Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

no text concepts found

Transcript

Reteach Workbook PUPIL EDITION G ra d e 4 Orlando • Boston • Dallas • Chicago • San Diego www.harcourtschool.com Copyright © by Harcourt, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Duplication of this work other than by individual classroom teachers under the conditions specified above requires a license. To order a license to duplicate this work in greater than classroom quantities, contact Customer Service, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. Telephone: 1-800-225-5425. Fax: 1-800-874-6418 or 407-352-3445. HARCOURT and the Harcourt Logo are trademarks of Harcourt, Inc. Printed in the United States of America ISBN 0-15-320444-3 2 3 4 5 6 7 8 9 10 022 2004 2003 2002 2001 © Harcourt Permission is hereby granted to individual teachers using the corresponding student’s textbook or kit as the major vehicle for regular classroom instruction to photocopy complete pages from this publication in classroom quantities for instructional use and not for resale. CONTENTS Unit 1: UNDERSTAND NUMBERS AND OPERATIONS Chapter 1: Place Value and Number Sense 1.1 Benchmark Numbers . . . . . . . . . . . . . 1 1.2 Understand Place Value . . . . . . . . . . 2 1.3 Place Value Through Hundred Thousands . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Place Value Through Millions . . . . . 4 1.5 Problem Solving Skill: Use a Graph . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 2: Compare and Order Numbers 2.1 Compare Numbers . . . . . . . . . . . . . . 6 2.2 Order Numbers . . . . . . . . . . . . . . . . . 7 2.3 Problem Solving Strategy: Make a Table . . . . . . . . . . . . . . . . . . . . 8 2.4 Round Numbers . . . . . . . . . . . . . . . . . 9 Chapter 3: Add and Subtract Greater Numbers 3.1 Estimate Sums and Differences . . . 10 3.2 Use Mental Math Strategies . . . . . . 11 3.3 Add and Subtract 4-Digit Numbers . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Subtract Across Zeros . . . . . . . . . . . 13 3.5 Add and Subtract Greater Numbers . . . . . . . . . . . . . . . . . . . . . . 14 3.6 Problem Solving Skill: Estimate or Find Exact Answers . . . . . . . . . . . . . . 15 Chapter 4: Algebra: Use Addition and Subtraction 4.1 Expressions . . . . . . . . . . . . . . . . . . . . 16 4.2 Use Parentheses . . . . . . . . . . . . . . . . 17 4.3 Match Words and Expressions . . . . 18 4.4 4.5 4.6 4.7 Use Variables . . . . . . . . . . . . . . . . . . . 19 Find a Rule . . . . . . . . . . . . . . . . . . . . 20 Add Equals to Equals . . . . . . . . . . . . 21 Problem Solving Strategy: Make a Model . . . . . . . . . . . . . . . . . 22 Unit 2: DATA, GRAPHING, AND TIME Chapter 5: Collect and Organize Data 5.1 Collect and Organize Data . . . . . . . 23 5.2 Find Median and Mode . . . . . . . . . 24 5.3 Line Plot . . . . . . . . . . . . . . . . . . . . . . . 25 5.4 Stem-and-Leaf Plot . . . . . . . . . . . . . 26 5.5 Compare Graphs . . . . . . . . . . . . . . . 27 5.6 Problem Solving Strategy: Make a Graph . . . . . . . . . . . . . . . . . . 28 Chapter 6: Analyze and Graph Data 6.1 Double-Bar Graphs . . . . . . . . . . . . . 29 6.2 Read Line Graphs. . . . . . . . . . . . . . . 30 6.3 Make Line Graphs . . . . . . . . . . . . . . . 31 6.4 Choose an Appropriate Graph . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.5 Problem Solving Skill: Draw Conclusions . . . . . . . . . . . . . . . . . . . . 33 Chapter 7: Understand Time 7.1 Before and After the Hour . . . . . . . 34 7.2 A.M. and P.M. . . . . . . . . . . . . . . . . . . 35 7.3 Elapsed Time . . . . . . . . . . . . . . . . . . 36 7.4 Problem Solving Skill: Sequence Information . . . . . . . . . . . . . . . . . . . . 37 7.5 Elapsed Time on a Calendar . . . . . 38 Unit 3: MULTIPLICATION AND DIVISION FACTS Chapter 8: Practice Multiplication and Division Facts 8.1 Relate Multiplication and Division . . . . . . . . . . . . . . . . . . . 39 8.2 Multiply and Divide Facts Through 5 . . . . . . . . . . . . . . . . . . . . . 40 8.3 Multiply and Divide Facts Through 10 . . . . . . . . . . . . . . . . . . . . . 41 8.4 Multiplication Table Through 12 . . . . . . . . . . . . . . . . . . . . . 42 8.5 Multiply 3 Factors . . . . . . . . . . . . . . 43 8.6 Problem Solving Skill: Choose the Operation . . . . . . . . . . . . . . . . . 44 Chapter 9: Algebra: Use Multiplication and Division Facts 9.1 Expressions with Parentheses . . . . 45 9.2 Match Words and Expressions . . . 46 9.3 Multiply Equals by Equals . . . . . . . 47 9.4 Expressions with Variables . . . . . . . 48 9.5 Equations with Variables . . . . . . . . 49 9.6 Find a Rule . . . . . . . . . . . . . . . . . . . . 50 9.7 Problem Solving Strategy: Work Backward . . . . . . . . . . . . . . . . . 51 Unit 4: MULTIPLY BY 1- AND 2-DIGIT NUMBERS Chapter 10: Multiply by 1-Digit Numbers 10.1 Mental Math: Multiplication Patterns . . . . . . . . . . . . . . . . . . . . . . . 52 10.2 Estimate Products . . . . . . . . . . . . . . 53 10.3 Model Multiplication . . . . . . . . . . . 54 10.4 Multiply 3-Digit Numbers . . . . . . . 55 10.5 Multiply 4-Digit Numbers . . . . . . . 56 10.6 Problem Solving Strategy: Write an Equation . . . . . . . . . . . . . . 57 Chapter 11: Understand Multiplication 11.1 Mental Math: Patterns with Multiples . . . . . . . . . . . . . . . . . 58 11.2 Multiply by Multiples of 10 . . . . . . 59 11.3 Estimate Products . . . . . . . . . . . . . . 60 11.4 Model Multiplication . . . . . . . . . . . 61 11.5 Problem Solving Strategy: Solve a Simpler Problem . . . . . . . . . . . . . . . 62 Chapter 12: Multiply by 2-Digit Numbers 12.1 Multiply by 2-Digit Numbers . . . . . 63 12.2 More About Multiplying by 2-Digit Numbers . . . . . . . . . . . . . . . 64 12.3 Multiply Greater Numbers . . . . . . 65 12.4 Practice Multiplication . . . . . . . . . . 66 12.5 Problem Solving Skill: Multistep Problems . . . . . . . . . . . . . . . . . . . . . . 67 Unit 5: DIVIDE BY 1-AND 2-DIGIT DIVISORS Chapter 13: Understand Division 13.1 Divide with Remainders . . . . . . . . . 68 13.2 Model Division . . . . . . . . . . . . . . . . . 69 13.3 Division Procedures . . . . . . . . . . . . 70 13.4 Problem Solving Strategy: Predict and Test . . . . . . . . . . . . . . . . . . . . . . . 71 13.5 Mental Math: Division Patterns . . . . . . . . . . . . . . . . . . . . . . . 72 13.6 Estimate Quotients . . . . . . . . . . . . . 73 Chapter 14: Divide by 1-Digit Divisors 14.1 Place the First Digit . . . . . . . . . . . . . 74 14.2 Divide 3-Digit Numbers . . . . . . . . . 75 14.3 Zeros in Division . . . . . . . . . . . . . . . 76 14.4 Divide Greater Numbers . . . . . . . . 77 14.5 Problem Solving Skill: Interpret the Remainder . . . . . . . . . . . . . . . . . 78 14.6 Find the Mean . . . . . . . . . . . . . . . . . 79 Chapter 15: Divide by 2-Digit Divisors 15.1 Division Patterns to Estimate . . . . . . . . . . . . . . . . . . . . . . 80 15.2 Model Division . . . . . . . . . . . . . . . . . 81 15.3 Division Procedures . . . . . . . . . . . . 82 15.4 Correcting Quotients . . . . . . . . . . . 83 15.5 Problem Solving Skill: Choose the Operation . . . . . . . . . . . . . . . . . 84 Chapter 16: Patterns with Factors and Multiples 16.1 Factors and Multiples . . . . . . . . . . . 85 16.2 Factor Numbers . . . . . . . . . . . . . . . . 86 16.3 Prime and Composite Numbers . . . . . . . . . . . . . . . . . . . . . . 87 16.4 Find Prime Factors . . . . . . . . . . . . . . 88 16.5 Problem Solving Strategy: Find a Pattern . . . . . . . . . . . . . . . . . . 89 Unit 6: FRACTIONS AND DECIMALS Chapter 17: Understand Fractions 17.1 Read and Write Fractions . . . . . . . 90 17.2 Equivalent Fractions . . . . . . . . . . . . 91 17.3 Equivalent Fractions . . . . . . . . . . . . 92 17.4 Compare and Order Fractions . . . . . . . . . . . . . . . . . . . . . . 93 17.5 Problem Solving Strategy: Make a Model . . . . . . . . . . . . . . . . . 94 17.6 Mixed Numbers . . . . . . . . . . . . . . . . 95 Chapter 18: Add and Subtract Fractions and Mixed Numbers 18.1 Add Like Fractions . . . . . . . . . . . . . . 96 18.2 Subtract Like Fractions . . . . . . . . . . 97 18.3 Add and Subtract Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . 