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Unit 1: WHOLE NUMBERS AND DECIMALS

Chapter 1: Place Value, Addition, and Subtraction

1.1 Place Value through Billions . . . . . . . . . . . . . . . . . . . . . . . . . . . . RW11.2 Compare and Order Whole

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . RW21.3 Round Whole Numbers . . . . . . . . RW31.4 Estimate Sums and Differences . . . . . . . . . . . . . . . . . . . . . . . RW41.5 Add and Subtract Whole Numbers

RW51.6 Mental Math: Addition

and Subtraction . . . . . . . . . . . . . . . . . RW61.7 Algebra: Addition and Subtraction

Expressions . . . . . . . . . . . . . . . . . . . . . . . RW71.8 Problem Solving Workshop

Strategy: Find a Pattern . . . . . . . RW8

Chapter 2: Place Value: Understand Decimals2.1 Decimal Place Value . . . . . . . . . . . . RW92.2 Model Thousandths . . . . . . . . . . RW102.3 Equivalent Fractions . . . . . . . . . . RW112.4 Change to Tenths and Hundredths

RW122.5 Compare and Order Decimals . . . . . . . . . . . . . . . . . . . . . . . . RW132.6 Problem Solving

Workshop Strategy: Draw a Diagram . . . . . . . . . . . . . . RW14

Chapter 3: Multiply Whole Numbers3.1 Mental Math: Patterns

in Multiples . . . . . . . . . . . . . . . . . . . . RW153.2 Estimate Products. . . . . . . . . . . . . RW163.3 The Distributive Property . . . RW173.4 Multiply by 1-Digit Numbers . . . . . . . . . . . . . . . . . . . . . . . RW183.5 Multiply by 2-Digit Numbers . . . . . . . . . . . . . . . . . . . . . . . RW193.6 Practice Multiplication . . . . . . . RW203.7 Problem Solving Workshop

Strategy: Predict and Test . . . RW21

Chapter 4: Divide by 1- and 2-Digit Divisors4.1 Estimate with 1-Digit Divisors . . . . . . . . . . . . . . . . . . . . . . . . . RW224.2 Divide by 1-Digit

Divisors . . . . . . . . . . . . . . . . . . . . . . . . . RW234.3 Algebra: Patterns in Division . . . . . . . . . . . . . . . . . . . . . . . . . RW244.4 Estimate with 2-Digit Divisors . . . . . . . . . . . . . . . . . . . . . . . . . RW254.5 Divide by 2-Digit

Divisors . . . . . . . . . . . . . . . . . . . . . . . . . RW264.6 Correcting Quotients . . . . . . . . . RW274.7 Practice Division . . . . . . . . . . . . . . RW284.8 Problem Solving

Workshop Skill: Interpret the Remainder . . . . . . . . . . . . . . . . RW29

4.9 Algebra: Multiplication and Division Expressions . . . . . . . . . . RW30

Unit 2: NUMBER THEORY AND FRACTION CONCEPTS

Chapter 5: Number Theory5.1 Prime and Composite Numbers . . . . . . . . . . . . . . . . . . . . . . . RW315.2 Problem Solving

Workshop Strategy: Makean Organized List . . . . . . . . . . . . . RW32

5.3 Introduction to Exponents . . . . . . . . . . . . . . . . . . . . . . RW33

5.4 Exponents and Square Numbers . . . . . . . . . . . . . . . . . . . . . . . RW34

5.5 Prime Factorization . . . . . . . . . . RW35

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Chapter 6: Fraction Concepts6.1 Equivalent Fractions . . . . . . . . . . RW366.2 Simplest Form . . . . . . . . . . . . . . . . . RW376.3 Understand Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . RW386.4 Compare and Order

Fractions and Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . RW39

6.5 Problem Solving Workshop Strategy: Make a Model . . . . . . . . . . . . . . . . . RW40

6.6 Relate Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . . . . RW416.7 Use a Number Line . . . . . . . . . . . RW42

Unit 3: FRACTION OPERATIONS

Chapter 7: Add and Subtract Like Fractions7.1 Model Addition

and Subtraction . . . . . . . . . . . . . . . RW437.2 Add and Subtract Like

Fractions . . . . . . . . . . . . . . . . . . . . . . . RW447.3 Add and Subtract Like

Mixed Numbers . . . . . . . . . . . . . . . RW457.4 Subtraction with Renaming . . . . . . . . . . . . . . . . . . . . . . RW467.5 Problem Solving Workshop

Strategy: Work Backward . . . RW47

Chapter 8: Add and Subtract Unlike Fractions

8.1 Model Addition of Unlike Fractions . . . .RW48

8.2 Model Subtraction of Unlike Fractions . . . . . . . . . . . . . . . . . . . . . . . RW49

8.3 Estimate Sums and Differences . . . . . . . . . . . . . . . . . . . . . RW508.4 Use Common Denominators . . . . . . . . . . . . . . . . . RW518.5 Add and Subtract

Fractions . . . . . . . . . . . . . . . . . . . . . . . RW528.6 Problem Solving

Workshop Strategy: Compare Strategies . . . . . . . . . . RW53

Chapter 9: Add and Subtract Mixed Numbers

9.1 Model Addition of Mixed Numbers . . . . . . . . . . . . . . . RW54

9.2 Model Subtraction of Mixed Numbers . . . . . . . . . . . . . . . RW55

9.3 Record Addition and Subtraction . . . . . . . . . . . . . . . RW56

9.4 Subtraction with Renaming . . . . . . . . . . . . . . . . . . . . . . RW579.5 Problem Solving Workshop

Skill: Sequence Information . . . . . . . . . . . . . . . . . . . . RW58

Chapter 10: Multiply and Divide Fractions10.1 Model Multiplication of Fractions . . . . . . . . . . . . . . . . . . . . . . . RW5910.2 Record Multiplication

of Fractions . . . . . . . . . . . . . . . . . . . . RW6010.3 Multiply Fractions and

Whole Numbers . . . . . . . . . . . . . . . RW6110.4 Multiply with Mixed

Numbers . . . . . . . . . . . . . . . . . . . . . . . RW6210.5 Model Fraction Division . . . . . RW6310.6 Divide Whole Numbers by

Fractions . . . . . . . . . . . . . . . . . . . . . . . RW6410.7 Divide Fractions . . . . . . . . . . . . . . . RW6510.8 Problem Solving Workshop

Skill: Multistep Problems . . . . RW66

Unit 4: DECIMAL OPERATIONS

Chapter 11: Add and Subtract Decimals11.1 Round Decimals . . . . . . . . . . . . . . . RW6711.2 Add and Subtract

Decimals . . . . . . . . . . . . . . . . . . . . . . . . RW6811.3 Estimate Sums and

Differences . . . . . . . . . . . . . . . . . . . . . RW6911.4 Mental Math: Add and

Subtract . . . . . . . . . . . . . . . . . . . . . . . . RW7011.5 Problem Solving Workshop

Skill: Estimate or Find Exact Answer . . . . . . . . . . . . . . . . . . RW71


Chapter 12: Multiply Decimals12.1 Model Multiplication by a Whole Number . . . . . . . . . . . . . RW7212.2 Algebra: Patterns in

Decimal Factors and Products . . . . . . . . . . . . . . . . . . . . . . . . RW73

12.3 Model Multiplication by a Decimal . . . . . . . . . . . . . . . . . . . RW7412.4 Estimate Products. . . . . . . . . . . . . RW7512.5 Place the Decimal

Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . RW7612.6 Zeros in the Products . . . . . . . . RW7712.7 Problem Solving Workshop

Skill: Choose the Operation . . . . . . . . . . . . . . . . . . . . . . RW78

Chapter 13: Divide Decimals13.1 Divide Decimals by Whole Numbers . . . . . . . . . . . RW7913.2 Estimate Quotients . . . . . . . . . . . RW8013.3 Divide Decimals by Whole

Numbers . . . . . . . . . . . . . . . . . . . . . . . RW8113.4 Divide Decimals

By Decimals . . . . . . . . . . . . . . . . . . . . RW8213.5 Decimal Patterns with

Powers of 10 . . . . . . . . . . . . . . . . . . RW8313.6 Division of Decimals

by Decimals . . . . . . . . . . . . . . . . . . . . RW8413.7 Problem Solving Workshop

Skill: Evaluate Answers for Reasonableness . . . . . . . . . . . RW85

Unit 5: ALGEBRA AND PERCENT

Chapter 14: Algebra: Expressions and Equations

14.1 Write Expressions . . . . . . . . . . . . . RW8614.2 Evaluate Expressions . . . . . . . . . RW8714.3 Write Equations . . . . . . . . . . . . . . . RW8814.4 Solve Equations . . . . . . . . . . . . . . . RW8914.5 Use the Distributive

Property . . . . . . . . . . . . . . . . . . . . . . . . RW9014.6 Mental Math: Use the

Properties . . . . . . . . . . . . . . . . . . . . . . RW9114.7 Problem Solving Workshop

Strategy: Write an Equation . . . . . . . . . . . . . . . . . . . . . . . RW92

Chapter 15: Algebra: Integers15.1 Understand Integers . . . . . . . . . RW9315.2 Compare and Order

Integers. . . . . . . . . . . . . . . . . . . . . . . . . RW9415.3 Model Integer Addition . . . . . RW9515.4 Record Integer

Addition . . . . . . . . . . . . . . . . . . . . . . . . RW9615.5 Model Integer Subtraction . . . . . . . . . . . . . . . . . . . . RW9715.6 Record Integer

Subtraction . . . . . . . . . . . . . . . . . . . . RW9815.7 Problem Solving Workshop

Strategy: Compare Strategies . . . . . . . . . . . . . . . . . . . . . . RW99

Chapter 16: Percent16.1 Understand Percent . . . . . . . . . RW10016.2 Fractions, Decimals, and

Percents . . . . . . . . . . . . . . . . . . . . . . . RW10116.3 Use a Number Line . . . . . . . . . . RW10216.4 Model Percent of a Number . . . . . . . . . . . . . . . . . . . . . RW10316.5 Percent Problems . . . . . . . . . . . . RW10416.6 Problem Solving Workshop

Strategy: Make a Graph . . . . RW10516.7 Compare Data Sets . . . . . . . . . . RW106

Unit 6: GEOMETRY

Chapter 17: Geometric Figures17.1 Points, Lines, and

Angles . . . . . . . . . . . . . . . . . . . . . . . . . RW10717.2 Measure and Draw

Angles . . . . . . . . . . . . . . . . . . . . . . . . . RW10817.3 Construct Parallel and

Perpendicular Lines . . . . . . . . . RW10917.4 Polygons . . . . . . . . . . . . . . . . . . . . . . RW11017.5 Sum of the Angles . . . . . . . . . . . RW11117.6 Problem Solving Workshop

Skill: Identify Relationships . . . . . . . . . . . . . . . . . RW112

17.7 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . RW11317.8 Construct Polygons . . . . . . . . . . RW11417.9 Congruent and Similar

Figures . . . . . . . . . . . . . . . . . . . . . . . . . RW115


Chapter 18: Plane and Solid Figures18.1 Triangles. . . . . . . . . . . . . . . . . . . . . . . RW11618.2 Quadrilaterals . . . . . . . . . . . . . . . . RW11718.3 Draw Plane Figures . . . . . . . . . . RW11818.4 Solid Figures . . . . . . . . . . . . . . . . . . RW11918.5 Nets for Solid

Figures . . . . . . . . . . . . . . . . . . . . . . . . . RW12018.6 Problem Solving Workshop

Strategy: Solve a Simpler Problem . . . . . . . . . . . . . . . . . . . . . . . RW121

18.7 Draw Solid Figures from Different Views . . . . . . . RW122

Chapter 19: Geometry and the Coordinate Plane

19.1 Algebra: Graph Ordered Pairs . . . . . . . . . . . . . . . . . RW12319.2 Algebra: Graph

Relationships . . . . . . . . . . . . . . . . . RW12419.3 Algebra: Graph Integers

on the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . RW125

19.4 Linear Functions . . . . . . . . . . . . . RW12619.5 Write and Graph Equations . . . . . . . . . . . . . . . . . . . . . RW12719.6 Problem Solving Workshop

Skill: Relevant or Irrelevant Information . . . . . . . . . . . . . . . . . . . RW128

Unit 7: MEASUREMENT

Chapter 20: Measurement and Perimeter20.1 Length . . . . . . . . . . . . . . . . . . . . . . . . . RW12920.2 Estimate Perimeter . . . . . . . . . . RW13020.3 Find Perimeter . . . . . . . . . . . . . . . RW13120.4 Algebra: Perimeter

Formulas . . . . . . . . . . . . . . . . . . . . . . RW13220.5 Algebra: Use Perimeter

Formulas . . . . . . . . . . . . . . . . . . . . . . RW13320.6 Problem Solving Workshop

Skill: Make Generalizations . . . . . . . . . . . . . . RW134

Chapter 21: Area21.1 Estimate Area . . . . . . . . . . . . . . . . RW13521.2 Algebra: Area of Rectangles . . . . . . . . . . . . . . . . . . . . RW13621.3 Algebra: Relate Perimeter

and Area . . . . . . . . . . . . . . . . . . . . . . RW13721.4 Problem Solving Workshop

Strategy: Compare Strategies . . . . . . . . . . . . . . . . . . . . . RW138

21.5 Model Area of Triangles . . . . RW13921.6 Algebra: Area of

Triangles. . . . . . . . . . . . . . . . . . . . . . . RW14021.7 Algebra: Area of

Parallelograms . . . . . . . . . . . . . . . RW141

Chapter 22: Surface Area and Volume22.1 Surface Area . . . . . . . . . . . . . . . . . . RW14222.2 Estimate Volume . . . . . . . . . . . . . RW14322.3 Algebra: Find

Volume . . . . . . . . . . . . . . . . . . . . . . . . RW14422.4 Relate Perimeter, Area,

and Volume . . . . . . . . . . . . . . . . . . . RW14522.5 Problem Solving Workshop

Strategy: Write an Equation . . . . . . . . . . . . . . . . . . . . . . RW146

Unit 8: DATA AND GRAPHS

Chapter 23: Analyze Data23.1 Collect and Organize

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . RW14723.2 Find the Mean . . . . . . . . . . . . . . . RW14823.3 Find the Median and

Mode . . . . . . . . . . . . . . . . . . . . . . . . . . RW14923.4 Compare Data . . . . . . . . . . . . . . . . RW15023.5 Analyze Graphs . . . . . . . . . . . . . . RW15123.6 Problem Solving Workshop

Strategy: Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . RW152

Chapter 24: Display and Interpret Data24.1 Make Histograms . . . . . . . . . . . . RW15324.2 Make Stem-and-Leaf

Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . RW15424.3 Make Line Graphs . . . . . . . . . . . RW15524.4 Problem Solving Workshop Skill: Draw

Conclusions . . . . . . . . . . . . . . . . . . . RW15624.5 Choose the Appropriate

Graph . . . . . . . . . . . . . . . . . . . . . . . . . . RW157


Name

© Harcourt • Grade 5Reteach the Standards

NS 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, and fractions, and percents. They understand the relative magnitudes of numbers.

RW1

Place Value Through BillionsYou can use the place-value chart to help you read and write whole numbers and find the value of a digit. A period is a group of three digits. The four periods shown in the place-valuechart below are ones, hundreds, millions, and billions.

Billions Millions Thousands Ones

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8 6 8 7 1 0 4 9 0 2

Write 8,687,104,902 in standard form, word form, and expanded form.

Standard Form: Each period is separated by a comma. 8,687,104,902

Word Form: Write the word name for the numbers in each period followed by the name of each period. eight billion, six hundred eight-seven million, one hundred four thousand, nine hundred two

Expanded Form: Multiply each digit by its place value and write an addition expression.

8,000,000,000 + 600,000,000 + 80,000,000 + 7,000,000 + 100,000 + 4,000 + 900 + 2

To find the value of a digit, mulitiply the digit by its place value. In 8,687,104,902 in digit 6 is equal to 6 � 100,000,000 or 600,000,000.

Write the value of the underlined digit.

1. 433,173,983,021

2. 275,487,601,035 3. 25,283,998,060

4. 809,237,228,771

5. 621,389,007,718

6. 51,906,200,141

Write each number in two other forms. 7. 3,209,003,812

8. 5,000,000,000 + 200,000,000 + 800,000, + 6,000 + 500 + 20

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© Harcourt • Grade 5Reteach the StandardsRW2

NS 1.0 Students compute with very large and very small numbers, postive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magitudes of numbers.

Compare and Order Whole NumbersA place-value chart can help you compare whole numbers. Compare the digits from left to right.

Compare 2,306,821 and 2,310,084. Write <, >, or =.

Compare the digits in the millions place. 2 = 2

Compare the digits in the hundred thousands place. 3 = 3

Compare the digits in the ten thousands place. 0 < 1

Since 0 ten thousands is less than 1 ten thousand, 2,310,084 is less than 2,310,084.

So, 2,306,821 < 2,310,084.

Compare. Write <, >, or = for each .

1. 2,518 2,815 2. 130,870 130,870 3. 5,266,918 5,264,613

4. 525,100 625,100 5. 670,430 640,470 6. 13,275,104 13,276,819

7. 962,338 962,338 8. 18,181 18,818 9. 72,345,995 72,345,795

Name the greatest place-value position where the digits differ.

Name the greater number.

10. 3,218; 3,208 11. 270,908; 270,608 12. 8,306,722; 8,360,272

13. 3,541,320; 3,541,230 14. 324,060; 326,040 15. 12,452,671; 12,543,671

Millions Period Hundreds Period Ones Period

2 3 0 6 8 2 1

2 3 1 0 0 8 4

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REMEMBER:

> means greater than< means less than= means equal to


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NS 1.1 Estimate round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.

Round Whole NumbersYou can round whole numbers by using the rounding rules.

Round 12,694,022 to the nearest million.

Step 1: Underline the digit in the place to which you want to round.

12,694,022

millions place

Step 2: Compare the digit to the right of the underlined digit to 5. 12,694,022

6 > 5, so you round up.

The 2 increases by 1 to a 3.

Step 3: Rewrite all digits to the right of the underlined digit as zeros.

So, 12,694,022, rounded to the nearest million, is 13,000,000.

12,694,022

13,000,000

Round each number to the place of the underlined digit.

1. 136,237,015 2. 35,211 3. 83,445,182 4. 355,264,319

5. 6,024 6. 35,118,247 7. 341,618,915 8. 849,207,284

9. 888,999,211 10. 67,704,257 11. 517,218,137 12. 487,293,618

Name the place to which each number was rounded.

13. 52,398 to 52,000 14. 736,147 to 740,000 15. 6,234,581 to 6,234,600

16. 216,593 to 200,000 17. 345,591 to 346,000 18. 3,517,004 to 4,000,000

Round down: If the digit to the right is less than 5, the underlined digit stays the same.Round up: If the digit to the right is 5 or greater increase by 1.


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NS 1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.

1. 2.

3. 4. 5. 6. 7.

8. 9. 10. 11. 12.

13. 35,229 + 19,111 + 3,502

14. 5,238,889 – 2,394,109

Estimate Sums and DifferencesYou can estimate to find an answer that is close to the exact answer.

You can use rounding to estimate. A number line can help you round. Estimate. 4,829 � 2,325

So, 4,829 � 2,325 is about 3,000.

Estimate. 25,902 � 18,188 � 3,502

Round each number to the nearest thousand.

So, 25,902 � 18,188 � 3,502 is about 48,000.

Estimate the sum or difference.

4,829

� 2,325

__

5,000

� 2,000

__

3,000

25,902 18,188� 3,502

26,000 18,000� 4,000

48,000

294,322

� 163,582

__

�

925,461

� 173,509

__

�

� �

529,617

� 237,510

__

223,873

� 78,905

__

72,543

� 29,583

__

56,108

� 42,336

__

$8,423

� 1,825

__

$63,895

� 37,228

__

773,645

� 135,710

__

95,223

� 103,229

__

745,556

� 132,881

__

554,903

� 125,318

__

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Name


NS 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers.

1.

2.

3.

4.

5.

6.

7.

8.

9. 259,562 487,018 + 241,393

10.

11.

12.

13. 18,275 + 5,225 + 3,093

14. 2,705,243 – 1,192,013

Add and Subtract Whole NumbersYou can add or subtract to find an exact number.

Estimates will help you determine if you have a reasonable answer.

Add 789,039 1 325,155.

Estimate. Round to the nearest

hundred thousand.

Align the digits.

Start adding from the right, regrouping as needed.

Estimate. Then find the sum or difference.

5,382

� 8,723

__

33,617

� 29,218

__

306,657

� 182,322

__

129,336

� 647,273

__

3,017,451

� 834,319

__

376,217

� 53,278

__

35,137

� 21,328

__

1,330,316

� 265,594

__

4,678,128

� 2,119,625

___

319,007

� 227,242

__

603,438

� 617,634

__

7 8 9, 0 3 9

� 3 2 5, 1 5 5

1 1 1

1, 1 1 4, 1 9 4

789,039

� 325,155

800,000

� 300,000

1,100,000

Find the sum.

So, 789,039 � 321,155 � 1,114,194.

Since 1,114,194 is close to the estimate of 1,100,000, it is reasonable.

��

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Name

� 50

18 � (22 � 34) � (18 � ) � 34

18 � 56 � � 34

74 �

� �



Mental Math Addition and SubtractionYou can use the properties of addition to help you add mentally.

1.

Use mental math strategies to find the sum or difference.

Complete. Name the property or strategy.

The Commutative Property of Addition is that addends may be added in any order without changing the sum.

2.

3. 4.

The Associative Property of Addition is that you may group addends differently without changing the sum.

You can also use a strategy called compensation to help you find sums and differences.

Use compensation to add. Add 1 to change 29 to a multiple of 10. Then adjust 37 by subtracting 1 to keep the balance.

Use compensation to subtract. Subtract 5 to change 125 to a multiple of 10. Then subtract 5 from 148 to keep the balance.

19 � 52 � 31 � 19 � 31 � 52

� 50 � 52

2 � (3 � 5) � (2 � 3) � 5

2 � 8 � 5 � 5

10 � 10

29 � 37 �(29 � 1) � (37 � 1)

�30 � 36�66

148 � 125 � (148 � 5) � (125 � 5)

� 143 � 120� 123

14 � 36� 36 �

128 � 95 � (128 � 5) � (95 � )

� 123 �

238 � 117 � (238 � 2) � (117 � )

� 240 �

� 102

8. 9. 10. 571 � 328 �248 � 118 � 132 �(116 � 363) � 214 �

5. 6. 7. 274 � 139 �52 � 18 �43 � 27 �


Name


AF 1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.

Addition and Subtraction ExpressionsA numerical expression has only number and operation signs.

