months?


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Estimate by rounding to the indicated place value.

1. 70.27 + 15.36; ones 2. 84.37 – 21.82; tenths Estimate each product or quotient. 3. 27.25 × 8.7 4. 44.52 ÷ 3.27

85 62.6

270

Course 1


15

Problem of the Day

Find a three-digit number that rounds to 440 and includes a digit that is the quotient of 24 and 3. Is there more than one possible answer? Explain your thinking.

438; no; the numbers that round to 440 are 435-444, 24 divided by 3 is 8, and 438 is the only number with 8 as a digit.

Course 1


Learn to add and subtract decimals.

Course 1


Course 1


Estimating before you add or subtract will help you check whether your answer is reasonable.

Helpful Hint

Course 1


Elise Ray’s Scores Event Points

Floor exercise 9.8

Balance beam 9.7

Vault 9.425

Uneven bars 9.85

American gymnast Elise Ray won the 2000 U.S. Championships in the all-around, uneven bars, and floor-exercise events.

To find the total number of points, you add all of the scores.

Course 1


Additional Example 1A: Sports Application

A. What was Elise Ray’s total for the events other than the floor exercise?

Estimate by rounding to the nearest whole number.

9.7 + 9.425 + 9.85

The total is about 29 points. 10 + 9 + 10 = 29

Find the sum of 9.7, 9.425, and 9.85.

Course 1


Additional Example 1A Continued

Add.

9.700

9.425 +9.850 28.975

Align the decimal points.

Use zeros as placeholders.

Add. Then place the decimal point.

Since 28.975 is close to the estimate of 29, the answer is reasonable. Elise Ray’s total for the events other than the floor exercise was 28.975.

Course 1


Additional Example 1B: Sports Application

B. How many more points did Elise need on the vault to have a perfect score of 10?

10.000 -9.425 0.575

Align the decimal points. Use zeros as placeholders. Subtract. Then place the decimal point.

Elise needed another 0.575 points to have a perfect score.

Find the difference between 10 and 9.425.

Course 1



A. What was Elise Ray’s total for the events other than the vault exercise?

Estimate by rounding to the nearest whole number. 9.8 + 9.7 + 9.85

The total is about 30 points. 10 + 10 + 10 = 30

Find the sum of 9.8, 9.7, and 9.85.

Course 1


Try This: Example 1A Continued

Add 9.800

9.700 +9.850 29.350

Align the decimal points.

Use zeros as placeholders.

Add. Then place the decimal point.

Since 29.350 is close to the estimate of 30, the answer is reasonable. Elise Ray’s total for the events other than the vault exercise was 29.350.

Course 1



B. How many more points did Elise need on the uneven bars to have a perfect score of 1

10.00 -9.85 0.15

Align the decimal points. Use zeros as placeholders. Subtract. Then place the decimal point.

Elise needed another 0.15 points to have a perfect score.

Find the difference between 10 and 9.85.

Course 1


Additional Example 2: Using Mental Math to Add and Subtract Decimals

Find each sum or difference.

A. 1.8 + 0.2

B. 4 – 0.7

Think: 0.8 + 0.2 = 1. 1.8 + 0.2 = 2

Think: What number added to 0.7 is 1?

4 – 0.7 = 3.3 0.7 + 0.3 = 1 So 1 – 0.7 = 0.3

Course 1


Try This: Example 2


A. 1.6 + 0.4

B. 6 – 0.3

Think: 0.6 + 0.4 = 1. 1.6 + 0.4 = 2.0

Think: What number added to 0.3 is 1?

6 – 0.3 = 5.7 0.3 + 0.7 = 1 So 1 – 0.3 = 0.7

Course 1


Additional Example 3A: Evaluating Decimal Expressions

Evaluate 6.73 – x for each value of x.

A. x = 3.8

Substitute 3.8 for x. 6.73 – x 6.73 – 3.8 6.73 Align the decimal points.

– 3.80 Use a zero as a placeholder.

2.93 Subtract. Place the decimal point.

Course 1


Additional Example 3B: Evaluating Decimal Expressions


B. x = 2.9765

Substitute 2.9765 for x. 6.73 – x 6.73 – 2.9765 6.7300 Align the decimal points.

–2.9765 Use a zero as a placeholder.


Course 1




A. x = 3.8

Substitute 3.8 for x. 7.58 – x 7.58 – 3.8 7.58 Align the decimal points. –3.80 Use a zero as a placeholder. 3.78 Subtract. Place the decimal point.

Course 1




B. x = 2.9765

Substitute 2.9765 for x. 8.17 – x 8.17 – 2.9765 8.1700 Align the decimal points. – 2.9765 Use a zero as a placeholder.


Lesson Quiz


1. 8.3 + 2.7

2. 9.7 – 4

3. 22.6 + 8.4

4. Evaluate 12.76 – x for x = 8.41.

5. During an ice-skating competition, Dawn received the following scores: 4.8, 5.2, 5.4. What was Dawn’s total score?

5.7

11


31

4.35

Course 1


15.4

3-4 Decimals and Metric Measurement

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Multiply.

84

87.2 4.2

Course 1


73,200 920

1. 8.4 × 10

2. 8.72 × 10 3. 0.42 × 10 4. 732 × 100 5. 9.2 × 100

Problem of the Day

A nurse must administer a 4 mL dose of a drug daily to a patient. If there is one liter of this drug on hand, will it last the patient 150 days? If so, how much will remain? If not, how much more will be needed?

0.4L will remain.

Course 1


Learn to multiply and divide decimals by powers of ten and to convert metric measurements.

Course 1


Course 1


Additional Example 1A: Multiplying and Dividing by Powers of Ten

Multiply.

A. 7,126 × 1,000 7,126.000

= 7,126,000

There are 3 zeros in 1,000. To multiply, move the decimal point 3 places right. Write 3 placeholder zeros.

Course 1

3-4 Decimals and Metric Measurement Additional Example 1B & 1C: Multiplying and

Dividing by Powers of Ten

Divide.

B. 7,126 ÷ 1,000

C. 46.34 ÷ 104

7,126.

= 7.126

There are 3 zeros in 1,000. To divide, move the decimal point 3 places left.

0046.34

= 0.004634

The power of 10 is 4. Move the decimal point 4 places left. Write placeholder zeros.

Course 1



Multiply.

A. 4,639 × 1,000

4,639.000

= 4,639,000

There are 3 zeros in 1,000. To multiply, move the decimal point 3 places right. Write 3 placeholder zeros.

Course 1


Try This: Example 1B & 1C

Divide.

B. 4,639 ÷ 1,000

C. 32.08 ÷ 104

4,639.

= 4.639

There are 3 zeros in 1,000. To divide, move the decimal point 3 places left.

0032.08

= 0.003208

The power of 10 is 4. Move the decimal point 4 places left. Write placeholder zeros.

Course 1


Unit Abbreviation Approximate Comparison

Length

Kilometer km Length of 10 football fields

Meter m Width of a door

Centimeter cm Width of your little finger

Millimeter mm Thickness of a dime

Course 1



Mass Kilogram kg Mass of a

textbook

Gram g Mass of a small paperclip

Course 1



Capacity Liter L Filled bottle of

sparkling water

Milliliter mL Half-filled eyedropper

Course 1

3-4 Decimals and Metric Measurement Additional Example 2A & 2B: Choosing

Appropriate Units

Use the abbreviation for the most appropriate metric unit.

A. A soda can is about 12 ____ tall.

B. The mass of a pen is about 5 ____.

A soda can is about 12 cm tall.

Think: A soda can is about the height of 12 little-finger widths.

?

The mass of a pen is about 5 g.

Think: A pen has a mass of about 5 small paper clips.

?

Course 1


Additional Example 2C: Choosing Appropriate Units


C. A sip of water is about 3 ____.

?

A sip of water is about 3 mL.

Think: A sip of water is about 3 half-filled eyedroppers.

Course 1




A. A one-car garage door is about 4 ____ wide.

B. The mass of an eraser is about 7 ____.

A garage door is about 4 m wide.

Think: A garage door is about the width of 4 door widths.

?

The mass of an eraser is about 7 g.

Think: An eraser has a mass of about 7 small paper clips.

?

Course 1




C. A telephone directory weighs about 3 ____.

?

A telephone directory weighs about 3 kg .

Think: A telephone directory is about 3 textbooks.

Course 1


1,000 100 10 1 0.1 0.01 0.001

Thousands Hundreds Tens Ones Tenths Hundredths Thousandths

Kilo- Hecto- Deca- Base unit Deci- Centi- Milli-

In the metric system, each unit of measure is ten times greater than the unit to its right in a place-value chart.

To convert units within the metric system, multiply or divide by powers of ten.

Course 1

3-4 Decimals and Metric Measurement Additional Example 3A: Converting Within the

Metric System

Convert each measure.

A. The height of a door is about 2 m. 2 m = ___ km

2 m = (2 ÷ 1000)km 1 km = 1,000 m, so divide by 1,000.

?

2 m = 0.002 km Move the decimal point 3 places left.

To convert to smaller units, multiply.

To convert to larger units, divide.

Helpful Hint

Course 1


Additional Example 3B: Converting Within the Metric System

Convert the measure.

B. The width of a door is about 0.85 m. 0.85 m = ___ cm

?

0.85 m = (0.85 × 100)cm 1 m = 100 cm, so multiply by 100.

0.85 m = 85 cm Move the decimal point 2 places right.

Course 1




A. The height of a building is about 30 m. 30 m = ___ km

30 m = (30 ÷ 1000)km 1 km = 1,000 m, so divide by 1,000.

?

30 m = 0.03 km Move the decimal point 3 places left.

Course 1



Convert the measure.

B. The width of a chair is about 0.98 m. 0.98 m = ___ cm ?

0.98 m = (0.98 × 100)cm 1 m = 100 cm, so multiply by 100.

0.98 m = 98 cm Move the decimal point 2 places right.

Lesson Quiz: Part 1

7.15731 1,604


m

cm

Course 1


Multiply or divide.

1. 16.04 × 102 2. 715.731 ÷ 100


3. The length of a soccer field is about 100 ___.

4. The width of a computer screen is about 23 ___.

?

?

Lesson Quiz: Part 2


5. The width of film for slides is 35 mm. 35 mm = ____ m.

6. The weight of an adult is about 70 kg. 70 kg = ____ g.

0.035


70,000

Course 1


?

?


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Multiply.

1. 724 × 102

2. 837 × 10 3. 632.9 × 100 4. 18,256 × 10 5. 10 × 10 × 10

72,400 8,370 63,290

Course 1


182,560 1,000

Problem of the Day

A rope ladder is hanging from the back of a yacht. At 10:00 A.M., the water reaches the third step on the ladder. If every 2 hours the water rises the height of two steps, at what step would the water level be at 5:00 P.M.? Explain.

the third, because the boat will rise with the water

Course 1


Learn to write large numbers in scientific notation.

Course 1


Vocabulary scientific notation


Course 1


Course 1


Scientific notation is a shorthand method for writing large numbers like 3,456,000.

Course 1


To write the number 3,456,000 in scientific notation, do the following:

3,456,000 Move the decimal point left to form a number that is greater than 1 and less than 10.

3,456,000 Multiply that number by a power of ten.

3.456 × 106 The power of 10 is 6, because the decimal point is moved 6 places left.

Course 1


The number of zeros in the power of ten, or the exponents in the power of ten, tells you how many places to move the decimal point

Remember!

Course 1


A number written in scientific notation has two parts that are multiplied.

3.456 × 106

The first part is a number that is greater than 1 and less than 10.

The second part is a power of 10.

Course 1

3-5 Scientific Notation Additional Example 1A & 1B: Writing Numbers

in Scientific Notation

Write each number in scientific notation.

A. 6,000,000

B. 411,000

6,000,000 Move the decimal point 6 places left. The power of 10 is 6.

6,000,000 = 6 × 106

411,000 Move the decimal point 5 places left. The power of 10 is 5.

411,000 = 4.11 × 105

Course 1


Additional Example 1C: Writing Numbers in Scientific Notation

Write the number in scientific notation.

C. 79,000,000


79,000,000 = 7.9 × 107

Course 1




A. 8,000,000

B. 587,000


8,000,000 = 8 × 106

587,000 Move the decimal point 5 places left. The power of 10 is 5.

587,000 = 5.87 × 105

Course 1



Write the number in scientific notation.

C. 13,000,000


13,000,000 = 1.3 × 107

Course 1


You can write a large number written in scientific notation in standard form. Look at the power of 10 and move the decimal point that number of places to the right.

Course 1

3-5 Scientific Notation Additional Example 2A & 2B: Writing Numbers

in Standard From

Write each number in standard form.

A. 6.2174 × 103

B. 9.5 × 108

The power of 10 is 3.

6.2174 × 103 = 6,217.4

6.2174


9.50000000

Move the decimal point 3 places right.

Move the decimal point 8 places right. Use zeros as placeholders.

9.5 × 108 = 950,000,000

Course 1

3-5 Scientific Notation Additional Example 2C: Writing Numbers in

Standard From

Write the number in standard form.

C. 4.83 × 105


4.83 × 105 = 483,000

4.83000 Move the decimal point 5 places right. Use zeros as placeholders.

Course 1

3-5 Scientific Notation Try This: Example 2A & 2B

Write each number in standard form.

A. 8.1974 × 103

B. 2.3 × 106


8.1974 × 103 = 8,197.4

8.1974

The power of 10 is 6. 2.300000

Move the decimal point 3 places right.

Move the decimal point 6 places right. Use zeros as placeholders.

2.3 × 106 = 2,300,000

Course 1



Write the number in standard form.

C. 9.77 × 105


9.77 × 105 = 977,000

9.77000 Move the decimal point 5 places right. Use zeros as placeholders.

Lesson Quiz


1. 6,300

2. 70,400,000 Write each number in standard form. 3. 7.241 × 104

4. 8.2137 × 107

5. A Wall Street report indicated that a fast-moving stock had sold 3,295,000 shares. Write this number in scientific notation.


72,410

82,137,000

Course 1


6.3 × 103

7.04 × 107

3.295 × 106


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Multiply.

27,840 754,400 38,060

Course 1


120,700 14,112 62,760

1. 87 × 320 2. 943 × 800 3. 3,806 × 10 4. 1,207 × 100 5. 72 × 196 6. 120 × 523

Problem of the Day

Carmen and Rita sold homemade oatmeal cookies. After 8 days, they had sales totaling $70. Each day, their sales were $0.50 higher than the previous day. What were their sales on the first day?

$7.00

Course 1


Learn to multiply decimals by whole numbers and by decimals.

Course 1


Course 1

3-6 Multiplying Decimals Additional Example 1: Science Application

Something that weighs 1 lb on Earth weighs 0.17 lb on the Moon. How much would a 4 lb dumbbell weigh on the Moon?

4 × 0.17

0.17 You can think of multiplication by a whole number as a repeated addition. 0.17

0.17 0.17 + _____ 0.68

Course 1

3-6 Multiplying Decimals Additional Example 1 Continued


You can also multiply as you would with whole numbers. Place the decimal point by adding the number of decimal places in the numbers multiplied.

0.17 4 × _____ 0.68

2 decimal places + 0 decimal places

2 decimal places

A 4 lb dumbbell on Earth weighs 0.68 lb on the Moon.

Course 1

3-6 Multiplying Decimals Try This: Example 1


7 × 0.17 0.17 You can think of multiplication by a

whole number as a repeated addition.

0.17 0.17 0.17

+ _____ 1.19

0.17 0.17

0.17

Course 1

3-6 Multiplying Decimals Try This: Example 1 Continued


You can also multiply as you would with whole numbers. Place the decimal point by adding the number of decimal places in the numbers multiplied.

0.17 7 × _____ 1.19


2 decimal places

A 7 lb dumbbell on Earth weighs 1.19 lb on the Moon.

Course 1

3-6 Multiplying Decimals Additional Example 2A: Multiplying a Decimal by

a Decimal

Find the product.

A. 0.3 × 0.4 Multiply. Then place the decimal point.

0.3 0.4 ×

1 decimal place + 1 decimal place

2 decimal places 0.12

Course 1

3-6 Multiplying Decimals Additional Example 2B: Multiplying a Decimal by

a Decimal

B. 0.07 × 0.8

Multiply. Then place the decimal point.

0.07 0.8 ×

2 decimal places + 1 decimal place

3 decimal places; use a placeholder zero

0.056

Estimate the product. 0.8 is close to 1.

0.07 × 1 = 0.07

0.056 is close to the estimate of 0.07. The answer is reasonable.

Course 1

3-6 Multiplying Decimals Additional Example 2C: Multiplying a Decimal by

a Decimal

C. 1.34 × 2.5


1.34 2.5 ×


3 decimal places

670

Estimate the product. Round each factor to the nearest whole number.

1 × 3 = 3

3.350 is close to the estimate of 3. The answer is reasonable.

2680 3.350

Course 1



Find each product.

A. 0.5 × 0.2


0.5 0.2 ×

1 decimal place + 1 decimal place

2 decimal places 0.10

Course 1



B. 0.03 × 0.9


0.03 0.9 ×


3 decimal places; use a placeholder zero

0.027

Estimate the product. 0.9 is close to 1.

0.03 × 1 = 0.03

0.027 is close to the estimate of 0.03. The answer is reasonable.

Course 1



C. 3.80 × 3.3


3.80 3.3 ×


3 decimal places

1140

Estimate the product. Round each factor to the nearest whole number.

4 × 3 = 12

12.54 is close to the estimate of 12. The answer is reasonable.

11400 12.540

Course 1



Evaluate 5x for each value of x.

A. x = 3.062

3.062 5 ×

3 decimal places

+ 0 decimal places 3 decimal places 15.310

Substitute 3.062 for x. 5x = 5(3.062)

These notations all mean multiply 3 times x.

3 • x 3x 3(x)

Remember!

Course 1



B. x = 4.79

4.79 5 ×

2 decimal places



Course 1




A. x = 2.012

2.012 5 ×

3 decimal places



Course 1



B. x = 6.22

6.22 5 ×

2 decimal places



Lesson Quiz Find each product.

1. 0.8

2. 0.006 × 0.07


0.00042 0.056


21.64 6.432

Course 1


× 0.07

$87.40

3. x = 2.705 4. x = 0.804

5. “Pick your own” peaches sell for $0.95 per pound. You picked 92 pounds of peaches. How much were you charged?


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Divide.

1. 56,000 ÷ 8 2. 5,219 ÷ 17 3. 9,180 ÷ 12

7,000 307 765

Course 1


Problem of the Day

In his pocket, Bill has $0.77 made up of 10 coins. What are the coins?

1 quarter, 3 dimes, 4 nickels, and 2 pennies

Course 1


Learn to divide decimals by whole numbers.

Course 1


Course 1


Additional Example 1A: Dividing a Decimal by a Whole Number

Find the quotient.

A. 0.84 ÷ 3

3 0.84 – 6

24 – 24

0

Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers.

0. 2 8

Course 1


Additional Example 1B: Dividing a Decimal by a Whole Number

Find the quotient.

B. 3.56 ÷ 4

4 3.56 – 32

36 – 36

0


0. 8 9

Course 1



Find each quotient.

A. 0.72 ÷ 3

3 0.72 – 6

12 – 12

0


0. 2 4

Course 1



Find the quotient.

B. 2.96 ÷ 4

4 2.96 – 28

16 – 16

0


0. 7 4

Course 1



Evaluate 0.936 ÷ x for each given value of x.

A. x = 9

9 0.936 – 9

3 – 0

36

Substitute 9 for x. 0. 1 0

0.936 ÷ x 0.936 ÷ 9

– 36 0

4 Sometimes you need to use a zero as a placeholder. 9 > 3, so place a zero in the quotient and divide 9 into 36.

Course 1




B. x = 18

18 0.936 – 0

93 – 90

36


0.936 ÷ x 0.936 ÷ 18

– 36 0


Course 1




A. x = 6

6 0.636 – 6

3 – 0

36


0.636 ÷ x 0.636 ÷ 6

– 36 0


Course 1




B. x = 12

12 0.636 – 0

63 – 60

36


0.636 ÷ x 0.636 ÷ 12

– 36 0


Course 1


Multiplication can “undo” division. To check your answer to a division problem, multiply the divisor by the quotient.

Remember!

divisor dividend quotient

Course 1


Additional Example 3: Consumer Application Jodi and three of her friends are making a tile design. The materials cost $10.12. If they share the cost equally, how much should each person pay?

4 $10.12 – 8

21 – 20

12

$2. 5 3

$10.12 should be divided into four equal groups. Divide $10.12 by 4.

– 12 0

Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers. Check 2.53 × 4 = 10.12 Each person should pay $2.53.

Course 1

3-7 Dividing Decimals by Whole Numbers Try This: Example 3

Ty and three of his friends are building a fence. The lumber cost $21.44. If they share the cost equally, how much should each person pay?

4 $21.44 – 20

14 – 12

24

$5. 3 6

$21.44 should be divided into four equal groups. Divide $21.44 by 4.

– 24 0

Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers. Check 5.36 × 4 = 21.44 Each person should pay $5.36.

Lesson Quiz

Find each quotient.

1. 3.12 ÷ 8 2. 5.68 ÷ 8

Evaluate the expression 1.25 ÷ x for the given value of x.

3. x = 5 4. x = 25

5. The tennis team is having 3 of their tennis rackets restrung. The total cost is $54.75. What is the average cost per racket?

0.71 0.39


0.25 0.05

Course 1


$18.25


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Divide.

1. 4.8 ÷ 2 2. 16.1 ÷ 7 3. 0.36 ÷ 3 4. 25.28 ÷ 4 5. 6.25 ÷ 5

2.4 2.3 0.12

Course 1


6.32 1.25

Problem of the Day

In the following magic square, 3.375 is the product of the numbers in every row, column, and diagonal. Fill in the missing numbers.

Course 1


4.5 0.25

0.5

3

1 1.5 2.25

0.75 9

Learn to divide whole numbers and decimals by decimals.

Course 1


Course 1


Multiplying the divisor and the dividend by the same number does not change the quotient.

42 ÷ 6 = 7

×10 × 10

420 ÷ 60 = 7

× 10 × 10

4,200 ÷ 600 = 7

Helpful Hint

Course 1


Additional Example 1A: Dividing a Decimal by a Decimal

Find each quotient.

A. 5.2 ÷ 1.3

1.3 5.2 Multiply the divisor and dividend by the same power of ten.

There is one decimal place in the divisor. Multiply by 101, or 10.

Think: 1.3 x 10 = 13 5.2 x 10 = 52

Divide.

13 52 4

–52 0

Course 1


Additional Example 1B: Dividing a Decimal by a Decimal

B. 61.3 ÷ 0.36

0.36 61.3

Think: 0.36 x 100 = 36 61.3 x 100 = 6,130

Make the divisor a whole number by multiplying the divisor and dividend by 102, or 100.

36 6,130.000

Course 1


Additional Example 1B Continued

Divide. 36 6,130.000

7

-36

1

253 252

10 -0

0

100

.2

-72

-252

7

280 -252

280

7

28

170.277… = 170.27 __

When a repeating pattern occurs, show three dots or draw a bar over the repeating part of the quotient.

Course 1



Find each quotient.

A. 4.8 ÷ 1.2

1.2 4.8 Multiply the divisor and dividend by the same power of ten.

There is one decimal place in the divisor. Multiply by 10 , or 10.

Think: 1.2 x 10 = 12 4.8 x 10 = 48

1

Divide.

12 48 4

-48 0

Course 1



B. 51.2 ÷ 0.24

0.24 51.2

Think: 0.24 x 100 = 24 51.2 x 100 = 5,120

Make the divisor a whole number by multiplying the divisor and dividend by 102, or 100.

24 5120.000

Course 1


Try This: Example 1B Continued

Divide. 24 5,120.00

1

-48

2

32 24

80 -72

3

80

.3

-72

-72 80

3

213.33… = 213.33 __

When a repeating pattern occurs, show three dots or draw a bar over the repeating part of the quotient.

8

Course 1



After driving 216.3 miles, the Yorks used 10.5 gallons of gas. On average, how many miles did they drive per gallon of gas?


The answer will be the average number of miles per gallon.


They drove 216.3 miles. They used 10.5 gallons of gas.

Course 1


2 Make a Plan Solve a simpler problem by replacing the decimals in the problem with whole numbers.

If they drove 10 miles using 2 gallons of gas, they averaged 5 miles per gallon. You need to divide miles by gallons to solve the problem.

Solve 3

10.5 216.3

Multiply the divisor and dividend by 10. Think: 10.5 x 10 = 105 216.3 x 10 = 2,163 Place the decimal point in the quotient. Divide.

The York family averaged 20.6 miles per gallon.

Course 1


Look Back 4

Use compatible numbers to estimate the quotient.

216.3 ÷ 10.5 220 ÷ 11 =20

The answer is reasonable since 20.6 is close to the estimate of 20.

Course 1


Try This: Example 2

After driving 191.1 miles, the Changs used 10.5 gallons of gas. On average, how many miles did they drive per gallon of gas?


The answer will be the average number of miles per gallon.


They drove 191.1 miles. They used 10.5 gallons of gas.

Course 1


2 Make a Plan Solve a simpler problem by replacing the decimals in the problem with whole numbers.

If they drove 10 miles using 2 gallons of gas, they averaged 5 miles per gallon. You need to divide miles by gallons to solve the problem.

Solve 3

10.5 191.1

Multiply the divisor and dividend by 10. Think: 10.5 x 10 = 105 191.1 x 10 = 1,911 Place the decimal point in the quotient. Divide.

The York family averaged 18.2 miles per gallon.

Course 1


Look Back 4

Use compatible numbers to estimate the quotient.

191.1 ÷ 10.5 190 ÷ 10 = 19

The answer is reasonable since 18.2 is close to the estimate of 19.

Lesson Quiz

Find each quotient.

1. 49 ÷ 0.7

2. 21.63 ÷ 2.1

3. 38.43 ÷ 6.1

4. 16.9 ÷ 5.2

5. John spent $13.44 renting 4 videos for the weekend. What was the cost per video?

10.3

70


6.3

3.25

Course 1


$3.36


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Divide.

1. 15.264 ÷ 3 2. 3.78 ÷ 3 3. 342 ÷ 7.6 4. 28.32 ÷ 4.8

5.088 1.26

45

Course 1


5.9

Problem of the Day

Divide your age in months by 12. What does the quotient tell you?

my age in years

Course 1


Learn to solve problems by interpreting the quotient.

Course 1


Course 1


To divide decimals, first write the divisor as a whole number. Multiply the divisor and dividend by the same power of ten.

Remember!

Course 1

3-9 Interpret the Quotient Additional Example 1: Measurement Application Suppose Mark wants to make bags of slime. If each bag of slime requires 0.15 kg of corn starch and he has 1.23 kg, how many bags of slime can he make? The question asks how many whole bags of slime can be made when the corn starch is divided into groups of 0.15 kg.

1.23 ÷ 0.15 = ? 1.23 ÷ 0.15 = 8.2

Think: The quotient shows that there is not enough to make 9 bags of slime that are 0.15 kg each. There is only enough for 8 bags. The decimal part of the quotient will not be used in the answer. Mark can make 8 bags of slime.

Course 1

3-9 Interpret the Quotient Try This: Example 1

Suppose Antonio wants to make bags of slime. If each bag of slime requires 0.15 kg of corn starch and he has 1.44 kg, how many bags of slime can he make? The question asks how many whole bags of slime can be made when the corn starch is divided into groups of 0.15 kg.

1.44 ÷ 0.15 = ? 1.44 ÷ 0.15 = 9.6

Think: The quotient shows that there is not enough to make 10 bags of slime that are 0.15 kg each. There is only enough for 9 bags. The decimal part of the quotient will not be used in the answer. Antonio can make 9 bags of slime.

Course 1

3-9 Interpret the Quotient Additional Example 2: Photography Application

There are 237 students in the seventh grade. If Mr. Jones buys rolls of film with 36 exposures each, how many rolls will he need to take every student’s picture? The question asks how many rolls are needed to take a picture of every one of the students.

Think: 6 rolls of film will not be enough to take every student’s picture. Mr. Jones will need to buy another roll of film The quotient must be rounded up to the next highest whole number. Mr. Jones will need 7 rolls of film.

237 ÷ 36 = 6.583 __

Course 1

3-9 Interpret the Quotient Try This: Example 2

There are 342 students in the seventh grade. If Ms. Tia buys rolls of film with 24 exposures each, how many rolls will she need to take every student’s picture? The question asks how many rolls are needed to take a picture of every one of the students.

Think: 14 rolls of film will not be enough to take every student’s picture. Ms. Tia will need to buy another roll of film. The quotient must be rounded up to the next highest whole number. Ms. Tia will need 15 rolls of film.

342 ÷ 24 = 14.25

Course 1


Additional Example 3: Social Studies Application

Gary has 42.25 meters of rope. If he cuts it into 13 equal pieces, how long is each piece?

The question asks how long each piece will be when the rope is cut into 13 pieces.

Think: The question asks for an exact answer, so do not estimate. Use the entire quotient.

Each piece will be 3.25 meters long.

42.25 ÷ 13 = 3.25

Course 1


Try This: Example 3

Ethan has 64.20 meters of rope. If he cuts it into 15 equal pieces, how long is each piece?

The question asks how long each piece will be when the rope is cut into 15 pieces.

Think: The question asks for an exact answer, so do not estimate. Use the entire quotient.

Each piece will be 4.28 meters long.

64.20 ÷ 15 = 4.28

Course 1


When the question asks You should

How many whole groups can be made when you divide?

Drop the decimal part of the quotient.

How many whole groups are needed to put all items from the dividend into a group?

Round the quotient up to the next highest whole number.

What is the exact number when you divide?

Use the entire quotient as the answer.

Lesson Quiz Solve. 1. The cross-country team’s goal is to run 26.25 mi next week. If they run only 5 days next week, how many miles would they have to run each day? 2. Shannon is having a surprise party for her parents. She wants to invite 22 friends. Invitations come in packages of 8. How many packages does Shannon need to buy?

3

5.25 mi


Course 1



Course 1

Warm up

Lesson Presentation

Problem of the Day

Warm Up Solve.

1. x – 3 = 11 2. 18 = x + 4 3. = 42 4. 2x = 52 5. x – 82 = 172

x = 14

Course 1


x 7

x = 14

x = 294

x = 26

x = 254

Problem of the Day

Find the missing entries in the magic square. 11.25 is the sum of every row, column, and diagonal.

3

Course 1


3.75 5.25

6

6.75 1.5

2.25

0.75 4.5

Learn to solve equations involving decimals.

Course 1


Course 1


You can solve equations with decimals using inverse operations just as you solved equations with whole numbers.

$45.20 + m = $69.95 –$45.20 –$45.20

m = $24.75

Course 1


Use inverse operations to get the variable alone on one side of the equation.

Remember!

Course 1


Additional Example 1A: Solving One-Step Equations with Decimals


A. k – 6.2 = 9.5

k – 6.2 = 9.5 6.2 is subtracted from k. Add 6.2 to both sides to undo the subtraction.

+ 6.2 + 6.2 k = 15.7

Check k – 6.2 = 9.5

Substitute 15.7 for k in the equation. 15.7 – 6.2 = 9.5

?

9.5 = 9.5 ?

15.7 is the solution.

Course 1


Additional Example 1B: Solving One-Step Equations with Decimals

Solve the equation. Check your answer. B. 6k = 7.2

6k = 7.2 k is multiplied by 6. Divide both sides by 6 to undo the multiplication.

k = 1.2

Check 6k = 7.2

Substitute 1.2 for k in the equation. 6(1.2) = 7.2

?

7.2 = 7.2 ?


6 6

6k = 7.2

Course 1

3-10 Solving Decimal Equations Additional Example 1C: Solving One-Step

Equations with Decimals


C. = 0.6

= 0.6

m is divided by 7.


m = 4.2

Check

Substitute 4.2 for m in the equation.

0.6 = 0.6 ?


· 7

m 7

m 7

· 7

= 0.6 m 7

= 0.6 4.2 7

?

Course 1




A. n – 3.7 = 8.6

n – 3.7 = 8.6 3.7 is subtracted from n. Add 3.7 to both sides to undo the subtraction.

+ 3.7 + 3.7 n = 12.3

Check n – 3.7 = 8.6

Substitute 12.3 for n in the equation. 12.3 – 3.7 = 8.6

?

8.6 = 8.6 ?


Course 1




B. 7h = 8.4

7h = 8.4 h is multiplied by 7. Divide both sides by 7 to undo the multiplication.

h = 1.2

Check 7h = 8.4

Substitute 1.2 for h in the equation. 7(1.2) = 8.4

?

8.4 = 8.4 ?


7 7

7h = 8.4

Course 1




C. = 0.3

= 0.3

w is divided by 9.


w = 2.7

Check

Substitute 2.7 for w in the equation.

0.3 = 0.3 ?


· 9

w 9

w 9

· 9

= 0.3 w 9

= 0.3 2.7 9

?

Course 1


The area of a rectangle is its length times its width.

A = lw

Remember!

w

l

Course 1


Additional Example 2A: Measurement Application

Solve the equation. Check your answer. A. The area of Emily’s floor is 33.75 m2. If its length is 4.5 meters, what is its width?

33.75 = 4.5w

Write the equation for the problem. Let w be the width of the room.

Divide both sides by 4.5 to undo the multiplication.

7.5 = w 4.5 4.5

33.75 = 4.5 · w

area = length · width

33.75 = 4.5w

The width of Emily’s floor is 7.5 meters.

Course 1


Additional Example 2B: Measurement Application

Solve the equation. Check your answer. B. If carpet costs $23 per square meter, what is the total cost to carpet the floor?

Let C be the total cost. Write the equation for the problem.

Multiply. C = 776.25

C = 33.75 · 23

total cost = area · cost of carpet per square meter

The cost of carpeting the floor is $776.25.

Course 1



Solve the equation. Check your answer. A. The area of Yvonne’s bedroom is 181.25 ft2. If its length is 12.5 feet, what is its width?

181.25 = 12.5w

Write the equation for the problem. Let w be the width of the room.

Divide both sides by 12.5 to undo the multiplication.

14.5 = w 12.5 12.5

181.25 = 12.5 · w

area = length · width

181.25 = 12.5w

The width of Yvonne’s bedroom is 14.5 feet.

Course 1



Solve the equation. Check your answer. B. If carpet costs $4 per square foot, what is the total cost to carpet the bedroom?

Let C be the total cost. Write the equation for the problem.

Multiply. C = 725

C = 181.25 · 4

total cost = area · cost of carpet per square foot

The cost of carpeting the bedroom is $725.

Lesson Quiz

Solve each equation. Check your answer.

1. x – 3.9 = 14.2

2. = 8.3

3. x – 4.9 = 16.2

4. 7x = 47.6

5. The area of the floor in Devon’s room is 35.7 m2. If the width is 4.2 m, what is the length of the bedroom?


Course 1


x 4

x = 18.1

x = 33.2

x = 21.1

x = 6.8

8.5 m

4-1 Divisibility

Course 1

Warm Up

Problem of the Day

Lesson Presentation

Warm Up

1. 20 2. 48

1 x 20, 2 x 10, 4 x 5

1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8

Course 1

4-1 Divisibility

Write each number as a product of two whole numbers in as many ways as possible.

Problem of the Day

In this magic square, every row, column, and diagonal has the same sum, 34. Complete the square using the whole numbers from 1 to 16.

Course 1

4-1 Divisibility

6 7 10 11

16 3 2

14 1 15 4 9 12

8 13

5

Possible answer:

Learn to use divisibility rules.

Course 1

4-1 Divisibility

Vocabulary

divisibility composite number prime number


Course 1

4-1 Divisibility

Course 1

4-1 Divisibility

A number is divisible by another number if the quotient is a whole number with no remainder.

42 ÷ 6 = 7 Quotient

Course 1

4-1 Divisibility

Divisibility Rules A number is divisible by. . . Divisible Not Divisible

2 if the last digit is even (0, 2, 4, 6, or 8). 3,978 4,975

3 if the sum of the digits is divisible by 3. 315 139

4 if the last two digits form a number divisible by 4.

8,512 7,518

5 if the last digit is 0 or 5. 14,975 10,978

6 if the number is divisible by both 2 and 3 48 20

9 if the sum of the digits is divisible by 9. 711 93

10 if the last digit is 0. 15,990 10,536

Course 1

4-1 Divisibility

Additional Example 1A: Checking Divisibility

Tell whether 462 is divisible by 2, 3, 4, and 5.

2

3

4

5 Not divisible

So 462 is divisible by 2 and 3.

The last digit, 2, is even.

The sum of the digits is 4 + 6 + 2 = 12. 12 is divisible by 3.

The last two digits form the number 62. 62 is not divisible by 4.

Divisible

Divisible

Not divisible

The last digit is 2.

Course 1

4-1 Divisibility

Additional Example 1B: Checking Divisibility

Tell whether 540 is divisible by 6, 9, and 10.

6

9

10

So 540 is divisible by 6, 9, and 10.

The number is divisible by both 2 and 3.



Divisible

Divisible

Divisible

Course 1

4-1 Divisibility


Tell whether 114 is divisible by 2, 3, 4, and 5.

2

3

4

5 Not Divisible

So 114 is divisible by 2 and 3.

The last digit, 4, is even.


The last two digits form the number 14. 14 is not divisible by 4.

Divisible

Divisible

Not Divisible


Course 1

4-1 Divisibility


Tell whether 810 is divisible by 6, 9, and 10.

6

9

10

So 810 is divisible by 6, 9, and 10.

The number is divisible by both 2 and 3.



Divisible

Divisible

Divisible

Course 1

4-1 Divisibility

Any number greater than 1 is divisible by at least two numbers—1 and the number itself. Numbers that are divisible by more than two numbers are called composite numbers.

A prime number is divisible by only the numbers 1 and itself. For example, 11 is a prime number because it is divisible by only 1 and 11. The numbers 0 and 1 are neither prime nor composite.

Course 1

4-1 Divisibility

Click to see which numbers from 1 through 50 are prime.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Tell whether each number is prime or composite.

Additional Example 2A & 2B: Identifying Prime and Composite Numbers

Course 1

4-1 Divisibility

A. 23 divisible by 1, 23 prime

B. 48 divisible by 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. composite

Additional Example 2C & 2D: Identifying Prime and Composite Numbers

Course 1

4-1 Divisibility

C. 31 divisible by 1, 31 prime

D. 18 divisible by 1, 2, 3, 6, 9, 18 composite




Course 1

4-1 Divisibility

A. 27 divisible by 1, 3, 9, 27 composite

B. 24 divisible by 1, 2, 3, 4, 6, 8, 12, 24 composite

Try This: Example 2C & 2D

Course 1

4-1 Divisibility

C. 11 divisible by 1, 11 prime

D. 8 divisible by 1, 2, 4, 8 composite


Lesson Quiz Tell whether each number is divisible by 2, 3, 4, 5, 6, 9, and 10.

1. 256

2. 720

3. 615


4. 47 5. 38

divisible by 2, 3, 4, 5, 6, 9, 10

divisible by 2, 4


divisible by 3, 5

Course 1

4-1 Divisibility

prime composite

4-2 Factors and Prime Factorization

Course 1

Warm Up

Problem of the Day

Lesson Presentation

Warm Up Identify each number as prime or composite.

1. 19 2. 82 3. 57 4. 85 5. 101 6. 121

prime composite

composite

Course 1


composite

composite prime

Problem of the Day

At the first train stop, 7 people disembarked. At the second stop, 8 people disembarked. At the fourth stop the last 6 people disembarked. If there were 28 people on the train before the first stop, how many people left at the third stop?

7 people left at the third stop

Course 1


Learn to write prime factorizations of composite numbers.

Course 1


Vocabulary

factor prime factorization


Course 1


Course 1


Whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors.

2 3 6 =

Factors Product

2 6 3 ÷ = 6 ÷ 2 = 3

6 is divisible by 3 and 2.

Course 1


When the pairs of factors begin to repeat, then you have found all of the factors of the number you are factoring.

Helpful Hint

Course 1


Additional Example 1A: Finding Factors List all factors of each number.

A. 16 Begin listing factors in pairs.

16 = 1 • 16 16 = 2 • 8

1 is a factor. 2 is a factor.

16 = 4 • 4 3 is not a factor. 4 is a factor. 5 is not a factor. 6 is not a factor. 7 is not a factor.

16 = 8 • 2 8 and 2 have already been listed so stop here.

The factors of 16 are 1, 2, 4, 8, and 16.

1 2 4 4 16 8 You can draw a diagram to illustrate the factor pairs.

Course 1


Additional Example 1B: Finding Factors List all factors of each number.

B. 19 Begin listing all factors in pairs.

19 = 1 • 19 19 is not divisible by any other whole number.

The factors of 19 are 1 and 19.

Course 1



List all factors of each number.

A. 12 Begin listing factors in pairs.

12 = 1 • 12 12 = 2 • 6

1 is a factor. 2 is a factor. 3 is a factor. 4 and 2 have already been listed so stop here.

12 = 4 • 3

1 2 4 3 12 6

The factors of 12 are 1, 2, 3, 4, 6, and 12

12 = 3 • 4

You can draw a diagram to illustrate the factor pairs.

Course 1



List all factors of each number.

B. 11 Begin listing all factors in pairs.

11 = 1 • 11 11 is not divisible by any other whole number.

The factors of 11 are 1 and 11.

Course 1


You can use factors to write a number in different ways.

Factorization of 12

2 • 6 1 • 12 3 • 4 3 • 2 • 2

The prime factorization of a number is the number written as the product of its prime factors.

Notice that these factors are all prime.

Course 1


You can use exponents to write prime factorizations. Remember that an exponent tells you how many times the base is a factor.

Helpful Hint

Course 1


Additional Example 2A: Writing Prime Factorizations

Write the prime factorization of each number.

A. 24 Method 1: Use a factor tree.

Choose any two factors of 24 to begin. Keep finding factors until each branch ends at a prime factor.

24

2 12 • 6

2

2 • 3 •

24

6 4 •

3 2 2 2

24 = 2 • 2 • 2 • 3

24 = 3 • 2 • 2 • 2

The prime factorization of 24 is 2 • 2 • 2 • 3, or 2 • 3 .

• •

3

Course 1


Additional Example 2B: Writing Prime Factorizations

Write the prime factorization of each number. B. 42

Method 1: Use a ladder diagram.

Choose a prime factor of 42 to begin. Keep dividing by prime factors until the quotient is 1.

3 42

2

1

14

7 7

42 = 3 • 2 • 7

The prime factorization of 42 is 2 • 3 • 7.

2 42

3

1

21

7 7

42 = 2 • 3 • 7

Course 1


In Example 2, notice that the prime factors may be written in a different order, but they are still the same factors. Except for changes in the order, there is only one way to write the prime factorization of a number.

Course 1



Write the prime factorization of each number. A. 28

Method 1: Use a factor tree.

Choose any two factors of 28 to begin. Keep finding factors until each branch ends at a prime factor.

28

2 14 • 7 2 •

28

7 4 • 2 2

28 = 2 • 2 • 7 28 = 7 • 2 • 2

The prime factorization of 28 is 2 • 2 • 7, or 22 • 7 .

•

Course 1




B. 36 Method 1: Use a ladder diagram.

Choose a prime factor of 36 to begin. Keep dividing by prime factors until the quotient is 1.

3 36

2

1

12

6 2

36 = 3 • 2 • 2 • 3

The prime factorization of 36 is 3 • 2 • 2 • 3, or 32 • 23.

3 36

3

1

12

2 4

36 = 3 • 3 • 2 • 2

3 3 2 2

Lesson Quiz

List all the factors of each number.

1. 22

2. 40

3. 51

1, 2, 4, 5, 8, 10, 20, 40

1, 2, 11, 22


1, 3, 17, 51

Course 1



4. 32

5. 120

25

23 × 3 × 5


Course 1

Warm Up

Problem of the Day

Lesson Presentation

Warm Up Write the prime factorization of each number.

1. 14 3. 63 2. 18 4. 54

2 × 7 32 × 7

2 × 32

Course 1


2 × 33

Problem of the Day

In a parade, there are 15 riders on bicycles and tricycles. In all, there are 34 cycle wheels. How many bicycles and how many tricycles are in the parade?

11 bicycles and 4 tricycles

Course 1


Learn to find the greatest common factor (GCF) of a set of numbers.

Course 1


Vocabulary

greatest common factor (GCF)


Course 1


Course 1


Factors shared by two or more whole numbers are called common factors. The largest of the common factors is called the greatest common factor, or GCF.

Factors of 24:

Factors of 36:

Common factors:

1, 2, 3, 4, 6, 8,

1, 2, 3, 4, 6,

The greatest common factor (GCF) of 24 and 36 is 12.

Example 1 shows three different methods for finding the GCF.

1, 2, 3, 4, 6, 9, 12, 12, 18,

24 36

12

Course 1


Additional Example 1A: Finding the GCF

Find the GCF of each set of numbers.

A. 28 and 42

Method 1: List the factors.

factors of 28:

factors of 42:

1, 2, 14, 7, 28

7, 1,

4,

3, 2, 42 6, 21, 14,

List all the factors.

Circle the GCF.

The GCF of 28 and 42 is 14.

Course 1


Additional Example 1B: Finding the GCF


B. 18, 30, and 24

Method 2: Use the prime factorization.

18 =

30 =

24 =

2

5 •

3

2

2

3

2

3

2 3


Find the common prime factors.

The GCF of 18, 30, and 24 is 6.

•

•

•

•

•

•

Find the product of the common prime factors.

2 • 3 = 6

Course 1


Additional Example 1C: Finding the GCF


C. 45, 18, and 27


3 3

5 2 3

45 18 27 Begin with a factor that divides into each number. Keep dividing until the three have no common factors.

Find the product of the numbers you divided by.

3 • 3 =


9

15 6 9

Course 1




A. 18 and 36

Method 1: List the factors.

factors of 18:

factors of 36:

1, 2, 9, 6, 18

6, 1,

3,

3, 2, 36 4, 12, 9,

List all the factors.

Circle the GCF.


18,

Course 1




B. 10, 20, and 30

Method 2: Use the prime factorization.

10 =

20 =

30 =

2

2 •

3

2

5

2

5

5


Find the common prime factors.


•

•

•

•

Find the product of the common prime factors.

2 • 5 = 10

Course 1




C. 40, 16, and 24


2 2 40 16 24 Begin with a factor that divides into

each number. Keep dividing until the three have no common factors.

Find the product of the numbers you divided by.

2 • 2 • 2 =


8

20 8 12

5 2 3 10 4 6 2

Course 1

4-3 Greatest Common Factor Additional Example 2:

Problem Solving Application Jenna has 16 red flowers and 24 yellow flowers. She wants to make bouquets with the same number of each color flower in each bouquet. What is the greatest number of bouquets she can make?

The answer will be the greatest number of bouquets 16 red flowers and 24 yellow flowers can form so that each bouquet has the same number of red flowers, and each bouquet has the same number of yellow flowers.


2 Make a Plan You can make an organized list of the possible bouquets.

Course 1


Solve 3

The greatest number of bouquets Jenna can make is 8.

