Notes7th Grade Math7th Grade Math

McDowellMcDowell

Order of Operations 9/11

PEMDASLR

Please Excuse My Dear Aunt Sally’s Last Request

Parenthesis – Not just parenthesis• Any grouping symbol

– Brackets– Fraction bars– Absolute Values

Example 2+4

2

6

2

Simplify the top of the fraction 1st

Then divide

3

Exponents Simplify all possible exponents

Multiplication

And

Division

Do multiplication and division in order from left to right

Don’t do all multiplication and then all division

Remember division is not commutative

Addition

And

Subtraction

Do addition and subtraction in order from left to right

Don’t do all addition and then all subtraction

Remember subtraction is not commutative

You Try1. 3 + 15 – 5 22. 48 8 – 1

3. 3[ 9 – (6 – 3)] – 10

4. 16 + 24

30 - 22

Exponents 9/11

Exponents Show repeated multiplication

baseexponent

The number being multiplied

The number of times to multiply the base

Base

Exponent

Example 2³

2 x 2 x 2

4 x 2

8

Expanded

NotationWhen a repeated

multiplication problem is written out long

Exponential

Notation

When a repeated multiplication problem is written out using powers

3 x 3 x 3 x 3

34

Example (-2)²

-2²

-2 x –24

-1 x 2²-1 x 2 x 2-1 x 4-4

Examples (12 – 3)² (2² - 1²)

(-a)³ for a = -3

5(2(3)² – 4)³

Scientific Notation 9/14

Powers

Of

Ten

Factors 10 10x10 10x10x10 10x10x10x10

Product 10 100 1,000 10,000

Power 101 102 103 104

# of 0s 1 2 3 4

You Try Fill in the chartFactors 10x10x10x10x10x10x10

Product 10,000,000 100,000,000,000,000

Power 107 1010

# of 0s 10 14

Scientific

Notation

Looks like:

2.4 x 104

A short way to write really big or really small numbers using factors

The other factor will be less than 10 but greater than one

1 < factor < 10

And will usually have a decimal

One factor will always be a power of ten: 10n

The first factor tells us what the number looks like

The exponent on the ten tells us how many places to move the decimal point

4.6 x 106

4600000

Example

Move the decimal 6 hops to the right

4.600000 Rewrite

Convert between scientific notation and expanded notation

Write in expanded notation

1. 2.3 x 103

2. 5.76 x 107

Answers

1. 2,300

2. 57,600,000

You Try

13,700,000

1.3 x 107

Example

Figure out how many hops left it takes to get a factor between 1 and 10

1.3,700,000 Rewrite: the number of hops is your exponent

Convert between expanded notation and scientific notation

Write in scientific notation

1. 340,000,000

2. 98,200

Answers

1. 3.4 x 108

2. 9.82 x 104

You Try

Factor Trees and GCF 9/15

Prime

Numbers

Integers greater than one with two positive factors

1 and the original number

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

Integers greater than one with more than two positive factors

Composite

Numbers

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, . . .

Factor Trees

A way to factor a number into its prime factors

Steps

Is the number even or odd?If even: divide by 2If odd: divide by 3, 5, 7, 11,

13 or another prime numberWrite down the prime factor and

the new numberIs the new number prime or composite?

Is the number prime or composite?

If Composite:If prime: you’re done

Example Find the prime factors of99

even or odddivide by 3

3 33even or odd

divide by 3

prime or composite

prime or composite

3 11 prime or composite

The prime factors of 99: 3, 3, 11

Example Find the prime factors of12

even or odddivide by 2

2 6even or odd

divide by 2

prime or composite

prime or composite

2 3 prime or composite

The prime factors of 12: 2, 2, 3

You Try Find the prime factors of

1. 8

2. 15

3. 82

4. 124

5. 26

GCF 9/15

GCF Greatest Common Factor

the largest factor two or more numbers have in common.