Write a number expression. Then find the value. Tell what the value represents.

Julie saved $52. Then she spent $8.

• Identify the operation, or operations, used. Use clue words to help you write expressions.

• Translate the words into a numerical expression. The word “spent” tells you to subtract.

Julie’s saved amount Amount she spent

1.

Tell what operation you would use to write each expression. Then write the expressions.

Write a numerical expression. Then find the value. Tell what the value represents.

So, the numerical expression is 52 � 8.

• Find the value of the expression. 52 � 8 � 44, or $44

So, the value, $44, represents the amount Julie has left.

A variable is a letter or symbol that represents one or morenumbers in an algebraic expression.

Tell what operation you would use to write each expression. Then write

the express.

8 increased by a number

• The word “increased” tells you to add. • Choose a variable. Let n equal the amount 8 is increased by. • Write an algebraic expression 8 � n.

So, you would use addition to write to 8 � n.

2. 3.Jerry had an average of 87. After a test, his average increased by 4.

The difference of 87 and 24.

The sum of 36 and 43 decreased by 7.

4. 5. 6.6 more than 13 15 less than 42 35 decreased 21

52 � 8


Name



Problem Solving Workshop Strategy: Find a PatternThe Redwood Club is making a quilt. So far, the quilt

has this design. If the pattern continues, what design

will the twelfth row of the quilt have?

Read to Understand

1. Write the question as a fill-in-the-blank sentence.

2.Plan

How can finding a pattern help you solve the problem?

3.Solve

Solve the problem. Describe the strategy you used.

4. Write your answer in a complete sentence.

Check

5. Is there another strategy you could use to solve the problem?

Solve by finding a pattern.

Hannah saved $105 to plant and maintain some flowers. After one week she had $94 left. After two weeks she had $83 left. After three weeks she had $72 left. If this pattern continues how much will Hannah have after 7 weeks?

A group of coast redwood trees produces 3 million seeds in one year. If 500,000 seeds weigh 4 pounds, how much will 3 million seeds weigh?

7.6.


Name

© Harcourt • Grade 5

RW9 Reteach the Standards

NS 1.0 Students compute with very large and very small numbers, positive integers, decimals and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitude of numbers.

Decimal Place ValueYou can use a place-value chart to write a decimal in standard form, expanded form, and word form.

Ones Tenths Hundredths

8 6 1

Standard Form: 8.61

Expanded Form: Multiply each digit by its place value to write the expanded form.(8 � 1) � (6 � 0.1) � (1 � 0.01) � 8 � 0.6 � 0.01

Word Form: Step 1 Write the word name for the number to the leftof the decimal point, eight.

Step 2 Write the word and for the decimal point.

Step 3 Write the word name for the number to the right of the decimal point, sixty-one, followed by the name of the last place, hundredths.

8.61 in word form is eight and sixty-one hundredths

To find the value of an underlined digit, multiply the digit by its place value.In 7.92, the digit 9 is equal to 9 � 0.1 or 0.9.

1. 2. 3. 4.6.84 9.37 0.12 0.59


5. 6. 7. 8.0.96 0.53 1.08 13.78

9. 10. 11. 12.8.64 1.07 12.73 27.91

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RW10 Reteach the Standards NS 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitude of numbers.

How many columns in the large square are completely shaded? 7How many tenths are shaded? 7 tenths or 0.7How many small squares in the next column are shaded? 2How many hundredths are shaded? 2 hundredths or 0.02How many rectangles are shaded in the small square? 5How many thousandths are shaded? 5 thousandths 0.005What is the decimal? 0.725

Model ThousandthsYou can use a place-value chart to write a decimal in standard form, expanded form, and word form.

Ones Tenths Hundredths

1 0 1 4

Standard Form: 1.014

Expanded Form: Multiply each digit by its place value to write expanded form.1 � 0.01 � 0.004

Word Form: Write the word name for the number followed by the name of the last place.Write the word and for the decimal. one and fourteen thousandths

To find the value of an underlined digit, multiply the digit by itsplace value.In 1.014, the digit 4 is equal to 4 � 0.001 or 0.004.

Write the decimal shown by the shaded part of the model.


2. 3. 4.0.502 0.816 7.039 8.264

Thousands

1.


Name


NS 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand relative magnitudes of numbers.


Equivalent DecimalsEquivalent decimals are different name for the same number or amount. You can use decimal squares to find equivalent decimals.

Are 0.42 and 0.420 equivalent?

The model shows 100 equal parts. Each part is 1 hundredth of the model.

42 of the parts are shaded.

This model shows 42 hundredths, or 0.42.

Now divide each of the 100 parts into 10 equal parts. The model at the right shows what each small square would look like.

The model shows 1,000 equal parts. Each part represents 1 thousandth, or 0.001 of the model.

There are 420 shaded parts.

This model shows 420 thousandths, or 0.420.

Each model shows the same amount shaded. So, 0.42 and 0.420 are equivalent.

Write equivalent or not equivalent to describe each pair of decimals.

1. 0.23 and 0.230 2. 0.51 and 0.500 3. 0.680 and 0.68

4. 4.87 and 4.870 5. 9.87 and 9.78 6. 1.11 and 1.111

7. 0.8300.8030.83

8. 0.930.0930.930

9. 1.0071.0701.07

Write the two decimals that are equivalent.

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RW12 Reteach the Standards NS 1.2 Interpret percents as a part of a hundred;

find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

Change to Tenths and HundredthsYou can change a decimal as fraction or mixed number withtenths or hundreths.

Write 0.80 as a fraction with tenths and hundredths.

Write each decimal as a fraction or mixed number with tenths and hundredths.

Model Fraction

Tenths

Hundredths

8 shaded parts

______________ 10 parts

� 8 ___ 10

80 shaded parts

_______________ 100 parts � 80 ____ 100

1. 1.2 2. 0.50 3. 2.60 4. 1.6

5. 0.70 8. 4.807. 7.56. 0.9

9. 11. 12.10. 0.403.3 3.1 0.20

13. 14. 15.6.8 5.10 0.4 16. 0.30


Name

•


NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. RW13

Compare and Order DecimalsA place-value chart can help you compare decimals. You may need to add zeros so you can compare the same number of digits in each decimal. Compare the digits from left to right.

Compare 0.28 and 0.208. Write <, >, or =.

Ones Tenths Hundredths Thousandths

0 2 8 0

0 2 0 8

Compare the digits in the ones place. 0 = 0

Compare the digits in the tenths place. 2 = 2

Compare the digits in the hundredths place. 8 > 0

Since 8 tenths is greater than 0 tenths, 0.28 is greater than 0.208.

So, 0.28 > 0.208.


1. 9.39 9.9 2. 0.308 0.30 3. 7.245 7.254

4. 8.16 8.106 5. 0.69 0.89 6. 1.83 1.833

7. 0.603 0.6 8. 0.78 7.8 9. 4.71 4.071

Order from least to greatest.

10. 0.614, 0.641, 0.64 11. 1.576, 1.765, 1.567 12. 3.971, 3.9, 4, 3.901

13. 3.08, 3.801, 3.8 14. 0.159, 0.154, 0.14

15. 7, 6.99, 6.099, 7.001

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Name


RW14 Reteach the Standards NS 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitude of numbers.

Problem Solving Workshop Strategy: Draw a DiagramJessica, her brother Sam, her mother Nan, and her father Bill

are the first four people in line at lunch. Sam is not first in

line. Jessica has at least two people ahead of her. Bill is third.

Give the order of the four in line.

Read to Understand

Plan

Solve

Check

Write the question as a fill-in-the-blank sentence.1.

How can drawing a diagram help you solve the problem?2.

Solve the problem. Describe the strategy you used.3.

What is the order of the four people in line?

5. Is there another strategy you could use to solve the problem? Explain.

4.

Draw a diagram to solve.

6. 7.There were 63 people in line at the movies. Then 7 left, and 3 times that number joined the line. How many people are in the line now?

Four students are standing in a line. Lea is not first in line. Kyle has at least two people behind him. Brooke is second in line. Shawn is behind Lea. Give the order of the four in line.


Name


NS 1.1: Estimate, round and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.

Mental Math Patterns in Multiples

Multiply. 4 � 3,000

You can use basic multiplication facts and patterns to find the product then you multiply by a multiple of 10.

4 � 3 � 4 � 3 ones � 12 ones � 12 basic fact

4 � 30 � 4 � 3 tens � 12 tens � 120 basic fact times 10

4 � 300 � 4 � 3 hundreds � 12 hundreds � 1,200 basic fact times 100

4 � 3,000 � 4 � 3 thousands � 12 thousands � 12,000 basic fact times 1,000

Find the missing numbers.

Find the product.

7. 2 � 80

1. 5 � 3 �

5 � 30 �

5 � 300 �

2. 9 � 4 �

9 � 40 �

9 � 400 �

3. 6 � 2 �

6 � 20 �

6 � 200 �

4. 4 � 5 �

4 � 50 �

4 � 500 �

5. 2 � 9 �

2 � 90 �

2 � 900 �

6. 7 � 1 �

7 � 10 �

7 � 100 �

8. 3 � 600 9. 60 � 50 10. 20 � 700 11. 9 � 4,000

12. 10 � 60 13. 50 � 5,000 14. 100 � 10 15. 7,000 � 40 16. 9,000 � 20

17. 80 � 6,000 18. 5 � 200 19. 800 � 80 20. 900 � 100 21. 300 � 500

22. 7 � 6,000 23. 8 � 300 24. 20 � 5,000 25. 800 � 400 26. 70 � 100


Name


NS1.1: Estimate, round and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.

Estimate ProductsYou can use round numbers and use basic multiplication facts to estimate products. Estimate. 489 � 32

STEP 1: Round both factors to the greatest place.

489 is closer to 500 than 400. 489 rounds to 500.

32 is closer to 30 than 40. 32 rounds to 30.

STEP 2: Use basic multiplication facts and patterns to find the product of the rounded factors.

489 � 32

500 � 30 � 15,000

Remember, if the digit to the right of the greatest place is 0–4, round down. If the digit is 5–9, round up.

Round each factor and estimate the product.

Estimate the product.

1. 52 � 31 2. 731 � 47

3. 63 � 63 4. 512 � 49

� � � �

� � � �

5. 456 � 76 6. 79 � 61 7. 53 � 1,299 8. 26 � 725 9. 71 � $9.58

10. 44 � 260 11. 489 � 706 12. 3,485 � 59 13. 45 � 914 14. 38 � 4,118


Name

Step 2: Slide apart the array into two smaller arrays for products you know.

Step 3: Add the products of each array to find the product of the original array.

5 � 10 � 50 5 � 9 � 45

50 � 45 � 95

So, 5 � 19 � 95.

© Harcourt • Grade 5Reteach the StandardsRW17 AF1.3: Know and use the distributive property in

equations and expressions with variables.

Estimate. Then find the product.

2. 7 � 64 �1. 8 � 23 � 3. 35 � 9 �

4. 57 � 6 � 5. 84 � 4 � 6. 23 � 12 �

7. 36 � 21 � 8. 18 � 25 � 9. 57 � 13 �

The Distributive PropertyThe Distributive Property states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products.Multiply. 5 � 19

You can use counters to solve the problem.

Step 1: Make an array to model 5 � 19.


Name

H T O H T O H T O

4

x

1

9 8

2

6

1

4

x

1

9

9

8

2

6

1

4

x

9

1

9

9

8

2

6



Multiply by 1-Digit NumbersYou can use a place-value chart to help you multiply by

1-digit numbers.

Multiply. 2 � 498Estimate: 2 � 500 � 1,000

Step 1: Multiply the ones. Step 2: Multiply the tens. Step 3: Multiply the hundreds.

2 � 8 ones � 16 ones16 ones � 1 ten 6 ones

2 � 9 tens � 18 tens18 tens � 1 ten � 19 tens19 tens � 1 hundred 9 tens

2 � 4 hundreds � 8 hundreds8 hundreds � 1 hundred � 9 hundreds

Record the 6 in the ones place. Write the 1 above the tens place.

Record the 9 in the tens place. Write the 1 above the hundreds place.

Record the 9 in the hundreds place.


1. 42

� 32. 65

� 43. 604

� 5

4. 532

� 6

5. 745

� 3

6. 464

� 77. 764

� 8

8. 1,208

� 9

9. 3,045

� 8

10. 4,365

� 6


Name

Step 3: Add the partialproducts.

Tt Th H T O Tt Th H T O Tt Th H T O


NS 1.0: Students compute with very large and small numbers, positive integers, decimals, and fractions, and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers.

Multiply by 2-Digit NumbersYou can use a place-value chart and regrouping to help you multiply by 2-digit numbers.

Multiply. 43 � 634Estimate: 40 � 600 � 24,000

Step 1: Multiply the first factor by the ones digit in the second factor.

Step 2: Multiply the first factor by the tens digit in the second factor.


1. 253

� 17

2. 439

� 563. $324

� 45

4. 576

� 43

5. 805

� 62

6. 287

� 387. 392

� 81

8. 466

� 29

9. 507

� 54

10. 189

� 86

11. 237 � 16 � 12. 407 � 28 � 13. 683 � 53 �

Think: 27,262 is close to the estimate of 24,000 so the answer is reasonable.

Think:

40 � 634 � 25,360

Think:

3 � 634 � 1,092

1 1

6 3 4

� 4 3

1 9 0 2

+2 5 3 6 0

11 11

6 3 4

� 4 3

1 9 0 2

2 7 2 6 2

+2 5 3 6 0

11 11

6 3 4

� 4 3

1 9 0 2


Name


NS1.0: Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers.

Practice MultiplicationYou can use the expanded form of each factor to help you multiply.

Multiply. 62 � 573

Estimate: 60 � 600 � 36,000

Step 1: Write each factor in expanded form.

62 � 60 � 2

573 � 500 � 70 � 3

Step 2: Multiply.

500 � 70 � 3� 60 � 2

61 4 0

1 0 0 01 8 0

4 2 0 03 0 0 0 03 5 5 2 6

Step 3: Check for reasonableness. 35,526 is close to 36,000, so the answer is reasonable.


Think:2 � 3 � 62 � 70 � 1402 � 500 � 1,00060 � 3 � 18060 � 70 � 4,20060 � 500 � 30,000

6. 283 � 16 �

1. 341 � 6

2. 509 � 4

3. 5,031 � 7

5. 669 � 5 � 7. 542 � 91 �

8. 3,608� 54 � 9. 9,135 � 37 � 10. 8,230 � 89 �

4. 2,378 � 9

�


Name



Problem Solving Workshop Strategy: Predict and TestLloyd is making wallets and bookmarks at camp.

Bookmark kits cost $3 and wallet kits cost $8. Lloyd

spent $34 on kits. How many bookmarks and how many

wallets is he planning to make?

Read to Understand

2.Plan

How can writing using the strategy predict and test help you solve the problem?

3.Solve

Solve the problem.

4. Write your answer in a complete sentence.

Check

5. How can you check your answer? Does your answer make sense for the problem?

Predict and test to solve.

Drama lessons are $25 each. Dancing lessons are $22 each. Jason expects his lessons to cost $532. How many of each type of lesson is he planning to take?

A test has 50 problems on it. For every correct answer, 2 points are given. For each incorrect answer, 1 point is subtracted. Sue scored 85 points. How many problems did she miss?

7.6.

1. Write the question you need to answer as a fill-in-the-blank sentence.

Predict Test Revise


Name

© Harcourt • Grade 5Reteach the StandardsRW22NS 1.1 Estimate, round, and manipulate very large (e.g.,

millions) and very small (e.g., thousandths) numbers.

Estimate with 1-Digit DivisorsCompatible numbers are numbers that are easy to work with mentally. In division, one compatible number divides evenly into the other. Think of the multiples of a number to help you find compatible numbers.

Estimate the quotient of .

Step 1: Think of the multiples of 6:

Find multiples that are close to the first 2 digits of the dividend.

18 and 24 are both close to 19. You can use either number, or both numbers to find a range.

Step 2: Estimate using the compatible numbers.

1,902 � 6

1,800 � 6 = 300

So, 1,800 � 6 is about 300.

6 12 18 24 30 36 42 48 54

Estimate the quotient.

1.

2.

3.

4. 265 � 4 5. 344 � 8 6. 4,860 � 5

7.

8.

9.

10. 499 � 7 11. 345 � 6 12. 3,918 � 8

3 � � 252 6 � � 546 4 � � 1,534

5 � � 314 2 � � 157 8 � � 5,928

F F

6 � � 1,902


Name


NS 2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.

Name the position of the first digit of the quotient. Then find the first digit.

1.

2.

3.

Divide.

6.

7.

8.

Divide by 1-Digit DivisorsYou can use estimation to help you place the first digit in the quotient. Then, you can follow steps to divide.

Divide.

Step 1: Use estimation to place the first digit.

562 � 6

540 � 6 = 90

So, the first digit is in the tens place.

Step 2: Divide the 56 tens.

Think:

Multiply: 6 � 9 = 54

Subtract: 56 – 54 = 2

Compare: 2 < 6

Step 3: Bring down the 2 ones. Then divide the 22 ones.

Think:


Subtract: 22 – 18 = 4

Compare: 4 < 6

Write the remainder to the right of the whole number part of the quotient.

F F

9

So, 562 � 6 = 93 r4

6 � � 562

6 � � 562 � 54 ___

2

93 r46 � � 562 � 54 ___

2 2

� 18 ___ 4

6 � � 56

6 � � 22

7 � � 456 3 � � 741 4 � � 9,449

7 � � 297 8 � � 136 4 � � 8,659


Name


NS2.2 Demonstrate proficiency with division, including divison with positive decimals and long division with multidigit divisors.

Algebra: Patterns in DivisionFind 36,000 � 60.

You can use basic facts and patterns to find quotients.

Step 1: Find the basic fact.

36,000 � 60

So, the basic fact is 36 � 6 = 6.

Step 2: Cross out zeros in the divisor. For each zero crossed out in the divisor, cross out a zero in the dividend.

36,000 � 60

Step 3: Insert the number of zeros left in the dividend to the right of the quotient of the basic fact.

36,000 � 60= 600

So, 36,000 � 60 = 600.

Use basic facts and patterns to find the quotient.

1. 40 � 2 2. 160 � 8 3. $270 � 90 4. 420 � 6

5. 500 � 50 6. 120 � 40 7. 480 � 6 8. 560 � 70

9. 210 � 3 10. $300 � 10 11. 630 � 90 12. 540 � 60

13. 6,300 � 7 14. 6,000 � 2 15. 3,000 � 30 16. $4,500 � 50

17. 8,000 � 10 18. 1,400 � 7 19. $2,400 � 30 20. 5,600 � 8


Name

© Harcourt • Grade 5Reteach the StandardsRW25NS 1.1 Estimate, round, and manipulate very large (e.g.,


Estimate the quotient of using two sets of compatible numbers.

Step 1: Round the divisor to the nearest ten. 49 rounds to 50.

Step 2: Find two numbers close to the dividend that are compatible with the rounded divisor.

400 and 500 are both close to 427 and easy to divide by 50.

Step 3: Divide the compatible numbers to find the estimates. 400 � 50 = 8 500 � 50 = 9

So, 8 and 9 are reasonable estimates of the quotient.

Estimate with 2-Digit DivisorsRemember that compatible numbers are numbers that are easy to work with mentally. In division, one compatible number divides evenly into the other. Think of the multiples of a number to help you find compatible numbers.

Write two pairs of compatible numbers for each. Then give two

possible estimates.

1.

2.

3.

4. 409 � 63

5. 478 � 19

6. 7,145 � 31


7.

8.

9.

42 � � 157 73 � � 268 54 � � 343

12 � � 622 34 � � 293 81 � � 738

Remember to use basic facts and patterns to divide.

49 � � 427


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Step 1: Use estimation to place the first digit. Remember to use compatible numbers to estimate.

So, the first digit is in the hundreds place.

Step 2: Divide the 54 hundreds.

Step 3: Bring down the 8 tens. Then divide the 168 tens.

Step 4: Bring down the 6 ones. Then divide the 166 ones.

Think:

Multiply: 19 � 8 = 152

Subtract: 168 – 152 = 16

Compare: 16 < 19

1. 2. 3.

4. 5. 6.

Divide by 2-Digit DivisorsYou can use estimation to help you place the first digit in the quotient. Then, you can follow steps to divide.

Divide. 19 � � 5,486

20 � � 6,000 300

19 � � 5,486 28

38 _____ 168

� 152 ____ 16

F

19 � � 5,486 2

� 38 ___ 16

Think:

Multiply: 19 � 2 = 38

Subtract: 54 – 38 = 16

Compare: 16 < 19

Think:

Multiply: 19 � 8 = 152

Subtract: 166 – 152 = 14

Compare: 14 < 19

Write the remainder to the right of the whole number part of the quotient.

19 � � 5,486 288 r14

38 _____ 168

� 152 ____ 16

6

�15214

F

52 � � 612

43 � � 6,413

63 � � 917

27 � � 4,684

24 � � 608

89 � � 1,597

Divide. Check your answer.

� �


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Correcting QuotientsEstimates can help you identify the first digit in the quotient, but sometimes you will need to correct the quotient.

Divide.

Step 1: Write two pairs of compatible numbers, and estimate the answer.

Step 2: Use one of your estimates as the quotient.

Think: The dividend, 3,892, is closer to 4,000 than 3,500. So, try using 8 as the first digit in the quotient.

Step 3: Divide.

Since 4 < 48, the estimate is just right.

Divide.

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.5,216 � 39 4,984 � 76 738 � 41 3,427 � 17 751 � 92

Write low, high, or just right for each estimate.

So, 3,892 � 48 = 81 r4

48 � � 3,892

50 � � 3,500 50 � � 4,000 70 80

48 � � 3,892 81 r4

�38452

�484

F

58 � � 1,325 37 � � 241 29 � � 2,276 82 � � 910 63 � � 3,784 30 6 80 4 60

24 � � 217 37 � � 4,819 71 � � 488 43 � � 9,189 16 � � 845


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Practice DivisionWhen you divide, it helps to remember that division is an operation that tells the number of equal groups, or the number in each equal group.

Divide.

Step 1: Use estimation to place the first digit.

226 � 3

210 � 3 = 70

So, the first digit is in the tens place.

Step 2: Divide the tens.


Subtract: 22 – 21 = 1

Compare: 1 < 3

Step 3: Divide the ones.


Subtract: 16 – 15 = 1

Compare: 1 < 3

Write 1 as the remainder.

F F

7

So, 226 � 3 � 75 r1

Divide. Multiply to check your answer.

1.

2.

3.

4.

5.