98 18.4 Problem Solving Skill: Choose the Operation . . . . . . . . . . . . . . . . . 99 18.5 Add Unlike Fractions . . . . . . . . . . 100 18.6 Subtract Unlike Fractions . . . . . . . 101 Chapter 19: Understand Decimals 19.1 Relate Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . 102 19.2 Decimals Greater Than 1 . . . . . . . . 103 19.3 Equivalent Decimals . . . . . . . . . . . 104 19.4 Compare and Order Decimals . . . 105 19.5 Problem Solving Strategy: Use Logical Reasoning . . . . . . . . . . . . . 106 19.6 Relate Mixed Numbers and Decimals . . . . . . . . . . . . . . . . . . . . . 107 Chapter 20: Add and Subtract Decimals 20.1 Round Decimals . . . . . . . . . . . . . . . 108 20.2 Estimate Sums and Differences . . . . . . . . . . . . . . . . . . . 109 20.3 Add Decimals . . . . . . . . . . . . . . . . . 110 20.4 Subtract Decimals . . . . . . . . . . . . . . 111 20.5 Add and Subtract Decimals . . . . . 112 20.6 Problem Solving Skill: Evaluate Reasonableness of Answers . . . . . 113 Unit 7: MEASUREMENT, ALGEBRA, AND GRAPHING Chapter 21: Customary Measurement 21.1 Choose the Appropriate Unit . . . 114 21.2 Measure Fractional Parts . . . . . . . . 115 21.3 Algebra: Change Linear Units . . . . 116 21.4 Capacity . . . . . . . . . . . . . . . . . . . . . . 117 21.5 Weight . . . . . . . . . . . . . . . . . . . . . . . 118 21.6 Problem Solving Strategy: Compare Strategies . . . . . . . . . . . . 119 Chapter 22: Metric Measurement 22.1 Linear Measure . . . . . . . . . . . . . . . . 120 22.2 Algebra: Change Linear Units . . . . 121 22.3 Capacity . . . . . . . . . . . . . . . . . . . . . . 122 22.4 Mass . . . . . . . . . . . . . . . . . . . . . . . . . 123 22.5 Problem Solving Strategy: Draw a Diagram . . . . . . . . . . . . . . . . . . . . . 124 Chapter 23: Algebra: Explore Negative Numbers 23.1 Temperature: Fahrenheit . . . . . . . . 125 23.2 Temperature: Celsius . . . . . . . . . . . 126 23.3 Negative Numbers . . . . . . . . . . . . . 127 23.4 Problem Solving Skill: Make Generalizations . . . . . . . . . . . . . . . 128 Chapter 27: Solid Figures and Volume 27.1 Faces, Edges, and Vertices . . . . . . 145 27.2 Patterns for Solid Figures . . . . . . . 146 27.3 Estimate and Find Volume of Prisms . . . . . . . . . . . . . . . . . . . . . 147 27.4 Problem Solving Skill: Too Much/ Too Little Information . . . . . . . . . 148 Chapter 24: Explore the Coordinate Grid 24.1 Use a Coordinate Grid . . . . . . . . . 129 24.2 Length on the Coordinate Grid . . . . . . . . . . . . . . . . . . . . . . . . . . 130 24.3 Use an Equation . . . . . . . . . . . . . . . . 131 24.4 Graph an Equation . . . . . . . . . . . . . 132 24.5 Problem Solving Skill: Identify Relationships . . . . . . . . . . . . . . . . . . 133 Chapter 28: Measure and Classify Plane Figures 28.1 Turns and Degrees . . . . . . . . . . . . . 149 28.2 Measure Angles . . . . . . . . . . . . . . . 150 28.3 Circles . . . . . . . . . . . . . . . . . . . . . . . . 151 28.4 Circumference . . . . . . . . . . . . . . . . . 152 28.5 Classify Triangles . . . . . . . . . . . . . . 153 28.6 Classify Quadrilaterals . . . . . . . . . 154 28.7 Problem Solving Strategy: Draw a Diagram . . . . . . . . . . . . . . . . 155 Unit 8: GEOMETRY Chapter 25: Plane Figures 25.1 Lines, Rays, and Angles . . . . . . . . . 134 25.2 Line Relationships . . . . . . . . . . . . . 135 25.3 Congruent Figures and Motion . . . . . . . . . . . . . . . . . . . . . . . 136 25.4 Symmetric Figures . . . . . . . . . . . . . 137 25.5 Problem Solving Strategy: Make a Model . . . . . . . . . . . . . . . . . 138 Chapter 26: Perimeter and Area of Plane Figures 26.1 Perimeter of Polygons . . . . . . . . . . 139 26.2 Estimate and Find Perimeter . . . . . . . . . . . . . . . . . . . . . 140 26.3 Estimate and Find Area . . . . . . . . . 141 26.4 Relate Area and Perimeter . . . . . . 142 26.5 Relate Formulas and Rules . . . . . . 143 26.6 Problem Solving Strategy: Find a Pattern . . . . . . . . . . . . . . . . . 144 Unit 9: PROBABILITY Chapter 29: Outcomes 29.1 Record Outcomes . . . . . . . . . . . . . 156 29.2 Tree Diagrams . . . . . . . . . . . . . . . . . 157 29.3 Problem Solving Strategy: Make an Organized List . . . . . . . . 158 29.4 Predict Outcomes of Experiments . . . . . . . . . . . . . . . . . . . 159 Chapter 30: Probability 30.1 Probability as a Fraction . . . . . . . . 160 30.2 More About Probability . . . . . . . . 161 30.3 Test for Fairness . . . . . . . . . . . . . . . 162 30.4 Problem Solving Skill: Draw Conclusions . . . . . . . . . . . . . . . . . . . 163 LESSON 1.1 Name Benchmark Numbers You can use a known number, called a benchmark, to help you estimate another number that is difficult to count or measure. • Is the tree about 20 or about 200 feet tall? You know that the man is 6 feet tall. So, 6 is the benchmark. Look at the tree. It looks like it is a little more than 3 times as tall as the man. Think: 3 × 6 = 18 6 ft So, the tree is about 20 feet tall. Use the benchmark to decide which is the more reasonable estimate for the unknown amount. Circle that number. 1. number of beans in jar 2. pounds of potatoes in the box 50 beans 30 lb 100 3. or 200 gallons of water in the pool 100 4. or 1,000 paper clips in the container © Harcourt 200 clips 400 gallons 800 or 1,200 100 or 1,000 Reteach RW1 LESSON 1.2 Name Understand Place Value The value of a digit in a number depends on its position within the number. Thousands Hundreds 3, The value of the digit 7 in 3,750 is 700 because it is in the hundreds place. Tens Ones 5 0 7 3 × 1,000 7 × 100 5 × 10 0 × 0 To find the value of the digit 5, first decide the place-value position of the 5. The 5 is in the place. 3,000 700 50 0 5 10 50, so the 5 has a value of 50. To increase a number by 5,000, increase the digit in the thousands place by 5. 3,750 increased by 5,000 is 8,750. Notice that the only change was the increase by 5 in the thousands place. Look at the number 48,792. 1. In what place-value position is the 4? 2. Complete to find the value of the 8. 8 3. What is the value of the 9? 4. What is the value of the 2? For 5–10, use the numbers at the top of each column. 88,203 to 81,203 In what place value was a digit changed? 5. 6. Was the digit increased or decreased? 7. 8. What is the change in the value of the number? 9. 10. RW2 Reteach © Harcourt 45,893 to 85,893 LESSON 1.3 Name Place Value Through Hundred Thousands In the place-value system, each place is ten times as great as the place to its right. Complete the table. THOUSANDS Period Place Hundreds Tens Ones Hundreds 1,000 Value Think: ONES 100 thousands 10 thousands 1 thousand 100 ones Tens Ones 10 1 10 ones 1 one To write a number in word form, write the number in each period and follow it with the name of the period. Do not name the ones period. Remember to use commas to separate each period of numbers. 367,412 is three hundred sixty-seven thousand, four hundred twelve. Write each number in word form. 1. 56,398 2. 308,481 To write a number in expanded form, write the sum of the value of each digit. Do not include digits for which the value is 0. © Harcourt 367,012 300,000 60,000 7,000 10 2 Write each number in expanded form. 3. 458,913 4. 290,384 Reteach RW3 LESSON 1.4 Name Place Value Through Millions The period after the thousands period is the millions period. Write the values that complete the table. MILLIONS Period Place Hundreds Tens Ones Value Think: 100 million THOUSANDS 10 million Hundreds Tens ONES Ones Hundreds 1,000,000 10,000 100 1 million 100 10 1 thousand thousand thousand 100 ones Tens Ones 10 ones 1 one To write a number given in word form with the millions period, use a place-value chart. Fill in the place-value chart to find the standard form of three hundred fifty-seven million, two hundred twelve thousand, one hundred eight. MILLIONS Period THOUSANDS Place Hundreds Tens Ones Value _______ 5 7, Hundreds Tens _______ _______ ONES Ones Hundreds Tens Ones 2, _______ 0 _______ Write the value of the bold digit. 1. 32,404,202 2. 489,304,203 3. 44,203,203 4. 5,306,200 Complete. 