reteaching 1-1 · •to estimate using compatible numbers, change the numbers to ones that are easy...

Course 1 Lesson 1-1 Reteaching12

Reteaching 1-1 Understanding Whole Numbers

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• Standard form: 4,201,578

• To find the value of a digit, multiply the digit by its placevalue.

4 stands for 4 � 1,000,000, or 4,000,000.

• 5 stands for 5 � 100, or 500.

Write each number in standard form.

1. six thousand one hundred four 2. fifteen million twenty-one thousand

3. sixty thousand one hundred twelve 4. 2 billion, 9 million, 6 thousand, 1

5. seventeen thousandths 6. twenty-nine hundredths

7. eight thousand two hundred ninety 8. one billion thirty thousand fifty

Use R or S to complete each statement.

9. 523 567 10. 1,292 1,192 11. 47 45

12. 9,120 912 13. 53,010 53,100 14. 4,293 4,239

15. 783 738 16. 4,121 4,212 17. 35,423 34,587

Write in order from least to greatest.

18. 782, 785, 783, 790 19. 1,240; 1,420; 1,346; 1,364

20. 6,214; 6,124; 6,421; 6,241 21. 92,385; 92,835; 93,582; 93,258

22. 45,923; 54,923; 45,932; 54,932 23. 1,111; 1,011; 1,101; 1,110

29100

171000

201 thousand

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Thousands OnesMillions

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5784 million

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Reteaching 1-2 Estimating With Whole Numbers

• To estimate by rounding, first round each number, then add to find an estimated sum. This kindof estimation is usually used with addition and subtraction.

73 � 28 � ?

70 � 30 � ? 3 � 5, so round 73 down to 70.8 � 5, so round 28 up to 30.

70 � 30 � 100

• To estimate using compatible numbers, change the numbers to ones that are easy to computementally. This kind of estimation is usually used with multiplication and division.

4 � 26 � ?

4 � 25 � ? Change 24 to 25 because 4 and 25 can be multiplied mentally.

4 � 25 � 100

Estimate by first rounding each number.

1. 695 � 523 2. 753 � 137 3. 79 � 113 � 54

4. 23 � 98 � 439 5. 4,023 � 6,983 6. 8,201 � 1,183 � 53

Estimate using compatible numbers.

7. 357 � 46 8. 4 � 94 9. 3,934 � 4

10. 487 � 55 11. 223 � 16 12. 147 � 12

Estimate.

13. 57 � 18 � 112 14. 284 � 29 15. 547 � 43 � 12


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The commutative properties state that changing the order of addends or factors in amultiplication or addition expression does not change the sum or the product.Examples: 5 � 1 � 1 � 5 4 � 7 � 7 � 4

The associative properties state that changing the grouping of addends or factors in amultiplication or addition expression does not change the sum or the product.Example: (9 � 3) � 7 � 9 � (3 � 7) 2 � (5 � 17) � (2 � 5) � 17

The identity properties state that adding 0 to a number or multiplying a number by 1will not change that number.Example: 32 � 0 � 32 15 � 1 � 15

You can use these properties to simplify mathematical phrases.Example:20 � (3 � 5) � 120 � (5 � 3) � 1 Commutative Property of Multiplication(20 � 5) � 3 � 1 Associative Property of Multiplication100 � 3 � 1 Simplify.300 � 3 Identity Property of Multiplication900 Simplify.

Name the property being used.

1. 4 � 12 � 25 � 4 � 25 � 12

2. (13 � 2) � 5 � 13 � (2 � 5)

3. 6 � 7 � 0 � 49 � 6 � 7 � 49

Solve using mental math.

4. 16 � 23 � 4 5. (10 � 3) � 25

6. 17 � 19 � 3 7. 25 � 14 � 25

8. 50 � 17 � 2 9. 11 � 10 � 3

Reteaching 1-3 Properties of Numbers

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To find the value of an expression follow the order of operations.

First, do all operations inside parentheses.Next, multiply and divide from left to right.Then, add and subtract from left to right.

Example 1 Find the value of 6 � (3 � 4) � 2.

Work inside parentheses. (3 � 4) ≠ 7

6 � 7 � 2

Multiply next. 7 � 2 ≠ 14

6 � 14

Then, add.

6 � 14 � 20

Example 2 Compare 10 � (6 � 2) � 1 and (10 � 6) � 2 � 1.

First, find the value of each expression. Then, use �, =, or � to compare.10 � (6 � 2) � 1 (10 � 6) � 2 � 1 8 � 310 � 3 � 1 4 � 2 � 1 So,

7 � 1 2 � 1 10 � (6 � 2) � 1 � (10 � 6) � 2 � 10.8 3

Find the value of each expression.

1. 3 � (4 � 1) � 2 2. 24 � (5 � 3) � 2

a. 4 � 1 � a. 5 � 3 �

b. � 2 � b. 24 � �

c. 3 � � c. � 2 �

3. 2 � 6 � 3 � 3 � 4. (6 � 2) � 3 � 4 �

Use R, ≠, or S to complete each statement.

5. 9 � 3 � 4 9 � (3 � 4) 6. (12 � 4) � 3 12 � (4 � 3)

7. 6 � 3 � 4 � 2 (6 � 3) � 4 � 2 8. 3 � (12 � 5) � 2 3 � 12 � (5 � 2)

9. 15 � (12 � 3) (15 � 12) � 3 10. 8 � 2 � (9 � 7) 8 � (2 � 9) � 7

3

2

1

Reteaching 1-4 Order of Operations

• Standard form: 2.369

• To find the value of a digit, multiply the digit by its place value.

9 stands for 9 � 0.001 or 0.009.

• Expanded form:2.369 � 2 � 0.3 � 0.06 � 0.009

• To round to the nearest tenth, look at the value to the right (hundredths).2.369 because 6 � 5, round up.2.4

Write each decimal in words.

1. 0.2 2. 0.15 3. 0.29

4. 0.11 5. 0.60 6. 0.9

Write each decimal in standard form.

7. seven tenths 8. one tenth 9. four hundredths

10. seven hundredths 11. twenty-two hundredths 12. forty-six hundredths

Write each decimal in expanded form.

13. 3.6 14. 4.72

15. 1.283 16. 21.5

Round each decimal to the underlined place.

17. 78.632 18. 9.1746

Find the value of the digit 6 in each number.

19. 10.3678 20. 145.620


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Reteaching 1-5 Understanding Decimals

2 and 369 thousandths

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Reteaching 1-6 Comparing and Ordering Decimals

Use �, � , or � to show how 4.092 and 4.089 compare.

Write the numbers on grid paper with the decimal points lined up.

Compare digits in the greatest place. Move to the right until you find digits that are not the same.

4 ones � 4 ones0 tenths � 0 tenths9 hundredths � 8 hundredths

So, 4.092 � 4.089.

To order numbers from least to greatest:

Write the numbers on grid paper (decimal points lined up) and compare.

Then arrange the numbers from least to greatest.

4.089, 4.09, 4.092

Use R, ≠, or S to complete each statement.

1. 0.01 0.15 2. 0.25 0.21 3. 0.30 0.26

4. 0.10 0.12 5. 0.35 0.34 6. 0.1 0.4

7. 34.4 34.40 8. 0.207 0.27 9. 0.08 0.40

Use place value to order the decimals from least to greatest.

10. 3.46, 3.64, 3.59 11. 22.97, 21.79, 22.86 12. 43, 43.22, 43.022

13. 10.02, 10.2, 1.02 14. 1.09, 1.9, 1.1 15. 7.54, 75.4, 7.4

Order each set of numbers on a number line.

16. 0.67, 0.7, 0.6 17. 0.03, 0.29, 0.019 18. 8.36, 8.01, 8.1

8.18.01 8.368.18 8.2 8.30 0.1 0.2 0.3

0.019 0.03 0.290.6 0.7

0.80.70.6

0.67

2

1

2

1

80. 94

90. 24

80. 94

90. 24

90.4


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Add 3.25 � 12.6 � 18.93.

First estimate. 3.25 312.6 13

� 18.93 1935

Then follow these steps.

Line up the decimal points. Add as you would add Place the decimal point.Write in any needed zeros. whole numbers. Regroup

when needed.

3.25 3.25 3.2512.60 12.60 12.60

� 18.93 � 18.93 � 18.9334 78 34.78

To subtract decimals, follow similar steps. Work from right to left andregroup when needed. Place the decimal point to complete thesubtraction.

First estimate and then find each sum.

1. 0.9 � 6.7 2. 3.1 � 9.4 3. 4.88 � 8.19

Estimate Estimate Estimate

Sum Sum Sum

Use mental math to find each sum.

4. 14.05 � 9.75 5. 6 � 0.22 � 0.78 6. 9.104 � 5.01 � 7.99

First estimate and then find each difference.

7. 8.5 - 4.2 8. 7.2 - 3.05 9. 5.07 - 2.8


Difference Difference Difference

10. 6.347 - 2.986 11. 14.2 - 9.86 12. 13.45 - 5.001


Difference Difference Difference

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Reteaching 1-7 Adding and Subtracting Decimals

Compare to your estimate.

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Reteaching 1-8 Multiplying Decimals

Multiply 0.3 � 1.4. This drawing can help you find 0.3 � 1.4.

Each small square is 1 hundredth or 0.01.Each column or row is 10 hundredthsor 1 tenth or 0.1.

Compare the result from the model to the result of multiplying the factors.

0.3 1 decimal place� 1.4 +1 decimal place

1 2+ 0 3 0

0.42 2 decimal places

Write a multiplication statement to describe each model.

1. 2.

For each product, place the decimal point in the correct place.

3. 0.9 4. 3.1 5. 6.22�2.8 �77 � 8

252 2387 4976

Find each product.

6. 1.6 7. 8.12 8. 12.3�3.7 �59 �6.1

9. 23.4 10. 4.8 11. 9.2�5.2 �42 �12.4

Shade 3 rows across to represent 0.3.

Shade 14 columns down to represent 1.4.

The area where the shading overlaps is 42 hundredths or 0.42.

0.3 � 1.4 � 0.42

When multiplying decimals, first multiply thefactors as though they are whole numbers.Then add the number of decimal places in eachfactor to find the number of decimal places in theproduct.

3

2

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Find the quotient 1.52 � 0.4.

You can use a model to estimate the quotient.

Draw a model for 1.52.

Since each square is 0.01, 40 squares represent 0.4.Circle groups of 0.4.

There are close to four groups of 0.4. The quotient is about 4.

Multiply the dividend and Divide as with whole Place the decimal point in thedivisor by 10 so that the numbers. quotient above its place in thedivisor is a whole number. dividend. Insert zeros as

placeholders if necessary.

3 8 3.8

�1 2 �1 232 32

�32 �320 0

3.8 is close to 4.

Use the model to find each quotient.

1. 2. 3.

0.8 � � 20 0.70 � 0.1 � � 0.05 �

Estimate, then find each quotient.

4. 5. 6. 8.4 � 6

7. 8. 12.96 � 5 9. 3 � 0.12

10. 11. 78 � 15.6 12. 0.9q1.351.5q84

8q27

4q2.683q1.35

4q15.24q15.20.4q1.52

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Reteaching 1-9 Dividing Decimals


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• The mean of a set of data is the sum of the values divided by the number of data items.87 � 87 � 88 � 90 � 90 � 91 � 93 � 110 � 736736 � 8 � 92The mean math test grade is 92.

• An outlier is a data item that is much greater or less than the other data items. If a data set has anoutlier, then the mean may not describe the data very well.

The outlier in the Math Test Grades is 110, because it is significantly higher than the other scores. Inthis case, omitting the outlier from the mean calculation lowers the mean by about 3 points.

87 � 87 � 88 � 90 � 90 � 91 � 93 � 626626 � 7 � 89.4

Find the mean of each data set.

1. 8, 6, 5, 9, 7, 13

2. 9, 12, 14, 6, 8, 5

Identify the outlier in each set of data. What effect does that outlierhave on the mean?

3. 94, 77, 37, 80, 74, 94, 87, 85 4. 13, 10, 9, 15, 11, 29, 12, 10

5. 378; 433; 364; 418; 2,877; 408 6. 4,333; 4,290; 4,315; 587; 4,100

Reteaching 2-1 Mean

Math Test Grades

Name GradeSharon 87Rashid 91Durrin 88Nicole 90Terry 90Mei-lin 93Kevin 87Carlos 110

• The median of a data set is the middle value when the data are arranged in numerical order.When the temperatures are arranged in order from least to greatest, there are two middle numbers.

35°, 40°, 45°, 50°, 60°, 70°, 75°, 80°, 98°, 98°, 100°, 105°

To find the median, add the two middle numbers and divide the total by 2.

70 � 75 � 145

145 � 2 � 72.5

The median temperature is 72.5° Fahrenheit.

• The mode is the data item(s) that appears most often.For this data, 98° is the mode.

• The mean, median, and mode of a data set are each a different way to measure the data. The mean is not always the best measure. The median works well for analyzing data sets with outliers. Themode is useful when the data items are repeated or are not numerical.

Find the median of each data set.

1. 5, 4, 7, 9, 8 2. 9, 12, 14, 6, 8, 5

3. 12, 16, 19, 14, 14, 18 4. 2.2, 3.6, 2.4, 2.5, 3.9, 3.4, 3.1

Find the mode of each data set.

5. 3, 4, 5, 5, 3, 5, 4, 2 6. blue, red, red, yellow, blue, red

7. 33, 35, 34, 33, 35, 33 8. girl, boy, boy, girl, girl, girl

Does the mean, the median, or the mode best describe the averagemonthly temperatures listed below?

9. 35°, 45°, 50°, 70°, 75°, 98°, 100°, 105°, 98°, 80°, 60°, 40°

10. 29°, 28°, 30°, 33°, 65°, 70°, 70°, 98°, 74°, 68°, 60°, 40°

Reteaching 2-2 Median and Mode


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Average Monthly Temperature

Month Temperature (in

degrees Fahrenheit)January 35February 45March 50April 70May 75June 98July 100August 105September 98October 80November 60December 40


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Sixteen students were asked to name Students compared the number of books their favorite school day. A frequency they carry to school. A line plot can be table can be used to organize their used to show this data. Each ✗responses. represents one student.