Red Yellow Bouquets 2 3 RR

YYY

16 red, 24 yellow:

Every flower is in a bouquet

RR

YYY

RR

YYY

RR

YYY

RR

YYY

RR

YYY

RR

YYY

RR

YYY

Look Back 4 To form the largest number of bouquets, find the GCF of 16 and 24. factors of 16:

factors of 24:

1, 4, 2, 16 8,

1, 3, 24 8, 2, 4, 6, 12,


Course 1


Try This: Example 2 Peter has 18 oranges and 27 pears. He wants to make fruit baskets with the same number of each fruit in each basket. What is the greatest number of fruit baskets he can make?

The answer will be the greatest number of fruit baskets 18 oranges and 27 pears can form so that each basket has the same number of oranges, and each basket has the same number of pears.


2 Make a Plan

You can make an organized list of the possible fruit baskets.

Course 1


Solve 3

The greatest number of baskets Peter can make is 9.

Oranges Pears Bouquets

2 3 OO

PPP

18 oranges, 27 pears:

Every fruit is in a basket

OO

PPP

OO

PPP

OO

PPP

OO

PPP

OO

PPP

OO

PPP

OO

PPP

Look Back 4 To form the largest number of bouquets, find the GCF of 18 and 27. factors of 18:

factors of 27:

1, 3, 2, 18 6,

1, 9, 3, 27


OO

PPP

9,

Lesson Quiz: Part 1

1. 18 and 30

2. 20 and 35

3. 8, 28, 52

4. 44, 66, 88

5

6


4

Course 1


22

Find the greatest common factor of each set of numbers.

Lesson Quiz: Part 2

5. Mrs. Lovejoy makes flower arrangements. She has 36 red carnations, 60 white carnations, and 72 pink carnations. Each arrangement must have the same number of each color. What is the greatest number of arrangements she can make if every carnation is used?


Course 1


Find the greatest common factor of the set of numbers.

12 arrangements

4-4 Decimals and Fractions

Course 1

Warm Up

Problem of the Day

Lesson Presentation

Warm Up Find the GCF of each set of numbers.

1. 15, 24

2. 30, 60

3. 54, 102

4. 12, 16, 24

3

30

6

Course 1


4

Problem of the Day

Write the day of the month as the product of two factors.

Possible answer: 12th of the month, 6 × 2

Course 1


Learn to convert between decimals and fractions.

Course 1


Vocabulary mixed number terminating decimal repeating decimal


Course 1


Course 1


A number that contains both a whole number greater than 0 and a fraction, such as 1 , is called a mixed number. 3

4 __

__ 4 1 1 1 __

2 1

1 __ 4 3

2 __ 4 1

2 __ 2 1

Mixed numbers 12

14

34

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

Course 1


Additional Example 1A: Writing Decimals as Fractions or Mixed Numbers

Write each decimal as a fraction or mixed number.

A. 0.67

Identify the place value of the digit farthest to the right.

The 7 is in the hundredths place, so use 100 as the denominator.

0.67

100 ____ 67 Remember!

One

s

Tent

hs

Hun

dred

ths

Thou

sand

ths

Place Value

Course 1


Additional Example 1B: Writing Decimals as Fractions or Mixed Numbers


B. 5.9

Identify the place value of the digit farthest to the right. Write the whole number, 1. The 9 is in the tenths place, so use 10 as the denominator.

5.9

10 ___ 9 5

Course 1




A. 0.73

Identify the place value of the digit farthest to the right.

The 7 is in the hundredths place, so use 100 as the denominator.

0.73

100 ____ 73

Course 1




B. 4.8

Identify the place value of the digit farthest to the right. Write the whole number, 1. The 8 is in the tenths place, so use 10 as the denominator.

4.8

10 ___ 8 4

Course 1


Additional Example 2A: Writing Fractions as Decimals

Write each fraction or mixed number as a decimal.

A.

Divide 3 by 20. Add zeros after the decimal point.

0 20 ____ 3

The remainder is 0.

.00 - 20

.1 3 20

100 - 100

5

0

20 ____ 3 = 0.15

Course 1


To write a repeating decimal, you can show three dots or draw a bar over the repeating part: 0.666… = 0.6

Writing Math

Course 1


Additional Example 2B: Writing Fractions as Decimals

A.

Divide 1 by 3. Add zeros after the decimal point.

0 3 __ 1

The three repeats in the quotient.

.000 - 9

.3 1 3

10 - 9

3

10

=

6

- 9

3

1 3 __ 1

6 6.333… = 6.3 _


Course 1




A. Divide 5 by 20. Add zeros after the decimal point.

0 20 ____ 5

The remainder is 0.

.00 - 40

.2 5 20

100 - 100

5

0

20 ____ 5 = 0.25

Course 1




B. Divide 2 by 3. Add zeros after the decimal point.

0 3 __ 2

The six repeats in the quotient.

.000 - 18

.6 2 3

20 - 18

6

20

=

7

- 18

6

2 3 __ 2

7 7.666… = 7.6 _

Course 1


A terminating decimal, such as 0.75, has a finite number of decimal places. A repeating decimal, such as 0.666…, has a block of one or more digits that repeat continuously

Common Fractions and Equivalent Decimals

0.2 0.25 0.3 0.4 0.5 0.6 0.6 0.75 0.8

1 3 __ 1

5 __ 1

4 __ 2

5 __ 1

2 __ 3

5 __ 2

3 __ 3

4 __ 4

5 __

Course 1


Additional Example 3: Comparing and Ordering Fractions and Decimals

Order the fractions and decimals from least to greatest.

, 0.8, 7 __ 10

3 __ 4

First rewrite the fractions as a decimal. = 0.75 = 0.7

Order the three decimals.

3 __ 4

0

7 __ 10

__ 10 7

3 __ 4

0.75 0.8 0.7

The numbers in order from least to greatest are , , and 0.8.

7 10

3 4

Course 1


Try This: Example 3

Order the fractions and decimals from least to greatest.

, 0.35, 1 __ 4

1 __ 2

First rewrite the fractions as decimals. = 0.5 = 0.25

Order the three decimals.

1 __ 2

0

1 __ 4

1 __ . 2

The numbers in order from least to greatest are , 0.35, and 1 __ 4

1 __ 2

0.35 0.5 0.25

1 __ 4

Lesson Quiz: Part 1 Write each decimal as a fraction or mixed number.

1. 0.24

2. 6.75


3. 2 4.


2.6 0.875

Course 1


3 5

__ 8 __ 7

6 25

____

6 3 4

__

Lesson Quiz: Part 2

5. Order the fractions and decimals from least to greatest.

0.38, ,

6. Jamal ran the following distances on three different days: 0.87 mile, mile, and 0.13 mile. Order the distances from least to greatest.

0.13, , 0.87


Course 1

4-4 Decimals and Fraction

3 4 __ 1

5 __

2 3 __

1 5 __ , 0.38, 3

4 __

2 3 __


Course 1

Warm Up

Problem of the Day

Lesson Presentation

Warm Up List the factors of each number.

1. 8

2. 10

3. 16

4. 20

5. 30

1, 2, 4, 8

1, 2, 5, 10

1, 2, 4, 8, 16

Course 1


1, 2, 4, 5, 10, 20

1, 2, 3, 5, 6, 10, 15, 30

Problem of the Day

John has 3 coins, 2 of which are the same. Ellen has 1 fewer coin than John, and Anna has 2 more coins than John. Each girl has only 1 kind of coin. Who has coins that could equal the value of a half-dollar? Ellen and Anna

Course 1


Learn to write equivalent fractions.

Course 1


Vocabulary equivalent fractions simplest form


Course 1


Course 1


Fractions that represent the same value are equivalent fractions. So , , and are equivalent fractions.

= =

1 2 __ 2

4 __ 4

8 __

12

24

48

Course 1


Additional Example 1: Finding Equivalent Fractions

Find two equivalent fractions for . 10 12 ___

10 12 ___ 5

6 __ 15

18 ___

10 12 ___ 15

18 ___ 5

6 __

= =

So , , and are all equivalent fractions.

Course 1


Try This: Example 1

Find two equivalent fractions for . 4 6 __

4 6

__ 2 3 __ 8

12 ___

4 6

__ 8 12 ___ 2

3 __

= =

So , , and are all equivalent fractions.

Course 1


Additional Example 2A: Multiplying and Dividing to Find Equivalent Fractions

Find the missing number that makes the fractions equivalent.

A. 3 5 __

20 ___ =

3 5

______

In the denominator, 5 is multiplied by 4 to get 20.

• 4 • 4

Multiply the numerator, 3, by the same number, 4.

= 12 20

____

So is equivalent to . 3 5

__ 12 20 ___

3 5

__ 12 20 ___ =

Course 1


Additional Example 2B: Multiplying and Dividing to Find Equivalent Fractions


B. 4 5 __ 80 ___ =

4 5 ______

In the numerator, 4 is multiplied by 20 to get 80.

• 20 • 20

Multiply the denominator by the same number, 20.

= 80 100 ____


__ 80 100 ___

4 5

__ 80 100 ___ =

Course 1



Find the missing number that makes the fraction equivalent.

A. 3 9 __

27 ___ =

3 9

______

In the denominator, 9 is multiplied by 3 to get 27.

• 3 • 3

Multiply the numerator, 3, by the same number, 3.

= 9 27

____


__ 9 27 ___

3 9

__ 9 27 ___ =

Course 1



Find the missing number that makes the fraction equivalent.

B. 2 4 __ 40 ___ =

2 4 ______

In the numerator, 2 is multiplied by 20 to get 40.

• 20 • 20

Multiply the denominator by the same number, 20.

= 40 80

____


__ 40 80 ___

2 4

__ 40 80 ___ =

Course 1


Every fraction has one equivalent fraction that is called the simplest form of the fraction. A fraction is in simplest form when the GCF of the numerator and the denominator is 1.

Example 3 shows two methods for writing a fraction in simplest form.

Course 1


Additional Example 3A: Writing Fractions in Simplest Form

Write the fraction in simplest form.

A. 20 48 ___

The GCF of 20 and 48 is 4, so is not in simplest form.

20 48 ___

Method 1: Use the GCF. 20 48 _______ ÷ 4

÷ 4 Divide 20 and 48 by their GCF, 4. = 5

12 __

Course 1


Additional Example 3A: Writing Fractions in Simplest Form



Use a ladder. Divide 20 and 48 by any common factor (except 1) until you cannot divide anymore

20 48 ___

2 20/48 2 10/24

5/12 5 12 ___

So written in simplest form is .

Method 2 is useful when you know that the numerator and denominator have common factors, but you are not sure what the GCF is.

Helpful Hint

Course 1


Additional Example 3B: Writing Fractions in Simplest Form


B. 7 10 ___

The GCF of 7 and 10 is 1 so is already in simplest form.

7 10 ___

Course 1




A. 12 16 ___

The GCF of 12 and 16 is 4, so is not in simplest form.

12 16 ___

Method 1: Use the GCF. 12 16 _______ ÷ 4

÷ 4 Divide 12 and 16 by their GCF, 4. = 3

4 __

Course 1





Use a ladder. Divide 20 and 48 by any common factor (except 1) until you cannot divide anymore

2 12/16 2 6/8

3/4

12 16 ___ 3

4 ___

So written in simplest form is .

Course 1




B. 3 10 ___

The GCF of 3 and 10 is 1, so is already in simplest form.

3 10 ___

Lesson Quiz Write two equivalent fractions for each given fraction.

1. 2.


3. 4.

Write each fraction in simplest form.

5. 6.


6

Course 1


4 10 ___ 7

14 ___

2 7

__ ___ 21

= 4 15 __ ___ 20 =

4 8

__ 7 49 ___ 1

7 ___ 1

2 __

75

1 2

___ 14 28 ___ , 8

20 ___ 2

5 ___ ,

Possible answers


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the missing number that makes the fraction equivalent.

1. 2. 3. 4.

9 8

15

Course 1


3 5

__ 15 =

3 4

__ 20

=

2 3

__ ___ 12 =

5 8

__ 24 = 15

Course 1


Problem of the Day

From 4:00 to 5:30, Carlos, Lisa, and Toni took turns playing the same computer game. Carlos played for hour, and Lisa played for hour. For how many minutes did Toni play the game?

3 4

___

1 2

___

15 minutes

Course 1


Learn to use pictures and number lines to compare and order fractions.

Course 1


Vocabulary like fractions unlike fractions common denominator

Course 1


When you are comparing fractions, first

check their denominators. When

fractions have the same denominator,

they are called like fractions. For

example, and are like fractions. 6 7

___ 4 7

___

Course 1


Additional Example 1A: Comparing Like Fractions

Compare. Write <, >, or =.

6 7 __ 4 __ A.

7

6 7

__ __ 7 4 >

From the model, > . 6 7

__ __ 7 4

Course 1


Additional Example 1B: Comparing Like Fractions


1 9 __ 5 __ B.

9

1 9

__ __ 9 5 <

From the model, < . 1 9

__ __ 9 5

Course 1




4 6 __ 5 __ A.

6

4 6

__ __ 6 5 <

From the model, < . 4 6

__ __ 6 5

Course 1




3 9 __ 2 __ B.

9

3 9

__ __ 9 2 >

From the model, > . 3 9

__ __ 9 2

Course 1


When two fractions have different denominators, they are called unlike fractions. To compare unlike fractions, first rename the fractions so they have the same denominator. This is called finding a common denominator.

Course 1


Additional Example 2: Cooking Application

2 3

__

3 4

__ Compare and . 2 3

__

3 4

__

Find a common denominator by multiplying the denominators.

3 • 4 = 12

Ray has cup of nuts. He needs cup to

make cookies. Does he have enough nuts for the recipe?

Course 1



Find equivalent fractions with 12 as the denominator.

2 3

__ 12 __ =

2 3 ______

• 4 = • 4 __ 12 8

3 4

__ 12 __ =

3 4 ______

• 3 • 3 __

12 9

2 3

__ 3 4

__ __ 12 8 =

=

= __ 12 9

Course 1



Compare the like fractions. __ 12 8 __

12 9 2

3 __ 3

4 __ < , so < .

Since cup is less than cup, he does not

have enough.

2 3

__ 3 4

__

Course 1


Try This: Example 2

1 3

__

1 4

__ Compare and . 1 3

__

1 4

__

Find a common denominator by multiplying the denominators.

3 • 4 = 12

Trevor has cup of soil. He needs cup to

fill a small planter. Does he have enough soil to fill the planter?

Course 1


Find equivalent fractions with 12 as the denominator.

1 3

__ 12 __ =

1 3 ______

• 4 = • 4 __ 12 4

1 4

__ 12 __ =

1 4 ______

• 3 • 3 __

12 3

1 3

__ 1 4

__ __ 12 4 =

=

= __ 12 3


Course 1


Compare the like fractions.

__ 12 4 __

12 3 1

3 __ 1

4 __

> , so > .

Since cup is more than cup, he does have

enough.

1 3

__ 1 4

__


Course 1


Additional Example 3: Ordering Fractions

Order , , and from least to greatest. 4 5 __

4 5

_______

2 3 __

2 3

_______ 1 3

_______ • 3 • 3

• 5 • 5

• 5 • 5 10

15 __ 12

15 __ = = = 5

15 __ Rename with like

denominators.

1 3 __

1 3

__ 2 3

__ The fractions in order from least to greatest are , , . 4

5 __

5 15

1 3

10 15

2 3

0 1

12 15

4 5

Course 1


Numbers increase in value as you move from left to right on a number line.

Remember!

Course 1


Try This: Example 3

Order , , and from least to greatest. 4 7 __

4 7

_______

3 4 __

3 4

_______ 1 4

_______ • 4 • 4

• 7 • 7

• 7 • 7 21

28 __ 16

28 __ = = = 7

28 __ Rename with like

denominators.

1 4 __

1 4

__ 4 7

__ The fractions in order from least to greatest

are , , . 3 4

__

7 28

4 7

16 28

3 4

21 28

1 4

0 1

Lesson Quiz


1. 2.

3. You drilled three holes in a piece of wood. The diameters of the holes are , , and inches. Which hole is the largest?

Order the fractions from least to greatest.

4. 5.

=


Course 1


7 8

__,

__ 8 4 > 5

8 __

16 9 __

1 8

__ 3 8

__ 3 16 __

3 6

__

5 8

__ , 2 3

__ 8 __, 5 2

3 __, 7

8 __ 3

4 __ , 5

8 __ , 5

6 __

8 5 5

6 __ __, 3

4 __ ,

3 8

__

4-7 Mixed Numbers and Improper Fractions

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Order the fractions from least to greatest.

1. 2. 3.

Course 1


2 9

__ 1 6

__ 2 3

__

7 12 __ 5

6 __ 2

3 __

5 8

__ 1 2

__ 4 11 __

, ,

, ,

, , 5 8

__ 1 2

__ 4 11 __ , ,

2 3

__ 5 6

__ 7 12 __ , ,

2 9

__ 2 3

__ 1 6

__ , ,

Problem of the Day

Two numbers have a product of 48. When the larger number is divided by the smaller, the quotient is 3. What are the numbers? 4 and 12

Course 1


Learn to convert between mixed numbers and improper fractions.

Course 1


Course 1


Vocabulary improper fraction proper fraction

Course 1


An improper fraction is a fraction in which the numerator is greater than or equal to the denominator such as .

11 4 __

is read as “eleven-fourths.”

Reading Math 11 4

__

Course 1


Whole numbers can be written as improper fractions. The whole number is the numerator, and the denominator is 1. For example, 7 = .

When the numerator is less than the denominator, the fraction is called a proper fraction.

7 1 __

Course 1


Improper and Proper Fractions Improper Fractions • Numerator equals denominator fraction is equal to 1 •Numerator greater than denominator fraction is greater than 1

Proper Fractions •Numerator less than denominator fraction is less than 1

3 3 __

2 5 __

9 5 __

=

>

1 102 102 ____

13 1 __

102 351 ____

= 1

1 > 1

< 1 < 1

You can write an improper fraction as a mixed number.

Course 1


Additional Example 1: Application

Ella hiked for hours yesterday. Write as a mixed number.

9 4 __ 9

4 __

Method 1: Use a model.

Draw squares divided into fourth sections. Shade 9 of the fourth sections.

1 4 __

1 4

1 4

1 4

1 4

There are 2 whole squares and 1 fourth square, or 2 squares, shaded.

1 4

1 4

1 4

1 4

1 4

1 4

1 4

1 4

1 2 1 4 __

Course 1



Method 2: Use division. 1 4 __

Divide the numerator by the denominator.

To form the fraction part of the quotient, use the remainder as the numerator and the divisor as the denominator.

4 9 - 8

2

1

Ella hiked for 2 hours. 1 4 __

Course 1


Try This: Example 1

Arnold biked for hours yesterday. Write

as a mixed number.

7 4 __ 7

4 __

Method 1: Use a model. Draw squares divided into fourth sections. Shade 7 of the fourth sections.

3 4 __

1 4

1 4

1 4

1 4

There is 1 whole square and 3 fourth squares, or 1 squares, shaded.

1 4

1 4

1 4

1 4

1 3 4 __

Course 1


Try This 1 Continued

Method 2: Use division.

3 4 __


To form the fraction part of the quotient, use the remainder as the numerator and the divisor as the denominator.

4 7 - 4

1

3

Arnold biked for 1 hours. 3 4 __

Course 1


Additional Example 2: Writing Mixed Numbers as Improper Fractions

Write 3 as an improper fraction. 2 3 __

Method 1: Use a model. You can draw a diagram to illustrate the whole and fractional parts.

There are 11 thirds or . 11 3 __ Count the thirds in the diagram.

Course 1


Additional Example 2 Continued Method 2: Use multiplication and addition. When you are changing a mixed number to an improper fraction, spiral clockwise as shown in the picture. The order of operations will help you remember to multiply before you add.

2 3 __ 3 =

(3 • 3) + 2 __________ 3

Multiply the whole number by the denominator and add the numerator.

Keep the same denominator.

= 9 + 2 _____ 3

= 11 3 __

Multiply.

Then add.

2 3 3

Course 1


Try This: Example 2

Write 4 as an improper fraction. 1 3 __

Method 1: Use a model. You can draw a diagram to illustrate the whole and fractional parts.

There are 13 thirds, or . 13 3 __ Count the thirds in the diagram.

Course 1


Try This: Example 2 Continued Method 2: Use multiplication and addition. When you are changing a mixed number to an improper fraction, spiral clockwise as shown in the picture. The order of operations will help you remember to multiply before you add.

1 3 __ 4 =

(3 • 4) + 1 __________ 3

Multiply the whole number by the denominator and add the numerator.

Keep the same denominator.

= 12 + 1 ______ 3

= 13 3 __

1 3 __ 4

Multiply.

Then add.

Lesson Quiz Write each mixed number as an improper fraction.

1. 3 2. 6

3. 10 4. 4

5. Janet watches hours of television per day. Write as a mixed number.


Course 1


3 5 __ 1

9 __

1 2 __ 1

8 __

7 4 __

7 4 __

55 9 __ 18

5 __

21 2 __ 33

8 __

3 4 __ 1


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Course 1


Warm Up Write each mixed number as an improper fraction.

1. 6 2. 2 3. 3 4. 8 5. 9 6. 4

1 3

__ 1 5

__

1 2

__

8 11 __

2 7

__

4 5

__

19 3

__ 11 5

__

23 7

__ 17 2

__

49 5

__ 52 11 __

Problem of the Day

Marge has six coins that total $1.15. She cannot make change for $1.00, a half-dollar, a quarter, a dime, or a nickel. What coins does she have? half dollar, quarter, 4 dimes

Course 1


Learn to add and subtract fractions with like denominators.

Course 1



1 4 __

1 4 __

1 2 __

1 2 __

1 4 __

2 4 __

+

1 4 __ 1

4 __ + =

=

After 2 hours inch of snow fell

Add the numerators. Keep the same denominator

Write your answer in simplest terms.

Snow was falling at a rate of inch per hour. How

much snow fell after two hours? Write your answer in simplest form.

+ =

Course 1


Try This: Example 1

1 8 __

1 8 __

1 4 __

1 8 __

1 8 __

2 8 __

+

1 8 __ 1

8 __ + =

=

After 2 hours inch of rain fell.

Add the numerators. Keep the same denominator.

Write your answer in simplest terms.

Rain was falling at a rate of inch per hour. How

much rain fell after two hours? Write your answer in simplest form.

+ =

Course 1


Additional Example 2A: Subtracting Like Fractions and Mixed Numbers

Subtract. Write the answer in simplest form. A. 1 –

5 5 __ 3

5 __

3 5 __

2 5 __ = –

To get a common denominator, rewrite 1 as a fraction with a denominator of 5.

Subtract the numerators. Keep the same denominator.

= ―

Course 1


Additional Example 2B: Subtracting Like Fractions and Mixed Numbers

Subtract. Write the answer in simplest form. B. 5 – 2

1 12 __

4 12 __

– Subtract the fractions.

Then subtract the whole numbers.

= -

5 12 __

5 12 __ 5 2

1 12 __

3

3 1 3 __ Write your answer in simplest form.

Course 1



Subtract. Write the answer in simplest form. A. 1 –

6 6 __ 2

6 __

2 6 __

4 6 __ = –

To get a common denominator, rewrite 1 as a fraction with a denominator of 6. Subtract the numerators. Keep the same denominator.

= –

Course 1



4 12 __

3 12 __

– Subtract the fractions.

Then subtract the whole numbers.

= -

7 12 __

7 12 __ 4 2

4 12 __

2

2 1 4 __ Write your answer in simplest form.

Subtract. Write the answer in simplest form. B. 4 – 2

Course 1


Evaluate the expression for x = . Write each answer in simplest form.

5 9 __ A.

2 9 __

5 9 __

5 9 __

–

–

– x

x

2 9 __ 3

9 __ =

1 3 __ =

Write the expression.

Substitute for x and subtract the

numerators. Keep the same denominator.

Write your answer in simplest form.

2 9 __

Course 1


Additional Example 3A: Evaluating Expressions with Fractions


4 9 __ B.

2 9 __

4 9 __

4 9 __

+

+

+ x

x

2 9 __ 6

9 __ =

2 3 __

=


Substitute for x. Add the

fractions. Then add the whole numbers.


2 9 __

2

2

2 2

2 , or 8 3 __

Course 1


Additional Example 3B: Evaluating Expressions with Fractions


8 9 __ C.

2 9 __

8 9 __

8 9 __

+

+

+ x

x

2 9 __ 10

9 __ =


Substitute for x and subtract

the numerators. Keep the same denominator.

2 9 __

1 , or 1 9 __

Course 1


Additional Example 3C: Evaluating Expressions with Fractions



4 6 __ A.

2 6 __

4 6 __

4 6 __

–

–

– x

x

2 6 __ 2

6 __ =

1 3 __ =


Substitute for x and subtract the

numerators. Keep the same denominator. Write your answer in simplest form.

2 6 __

Course 1




1 6 __ B.

2 6 __

1 6 __

1 6 __

+

+

+ x

x

2 6 __ 3

6 __ =

1 2 __

=



the numerators. Keep the same denominator. Write your answer in simplest form.

2 6 __

3

3

3 3

3

Course 1




5 6 __ C.

2 6 __

5 6 __

5 6 __

+

+

+ x

x

2 6 __ 7

6 __ =



the numerators. Keep the same denominator.

2 6 __

1 , or 1 6 __

Course 1


Lesson Quiz Add or subtract. Write each answer in simplest form.

1. 1 – 2. 1 –

4. Evaluate – x for x = .

9 11 __

3 10 __

8 9 __

1 10 __

6 21 __ 16

21 ___

3 4 __

2 11 __ 1

9 __

2 5 __

10 21 __

6

3 miles

3. 2 + 4

5. Weston walks of a mile every day. How far

will he have walked in 4 days?

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Multiply.

30 96 144

Course 1


66 120

1. 15 × 2

2. 12 × 8 3. 9 × 16 4. 6 × 11

5. 8 × 15

Problem of the Day

The length of three dowels are 13 inches, 27 inches, and 19 inches. How can you use the three dowels to mark off a length of 5 inches? 13 + 19 – 27

Course 1


Learn to multiply fractions by whole numbers.

Course 1


Course 1


Recall that multiplication by a whole number can be represented as repeated addition. For example, 4 • 5 = 5 + 5 + 5 + 5. You can multiply a whole number by a fraction using the same method.

+ + =

3 • = 3 4 __ 1

4 __

Course 1


3 1

__ 1 4

__ 3 • 1 1 • 4

______ 3 4

__ •

There is another way to multiply with fractions. Remember that a whole number can be written as an improper fraction with 1 in the denominator. So 3 = . 3

1 __

= = Multiply numerators Multiply denominators

Course 1


Additional Example 1A: Multiplying Fractions and Whole Numbers

Multiply. Write the answer in simplest form.

1 9

__

1 9

__ 7 1

__ 7 • 1 1 • 9

______

7 9

__

• =

=

Write 7 as a fraction. Multiply numerators and denominators.

A. 7 •

Course 1


Additional Example 1B: Multiplying Fractions and Whole Numbers


1 8

__

1 8

__ 6 1

__ 6 • 1 1 • 8

______

6 8

__

• =

=


3 4

__ = Write your answer in simplest form.

B. 6 •

Course 1


Additional Example 1C: Multiplying Fractions and Whole Numbers


2 3

__

2 3

__ 8 1

__ 8 • 2 1 • 3

______

16 3

___

• =

=


1 3

__ or 5

C. 8 •

Course 1




1 5

__

1 5

__ 4 1

__ 4 • 1 1 • 5

______

4 5

__

• =

=


A. 4 •

Course 1




1 6

__

1 6

__ 3 1

__ 3 • 1 1 • 6

______

3 6

__

• =

=


1 2

__ = Write your answer in simplest form.

B. 3 •

Course 1




3 4

__

3 4

__ 9 1

__ 9 • 3 1 • 4

______

27 4

___

• =

=


3 4

__ or 6

C. 9 •

Course 1


Additional Example 2A: Evaluating Fraction Expression

Evaluate 4x for the value of x. Write the answer in simplest form.

1 10

___

4 1

__

1 10

___

4 10

___

•

=

Write the expression. 4x

4

• 1 10

___

= 2 5

__

Multiply.

Substitute for x. 1 10

___


A. x =

Course 1


Additional Example 2B: Evaluating Fraction Expression

Evaluate 4x for the value of x. Write the answer in simplest form.

3 8

__

4 1

__

3 8

__

12 8

___

•

=


4

• 3 8

__

= 3 2

__

Multiply.


__

Write your answer in simplest form. 1 2

__ ,or 1

B. x =

Course 1


Try This: Example 2A Evaluate 3x for the value of x. Write the answer in simplest form.

1 9

__

3 1

__

1 9

__

3 9

__

•

=


3

• 1 9

__

= 1 3

__

Multiply.


__


A. x =

Course 1


Try This: Example 2B Evaluate 3x for the value of x. Write the answer in simplest form.

4 7

__

3 1

__

4 7

__

12 7

___

•

=


3

• 4 7

__

=

Multiply.


__

Write your answer in simplest form. 5 7

__ 1

B. x =

Course 1


Sometimes the denominator of an improper fraction will divide evenly into the numerator. When this happens, the improper fraction is equivalent to a whole number, not a mixed number.

This makes sense if you remember that the fraction bar means “divided by”.

12 3

___ = 4

Course 1


Additional Example 3: Application There are 25 students in the music club. Of those

students, are also in the band. How many music

club students are in the band?

3 5

__

5 3 __ To find of 25, multiply.

25 1

___ 3 5

__ • 25 = • 3 5

__

75 5

___ =

= 15

There are 15 music club students in the band.

Course 1


Try This: Example 3 There are 18 saltwater fish in the aquarium. Of those

fish, are blue. How many fish in the aquarium

are blue?

2 3

__

2 3

__

To find of 18, multiply.

18 1

___ 2 3

__ • 18 = • 2 3

__

36 3

___ =

= 12

There are 12 blue fish in the aquarium.

Course 1


Lesson Quiz

Multiply. Write each answer in simplest form.

1. 10 • 2. 6 •

Evaluate 6x for each value of x. Write your answer in simplest form.

3. x = 4. x =

5. Alicia spent 15 minutes making a pizza. Of those minutes were spent rolling out the crust. How many

minutes did Alicia spend rolling out the crust?

2 5

__ 1 10

__

1 4

__ 5 6

__

1 3

__

5 minutes

5

4 3 5

__

, or 1 1 2

__ 3 2

__


Course 1

Warm Up

Problem of the Day Lesson Presentation

Warm Up

1. What is of 12? 2. What is of 100? 3. What is of 120? 4. What is of 100? 5. What is of 480?

6

50

60

Course 1


1 2 __

1 2 __

1 2 __

1 4 __

1 4 __

25

120

Problem of the Day

Your favorite uncle left one-fifth of his estate to each of his three children and the rest to his favorite charity. If his estate was worth $105,000, how much was given to charity? $42,000

Course 1


Learn to multiply fractions.

Course 1


Course 1


On average, people spend of their lives asleep. About

of the time they sleep, they dream. What fraction of

a lifetime does a person typically spend dreaming?

1 3 __

1 3 __

1 4 __

1 4 __

1 4 __ 1

3 __

1 4 __ 1

3 __ • =

1 12 __

One way to find of is to make a model. Find of .

Course 1


You can also multiply fractions without making a model.

1 4 __ 1

3 __ •

1 • 1 4 • 3

_____

1 12 __

=

=

Multiply the numerators. Multiply the denominators.

A person typically spends of his or her lifetime dreaming.

1 12 __

Course 1


You can look for a common factor in a numerator and a denominator to determine whether you can simplify before multiplying.

Helpful Hint

Course 1


Additional Example 1A: Multiplying Fractions Multiply. Write the answer in simplest form.

A.

1 4 __ 2

5 __ • _____ =

=

Multiply numerators. Multiply denominators.

2 20 __

1 • 2 4 • 5

1 4 __ •

2 5 __


The answer is in simplest form. = 1 10 __

Course 1


Additional Example 1B: Multiplying Fractions Multiply. Write the answer in simplest form.

B.

5 7 __ 4

15 __ •

_____

=

=

Use the GCF to simplify the fractions before multiplying. The greatest common factor of 5 and 15 is 5.

4 21 __

1 • 4 7 • 3

5 7 __ •

4 15 __


The answer is in simplest form.

1 7 __ •

4 3 __ 1

3

=

Course 1


Additional Example 1C: Multiplying Fractions Multiply. Write the answer in simplest form.

C.

4 9 __ 6

10 __ • ______ =

=


24 90 __

4 • 6 9 • 10

4 9 __ •

6 10 __



Course 1


Try This: Example 1A Multiply. Write the answer in simplest form.

A.

1 2 __ 3

6 __ • _____ =

=


3 12 __

1 • 3 2 • 6

1 2 __ •

3 6 __



Course 1


Try This: Example 1B Multiply. Write the answer in simplest form.

B.

3 7 __ 5

9 __ •

_____

=

=

Use the GCF to simplify the fractions before multiplying. The greatest common factor of 3 and 9 is 3.

5 21 __

1 • 5 7 • 3

3 7 __ •

5 9 __



1 7 __ •

5 3 __ 1

3

=

Course 1


Try This: Example 1C Multiply. Write the answer in simplest form.

C.

3 7 __ 5

15 __ • ______ =

=


15 105 ____

3 • 5 7 • 15

3 7 __ •

5 15 __



Course 1

5-1 Multiplying Fractions Additional Example 2A: Evaluating Fraction

Expressions 2 5 __

1 3 __ b • 2

5 __

2 5 __ 1

3 __ •

_____ 1 • 2 3 • 5

2 15 __

Substitute for b. 1 3 __

Multiply.


A. b =

Evaluate the expression b • for each value of b. Write the answer in simplest form.

Course 1


Additional Example 2B: Evaluating Fraction Expressions

B. b = 3 8 __ b •

2 5 __

2 5 __ 3

8 __ •

_____ 3 • 1 4 • 5

3 20 __


Use the GCF to simplify.


1

4

2 5 __ 3

8 __ •

Multiply.

Course 1


Additional Example 2C: Evaluating Fraction Expressions

C. b = 5 7 __ b • 2

5 __

2 5 __ 5

7 __ •

_____ 5 • 2 7 • 5 10 35 __


Multiply.


The answer is in simplest form. 2 7 __

Course 1


Try This: Example 2A Evaluate the expression c • for each value of c.

Write the answer in simplest form.

A. c =

1 4 __

1 7 __ c • 1

4 __

1 4 __ 1

7 __ •

_____ 1 • 1 7 • 4

1 28 __

Substitute for c. 1 7 __

Multiply.


Course 1



B. c = 2 7 __ c •

1 4 __

1 4 __ 2

7 __ •

_____ 1 • 1 7 • 2

1 14 __


Use the GCF to simplify.


1

2

1 4 __ 2

7 __ •

Multiply.

Course 1



C. c = 4 7 __ c • 1

4 __

1 4 __ 4

7 __ •

_____ 4 • 1 7 • 4

4 28 __


Multiply.


The answer is in simplest form. 1 7 __

Lesson Quiz


1. 2.

Evaluate the expression x • for each value of x. Write the answer in simplest form.

3. x = 4. x =

5. At a particular college of the students take a math class. Of these students take basic algebra. What fraction of all students take basic algebra?


Course 1


3 5 __ 1

4 __ •

5 7 __ 3

10 __ •

1 8 __

4 5 __ 1

8 __

2 5 __

1 4 __

3 20 __ 3

14 __

1 10 __ 1

64 __

1 10 __

5-2 Multiplying Mixed Numbers

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Multiply.

1. 2. 3. 4.

Course 1


3 8 __ 5

8 __ • 2

• 4 5

__ 3 8

__

3 11 __

1 2 __ 1

3 __ •

•

1 6

__ 3 10 __

6 11 __ 15

64 __

Problem of the Day

Tracy and Zachary are sharing a large pizza. Tracy cuts the pizza in half. Zachary says he cannot eat that much. Zachary cuts his half into fourths and eats one piece. What fraction of the whole pizza does Zachary not eat?

Course 1


7 8 __

Learn to multiply mixed numbers.

Course 1


Course 1


To write a mixed number as an improper fraction, start with the whole number, multiply by the denominator, and add the numerator. Use the same denominator.

Remember!

1 5

__ 1 1 • 5 + 1 5

_________ 6 5 __ = =

Course 1


Additional Example 1A: Multiplying Fractions and Mixed Numbers


A. • 1 1 4

__ 1 3

__

Write 1 as an improper fraction. 1 = 4 3 __ 1

3 __ 1

3 __ 1

4 __ 4

3 __ •

1 • 4 4 • 3

_____ Multiply numerators. Multiply denominators.

4 12 __

1 3 __ Write the answer in simplest form.

Course 1


Additional Example 1B: Multiplying Fractions and Mixed Numbers


B. 3 • 1 2

__ 4 5

__


2 __ 1

2 __ 7

2 __ 4

5 __ •

7 • 4 2 • 5


28 10 __

14 5 __ Write the answer in simplest form. You

can write the answer as a mixed number. 4 5 __ = 2

Course 1


Additional Example 1C: Multiplying Fractions and Mixed Numbers


C. • 2 12 13 __ 3

8 __


8 __ 3

8 __ 12

13 __ 19

8 __ •

3 • 19 13 • 2 ______

Use the GCF to simplify before multiplying.

57 26 __ You can write the answer as a mixed number. 5

26 __ = 2

3

2

12 13 __ 19

8 __ •

Course 1




A. • 1 1 9

__ 1 2

__


2 __ 1

2 __ 1

9 __ 3

2 __ •

1 • 3 9 • 2


3 18 __

1 6 __ Write the answer in simplest form.

Course 1




B. 2 • 2 3

__ 5 6

__


3 __ 2

3 __ 8

3 __ 5

6 __ •

8 • 5 3 • 6


40 18 __

20 9 __ Write the answer in simplest form. You

can write the answer as a mixed number. 2 9 __ = 2

Course 1




C. • 3 14 15 __ 2

7 __


7 __ 2

7 __ 14

15 __ 23

7 __ •

2 • 23 15 • 1 ______

Use the GCF to simplify before multiplying.

46 15 __ You can write the answer as a mixed

number. 1 15 __ = 3

2

1

14 15 __ 23

7 __ •

Course 1


Additional Example 2A: Multiplying Mixed Numbers

Find each product. Write the answer in simplest form. A. 1 • 2 2

3 __ 1

7 __

5 3 __ 15

7 __ •

5 • 15 3 • 7

______ Multiply numerators. Multiply denominators.

75 21 __

12 21 __ You can write the improper fraction as a

mixed number.

Write the mixed numbers as improper

fractions. 1 = 2 = 5 3 __ 15

7 __ 2

3 __ 1

7 __

3

Simplify. 4 7 __ 3

Course 1


Additional Example 2B: Multiplying Mixed Numbers

Find each product. Write the answer in simplest form. B. 1 • 2 3

8 __ 2

5 __

11 8 __ 12

5 __ •

11 • 3 2 • 5

______

Use the GCF to simply before multiplying.

33 10 __



fractions. 1 = 2 = 11 8 __ 12

5 __ 3

8 __ 2

5 __

Simplify. 3 10 __ 3

11 8 __ 12

5 __ •

3

2

Course 1


Additional Example 2C: Multiplying Mixed Numbers

Find each product. Write the answer in simplest form. C. 2 • 4 2

5 __

Use the Distributive Property.

Add.

4 5 __ 8 +

2 • 4 2 5 __

2 • (4 + ) 2 5 __

(2 • 4) + ( 2 • )

2 1 __

Multiply.

(2 • 4) + ( • ) 2 5 __

2 5 __

4 5 __ 8

Course 1


Try This: Example 2A Find each product. Write the answer in simplest form.

A. 1 • 2 3 4

__ 1 6

__

7 4 __ 13

6 __ •

7 • 13 4 • 6

______ Multiply numerators. Multiply denominators.

91 24 __

19 24 __ You can write the improper fraction as a

mixed number.


fractions. 1 = 2 = 7 4 __ 13

6 __ 3

4 __ 1

6 __

3

Course 1


Try This: Example 2B Find each product. Write the answer in simplest form.

B. 1 • 3 2 9

__ 3 5

__

11 9 __ 18

5 __ •

11 • 2 1 • 5

______

Use the GCF to simply before multiplying.

22 5 __



fractions. 1 = . 3 = 11 9 __ 18

5 __ 2

9 __ 3

5 __

Simplify. 2 5 __ 4

11 9 __ 18

5 __ •

2

1

Course 1


Try This: Example 2C Find each product. Write the answer in simplest form.

C. 3 • 6 1 5

__

Use the Distributive Property.

Add.

3 5 __ 18 +

3 • 6 1 5 __

3 • (6 + ) 1 5 __

(3 • 6) + ( 3 • )

3 1 __

Multiply.

(3 • 6) + ( • ) 1 5 __

1 5 __

3 5 __ 18

Lesson Quiz


1. 3 2. 1

Find each product. Write the answer in simplest form.

3. 2 • 1 4. 5 • 3

5. A nurse gave a patient 3 tablets of a medication. If each

tablet contained grain of the medication, how much

medication did the patient receive?

2


Course 1


5 8 __ 1

5 __ •

5 7 __ 7

10 __

•

1 4 __ 1

5 __ 1

5 __ 16

1 2 __

1 20 __

1 3 14 __

2 3 4 __

grain 7 40 __


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Multiply. Write each answer in simplest form.

1. 2 • 2 2. 1 • 3 3. 3 • 3 4. 2 • 1

5

or 5

10

Course 1


1 5 __

1 2 __

1 2 __

1 2 __

1 3 __

1 4 __

21 4 __ 1

4 __

or 2 11 4 __ 3

4 __

Problem of the Day

Rachel is building a doghouse. She needs to cut a 12-foot-long board into lengths of 65 in. How many lengths can she cut, and will there be any wood left? If so, how much? 2 cuts; yes, 14 in.

Course 1


Learn to divide fractions and mixed numbers.

Course 1


Vocabulary reciprocal


Course 1


Course 1


Reciprocals can help you divide by fractions. Two numbers are reciprocals if their product is 1.

Course 1


Additional Example 1A: Finding Reciprocals

Find the reciprocal.

A. 1 9

__

1 9 __ • = 1 Think: of what number is 1?

1 9 __ 1

9 __

1 9 __ • 9 = 1 1 9 __ of is 1. 1

9 __ 9

1 __

The reciprocal of is 9. 1 9 __

Course 1


Additional Example 1B: Finding Reciprocals


B. 2 3

__


2 3 __

3 2 __ 2

3 __ • = = 1 6

6 __ of is 1. 2

3 __ 3

2 __

The reciprocal of is . 2 3 __ 3

2 __

Course 1


Additional Example 1C: Finding Reciprocals


C. 3 1 5

__

16 5 __ • = 1 Write 3 as . 1

5 __

5 16 __ 16

5 __ • = = 1 80

80 __ of is 1. 16

5 __ 5

16 __


16 __

16 5 __

Course 1




A. 1 4

__


1 4 __

1 4 __ • 4 = 1 of is 1. 1

4 __ 4

1 __

The reciprocal of is 4. 1 4 __

Course 1




B. 4 5

__


4 5 __

5 4 __ 4

5 __ • = = 1 20

20 __ of is 1. 4

5 __ 5

4 __


4 __

Course 1




C. 4 1 8

__

33 8 __ • = 1 Write 4 as . 1

8 __

8 33 __ 33

8 __ • = = 1 264

264 ___ of is 1. 33

8 __ 8

33 __


33 __

33 8 __

Course 1


Look at the relationship between the

fractions and . If you switch the

numerator and denominator of a fraction,

you will find its reciprocal. Dividing by a

number is the same as multiplying by its

reciprocal.

3 4 __ 4

3 __

24 ÷ 4 = 6 24 • = 6 1 4 __

Course 1


Additional Example 2A: Using Reciprocals to Divide Fractions and Mixed Numbers

Divide. Write each answer in simplest form.

A. ÷ 7 8 7

__

8 7 __ ÷ 7 = •

Rewrite as multiplication using the

reciprocal of 7, . 1 7 __

Multiply by the reciprocal.

8 7 __ 1

7 __

8 • 1 7 • 7 ____ =


Course 1


Additional Example 2B: Using Reciprocals to Divide Fractions and Mixed Numbers

B. ÷ 5 6

__

5 6 __ ÷ = •


reciprocal of , . 3 2 __

Simplify before multiplying.

5 6 __ 3

2 __

5 • 3 6 • 2 ____ =

You can write the answer as a mixed number.

= 5 4 __

2 3

__

2 3 __ 2

3 __

1

2

Multiply.

= 1 1 4 __

Course 1


Additional Example 2C: Using Reciprocals to Divide Fractions and Mixed Numbers

C. 2 ÷ 1 3 4

__

Write the mixed numbers as

improper fractions. 2 =

and 1 = 13 12 __

Rewrite as multiplication.

11 • 12 4 • 13 ______

=


=

33 13 __

1 12 __

3 4 __

1

3 Simplify before multiplying.

= 2 7 13 __

2 ÷ 1 = ÷ 3 4 __ 1

12 __ 11

4 __ 13

12 __ 11

4 __

1 12 __

11 4 __ 12

13 __ •

Multiply. =

Course 1




A. ÷ 3 2 3

__

2 3 __ ÷ 3 = •


reciprocal of 3, . 1 3 __

Multiply by the reciprocal.

2 3 __ 1

3 __

2 • 1 3 • 3 ____ =


Course 1



B. ÷ 7 10 __

7 10 __ ÷ = •


reciprocal of , . 5 1 __

Simplify before multiplying.

7 10 __ 5

1 __

7 • 5 10 • 1 ____ =


= 7 2 __

1 5

__

1 5 __ 1

5 __

1

2

Multiply.

= 3 1 2 __

Course 1



C. 3 ÷ 1 2 3

__

Write the mixed numbers as

improper fractions. 3 =

and 1 = . 10 9 __

Rewrite as multiplication.

11 • 9 3 • 10 ______

=


=

33 10 __

1 9

__

2 3 __

1

3 Simplify before multiplying.

= 3 3 10 __

3 ÷ 1 = ÷ 2 3 __ 1

9 __ 11

3 __ 10

9 __ 11

3 __

1 9 __

11 3 __ 9

10 __ •

Multiply. =

Lesson Quiz


1. 2.


3. ÷ 20 4. 3 ÷ 2

5. Rhonda put 2 pounds of pecans into a -pound bag. How many bags did Rhonda fill?


Course 1


1 11 __ 8

13 __

4 7 __ 1

2 __ 1

2 __

3 4 __ 1

4 __

11 1 __ 13

8 __

1 35 __ 7

5 __ 2

5 __ , or 1

11 bags

5-4 Solving Fraction Equations: Multiplication and Division

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Solve.

1. x – 5 = 17 2. 5x = 125 3. x + 12 = 86 4. 9x = 108

x = 22

x = 25

x = 74

Course 1


x = 12

Problem of the Day

Stephen forgot his locker number, but he remembered that the sum of the digits is 11 and that the digits are all odd numbers. The locker numbers are from 1 to 120. What is Stephen’s locker number? 119

Course 1


Learn to solve equations by multiplying and dividing fractions.

Course 1


Dividing by a number is the same as multiplying by its reciprocal.

Remember!

Course 1


Additional Example 1A: Solving Equations by Multiplying and Dividing

Solve the equation. Write the answer in simplest form.