349,305,203 three hundred forty-nine two hundred three 6. ,three hundred five 62,617,204 -two million, six hundred hundred four RW4 Reteach thousand, , © Harcourt 5. LESSON 1.5 Name Problem Solving Skill Use a Graph A pictograph is a graph that uses pictures to represent numbers in order to compare information. The pictures are related to the data. Box Office Video records their national rentals each week using a pictograph. Each picture of a VCR represents 1,000,000 rentals. BOX OFFICE VIDEO NATIONAL RENTALS Week 6 Week 5 Week 4 Week 3 Week 2 Week 1 Key: Each = 1,000,000 rentals. The graph shows four VCR symbols during Week 1. So, 4 1,000,000 4,000,000 video rentals were made during Week 1. Half of a symbol represents 500,000 video rentals. © Harcourt So, during Week 2 there were 4,000,000 500,000 4,500,000 rentals made. 1. What other symbol might Box Office Video have chosen to represent 1,000,000 videos? 2. During which week were the greatest number of rentals made? the fewest? 3. How many rentals were made during Week 3? 4. How many rentals were made during Week 1? 5. During Week Five, 7,000,000 videos were rented. Add this information to the graph. 6. During Week Six, 6,500,000 videos were rented. Add this information to the graph. Reteach RW5 LESSON 2.1 Name Compare Numbers Which number is greater, 3,491 or 3,487? Step 1 Write one number under the other. Step 2 Compare the digits, beginning with the greatest place-value position. Step 3 Circle the first digits that are different. ts ts gi igi i d d e e m m sa sa ↓ ↓ 3,487 3,491 Think: Since 8 tens < 9 tens, 3,487 < 3,491. Circle the greater number in each pair of numbers. Then write < or >. 627 637 2. 627 4. 637 7. 5. 312,629 54,329 28,617 54,329 10. 28,617 327,203 327,320 327,320 6,410 6,503 6,410 RW6 6,503 Reteach 3,200 998 4,423 1,062 1,028 40,894 763,492 739,814 763,492 15. 44,988 40,890 40,894 40,890 12. 24,318 45,225 44,988 45,225 9. 1,062 1,028 25,614 24,318 25,614 6. 4,408 4,423 4,408 14. 3,285 1,456 998 1,456 11. 3. 2,400 3,200 2,400 8. 327,203 13. 3,296 314,619 312,629 314,619 3,296 3,285 739,814 5,301 5,401 5,301 5,401 © Harcourt 1. LESSON 2.2 Name Order Numbers Abraham is buying a new car. He has prices from 3 dealers. Dealer Price Autoworld $25,614 Car Land $28,327 Cars & More $28,619 List the prices in order from greatest to least. Step 1 Compare the digits in each place-value position, beginning with the ten thousands. Step 2 Compare the digits in the next place-value position. $28,327 ↓ $25,614 ↓ $25,614 $28,327 $28,327 $28,619 $28,619 Think: 5 ten thousands 8 ten thousands, so 25,614 is the least number. ↓ ↓ Step 3 Compare $28,327 and $28,619. ↓ $28,619 Think: 6 hundreds 3 hundreds, so $28,619 $28,327. So, the prices in order from greatest to least are $28,619; $28,327; $25,614. © Harcourt Circle the greatest number and underline the least number. 1. 738 689 691 2. 3,670 3,760 3,761 3. 14,285 14,119 14,148 4. 252,861 251,312 253,112 5. 6,228,119 6,282,113 6,228,214 6. 6,260 62,260 26,260 7. 7,932 8,239 7,392 8. 52,441 52,414 52,144 9. 10,804 107,408 107,084 Reteach RW7 LESSON 2.3 Name Problem Solving Strategy Make a Table A table can help organize information to make it easier to answer questions. Students are doing the long jump. Emil jumps 157 centimeters, Rob jumps 135 centimeters, Kate jumps 135 centimeters, Hayley jumps 148 centimeters, and Kyle jumps 105 centimeters. Find out how their jumps compare. Make a table to show how far each student jumped. Name Length of Jump Use the table to answer 1–8. jump? 3. Which two students jumped the same distance? 5. Look at the length of Hayley’s jump. What digit is in the tens place? 7. Pete jumps 152 centimeters. In what place is Pete? RW8 Reteach 2. Which student had the shortest jump? 4. Who jumped farther, Rob or Hayley? 6. Whose jump is closest to 100 centimeters? © Harcourt 1. Which student had the longest 8. Whose jump is 3 less than 160 centimeters? LESSON 2.4 Name Round Numbers Round 314,289 to the nearest ten thousand. Follow these steps to round a number to a given place value: • Circle the digit to be rounded. 314,289 • Underline the digit to its right. 314,289 • If the underlined digit is 5 or more, the circled digit increases by 1. If the underlined digit is less than 5, the circled digit stays the same. 4 < 5, so 1 stays the same • Write zeros in all places to the right of the circled digit. 310,000 Use the steps above to help round each number to the place of the circled digit. 1. 243, 873 2. 2,375,467 3. 54,239 4. 5,394,372 5. 25,658,345 6. 43,872 7. 34,456,379 8. 359,752 Round each number to the nearest ten thousand. © Harcourt 9. 2,418,652 10. 65,528,514 11. 651,842 12. 896,328 15. 65,978,154 16. 658,548 Round each number to the nearest million. 13. 25,548,475 14. 6,514,879 Reteach RW9 LESSON 3.1 Name Estimate Sums and Differences When you estimate, you find an answer that is close to the exact answer. You can use rounding to help estimate a sum or difference. Round 4,782 to the nearest thousand. Follow these rules: 4,782 • Circle the digit to be rounded. 4,782 • Underline the digit to its right. 7>5 • If the underlined digit is 5 or more, the circled digit increases by 1. If the underlined digit is less than 5, the circled digit stays the same. 5,000 • Write zeros in all places to the right of the circled digit. So, 4,782 rounded to the nearest thousand is 5,000. Circle the digit to be rounded. Underline the digit to its right. Round 649,385 to the given place value. 1. nearest ten thousand 2. nearest hundred 649,385 3. 649,385 nearest hundred thousand 649,385 Round to the greatest place value. 2,617 5. 304,921 6. 89,314 Round each number to the greatest place value. Write the estimated sum or difference. 7. 397,216 → − 149,348 → → RW10 Reteach 8. 652,381 → + 348,927 → + → © Harcourt 4. LESSON 3.2 Name Use Mental Math Strategies Mental math strategies help you compute without using pencil and paper. Use the break apart strategy to find the sums and differences. 1. 3. 48 27 2. 19 58 Add the tens. Add the tens. Add the ones. Add the ones. Add the sums. Add the sums. 59 37 4. 98 22 Subtract the tens. Subtract the tens. Subtract the ones. Subtract the ones. Add the differences. Add the differences. Use the make a ten strategy to find the sums and differences. 5. © Harcourt 7. 89 24 6. 82 48 Change one number to a multiple of 10. Change the number being subtracted to a multiple of 10. Adjust the other number. Adjust the other number. Add. Subtract. 42 63 8. 92 19 Change one number to a multiple of 10. Change the number being subtracted to a multiple of 10. Adjust the other number. Adjust the other number. Add. Subtract. Reteach RW11 LESSON 3.3 Name Add and Subtract 4-Digit Numbers You may need to regroup when adding or subtracting. A grid can help you to regroup neatly. Record your regroupings in the top 2 rows. + 1 4, 2, 7, 5 6 1 1 6 1 8 7 8 5 3 $4, $2, $1, 12 2 3 6 6 10 0 1 1 9 12 2 8 4 Copy each problem into the grid. Find the sum or difference. Remember to leave the first 2 rows of boxes empty for regrouping. 1. $2,456 $1,782 2. + 4. 3. $4,211 $2,732 5. 8,742 2,178 6. $7,419 $5,297 9. 5,320 6,942 © Harcourt 8. 6,163 5,254 + + 3,877 4,399 2,631 1,975 + + + 7. 2,312 1,452 + + RW12 Reteach + LESSON 3.4 Name Subtract Across Zeros When you add or subtract numbers with many digits, it’s important that you write the numbers carefully so that digits with the same place value line up. This also helps you write regrouped digits in the correct places. 3 4 3 9 10 0, 2, 7, 9 10 0 9 0 9 10 0 6 3 10 0 3 7 Copy each problem into the grid. Find the difference. Remember to leave the first 2 rows of boxes empty for regrouping. 1. 900 831 2. 5,002 2,764 3. 7,001 3,208 , 4. , 6,000 1,955 5. 80,000 31,523 6. 10,000 4,661 , © Harcourt 7. 40,000 20,118 8. 70,000 55,049 9. 60,000 33,604 Reteach RW13 LESSON 3.5 Name Add and Subtract Greater Numbers When you’re adding or subtracting greater numbers, a grid can help you make sure you’re lining up the digits correctly. Record your regroupings in the top 2 rows. 6 3 3 5 1 4 3 4, 2, 1, 12 2 8 4 0 1 0 0 17 7 9 8 Copy each problem into the grid. Find the sum or difference. Remember to leave the first two rows of boxes empty for regrouping. 1. 352,134 277,679 2. 312,852 291,785 3. 