To make a frequency table: To use a line plot to find the range:

Organize the data by completing the frequency table. Then use the data to complete the line plot.

1. ages of middle school students:11, 12, 12, 12, 12, 12, 13, 13, 13, 13,13, 13, 14, 14, 14

The range is .

11

12

13

14

Age Tally Frequency

Reteaching 2-3 Organizing and Displaying Data

FavoriteSchool Day Tally Frequency

Monday ❘ ❘ ❘ 3Tuesday ❘ ❘ 2Wednesday ❘ ❘ ❘ 3Thursday ❘ ❘ ❘ ❘ 4Friday ❘ ❘ ❘ ❘ 4

Number of Books Carried to School

1

✗✗

2 3

✗✗

4

✗✗✗

5

✗✗✗✗

6

✗

List all the choices.

Mark a tally for each student’s response.

Total the tallies for each choice.

Subtract the least value from the greatestvalue along the horizontal line.

The range is 6 � 1 or 5 books.2

1

3

2

1

11 12 13 14

Ages of Middle School Students


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A bar graph uses vertical or horizontal bars to display numerical information.The length of the bars tell you the numbers they represent. To read the graph at the right, first read the horizontalaxis. Then read from the top of a bar to the vertical axis. This graph shows that inSeptember, an English class read 25 books.

A line graph is a graph that uses a seriesof line segments to show changes in data. Usually, a line graph shows changes over time. For example, it mayshow a pattern of increase, decrease, or no change over time.

Use the bar graph above for Exercises 1–2.

1. How many books were read in 2. In which two months did students read the December? same number of books?

Use the tables at the right for Exercises 3–4.

3. Would you use a bar graph, a histogram,or a line graph to display the data in theSports table? Explain your choice.

4. Would you use a bar graph or a linegraph for the Population data? Explain.

Reteaching 2-4 Bar Graphs and Line Graphs

1 2 3 4Week

Amount Earned fromAfter-School Job

Ear

nin

gs

(do

llars

)20

15

10

5

0

Favorite SportsSport Number Answering

Wrestling 290Football 50Basketball 520Baseball 130

Population of SpringdaleYear Population1970 45,0001980 62,0001990 68,000

Books Read by an English Class

Month

Sep. Oct. Nov. Dec. Jan.

Nu

mb

er o

f B

oo

ks

40

30

20

10

0


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Party Pals has party equipment for rent. The company uses a spreadsheetto keep track of the number of hours its equipment is rented.

• A cell is a box in a spreadsheet where a particular row andcolumn meet. In column C, you find the cells C1, C2, C3, and soon. C2 shows 8. Its value is 8 hours.

• Missing values can be found by telling the spreadsheet whatcalculation to do. The value for cell E3 can be found by using theformula � B3 � C3 � D3.

Use the spreadsheet above for Exercises 1–11. Identify the cell(s) thatindicate each category.

1. machines 2. rental times for 3. total rental rented Saturday hours for the

Popcorner

Write the value for the given cell.

4. B5 5. C4 6. D3 7. A3

Write a formula to find each quantity.

8. the total in cell E4 9. the total in cell E5

10. the total in cell E2 11. the mean rental time in cell F5

Reteaching 2-5 Using Spreadsheets to Organize Data

Party PopcornerJuice FountainSoft Ice Cream MachinePretzel Oven

Machine

2345

1 Fri.

4081

Sat.

8396

Sun.

6345

TotalA B C D E

MeanRentalTime

F


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A stem-and-leaf plot is a graph that uses the digits of each number toshow the shape of the data. Each data value is broken into a “stem”on the left and a “leaf” on the right. A vertical segment separates thestems from the leaves. To read the data, combine the stem with eachleaf in the same row.

Example: Make a stem-and-leaf diagram of the data showing minutes spent eating lunch.

Decide what the stem of the diagram willrepresent. Since these data are two-digit numbers,the stem will be the tens digits and the leaves willbe the ones digits.

Write the tens digits in order in the lefthandcolumn of the diagram. Then write each leaf at theright of its stem as they occur in the problem.

Complete the second stem-and-leaf diagram, withthe leaves in order from least to greatest.

For Exercises 1–3, use the stem-and-leaf plot at the right.

1. What does 1 | 8 represent?

2. How many people were older than 40?

3. How many people were at the poetry reading?

3

2

1

Reteaching 2-6 Stem-and-Leaf Plots

Minutes Spent Eating Lunch

46, 35, 12, 37, 28, 10, 22, 54, 19, 13, 46, 51

Lunch Times

stem leaf

1 2 0 9 3

2 8 2

3 5 7

4 6 6

5 4 1

2

Lunch Times

stem leaf

1 0 2 3 9

2 2 8

3 5 7

4 6 6

5 1 4

3

Ages of PeopleAttending a Poetry

Reading

stem leaf

0 5 8 8 9

1 8 8 9 9

2 3 5 5 6 8 8 9

3 2 2 7 8

4 1 3

5 2 8

6 4 4 6 7 8

Key: 0 | 8 means 8years old.

Key: 1 | 0 means 10


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Data can be displayed on graphs in ways that are misleading.

The horizontal scales make these line graphs seem different.

As the numbers are moved farther apart, it appears that the change over time is less.

These bar graphs may seem different because of how the vertical scales are drawn.

The break in the vertical scale makes the differences seem greater than they really are.

Use the graphs above for Exercises 1–3.

1. Which graph might be used to convince someone that the priceof pizza has risen too quickly over the years?

2. Which graph would Car Company X use to show that its carslast longer than the competition?

3. Which graph of cars still on the road after 10 years would CarCompany Z prefer?

4. On a science exam, six students scored a mean of 75. Theirscores were 88, 90, 12, 85, 87, and 88. Why might the mean bemisleading?

Reteaching 2-7 Misleading Graphs and Statistics

’60 ’70 ’80 ’90Year

Price of a Slice of Pizza

Pri

ce (

do

llars

) 1.501.251.00.75.50.25

0

B

Cars on the RoadAfter 10 Years

Car CompanyX Y ZN

um

ber

of

Car

s 4,000

3,000

2,000

1,000

0

C Cars on the RoadAfter 10 Years

Car CompanyX Y ZN

um

ber

of

Car

s 4,000

3,000

2,000

D

’60

’70

’80

’90

Year

Price of a Sliceof Pizza

Pri

ce (

do

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) 1.501.251.00.75.50.25

0

A


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Each number in a pattern is a term. Find the next three termsin the pattern.

3, 9, 15, 21, ?, ?, ?

Look at how the second term can be found from the first.

3, 9, 15, 21 or 3, 9, 15, 21

Look at how the third term can be found from the second.

3, 9, 15, 21 or 3, 9, 15, 21

Try adding 6 to the third term.

3, 9, 15, 21

Now you can write a rule to describe the pattern. The rule isStart with the number 3 and add 6 repeatedly.

The next three terms in the pattern are 27, 33, and 39.

Write a rule for each number pattern. Then write the next three terms.

1. 2, 5, 8, 11, , , 2. 3, 6, 12, 24, , ,

3. 9, 18, 27, 36, , , 4. 64, 56, 48, 40, , ,

5. 1, 4, 16, 64, , , 6. 75, 70, 65, 60, , ,

15 1 6 5 211 6 1 6 1 6

9 1 6 5 151 6 1 69 � 3 is not 15.3 3

3 1 6 5 91 63 3 3 5 93 3

Reteaching 3-1 Describing a Pattern

3, 9, 15, 21, 27, 33, 39

+ 6 + 6 + 6 + 6 + 6 + 6


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Numerical expressions are made up of numbers Example 1:and operation symbols. 6 + 3 9 � 2 + 1

Algebraic expressions contain one or more variables. Example 2:A variable is a letter that stands for an unknown number. x + 4 � 2 a - b

You can model algebraic expressions using algebra tiles.

3 � p � 4 � 1 � 3p + 4

You can evaluate the algebraic expression 3p + 4 if you know a valuefor p. Evaluate this expression for p � 6.

Think of each rectangle as having a Then use the order of operations tovalue of 6. evaluate.

3p + 4 for p = 6 means 3 � 6 + 4 3p + 4 = 3 � 6 + 4= 18 + 4= 22

Write an algebraic expression for each model.

1. 2. 3.

Evaluate each expression.

4. 7c for c = 6 5. 3t - 4 for t = 8

7 � = 3 � - 4 =

6. 15 + m for m = 6 7. 2x + 1 for x = 3

8. 5y - 10 for y = 6 9. 3(4h) for h = 2

10. a - b for a = 5 11. x + 2y for x = 3and b = 4 and y = 2

21

The 3 shaded rectangles represent3 of the same variable. The 4 whitesquares represent 4 ones.

Reteaching 3-2 Variables and Expressions


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These terms are used to describe mathematical operations.

You can use the terms above to write word phrases for algebraicexpressions, and algebraic expressions for word phrases.

Word Phrase Algebraic Expression

the sum of m and 17 S m + 17the difference of x and 12 S x - 123 times w S 3wthe quotient of q and 6 S q � 6

Write a word phrase for each expression.

1. n � 8 2. 3x 3. w � 2

Write an expression for each word phrase.

4. 6 increased by y 5. the quotient of 8 and e

6. 3 less than h 7. 4 times w

8. the difference of s and 8 9. r divided by 2

10. 5 more than n 11. the product of 6 and m

Reteaching 3-3 Writing Algebraic Expressions

Addition Subtraction Multiplication Division

sum difference product quotient ofmore than less than times divided by

increased by fewer than multiplied bytotal decreased by

added to

SSSS


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Reteaching 3-4 Using Number Sense to Solve One-Step Equations

One way to solve some equations is to use mental math.

Example 1: Find the solution to the equation. Example 2: Find the solution to the equation.a + 5 = 10 y - 9 = 15

What you think: What you think:If I add 5 to 5, the sum is 10. If I subtract 9 from 24, the difference is 15,5 + 5 = 10 24 - 9 = 15So, a = 5. So, y = 24.

Example 3: Find the solution to the equation. Example 4: Find the solution to the equation.w � 5 = 100 4w = 24

What you think: What you think:w � 5 means w divided by 5. 4w means 4 times w.I know that 500 � 5 = 100. I know that 4 � 6 = 24.500 � 5 = 100

So, w = 500. So, w = 6.

Use mental math to solve each equation.

1. 4q = 12 2. 3w = 15

3. h + 7 = 16 4. k + 2 = 8

5. n � 3 = 12 6. m � 2 = 10

7. y - 8 = 12 8. w - 5 = 8

Tell whether each equation is true or false.

9. 18 + 25 = 43 10. 1,100 � 200 = 900

11. 16 � 4 = 32 12. 18 = 9 � 2

13. 77 � 12 = 99 14. 2 � 9 = 81


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Reteaching 3-5 Solving Addition Equations

In the equation x � 5 � 33, 5 is added to the variable. To solve theequation, undo the operation to get the x alone on one side of theequal sign. Undo addition by subtracting.

Solve x � 5 � 33.

Solve x � 5 � 33

x � 5 � 5 � 33 � 5 Subtract 5 from each side toundo the addition and get xby itself.

x � 28 Simplify.

Check x � 5 � 33 Check your solution in theoriginal equation.

28 � 5 � 3 Substitute 28 for x.

33 � 33 ✓

Drawing a diagram can help you writean equation to solve a problem.

Solve each equation. Then check each solution.

1. Solve: x � 5 � 33 Check: x � 5 � 33

x � 5 �___� 33 � ___ ___ � 5 � 33

x � ___ ___ � 33

2. 19 � t � 51 3. 60 � n � 30

4. 86 � m � 107 5. w � 349 � 761

Draw a diagram to model the situation. Then write and solve an equation.

6. A car dealer purchased a car for $2,000 and then sold it for $3,200. What was the profit?

S

S

S

S

Whole

Part Part

33

x 5


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Reteaching 3-6 Solving Subtraction Equations

S

S

S

S

In some equations, a number is subtracted from the variable.To solve, undo the subtraction by adding the same amount to both sides of the equation.

Solve r � 13 � 9.

Solve r � 13 � 9

r � 13 � 13 � 9 � 13 Add 13 to each side to undo thesubtraction and get r by itself.

r � 22 Simplify.

Checkr � 13 � 9 Check your solution in theoriginal equation.

22 � 13 � 9 Substitute 22 for r.

9 � 9 ✓

Drawing a diagram can help you writean equation to solve a problem.


1. Solve: 8 � x � 21 Check: 8 � x � 21

8 �___ � x � 21 � ___ 8 � ___ � 21

___ � x 8 � ___

2. p � 11 � 12 3. 71 � b � 29

4. y � 53 � 30 5. 75 � d � 127

Draw a diagram to model the situation. Then write and solve an equation.

6. There were 18 friends at Luke’s birthday party. Seven of thefriends he invited were not able to attend. How many friends did he invite?

Whole

Part Part

r

13 9


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Reteaching 3-7 Solving Multiplication and Division Equations

To solve a multiplication or division equation, undo the operationto get the variable alone. Multiplication and division are inverseoperations, which means that they undo each other.

To solve a multiplication equation, divide. To solve a division equation, multiply.

4w � 56 y � 7 � 13

4w � 4 � 56 � 4 y � 7 � 7� 13 � 7

w � 14 y � 91

Check 4 � 14 � 56 Check 91 � 7 � 13

56 � 56 ✓ 13 � 13 ✓

Drawing a diagram can help you solve a problem.

State whether the given number is a solution to the equation.

1. 3g � 56; g � 12 2. t � 8 � 2; t � 4 3. 6 � r � 9; r � 54


4. 4y � 64 5. n � 9 � 12

4y � ___ � 64 � ___ n � 9 � ___ � 12 � ___

y � ___ n � ___

Check: Check:

6. 23p � 115 7. z � 11 � 11

8. 66 � 3s 9. 34 � t � 14

Divide each side by 4.

Simplify.

Multiply each side by 7.

Simplify.

SS

SS

Check your solution.Replace w with 14.