6. 115 � 6

7. 935 � 4

8. 2,198 � 7

9. 9,217 � 7

10. 8,032 � 4

3 � � 226

� 21 ___ 1

3 � � 226

75 r13 � � 226

21 _____ 16

�15 ____ 1

7 � � 219 9 � � 326 5 � � 6,221 9 � � 3,504 6 � � 3,167

�


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Problem Solving Workshop Skill: Interpret the RemainderA total of 75 fifth graders went on a field trip to the local recycling center. The

school is providing mininvans to take the students to the center. If each minivan

holds 9 students, how many minivans are needed?


2. First, divide to find the quotient and remainder, if there is one.

3. Circle the way you would interpret the remainder. Then explain why.

A I will add 1 to the quotient

B I will use the quotient and write the remainder as a fraction

C I will only use the remainder

D I will only use the quotient

4. How many minivans are needed for the field trip to the recycling center?

5. How can you check to see if your answer is reasonable?

6. A group of hikers wants to travel 1,250 miles on the Appalachian Trail. They plan to hike 15 miles each day. Exactly how

long will it take them to hike the entire distance?

7. Rosa and her family want to hike 12 miles per day along a 165-miles long trail. How many days will Jessie and her family hike exactly 12 miles?

9 � � 75


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Algebra: Multiplication and Division ExpressionsAn expression has numbers, operation signs, and sometimes variables. An expression does not have an equal sign.

An algebraic expression is an expression with at least one variable. A variable is a letter or symbol that stands for one or more numbers.

Write an algebraic expression for the phrase.24 models displayed equally on s shelves.

number of models variable to show the number of shelves

You can evaluate the expression by replacing the variable with a given number and then finding the value of the expression.

Evaluate the expression above if s = 4.

Step 1 Replace the variable (s) with the given value (4).

24 � s

24 � 4

Step 2 Evaluate the numerical expression.

24 � 4 6

So, there are 6 models on each shelf.

Write an algebraic expression for each phrase.

1. 48 markers shared equally among x friends.

2. 9 groups of y apples 3. 14 books placed equally on z shelves

4. 15 � a 5. 72 � b 6. 26 � a 7. 132 � a 8. 8 � b

9. 46 – b 10. b � 73 11. 152a 12. a � 4 13. 18b

14. 108 � b 15. 9b 16. 456 � a 17. 336 � b 18. 81 – a

24 � s E

use division to find the models on each shelf

Evaluate each expression if a = 4 and b = 24.


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NS 1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3).

Prime and Composite NumbersA number is prime if it has exactly two factors—1 and itself.

A number is composite if it has more than two factors.

The number 1 is neither prime nor composite.

You can make arrays to find if a number is prime or composite. An array is an arrangement of objects in rows and columns. A number with exactly two arrays is prime. A number with more than two arrays is composite.

Is the number 32 prime or composite?

The number 32 is composite.

Write prime or composite.

1. 15 2. 19 3. 11

4. 38 5. 45 6. 24

32 � 1 16 � 2

1 � 32

2 � 16

8 � 4

4 � 8


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Problem Solving WorkshopStrategy: Make an Organized ListToni and Hector are weaving potholders using the colors

red, white, green, and blue. If they only use two colors for

each potholder, how many ways can they combine two colors?

Read to Understand

2.Plan

How can an organized list help you solve the problem?

3.Solve

Complete the list to answer the question. Do not

list any combination more than once. For example,

red and white is the same as white and red.

4. Answer the question. How many ways can they combine two colors?

Check

5. Look back at the problem. Does the answer make sense? Explain.

Use an organized list to solve.

Elijah has scouts every Tuesday and soccer practice every three days. On May 3 he has scouts and soccer practice. What other days in May does he have both scouts and soccer practice?

6.

1. Write the question as a fill-in-the blank sentence.

Color Combinations

red and white; red and ; red and

white and ; white and

green and


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NS 1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication.

Introduction to ExponentsAn exponent is a number that tells how many times another number, the base, is used as a factor.

10 3 = 10 � 10 � 10 The base, 10 , is used as a factor 3 times.

What is the value of 102?

What is the base? What is the exponent?

10 2

How many times do you use the base as a factor?

2What expression shows 10 used as a factor 2 times?

10 � 10

What is the value of 10 � 10?

10 � 10 = 100

So 102 = 100.

What is 10 � 10 � 10 � 10 written in exponent form?

What number is used as a factor?

10, so write 10 as the base.

How many times is it used as a factor?

4, so write 4 as the exponent.

So, 10 � 10 � 10 � 10 = 104.

Find the value.

1. 104 2. 109 3. 106

Write in exponent form. Then find the value.

4. 10 � 10 � 10 � 10 � 10 5. 10 � 10 6. 10 � 10 � 10 � 10 � 10 � 10 � 10

103 exponent = number of times to use the base as a factor

base

}


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NS 1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication.

Exponents and Square NumbersYou already know that an exponent is a number that tells how many times the base is used as a factor.

In the last lesson, the base was always 10. The base does not have to be 10, though.

Write 3 � 3 � 3 � 3 � 3 � 3 in exponent form.

3 is the repeated factor, so 3 is the base.

The base is repeated 6 times, so 6 is the exponent.

3 � 3 � 3 � 3 � 3 � 3 = 36

base 36

A base with an exponent can be written in words.

Write 36 in words.

The exponent 6 means “the sixth power.”

36 in words is “the sixth power of three.”

There are two ways to write the word form for an exponent of 2 or 3.

Write 4 � 4 and 6 � 6 � 6 in exponent form and in words.

Exponent form: 42

Words: the second power or four

OR four squared

Exponent form: 63

Words: the third power of six

OR six cubed

Write in exponent form and then write in words.

1. 5 � 5 � 5 � 5 � 5 � 5 � 5 2. 8 � 8 � 8 3. 2 � 2 � 2 � 2 � 2

Find the value.

4. 27 5. 123 6. 192

exponent


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Prime FactorizationPrime factorization is a way to show a composite number as the product of prime factors.

A factor tree is a diagram that shows the prime factorization of a composite number.

What is the prime factorization of 12?

The prime factorization of 12 is 2 � 2 � 3.

You can use exponents in prime factorization for factors that appear two or more times.

Rewrite 2 � 2 � 3 using exponents.

2 � 2 � 3 = 22 � 3

Find the prime factorization. You may use a factor tree.

1. 12 2. 20 3. 90

Rewrite the prime factorization using exponents.

4. 2 � 3 � 5 � 5 5. 3 � 5 � 3 � 5

6. 2 � 5 � 3 � 5 � 2 � 5

Find the number for each prime factorization.

7. 52 � 7

8. 2 � 3 � 5 � 2 � 7 × 3

9. 23 × 3 × 112

12

4 � 3

2 � 2 � 3

Write the number being factored at the top.

Write the number as a product of two factors.

Write each composite number as a product of prime factors.


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NS 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

Equivalent FractionsEquivalent fractions are fractions that have the same value or

amount as each other. For example, and are equivalent

fractions because they are equal in value.

3 __ 6

1 __ 2

Write an equivalent fraction for .

An important rule to remember when working with fractions is: “What you do to the top (numerator) you must do to the bottom (denominator).” Keep this rule in mind when writing equivalent fractions.

Write an equivalent fraction.

1.

2.

3.

4.

5.

6.

7.

8.

9. 10.

11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

3 __ 5

To write an equivalent fraction for :

1. Choose a number to multiply with. Let’s use 2.2. Multiply the top (numerator) by 2. 3 � 2 = 63. Multiply the bottom (denominator) by 2. 5 � 2 = 104. So, an equivalent fraction for is 6 over 10, or .

3 __ 5

6 ___ 10

3 __ 5

1 __ 2 1 __

6 2 __

5 2 __

7 3 __

4

2 __ 3 4 __

6 3 __

9 2 __

4 5 __

7

3 __ 8 4 __

7 3 __

6 6 __

8 1 ___

12

4 ___ 10

5 __ 9 6 ___

12 7 ___

10 4 __

5

2 __ 8 8 ___

10 3 ___

11 2 ___

12 6 __

6


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NS 2.3 Solve simple problems, including ones arising in concrete situations, involving addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less) and express answers in the simplest form.

Simplest FormA fraction is in simplest form when the numerator and denominator only have 1 as a common divisor. You can write a fraction in simplest form by dividing the numerator and denominator by the greatest number that will divide into both the numerator and denominator. This number is the greatest common factor (GCF).

Name the GCF of the numerator and denominator for .

One way to find to the greatest common factor (GCF) of the numerator and denominator is to list the factors of the lesser number and eliminate the ones that aren’t common.

1. Since 8 is the lesser number of 8 and 20, list the factors of 8: The factors of 8 are 1, 2, 4, and 8.2. Now eliminate the numbers that are not factors of 20: Is 8 a factor of 20? No. Is 4 a factor of 20? Yes.

3. 4 is the greatest common factor of 8 and 20.

8 ___ 20

Name the GCF of the numerator and denominator.

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Write each fraction in simplest form.

11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

4 __ 8 6 __

9 3 ___

12 8 ___

16 6 ___

18

20 ___ 90

21 ___ 35

30 ___ 45

4 __ 7

5 ___ 10

2 __ 4 3 __

9 5 ___

18 4 ___

12 16 ___

24

8 __ 9 25 ___

30 18 ___

27 28 ___

49 13 ___

13


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Understand Mixed NumbersA mixed number is any fraction greater than 1 that contains a whole number and a fraction. An improper fraction is any fraction greater than 1 that does not contain a whole number.

Write the fraction as a mixed number.

Write each mixed number as a fraction. Write each fraction

as a mixed number.

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

Step 1. Divide . The answer is 5 r 2.

Step 2. The whole number part is 5.

Step 3. For the fraction part, the remainder is the numerator and the divisor is the denominator. The fraction is .

Step 4. Add the whole number and fraction to get .

To change to a mixed number, divide the denominator into the numerator.

17 ___ 3

17 ___ 3

5 2 __ 3

2 __ 3

3 � � 17

7 __ 4 11 ___

2 12 ___

5 1 5 __

6 2 1 __

3

19 ___ 6 3 3 __

8 5 2 __

5 29 ___

9 18 ___

7

17 ___ 10

6 4 __ 5 4 7 ___

12 23 ___

3 7 1 __

8

35 ___ 11

9 9 ___ 10

5 5 __ 8

2 4 __ 9

65 ___ 6

47 ___ 4 19 ___

2 7 4 __

5 83 ___

9 6 3 ___

10


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Compare and Order Fractions and Mixed NumbersFractions and mixed numbers can be compared by using fraction bars. You can also use common multiples to find common denominators for both fractions and then compare the numerators.

Compare. Write <, >, or = for .


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

Use fraction bars to compare. The row with five -bars is longer than the row with

three -bars, so is less than ( < ) .

4 __ 9 5 __

9

5 __ 7 3 __

5

3 1 __ 8 3 ___

24

6 6 __ 7 6 11 ___

14

7 ___ 12

15 ___ 24

7 ___ 12

3 ___ 10

7 ___ 20

4 11 ___ 16

4 5 __ 8

2 7 ___ 10

2 8 ___ 15

1 1 __ 3

1 8 ___ 21

2 2 __ 3

2 5 __ 6

1 __ 2

5 __ 6

4 __ 5

10 ___ 11

2 __ 8

1 __ 4

13 ___ 15

4 __ 5

2 __ 3

12 ___ 18

3 __ 8 5 ___

12

3 __ 8

1 __ 8

1 ___ 12

5 ___ 12

3 � 3 _____ 8 � 3

� 9 ___ 24

5 � 2 ______ 12 � 2

� 10 ___ 24

1. You can also compare the fractions using common denominators. First, find a common multiple of the denominators, 8 and 12.Multiples of 8: 8, 16, 24, 32, 40, 48, … Multiples of 12: 12, 24, 36, 48, 60, … Let’s use 24.

2. 8 � 3 = 24, so 12 � 2 = 24, so

3. Now compare the numerators: 9 < 10, so < . 3 __ 8

5 ___ 12

3 ___ 12


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Problem Solving Strategy: Make a ModelAmber, Marcus, Paul, and Shelly lined up to make their

jumps. Shelly was not first. Amber has at least two people

ahead of her. Paul is third. Give the order of the four.

Read to Understand

2.Plan

What strategy can you use to solve the problem?

3.Solve

How can you use the strategy to solve the problem?

4. Finish the model. Give the order of the four people.

Check

5. What other strategy could you use to solve the problem?

Make a table to solve.

Bernice threw the shot put

feet. Terry threw the shot

put ft and Carla threw for

ft. Who threw the shot put the

longest distance? Who threw for

the shortest distance?

Jeremy, Alexis and Travis are taller than Ruth. Jeremy is taller than two people. Travis is not the tallest. Give the order of the four from shortest to tallest.

7.6.

What are you asked to do?

10 5 __ 9

10 4 __ 7 10 2 __

5

Paul Amber

1 2 3 4

1.

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NS 1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

Relate Fractions and DecimalsYou can represent a fraction as a decimal and vice versa. For

example, 0.5 is the decimal equivalent for .

Write the fraction as a decimal.

Step 1. Use a hundredths grid to write a decimal. Since the denominator of the fraction is not a power of 10, multiply the numerator and denominator by 5.

Step 2. Use a hundredths grid and shade 55 squares.

Step 3. Since 55 hundredths squares are shaded, the decimal is 0.55.

So, = 0.55.

11 ___ 20

1 __ 2

11 ___ 20

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

1. 0.8 2. 3. 0.78 4. 0.45 5.

6. 7. 0.7 8. 0.92 9. 10. 0.35

11. 12. 13. 0.4 14. 15.

16. 0.58 17. 18. 0.36 19. 20. 0.32

11 � 5 ______ 20 � 5

� 55 ____ 100

3 __ 4 66 ____

100

8 ___ 16

43 ____ 100

1 __ 4 1 ___

20 22 ___

25 16 ___

25

9 ___ 10

7 ___ 35


Name

Identify a decimal and a fraction for point A.

Step 1. Count how many times the number line is divided from 0 to 1.

Step 2. Count how many twentieths Point A is from 0.

Step 3. Put the answer over 20 to make a fraction.

Step 4. Simplify. Divide the numerator and denominator by the GCD.

Step 5. Divide the denominator into the numerator to convert the fraction into a decimal.

There are 20 segments dividing the line from 0 to 1, so the number line is divided into twentieths.

Point A is 16 segments from 0.

A

16 ___ 20

16 � 4 ______ 20 � 4

� 4 __ 5

5 � � 4 � 0.8

Identify a decimal and a fraction for the point.

1. Point C 2. Point A 3. Point E 4. Point B 5. Point D

For 6–10, locate each mixed number or decimal on one number line.

Then write the numbers in order from least to greatest.

6. 1.4 7. 8. 1.85 9. 10. 1.551 3 ___ 10

1 3 __ 4

A B C D E



Use a Number LineWhen identifying points on a number line, use benchmark fractions for reference. Benchmark fractions are familiar fractions that are sometimes labeled on the line.


Name


NS 2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers like or unlike denominators of 20 or less, and express answers in the simplest form.

There are 4 tenths shaded, and . So .

Find the sum or difference. Write it in simplest form.

1. 2. 3. 4.

Model Addition and SubtractionYou can use rectangles to add fractions.

Add.

Step 1

The denominator of each fraction is 8. Divide a rectangle into 8 equal parts.

Step 2

The first addend is 3 __ 8 . Shade 3 of the 8 parts.

Step 3

The second addend is 2 __ 8 . Shade 2 more of the 8 parts.

Step 4

Count the number of shaded eighths.

There are 5 eighths shaded, so .

You can also use rectangles to subtract fractions.

Subtract.

Step 1

The denominator of each fraction is 10. Divide a rectangle into 10 equal parts.

Step 2

The first fraction

is 9 ___ 10 . Shade 9 of the

10 parts.

Step 3

The subtracted

fraction is 5 ___ 10 .

Cross off 5 tenths.

Step 4

Count the number of shaded tenths left.

3 __ 8 � 2 __

8 � 5 __

8

9 ___ 10

� 5 ___ 10

4 ___ 10

� 2 __ 5

9 ___ 10

� 5 ___ 10

� 2 __ 5

1 __ 6 � 5 __

6 3 ___

10 � 5 ___

10 4 __

5 � 3 __

5 5 __

6 � 3 __

6

3 __ 8 � 2 __

8


Name


NS 2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form.

Add and Subtract Like FractionsYou can use a number line to add fractions.

Add.

Step 1

The denominator of each fraction is 8. Divide a number line into 8 equal parts.

Step 2

The first addend is 3 __ 8 . Start at 0 and jump 3 eighths.

Step 3

The second addend is 2 __ 8 . Start at 3 eighths and jump 2 more eighths.

You land at 5 eighths, so .

You can also use a number line to subtract fractions.

Subtract.

Step 1

The denominators are 10. Divide a number line into 10 equal parts.

Step 2

The first fraction is .

Start at 0. Jump 9 tenths.

Step 3

The subtracted fraction is .

Jump back 5 tenths.

You land at 4 tenths and so, .


1. 2. 3.

3 __ 8 � 2 __

8

3 __ 8 � 2 __

8 � 5 __

8

9 ___ 10

� 5 ___ 10

9 ___ 10

5 ___ 10

4 ___ 10

� 2 __ 5 9 ___

10 � 5 ___

10 � 2 __

5

1 __ 6 � 5 __

6 3 ___

10 � 5 ___

10 4 __

5 � 3 __

5


Name



Add and Subtract Like Mixed NumbersYou can use rectangles to add and subtract mixed numbers.

Add.

Step 1

Represent 2 2 __ 3 with

rectangles.

Step 2

Represent 1 2 __ 3

with rectangles.

Step 3

Combine.

Step 4

Record.

3 4 __ 3

� 4 1 __ 3

So, .

Subtract.

Step 1

Represent 3 3 __ 4 with

rectangles.

Step 2

Cross out 2 1 __ 4

rectangles.

Step 3

Record The rectangles and their parts that are not crossed out.

Step 4

Simplify.

So, .

1 2 __ 4

1 2 __ 4

� 1 1 __ 2

2 2 __ 3 + 1 2 __

3

2 2 __ 3 + 1 2 __

3 � 4 1 __

3

3 3 __ 4 � 2 1 __

4

3 3 __ 4 � 2 1 __

4 � 1 1 __

2


1. 2. 3. 1 5 __ 6 � 1 1 __

6 2 3 __

4 � 1 3 __

4 3 5 __

8 � 1 3 __

8


Name



So, .

Find the difference. Write it in simplest form.

1. 2. 3.

Subtraction with RenamingYou can rename a mixed number as a fraction to subtract.

Subtract.

Step 1Use rectangles to represent the first mixed number. Shade rectangles to represent 4 3 __

8 .

Step 2

You cannot subtract 3 __ 8 – 5 __ 8 .

Rename 1 unit rectangle as 8 __ 8 , so,

4 3 __ 8 = 3 11 ___

8 .

Step 3

To subtract 1 5 __ 8 , cross out 1 unit

rectangle and 5 eighths.

Step 4Count the remaining units and eighths and record the number.

2 units + 6 eighths =

Step 5Simplify.

4 3 __ 8 � 1 5 __

8

2 6 __ 8

2 6 __ 8

� 2 3 __ 4

4 3 __ 8 � 1 5 __

8 � 2 3 __

4

2 3 __ 5 � 1 4 __

5 5 1 __

4 � 2 3 __

4 4 1 __

6 � 2 5 __

6


Name



Problem Solving Workshop Strategy: Work BackwardTwenty minutes before the puppet show’s starting time, students were still working on

the stage. Molly spent 12.5 minutes stapling the skirt and then passed the stapler to

Jordan, who spent 6.75 minutes stapling the roof. By the time they were finished, how

many minutes were there until show time?

Read to Understand

2.

Plan

What part of the problem is unknkown?

3.

Solve

How can you work backward to solve the problem?

4. Show how you solve the problem.

Check

5. How can you check to see that your answer is reasonable?

Work backward to solve.

Jill spent $4.50 for lunch and $32.95 on shoes. She had $8.10 left. How much did she start with?

Hank boarded the bus 30 minutes before school started and rode 10 minutes. It was 8:20 A.M when he arrived at school. When does school start?

7.6.

1. List the events from start to finish.


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Model Addition of Unlike FractionsYou can use fraction circles to help you add fractions with unlike denominators. Trade fractions circle pieces of fractions with unlike denominators for equivalent pieces of fractions with like denominators.

Add.

Step 1

Use a 1 __ 4 piece and two

1 ___ 12

pieces to model the fractions

with unlike denominators.

Step 2

Trade each 1 __ 4 piece for

three 1 ___ 12

pieces.

Step 3

Add the fractions withlike denominators.

So, .

Find each sum. Write it in simplest form.

1. 2. 3. 2 __ 3 � 1 __

4 3 __

8 � 3 __

4 5 __

6 � 7 ___

12

1 __ 4 � 2 ___

12

1 __ 4 � 2 ___

12 � 3 ___

12 � 2 ___

12

3 ___ 12

� 2 ___ 12

� 5 ___ 12

1 __ 4 � 2 ___

12 � 5 ___

12


Name



Model Subtraction of Unlike FractionsYou can use fraction circles to help you subtract fractions with unlike denominators. Trace fraction circle pieces of fractions with unlike denominators for equivalent pieces of fractions with like denominators.

Subtract.

Step 1

Use a 1 __ 2 piece to model

the first fraction.

Step 2

Trade the 1 __ 2 piece for

five 1 ___ 10

pieces.

Step 3

Subtract the fractions withlike denominators.

So, .

Find each difference. Write it in simplest form.

1. 2. 3. 2 __ 3 � 1 __

4 7 __

8 � 1 __

2 5 __

6 � 1 __

3

1 __ 2 � 1 ___

10 � 5 ___

10 � 1 ___

10

1 __ 2 � 1 ___

10 � 4 ___

10 , or 2 __

5

5 ___ 10

� 1 ___ 10

� 4 ___ 10

1 __ 2 � 1 ___

10


Name

© Harcourt • Grade 5Reteach the StandardsRW50 NS 1.1 Estimate, round, and manipulate very large (e.g.,


Estimate Sums and DifferencesYou can round fractions to 0, to 1 __

2 ,or to 1 estimate sums and differences.

Estimate the sum.

Use the number line to estimate whether each fraction is closest to 0, to1 __ 2 ,or to 1. Then find the sum or difference. The first one is done for you.

1. 2. 3.

4.

5. 6.

4 __ 6 � 1 __

9

Step 1

Find 4 __ 6 on the number line.

Is it closest to 0, 1 __ 2 ,or 1?

The fraction 4 __ 6 is closest to 1 __

2 .

Step 2

Find 1 __ 9 on the number line.

Is it closest to 0, 1 __ 2 ,or 1?

The fraction 1 __ 9 is closest to 0.