994,117 504,334 , 4. 132,741 83,296 5. 514,902 423,978 6. 788,435 662,929 , , 225,068 102,266 8. 734,668 519,862 9. 412,396 325,052 © Harcourt 7. , RW14 Reteach LESSON 3.6 Name Problem Solving Skill Estimate or Find Exact Answers When solving a word problem, you must decide if you need to estimate or give the exact answer. Read the problem carefully. Words and phrases like about how much and estimate indicate that an estimate will do. Jennifer and Kyle are planning a trip to the art museum. Admission to the museum is $6.95, lunch costs $3.80, and a drink costs $0.85. About how much money should each bring for the trip? Is an exact answer required? No, the words about how much tell you that an estimate will do. © Harcourt Tell whether an estimate or an exact answer is needed. Solve. 1. About how much money should Jennifer bring on the trip to the art museum? 2. Kyle brought a $20.00 bill to the museum. How much change should he get after buying his admission? 3. There will be 120 students going on the field trip. Permission slips have been turned in by 87 students. How many permission slips still need to be turned in? 4. Greg has 504 baseball cards and 398 football cards. About how many cards does Greg have altogether? 5. On vacation, the Clarks drove 1,225 miles to visit relatives. Then they drove 488 miles to the beach and 1,348 miles home. How many miles did the Clarks drive on their vacation? 6. At the sports awards luncheon, there will be 52 soccer players, 38 baseball players, 44 basketball players, and 28 cheerleaders. About how many people will attend? Reteach RW15 LESSON 4.1 Name Expressions An expression is a part of a number sentence that has no equal sign. Expressions can include many different operations. Find the value 4 (10 9). 41 Always do the problem inside parentheses first. Subtract 9 from 10 to get 1. 5 Find the final answer. Add or subtract the numbers in the parentheses. Write that sum or difference. 1. 5 (12 9) 2. (12 13) 10 3. 20 (3 16) 5 10 4. (28 12) 7 20 7 Find the value of each expression. 5. 7 (2 4) 6. 7 8. 13 (3 6) (13 9) 5 9. 48 (2 11) 11 (55 17) 4 10. 4 12. (65 43) 9 48 14. (21 17) 11 13 5 11. 7. 40 (18 7) 40 13. (15 7) 2 8 9 56 (12 9) 15. 16. 12 68 (14 13) 68 © Harcourt 56 12 (3 4 1) 28 17. (16 132 5) 48 48 RW16 Reteach 18. 59 (15 12) 59 19. 1,210 (234 761) 1,210 LESSON 4.2 Name Use Parentheses An expression can have different values depending on where you place the parentheses. 14 (3 9) 14 12 (14 3) 9 ← Numbers are the same. Parentheses in different places. 11 9 ← Problem in parentheses solved. 2 20 ← Different answers. Show how placing the parentheses differently can give two different answers. 1. 16 9 2 16 9 2 2. 12 5 3 12 5 3 Choose the expression that shows the given value. Write a or b. 3. 6 4. 13 5. 45 a 22 (10 6) a (25 8) 4 a 56 (20 9) b (22 10) 6 b 25 (8 4) b (56 20) 9 Find the value that gives the expression a value of 16. 6. (8 7) 7. (15 )5 8. 30 (9 ) Show where the parentheses should be placed to make the expression equal to the given value. © Harcourt 9. 15 10 1; 6 10. 40 15 6; 31 11. 20 5 6; 9 14. 954=26 Place the parentheses so that the equations are true. 12. 541=33 13. 10 6 3 = 12 − 5 Reteach RW17 LESSON 4.3 Name Match Words and Expressions Marni had 12 pairs of shoes in her closet. She gave 3 pairs to her little sister and 4 pairs to her cousin. How many pairs of shoes does she have left? Write an expression using parentheses to match the meaning of the words. Use parentheses to group together a problem which should be solved first. 12 (3 4) ↑ number of pairs of shoes Marni had ↑ total number of pairs of shoes she gave away Add 3 4 to find the total number of pairs of shoes Marni gave away. Then subtract this number from 12. So, Marni has 5 pairs of shoes left. Write an expression that matches the words. 1. Mary earned $6 washing cars. She earned $4 more baby-sitting, and then spent $3 on a snack. ($ $ ↑ amount of money earned )$ ↑ amount of money spent 2. Our class began the year with 23 students. Two students moved away, and 3 new students enrolled. ) ↑ ↑ number of students number of after 2 moved away new students ( 3. There are 17 marbles in a jar. Ten are removed and 6 are added. 4. Kyra cares for 3 injured birds. She adopts 4 more and then gives away 5. 5. Dan’s father bought 6 pears and 9 apples at the grocery store. The family ate 4 pieces of fruit. 6. There are 28 people on the bus. At the Main Street stop, 4 more get on and 6 get off. 7. Aidan had 17 toy race cars. He gave 3 to his friend and bought 5 more. 8. Ricky brought a package of 12 snack bars on a bike trip. He gave 3 to his friends and ate 2. RW18 Reteach © Harcourt Write an expression for the words. LESSON 4.4 Name Use Variables Use variables to hold the place of unknown numbers. Unless the directions specify a certain letter to use as a variable, choose any letter you want. Write an expression for the number of pens in the desk. There were some pens in the desk.→ Use p to stand for the unknown number of pens in the desk. 9 pens were added. → Add 9 to p. p9 Write an equation to represent the situation. Robin researched some animals for her Science report. → Use a to stand for the unknown number of animals Robin researched. She researched three more. → Add 3 to a. Robin researched a total of 13 animals. → Make a 3 equal to 13. a 3 13 Write an expression. Choose a variable for the unknown. 1. There were some ducks in the 2. Rosie counted some stars in the pond. Three of the ducks went sky. An hour later, there were onto the shore. 20 more. © Harcourt 3. Some volleyball teams were in the gym. Twelve teams went outside to practice. 4. There were some snow shovels in the shed. Rob returned 4 shovels to the shed. Write an equation. Choose a variable for the unknown. 5. Paula baked 36 chocolate chip 6. Cindy began the day with 24 cookies. She baked some more. pencils. She gave some away to Now she has 61 cookies. her friends. Now she has 6 pencils. Reteach RW19 LESSON 4.5 Name Find a Rule The table shows inputs and outputs. Find a rule which will give you the output value when you use the input value. Input Output n b 2 8 4 10 3 9 10 16 Think: What can you add to 2 to get 8? Possible rule: Add 6. Test your rule on each row of the input/output table. If your rule holds true, you can write an equation for the rule. Because you add 6 to every input to get the correct output, the equation is n 6 b. Complete the output for the given rule. 1. Rule: Add 5. Input 2. Output Rule: Subtract 7. Input 3. Output Rule: Add 9. Input 7 12 13 9 20 10 3 9 2 8 11 6 Output 4. RW20 Input Output a 5. b Input x Output y 3 15 20 5 17 12 8 Reteach 6. Input r Output p 12 14 4 10 2 56 46 24 25 17 17 7 20 9 1 22 12 © Harcourt Find the rule. Then write the rule as an equation. LESSON 4.6 Name Add Equals to Equals An equation is like a balanced scale. Both sides of an equation have the same value, just like both sides of a balanced scale have the same weight. If you add two blocks to each side of the scale, it will still be balanced. If you add five blocks to one side, you must add five blocks to the other side for it to balance. What if Walt placed six more blocks on the left side of the scale below? How many blocks must he place on the other side to keep the scale balanced? 2. What can you do to balance the scale below? 3. Tricia is trying to balance the scale below, but she has no more blocks. How can she balance the scale? 4. How can you balance this scale? © Harcourt 1. Reteach RW21 LESSON 4.7 Name Problem Solving Strategy Make a Model Sometimes it helps to make a model of a problem. Simple objects like counters can represent part of the problem. Example Greta and Nigel are playing a game in which the players earn and lose points. After Round 3, Greta has 10 points and Nigel has 7 points. In Round 4, Greta earns 4 points and Nigel loses 2 points. In Round 5, Greta loses 3 points and Nigel gains 5 points. Which player has more points after Round 5? Use counters to model the problem. Each counter represents 1 point. The counters represent the number of points for each player after 3 rounds. Greta Nigel To model Round 4, give Greta 4 more points and take away 2 of Nigel’s points. Greta Nigel To model Round 5, take away 3 of Greta’s points and give Nigel 5 points. Greta Nigel So, Greta has 11 points and Nigel has 10 points. Greta has more points. 