S Check your solution.Replace y with 91.

S

y

13 13 13 13 13 13 13

56

w w w w


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The Distributive Property allows you to break The Distributive Property may also help you tonumbers apart to make mental math easier. simplify an expression.

Multiply 9 � 24 mentally. (8 � 7) + (8 � 3) = 8 � (7 + 3)Think: 9 � 24 = 9 � (20 � 4) = 8 � 10

= (9 � 20) + (9 � 4) = 80= 180 + 36= 216

Use the Distributive Property to find the missing numbers in the equation.

1. (6 � ) - ( � 3) = 6 � (5 - 3) 2. 4 � ( - 3) = ( � 9) - (4 � )

3. ( � 7) - (6 � ) = 6 � (7 - 5) 4. � (12 + 8) = (6 � ) + ( � 8)

Use the Distributive Property to rewrite and simplify each expression.

5. (2 � 7) + (2 � 5) 6. 8 � (60 - 5)

7. (7 � 8) - (7 � 6) 8. (12 � 3) + (12 � 4)

Use the Distributive Property to simplify each expression.

9. 3 � 27 10. 5 � 43 11. 8 � 59

12. 7 � 61 13. 5 � 84 14. 6 � 53

8 � 78 88 � 3

37

9 � 209

420

99 � 4

Reteaching 3-8 The Distributive Property


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A number is divisible by a second number if the second numberdivides into the first with no remainder. Here are some rules.

Circle the numbers in each row that are divisible by the number at the left.

1. 2 8 15 26 42 97 105 218

2. 5 14 10 25 18 975 1,005 2,340

3. 10 100 75 23 60 99 250 655

4. 3 51 75 12 82 93 153 274

5. 9 27 32 36 108 126 245 387

Use mental math to determine if the first number is divisible by thesecond.

6. 185; 5 7. 76,870; 10 8. 456; 3

9. 35,994; 2 10. 12,866; 9 11. 151,002; 9

12. 6,888; 2 13. 31,067; 5 14. 901,204; 3

15. 2,232; 3 16. 45,812; 9 17. 3,090; 10

18. 312; 9 19. 1,933; 3 20. 28,889; 2

Test each number for being divisible by 2, 5, or 10. Some numbersmay be divisible by more than one number.

21. 800 22. 65 23. 1, 010

Name Class Date

Reteaching 4-1 Divisibility and Mental Math

Last Digit of a Number The Number Is Divisible by Examples

any 1 any number

0, 2, 4, 6, 8 2 10; 24; 32; 54; 106; 138

0, 5 5 10; 25; 70; 915; 1,250

0 10 10; 20; 90; 500; 4,300

The Sum of the The Number IsDigits Divisible by Examples

is divisible by 3 3 843 S 8 � 4 � 3 = 15 281 R0and 15 � 3 = 5

is divisible by 9 9 2,898 S 2 � 8 � 9 � 8 = 27 322 R0and 27 � 9 = 3 9q2,898

3q843


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Reteaching 4-2 Exponents

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

3 � 3 � 3 � 3 shows the number 3 is used as a factor 4 times.

3 � 3 � 3 � 3 can be written 34.

In 34, 3 is the base and 4 is the exponent.

Read 34 as “three to the fourth power.”

• To simplify a power, first write it as a product.

25 = 2 � 2 � 2 � 2 � 2 = 32

• When you simplify expressions with exponents, do all operationsinside parentheses first. Then simplify the powers.

Example: 30 - (2 + 3)2 = 30 - 52

= 30 - 25= 5

Name the base and the exponent.

1. 36 2. 62 3. 84

base base base

exponent exponent exponent

Write each expression using an exponent. Name the base and the exponent.

4. 9 � 9 � 9 5. 6 � 6 � 6 � 6 6. 1 � 1 � 1 � 1 � 1

Simplify each expression.

7. 62 8. 35 9. 104

10. 42 + 52 11. 2 � 6 - 23 12. 62 + 42

13. 5 + 52 - 2 14. 24 � 4 + 24 15. 9 + (40 � 23)

16. (42 + 4) � 5 17. 10 � (30 - 52) 18. 12 + 18 � 32


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A prime number has exactly two factors, thenumber itself and 1.

5 � 1 = 55 is a prime number.

A composite number has more than two factors.1 � 6 = 62 � 3 = 6

1, 2, 3, and 6 are factors of 6.6 is a composite number.

The number 1 is neither prime nor composite.

Every composite number can be written as aproduct of prime numbers.

6 = 2 � 38 = 2 � 2 � 2

12 = 2 � 2 � 3

Factors that are prime numbers are called primefactors. You can use a factor tree to find primefactors. This one shows the prime factors of 50.

50 = 2 � 5 � 5 is the prime factorization of 50.

50

252

5 5

Reteaching 4-3 Prime Numbers and Prime Factorization

Tell whether each number is prime or composite. Explain.

1. 21 2. 43 3. 53 4. 74

5. 54 6. 101 7. 67 8. 138

9. 83 10. 95 11. 41 12. 57

Complete each factor tree.

13. 14. 15.

Find the prime factorization of each number.

16. 21 17. 48 18. 81

19. 63 20. 100 21. 103

120

10

64

8

4

60

106


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Reteaching 4-4 Greatest Common Factor

List the factors of 12 and 18.

12: 1, 2, 3, 4, 6, 1218: 1, 2, 3, 6, 9, 18

Find the common factors.

12: 1 , 2 , 3 , 4, 6 , 1218: 1 , 2 , 3 , 6 , 9, 18

The common factors are 1, 2, 3, and 6.

Name the greatest common factor: 6.

Draw factor trees.

Write each prime factorization.Identify common factors.

12: 2 � 2 � 318: 2 � 3 � 3

Multiply the common factors. 2 � 3 = 6.The GCF of 12 and 18 is 6.

3

2

18

92

3 3

12

4 3

22

1

3

2

1

You can find the greatest common factor (GCF) of 12 and 18 using adivision ladder, factor trees, or by listing the factors. Two of thesemethods are shown.

List the factors to find the GCF of each set of numbers.

1. 10: 2. 14: 3. 9:

15: 21: 21:

GCF: GCF: GCF:

4. 12: 5. 15: 6. 15:

13: 25: 18:

GCF: GCF: GCF:

7. 36: 8. 24:

48: 30:

GCF: GCF:

Find the GCF of each set of numbers.

9. 21, 60 10. 15, 45

11. 54, 60 12. 20, 50

13. 36, 40 14. 48, 72


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Reteaching 4-5 Equivalent Fractions

Equivalent fractions are fractions that name thesame amount.

To find equivalent fractions, multiply or dividethe numerator and denominator by the samenumber.

� 2 � 2

= =

� 2 � 2

So, = .

To write a fraction in simplest form, divide thenumerator and denominator by their greatestcommon factor.

Example: Write in simplest form.

Find the greatest common factor.

8: 1, 2, 4, 812: 1, 2, 3, 4, 6, 12

The GCF is 4.

Divide the numerator and denominator bythe GCF.

� 4

=

� 4

in simplest form is .23

812

23

812

2

1

812

410

25

25

410

410

25

Write two fractions equivalent to each fraction.

1. 2. 3.

4. 5. 6.

State whether each fraction is in simplest form. If it is not, write it insimplest form.

7. 8.

9. 10.

Write each fraction in simplest form.

11. 12. 13.

14. 15. 16.

17. 18. 19.

20. There are 420 girls out of 1,980 people attending a state fair. Insimplest form, what fraction of the people attending are girls?

30225

2233

749

1216

130170

39

5664

10200

1224

1430

1522

815

1215

15

36

311

78

37

56

1518

1012

933

622

12

612

525

210

921

614

2832

1416

45

715

12

1317

733

23

215

1317

34

120

78


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Write each mixed number as an improper fraction.

1. 2. 3.

4. 5. 6.

7. 8. 9.

On a separate sheet of paper, draw a model of a 4-inch ruler marked off in eighths. Find and label each measurement on your ruler.

10. 3 11. 2 12. 3

13. 14. 15.

Write each improper fraction as a mixed number in simplest form.

16. 17. 18.

19. 20. 21.

22. 23. 24. 249

298

95

175

203

148

83

72

98

31421

2134

12

68

58

8 162

373

38

4 459

133

410

6 585

342

27

Name Class Date

Reteaching 4-6 Mixed Numbers and Improper Fractions

To write a mixed number as an improper fraction:

Multiply thewhole number bythe denominator.

Add this productto the numerator.

Write this sumover the denominator.

To write an improper fraction as a mixed number:

Divide the = 2 remainder 4numerator by the denominator.

Write the =

remainder over the denominator.

Simplify, if =

possible.

208 5 2

12

2 123

2 482

2081

3

12

3 58 5 29

8

321

167

234

538

3410

283

245

278

177

496

118 3

12 2

23

134 6

23 3

25

145 3

58 2

23


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Reteaching 4-7 Least Common Multiple

Find the least common multiple (LCM) of 8 and 12.

Begin listing multiples of each number.

8: 8, 16, 24, 32, 40

12: 12, 24

Continue the lists until you find the first multiple that is commonto both lists. That is the LCM.

The least common multiple of 8 and 12 is 24.

List multiples to find the LCM of each pair of numbers.

1. 4: 2. 6:

5: 7:

LCM: LCM:

3. 9: 4. 10:

15: 25:

LCM: LCM:

5. 8: 6. 8:

24: 12:

LCM: LCM:

7. 4: 8. 15:

7: 25:

LCM: LCM:

Use prime factorization to find the LCM of each set of numbers.

9. 9, 21 10. 6, 8

11. 18, 24 12. 40, 50

2

1


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Reteaching 4-8 Comparing and Ordering Fractions

Compare each pair of numbers using R, ≠, or S.

1. � 2. � 3. �

4. � 5. � 6. �

7. � 8. � 9. �

Order each set of numbers from least to greatest.

10. , , 11. , , 12. , ,

13. , , 14. , , 15. , ,

16. Suzanne swims 1 miles. Eugene swims 1 miles. Who swims farther? Show your work.512

19

1316

34

78

38

35

12

712

23

35

23

512

12

23

56

58

12

58

34

2 352

59

710

23

79

45

39

13

58

37

46

23

411

36

78

56

13

29

Compare Fractions

Example 1: Compare and .

Find the LCD of the denominators 4 and 10:

4 = 2 � 2

10 = 2 � 5

LCD = 2 � 2 � 5= 20

Write equivalent fractions:

� 5 � 2

= =

� 5 � 2

Compare: , or

Order Fractions

Example 2: Order from least to greatest: , , .

Find the LCD of the denominators 3, 8, and 4:

3 = 3

8 = 2 � 2 � 2

4 = 2 � 2

LCD = 2 � 2 � 2 � 3= 24

Write equivalent fractions:

� 8 � 3 � 6

= = =

� 8 � 3 � 6

Order:

15 , 16 , 18

, or 58 , 23 , 3

41524 , 16

24 , 1824

3

1824

34

1524

58

1624

23

2

1

34

58

23

34 . 7

10

1520 . 14

203

1420

710

1520

34

2

1

710

34

To compare and order fractions, use the least common denominator (LCD).The LCD is the least common multiple (LCM) of the original denominators.

34

58

12

436

1536

1536

512

436

19

56

23

58

23

12

512

23

35

712

35

12

38

78

1316

34


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Reteaching 4-9 Fractions and Decimals

Example 1: Write 0.320 as a fraction in simplestform.

Read. “320 thousandths”

Write.

Simplify.

0.320 =

Example 2: Write 6.95 as a mixed number insimplest form.

Read. “6 and 95 hundredths”

Write.

Simplify.

6.95 =

Example 3: Write and as decimals.

Divide the numerator by the denominator.Insert zeros if needed.

. . .

= 0.2 = 0.666 . . . =

0.2 is a terminating decimal because there is noremainder.

0.666 . . . is a repeating decimal because theremainder repeats. Write it as .0.6

0.623

15

d The digitsrepeatbecausetheremainder

d repeats.

0.66603q2.000021.8000

200021800

200

0.25q1.0

23

15

61920

6 95100 5 619

203

6 951002

1

825

3201,000 5 320 4 40

1,000 4 40 5 8253

3201,0002

1

Write each decimal as a fraction or mixed number in simplest form.

1. 0.8 2. 0.55 3. 1.25

4. 3.375 5. 0.125 6. 1.32

7. 0.084 8. 0.006 9. 0.65

Write each fraction or mixed number as a decimal.

10. 11. 12.

13. 14. 15.

16. 17. 18.

State whether each fraction is less than, equal to, or greater than 0.50.Show your work.

19. 20. 21.

22. 23. 24. 820

1113

78

16

2040

13

1252

2812

9

711

49

1925

720

16

1320

45

1120 11

4

3 38

18 1 8

2521250

3500

1320


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Estimate each sum or difference. Use the benchmarks 0, , and 1.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21.

Solve.

6 9

11 2 1185

89 1 3

2138

112 2 8

10

3 89 1 4

8911

5 2 9106

18 1 2

910

1314 2 4

71112 2 9

1067 1 2

3

118 1 2

101

10 1 18

59 1 1

15

710 1 3

2467 1 4

54

10 2 215

12 1 1

182125 1 1

658 2 3

7

1516 2 1

99

10 1 45

78 1 1

16

12

Reteaching 5-1 Estimating Sums and Differences

Estimate sums and differences of mixednumbers by rounding to the nearest wholenumber.

Subtract:

So, . So, .

3 � 2 � 1

3 16 2 1 9

10 < 1

1 910 < 23

16 < 3

910 < 11

6 < 0

3 16 2 1 9

10

Estimate sums and differences of fractions byusing a benchmark. A benchmark is a numberthat is close to a fraction and is easy to use whenyou estimate.

Estimate: by using the benchmarks 0, , or 1.

45 1 3

8 < 112

1 1 12 5 11

2

12

45 1 3

8

12

1

38

45

22. Katrina has a -foot roll of ribbon. Sheneeds 2 strips of ribbon that each measure

-feet long. Does she need more ribbon?