Step 3

To estimate the sum 4 __ 6 � 1 __

9 ,

add the two rounded numbers.

So, 4 __ 6 � 1 __

9 is about 1 __

2 .

1 __ 2

� 0 � 1 __ 2

4 __ 6 � 1 __

8

1 __ 2 � 0

2 __ 6

� 7 __ 8

4 __ 6 � 3 __

8

5 __ 6

� 3 __ 8

� �

7 __ 8

� 5 __ 6

1 __ 6

� 7 __ 8

� � �

1 __ 2


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Use Common DenominatorsTo add or subtract unlike fractions, you need to rename them as like fractions. You can do this by making a list of equivalent fractions. When you find two fractions with the same denominator, they are like fractions.

Example 1: Add. 5 ___ 12

� 1 __ 3

Step 1 Write equivalent fractions for 5 ___ 12

. 5 ___ 12

, 10 ___ 24

, 15 ___ 36

, 20 ___ 48

Step 2 Write equivalent fraction for 1 __ 3

. 1 __ 3

, 2 __ 6

, 3 __ 9

, 4 ___ 12

Step 3 Rewrite the problem using the equivalent fractions.

Then add.

Example 2: Subtract 9 ___ 10

� 1 __ 2

Step 1 Write equivalent fractions for 9 ___ 10

. 9 ___ 10

, 18 ___ 20

, 27 ___ 30

, 36 ___ 40

Step 2 Write equivalent fraction for 1 __ 2

. 1 __ 2

, 2 __ 4

, 3 __ 6

, 4 __ 8

, 5 ___ 10

Step 3 Rewrite the problem using the equivalent fractions.

Then subtract.

5 ___ 12

� 1 __ 3 � 5 ___

12 � 4 ___

12 � 9 ___

12 , or

3 __ 4

9 ___ 10

� 1 __ 2 � 9 ___

10 � 5 ___

10 � 4 ___

10 , or 2 __

5

Stop when you find2 fractions with like denominators.

Stop when you find2 fractions with like denominators.


1. 2. 3. 4.

5. 6. 7. 8.

3 __ 5 � 1 __

3

7 __ 8 � 1 __

4

1 __ 2 � 2 __

5

3 __ 4 � 2 __

3

1 __ 4

� 1 __ 6

9 ___ 10

� 4 __ 5

1 __ 5

� 3 __ 4

8 __ 9

� 5 __ 6


Name



1. 1 __ 4 � 3 __

8 � 2. 2 __

3 � 1 __

2 � 3. 1 __

3 � 5 __

6 �

4. 5 __ 8 � 1 __

4 � 5. 3 __

5 � 3 ___

10 � 6. 3 __

4 � 1 __

2 �

Add and Subtract FractionsYou can use the least common denominator toadd and subtract unlike fractions.

Subtract.

• List the multiples of 3: 3, 6, 9, 12, 15

the denominators 3 and 4. 4: 4, 8, 12, 16, 20

• The least common multiple

is 12. The least common

denominator of 2 __ 3 and 1 __

4 is 12.

• Write equivalent fractions.

Then subtract.

So, 2 __ 3 � 1 __

4 � 5 ___

12 .

Add. 5 __ 8 � 1 __

3

• List the multiples of 8: 8, 16, 24, 32

the denominators 8 and 3. 3: 3, 6, 9, 12, 15, 18, 21, 24

• The least common multiples

is 24. The least common

denominator of 5 __ 8 and 1 __

3 is 24.

• Write equivalent fractions.

Then add.

So, 5 __ 8 � 1 __

3 � 23 ___

24 .

2 __ 3 � 1 __

4

2 __ 3

� 2 � 4 ______ 3 � 4

� 8 ___ 12

� 1 __ 4

� 1 � 3 ______ 4 � 3

� � 3 ___ 12

5 ___ 12

5 __ 8

� 5 � 3 ______ 8 � 3

� 15 ___ 24

� 1 __ 3

� 1 � 8 ______ 3 � 8

� � 8 ___ 24

23 ___ 24



Name



Problem Solving Workshop Strategy: Compare StrategiesDavid bought some supplies for the science project. He spent $3.99 for poster board,

$1.24 for a glue stick, and $4.55 for color pencils. If David had $1.57 when he left the

store, how much money did he have before his purchases?

Read to Understand

2.

Plan

How can comparing strategies help you solve the problem?

3.Solve

4. Solve the problem. Describe the strategy you used.

Check

5. How can you check your answer?

Compare strategies to solve.

During January and February, Jeff

grew a total of 7 __ 8 inch. In February he

grew 1 __ 4 inch. How much did Jeff grow

in January?

Central School has 5th-, 6th-, 7th-, and

8th-graders. If 3 __ 8 are 6th-graders, 1 __ 4 are

7th-graders, and 1 __ 6 are 8th-graders,

what fraction are 5th-graders?

7.6.


Solve the problem. Describe the strategy you used.

Make a Model

Work Backward


Name

RW54 NS 2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denomination of 20 or less), and express answers in the simplest form.


Reteach the Standards

Model Addition of Mixed NumbersFraction bars can help you add mixed numbers.

Add.

STEP 1 Model each addend.

STEP 2 Use like fraction bars to model and .

STEP 3 Count like bars to find the sum.

Three 1 bars and five bars is .

STEP 4 Rename .

5 _ 4 � 1

1 _ 4, so 3

5 _ 4 � 3 � 1

1 _ 4 � 4

1 _ 4.

So, 1 3

_ 4 � 2

1 _ 2 � 4

1 _ 4.

Use fraction bars to find the sum. Write it in simplest form.

1 3 _ 4 � 2

1 _ 2.

1 3

_ 4

2 1 _ 2

3 _ 4

1 _ 2

1 3

_ 4

2 2 _ 4

1 _ 4 3

5 _ 4

5

_ 4

1 1

_ 4

3 5 _ 4

1. 5 1 __ 10 � 2

3 __ 10 � 2. 2

3 _ 8 � 1

1 _ 4 � 3. 3

1 _ 2 � 4

1 _ 5 �

4. 4 2 _ 3

� 2 3 _ 4

5. 5 5

_ 8

� 4 1

_ 2

6. 2 5

_ 6

� 3 1

_ 6

7. 3 7 __ 10

� 6 5 _ 4


Name




Model Subtraction of Mixed NumbersFraction bars can help you subtract mixed numbers.

Subtract.

STEP 1 Draw a picture to show .

STEP 2 Subtract 1 1 __ 8 . Rename the fourths using bars.

STEP 3 Cross out bars to subtract. Count the bars that are left.

So, .

Use fraction bars, or draw a picture to find the difference.

Write it in simplest form.

2. 3 4

_ 5 � 2

1 __ 10 �1. 6

1 _ 3 � 4

1 _ 6 � 3. 6

3 _ 4 � 4

1 _ 8 �

4. 4 2 _ 5 � 1

3 __ 10 � 5. 5

5 _ 6 � 2

1 _ 4 � 6. 2

2 _ 3 � 1

5 __ 12 �

2 3 __ 4 � 1 1 __

8

2 3 __ 4

1 __ 8

2 3 __ 4 � 2 6 __

8

2 6 __ 8 � 1 1 __

8 � 1 5 __

8 .

2 3 __ 4 � 1 1 __

8 � 1 5 __

8

2 6 __ 8

2 3 __ 4


Name




Record Addition and SubtractionWhen you add or subtract mixed numbers, you may need torename the fractions as fractions with a common denominator.

Subtract .

Step 1 Model .

Step 2 The LCD for and is fourths, so rename as .

Step 3 Since there are not enough fourths to subtract from rename as .

Step 4 Subtract.

The answer is already in simplest form.

So,


1. 2 2 _ 9 � 4

1 _ 6 � 3. 11

7 _ 8 � 9

5 _ 6 � 4. 18

3 _ 5 � 14

1 _ 2 � 2. 10

5 _ 6 � 5

3 _ 4 �

2 1 __ 2 � 1 3 __

4

2 1 __ 2

1 __ 2 3 __

4 2 1 __

2 2 2 __

4

1 3 __ 4

2 1 __ 2

2 1 __ 2

1 6 __ 4

2 1 __ 2 � 1 3 __

4 � 3 __

4 .


Name




Subtraction with RenamingWhen you subtract mixed numbers you may need to rename the whole numbers.

Add.

2. 5 3 __ 10 � 3

7 __ 10 �1. 3

1 _ 8 � 1

5 _ 8 � 3. 4

1 _ 3 � 1

5 _ 6 �

4. 6 1 _ 3 � 2

3 _ 4 � 5. 5

1 _ 6 � 2

5 _ 6 � 6. 4

1 _ 5 � 1

3 _ 5 �

5 3 __ 4 � 1 1 __

3

Step 1 Use the least common multiple of 4 and 3 to rename and as fractions with common denominators.

5 3 __ 4

1 1 __ 3

4: 4, 8, 12, 16, 20 . . .3: 3, 6, 9, 12, 15 . . . .

The LCD is 12.

Step 2 Rename the fractions.

3 __ 4 � 3 � 3 ______

4 � 3 � 9 ___

12

1 __ 3 � 1 � 4 ______

3 � 4 � 4 ___

12

So, 5 3 __ 4

� 5 9 ___ 12

So, 1 1 __ 3

� 1 4 ___ 12

Step 3 Add the fractions.

5 9 ___ 12

�1 4 ___ 12

__

13 ___ 12

Step 4 Add the whole numbers. Write the answer in simplest form.

Use fraction bars to find the difference. Write it

in simplest form.

5 3 __ 4 � 5 9 ___

12

1 1 __ 3 � 1 4 ___

12 �

6 13 ___ 12

� 7 1 ___ 12

So, 5 3 __ 4 � 1 1 __

3 � 7 1 ___

12


Name




Problem Solving WorkshopSkill: Sequence Information

9. In July, the lake’s level was 12,053 feet. During June, the lake’s level fell 2 feet. During May, the lake’s level rose 6 feet. What was the lake’s level in May?

8. Lance bikes a total of 7 miles. He bikes 3 miles from his house to the trail. He stops at the store before biking 1 miles home. How far is the trail from the store?

Use fraction bars to find the difference. Write it in simplest form.

Allison walked 4 miles around town. She walked 1 miles from home tothe park, and then walked to Jessie’s house before going home. Allisonlives 1 miles from Jessie. How far does Jessie live from the park?

1. What are you aked to find?

2. What did Allison do first?

3. What did she do next?

4. What did she do last?

5. Complete the table. List the things Allison did in order of when they happened. Find the missing information. The first row has been completed for you.

Allison walked from to . Distance

fi rst home to the park 1 1 __ 2 mi

next

last

6. What is the answer to the question?


1 _ 2

3 _ 4

7 _ 8

5 _ 8

3 _ 4

1 _ 2 3 _ 4

5 _ 8


Name

© Harcourt • Grade 5Reteach the StandardsRW59 NS 2.4 Understand the concept of multiplication and

division of fractions.

Model Multiplication of FractionsYou can use grid paper to model multiplication of fractions. You will use the grid to make and shade a rectangle.

Find the product:

Use the denominator of the first factor, 8, to make columns.Use the denominator of the second factor, 3, to make rows.

• The area of the rectangle is 8 � 3, or 24 square units.

Next, shade of the first row.

Shade of the rows in the same way as the first row.

• The area of the shaded rectangle is 3 � 2, or 6 square units.

• The fraction of the rectangle that is shaded is , or .

So, .

Use the model to find the product.

1.

2. 3. 4.

Find the product.

5.

6. 7. 8.

9. 10. 11. 12.

3 __ 8 2 __

3 �

3 __ 8

2 __ 3

6 ___ 24

1 __ 4

3 __ 8 2 __

3 � � 1 __

4

1 __ 2 � 1 __

3 = 1 __

6 � 3 __

4 = 3 __

5 � 2 __

3 � 1 __

3 � 1 __

3 �

1 __ 5 � 2 __

3 � 3 __

4 � 2 __

5 � 5 __

8 � 4 __

5 � 3 __

5 � 1 __

3 �

2 __ 3 � 1 __

5 � 3 __

4 � 3 __

4 � 2 __

3 � 7 __

8 � 2 __

3 � 5 __

6


Name



Record Multiplication of FractionsYou can use an area model to record multiplication of fractions.

Multiply.

Look at the denominators. They tell you to make a rectangle with 9 columns and 4 rows.

• The rectangle has an area of 36 square units.

Look at the numerators. They tell you to shade a rectangle that has 8 columns and 3 rows.

• The shaded rectangle has an area of 24 square units.

Use the two areas to write a fraction.

•

So, .

Find the product. Write it in simplest form.

8 __ 9 � 3 __

4

8 __ 9 � 3 __

4 � 2 __

3

area of shaded rectangle

_______________________ area of big rectangle

� 24 ___ 36

� 24 � 12 ________ 36 � 12

� 2 __ 3

1. 2. 3.

4. 5. 6.

7. 8. 9.

3 __ 5 � 5 __

6 = 4 __

9 � 3 __

8 = 1 __

2 � 6 __

7 =

2 __ 3 � 9 ___

10 = 5 __

8 � 2 __

3 = 2 __

3 � 4 __

5 =

1 __ 3 � 2 __

5 = 9 ___

10 � 5 ___

12 = 1 __

8 � 4 __

9 =


Name

© Harcourt • Grade 5Reteach the StandardsRW61NS 2.4 Understand the concept of multiplication and


Multiply Fractions and Whole NumbersYou can use a model to multiply a fraction and a whole number.

Multiply.

Step 1 Draw 10 rectangles.

Step 2 The denominator of the fraction 4 __ 5 is 5.

This means there are 5 equal parts, so divide the rectangles into 5 equal groups.

Step 3 The numerator of the fraction 4 __ 5 is 4.

This means there are 4 parts given, so shade 4 of the groups.

Step 4 Count the shaded rectangles. There are 8 rectangles.

So, 10 � 4 __ 5 � 8.

10 � 4 __ 5

1. 2.

3. 4. 5.

6. 7. 8.

5 � 3 __ 4 � 5 � 3 �

2 __ 5

� 4 � 4 � 2 �

2 � 1 __ 4 � 2 __

3 � 5 � 9 � 1 __

5 �

1 __ 3 � 6 � 1 __

4 � 10 � 2 � 5 __

6 �

Find the product.


Name



Multiply with Mixed NumbersYou can use a multiplication square to multiply mixed numbers.

Multiply.

Step 1

Write each mixed number as a fraction.

First multiply the denominator by the wholenumber, then add the numerator.

Step 2Multiply the fractions.

Step 3Write the product as a mixed number insimplest form

So,

1. 2. 3.

4. 5. 6.

7. 8. 9.

2 1 __ 2 � 3 1 __

5 � 3 __

8 � 2 2 __

3 � 2 5 __

8 � 1 1 __

2 �

1 4 __ 5 � 2 1 __

5 � 1 3 __

4 � 1 2 __

5 � 2 2 __

3 � 2 2 __

3 �

1 1 __ 3 � 1 __

2 � 3 5 __

6 � 1 1 __

3 � 1 5 __

8 � 3 1 __

3 �

Find the product.

1 1 __ 2 � 2 1 __

4 .

2 1 __ 4

� (4 � 2) � 1

___________ 4

� 9 __ 4

1 1 __ 2

� (2 � 1) � 1

___________ 2

� 3 __ 2

3 __ 2

� 9 __ 4

� 27 ___ 8

1 1 __ 2 � 2 1 __

4 � 3 3 __

8 .

27 ___ 8

� 3 3 __ 8


Name



Model Fraction DivisionYou can use pictures to model division of fractions.

Divide.

Step 1 Draw one whole rectangle and shade one fifth of it

Step 2 Divide the rectangle into tenths.

Step 3 Count the number of shaded tenths.

There are 2 tenths in 1 __ 5 . So, 1 __ 5 � 1 ___ 10 � 2.

Find the quotient. Record a number sentence.

1. 2. 3. 4.

5.

6.

7.

8.

1 � 1 __ 4 2 � 1 __

3 3 � 1 __

2 2 � 1 __

4

1 � 1 __ 5 3 � 1 __

3 2 � 1 __

5 3 � 1 __

4

1 __ 5 � 1 ___ 10 .

1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10


Name



Divide Whole Numbers by FractionsDivide.

Step 1Rewrite as a division sentence.

Step 2

Use the reciprocal of the divisor to write amultiplication problem.

Step 3Multiply

So,

1. 2. 3. 4.

5. 6. 7. 8.

1 � 3 __ 4 3 � 3 __

5 6 � 2 __

3 4 � 1 __

2

4 � 2 __ 5 4 � 2 __

3 3 � 6 __

7 1 � 5 __

8

Find the quotient. Write it in simplest form.

4 � 7 __ 8 .

4 __ 1

� 7 __ 8

4 __ 1

� 8 __ 7

4 __ 1

� 8 __ 7

� 32 ___ 7

� 4 4 __ 7

4 � 7 __ 8 � 4 4 __

7

Think: Write 4 as 4 __ 1 .

Think: The reciprocal

of 7 __ 8 is 8 __ 7 .


Name



Divide FractionsWe can rewrite all division problems as multiplication problems.

Question: How many are in ?

Divide.

Write the division problem as a multiplication problem.

Step 1 Locate the divisor. Step 2 Write the reciprocal of the divisor.

Step 3 Write the multiplication problem.

The divisor is always the second number.

The divisor in is .

To write the reciprocal, switch the numerator and denominator of the divisor.

The reciprocal of is .

Replace the divisor with its reciprocal and change � to �.

2 1 __ 4 � 3 __

5 3 __

5 3 __

5 5 __

3 9 __

4 � 3 __

5 � 9 __

4 � 5 __

3

3 __ 5 2 1 __

4

2 1 __ 4 � 3 __

5

So, .

Write a division sentence for each model.

1. 2.

Divide. Write the answer in simplest form. 3.

4.

5.

6.

7.

8.

9.

10.

2 1 __ 4 � 3 __

5 � 2 1 __

4 � 5 __

3

2 1 __ 8 � 3 __

4 5 __

9 � 5 __

6 4 3 ___

10 � 1 __

5 3 2 __

7 � 5 ___

14

5 3 __ 4 � 1 5 __

8 4 1 __

2 � 6 2 __

3 10 � 2 ___

15 2 1 ___

12 � 4 2 __

3


Name



Problem Solving Workshop Skill: Multistep ProblemsEllie has -yard of ribbon to make hair bows. She used -yard

of ribbon for each bow. She gave of the bows to her

friends. How many bows did Ellie give to her friends?

Read to Understand

2.Plan

What must you determine before you can solve the problem?

3.Solve

What steps will you use to solve the problem? Show how to solve the problem in the space provided.

4. How many bows did Ellie give to her friends?

Check


Describe the steps required to solve the problem. Then solve.

Twelve members of Sam’s scout troop do not own a pet. This is of the entire troop. How many do own a pet?

Mollie stretches for hour, cools down hour, and jogs the rest of the hour. What part of the hour does she jog?

7.6.

1. What does the problem ask you to find?

2 __ 3 1 __

6

1 __ 2

3 __ 4

1 __ 6

1 ___ 12


Name


NS1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.

Underline: 0.807

tenths place

Round DecimalsYou can use the same rules you learned for rounding whole numbers to round decimals.

Step 1: Underline the digit to the place to which you want to round.

Step 2: Compare the digitto the right of theunderlined digit to 5 using the rounding rules.

Step 3: Rewrite all digits to the right of the underlined digit as zeros. An equivalent decimal can be written by leaving off the trailing zeros.

ROUNDING RULES:

• If the digit to the right is less than 5, the underlined digit stays the same.

• If the digit to the right is greater than or equal to 5, the underlined digit increases by 1.

Round each number to the place of the underlined digit.

1. 7.325 2. 9.028 3. 108.108 4. 26.199

5. 12.63

6. 11.323

7. 4.289 8. 7.547

9. 0.964 10. 20.595 11. 6.89 12. 32.514

Name the place to which each number was rounded.

13. 12.35 to 12.4 14. 0.428 to 0.43 15. 9.462 to 9.46

�Compare: 0 < 50 is less than 5, sothe digit stays the same.

Rewrite: 0.800 or 0.8

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Name


NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

1. 2. 3.

5. 6. 7. 8.

Add and Subtract DecimalsTo add or subtract decimals, line up the decimal points in the numbers. Estimate the answer first. It will help you determine if your answer is reasonable.

Subtract. 13.04 – 0.95

Estimate. Subtract.

13.04

� 0.95

13

� 1

12

�� 1 3. 0 4

� 0. 9 5

1 2. 0 9

The answer should be about 12. 12.09 is close to the estimate.So, the answer is reasonable.

Find the sum or difference.

1 5 8

� 4 5 3

1 8 5 2

� 3 7 3

6 3 9

2 1 8

� 7 8 5

0. 4 5

� 0. 7

4

� 6. 8

2. 9

� 0. 6 3

2 1. 4

� 1. 3 3

��

��


Name

© Harcourt • Grade 5Reteach the StandardsRW69 NS1.1 Estimate, round, and manipulate very large

(e.g. millions) and very small (e.g. thousandths) numbers. also NS2.0, MR2.1, MR2.5, MR3.1

Estimate Decimal Sums and DifferencesYou can estimate to find an answer that is close to the exact answer. You can use the rounding rules to help you estimate sums or differences.

E

Example 1

Estimate by rounding. 1.245 + 0.79 + 0.42

First, decide what place to round to. Since two of the addends have digits to the hundredths, tenths would be a good place to round to.

Round and add to estimate.

Example 2

Estimate by rounding. 0.35 – 0.128

Round and subtract to estimate.

Estimate by rounding.

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 0.427 + 0.711 12. 61.05 – 18.63

13. 40.512 + 30.399

Remember the rounding rules:

If the digit to the right of the place you are rounding to is:• less than 5, round down.

• greater than or equal to 5, round up.

1.247 1.2

0.82 0.8

3.4 + 3.4

5.4

+

EE

0.78 0.8

0.305 0.3

0.5

� �

EE

$29.38

� $42.75

__

7.6

� 2.15

__

0.443

�0.207

__

2.083 0.56� 0.41

1.731.4

+ 3.238

3.28

� 0.86

__

$23.07

� $7.83

__

51.234 � 28.4

6.17 � 3.5

15.27 � 41.8


Name

RW70NS 2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals.


Mental Math: Add and SubtractYou can use mental math to find decimal sums.

Add. 1.5 � 2.5 � 4.5

Add each number from left to right, one place-value at a time.

1.5 � 2.5 � 4.5 Use 1.5 and start with the next number to the right.

1.5 � 0.5 � 4.5 Add the ones of 2.5 to 1.5.

3.5 � 0.5 � 4.5 Add the tenths of 0.5 to 3.5.

4.0 � � 4.5 Add the ones of 4.5 to 4.0.