1. Chris bought 2 dogs. Sparky weighed 12 pounds and Fido weighed 15 pounds. During the first month, Sparky gained 2 pounds and Fido gained 1 pound. During the second month, Sparky lost 2 pounds and Fido gained 3 pounds. How much does each dog weigh after 2 months? RW22 Reteach 2. Roger was collecting shells on the beach to glue onto a picture frame. He found 11 shells. His sister gave him 7 more and he threw 3 back into the ocean. He gave 2 shells to his mother. How many shells does Roger have now? © Harcourt Make a model and solve. LESSON 5.1 Name Collect and Organize Data Mr. Cohen used tally marks to keep a record of the number of gerbils that he sold each week in his pet shop. Week Gerbils Sold 1 2 3 4 A frequency table helps you organize the data from a tally table. The frequency column shows the total for each week. The cumulative frequency column shows a running total of the number sold. Use the tally table to complete the frequency table below. NUMBER OF GERBILS SOLD Week Frequency (Gerbils Sold) Cumulative Frequency 1 Cumulative Frequency: ← sold in Week 1 2 ← sold in Weeks 1 and 2 3 ← sold in Weeks 1, 2, and 3 4 ← sold in Weeks 1, 2, 3, and 4 © Harcourt For 1–3, use the frequency table. 1. During which week did Mr. Cohen sell the most gerbils? 2. During which two weeks did Mr. Cohen sell the same number of gerbils? 3. How many gerbils did Mr. Cohen sell in all? Reteach RW23 LESSON 5.2 Name Find Median and Mode When a list of data is written in order from least to greatest, the number in the middle is called the median. The number that occurs most often is called the mode. John’s scores on his math tests Monday through Friday were: 72, 88, 95, 65, and 88. Step 1: List the scores in order from least to greatest. 65, 72, 88, 88, 95 Step 2: Cross out one from each end until only one number is left. 65, 72, 88, 88, 95 Step 3: The number in the middle is the median. 65, 72, 88, 88, 95 Step 4: The number that occurs most often is the mode. 65, 72, 88, 88, 95 In this case, 88 is both the mode and the median. 1. In the art classes there are 4 nine-year-olds, 5 ten-year-olds, 6 elevenyear-olds, and 8 twelve-year-olds. 2. In Hawaii the temperatures for a given week were 80, 84, 79, 90, 86, 84, and 89. 3. The car traveled 250 miles the first day, 600 the second day, 455 the third day, 320 the fourth day, and 800 the fifth day. 4. The baseball team sold 80 tickets on Monday, 66 on Tuesday, 55 on Wednesday, 68 on Thursday, and 80 on Friday. RW24 Reteach © Harcourt List the data from least to greatest. Find the median and the mode. LESSON 5.3 Name Line Plot A line plot is a graph that shows the frequency of data along a number line. You can use a line plot to find the range. The range is the difference between the greatest and the least values in a set of data. ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ 0 1 ✗ ✗ ✗ ✗ ✗ ✗ ✗ 2 3 4 Homerun Total 5 6 How many players hit 3 home runs? Step 1 Locate 3 on the diagram. Step 2 Count the number of Xs above the number 3. Three players hit 3 home runs. For 1–4, use the line plot. © Harcourt 1. players hit 1 home run. 2. More players hit number of home runs. home runs than any other 3. The same number of players hit home runs. 4. An outlier is a value that is seperated from the rest of the data. The outlier in the line plot above is . 5. Four new players joined the team for the 2000-2001 season. Use the data in the table to make a line plot. and Homerun Total Number of Homeruns 0 1 2 3 4 5 Number of Players 0 2 4 7 2 1 6 Reteach RW25 LESSON 5.4 Name Stem-and-Leaf Plot Becky measured the height in inches of each sunflower in her garden. She made a list of the heights. 20 35 12 23 27 18 inches inches inches inches inches inches 14 12 32 24 27 33 inches inches inches inches inches inches 17 15 27 21 33 inches inches inches inches inches • To organize the data in a stem-and-leaf plot write the numbers in order from least to greatest. Group the data in rows according to the digit in the tens place. 12, 12, 14, 15, 17, 18 20, 21, 23, 24, 27, 27, 27 32, 33, 33, 35 • The stem is the tens digit. It is written only once. The leaves are the ones digits. Stem (Tens) • 1 2 3 1 2 means 12. 3 5 means 35. Leaves (Ones) 2 2 4 5 7 8 0 1 3 4 7 7 7 2 3 3 5 The shortest sunflower is 12 inches. The tallest sunflower is 35 inches. 1. How many sunflowers are 33 inches tall? 2. How many sunflowers are 20 inches or taller? 3. The mode is the number that occurs most often in a set of data. What is the mode for the sunflower heights? 4. The median is the middle number in an ordered set of numbers. What is the median sunflower height? 5. The range is the difference between the greatest and the least values in a data set. What is the range of the sunflower heights? RW26 Reteach © Harcourt For 1–5, use the stem-and-leaf plot. LESSON 5.5 Name Compare Graphs Tom and his friends earned money cutting lawns during the summer. Lawns Cut This Summer Number of Lawns Tom 6 Jack 10 Barb 8 Kim 4 Lia 3 The interval of a scale is the difference between any two numbers. The interval of this graph is 2. To make a bar graph with a different interval, you would follow these steps. Lawns Cut This Summer Number of Lawns Cut Name The scale is the series of numbers placed at fixed distances. The scale of this graph is 0 -12. 12 10 8 6 4 2 0 Tom Jack Barb Kim Name Lia Step 1 Choose an interval for the graph. Label the graph. Step 2 Draw the bars. Step 3 Check to see if your graph is correct. Use the table above to complete the bar graphs. 1. What is the scale and the interval of the graph on the bottom left? 2. What is the scale and the interval of the graph on the bottom right? Lawns Cut This Summer Tom Tom Jack Jack Name Name © Harcourt Lawns Cut This Summer Barb Barb Kim Kim Lia Lia 0 1 2 3 4 5 6 7 8 9 10 11 12 Number of Lawns Cut 0 2 4 6 8 10 12 Number of Lawns Cut Reteach RW27 LESSON 5.6 Name Problem Solving Strategy Make a Graph The table below shows how many birthdays are in each month for Mr. Robinson’s class. Month Number of Birthdays January 3 February 5 March 2 April 6 May 0 June 2 July 1 August 1 September 1 October 3 November 4 December 0 Follow these hints to complete the bar graph below: Step 1: On the side, write: Number of Birthdays. Step 2: What label should you write on the bottom? Step 3: What is the title of the graph? Step 4: What should the length of each bar show? 7 6 5 4 2 1 0 RW28 Reteach © Harcourt 3 LESSON 6.1 Name Double-Bar Graphs Peter made a table of the number of shells that he collected on two beaches. Then he started a double-bar graph to compare the data. Each bar represents the number of each kind of shell collected. SHELLS COLLECTED ON WEST AND DUNE BEACHES Type of Shell West Beach Dune Beach Clam 6 8 Oyster 9 1 Scallop 10 2 6 6 Periwinkle © Harcourt Number of Shells 12 Key 10 West Beach 8 6 4 2 0 Dune Beach Clam 1. What scale did Peter choose? 2. What is the interval of the scale? 3. Complete Peter’s graph. Remember to include bars, a title, and labels. For 4–5, use the graph. 4. On which beach were more oyster shells collected? 5. On which beach were equal numbers of clam shells and periwinkle shells collected? Reteach RW29 LESSON 6.2 Name Read Line Graphs Mr. Lowe made this line graph to show the number of books he sold each month for the first seven months of the year. This line graph shows how book sales change over time. BOOKS SOLD JANUARY THROUGH JULY Number of Books 500 300 200 100 0 • • 400 • • • • • Jan Feb Mar Apr May Jun Month Jul How many books were sold in April? Step 1 Find the line labeled Apr. Put your finger on the line. Follow that line up to the point ( • ). Step 2 Move your finger along the line to the left to find the number of books sold in April. Mr. Lowe sold 400 books during April. For 1–5, use the graph. Mr. Lowe sold the greatest number of books during and . 2. During June, Mr. Lowe sold about 3. The difference in the number of books sold in February and April was about books. © Harcourt 1. . 