23. Ricardo jogs miles on Monday and miles on Wednesday. Estimate the total

number of miles he jogs.2

15

3 34

3 56

7 12

38 < 1

2

45 < 1

112

12

12

12

12

12

12


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Find each sum.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Find each difference.

10. 11. 12.

13. 14. 15.

16. 17. 18.

Find each sum or difference.

19. 20. 21.

22. 23. 24.

25. For school photos, of the students choose to have a bluebackground, of the students choose to have a purplebackground, and of the students choose to have a graybackground. What portion of the students choose to haveanother background color?

15

25

15

6280 2 10

80 2 580

810 2 2

10 2 110

1011 2 Q 2

11 1 411R

25 2 2

5 1 55

10100 1 20

100 1 90100

27 1 2

7 2 17

910 2 4

107

12 2 612

510 2 1

10

46 2 2

68

10 2 610

712 2 1

12

34 2 1

49

10 2 310

68 2 3

8

38 1 6

838 1 9

858 1 1

8

612 1 3

123

10 1 210

610 1 5

10

312 1 3

1246 1 1

615 1 3

5

Reteaching 5-2 Fractions With Like Denominators

Add:

Combinenumerators over the denominator.

Add numerators.

Simplify, if possible.

Subtract:

Combine numerators over the denominator.

Subtract numerators.

Simplify, if possible.

710 2 2

10 5 12

5 123

5 5102

710 2 2

10 5 7 2 210

1

710 2 2

10

16 1 3

6 5 23

5 233

5 462

16 1 3

6 5 1 1 36

1

16 1 3

6

12

56

45

34

121 1

10

11811

234

12

35

38

13

15

12

12

112

25

115

37

4780

12

411

15


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To add or subtract fractions with unlike denominators, you can useequivalent fractions.


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21. 18 1 1

535 2 1

258 1 2

3

38 2 1

326 1 3

478 2 3

10

112 1 1

1035 1 1

678 2 1

4

56 1 1

37

12 2 13

1516 2 1

4

58 1 1

434 2 5

1212 1 7

10

23 1 5

99

10 1 12

78 2 1

2

16 1 1

31116 2 5

1612 1 3

4

Reteaching 5-3 Fractions With Unlike Denominators

Example 1: Find .

Find the least common denominatorof 6 and 2.

The LCD is 6.

Write equivalent fractions using the LCD.

Add. Write the sumin simplest form.

Example 2: Find .

Find the least common denominatorof 5 and 3.

The LCD is 15.

Write equivalent fractions using the LCD.

Subtract. Write thedifference insimplest form.

45 2 1

3 5 715

3

13 5 1 3 5

3 3 5 5 515

45 5 4 3 3

5 3 3 5 1215

2

1

45 2 1

3

56 1 1

2 5 113

3

12 5 1 3 3

2 3 3 5 36

56 5 5

6

2

1

56 1 1

2

5 113

5 43

5 86

5 5 1 36

56 1 1

2 5 56 1 3

6

5 715

5 12 2 515

45 2 1

3 5 1215 2 5

15

12

3811

4

12912

538

78

1311

5

116

14

1116

1160

2330

58

1241 1

122340

1340

1101 7

24


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Find each sum.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18. 8 1

16 1 4 589

16 1 11

36 1 3 25

4 25 1 6

7103

23 1 5

566

34 1 8

45

125 1 9

237

23 1 8

5613

4 1 4 78

9 38 1 2

128

314 1 2

177

29 1 14

9

2 7

10 1 3 157

34 1 2

188

25 1 4

110

2 12 1 411

3 1 3 134

47 1 11

7

Reteaching 5-4 Adding Mixed Numbers

Some mixed numbers can be added mentally.

Example 1: Find .

Add the whole numbers.

5 � 2 � 7

Add the fractions.

Combine the two parts.

Or, you can follow these steps.

Example 2: Find .

Write with a common denominator.

Add the whole numbers.Add the fractions.

Rename as .

4 45 1 2

910 5 7

710

5 7 7

107 710617

103

5 6 17102

4 45 1 2

910 5 4

810 1 2

910

1

4 45 1 2

910

5 14 1 2

18 5 7

38

7 1 38 5 7

38

3

14 1 1

8 5 28 1 1

8 5 38

2

1

5 14 1 2

18

121116101

2925

11 11091

2151120

1115161

2658

117810 5

14823

5 91097

81212

61242

3557


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Reteaching 5-5 Subtracting Mixed Numbers


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18. 4 1

10 2 2 4515

512 2 8

137

25 2 2

14

12 34 2 6

189

512 2 5

346

18 2 3

116

8 25 2 3

235

716 2 11

210 23 2 7

56

7 57 2 4

13146

58 2 2

3412

13 2 9

23

14 13 2 8

148

12 2 3

149

78 2 7

34

6 23 2 2

163

34 2 11

27 7

10 2 2 3

10

Some mixed numbers can be subtracted mentally.

Example 1: Find .

Subtract the whole numbers.

5 � 2 � 3

Then, subtract the fractions.

Combine the two parts.

Sometimes you must rename the first fractionbefore subtracting.

Example 2: Find .

Write with a common denominator.

Rename .

Subtract the whole numbers.Then, subtract the fractions.Simplify, if necessary.

6 12 2 2

34 5 3

34

5 3 343

5 5 64 2 2

346

242

6 12 2 2

34 5 6

24 2 2

34

1

6 12 2 2

34

5 23 2 2

16 5 3

12

3 1 12 5 3

12

3

23 2 1

6 5 46 2 1

6 5 36 5 1

2

2

1

5 23 2 2

16

1 3107 1

125 320

65832

33 116

41115315

16256

2111437

8223

6 11251

4218

41221

4525


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You can use mental math to solve addition and subtraction equationsthat involve fractions or mixed numbers. To solve equations involvingfractions with unlike denominators, you need to change the fractionsto equivalent fractions with like denominators.

Solve .

Add to each side.

x Write the sum.

The LCD is 16. Write as .

Simplify.

Solve each equation.

1.

What number plus equals ? So, x � .

Show that the equation is true.

2.

What is the least common multiple of 3 and 9?

Rewrite the equation using like denominators.

What number minus equals ?

What number minus 3 equals 4? So, x � .

Show that the equation is true.

Solve each equation.

3. x� 4. y�

5. c� 6. r�

7. b� 8. s�

9. d� 10. f�756 2 f 5 7

12d 1 513 5 8 7

12

s 2 12 5 1

61

12 1 b 5 14

512 1 r 5 7

37

10 2 c 5 25

y 2 58 5 1

814 1 x 5 3

4

29

39

x 2 313 5 42

9

45

15

x 1 15 5 4

5

5 21116

616

385 2 5

16 1 616

5 2 516 1 3

8

381 3

81 38

x 2 38 5 2 5

16

x 2 38 5 2 5

16

Reteaching 5-6 Equations With Fractions

45 5 4

535 1 1

5 5 45

35

35

x 2 339 5 42

959

29

29

29

39

59

68 5 3

424 5 1

22312 5 111

12310

146 5 12

32

12 5 16

7 312 5 71

43 312 5 31

4

59


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Find the elapsed time between 6:15 A.M. and 11:10 A.M.

1. Set up as subtraction. 2. Rename 11:10 as 10:70. 3. Subtract.

11:10 11:10 S 10:70 10:70�6:15 �6:15 S �6:15 �6:15

4:55

The elapsed time is 4 hours 55 minutes.

You can find elapsed time from a schedule.

For travel time, find the elapsed timebetween 7:09 A.M. and 11:02 A.M.

11:02 � 7:09 � 3 hours 53 minutes

Find the total time.

1. 25 min � 55 min 2. 39 min � 10 min � 11 min

Estimate each elapsed timme to the nearest hour.

3. from 4:15 A.M. TO 11:00 A.M. 4. from 9:59 A.M. TO 7:46 P.M.

Find the elapsed time.

5. 6:45 P.M. and 9:20 P.M. 6. 9:36 A.M. and 11:50 A.M.

7. 5:45 A.M. and 11:30 A.M. 8. 3:11 P.M. and 10:40 P.M.

9. 8:15 A.M. and 10:09 P.M. 10. 1:00 A.M. and 7:28 P.M.

Use the schedule to answer the following questions.

11. How much time do you haveto get to the game?

12. How long is the game?

Reteaching 5-7 Measuring Elapsed Time

Leave ArriveBoston 7:09 A.M. New York 11:02 A.M.

Leave for game 6:15 P.M.Game begins 7:35 P.M.Game ends 10:20 P.M.


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You can model of .

Show . Divide into thirds. Shade of the .

of

Or you can use multiplication.

of

Write the multiplication problem each model represents.

1. 2.

Find each product.

3. of 4. 5. 6.

7. of 8. 9. of 10.

11. of 8 12. of 2 13. of 4 14.

15. Every day you eat cup of cereal. Your brother eats 5 times as much. How many cups of cereal does your brother eat?

14

34 ? 2

559

13

29

34 3 1

9710

38

56 3 3

413

710

34 ? 4

758 ? 62

7 3 12

23

19

5 16

5 212

5 2 3 13 3 4

14 5 2

3 3 14

23

14 5 2

12 5 16

23

14

23321

41

14

23

Reteaching 6-1 Multiplying Fractions

12 3 2

3 5 26 5 1

334 3 1

2 5 38

114

31022

92317

9

112

2180

58

730

3733

417

227


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Find each product.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20. 316 3 11

71 310 3 26

7123 3 71

2325 3 21

2

17 3 13

5125 3 41

3123 3 31

2313 3 3 3

10

17 3 13

414 3 13

5514 3 22

7156 3 21

4

217 3 42

312 3 33

5513 3 17

8218 3 22

3

15 3 27

9317 3 24

5225 3 41

2114 3 22

3

Reteaching 6-2 Multiplying Mixed Numbers

3

1

d 3 � 12d 7 � 1

217 3 22

5 5 517

517

367

157 3 12

5

157 3 12

5

217 3 22

5 Multiply:

1 7

d 1 � 7d 1 � 2

23 3 51

4 5 312

312

72

23 3 21

4

23 3 21

4

23 3 51

4

31435

7121281

2

8356 1

15556

14

2541

8

145

59

84584

5104531

3

Solve.

21. Estimate the area of a window pane that hasdimensions by inches.111

4618

22. A hamster is inches long. A rabbit is times as long as the hamster. How long is

the rabbit?31

2

212

34

1 2

Change to improperfractions.

Simplify.

Multiply.

Simplify.4

3

2

1

Example 1: Multiply: Example 2:


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Find .

The reciprocal of is .

Multiply 8 by the reciprocal.

Find .


Multiply by the reciprocal.

49 4 8

15 5 56

49 4 8

15 5 49 3 15

8 5 49 3 15

8 5 1 3 53 3 2 5 5

6

492

158

815

158

8151

49 4 8

15

8 4 45 5 10

8 4 45 5 8 3 5

4 5 81 3 5

4 5 2 3 51 3 1 5 10

2

54

45

54

451

8 4 45

Reteaching 6-3 Dividing Fractions

Write the reciprocal of each number.

1. 2. 3.

4. 5. 14 6. 18

7. 8. 9.

10. 11. 12.

Find each quotient.

13. 14. 15.

16. 17. 18.

19. 20. 21.

22. 23. 24.

25. 26. 27. 58 4 103

5 4 5910 4 3

25 4 3

412 4 47

8 4 23

47 4 27

8 4 79

45 4 4

7

14 4 565 4 2

36 4 25

9 4 347 4 7

82 4 23

315

27

1112

97

311

59

89

120

53

14

116

325

310

815

181 5

16

2711

8125

164571

2

153

72

1211

79

113

95

118

114

98

35

2

1

1 5

3 2


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Reteaching 6-4 Dividing Mixed Numbers

Example 1: Estimate .

Round mixed numbers tonearest whole number.

36 � 6 = 6 Find the quotient of therounded values.

Example 2: Find .

Write each mixed number as an improperfraction.


Multiply by the reciprocal.

513 4 22

5 5 229

163 4 12

5 5 163 3 5

12 5 4 3 53 3 3 5 20

9 5 229

1633

512

125

512

1252

513 4 22

5 5 163 4 12

5

1

513 4 22

5

3613 4 57

8

3613 4 57

8

Estimate each quotient.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Find each quotient.

13. 14. 15.

16. 17. 18.

19. 20. 21. 1 110 4 15

61829 4 11

21023 4 41

3

116 4 22

37512 4 51

2618 4 22

4

323 4 11

210018 4 61

4212 4 1

4

578 4 21

2523 4 13

4612 4 21

6

2 710 4 4

55 4 11512 4 61

2

9 4 31361

4 4 21251

3 4 223

5023 4 26

71923 4 38

91489 4 51

5

3512 4

272 613

71613 8

112 920

249

4

3


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Name Class Date

art t/k

art t/k

When solving multiplication equations, it may help to first find thenumerator of the missing value and then the denominator. If theequation includes whole numbers or mixed numbers, you may need torewrite these numbers as fractions.

Solve: .

Think: What number times equals ?

Then use mental math to find the numerator.

Use mental math to find the denominator.

Check to see that the equation is true. ✓

So, .

Solve each equation. Check the solution.

1. 2.

x = x =

3. 4.

x = x =

5. 6.

x = x =

7. 8.

x = x =

54x 5 35

4883x 5 16

27

95x 5 9

2532x 5 6

10

38x 5 18

1613x 5 4

34x 5 9

2023x 5 8

15

x 5 25

25 3 2

5 5 4254

25 3 2

? 5 4253

25 3 ?

? 5 4252

425

251

25x 5 4

25

Reteaching 6-5 Solving Fraction Equations by Multiplying

712

29

15

25

62 5 3

35

45

9. A wrestler weighed 112 pounds before the state meet. After the meet, the wrestlerweighed of his original weight. How muchdid the wrestler weigh after the meet?