A. j = 25 3 5

__

j ÷ = 25 ÷ 3 5 __ 3

5 __ 3

5 __

j • = 25 • 3 5 __ 5

3 __ 5

3 __

j = 25 • 5 1 • 3

_____

j = 25 • 5 3 __

j = , or 41 125 3

___ 2 3 __

Divide both sides of the equation by . 3 5 __

Multiply by , the reciprocal of . 3 5 __ 5

3 __

Course 1


Additional Example 1B: Solving Equations by Multiplying and Dividing


B. 7x = 2 5

__

7x 1 __ 2

5 __ 1

7 __ 1

7 __ • = •

x = 2 • 1 5 • 7 ____

x = 2 35 __

Multiply both sides by the reciprocal of 7.


Course 1


Additional Example 1C: Solving Equations by Multiplying and Dividing


C. = 6 5y 8

__

5y 8 __ 6

1 __ 5

8 __ 5

8 __ ÷ = ÷

y = , or 9 48 5 __

Divide both sides by . 5 8 __

5y 8 __ 6

1 __ 8

5 __ 8

5 __ • = •

3 5 __

5 8 Multiply by the reciprocal of . __

Course 1


Try This: Example 1A Solve the equation. Write the answer in simplest form.

A. j = 19 3 4

__

j ÷ = 19 ÷ 3 4 __ 3

4 __ 3

4 __

j • = 19 • __ 3 4 __ 4

3 4 3 __

j = 19 • 4 1 • 3

_____

4 j = 19 • 3 __

76 ___ 1 3

j = , or 25 3 __

Divide both sides of the equation by . 3 4 __

Multiply by , the reciprocal of . 3 4 __ 4

3 __

Course 1


Try This: Example 1B Solve the equation. Write the answer in simplest form.

B. 3x = 1 7

__

3x 1 __ 1

7 __ 1

3 __ 1

3 __ • = •

x = 1 • 1 7 • 3 ____

x = 1 21 __

Multiply both sides by the reciprocal of 3.


Course 1


Try This: Example 1C Solve the equation. Write the answer in simplest form.

C. = 4 6y 7

__

6y 7 __ 4

1 __ 6

7 __ 6

7 __ ÷ = ÷

y = , or 4 28 6 __

Divide both sides by .

Multiply by the reciprocal of .

6 7 __

6y 7 __ 4

1 __ 7

6 __ 7

6 __ • = •

2 3 __

6 7 __

Course 1



2 3

__

Course 1


Dexter makes of a recipe, and he uses 12 cups of powdered milk. How many cups of powdered milk are in the recipe?



The answer will be the number of cups of powdered milk in the recipe.


Course 1


2 3 __ • He makes of the recipe.

• He uses 12 cups of powdered milk.

2 Make a Plan

You can write and solve an equation. Let x represent the number of cups in the recipe.

Course 1


He uses 12 cups, which is two-thirds of the amount of the recipe.

2 3 __ 12 = x


Solve 3 2 3 __ 12 = x

12 • = x • 3 2 __ 2

3 __ 3

2 __

12 1 __ 3

2 __ • = x

6

1 18 = x

There are 18 cups of powdered milk in the recipe.

Multiply both sides by , the

reciprocal of .

3 2 __

2 3 __

Simplify. Then multiply.

Course 1



Look Back 4

Check 12 = x 2 3 __

2 3 __ 12 = ? (18)

36 3 __ 12 = ?

12 = 12 ?

Substitute 18 for x.

Multiply and simplify.

18 is the solution.

Course 1



12 1

Try This: Example 2

2 5

__

Course 1


Rich makes of a recipe, and he uses 8 cups of wheat flour. How many cups of wheat flour are in the recipe?



The answer will be the number of cups of wheat flour in the recipe.

2 5 __

Course 1



• He makes of the recipe.

• He uses 8 cups of powdered milk.

2 Make a Plan

You can write and solve an equation. Let y represent the number of cups in the recipe.

2 5 __

Course 1


He uses 8 cups, which is two-fifths of the amount of the recipe.

8 = y


Solve 3 2 5 __ 8 = y

8 • = y • 5 2 __ 2

5 __ 5

2 __

8 1 __ 5

2 __ • = y

4

1 20 = y

There are 20 cups of wheat flour in the recipe.

Multiply both sides by , the

reciprocal of .

5 2 __

2 5 __

Simplify. Then multiply.

Course 1



Look Back 4

Check 8 = y 2 5 __

2 5 __ 8 = ? (20)

40 5 __ 8 = ?

8 = 8 ?

Substitute 20 for y.

Multiply and simplify.

20 is the solution.

Course 1



8 1

Lesson Quiz Solve each equation. Write the answer in simplest form.

1. 3x = 2. x = 4

3. x = 14 4. = 9

5. Rebecca used 3 pt of paint to paint of the trim in her bedroom. How many pints will Rebecca use for the trim in the entire bedroom?


1 8 __ 1

4 __

3 7 __ y

7 __

1 4 __

x = 16 1 24 __ x =

98 3 __ 2

3 __ x = or 32 y = 63

12

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write the first five multiples of each number.

1. 5

2. 6

3. 10

4. 12

5, 10, 15, 20, 25

6, 12, 18, 24, 30

10, 20, 30, 40, 50

Course 1


12, 24, 36, 48, 60

Problem of the Day

Greg, Sam and Mary all work at the same high school. One of them is a principal, one of them is a teacher, and one of them is a janitor. Sam is older than Mary. Mary does not live in the same town as the principal. The teacher, the oldest of the three, often plays golf with Greg. What is each person’s job. Greg, principal; Sam, teacher; Mary, janitor

Course 1


Learn to find the least common multiple (LCM) of a group of numbers.

Course 1


Vocabulary

least common multiple (LCM)


Course 1


Course 1


Additional Example 1: Consumer Application English muffins come in packs of 8, and eggs in cartons of 12. If there are 24 students, what is the least number of packs and cartons needed so that each student has a muffin sandwich with one egg and there are no muffins left over?

There are 24 English muffins and 24 eggs.

So 3 packs of English muffins and 2 cartons of eggs are needed.

Draw muffins in groups of 8. Draw eggs in groups of 12. Stop when you have drawn the same number of each.

Course 1

5-5 Least Common Multiple Try This: Example 1

There are 18 dog cookies and 18 bones.

So 3 packages of dog cookies and 2 bags of bones are needed.

Dog cookies come in packages of 6, and bones in bags of 9. If there are 18 dogs, what is the least number of packages and bags needed so that each dog has a treat box with one bone and one cookie and there are no bones or cookies left over?

Draw cookies in groups of 6. Draw bones in groups of 9. Stop when you have drawn the same number of each.

Course 1


The smallest number that is a multiple of two or more numbers is the least common multiple (LCM). In Additional Example 1, the LCM of 8 and 12 is 24.

Course 1


Additional Example 2A: Using Multiples to Find the LCM

Find the least common multiple (LCM).

Method 1: Use a number line. A. 3 and 4

Use a number line to skip count by 3 and 4.

0 2 4 6 8 10 12

The least common multiple (LCM) of 3 and 4 is 12.

Course 1


Additional Example 2B: Using Multiples to Find the LCM


Method 2: Use a list.

4: 4, 8, 12 , 16, 20, 24, 28, 32, 36, 40, 44, . . .

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, . . .

8: 8, 16, 24, 32, 40, 48, . . .

LCM: 40

List multiples of 4, 5, and 8.

Find the smallest number that is in all the lists.

B. 4, 5, and 8

Course 1


The prime factorization of a number is the number written as a product of its prime factorization.

Remember!

Course 1


Additional Example 2C: Using Multiples to Find the LCM


Method 3: Use prime factorization. C. 6 and 20

6 = 2 • 3 20 = 2 • 2 • 5

Write the prime factorization of each number in exponential form.

Line up the common factors. 2 • 3 • 2 • 5

2 • 3 • 2 • 5 = 60 To find the LCM, multiply one number from each column.

LCM: 60

Course 1


Additional Example 2D: Using Multiples to Find the LCM


D. 15, 6, and 4

15 = 3 • 5

4 = 2


2

3 • 5 • 2 2

3 • 5 • 2 = 60 2

To find the LCM, multiply each prime factor once with the greatest exponent used in any of the prime factorizations.

6 = 3 • 2

LCM: 60

Course 1


Try This: Example 2A Find the least common multiple (LCM).

Method 1: Use a number line. A. 2 and 3

Use a number line to skip count by 2 and 3.

0 1 2 3 4 5 6

The least common multiple (LCM) of 2 and 3 is 6.

Course 1


Try This: Example 2B Find the least common multiple (LCM).

Method 2: Use a list. B. 3, 4, and 9

3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, . . .

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …

9: 9, 18, 27, 36, 45, . . .

The least common multiple of 3, 4, and 9 is 36.

List multiples of 3, 4, and 9.

Find the smallest number that is in all the lists.

Course 1


Try This: Example 2C Find the least common multiple (LCM).

Method 3: Use prime factorization. C. 4 and 10

4 = 2 • 2 10 = 2 • 5


Line up the common factors. 2 • 2 • 5

2 • 2 • 5 = 20 To find the LCM, multiply one number from each column.

LCM: 20

Course 1


Try This: Example 2D Find the least common multiple (LCM).

D. 12, 6, and 8

12 = 22 • 3

6 = 2 • 3 Write the prime factorization of each number in exponential form.

23 • 3

23 • 3 = 24

To find the LCM, multiply each prime factor once with the greatest exponent used in any of the prime factorizations.

8 = 23

LCM: 24

Course 1


Lesson Quiz Find the least common multiple (LCM).

1. 6, 14 2. 9, 12

3. 5, 6, 10 4. 12, 16, 24, 36

5. Two students in Mrs. Albring’s preschool class are stacking blocks, one on top of the other. Reece’s blocks are 4 cm high and Maddy’s blocks are 9 cm high. How tall will their stacks be when they are the same height for the first time?

36 cm

42 36

30 144


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write each sum or difference in simplest form.

1. + 2. – 3. 2 + 1 4. 5 –1

Course 1


3 8 __ 5

8 __ 9

10 __ 3

10 __

1 4 __ 1

4 __ 8

9 __ 2

9 __

1 3 5 __

1 2 __ 3

2 3 __ 4

Problem of the Day

Students at a school dance formed equal teams to play a game. When they formed teams of 3, 4, 5, or 6, there was always one person left. What is the smallest number of people that there could have been at the dance? 61, the LCM of all the numbers, plus 1

Course 1


Learn to estimate sums and differences of fractions and mixed numbers.

Course 1


You can estimate fractions by rounding to 0, , or 1. 1 2 __

1

5 3 1 1

3

7

1 2 __

8 __

8 __

4 __

8 __

4 __

8 __

0 •

The fraction rounds to 1. 3 4 __

Course 1


closer to 0

Each numerator is much less than half the denominator, so the fractions are close to 0.

closer to 1

Each numerator is about the same as the denominator, so the fractions are close to 1.

• •

1 2 __

0 1

1 5 __ 2

11 __ 2

15 __ 4

7 __ 5

11 __ 6

7 __ 9

10 __ 16

19 __ 9

20 __

1 2 __

You can round fractions by comparing the numerator and denominator.

Course 1


closer to

Each numerator is about half the denominator, so the fractions are close to .

1 2 __

Additional Example 1A: Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1.

A. +

1 2

__

6 7

__ 3 8

__

+ 6 7 __ 3

8 __

1 + = 1 1 2 __

6 7 __ 3

8 __ + is about 1 .

1 2 __

Think: rounds to 1 and rounds to . 6 7 __ 3

8 __ 1

2 __

Course 1


1 2 __

Additional Example 1B: Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1.

B. –

1 2

__

9 10 __ 7

8 __

– 9 10 __ 7

8 __

1 – 1 = 0

9 10 __ 7

8 __ – is about 0.

Think: rounds to 1 and rounds to 1. 9 10 __ 7

8 __

Course 1


Try This: Example 1A Estimate each sum or difference by rounding to 0, , or 1.

A. +

1 2

__

5 6

__ 3 7

__

+ 5 6 __ 3

7 __

1 + = 1 1 2 __

5 6 __ 3

7 __ + is about 1 .

1 2 __

Think: rounds to 1 and rounds to . 5 6 __ 3

7 __ 1

2 __

Course 1


1 2 __

Try This: Example 1B Estimate each sum or difference by rounding to 0, , or 1.

B. –

1 2

__

13 19 __ 2

11 __

1 – 0 = 1

13 19 __ 2

11 __ – is about 1.

Think: rounds to 1 and rounds to 0. 13 19 __ 2

11 __

Course 1


– 2 11 __ 13

19 __

You can also estimate by rounding mixed numbers. You compare each mixed number to the

two nearest whole numbers and the nearest .

Does 10 round to 10, 10 , or 11?

1 2 __

3 10 __ 1

2 __

10 10 10 10 10 10 10 6 10 __ 2

10 __ 1

10 __ 4

10 __ 7

10 __ 9

10 __ 8

10 __

10 10 10 11 3 10 __ 1

2 __

•

The mixed number 10 rounds to 10 . 3 10 __ 1

2 __

Course 1


Additional Example 2: Sports Application The table shows the distances Tosha walked.

A. About how far did Tosha

walk on Tuesday and Thursday? Day

Tuesday

Thursday

Saturday

Sunday

Tosha’s Walking Distances

Distance (mi)

5

4

8

6

1 10 __

7 8 __

3 7 __

9 10 __

1 10 __ 7

8 __ 5 + 4

5 + 5 = 10

She walked about 10 miles on Tuesday and Thursday.

Course 1


Additional Example 2: Sports Application The table shows the distances Tosha walked.

B. About how much farther did

Tosha walk on Sunday than

Thursday?

Day

Tuesday

Thursday

Saturday

Sunday


Distance (mi)

5

4

8

6

1 10 __

7 8 __

3 7 __

9 10 __

9 10 __ 7

8 __ 8 – 4

9 – 5 = 4

She walked about 4 miles farther on Sunday than on Thursday.

Course 1


Additional Example 2: Sports Application

The table shows the distances Tosha walked.

C. Estimate the total distance

Tosha walked on Thursday,

Saturday, and Sunday.

1 2 __

7 8 __ 4 + 6 + 8

She walked about 20 miles on Thursday, Saturday, and Sunday.

9 10 __ 3

7 __

1 2 __ 5 + 6 + 9 = 20

1 2 __

Course 1


Day

Tuesday

Thursday

Saturday

Sunday


Distance (mi)

5

4

8

6

1 10 __

7 8 __

3 7 __

9 10 __

Try This: Example 2A The table shows the distances Jerry roller skated.

A. About how far did Jerry

skate on Tuesday and Sunday?

1 5 __ 6

7 __ 3 + 2

3 + 3 = 6

He roller skated about 6 miles on Tuesday and Sunday.

Course 1


Jerry’s Roller Skating Distances

Distance (mi)

3

6

2

8

1 5 __

3 7 __

1 7 __

6 7 __

Day

Tuesday

Thursday

Saturday

Sunday

Try This: Example 2B The table shows the distances Jerry roller skated.

B. About how much farther did

Jerry skate on Saturday than

Tuesday? 1 7 __ 1

5 __ 8 – 3

8 – 3 = 5

He roller skated about 5 miles farther on Saturday than Thursday.

Course 1



Distance (mi)

3

6

2

8

1 5 __

3 7 __

1 7 __

6 7 __

Day

Tuesday

Thursday

Saturday

Sunday


The table shows the distances Jerry roller skated.

C. Estimate the total distance

Tosha walked on Tuesday,

Thursday, and Sunday.

1 2 __

He roller skated about 12 miles on Tuesday, Thursday, and Sunday.

1 5 __ 3 + 6 + 2 6

7 __ 3

7 __

1 2 __ 3 + 6 + 3 = 12

1 2 __

Course 1



Distance (mi)

3

6

2

8

1 5 __

3 7 __

1 7 __

6 7 __

Day

Tuesday

Thursday

Saturday

Sunday

Lesson Quiz: Part 1

1. – 2. +

3. – 4. +


9 10 __ 2

5 __ 3

8 __ 8

9 __

10 11 __ 8

9 __ 1

4 __ 8

15 __

1 2 __ 1

2 __ 1

0 1

Course 1


Estimate each sum or difference by rounding to 0, , or 1.

1 2

__

Lesson Quiz: Part 2


Week Pounds Picked Up

1 18

2 16

3 20

1 2 __

1 3 __

9 10 __

5. The conservation club picked up trash along the road for three weeks. The table shows the number of pounds of trash they collected. About how many pounds did they collect in weeks 2 and 3?

37 pounds

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Course 1


Warm Up Add. Write each answer in simplest form.

1. + 2. + 3. + 4. +

1 7 __ 1

7 __ 8

15 __ 3

15 __

7 9 __ 2

9 __ 11

20 __ 4

20 __

2 7 __ 11

15 __

1 3 4 __

Problem of the Day

If it takes 12 minutes to cut a pipe into three pieces, how long would it take to cut the pipe into four pieces. 18 minutes

Course 1


Learn to add and subtract fractions with unlike denominators.

Course 1


Vocabulary least common denominator (LCD)


Course 1


The Pacific Ocean covers of Earth’s surface. The Atlantic

Ocean covers of Earth’s surface. To find the fraction of the

Earth’s surface that is covered by both oceans, you can add

and , which are unlike fractions.

1 3 __

1 3 __

1 5 __

1 5 __

Course 1


Fractions that represent the same value are equivalent.

Remember!

Course 1


To add or subtract unlike fractions, first rewrite them as equivalent fractions with a common denominator.

Additional Example 1: Cooking Application Mark made a pizza with pepperoni covering of the

pizza and onions covering another . What fraction of

the pizza is covered by pepperoni or onions?

1 4

__

1 3

__

1

__

1 4 __

1 3 __

_____ + Find a common denominator for 4 and 3.

Course 1


?

1 4

__ 1 3

__ 1 4

__ 1 3

__ Add +


Write equivalent fractions with 12 as the common denominator.

1 4 __

1 3 __

_______ +

3 12 __

4 12 __

_____ Add the numerators. Keep the common denominator.

1 4

__ 1 3

__ 7 12 __

1 12

1 12

1 12

1 12

1 12

1 12

1 12

The pepperoni or onions cover of the pizza. 7 12 __

Course 1


Try This: Example 1 Tori made a pizza with peppers covering of the

pizza and ham covering another . What

fraction of the pizza is covered by ham or peppers?

1 3

__

1 2

__

1 3

__ 1 2

__

? 1 3 __

1 2 __

_____ + Find a common denominator for 3 and 2.

Course 1


1 3

__ 1 2

__ Add +


Write equivalent fractions with 6 as the common denominator.

1 3 __

1 2 __

_______ +

2 6 __

3 6 __

_____ Add the numerators. Keep the common denominator.

5 6 __

1 3

__ 1 2

__

1 6

1 6

1 6

1 6

1 6

The peppers or ham cover of the pizza. 5 6 __

Course 1


You can use any common denominator or the least common denominator to add and subtract unlike fractions. The least common denominator (LCD) is the least common multiple of the denominators.

Course 1


Additional Example 2A: Adding and Subtracting Unlike Fractions

Add or Subtract. Write each answer in simplest form.

Method 1: Multiplying denominators.

A. – 7 10 __ 3

8 __

– 7 10 __ 3

8 __

– 56 80 __ 30

80 __

Multiply the denominators. 10 • 8 = 80

Write equivalent fractions.

Subtract.


26 80 __

13 40 __

Course 1


Additional Example 2B: Adding and Subtracting Unlike Fractions


Method 2: Use the LCD.

B. – 11 12 __ 3

8 __

– 11 12 __ 3

8 __

– 22 24 __ 9

24 __

The LCD is 24.


Subtract. 13 24 __

Course 1


Additional Example 2C: Adding and Subtracting Unlike Fractions


Method 3: Use mental math.

C. + 5 16 __ 1

8 __

+ 5 16 __ 1

8 __

+ 5 16 __ 2

16 __

Think: 16 is a multiple of 8, so the LCD is 16. Rewrite with a denominator of 16.

Add. 7 16 __

1 8 __

Course 1


Additional Example 2D: Adding and Subtracting Unlike Fractions



D. – 5 16 __ 1

8 __

– 5 16 __ 1

8 __

– 5 16 __ 2

16 __


Subtract. 3 16 __

Course 1


1 8 __

Try This: Example 2A Add or Subtract. Write each answer in simplest form.

Method 1: Multiplying denominators.

A. – 8 10 __ 2

6 __

– 8 10 __ 2

6 __

– 48 60 __ 20

60 __

Multiply the denominators. 10 • 6 = 60


Subtract.


28 60 __

7 15 __

Course 1


Try This: Example 2B Add or Subtract. Write each answer in simplest form.

Method 2: Use the LCD.

B. – 4 7

__ 1 3

__

– 4 7 __ 1

3 __

– 12 21 __ 7

21 __

The LCD is 21.


Subtract. 5 21 __

Course 1


Try This: Example 2C Add or Subtract. Write each answer in simplest form.


C. + 3 12 __ 1

6 __

+ 3 12 __ 1

6 __

+ 3 12 __ 2

12 __


Add. 5 12 __

1 6 __

Course 1


Try This: Example 2D Add or Subtract. Write each answer in simplest form.


D. – 3 12 __ 1

6 __

– 3 12 __ 1

6 __

– 3 12 __ 2

12 __


Subtract. 1 12 __

1 6 __

Course 1


Lesson Quiz Add or Subtract. Write each answer in simplest form.

1. + 2. –

3. – 4. +

5. Bonnie is making oatmeal bars and the recipes calls for cup of brown sugar. If she has cup of brown sugar, how much more does she need?


3 8 __ 1

6 __ 1

2 __ 1

8 __

5 12 __ 1

6 __ 1

6 __ 1

8 __

3 4 __

1 3 __

13 24 __ 3

8 __

1 4 __ 7

24 __

5 12 __ cup

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Course 1


Warm Up Add. Write each answer in simplest form.

1. – 2. + 3. + 4. –

7 12 __ 1

2 __ 3

11 __ 2

3 __

1 6 __ 3

4 __ 11

12 __ 1

3 __

1 12 __ 31

33 __

11 12 __ 7

12 __

Course 1


Problem of the Day

The sum of every row, column and diagonal is the same. Complete the magic square.

1

3 4 __ 11

12 ___ 1 1 3

__

1 1 6 __

12 1 1 __

5 6 __

2 3 __ 1

4 __ 1

Learn to add and subtract mixed numbers with unlike denominators.

Course 1


Course 1


Additional Example 1A: Adding and Subtracting Mixed Numbers

Find the sum. Write the answer in simplest form.

1 8

__ 5 6

__

3 1 8 __ 3 6

48 __ Multiply the denominators.

8 · 6 = 48

+1 5 6 __ + 1 40

48 __

______ ______ Write equivalent fractions with a denominator of 48.

4 = 4 46 48 __ 23

24 __ Add the fractions and then

whole numbers, and simplify.

A. 3 + 1

Course 1


Additional Example 1B: Adding and Subtracting Mixed Numbers

Find the difference. Write the answer in simplest form.

2 3

__ 1 4

__

5 2 3 __ 5 8

12 __ The LCD of the denominators is 12.

–1 1 4 __ – 1 3

12 __


4 5 12 __ Subtract the fractions and

then the whole numbers.

B. 5 – 1

Course 1


Additional Example 1C: Adding and Subtracting Mixed Numbers

Find the sum. Write the answer in simplest form.

1 2

__ 4 5

__

2 1 2 __ 2 5

10 __ The LCD of the denominators

is 10.

+4 4 5 __ + 4 8

10 __


6 = 7 13 10 __ 3


the whole numbers.

6 = 6 + 1 13 10 __ 3

10 __

C. 2 + 4

Course 1


Additional Example 1D: Adding and Subtracting Mixed Numbers

Find the difference. Write the answer in simplest form.

2 3

__ 2 9

__

6 2 3 __ 6 6

9 __ Think: 9 is a multiple of 3, so 9

is the LCD. –3 2

9 __ – 3 2

9 __




D. 6 – 3

Course 1


Try This: Example 1A Find the sum. Write the answer in simplest form.

1 4

__ 1 6

__

2 1 4 __ 2 6

24 __ Multiply the denominators.

4 · 6 = 24

+3 1 6 __ + 3 4

24 __


5 = 5 10 24 __ 5


whole numbers, and simplify.

A. 2 + 3

Course 1


Try This: Example 1B Find the difference. Write the answer in simplest form.

1 2

__ 3 7

__

4 1 2 __ 4 7


–1 3 7 __ – 1 6

14 __




B. 4 – 1

Course 1


Try This: Example 1C Find the sum. Write the answer in simplest form.

7 8

__ 2 3

__

2 7 8 __ 2 21


+4 2 3 __ + 4 16

24 __


6 = 7 37 24 __ 13

24 __ Add the fractions and then the

whole numbers. 6 = 6 + 1 37 24 __ 13

24 __

C. 2 + 4

Course 1


Try This: Example 1D Find the difference. Write the answer in simplest form.

3 4

__ 1 8

__

7 3 4 __ 7 6

8 __ Think: 8 is a multiple of 4, so 8

is the LCD.

–2 1 8 __ – 2 1

8 __

______ ______

Write equivalent fractions with a denominator of 8.



D. 7 – 2

Course 1


Additional Example 2: Measurement Application The length of Jen’s kitten’s body is 10 inches. Its tail is 5

inches long. What is the total length of its body and tail?

1 4

__ 1 8

__

1 4

__ 1 8

__

10 1 4 __ 10 2

8 __ Find a common denominator. Write

equivalent fractions with the LCD, 8, as the denominator.

+5 1 8 __ + 5 1

8 __

______ ______ Add the fractions and then the whole numbers. 15

3 8 __

The total length of the kitten’s body and tail is 15 inches.

You can use mental math to find an LCD. Think: 8 is a multiple of 4 and 8. Helpful Hint

3 8 __

Add 10 + 5

Course 1


Try This: Example 2 The length of Regina’s mouse’s body is 2 inches. Its tail is 2

inches long. What is the total length of its body and tail?

2 3

__ 1 6

__

2 3

__ 1 6

__

2 2 3 __ 2 4

6 __ Find a common denominator. Write

equivalent fractions with the LCD, 6, as the denominator.

+2 1 6 __ + 2 1

6 __

______ ______

Add the fractions and then the whole numbers.

4 5 6 __

The total length of the mouse’s body and tail is 4 inches. 5 6 __

Add 2 +2

Lesson Quiz

Find each sum or difference. Write the answer in simplest form.

1. 7 – 4 2. 9 + 7

3. 6 + 1 4. 10 – 4

5. Miles worked 5 hours on Friday and 8 hours on

Saturday. How many total hours did he work?

11 12 __ 2

3 __ 1

6 __ 3

8 __

7 8 __ 3

10 __ 2

5 __ 1

6 __

2 3 __ 3

4 __

3 1 4 __ 16 13

24 __

8 7 40 __ 6 7

30 __

14 hours 5 12 __

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Course 1


Warm Up Add or Subtract.

1. 3 + 1 2. 7 + 3. 8 – 2 4. 6 – 1

2 5 __ 3

5 __ 9

11 __ 11

11 __

4 5 __ 1

4 __ 1

3 __ 1

9 __

5 8

6 5

9 11 __

11 20 __ 2

9 __

Problem of the Day

Complete the magic square so that every row, column, and diagonal, has the same sum.

Course 1


1 2 1 1

1 1 2

7 8 __

5 8 __

5 8 __ 1

2 __

1 2 __ 1 7

8 __

3 4 __ 1 1

8 __ 1

1 4 __

3 8 __

3 8 __ 1

8 __ 3

4 __

1 4 __

2

Learn to rename mixed numbers to subtract.

Course 1


Course 1


Jimmy and his family planted a tree when it was 1 ft

tall. Now the tree is 2 ft tall. How much has the tree

grown since it was planted?

3 4 __

1 4 __

1 4 __ 3

4 __ The difference in the heights is 2 – 1 .

1 4 __

3 4 __ 1

4 __

You will need to rename 2 because the fraction

in 1 is greater than .

1 4 __ Divide one whole of 2 into fourths.

Course 1


2 1 4 __ 1 5

4 __

–1

1 4 __

– 1 ______ ______

= 2 4 __ 1

2 __

3 4 __

Rename 2 as 1 . 3 4 __

5 4 __

The tree has grown ft since it was planted. 1 2 __

1 4 1 1

4 1 4

1 4

1 4

1 1 4

1 4

1 4

1 4

1

1

1 4 1 4

Course 1


Additional Example 1A: Renaming Mixed Numbers Subtract. Write each answer in simplest form.

1 6

__ 5 6

__

7 1 6 __ 6 7

6 __ Rename 7 as 6 + 1 = 6 + + .

–2 5 6 __ –2 5

6 __

______ ______

4 2 6 __

Subtract the fractions and then the whole numbers.

1 6 __ 1

6 __ 6

6 __ 1

6 __

Write the answer in simplest form. = 4 1 3 __

A. 7 – 2

Course 1


Additional Example 1B: Renaming Mixed Numbers Subtract. Write each answer in simplest form.

2 5

__ 7 10 __

8 4 10 __ 7 14

10 __

Rename 8 as 7 + 1 = 7 + + . –6 7 10 __ –6 7

10 __

______ ______

1 7 10 __ Subtract the fractions and then the

whole numbers.

4 10 __ 4

10 __ 10

10 __ 4

10 __

10 is a multiple of 5, so 10 is a common denominator.

B. 8 – 6

Course 1


Additional Example 1C: Renaming Mixed Numbers Subtract. Write each answer in simplest form.

2 3

__

6 5 3 3 __

–3 2 3 __ –3 2

3 __

______ ______

2


Write 6 as a mixed number with a denominator of 3. Rename 6 as 5 + . 3

3 __

1 3 __

C. 6 – 3

Course 1


Additional Example 1D: Renaming Mixed Numbers Subtract. Write each answer in simplest form.

1 3

__ 3 4

__

5 4 12 __ 4 16

12 __

–2 9 12 __ –2 9

12 __

______ ______


whole numbers.

Rename 5 as 4 + 1 = 4 + + . 4 12 __ 4

12 __ 12

12 __ 4

12 __

The LCM of 4 and 3 is 12.

D. 5 – 2

Course 1


Try This: Example 1A Subtract. Write each answer in simplest form.

1 8

__ 5 8

__

5 1 8 __ 4 9

8 __ Rename 5 as 4 + 1 = 4 + + .

–2 5 8 __ –2 5

8 __

______ ______

2 4 8 __


1 8 __ 1

8 __ 4

4 __ 1

8 __

Write the answer in simplest form. = 2 1 2 __

A. 5 – 2

Course 1


Try This: Example 1B Subtract. Write each answer in simplest form.

1 6

__ 7 12 __

7 2 12 __ 6 14

12 __

Rename 7 as 6 + 1 = 6 + + . –3 7 12 __ –3 7

12 __

______ ______


whole numbers.

2 12 __ 2

12 __ 12

12 __ 2

12 __

12 is a multiple of 6, so 12 is a common denominator.

B. 7 – 3

Course 1


Try This: Example 1C Subtract. Write each answer in simplest form.

3 4

__

5 4 4 4 __ Write 5 as a mixed number with a

denominator of 4. Rename 5 as 4 + .

–2 3 4 __ –2 3

4 __

______ ______

2


4 4 __

1 4 __

C. 5 – 2

Course 1


Try This: Example 1D Subtract. Write each answer in simplest form.

1 2

__ 2 3

__

8 3 6 __ 7 9

6 __

Rename 8 as 7 + 1 = 7 + + . –4 4 6 __ –4 4

6 __

______ ______


whole numbers.

3 6 __ 3

6 __ 6

6 __ 3

6 __

The LCM of 2 and 3 is 6.

D. 8 – 4

Course 1


Additional Example 2A: Measurement Application Li is making a quilt. She needs 15 yards of fabric.

A. Li has 2 yards of fabric. How many more yards does she need?

3 4

__



–2 3 4 __ –2 3

4 __

______ ______

12


4 4 __

1 4 __

Li needs another 12 yards of fabric. 1 4 __

Course 1


Additional Example 2B: Measurement Application

B. If Li uses 11 yards of fabric, how much of the 15

yards will she have left?

1 6

__



–11 1 6 __ –11

1 6 __

______ ______

3


6 6 __

5 6 __

Li will have 3 yards of fabric. 5 6 __

Course 1


Try This: Example 2A Peggy is making curtains. She needs 13 feet of fabric.

A. Peggy has 4 yards of fabric. How many more yards does she need?

5 6

__



–4 5 6 __ –4 5

6 __

______ ______

8


6 6 __

1 6 __

Peggy needs another 8 yards of fabric. 1 6 __

Course 1



B. If Peggy uses 9 yards of fabric, how much of the 13

yards will she have left?

1 4

__



–9 1 4 __ –9 1

4 __

______ ______

3


4 4 __

3 4 __

Peggy will have 3 yards of fabric left. 3 4 __

Lesson Quiz


Course 1


Subtract. Write each answer in simplest form.

1. 7 – 3 2. 8 – 3

3. 7 – 5 4. 10 – 4

5. Irena put 10 cups of food in the bird feeder. The birds ate 3 cups. How many cups are left?

4 7 __ 6

7 __ 7

10 __ 4

5 __

2 9 __ 3

8 __ 9

16 __

3 8 __

3 5 7 __ 4 9

10 __

1 7 9 __

5 13 16 __

6 cups 5 8 __


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Solve.

1. x – 15 = 9

2. x + 21 = 34

3. 17 = x – 11

4. 22 = x – 34

x = 24 x = 13

x = 28

Course 1


x = 56

Problem of the Day

If a newborn baby weighs two pounds plus three-fourths its own weight, how much does it weigh? 8 pounds

Course 1


Learn to solve equations by adding and subtracting fractions.

Course 1


Additional Example 1A: Solving Equations by Adding and Subtracting

Solve the equation. Write the solution in simplest form.

3 5

__

x + 5 = 14 3 5 __

3 5 __ – 5 – 5 3

5 __

______ ______

x = 13 – 5 3 5 __ 5

5 __

x = 8 2 5 __

Subtract 5 from both sides to undo addition.

3 5 __

Rename 14 as 13 . 5 5 __

Subtract.

A. x + 5 = 14

Course 1


Additional Example 1B: Solving Equations by Adding and Subtracting


2 9

__

3 = x – 4 2 9 __

1 3 __ + 4 + 4 1

3 __

______ ________

3 + 4 = x 3 9 __ 2

9 __

7 = x 5 9 __

Add 4 to both sides to undo the

subtraction.

1 3 __

Find a common denominator.

Add.

1 3

__

1 3 __

B. 3 = x – 4

Course 1


Additional Example 1C: Solving Equations by Adding and Subtracting


1 6

__

6 = m + 1 6 __

7 12 __ – – 7

12 __

______ ______

6 – = m 7 12 __ 2

12 __

5 = m 7 12 __

Subtract from both sides to undo the

addition.

7 12 __


Subtract.

7 12 __

7 12 __

5 – = m 7 12 __ 14

12 __ Rename 6 as 5 + .

2 12 __ 12

12 __ 2

12 __

C. 6 = m +

Course 1


Additional Example 1D: Solving Equations by Adding and Subtracting


4 5

__

w – = 3 4 5 __

4 5 __ + + 4

5 __

_____ _____

w = 3 + 4 5 __ 3

10 __

w = 3 + 3 10 __

Add to both sides to undo addition. 4 5 __


Add.

3 10 __

3 10 __

8 10 __

w = 3

w = 4

11 10 __

1 10 __

3 = 3 + 1 . 11 10 __ 1

10 __

D. w – = 3

Course 1


Try This: Example 1A Solve the equation. Write the solution in simplest form.

5 8

__

x + 3 = 12 5 8 __

5 8 __ – 3 – 3 5

8 __

______ ______

x = 11 – 3 5 8 __ 8

8 __

x = 8 3 8 __

Subtract 3 from both sides to undo addition.

5 8 __

Rename 12 as 11 . 8 8 __

Subtract.

A. x + 3 = 12

Course 1


Try This: Example 1B Solve the equation. Write the solution in simplest form.

3 8

__

2 = x – 3 3 8 __

1 4 __ + 3 + 3 1

4 __

______ ______

2 + 3 = x 2 8 __ 3

8 __

5 = x 5 8 __


subtraction.

1 4 __


Add.

1 4

__

1 4 __

B. 2 = x – 3

Course 1


Try This: Example 1C Solve the equation. Write the solution in simplest form.

2 5

__

5 = m + 2 5 __

7 10 __ – – 7

10 __

______ ______

5 – = m 7 10 __ 4

10 __

4 = m 7 10 __

Subtract from both sides to undo the

addition.

7 10 __


Subtract.

7 10 __

7 10 __

4 - = m 7 10 __ 14

10 __ Rename 5 as 4 + .

4 10 __ 10

10 __ 4

10 __

C. 5 = m +

Course 1


Try This: Example 1D Solve the equation. Write the solution in simplest form.

1 3

__

w – = 5 1 3 __

1 3 __ + + 1

3 __

_____ _____

w = 5 + 1 3 __ 5

6 __

w = 5 + 5 6 __

Add to both sides to undo addition. 1 3 __


Add.

5 6

__

5 6 __

2 6 __

w = 5

w = 6

7 6 __

1 6 __

5 = 5 + 1 . 7 6 __ 1

6 __

D. w – = 5

Course 1


Additional Example 2: Application 1 4

__ 1 2

__

d – 17 = 85 1 2 __

1 2 __ + 17 + 17

1 2 __

________ ________

1 4 __

d = 102 3 4 __

Let d represent the weight of Ian’s dog.


subtraction.

1 2 __

Linda’s dog weighs 85 pounds. If Linda’s dog weighs 17 pounds less than Ian’s dog, how much does Ian’s dog weigh?

Course 1



Check

d – 17 = 85 1 2 __ 1

4 __

3 4 __ 102 – 17 = 85

1 2 __ 1

4 __ ?

3 4 __ 102 – 17 = 85

2 4 __ 1

4 __ ?

85 = 85 ? 1 4 __ 1

4 __

Substitute 102 for d. 3 4 __


102 is the solution. 3 4 __

Ian’s dog weighs 102 pounds. 3 4 __

Course 1


Try This: Example 2 2 3

__

1 6

__

c – 4 = 13 1 6 __

1 6 __ + 4 + 4

1 6 __

_______ ________

2 3 __

c = 17 5 6 __

Let c represent the weight of Vicki’s cat.


subtraction.

1 6 __

Jimmy’s cat weighs 13 pounds. If Jimmy’s cat weighs

4 pounds less than Vicki’s cat, how much does Vicki’s

cat weigh?

Course 1


Try This: Example 2

Check

c – 4 = 13 1 6 __ 2

3 __

5 6 __ 17 – 4 = 13

1 6 __ 2

3 __ ?

5 6 __ 17 – 4 = 13

1 6 __ 4

6 __ ?

13 = 13 ? 4 6 __ 4

6 __

Substitute 17 for c. 5 6 __


17 is the solution. 5 6 __

Vicki’s cat weighs 17 pounds. 5 6 __

Course 1


Solve each equation. Write the solution in simplest form.

1. x – 3 = 2 2. x + 6 = 11

3. 1 = x – 2 4. 5 = x –

5. Marco needs to work 40 hours per week. He has worked 31 hours so far this week. How many hours does he need to work on Friday to meet his 40-hour requirement?

Lesson Quiz

3 4 __ 5

12 __ x = 6 1

6 __ 5

8 __

7 10 __ 1

2 __ 3

4 __

5 8 __

x = 4 3 8 __

x = 4 1 2 __ x = 6 1

4 __

8 hours 3 8 __

4 5 __

Course 1


6-1 Make a Table

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write the values in simplest form. 1. + 2. – 3. ÷ 4. 5 · 2

Course 1

6-1 Make a Table

13 58

58

13

78

14

56

5 12

23 24

5 24 5 8 2

1 8 13

Problem of the Day

If February 1 falls on a Tuesday, then March 1 falls on what day of the week? Tuesday or Wednesday, depending on whether or not it is a leap year.

Course 1

6-1 Make a Table

Learn to use tables to record and organize data.

Course 1

6-1 Make a Table

Course 1

6-1 Make a Table

Additional Example 1: Application Use the audience data to make a table. Then use your table to describe how attendance has changed over time.

Date People in Audience May 1 May 2 May 3

From the table you can see that the number of people in the audience increased from May 1 to May 3.

Make a table. Write the dates in order so that you can see how the attendance changed over time.

On May 1, there were 275 people in the audience at the school play. On May 2, there were 302 people. On May 3 there were 322 people.

275 302 322

Course 1

6-1 Make a Table

Try This: Example 1 Use the audience data to make a table. Then use your table to describe how attendance has changed over time.

Date People in Audience April 1 May 1 June 1

From the table you can see that the number of people in the audience decreased from April 1 to June 1.

Make a table. Write the dates in order so that you can see how the attendance changed over time.

On April 1, there were 212 people at the symphony. On May 1, there were 189 people. On June 1 there were 172 people.

212 189 172

Course 1

6-1 Make a Table

Additional Example 2: Organizing Data in a Table Use the temperature data to make a table. Then use your table to find a pattern in the data and draw a conclusion.

At 3 A.M., the temperature was 53°F. At 5 A.M., it was 52°F. At 7 A.M., it was 50°F. At 9 A.M., it was 53°F. At 11 A.M., it was 57°F.

53

50 52

Time Temperature (°F) 3 A.M.

5 A.M.

7 A.M.

9 A.M.

11 A.M. 53 57

The temperature dropped until 7 A.M., then it rose. One conclusion is that the low temperature on this day was 50°F.

Course 1

6-1 Make a Table

Try This: Example 2 Use the temperature data to make a table. Then use your table to find a pattern in the data and draw a conclusion.

The temperature dropped until 6 A.M., then it rose. One conclusion is that the low temperature on this day was 44° F.

At 2 A.M., the temperature was 48°F. At 4 A.M., it was 46°F. At 6 A.M., it was 44°F. At 8 A.M., it was 47°F. At 10 A.M., it was 51°F.

48

44 46

Time Temperature (°F) 2 A.M.

4 A.M.

6 A.M.

8 A.M.

10 A.M.

47 51

Lesson Quiz: Part 1

1. Humans have the following approximate heart rates at the ages given: newborn, 135 beats per minute (bpm); 2 years old, 110 bpm; 6 years old, 95 bpm; 10 years old, 87 bpm; 20 years old, 71 bpm; 40 years old, 72 bpm; and 60 years old, 74 bpm. Use this data to make a table.


Course 1

6-1 Make a Table

Age Heart rate newborn 135 bpm

2 110 bpm 6 95 bpm 10 87 bpm 20 71 bpm 40 72 bpm 60 74 bpm

Lesson Quiz: Part 2

2. Use the data from problem 1 to estimate how many times per minute an 8-year-old’s heart beats. 91


Course 1

6-1 Make a Table

Age Heart rate newborn 135 bpm

2 110 bpm 6 95 bpm 10 87 bpm 20 71 bpm 40 72 bpm 60 74 bpm

6-2 Range, Mean, Median, and Mode

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Order the numbers from least to greatest. 1. 8, 6, 25, 7, 4, 12 2. 60, 11, 27, 45, 32 Divide 3. 720 ÷ 4 4. 760 ÷ 10

4, 6, 7, 8, 12, 25

11, 27, 32, 45, 60

180

Course 1


76

Problem of the Day

Ms. Red, Ms. Blue, and Ms. Green attended a holiday party. They each wore a different-colored dress, red, blue, or green. Ms. Red said to the lady wearing the blue dress, “Did you notice that none of us is wearing a dress with the color that corresponds to our name?” Who wore which color dress? Ms. Red: green; Ms. Blue: red; Ms. Green: blue

Course 1


Learn to find the range, mean, median, and mode of a data set.

Course 1


Vocabulary range mean median mode


Course 1


Course 1

6-2 Range, Mean, Median, and Mode Some descriptions of a set of data are called the range, mean, median, and mode.

•The range is the difference between the least and greatest values in the set.

•The mean is the sum of all the items, divided by the number of items in the set. (The mean is sometimes called the average.)

•The median is the middle value when the data are in numerical order, or the mean of the two middle values if there are an even number of items.

•The mode is the value or values that occur most often. There may be more than one mode for a data set. When all values occur an equal number of times, the data set has no mode.

Course 1


Additional Example 1A: Finding the Range, Mean, Median, and Mode of a Data Set

Depths of Puddles (in.) 5 8 3 5 4 2 1

Start by writing the data in numerical order. 1, 2, 3, 4, 5, 5, 8 range: 8 – 1 = 7

mean: 1 + 2 + 3 + 4 + 5 + 5 + 8 = 28

28 ÷ 7 = 4 median: 4 mode: 5

The range is 7 in.; the mean is 4 in.; the median is 4 in.; and the mode is 5 in.

Subtract least value from greatest value.

Add all values. Divide the sum by the number of items.

There are an odd number of items, find the middle value. 5 occurs most often.

Find the range, mean, median, and mode of each data set.

A.

Course 1


Additional Example 1B: Finding the Range, Mean, Median, and Mode of a Data Set

Number of Points Scored 96 75 84 7

Write the data in numerical order. 7, 75, 84, 96 range: 96 – 7 = 89 mean: 96 + 75 + 84 + 7

4 = 65.5

median: 7, 75, 84, 96

74 + 84 2

= 79.5

mode: none

The range is 89 points; the mean is 65.5 points; the median is 79.5 points; and there is no mode.

There are an even number of items, so find the mean of the two middle values.

B.

Course 1


Try This: Example 1A Find the range, mean, median, and mode of each data set.

A. Rainfall per month (in.) 1 2 10 2 5 6 9

Start by writing the data in numerical order. 1, 2, 2, 5, 6, 9, 10 range: 10 – 1 = 9

mean: 1 + 2 + 2 + 5 + 6 + 9 + 10 = 35

35 ÷ 7 = 5 median: 5 mode: 2

The range is 9 in.; the mean is 5 in.; the median is 5 in.; and the mode is 2 in.

Subtract least value from greatest value.

Add all values.

Divide the sum by the number of items. There are an odd number of items, find the middle value. 2 occurs most often.

Course 1



B. Number of Points Scored 53 26 47 12

Write the data in numerical order. 12, 26, 47, 53 range: 53 – 12 = 41 mean: 53 + 26 + 47 + 12

4 = 34.5

median: 12, 26, 47, 54

26 + 47 2

= 36.5

mode: none

The range is 41 points; the mean is 34.5 points; the median is 36.5 points; and there is no mode.

There are an even number of items, so find the mean of the two middle values.

Lesson Quiz

Use the following data set: 18, 20, 56, 47, 30, 18, 21.

1. Find the range. 2. Find the mean.

3. Find the median. 4. Find the mode.

5. Bonnie ran a mile in 8 minutes, 8 minutes, 7 minutes, 9 minutes, and 8 minutes. What was her mean time?

38


Course 1


30

21 18

8 minutes


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Use the numbers to answer the questions. 146, 161, 114, 178, 150, 134, 172, 131, 128 1. What is the greatest number? 2. What is the least number? 3. How can you find the median?

178

114

Order the numbers and find the middle value.

Course 1


Problem of the Day

Ms. Green has 6 red gloves and 10 blue gloves in a box. She closes her eyes and picks some gloves. What is the least number of gloves Ms. Green will have to pick to ensure 2 gloves of the same color? 3

Course 1


Learn the effect of additional data and outliers.

Course 1


Vocabulary outlier


Course 1


Course 1


The mean, median, and mode may change when you add data to a data set.