8.5 Add the tenths of 0.5 to 8.5.

The sum is 8.5.

You can use mental math to find decimal differences.

Subtract. 5.50 � 3.25

Subtract each number from left to right, one place-value at a time.

5.50 � 3.25

2.50 � 0.25 Subtract the ones of 3.25 from 5.50.

2.30 � 0.05 Subtract the tenths of 0.25 from 2.50.

2.25 Subtract the hundredths of 0.05 from 2.30.

The difference is 2.25.

Use mental math to find the sum or difference.

1. 52.09 � 3.8 2. 4.35 � 0.64 � 2 3. 1.72 � 3.13 � 2.11

4. 28.77 � 3.24 5. 16.56 � 4.3 6. 8.91 � 8.51

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Problem Solving Workshop Skill: Estimate or Find Exact AnswerIn a baseball-throwing contest, a score of 50 meters or more is

needed to advance to the final round. Jenna’s first two throws were

16.64 meters and 15.33 meters. How long does her last throw need

to be for her to advance to the final round?

1. What does the problem ask you to find?

2. Do the judges rely on estimates or exact answers? Explain how you know.

3. Will you estimate or find an exact answer to this problem?Explain your choice.

4. Show how you solve the problem in the space below.

5. What is the least distance Jenna must throw to advance to the final round?

Tell whether you need an estimate or an exact answer.

Then solve the problem.

A waitress charged Hannah $6.75 forlunch. Hannah wants to tip thewaitress 20% of $6.75. How muchshould she leave?

Lance is purchasing 4 new tires for hiscar. Each tire costs $110.60. What isthe total cost?

7.6.


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Find the product.


NS 2.1 Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify reasonableness of the results.

Model Multiplication by a Whole NumberAlong with hundredths models, you can also use rectangles and squares to represent multiplication expressions.

Use the models to find the product.

0.24 � 3

Use rectangles to represent tenths and squares to represent hundredths.

So, 0.24 � 3 � 0.72

So.

� 3 �

(0.2 � 0.04) � 3 (0.2 � 3) � (0.04 � 3)

Use models to find the product.

1. 0.41 � 2 � 2. 5 � 0.36 �

3. 4 � 0.12 4. 0.09 �6 5. 3 �0.32 6. 7 �0.25

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Name

Number of

Zeros in Whole

Number

Number of Places

to Move Decimal

Point

Answer

1. 017 � 1

1. 017 � 10

1. 017 � 100

1. 017 � 1,000

0

1

2

3

0

1

2

3

1. 017

10.17

101.7

1, 017


NS 2.1 Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

Algebra: Patterns in Decimal Factors and ProductsWhen you multiply a decimal number by 10, 100, 1,000 or 10,000, first count the number of zeros in the whole number. In your answer, move the decimal point one place to the right for every zero that you had counted.

You can use these patterns to find products.

Use patterns to find the product.

1. 0.185 � 1 � 0.185 � 10 � 0.185 � 100 � 0.185 � 1,000 �

2. 6.37 � 1 � 6.37 � 10 � 6.37 � 100 � 6.37 � 1,000 �

4. 0.008 � 1 � 0.008 � 10 � 0.008 � 100 � 0.008 � 1,000 �

5. $3.75 � 1 � $3.75 � 10 � $3.75 � 100 � $3.75 � 1,000 �

7. 1.313 � 10 � 1.313 � 100 � 1.313 � 1,000 � 1.313 � 10,000 �

8. $0.25 � 10 � $0.25 � 100 � $0.25 � 1,000 � $0.25 � 10,000 �


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NS 2.1 Add, subtract, multiply and divide with decimals; add with negative numbers; subtract positive integers from negative integers; and verify reasonableness of the results

Model Multiplication by a DecimalWhen you use models to multiply decimals, remember that each square in the hundredths grid represents 0.01. So, 25 square represents 0.25. One full grid represents the whole number, one.

0.7 � 0.70, or 70 squares Show 0.4 or 0.40 of 0.7 or 0.70.

28 squares, or 0.28 is the area where the shading overlaps.

Find the product.

1. 0.6 � 0.8 � 2. 0.5 � 0.5 � 3. 0.3 � 0.9 �

4. 0.4 � 0.2 � 5. 1.1 � 0.7 � 6. 1.6 � 0.8 �

Make a model to find the product.

7. 0.4 � 0.8 8. 0.9 � 0.2

Multiply. 0.4 � 0.7


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NS 1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.

Estimate ProductsWhen estimating to find the product of two decimals, use a range to determine if your answer is reasonable.Estimate the product.

0.64 � 0.35

0.64 � 0.35

Round 0.64 to 0.6, round 0.35 to 0.4. The product of these two estimates is 0.24.

Now use a range to determine if your answer is reasonable.

To find the lower part of the range, round both numbers down to 0.6 and 0.3. Multiply to result in the product 0.18.

To find the upper part of the range, round both numbers up to 0.7 and 0.4. Multiply to result in the product 0.28.

Your original estimated answer should fall between the other two estimates in the range. Does 0.24 fall between 0.18 and 0.28? Yes, therefore this estimate is reasonable.

Estimate the product.

1. 4.25 � 7.82 2. $26.83 � 11 3. 3.3 � 9.4 4. 1.6 � 8.8

5. 6.71 � 4.22 6. 29.3 � 0.31 7. $8.54 � 9 8. 7.474 � 12.08

9. $0.39 � 291 10. 6.6 � 8.2

13. 487.66 � 2.12

11. 7.6 � 9.217 12. 70.33� 0.89

14. 300.59 � 0.3 15. 0.409 � 1.47 16. $23.56 � 7


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NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonable ness of the results.

Place the Decimal PointWhen you multiply decimals, first multiply the factors as whole numbers. Use estimation to figure out where to place the decimal point in your answer. You can do this by estimating your factors. Multiply your estimated factors and place the decimal point in the answer. Your estimate should show that your actual answer is reasonable.


0.34 � 0.65

Multiply as whole numbers.

34 � 65

2,210

Now estimate the factors.

The decimal 0.34 will round to 0.3 and 0.65 will round to 0.7. Multiply.

0.3� 0.70.21

You know your actual answer should be close to 0.21, so the actual answer is 0.221.


721. � 1.3

0.92. � 0.4

0.53. � 0.7

3.64. � 0.8

4.15. � 5.3

7.96. � 6.6

12.4 7. � 8.5

3.43 8. � 9.3

60.2 9. � 2.6

5.9410. � 0.07

11. 194.6 � 0.2 12. 0.381 � 14 13. $4.50 � 9.5 14. 8.266 � 3

15. 6.95 � 5.3 16. 0.08 � 2 17. 1.11 � 1.1 18. $12.74 � 36.5


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© Harcourt • Grade 5Reteach the StandardsRW77 NS 2.1 Add, subtract, multiply, and divide with

decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

0.0711. � 0.6

Zeros in the ProductAfter you have multiplied decimals like they were whole numbers,start from the right end of the product and count to the left the total number of decimal places in the factors.Find the number of decimal places in the product.

0.05 � 0.003 How many decimal plces in 0.05? 2 (Remember, start counting from the right.)

How many decimal places in 0.003? 3 Add both answer together. How many total decimal places? 5

Find the product.

0.039. � 3

0.410. � 0.2

0.0913. � 0.05

1. 6 � 0.04 2. 0.002 � 7 3. 0.3 � 0.08 4. 1.5 � 0.06

5. 0.01 � 0.009 6. 0.003 � 0.087 7. 1.29 � 0.004 8. 0.675 � 0.00002

0.94 16. � 0.001

$7.11 15. � 0.008

1.022 14. � 0.06

0.0812. � 0.03

First multiply as whole numbers.

0. 0 0 3� 0. 0 5

0 0 0 0 1 5

You will have to count 5 five places to the left from the right of your answer to place the decimal point in its proper place.

So, 0.05 � 0.003 � 0.00015

Find the number of decimal places in the product.


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NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify reasonableness of the results.

Problem Solving Workshop Skill: Choose the OperationSuppose the total cost of a class trip, including transportation and lunch, is $278.00. Kelly’s teacher collects $7.50 from each of Kelly’s 20 classmates as well as from Kelly herself. Kelly’s teacher will use the money from the class trip fund to pay for the remaining cost. How much money will Kelly’s teacher need to take from the class trip fund?

1. What are you asked to find?

2. Which operations will you use to solve this problem?

3. Solve each operation separately.

$7.50 �21

4. How much money will Kelly’s teacher need to take from the class trip fund?

Tell which operations you would use to solve. Then solve the problem.

5. FAST FACT In 1940, Americans ate an average of 92.4 lb of red meat. They ate 1.21 times this amount of red meat in 2003. In 1940, Americans ate an average of 12.3 lb of poultry, and 5.79 times this amount of poultry in 2003. How many total pounds of poultry and red meat did the average American eat in 2003?

6. The population of California in 1990 was 29.8 million people. The population of California in 2000 was 1.14 times that of the Californian population in the year 1990. How many more people lived in California in 2000 than in 1990?

$278.00 � �


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Use decimal models or play money to model the quotient.

Record your answer.

1. 0.36 � 6 =

2. 1.95 � 3 =

3. $9.75 � 5 = 4. 9.44 � 4 =

5. 6.03 � 9 = 6. $7.32 � 6 = 7. 7.63 � 7 = 8. $3.04 � 8 =

9. 0.94 � 2 =

10. 8.56 � 4 =

11. $9.18 � 9 =

12. 2.15 � 5 =

13. $9.99 � 3 =

14. 4.88 � 8 =

15. 5.53 � 7 =

16. 8.28 � 6 =

Just as you did with whole numbers, you can use decimal models to divide decimals into groups.

Use decimal models to model the

quotient. Record your answer.

0.28 � 4

Show 0.28 using a decimal model.

Now divide the squares into four groups of the same size.

There are four groups of 7 squares. Since we are using hundredths models, the answer must be in hundredths.

So, 0.28 � 4 � 0.07.

Divide Decimals by Whole Numbers


Name

© Harcourt • Grade 5Reteach the StandardsRW80NS 1.1 Estimate, round, and manipulate very large

(e.g., millions) and very small(e.g., thousandths) numbers.

When dividing a decimal by a two-digit number, use compatible numbers for the dividend and divisor. Compatible numbers are numbers that are easy to divide.


3.409 � 83

Estimate Quotients

Think of some compatible numbers near 3.409 and 83:

• 3.2 and 80 are compatible.3.2 � 80 = 0.04

• 3.6 and 60 are compatible.3.6 � 60 = 0.03

Which compatible numbers are closer to the original problem?

• 3.2 and 80 are closer to 3.409 and 83.

• So, an appropriate estimate is 0.04.


1. 23.7 � 9 = 2. 46.8 � 5 = 3. 94.2 � 6 = 4. 87.5 � 3 =

5. 5.3 � 8 = 6. 6.4 � 9 = 7. 0.312 � 5 = 8. 503.1 � 4 =

9. 65.8 � 22 = 10. 91.7 � 18 = 11. 45.43 � 36 = 12. 72.89 � 49 =

Find two estimates for the quotient.

13. 47.8 � 59 = 14. 8.91 � 27 = 15. 7.42 � 35 = 16. 6.16 � 93 =


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Divide Decimals by Whole NumbersWhen dividing decimals, you can use fractions to represent decimals.

Use fractions to find the quotient.

5 � � 3.15

5 � � 34.15

• Rewrite each number as a fraction.

• Write 5 as a fraction: 5 = 5 __ 1

• Write 3.15 as a fraction: 3.15 = 315 ____ 100

• 5 � � 3.15 = 315 ____ 100

� 5 __ 1

• Use the reciprocal of 5 __ 1 to multiply: 315 ____ 100 � 1 __

5 = 315 ____ 500 .

• Simplify: 315 � 5 ________ 500 � 5 = 63 ____ 100

• Write 63 ____ 100 as a decimal: 63 ____ 100 = 0.63

So, 5 � � 3.15 = 0.63


1. 2. 3.

Write the quotient correctly.

4. 5. 6.

2 � � 85.12 17 � � $99.28

47 � � 164.5 9 � � 78.84 6 � � 1.71 35 876 0285


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Divide Decimals by DecimalsModels such as hundredths grids and play money show how to divide decimals into groups.

Use a model to find the quotient.

0.24 � 0.03

Use a hundredths model to shade 24 hundredths.

Divide the shaded 24 hundredths sothat there are3 hundredths ineach group.

There are 8 groups of 3 hundredths in24 hundredths, so 0.24 � 0.03 = 8.

Use a model to find the quotient.

1. 6.3 � 0.9 = 2. 2.5 � 0.5 = 3. 1.4 � 0.2 =

4. 7.2 � 0.8 = 5. 0.32 � 0.08 = 6. 0.15 � 0.03 =


7. 2.8 � 0.7 = 8. 4.5 � 0.9 = 9. 3.6 � 0.6 =

10. 2.4 � 0.4 = 11. 0.48 � 0.06 = 12. 0.64 � 0.8 =


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Divide Patterns with Powers of 10If you’re having trouble dividing decimals, you can make them wholenumbers. This can make the division easier.

Divide.

0.54 � 0.6

It is often easier to divide by a whole number than by a decimal. You can change a decimal to a whole number by multiplying by 10.

• The divisor is 0.6. Multiply 0.6 by 10 to get 6, which is a whole number

• The dividend is 0.54. Multiply 0.54 by 10 to get 5.4.

• Divide the whole number divisor into the dividend: 5.4 � 6 = 0.9.

So, 0.54 � 0.6 = 0.9

Divide.

1. 0.16 � 0.4 = 1.6 � 4 =

2. 0.12 � 0.2 = 1.2 � 2 =

3. 0.15 � 0.3 =1.5 � 3 =

4. 0.35 � 0.5 = 5. 0.42 � 0.6 = 6. 0.81 � 0.9 =

7. 0.32 � 0.8 = 8. 0.36 � 0.6 =

9. 4 � 0.5 =

10. 8 � 0.4 = 11. 9 � 0.03 = 12. 0.63 � 0.07 =


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Divide. Use multiplication to check your answer.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Division of Decimals by DecimalsJust as you changed decimals into whole numbers when working with patterns of 10, you can also change the divisor (the number on the outside) into a whole number.

Divide. Use multiplication to check your answer.

Change 2.6 to a whole number by multiplying by 10.

Multiply 8.84 by 10.

Divide

2.6 �10 = 26

8.84 �10 = 88.4

Multiply to check:

Does 2.6 � 3.4 = 8.84? 8.84 = 8.84

Yes, the answer is correct. 3.4

313.2 � 36 � 40.59 � 0.9 � 89.18 � 7 �

86.18 � 6.2 � 216.2 � 9.4 � $6.57 � 0.73 �

2.6 � � 8.84

26 � � 88.4 26 � � 88.4

0.23 � � 1.38 1.8 � � 13.68 4.1 � � 9.84


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1. What are you asked to find? 2. Estimate the total amount of fruit Brittany bought. Round each

amount to the nearest whole number.

3. Compare your estimate with Brad and Britney’s answers.

4. Whose answer is reasonable? Explain.

Problem Solving Workshop Skill: Evaluate Answers for ReasonablenessWhen solving a problem, ask yourself, “Does the answer make sense?” Should a car cost $20.99? Should a can of beans cost $10.50? Check your answers to make sure they are reasonable.

Britney bought 0.97 kilogram of apples, 1.05 kilograms of

bananas, and 0.57 kilogram of oranges. Britney says she

bought 25.9 kilograms of fruit. Brad says that Britney bought

2.59 kilograms of fruit. Use estimation to find whose answer

is reasonable. Explain.

USE DATA For 5-6, use the table.

5. Buddy says he was faster than Brent by an average speed of 19.18 mph. Brent says Buddy was faster by 9.18 mph. Whose

answer is reasonable?

6. For the first three laps of the race, Susie says her average speed faster than Danielle’s total

average speed. Susie drove 96.4 mph, 88.5 mph, and 83.9 mph. Is Susie’s answer

reasonable?

0.97 G

1.05 G

+ 0.57 G

Middleville Auto Race

Driver

Average

Speed per

Lap

Number

of Laps

Finished

Buddy 98.65 mph 21Danielle 91.43 mph 43

Rico 99.10 mph 45

Susie 73.25 mph 50Brent 89.47 mph 50


Name

1. Desiree ran 2.3 miles yesterday and another 1.8 miles today.

2. Brian bought three packs of mints with a number of mints in each pack.

3. Jean studied another 2 2 _ 3 hours today.

4. Mr. Strait has 17 pens and pencils. He lent eight and got back five.

5. Gilberto is 2 years older than two times his brother’s age.

6. Lacy caught six fish and threw four of them back.

7. Muhammad has $37.45 in the bank. He deposited $6.25.

8. The distance to the library is 2 1 _ 4 miles farther than the distance to the school.

9. Jodi has 4 7 _ 8 times more inches of yarn than Daphne.



Write ExpressionsAn expression is a number sentence that represents a situation. A numerical expression uses numbers and operations. An algebraic expression uses variables for unknown amounts.

Write an expression for the situation.

If you use a variable, explain what the variable represents.

Darryl is two years

less than three times

Judy’s age.

Since we have anunknown amount (Judy’sage), we are going to use an algebraic expression. Let’s use j for Judy’s age.

“Darryl is two years less

than” that means we’re going to subtract 2 from the age. “three times Judy’s age” that means we’re multiplying by 3. The expression is 3 j – 2, where j = Judy’s age.

Write an expression for each situation.

If you use a variable, explain what the variable represents.


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Evaluate ExpressionsWhen evaluating an expression with more than one operation, you must follow the order of operations. You can use the saying, “Please Meet Dear Aunt Sally” to help you remember the order of 1) Parentheses, 2) Multiply or Divide (from left to right), and 3) Add or Subtract (from left to right).

Evaluate the expression.

(18 � 9) � 3 + 11Follow the order of operations:

1) Parentheses (18 � 9) � 3 + 11 = 2 � 3 + 11

2) Multiply or Divide 2 � 3 + 11 = 6 + 11

3) Add or Subtract 6 + 11 =

Evaluate each expression.

1. 33 � (48 � 4) � 7 2. 81 + (5 � 6)

3. 9 + 13 � (8 + 2) 4. 52 – (18 � 9) × 4

5. 48 – 16 � 3 + 3 6. (29 – 8) � 7 + 6 7. 60 – (54 � 9) 8. (11 + 3) � 1 + 7

9. 10 � (25 – 9) – 8 10. 72 – 6 � 2 + 4 11. 43 + 4 � 5 12. 98 + 24 � (4 � 1)

Evaluate the algebraic expression for the given value of the variable.

13. 2 + 15 � n – 3 if n = 3

14. 16w – 5 + 4 if w = 4

15. (r + 9) + (6.1 � 4) if r = 1.3

16. 96 � 12 + × – 4.5 if × = 5.9

17. 8 + 4n + 10 if n = 2.3

18. ( + k) + 7 �

5 if k =

19. 2 + z + (8 � 1) if z =

20. + 14 if m = 30

17

2 1 __ 8

3 3 __ 8

1 4 __ 5

m __ 6

So, (18 � 9) � 3 � 11 � 17.


Name

© Harcourt • Grade 5Reteach the StandardsRW88AF 1.2 Use a letter to represent an unknown

number; write and evaluate simple algebraic expressions in one variable by substitution.

Write EquationsAn equation is a number sentence that shows that two quantities are equal. Like an expression, an equation has numbers, operation signs, and sometimes variables. An equation is different from an expression because an equation does have an equal sign.

Write an equation.

Tia has 5 fewer keys than Omar. If Tia has 7 keys, how many does Omar have?

Step 1Write a word equation.

The number of keys Omarhas minus 5 equals 7.

Step 2Choose a variable.

Let k stand for the number of keys Omar has.

Step 3Write the equation.

k � 5 = 7

Write an equation for each. Tell what the variable represents.

1. Mary Beth’s mother made 24 blueberry muffins. Mary Beth’s family ate some. Now there are 16 muffins left. How

many did Mary Beth’s family eat?

2. Jerrod saved $85 to buy a new jacket. After he bought the jacket he had $18 left. How much did the jacket cost?

3. Ryan studied 45 minutes for his science test. He studied 3 times longer for his science test than his spelling test. How long did Ryan study for his spelling test?

4. Mrs. Greene planted 18 tomato seedlings and some cucumber

seedlings. She planted 31 seedlings altogether. How many cucumber

seedlings did she plant?

5. The bookstore gets a delivery of 540 books. Thirty books are packed into each carton. How many cartons of

books does the bookstore get?

6. Pat is on a bike tour that is 34 miles long. He is currently halfway through the tour. How many miles has Pat

ridden so far?


Name

© Harcourt • Grade 5Reteach the StandardsRW89AF 1.2 Use a letter to represent an unknown

number; write and evaluate simple algebraic expressions in one variable by substitution.

Solve EquationsAn equation is a number sentence with a missing number. The unknown number is represented by a letter, or variable. When you determine the variable, that value is the solution.

Which of the numbers 5, 8, or 10 is the solution of the equation, s � 5 = 2?

To find the solution for s � 5 = 2, replace the variable, s, with each of the possible answers:

Which of the numbers 5, 8, or 10 is the solution of the equation?

1. 17 – x = 9 2. 15 � m = 150 3. 13 + r = 21

4. 60 � q = 12 5. z � 5 = 40 6. – s =

Use mental math to solve each equation. Check your solution.

7. 16 + n = 25 8. 27 – s = 14 9. d + 7 = 12

10. g – 8 = 12 11. 4 � k = 28 12. 63 � f = 9

13. w � 15 = 2 14. h � 13 = 26 15. – a =

Is 5 a solution? Is 8 a solution? Is 10 a solution?

s � 5 = 2 s � 5 = 2 s � 5 = 2(5) � 5 ?= 2 (8) � 5 ?= 2 (10) � 5 ?= 2 1 � 2 1.6 � 2 2 � 2 No, 5 is not No, 8 is not Yes, 10 is a solution. a solution. a solution.

16 3 __ 4

6 3 __ 4

7 2 __ 3

5 1 __ 3


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© Harcourt • Grade 5Reteach the StandardsRW90AF 1.3 Know and use the distributive property in

equations and expressions with variables.

Use the Distributive PropertyYou can also use the Distributive Property to find missing numbers and variables.

Use the Distributive Property to find the value of the expression.

4 � (n + 6) if n = 20

The Distributive Property follows the rule: a � (b + c) = (a � b) + (a � c).Use the property by replacinga, b, and c with values in the expression.

Use the Distributive Property to find the value of the expression.