10. A number divided by 4 equals 10 . What isthe number?

18

4950

12

14

12


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Choose an appropriate customary unit of measure to describe the following:

length of a train engine weight of a train engine

You need a unit to measure length. You need a unit to measure weight.A train engine will be quite long, Since a train engine will be quite heavy,so choose feet or yards. choose tons.

amount of liquid in a large bucket length of a CD case

You need a unit to measure capacity. You need a unit to measure length.A bucket is likely to contain quite A CD case is quite small, so a bit of water, so choose quarts or gallons. choose inches.

weight of a bale of straw amount of liquid in a bottle of eye drops

You need a unit to measure weight. You need a unit to measure capacity.A bale of straw is heavy, so choose pounds. Since a bottle of eye drops will be very

small, so choose fluid ounces.

Choose an appropriate unit for each measurement. Explain.

1. the length of a garden 2. the length of a hummingbird

Choose an appropriate unit for each weight. Explain.

3. the weight of a letter 4. the weight of steel girders

Choose an appropriate unit for each capacity. Explain.

5. a pitcher of juice 6. the water in an aquarium

Compare using R, N, or S.

7. weight of a tank 100 pounds 8. length of a TV remote 5 inches

Reteaching 6-6 The Customary System


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Complete the statement: c =9 fl oz

Find the relationship between cups and fluidounces: 1 c = 8 fl oz

Since there are 8 fluid ounces in each cup,multiply the number of cups by 8.

= 45

45 fl oz

Subtract: 9 ft 8 in. - 2 ft 11 in.

Find the relationship between feet andinches: 1 ft = 12 in.

Use the relationship to rename9 feet 8 inches. as 8 feet 20 inches.

Subtract.

9 ft 8 in. 8 ft 20 in.- 2 ft 11 in. - 2 ft 11 in.

6 ft 9 in.

9 ft 8 in. - 2 ft 11 in. = 6 ft 9 in.

3

2

1

558 c 5

5 458 3 8

1

558 3 8 5 45

8 3 8

2

1

558

Reteaching 6-7 Changing Units in the Customary System

To compare amounts, first change them to the same unit.

Compare: 25 fl oz 9 3 c 25 fl oz 9 24 fl oz25 fl oz > 24 fl oz25 fl oz > 3 c

Complete each statement.

1. 12 ft =9 yd 2. 32 qt =9 gal 3. mi =9 ft

4. 15 pt =9 qt 5. 440 yd =9 mi 6. T =9 lb

Add or subtract.

7. 3 pt 1 c 8. 4 yd 1 ft 9. 5 lb 20 oz+ 4 pt 1 c - 1 yd 2 ft + 8 lb 12 oz

Use R, �, or S to complete each statement.

10. 43 in. 4 ft 11. gal 136 c 12. 108 in. yd

13. lb 40 oz 14. 7,000 lb t 15. pt 3 qt

16. A semi-truck can hold 8,500 pounds of cargo. How many tons can

it hold?

51231

4212

31281

2

212

112

1471

2

414

1

1

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Write three different ratios equal to each ratio.

1. 2. 1 : 3 3. 3 to 4 4. 5 : 8

5. 2 to 7 6. 7. 12 to 20 8. 6 : 16

Write each ratio in simplest form.

9. 32 : 16 10. 11. 12. 60 : 25

13. 14. 60 : 180 15. 16. 80 : 20

Find the value that makes the ratios equal.

17. 3 : 4, 9 : 16 18. 20 to 25, 40 to 9

19. 7 : 10, 9 : 100 20. 1 to 8, 9 to 24

21. 5 : 100, 25 : 9 22. , 9280756

75120

2540

3650

1424

15

25

Reteaching 7-1 Ratios

A ratio is a comparison of two numbers bydivision. Each number in a ratio is called a term.You can write a ratio in three different ways.For example, the ratio 4 to 5 can be written:

4 to 5

4 : 5

Equal ratios name the same number. They havethe same simplest form.

• To find equal ratios, multiply or divide boththe numerator and denominator of a ratio bythe same number.

Find a ratio equal to .

is equal to .

Find the simplest form for the ratio .

is the simplest form for .1620

45

1620 5 16 4 4

20 4 4 5 45

1620

47

814

47 5 4 3 2

7 3 2 5 814

47

45

58

58

1825

712

2464

1848

1232

4880

3660

2440

420

315

210

828

621

414

2032

1524

1016

1520

912

68

412

39

26

820

615

410

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Reteaching 7-2 Unit Rates

A rate is a ratio that compares quantities that are measured indifferent units. Suppose a sprinter runs 100 yards in 10 seconds.

compares yards to seconds.

A unit rate compares a quantity to one unit of another quantity.

You can find the unit rate by dividing by the denominator.

10 yards per second is the sprinter’s unit rate.

Find the unit price for each situation.

1. $70 for 10 shirts 2. $150 for 3 games 3. $20 for 5 toys

4. $120 for 6 shirts 5. $45 for 5 boxes 6. $132 for 3 books

7. $100 for 5 rackets 8. $56 for 7 hours 9. $1.98 for 6 cans

Write the unit rate as a ratio. Then find an equal ratio.

10. The cost is $4.25 for 1 item. Find the cost of 5 items.

11. There are 7 cheerleaders in a squad. Find the number of cheerleaders on 12 squads.

12. The cost if $10.10 for 1 item. Find the cost of 10 items.

13. There are 2.54 centimeters per one inch. Find the number of centimeters in 5 inches.

For Exercises 14–16, tell which unit rate is greater.

14. Dillan scores 24 points in 2 games. Eric scores 40 points in 4 games.

15. A fern grows 4 inches in 2 months. A tree grows 6 inches in 4 months.

16. Tyler jogs 4 miles in 32 minutes. Joey jogs 2 miles in 18 minutes.

100 yd 4 1010 s 4 10 5

10 yd1 s

100 yd10 s

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Reteaching 7-3 Understanding Proportions

Do the ratios in each pair form a proportion?

1. , 2. , 3. ,

4. , 5. , 6. ,

7. , 8. , 9. ,

Find the value that completes each proportion.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. A basketball player bounces the ball one time for every threesteps. How many times will the player bounce the ball fortwelve steps?

20. Four laps around the track equals one mile. How many milesdoes sixteen laps equal?

189

5 29

459 5 25

9

630 5

9

90

9

25 5 1450

119

5 3315

29

5 1050

921 5

9

78

16 5 49

610 5 3

9

13

824

1920

910

53

35

2430

1824

4977

711

366

549

4550

3540

45

1520

416

25100

A proportion is an equation stating that two ratiosare equal.

Does ?

First, simplify each fraction.

, and

So, .

You can use mental math to solve proportions.

Use mental math to solve .

3 � 5 � 15, so 5 � 5 � 25

So, .35 5 15

25

35 5 15

9

410 5 6

15

615 5 2

5410 5 2

5

410 5 6

15

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Does each pair of ratios form a proportion?

1. , 2. , 3. ,

4. , 5. , 6. ,

7. , 8. , 9. ,

Solve each proportion.

10. 11. 12.

13. 14. 15.

16. 17. 18. 1114 5 33

y6024 5 w

1256 5 h

36

1832 5 27

m1612 5 r

18t

24 5 38

306 5 a

99n 5 27

3n5 5 2

10

810

4050

1421

47

17

568

1920

910

1520

1216

2560

1545

35

68

104

52

814

47

Reteaching 7-4 Using Cross Products

If two ratios are equal, they form a proportion.

Equal ratios have equal cross products.

5 � 2 � 101 � 10 � 10

Equal cross products also show that a proportionis true.

6 � 3 � 181 � 18 � 18

The cross products are equal, so the ratios areequal and form a proportion.

You can find the missing term in a proportion byusing cross products.

Solve .

Write the cross products. 4 � n = 7 � 12

Simplify. 4n = 84

Divide by 4.

Simplify. n = 214

4n4 5 84

43

2

1

47 5 12

n

16 5 3

18

15 5 2

10

15 5 2

10

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The scale drawing at the right shows a game field at Weld MiddleSchool. The scale is a ratio that compares length on a drawing to theactual length. Here, every inch equals 36 yards on the actual field.

You can write the scale as a ratio in fraction form:

To find the actual length of the field:

Measure the scale drawing. 3 in.

Write the scale as a ratio.

Use the scale ratio in a proportion.

Write cross products. 1 � n = 3 � 36

Solve for n. n = 108

The actual length is 108 yards.

Use the scale drawing above to find the actual size.

1. Find the width of the field. 2. Find the perimeter of the field.

3. Find the measure of the shorter side of 4. Find the distance from the center spot to the penalty area. the front of the goal.

5. Brian kicks the ball from the penalty kick 6. Kaitlin makes a direct kick from the spot line to the opposite goal area. About how marked X. She scores by getting the ball into far does he kick the ball? the goal nearest her. About how far does she

kick the ball?

Write each scale as a ratio.

7. a 12-inch model of a 60-foot boat 8. a 6-inch drawing of an 18-inch TV

9. a 4-centimeter model of a 10. a 9-inch drawing of a 54-foot garden28-centimeter hammer

5

4

136 5 3

n3

1362

1

drawing (in.)actual (yd) 5 1

36

Reteaching 7-5 Scale Drawings

scale: 1 in. : 36 yd

Goal

X

Goal Area

HalfwayLine

PenaltyKick Line

Z

Penalty KickCircle

CenterCircle

CenterSpot

PenaltyArea

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Write each percent as a decimal and as a fraction in simplest form.

1. 30% 2. 14% 3. 16% 4. 5%

5. 92% 6. 80% 7. 21% 8. 38%

Write each fraction or decimal as a percent.

9. 10. 0.85 11. 0.16 12.

13. 14. 15. 0.64 16. 0.008

17. 18. 19. 0.32 20. 0.07

21. 22. 23. 0.010 24. 0.604550

13100

615

920

110

7200

540

1725

Reteaching 7-6 Percents, Fractions, and Decimals

• To write a percent as a fraction in simplestform, first write a fraction with a denominatorof 100. Then simplify.

74% �

• To write a percent as a decimal, first write afraction with a denominator of 100. Then writethe decimal.

74% � � 0.74

• To write a decimal as a percent, move the deci-mal point two places to the right.

0.23 � 23%

Here are two ways to write a fraction as a percent.

• Write an equivalent fraction with a denominator of 100, then write the percent.

� 15%

• Divide the numerator by the denominator.

� � 37.5%

So, � 37.5%.38

0.3758q3.00022.400

6002.560

402.40

0

38

320 5 15

100

74100

74100 5 37

50

cMove the decimal pointtwo places to the right.

1950

21100

45

2325

120

425

750

310

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Find each answer using mental math.

1. 45% of 60 2. 60% of 160

3. 90% of 80 4. 35% of 60

Find each answer using a proportion.

5. 40% of 60 6. 85% of 300

7. 22% of 500 8. 37% of 400

Find each answer.

9. 25% of 100 10. 70% of 70

11. 75% of 40 12. 80% of 50

13. 24% of 80 14. 45% of 90

Reteaching 7-7 Finding a Percent of a Number

You can find 70% of 90 using different methods.

Use mental math.

Write the percent as a fraction in simplestform.

70% �

Multiply by the fraction.

70% of 90 � 63.

Use a proportion.

Write a proportion.

Write cross products and simplify.

100 � c � 70 � 90100c � 6,300

Solve.

c �

c � 63

70% of 90 � 63.

6,300100

3

2

70100 5 c

90

1

710 3 90

1 5 63010 5 63

2

70100 5 7

10

1

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Reteaching 7-8 Circle Graphs

A circle graph is a graph of data where theentire circle represents the whole. Each wedgein the circle represents part of the whole.The graph at the right shows that 27 people inthe survey speak English at home.

Use the circle graph above for Exercises 1–4.

1. Which 2 languages did most people 2. How many people spoke Polish?surveyed speak?

3. How many more people spoke 4. Did more people speak Polish or Spanish than Polish? Chinese?

Sketch a circle graph for the given percents.

5. 65%, 35% 6. 26%, 62%, 12%

7. 10%, 90% 8. 30%, 50%, 20%

Shade the circle graphs with a section equal to the given fraction.

9. 10. 23

14

Languages Spoken at Home

English27

Polish12

Spanish24

Chinese6

Others3

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You can estimate a percent of a number using mental math.

Example: Estimate 19% of $83.

Round to convenient numbers.

20% of 80

Find 10% of 80.

10% of 80 � 8.

20% of 80 is 2 times as much.

20% of 80 is 2 � 8, or 16.

19% of 83 is about 16.

Estimate each amount.

1. 50% of 41 2. 20% of 99 3. 40% of 59

4. 1% of 94 5. 70% of 498 6. 15% of 172

7. 90% of 81 8. 30% of 60 9. 40% of 23

10. 20% of 178 11. 25% of 216 12. 50% of 77

13. A baseball glove is on sale for 75% off the original price of$96.25. Estimate the sale price of the glove.

3

2

1

Reteaching 7-9 Estimating With Percents

2


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Read each statement. Write true or false.

1. A line has two endpoints. 2. A plane has only two points.

3. A segment is part of a line. 4. A plane is flat.

5. A ray has no beginning or end. 6. A plane contains only one line.

7. Parallel segments do not intersect. 8. Skew lines intersect.

Match each figure with its name.

9. 10.

11.12.

Reteaching 8-1 Points, Lines, Segments, and Rays

Each point F, G, and H,indicates an exact locationin space.

Line KM ( ) is a seriesof points that extends intwo opposite directionswithout end.

and are parallellines. They are in the sameplane but do not intersect.They have no points incommon.

*UV)*

ST)

*KM

)

Plane FGH is flat andextends indefinitely assuggested by the arrows.

Segment LM ( ) is partof . The points L andM are endpoints of .

Ray LM ( ) is part of aline. Point L is its onlyendpoint.

Points on the same lineare collinear. Points S andT are collinear.

Skew lines are neitherparallel nor intersecting.

LM)

LM

*KM

)LM

F

H G

K L M

S T

U V

a. ray

b. plane

c. line

d. segment

Reteaching Course 1 Lesson 8-2 245

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You can classify an angle according to its measure in degrees.

Classifying can help you estimate and measure the size of &RST.

&RST is an obtuse angle.It is greater than 90� but less than 180�.A good estimate is 135�.

To find the measure of &RST:

Extend the sides so they reach the scales on the protractor.