Course 1


Additional Example 1A & 1B: Sports Application A. Find the mean, median, and mode of the data in the table.

EMS Football Games Won Year 1998 1999 2000 2001 2002

Games 11 5 7 5 7

mean = 7 modes = 5, 7 median = 7

B. EMS also won 13 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode.

mean = 8 modes = 5,7 median = 7

The mean increased by 1, the modes remained the same, and the median remained the same.

Course 1


Try This: Example 1A & 1B A. Find the mean, median, and mode of the data in the table.

MA Basketball Games Won Year 1998 1999 2000 2001 2002

Games 13 6 4 6 11

mean = 8 mode = 6 median = 6

B. MA also won 15 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode.

mean = 9 mode = 6 median = 8

The mean increased by 1, the mode remained the same, and the median increased by 2.

Course 1


An outlier is a value in a set that is very different from the other values.

Course 1


Additional Example 2: Application Ms. Gray is 25 years old. She took a class with students who were 55, 52, 59, 61, 63, and 58 years old. Find the mean, median, and mode with and without Ms. Gray’s age.

mean ≈ 53.3 no mode median = 58

mean = 58 no mode median = 58.5

When you add Ms. Gray’s age, the mean decreases by about 4.7, the mode stays the same, and the median decreases by 0.5. The mean is the most affected by the outlier. The median is closer to most of the students’ ages.

Data with Ms. Gray’s age:

Data without Ms. Gray’s age:

Course 1


Try This: Example 2 Ms. Pink is 56 years old. She volunteered to work with people who were 25, 22, 27, 24, 26, and 23 years old. Find the mean, median, and mode with and without Ms. Pink’s age.

mean = 29 no mode median = 25

mean = 24.5 no mode median = 24.5

When you add Ms. Gray’s age, the mean increases by 4.5, the mode stays the same, and the median increases by 0.5. The mean is the most affected by the outlier. The median is closer to most of the students’ ages.

Data with Ms. Pink’s age:

Data without Ms. Gray’s age:

Course 1


Additional Example 3: Describing a Data Set The Yorks are shopping for skates. They found 8 pairs of skates with the following prices:

$35, $42, $75, $40, $47, $34, $45, $40

What are the mean, median, and mode of this data set? Which statistic best describes the data set?

mean = $44.75 mode = $40 median = $41

The median price is the best description of the prices. Most of the skates cost about $41.

The mean is higher than most of the prices because of the $75 skates.

Course 1


Try This: Example 3 The Oswalds are shopping for gloves. They found 8 pairs of gloves with the following prices:

$17, $15, $39, $12, $13, $16, $19, $15

What are the mean, median, and mode of this data set? Which statistic best describes the data set?

mean = $18.25 mode = $15 median = $15.50

The median price is the best description of the prices. Most of the gloves cost about $15.50.

The mean is higher than most of the prices because of the $39 gloves.

Course 1


Some data sets do not contain numbers. For example, the circle graph shows the result of a survey to find people’s favorite color.

When it does not contain numbers the only way to describe the data set is with the mode. You cannot find a mean or a median for a set of colors.

The mode for this data set is blue. Most people in this survey chose blue as their favorite color.

Lesson Quiz

At the college bookstore, your brother buys 6 textbooks at the following prices: $21, $58, $68, $125, $36, and $140.

1. Find the mean.

2. Find the median.

3. Find the mode.

4. Your brother signs up for an additional class, and the textbook costs $225. Recalculate the mean, including the extra book.

$63

$74.67


none

$96.14

Course 1


6-4 Bar Graphs

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Use the following data set. 45 55 58 63 63 37 76 46 34

1. What is the mean of the data? 2. What is the median of the data? 3. What is the mode of the data?

53 55

63

Course 1

6-4 Bar Graphs

Problem of the Day

The distance around the bases is 4 × 90 feet. How many runs does a baseball team need to score before the scoring base runners have covered a mile? (1 mile = 5,280 feet) 15 runs

Course 1

6-4 Bar Graphs

Learn to display and analyze data in bar graphs.

Course 1

6-4 Bar Graphs

Vocabulary bar graph double-bar graph


Course 1

6-4 Bar Graphs

Course 1

6-4 Bar Graphs

A bar graph can be used to display and compare data. A bar graph displays data with vertical or horizontal bars.

Course 1

6-4 Bar Graphs

Additional Example 1A: Reading a Bar Graph

Use the bar graph to answer each question. A. Which biome in the

graph has the lowest average summer temperature?

Find the lowest bar.

The coniferous forest has the least average summer temperature.

Course 1

6-4 Bar Graphs

Additional Example 1B: Reading a Bar Graph

Use the bar graph to answer each question. B. Which biomes in the

graph have an average summer temperature of 30°C or greater? Find the bar or bars whose heights measure 30 or more than 30. The grassland and the rain forest have average summer temperatures of 30°C or greater.

Course 1

6-4 Bar Graphs


Use the bar graph to answer each question.

A. Which biome in the graph has the highest average summer temperature? Find the highest bar.

The rain forest has the highest average summer temperature.

Course 1

6-4 Bar Graphs



B. Which biomes in the graph have an average summer temperature of 25°C or greater? Find the bar or bars whose heights measure 25 or more than 25. The deciduous forest, the grassland, and the rain forest have average summer temperatures of 25°C or greater.

Course 1

6-4 Bar Graphs

Additional Example 2: Making a Bar Graph

Use the given data to make a bar graph. Magazine Subscriptions Sold Grade 6 Grade 7 Grade 8

258 597 374

Step 1:Find an appropriate scale and interval. The scale must include all of the data values. The interval separates the scale into equal parts.

Step 2:Use the data to determine the lengths of the bars. Draw bars of equal width. The bars cannot touch.

Step 3: Title the graph and label the axes.

Course 1

6-4 Bar Graphs

Try This: Example 2

Use the given data to make a bar graph. Tickets Sold

Grade 6 Grade 7 Grade 8 310 215 285

Step 1:Find an appropriate scale and interval. The scale must include all of the data values. The interval separates the scale into equal parts.

Step 2:Use the data to determine the lengths of the bars. Draw bars of equal width. The bars cannot touch.

Step 3: Title the graph and label the axes.

Grade 6

Grade 7

Grade 8

0

50

100

150

200

250

300

350Tickets Sold

Grade

Tick

ets

Course 1

6-4 Bar Graphs

A double-bar graph shows two sets of related data.

Course 1

6-4 Bar Graphs Additional Example 3: Problem Solving Application

Make a double-bar graph to compare the data in the table. Club Memberships

Club Art Music Science Boys 12 6 16 Girls 8 14 4

1 Understand the Problem You are asked to use a graph to compare the data given in the table. You will need to use all of the information given.

2 Make a Plan You can make a double-bar graph to display the two sets of data.

Course 1

6-4 Bar Graphs


Solve 3 Determine appropriate scales for both sets of data.

Use the data to determine the lengths of the bars. Draw bars of equal width. Bars should be in pairs. Use a different color for boy memberships and girl memberships. Title the graph and label both axes.

Include a key to show what each bar represents.

Look Back 4 You could make two separate graphs, one of boy memberships and one of girl memberships. However, it is easier to compare the two data sets when they are on the same graph.

Course 1

6-4 Bar Graphs Try This: Example 3

Make a double-bar graph to compare the data in the table.

Club Memberships Club Band Chess Year Book Boys 9 14 16 Girls 11 7 15

1 Understand the Problem You are asked to use a graph to compare the data given in the table. You will need to use all of the information given.

2 Make a Plan You can make a double-bar graph to display the two sets of data.

Course 1

6-4 Bar Graphs


Solve 3 Determine appropriate scales for both sets of data.

Use the data to determine the lengths of the bars. Draw bars of equal width. Bars should be in pairs. Use a different color for boy memberships and girl memberships. Title the graph and label both axes.

Include a key to show what each bar represents.

Look Back 4 You could make two separate graphs, one of boy memberships and one of girl memberships. However, it is easier to compare the two data sets when they are on the same graph.

02468

10121416

Chess Band Year Book

BoysGirls

Club

Club Memberships

Mem

bers

hips

Lesson Quiz: Part 1


1. Which animal was least popular among students?

2. Which pet was more popular to twice as many students as rabbits were?

dog

bird


Course 1

6-4 Bar Graphs

Student Pet Survey

Lesson Quiz: Part 2

3. Make a bar graph of this data.


Course 1

6-4 Bar Graphs

Number of Daily Servings

Grains = 6

Fruit = 2

Meat = 2

Milk = 3

Vegetables = 3

Number of Daily Servings

01234567

Grains

Fruit

Meat

Milk

Vegeta

bles

Daily

Ser

ving

s

GrainsFruitMeatMilkVegetables


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Create a bar graph of the data.

Course 1


Favorite rides at fair: Ferris wheel = 5, loop the loop = 4, merry-go-round = 3, bumper cars = 7, sit and spin = 9

Favorite Rides at Fair

0

2

4

6

8

10

Ferris Wheel Loop the loop Merry-go-round

Bumper cars Sit and spin

Problem of the Day

A set of 7 numbers has a mean of 36, a median of 37, a mode of 37, and a range of 6. What could the 7 numbers be?

Possible answer: 33, 33, 36, 37, 37, 37, 39

Course 1


Learn to organize data in frequency tables and histograms.

Course 1


Vocabulary frequency table cumulative frequency histogram


Course 1


Course 1

6-5 Frequency Tables and Histograms Additional Example 1: Making a Tally Table

Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do most students in Mr. Ray’s class have?

whorl loop whorl loop

arch arch loop whorl

loop arch whorl arch

arch whorl arch loop

Make a tally table to organize the data.

Course 1

6-5 Frequency Tables and Histograms Additional Example 1 Continued

Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do more students in Mr. Ray’s class have?





Step 1: Make a column for each fingerprint pattern.

Course 1








Step 2: For each fingerprint, make a tally mark in the appropriate column.

Course 1









Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l

Course 1







Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l

Most students in Mr. Ray’s class have an arch fingerprint.

Course 1


A group of four tally marks with a line through it means five. llll = 5 llll llll = 10

Reading Math

Course 1

6-5 Frequency Tables and Histograms Try This: Example 1

Students in Ms. Gracie’s class recorded their fingerprint patterns. Which type of pattern do more students in Ms. Gracie’s class have?

Make a tally table to organize the data.


arch whorl loop whorl

loop whorl whorl arch


Course 1

6-5 Frequency Tables and Histograms Try This: Example 1 Continued







Course 1









Course 1









Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l l

Course 1







Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l l

Most students in Ms. Gracie’s class have a whorl fingerprint.

Course 1


A frequency table tells the number of times an event, category, or group occurs. The cumulative frequency column shows a running total of all frequencies.

Course 1

6-5 Frequency Tables and Histograms Additional Example 2: Making a Cumulative

Frequency Table

Step 1: Make a row for each pattern.

Step 2: The frequency is how many times each pattern occurred.

Number of Fingerprint Patterns Fingerprint Pattern Frequency Cumulative Frequency

Whorl Arch Loop

Step 3: Find the cumulative frequency for each row by adding all frequency values above or in that row.

6 5

11 16

5 5

Use the tally table in Additional Example 1 to make a cumulative frequency table.

Course 1

6-5 Frequency Tables and Histograms Try This: Additional Example 2

Step 1: Make a row for each pattern.

Step 2: The frequency is how many times each pattern occurred.

Number of Fingerprint Patterns Fingerprint Pattern Frequency Cumulative Frequency

Whorl Arch Loop

Step 3: Find the cumulative frequency for each row by adding all frequency values above or in that row.

4 5

11 16

7 7

Use the tally table in Try This: Example 1 to make a cumulative frequency table.

Course 1


Additional Example 3: Making a Frequency Table with Intervals

Use the data in the table to make a frequency table with intervals.

Number of Pages Read per Student Last Weekend

12 15 40 19 7 5 22 34 37 18

Step 1: Choose equal intervals.

Step 2: Find the number of data values in each interval. Write these numbers in the “Frequency” row.

Course 1



Number of Pages Read per Student Last Weekend Number 1–10 11–20 21–30 31–40

Frequency

This table shows that 2 students read between 1 and 10 pages, 4 students read between 11 and 20 pages, and so on.

2 4 1 3


12 15 40 19 7 5 22 34 37 18

Course 1


Try This: Example 3

Use the data in the table to make a frequency table with intervals.


17 29 9 19 7 5 27 34 21 38

Step 1: Choose equal intervals.

Step 2: Find the number of data values in each interval. Write these numbers in the “Frequency” row.

Course 1




Number 1–10 11–20 21–30 31–40 Frequency

This table shows that 3 students read between 1 and 10 pages, 2 students read between 11 and 20 pages, and so on.

3 2 3 2


17 29 9 19 7 5 27 34 21 38

Course 1


A histogram is a bar graph that shows the number of data items that occur within each interval.

Course 1


Additional Example 4: Making a Histogram Use the frequency table in Additional Example 3 to make a histogram.

Step 1: Choose an appropriate scale and interval.

Step 2: Draw a bar for the number of students in each interval. The bars should touch but not overlap. Step 3: Title the graph and label the axes.

Course 1

6-5 Frequency Tables and Histograms Try This: Example 4

Use the frequency table in Try This: Example 3 to make a histogram.

Step 1: Choose an appropriate scale and interval.

Step 2: Draw a bar for the number of students in each interval. The bars should touch but not overlap. Step 3: Title the graph and label the axes.

0

1

2

3

4

1- 10 11- 20 21- 30 31- 40


Number of Pages

Stu

den

ts

Lesson Quiz: Part 1

1. Students listed the number of days they spent on vacation in one year. Make a tally table with intervals of 5.

2, 18, 5, 15, 7, 10, 1, 10, 4

16, 7, 11, 17, 3, 8, 14, 13, 10


Course 1


Number of Days Spent on Vacation 1–5 6–10 11–15 16–20 |||| |||| | |||| |||

Lesson Quiz

2. Use your tally table from problem 1 to make a frequency and cumulative frequency table.


Course 1


Number of Days Spent on Vacation

Number of Days Frequency Cumulative Frequency

1–5 5 5 6–10 6 11 11–15 4 15 16–20 3 18

6-6 Ordered Pairs

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up 24 35 47 67 36 22 80 41 Use the set of data above to find the following:

1. the mean 2. the median 3. the mode 4. the range

44 38 none

Course 1

6-6 Ordered Pairs

47

Problem of the Day

What is the largest 6-digit number with each digit different and no digit a prime number? 986,410

Course 1

6-6 Ordered Pairs

Learn to graph ordered pairs on a coordinate grid.

Course 1

6-6 Ordered Pairs

Vocabulary coordinate grid ordered pair


Course 1

6-6 Ordered Pairs

Course 1

6-6 Ordered Pairs

Cities, towns, and neighborhoods are often laid out on a grid. This makes it easier to map and find locations.

A coordinate grid is formed by horizontal and vertical lines and is used to locate points.

Each point on a coordinate grid can be located by using an ordered pair of numbers, such as (4, 6). The starting point is (0, 0).

• The first number tells how far to move horizontally from (0, 0).

• The second number tells how far to move vertically.

Course 1

6-6 Ordered Pairs

Additional Example 1A: Identifying Ordered Pairs Name the ordered pair for the location.

0 1 2 3 4 5

1 2 3 4

Gym

Theater Park

A. Gym

Start at (0, 0). Move right 2 units and then up 3 units.

The gym is located at (2, 3).

Course 1

6-6 Ordered Pairs

Additional Example 1B: Identifying Ordered Pairs Name the ordered pair for the location.

B. Theater Start at (0, 0). Move right 3 units and then up 1 unit.

The theater is located at (3, 1).

0 1 2 3 4 5

1 2 3 4

Gym

Theater Park

Course 1

6-6 Ordered Pairs

Additional Example 1C: Identifying Ordered Pairs Name the ordered pair for the location.

C. Park Start at (0, 0). Move right 4 units and then up 2 units.

The park is located at (4, 2).

0 1 2 3 4 5

1 2 3 4

Gym

Theater Park

Course 1

6-6 Ordered Pairs

Try This: Example 1A Name the ordered pair for each location.

0 1 2 3 4 5

1 2 3 4 Library

Mall School

A. Library

Start at (0, 0). Move right 1 unit and then up 4 units. The library is located at (1, 4).

Course 1

6-6 Ordered Pairs

Try This: Example 1B Name the ordered pair for the location.

B. Mall Start at (0, 0). Move right 4 units and then up 1 unit.

The mall is located at (4, 1).

0 1 2 3 4 5

1 2 3 4 Library

Mall School

Course 1

6-6 Ordered Pairs

Try This: Example 1C Name the ordered pair for the location.

C. School Start at (0, 0). Move right 2 units and then up 2 units.

The school is located at (2, 2).

0 1 2 3 4 5

1 2 3 4 Library

Mall School

Course 1

6-6 Ordered Pairs

Additional Example 2: Graphing Ordered Pairs

Graph and label each point on a coordinate grid.

A. L (3, 5) Start at (0, 0).

1 2 3 4 5 6 0 1 2 3 4 5 6

Move right 3 units. Move up 5 units.

L

B. M (4, 0) Start at (0, 0). Move right 4 units. Move up 0 units.

M

Course 1

6-6 Ordered Pairs

Try This: Example 2


A. T (2, 6) Start at (0, 0).

1 2 3 4 5 6 0 1 2 3 4 5 6

Move right 2 units. Move up 6 units.

T

B. V (5, 0) Start at (0, 0). Move right 5 units. Move up 0 units.

V

Lesson Quiz

Give the ordered pair for each point.

1. A

2. B

3. C

4. D


5. F(7, 2) 6. G(1, 7)

(6, 1)

(4, 6)


(1, 4)

(2, 1)

Course 1

6-6 Ordered Pairs

F

G

6-7 Line Graphs

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Describe how to graph each point on a coordinate grid.

1. (4, 5) 2. (0, 2) 3. (3, 0)

right 4, up 5 up 2

right 3

Course 1

6-7 Line Graphs

Problem of the Day

Study the first two columns to determine a pattern to help fill in the blank square at the bottom.

6

Course 1

6-7 Line Graphs

4 49 6 11 134 12 7 85

Learn to display and analyze data in line graphs.

Course 1

6-7 Line Graphs

Vocabulary line graph double-line graph


Course 1

6-7 Line Graphs

Course 1

6-7 Line Graphs

Data that shows change over time is best displayed in a line graph. A line graph displays a set of data using line segments.

Course 1

6-7 Line Graphs

Additional Example 1: Making a Line Graph

Use the data in the table to make a line graph.

Population of New Hampshire

Year Population

1650 1,300

1670 1,800

1690 4,200

1700 5,000

Course 1

6-7 Line Graphs

Because time passes whether or not the population changes, time is independent of population. Always put the independent quantity on the horizontal axis.

Helpful Hint

Course 1

6-7 Line Graphs


Step 1: Place year on the horizontal axis and population on the vertical axis. Label the axes.

Step 2: Determine an appropriate scale and interval for each axis.

Step 3: Mark a point for each data value. Connect the points with straight lines.

Step 4: Title the graph. 0 1650 1670 1690 1700

1,000 2,000 3,000 4,000

5,000 6,000

Pop

ula

tion

Year

Population of New Hampshire

Course 1

6-7 Line Graphs

Try This: Example 1

Use the data in the table to make a line graph.

School District Enrollment

Year Population

1996 2,300

1998 2,800

2000 5,200

2002 6,000

Course 1

6-7 Line Graphs


Step 1: Place year on the horizontal axis and number of students on the vertical axis. Label the axes.

Step 2: Determine an appropriate scale and interval for each axis.

Step 3: Mark a point for each data value. Connect the points with straight lines.

Step 4: Title the graph. 0 1996 1998 2000 2002

1,000 2,000 3,000 4,000

5,000 6,000

Nu

mb

er o

f S

tud

ents

Year

School District Enrollment

Course 1

6-7 Line Graphs

Additional Example 2: Reading a Line Graph

Use the line graph to answer each question. A. In which year did

CDs cost the most? 2002

B. About how much did CDs cost in 2000? $15

C. Did CD prices increase or decrease from 1999 through 2002? They increased.

Course 1

6-7 Line Graphs

Try This: Example 2

Use the line graph to answer each question. A. In which year did

CDs cost the least? 1999

B. About how much did CDs cost in 1999? $13

C. Did CD prices increase or decrease from 2001 to 2002? They increased.

Course 1

6-7 Line Graphs

Line graphs that display two sets of data are called double-line graphs.

Course 1

6-7 Line Graphs

Additional Example 3: Making a Double-Line Graph

Use the data in the table to make a double-line graph.

Stock Prices 1985 1990 1995 2000

Corporation A $16 $20 $34 $33 Corporation B $38 $35 $31 $21

Use different colors of lines to connect the stock values so you will easily be able to tell the data apart.

Helpful Hint

Course 1

6-7 Line Graphs


Step 1: Determine an appropriate scale and interval.

Step 2: Mark a point for each Corporation A value and connect the points.

Step 3: Mark a point for each Corporation B value and connect the points.

Step 4: Title the graph and label both axes. Include a key.

$0 1985 1990 1995 2000

$10 $20 $30 $40

Pri

ce o

f S

tock

Year

Stock Prices

Corp. A

Corp. B

Course 1

6-7 Line Graphs

Try This: Example 3

Use the data in the table to make a double-line graph.

Stock Prices 1985 1990 1995 2000

Corporation C $8 $16 $20 $28 Corporation D $35 $22 $14 $7

Course 1

6-7 Line Graphs


Step 1: Determine an appropriate scale and interval.

Step 2: Mark a point for each Corporation C value and connect the points.

Step 3: Mark a point for each Corporation D value and connect the points.

Step 4: Title the graph and label both axes. Include a key.

$0 1985 1990 1995 2000

$10 $20 $30 $40

Pri

ce o

f S

tock

Year

Stock Prices

Corp. C

Corp. D

Lesson Quiz: Part 1 1. Use the data to make a line graph.


Course 1

6-7 Line Graphs

Number of Aluminum Cans Collected Mon Tue Wed Thu Fri 100 150 200 125 175

0

50 100

150 200

250

Nu

mb

er o

f C

ans

Aluminum Cans Collected

M T W Th F

Lesson Quiz: Part 2 Use the line graph to answer each question.

2. Which plant was taller on Tuesday?

3. Which plant grew more between Thursday and Friday?

4. Which plant grew the most in one week?

Each grew the same amount.

A


A

Course 1

6-7 Line Graphs


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Use the data below to answer each question. 20 21 23 24 27 33 34 35 36 38 40 41 42 43 46 52 53

1. What is the median? 2. What is the mode? 3. What is the range?

36 none 33

Course 1


Problem of the Day

Nine students in a group found that their mean score was 86 for the first math test. On the next test, each student in the group scored 7 points higher than on the first test. What was their mean score for the two tests? 89.5

Course 1


Learn to recognize misleading graphs.

Course 1


Course 1


Additional Example 1A: Misleading Bar Graphs

A. Why is this bar graph misleading? Because the lower part of the vertical axis is missing, the differences in prices are exaggerated.

Course 1


Additional Example 1B: Misleading Bar Graphs

B. What might people believe from the misleading graph?

People might believe that Cars B and C cost 1 – 2

times as much as Car A. In reality, Cars B and C are only a few thousand dollars more than Car A.

1 2 __ 1

2 __

Course 1



A. Why is this bar graph misleading?

The vertical axis begins at 530 rather than 0.

600

550 540 530

580 570 560

Dol

lars

Money Raised

4th graders 5th graders 6th graders

Course 1



B. What might people believe from the misleading graph? That the 5th graders have raised twice as much money as the 4th graders.

600

550 540 530

580 570 560

Dol

lars

Money Raised

4th graders 5th graders 6th graders

Course 1


Additional Example 2A: Misleading Line Graphs

A. Why are these line graphs misleading?

If you look at the scale for each graph, you will notice that the April graph goes from 54° to 66° and the May graph goes from 68° to 80°.

Course 1


Additional Example 2B: Misleading Line Graphs

B. What might people believe from these misleading graphs? People might believe that the temperatures in May were about the same as the temperatures in April. In reality, the temperatures in April were about 15 degrees lower.

Course 1


Additional Example 2C: Misleading Line Graphs

C. Why is this line graph misleading?

The scale goes from $0 to $80, and then increases by $5.

Course 1



A. Why are these line graphs misleading?

If you look at the scale for each graph, you will notice that the September Graph goes from 85° to 70° and the November graph goes from 65° to 50°.

0102030405060708090

1 2 3 40

10

20304050

6070

1 2 3 4Week Week

Tem

pera

ture

(°F

)

Tem

pera

ture

(°F

) September November

Course 1



B. What might people believe from these misleading graphs?

People might believe that the temperatures in September were about the same as the temperatures in November. In reality, the temperatures in September were about 20 degrees higher.

0102030405060708090

1 2 3 40

10

20304050

6070

1 2 3 4Week Week

Tem

pera

ture

(°F

)

Tem

pera

ture

(°F

) September November

Course 1



C. Why is this line graph misleading?

The scale goes from $0 to $50, and then increases by $10.

0

50

60

70 80

1985 1990 1995 2000

Stock prices

Year

Pric

e of

sto

ck ($

) Corp. C

Corp. D

Lesson Quiz

1. Why might this line graph be misleading?

2. What might people believe from the graph?

Possible answer: that there were hardly any visitors on Monday

The scale does not start at zero.


Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up A set of data ranges from 12 to 86. What intervals would you use to display this data in a histogram with four intervals?

Possible answer: 10–29, 30–49, 50–69, 70–89

Course 1


Problem of the Day

What is the least number that can be divided evenly by each of the numbers 1 through 12? 27,720

Course 1


Learn to make and analyze stem-and-leaf plots.

Course 1


Vocabulary stem-and-leaf plot


Course 1


Course 1


A stem-and-leaf plot shows data arranged by place value. You can use a stem-and-leaf plot when you want to display data in an organized way that allows you to see each value.

Course 1


Additional Example 1: Creating Stem-and-Leaf Plots

Use the data in the table to make a stem-and-leaf plot.

Test Scores 75 86 83 91 94 88 84 99 79 86

Step 1: Group the data by tens digits.

Step 2: Order the data from least to greatest.

75 79 83 84 86 86 88 91 94 99

Course 1


To write 42 in a stem-and-leaf plot, write each digit in a separate column.

4 2

Helpful Hint

Stem Leaf

Course 1


Additional Example 1 Continued Step 3: List the tens digits of

the data in order from least to greatest. Write these in the “stems” column.

Step 4: For each tens digit, record the ones digits of each data value in order from least to greatest. Write these in the “leaves” column.

Step 5: Title the graph and add a key.

Stems Test Scores

Key: 7 5 means 75

Leaves

7 8 9

5 9 3 4 6 6 8 1 4 9

75 79 83 84 86 86 88 91 94 99

Course 1


Try This: Example 1

Use the data in the table to make a stem-and-leaf plot.

Test Scores 72 88 64 79 61 84 83 76 74 67

Step 1: Group the data by tens digits.

Step 2: Order the data from least to greatest.

61 64 67 72 74 76 79 83 84 88

Course 1


Try This: Example 1 Cont.

Step 3: List the tens digits of the data in order from least to greatest. Write these in the “stems” column.

Step 4: For each tens digit, record the ones digits of each data value in order from least to greatest. Write these in the “leaves” column.

Step 5: Title the graph and add a key.

Stems Test Scores

Key: 6 1 means 61

Leaves

6 7 8

1 4 7 2 4 6 9 3 4 8

61 64 67 72 74 76 79 83 84 88

Course 1

6-9 Stem-and-Leaf Plots Additional Example 2: Reading Stem-and-Leaf

Plots

Find the least value, greatest value, mean, median, mode, and range of the data.

Stems

Leaves 4 5 6

0 0 1 5 7 1 1 2 4 3 3 3 5 9 9

7 8 9

0 4 4 3 6 7 1 4

The least stem and least leaf give the least value, 40.

The greatest stem and greatest leaf give the greatest value, 94.

Use the data values to find the mean (40 + … + 94) ÷ 23 = 64.

Key: 4 0 means 40

Course 1



The median is the middle value in the table, 63.

To find the mode, look for the number that occurs most often in a row of leaves. Then identify its stem. The mode is 63.

The range is the difference between the greatest and the least value. 94 – 40 = 54.

Stems

Leaves 4 5 6

0 0 1 5 7 1 1 2 4 3 3 3 5 9 9

7 8 9

0 4 4 3 6 7 1 4

Key: 4 0 means 40

Course 1


Try This: Example 2

Find the least value, greatest value, mean, median, mode, and range of the data.

Stems

Leaves 3 4 5

0 2 5 6 8 1 1 3 4 4 5 6 9 9 9

6 7 8

1 2 4 5 6 9 1 5

The least stem and least leaf give the least value, 30.

The greatest stem and greatest leaf give the greatest value, 85.

Use the data values to find the mean (30 + … + 85) ÷ 23 = 55.

Key: 3 0 means 30

Course 1



The median is the middle value in the table, 56.

To find the mode, look for the number that occurs most often in a row of leaves. Then identify its stem. The mode is 59.

The range is the difference between the greatest and the least value. 85 – 30 = 55.

Stems

Leaves 3 4 5

0 2 5 6 8 1 1 3 4 4 5 6 9 9 9

6 7 8

1 2 4 5 6 9 1 5

Key: 3 0 means 30

Lesson Quiz: Part 1

1. Make a stem-and-leaf plot of the data.

42 36 40 31 29 49 21 28 52 27 22 35 30 46 34 34


Course 1


2 1 2 7 8 9

3 0 1 4 4 5 6

4 0 2 6 9

5 2

Stems

Leaves

Key: 3 | 0 means 30

Lesson Quiz: Part 2

Find each value using the stem-and-leaf plot.

2. What is the least value?

3. What is the mean?

4. What is the median?

5. What is the mode?

34.75

21


34

34

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up What geometry term might you associate with each object?

1. a string on a guitar

2. a window

3. the tip of a pencil

4. a sheet of paper

line segment

plane or rectangle

point

Course 1


plane or rectangle

Problem of the Day

Draw a clock face that includes the numerals 1–12. Draw two lines that do not intersect and that separate the clock face into three parts so that the sums of the numbers on each part are the same.

Course 1


Learn to describe figures by using the terms of geometry.

Course 1


Vocabulary point line plane line segment ray


Course 1


Course 1


The building blocks of geometry are points, lines, and planes.

A plane is a flat surface that plane LMN, extends without end in all plane MLN, directions. plane NLM

A plane is named by three points on the plane that are not on the same line.

M

N

L

A point is an exact location. P point P, P

A point is named by a capital letter.

A line is a straight path that line AB, AB, extends without end in A B line BA, BA opposite directions.

A line is named by two points on the line.

Course 1


Additional Example 1A &1B: Identifying Points, Lines, and Planes

Use the diagram to name each geometric figure.

A. three points M, N, and P Five points are labeled: points M, N, P, Q, and R.

B. two lines PR and NR You can also write RP and RN.

Q

N

P R M

Course 1


Additional Example 1C & 1D: Identifying Points, Lines, and Planes


C. a point shared by two lines point R

D. a plane plane QRM Use any three points in the plane that are not on the same line. Write the three points in any order.

Point R is a point on PR and NR.

Q

N

P R M

Course 1


Try This: Example 1A &1B Use the diagram to name each geometric figure.

A. three points S, T, and U Five points are labeled: points S, T, U, V, and W.

B. two lines UW and SW You can also write WU and WS.

V

S

U W T

Course 1


Try This: Example 1C & 1D Use the diagram to name each geometric figure.

C. a point shared by two lines point W

D. a plane plane VUT Use any three points in the plane that are not on the same line. Write the three points in any order.

Point W is a point on WS and WU. V

S

U W T

Course 1


A line segment is a line segment XY, XY, made of two endpoints Y line segment YX, YX and all the points between X the endpoints.

A line segment is named by its endpoints.

A ray has one endpoint. ray JK, JK From the endpoint, the ray J K extends without end in one direction only.

A ray is named by its endpoint first followed by another point on the ray.

Course 1


Additional Example 2A & 2B: Identifying Line Segments and Rays

Use the diagram to give a possible name to each figure.

A. three different line segments

B. three ways to name the line

You can also write BA, CB, and CA.

AB, BC, and AC You can also write BA, CB, and CA.

AB, BC, and AC C

A B

Course 1


Additional Example 2C & 2D: Identifying Line Segments and Rays

Use the diagram to give a possible name to each figure. C. six different rays

D. another name for ray AB

A is still the endpoint. C is another point on the ray.

AB, AC, BC, CB,

CA, and BA

AC

C

A B

Course 1


Try This: Example 2A & 2B Use the diagram to give a possible name to each figure.

A. three different line segments

B. three ways to name the line

You can also write ED, FE, and FD.

DE, EF, and DF You can also write ED, FE, and FD.

DE, EF, and DF F

D E

Course 1


Try This: Example 2C & 2D Use the diagram to give a possible name to each figure.

C. six different rays

D. another name for ray DE

D is still the endpoint. F is another point on the ray.

DE, EF, DF, FE,

FD, and ED

DF

F

D E

Lesson Quiz


1. three points

2. two lines

3. a point shared by a line and a ray

4. a plane

Possible answer: A, B, C


Possible answer: F

Possible answer: plane AEC

Course 1


Possible answer: AD, CE

7-2 Angles

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. Draw two points. Label one point A and the other point B. 2. Draw a line through points A and B. 3. Draw a ray with A as an endpoint and C as a point on the ray. 4. Name all the rays in your drawing.

Course 1

7-2 Angles

AB, BA, and AC

A B

C

Problem of the Day

The measure of Jack’s angle is twice that of Amy’s and half that of Nate’s. The sum of the measures of Amy’s and Trisha’s angles is equal to the sum of the measures of Jack’s and Nate’s angles. The sum of the measures of all the angles is equal to 180°. What is the measure of each student’s angle? Jack’s angle: 30°; Nate’s angle: 60°; Amy’s angle: 15°; Trisha’s angle: 75°

Course 1

7-2 Angles

Learn to name, measure, classify, estimate, and draw angles.

Course 1

7-2 Angles

Vocabulary angle vertex acute angle right angle obtuse angle straight angle


Course 1

7-2 Angles

Course 1

7-2 Angles

An angle is formed by two rays with a common endpoint, called the vertex. An angle can be named by its vertex or by its vertex and a point from each ray. The middle point in the name should always be the vertex.

Angles are measured in degrees. The number of degrees determines the type of angle. Use the symbol ° to show degrees: 90° means “90 degrees.”

Course 1

7-2 Angles

An acute angle measures less than 90°.

A right angle measures exactly 90°.

Course 1

7-2 Angles

An obtuse angle measures more than 90° and less than 180°.

A straight angle measures exactly 180°.

Course 1

7-2 Angles Additional Example 1: Measuring an Angle with a

Protractor Use a protractor to measure the angle. Tell what type of angle it is.

• Place the center point of the protractor on the vertex of the angle.

G H

F

Course 1

7-2 Angles Additional Example 1 Continued

Use a protractor to measure the angle. Tell what type of angle it is.

• Place the protractor so that ray GH passes through the 0° mark.

G H

F

Course 1

7-2 Angles

• Using the scale that starts with 0° along ray GH, read the measure where ray GF crosses.

Additional Example 1 Continued Use a protractor to measure the angle. Tell what type of angle it is.

G H

F

Course 1

7-2 Angles

• The measure of FGH is 120°. Write this as m FGH = 120°.


G H

F

Course 1

7-2 Angles

• Since 120° > 90° and 120° < 180°, the angle is obtuse.


G H

F

Course 1

7-2 Angles Try This: Example 1


H I

G

• Place the center point of the protractor on the vertex of the angle.

Course 1

7-2 Angles Try This: Example 1 Continued


• Place the protractor so that ray HI passes through the 0° mark.

H I

G

Course 1



H I

G

• Using the scale that starts with 0° along ray HI, read the measure where ray HI crosses.

Course 1



H I

G

• The measure of GHI is 70°. Write this as m GHI = 70°.

Course 1



H I

G

• Since 70° < 90°, the angle is acute.

Course 1

7-2 Angles

Additional Example 2: Drawing an Angle with a Protractor

Use a protractor to draw an angle that measures 80°.

• Draw a ray on a sheet of paper.

Course 1

7-2 Angles

Additional Example 2 Continued Use a protractor to draw an angle that measures 80°.

• Place the center point of the protractor on the endpoint of the ray. • Place the protractor so that the ray passes through the 0° mark.

Course 1

7-2 Angles

Additional Example 2 Continued Use a protractor to draw an angle that measures 80°.

• Make a mark at 80° above the scale on the protractor.

• Use a straightedge to draw a ray from the endpoint of the first ray through the mark you make at 80°.

Course 1

7-2 Angles

Try This: Example 2 Use a protractor to draw an angle that measures 45°.

• Draw a ray on a sheet of paper.

Course 1

7-2 Angles

• Place the center point of the protractor on the endpoint of the ray. • Place the protractor so that the ray passes through the 0° mark.

Try This: Example 2 Continued Use a protractor to draw an angle that measures 45°.

Course 1

7-2 Angles

• Make a mark at 45° above the scale on the protractor.

• Use a straightedge to draw a ray from the endpoint of the first ray through the mark you make at 45°.

Try This: Example 2 Continued Use a protractor to draw an angle that measures 45°.

Course 1

7-2 Angles

To estimate the measure of an angle, compare it with an angle whose measure you already know. A right angle has half the measure of a straight angle. A 45° angle has half the measure of a right angle.

Course 1

7-2 Angles

Additional Example 3: Estimating Angle Measures

Estimate the measure of the angle, and then use a protractor to check the reasonableness of your estimate.

Think: The measure of the angle is about halfway between 90° and 180°. A good estimate would be 135°.

The angle measures 131°, so the estimate is reasonable.

Course 1

7-2 Angles

Try This: Example 3 Estimate the measure of the angle, and then use a protractor to check the reasonableness of your estimate.

Think: The measure of the angle is a little more than halfway between 0° and 90°. A good estimate would be 50°.

The angle measures 52°, so the estimate is reasonable.

Lesson Quiz

Use a protractor to draw an angle with the given measure. Tell what type of angle it is.

1. 140°

2. 20°

3. Draw a right angle.

4. Is the angle shown closer to 30° or 120°?

acute

obtuse


30°

Course 1

7-2 Angles


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Identify the type of angle.

1. 70°

2. 90°

3. 140°

4. 180°

acute

right

obtuse

Course 1


straight

Problem of the Day

A line forms an angle of 57° with the vertical axis. What angle does the line form with the horizontal axis? 33° or 147°

Course 1


Learn to understand relationships of angles.

Course 1


Vocabulary congruent vertical angles adjacent angles complementary angles supplementary angles


Course 1


Course 1


When angles have the same measure, they are said to be congruent.

Vertical angles are formed opposite each other when two lines intersect. Vertical angles have the same measure, so they are always congruent. MRP and NRQ are vertical angles.

MRN and PRQ are vertical angles

M N 160°

160°

20° 20° R

P Q

Course 1

7-3 Angle Relationships Adjacent angles are side by side and have a common vertex and ray. Adjacent angles may or may not be congruent.

MRN and NRQ are adjacent angles. They share vertex R and RN.

NRQ and QRP are adjacent angles. They share vertex R and RQ.

M N 160°

160°

20° 20° R

P Q

Course 1


Additional Example 1A: Identifying Types of Angle Pairs

Identify the type of each angle pair shown.

A.

5 and 6 are opposite each other and are formed by two intersecting lines.

They are vertical angles.

5 6

Course 1


Additional Example 1B: Identifying Types of Angle Pairs


B. 7 and 8 are side by side and

have a common vertex and ray.

They are adjacent angles.

7 8

Course 1


Try This: Additional Example 1A


A.

3 and 4 are side by side and have a common vertex and ray.

They are adjacent angles.

3 4

Course 1


Try This: Additional Example 1B


B.

7 and 8 are opposite each other and are formed by two intersecting lines.

They are vertical angles.

7

8

Course 1


65° + 25° = 90°

LMN and NMP are complementary.

Complementary angles are two angles whose measures have a sum of 90°.

P

N

M

L

25° 65°

Course 1


Supplementary angles are two angles whose measures have a sum of 180°.

65° + 115° = 180°

GHK and KHJ are supplementary.

J

K

H 115° 65°

G

Course 1


Additional Example 1A: Identifying an Unknown Angle Measure

Find each unknown angle measure.

71° + a = 90° –71° –71° a = 19°

The sum of the measures is 90°.

a

71°

A. The angles are complementary.

Course 1


Additional Example 1B: Identifying an Unknown Angle Measure


125° + b = 180° –125° –125° b = 55°


b 125°

B. The angles are supplementary.

Course 1


Additional Example 1C: Identifying an Unknown Angle Measure


c = 82° Vertical angles are congruent.

c 82°

C. The angles are vertical angles.

Course 1


Additional Example 1D: Identifying an Unknown Angle Measure


x + y + 80° = 180° –80° –80°

x + y = 100°


D. JKL and MKN are congruent.

x = 50° and y = 50° Each angle measures half of 100°.

x 80°

K

M L

N J y

Course 1




65° + d = 90° –65° –65° d = 25°


d

65°

A. The angles are complementary.

Course 1




145° + s = 180° –145° –145° s = 35°


s 145°

B. The angles are supplementary.

Course 1




t = 32° Vertical angles are congruent.

t

32°

C. The angles are vertical angles.

Course 1


Try This: Example 1D


x + y + 50° = 180° –50° –50°

x + y = 130°


D. ABC and DBE are congruent.

x = 65° and y = 65° Each angle measures half of 130°.

x 50°

B

D C

E A y

Lesson Quiz

Give the complement of each angle.

1. 70° 2. 42°

Give the supplement of each angle.

3. 120° 4. 17°

5. Identify the type of angle pair shown.

48° 20°


60° 163°

Course 1


adjacent

7-4 Classifying Lines

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Give the complement and supplement of each angle.

1. 80°

2. 64°

3. 15°

10°, 100°

26°, 116°

75°, 165°

Course 1


Problem of the Day

Draw three points that are not in a straight line. Label them A, B, and C. How many different lines can you draw that contains two of the points? Name the lines.

Course 1


3; AB, AC, BC

Learn to classify the different types of lines.

Course 1


Vocabulary parallel lines perpendicular lines skew lines


Course 1


Course 1


Intersecting lines are lines that cross at one common point.

Parallel lines are lines in the same plane that never intersect.

Line YZ intersects line WX.

YZ intersects WX.

Line AB is parallel to line ML.

AB ML.

Y W

Z X

B

A

M

L

Course 1


Perpendicular lines intersect to form 90° angles, or right angles.

Line RS is perpendicular to line TU.

RS TU. S

R

T U

The square inside a right angle shows that the rays of the angle are perpendicular.

Writing Math

Course 1


Skew lines are lines that lie in different planes. They are neither parallel nor intersecting.

Line AB and line ML are skew.

AB and ML are skew.

M

L

A B

Course 1


Additional Example 1A: Classifying Pairs of Lines

Classify each pair of lines.

A.

The lines intersect to form right angles.

They are perpendicular.

Course 1


Additional Example 1B: Classifying Pairs of Lines


B.

The lines are in different planes and are not parallel or intersecting.

They are skew.

Course 1


Additional Example 1C: Classifying Pairs of Lines


C.

The lines are in the same plane. They do not appear to intersect.

They are parallel.

Course 1


Additional Example 1D: Classifying Pairs of Lines


D.

The lines cross at one common point.

They are intersecting.

Course 1




A.

The lines are in the same plane. They do not appear to intersect.

They are parallel.

Course 1




B.

The lines intersect to form right angles.

They are perpendicular.

Course 1


Try This: Additional Example 1C


C.

The lines are in different planes and are not parallel or intersecting.

They are skew.

Course 1


Try This: Additional Example 1D


D.


They are intersecting.

Course 1



The handrails on an escalator are in the same plane. What type of line relationship do they represent?

The handrails are in the same plane and do not intersect.

The lines are parallel.

Course 1


Try This: Example 2

The roads are in the same plane. What type of line relationship do they represent?


The lines are intersecting.

Lesson Quiz

1. Sketch a pair of perpendicular lines.

2. Sketch a pair of parallel lines.

Use the figure to classify the lines.

3. AD and BC

4. AD and CF

5. AB and BG

parallel


skew

perpendicular

Course 1


7-5 Triangles

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. What are two angles whose sum is 90°? 2. What are two angles whose sum is 180°? 3. A part of a line between two points is called a _________. 4. Two lines that intersect at 90° are ______________.

complementary angles

supplementary angles

segment

Course 1

7-5 Triangles

perpendicular

Problem of the Day

Find the total number of shaded triangles in each figure.

3 6 10

Course 1

7-5 Triangles

Learn to classify triangles and solve problems involving angle and side measures of triangles.

Course 1

7-5 Triangles

Vocabulary acute triangle obtuse triangle right triangle scalene triangle isosceles triangle equilateral triangle


Course 1

7-5 Triangles

Course 1

7-5 Triangles

A triangle is a closed figure with three line segments and three angles. Triangles can be classified by the measures of their angles. An acute triangle has only acute angles. An obtuse triangle has one obtuse angle. A right triangle has one right angle.

Acute triangle Obtuse triangle Right triangle

Course 1

7-5 Triangles

To decide whether a triangle is acute, obtuse, or right, you need to know the measures of its angles.

The sum of the measures of the angles in any triangle is 180°. You can see this if you tear the corners from a triangle and arrange them around a point on a line.

By knowing the sum of the measures of the angles in a triangle, you can find unknown angle measures.

Course 1

7-5 Triangles


D

E

F

To classify the triangle, find the measure of D on the trophy.

So the measure of D is 90°. Because DEF has one right angle, the trophy is a right triangle.

Subtract the sum of the known angle measures from 180°

m D = 180° – (38° + 52°)

m D = 180° – 90°

m D = 90°

Sara designed this triangular trophy. The measure of E is 38°, and the measure of F is 52°. Classify the triangle.

Course 1

7-5 Triangles

Try This: Example 1

D

E

F

To classify the triangle, find the measure of D on the trophy.

So the measure of D is 136°. Because DEF has one obtuse angle, the trophy is an obtuse triangle.

Subtract the sum of the known angle measures from 180°

m D = 180° – (22° + 22°)

m D = 180° – 44°

m D = 136°

Sara designed this triangular trophy. The measure of E is 22°, and the measure of F is 22°. Classify the triangle.

Course 1

7-5 Triangles

You can use what you know about vertical, adjacent, complementary, and supplementary angles to find the measures of missing angles.

Course 1

7-5 Triangles

Additional Example 2A: Using Properties of Angles to Label Triangles

Use the diagram to find the measure of each indicated angle.

QTR and STR are supplementary angles, so the sum of m QTR and m STR is 180°.

m QTR = 180° – 68°

= 112°

P

S R

Q

T

68°

55°

A. QTR

Course 1

7-5 Triangles

Additional Example 2B: Using Properties of Angles to Label Triangles

QRT and SRT are complementary angles, so the sum of m QRT and m SRT is 90°. m SRT = 180° – (68° + 55°)

= 180° – 123° = 57°

m QRT = 90° – 57° = 33°

P

S R

Q

T

68°

55°

B. QRT

Course 1

7-5 Triangles


Use the diagram to find the measure of each indicated angle.