1. 6 � (9 + n) if n = 10 =

(6 � 9) + (6 � ) =

+ =

2. 8 � (25 – n) if n = 4

= (8 � 25) – (8 � )

= – =

3. 2 � (n + 13) if n = 44

4. z � (7 – 1) if z = 20 5. 11 � (h – 3) if h = 31 6. m � (5 – 2) if m = 15

7. 9 � (p + 50) if p = 4 8. 6 � (8 + s) if s = 14 9. 3 � (68 – g) if g = 50

What value for the variable makes the equation true?

10. 2 � 76 = (2 � 70) + (2 � x) 11. 4 � 18 = (4 � 10) + (4 � s)

12. 9 � 54 = (9 � 50) + (9 � r) 13. 6 � 33 = (6 � 30) + (6 � w)

14. 7 � 42 = (7 � 40) + (7 � g) 15. 5 � 81 = (5 � 80) + (5 � h)

1. Replace n with 20.

2. Rewrite.

3. Multiply.

4. Add.

The answer is 104.

1. 4 � (20 � 6)

2. (4 � 20) � (4 � 6)

3. (80) � (24)

4. 80 � 24 = 104


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Mental Math: Use the PropertiesYou can use properties and mental math to help you solve problems.

The Associative Property states that you may group addends or factors differently without changing the value of the sum or product.

7 + (8 + 4) = (7 + 8) + 47 + 12 = 15 + 4 19 = 19

The Commutative Property states that addends may be added in any order or factors may be multiplied in any order without changing the value of the sum or product.

2 � 17 � 5 = 2 � 5 � 17 = 10 � 17 = 170

The Distributive Property states that you can break apart a factor to multiply.You can use addition or subtraction to break apart a factor.

7 � 42 = 7 � (40 + 2) = (7 � 40) + (7 � 2) = 280 + 14 = 294

Use properties and mental math to find the value.

1. 27 + 26 + 33 2. 4 � 27 3. (24 + 19) + 16

4. 7 � 5 � 3 5. 21 + 39 + 38 6. 6 � 45

7. 3 � 9 � 2 8. 3 � 360 9. 62 + 28 + 17

10. 7 � 3 � 9 11. 37 � 5 12. 43 + (47 + 46)

13. 73 � 3 14. 38 + 41 + 22 15. 12 � 4 � 5

16. 82 � 8 17. 14 + 36 + 27 18. 19 � 7


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Problem Solving Strategy: Write an EquationIn 1987, Yuri Romanenko set a record for the longest space

flight. In 1995, Valery Polyakov beat that record by 113 days.

If Polyakov spent 439 days in space, how many days did

Romanenko spend in space?

Read to Understand

Read the question. What are you asked to find?

2.Plan

What strategy can you use to solve the problem?

3.Solve

Choose a variable. Complete the equation below to solve the problem.

4. Solve the equation. How many days did Romanenko spend in space?

Check



Shannon has 250 marbles in several

jars. Each jar has 50 marbles. How

many jars does Shannon have?

Philippe divided his marbles evenly

into six jars. Each jar has 80 marbles.

How many marbles does Philippe

have?

7.6.

113 + =

1.


Name

RW93 NS 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. © Harcourt • Grade 5


Understand IntegersIntegers can be graphed on a number line.Positive integers are always to the right of 0.Negative integers are always to the left of 0.The greatest negative integer is �1.The least positive integer is �1.Numbers with no sign in front of them are positive integers.

Identify the integers graphed on the number line.

• The point that is 6 units left of 0 represents �6.• The point that is on 0 represents 0.• The point that is 4 units right of 0 represents �4.So, the integers are �6, 0, and �4.You can use this chart as a guide for usingpositive or negative integers to describea situation.

Write an integer that describes

120 feet below sea level.

Sea level is at 0 feet, so below sea levelwould be a negative integer.So, the integer for 120 feet below sea level is �120.

Write an integer that describes a profit of $800.

No profit or loss is at $0, and profit is a positive word.So, the integer for a profit of $800 is �800, or just 800.

Some words used with negative

integers

Some words used with positive

integers

lossdecreasebehind

backwardbelowdownunder

withdrawalto the left

gain or profitincreaseahead

forwardabove

upover

depositto the right

Identify the integers graphed on the number line.

1.

Write an integer to represent each situation.

2. a gain of 12 yards 3. 7 degrees below freezing 4. 8 floors up

negative integers positive integers

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RW94 NS 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

Two integers are equal only when they have the same sign and same number.


Compare and Order IntegersYou can use a number line to compare and order integers.Compare �4 and �9. Use <, >, or �.Graph the integers on a number line.Since �4 is right of �9, �4 > �9

Compare �4 and �9. Use <, >, or �.

Graph the integers on a number line.Since �4 is right of �9, �4 < �9.

Compare. Write <, >, or �.

1.


Graph the integers on a number line.Since �4 is right of �9, �4 > �9.


Graph the integers on a number line.Since �4 is right of �9, �4 < �9.

2. 3.

4. 5. 6.

7. 8. 9.

�7 �1 �9 �8 �4 0

�6 �6 �2 �10 �2 �6

�1 �1 �5 �6 �3 �3


Name

RW95 NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative, integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

So, �4 � �4 � 0.

Find �4 � �4.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Model Integer AdditionYou can use a number line to add integers.A positive integer tells you to move right from a position.A negative integer tells you to move left from a position.

Step 1Start at 0 on a number line.

Use a number line to find the sum.

1. 2. 3. �5 � �1 �9 � �4 �2 � �7

Step 2The first integer is negative.From 0, move 4 spaces left.

Step 3The second integer is positive.

From �4, move 4 spaces right.

Step 4The ending place is 0.


Name

RW96 NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative, integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

Add. �6 � �4


Record Integer AdditionYou can use a number line to represent integer addition.

Step 1

Draw a number line representing �10 to �10.

Find the sum.

1. 2. 3. �6 � �3 �2 � �7 �10 � �10

Step 2Begin at 0 and move 6 spaces to the right.

Step 3

Adding �4 means move to the left. Begin at �6 and move 4 spaces to the left.

Step 4

Record where you stopped on the number line: �2.

4. 5. 6. �2 � �3 �7 � �3 �11 � �9

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Name

RW97 NS 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

Find �9 � �4.


Model Integer SubtractionYou can use a number line to subtract integers.A positive integer tells you to move right from a position.A negative integer tells you to move left from a position.

Step 1.

Start at 0 on a number line.

Use a number line to find the difference.

1. 2. 3. �8 � �1 �6 � �2 �3 � �9

Step 2.

The first integer is negative.From 0, move 9 spaces left.

Step 3.

Move 4 units to the left to subtract �4.

From �9, move 4 spaces left.

Step 4.

The ending place is left of 0 so the sum is

negative. Count the number of spaces it is

from 0.

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

The ending place is 13 spaces left of 0,

so the integer is �13.

So, �9 � �4 � �13

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Name

RW98 NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive

integers from negative integers; and verify the reason-ableness of the results.

Subtract. �2 � �2.


Record Integer SubtractionYou can use a number line to represent integer subtraction.

Step 1.

Draw a number line representing �10 to �10.

Use a number line to find the difference.

1. 2. 3. �5 � �2 �2 � �6 �7 � �3

Step 2.

Begin at 0 and move 2 spaces to the right.

Step 3.

Subtracting positive 2 means move to the left. Begin at �2 and move 2 spaces to the left.

Step 4.

Record where you stopped on the number line: 0.

So, �2 � �2 � 0


Name

RW99 NS 1.5 Identify and represent on a number line decimals, fraction, mixed numbers, and positive

and negative integers.

The lowest recorded temperature in Laguna Beach is

-6ºC. The highest recorded temperature is 48ºC higher.

What is the highest recorded temperature in Laguna Beach?

Read to Understand

Plan

Solve

Check

1. What do you visualize when you read the problem?

2. Describe a diagram you could use to solve the problem.

3. Describe a model you could use to solve the problem.

4. Select a strategy and solve the problem.

5. How did the strategy help you solve the problem? Explain.


Problem Solving Workshop Strategy:Compare Strategies:

Draw a diagram or make a model to solve.

6. Tuesday’s high temperature was 4ºF.Wednesday’s high temperature was 11ºF colder. What was the high temperature on Wednesday?

7. Cliff went scuba diving. He dove 22 feet below sea level. Later that week, he went rock climbing and climbed to 428 feet above sea level. What is the difference in elevations?


Name

© Harcourt • Grade 5Reteach the StandardsRW100 NS 1.2 Interpret percents as a part of a hundred;


Understand PercentYou can represent part of the whole by using a percent.Percent means “per hundred.” 100 percent is the whole.

The 10 � 10 grid has 100 squares. Each square represents 1 percent.

60 out of 100, or 60 ____ 100 , of the squares are shaded.

0.60 of the squares are shaded.60% of the squares are shaded.

Write a ratio and a percent to represent the shaded part.

1.

2. 3.

Write a decimal and a percent to represent the shaded part.

4. 5. 6.

54 out of 100, or 54 ____ 100 , of the squares are shaded.

0.54 of the squares are shaded. 54% of the squares are shaded.


Name


NS 1.2 Interpret percents as a part of a hundred;find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

Fractions, Decimals, and Percent

Percents can be written as decimals, or as fractions with 100 as the denominator.

• 98% means ninety-eight hundredths, or 0.98

• 98% also means 98 ___ 100

To write a fraction in simplest form, divide the numerator and the denominator by the same number keep doing this until 1 is the only common factor.

• 98% � 98 ____ 100 � 98 � 2 ________ 100 � 2 � 49 ___ 50

So, 98% � 0.98 � 49 ___ 50

1. Write 3 __ 4 as a percent. 3 �

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

2. 80% 3. 35%

4. 40% 5. 76%

Write each fraction or decimal as a percent.

6. 3 __ 5 7. 0.45

8. 9 ___ 10 9. 0.10

4 �� 75 ____

100 �


Name

0

A B C

© Harcourt • Grade 5Reteach the StandardsNS 1.5 Identify and represent on a number line

decimals, fractions, mixed numbers, and positive and negative integers.

RW102

Use a Number LineNumber lines are not just for whole numbers. You can locate percents, fractions, and decimals on number lines, too.

1.

For 2-6, make a number line. Then, locate each quantity on the number line.

Locate 3 __ 4 on a number line.

Draw a number line divided into hundredths.

Label 0, 0.50, and 1 on the number line.

Then locate 0.10 on the number line.

Write the letter that represents each quantity on the number line.

0.6 20% 2 __ 5

2. 3. 4. 5. 6.1 __ 4 0.70 0.45 95% 3 ___

10

Locate 0.10 on a number line.

1 _ 2 10

3 __ 4

is halfway between 1 __ 2

and 1

10 0.50

count by hundredths


Name


NS 1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

RW103

Model Percent of a Number You can use two-color counters to model percent of a number.

1.

What is 90% of 10?

Show 10 counters. Each counter represents 10%.

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Each counter also represents 1, since 10 � 1 �10.

1 2 3 4 5 6 7 8 9 10

Flip over 90% of the counters.

10% 20%

Count by 1s to find 90% of 10.

So, 90% of 10 is 9.What is 25% of 24?

Show 24 counters.

Since 25% � 1 __ 4 , separate the counters into 4 equal

groups.

1 2

Change the color of the counters in 1 of the 4 groups. Count the counters whose color you changed.

So, 25% of 24 is 6.

Find the percent of each number. Use counters to solve.

2. 3. 4.

Write each percent as a fraction. Then, use counters to find the percent of each

number.

5. 6. 7. 8.

40% of 80 90% of 20 50% of 12 75% of 32

40% of 35 50% of 26 25% of 20 75% of 28


Name

1 2 3 4 5 6 7 8 9 10

0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8� 0.80

1. 2. 3.

© Harcourt • Grade 5Reteach the Standards NS 1.2 Interpret percents as a part of a hundred;


RW104

Percent Problems You can use multiplication to find percent of a number.

What is 2% of 80?

Think: 2% of 80 means 2% � 80.

Write: 2% as a decimal. 2% � 0.02

Multiply. 2% � 80 � 0.02 � 80 � 1.6

So, 2% of 80 is 1.6.

You can use equivalent fractions to find what percent one number is of another number.

What percent of 10 is 2?

Write a fraction.

Write an equivalent fraction with a denominator of 100.

Write a percent.

So, 2 is 20% of 10.

You can make a table to find a number when you know the percent of the number and the percent.

8 is 80% of what number?

Think: 80% � 0.80 0.80 � � 8

Make a table to show 0.80 � .

Since 10 � 0.80 � 8, 8 is 80% of 10.

Solve each percent problem.

What is 35% of 60? What percent of 24 is 6? 2 is 40% of what number?

2 ___ 10

2 ___ 10

� 2 � 10 _______ 2 � 10

� 20 ____ 100

20 ____ 100

� 20%


Name


SDAP 1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.

Problem Solving Workshop Strategy: Make a GraphOf the top 10 prizes awarded at a state fair, 3 out of 10 went to

Intermediates. How does the number of winners in the Intermediate group

compare to the other groups?

Read to Understand


2.Plan

How can making a graph help you solve the problem?

3.Solve

What type of graph would best display the data?

4. Make a graph to solve the problem.

Answer the question.

Check

5. What other strategy could you use?


When 10 students were asked to name their favorite part of California, 5 said the beach, 2 said the mountains, and 3 said the forest. Which part did the most students choose?

Over the past 5 years, the 4-H club at Lisa’s school had 10, 12, 15, 20, and 28 members. How did 4-H club membership change from the first year to the fifth year?

7.6.


Name

Legoland25%

Exploratorium45%

Mt Lassen15%

Aquarium15%

Aquarium20%

Exploratorium25%

Mt Lassen10%

Legoland45%

Reteach the StandardsRW106 SDAP 1.3 Use fractions and percentages to compare data sets of different sizes.

Compare Data SetsThe graphs show field trip choices of two classes. Which class had more students vote for Mt. Lassen?

Find the percent for Mt. Lassen on each graph. 15% 10%

Write each percent as a decimal. 0.15 0.10

Multiply by the number of students in class. = ■ = ■

The answer is the number of votes. 0.15 × 20 = 3 0.10 × 40 = 4

So, 3 students from Class 5A and 4 students from Class 5B voted for Mt. Lassen. Class 5B had more votes.

Since 4 – 3 = 1, Class 5B had 1 more vote.

Class 5A Class 5B

Class 5A Field Trip Choices Class 5B Field Trip Choices

20 students 40 students

1. Complete the steps shown at the right. Which class had more votes for Legoland?

5A

25% = 0.25

0.25 × 20 =

5B

45% = 0.45

0.45 × 40 =

USE DATA For 2–3, use the circle graphs.

2. Whose school had more sparrows? How many more?

3. How many more cardinals are there at Jeff’s school than at Edna’s school?

Edna’s School 50 Birds Jeff’s class 60 Birds



Name

RW107 MG 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and

triangles by using appropriate tools (e.g., straight-edge, ruler, compass, protractor, drawing software).

A

C

E

F

G

H

D

J

B


Points, Lines, and AnglesIn geometry, objects have special names.

A point marks an exact location.

A plane is an endless flat surface.

A line is an endless path. It has no endpoints and can be named by any two points on the line, such as Line AJ or Line CD.

A line segment is part of a line that includes two endpoints and all of thepoints between them.

�BHC is a right angle that measures exactly 90º.

�CHD is a straight angle thatmeasures exactly 180º.

�ABH is an acute angle thatmeasures less than 90º.

�HBJ is an obtuse angle thatmeasures more than 90º.

Line AJ and Line BF cross at onepoint, so they are intersecting.

Line CD and Line BF intersect to formfour right angles, so they are perpendicular.

Line CD and Line EG are the samedistance from each other and willnever intersect, so they are parallel.

1. intersecting lines 2. line 3. line segment

For 1 –3, use the figure above. Name an example of each.

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RW108 MG 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles

by using appropriate tools (e.g., straightedge, ruler, com-pass, protractor, drawing software).

T

U

W

Y

ZX

T

U

W

Y

ZX


Measure and Draw AnglesYou can use benchmarks to estimate angle measures.

0° 45° 90° 135° 180°

Estimate the measure of �UXY.

Think: �UXY appears to be greater than 45˚.�UXY appears to be less than 135˚.

Compare �UXY to the benchmark 90˚.

The measure of �UXY is about 90˚.

Use a protractor to measure �UXY.

Step 1 Think: Point X is the vertex. Rays

_ › XY and

_ › XU form the sides of an angle.

Step 2 Place the center point of the protractor on the vertex, point X, so that

_ › XU passes

through 0˚.

Step 3 Measure �UXY. So, �UXY is 105˚.

Estimate the measure of each angle. Then use a protractor to find the measure.

1. �JXK 2. �KXN 3. �MXN 4. �KXL

W Y

ZXT

U


Name


by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).

You can construct perpendicular lines using a straightedgeand compass.

Perpendicular lines

STEP 1

Draw a line with point M near the center.

STEP 2

Place the point of the compass at point M. Draw arcs that intersect the line on each side of the point. Do not change the compass opening between arcs. Label the points L and N.

STEP 3

Open the compass wider than it was in step 2. Place its point on points L and N.Draw two intersecting arcs above point M.Label point P.

STEP 4

Draw PM.

PM is perpendicular to LN.

M

L M N

L M N

P

L M N

P

J

K

G H


Construct Parallel and Perpendicular Lines

Use a compass and a straightedge to construct a line that

is perpendicular to each line.

1. 2.


Name

RW110 MG 2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and

solid geometric fi gures.

A polygon is a closed plane figure formed by three or morestraight sides that are connected line segments. A regular

polygon has sides that are all the same length and angles thatare all the same measure. A polygon that is not regular has sides and angles that are not the same measure, but is stillnamed by its number of sides and angles.

Classify the polygon below.Polygon

Number of Sides and

Angles

Triangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

Octagon 8

Decagon 10

How many sides does this polygon have?

How many angles does this polygon have?

Name the polygon.

What type of angles does this polygon have?

Are the sides all the same length?

Is the pentagon a regular pentagon?

So, the polygon above is a pentagon that is not regular.

5 sides

5 angles

Pentagon

2 right and 3 obtuse

No

No


Polygons

Name each polygon and tell whether it is regular or not regular.

1. 2. 3. 4.


Name

RW111 MG 2.2 Know that the sum of the angles of any triangle is 180º and the sum of the angles of any

quadrilateral is 360º and use this information tosolve problems.

The sum of the angles in any triangle is always 180º.The sum of the angles in any quadrilateral is always 360º.

Tell if the given angle measures would form the figure named.

Quadrilateral: 110º, 110º, 70º, 70º

• Add the angle measures together. 110º � 110º � 70º � 70º � 360º

• Does the sum equal 360º? Yes

So, angles measuring 110º, 110º, 70º, 70º would form a quadrilateral.

Quadrilateral: 90º, 45º, 90º, 135º

• Add the angle measures together. 90º � 45º � 90º � 135º � 360º

• Does the sum equal 360º? Yes

So, angles measuring 90º, 45º, 90º, 135º would form a quadrilateral.

Triangle: 30º, 60º, 80º

• Add the angle measures together. 30º � 60º � 80º � 170º

• Does the sum equal 180º? No

So, angles measuring 30º, 60º, 80º would not form a triangle.


Sum of the Angles

Tell if the given angle measures would form the figure named.

1. triangle; 30º, 40º, 70º



4. quadrilateral; 90º, 60º, 70º, 100º



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RW112 MG 2.2 Know that the sum of the angles of any triangles is 180º and the sum of the angles of any quadrilateral is 360º and use this information to solve problems.


Problem Solving Workshop Skill: Identify RelationshipsConnect the vertices within a square, a regular pentagon, and

a regular hexagon. A square and a regular pentagon are shown

at the right. Count the lines within each figure. Predict

how many lines you would draw within a regular 7-sided figure.

1. What are you asked to do?

2. Connect the vertices of the hexagon at the right.How many lines did you draw?

3. Use the table to help you find a pattern and make a prediction.

Number of Sides 4 5 6 7

Number of Lines

Connecting Vertices2 5 9 ?

4. Predict how many lines will be drawn for a 7-sided figure.


+3 +4 +?

6. Predict how many lines will be drawn to connect the vertices for 8-, 9-, and 10- sided figures.

7. The distance around a regular hexagon with 1-inch sides is 6 inches. The

distance around a regular hexagon with 2-inch sides is 12 inches, and the distance around a regular hexagon with 3-inch sides is 18 inches. What is the distance around a regular hexagon with 6-inch sides?

Identify the relationship. Then solve.

Pattern: The number of lines connecting the vertices of a regular polygon increases by �3, �4, �?

How many more lines would there be connecting the vertices of a 7-sided figure than a hexagon?


Name

RW113 MG 2.0 Students identify, describe, and classify the properties of, and the relationships between,

plane and solid geometric fi gures.

A circle is a closed plane figure whose points are the samedistance from its center. A circle is named by its center point.

Look at circle T at the right.

A radius of a circle connects the center of the circle with anypoint on the circle. TU is a radius. TS is also a radius.

A diameter of a circle passes through the center and has its endpoints on the circle. RU is a diameter.

S

R

TU

The length of a diameter is always twice the length of a radius.

A chord has its endpoints on the circle. SU is a chord.

You can draw a circle if you know the length of a radius. You need a compass and a straightedge.

Draw circle V with a radius of 4 cm.

4 cmV

V


STEP 1

Draw and label point V.

V •

STEP 2

Open your compass to

4 cm. Place the point of

the compass on point V.

STEP 3

Draw circle V so that point

V is its center.

Circles

For 1–4, use the circle at the right.

1. Name the circle. 2. Name a diameter. E

F

X

H

3. Name a chord. 4. Name a radius.

Complete 5–6. Then use a compass to draw each circle.

Draw and label the measurements.

5. radius � 1.5 cm diameter �

6. radius � diameter � 4 cm


Name


by using appropriate tools (e.g., straightedge, ruler, com-pass, protractor, drawing software).

You can use a compass and a straightedge to constructequilateral triangles.

Draw an equilateral triangle inside with sides that have a measure of 5 inches.

STEP 1

Draw a line. Mark and label point A at one end of the line.

STEP 2

Open the compass to 5 inches and place the compass point on point A to mark another point on the line. Label this point B.

STEP 3

Keep the compass opened to 5 inches and place the compass point on point A. Draw an arc above the line. Do the same from point B. Where the two arcs intersect, mark and label point C.

STEP 4

Use a straightedge to draw AC and BC.

A A B

A B A B

CC


Construct Polygons

Use a compass and straightedge to construct each triangle.

1. equilateral triangle with sides that have a measure of 3 centimeters.

2. equilateral triangle with sides that have a measure of 2 centimeters.


Name

RW115 MG 2.0 Students identify, describe, and classify the

properties of, and the relationships between plane and solid geometric fi gures.