Place the protractor over the angle as shown. is on zero.

Read the inner scale.

The measure of &RST is 130�. Since &RST is obtuse, it cannot be 50�.

Classify each angle as acute, right, obtuse, or straight.

1. 2. 3.

Use a protractor to measure each angle.

4. 5. 6.

3

ST)

2

1

Reteaching 8-2 Angles


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Find the value of x in each figure.

1. 2. 3.

Use the diagram at the right. Complete each sentence with complementary,supplementary, or congruent. Some exercises use more than one word.

4. Angles 1 and 2 are

.


.


.


.


.


.

90°x°65° x°35°

x°

Reteaching 8-3 Special Pairs of Angles

Complementary angles:sum of measures = 90�.

Example 1: Find thecomplement of &HFG.

x + 65� = 90�

x = 25�

&EFH has a measure of 25�.

Supplementary angles:sum of measures = 180�.

Example 2: Find thesupplement of &USR.

x + 130� = 180�

x = 50�

&RST has a measure of 50�.

Congruent angleshave the same measure.

Example 3: &ABC and &BDEhave the same measure; theyare congruent.

E

CA

B

DU S T

R

130°

HE

65°

F G

8

34

5

7

216

91213 11

1014


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Triangles can be classified by the measures of their angles.

Triangles can be classified by the number of congruent sides.

Classify each triangle as acute, right, or obtuse.

1. 2. 3. 4.

Classify each triangle by its angles.

5. 90�, 40�, 50� 6. 38�, 72�, 70� 7. 115�, 30�, 35� 8. 70�, 60�, 50�

Classify each triangle by its sides.

9. 10. 11. 12.

Reteaching 8-4 Classifying Triangles


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A polygon is a closed figure formed by three or more line segments that do not cross.

Polygons can be named according to the number of sides.

Write all the possible names for each quadrilateral. Choose fromparallelogram, rectangle, rhombus, square, and trapezoid.

1. 2. 3. 4.

Identify each polygon according to the number of sides.

5. 6. 7. 8.

9. 10. 11. 12.

Reteaching 8-5 Exploring and Classifying Polygons


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For each triangle, tell whether it is congruent to triangle RST.

1. 2. 3.

Tell whether the figures are congruent or similar. If necessary,measure angles with a protractor.

4. 5. 6.

4 7

7

7 7

4

4 4

6

6

9 3 4.5

3

3

3

3 3 3 3

33

A C

B

3 5

4

D F

E

10 10

4

N

P

O

46

8

R T

S

4 6

8

Reteaching 8-6 Congruent and Similar Figures

Congruent figures have the same size and shape.Matching sides and matching angles arecongruent. These are corresponding parts. Here,quadrilaterals ABCD and EFGH are congruent.

Similar figures have the same shape, but may not bethe same size. They have congruent correspondingangles and proportional corresponding sides. Here,triangles RST and UVW are similar.

R

U4

2

6

3

8 4ST VW

A

D

E

HC

B

G

F

4

3 3

66

2 2

4


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A figure has line symmetry if you can fold it in half so that the twohalves match exactly. The line is called a line of symmetry.

This figure has line symmetry. This figure does not have line symmetry.Trace the figure and the line through it. Trace the figure and the line through it.Cut out the figure and fold it on the line. Cut out the figure and fold it on the lineThe two halves match exactly. (or on any other line). The two halves do

not match exactly.

Some figures have many lines of symmetry. Draw a circle and try tofind all the lines of symmetry.

Does the figure have line symmetry? Write yes or no. If yes, trace thefigure and draw all the lines of symmetry.

1. 2. 3.

4. 5. 6.

Can you find a line of symmetry for each word? Write yes or no. Ifyes, copy the word and draw the line of symmetry.

7. DAD 8. HAH 9. DAY 10. COB

Reteaching 8-7 Line Symmetry


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In a translation, or slide, every point of a figuremoves the same distance and in the samedirection.

In a reflection, or flip, a figure is flipped across a line. The new figure is a mirror image of theoriginal figure.

In a rotation, a figure is turned, or rotated about a point. You can describe a rotation in terms of degrees. The triangle has been rotated 90�

clockwise.

Draw a translation of each triangle.

1. 2.

Copy each triangle. Draw its reflection over the given line.

3. 4.

Circle all rotations of the first shape. State the number of degrees youmust rotate the shape.

5. a. b. c.

Reteaching 8-8 Exploring Transformations


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The standard unit of length in the metric system is the meter.

A length can be named using different metric units. Thepoint marked on the ruler is 2.7 centimeters.

Since each centimeter is 10 millimeters, the point is also27 millimeters.

In the metric system, solids can be measured in units of mass.

The standard unit of mass is the gram.• The mass of a vitamin pill may be measured in

milligrams.• A thumbtack has a mass of about 1 gram.• A full liter bottle of soda has a mass of about

1 kilogram.

Liquids are measured in units of capacity.

The standard unit of capacity is the liter.• The capacity of a soup spoon is measured in milliliters.• A 1-liter soda bottle can fill about four average-sized

glasses.• Water in a river is measured in kiloliters.

Choose an appropriate metric unit of length.

1. distance across the end of a pencil

2. length of a thumb

3. distance from your home to Australia

4. width of a swimming pool

Choose an appropriate metric unit of mass.

5. the mass of a tooth

6. the mass of a puppy

Choose an appropriate metric unit of capacity.

7. the capacity of a bucket of water

8. the amount of water in a pond

1

centimeters

2 3 4 5

Reteaching 9-1 Metric Units of Length, Mass, and Capacity

millimeter (mm) � 0.001 metercentimeter (cm) � 0.01 metermeter (m) � 1 meterkilometer (km) � 1,000 meters

milligram (mg) � 0.001 gramgram (g) � 1 gramkilogram (kg) � 1,000 grams

milliliter (mL) � 0.001 literliter (L) � 1 literkiloliter (kL) � 1,000 liters


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Multiply to convert from larger units tosmaller units.

Convert 4.7 kilometers to meters.

• A kilometer is larger than a meter. Multiply.

• Since 1 km � 1,000 m, multiply by 1,000.

4.7 � 1,000 � 4,7004.7 km � 4,700 m

• Or use mental math. Multiply by 1,000 bymoving the decimal point three places tothe right.

4.7 S 4,700

Divide to convert from smaller units tolarger units.

Convert 347 milliliters to liters.

• A milliliter is smaller than a liter. Divide.

• Since 1,000 mL � 1 L, divide by 1,000.

347 � 1,000 � 0.347347 mL� 0.347 L

• Or use mental math. Divide by 1,000 bymoving the decimal point three places tothe left.

347 S 0.347

Reteaching 9-2 Converting Units in the Metric System

Convert each measurement to meters.

1. 2.5 km 2. 371 cm 3. 490 mm

4. 48 cm 5. 4 km 6. 1,500 mm

Convert each measurement to liters.

7. 0.6 kL 8. 799 cL 9. 0.9 mL

10. 35.6 mL 11. 0.006 kL 12. 1.8 cL

Convert each measurement to grams.

13. 4 kg 14. 661 cg 15. 1,500 mg

16. 2 cg 17. 1.95 kg 18. 2.3 mg

Convert each measurement.

19. 19 mL � L 20. 5.5 kg � g 21. 4.9 cL � L

22. 730 mg � g 23. 0.06 kL � L 24. 2,540 mm � cm

The most common metric units use the prefixes kilo-, centi-, and milli-.

Prefix Meaning Examples

kilo- 1,000 kilometer (1,000 m), kilogram (1,000 g), kiloliter (1,000 L)

centi- or 0.01 centimeter (or 0.01 m), centigram (or 0.01 g), centiliter (or 0.01 L)

milli- or 0.001 millimeter (or 0.001 m), milligram (or 0.001 g), milliliter (or 0.001 L)11,000

1100


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Perimeter

The perimeter of a figure is the sum of thelengths of its sides. Opposite sides of a rec-tangle are equal. To find the perimeter, add the2 lengths (�) and the 2 widths (w).

P � � � � � w � w or P � 2� � 2w

Find the perimeter.

P � 2� � 2w� 2(14) � 2(9)� 28 � 18 � 46

The perimeter is 46 centimeters.

Area

The area of a figure is the number of squareunits needed to cover the figure. To find thearea of a rectangle, multiply the length (�) andthe width (w).

A � � � w

Find the area.

A � � � w� 6 � 5� 30

The area is 30 square meters.

6 m

5 mw

�

14 cm

�

9 cmw

Reteaching 9-3 Perimeters and Areas of Rectangles

Estimate the area of each figure. Each square represents 1 square inch.

1. 2. 3.

Find the perimeter and area of each rectangle or square.

4. � � 12 cm, w � 2 cm 5. � � 9 ft, w � 7.5 ft

6. � � 2.5 m, w � 2.5 m 7. � � 5.5 in., w � 5.5 in.

8. � � 6.2 in., w � 3.4 in. 9. � � 4.5 ft, w � 0.75 ft

10. What is the area of a square with a perimeter of 60 meters?

12


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Parallelogram

To find the area of a parallelogram, multiplybase times height.

Find the area of the parallelogram.

A � b � h� 3 � 6� 18

The area is 18 square centimeters.

Triangle

The area of a triangle is times the base timesthe height.

Find the area of the triangle.

A � � b � h

� � 3 � 6

� 9

The area is 9 square centimeters.

12

12

b � 3 cm

h � 6 cm

A 5 12 b 3 h

12

b � 3 cm

h � 6 cm

A 5 b 3 h

Reteaching 9-4 Areas of Parallelograms and Triangles

Find the area of each parallelogram.

1. b � 6 ft, h � 8 ft 2. b � 12 in., h � 9 in. 3. b � 6 yd, h � 12 yd

4. b � 2.8 in., h � 3.4 in. 5. b � 31 yd, h � 19 yd 6. b � 4.5 m, h � 4.5 m

7. b � 15 cm, h � 7 cm 8. b � 8.3 ft, h � 11.7 ft 9. b � 14.4 m, h � 6.5 m

Find the area of each triangle.

10. b � 8 cm, h � 14 cm 11. b � 7 in., h � 18 in. 12. b � 11 m, h � 4.6 m

13. b � 6.4 ft, h � 3.5 ft 14. b � 104 in., h � 55 in. 15. b � 5.9 cm, h � 4.2 cm

16. b � 1.7 m, h � 3.3 m 17. b � 5.8 yd, h � 5.8 yd 18. b � 8.6 in., h � 0.8 in.


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Parts of a Circle

Point O is the center of the circle.AB—

is a diameter.OA—

is a radius. OP—

is also a radius.

In any circle, the length of the diameter is twicethe length of the radius.

d = 2r

The radius is half the diameter.

r =

CD—

and EF—

are chords.A diameter of a circle is the longest chord.

Circumference of a Circle

Circumference is the distance around a circle.

To find circumference:

• Multiply p times the diameter.

C = pd

• Or multiply p times twice the radius.

C = 2pr

To estimate the circumference of a circle,use 3 for p.

Estimate the circumference of a circle.

C � 3d

= 3 � 8

= 24

The circumference is about 24 centimeters.

8 cm

C

DO

FE

d2

P

OBA

Reteaching 9-5 Circles and Circumference

List each of the following for circle Q.

1. one diameter 2. three chords 3. three radii

Find the unknown length for a circle with the given dimension.

4. r = 8 cm 5. d = 110 in. 6. d = 48 ft

d = r = r =

Use 3 for π to estimate the circumference of a circle with the givenradius or diameter.

7. r = 12 in. 8. d = 15 yd 9. d = 7 m

10. d = 13 ft 11. r = 21 yd 12. r = 19 cm

T

XQ

YZ

QYQXQTTYZYTZTY


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The formula for the area of a circle is:

Area = p � radius � radius

A = p � r � r

A = pr2

Find the area of the circle.

A = pr2

= p � 42

= p � 4 � 4= p � 16� 50.27

The area is about 50.27 square meters.

Use for p when the radius or diameter of acircle is a multiple of 7 or a fraction.

Find the area of the circle.

d = 14 m, so r = 7 mA = pr2

� � 72

= � 7 � 7

= � 7 � 7= 154

The area is about 154 square meters.

227

227

227

d � 14 m

227

r � 4 m

Reteaching 9-6 Area of a Circle

Find the area of each circle. Round to the nearest tenth or use for π where appropriate.

1. 2. 3.

4. r = 8 cm 5. r = 13 in. 6. d = 28 m 7. d = 24 ft

8. r = m 9. d = 13 cm 10. r = ft 11. d = 29 in.

Find the area of each circle to the nearest tenth. Use 3.14 for π.

12. d = 23 ft 13. r = 2.5 yd 14. d = 48 in. 15. r = 19 cm

1 31131

2

12 m2 m3 m

227

1

1


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Prisms and pyramids are three-dimensional figures. Their parts havespecial names.

• Faces—flat surface on a prism or pyramid

• Edge—segment where two faces meet

• Vertex—point where edges meet

Prisms and pyramids can be named by the shape of their bases.

edge

base vertex

face

Prism

• has two bases congruent and parallel toone another

6 faces12 edges8 vertices

The bases are rectangles.This prism is a rectangular prism.

Pyramid

• has one base; other faces are triangles

5 faces8 edges5 vertices

The base is a square.This pyramid is a square pyramid.

base

top base

Name each three-dimensional figure.

1. 2.

3. 4.

5. How many faces, edges, and vertices does a pentagonal prism have?

Reteaching 9-7 Three-Dimensional Figures and Spacial Reasoning


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The surface area of a rectangular prism is the sum of the areasof the faces. You can use nets to find surface area.

Find the surface area of the prism.

area of top = area of bottomarea of front = area of back

area of right side = area of left side

Find the area of Find the area of Find the area ofthe top. the front. the right side.A = � � w A = � � h A = w � h= 14 � 6 = 14 � 4 = 6 � 4= 84 cm2

= 56 cm2= 24 cm2

Add.84 + 84 + 56 + 56 + 24 + 24 = 328

The surface area of the prism is 328 square centimeters.

Find the surface area of each prism.