MNO and PNO are supplementary angles, so the sum of m MNO and m PNO is 180°.

m MNO = 180° – 44°

= 136°

L

P O

M

N

44°

60°

A. MNO

Course 1

7-5 Triangles


MON and PON are complementary angles, so the sum of m MON and m PON is 90°.

m PON = 180° – (44° + 60°) = 180° – 104° = 76°

m MON = 90° – 76° = 14°

L

P O

M

N

44°

60°

B. MON

Course 1

7-5 Triangles

Triangles can be classified by the lengths of their sides. A scalene triangle has no congruent sides. An isosceles triangle has at least two congruent sides. An equilateral triangle has three congruent sides.

Course 1

7-5 Triangles

Additional Example 3: Classifying Triangles by Lengths of Sides

Classify the triangle. The sum of the lengths of the sides is 19.5 in.

c = 6.5

6.5 in.

N L c

M

c + (6.5 + 6.5) = 19.5 c + 13 = 19.5

c + 13 – 13 = 19.5 – 13

6.5 in.

Side c is 6.5 inches long. Because LMN has three congruent sides, it is equilateral.

Course 1

7-5 Triangles

Try This: Example 3

Classify the triangle. The sum of the lengths of the sides is 21.6 in.

d = 7.2

7.2 in.

C A d

B

d + (7.2 + 7.2) = 21.6 d + 14.4 = 21.6

d + 14.4 – 14.4 = 21.6 – 14.4

7.2 in.

Side d is 7.2 inches long. Because ABC has three congruent sides, it is equilateral.

Lesson Quiz

If the angles can form a triangle, classify the triangle as acute, obtuse, or right.

1. 37°, 53°, 90° 2. 65°, 110°, 25°

3. 61°, 78°, 41° 4. 115°, 25°, 40°

The lengths of three sides of a triangle are given. Classify the triangle.

5. 12, 16, 25 6. 10, 10, 15

not a triangle right


acute obtuse

Course 1

7-5 Triangles

scalene isosceles

7-6 Quadrilaterals

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up The lengths of three sides of a triangle are given. Classify the triangle.

1. 12, 12, 12

2. 18, 10, 14

3. 15, 15, 26

4. 23, 36, 16

equilateral

scalene

isosceles

Course 1

7-6 Quadrilaterals

scalene

Problem of the Day

How many different rectangles are in the figure.

10; if the rectangles are marked 1-5 clockwise from upper left, they are 1; 2; 3; 4; 5; 1-2; 3-4; 4-5; 3-5; 1-2-3-4-5.

Course 1

7-6 Quadrilaterals

Learn to identify, classify, and compare quadrilaterals.

Course 1

7-6 Quadrilaterals

Vocabulary quadrilateral parallelogram rectangle rhombus square trapezoid


Course 1

7-6 Quadrilaterals

Course 1

7-6 Quadrilaterals

A quadrilateral is a plane figure with four sides and four angles.

Five special types of quadrilaterals and their properties will be shown in this lesson. The same mark on two or more sides of a figure indicates that the sides are congruent.

Course 1

7-6 Quadrilaterals

Parallelogram

Opposite sides are parallel and congruent. Opposite angles are congruent.

Rectangle Parallelogram with four right angles.

Course 1

7-6 Quadrilaterals

Rhombus Parallelogram with four congruent sides.

Square Rectangle with four congruent sides.

Course 1

7-6 Quadrilaterals

Trapezoid Quadrilateral with exactly two parallel sides. May have two right angles.

Course 1

7-6 Quadrilaterals

Additional Example 1A: Naming Quadrilaterals

Give the most descriptive name for each figure.

The figure is a quadrilateral and a trapezoid.

Trapezoid is the most descriptive name.

A.

Course 1

7-6 Quadrilaterals

Additional Example 1B: Naming Quadrilaterals


B.

The figure is a quadrilateral, a parallelogram, and a rectangle.

Rectangle is the most descriptive name.

Course 1

7-6 Quadrilaterals

Additional Example 1C: Naming Quadrilaterals


C. The figure is a plane figure with three congruent sides.

Equilateral triangle is the most descriptive name.

Course 1

7-6 Quadrilaterals

Additional Example 1D: Naming Quadrilaterals


D. The figure is a quadrilateral, parallelogram, and rhombus.

Rhombus is the most descriptive name.

Course 1

7-6 Quadrilaterals

Try This: Additional Example 1A


A. The figure is a quadrilateral and a trapezoid.

Trapezoid is the most descriptive name.

Course 1

7-6 Quadrilaterals

Try This: Additional Example 1B


B.

The figure is a quadrilateral.

Parallelogram is the most descriptive name.

Course 1

7-6 Quadrilaterals

Try This: Additional Example 1C


C. The figure is a plane figure, but it has more than 4 sides.

This figure is not a quadrilateral. It is a hexagon.

Course 1

7-6 Quadrilaterals

Try This: Additional Example 1D


D. The figure is a quadrilateral and a parallelogram.

Parallelogram is the most descriptive name.

Course 1

7-6 Quadrilaterals

You can draw a diagram to classify quadrilaterals based on their properties.

Course 1

7-6 Quadrilaterals

Additional Example 2A & 2B: Classifying Quadrilaterals

Complete each statement.

A. A rectangle can also be called a ________.

B. A parallelogram cannot be a __________.

A rectangle has opposite sides that are parallel; it can be called a parallelogram.

?

?

A parallelogram has opposite sides that are parallel; it cannot be called a trapezoid.

Course 1

7-6 Quadrilaterals


Complete each statement.

A. A rhombus with four right angles is a ________.

B. A trapezoid has exactly _______ parallel sides, and may have _____ right angles.

A rhombus has four congruent sides, and the opposite sides are parallel. If it has four right angles, then it is a square.

?

A trapezoid is a quadrilateral with exactly two parallel sides and it may have two right angles.

? ?

Lesson Quiz Complete each statement.

1. A quadrilateral with four right angles is a _________.

2. A parallelogram with four right angles and four congruent sides is a _______.

3. A figure with 4 sides and 4 angles is a ______.

4. Give the most descriptive name for this quadrilateral.

square

square or rectangle


quadrilateral

trapezoid

Course 1

7-6 Quadrilaterals

?

?

?

7-7 Polygons

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up True or false?

1. Some trapezoids are parallelograms. 2. Some figures with 4 right angles are

squares.

false

true

Course 1

7-7 Polygons

Problem of the Day

Four square tables pushed together can seat either 8 or 10 people. How many people could 12 square tables pushed together seat?

14, 16, 18, or 26 people

Course 1

7-7 Polygons

Learn to identify regular and not regular polygons and to find the angle measures of regular polygons.

Course 1

7-7 Polygons

Vocabulary polygon regular polygon


Course 1

7-7 Polygons

Course 1

7-7 Polygons

Triangles and quadrilaterals are examples of polygons. A polygon is a closed plane figure formed by three or more line segments. A regular polygon is a polygon in which all sides are congruent and all angles are congruent.

Polygons are named by the number of their sides and angles.

Course 1

7-7 Polygons

Course 1

7-7 Polygons

Additional Example 1A: Identifying Polygons

Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.

A. The shape is a closed plane figure formed by three or more line segments. polygon

There are five sides and five angles. pentagon

All 5 sides do not appear to be congruent. Not regular

Course 1

7-7 Polygons

Additional Example 1B: Identifying Polygons


B.

There are eight sides and eight angles. octagon The sides and angles appear to be congruent. regular

The shape is a closed plane figure formed by three or more line segments. polygon

Course 1

7-7 Polygons



A. There are four sides and four angles. quadrilateral

The sides and angles appear to be congruent. regular

Course 1

7-7 Polygons



B. There are four sides and four angles.

quadrilateral

The sides and angles appear to be congruent.

regular

Course 1

7-7 Polygons

The sum of the interior angle measures in a triangle is 180°, so the sum of the interior angle measures in a quadrilateral is 360°.

Course 1

7-7 Polygons


Malcolm designed a wall hanging that was a regular 9-sided polygon (called a nonagon). What is the measure of each angle of the nonagon?

1 Understand the Problem The answer will be the measure of each angle in a nonagon.


• A regular nonagon has 9 congruent sides and 9 congruent angles.

Course 1

7-7 Polygons

2 Make a Plan Make a table to look for a pattern using regular polygons.

Solve 3 Draw some regular polygons and divide each into triangles.


Course 1

7-7 Polygons


720°

Course 1

7-7 Polygons


The number of triangles is always 2 fewer than the number of sides.

A nonagon can be divided into 9 – 2 = 7 triangles.

The sum of the interior angle measures in a nonagon is 7 × 180° = 1,260°.

So the measure of each angle is 1,260° ÷ 9 = 140°.

Course 1

7-7 Polygons Additional Example 2 Continued

Look Back 4

Each angle in a nonagon is obtuse. 140° is a reasonable answer, because an obtuse angle is between 90° and 180°.

Course 1

7-7 Polygons

Try This: Additional Example 2

Sara designed a picture that was a regular 6-sided polygon (called a hexagon). What is the measure of each angle of the hexagon?

1 Understand the Problem The answer will be the measure of each angle in a hexagon.


• A regular hexagon has 6 congruent sides and 6 congruent angles.

Course 1

7-7 Polygons

2 Make a Plan Make a table to look for a pattern using regular polygons.

Solve 3 Draw some regular polygons and divide each into triangles.


Course 1

7-7 Polygons


Course 1

7-7 Polygons

The number of triangles is always 2 fewer than the number of sides. A hexagon can be divided into 6 – 2 = 4 triangles.

The sum of the interior angles in a octagon is 4 × 180° = 720°.

So the measure of each angle is 720° ÷ 6 = 120°.


Course 1

7-7 Polygons

Look Back 4

Each angle in a hexagon is obtuse. 120° is a reasonable answer, because an obtuse angle is between 90° and 180°.


Lesson Quiz

1. Name each polygon and tell whether it appears to be regular or not regular.

2. What is the measure of each angle in a regular dodecagon (12-sided figure)?

150°

nonagon, regular; octagon, not regular


Course 1

7-7 Polygons

7-8 Geometric Pattern

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Divide.

1. What is the sum of the angle measures in a quadrilateral?

2. What is the sum of the angle measures

in a hexagon? 3. What is the measure of each angle in a

regular octagon?

360°

720°

135°

Course 1


Problem of the Day

Which three letters come next in the following series: W, T, L, C, N, I, . . .

T, F, S; the letters are the initial letters of the words in the question.

Course 1


Learn to recognize, describe, and extend geometric patterns.

Course 1


Course 1


Additional Example 1A: Extending Geometric Patterns

Identify a possible pattern. Use the pattern to draw the next figure.

A.

Each circle has one more dot than the one to its left. The dots are positioned around the circle from top to bottom and right to left.

So the next figure might be .

Course 1


Additional Example 1B: Extending Geometric Patterns


B. Each figure has one or two more circles than the figure to its left.


Course 1




A. Each square has one more dot than the one to its left. The dots are positioned around the square from top to bottom and right to left.


Course 1




B. Each figure has two more triangles than the figure before it. The triangles within the figure are shaded one at a time from left to right.


Course 1


Additional Example 2A: Completing Geometric Patterns

Identify a possible pattern. Use the pattern to draw the missing figure.

A. The first figure has 1 right triangle. The second figure has 2 right triangles arranged counterclockwise. The fourth figure has 4 right triangles.

So the missing figure might be .

Course 1


Additional Example 2B: Completing Geometric Patterns


B.

The first figure has 1 dot. The second figure is double the first figure, vertically. The third figure is double the second figure, horizontally. The fourth figure could be double the third figure, vertically, and then the fifth figure will be double the fourth figure, horizontally.

So the missing figure might be

.

Course 1




A. Each figure has 1 more square than the previous figure.


?

Course 1




B. Each figure is a right triangle. The first figure has 2 red triangles along the base. The third figure has 4 red triangles, and the last figure has 5.

?


Course 1


Additional Example 3: Art Application

Travis is painting a platter. Identify a pattern that Travis is using and draw what the finished platter might look like.

The pattern from inside to outside is narrow stripe, narrow stripe, wide stripe, narrow stripe, narrow stripe, wide strip. The color pattern from inside to outside is brown, green, brown, green.

Following this pattern, the finished platter might look like this platter.

Course 1


Try This: Example 3

Nancy is designing a plate. Identify a pattern that Nancy is using and draw what the finished plate might look like.

The pattern from inside to outside is narrow stripe, wide stripe, narrow stripe, wide stripe. The color pattern from inside to outside is dark blue, light blue, dark blue, light blue.

Following this pattern, the finished plate might look like this plate.

Lesson Quiz



Course 1


Possible answer:

7-9 Congruence

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. A straight angle has what degree measure? 2. How do you name a line?

180°

Name any two points on a line.

Course 1

7-9 Congruence

Problem of the Day

The sum of two decimals is 9.3; their difference is 4.3, and their product is 17.00. What are they? 2.5, 6.8

Course 1

7-9 Congruence

Learn to identify congruent figures and to use congruence to solve problems.

Course 1

7-9 Congruence

Course 1

7-9 Congruence Additional Example 1A: Identifying Congruent

Figures

These figures do not have the same shape and they are not the same size.

These figures are not congruent.

Decide whether the figures in each pair are congruent. If not, explain.

A.

Course 1

7-9 Congruence

Additional Example 1B: Identifying Congruent Figures


B. These figures have the same shape size and size.

These figures are congruent.

Course 1

7-9 Congruence

Additional Example 1C: Identifying Congruent Figures


C. Each figure is a trapezoid. The corresponding sides are 5.7 in., 7.5 in., 5 in., and 10 in.


Course 1

7-9 Congruence

Additional Example 1D: Identifying Congruent Figures


D. Each figure is a rectangle. Each short side of each rectangle measure 4 cm. Each long side of each rectangle measures 9 cm.


Course 1

7-9 Congruence



A. These figures do not have the same shape and they are not the same size.

These figures are not congruent.

Course 1

7-9 Congruence



B.

These figures have the same shape size and size.


Course 1

7-9 Congruence



C. Each figure is a triangle. The corresponding sides are 3 cm, 4 cm, and 5 cm.


3 cm

3 cm

4 cm

4 cm 5 cm

5 cm

Course 1

7-9 Congruence



D.

Each figure is a rectangle. Each short side of each rectangle measures 5 in. Each long side of each rectangle measures 12 in. These figures are congruent.

12 in. 12 in.

5 in.

5 in.

Course 1

7-9 Congruence Additional Example 2: Consumer Application Jodi needs a sleeping pad that is congruent to her sleeping bag. Which sleeping pad should she buy?

Both sleeping pads are trapezoids. Only sleeping pad B is the same size as the sleeping bag.

Which sleeping pad is the same size and shape as the sleeping bag?

Sleeping pad B is congruent to the sleeping bag.

Course 1

7-9 Congruence

Try This: Example 2

Jodi needs a frame that is congruent to her flag. Which frame should she buy?

Both frames are triangles. Only Frame B is the same size as the flag.

Which frame is the same size and shape as the flag?

Frame B is congruent to the flag.

11.2 in.

7.8 in.

4 in. 10 in. 11.2 in.

7.8 in. 9.2 in.

4 in. 4 in.

Flag Frame A Frame B

Lesson Quiz True or false?

1. If two figures have sides of the same length, they are congruent.

2. Congruent figures have angle measures that are equal.

3. Decide whether the figures are congruent. If not, explain.

true

false


No, both are quadrilaterals but they are not the same size or shape.

Course 1

7-9 Congruence


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Tell whether the figures described are congruent.

1. two triangles, each with sides measuring 24 cm, 32 cm, and 40 cm 2. a square with sides of length 22 cm and a rectangle with side lengths 12 cm and 11 cm

congruent

not congruent

Course 1


Problem of the Day

Imagine that each figure is folded so that point A lies on point B. What figures would be formed?

trapezoid, right triangle

Course 1


Learn to use translations, reflections, and rotations to transform geometric shapes.

Course 1


Vocabulary transformation translation rotation reflection line of reflection


Course 1


A rigid transformation moves a figure without changing its size or shape. So the original figure and the transformed figure are always congruent.

The illustrations of the alien will show three transformations: a translation, a rotation, and a reflection. Notice the transformed alien does not change in size or shape.

Course 1


A translation is the movement of a figure along a straight line.

Only the location of the figure changes with a translation.

Course 1


A rotation is the movement of a figure around a point. A point of rotation can be on or outside a figure.

Course 1


The location and position of a figure can change with a rotation.

When a figure flips over a line, creating a mirror image, it is called a reflection. The line the figure is flipped over is called line of reflection.

The location and position of a figure change with a reflection.

Course 1


Additional Example 1A: Identifying Transformations

Tell whether each is a translation, rotation, or reflection.

A.

The figure is flipped over a line.

It is a reflection.

Course 1


Additional Example 1B: Identifying Transformations


The figure is moved along a line.

It is a translation.

Course 1


B.

Additional Example 1C: Identifying Transformations


C. The figure moves around a point.

It is a rotation.

Course 1




A.

The figure is flipped over a line.

It is a reflection.

Course 1




B.

The figure moves around a point.

It is a rotation.

Course 1




C.

The figure is moved along a line.

It is a translation.

Course 1


A full turn is a 360°

rotation. So a turn

is 90°, and a turn

is 180°.

1 2 __

1 4 __

90°

180°

360°

Course 1


Additional Example 2A: Drawing Transformations

Draw each transformation.

A. Draw a 180° rotation about the point shown.

Trace the figure and the point of rotation.

Place your pencil on the point of rotation.

Rotate the figure 180°.

Trace the figure in its new location.

Course 1


Additional Example 2B: Drawing Transformations


B. Draw a horizontal reflection.

Trace the figure and the line of reflection.

Fold along the line of reflection.


Course 1




A. Draw a 180° rotation about the point shown.

Trace the figure and the point of rotation.

Place your pencil on the point of rotation.

Rotate the figure 180°.


Course 1




B. Draw a horizontal reflection.

Trace the figure and the line of reflection.

Fold along the line of reflection.


Course 1


Lesson Quiz

1. Tell whether the figure is translated, rotated, or reflected.

2. Draw a vertical reflection of the first figure in problem 1.

rotated


Course 1


7-11 Symmetry

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up True or false?

1. In a reflection the original figure is flipped over a line of symmetry.

2. In a translation the original figure slides

along a straight line and is flipped. 3. In a rotation a figure is moved around a

point.

true

false

true

Course 1

7-11 Symmetry

Problem of the Day

Name four plane figures that can be rotated 180° around a center point and look the same after the rotation as they did before. Possible answer: rectangle, square, parallelogram, regular hexagon

Course 1

7-11 Symmetry

Learn to identify line symmetry.

Course 1

7-11 Symmetry

Vocabulary line symmetry line of symmetry


Course 1

7-11 Symmetry

A figure has line symmetry if it can be folded or reflected so that the two parts of the figure match, or are congruent. The line of reflection is called the line of symmetry.

Course 1

7-11 Symmetry

Additional Example 1A: Identifying Lines of Symmetry

Determine whether each dashed line appears to be a line of symmetry.

A. The two parts of the figure appear to match exactly when folded or reflected across the line.

The line appears to be a line of symmetry.

Course 1

7-11 Symmetry

Additional Example 1B: Identifying Lines of Symmetry


B. The two parts of the figure do not appear congruent.

The line does not appear to be a line of symmetry.

Course 1

7-11 Symmetry



A.

The two parts of the figure do not appear congruent.

The line does not appear to be a line of symmetry.

Course 1

7-11 Symmetry



B. The two parts of the figure appear to match exactly when folded or reflected across the line.

The line appears to be a line of symmetry.

Course 1

7-11 Symmetry

Some figures have more than one line of symmetry.

Course 1

7-11 Symmetry

Additional Example 2A: Finding Multiple Lines of Symmetry

Find all of the lines of symmetry in the regular polygon.

A. Trace the figure and cut it out.

Fold the figure in half in different ways.

Count the lines of symmetry.

5 lines of symmetry

Course 1

7-11 Symmetry

Additional Example 2B: Finding Multiple Lines of Symmetry


B.


8 lines of symmetry

Course 1

7-11 Symmetry

Additional Example 2C: Finding Multiple Lines of Symmetry


C.


4 lines of symmetry

Course 1

7-11 Symmetry



A.

1 line of symmetry


Course 1

7-11 Symmetry



B.


4 lines of symmetry

Course 1

7-11 Symmetry



C. Count the lines of symmetry.

4 lines of symmetry

Course 1

7-11 Symmetry

Additional Example 3A: Social Studies Application

Find all of the lines of symmetry in the flag.

A. Alaska

There are no lines of symmetry.

Course 1

7-11 Symmetry

Additional Example 3B: Social Studies Application


B. Arizona

1 line of symmetry Course 1

7-11 Symmetry

Additional Example 3C: Social Studies Application


C. Colorado


7-11 Symmetry

Additional Example 3D: Social Studies Application


D. New Mexico

2 lines of symmetry Course 1

7-11 Symmetry


2 lines of symmetry

Course 1

7-11 Symmetry

Find all of the lines of symmetry in each design.

There are no lines of symmetry. Course 1

7-11 Symmetry




7-11 Symmetry



2 lines of symmetry Course 1

7-11 Symmetry



Lesson Quiz

Does the described figure have line symmetry?

1. right isosceles triangle

2. rectangle

Do the following capital letters of the alphabet have symmetry? If so, is the line of symmetry horizontal or vertical?

3. H 4. R

yes

yes


yes, vertical and horizontal no

Course 1

7-11 Symmetry

7-12 Tessellations

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Do the following letters of the alphabet have symmetry, and if so, is the line of symmetry horizontal or vertical?

1. X 2. P 3. C 4. T

yes; both no yes; horizontal

Course 1

7-12 Tessellations

yes; vertical

Problem of the Day

If either diagonal is drawn in a figure, the figure is divided into two congruent triangles. What is the most general classification of this figure? parallelogram

Course 1

7-12 Tessellations

Learn to identify tessellations and shapes that can tessellate.

Course 1

7-12 Tessellations

Vocabulary tessellation


Course 1

7-12 Tessellations

A tessellation is a repeating arrangement of one or more shapes that completely covers a plane with no gaps and no overlaps.

Although most tessellations are made by humans, a few occur in nature. Honeycombs are naturally occurring tessellations of regular hexagons.

Course 1

7-12 Tessellations

Additional Example 1A: Identifying Polygons that Tessellate the Plane

Identify whether the polygon can tessellate the plane. Make a drawing to show your answer.

A. The trapezoids cover the plane without any gaps or overlaps.

The trapezoid can tessellate the plane.

Course 1

7-12 Tessellations

Additional Example 1B: Identifying Polygons that Tessellate the Plane


B. There are gaps between the pentagons.

The pentagon cannot tessellate the plane.

Course 1

7-12 Tessellations



A. The triangles cover the plane without any gaps or overlaps.

The triangle can tessellate the plane.

Course 1

7-12 Tessellations



B.

There are no gaps between the parallelograms.

The parallelogram can tessellate the plane.

Course 1

7-12 Tessellations

Additional Example 2A: Identifying Nonpolygons That Tessellate the Plane

Identify whether the shape can tessellate the plane. Make a drawing to show your answer.

The shapes cover the plane without any gaps or overlaps.

This shape can tessellate the plane.

Course 1

7-12 Tessellations

A.

Additional Example 2B: Identifying Nonpolygons That Tessellate the Plane


B.

The shapes cover the plane without any gaps or overlaps.


Course 1

7-12 Tessellations



A. There are gaps between the shapes.

This shape cannot tessellate the plane.

Course 1

7-12 Tessellations



B.

The shapes cover the plane with any gaps or overlaps.


Course 1

7-12 Tessellations

Lesson Quiz

True or false?

1. A regular hexagon can tessellate the plane.

2. A regular octagon can tessellate the plane.

3. A square can tessellate the plane.

4. A regular pentagon can tessellate the plane.

false

true


true

false

Course 1

7-12 Tessellations


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write each fraction in simplest form.

1. 2. 3. 4.

Course 1


2 6 __ 1

3 __ 4

16 __ 1

4 __

25 70 ___ 5

14 __ 6

52 __ 3

26 __

Problem of the Day

What three consecutive odd numbers are factors of 105?

3, 5, 7

Course 1


Learn to write ratios and rates and to find unit rates.

Course 1


Vocabulary ratios equivalent ratios rate unit rate


Course 1


Course 1


You can compare the different groups by using ratios. A ratio is a comparison of two quantities using division.

For a time, the Boston Symphony Orchestra was made up of 95 musicians.

Violins 29 Violas 12 Cellos 10 Basses 9 Flutes 5 Trumpets 3 Double reeds 8 Percussion 5 Clarinets 4 Harp 1 Horns 6 Trombones 3

Course 1


For example, you can use a ratio to compare the number of violins (29) with the number of violas (12). This ratio can be written in three ways.

Notice that the ratio of violins to violas, is

different from the ratio of violas to violins, . The order of the terms is important.

29 12 ___

29 12 ___

12 29 ___

29 to 12 29:12 Terms

Ratios can be written to compare a part to a part, a part to the whole, or the whole to a part.

Course 1


Read the ratio as “twenty-nine to twelve.”

Reading Math 29

12 ___

Course 1


Additional Example 1A: Writing Ratios

Use the table to write the ratio.

A. cats to rabbits

Animals at the Vet Cats 5 Dogs 7 Rabbits 2

or 5 to 2 or 5:2 5 2 __ Part to part

Course 1


Additional Example 1B: Writing Ratios


B. dogs to total number of pets

Animals at the Vet Cats 5 Dogs 7

Rabbits 2

or 7 to 14 or 7:14 7 14

__ Part to whole

Course 1


Additional Example 1C: Writing Ratios


C. total number of pets to cats

Animals at the Vet Cats 5 Dogs 7

Rabbits 2

or 14 to 5 or 14:5 14 5

__ Whole to part

Course 1




A. birds to total number of pets

Animals at the Vet Birds 6

Hamsters 9 Snakes 3

or 6 to 18 or 6:18 6 18

__ Part to whole

Course 1




B. snakes to birds

or 3 to 6 or 3:6 3 6

__ Part to part


Hamsters 9 Snakes 3

Course 1




C. total number of pets to hamsters

or 18 to 9 or 18:9 18 9

__ Whole to part


Hamsters 9 Snakes 3

Course 1


Equivalent ratios are ratios that name the same comparison. You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number.

Course 1


Additional Example 2: Writing Equivalent Ratios

Write three equivalent ratios to compare the number of diamonds to the number of spades in the pattern.

There are 3 diamonds and 6 spades.

number of diamonds number of spades =

3 6 __

There is 1 diamond for every 2 spades.

3 6 __ =

3 ÷ 3 6 ÷ 3 ____ =

1 2 __

3 6 __ = 9

18 __ =

If you triple the pattern, there will be 9 diamonds for 18 spades.

So , , and are equivalent ratios. 3 6 __ 1

2 __ 9

18 __

3 • 3 6 • 3

Course 1

8-1 Ratios and Rates Try This: Example 2

Write three equivalent ratios to compare the number of triangles to the number of hearts in the pattern.

There are 3 triangles and 9 hearts.

number of triangles number of hearts =

3 9 __

There is 1 triangle for every 3 hearts.

3 9 __ =

3 ÷ 3 9 ÷ 3 ____ =

1 3 __

3 9 __ = 9

27 __ =

If you triple the pattern, there will be 9 triangles for 27 hearts.

So , , and are equivalent ratios. 3 9 __ 1

3 __ 9

27 __

3 • 3 9 • 3

Course 1


A rate compares two quantities that have different units of measure. Suppose a 2-liter bottle of soda costs $1.98.

When the comparison is to one unit, the rate is called a unit rate. Divide both terms by the second term to find the unit rate.

When the prices of two or more items are compared, the item with the lowest unit rate is the best deal.

rate = price number of liters _____________ $1.98

2 liters ________ = $1.98 for 2 liters

unit rate = $1.98 2 _____ $1.98 ÷ 2

2 ÷ 2 ________ = $0.99 for 1 liter = $0.99

1 _____

Course 1


Additional Example 3: Consumer Application

A 3-roll pack of paper towels costs $2.79. A 6-roll pack of the same paper towels costs $5.46. Which is the better deal?

Write the rate. $2.79 3 rolls _____

$2.79 ÷ 3 3 rolls ÷ 3 _________ Divide both

terms by 3.

$0.93 1 roll _____ $0.93 for 1 roll.

Write the rate. $5.46 6 rolls _____

$5.46 ÷ 6 6 rolls ÷ 6 _________ Divide both

terms by 6.

$0.91 1 roll _____ $0.91 for 1 roll.

The 6-roll pack of paper towels is the better deal.

Course 1


Try This: Example 3

A 3-pack of juice boxes costs $2.10. A 9-pack of the same juice boxes costs $5.58. Which is the better deal?

Write the rate. $2.10 3 pack _____

$2.10 ÷ 3 3 pack ÷ 3 _________ Divide both

terms by 3.

$0.70 1 box _____ $0.70 for 1 juice

box.

Write the rate. $5.58 9 pack _____

$5.58 ÷ 9 9 pack ÷ 9 _________ Divide both

terms by 9.

$0.62 1 box _____ $0.62 for 1 juice

box.

The 9-pack of juice boxes is the better deal.

Lesson Quiz

Use the table to

write each ratio.

1. tulips to daffodils

2. crocuses to total number of bulbs

3. total bulbs to lilies

4. A dozen eggs cost $1.25 at one market. At a competing market, 1 dozen eggs cost $2.00. Which is the better buy?

20:51

9:17


51:5

eggs from the first market

Course 1


1 2 __

8-2 Proportions

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Use the table to write each ratio.

1. giraffes to monkeys 2. polar bears to all bears 3. monkeys to all animals 4. all animals to all bears

2:17 4:7

17:26

Course 1

8-2 Proportions

Brown bears 3 Giraffes 2 Monkeys 17 Polar Bears 4

26:7

Problem of the Day

A carpenter can build one doghouse in one day. How many doghouses can 12 carpenters build in 20 days? 240

Course 1

8-2 Proportions

Learn to write and solve proportions.

Course 1

8-2 Proportions

Vocabulary proportion


Course 1

8-2 Proportions

Course 1

8-2 Proportions

A proportion is an equation that shows two equivalent ratios.

Read the proportion = as “two is to

one as four is to two.”

2 1 __ 4

2 __ = 4

2 __ 8

4 __ = 2

1 __ 6

3 __ =

2 1 __ 4

2 __

Course 1

8-2 Proportions

Additional Example 1: Modeling Proportions

Write a proportion for the model.

number of hearts number of stars

4 8 __ =

First write the ratio of hearts to stars.

Next separate the hearts and stars into two equal groups.

Course 1

8-2 Proportions


Now write the ratio of hearts to stars in each group.

number of hearts number of stars

2 4 __ =

A proportion shown by the model is = . 4 8 __ 2

4 __

Course 1

8-2 Proportions

Try This: Example 1

Write a proportion for the model.

number of faces number of moons

2 6 __ =

First write the ratio of faces to moons.

Next separate the faces and moons into two equal groups.

Course 1

8-2 Proportions

Try This: Example 1

Now write the ratio of hearts to stars in each group.

number of faces number of moons

1 3 __ =

A proportion shown by the model is = . 2 6 __ 1

3 __

Course 1

8-2 Proportions

CROSS PRODUCTS

Cross products in proportions are equal 3 5 __ 9

15 ___ = 4

8 __ 2

4 __ = 9

6 __ 3

2 __ = 14

7 __ 2

1 __ =

8 • 2 = 4 • 4 5 • 9 = 3 • 15

16 = 16 45 = 45

6 • 3 = 9 • 2

18 = 18

7 • 2 = 14 • 1

14 = 14

Course 1

8-2 Proportions

Additional Example 2: Using Cross Products to Complete Proportions

Find the missing value in the proportion.

6n 6

___

5 6 __ n

18 __ = Find the cross products.

6 • n = 5 • 18 The cross products are equal.

6n = 90 n is multiplied by 6.

90 6

___ = Divide both sides by 6 to undo the multiplication.

n = 15

Course 1

8-2 Proportions

Try This: Example 2

Find the missing value in the proportion.

5n 5

___

3 5 __ n

15 __ = Find the cross products.

5 • n = 3 • 15 The cross products are equal.

5n = 45 n is multiplied by 5.

45 5


n = 9

Course 1

8-2 Proportions

In a proportion, the units must be in the same order in both ratios.

Helpful Hint

tsp lb

___ tsp lb

___ =

lb tsp

___ lb tsp

___ = or

Course 1

8-2 Proportions Additional Example 3: Measurement Application

According to the label, 1 tablespoon of plant fertilizer should be used per 6 gallons of water. How many tablespoons of fertilizer would you use for 4 gallons of water?

f 4 gal

____ 1 tsp 6 gal

_____ = Let f be the amount of fertilizer for 4 gallons of water.

f 4 gal

____ 1 tsp 6 gal

_____ = Write a proportion.

6 • f = 1 • 4 The cross products are equal.

Course 1

8-2 Proportions


f is multiplied by 6.



6f = 4

6f 6

___ 4 6

__ =

f = tbsp 2 3

__

You would use tbsp of fertilizer for 4 gallons of water.

2 3

__

Course 1

8-2 Proportions

Try This: Example 3

According to the label, 3 tablespoon of plant fertilizer should be used per 9 gallons of water. How many tablespoons of fertilizer would you use for 2 gallons of water?

f 2 gal

____ 3 tsp 9 gal

_____ = Let f be the amount of fertilizer for 2 gallons of water.

f 2 gal

____ 3 tsp 9 gal

_____ = Write a proportion.

9 • f = 3 • 2 The cross products are equal.

Course 1

8-2 Proportions


f is multiplied by 9.

Divide both side by 9 to undo the multiplication.


9f = 6

9f 9

___ 6 9

__ =

f = tbsp 2 3

__

You would use tbsp of fertilizer for 2 gallons of water.

2 3

__

Lesson Quiz

1. Write a proportion for the model.

Find the missing value in each proportion.

2. = 3. =

4. The label on a bottle of salad dressing states that there are 3 grams of fat per tablespoon. If you use 3 tablespoons, how many grams of fat would you be getting?

p = 30 x = 25


9 g

Course 1

8-2 Proportions

2 6

__ 1 3

__ =

9 5

__ 45 x

__ p 36

__ 5 6

__


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the missing value in each proportion.

1. = 2. = 3. = 4. =

x = 6

x = 25

x = 35

Course 1


4 1

__ 24 x

__

x 30

__ 5 6

__

14 16

__ x 40

__

8 x

__ 12 18

__ x = 12

Problem of the Day

There are 4 ounces in a gill. There are 4 gills in a pint. There are 8 pints in a gallon. How many ounces are the same as the total of 3 gallons, 3 pints, 3 gills, and 3 ounces? 447

Course 1


Learn to use proportions to make conversions within the customary system.

Course 1


Course 1


Additional Example 1A: Using Proportions to Convert Measurements

A. Sonja went hiking for 4 hours. For how many minutes did she go hiking?

4 hr x min

____ 1 hr 60 min

_____ = 1 hour is 60 minutes. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. 60 • 4 = 1 • x

240 = x

Sonja hiked for 240 minutes.

Course 1


Additional Example 1B: Using Proportions to Convert Measurements

B. Mr. Lee is 72 inches tall. Find his height in feet.

x ft 72 in.

____ 1 ft 12 in

____ = 1 foot is 12 inches. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. 12 • x = 1 • 72 x is multiplied by 12. 12x = 72

Mr. Lee is 6 feet tall.

12x 12

___ 72 12


x = 6

Course 1


Additional Example 1C: Using Proportions to Convert Measurements

C. Hunter used 56 cups of water to wash his car. How many gallons of water did he use?

x gal 56 cups

______ 1 gal 16 cups

_______ = 1 gallon is 16 cups. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. 16 • x = 1 • 56 x is multiplied by 16. 16x = 56

Hunter used 3.5 gallons of water to wash his car.

16x 16

___ 56 16


x = 3.5

Course 1



A. Helga went biking for 3 hours. For how many minutes did she go biking?

3 hr x min

____ 1 hr 60 min

_____ = 1 hour is 60 minutes. Write a proportion. Use a variable for the value you are trying to find.

The cross products are equal. 60 • 3 = 1 • x

180 = x

Helga biked for 180 minutes.

Course 1



B. Ms. Hagan is 60 inches tall. Find her height in feet.

x ft 60 in.

____ 1 ft 12 in.

____ = 1 foot is 12 inches. Write a proportion. Use a variable for the value you are trying to find.

The cross products are equal. 12 • x = 1 • 60

x is multiplied by 12. 12x = 60

Ms. Hagan is 5 feet tall.

12x 12

___ 60 12


x = 5

Course 1



C. Seymour used 40 cups of water to wash his dog. How many gallons of water did he use?

x gal 40 cups

______ 1 gal 16 cups

_______ = 1 gallon is 16 cups. Write a proportion. Use a variable for the value you are trying to find.

The cross products are equal. 16 • x = 1 • 40

x is multiplied by 16. 16x = 40

Seymour used 2.5 gallons of water to wash his dog.

16x 16

___ 40 16


x = 2.5

Course 1


Lesson Quiz

Find the missing value.

1. in. = 6 yd

2. 24 pt = gal


3. 42 oz 2 lb.

4. 7,920 ft 1.5 mi

5. If your family is going to take a 3-week vacation, how many days will you be gone?

3

216


>

=

1 2

__

21 days Course 1


8-4 Similar Figures

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Fill in the missing value.

1. c = 2 qt

2. 180 in. = yd

3. 3 tons = lb

4. min = 2,760 s

8

5

6,000

Course 1

8-4 Similar Figures

45

Problem of the Day

How many 8 in. by 10 in. rectangular tiles would be needed to cover a 16 ft by 20 ft floor? 576

Course 1

8-4 Similar Figures

Learn to use ratios to identify similar figures.

Course 1

8-4 Similar Figures

Vocabulary similar corresponding sides corresponding angles


Course 1

8-4 Similar Figures

Course 1

8-4 Similar Figures

Two or more figures are similar if they have exactly the same shape. Similar figures may be different sizes. Similar figures have corresponding sides and corresponding angles.

• Corresponding sides have lengths that are proportional.

• Corresponding angles are congruent.

Course 1

8-4 Similar Figures

A D

B C

3 cm

2 cm 2 cm

3 cm

W Z

X Y

9 cm

9 cm

6 cm 6 cm

Corresponding sides: Corresponding angles:

AB corresponds to WX.

AD corresponds to WZ. CD corresponds to YZ. BC corresponds to XY.

A corresponds to W. B corresponds to X. C corresponds to Y. D corresponds to Z.

Course 1

8-4 Similar Figures

A D

B C

3 cm

2 cm 2 cm

3 cm

W Z

X Y

9 cm

9 cm

6 cm 6 cm

In the rectangles above, one proportion is

= , or = . AB WX

AD WZ

2 6

3 9

If you cannot use corresponding side lengths to write a proportion, or if corresponding angles are not congruent, then the figures are not similar.

Course 1

8-4 Similar Figures Additional Example 1: Finding Missing

Measures in Similar Figures

111 y

___ 100 200

____ = Write a proportion using corresponding side lengths. The cross products are equal. 200 • 111 = 100 • y

The two triangles are similar. Find the missing length y and the measure of D.

Course 1

8-4 Similar Figures


y is multiplied by 100. 22,200 = 100y

22,200 100

______ 100y 100

____ = Divide both sides by 100 to undo the multiplication.

222 mm = y

Angle D is congruent to angle C, and m C = 70°.

m D = 70°

The two triangles are similar. Find the missing length y and the measure of D.

Course 1

8-4 Similar Figures

Try This: Example 1

52 y

___ 50 100

____ = Write a proportion using corresponding side lengths.

The cross products are equal. 100 • 52 = 50 • y

A B 60 m 120 m

50 m 100 m

y 52 m

65°

45°

The two triangles are similar. Find the missing length y and the measure of B.

Course 1

8-4 Similar Figures


y is multiplied by 50. 5,200 = 50y

5,200 50

_____ 50y 50


104 m = y

Angle B is congruent to angle A, and m A = 65°.

m B = 65°

The two triangles are similar. Find the missing length y and the measure of B.

Course 1

8-4 Similar Figures

Additional Example 2: Problem Solving Application This reduction is similar to a picture that Katie painted. The height of the actual painting is 54 centimeters. What is the width of the actual painting?

1 Understand the Problem The answer will be the width of the actual painting. List the important information: • The actual painting and the reduction above are similar. • The reduced painting is 2 cm tall and 3 cm wide. • The actual painting is 54 cm tall.

Course 1

8-4 Similar Figures


Draw a diagram to represent the situation. Use the corresponding sides to write a proportion.

2 Make a Plan

Reduced Actual

2 54

3 w

Course 1

8-4 Similar Figures


Solve 3

54 • 3 = 2 • w

162 = 2w

162 2

____ 2w 2

___ =

81 = w

The width of the actual painting is 81 cm.

Write a proportion.

The cross products are equal.

w is multiplied by 2.


3 cm w cm

2 cm 54 cm

_____ =

Course 1

8-4 Similar Figures


Look Back 4 Estimate to check your answer. The ratio of the heights is about 2:50 or 1:25. The ratio of the widths is about 3:90, or 1:30. Since these ratios are close to each other, 81 cm is a reasonable answer.

Course 1

8-4 Similar Figures

Try This: Example 2

This reduction is similar to a picture that Marty painted. The height of the actual painting is 39 inches. What is the width of the actual painting?

1 Understand the Problem The answer will be the width of the actual painting. List the important information: • The actual painting and the reduction above are similar. • The reduced painting is 3 in. tall and 4 in. wide. • The actual painting is 39 in. tall.

4 in.

3 in.

Course 1

8-4 Similar Figures


Draw a diagram to represent the situation. Use the corresponding sides to write a proportion.

2 Make a Plan

Reduced Actual

3 39

4 w

Course 1

8-4 Similar Figures


Solve 3 4 in w in

____ 3 in 39 in

_____ =

39 • 4 = 3 • w

156 = 3w

156 3

____ 3w 3

___ =

52 = w

The width of the actual painting is 52 inches.

Write a proportion.


w is multiplied by 3.


Course 1

8-4 Similar Figures


Look Back 4 Estimate to check your answer. The ratio of the heights is about 4:40, or 1:10. The ratio of the widths is about 5:50, or 1:10. Since these ratios are the same, 52 inches is a reasonable answer.

Lesson Quiz These two triangles are similar.

1. Find the missing length x.

2. Find the measure of J.

3. Find the missing length y.

4. Find the measure of P.

5. Susan is making a wood deck from plans for an 8 ft by 10 ft deck. However, she is going to increase its size proportionally. If the length is to be 15 ft, what will the width be?

36.9°

30 in.


4 in.

90°

Course 1

8-4 Similar Figures

12 ft


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the missing value in each proportion.

1. = 2. = 3. = 4. =

t = 15

k = 5

n = 56

Course 1


6 t

__ 18 45

__

k 19

__ 20 76

__

6 8

__ 42 n

__

21 11

__ x 44

__ x = 84

Problem of the Day

Bryce, Kate, and Annie have drawn rectangles. Each side of Bryce’s rectangle is twice the size of one side of Kate’s. The same side of Kate’s rectangle is congruent to one side of Annie’s. One side of Annie’s rectangle is congruent to one side of Bryce’s. Which two rectangles could be congruent? Kate’s and Annie’s

Course 1


Learn to use proportions and similar figures to find unknown measures.

Course 1


Vocabulary indirect measurement


Course 1


Course 1


One way to find a height that you cannot measure directly is to use similar figures and proportions. This method is called indirect measurement.

Course 1


Additional Example 1: Using Indirect Measurement

Use the similar triangles to find the height of the tree.

2 7

__ 6 h

__ =

h • 2 = 6 • 7

2h = 42

2h 2

___ 42 2

___ =

h = 21

Write a proportion using corresponding sides.


h is multiplied by 2.

Divide both sides by 2 to undo multiplication.

The tree is 21 feet tall.

Course 1


Try This: Example 1

Use the similar triangles to find the height of the tree.

3 9

__ 6 h

__ =

h • 3 = 6 • 9

3h = 54

3h 3

___ 54 3

___ =

h = 18





The tree is 18 feet tall.

3 ft. 9 ft.

6 ft. h

Course 1


Additional Example 2: Measurement Application

A rocket casts a shadow that is 91.5 feet long. A 4-foot model rocket casts a shadow that is 3 feet long. How tall is the rocket?

91.5 3

____ h 4

__ =

4 • 91.5 = h • 3

366 = 3h

366 3

___ 3h 3

___ =

122 = h





The rocket is 122 feet tall.

Course 1


Try This: Example 2

A building casts a shadow that is 72.5 feet long when a 4-foot model building casts a shadow that is 2 feet long. How tall is the building?

72.5 2

____ h 4

__ =

4 • 72.5 = h • 2

290 = 2h

290 2

___ 2h 2

___ =

145 = h

Write a proportion using corresponding sides. The cross products are equal.

h is multiplied by 2. Divide both sides by 2 to undo multiplication.

The building is 145 feet tall.

h

72.5 ft 2 ft

4 ft

Lesson Quiz

1. On a sunny day, a telephone pole casts a shadow 21 ft long. A 5-foot-tall mailbox next to the pole casts a shadow 3 ft long. How tall is the pole?

2. On a sunny afternoon, a goalpost casts a 75 ft shadow. A 6.5 ft football player next to the goal post has a shadow 19.5 ft long. How tall is the goalpost?

25 feet

35 feet


Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the unknown heights.

1. A tower casts a 56 ft shadow. A 5 ft girl next to it casts a 3.5 ft shadow. How tall is the tower?

2. On a sunny day, a 50 ft silo casts a 10 ft

shadow. The barn next to the silo casts a shadow that is 4 ft long. How tall is the barn?

80 ft

20 ft

Course 1


Problem of the Day

Hal runs 4 miles in 32 minutes. Julie runs 5 miles more than Hal runs. If Julie runs at the same rate as Hal, for how many minutes will Julie run? 72 minutes

Course 1


Learn to read and use map scales and scale drawings.

Course 1


Vocabulary scale drawing scale


Course 1


Course 1

8-6 Scale Drawings and Maps The map shown is a scale drawing. A scale drawing is a drawing of a real object that is proportionally smaller or larger than the real object. In other words, measurements on a scale drawing are in proportion to the measurements of the real object.

A scale is a ratio between two sets of measurements. In the map above, the scale is 1 in:100 mi. This ratio means that 1 inch on the map represents 100 miles.

Course 1

8-6 Scale Drawings and Maps Additional Example 1: Finding Actual Distances The scale on a map is 4 in: 1 mi. On the map, the distance between two towns is 20 in. What is the actual distance?

20 in. x mi

_____ 4 in. 1 mi

____ =

1 • 20 = 4 • x 20 = 4x 20 4

___ 4x 4

___ =

5 = x

Write a proportion using the scale. Let x be the actual number of miles between the two towns. The cross products are equal. x is multiplied by 4. Divide both sides by 4 to undo multiplication.

The actual distance between the two towns is 5 miles.

Course 1


In Additional Example 1, think “4 inches is 1 mile, so 20 inches is how many miles?” This approach will help you set up proportions in similar problems.

Helpful Hint

Course 1


Try This: Example 1

18 in. x mi

_____ 3 in. 1 mi

____ =

1 • 18 = 3 • x 18 = 3x 18 3

___ 3x 3

___ =

6 = x

Write a proportion using the scale. Let x be the actual number of miles between the two cities. The cross products are equal. x is multiplied by 3. Divide both sides by 3 to undo multiplication.