Steps toFindingGCF

1. Find the prime factors of each number or expression

3. Pick out the prime factors that match

2. Compare the factors

4. Multiply them together

Example Find the GCF of 126 and 130

130126

2 63

3

2

5

75

21 15

5 3

The common factors are 2, 3

3 7

2 x 3The GCF of 126 and 130 is 6

You Try Work Book

p 47 # 1-9

p 48 # 3-33 3rd

LCM 9/16

LCM Least common multiple

The smallest number that is a multiple of both numbers

Steps

To

Find

LCM

1. Make a multiplication table for each number

2. Compare the multiplication tables

3. Pick the smallest number that both (all) tables have

Example Find the LCM of 8 and 3

1 2 3 4 5 6 7 8

8 16 24 32 40 48 56 64

3 6 9 12 15 18 21 24

1. Make a mult table

2. Compare

3. Find the smallest match

The LCM of 8 and 3 is 24

You Try Find the LCM between

1. 2 and 5

2. 9 and 7

Simplifying Fractions 9/16

Simplest form When the numerator and denominator have no common factors

Simplifying fractions

1. Find the GCF between the numerator and denominator

2. Divide both the numerator and denominator of the fraction by that GCF

Example Simplify 2852

28s Prime factors: 2, 2, 752s Prime factors: 2, 2, 13

Use a factor tree to find the prime factors of both numbers and then the GCF

GCF: 2 x 24

2852

4

4

= 7 13

You Try Write each fraction in simplest form

1. 27/30

2. 12/16

½ and 2/4 are equivalent fractions

Fractions that represent the same amount

Equivalent fractions

Making

Equivalent

Fractions

1. Pick a number

2. Multiply the numerator and denominator by that same number

58

x 3x 3

= 15 24

You Try Find 3 equivalent fractions to

611

Are the

Fractions

equivalent?

1. Simplify each fraction

2. Compare the simplified fraction

3. If they are the same then they are equivalent

You try Work Book

p 49 #1-17 odd

Least common Denominator 9/17

Common

DenominatorWhen fractions have the same denominator

Steps to

Making

Common

Denominators

1. Find the LCM of all the denominators

2. Turn the denominator of each fraction into that LCM using multiplication

Remember: what ever you multiply by on the bottom, you have to multiply by on the top!

Example Make each fraction have a common denominator

5/6, 4/9 Find the LCM of 6 and 9

6 12 18 24 30 36 42 489 18 27 36 45 64 73 82

Multiply to change each denominator to 18

5 x 36 x 3

= 15 18

= 8 18

4 x 29 x 2

You try What are the least common denominators?

1. ¼ and 1/3

2. 5/7 and 13/12

Comparing

And

Ordering

fractions

Manipulate the fractions so each has the same denominator

Compare/order the fractions using the numerators (the denominators are the same)

ProjectGroup work!

1. Get into groups of 3 or 4

2. Pick 3 or 4 different fractions

3. Each person make a picture of their fraction

4. Get together as a group and put the fractions/pictures in order from least to greatest

You try Workbook

p 51 #1-17 oddp 52 #3-36 3rd

Mixed Numbers and Improper Fractions 9/18

Improper fractions

When the numerator is bigger than the denominator

74

Represents more than 1

You try http://www.youtube.com/user/MathRaps#play/uploads/3/VZQDvb5Yjvw

Mixed

NumbersThe sum of a whole number

and a fraction

1 + ¾

1¾ =

74

1¾

ConvertingA MixedNumberTo anImproper

MAD face

Multiply the denominator by the whole number then add the product to the numeratorThat is the new numerator—keep the old denominator

Multiply, Add, keep the Denominator

6 x 5 + 3 630 + 3 6

33 6

You try Convert to an improper fraction

1.

2.

Divide the numerator by the dominator

ConvertingAnImproperTo aMixed #

The quotient is the whole number

The remainder is the new numerator

Keep the same denominator

Example Convert 26 to a mixed number 3

3 268

-242

R 2

You try Convert each improper fraction to a mixed number

1. 14 3

2. 25 5

Fractions and Decimals 9/21

Terminating

Decimala decimal that ends

1.25

When the same group of numbers continues to repeat forever

4.33333333333

4.3

Repeating

Decimal

Converting FractionsTodecimals

Divide the numerator by the denominator

516

Insert the decimal and some place holders to divide

You try Convert each fraction to a decimal. Determine if the decimal is a terminating decimal or a repeating decimal

1. 3 5

2. 1 6

Converting decimals to fractions

Remember place values

Converting decimals to fractions

Find the place value of the terminating decimal

Place the numbers after the decimal over the place value

Keep whole numbers as whole numbers

Simplify the fraction to lowest terms

Example 0.925 The 5 is in the thousandths place so 1000 is the denominator

9251000 Simplify

925 251000 25

3740

You try Convert each decimal to a fraction

1. 0.05

2. 4.7

3. 0.84

Number Sets 9/22

Whole

Numbers 0, 1, 2, 3, . . .

for short

Also known as the counting numbers

Natural

Numbers

1, 2, 3, 4, . . .

Integers Positive and negative whole numbers

for short

. . . –2, -1, 0, 1, 2, . . .

Rational

NumbersNumbers that can be written as fractions

for short

½, ¾, -¼, 1.6, 8, -5.92

You Try Copy and fill in the Venn Diagram that compares Whole Numbers, Natural Numbers, Integers, and Rational Numbers

Whole #s

Ordering

Rational

Numbers

Two Options

1. Change each number to a decimal and compare

2. Write each number as a fraction with a common denominator and compare

You Try Order from least to greatest1. 2.7, -0.3, -4/11

2. -5/6, 2.2, -0.5

3. 2.56, -2.5, 24/10