1. 2. 3.

4. 5. 6.

5.8 mm

6.2 mm

6.2 mm

20.2 mm

5.78 in.23.7 in.

6.92 in.9.2 cm

11.1 cm7.5 cm

5.3 cm

6 cm

8 cm

10 cm

10 cm2 cm2 cm

6 cm

4 m

4 m

4 m

4

321

� � 14 cm

h � 4 cmw � 6 cm

Reteaching 9-8 Surface Areas of Prisms

14 cm

4 cm

6 cm

Top

Front

Bottom

Back

Rightside

Leftside

6 cm

4 cm


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Volume is the number of cubic units needed to fill the space inside a three-dimensional figure. It ismeasured in cubic units.

Find the volume of the rectangular prism.

Volume = Area of base � height

V = B � h= � � w � h= 8 � 6 � 4= 192

The volume is 192 cubic centimeters.

Find the volume of each rectangular prism.

6 cm

8 cm

4 cm

Reteaching 9-9 Volumes of Rectangular Prisms

1. 2. 3.

4. 5. 6.

Find the volume of each rectangular prism with the given dimensions.

7. � = 6 in., w = 9 in., h = 3 in. 8. � = 3.5 cm, w = 1.5 cm, h = 7 cm

9. � = 16 mm, w = 18 mm, h = 2.5 mm 10. � = 5 m, w = 6.2 m, h = 3.9 m

28.93 mm

63.65 mm

31.22 mm

3.9 cm

18.7 cm

4.2 cm

2.3 m

0.7 m3.1 m

10 cm6 cm

3 cm

15 m

6 m

2 m4 cm

4 cm

4 cm


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Reteaching 9-10 Surface Areas and Volumes of Cylinders

Find the surface area and volume of each of the following cylinders.Round to the nearest whole unit.

1. 3.

Surface area: Surface area:

Volume: Volume:

2. 4.

Surface area: Surface area:

Volume: Volume:

2.66 ft

4.82 ft

6 in.

16 in.

5.8 cm

9.3 cm

12 m

12 m

The surface area of a cylinder is the sum of theareas of the faces. A cylinder has two circularfaces and a rectangular face.

You can find the surface area of a cylinder usingthe following formulas:

�r2 = area of circlel × w = area of a rectangle

2(�r2) + (l × w) = surface area of a cylinder

In a cylinder, the width of the rectangular face isequal to the circumference of the circular base,or 2�r. The length is the height of the cylinder,or h. Substitute these variables into the formula:

SA = 2(�r2) + (2�rh)

The volume of a cylinder is calculated bymultiplying the area of one of the circular basesby the height of the cylinder.

V = �r2 × h


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Your choices for your new car are an exterior color of white, blue, orblack, and an interior of fabric or leather.

Draw a tree diagram to find the number of possible outcomes. Show your work.

1. Marva can have a small, medium, or large salad. She can haveItalian, French, or Russian dressing on it. How many choices doesshe have?

2. You flip a coin three times. How many possible outcomes arethere?

Use the counting principle to find the total number of outcomes.

3. There are 4 kinds of fruit, 2 kinds of 4. There are 4 choices for skis, 2 choicescereal, and 2 kinds of milk. How many for bindings, and 5 choices for boots.ways can a bowl of cereal, fruit, and How many ways can skis, bindings,milk be chosen? and boots be chosen?

5. There are 3 pairs of jeans, 2 vests, and 6. There are 10 yogurt flavors, 4 syrups, and5 shirts. How many ways can jeans, a 5 toppings. How many ways can one flavor,vest, and a shirt be chosen? one syrup, and one topping be chosen?

Reteaching 10-1 Tree Diagrams and the Counting Principle

A tree diagram shows all possible choices. Eachbranch shows one choice.

The tree diagram shows 6 possible outcomes.

You can use the counting principle to find thetotal number of choices.

Exterior Interior Totalchoices choices

3 � 2 = 6

There are 6 possible choices for your car.

When there are m ways of makingone choice and n ways of making asecond choice, then there are m � nways to make the first choicefollowed by the second choice.blue

black

white

interior

fabricleather

fabricleather

fabricleather

outcome

white fabricwhite leather

blue fabricblue leather

black fabricblack leather

color


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The probability of an event is a number that describes how likely it isthat the event will occur. When the outcomes are equally likely, theprobability of an event is the following ratio.

P(event) �

Find the probability of choosing the red chipif the chips are placed in a bag and mixed.

P(red) � =

The probability of choosing the red chip is .

• If an event is impossible, its probability is 0. The probability ofdrawing an 11 from cards numbered 1 to 10 is impossible.

• If an event is equally likely, its probability is or 0.5. The

probability that you will draw an even-number card from cards

numbered 1 to 10 is equally likely.

• If an event is certain, its probability is 1. The probability that youwill draw a card from 1 to 10 from a set of cards numbered 1 to 10is certain.

Find the probability of each event.

1. You pick a vowel at random from 2. You pick a weekend day at random the letters in EVENT. from days of the week.

3. You pick a month at random that 4. A spinner is labeledbegins with the letter J. 1–6. You spin 1 or 5.

5. You pick an odd 6. You pick a word at random number at random from with four letters from this75 to 100. sentence.

7. You have a birthday 8. A number cube is on February 30. tossed. You toss a 1, 3, or 5.

Each of the 6 letters in the word EQUALS is put on a slip of paper. One slip isselected at random. Classify each event as impossible, equally likely, or certain.

9. P(consonant) 10. P(D) 11. P(vowel)

12

15

15

number of favorable outcomestotal number of outcomes

number of favorable outcomestotal number of outcomes

Reteaching 10-2 Probability

Red Blue Blue Blue Blue

25

27

14

13

26

12

35

12

12

12


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You and your friend play a game. You win if you roll a number cube and it lands on 5. Your friend wins if she rolls a number cube and it lands on a factor of 6.

1. What is the probability that you win?

2. What is the probability that your friend wins?

3. Is the game fair? Explain your reasoning.

The line plot shows the results of rolling a number cube 20 times.Find the experimental probability.

4. P(6)

5. P(less than 4)

6. P(greater than 3)

7. P(even number)

8. P(prime number)

Reteaching 10-3 Experimental Probability

A fair game generates equally likely outcomes.To decide whether a game is fair:

Make a list of all the possible outcomes ofthe game.

Determine whether each player has aboutthe same probability of winning.

The Spinner GameSpin a spinner with equal-size sectionsnumbered 1–8. Player A wins on a multiple of 2or 3. Player B wins on any other number. Is thisgame fair?

Possible outcomes: 1, 2, 3, 4, 5, 6, 7, 8.

Player A wins with a 2, 3, 4, 6, and 8.Player B wins with a 1, 5, 7.

P(A winning) = , P(B winning) =

The game is not fair.

You can use the results of playing a game tofind the experimental probability of each playerwinning.

Example: Two players played a game 25 times.Player A won 15 times and player B won10 times.

P(A wins) �

�

�

P(B wins) �

�

� 25

1025

number of times B wontotal games played

35

1525

number of times A wontotal games played

38

58

2

1

2

1

4 5 6321

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

✗

14

520

920

16

23

46

1120

35

122015


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Sometimes you cannot survey an entire population. Instead, yousurvey a sample—a part of the population that can be used to makepredictions about the entire population.

A city has 5,000 sixth graders. To get an estimate of the number ofsixth graders who ride bicycles to school, a random sample was used.Of the 200 sixth graders chosen, 40 said they ride bicycles to school.Predict the number of sixth graders out of 5,000 who ride bicycles to school.

Write a proportion.

Solve.

200n = 200,000

n = 1,000

The sample suggests that 1,000 sixth graders ride bikes to school.

Identify the sample size. Then make a prediction for the population.

n 5200,000

200

200 3 n 5 40 3 5,0002

40200 5 n

5,0001

Reteaching 10-4 Making Predictions From Data

1. How many in a class of 100 students preferbanana yogurt to other flavors? Of 10students asked, 6 prefer banana.

Sample size

Prediction for population

3. Of 600 first graders in the school district,109 are not yet reading. How many firstgraders out of 180,000 in the entire state arenot yet reading?

Sample size


5. How many in a town of 500 students walkto school? Of 100 students asked, 32 walk toschool.

Sample size


2. How many of the 1,900 joggers seen at thepark like to run at 6 A.M.? Of 190 joggersasked, 35 like to run at 6 A.M.

Sample size


4. Out of 300 families, 150 read the morning newspaper. There are 2,400families in town. How many read the morning newspaper?

Sample size


6. Out of 150 families, 16 drive sport utilityvehicles (SUVs). How many of the 4,500families in the county drive SUVs?

Sample size



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Decide whether or not the events are independent. Write yes or no.

1. You pick a green ball from the bag above. You keep the ball out, and pick a red ball.

2. You toss five coins.

3. You get three 4’s on three rolls of a number cube.

4. You select a colored marker from a package of colored markers.Your teacher selects one after you.

Find the probability for each situation.

5. A letter is chosen from the words 6. A letter is chosen from the wordsBABY GIRL and then replaced. BABY BOY and then replaced.

Find P(Y, then R). Find P(B, then B).

A number cube is rolled and a coin is tossed. Find the probability ofeach event.

7. the number 6 and tails 8. an even number and heads

9. a number less than 1 and heads 10. an odd number and tails

Reteaching 10-5 Independent Events

Events are independent when the occurrence ofone event does not affect the other event. If twoevents are independent, the probability that bothwill occur is the product of their probabilities.

Example 1: A ball is drawn from the bag, its colornoted, and then put back into the bag.Then anotherball is drawn. Are the two events independent?

The events are independent because the color ofthe first ball drawn does not affect the color ofthe next ball drawn.

A compound event consists of two or moreseparate events.

Example 2: Find the probability that a red ball isdrawn, replaced, and then a blue ball is drawn atrandom.

• Probability of drawing a red ball is .• Probability of drawing a blue ball is .

P(red) � P(blue)

The probability of drawing a red ball, replacingit, and then drawing a blue ball at random is .2

25

5 225

5 25 3 1

5

15

25

Red Red

Green

Blue

Green

949

164

112

14

14


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The numbers . . . -3, -2, -1, 0, +1, +2, +3, . . . are integers.Integers are the set of positive whole numbers, their opposites, and 0.

The absolute value of a number is its distance from 0 on a numberline. ∆-4« = 4. Opposite integers, like -4 and 4, are the same distancefrom 0.

You can use integers to represent real-world situations. On theFahrenheit thermometer to the right, the temperature reads 5° belowzero. The integer –5 can be used to represent this situation.

Write the opposite of each integer.

1. 7 2. -212 3. 49

4. 1,991 5. -78 6. 16

Find each absolute value.

7. ∆-2« 8. ∆-100« 9. ∆-16«

10. ∆16« 11. ∆12« 12. ∆75«

13. spend $20 14. ride to the 12th floor on an elevator

15. 8° below 0° Centigrade 16. dive 10 feet below the water’s surface

17. earn $15 18. gain of 1,400 feet in elevation

Reteaching 11-1 Exploring Integers

0 1 2 3 4 5 6 7 8 9 10�1�2�3�4

0 4�4

�5�6�7�8�9�10

30

°F

20

10

0

10

20

30

40


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You can use a number line to compare integers. For two integers on anumber line, the greater integer is farther to the right.

Compare �2 and 1.

Locate �2 and 1 on the number line.

Find that 1 is farther to the right.

Write 1 > �2 (1 is greater than �2),or –2 < 1 (�2 is less than 1).

Compare, using < or >.

1. 7 �5 2. �9 �5 3. 6 �6 4. �12 0

5. �33 0 6. �11 �13 7. �5 4 8. �3 �2

Order each set of integers from least to greatest.

9. �7, �9, �19, �8 10. 1, �5, 6, 8, �2

11. 5, �31, �4, �10 12. �2, �22, 10, �7

Write an integer that is located on a number line between the given integers.

13. �3, , 8 14. �24, , 22 15. �5, , 9

16. 0, , 4 17. �2, , 2 18. �17, , �15

Complete with an integer that makes the statement true.

19. �10 > 20. 0 > 21. �2 >

22. 5 < 23. �7 < 24. �36 <

3

2

1

Reteaching 11-2 Comparing and Ordering Integers

0 1 2 3 4 5�1�2�3�4�5


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Reteaching 11-3 Adding Integers

You can add integers on a number line.

Example 1: Find 4 + 3.

Start at 0. Move 4 units right and then 3 unitsright.

4 + 3 = 7

Example 2: Find -3 + -2.

Start at 0. Move 3 units left and then 2 units left.

-3 + (-2) = -5

Example 3: Find 5 + (-3).

Start at 0. Move 5 units right and then 3 unitsleft.

5 + (-3) = 2

Example 4: Find -4 + 1.

Start at 0. Move 4 units left and then 1 unit right.

-4 + 1 = -3

�2 0 2 4 6 8 10�8�10 �6�4

�2 0 2 4 6 8 10�8�10 �6�4

�2 0 2 4 6 8 10�8�10 �6�4

�2 0 2 4 6 8 10�8�10 �6�4

Use the number line to find each sum.

1. -4 + (-8) 2. 4 + (-1) 3. -6 + 8

4. -7 + 3 5. -5 + 8 6. 3 + 5

7 -3 + (-5) 8. 3 + (-5) 9. -3 + 5

Find each sum.

10. -14 + (-5) 11. 5 + (-12) 12. -9 + 9

13. 18 + (-18) 14. 0 + (-4) 15. 6 + 0

16. 15 + (-15) 17. -12 + 0 18. -9 + 10

19. 12 + (-11) 20. -12 + 11 21. 2 + (-10)

�20 �15 �10 �5 0 5 10 15 20


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Reteaching 11-4 Subtracting Integers

To subtract an integer, add the opposite.

Example 1: Subtract 5 - 8.

Add the opposite: 5 + (-8)

5 - 8 = -3

�2 0 2 4 6 8 10�8�10 �6�4

Example 2: Subtract 2 - (-4).

Add the opposite: 2 + 4

2 - (-4) = 6

�2 0 2 4 6 8 10�8�10 �6�4

Use a number line. Find each difference.