The actual distance between the two cities is 6 miles.

The scale on a map is 3 in: 1 mi. On the map, the distance between two cities is 18 in. What is the actual distance?

Course 1

8-6 Scale Drawings and Maps Additional Example 2A: Astronomy Application A. If a drawing of the planets were made using the scale 1 in:30 million km, the distance from Mars to Jupiter on the drawing would be about 18.3 in. What is the actual distance between Mars to Jupiter?

18.3 in. x million km

_________ 1 in. 30 million km

___________ =

30 • 18.3 = 1 • x

549 = x

Write a proportion. Let x be the actual distance from Mars to Jupiter.


The actual distance from Mars to Jupiter is about 549 million km.

Course 1

8-6 Scale Drawings and Maps Additional Example 2B: Astronomy Application B. The actual distance from Earth to Mars is about 78 million kilometers. How far apart should Earth and Mars be drawn?

x in. 78 million km

__________ 1 in. 30 million km

___________ =

30 • x = 1 • 78

x = 2

Write a proportion. Let x be the distance from Earth to Mars on the drawing. The cross products are equal.

Earth and Mars should be drawn 2 inches apart.

30x = 78 30x 30

___ 78 30

___ =

x is multiplied by 30. Divide both sides by 30 to undo multiplication.

3 5 __

3 5 __

Course 1

8-6 Scale Drawings and Maps Try This: Additional Example 2A A. If a drawing of the planets were made using the scale 1 in:15 million km, the distance from Mars to Venus on the drawing would be about 8 in. What is the actual distance from Mars to Venus?

8 in. x million km

_________ 1 in. 15 million km

___________ =

15 • 8 = 1 • x

120 = x

Write a proportion. Let x be the distance from Mars to Venus.


The actual distance from Mars to Venus is about 120 million km.

Course 1

8-6 Scale Drawings and Maps Try This: Example 2B B. The distance from Earth to the Sun is about 150 million kilometers. How far apart should Earth and the Sun be drawn?

x in. 150 mil km

________ 1 in. 15 mil km

________ =

15 • x = 1 • 150

x = 10

Write a proportion. Let x be the distance from Earth to the Sun on the drawing. The cross products are equal.

Earth and the Sun should be drawn 10 inches apart.

15x = 150 15x 15

___ 150 15

____ =

x is multiplied by 15.


Lesson Quiz

On a map of the Great Lakes, 2 cm = 45 km. Find the actual distance of the following, given their distances on the map.

1. Detroit to Cleveland = 12 cm

2. Duluth to Nipigon = 20 cm

3. Buffalo to Syracuse = 10 cm

4. Sault Ste. Marie to Toronto = 33 cm

450 km

270 km


225 km

742.5 km

Course 1


8-7 Percents

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write each fraction as a decimal.

1. 2. Write each decimal as a fraction. 3. 0.375 4. 0.05

0.75 0.9

Course 1

8-7 Percents

3 4 __ 9

10 __

3 8 __ 1

20 __

Problem of the Day

Wally wanted to change the scale of a drawing from 1 in. = 2 ft to 1 in. = 10 ft. The scale height of a building in the first drawing is 25 in. How high is the building in the new drawing? 5 in.

Course 1

8-7 Percents

Learn to write percents as decimals and as fractions.

Course 1

8-7 Percents

Vocabulary percent


Course 1

8-7 Percents

Course 1

8-7 Percents

Most states charge sales tax on items you purchase. Sales tax is a percent of the item’s price. A percent is a ratio of a number to 100.

You can remember that percent means “per hundred.” For example, 8% means “8 per hundred,” or “8 out of 100.”

Course 1

8-7 Percents

If a sales tax rate is 8%, the following statements are true:

• For every $1.00 you spend, you pay $0.08 in sales tax.

• For every $10.00 you spend, you pay $0.80 in sales tax.

• For every $100 you spend, you pay $8 in sales tax.

Because percent means “per hundred,” 100% means “100 out of 100.” This is why 100% is often used to mean “all” or “the whole thing.”

Course 1

8-7 Percents Additional Example 1: Modeling Percent

Use a 10-by-10-square grid to model 17%.

A 10-by-10 square grid has 100 squares.

17% means “17 out of 100”

or .

Shade 17 squares out of 100 squares.

17 100

___

Course 1

8-7 Percents Try This: Example 1

Use a 10-by-10-square grid to model 26%.

A 10-by-10 square grid has 100 squares.

26% means “26 out of 100”

or .

Shade 26 squares out of 100 squares.

26 100

___

Course 1

8-7 Percents Additional Example 2: Writing Percents as

Fractions

Write 35% as a fraction in simplest form.

Write the percent as a fraction with a denominator of 100.


35% = 35 100

___

35 ÷ 5 100 ÷ 5

_______ = 7 20 __

Written as a fraction, 35% is . 7 20 __

Course 1


Write 65% as a fraction in simplest form.



65% = 65 100

___

65 ÷ 5 100 ÷ 5

_______ = 13 20 __


Course 1

8-7 Percents Additional Example 3: Life Science Application

Janell is an ice skater with 20% body fat. Write 20% as a fraction in simplest form.



20% = 20 100

___

20 ÷ 20 100 ÷ 20

_______ = 1 5 __


Course 1


Timmy is a football player with 10% body fat. Write 10% as a fraction in simplest form .



10% = 10 100

___

10 ÷ 10 100 ÷ 10

_______ = 1 10

__


Course 1

8-7 Percents

Additional Example 4: Writing Percents as Decimals

Write 56% as a decimal. Write the percent as a fraction with a denominator of 100.

Write the fraction as a decimal.

56% = 56 100

___

Written as a fraction, 56% is 0.56.

100 56.00 0.56

–500 600

–600 0

To divide by 100, move the decimal point two places to the left. 56 ÷ 100 = 0.56

Remember!

Course 1


Write 32% as a decimal.



32% = 32 100

___

Written as a fraction, 32% is 0.32.

100 32.00 0.32

–300 200

–200 0

Course 1

8-7 Percents Additional Example 5: Application

Water made up 85% of the fluids that Kirk drank yesterday. Write 85% as a decimal.



85% = 85 100

___

85 ÷ 100 = 0.85

Written as a decimal, 85% is 0.85.

Course 1


Water made up 95% of the fluids that Lisa drank yesterday. Write 95% as a decimal.



95% = 95 100

___

95 ÷ 100 = 0.95

Written as a decimal, 95% is 0.95.

Lesson Quiz

Write each percent as a fraction in simplest form.

1. 52% 2. 29%

Write each percent as a decimal.

3. 17% 4. 86%

5. A store clerk has an 8% sales increase. Write the increase as a fraction in simplest form and as a decimal.


, 0.08

Course 1

8-7 Percents

13 25 __ 29

100 ___

0.17 0.86

2 25 __


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write each percent as a fraction in simplest form.

1. 37% 2. 78% Write each percent as a decimal. 3. 59% 4. 7% 0.59 0.07

Course 1


37 100

___ 39 50

___

Problem of the Day

Jennifer gave Karen 50% of her comic books. Karen gave Jack 50% of the books she got from Jennifer. Jack gave Lucas 50% of the books he got from Karen. Lucas got 4 comic books. What percent of Jennifer’s comic books does Lucas have? 12.5%

Course 1


Learn to write decimals and fractions as percents.

Course 1


Course 1

8-8 Percents, Decimals, and Fractions Additional Example 1A: Writing Decimals as

Percents

Write each decimal as a percent.

Method 1: Use place value.

A. 0.7

Write the decimal as a fraction.

Write an equivalent fraction with 100 as the denominator.

0.7 = 7 10

__

7 • 10 10 • 10

_______ = 70 100

___

70 100

___ = 70% Write the numerator with a percent symbol.

Course 1

8-8 Percents, Decimals, and Fractions Additional Example 1B: Writing Decimals as

Percents



B. 0.16

Write the decimal as a fraction. 0.16 = 16 100

__

16 100


Course 1

8-8 Percents, Decimals, and Fractions Additional Example 1C: Writing Decimals as

Percents


Method 2: Multiply by 100.

C. 0.4118

Multiply by 100. 0.4118 • 100

41.18% Add the percent symbol.

Course 1

8-8 Percents, Decimals, and Fractions Additional Example 1D: Writing Decimals as

Percents



D. 0.067

Multiply by 100. 0.067 • 100


Course 1

8-8 Percents, Decimals, and Fractions Try This: Example 1A



A. 0.9

Write the decimal as a fraction.


0.9 = 9 10

__

9 • 10 10 • 10

_______ = 90 100

___

90 100


Course 1

8-8 Percents, Decimals, and Fractions Try This: Example 1B



B. 0.31

Write the decimal as a fraction. 0.31 =

31 100


31 100

Course 1

8-8 Percents, Decimals, and Fractions Try This: Example 1C



C. 0.1225

Multiply by 100. 0.1225 • 100


Course 1

8-8 Percents, Decimals, and Fractions Try This: Example 1D



D. 0.023

Multiply by 100. 0.023 • 100


Course 1


When the denominator is a factor of 100, it is often easier to use method 1. When the denominator is not a factor of 100, it is usually easier to use method 2.

Helpful Hint

Course 1

8-8 Percents, Decimals, and Fractions Additional Example 2A: Writing Fractions as

Percents


9 • 4 25 • 4

______ = 36 100

___

36 100


9 25

__

Write each fraction as a percent.

Method 1: Write an equivalent fraction with a denominator of 100.

A.

Course 1

8-8 Percents, Decimals, and Fractions Additional Example 2B: Writing Fractions as

Percents


0.15 = 15% Multiply by 100 by moving the decimal point right two places. Add the percent symbol.

3 20

__

100

20 3.00 0.15

–20

–100 0

Method 2: Use division to write the fraction as a decimal.

B.

Course 1

8-8 Percents, Decimals, and Fractions Try This: Example 2A


Method 1: Write an equivalent fraction with a denominator of 100.

A.


7 • 2 50 • 2

______ = 14 100

___

14 100


7 50

__

Course 1




0.16 = 16% Multiply by 100 by moving the decimal point right by two places. Add the percent symbol.

8 50

__

50 8.00 0.16

Method 2: Use division to write the fraction as a decimal.

B.

Course 1



One year, of people with home offices

were self-employed. What percent of people with home offices were self employed?

Write an equivalent fraction with a denominator of 100.

7 • 4 25 • 4

______ = 28 100

___

28 100

___ = 28%

7 25

__

7 25

__

28% of people with home offices were self-employed.

Course 1

8-8 Percents, Decimals, and Fractions Try This: Example 3

One year, of people who bought new cars

bought cars that were yellow. What percent of people that bought new cars bought yellow cars?

Write an equivalent fraction with a denominator of 100.

3 • 2 50 • 2

______ = 6 100

___

6 100

___ = 6%

3 50

__

3 50

__

6% of people with new cars bought yellow cars.

Course 1


Lesson Quiz


1. 0.26 2. 0.419


3. 4.

5. About of all the students at a local high

school own their own car. What percent is this?

41.9% 26%


20% 56.25%

Course 1


1 5

__ 9 16

__

1 16

__

6.25%


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write each decimal as a percent and fraction.

1. 0.38 2. 0.06 3. 0.2

38%,

6%,

20%,

Course 1


19 50

__

3 50

__

1 5

__

Problem of the Day

Lucky Jim won $16,000,000 in a lottery. Every year for 10 years he spent 50% of what was left. How much did Lucky Jim have after 10 years? $15,625

Course 1


Learn to find the missing value in a percent problem.

Course 1


Course 1


To find the percent of one number is of another, use this proportion:

% 100 = is

of

If you are looking for 45% of 420, 45 replaces the percent sign and 420 replaces “of.” The first denominator, 100, always stays the same. The “is” part is what you have been asked to find.

Course 1



First estimate your answer. Think: 35% = ,

which is close to , and 560 is close to 600. So

about of the students participate in after-school sports.

35 100

___

1 3

__

1 3

__

1 3

__ This is the estimate. • 600 = 200

Think: “35 out of 100 is how many out of 560?” Helpful Hint

There are 560 students in Ella’s school. If 35% of the students participate in after-school sports, how many students participate in after-school sports?

Course 1


Additional Example 1 Continued Now solve:

s 560

___ 35 100

___ =

100 • s = 35 • 560 100s = 19,600 100s 100

____ 19,600 100

_____ =

s = 196

Let s represent the number of students who participate in after-school sports. The cross products are equal. s is multiplied by 100. Divide both sides of the equation by 100 to undo multiplication.

Since 196 is close to your estimate of 200, 196 is a reasonable answer.

196 students participate in after-school sports.

Course 1


Try This: Example 1

There are 480 students in Tisha’s school. If 70% of the students participate in the fundraising program, how many students participate in the fundraising program?

First estimate your answer. Think: 70% = ,

which is close to , and 480 is close to 500. So

about of the students participate in after

school sports.

70 100

___

3 4

__

3 4

__

This is the estimate. 3 4

__ • 500 = 375

Course 1


Try This: Example 1 Continued Now solve:

s 480

___ 70 100

___ =

100 • s = 70 • 480 100s = 33,600 100s 100

____ 33,600 100

_____ =

s = 336

Let s represent the number of students who participate in after-school sports. The cross products are equal. s is multiplied by 100. Divide both sides of the equation by 100 to undo multiplication.

Since 336 is close to your estimate of 375, 336 is a reasonable answer.

336 students participate in the fundraising program.

Course 1



Johan is 25% of the way through his exercises. If he has exercised for 20 minutes so far, how much longer does he have to work out?

is of

__ % 100

___ =

100 • 20 = 25 • m

25% of the exercises are completed, so 20 minutes is 25% of the total time needed.


20 m

__ 25 100

___ = Set up a proportion. The “of” part is what you have been asked to find.

Course 1



2,000 = 25m

2,000 25

_____ 25m 25

____ =

80 = m

m is multiplied by 25.


The time needed for the exercises is 80 min. So far, the exercises have taken 20 min. Because 80 – 20 = 60, Johan will be finished in 60 min.

Course 1


Try This: Example 2

Phil is 30% of the way through his homework. If he has worked for 15 minutes so far, how much longer does he have to work?

is of

__ % 100

___ =

100 • 15 = 30 • m

30% of the exercises are completed, so 15 minutes is 30% of the total time needed.


15 m

__ 30 100

___ = Set up a proportion. The “of” part is what you have been asked to find.

Course 1



1,500 = 30m

1,500 30

_____ 30m 30

____ =

50 = m

m is multiplied by 30.


The time needed for the homework is 50 min. So far, the homework has taken 15 min. Because 50 – 15 = 35, Phil will be finished in 35 min.

Course 1


Instead of using proportions, you can also multiply to find a percent of a number.

Course 1

8-9 Percent Problems Additional Example 3: Multiplying to Find a

Percent of a Number

36% = 0.36 0.36 • 50

Write the percent as a decimal. Multiply using the decimal.

18 So 18 is 36% of 50.

17% = 0.17 0.17 • 95


16.15 So 16.15 is 17% of 95.

A. Find 36% of 50.

B. Find 17% of 95.

Course 1

8-9 Percent Problems Try This: Example 3

20% = 0.20 0.20 • 70


14 So 14 is 20% of 70.

12% = 0.12 0.12 • 83


9.96 So 9.96 is 12% of 83.

A. Find 20% of 70.

B. Find 12% of 83.

Lesson Quiz

1. Find 28% of 310.

2. Find 70% of 542.

3. Martha is taking a 100-question test. She has completed 60% of the test in 45 minutes. How much longer will it take her to finish the test?

4. Crystal has a collection of 72 pennies. If 25% of them are Canadian, how many Canadian pennies does she have?

379.4

86.8


30 min

18

Course 1


8-10 Using Percents

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the percent of each number.

1. 75% of 300 2. 93% of 56 3. 32% of 128 4. 9% of 60

225 52.08 40.96

Course 1

8-10 Using Percents

5.4

Problem of the Day

A chessboard is 8 squares wide by 8 squares long. Each player has 8 pawns, 1 king, and 7 other pieces. At the start of a game, all the pieces are on the board, 1 piece per square. What percent of the total number of squares have a chess piece? 50%

Course 1

8-10 Using Percents

Learn to solve percent problems that involve discounts, tips, and sales tax.

Course 1

8-10 Using Percents

Vocabulary discount tip sales tax


Course 1

8-10 Using Percents

Common Uses of Percents Discounts A discount is an amount that is

subtracted from the regular price of an item. discount = price • discount rate total cost = price – discount

Tips A tip is an amount added to a bill for service. tip = bill • tip rate total cost = bill + tip

Sales tax Sales tax is an amount added to the price of an item.

sales tax = price • sales tax rate total cost = price + sales tax

Course 1

8-10 Using Percents

Additional Example 1: Finding Discounts A clothing store is having a 10% off sale. If Angela wants to buy a sweater whose regular price is $19.95, about how much will she pay for the sweater after the discount? Step 1: First round $19.95 to $20. Step 2: Find 10% of $20 by multiplying 0.10 • 20. (Hint: Moving the decimal point one place left is a shortcut.) 10% of 20 = 0.10 • $20 = $2.00

The approximate discount is $2.00. Subtract this amount from $20.00 to estimate the cost of the sweater. $20.00 – $2.00 = $18.00 Angela will pay about $18.00 for the sweater.

Course 1

8-10 Using Percents

To multiply by 0.10, move the decimal point one place left.

Remember!

Course 1

8-10 Using Percents

Try This: Example 1 A fishing store is having a 10% off sale. If Gerald wants to buy a fishing pole whose regular price is $39.95, about how much will he pay for the pole after the discount? Step 1: First round $39.95 to $40. Step 2: Find 10% of $40 by multiplying 0.10 • 40.

10% of 40 = 0.10 • $40 = $4.00

The approximate discount is $4.00. Subtract this amount from $40.00 to estimate the cost of the pole. $40.00 - $4.00 = $36.00

Gerald will pay about $36.00 for the fishing pole.

Course 1

8-10 Using Percents

When estimating percents, use percents that you can calculate mentally.

• You can find 10% of a number by moving the decimal point one place to the left.

• You can find 1% of a number by moving the decimal point two places to the left.

• You can find 5% of a number by finding one-half of 10% of the number.

Course 1

8-10 Using Percents

Additional Example 2: Finding Tips

Ben’s dinner bill is $7.85. He wants to leave a tip that is 15% of the bill. About how much should his tip be? Step 1: First round $7.85 to $8. Step 2: Think: 15% = 10% + 5%

Step 3: 5% = 10% ÷ 2 = $0.80 ÷ 2 = $0.40

Step 4: 15% = 10% + 5%.

Ben should leave about $1.20 as a tip.

= $0.80 + $0.40 = $1.20

10% of $8 = 0.10 • $8 = $0.80

Course 1

8-10 Using Percents

Try This: Example 2

Lita’s dinner bill is $11.95. She wants to leave a tip that is 15% of the bill. About how much should her tip be? Step 1: First round $11.95 to $12. Step 2: Think: 15% = 10% + 5%

Step 3: 5% = 10% ÷ 2 = $1.20 ÷ 2 = $0.60

Step 4: 15% = 10% + 5%.

Lita should leave about $1.80 as a tip.

= $1.20 + $0.60 = $1.80

10% of $12 = 0.10 • $12 = $1.20

Course 1

8-10 Using Percents

Additional Example 3: Finding Sales Tax Ann is buying a dog bed for $29.75. The sales tax rate is 7%. About how much will the total cost of the dog bed be? Step 1: First round $29.75 to $30. Step 2: Think: 7% = 7 • 1%

1% of $30 = 0.01 • $30 = $0.30 Step 3: 7% = 7 • 1%.

= 7 • $0.30 = $2.10. The approximate sales tax is $2.10. Add this amount to $30 to estimate the total cost of the dog bed.

$30 + $2.10 = $32.10

Ann will pay about $32.10 for the dog bed. Course 1

8-10 Using Percents

Try This: Example 3 Erik is buying a blanket for $19.83. The sales tax rate is 8%. About how much will the total cost of the blanket be? Step 1: First round $19.83 to $20. Step 2: Think: 8% = 8 • 1%

1% of $20 = 0.01 • $20 = $0.20 Step 3: 8% = 8 • 1%

= 8 • $0.20 = $1.60. The approximate sales tax is $1.60. Add this amount to $20 to estimate the total cost of the blanket.

$20 + $1.60 = $21.60

Erik will pay about $21.60 for the blanket. Course 1

8-10 Using Percents

Lesson Quiz

1. Sean’s new jeans are priced at $29.97, but the sale sign reads, “Take 15% off.” About how much will the jeans cost after the discount?

2. The bill for a family dinner comes to $56.78. About how much would a 20% tip be?

3. The price on a book is $12.99. If sales tax is 4%, about how much will its total cost be?

4. Megan wants a new bike. She is happy to see a sign that reads, “All bikes 10% off.” If the original price of the bike was $159.90 and sales tax is 6%, about how much will the total cost of the bike be?

About $11.50

About $25.50


About $13.50

About $150 Course 1

8-10 Using Percents


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Add or subtract.

1. 16 + 25 2. 84 – 12 3. Graph the even numbers from 1 to 10 on

a number line.

41 72

Course 1


0 1 2 3 4 5 6 7 8 9 10

Problem of the Day

Carlo uses a double-pan balance and three different weights to weigh bird seed. If his weights are 1 lb, 2 lb, and 5 lb, what whole pound amounts is he able to weigh? 1, 2, 3, 5, 6, 7, and 8 lb

Course 1


Learn to identify and graph integers, find opposites, and find the absolute value of an integer.

Course 1


Vocabulary positive number negative number opposites integer absolute value


Course 1


Course 1


Positive numbers are greater than 0. They may be written with a positive sign (+), but they are usually written without it.

Negative numbers are less than 0. They are always written with a negative sign (–).

Course 1


Additional Example 1A & 1B: Identifying Positive and Negative Numbers in the Real World

Name a positive or negative number to represent each situation.

A. a jet climbing to an altitude of 20,000 feet

B. taking $15 out of the bank

Positive numbers can represent climbing or rising.

+20,000

Negative numbers can represent taking out or withdrawing. –15

Course 1




C. 7 degrees below zero

Negative numbers can represent values below or less than a certain value. –7

Course 1


Try This: Additional Example 1A & 1B


A. 300 feet below sea level

B. a hiker hiking to an altitude of 4,000 feet

Negative numbers can represent values below or less than a certain value. –300

Positive numbers can represent climbing or rising. +4,000

Course 1



Name a positive or negative number to represent each situation?

C. spending $34

Negative numbers can represent losses or decreases. –34

Course 1


You can graph positive and negative numbers on a number line. On a number line, opposites are the same distance from 0 but on different sides of 0. Integers are the set of all whole numbers and their opposites.

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

Opposites

Positive Integers Negative Integers

0 is neither negative nor positive.

Course 1


The set of whole numbers includes zero and the counting numbers.

{0, 1, 2, 3, 4, …}

Remember!

Course 1


Additional Example 2A & 2B: Graphing Integers

Graph each integer and its opposite on a number line.

A. +2

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

B. –3

–2 is the same distance from 0 as +2.

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

+3 is the same distance from 0 as –3.

Course 1


Additional Example 2C: Graphing Integers


C. +1

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5


Course 1




A. +5

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

B. –4


–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

+4 is the same distance from 0 as –4.

Course 1




C. +4

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5


Course 1


|–3| = 3 |3| = 3

The absolute value of an integer is its distance from 0 on a number line. The symbol for absolute value is | |.

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

3 units 3 units

Course 1


• Absolute values are never negative.

• Opposite integers have the same absolute value.

• |0| = 0

Read |3| as “the absolute value of 3.”

Read |–3| as “the absolute value of negative 3.”

Reading Math

Course 1


Additional Example 3A & 3B: Finding Absolute Value

Use the number line to find the absolute value of each integer.

A. |–4|

B. |2|

–4 is 4 units from 0, so |–4| = 4. 4

2 is 2 units from 0, so |2| = 2. 2

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

Course 1



Use the number line to find the absolute value of each integer.

A. |–3|

B. |1|

–3 is 3 units from 0, so |–3| = 3. 3

1 is 1 unit from 0, so |1| = 1. 1

–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5

Lesson Quiz Name a positive or negative number to represent each situation.

1. saving $15

2. 12 feet below sea level

3. What is the opposite of –6?

4. What is the absolute value of –12?

5. When the Swanton Bulldogs football team passed the football, they gained 25 yards. Write an integer to represent this situation.

–12

+15


6

12

Course 1


+25

9-2 Comparing and Ordering Integers

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Compare. Write <, >, or =.

1. 8,426 8,246

2. 9,625 6,852

3. 2,071 2,171

4. 2,250 2,250

>

>

<

Course 1


=

Problem of the Day

Four friends are waiting in line at the amusement park. Jenna is in front of Kyle. Kyle is behind Gary and in front of Maggie. Gary is first. In what order are they waiting?

Gary, Jenna, Kyle, Maggie

Course 1


Learn to compare and order integers.

Course 1


Course 1


Numbers on a number line increase in value as you move from left to right.

Remember!

Course 1


Additional Example 1: Comparing Integers Use the number line to compare each pair of integers. Write < or >.

A. –2 2

B. 3 –5

C. –1 –4

–5 –4 –3 –2 –1 0 1 2 3 4 5

–2 is to the left of 2 on the number line. –2 < 2

3 > –5 3 is to the right of –5 on the number line.

–1 is to the right of –4 on the number line. –1 > –4

Course 1


Try This: Example 1

Use the number line to compare each pair of integers. Write < or >.

A. –2 1

B. 2 –3

C. –3 –4

–5 –4 –3 –2 –1 0 1 2 3 4 5

–2 is to the left of 1 on the number line. –2 < 1

2 > –3 2 is to the right of –3 on the number line.

–3 is to the right of –4 on the number line. –3 > –4

Course 1

9-2 Comparing and Ordering Integers Additional Example 2: Ordering Integers

Order the integers in each set from least to greatest.

A. –2, 3, –1

B. 4, –3, –5, 2

–3 –2 –1 0 1 2 3

Graph the integers on the same number line.

Then read the numbers from left to right: –2, –1, 3.

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6


Then read the numbers from left to right: –5, –3, 2, 4.

Course 1

9-2 Comparing and Ordering Integers Try This: Example 2

Order the integers in each set from least to greatest.

A. –2, 2, –3

B. 6, –2, 5, –3

–3 –2 –1 0 1 2 3


Then read the numbers from left to right: –3, –2, 2.

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6


Then read the numbers from left to right: –3, –2, 5, 6.

Course 1




The answer will be the player with the lowest score. List the important information:

• Craig scored +2.

• Cameron scored +3.

• Rob scored –1.

In a golf match, Craig scored +2, Cameron scored +3, and Rob scored –1. Who won the golf match?

Course 1



2 Make a Plan

You can draw a diagram to order the scores from least to greatest.

Solve 3 Draw a number line and graph each player’s score on it.

–3 –2 –1 0 1 2 3 • • •

Rob’s score, –1, is farthest to the left, so it is the lowest score. Rob won the golf match.

Course 1



Negative integers are always less than positive integers, so neither Craig nor Cameron won the golf match.

Look Back 4

Course 1


Try This: Example 3


The answer will be the player with the lowest score. List the important information:

• Melissa scored +6.

• Trista scored –3.

• Alyssa scored –1.

In a golf match, Melissa scored +6, Trista scored –3, and Alyssa scored –1. Who won the golf match?

Course 1



2 Make a Plan

You can draw a diagram to order the scores from least to greatest.

Solve 3 Draw a number line and graph each player’s score on it.

• • •

Trista’s score, –3, is farthest to the left, so it is the lowest score. Trista won the golf match.

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

Course 1


Negative integers are always less positive integers, so Melissa cannot be the winner. Since Trista’s score of -3 is less than Alyssa’s score of -1, Trista won.

Look Back 4


Lesson Quiz Order the integers in each set from least to greatest.

1. –3, 7, 4

2. –11, 2, 5, –15


3. –3 4 4. –12 –10

5. A location in Carlsbad Caverns is 752 ft below sea level, and another location is 910 ft below sea level. Which location is closer to sea level?

–15, –11, 2, 5

–3, 4, 7


< <

Course 1


the location at –752 feet


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Use the number line to compare each pair of integers. Write < or >.

1. 7 –7 2. –8 –3 3. 0 –4 4. –2 –5

> < >

Course 1


>

Problem of the Day

While delivering pizza, Christian drove 4 miles south, 6 miles west, 2 miles north, 8 miles east, and then 2 miles north. How far is Christian from where he started? 2 miles

Course 1


Learn to locate and graph points on the coordinate plane.

Course 1


Vocabulary coordinate plane axes x-axis y-axis quadrants origin coordinates x-coordinate y-coordinate


Course 1


Course 1


A coordinate plane is formed by two number lines in a plane that intersect at right angles. The point of intersection is the zero on each number line.

• The two number lines are called the axes.

Course 1



• The horizontal axis is called the x-axis.

Course 1



• The vertical axis is called the y-axis.

Course 1



• The two axes divide the coordinate plane into four quadrants.

Course 1



• The point where the axes intersect is called the origin.

Course 1


Additional Example 1: Identifying Quadrants

Name the quadrant where each point is located.

A. X

B. Y

C. S

Quadrant IV

Quadrant III

Quadrant II

Course 1


Try This: Example 1

Name the quadrant where each point is located.

A. V

B. Z

C. T

Quadrant I

y-axis

Quadrant IV

Points on the axes are not in any quadrant. Helpful Hint

no quadrant


Course 1


An ordered pair gives the location of a point on a coordinate plane. The first number tells how far to move right (positive) or left (negative) from the origin. The second number tells how far to move up (positive) or down (negative).

The numbers in an ordered pair are called coordinates. The first number is called the x-coordinate. The second number is called the y-coordinate.

The ordered pair for the origin is (0,0).

Course 1


Additional Example 2A: Locating Points on a Coordinate Plane

Give the coordinates of each point.

A. X

From the origin, X is 4 units right and 1 unit down. (4, –1)

Course 1


Additional Example 2B: Locating Points on a Coordinate Plane


B. Y

(–2, –3)

From the origin, Y is 2 units left, and 3 units down.

Course 1


Additional Example 2C: Locating Points on a Coordinate Plane


C. S

From the origin, S is 3 units left, and 3 units up. (–3, 3)

Course 1




A. V

From the origin, V is 4 units right and 2 units up. (4, 2)

Course 1




B. Z

(0, 4)

From the origin, Z is 0 units right, and 4 units up.

Course 1




C. T

From the origin, T is 1 unit right, and 3 units down. (1, –3)

Course 1


Additional Example 3A & 3B: Graphing Points on a Coordinate Plane

Graph each point on a coordinate plane.

A. M(4, 3)

B. Q (–4, 1)

x

y

–4 –2 0 2 4

4

2

–2

–4

From the origin, move 4 units right, and 3 units up.

M

Q

From the origin, move 4 units left, and 1 unit up.

Course 1


Additional Example 3C: Graphing Points on a Coordinate Plane


C. R(0, –3)

From the origin, move 3 units down.

x

y

–4 –2 0 2 4

4

2

–2

–4 R

Course 1




A. L(3, 4)

B. M (–3, –3)

x

y

–4 –2 0 2 4

4

2

–2

–4

From the origin, move 3 units right, and 4 units up.

L

M From the origin, move 3 units left, and 3 units down.

Course 1




C. P(1, –2)

From the origin, move 1 unit right and 2 units down.

x

y

–4 –2 0 2 4

4

2

–2

–4

P

Lesson Quiz

Name the quadrant where each ordered pair is located.

1. (3, –5) 2. (–4, –2)

3. (6, 2) 4. (–7, 9)


5. A 6. B

7. C 8. D

III IV


I II

Course 1


(–2, 3) (1, 2)

(0, –2) (–2, –3)

9-4 Adding Integers

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Add.

1. 2 + 3 2. 7 + 4 3. 8 + 5 4. 17 + 12 Name the integer that corresponds to each point. 5. A 6. B

5 11

–3

Course 1

9-4 Adding Integers

29 13

5

Problem of the Day

Karen earned 40 points on part I of a test, 26 points on part II, and 25 points on part III. What number of points did Karen earn on parts I and III combined?

65

Course 1

9-4 Adding Integers

Learn to add integers.

Course 1

9-4 Adding Integers

Course 1

9-4 Adding Integers

Adding Integers on a Number Line

Move right on a number line to add a positive integer. Move left on a number line to add a negative integer.

Course 1

9-4 Adding Integers

Parentheses are used to separate addition, subtraction, multiplication, and division signs from negative integers.

–2 + (–5) = –7

Writing Math

Course 1

9-4 Adding Integers

Additional Example 1A & 1B: Writing Integer Addition

Write the addition modeled on each number line.

A.

B.

–1 0 1 2 3 4 5 6

5 + (–4)

The addition modeled is 5 + (–4) = 1.

–5 –4 –3 –2 –1 0 1 2 3 4

(–4) +7

The addition modeled is –4 + 7 = 3.

Course 1

9-4 Adding Integers

Additional Example 1C: Writing Integer Addition


C.

3 + (–5)

The addition modeled is 3 + (–5) = –2.

–4 –3 –2 –1 0 1 2 3 4

Course 1

9-4 Adding Integers



A.

B.

6 + (–3)

The addition modeled is 6 + (–3) = 3.

(–2) +3

The addition modeled is –2 + 3 = 1.

–1 0 1 2 3 4 5 6

–5 –4 –3 –2 –1 0 1 2 3 4

Course 1

9-4 Adding Integers



C.

4 + (–7)

The addition modeled is 4 + (–7) = –3.

–4 –3 –2 –1 0 1 2 3 4

Course 1

9-4 Adding Integers

Additional Example 2A: Adding Integers

Find the sum.

A. –3 + (–2)

–6 –5 –4 –3 –2 –1 0 1

–3 + (–2)

–3 + (–2) = –5

Think:

Course 1

9-4 Adding Integers

Additional Example 2B: Adding Integers

Find the sum.

B. 6 + (–8)

6 + (–8) = –2

+(–8) 6

–3 –2 –1 0 1 2 3 4 5 6 7

Think:

Course 1

9-4 Adding Integers


Find the sum.

A. –2 + (–4)

–6 –5 –4 –3 –2 –1 0 1

–2 + (–4)

–2 + (–4) = –6

Think:

Course 1

9-4 Adding Integers


Find the sum.

B. 3 + (–6)

3 + (–6) = –3

+ (–6) 3

–3 –2 –1 0 1 2 3 4 5 6 7

Think:

Course 1

9-4 Adding Integers

Additional Example 3: Evaluating Integer Expressions

Evaluate y + (–2) for each value of y.

A. y = 7

B. y = 1

y + (–2) Write the expression. 7 + (–2) Substitute 7 for y.

5 Add.

y + (–2) Write the expression. 1 + (–2) Substitute 1 for y.

–1 Add.

Course 1

9-4 Adding Integers

Try This: Example 3

Evaluate z + (–4) for each value of z.

A. z = 2

B. z = 3

z + (–4) Write the expression. 2 + (–4) Substitute 2 for z. –2 Add.

z + (–4) Write the expression. 3 + (–4) Substitute 3 for z. –1 Add.

Course 1

9-4 Adding Integers


A sunken ship is 12 m below the sea level. A search plane flies 35 m above the sunken ship. How far above the sea is the plane?

The ship is 12 m below the sea level and the plane is 35 m above the ship.

–12 + 35

23

The plane is 23 m above the sea.

Course 1

9-4 Adding Integers

Try This: Example 4

A sunken ship is 33 m below the sea level. A search plane flies 54 m above the sunken ship. How far above the sea is the plane?

The ship is 33 m below the sea level and the plane is 54 m above the ship.

–33 + 54

21

The plane is 21 m above the sea.

Lesson Quiz Find each sum.

1. 7 + (–3)

2. –5 + 2

3. –8 + (–4)

4. Evaluate x + 5 for x = –4.

5. At midnight on a winter night, the temperature was –12°F. By 10 A.M. the temperature had risen 42°F. What was the new temperature?

–3

4


–12

1

Course 1

9-4 Adding Integers

30°F


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find each sum.

1. –4 + 6 2. 5 + 12 3. –3 + 3 4. –8 + 9 Give the opposite of each number. 5. 7 6. –8 7. –3 8. 1

2 17 0

Course 1


1

–7 8 3 –1

Problem of the Day Jamal knows his average score for 3 science tests. If he adds the average and the amount that each score differs from the average, what will his sum be?

Course 1


The average

Learn to subtract integers.

Course 1


Course 1


Subtracting Integers on a Number Line Move left on a number line to subtract a positive integer. Move right on a number line to subtract a negative integer.

Course 1


Additional Example 1: Writing Integer Subtraction

Write the subtraction modeled on each number line.

A.

B.

–8 –7 – 6 –5 –4 –3 –2 –1 0 1

–3 –4

The subtraction modeled is –3 – 4 = –7.

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5

(–5) – (–9)

The addition modeled is –5 – (–9) = 4.

Course 1


Try This: Example 1

Write the subtraction modeled on each number line.

A.

B.

–5 –2

The subtraction modeled is –5 – 2 = –7.

(–3) –(–5)

The addition modeled is –3 – (–5) = 2.

–8 –7 – 6 –5 –4 –3 –2 –1 0 1

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5

Course 1


Additional Example 2A: Subtracting Integers

Find the difference.

A. 4 – 6

4 – 6 = –2

(–6) 4

–3 –2 –1 0 1 2 3 4 5

Course 1


Additional Example 2B: Subtracting Integers


B. 3 – (–3)

–(–3) 3

3 – (–3) = 6

–1 0 1 2 3 4 5 6 7

Course 1




A. 2 – (–5)

–1 0 1 2 3 4 5 6 7

– (–5) 2

2 – (–5) = 7

Course 1




B. 3 – 6

3 – 6 = –3

(–6) 3

–3 –2 –1 0 1 2 3 4 5

Course 1



Evaluate a – 4 for each value of a.

A. a = 2

B. a = 8

a – 4 Write the expression. 2 – 4 Substitute 2 for a. –2 Subtract.

a – 4 Write the expression. 8 – 4 Substitute 8 for a. 4 Subtract.

Course 1


Try This: Example 3

Evaluate b – 6 for each value of b.

A. b = 8

B. b = 12

b – 6 Write the expression. 8 – 6 Substitute 8 for b. 2 Subtract.

b – 6 Write the expression. 12 – 6 Substitute 12 for b.

6 Subtract.

Lesson Quiz

Find each difference.

1. 8 – 5 2. –9 – (–3)

Evaluate x – (–7) for each value of x.

3. x = –2 4. x = 5

5. The peak of Mt. Everest is approximately 26,544 ft above sea level. The shore of the Dead Sea is about 1,200 ft below sea level. How much higher is the peak of Mt. Everest than the shore of the Dead Sea?

–6 3


5

27,744 ft Course 1


12


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find each product.

1. 8 • 4 2. 7 • 12 3. 3 • 9 4. 6 • 5 5. 80 • 6 6. 50 • 6 7. 40 • 90 8. 20 • 700

32 84 27

Course 1


30 480 300 3,600 14,000

Problem of the Day

Catherine has $14.00 and earns $12.00 for each lawn she mows. If she mows 4 lawns and buys 5 DVDs that cost $11.95 each, including tax, how much money does she have left?. $2.25

Course 1


Learn to multiply integers.

Course 1


Course 1

9-6 Multiply Integers

Numbers 3 • 2 –3 • 2

Words 3 groups of 2 the opposite of 3 groups of 2

Addition 2 + 2 + 2 –(2 + 2 + 2)

Product 6 –6

Course 1


Numbers 3 • (–2) –3 • (–2)

Words 3 groups of –2 the opposite of 3 groups of –2

Addition (–2) + (–2) + (–2) –[(–2) + (–2) + (–2)]

Product –6 6

Course 1


Additional Example 1A & 1B: Multiplying Integers

Find each product.

A. 5 • 2

B. 4 • (–5)

5 • 2 = 10 Think: 5 groups of 2.

4 • (–5) = –20 Think: 4 groups of –5.

To find the opposite of a number, change the sign. The opposite of 6 is –6. The opposite of –4 is 4.

Remember!

Course 1


Additional Example 1C & 1D: Multiplying Integers

Find each product.

C. –3 • 2

D. –2 • (–4)

–3 • 2 = –6 Think: the opposite of 3 groups of 2.

–2 • (–4) = 8 Think: the opposite of 2 groups of –4.

Course 1



Find each product.

A. 3 • 4

B. 2 • (–7)

3 • 4 = 12 Think: 3 groups of 4.

2 • (–7) = –14 Think: 2 groups of –7.

Course 1



Find each product.

C. –5 • 3

D. –4 • (–6)

–5 • 3 = –15 Think: the opposite of 5 groups of 3.

–4 • (–6) = 24 Think: the opposite of 4 groups of –6.

Course 1


MULTIPLYING INTEGERS If the signs are the same, the product is positive. 4 • 3 = 12 –6 • (–3) = 18

If the signs are different, the product is negative. –2 • 5 = –10 7 • (–8) = –56

The product of any number and 0 is 0.

0 • 9 = 0 (–12) • 0 = 0

Course 1



Evaluate –7x for each value of x.

A. x = –3

B. x = 5

–7x Write the expression. –7 • (–3) Substitute –3 for x. 21 The signs are the same, so the answer

is positive.

–7x Write the expression. –7 • 5 Substitute 5 for x. –35 The signs are different,

so the answer is negative. –7x means –7 • x.

Remember!

Course 1


Try This: Example 2

Evaluate –4y for each value of y.

A. y = – 2

B. y = 7

–4y Write the expression. –4 • (–2) Substitute –2 for y.

8 The signs are the same, so the answer is positive.

–4y Write the expression. –4 • 7 Substitute 7 for y. –28 The signs are different, so the answer

is negative.

Lesson Quiz Find each product.

1. 6 • (4) 2. 3 • (–2)

3. –9 • (–2) 4. –6 • 5

5. Evaluate 3y for y = –7.

6. During a football game, Raymond’s team lost 6 yards on each of 3 plays and gained 8 yards on each of two plays. What integer represents the total change in the team’s position?

–6 24


18 –30

Course 1


–21

–2


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find each quotient.

1. 18 ÷ 2 2. 42 ÷ 7

3. 56 ÷ 8 4. 24 ÷ 6

5. 3,600 ÷ 4 6. 540 ÷ 60

9 6 7

Course 1


4

900 9

Problem of the Day

Hank wanted to record the number of marbles he lost to his friend Marcus each day for 5 days. He forgot to record one day, but for the other days he wrote 8, 2, 3, and 4. The average number he lost was 4. What number did he forget to write? 3

Course 1


Learn to divide integers.

Course 1


Course 1


Multiplication and division are inverse operations. To solve a division problem, think of the related multiplication.

Course 1


Additional Example 1: Dividing Integers

Find each quotient.

A. –30 ÷ 6

B. –42 ÷ (–7)

Think: What number times 6 equals –30?

–5 • 6 = –30, so –30 ÷ 6 = –5.

Think: What number times –7 equals –42?

6 • (–7) = –42, so –42 ÷ (–7) = 6.

Course 1


Try This: Example 1

Find each quotient.

A. –15 ÷ 5

B. –36 ÷ (–6)

Think: What number times 5 equals –15?

–3 • 5 = –15, so –15 ÷ 5 = –3.

Think: What number times -6 equals –36?

6 • (–6) = –36, so –36 ÷ (–6) = 6.

Course 1


Dividing Integers If the signs are the same, the product is positive. 24 ÷ 3 = 8 –6 ÷ (–3) = 2

If the signs are different, the quotient is negative. –20 ÷ 5 = –4 72 ÷ (–8) = –9

Zero divided by any integer equals 0.

= 0 = 0 0 14 __ 0

–11 __

You cannot divide any integer by 0.

Because division is the inverse of multiplication, the rules for dividing integers are the same as the rules for multiplying integers.

Course 1


Additional Example 2A: Evaluating Integer Expressions

Evaluate for each value of d.

A. d = 16


= 16 ÷ 4 Substitute 16 for d.

= 4 The signs are the same, so the answer is positive.

d 4 __

d 4 __

16 4 __

Course 1


Additional Example 2B: Evaluating Integer Expressions


B. d = –24


= –24 ÷ 4 Substitute –24 for d.

= –6 The signs are different, so the answer is negative.

d 4 __

d 4 __

-24 4 ___

Course 1


Additional Example 2C: Evaluating Integer Expressions


C. d = –12


= –12 ÷ 4 Substitute –12 for d.


d 4 __

d 4 __

-12 4 ___

Course 1



Evaluate for each value of c.

B. c = –15


= –15 ÷ 3 Substitute –15 for c.


c 3 __

c 3 __

-15 3 ___

Course 1




B. c = 12


= 12 ÷ 3 Substitute 12 for c.

= 4

c 3 __

c 3 __

12 3 ___

The signs are the same, so the answer is positive.

Course 1




C. c = 27


= 27 ÷ 3 Substitute 27 for c.

= 9 The signs are the same, so the answer is positive.

c 3 __

c 3 __

27 3 __

Lesson Quiz Find each quotient.

1. 18 ÷ (–3) 2. –36 ÷ (–6)

3. –64 ÷ (–8) 4. –45 ÷ 3

Evaluate for each value of x.

5. x = –12 6. x = 20

7. Cameron’s mom has $3,200 in her bank account. If she uses the money to make monthly rent payments of $375, how many whole payments can she make?

6 –6


8 –15

Course 1


–3 5

8

x 4 __

9-8 Solving Integer Equation

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Use mental math to find each solution.

1. 6 + x = 12 2. 3x = 15 3. = 4 4. x – 8 = 12

x = 6 x = 5

x = 20

Course 1


x = 20 x 5 __

Problem of the Day

Marie spent $15 at the fruit stand, buying peaches that cost $2 per container and strawberries that cost $3 per container. She bought the same number of containers of each fruit. How many containers each of peaches and strawberries did she buy? 3

Course 1


Learn to solve equations containing integers.

Course 1


Course 1


A. Solve –8 + y = –13. Check your answer.

–8 + y = –13 –8 is added to y.

+ 8 + 8 Subtracting –8 from both sides to undo the addition is the same as adding +8.

y = –5 Check –8 + y = –13 Write the equation.

Substitute –5 for y. –5 is a solution.

–8 + (–5) = –13 ?

–13 = –13 ?

Additional Example 1A: Adding and Subtracting to Solve Equations

Course 1


To solve the equation on the previous slide using algebra tiles, you can add four tiles to both sides and then remove pairs of red and yellow tiles. This is because subtracting a number is the same as adding its opposite.

Helpful Hint

Course 1


Additional Example 1B: Adding and Subtracting to Solve Equations

B. n – 2 = –8. n – 2 = –8 2 is subtracted from n.

+ 2 + 2 Add 2 to both sides to undo the subtraction.

n = –6

Check n – 2 = –8 Write the equation.

Substitute –6 for n. –6 is a solution.

–6 – 2 = –8 ?

–8 = –8 ?

Course 1



A. Solve –2 + y = –7. Check your answer.

–2 + y = –7 –2 is added to y.

+ 2 + 2 Subtracting –2 from both sides to undo the addition is the same as adding +2.

y = –5

Check –2 + y = –7 Write the equation.

Substitute –5 for y. –5 is a solution.

–2 + (–5) = –7 ?