A

B

F

E

D

C

Congruent and Similar FiguresFigures can be congruent, similar, both, or neither.Congruent figures have the same size and thesame shape. Similar figures have the same shape,but may or may not be the same size. Corresponding

angles and corresponding sides are in the samerelated position in different angles.

Tell whether the two figures at the right appear to be congruent,

similar, or neither.

The figure on the left appears to have greater heightthan the figure on the right.

Think, do the figures appear to be the same shape? • No

Think, do the figures appear to be the same size? • No

So, the two figures are neither.

Identify the angle that corresponds to /C.

Figures ABC and DEF are congruent.

/A corresponds to /D and/B corresponds to /E.

So, /C corresponds to /F.


Write whether the two figures appear to be congruent, similar, or neither.

1. 2. 3. 4.

Identify the corresponding side or angle.

5. EF 6. GH 7. /G 8. /MH

FE

G

J K

ML


Name

10 m

10 m

4 m


MG 2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.

TrianglesYou can classify triangles either according to their sides or according to their angles.

Use a ruler to compare side lengths.

Use the corner of a paper to classify angles.

equilateral triangle

All sides are the same length.

acute triangle

All three angles are acute.

isosceles triangle

Two sides are the same length.

obtuse triangle

One angle is obtuse and the other two angles are acute.

scalene triangle

All sides are of different lengths.

right triangle

One angle is right and the other two angles are acute.

Classify the triangle to the right according to its sides.

It has two equal side lengths.

The triangle is an isosceles triangle.

Classify the triangle to the right according to its angles.

It has one right angle.

The triangle is a right triangle.

Classify each triangle. Write isosceles, scalene, or equilateral.

1. 2. 3.

Classify each triangle. Write acute, right, or obtuse.

4. 5. 6.


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MG 2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.

QuadrilateralsYou can use this chart to classify quadrilaterals.

Classify each figure in as many ways as possible.

Write quadrilateral, parallelogram, square, rectangle, rhombus, or trapezoid.

1. 2.

Describe each quadrilateral using parallel, perpendicular, and congruent.

3. 4.

quadrilateral

4 sidesparallelogram

quadrilateral opposite sides are parallel opposite sides congruent

trapezoid

quadrilateralone pair of parallel sides

rectangle

parallelogram4 right angles

rhombus

parallelogram4 congruent sides

square

rhombusrectangle

Describe the figure.

It has no right angles, so no sides are perpendicular.

Two sides of the figure are congruent.

The figure has 4 sides, so it is a quadrilateral.

The figure has one pair of parallel sides, so it is a trapezoid.


Name


MG 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).

Draw Plane FiguresYou can draw a plane figure based on a description.

Use a protractor and ruler to draw a quadrilateral. Then identify the

quadrilateral by its best name.

One pair of opposite sides measures 3 centimeters. The other pair of

opposite sides measures 4 centimeters. One pair of opposite angles

measures 35°. The other pair measures 145°.

Step 1

Draw a ray.

Step 2

Use a protractor to draw a 35° angle.

Step 3

Measure and mark 4 cm along one ray and 3 cm along the other ray.

Step 4

Measure a 145° angle with a vertex at each mark. Extend rays to meet.

Step 5

Check all the measures against the given information.

Both pairs of opposite sides are parallel and congruent.

The quadrilateral is a parallelogram.

Use a protractor and ruler. Then classify the figure.

1. triangle: 2 angles that measure 30°; one side between the angles that measures 4 centimeters

Classify the triangle according to its

sides:

angles:

2. quadrilateral: four 3-centimeter sides; 1 pair of angles measuring 60°, the other pair measuring 120°

What name best describes this quadrilateral?


Name


MG 2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures.

Solid FiguresYou can identify a polyhedron, or solid figure with faces that are polygons, by the shape of its faces.

A pyramid has more than 3 triangular faces.

A prism has 3 or more rectangular faces.

triangular

pyramid

All faces are triangles.

trianglular

prism

The two bases are triangles.

rectangular

pyramid

The one base is a rectangle.

rectangular

prism

All faces are rectangles.

square

pyramid

The one base is a square.

cube All faces are squares.

pentagonal

pyramid

The one base is a pentagon.

pentagonal

prism

The two bases are pentagons.

hexagonal

pyramid

The one base is a hexagon.

hexagon

prism

The two bases are hexagons.

A solid figures with curves is not a polyhedron.cone The one base

is a circle.cylinder The two

bases are circles.

sphere shape of a ball and has no base

Classify the figure.

The figure has lots of triangular faces that meet in a point.The figure is a pyramid.

The one face that is not a triangle is a rectangle.

So, the figure is a rectangular pyramid.

Classify each solid figure. Write prism, pyramid, cone, cylinder, or sphere.

1. 2. 3. 4.


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MG 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.

Nets for Solid FiguresYou can identify solid figures by their nets. A net can be cut out, folded, and taped together to form a polyhedron.

A pyramid must have more than 3 faces that are triangles.

A prism must have 3 or more faces that are rectangles.

Name the solid figure for the net.

The net has no rectangles, so it is not a prism.

The net has more than 3 faces that are triangles, so it must be a pyramid.

All faces are triangles, so the net is for a triangular pyramid.

Name the solid figure for the net.

The net has no triangles, so it is not a pyramid.

The net has 3 or more faces that are rectangles, so it must be a prism.

The faces that are not rectangles are pentagons.

The net is for a pentagonal prism.

Name each solid figure. Write the letter for the net that can be used

to form each type of solid figure.

1. 2. 3. 4.

a. b. c. d.


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MG 2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures.

Problem Solving Workshop Strategy : Solve a Simpler ProblemSharon has measured a castle to be 250 feet high. If her house is 3 __ 5 as tall, how tall is Sharon’s house?

Read to Understand

2.Plan

What operation will you use to solve the problem?

3.

Solve

What numbers might be easier to work with?

4. Show how you use the strategy to solve the problem.

Check

5. Look back at the problem. Does the answer make sense for the problem?

Explain.

Solve a simpler problem.

Beth paints the inside and outside walls of a hexagonal prism barn. Each wall requires 42 gallons of paint. How much paint will Beth need?

A model of a locomotive train is 1 ____ 120

the size of an actual train. The

caboose model is 3 inches long. How

long, in feet, is the actual train?

7.6.

1. What strategy can you use to solve the problem?


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© Harcourt • Grade 5Reteach the StandardsRW122 MG 2.3 Visualize and draw two-dimensional views of

three-dimensional objects made from rectangular solids.

1. 2.

3. 4.

Draw Solid Figures from Different ViewsYou can use this chart to help you identify solids from different views.

Identify the solid figure that has the given views.

The front and side views are triangles, so the figure is a pyramid or cone.

The top view is a triangle. All inside segments meet at a point.

So the figure is a triangular pyramid.

top view front view side view

prism or cylinder shape of base rectangle rectangle

pyramid or cone shape of base with a point in the center triangle triangle

sphere circle circle circle

top front side

top front side

Identify the solid figure that has the given views.

top front side

top front side top front side


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SDAP 1.5 Know how to write ordered pairscorrectly; for example, (x, y).



Algebra: Graph Ordered PairsThe coordinate grid at the right is formed bytwo rays. The x-axis is horizontal. The y-axis isvertical.An ordered pair names the location of a pointon a coordinate grid. What ordered pair names the location of point H?

Start at (0,0).Move 5 units along the x-axis.This is the x-coordinate.

Move 7 units vertically. Thisis the y-coordinate.

Ordered pairs are written (x,y). The x-coordinate is first and the y-coordinate is second.

(5,7) names the location of point H.

1. Use the coordinate grid. Start at (0,0).Move 3 units to the right and 5 units up.What point is at (3,5)?

Use the coordinate grid. Write an ordered

pair for each point.

2. A 3. F 4. W

5. Q 6. B 7. J

Graph and label the following points on

the coordinate grid.

8. R at (6, 0) 9. P at (9, 5) 10. C at (0, 5) 11. M at (8, 4)

12. H at (5, 1) 13. S at (7, 9) 14. E at (2, 8) 15. L at (6, 7)

F

B

A

Q

J

W

Z

H


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RW124 SDAP 1.5 Know how to write ordered pairs correctly; for example, (x, y).

Number of Squares, x 1 2 3

Number of Sides, y 4 8 12

Number of faces, x 6 12 18

Number of cubes, y 1 2 3

Number of Triangles, x 1 2 3 4

Number of Vertices, y 3 6 9 12

“Number of Squares, x”

“Num

ber o

f Side

s,y”



Algebra: Graph RelationshipsYou can graph ordered pairs to show relationships between two amounts. For example, you can show the relationship between a number of squares and the number of sides.

A table can show the relationship.

1.

Write the ordered pairs. Then graph them.

Each column can be written as an ordered pair (x,y):1 square, 4 sides: (1,4) 2 squares, 8 sides: (2,8) 3 squares, 8 sides: (3,12)

You can graph the ordered pairs for therelationship shown in the table.

For this relationship, the y-coordinate is always4 times the x-coordinate.

2.


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RW125 AF 1.4 Identify the graph and ordered pairs in the four quadrants of the coordinate plane.

axis

axis

(+,+)

(+,–)

(–,+)

(–,–)


Algebra: Graph Integers on the Coordinate Plane

1.

7.

For 2–9, identify the ordered pair for each point.

To graph (23, 2), you would start at (0,0) and the go

to 23 and to 2

2.

4.

3.

5. 6.

8. 9.

S N

D Z U

E Q H

Just as a coordinate grid is formed by two perpendicular rays, a coordinate plane is formed by two perpendicular lines.

The horizontal line is the x-axis. The vertical line is the y-axis.

The origin is the place where the x- and y-axis intersect.

The x-coordinates to the right of the y-axis are positive integers. The x-coordinates to the left of the y-axis are negative integers.

The y-coordinates above the x-axis are positive integers. The y-coordinates below the x-axis are negative integers.

What ordered pair names the location

of point P on the coordinate plane?

• Point P is 2 units left of the y-axis, so the x-coordinate is 22.

• Point P is 4 units above the x-axis, so the y-coordinate is 4.

So, the ordered pair (22, 4) names the location of point P.

What ordered pair names the location of point B?

• Point B is 3 units right of the y-axis, so the x-coordinate is 3.

• Point P is 5 units below the x-axis, so the y-coordinate is 25.

So, the ordered pair (3, 25) names the location of point B.


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RW126 AF 1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid. © Harcourt • Grade 5


Linear Functions

1.

Find the rule to complete the function table. Then write the rule as an equation.

Find the rule. Then complete theequation and the function table.

Rule: Add .

Equation: y � � 7

Sometimes one quantity depends on another quantity. This relationship is called a function. You can show the relationship in a function table or by using an equation.

Shanti earns $4 dollars each hour she baby-sits.

You can make a function table:

Number of hours, h 1 2 3 4

Dollars earned, d 4 8 12 16 output

inputThe rule for the function table is multiply the number of hours, h, by 4.

You can use the rule for the function table to write an equation:dollars earned (d) � number of hours (h) � 4

or

d � h � 4

You can use the equation to find unknown values.How much will Shanti earn for 7 hours of baby-sitting?

d � h � 4

d � 7 � 4 Substitute the number of hours, 7, for h.

d � 28 Shanti will earn $28 for 7 hours of baby-sitting.

input, x 1 2 4

output, d 8 10 11

2.input, x 1 2 3 4 5 6

output, y 2 4 6 8

3.input, x 30 20 15 10 5

output, y 6 5 4 2 1

4. 5.input, x 15 14 13 12 10

output, y 7 5 4 3 2

input, x 5 7 9 11 13

output, y 18 30 42 54 66


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RW127 AF 1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid © Harcourt • Grade 5


Algebra: Write and Graph Equations

1.

Find the rule to complete the function table. Then write an equation.

Complete the function table. Then graph the points in the coordinate plane.

Rule: Subtract 3.

Equation: y 5 x 2

2.input, x 1 2 3 4 5

output, y 5 10 15 25

3.input, x 0 1 2 3 4

output, y 1 4 10 13

You can use a function table to show a relationship between two quantities. Anequation can also show this relationship.

When you graph an equation, you write orderedpairs and graph them on a coordinate plane.How can you graph the rule, add 4.

Step 1

Write an equation for the rule

Add 4.y 5 x 1 4 y equals 4 added to x.

Step 2

Make a function table showingthe relationshipy 5 x 1 4

input, x 21 0 1 2

output, y 3 4 5 6 1 4

Step 4

Graph the ordered pairs.Step 3

Write the data in the table as orderedpairs. The x-coordinates are in the toprow and the y-coordinates are in thebottom row.

(21,3)

(0,4)

(1,5)

(2,6)

Check that each orderedpair follows the rule. Is they-coordinate 4 more thanits x-coordinate?

input, x 3 4 5

output, y 1 2 3


Name

RW128 AF 1.4 Identify the graph and ordered pairs in the four quadrants of the coordinate plane.

Whitewater Park

City Park

Pasta Bowl

Burger Bistro


Problem Solving Workshop Skill:Relevant or Irrelevant Information

For 5-6, use the map. Tell the relevant information and

solve.

1. What are the coordinates of The Pasta Bowl?

Inez wants to open a restaurant that is 10 blocks

directly south of The Pasta Bowl, and has an

x-coordinate that is 4 blocks west of the Burger

Bistro. What will the coordinates of her restaurant be?

2. What coordinates name the location 10blocks directly south of The Pasta Bowl?Is the x-coordinate 4 blockswest of the Burger Bistro?

What will the coordinates of Inez’s restaurant be?3.

4. What information from the problem was relevant, and what informationwas irrelevant? Why?

5. Jean Paul is making a map of parks in histown. City Park is located at (¯3,2). River Parkis located 5 blocks south and 8 blocks east ofCity Park. Meadow Park is 3 blocks west and 5 blocks south of City Park. Applewood Parkis 5 blocks directly south of River Park. Whatare the coordinates of Applewood Park?

Whitewater Park is located at (5,6) on Jean Paul’s map. LakeshorePark is 12 blocks directly west of Whitewater Park. Bald Mountain is12 blocks directly south of Lakeshore Park. Eagle Ridge is 6 blockseast and 3 blocks north of Bald Mountain. What are the coordinates ofBald Mountain?

6.


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© Harcourt • Grade 5RW129 AF 1.2 Use a letter to represent an unknown

number; write and evaluate simple algebraicexpressions in one variable by substitution.


LengthYou can use this chart to convert customarymeasures of length.

Change 2 miles to yards.

Yard is a smaller unit than mile, so multiply.

The chart shows 1,760 yards per mile, so find 1,760 � 2.

1,760 � 2 � 3,520 yd

There are 3,520 yards in 2 miles.

You can use this chart to convertmetric measures of length.

Change 600,000 centimeters

to kilometers.

Meter is a larger unit than centimeter,so divide.

The chart shows 100 centimeters for 1 meter.

600,000 cm � 100 � 6,000 m

The chart shows 1,000 meters for 1 kilometer.

6,000 m � 1,000 � 6 km

So, there are 6 kilometers in 600,000 centimeters.

Change the given units.

1. 60 in. � ft 2. 4 yd � ft 3. 3 mi � ft

4. 51 ft � yd 5. 2 yd � in. 6. 4 km � m

7. 8,000 cm � m 8. 58 cm � mm 9. 40,000 mm � m

Metric Measures of Length

1

kilometer1 meter

1

centimeter

1

millimeter

1.000 meters

100 centimeters

10millimeters

Customary Measures of Length

1 mile 1 yard 1 foot 1 inch

1,760 yards

5,280 feet

3 feet36 inches

12 inches

� Multiply to convert larger units to smaller units.Divide to convert smaller units to larger units. �

� Multiply to convert larger units to smaller units.Divide to convert smaller units to larger units. �

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© Harcourt • Grade 5RW130 MG 1.4 Differentiate between, and use appropriate

units of measures for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).


Estimate PerimeterThe perimeter of a figure is the distance around the figure. You can use a piece of string to estimate the perimeter of a figure.

Estimate the perimeter of the triangle in centimeters..

Step 1 Lay a piece around the figure.

Step 2 Cut the string where it meets itself.

Step 3 Lay the string in a straight line and measure its length with a centimeter ruler.

The string is about 15 centimeters long, so the perimeter ofthe triangle is about 15 centimeters.

Estimate the perimeter of the polygon in centimeters.

1. 2.

3. 4.

centimeters

13 14 151 2 3 4 5 6 7 8 9 10 11 12

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Name




Find PerimeterSince opposite sides of a parallelogram are equal, you can use addition to find the perimeter of a parallelograms.

2 sides are 8 ft and 2 sides are 4 ft.Add 8 � 8 � 4 � 4 � 24

So, the perimeter of the parallelogram is 24 ft.

Since the sides of a regular polygon are equal, you can use multiplication to find the perimeter of a regular polygon.

Each of the 3 sides are 6 in. long.Multiply 6 � 3 � 18

So, the perimeter of the triangle is18 in.

Find the perimeter of each polygon.

1. 2. 3.

4. 5. 6.

4 ft

8 ft

6 in. 6 in.

6 in.

4 in. 4 in.

5 in.

7 in.

5 in.

3 ft

4 ft 5 ft

6 ft

7 cm

4 cm

2 cm

7 ft

7 ft3 m


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Perimeter FormulasSince the sides of a regular polygon areequal, you can use a formula to find theperimeter of a regular polygon.

Find the perimeter of each regular polygon by

using a formula.

• Perimeter (P) � (number of sides) � S

• P � 5 � 8

• P � 40

So, the perimeter of the pentagon is 40 cm.

• Perimeter (P) � (number of sides) � S

• P � 4 � 9

• P � 36

So, the perimeter of the square is 36 in.

Find the perimeter of each polygon.

1. 2. 3.

4. 5. 6.

6 ft

8 cm 8 cm

8 cm8 cm

8 cm

9 in.

9 in.

9 in. 9 in.

6 ft

8 ft

4.5 m 3.6 cm3.6 cm

5.7 cm5.7 cm

6 in.

3 in. 3 yd

5 yd

2 yd

8 yd

10 in.

3 in.

10 in.

3 in.


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Use Perimeter FormulasThe perimeter of a polygon is the sum of the lengths of its sides.

If you know the perimeter of a polygon and the lengths of all but one side,you can find the unknown length.

The perimeter of the figure

is 98 in.

Find the unknown length, f.

Step 1

Write the formula using a variable, one for each side.

Step 2

Replace the variables with the knownlengths. Add.

Step 3

Solve for f.

So, the unknown length is 9 1 __ 4 in.

The perimeter is given. Find the unknown length.

1. P � 89 ft 2. P � 95 in. 3. P � 14.8 m

P � a � b � c � d � e � f

98 � 7 � 35 � 14 � 15 3 __ 4

� 7 � f

98 � 78 3 __ 4

� f

98 � 78 3 __ 4

� f

9 1 __ 4

� f

35 in.

15 3 _ 4 in.

14 in. 7 in.

f

7 in.

3.9 m

1.5 m

1.5 m

2 m

r

3.5 m20 in.

12 in.

27 in.

12 in.

w22 ft

m

18 ft

25 ft


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Problem Solving Workshop Strategy:Make GeneralizationsTwo tissue boxes are congruent cubes. If the perimeter of the base of one

tissue box is 16 in., what is the length of one side of the base of the other

tissue box?

1. What are you asked?

2. How does knowing that the two tissue boxes are congruent help you?

3. How does knowing the shape of the boxes help you?

4. Which formula would you use to solve the problem?

5. What is the length of one side of the base of the other tissue box?

Make generalizations to solve.

6. The lid of a designer box is in the shape of a regular hexagon. The perimeter is 84 inches. What is the length of each side of the lid?

7. Jen has 96 feet of fencing to make a rectangular shaped pen in her backyard. The width of the pen is 15 feet. What is the length of the pen?


Name

© Harcourt • Grade 5Reteach the StandardsRW135MG 1.0 Students understand and compute the volumes

and areas of simple objects.

Estimate AreaYou can use centimeter grid paper to estimate area of a figure.

Estimate the area of the figure.

Step 1 Count the number of partial squares. Skip squares with only a tiny corner.

Row 1: 4

Row 2: 2

Row 3: 4

Total: 10

Step 2 Divide the sum by 2. 10 � 2 � 5

Step 3 Count the number of full squares, including those missing only a tiny corner.

Row 1: 1

Row 2: 3

Row 3: 1

Total: 5

Step 4 Add the values from Steps 2 and 3. 5 � 5 � 10

Estimate the area of the figure. Each square on the grid is 1 in.2.

1. 2. 3.

1

2

3


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© Harcourt • Grade 5Reteach the StandardsRW136 MG 1.4 Differentiate between, and use appropriate

units of measures for two- and three-dimensional objects (i.e. find the perimeter, area, volume).

Area of RectanglesYou can find the area of a square or rectangle by drawing a grid over it and counting the squares.

Find the area of the figure.

One part of the figure is a 4 ft-by-4 ft square.

To find the area of the square, draw 4 columns and 4 rows insidethe square.

The square has 4 columns of 4 squares each. So, you can count by multiplying 4 � 4 to get 16 square feet.

The other part of the figure is a 6 ft-by-11 ft rectangle.

To find the area of the rectangle, draw 11 columns and 6 rows inside the rectangle.

The rectangle has 11 columnsof 6 squares each. So, youcan count by multiplying 11 � 6 to get 66 square feet.

To find the total area, add the area of the square to the area of the rectangle.

16 � 66 � 82

So, the area of the figure is 82 ft2.

Find the area of each figure.

1. 2. 3.

The formula for the area of a rectangle with length l units and width w units is

A � l � w � lw

The formula for the area of a square with side s units is

A � s2

4 ft

4 ft

11 ft

6 ft

5 in.

5 in.

5 in.

12 in.

10 yd

4 yd

8 cm

8 cm


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units of measures for, two- and three-dimensional objects(i.e., find the perimeter, area, volume).

Relate Perimeter and AreaYou can use grid paper to relate perimeter and area.

A rectangle has a perimeter of 16 feet. What are the length and width of the

rectangle with the greatest possible area?

Step 1

Divide the perimeter by 2. The length and width of a rectangle whose perimeter is 16 have a sum of 8.

Step 2

Draw a 1-row rectangle with a perimeter of 16 units on grid paper. Imagine it as a tabletop and each place-setting is 1 unit wide. Place a dot around the “table” for each person.

The perimeter, 16 units, is the number of people who can sit around the table. The area of the table at the right is 7 units2, the number of squares on the tabletop.

Step 3

Make as many different rectangles as possible. Find the area of each. Complete a table like the one at the right. Stop when you realize you have found the greatest area.

The rectangle with the greatest area is 4 ft x 4 ft.

The rectangle with the greatest area for a given perimeter

is a square.

The rectangle with the least perimeter for a given area

is also a square.