1. 3 - (-6) 2. 2 - (-4) 3. -1 - 2

4. -3 - (-5) 5. -8 - (-3) 6. 4 - (-4)

7. -8 - 2 8. 8 - (-2) 9. -8 - (-2)

10. -7 - 4 11. -10 - 2 12. -5 - (-5)

13. -5 - 6 14. 9 - (-3) 15. -11 - (-6)


16. 15 - (-4) 17. -12 - 3 18. 21 - (-7)

19. 3 - (-12) 20. -2 - 10 21. -13 - 13

22. 5 - (-5) 23. 18 - (-10) 24. -7 - (-13)

25. 14 - 16 26. 3 - 15 27. -6 - (-9)

�20 �15 �10 �5 0 5 10 15 20


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When two integers have like signs, the product will always be positive.

Both integers are positive: 3 � 4 = 12Both integers are negative: -3 � (-4) = 12

When two integers have different signs, the product will always be negative.

One integer positive, one negative: 3 � (-4) = -12One integer negative, one positive: -3 � 4 = -12

Find each product.

1. 7 � (-4) 2. -5 � (-9) 3. -11 � 2

4. 8 � (-9) 5. 15 � (-3) 6. -7 � (-6)

7. -12 � 6 8. 13 � (-5) 9. -10 � (-2)

10. A dog lost 2 pounds per week three weeks in a row. What integer expresses the total change in the dog’s weight?

Find each quotient.

11. 18 � (-6) 12. -35 � (-7) 13. -15 � 3

14. 28 � (-4) 15. 25 � (-5) 16. -27 � (-9)

17. -12 � 4 18. 33 � (-11) 19. -50 � (-2)

Reteaching 11-5 Multiplying Integers

Example 1: Find -8 � 3.

Determine the product.8 � 3 = 24

Determine the sign of the product. Sinceone integer is negative and one is positive,the product is negative.

So -8 � 3 = -24.

Example 2: Find (-10) � (-20).

Determine the product.10 � 20 = 200

Determine the sign of the product. Sinceboth integers are negative, the product ispositive.

So (-10) � (-20) = 200.3

2

1

3

2

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When two integers have like signs, the quotient will always be positive.

Both integers are positive: 8 � 2 = 4Both integers are negative: -8 � (-2) = 4

When two integers have different signs, the quotient will always be negative.

One integer positive, one negative: 8 � (-2) = -4One integer negative, one positive: -8 � 4 = -2

Find each quotient.

1. 18 � (-6) 2. -35 � (-7) 3. -15 � 3

4. 28 � (-4) 5. 25 � (-5) 6. -27 � (-9)

7. -12 � 4 8. 33 � (-11) 9. -50 � (-25)

Find the rate of change for each situation.

10. The water level in a lake rises 12 inches in 4 days.

11. The temperature drops 40° as you rise 4 kilometers into the air.

12. A dog grows 24 inches in 12 months.

13. A diver descends 120 feet in 6 minutes.

14. A ship sinks 90 feet in 10 seconds.

Reteaching 11-6 Dividing Integers

Example 1: Find -24 � 8.

Determine the quotient.24 � 8 = 3

Determine the sign of the quotient. Sinceone integer is negative and one is positive,the quotient is negative.

So, -24 � 8 = 3.

Example 2: Find 35 � (-7).

Determine the quotient.35 � 7 = 5

Determine the sign of the quotient. Sinceone integer is positive and one is negative,the quotient is negative.

So, 35 � (-7) = -5.3

2

1

3

2

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You can solve equations that contain integers using the same methodsyou used to solve equations with whole numbers.

Solve the equations. Check the solution.

Solve. y � 6 � �18

y � 6 � 6 � �18 � 6 Add 6 to each side to undo the subtraction.

y � �12 Simplify.

Check (�12) � 6 � �18 Check by replacing y with �12.

Solve. �4x � �16

�4x � (�4) � �16 � (�4) Divide each side by �4 to undo the multiplication.

x � 4 Simplify.

Check. �4 ? 4 � �16 Check by replacing x with 4.


1. Solve.

n � 6 � 36

n � 6 � 6 � 36 � 6 Subtract 6 from each side to undo the addition.

n � Simplify.

Check.

� 6 � 36 Check by replacing n with your solution.

2. r � 10 � �33 3. c � 13 � �3 4. 9k � �108

5. �6r � 96 6. �11 � s � –1 7. b � (�3) � �18

Reteaching 11-7 Solving Equations with Integers


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Example: Graph (2, -4).

• 2 is the x-coordinate. It tells how farto move left or right from the origin.

• -4 is the y-coordinate. It tells how farto move up or down from the origin.

Find the coordinates of point A.

Start at the origin.

How far left or right? 3 leftThe x-coordinate is -3.

How far up or down? 5 upThe y-coordinate is 5.

The coordinates of point A are (-3, 5).

Graph each point in a coordinate plane.

1. B (1, 6) 2. C (-4, -3)

3. D (0, 5) 4. E (-2, 2)

5. F (-1, -5) 6. G (6, -4)

7. H (5, 5) 8. J (4, 0)

9. K (-4, -4) 10. L (2, -3)

11. M (-2, 0) 12. N (5, -1)

13. P (0, -3) 14. Q (-4, 0)

Find the coordinates of each point.

15. R 16. S

17. T 18. U

Look at the coordinate grid above.

19. If you travel 7 units down from S, at which point will you be located?

20. If you travel 4 units right from T and 2 units down, at which point will you be located?

3

2

1

Reteaching 11-8 Graphing in the Coordinate Plane

O�2

�2

�4

�6

6

4

2

�4�6 2

2 rightOrigin

4 down(2, �4)

4 6x

A

y

O�2

�2

�4

�6

6

4

2

�4�6 2 6x

S

y

RU

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To find a balance, add the income (positivenumber) and the expenses (negative number).The sum is the balance.

• To find the balance for February, add

1,468 + (-695) = 773.

Lunch Express made a profit of $773.

• To find the balance for April, add

602 + (-655) = -53.

Lunch Express had a loss of $53.

To look for a trend in the data, draw a linegraph of the monthly balances.

• Balances range from -$53 to $1,073. Makethe vertical scale from -$200 to $1,100.Use intervals of $100.

• Use the horizontal scale for the months.

The trend was for increasing balances—until April.

$1,1001,000

900800700600500400300200100

0�100�200

Jan.B

alan

ceFeb. Mar.

MonthApr.

Lunch ExpressMonthly Balances

Reteaching 11-9 Application of Integers


1. -$9 + $17 2. $51 - $83 3. $42 - (-$18)

4. -$77 + $92 5. -$109 + $109 6. $28 - $4,310

7. -$156 + $429 8. $232 - (-$97) 9. -$401 - $582

10. A company earned $2,357 in January. The 11. Your bank account is overdrawn $31. The company earned $2,427 in February and bank charges $20 for being overdrawn.$1,957 in March. The company’s total You deposit $100. What is the balance of expenses for the first quarter were $4,594. your bank account?What was the company’s profit?

Balance Sheet for Lunch Express

Month Income Expenses

January $1,095 -$459

February $1,468 -$695

March $1,773 -$700

April $602 -$655


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A table or a graph can show how the input and output of a functionare related.

Reteaching 11-10 Graphing Functions

Make a table to show how number of feet is afunction of number of yards.

The table shows that for every yard, there are3 feet. You multiply the number of yards by 3to find the number of feet.

Use the values in the table to draw a graph ofthe function.

Locate the points from the table:(1, 3), (2, 6), (3, 9), (4, 12), (5, 15)

Draw a line through the points.2

1

Complete the table.

1. 2. 3.

Complete each table given the rule. Then graph some points for the function.

4. cups as a function of quarts 5. days as a function of weeks

Input (yards) Output (feet)

1 3

2 6

3 9

4 12

5 15

15

12

9

6

3

1 2 3 4 5Yards

Fee

t

Input Output

1 4

2 5

3 6

4

5

Input Output

4 2

6 4

8 6

10

12

Input Output

2 10

3 15

4 20

5

6


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Some equations contain two operations. To solve them, use inverseoperations to get the variable alone on one side of the equation.Begin by undoing addition or subtraction. Then undo multiplicationor division.

Example: Solve 2d � 1 � 9.

2d � 1 � 9

2d � 1 � 1 � 9 � 1 Subtract 1 from each side to undo the addition.

Divide each side by 2 to undo the multiplication.

d � 4 Simplify.

Check your work by substituting 4 for d2 ? 4 � 1 0 9 in the equation and solving.

9 � 9 Since 9 � 9, the solution is correct.

1. Solve 7x � 5 � 16.

a. What must you first do to both sides?

b. What must you next do to both sides?

c. What is the solution?

2. Solve 12 � � 8.

a. What must you first do to both sides?

b. What must you next do to both sides?

c. What is the solution?


3. 7y � 6 � 8 4. 81 � 3x � 6

5. � 10 � 15 6. 2f � 6 � 4

7. 4k � 20 � 24 8. � 100 � 120e5

c8

t5

2d2 5 8

2

Reteaching 12-1 Solving Two-Step Equations


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An inequality contains �, �, �, �, or 2. Unlike the equations youhave worked with, an inequality may have many solutions.

The solutions of an inequality are the values that make the inequalitytrue. They can be graphed on a number line. An open circle showsthat the number below it is not a solution. A closed circle shows thatthe number below it is a solution.

Example: Graph the inequality x � 4.

The inequality is read as “x is greater than 4.” Since all numbers tothe right of 4 are greater than 4, you can draw an arrow from 4 to theright. Since 4 is not greater than itself, use an open circle on 4.

1. Graph the inequality x � �4.

a. Write the inequality in words.

b. Will the circle at �4 be open or closed?

c. Graph the solution.

2. Graph the inequality x � �1.


b. Will the circle at �1 be open or closed?

c. Graph the solution.

3. Graph the inequality x � 4.


b. Will the circle at 4 be open or closed?

c. Graph the solution.–4 –3 5 64–2 –1 0 1 2 3

–4 –3–5 54–2 –1 0 1 2 3

–4 –3–5–6–7 –2 –1 0 1 2 3

6 7 8–2 –1 0 1 2 3 4 5

x � –2 (less than or equal to)

–5 –4 –3 –2 –1 0 1 2 3 4 5

x ≥ –2 (greater than or equal to)

–5 –4 –3 –2 –1 0 1 2 3 4 5

x < –2 (less than)

–5 –4 –3 –2 –1 0 1 2 3 4 5

x > –2 (greater than)

–5 –4 –3 –2 –1 0 1 2 3 4 5

Reteaching 12-2 Inequalities


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You can solve an inequality by using inverse operations to get thevariable alone.

Example 1: Solve x � 7 � 2. Then check the solution.

x � 7 � 2x � 7 � 7 � 2 � 7 Add 7 to both sides.

x � 9

Check. Test a number greater than 9 and another number less than 9.

Try 11. 11 � 7 � 2 Try 5. 5 � 7 � 24 � 2 false �2 � 2 true

Example 2: Solve a � 15 � 10. Then check the solution.

a � 15 � 10

a � 15 � 15 � 10 � 15 Subtract 15 from both sides.

a � �5

Test a number greater than �5 and another number less than �5.

Try 0. 0 � 15 � 10 Try �6. �6 � 15 � 1015 � 10 true 9 � 10 false

Solve each inequality.

1. x � 8 � 15 2. y � 2 � 8

3. a � 5 � �1 4. x � 10 � �11

5. y � 7 � 2 6. d � 18 � 2

7. 13 � c � 33 8. �12 � b � 4

9. 4 � w � 18 10. x � 15 � �9

Reteaching 12-3 Solving One-Step Inequalities


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Reteaching 12-4 Exploring Square Roots and Rational Numbers

A perfect square is the square of a whole number. The number 81 is aperfect square because it is the square of 9.

You can also say that 9 is the square root of 81, or . The squareroot of a given number is a number that, when multiplied by itself, isthe given number. You can use a calculator to find square roots.

Example 1

!81 5 9

a. Find .

Since 2 � 2 � 4, � 2.

b. Find .

� 8.6602540!75

!75

!4

!4

You can estimate square roots using perfect squares.

Example 2

Tell which two consecutive whole numbers is between.

49 � 52 � 64 Find perfect squares close to 52.

� � Write the square roots in order.

7 � � 8 Simplify.

is between 7 and 8.

Determine if each number is a perfect square.

1. 24 2. 36 3. 49 4. 121

Find each square root.

5. 6. 7.

8. 9. 10.

Use a calculator to tell whether each number is a perfect square.

11. 576 12. 1,200 13. 2,401

14. 900 15. 1,521 16. 1,875

Tell which two consecutive whole numbers the square root is between.Use a calculator to find each square root to the nearest tenth.

17. 18. 19.

20. 21. 22. !97!30!75

!63!88!42

!2,500!400!100

!4!25!9

!52

!52

!64!52!49

!52


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The hypotenuse of a right triangle is the side opposite the right angle and is the longest side. The other two sides are called legs. In the shown triangle, sides a and b are the legs. Side c is the hypotenuse.

The Pythagorean Theorem states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. This can be written algebraically as a2 � b2 � c2.

Example

Find the missing length.

a2 � b2 � c2 Use the Pythagorean Theorem to find the length of side c.

92 � 122 � c2 Substitute 9 for a and 12 for b.

81 � 144 � c2 Square 9 and 12.225 � c2 Add.

15 � c Find .

The length of the hypotenuse is 15 cm.

Find the missing side length of each right triangle.

1. 2. 3.

4. 5. 6.

7. An 8-foot ladder is leaning against a building. If the bottom of the ladder is 3 feet from the base of the building, how far up the building is the top of the ladder? Round to the nearest tenth of a foot.

4 3

x

45

53x

44

125

x

6 in.

8 in.

c

12 cm

c5 cm

16 in.

c12 in.

!225

Reteaching 12-5 Introducing the Pythagorean Theorem

c

12 cm

9 cm

a

b

c

Rightangle

reteaching 1-1 · •to estimate using compatible numbers, change the numbers to ones that are easy...

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