-7 = -7 ?

Course 1



B. n – 6 = –14. n – 6 = –14 –14 is added to n.

+ 6 + 6 Add 6 to both sides to undo the subtraction.

n = –8

Check

n – 6 = –14 Write the equation. Substitute –8 for n. –8 is a solution.

–8 – 6 = –14 ?

–14= –14 ?

Course 1


Additional Example 2A: Multiplying and Dividing to Solve Equations

A. 4m = –20

= m is multiplied by 4. Divide both sides by 4 to undo the multiplication.

m = –5

Check 4m = –20 Write the equation.

Substitute –5 for m. –5 is a solution.

4(–5) = –20 ?

–20 = –20 ?

4m 4 ___ -20

4 ___

Course 1


3 • = 3 • (–7) x is divided by 3. Multiply both sides by 3 to undo the division.

x = –21

Check

= –7 Write the equation.

Substitute –21 for x.

–21 is a solution.

–21 ÷ 3 = –7 ?

–7 = –7 ?

x 3 __

x 3 __

Additional Example 2B: Multiplying and Dividing to Solve Equations

B. = –7 x 3

Course 1



A. 3n = –15

= n is multiplied by 3. Divide both sides by 3 to undo the multiplication.

n = –5

Check

3n = –15 Write the equation.

Substitute –5 for n. –5 is a solution.

3(–5) = –15 ?

–15 = –15 ?

3n 3 ___ -15

3 ___

Course 1



B. = –6

4 • = 4 • (–6) y is divided by 4. Multiply both sides by 4 to undo the division.

y = –24

Check

= –6 Write the equation.

Substitute –24 for y.

–24 is a solution.

–24 ÷ 4 = –6 ?

–6 = –6 ?

y 4 __

y 4 __

y 4 __

Lesson Quiz

Solving each equation.

1. 5 + x = –6 2. y – 9 = –7

3. –6a = 24 4. = –9

5. A submarine captain sets the following diving course: dive 300 ft, stop, and then dive another 300 ft. If this pattern is continued, how many dives will be necessary to reach a location 3,000 ft below sea level?

a = –4

x = –11


x = –45

10

Course 1


y = 2 x

5 __


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. What figure has four equal sides and four right angles? 2. What figure has eight sides? 3. What figure has five sides?

square octagon

pentagon

Course 1


Problem of the Day

Find the greatest perimeter possible when 6 identical unit squares are arranged to form a closed figure. Adjacent squares must share an entire side. 14 units

Course 1


Learn to find the perimeter and missing side lengths of a polygon.

Course 1


Vocabulary perimeter


Course 1


The perimeter of a figure is the distance around it. To find the perimeter you can add the lengths of the sides.

Course 1


Additional Example 1: Finding the Perimeter of a Polygon

Find the perimeter of the figure.

2.8 + 3.6 + 3.5 + 3 + 4.3

Add all the side lengths.

The perimeter is 17.2 in.

Course 1


Try This: Example 1

Find the perimeter of the figure.

3.3 + 3.3 + 4.2 + 4.2 + 3

Add all the side lengths. 3.3 in. 3.3 in.

4.2 in. 4.2 in.

3 in.

The perimeter is 18 in.

Course 1


Course 1


Additional Example 2: Using a Formula to Find Perimeter

Find the perimeter P of the rectangle.

P = 2l+ 2w Substitute 12 for l and 3 for w. P = (2 • 12) + (2 • 3)

P = 24 + 6 P = 30

Multiply. Add.

The perimeter is 30 cm.

12 cm

3 cm

Course 1


Try This: Example 2

Find the perimeter P of the rectangle.

P = 2l + 2w Substitute 15 for l and 2 for w. P = (2 • 15) + (2 • 2)

P = 30 + 4 P = 34

Multiply. Add.

The perimeter is 34 cm.

15 cm

2 cm

Course 1


A. What is the length of side a if the perimeter equals 1,471 mm?

P = sum of side lengths

Use the values you know. 1,471 = 416 + 272 + 201 + 289 + a

1,471 = 1,178 + a

1,471 – 1,178 = 1,178 + a – 1,178

Add the known lengths.

Subtract 1,178 from both sides. 293 = a

Side a is 293 mm long.

Additional Example 3A: Finding Unknown Side Lengths and the Perimeter of a Polygon

Find each unknown measure.

Course 1


Additional Example 3B: Finding Unknown Side Lengths and the Perimeter of a Polygon


B. What is the perimeter of the polygon?

First find the unknown side length.

Find the sides opposite side b.

The length of side b = 50 + 22.

Side b is 72 cm long.

P = 72 + 55 + 50 + 33 + 22 + 22 P = 254

Find the perimeter. The perimeter of the polygon is 254 cm.

Course 1


72 cm

l = 4w Find the length.

l = (4 • 19)

l = 76

Substitute 19 for w.

Multiply.

Additional Example 3C: Finding Unknown Side Lengths and the Perimeter of a Polygon

C. The width of a rectangle is 19 cm. What is the perimeter of the rectangle if the length is 4 times the width? 4w

19 cm

Course 1


Additional Example 3C Continued

P = 2l + 2w

Substitute 76 and 19. P = 2(76) + 2 (19)

P = 152 + 38

P = 190

Multiply.

Add.

Use the formula for the perimeter of a rectangle.

The perimeter of the rectangle is 190 cm.

4w

19 cm

Course 1




A. What is the length of side a if the perimeter equals 1,302 mm?

P = sum of side lengths

Use the values you know. 1,302 = 212 + 280 + 250 + 240 + a

1,302 = 982 + a

1,302 – 982 = 982 + a – 982

Add the known lengths.

Subtract 982 from both sides. 320 = a

Side a is 320 mm long.

212 mm 280 mm

a 250 mm

240 mm

Course 1


First find the unknown side length. Find the sides opposite side b. The length of side b = 12 + 4.

Side b is 16 m long.

P = 14 + 12 + 8 + 4 + 6 + 16 P = 60

Find the perimeter.

The perimeter of the polygon is 60 m.



B. What is the perimeter of the polygon? 12 m

14 m

b m

8 m

6 m 4 m

16 m

Course 1


l = 5w Find the length. l = (5 • 13)

l = 65

Substitute 13 for w.

Multiply.


C. The width of a rectangle is 13 cm. What is the perimeter of the rectangle if the length is 5 times the width?

5w

13 cm

Course 1


Try This: Example 3C Continued

65 cm

13 cm

P = 2l + 2w

Substitute 65 and 13. P = 2(65) + 2 (13)

P = 130 + 26

P = 156

Multiply.

Add.

Use the formula for the perimeter of a rectangle.

The perimeter of the rectangle is 156 cm.

Course 1


Lesson Quiz

Find each perimeter.

1. 2.

3. What is the perimeter of a polygon with side lengths of 15 cm, 18 cm, 21 cm, 32 cm, and 26 cm?

4. What is the perimeter of a rectangle with length 22 cm and width 8 cm?

9 cm 4 ft


112 cm

60 cm

5 6 __

Course 1


10-2 Estimating and Finding Area

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. What is the perimeter of a square with side lengths of 15 in.? 2. What is the perimeter of a rectangle with length 16 cm and width 11 cm?

60 in.

54 cm

Course 1


Problem of the Day

Two wibbles equal four wabbles, and eight wabbles equal two bibbles. If a square has a perimeter of 16 wibbles, what is its area in square bibbles? 4 square bibbles

Course 1


Learn to estimate the area of irregular figures and to find the area of rectangles, triangles, and parallelograms.

Course 1


Vocabulary area


Course 1


The area of a figure is the amount of surface it covers. We measure area in square units.

Course 1


Additional Example 1: Finding the Area of an Irregular Figure

Find the area of the rectangle. Count full squares: 19 red squares. Count almost-full squares: 9 blue squares. Count squares that are about half-full: 4 green squares = 2 full squares. Do not count almost empty yellow squares. Add. 19 + 9 + 2 = 30

The area of the figure is about 30 mi . 2

Course 1


Try This: Example 1

Estimate the are of the figure.

Count full squares: 11 red squares. Count almost-full squares: 5 green squares. Count squares that are about half-full: 4 blue squares = 2 full squares. Do not count almost empty yellow squares. Add. 11 + 5 + 2 = 18

The area of the figure is about 18 mi . 2

= mi2

Course 1


Course 1


Additional Example 2: Estimating the Area of a Rectangle

Find the area of the figure.

Write the formula.

Substitute 15 for l. Substitute 9 for w.

A = lw

A = 15 • 9

A = 135

The area is about 135in2.

15 in.

9 in.

Course 1


Try This: Example 2

Find the area of the figure.

Write the formula.

Substitute 13 for l. Substitute 7 for w.

A = lw

A = 13 • 7

A = 91

The area is about 91 in2.

13 in.

7 in.

Course 1


You can use the formula for the area of a rectangle to write a formula for the area of a parallelogram. Imagine cutting off the triangle drawn in the parallelogram and sliding it to the right to form a rectangle.

The area of a parallelogram = bh. The area of a rectangle = lw.

The base of the parallelogram is the length of the rectangle. The height of the parallelogram is the width of the rectangle.

Course 1


Additional Example 3: Finding the Area of a Parallelogram

Find the area of the parallelogram.

Write the formula.

Multiply.

A = bh

2 ft 1 2

A = 2 • 1 1 2 __ 1

4 __ Substitute 2 for b

and 1 for h.

1 2 __

1 4 __

A = • 5 2 __ 5

4 __

A = or 3 25 8

___ 1 8 __

The area is 3 ft2. 1 8 __

1 4 1 ft

Course 1


Try This: Example 3

Find the area of the parallelogram.

Write the formula.

Multiply.

A = bh

A = 1 • 3 1 2 __ 1

2 __ Substitute 1 for b

and 3 for h.

1 2 __

1 2 __

A = • 3 2 __ 7

2 __

A = or 5 21 4

___ 1 4 __

The area is 5 ft2. 1 4 __

1 1 ft 2

1 3 ft 2

Course 1


You can make a parallelogram out of two congruent triangles.

The area of each triangle is half the area of the parallelogram, so the formula for the area of a

triangle is A = bh. 1 2 __

Course 1


Additional Example 4: Finding the Area of a Triangle

Find the area of the triangle.

Write the formula.

Substitute 9.6 for b. Substitute 2.9 for h.

Multiply.

The area is 13.92 cm2.

A = 13.92

A = bh 1 2 __

A = (9.6 • 2.9) 1 2 __

A = (27.84) 1 2 __

Course 1


Try This: Example 4

Find the area of the triangle.

Write the formula.

Substitute 8.5 for b. Substitute 3.4 for h.

Multiply.

The area is 14.45 cm2.

3.4 cm

8.5 cm

A = 14.45

A = bh 1 2 __

A = (8.5 • 3.4) 1 2 __

A = (28.9) 1 2 __

Course 1


Lesson Quiz

Find the area of each figure.

1. 2.

3. What is the area of a parallelogram with base 16 in. and height 10 in.?

4. What is the area of a triangle with base 10 in. and height 7 in.?


4.5 m 9.2 m

41.4 m2 24 ft2

160 in2

35 in2

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. What is the area of a rectangle with length 10 cm and width 4 cm?

2. What is the area of a parallelogram with

base 18 ft and height 12 ft? 3. What is the area of a triangle with base

16 cm and height 8 cm?

40 cm2

216 ft2

64 cm2

Course 1


Problem of the Day

Four squares are stacked in a tower. The bottom square is 12 inches on a side. The perimeter of each of the other squares is half of the one below it. What is the perimeter of the combined figure? 69 in.

Course 1


Learn to break a polygon into simpler parts to find its area.

Course 1


Additional Example 1A: Finding Areas of Composite Figures

Find the area of the polygon.

A.

Think: Break the polygon apart into rectangles. Find the area of each rectangle.

1.7 cm

4.9 cm 1.3 cm

2.1 cm

Course 1



A = lw A = lw A = 4.9 • 1.7 A = 2.1 • 1.3

Write the formula for the area of a rectangle. A = 8.33 A = 2.73

8.33 + 2.73 = 11.06 Add to find the total area. The area of the polygon is 11.06 cm2.

1.7 cm

4.9 cm

1.3 cm

2.1 cm

Course 1




B.

Think: Break the figure apart into a rectangle and a triangle.

Find the area of each polygon. Course 1



A = lw A = 28 • 24 A = 672 A = 168

672 + 168 = 840 Add to find the total area of the polygon.

The area of the polygon is 840 ft2.

A = bh 1 2 __

A = 28 • 12 1 2 __

Course 1




A.

Think: Break the polygon apart into rectangles. Find the area of each rectangle.

1.9 cm

5.5 cm 1.5 cm 2 cm

1.9 cm

1.5 cm

2 cm

3.4 cm

5.5 cm

Course 1


A = lw A = lw A = 5.5 • 1.9 A = 2 • 1.5

Write the formula for the area of a rectangle. A = 10.45 A = 3

10.45 + 3 = 13.45 Add to find the total area. The area of the polygon is 13.45 cm2.

Try This: Example 1A Continued 1.9 cm

5.5 cm 1.5 cm 2 cm

Course 1




B.

Think: Break the figure apart into a rectangle and a triangle.

Find the area of each polygon.

36 ft

22 ft

20 ft

20 ft

22 ft

22 ft

16 ft

Course 1


A = lw A = 22 • 20 A = 440 A = 176

440 + 176 = 616 Add to find the total area of the polygon.

The area of the polygon is 616 ft2.

A = bh 1 2 __

A = 22 • 16 1 2 __


20 ft 22 ft

22 ft

16 ft

Course 1


Additional Example 2: Art Application

Patrick made a design. All the sides are 5 inches long, except for two longer sides that are each 20 inches. All the angles are right angles. What is the area of the quilt design?

Think: Divide the design into 3 rectangles. Find the area of one rectangle that has a length of 20 in and a width of 5 in. Write the formula. A = lw

A = 20 • 5 = 100 3 • 100 = 300 Multiply to find the area of the 3

rectangles. The area of the design is 300 in2.

20 in.

20 in.

5 in.

Course 1


You can also use the formula A = s2 , where s is the length of a side, to find the area of a square.

Helpful Hint

Course 1


Try This: Example 2

Yvonne made quilt design. All the sides are 4 inches long, except for the two longer sides that are each 16 inches. All the angles are right angles. What is the area of the quilt design?

Think: Divide the quilt design into 10 squares. Find the area of one square that has a side length of 4 in.

Write the formula. A = lw A = 4 • 4 = 16 10 • 16 = 160

Multiply to find the area of the 10 squares.

The area of the quilt design is 160 in2.

4 in. 16 in.

16 in.

Course 1


Lesson Quiz

Find the area of the figure shown.


220 units2

Course 1


10-4 Comparing Perimeter and Area

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. What is the area of a figure made up of a rectangle with length 12 cm and height 4 cm and a parallelogram with length 12 cm and height 6 cm?

2. What is the area of a figure consisting of

a triangle sitting on top of a rectangle? The triangle has a base of 12 in. and height of 9 in., and the rectangle has a base of 12 in. and height of 5 in.

Course 1


120 cm2

114 in2

Problem of the Day

If sixteen people sit, evenly spaced, in a circle for story time, who sits directly across from person 5? person 13

Course 1


Learn to make a model to explore how area and perimeter are affected by changes in the dimensions of a figure.

Course 1


Additional Example 1: Changing Dimensions

Find how the perimeter and the area of the figure change when its dimensions change.

P = 29.2 in. A = 35 in.2

P = 14.6 in. A = 8.75 in.2

When the dimensions of the triangle are divided by 2, the perimeter is divided by 2, and the area is divided by 4, or 22.

Course 1


Divide each dimension by 2.

Try This: Example 1

Find how the perimeter and the area of the figure change when its dimensions change.

Multiply each dimension by 2.

4 in. 2 in.

8 in.

4 in.

P = 12 in. A = 8 in.2

P = 24 in. A = 32 in.2

When the dimensions of the rectangle are multiplied by 2, the perimeter is multiplied by 2, and the area is multiplied by 4, or 22.

Course 1



Draw a rectangle whose dimensions are 4 times as large as the given rectangle. How do the perimeter and area change?

Multiply each dimension by 4. P = 10 cm

A = 6 cm2 P = 40 cm

A = 96 cm2 When the dimensions of the rectangle are multiplied by 4, the perimeter is multiplied by 4, and the area is multiplied by 16, or 42.

3 cm 2 cm 8 cm

12 cm

Course 1


Try This: Example 2

Draw a rectangle whose dimensions are 2 times as large as the given rectangle. How do the perimeter and area change?

Multiply each dimension by 2. P = 16 cm

A = 15 cm2 P = 32 cm

A = 60 cm2 When the dimensions of the rectangle are multiplied by 2, the perimeter is multiplied by 2, and the area is multiplied by 4, or 22.

5 cm 3 cm 6 cm

10 cm

Course 1


Lesson Quiz

Find how the perimeter and area of the triangle change when its dimensions change.

The perimeter is multiplied by 2, and the area is multiplied by 4; perimeter = 24, area = 24; perimeter = 48, area = 96.


Course 1


10-5 Circles

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up The length and width of a rectangle are each multiplied by 5. Find how the perimeter and area of the rectangle change.

The perimeter is multiplied by 5, and the area is multiplied by 25.

Course 1

10-5 Circles

Problem of the Day

When using a calculator to find the height of a rectangle whose length one knew, a student accidentally multiplied by 20 when she should have divided by 20. The answer displayed was 520. What is the correct height? 1.3

Course 1

10-5 Circles

Learn to identify the parts of a circle and to find the circumference and area of a circle.

Course 1

10-5 Circles

Vocabulary circle center radius (radii) diameter circumference pi


Course 1

10-5 Circles

A circle is the set of all points in a plane that are the same distance from a given point, called the center.

Center

Course 1

10-5 Circles

A line segment with one endpoint at the center of the circle and the other endpoint on the circle is a radius (plural: radii).

Center Radius

Course 1

10-5 Circles

A chord is a line segment with both endpoints on a circle. A diameter is a chord that passes through the center of the circle. The length of the diameter is twice the length of the radius.

Center Radius

Diameter

Course 1

10-5 Circles

Additional Example 1: Naming Parts of a Circle

Name the circle, a diameter, and three radii.

N The circle is circle Z.

LM is a diameter.

ZL, ZM, and ZN are radii.

M

Z L

Course 1

10-5 Circles

Try This: Example 1

Name the circle, a diameter, and three radii.

The circle is circle D.

IG is a diameter.

DI, DG, and DH are radii.

G

H

D I

Course 1

10-5 Circles

The distance around a circle is called the circumference.

Center Radius

Diameter

Circumference

Course 1

10-5 Circles

The ratio of the circumference to the diameter, , is the same for any circle. This

ratio is represented by the Greek letter π, which is read “pi.”

C d

C d

= π

Course 1

10-5 Circles

The formula for the circumference of a circle is C = πd, or C = 2πr.

The decimal representation of pi starts with 3.14159265 . . . and goes on forever without repeating. We estimate pi using either 3.14

or . 22 7

Course 1

10-5 Circles

Additional Example 2A: Using the Formula for the Circumference of a Circle

Find the missing value to the nearest hundredth. Use 3.14 for pi.

A. d = 11 ft; C = ?

C = πd

C ≈ 3.14 • 11

C ≈ 34.54 ft

Write the formula.

Replace π with 3.14 and d with 11.

11 ft

Course 1

10-5 Circles

Additional Example 2B: Using the Formula for the Circumference of a Circle

Find each missing value to the nearest hundredth. Use 3.14 for pi.

B. r = 5 cm; C = ?

C = 2πr

C ≈ 2 • 3.14 • 5

C ≈ 31.4 cm

Write the formula.

Replace π with 3.14 and r with 5.

5 cm

Course 1

10-5 Circles

Additional Example 2C: Using the Formula for the Circumference of a Circle


C. C = 21.98 cm; d = ?

C = πd

21.98 ≈ 3.14d

7.00 cm ≈ d

Write the formula.

Replace C with 21.98 and π with 3.14.

21.98 3.14d _______ _______

3.14 3.14 ≈ Divide both sides by 3.14.

Course 1

10-5 Circles


Find the missing value to the nearest hundredth. Use 3.14 for pi.

A. d = 9 ft; C = ?

C = πd

C ≈ 3.14 • 9

C ≈ 28.26 ft

Write the formula.

Replace π with 3.14 and d with 9.

9 ft

Course 1

10-5 Circles



B. r = 6 cm; C = ?

C = 2πr

C ≈ 2 • 3.14 • 6

C ≈ 37.68 cm

Write the formula.

Replace π with 3.14 and r with 6.

6 cm

Course 1

10-5 Circles



C. C = 18.84 cm; d = ?

C = πd

18.84 ≈ 3.14d

6.00 cm ≈ d

Write the formula.

Replace C with 18.84 and π with 3.14.

18.84 3.14d _______ _______

3.14 3.14 ≈ Divide both sides by 3.14.

Course 1

10-5 Circles

The formula for the area of a circle is A = πr2.

Course 1

10-5 Circles

Additional Example 3: Using the Formula for the Area of a Circle

Find the area of the circle. Use for pi.

d = 42 cm; A = ?

Write the formula to find the area. A = πr2

r = d ÷ 2 r = 42 ÷ 2 = 21

The length of the diameter is twice the length of the radius.

Replace π with and r with 21. 22 7 __

A ≈ • 441 22 7 __ Use the GCF to simplify. 63

A ≈ 1,386 cm2 Multiply.

22 7

A ≈ • 212 22 7

1

42 cm

Course 1

10-5 Circles

Write the formula to find the area. A = πr2

r = d ÷ 2 r = 28 ÷ 2 = 14

The length of the diameter is twice the length of the radius.

Replace π with and r with 14. 22 7 __

A ≈ • 196 22 7 __ Use the GCF to simplify. 28

A ≈ 616 cm2 Multiply.

Try This: Example 3

Find the area of the circle. Use for pi.

d = 28 cm; A = ?

22 7

A ≈ • 142 22 7

1

28 cm

Course 1

10-5 Circles

Lesson Quiz

Find the circumference and area of each circle. Use 3.14 for π.

1. 2.

3. Find the area of a circle with a diameter of 20 feet. Use 3.14 for π.

C = 25.12 in.


C = 18.84 in.

8 in.

314 ft2

A = 50.24 in2 A = 28.26 in2

3 in.

Course 1

10-5 Circles

10-6 Solid Figures

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Solve. Use 3.14 for π.

37.68 in.

56.52 cm

Course 1

10-6 Solid Figures

452.16 ft2

1. The diameter of a circle is 12 in. What is the circumference?

2. The radius of a circle is 9 cm. What is its circumference?

3. Find the area of a circle with a 12 ft radius.

Problem of the Day

To measure the perimeter of her square patio, Becky used an old bicycle wheel with a 22 in. diameter. She rolled the wheel from one corner of the patio along the edge to the next. The wheel made 6.75 revolutions. What is the perimeter in feet of the patio? Use 3.14 for π. 155.43 ft

Course 1

10-6 Solid Figures

Learn to name solid figures.

Course 1

10-6 Solid Figures

Vocabulary polyhedron face edge vertex prism base pyramid cylinder cone


Course 1

10-6 Solid Figures

A polyhedron is a three-dimensional object, or solid figure, with flat surfaces, called faces, that are polygons.

When two faces of a solid figure share a side, they form an edge. On a solid figure, a point at which three or more edges meet is a vertex (plural: vertices).

Course 1

10-6 Solid Figures

Additional Example 1: Identifying Faces, Edges, and Vertices

Identify the number of faces, edges, and vertices on each solid figure.

A.

B.

5 faces 8 edges

5 vertices

7 faces 15 edges

10 vertices

Course 1

10-6 Solid Figures

Try This: Example 1

Identify the number of faces, edges, and vertices on each solid figure.

A.

B.

6 faces 12 edges

8 vertices

5 faces 9 edges

6 vertices

Course 1

10-6 Solid Figures

A prism is a polyhedron with two congruent, parallel bases, and other faces that are all parallelograms. A prism is named for the shape of its bases. A cylinder also has two congruent, parallel bases, but bases of a cylinder are circular. Cylinders are not polyhedra because not every surface is a polygon.

Course 1

10-6 Solid Figures

A pyramid has one polygon shaped base, and the other faces are triangles that come to a point. A pyramid is named for the shape of its base. A cone has a circular base and a curved surface that comes to a point. Cones are not polyhedra because not every surface is a polygon.

Course 1

10-6 Solid Figures

The point of a cone is called its vertex.

Helpful Hint

Course 1

10-6 Solid Figures

Additional Example 2A: Naming Solid Figures

Name the solid figures represented by the object.

A.

The figure is not a polyhedron.

There is a curved surface.

The figure represents a cylinder.

There are two congruent, parallel bases.

The bases are circles.

Course 1

10-6 Solid Figures

Additional Example 2B: Naming Solid Figures


B.

The figure is a polyhedron.

All the faces are flat and are polygons.

The figure is a triangular pyramid.

There is one base and the other faces are triangles that meet at a point, so the figure is a pyramid. The base is a triangle.

Course 1

10-6 Solid Figures

Additional Example 2C: Naming Solid Figures


C.



The figure is a rectangular prism.

There are two congruent, parallel bases, so the figure is a prism. The bases are rectangles.

Course 1

10-6 Solid Figures



A.



The figure is a square pyramid.

There is one base and the other faces are triangles that meet at a point, so the figure is a pyramid. The base is a square.

Course 1

10-6 Solid Figures



B.



The figure is a rectangular prism.

There are two congruent, parallel bases, so the figure is a prism. The bases are rectangles.

Course 1

10-6 Solid Figures



C.

The figure is not a polyhedron.

There is a curved surface.

The figure represents a cylinder.

There are two congruent, parallel bases.

The bases are circles.

Course 1

10-6 Solid Figures

Lesson Quiz

1. Identify the number of faces, edges, and vertices in the figure shown.

Identify the figure described

2. two congruent circular faces connected by a curved surface

3. one flat circular face and a curved lateral surface that comes to a point

cylinder

8 faces, 18 edges, and 12 vertices


cone Course 1

10-6 Solid Figures

10-7 Surface Area

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Identify the figure described.

1. two parallel congruent faces, with the other faces being parallelograms 2. a polyhedron that has a vertex and a face at opposite ends, with the other faces being triangles

prism

pyramid

Course 1

10-7 Surface Area

Problem of the Day

Which figure has the longer side and by how much, a square with an area of 81 ft2 or a square with perimeter of 84 ft?

A square with a perimeter of 84 ft; by 12 ft

Course 1

10-7 Surface Area

Learn to find the surface areas of prisms, pyramids, and cylinders.

Course 1

10-7 Surface Area

Vocabulary surface area net


Course 1

10-7 Surface Area

The surface area of a solid figure is the sum of the areas of its surfaces. To help you see all the surfaces of a solid figure, you can use a net. A net is the pattern made when the surface of a solid figure is layed out flat showing each face of the figure.

Course 1

10-7 Surface Area

Additional Example 1A: Finding the Surface Area of a Prism

Find the surface area S of the prism.

A. Method 1: Use a net.

Draw a net to help you see each face of the prism.

Use the formula A = lw to find the area of each face.

Course 1

10-7 Surface Area


A: A = 5 × 2 = 10

B: A = 12 × 5 = 60

C: A = 12 × 2 = 24

D: A = 12 × 5 = 60

E: A = 12 × 2 = 24

F: A = 5 × 2 = 10

S = 10 + 60 + 24 + 60 + 24 + 10 = 188 Add the areas of each face.

The surface area is 188 in2. Course 1

10-7 Surface Area

Additional Example 1B: Finding the Surface Area of a Prism

Find the surface area S of each prism.

B. Method 2: Use a three-dimensional drawing.

Find the area of the front, top, and side, and multiply each by 2 to include the opposite faces.

Course 1

10-7 Surface Area


Front: 9 × 7 = 63

Top: 9 × 5 = 45

Side: 7 × 5 = 35

63 × 2 = 126

45 × 2 = 90

35 × 2 = 70

S = 126 + 90 + 70 = 286 Add the areas of each face.

The surface area is 286 cm2.

Course 1

10-7 Surface Area


Find the surface area S of the prism.

A. Method 1: Use a net.

Draw a net to help you see each face of the prism.

Use the formula A = lw to find the area of each face.

3 in. 11 in.

6 in. 11 in.

6 in. 6 in. 3 in.

3 in.

3 in.

3 in.

A

B C D E F

Course 1

10-7 Surface Area


A: A = 6 × 3 = 18

B: A = 11 × 6 = 66

C: A = 11 × 3 = 33

D: A = 11 × 6 = 66

E: A = 11 × 3 = 33

F: A = 6 × 3 = 18

S = 18 + 66 + 33 + 66 + 33 + 18 = 234 Add the areas of each face.

The surface area is 234 in2.

11 in.

6 in. 6 in. 3 in.

3 in.

3 in.

3 in.

A

B C D E F

Course 1

10-7 Surface Area


Find the surface area S of each prism.

B. Method 2: Use a three-dimensional drawing.

Find the area of the front, top, and side, and multiply each by 2 to include the opposite faces.

6 cm 10 cm

8 cm

top front side

Course 1

10-7 Surface Area


Front: 10 × 8 = 80

Top: 10 × 6 = 60

Side: 8 × 6 = 48

80 × 2 = 160

60 × 2 = 120

48 × 2 = 96

S = 160 + 120 + 96 = 376 Add the areas of each face.

The surface area is 376 cm2.

6 cm 10 cm

8 cm

top front side

Course 1

10-7 Surface Area

The surface area of a pyramid equals the sum of the area of the base and the areas of the triangular faces. To find the surface area of a pyramid, think of its net.

Course 1

10-7 Surface Area

Additional Example 2: Finding the Surface Area of a Pyramid

Find the surface area S of the pyramid. S = area of square + 4 × (area of

triangular face)

S = 49 + 4 × 28 S = 49 + 112

Substitute.

S = s2 + 4 × ( bh) 1 2 __

S = 72 + 4 × ( × 7 × 8) 1 2 __

S = 161 The surface area is 161 ft2.

Course 1

10-7 Surface Area

Try This: Example 2

Find the surface area S of the pyramid.

S = area of square + 4 × (area of triangular face)

S = 25 + 4 × 25 S = 25 + 100

Substitute.

S = s2 + 4 × ( bh) 1 2 __

S = 52 + 4 × ( × 5 × 10) 1 2 __

S = 125 The surface area is 125 ft2.

5 ft

5 ft

10 ft

10 ft 5 ft

Course 1

10-7 Surface Area

The surface area of a cylinder equals the sum of the area of its bases and the area of its curved surface.

To find the area of the curved surface of a cylinder, multiply its height by the circumference of the base.

Helpful Hint

Course 1

10-7 Surface Area

Additional Example 3: Finding the Surface Area of a Cylinder

Find the surface area S of the cylinder. Use 3.14 for π, and round to the nearest hundredth.

S = area of lateral surface + 2 × (area of each base) Substitute. S = h × (2πr) + 2 × (πr2)

S = 7 × (2 × π × 4) + 2 × (π × 42)

ft

Course 1

10-7 Surface Area



S ≈ 7 × 8(3.14) + 2 × 16(3.14)

S ≈ 7 × 25.12 + 2 × 50.24

The surface area is about 276.32 ft2.

Use 3.14 for π.

S ≈ 175.84 + 100.48

S ≈ 276.32

S = 7 × 8π + 2 × 16π

Course 1

10-7 Surface Area

Try This: Example 3


S = area of lateral surface + 2 × (area of each base) Substitute. S = h × (2πr) + 2 × (πr2)

S = 9 × (2 × π × 6) + 2 × (π × 62)

6 ft

9 ft

Course 1

10-7 Surface Area



S ≈ 9 × 12(3.14) + 2 × 36(3.14)

S ≈ 9 × 37.68 + 2 × 113.04

The surface area is about 565.2 ft2.

Use 3.14 for π.

S ≈ 339.12 + 226.08

S ≈ 565.2

S = 9 × 12π + 2 × 36π

Course 1

10-7 Surface Area

Lesson Quiz

Find the surface area of each figure. Use 3.14 for π.

1. rectangular prism with base length 6 ft, width 5 ft, and height 7 ft

2. cylinder with radius 3 ft and height 7 ft

3. Find the surface area of the figure shown.


Course 1

10-7 Surface Area

214 ft2

188.4 ft2

208 ft2

10-8 Finding Volume

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the surface area of each figure. Use 3.14 for π.

Course 1

10-8 Finding Volume

432 in2

226.08 ft2

1. rectangular prism with base length 8 in., width 6 in., and height 12 in.

2. cylinder with diameter 8 ft and height 5 ft

Problem of the Day

A rectangular park is bordered by a 3-foot-wide sidewalk. The park, including the sidewalk, measures 125 ft by 180 ft. What is the area of the park, not including the sidewalk? 20,706 ft2

Course 1

10-8 Finding Volume

Learn to estimate and find the volumes of rectangular prisms and triangular prisms.

Course 1

10-8 Finding Volume

Vocabulary volume


Course 1

10-8 Finding Volume

Volume is the number of cubic units needed to fill a space.

Course 1

10-8 Finding Volume

It takes 10, or 5 · 2, centimeter cubes to cover the bottom layer of this rectangular prism.

There are 3 layers of 10 cubes each. It takes 30, or 5 · 2 · 3, cubes to fill the prism.

The volume of the prism is 5 cm · 2 cm · 3 cm = 30 cm3.

Course 1

10-8 Finding Volume

Additional Example 1: Finding the Volume of a Rectangular Prism

Find the volume of the rectangular prism.

V = lwh Write the formula.

V = 26 • 11 • 13 l = 26; w = 11; h = 13

Multiply. V = 3,718 in3

13 in.

26 in. 11 in.

Course 1

10-8 Finding Volume

Try This: Example 1

Find the volume of the rectangular prism.

V = lwh Write the formula.

V = 29 • 12 • 16 l = 29; w = 12; h = 16

Multiply. V = 5,568 in3

16 in.

29 in. 12 in.

Course 1

10-8 Finding Volume

To find the volume of any prism, you can use the formula V= Bh, where B is the area of the base, and h is the prism’s height. So, to find the volume of a triangular prism, B is the area of the triangular base and h is the height of the prism.

Course 1

10-8 Finding Volume

Additional Example 2A: Finding the Volume of a Triangular Prism

Find the volume of each triangular prism.

A.

V = Bh Write the formula. V = ( • 3.9 • 1.3) • 4 1

2 __ B = • 3.9 • 1.3; h = 4. 1

2 __

Multiply. V = 10.14 m3

Course 1

10-8 Finding Volume

Additional Example 2B: Finding the Volume of a Triangular Prism

Find the volume of the triangular prism.

B.

V = Bh Write the formula. V = ( • 6.5 • 7) • 6 1

2 __ B = • 6.5 • 7; h = 6. 1

2 __

Multiply. V = 136.5 ft 3

Course 1

10-8 Finding Volume



A.

V = Bh Write the formula. V = ( • 4.2 • 1.6) • 7 1

2 __ B = • 4.2 • 1.6; h = 7. 1

2 __

Multiply. V = 23.52 m3

1.6 m 7 m 4.2 m

Course 1

10-8 Finding Volume



B.

V = Bh Write the formula. V = ( • 4.5 • 9) • 5 1

2 __ B = • 4.5 • 9; h = 5. 1

2 __

Multiply. V = 101.25 ft3

4.5 ft 5 ft

9 ft

Course 1

10-8 Finding Volume


Suppose a facial tissue company ships 16 cubic tissue boxes in each case. What are the possible dimensions for a case of tissue boxes?


The answer will be all possible dimensions for a case of 16 cubic boxes.


• There are 16 tissue boxes in a case.

• The boxes are cubic, or square prisms.

Course 1

10-8 Finding Volume

You can make models using cubes to find the possible dimensions for a case of 16 tissue boxes.

2 Make a Plan


Course 1

10-8 Finding Volume

Solve 3

You can make models using cubes to find the possible dimensions for a case of 16 cubes.


The possible dimensions for a case of 16 cubic tissue boxes are the following: 1 • 1 • 16, 1 • 2 • 8, 1 • 4 • 4, and 2 • 2 • 4 .

Course 1

10-8 Finding Volume

Notice that each dimension is a factor of 16. Also, the product of the dimensions (length • width • height) is 16, showing that the volume of each case is 16 cubes.

Look Back 4


Course 1

10-8 Finding Volume

Try This: Example 3

Suppose a paper company ships 12 cubic boxes of envelopes in each case. What are the possible dimensions for a case of envelope boxes? 1 Understand the Problem

The answer will be all possible dimensions for a case of 12 cubic boxes.


• There are 12 envelope boxes in a case.

• The boxes are cubic, or square prisms.

Course 1

10-8 Finding Volume

You can make models using cubes to find the possible dimensions for a case of 12 boxes of envelopes.

2 Make a Plan


Course 1

10-8 Finding Volume

Solve 3 You can make models using cubes to find the possible dimensions for a case of 12 cubes.

The possible dimensions for a case of 24 cubic envelope boxes are the following: 1 • 1 • 12, 1 • 2 • 6, 1 • 3 • 4, and 2 • 2 • 3.


Course 1

10-8 Finding Volume

Notice that each dimension is a factor of 12. Also, the product of the dimensions (length • width • height) is 12, showing that the volume of each case is 12 cubes.

Look Back 4


Course 1

10-8 Finding Volume

Lesson Quiz

Find the volume of each figure.

1. rectangular prism with length 20 cm, width 15 cm, and height 12 cm

2. triangular prism with a height of 12 cm and a triangular base with base length 7.3 cm and height 3.5 cm

3. Find the volume of the figure shown.


3,600 cm3

153.3 cm3

38.13 cm3

Course 1

10-8 Finding Volume


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Find the volume of each figure described.

Course 1


359.04 cm3

1,320 cm3

1. rectangular prism with length 12 cm, width 11 cm, and height 10 cm

2. triangular prism with height 11 cm and triangular base with base length 10.2 cm and height 6.4 cm

Problem of the Day

The height of a box is half its width. The length is 12 in. longer than its width. If the volume of the box is 28 in , what are the dimensions of the box? 1 in. × 2 in. × 14 in.

3

Course 1


Learn to find volumes of cylinders.

Course 1


To find the volume of a cylinder, you can use the same method as you did for prisms: Multiply the area of the base by the height.

volume of a cylinder = area of base × height

The area of the circular base is πr2, so the formula is V = Bh = πr2h.

Course 1


Additional Example 1A: Finding the Volume of a Cylinder

Find the volume V of the cylinder to the nearest cubic unit.

A.

Write the formula. Replace π with 3.14, r with 4, and h with 7. Multiply. V ≈ 351.68

V = πr2h V ≈ 3.14 × 42 × 7

The volume is about 352 ft3. Course 1


Additional Example 1B: Finding the Volume of a Cylinder

B.

10 cm ÷ 2 = 5 cm Find the radius.

Write the formula. Replace π with 3.14, r with 5, and h with 11. Multiply. V ≈ 863.5

V = πr2h V ≈ 3.14 × 52 × 11

The volume is about 864 cm3. Course 1


Additional Example 1C: Finding the Volume of a Cylinder

C.

Find the radius. r = + 4 h 3 __

r = + 4 = 7 9 3 __ Substitute 9 for h.

Write the formula. Replace π with 3.14, r with 7, and h with 9. Multiply. V ≈ 1,384.74

V = πr2h V ≈ 3.14 × 72 × 9

The volume is about 1,385 in3. Course 1



Find the volume V of each cylinder to the nearest cubic unit.

A.

Multiply. V ≈ 565.2

The volume is about 565 ft3.

6 ft

5 ft

Write the formula. Replace π with 3.14, r with 6, and h with 5.

V = πr2h V ≈ 3.14 × 62 × 5

Course 1



B.

Multiply. V ≈ 301.44

8 cm ÷ 2 = 4 cm

The volume is about 301 cm3.

Find the radius.

8 cm

6 cm


V = πr2h V ≈ 3.14 × 42 × 6

Course 1



C.

Multiply. V ≈ 1230.88 The volume is about 1231 in3.

Find the radius. r = + 5 h 4 __

r = + 5 = 7 8 4 __ Substitute 8 for h.

r = + 5

h = 8 in

h 4


V = πr2h V ≈ 3.14 × 72 × 8

Course 1


Additional Example 2A: Application Ali has a cylinder-shaped pencil holder with a 3 in. diameter and a height of 5 in. Scott has a cylinder-shaped pencil holder with a 4 in. diameter and a height of 6 in. Estimate the volume of each cylinder to the nearest cubic inch.

A. Ali’s pencil holder

Write the formula. Replace π with 3.14, r with 1.5, and h with 5. Multiply. V ≈ 35.325

3 in. ÷ 2 = 1.5 in.

V ≈ 3.14 × 1.52 × 5

The volume of Ali’s pencil holder is about 35 in3.

Find the radius.

V = πr2h

Course 1


Additional Example 2B: Application

B. Scott’s pencil holder

Write the formula.

Multiply.

4 in. ÷ 2 = 2 in.

The volume of Scott’s pencil holder is about 75 in3.

Find the radius.

V = πr2h

Replace π with , r with

2, and h with 6.

22 7

__ V ≈ × 22 × 6 22

7 __

V ≈ = 75 528 7

___ 3 7

__

Course 1



Sara has a cylinder-shaped sunglasses case with a 3 in. diameter and a height of 6 in. Ulysses has a cylinder-shaped pencil holder with a 4 in. diameter and a height of 7 in. Estimate the volume of each cylinder to the nearest cubic inch.

A. Sara’s sunglasses case

Write the formula. Replace π with 3.14, r with 1.5, and h with 6. Multiply. V ≈ 42.39

3 in. ÷ 2 = 1.5 in.

V ≈ 3.14 × 1.52 × 6

The volume of Sara’s sunglasses case is about 42 in3.

Find the radius.

V = πr2h

Course 1



B. Ulysses’ pencil holder

Write the formula.

Multiply.

4 in. ÷ 2 = 2 in.

The volume of Scott’s pencil holder is about 75 in3.

Find the radius.

V = πr2h

Replace π with , r with

2, and h with 7.

22 7

__ V ≈ × 22 × 7 22

7 __

V = 88

Course 1


Additional Example 3A & 3B: Comparing Volumes of Cylinders

Find which cylinder has the greater volume.

Cylinder 1:

V ≈ 3.14 × 1.52 × 12 V = πr2h

V ≈ 84.78 cm3

Cylinder 2:

V ≈ 3.14 × 32 × 6 V = πr2h

V ≈ 169.56 cm3

Cylinder 2 has the greater volume because 169.56 cm3 > 84.78 cm3.

Course 1



Find which cylinder has the greater volume.

Cylinder 1:

V ≈ 3.14 × 2.52 × 10 V = πr2h

V ≈ 196.25 cm3

Cylinder 2:

V ≈ 3.14 × 22 × 4 V = πr2h

V ≈ 50.24 cm3

Cylinder 1 has the greater volume because 196.25 cm3 > 50.24 cm3.

10 cm 2.5 cm

4 cm

4 cm

Course 1


Lesson Quiz

Find the volume of each cylinder to the nearest cubic unit. Use 3.14 for π.


cylinder B

1,560.14 ft3

193 ft3

1017 ft3

1,181.64 ft3

Course 1


1. radius = 9 ft, height = 4 ft

2. radius = 3.2 ft, height = 6 ft

3. Which cylinder has a greater volume?

a. radius 5.6 ft and height 12 ft

b. radius 9.1 ft and height 6 ft


Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write each fraction as a decimal and as a percent.

1. 2. 3. 4.

0.75; 75%

0.50; 50%

Course 1


1 2 __

3 4 __

2 9 __

3 8 __ 0.375; 37.5%

0.2; 22.2% __

Problem of the Day

What fraction of the numbers from 0 to 99 are divisible by 3?.

or 34 100

___ 17 50

__

Course 1


Learn to estimate the likelihood of an event and to write and compare probabilities.

Course 1


Vocabulary probability


Course 1


Probability is the measure of how likely an event is to occur.

Course 1


Probabilities are written as fractions or decimals from 0 to 1 or as percents from 0% to 100%. The higher an event’s probability, the more likely that event is to happen.

• Events with a probability of 0, or 0%, never happen.

• Events with a probability of 1, or 100%, always happen.

• Events with a probability of 0.5, or 50%, have the same chance of happening as of not happening.

Course 1


Impossible Unlikely As likely as not Likely Certain

0 0.5 1

2 1 __ 0% 100%

50%

Course 1


Insert Lesson Title Here Additional Example 1A & 1B: Estimating the

Likelihood of an Event

Write impossible, unlikely, as likely as not, likely, or certain to describe each event.

A. You will roll an even number on a standard number cube.

B. The month of February has 28 days.

as likely as not

certain

A standard number cube is numbered from 1 to 6.

Helpful Hint

Course 1



Additional Example 1C & 1D: Estimating the Likelihood of an Event


C. This spinner lands on blue.

D. This spinner lands on an odd number.

unlikely

impossible

Course 1





A. You guess a number between 1 and 1000.

B. You will roll an odd number on a standard number cube.

unlikely

as likely as not

A standard number cube is numbered from 1 to 6.

Helpful Hint

Course 1





C. This spinner lands on green.

D. This spinner lands on an even number.

unlikely

certain

Course 1



Additional Example 2A & 2B: Writing Probabilities

A. The weather report gives a 75% chance of snow. Write this probability as a decimal and as a fraction.

75% = 0.75

B. The chance of being chosen is 0.8. Write this probability as a fraction and as a percent.

Write as a decimal.

75% = = 75 100 ___ 3

4 __ Write as a fraction in

simplest form.

0.8 = = 8 10 __ 4


simplest form. 0.8 = 80% Write as a percent.

Course 1



Additional Example 2C: Writing Probabilities

C. There is a chance of getting a ring. Write

this probability as a decimal and as a percent.

Write as a decimal.

Write as a percent.

7 50 __

= 7 ÷ 50 = 0.14 7 50 __

= = = 14% 7 • 2 50 • 2 _____ 7

50 __ 14

100 ___

In Example 2C, after you find the decimal form

of , you can use it to find the percent.

0.14 = 14%.

Helpful Hint

7 50 __

Course 1




A. The weather report gives a 25% chance of snow. Write this probability as a decimal and as a fraction.

25% = 0.25

B. The chance of being chosen is 0.6. Write this probability as a fraction and as a percent.

Write as a decimal.

25% = = 25 100 ___ 1


simplest form.

0.6 = = 6 10 __ 3


simplest form. 0.6 = 60% Write as a percent.

Course 1




C. There is a chance of getting a ring. Write

this probability as a decimal and as a percent.

Write as a decimal.

Write as a percent.

4 25 __

= 4 ÷ 25 = 0.16 4 25 __

= = = 16% 4 • 4 25 • 4 _____ 4

25 __ 16

100 ___

Course 1



Additional Example 3A: Comparing Probabilities

A. On a standard number cube, there is a 50%

chance of rolling a multiple of 2 and a 33 %

chance of rolling a multiple of 3. Is it more likely to roll a multiple of 2 or a multiple of 3?

It is more likely to roll a multiple of 2.

Compare: 33 % < 50% 1 3 __

1 3 __

Course 1



Additional Example 3B: Comparing Probabilities

B. When you spin a certain spinner, there is a 15% chance that it will land on yellow, a 15% chance it will land on green, and a 70% chance that it will land on purple. Is it more likely to land on purple or on green?