4. Mr. Lowe sold about more books in June than in January. 5. There was an increase in the number of books sold from . RW30 Reteach LESSON 6.3 Name Make Line Graphs JACOB’S SOCCER GOALS A line graph is used to display data which change over time. Make a line graph for the data at the right. 1. The greatest number of goals is 20 and the least is , so the scale should go from 0 to a little more than 20. Year Number of Goals 1998 6 1999 8 2000 9 2001 17 2002 20 Since the data are close together, an interval of would be appropriate. Fill in the missing parts of the line graph: Title: 21 15 12 3 0 © Harcourt 1998 2. To plot 6 goals for Jacob in 1998, move your finger or pencil straight up from 1998. Stop when you reach the line for 6. Make a dot for 6 goals for Jacob in 1998. 3. Plot the rest of the data in the table on the graph the same way. 4. Do you notice any trend in the number of soccer goals Jacob scored over the years? Reteach RW31 LESSON 6.4 Name Choose an Appropriate Graph A line graph shows change over time. Books Read Each Week Number of Books Kinds of Books Read • • • 1 2 3 Week y Bi ster og y ra p Sc hy ie nc An e im al • 60 50 40 30 20 10 0 M Number of Books 60 50 40 30 20 10 0 A bar graph compares data. 4 Kind of Book Use a line graph to show whether data increase or decrease. Use a bar graph to see how many of each type of book are in the data. A line plot shows how many times each number occurs in the data. A stem-and-leaf plot shows data arranged by place value. Number of Pages in 15 Books ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ 2 3 4 5 6 7 Stem Leaves 2 3 4 5 5 8 9 3 1 2 8 9 4 0 1 3 4 9 Number of Books Read Use a line plot to record how many students read books. Use a stem-and-leaf plot to organize a long list of numbers by grouping them. Write the kind of graph or plot you would choose. to show a decrease in absences during the school year 2. to organize the list of weights of the football team members 3. to compare the numbers of students signing up for hockey, football, and cheerleading 4. to record the number of students who have 0, 1, 2, 3, or 4 siblings © Harcourt 1. RW32 Reteach LESSON 6.5 Name Problem Solving Skill: Draw Conclusions Bobby saved his money for a new telescope. He put his money into a savings account at Colesville Federal Bank. BOBBY'S SAVING ACCOUNT 220 • 200 180 Amount of Money (in dollars) 160 • 140 120 • 100 • 80 40 • • 60 • • • • 20 0 Jan • • Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month The bank gives its customers a free poster for each month in which their balance is greater than $150.00. Since Bobby’s balance was greater than $150.00 in July and August, he received posters in those months. During which month was Bobby’s bank account balance the greatest? 2. During which month was Bobby’s bank account balance the least? 3. When did Bobby most likely buy his telescope? 4. When did Bobby most likely make a $30 withdrawal to buy a book about the planets? 5. During which months did Bobby most likely earn money at a part-time job? 6. About how much money was in Bobby’s account in October? © Harcourt 1. Reteach RW33 LESSON 7.1 Name Before and After the Hour Digital Clock Analog Clock The second hand shows the seconds and moves around the clock in 1 minute. The minute hand is the long hand. 11 12 1 2 3 4 10 9 8 7 6 5 2:35 The hour hand is the short hand. Read: 5:15 and 10 seconds, or 15 minutes and 10 seconds after five Read: 35 minutes after two Complete to write the time shown on the clock. Include the seconds. 1. 2. 11 12 1 11 12 1 2 3 4 10 9 8 7 6 5 7 6 5 and seconds 3. and seconds 4. 11 12 1 11 12 1 2 3 4 10 9 8 2 3 4 10 9 8 7 6 5 minutes and 2 3 4 10 9 8 7 6 5 seconds minutes and after seconds after Draw the hour hand and the minute hand to show the time. 6. 11 12 1 2 3 4 10 9 8 7. 11 12 1 11 12 1 2 3 4 10 9 8 7 6 5 7 6 5 :35 3:25 2 :35 8:05 2 2 3 4 10 9 8 7 6 5 :35 9:57 2 © Harcourt 5. Write second, minutes, hour, or day. 8. It took Joe 45 9. A snap of your fingers takes about 1 10. Anne’s piano lesson was 1 RW34 Reteach to wash and vacuum his car. . long. LESSON 7.2 Name A.M. and P.M. A.M. means “the time between P.M. means “the time between noon midnight and noon.” and midnight.” At 8:15 A.M. Kathy goes to school. At 8:15 P.M. Kathy goes to bed. Write A.M. or P.M. 1. George eats supper at 5:30 2. Carlos goes to school from 8:40 3. Maria has a violin lesson from 3:45 . to 2:40 . to 4:30 . Write the time, using A.M. or P.M. 4. the time the library opens 5. the time Kitty sees the sunset 11 12 1 11 12 1 2 3 4 10 9 8 7 6 5 6. 7 6 5 the time Lu takes a spelling test at school 7. the time Frank gets home from school 11 12 1 11 12 1 2 3 4 10 9 8 © Harcourt 7 6 5 the time Jack has soccer practice 9. the time Dad cooks breakfast 11 12 1 2 3 4 10 9 8 7 6 5 2 3 4 10 9 8 7 6 5 8. 2 3 4 10 9 8 11 12 1 2 3 4 10 9 8 7 6 5 Reteach RW35 LESSON 7.3 Name Elapsed Time Skip counting can help you find the elapsed time. What is the elapsed time from 7:20 A.M. to 9:40 A.M.? 11 12 1 10 2 hr 9 8 1 hr 11 12 1 2 3 4 2 3 4 10 9 8 7 6 5 7 6 5 20 15 It is 2 hours from 7:20 A.M. to 9:20 A.M. 10 5 It is 20 minutes from 9:20 A.M. to 9:40 A.M. So, it is 2 hours 20 minutes from 7:20 A.M. to 9:40 A.M. Find the elapsed time. from 12:30 A.M. to 12:45 A.M. 2. from 4:25 P.M. to 5:20 P.M. 11 12 1 11 12 1 2 3 4 10 9 8 7 6 5 3. 7 6 5 from 10:15 A.M. to 11:45 A.M. 4. from 11:15 P.M. to 2:40 A.M. 11 12 1 11 12 1 2 3 4 10 9 8 7 6 5 from 7:05 A.M. to 9:20 P.M. 6. from 6:35 A.M. to 6:45 P.M. 11 12 1 2 3 4 10 9 8 7 6 5 RW36 Reteach 2 3 4 10 9 8 7 6 5 5. 2 3 4 10 9 8 11 12 1 2 3 4 10 9 8 7 6 5 © Harcourt 1. LESSON 7.4 Name Problem Solving Skill: Sequence Information Zoo Schedule of Shows Kevin is going to be at the zoo from 8:30 A.M. until 1:30 P.M. He will be meeting his friends for lunch from 12:00 P.M. to 1:00 P.M. He has decided to see the 9:00–10:30 Manatee Madness show. He put this into his schedule. Kevin’s Schedule 9:00–10:30 Manatee Madness 10:00–10:45 Monkey Business 11:00–12:00 Birds of the Wild 12:00–12:30 Insect Round-Up 1:00–2:30 Manatee Madness 2:00–2:45 Monkey Businesss 3:00–4:00 Birds of the Wild 4:00–4:30 Insect Round-Up Sara’s Schedule 8:30 to 9:00 8:30 to 9:00 9:00 to 9:30 Manatee Madness 9:00 to 9:30 9:30 to 10:00 Manatee Madness 9:30 to 10:00 10:00 to 10:30 Manatee Madness 10:00 to 10:30 10:30 to 11:00 10:30 to 11:00 11:00 to 11:30 11:00 to 11:30 11:30 to 12:00 11:30 to 12:00 12:00 to 12:30 12:00 to 12:30 12:30 to 1:00 12:30 to 1:00 1:00 to 1:30 1:00 to 1:30 Put Kevin’s lunch with his friends into the schedule. 2. Kevin wants to see another show. Which show can he see? Put the show into Kevin’s schedule. 3. Can Kevin see another show? Explain. 4. Sara is also going to the zoo from 8:30 A.M. to 1:30 P.M. She wants to see three shows. Make a schedule for Sara. © Harcourt 1. Reteach RW37 LESSON 7.5 Name Elapsed Time on a Calendar You can use a calendar to find elapsed time. How many days will the Harvest Festival last? • Look at the October calendar. Find 9. • Count each day of the festival. So, the festival will last 10 days. Wilson Farm began advertising the Harvest Festival 5 weeks before the first day of the festival. On what date did Wilson Farm begin advertising? • Begin at October 9. It is a Thursday. • Use the calendars to count back 5 weeks, or 5 Thursdays. So, Wilson Farm began advertising September 4. For Problems 1–4, use the calendars. 2. 3. 4. On how many Saturdays will the Harvest Festival be held? Tom bought a pumpkin on the first Friday of the festival. How many weeks will it be from then until October 31? How many days is it from September 28 to the last day of the Harvest Festival? Wilson Farm began harvesting pumpkins 4 weeks before the beginning of the festival. On what date did the farm begin harvesting pumpkins? RW38 Reteach September Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Wed Thu Fri Sat 1 2 3 4 October Sun Mon Tue 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 © Harcourt 1. LESSON 8.1 Name Relate Multiplication and Division Multiplication and division are inverse operations. One operation undoes the other. You can use models to show this. 2 × 5 = 10 10 ÷ 5 = ? Divide 10 into 5 equal groups. 