If you cannot form a square, look for the shape closest to a square.

For the given perimeter, find the length and width of the

rectangle with the greatest area.

1. perimeter � 18 in. 3. perimeter � 30 km

perimeter length width area

16 1 7 716 2 6 1216 3 5 1516 4 4 1616 5 3 1516 6 2 12

You can use factors of a given area to find the length and width of rectangles.


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MG 1.4 Differentiate between, and use appropriate units of measures for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).

Problem Solving Workshop Strategy: Compare StrategiesA garden center has 5 rectangular flower boxes

displayed in a row. The first flower box is 24 in. long and

4 in. wide. Each flower box is the same length but 2 in.

wider than the previous flower box. What is the

perimeter of the fifth flower box?

Read to Understand

1. What do you visualize when you read the problem?

Plan

2. What strategies can help you solve this problem?

Solve

3. What are the measurements of each flowerbox?

4. What is the perimeter of the fifth flowerbox?

Check

5. What other strategy could you use? Explain how you would use it.

Solve.

6. A goat pen is 36 ft by 36 ft. A fence post is every 6 feet. How many fence posts are there?

7. A wall has 8 rows of bricks. The bottom row has 10 bricks. Each row has one less brick than the row below it. How many bricks are in the wall?


Name


MG 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram.)

Model Area of TrianglesYou can use a rectangle to model the area of a triangle.

Step 1

Draw any rectangle on grid paper. Make each side greater than 10 units.Find and record the area of the rectangle.

Step 2

Cut out the rectangle.

Step 3

Place a dot on one edge of the rectangle. Use a ruler to draw a line from the dot to each opposite corner. You have made a triangle.

Step 4

Fold the rectangle along two sides of the triangle.• Does it cover the triangle without overlapping?

Use the rectangle to

find the area of the

triangle.

Step 1

Find the area of the rectangle.

Area of rectangle � 5 � 10

Area of rectangle � 50 cm2

Step 2

Find half the area of the rectangle.

Area of triangle � 50 � 2

Area of triangle � 25 cm2

So, the area is 25 cm2.

Find the area of each triangle in cm2.

1.

2.

3.

The area of a triangle is

half the area of the

rectangle that surrounds

it.

10 cm

5 cm


Name


MG 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram).

Area of Triangles

You can use a formula to find the area of a triangle.

You can think of the length and width of a rectangle as the base and height of a triangle inside it.

If b is the length of the base and h is the height of atriangle, you can use the formula A � 1 __ 2 bh to find itsarea, A.

Find the area of the triangle.

Step 1 Draw a rectangle around the triangle.

Step 2 Count the units for the height and the units for the base.

height = 5 units

base = 8 units

Step 3 Replace 8 for b and 5 for h in the formula. A � 1 __ 2

bh

A � 1 __ 2

� 8 � 5

Step 4 Multiply. 1 __ 2

� 8 � 5 � 20

So, the area of the triangle is 20 units2.

Find the area of each triangle in square units.

1.

2.

3.

height

width

length = base

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MG 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram).

Algebra: Area of ParallelogramsYou can use grid paper and the base and heightof a parallelogram to find its area.

Find the area of the parallelogram.

Step 1 Draw a parallelogramon grid paper and cut it out.

Step 2 Draw a line segment to form aright triangle.

Step 3 Cut out the triangle on the bottomand move it to the topof the parallelogram toform a rectangle.

Step 4 Count the gridsquares to find the areaof the parallelogram.The base is 3 units andthe height is 5 units.

The formula for the area of a parallelogram with base, b, and height,h, is A � bh.

So, the area of the parallelogram is3 � 5, or 15 units2.

Find the area of each parallelogram.

1. 2. 3.

5

3

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MG 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.

Surface AreaYou can use a net to find the surface area of rectangular and triangular prisms.

You can find the surface area, the total area of the surface of a solid figure, by adding the area of each face.

Another way to find the surface area is to use a net.

Use the net to find the surface area of the

rectangular prism.

• What is the area of the rectangle of A, B, C, and E?

• What is the area of Rectangle D?

• What is the area of Rectangle F?

• What is the surface area of the rectangular prism?

Use the net to find the surface area of each prism

in square units.

1. 2.

2

4

5

A B C

D

E

F


Name


MG 1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm.3], cubic meter [m.3], cubic inch [in.3], cubic yard [yd.3], to compute the volume of rectangular solids.

Estimate VolumeJust as you use squares to estimate the area of a rectangle, you can use cubes to estimate the volume of a rectangular prism.

Area is measured in squares, or in square units.

Volume is measured in cubes, or in cubic units.

You can use the square face of a cubic unit to estimate the area of the base of a rectangular prism.

Estimate the volume of the rectangular prism

with the measurements 4.2 cm � 6.8 cm � 1.7 cm.

Step 1

Choose the 4.2 cm � 6.8 cm to be the base of the prism.

Step 2

Round to estimate the area of the base of the prism.

4.2 � 4 and 6.8 � 7

4 x 7 = 28

The area of the base of the prism is about 28 cm2.

Step 3

Round the third measurement. Multiply it by the area of the base.

1.7 � 2

28 � 2 = 56

The volume of the rectangular prism is about 56 cm3.

Estimate the volume of each rectangular prism with the given

measurements.

1. 3.8 in. � 5.1 in. � 3.4 in. 2. 5.5 cm � 4 cm � 4.9 cm 3. 8 yd � 9.3 yd � 2.6 yd

4. 10.6 m � 6.3 m � 9.9 m 5. 8.7 cm � 11.6 cm � 7 cm 6. 9 ft � 5 ft � 8 ft

4.2 cm6.8 cm

1.7 cm


Name


MG 1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm.3], cubic [m.3], cubic inch [in.3] cubic yard [yd.3] to compute the volume of rectangular solids.

Find VolumeTo find area of a rectangle, find the number of square units that fit in it. To find volume of a rectangular prism, find the number of cubic units that fit in it.

Find the volume of the rectangular prism.

Step 1

Remember: A rectangular prism has three dimensions: length, width, and height.

Find how many centimeter cubes will fit on the bottom face. Multiply the length and width of the bottom face.

5 � 2 � 10

So 10 is the number of centimeter cubes in the bottom layer.

Step 2

The height, 6 cm, tells how many layers of centimeter cubes the prism holds.

Multiply the number of cubes that fit on the bottom layer by the height of the prism.

10 � 6 � 60

The volume of the prism is 60 cubic centimeters, or 60 cm3.

Find the volume of each rectangular prism.1. 2. 3.

4 in.

5 in.5 in. 6 cm

8 cm

10 cm

5 m

3 m7 m

6 cm

5 cm

2 cm

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units of measure for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).

Relate Perimeter, Area, and VolumePerimeter is the distance around a flat figure.

Area is the space on a flat surface of a flat figure.

Volume is the amount of space a solid figure takes up.

Nel has an aquarium in the shape of a rectangular prism.

It is 4 ft tall, 8 ft wide, and 2 ft deep.

What unit should she use to measure the perimeter of the aquarium?

Perimeter is measured in units of length, such as inches, feet, yards, miles, centimeters, meters, and kilometers.

Nel should use feet to measure perimeter.

What unit should Nel use to measure the area of the aquarium?

Area is measured by using unit squares.

So, measure area in square units, such as square inches (in.2), square feet (ft2), square yards (yd2), square miles (mi2), square centimeters (cm2), square meters (m2), and square kilometers (km2).

Nel should use square feet, or ft2, to measure area.

What unit should Nel use to measure the volume of the aquarium?

Volume is measured by using unit cubes.

So, measure volume in cubic units, such as cubic inches (in.3), cubic feet (ft3), cubic yards (yd3), cubic centimeters (cm3), and cubic meters (m3).

Nel should use cubic feet, or ft3, to measure volume.

Write the units you would use to for measuring each.1. area of this parallelogram 2. volume of this

figure3. perimeter of this

regular pentagon

4 ft

8 ft2 ft

2 ft

8 ft

7 cm4 ft 5 cm

3 cm8 in.


Name


MG 1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm.3], cubic meter [m.3], cubic inch [in.3], cubic yard [yd.3] to compute the volume of rectangular solids.

Problem Solving Workshop Strategy: Write an EquationThe area of a triangular coffee table is 108 square

inches. The two sides that are against the wall meet at a

right angle. One side that is against the wall is 12 inches

long. What is the length of the other side that is against

the wall?Read to Understand

1. Describe what you are asked to find.

2. What is the shape of the triangle?

Plan

3. How can you use the given information to write an equation?

Solve

4. Show how you solve the problem.

Check

5. What other strategy can you use to check your answer?

Write an equation to solve.

6. A computer screen is 704 square centimeters. If it is 22 centimeters high, how wide is it?

7. Tamara paid $7 for lunch, including a tip. If the cost of her meal was $5.90, how much was the tip?


Name


Reteach the Standards SDAP 1.0 Students display, analyze, compare, and interpet different data sets, including data sets of different sizes. RW147

Collect and Organize DataA survey is a way to gather information about a group.When you are gathering information about a group,• the whole group is called the population,• and the people surveyed are called the sample.

1.

A playground maker wants to find out if children in grades 4–6 like their new

playground equipment. Tell whether each sample represents the population. If it

does not, explain.

A restaurant wants to find out how many children ages 8–12 like their new children’s menu. Which sample represents the population?

a. b. c.a random sample of 100 fifth graders

a random sample of 100 parents

a random sample of 100 children ages 8–12

This is not a good sample because it only includes fifth graders, not all children ages 8–12.

This is not a good sample because it asks parents, not children, about the menus.

This sample represents the population fairly.

A radio station wants to find out the favorite type of music of people that live in

Los Angeles, California. Tell whether each sample represents the population. If it

does not, explain.

2. 3.a random sample of 400 boys in grades 4–6

a random sample of children in grades 4–6

a random sample of 400 teachers

4. a random sample of 200 people who live in California.

a random sample of 200 adults who live in Los Angeles.

5. 6. a random sample of 200 people who live in Los Angeles.

A sample must fairly represent the population. In a random sample, everyone in the population has an equal change of being surveyed.

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© Harcourt • Grade 5RW148 Reteach the Standards SDAP 1.1 Know the concepts of mean, median, and

mode; compute and compare simple examples to show that they may differ.

Find the MeanThe mean is the average of a set of data.

Find the mean.

Find the mean for the data set:$125, $83, $134

Step 1: Divide the sum by the number of addends. There are 3 addends.

Example 1

$342 � 3 � $114

The mean is $114

$12583

134

$342

Example 2

Find the mean for the data set:8.1, 4.5, 7.6, 8.4, 7.4

Step 1:

Add to find the sum of the data set.Step 2:

Add to find the sum of the data set.8.1

4.57.6

8.4

7.4

_

36.0

Divide the sum by the number of addends. There are 5 addends.

Step 2:

36.0 � 5 � 7.2

The mean is 7.2.

1.

3.

5.

7.

9.

11.

2.

4.

6.

8.

10.

12.

$64, $48, $40, $52 75, 62, 82

9.8, 10.2, 11.6, 10.2, 9.7 21, 34, 45, 32

$142, $167, $171742, 684, 923, 867

10.5, 12.5, 11, 10, 13 23.6, 31.4, 22.8, 34, 27.2

$12, $8, $17, $9, $7

223, 425, 351, 215 1.2, 2.1, 3.3, 3.6, 1.8, 1.8

1,143; 986; 1,267


Name

© Harcourt • Grade 5RW149 Reteach the Standards SDAP 1.1 Know the concepts of mean, meadian, and

mode; compute and compare simple examples to show that they differ.

Find the Median and ModeFind the median and mode for the data set: 22, 18, 20, 16, 14, 16

Find the median and mode for each set of data.

The median is the middle number when a set of data is arranged in order.

Step 1: Order the data from least to greatest.

The mode is 16.

Step 1: Order the data from least to greatest.

Sometimes there is more than one mode, or no mode.

1.

3.

5.

7.

9.

11.

2.

4.

6.

8.

10.

12.

21, 15, 17, 21, 16 641, 874, 614, 755

7.8, 9.4, 10.6, 9.8, 7.8, 9.4 164, 215, 174, 174, 193

21, 24, 22, 24, 21, 31, 25$25, $36, $28, $27

350, 378, 350, 252, 275 43, 53, 63, 53

873, 954, 896, 941

99, 103, 126, 84, 99, 101 1,024; 973; 1,204

1.3, 1.55, 2.75, 1.3, 2.6

14, 16, 16, 18, 20, 22

Step 2: Find the middle number. Since there is an even number of data, the median is the mean of the two middle numbers.

( 16 � 18 ) � 2 � 17

The median is 17.

The mode is the number that occurs most often in a set of data.

14, 16, 16, 18, 20, 22

Step 2: Find the number that occurs most often.

You can compare the median and mode.The median, 17, is greater than the mode, 16.

Find the median and mode for each set of data. Tell how the median and the mode

compare.

13.

15.

14.

16.

63, 56, 63, 61, 58

14, 27, 13, 14, 23 $36, $42, $48, $36, $42

7.8, 7.9, 7.8, 7.7, 7.6


Name

© Harcourt • Grade 5Reteach the Standards SDAP 1.1 Know the concepts of mean, median, and

mode; compute and compare simple examples to show that they may differ.

RW150

Compare DataCompare the data sets. Tell how the data sets compare.You can compare data sets using the mean, range, and median.

1.

Compare the data sets. Tell how the data sets compare.

A: Number of model cars built

15 8 11 13 1612 15 7 18 9

B: Number of model cars built

15 9 10 11 1716 12 8 22 18

Mean: 124 � 10 � 12.4 Mean: 138 � 10 � 13.8

The mean for data set B is greater than the mean for data set A.

Range: 18 � 7 � 11 Range: 22 � 8 � 14

The range for data set B is greater than the range for data set A.

Median: (12 � 13) � 2 � 12.5 Median: (12 � 15) � 2 � 13.5

The median for data set B is greater than the median for data set A.

A: Weights of Boxes

11 13 15 17 1612 15 20 9 13

B: Weights of Boxes

22 13 15 16 1811 18 21 14 15

2.A: Songs Students Heard

3 5 2 1 0 9 3 12 8 3 4 7 6 2 4

B: Songs Students Heard

5 7 2 0 910 5 7 9 6


Name

Mon. Tues. Wed. Thurs. Fri.


AF 1.1 Use information taken from a graph or equation to answer questions about a problem situtation. RW151

Analyze GraphsGraphs can help you draw conclusions, answer questions, and make predictions about data.

Which student read the most books?The bar for Sally is the longest, so she read the most books.

A bar graph uses bars to display countable data. A bar graph is useful when comparing data by groups.

A pictograph displays countable data using pictures and symbols. Pictographs have a key to show how many each picture or symbol stands for.

How many fantasy books are in Mr. Li’s class?The key shows each symbol stands for 4 books. A half symbol stands for 2 books.(5 � 4) � 2 � 22 books

A circle graph shows how parts of data related to each other and to the whole.

A line graph shows how data changes over a period of time.

What kinds of animals does the pet store have the same number of?cats and dogs

How can you describe the trend in temperature from Wednesday to Friday?

The trend is increasing temperatures.

For 1–4, use the pictograph at the right.

Twenty-eight wins would be shown as Which sport had this number of wins?

1.

2.How many wins did baseball have?3. How many more wins did soccer have than

basketball? 4. What if football had 20 wins. How would this

amount be shown on the pictograph?

Sport Team Wins

Soccer

BaseballBasketball

Hockey

Key: = 8 wins

Pet Store Population

Birds, 20

Dogs, 10

Cats, 10

Reptiles, 5

Types of Books in Mr. Li’s Class

Fantasy

Mystery

Biography

Poetry

Key: = 4 books


Name


AF 1.1 Use informations taken from a graph or equation to answer questions about a problem situation.

Problem Solving Workshop Strategy: Use Logical Reasoning

RW152

Five students wrote a report about U.S. Presidents, 8 wrote a report

about U.S. First Ladies, and 3 wrote a report about both U.S.

Presidents and First Ladies. How many students wrote reports?

Solve

Read to Understand


2. Plan

How can drawing a diagram help you solve the problem?

3. Solve the problem using the Venn diagram.

Write your answer as a complete sentence.4.

Check

5. Explain one way you could check your solution.

Draw a Venn diagram to solve.

Eight students wrote reports about the inner planets and 9 students wrote reports about the outer planets. Four of these students wrote reports about both the inner planets and the outer planets. How many students wrote reports?

7.During a one-hour research period, 8 people used the Internet and 4people used an atlas. Two of these people used the Internet and an atlas. How many people used the Internet or an atlas during the research period?

6.


Name

876543210

6-7 8-9 10-11 12-13Ages (in years)

Ages of Students Using SchoolLaptop Computers

Num

ber

of

Lapto

ps

Height (in Inches) of Students


SDAP 1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.

Make HistogramsA histogram shows the number of times, or frequency, an event or data item occurs.

For 1–2, use the table.

1. Use 10 inches for each interval. List the intervals.

2. Make a histogram of the data.

Make a histogram of the data.

Which age group uses laptops the least?

• To make a histogram, you need to find a reasonable interval.

• The data in the table shows ages from 6 to 12.

• A reasonable interval would be 2 years, starting at 6.

• Make a frequency table to determine the number of times each age appears in the data.

• Make the histogram by first choosing an appropriate scale for the vertical axis. Label the axis.

• Use the frequency table to find how long to make each bar.

• Draw the bars touching, but not overlapping.

• Make each bar the same width.

• Label each bar with the frequency table interval and label the axis.

• So, the 6-7 year old age group uses laptops the least.

Ages of School Laptop Computer Users

11 12 10 9 8 12 11

12 10 8 7 13 7 6

10 7 12 12 9 10 9

Frequency Table

Interval 6-7 8-9 10-11 12-13Frequency 4 5 6 6

Heights (in inches) of Students

52 48 47 41 5460 42 46 39 5744 49 46 47 6143 63 49 56 5251 60 54 42 62


Name

Make Stem-and-Leaf PlotsA stem-and-leaf plot lists numbers from least to greatest by place value. It is especially helpful when dealing with many numbers.

Use the data to make a stem-and-leaf plot.

What are the lowest and highest scores?

• The table shows the numbers are in the 50s, 60s, 70s, and 80s. So the plot will have four stems: 5, 6, 7, and 8. Write the digits in the Stem column.

• Use the ones digit of each basketball score to fill in the leaves column. Arrange the leaves of each stem in order from least to greatest.

• The lowest score is 52.

• The highest score is 85.

For 1–2, use the data.

1. Use the data to make a stem-and-leaf plot.

2. Which age occurred most often?

For 3–4, use the data.

3. Use the data to make a stem-and-leaf plot.

4. What are the least and greatest yards rushed?

Basketball Team Scores

63 67 73 55 61 53 60 63

52 61 64 85 74 59 67 72

64 55 64 58 66 78 56 61

Basketball Team Scores

Stem Leaves

5 2 3 5 5 6 8 96 0 1 1 1 3 3 4 4 4 6 7 77 2 3 4 88 5

Ages of Carnival Customers

12 13 34 28 14 12 18 48

45 32 12 39 13 11 28 16

37 42 10 21 26 10 35 11

Yards Rushed by Football Players

47 63 67 72 44

62 55 49 56 70

63 44 53 46 67

58 71 77 61 63

© Harcourt • Grade 5Reteach the StandardsRW154SDAP 1.2 Organize and display single-variable data in

appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.


Name

Te

mp

era

ture

Sacramento, CA Average Highs

Month

March JulyJuneMayApril

80

60

40

200

100

San Diego, CA Average LowsJim’s Check-Up Heights

© Harcourt • Grade 5Reteach the StandardsRW155SDAP 1.4 Identify ordered pairs of data from a

graph and interpret the meaning of the data in terms of the situation depicted by the graph.

Make Line GraphsA line graph shows how a set of data changes over time.

Use the table to make a graph.

• To make a line graph, you need to determine the greatest number on the vertical scale.

• The greatest temperature in the table is 93.

• The scale should go from 0 to 100 using intervals of 20.

• Write the months along the horizontal axis using even spacing.

• Use the data to find where to graph each point.

• Connect the points to show the average temperature rising from March to July.

For 1–2, make a line graph for the data.

1. 2.

Sacramento, CA Average Highs

Month March April May June July

Temp. 64 71 80 87 93

Jim’s Height (inches)

Age 3 4 5 6 7Height 39 41 43 46 48

San Diego, CA Average Lows

Month March April May June July

Temp. 53 53 60 62 66


Name

Problem Solving Workshop Skill: Draw ConclusionsWhen answering questions involving graphs, use the information given to draw conclusions about what should happen next.

In 2005, the September rainfall was 0.10 inch. In October it was 0.46 inch, in November

it was 0.12 inch, and in December it was 0.33 inch. What conclusion can you draw

about the rainfall during this period?

1. What are you asked to find?

2. Using the information given, what conclusion can you draw about the rainfall from September to December?

3. How can you check if your conclusion is correct?

For 4–5, use the double-bar graph.

4. Volcanoes in this region are part of the Pacific Ring of Fire. What conclusions can you draw about this region?

5. What conclusion can you draw about the volcanoes in the Philippines versus those in Alaska?

100

Alaska Philippines MexicoRegion

Active

Inactive

Volcanoes of the World

Num

ber

of

Volc

anoes 90

80706050

30

100

20

40

© Harcourt • Grade 5Reteach the StandardsRW156SDAP 1.0 Students display, analyze, compare, and

interpret different data sets, including data sets of different sizes.


Name


SDAP 1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which type of graphs are appropriate for various data sets.

Choose the Appropriate GraphBar and double-bar graphs, line graphs, line plots and stem-and-leaf plots organize numerical data.

Tell which graph best displays Marta’s and Phil’s

test scores changing over time.

Double-bar Graph

A double-bar graph is used to compare two sets of data.

Line Graph

A line graph shows change over time.

Line Plot

A line plot is used to record data as it is collected.

Stem-and-leaf Plot

A stem-and-leaf plot is used to organize data by place value.

100

80

60

40

20

01 2 3 4 5

Test

Test Scores

Sco

re

Marta

Phil Sco

re

Test Scores

Tests

Marta

Phil

1 4 532

100

80

60

40

20

0

Choose the best type of graph or plot for the data. Explain your choice.

1. The populations of males and females in 6 major cities.

2. The daily growth of a sunflower in inches.

3. The scores of a basketball team for one season.

4. The temperature recorded every hour for 24 hours.

Test Scores

Test 1 2 3 4 5

Marta 88 85 92 95 98

Phil 83 93 85 80 92

Test Scores

Stem Leaves

8 0 3 5 5 8

9 2 2 3 5 8

A line graph best displays Marta’s and Phil’s test scores change over time.


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