It is more likely to land on purple than on green.

Compare: 70% > 15%

Course 1




A. When you spin a certain spinner, there is a 25% chance that it will land on purple, a 35% chance it will land on green, and a 40% chance that it will land on red. Is it more likely to land on red or on green?

It is more likely to land on red than on green.

Compare: 40% > 35%

Course 1




B. When you spin a certain spinner, there is a

9% chance that it will land on yellow, a 35 %

chance it will land on brown, and a 55 %

chance that it will land on blue. Is it more likely to land on brown or on yellow?

It is more likely to land on brown than on yellow.

Compare: 9% < 35 % 1 3 __

1 3 __

2 3 __

Course 1


Lesson Quiz Write impossible, unlikely, equally likely, likely, or certain to describe each event.

1. The sun will rise tomorrow.

2. You will roll 13 when rolling two dice.

3. There is a 0.125 chance of picking the winning ticket. Write this probability as a fraction and as a percent.

4. At Hamburger Hut, there is a 20% chance of getting a plastic dinosaur cup and a 35% chance of getting a plastic rabbit cup. It is less likely that you will receive a rabbit cup or dinosaur cup?

impossible

certain


dinosaur cup

, 12.5% 1 8 __

Course 1


11-2 Experimental Probability

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write impossible, unlikely, equally likely, likely, or certain to describe each event. 1. A particular person’s birthday falls on the first of a month. 2. You roll an odd number on a fair number cube. 3. There is a 0.14 probability of picking the winning ticket. Write this as a fraction and as a percent.

unlikely

equally likely

Course 1


, 14% 7 50 __

Problem of the Day

Max picks a letter out of this problem at random. What is the probability that the letter is in the first half of the alphabet?

57 101 ___

Course 1


Learn to find the experimental probability of an event.

Course 1


Vocabulary experiment outcome sample space experimental probability


Course 1


An experiment is an activity involving chance that can have different results. Flipping a coin and rolling a number cube are examples of experiments.

The different results that can occur are called outcomes of the experiment. If you are flipping a coin, heads is one possible outcome.

The sample space of an experiment is the set of all possible outcomes. You can use {} to show sample spaces. When a coin is being flipped, {heads, tails} is the sample space.

Course 1


Additional Example 1A: Identifying Outcomes and Sample Spaces

For each experiment, identify the outcome shown and the sample space.

A. Spinning two spinners

outcome shown: B1 sample space: {A1, A2, B1, B2}

Course 1


Additional Example 1B: Identifying Outcomes and Sample Spaces


B. Spinning a spinner

outcome shown: green sample space: {red, purple, green}

Course 1




A. Spinning two spinners

outcome shown: C3

sample space: {C3, C4, D3, D4}

C D 3 4

Course 1




B. Spinning a spinner

outcome shown: blue

sample space: {blue, orange, green}

Course 1


Performing an experiment is one way to estimate the probability of an event. If an experiment is repeated many times, the experimental probability of an event is the ratio of the number of times the event occurs to the total number of times the experiment is performed.

Course 1


The probability of an event can be written as P(event). P(blue) means “the probability that blue will be the outcome.”

Writing Math

Course 1


Additional Example 2: Finding Experimental Probability

For one month, Mr. Crowe recorded the time at which his train arrived. He organized his results in a frequency table.

Time 6:49-6:52 6:53-6:56 6:57-7:00

Frequency 7 8 5

Course 1



= 7 + 8

20 _____

= 15 20 ___ =

3 4 __

Before 6:57 includes 6:49-6:52 and 6:53-6:56.

P(before 6:57) ≈ number of times the event occurs total number of trials

___________________________

A. Find the experimental probability that the train will arrive before 6:57.

Course 1


Additional Example 2B: Finding Experimental Probability

= 8 20 ___ =

2 5 __

P(between 6:53 and 6:56) ≈ number of times the event occurs

total number of trials ___________________________

B. Find the experimental probability that the train will arrive between 6:53 and 6:56.

Course 1


Try This: Example 2

For one month, Ms. Simons recorded the time at which her bus arrived. She organized her results in a frequency table.

Time 4:31-4:40 4:41-4:50 4:51-5:00

Frequency 4 8 12

Course 1



= 4 + 8

24 _____

= 12 24 ___ =

1 2 __

Before 4:51 includes 4:31-4:40 and 4:41-4:50.

P(before 4:51) ≈ number of times the event occurs total number of trials

___________________________

A. Find the experimental probability that the bus will arrive before 4:51.

Course 1



= 8

24 ___ =

1

3 __

P(between 4:41 and 4:50) ≈ number of times the event occurs

total number of trials ___________________________

B. Find the experimental probability that the bus will arrive between 4:41 and 4:50.

Course 1


Additional Example 3: Comparing Experimental Probabilities

Erika tossed a cylinder 30 times and recorded whether it landed on one of its bases or on its side. Based on Erika’s experiment, which way is the cylinder more likely to land?

Outcome On a base On its side

Frequency llll llll llll llll llll llll l

Find the experimental probability of each outcome.

Course 1



= 21 30 ___ P(side) ≈

number of times the event occurs total number of trials

___________________________

Compare the probabilities. 9 30 ___ <

21 30 ___

It is more likely that the cylinder will land on its side.

= 9 30 ___ P(base) ≈


___________________________

Course 1


Additional Example 3: Comparing Experimental Probabilities

Chad tossed a cylinder 25 times and recorded whether it landed on one of its bases or on its side. Based on Chads’s experiment, which way is the cylinder more likely to land?

Outcome On a base On its side

Frequency llll llll llll llll llll

Find the experimental probability of each outcome.

Course 1



= 20 25 ___ P(side) ≈


___________________________

Compare the probabilities. 5 25 ___ <

20 25 ___

It is more likely that the cylinder will land on its side.

= 5 25 ___ P(base) ≈


___________________________

Course 1


Lesson Quiz: Part 1

1. The spinner below was spun. Identify the outcome shown and the sample space.

outcome: green; sample space: {red, blue, green, purple, yellow}


Course 1


Lesson Quiz: Part 2

2. Find the experimental probability that the spinner will land on blue.

3. Find the experimental probability that the spinner will land on red.

4. Based on the experiment, on which color will the spinner most likely land?


red

2 9 __

4 9 __

Sandra spun the spinner above several times and recorded the results in the table.

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Tim took one marble from a bag, recorded the color, and returned it to the bag. He repeated this several times and recorded the results. 1. Find the experimental probability that a marble selected from the bag will be green.

2. Find the experimental probability that a marble selected from the bag will not be yellow.

Course 1


3 5 __

4 5 __

Problem of the Day

What is the probability that the sum of four consecutive whole numbers is divisible by 4?. 0

Course 1


Learn to find the theoretical probability of an event.

Course 1


Vocabulary theoretical probability equally likely fair


Course 1


Another way to estimate probability of an event is to use theoretical probability. One situation in which you can use theoretical probability is when all outcomes have the same chance of occurring. In other words, the outcomes are equally likely.

Course 1


An experiment with equally likely outcomes is said to be fair. You can usually assume that experiments involving items such as coins and number cubes are fair.

Course 1


Additional Example 1A: Finding Theoretical Probability

A. What is the probability that this fair spinner will land on 3? There are three possible outcomes when spinning this spinner: 1, 2, or 3. All are equally likely because the spinner is fair.

P(3)= 3 possible outcomes _________________

There is only one way for the spinner to land on 3.

P(3)= 3 possible outcomes __________________ 1 way event can occur

= 1 3 __

Course 1


Additional Example 1B: Finding Theoretical Probability

B. What is the probability of rolling a number greater than 4 on a fair number cube?

There are six possible outcomes when a fair number cube is rolled: 1, 2, 3, 4, 5, or 6. All are equally likely. There are 2 ways to roll a number greater than 4:5 or 6.

P(greater than 4)= 6 possible outcomes _________________

P(greater than 4)= 6 possible outcomes ____________________ 2 ways events can occur

= 2 6 __

Course 1



A. What is the probability that this fair spinner will land on 1?

There are three possible outcomes when spinning this spinner: 1, 2, or 3. All are equally likely because the spinner is fair.

P(3)= 3 possible outcomes _________________

There is only one way for the spinner to land on 1.

P(3)= 3 possible outcomes __________________ 1 way event can occur

= 1 3 __

Course 1



B. What is the probability of rolling a number less than 4 on a fair number cube?

There are six possible outcomes when a fair number cube is rolled: 1, 2, 3, 4, 5, or 6. All are equally likely. There are 3 ways to roll a number greater than 4:3, 2 or 1.

P(less than 4)= 6 possible outcomes _________________

P(less than 4)= 6 possible outcomes ____________________ 3 ways events can occur = 1

2 __

Course 1


Think about a single experiment, such as tossing a coin. There are two possible outcomes, heads or tails. What is P(heads) + P(tails)?

Experimental Probability (coin tossed 10 times)

Theoretical Probability

H T

llll l llll P(heads) = P(tails) =

+

6 10 __ 4

10 __ 10

10 __ = = 1

6 10 __ 4

10 __ P(heads) = P(tails)=

1 2 __ 6

10 __

1 2 __ + 1

2 __ 2

2 __ = = 1

Course 1


No matter how you determine the probabilities, their sum is 1.

This is true for any experiment—the probabilities of the individual outcomes add to 1 (or 100%, if the probabilities are given as percents.)

Course 1


Additional Example 2: Finding Probabilities of Events not Happening

Suppose there is a 45% chance of snow tomorrow. What is the probability that it will not snow?

In this situation there are two possible outcomes, either it will snow or it will not snow. P(snow) + P(not snow) = 100%

45% + P(not snow) = 100% -45% -45%

P(not snow) = 55% _____ _____ Subtract 45%

from each side.

Course 1


Course 1


Try This: Example 2

Suppose there is a 35% chance of rain tomorrow. What is the probability that it will not rain?

In this situation there are two possible outcomes, either it will rain or it will not rain.

P(rain) + P(not rain) = 100%

35% + P(not rain) = 100%

P(not rain) = 65%

-35% -35% _____ _____ Subtract 35% from each side.

Lesson Quiz

Use the spinner shown for problems 1-3.

1. P(2)

2. P(odd number)

3. P(factor of 6)

4. Suppose there is a 2% chance of spinning the winning number at a carnival game. What is the probability of not winning?


98%

2 7 __

4 7 __

4 7 __

Course 1


11-4 Make an Organized List

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up A game die with eight sides numbered 1 through 8 is rolled. Find each probability.

1. P(1, 2, or 3) 2. P(even number) 3. P(number greater than 9)

Course 1


3 8 __

1 2 __

0

Problem of the Day

Sam and Pam can have an apple, an orange, or a pear. What is the probability that they will pick the same snack?

Course 1


1 3 __

Learn to make an organized list to find all possible outcomes.

Course 1


Course 1


When you have to find many possibilities, one way to find them all is to make an organized list. A tree diagram is one way to organize information.

Course 1

11-4 Make an Organized List Additional Example 1: Using a Tree Diagram Matt wants to take a 3-day weekend trip to visit his grandparents. He can take either Friday or Monday off from work, and he can either fly, drive, take a train, or take a bus. How many options are available to Matt?

Friday

Monday

fly drive take a train take a bus

Friday and fly Friday and drive Friday and take a train Friday and take a bus

fly drive take a train take a bus

Monday and fly Monday and drive Monday and take a train Monday and take a bus

Course 1



Follow each branch on the tree diagram to find all of the possible outcomes. There are 8 different weekend trip combinations available to Matt.

Course 1

11-4 Make an Organized List Try This: Example 1

For her work uniform, Missy has a choice of three colors of pants—black, khaki, or navy. She has four choices for shirt colors—red, white, green, and yellow. How many different uniforms can Missy wear?

black pants

khaki pants

red shirt white shirt green shirt yellow shirt

black pants and red shirt black pants and white shirt black pants and green shirt black pants and yellow shirt


khaki pants and red shirt khaki pants and white shirt khaki pants and green shirt khaki pants and yellow shirt

Course 1



Follow each branch on the tree diagram to find all of the possible outcomes. There are 12 different uniform combinations available to Missy.

navy pants

navy pants and red shirt navy pants and white shirt navy pants and green shirt navy pants and yellow shirt


Course 1



One girl and one boy will be chosen to go to the state science fair. The girl finalists are Alia, Brenda, Cathy, Deb, and Erika. The boy finalists are Frank, Greg, and Hal. How many different pairs of one girl and one boy can be formed?

Course 1




The answer will be the number of different pairs of one girl and one boy.


• There are five girls, A, B, C, D, and E.

• There are three boys, F, G, and H.

• Only one girl and one boy will be chosen.

Use each student’s first initial.

Course 1


You can make an organized list to keep track of the sequences.

2 Make a Plan


Course 1


Solve 3 • List all the pairs that begin with A. AF, AG, AH

• List all the pairs that begin with B. BF, BG, BH

• List all the pairs that begin with C. CF, CG, CH

• List all the pairs that begin with D. DF, DG, DH

• List all the pairs that begin with E. EF, EG, EH


There are 5 groups of 3 pairs.

3 + 3 + 3 + 3 + 3 = 15 There are 15 pairs of one girl and one boy.

Course 1


You could have made a list beginning with a boy’s name. There would be 3 groups of 5 pairs.

5 + 5 + 5 = 15

Look Back 4

Each list will have 15 pairs of one girl and one boy.


Course 1


Try This: Example 2

One girl and one boy will be chosen to go to the movie preview. The girl finalists are Fay, Gerri, Heidi, and Ingrid. The boy finalists are Kevin, Larry, and Marc. How many different pairs of one girl and one boy can be formed?

Course 1




The answer will be the number of different pairs of one girl and one boy.


• There are five girls, F, G, H, and I.

• There are three boys, K, L, and M.

• Only one girl and one boy will be chosen.

Use each student’s first initial.

Course 1


You can make an organized list to keep track of the sequences.

2 Make a Plan


Course 1


Solve 3 • List all the pairs that begin with F. FK, FL, FM

• List all the pairs that begin with G. GK, GL, GM

• List all the pairs that begin with H. HK, HL, HM

• List all the pairs that begin with I. IK, IL, IM


There are 4 groups of 3 pairs.

3 + 3 + 3 + 3 = 12

There are 12 pairs of one girl and one boy.

Course 1


You could have made a list beginning with a boy’s name. There would be 3 groups of 4 pairs.

4 + 4 + 4 = 12

Look Back 4

Each list will have 12 pairs of one girl and one boy.


Lesson Quiz

1. A baseball coach has 4 pitchers, 3 catchers, and 2 shortstops on his team. How many different combinations of players can he use for the positions?

2. You are taking a 5-question true/false test. How many possible combinations of answers are there?

3. You are planning a small game booth at the local street fair. You have a choice of 3 games and 4 different prizes. How many combinations of games and prizes are there?

10

24


12

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up A café offers a soup-and-sandwich combination lunch. You can choose tomato soup, chicken noodle soup, or clam chowder. You can choose a turkey, ham, veggie, or tuna sandwich. How many lunch combinations are there?

12

Course 1


Problem of the Day

Rory dropped a quarter, a nickel, a dime, and a penny. What is the probability that all four landed tails up?

1 16 __

Course 1


Learn to list all the outcomes and find the theoretical probability of a compound event.

Course 1


Vocabulary compound event


Course 1


A compound event consists of two or more single events. For example, the birth of one child is a single event. The births of four children make up a compound event.

Course 1


Additional Example 1A: Finding Probabilities of Compound Events

Jerome spins the spinner and rolls a fair number cube.

A. Find the probability that the number cube will show an even number and that the spinner will show a B.

Course 1



1 2 3 4 5 6 A 1, A 2, A 3, A 4, A 5, A 6, A B 1, B 2, B 3, B 4, B 5, B 6, B C 1, C 2, C 3, C 4, C 5, C 6, C

Number Cube

Spinner

First find all of the possible outcomes.

There are 18 possible outcomes, and all are equally likely.

Three of the outcomes have an even number and B: 2, B; 4, B; and 6, B.

Course 1



P(even, B)= 18 possible outcomes ____________________ 3 ways events can occur

= 3 18 ___

= 1 6 __ Write your answer in simplest

form.

Course 1


Additional Example 1B: Finding Probabilities of Compound Events

B. Find the probability that the number cube will show a 4 and that the spinner will show an A.

Only one outcome is 4, A.

P(4, A)= 18 possible outcomes ____________________ 1 way event can occur

= 1 18 ___

Course 1


Additional Example 1C: Finding Probabilities of Compound Events

C. In the experiment on page 575, what is the probability that the coin will show tails, the spinner will land on purple, and a green marble will be chosen?

There are 18 equally likely outcomes.

P(tails, purple, green)= 18 possible outcomes ____________________ 1 way event can occur

= 1 18 ___

Course 1



Kiki spins the spinner and rolls a fair number cube.

A. Find the probability that the number cube will show an odd number and that the spinner will show an A.

Course 1



1 2 3 4 5 6 A 1, A 2, A 3, A 4, A 5, A 6, A B 1, B 2, B 3, B 4, B 5, B 6, B C 1, C 2, C 3, C 4, C 5, C 6, C

Number Cube

Spinner

First find all of the possible outcomes.

There are 18 possible outcomes, and all are equally likely.

Course 1


Three of the outcomes have an odd number and BA: 1, A; 3, A; and 5, A.

P(odd, A)= 18 possible outcomes ____________________ 3 ways events can occur

= 3 18 ___

= 1 6 __ Write your answer in simplest

form.


Course 1



B. Find the probability that the number cube will show a 6 and that the spinner will show an C.

Only one outcome is 6, C.

P(6, C)= 18 possible outcomes ____________________ 1 way event can occur

= 1 18 ___

Course 1



C. In the experiment on page 575 in the student book, what is the probability that the coin will show heads, the spinner will land on orange, and a red marble will be chosen?

There are 18 equally likely outcomes .

P(heads, orange, red)= 18 possible outcomes ____________________ 1 way event can occur

= 1 18 ___

Course 1


Lesson Quiz An experiment involves one spin of the spinner and one flip of the coin. Find each probability.

1. What is the probability of the spinner landing on red and the coin landing tails up?

2. What is the probability of the spinner landing on an odd number and the coin landing heads up?

3. What is the probability of the spinner landing on an even number and the coin landing tails up?


1 6 __

1 3 __

1 6 __

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. Zachary rolled a fair number cube twice. Find the probability of the number cube showing an odd number both times.

2. Larissa rolled a fair number cube twice.

Find the probability of the number cube showing the same number both times.

Course 1


1 4 __

1 36 ___

Problem of the Day

The average of three numbers is 45. If the average of the first two numbers is 47, what is the third number?

41

Course 1


Learn to use probability to predict events.

Course 1


Vocabulary prediction


Course 1



A prediction is a guess about something in the future. A way to make a prediction is to use probability.

Course 1


Additional Example 1A: Using Probability to Make Prediction

A. A store claims that 78% of shoppers end up buying something. Out of 1,000 shoppers, how many would you predict will buy something?

You can write a proportion. Remember that percent means “per hundred.”

Course 1



100x 100 ____ 78,000

100 ______ =


x = 780

You can predict that about 780 out of 1,000 customers will buy something.

78 100 ___ x

1000 ____ =

Think: 78 out of 100 is how many out of 1,000.

100 • x = 78 • 1,000

100x = 78,000



Course 1


Additional Example 1B: Using Probability to Make Predictions

B. If you roll a number cube 30 times, how many times do you expect to roll a number greater than 2?

2 3 __ x

30 ___ =

Think: 2 out of 3 is how many out of 30.

3 • x = 2 • 30

3x = 60



P(greater than 2) = = 4 6 __ 2

3 __

Course 1




x = 20

You can expect to roll a number greater than 2 about 20 times.

3x 3 __ 60

3 __ =

Course 1



A. A store claims 62% of shoppers end up buying something. Out of 1,000 shoppers, how many would you predict will buy something?

You can write a proportion. Remember that percent means “per hundred.”

Course 1



100x 100 ____ 62,000

100 ______ =


x = 620 You can predict that about 620 out of 1,000 customers will buy something.

62 100 ___ x

1000 ____ =

Think: 62 out of 100 is how many out of 1,000.

100 • x = 62 • 1,000

100x = 62,000



Course 1



B. If you roll a number cube 30 times, how many times do you expect to roll a number greater than 3?

1 2 __ x

30 ___ =

Think: 1 out of 2 is how many out of 30.

2 • x = 1 • 30

2x = 30



P(greater than 3) = = 3 6 __ 1

2 __

Course 1




x = 15

You can expect to roll a number greater than 3 about 15 times.

2x 2 __ 30

2 __ =

Course 1



A stadium sell yearly parking passes. If you have a parking pass, you can park at that stadium for any event during that year.

The managers of the stadium estimate that the probability that a person with a pass will attend any one event is 50%. The parking lot has 400 spaces. If the managers want the lot to be full at every event, how many passes should they sell?

Course 1


1 Understand the Problem The answer will be the number of parking passes they should sell.


• P(person with pass attends event): = 50%

• There are 400 parking spaces

The managers want to fill all 400 spaces. But on average, only 50% of parking pass holders will attend. So 50% of pass holders must equal 400. You can write an equation to find this number.

2 Make a Plan

Course 1


Solve 3

50 100 ___ 400

x ____ =

Think: 50 out of 100 is 400 out of how many?

100 • 400 = 50 • x

40,000 = 50x



40,000 50 ______ 50x

50 ___ = Divide both sides by 50 to

undo the multiplication.

800 = x

The managers should sell 800 parking passes.

Course 1



If the managers sold only 400 passes, the parking lot would not usually be full because only about 50% of the people with passes will attend any one event. The managers should sell more than 400 passes, so 800 is a reasonable answer.

Look Back 4

Course 1


Try This: Example 2

A stadium sells yearly parking passes. If you have a parking pass, you can park at that stadium for any event during that year.

The managers estimate that the probability that a person with a pass will attend any one event is 60%. The parking lot has 600 spaces. If the managers want the lot to be full at every event, how many passes should they sell?

Course 1


1 Understand the Problem The answer will be the number of parking passes they should sell.


• P(person with pass attends event): = 60%

• There are 600 parking spaces

The managers want to fill all 600 spaces. But on average, only 60% of parking pass holders will attend. So 60% of pass holders must equal 600. You can write an equation to find this number.

2 Make a Plan

Course 1


Solve 3

60 100 ___ 600

x ____ =

Think: 60 out of 100 is 600 out of how many?

100 • 600 = 60 • x

60,000 = 60x



60,000 60 ______ 60x

60 ___ = Divide both sides by 60 to

undo the multiplication.

1000 = x

The managers should sell 1000 parking passes.

Course 1



If the managers sold only 600 passes, the parking lot would not usually be full because only about 50% of the people with passes will attend any one event. The managers should sell more than 600 passes, so 1000 is a reasonable answer.

Look Back 4

Course 1


Lesson Quiz: Part 1

1. The owner of a local pizzeria estimates that 72% of his customers order pepperoni on their on their pizza. Out of 250 orders taken in one day, how many would you predict to have pepperoni? 180


Course 1


Lesson Quiz: Part 2

2. A bag contains 9 red chips, 4 blue chips, and 7 yellow chips. You pick a chip from the bag, record its color, and put the chip back in the bag. If you do this 100 times, how many times do you expect to remove a yellow chip from the bag?

3. A quality-control inspector has determined that 3% of the items he checks are defective. If the company he works for produces 3,000 items per day, how many does the inspector predict will be defective?

35


90

Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Evaluate each expression for the given value of the variable.

1. 4x – 1 for x = 2 2. 7y + 3 for y = 5 3. x + 2 for x = –6 4. 8y – 3 for y = –2

7 38

–1

Course 1


1 2 __

–19

Problem of the Day

Maria rented an electric car at a rate of $40 per day and $0.15 per mile. She returned the car the same day, gave the rental clerk a $100 bill, and got $21.75 back in change. How far did Maria drive the car?

255 miles

Course 1


Learn to use data in a table to write an equation for a function and to use the equation to find a missing value.

Course 1


Vocabulary function input output


Course 1


A function is a rule that relates two quantities so that each input value corresponds exactly to one output value.

Course 1


Additional Example 1A: Writing Equations from Function Tables

x 3 4 5 6 7 10 y 13 16 19 22 25

y is 3 times x + 4.

y = 3x + 4

Compare x and y to find a pattern. Use the pattern to write an equation.

y = 3(10) + 4 Substitute 10 for x.

y = 30 + 4 = 34 Use your function rule to find y when x = 10.

Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x.

Course 1


When all the y-values are greater than the corresponding x-values, use addition and/or multiplication in your equation.

Helpful Hint

Course 1


Try This: Example 1

x 3 4 5 6 7 10 y 10 12 14 16 18

y is 2 times x + 4.

y = 2x + 4

Compare x and y to find a pattern. Use the pattern to write an equation.

y = 2(10) + 4 Substitute 10 for x.

y = 20 + 4 = 24 Use your function rule to find y when x = 10.

Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x.

Course 1


You can write equations for functions that are described in words.

Course 1


Additional Example 2: Translating Words into Math

The height of a painting is 7 times its width.

h = height of painting Choose variables for the equation.

h = 7w Write an equation.

Write an equation for the function. Tell what each variable you use represents.

w = width of painting

Course 1


Try This: Example 2

The height of a mirror is 4 times its width.

h = height of painting Choose variables for the equation.

h = 4w Write an equation.

Write an equation for the function. Tell what each variable you use represents.

w = width of painting

Course 1



The school choir tracked the number of tickets sold and the total amount of money received. They sold each ticket for the same price. They received $80 for 20 tickets, $88 for 22 tickets, and $108 for 27 tickets. Write an equation for the function.


The answer will be an equation that describes the relationship between the number of tickets sold and the money received.

Course 1


You can make a table to display the data. 2 Make a Plan

Solve 3 Let t be the number of tickets. Let m be the amount of money received.

t 20 22 27 m 80 88 108

m is equal to 4 times t. Compare t and m.

m = 4t. Write an equation.

Course 1


Substitute the t and m values in the table to check that they are solutions of the equation m = 4t.

Look Back 4

m = 4t (20,80)

80 = 4 • 20 ?

80 = 80 ?

m = 4t (22,88)

88 = 4 • 22 ?

88 = 88 ?

m = 4t (27, 108)

108 = 4 • 27 ?

108 = 108 ?

Course 1


Try This: Example 3

The school choir tracked the number of tickets sold and the total amount of money received. They sold each ticket for the same price. They received $60 for 20 tickets, $66 for 22 tickets, and $81 for 27 tickets. Write an equation for the function.


The answer will be an equation that describes the relationship between the number of tickets sold and the money received.

Course 1


You can make a table to display the data. 2 Make a Plan

Solve 3 Let t be the number of tickets. Let m be the amount of money received.

t 20 22 27 m 60 66 81

m is equal to 3 times t. Compare t and m.

m = 3t. Write an equation.

Course 1


Substitute the t and m values in the table to check that they are solutions of the equation m = 3t.

Look Back 4

m = 3t (20, 60)

60 = 3 • 20 ?

60 = 60 ?

m = 3t (22, 66)

88 = 3 • 22 ?

88 = 66 ?

m = 3t (27, 81)

81 = 3 • 27 ?

81 = 81 ?

Course 1


Lesson Quiz

1. Write an equation for a function that gives the values in the table below. Use the equation to find the value for y for the indicated value of x.

2. Write an equation for the function. Tell what each variable you use represents. The height of a round can is 2 times its radius.

h = 2r, where h is the height and r is the radius.

y = 3x; 21


Course 1


X 0 1 3 5 7 y 0 3 9 15

12-2 Graphing Functions

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Write an equation for each function. Tell what each variable you use represents.

1. The length of a wall is 4 ft more than three times the height. 2. The number of trading cards is 3 less than the number of buttons.

l = 3h + 4, where l is length and h is height.

c = b – 3, where c is the number of cards and b is the number of buttons.

Course 1


Problem of the Day

Steve saved $1.50 each week. How many weeks did it take him to save enough to buy a $45 skateboard?

30

Course 1


Learn to represent linear functions using ordered pairs and graphs.

Course 1


Vocabulary linear equation


Course 1


Additional Example 1: Finding Solutions of Equations with Two Variables

Use the given x-values to write solutions of the equation y = 4x + 2 as ordered pairs. x =1, 2, 3, 4. Make a function table by using the given values for x to find values for y.

x 4x + 2 y 1 4(1) + 2 6 2 4(2) + 2 10 3 4(3) + 2 14 4 4(4) + 2 18

Write these solutions as ordered pairs.

(x, y) (1, 6) (2, 10) (3, 14) (4, 18)

Course 1


Try This: Example 1

Use the given x-values to write solutions of the equation y = 3x + 2 as ordered pairs. x = 2, 3, 4, 5. Make a function table by using the given values for x to find values for y.

x 3x + 2 y 2 3(2) + 2 8 3 3(3) + 2 11 4 3(4) + 2 14 5 3(5) + 2 17


(x, y) (2, 8) (3, 11) (4, 14) (5, 17)

Course 1



Check if an ordered pair is a solution of an equation by putting the x and y values into the equation to see if they make it a true statement.

Course 1


Additional Example 2: Checking Solutions of Equations with Two Variables

Determine whether the ordered pair is a solution to the given equation.

(3, 21); y = 7x

y = 7x Write the equation.

21 = 7(3) ?

21 = 21 ? Substitute 3 for x and 21 for y.

So (3, 21) is a solution to y = 7x.

Course 1


Try This: Example 2

Determine whether the ordered pair is a solution to the given equation.

(4, 20); y = 5x

y = 5x Write the equation.

20 = 5(4) ?

20 = 20 ? Substitute 4 for x and 20 for y.

So (4, 20) is a solution to y = 5x.

Course 1



You can also graph the solutions of an equation on a coordinate plane. When you graph the ordered pairs of some functions, they form a straight line. The equations that express these functions are called linear equations.

Course 1



Additional Example 3: Reading Solutions on Graphs

Use the graph of the linear function to find the value of y for the given value of x.

x = 4

When x = 4, y = 2.

Start at the origin and move 4 units right. Move up until you reach the graph. Move left to find the y-value on the y-axis.

The ordered pair is (4, 2).

x

y

-4 -2 0 2 4

4

-2

-4

2

Course 1



Try This: Example 3

Use the graph of the linear function to find the value of y for the given value of x.

x = 2

When x = 2, y = 4.

Start at the origin and move 2 units right. Move up until you reach the graph. Move left to find the y-value on the y-axis.

The ordered pair is (2, 4).

x

y

-4 -2 0 2 4

4

-2

-4

2

Course 1


Additional Example 4: Graphing Linear Functions

Graph the function described by the equation. y = –x – 2

Make a function table.

x –x – 2 y

–1 –(–1) – 2 –1

0 –(0) – 2 –2

1 –(1) – 2 –3


(x, y) (–1, –1) (0, –2) (1, –3)

Course 1




x

y

Graph the ordered pairs on a coordinate plane.

-5 -4 -3 -2 -1 0 1 2 3 4 5

5 4 3 2 1

-5 -4 -3 -2 -1

Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y.

Course 1


Try This: Example 4

Graph the function described by the equation. y = –x – 4

Make a function table.

x –x – 4 y

–1 –(– 1) – 4 –3

0 –(0) – 4 –4

1 –(1) – 4 –5


(x, y) (–1, –3) (0, –4) (1, –5)

Course 1



Try This: Example 4

x

y

Graph the ordered pairs on a coordinate plane.

-5 -4 -3 -2 -1 0 1 2 3 4 5

5 4 3 2 1

-5 -4 -3 -2 -1

Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y.

Course 1


Lesson Quiz

1. Use the given x-values to write solutions as ordered pairs to the equation y = –3x + 1 for x = 0, 1, 2, and 3.

2. Determine whether (4, –2) is a solution to the equation y = –5x + 3.

3. Graph the function described by the equation y = –x + 3.

No, –2 ≠ –5(4) + 3

(0, 1), (1, –2), (2, –5), (3, –8)


Course 1


12-3 Graphing Translations

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

1. Use the given x-values to write solutions of the following equation as ordered pairs. y = 6x – 2 for x = 0, 1, 2, 3

2. Determine whether (3, –13) is a solution

to the equation y = –4x – 1.

(0, –2), (1, 4), (2, 10), (3, 16)

yes

Course 1

12-3 Graphing Translation

Problem of the Day

Samantha’s house is 3 blocks east and 5 blocks south of Tyra. If Tyra walks straight south and then straight east to Samantha’s house, does she walk more blocks east or more blocks south? How many more?

south; 2 blocks

Course 1


Learn to use translations to change the positions of figures on a coordinate plane.

Course 1


A translation is a movement of a figure along a straight line. You can translate a figure on a coordinate plane by sliding it horizontally, vertically, or diagonally.

Course 1


Additional Example 1: Translating Figures on a Coordinate Plane

Give the coordinates of the vertices of the figure after the given translation.

Translate triangle DEF 4 units left and 3 units up.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

F

D

E

y

x

Course 1



To move the triangle 4 units left, subtract 4 from each of the x-coordinates.

To move the triangle 3 units up, add 3 to each of the y-coordinates.

DEF D’E’F’ D(1, 4)

E(4, 2)

F(–3, –3)

D’(1 – 4, 4 + 3) D’(–3, 7)

E’(0, 5)

F’(–7, 0)

E’(4 – 4, 2 + 3)

F’(-3 – 4, –3 + 3)

Course 1



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

5 4 3

1

-2 -3

-5 -6

6

2

-1

-4

E’

F’

D’ y

x

F

D

E

Course 1


Try This: Example 1

Give the coordinates of the vertices of the figure after the given translation.

Translate triangle GHJ 3 units left and 3 units up.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

J

G H

y

x

Course 1


Try This: Example 1

To move the triangle 3 units left, subtract 3 from each of the x-coordinates.

To move the triangle 3 units up, add 3 to each of the y-coordinates.

GHJ G’H’J’ G(2, 4)

H(4, 4)

J(–3, –2)

G’(2 – 3, 4 + 3) G’(–1, 7)

H’(1, 7)

J’(–6, 1)

H’(4 – 3, 4 + 3)

J’(–3 – 3, –2 + 3)

Course 1



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

5 4 3

1

-2 -3

-5 -6

6

2

-1

-4

H’

J’

G’ y

x

J

G H

x

Course 1


Additional Example 2: Music Application

Members of a marching band begin in a trapezoid formation represented by trapezoid KLMN. Then they move 4 steps right and 5 steps down. Give the coordinates of the vertices of the trapezoid after such a translation.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

M

K L

y

x

N

-7 -8

Course 1



To move 4 steps right, add 4 to each of the x-coordinates.

To move 5 steps down, subtract 5 from each of the y-coordinates.

KLMN K’L’M’N’ K(–2, 1)

L(1, 1)

M(3, –3)

K’(–2 + 4, 1 – 5) K’(2, –4)

L’(5, –4)

M’(7, –8)

L’(1 + 4, 1 – 5)

M’(3 + 4, –3 – 5)

N(–4, –3) N’(0, –8) M’(–4 + 4, –3 – 5)

Course 1



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

M

K L

y

x

N

-7 M’

K’ L’

N’ -8

x

Course 1


Try This: Example 2

Members of a flag team begin in a trapezoid formation represented by trapezoid KLMN. Then they move 3 steps right and 2 steps down. Give the coordinates of the vertices of the trapezoid after such a translation.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

M

K L

y

x

N

Course 1


Try This: Example 2

To move 3 steps right, add 3 to each of the x-coordinates.

To move 2 steps down, subtract 2 from each of the y-coordinates.

KLMN K’L’M’N’ K(–1, 3)

L(1, 3)

M(3, –1)

K’(–1 + 3, 3 – 2) K’(2, 1)

L’(4, 1)

M’(6, –3)

L’(1 + 3, 3 – 2)

M’(3 + 3, –1 – 2)

N(–3, –1) N’(0, –3) M’(-3 + 3, –1 – 2)

Course 1



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

M

K L

y

N

x

M’

K’ L’

N’

Course 1


Lesson Quiz

Give the coordinates of the vertices of triangle ABC, with vertices A(-5, -4), B(-3, 2), and C(1, -3), after the given translations.

1. Translate triangle ABC 3 units up and 2 units right.

2. Translate triangle ABC 5 units down and 3 units left.

3. Translate triangle ABC 2 units down and 4 units right.

A’(-3, -1), B’(-1, 5), and C’(3,0)


A’(-8, -9), B’(-6, -3), and C’(-2,-8)

A’(-1, -6), B’(1, 0), and C’(5,-5)

Course 1


12-4 Graphing Reflections

Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up

A parallelogram has vertices (-4, 1), (0, 1), (-3, 4), and (1, 4). What are its vertices after it has been translated 4 units right and 3 units up? (0, 4), (4, 4), (1, 7), (5, 7)

Course 1


Problem of the Day

These are rits: 24042, 383, and 4994. These are not rits: 39239, 28, and 5505. Which of these are rits: 39883, 4040, and 101? Why?

101 is a rit because it is the same forward and backward.

Course 1


Learn to use reflections to change the positions of figures on a coordinate plane.

Course 1


-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

Additional Example 1: Reflecting Figures on a Coordinate Plane

Give the coordinates of the figure after the given reflection. Reflect parallelogram EFGH across the y-axis.

E F

G H

Course 1



To reflect the parallelogram across the y-axis, write the opposites of the x-coordinates.

The y-coordinates do not change.

EFGH E’F’G’H’

E(–4, 0) E’(4, 0)

F(0, 0) F’(0, 0) G(–2, –3) G’(2, –3)

H(–6, –3) H’(6, –3)

Course 1



Reflect parallelogram EFGH across the y-axis.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x E F

G H

E’ F’

G’ H’

Course 1


-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

Try This: Example 1

Give the coordinates of the figure after the given reflection. Reflect parallelogram EFGH across the x-axis.

E F

G H

Course 1



To reflect the parallelogram across the x-axis, write the opposites of the y-coordinates.

The x-coordinates do not change.


E(–4, 0) E’(–4, 0)

F(0, 0) F’(0, 0) G(–2, –3) G’(–2, 3)

H(–6, –3) H’(–6, 3)

Course 1



Reflect parallelogram EFGH across the x-axis.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x E F

G H

E’ F’

G’ H’

Course 1


Additional Example 2: Design Application A designer is using a stencil that is shaped like the figure below. A pattern is made by reflecting the figure across the x-axis. Give the coordinates of the vertices of the figure after the reflection.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

P

Q S

R

Course 1



To reflect the quadrilateral across the x-axis, write the opposites of the y-coordinates.

The x-coordinates do not change.

PQRS P’Q’R’S’ P(1, 7) P’(1, –7) Q(2, 1) Q’(2, –1) R(–1, 1) R’(–1, –1) S(–3, 3) S’(–3, –3)

Course 1



-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

P

Q S

R

P’

Q’ S’

R’

Course 1


Try This: Example 2

A designer is using a stencil that is shaped like the figure below. A pattern is made by reflecting the figure across the y-axis. Give the coordinates of the vertices of the figure after the reflection.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

P

Q S

R

Course 1



To reflect the quadrilateral across the y-axis, write the opposites of the x-coordinates.

The y-coordinates do not change.

PQRS P’Q’R’S’ P(–3, 7) P’(3, 7) Q(–2, 1) Q’(2, 1) R(–5, 1) R’(5, 1) S(–7, 3) S’(7, 3)

Course 1



-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

P

Q S

R Q’

P’

S’ R’

Course 1


Lesson Quiz

Give the coordinates of the vertices of each figure after the given reflection.

1. Reflect triangle ABC with vertices A(5, 1), B(2, 3), and C(4, 6) across the x-axis.

2. Reflect parallelogram PQRS with vertices P(–4, –4), Q(1, –4), R(–2, 2), and S(3, 2) across the y-axis.

P’(4, –4), Q’(–1, –4), R’(2, 2), and S’(–3, 2)

A’(5, –1), B’(2, –3), C’(4, –6)


Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up Parallelogram ABCD has vertices (-4, 1), (0, 1), (–3, 4), and (1, 4).

1. What are the vertices of ABCD after it has been reflected across the x-axis? 2. What are the vertices after is has been reflected across the y-axis?

(–4, –1), (0, –1), (–3, –4), (1, –4)

(4, 1), (0, 1), (3, 4), (–1, 4)

Course 1


Problem of the Day

If each of the capital letters of the alphabet is rotated a half turn around its center, which will look the same?

H, I, N, O, S, X, Z

Course 1


Learn to use rotations to change positions of figures on a coordinate plane.

Course 1


You can rotate a figure about the origin or another point on a coordinate plane.

Course 1


A rotation is the movement of a figure about a point. Rotating a figure “about the origin” means that the origin is the center of rotation.

Remember!

Course 1


-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

Additional Example 1: Rotating Figures on a Coordinate Plane

E F

G H

Course 1


Give the coordinates of the vertices of the figure after the given rotation. Rotate parallelogram EFGH clockwise 90° about the origin.


The new x-coordinates are the old y-coordinates.

The new y-coordinates are the opposites of the old x-coordinates.


E(–4, 0) E’(0, 4)

F(0, 0) F’(0, 0) G(–2, –3) G’(–3, 2)

H(–6, –3) H’(–3, 6)

Course 1



Rotate parallelogram EFGH 90° about the origin.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x E F

G H

F’

E’ G’

H’

Course 1


-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

Try This: Example 1

Give the coordinates of the vertices of the figure after the given rotation. Rotate parallelogram EFGH counterclockwise 180° about the origin.

E F

G H

Course 1



The new x-coordinates are the opposites of the old x-coordinates.

The new y-coordinates are the opposites of the old y-coordinates.


E(–4, 0) E’(4, 0)

F(0, 0) F’(0, 0) G(–2, –3) G’(2, 3)

H(–6, –3) H’(6, 3)

Course 1


-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x


Rotate parallelogram EFGH counterclockwise 180° about the origin.

E F

G H

E’ F’

G’ H’

Course 1


Additional Example 2A: Art Application

An artist draws this figure. She rotates the figure without changing its size or shape. Give the coordinates of the vertices of the figure after a clockwise rotation of 90° about the origin.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

Course 1


ABCDE A’B’C’D’E’

A(0, 0) A’(0, 0)

B(1, 4) B’(4, –1)

C(3, 5) C’(5, –3)

D(7, 3) D’(3, –7)


E(6, 0) E’(0, –6)



Course 1



Rotate the figure after a clockwise 90° about the origin.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

C’

D’

B’ A’

E’

Course 1


B. Give the coordinates of the vertices of the figure after a counterclockwise rotation of 180° about the origin.

Additional Example 2B: Art Application

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

Course 1






A(0, 0) A’(0, 0)

B(1, 4) B’(–1, –4)

C(3, 5) C’(–3, –5)

D(7, 3) D’(–7, –3)

E(6, 0) E’(–6, 0)

Course 1




-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

C’

D’ B’

A’ E’

Course 1



An artist draws this figure. She rotates the figure without changing its size or shape. Give the coordinates of the vertices of the figure after a clockwise rotation of 90° about the origin.

-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

Course 1



A(–6, 0) A’(0, 6)

B(–5, 4) B’(4, 5)

C(–3, 5) C’(5, 3)

D(1, 3) D’(3, –1)


E(0, 0) E’(0, 0)



Course 1




-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

C’

D’

B’ A’

E’

Course 1


B. Give the coordinates of the vertices of the figure after a counterclockwise rotation of 180° about the origin.


-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

Course 1






A(–6, 0) A’(6, 0)

B(–5, 4) B’(5, –4)

C(–3, 5) C’(3, –5)

D(1, 3) D’(1, –3)

E(0, 0) E’(0, 0)

Course 1




-8 -6 -4 -2 2 4 6 8

8

2

4

6

-2

-4

-6

-8

y

x

C

D B

A E

C’ D’ B’

A’ E’

Course 1


Lesson Quiz Give the coordinates of the vertices of the trapezoid after the given rotation.

1. Rotate trapezoid ABCD counterclockwise 180° about the origin.

2. Rotate trapezoid ABCD clockwise 90° about the origin.

A’(0, 5), B’(0, 0), C’(–3, 1), D’(–3, 4)

A’(5, 0), B’(0, 0), C’(1, 3), D’(4, 3)


Course 1



Course 1

Warm Up

Lesson Presentation

Problem of the Day

Warm Up ABCD has vertices (–4, 1), (0, 1), (–3, 4), and (1, 4). What are the vertices of ABCD after it has been rotated clockwise 180° about the origin?

(4, –1), (0, –1), (3, –4), (–1, –4)

Course 1


Problem of the Day

Alice was 4 feet tall. She took a bite of one side of the caterpillar’s mushroom and became 5 times as tall! Then she took a bite of the other side of the mushroom and became times as tall. She took a bite from each side two more times. How tall was Alice then?

1 4 __

7 feet 13 16 ___

Course 1


Learn to visualize and show the results of stretching or shrinking a figure.

Course 1


Additional Example 1A: Stretching Figures

Write the dimensions of each part of the figure. Stretch the figure as stated and give the new dimensions of each part.

A. Increase the horizontal dimensions of the face, eyes, and mouth by a factor of 3.

Course 1



Original Dimensions New Dimensions Face Vertical 8

Horizontal 9 Vertical 8 Horizontal 27

Eyes Vertical 2 Horizontal 2

Vertical 2 Horizontal 6

Mouth Vertical 2 Horizontal 5


Course 1


Original Dimensions

New Dimensions

Face Vertical 8 Horizontal 9






Additional Example 1B: Stretching Figures

B. Increase the vertical dimensions of the face, eyes, and mouth by a factor of 2.

Course 1



Write the dimensions of each part of the figure. Stretch the figure as stated and give the new dimensions of each part.

A. Increase the horizontal dimensions of the face, eyes, and mouth by a factor of 3.

Course 1



Original Dimensions New Dimensions Face Vertical 5

Horizontal 6 Vertical 5 Horizontal 18





Course 1


Original Dimensions

New Dimensions

Face Vertical 5 Horizontal 6







B. Increase the vertical dimensions of the face, eyes, and mouth by a factor of 2.

Course 1


Additional Example 2A: Shrinking Figures Write the dimensions of each figure. Shrink the figure as stated and give the new dimensions.

A. Decrease the vertical dimensions by

multiplying by . 1 3 __

Original Dimensions New Dimensions Vertical 12 Vertical 4 Horizontal 10 Horizontal 10

Course 1


Additional Example 2B: Shrinking Figures

B. Decrease the horizontal dimensions by


Original Dimensions

New Dimensions

Vertical 12 Vertical 12 Horizontal 10 Horizontal 5

Course 1



Write the dimensions of each figure. Shrink the figure as stated and give the new dimensions.

A. Decrease the vertical dimensions by


Original Dimensions New Dimensions Vertical 9 Vertical 3 Horizontal 6 Horizontal 6

Course 1



B. Decrease the horizontal dimensions by


Original Dimensions

New Dimensions

Vertical 9 Vertical 9 Horizontal 6 Horizontal 3

Course 1


Lesson Quiz: Part 1 Determine the vertical and horizontal dimensions of the figure. Stretch the figure as stated and give the new dimensions.

1. Increase the vertical dimensions of the shaded region by a factor of 4. Vertical 12 48; horizontal 9 no change


Course 1


Lesson Quiz: Part 2

2. Decrease the horizontal dimensions of the

shaded region by multiplying by .

Vertical 12no change; horizontal 93


1 3 __

Course 1


chapter 1 number toolbox

Documents