12 ÷ 3 = ? Divide 12 into 3 equal groups. 3 × 4 = 12 How many are in each group? 4 12 ÷ 3 = 4 How many are in each group? 2 10 ÷ 5 = 2 Circle equal groups to show the division. Complete the division equation. 1. 3 × 6 = 18 18 ÷ 6 = 2. 7 × 2 = 14 3. 14 ÷ 2 = 4 × 5 = 20 20 ÷ 5 = Write the fact family by writing two multiplication equations and two division equations which are shown by these models. 4. 6. © Harcourt 5. 7. What multiplication equation can you use to help you find 28 ÷ 4? 8. What multiplication equation can you use to help you find 12 ÷ 4? Reteach RW39 LESSON 8.2 Name Multiply and Divide Facts Through 5 You can use models to help you remember fact families. How many stars are in the array? How many rows are there? How many columns are there? Since there are 3 rows of 6 stars, multiply 3 6 to get the number of stars. 3 6 18 Use the model to write the other three facts in the fact family for 3, 6, and 18. Find the product or quotient. You may wish to use a model. 64 2. 28 3. 28 4 4. 36 9 5. 24 8 6. 62 7. 75 8. 55 9. 15 3 10. 95 11. 54 12. 30 5 © Harcourt 1. RW40 Reteach LESSON 8.3 Name Multiply and Divide Facts Through 10 The break apart strategy can help you find products of greater numbers. What is 6 9? The unshaded squares represent the multiplication problem 4 6. What is 4 6? What multiplication problem is represented by the light gray squares? What multiplication problem is represented by the darker gray squares? 9 6 (4 6) ( ↓ 9 6 24 So, 9 6 6) ( ↓ 6) ↓ . You can check your answer by counting the squares. © Harcourt Use shading and the break apart strategy to solve. 1. 78 2. 88 3. 76 4. 77 Reteach RW41 LESSON 8.4 Name Multiplication Table Through 12 Look at the 10 Facts Chart. 0 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 10 10 Facts Chart 0 10 20 30 40 50 60 70 80 90 100 110 120 What is the rule for finding the product of 10 and another number? You can use the basic facts for 10 and a model or other facts to help you find the product of factors greater than 10. 12 11 ? ↓ 10 2 What is 12 11? Step 1 Write an addition expression using break apart numbers. Step 2 Use the 10 Facts Chart and a model to find the products. (10 11) ↓ Step 3 Find the sum of the products. 110 (2 11) ? ↓ 132 22 So, 11 12 132. Complete each step. 12 12 ? ↓ 10 2 2. (10 12) (2 12) ? So, 12 12 RW42 Reteach 11 11 ? ↓ 10 (10 11) ( . So, 11 11 11) ? . © Harcourt 1. LESSON 8.5 Name Multiply 3 Factors The Grouping Property may help you when you multiply. Remember that when the grouping of factors changes, the product remains the same. Look at the example below which shows the Grouping Property. (3 2) 2 gives the same answer as 3 (2 2). Choose the way of grouping that is easier for you to multiply. What is 5 2 3? • Group the factors two ways. (5 2) 3 ? 5 (2 3) ? • Choose the way that is easier for you. • Solve. Think: 5 2 10. It’s easy to multiply tens. 10 3 30 5 6 30 © Harcourt Circle the grouping that is easier for you. Show how to solve it. 1. (2 2) 3 ? , or 2 (2 3) ? 2. (2 5) 3 ? , or 2 (5 3) ? 3. (4 2) 3 ? , or 4 (2 3) ? 4. (5 3) 3 ? , or 5 (3 3) ? 5. (2 2) 8 ? , or 2 (2 8) ? 6. (4 3) 3 ? , or 4 (3 3) ? 7. (5 2) 2 ? , or 5 (2 2) ? 8. (2 6) 1 ? , or 2 (6 1) ? Reteach RW43 LESSON 8.6 Name Problem Solving Skill Choose the Operation Add Join groups of different sizes There are 24 students in Mr. Clark’s class and 26 students in Ms. Young’s class. How many students are there in both classes? 24 26 50 Subtract Take away or compare groups How many more students are in Ms. Young’s class than in Mr. Clark’s class? 26 24 2 Multiply Join equal-sized groups Find how many in each group or how many groups 2 more students Mr. Clark gave 3 pencils to each of his 24 students. How many pencils in all did Mr. Clark give his students? 24 3 72 Divide 50 students 72 pencils Mr. Clark divided his class of 24 students into 4 equal groups to work on a science project. How many students were in each group? 24 4 6 6 students 1. Maria used 3 rolls of film on her vacation. Each roll had 12 exposures. How many photographs did she take? 2. Maria bought a photograph album for $7.00 and glue for $4.00. What change should she get from $20.00? 3. Maria put 7 photographs on each page of her album. How many pages did she use for her 84 photographs of the Rocky Mountains? 4. During a 4-day hiking trip, Maria hiked 5 km, 7 km, 9 km, and 10 km. What was the total length of her hike? RW44 Reteach © Harcourt Name the operation or operations you use. Then solve. LESSON 9.1 Name Expressions with Parentheses An expression is a part of a number sentence that does not contain an equal sign. If an expression contains parentheses, do the problem in the parentheses first. Evaluate (5 12) 20 ↓ ↓ 60 1. Do the problem in the parentheses first. Think: 5 12 60. 20 2. Write that answer and bring down the rest of the problem. 40 3. Solve the new problem. Solve the problem inside the parentheses and write the result on the line. 1. 3 (12 9) 2. (10 14) 4 3 3. 4 15 (2 3) 4. (4 6) 8 15 8 Solve the problem inside the parentheses and write the result on the line. Then find the final answer. 5. (4 7) 8 6. 20 (15 3) 7. 20 8 (36 4) 8 8 Find the value of the expression. © Harcourt 8. 11. 20 (14 2) (4 5) (14 7) 9. 12. 54 (18 2) 10. 4 (7 5) (81 9) 5 13. 50 (5 5) Reteach RW45 LESSON 9.2 Name Match Words and Expressions The words in a problem can help you decide which operation to choose. Words like total, and, altogether, and plus might tell you to add. Words like gave away, spent, and less than might tell you to subtract. Robin bought 2 books and 3 magazines. How many items did she buy altogether? Rick had 10 cookies and gave away 4. How many does he have left? 235 10 4 6 Words like each and per might tell you to multiply. Words like share, split, and divide might tell you to divide. Mary bought 3 sandwiches for $4.00 each. How much did she spend? Ike divided 52 cards among 4 friends. How many did each friend get? 3 $4.00 $12.00 52 4 13 Sometimes you have to use more than one operation to write an expression. Use parentheses to show the operation that should be done first. 50 Monique picked 50 apples. ↓ ↓ She gave away 5 apples to each original gave away of her 3 friends. number tells you to of apples subtract (3 5) ↓ number of apples given away 1. Tim had 4 pages with 6 stamps on each page and then his mother used 5 stamps. 2. Vicki had 85¢. Then she lost two dimes. a. (4 6) 5 a. (85 2) 10 b. 4 (6 5) b. 85 (2 10) RW46 Reteach © Harcourt Choose the expression that matches the words. Underline the words that tell you the operations to use. LESSON 9.3 Name Multiply Equals by Equals You can think of an equation as being like a balance. Look at the balance. It models the equation 6 3 2. Since there are 6 blocks on each side of the balance, it is level. When you multiply both sides of an equation by the same number, the equation stays true. How many blocks will be on the left-hand side if we multiply the number of blocks on that side by 3? How many blocks will be on the right-hand side if we multiply the number of blocks on that side by 3? Since both sides of the balance have 18 blocks, the balance is still level. 632 6 3 (3 2) 3 18 18 1. Original equation Multiply both sides by 3. Find the value of each side. Multiply both sides by 5. Find the value of each side. 2. Original equation Multiply both sides by 4. © Harcourt Find the value of each side. 3. 583 Original equation Original equation 5 (8 3) 25 4 nickels 2 dimes 20¢ 7429 Multiply both sides by 5. Find the value of each side. Reteach RW47 LESSON 9.4 Name Expressions with Variables A variable is a letter or symbol that stands for a number. When the number is given, replace the variable with that number. Find the value of 7 p if p 9. Replace p in 7 p with 9. 7 9 63. Variables are helpful because we don’t always have all the information needed to write an expression. Marguerite has 12 stuffed animals to share with her friends at her slumber party. How many animals will each girl get? The number of stuffed animals each girl gets depends on the number of girls at the party. Since that information is not given, use a variable to stand for that number. 12 g is the number of stuffed animals each girl will get. If there are 4 girls at the party, then g 4 and 12 g becomes 12 4, which is 3. Each girl will get 3 stuffed animals. If there are 6 girls at the party, then g 6 and 12 g becomes 12 6, which is 2. Each girl will get 2 stuffed animals. Find the value of the expression. 1. 4 n if n 5 2. 12 w if w 4 45 3. 30 pancakes divided by the number of people, p, at breakfast 4. 12 eggs times the number of egg cartons, c 5. a number of tennis balls, t, divided among 8 players 6. twice the number of football helmets, h RW48 Reteach © Harcourt Write an expression that matches the words. LESSON 9.5 Name Equations with Variables An equation is a number sentence. Equations are different from expressions because equations contain equal signs. Turn word sentences into number sentences by looking for important words. The words is or are tell you where the equal sign goes. Write an equation for the words. 4 trays with 5 glasses on each tray are the total number of glasses. ↓ ↓ ↓ 45 t t represents the total number of glasses on the trays. Write an equation using a variable. Tell what the variable represents. 1. A number of desks divided into 5 rows is 4 desks in each row. ↓ ↓ ↓ d d is the total number of desks. 2. 4 chairs at each of some tables are 48 chairs. ↓ ↓ ↓ t is A number of blankets in each of 7 stacks is 49 blankets. © Harcourt 3. Reteach RW49 LESSON 9.6 Name Find a Rule Find a rule for the input/output table. Look at the first row of numbers in the table. Think: What can you do to 5 to find 20? You could add 15 or multiply by 4. Look at the next row of the table. Think: What can you do to 6 to find 24? You could add 18 or multiply by 4. input output a b 5 20 6 24 2 8 10 40 3 12 Think: The rule multiply by 4 works for the first two rows. Check the rule on the other rows. Does 2 4 8? Yes Does 10 4 40? Yes Does 3 4 12? Yes; so, the rule is multiply by 4. a 4 b The input number, multiplied by 4, equals the output number. Write the answer. 1. What operations on 5 give a value of 10? a. b. 2. input output t input output w s b 15 5 5 40 9 3 2 16 12 4 4 32 18 6 8 64 24 8 9 72 Equation: RW50 Reteach 3. Rule: Equation: Rule: © Harcourt Find a rule, and write the rule as an equation. LESSON 9.7 Name Problem Solving Strategy Work Backward Sometimes you can work backward to solve a problem. Operation Opposite Operation Bruce collected some sand dollars. He gave 10 to his younger sister. His friend, Susan, gave him 7 sand dollars. He put 4 back in the water. He had 14 sand dollars left. How many did he collect? Add Subtract Subtract Add Divide Multiply Multiply Divide • First, write the steps of the problem in order from left to right. Use numbers and operations to describe the steps. Number of sand dollars collected. ? • → He gave 10 to sister. 10 Susan gave him 7. → 7 He put 4 back. → 4 Number left → 14 ← 14 Then, reverse the order and use the opposite operations. Find the answer. 21 ← 10 ← 7 ← 4 So, Bruce collected 21 sand dollars. Reverse the order and use opposite operations. Find the answer. 1. ? → 3 ← 2. ? → © Harcourt ← 3. → 8 ← 11 → ← → 11 ← 25 → → 27 ← 52 ← Sasha and Chuck played a number game. Sasha chose a number and subtracted 3. Next, she added 12. Then, she subtracted 20. Last, she added 11. The result was 30. What number did Sasha begin with? Reteach RW51 LESSON 10.1 Name Mental Math: Multiplication Patterns 60 is the product of 6 and 10. 6 10 60. You can use basic multiplication facts and patterns to find the product. Solve 60 2 ? . Step 1 Think of the basic fact. Multiply. 6 2 12 Step 2 Count the number of zeros in the factors. 60 has 1 zero. Step 3 Place the number of zeros in the factors at the end of the basic-fact product. The answer is 120. Write the basic multiplication fact. 1. 70 4 280 2. 2,000 2 4,000 3. 600 8 4,800 5. 77 6. 54 Complete each pattern. 7. 10. 33 30 3 70 7 50 4 300 3 700 7 500 4 3,000 3 7,000 7 5,000 4 62 8. 48 9. 39 60 2 40 8 30 9 600 2 400 8 300 9 6,000 2 4,000 8 3,000 9 3 18 6 180 6 1,800 3,000 6 RW52 Reteach 11. 75 12. 5 350 700 5 7,000 9 36 4 360 900 4 35,000 4 36,000 © Harcourt 4. LESSON 10.2 Name Estimate Products When you don’t need an exact answer, you can estimate by rounding the factors. To round: • Circle the digit to be rounded. • Underline the digit to its right. • If the underlined digit is 5 or greater, increase the circled digit by 1. Otherwise, the circled digit stays the same. Round to the greatest place value. 551 → 2 → 600 ← 2 1,200 Find the estimated product. • Write zeros in all places to the right of the circled digit. Round the larger factor to its greatest place value. Then estimate the product. 1. 4. © Harcourt 7. 10. 623 → 600 2 → 2 1,200 328 → 2. 5. 435 → 3. 575 → 4 → 2 → 944 → 6. 894 → 7 → 8 → 6 → $37 → 8. 155 → 9. $984 → 5 → 5 → 4 → 448 → 11. 637 → 5 → 9 → 12. $275 → 7 → Reteach RW53 LESSON 10.3 Name Model Multiplication 243 3 729 Step 1 Hundreds Ones Use base-ten blocks to model the number 243. Hundreds Step 2 Tens Tens Ones Use base-ten blocks to model the problem 3 243. Hundreds Tens 3 groups of 4 tens, or 12 tens 3 groups of 2 hundreds, or 6 hundreds Step 3 Ones 3 groups of 3 ones, or 9 ones Regroup the tens. Hundreds Tens Ones © Harcourt So, 3 243 729. Use base-ten blocks to model the multiplication and find the product. 124 4 135 3 1. RW54 2. Reteach 123 5 3. 232 3 4. LESSON 10.4 Name Multiply 3-Digit Numbers Break a 3-digit number down by place value to multiply it by another number. Multiply. 5 361 300 60 1 5 5 5 1,500 300 5 Add the products. 1,500 300 5 1,805 So, 5 361 1,805. 1. Multiply. 7 819 800 10 9 7 7 7 5,600 70 63 Add the products. 5,600 So, 7 819 5,733. Use the above method to find the products. 453 6 4. 357 6 6. 218 7 8. 627 3 © Harcourt 2. 3. 216 4 5. 846 2 7. 851 4 9. 524 5 Reteach RW55 LESSON 10.5 Name Multiply 4-Digit Numbers You can use place value and the expanded form of a number to help you find the product of a one-digit number and a four-digit number. EXAMPLE: Find the product 4 2,281. Step 1 Write the expanded form of the four-digit number. 2,281 2,000 200 80 1 Step 2 Multiply each addend by the one-digit number. Step 3 Add these numbers. EXPAND 2,000 200 80 1 MULTIPLY by 4 8,000 800 320 4 9,124 SUM So, 4 2,281 9,124. Use place value and the expanded form of the four-digit number to find the product. 3 3,418 EXPAND 3,000 400 10 8 2. MULTIPLY by 3 5 $3,067 EXPAND MULTIPLY by 5 SUM SUM 3. 9 2,847 EXPAND 4. MULTIPLY by 9 EXPAND SUM RW56 Reteach 7 7,209 © Harcourt 1. MULTIPLY by 7 SUM LESSON 10.6 Name Problem Solving Strategy Write an Equation A variable represents an unknown number. EXAMPLE: Mary has seven lambs. Each lamb produces 15 pounds of wool. How much wool do the lambs produce in all? Use the equation 7 15 n to answer the problem. Use the letter n to represent the unknown number, the total number of pounds of wool produced by the seven lambs. Since 7 15 105, the total pounds produced is n 105. © Harcourt Write an equation and solve. 1. In a tall building there are nine flights of stairs. If each flight of stairs has 21 steps, how many steps are there in the building? 2. June has 90 minutes before her piano lesson. If she does homework for 48 minutes, how much time will she have before her piano lesson? 3. How many days are in 52 weeks and one day? 4. In one hour, five squirrels each store 12 acorns. How many acorns do they store in all? 5. Peter divides 48 index cards into four stacks. How many cards are in each stack? Reteach RW57 LESSON 11.1 Name Mental Math: Patterns with Multiples A multiple is the product of a given whole number and another whole number. For example, 60 is a multiple of 6 and 10: 6 10 60. You can use basic multiplication facts and mental math to find the product when you multiply by a multiple of 10, 100, or 1,000. Solve 60 20 ? . Step 1 Think of the basic fact. Multiply. 6 2 12 Step 2 Count the number of zeros in the factors. 60 has 1 zero and 20 has 1 zero. Step 3 Write that number of zeros to the right of the basic-fact product. The answer is 1,200. Write the basic multiplication fact. 1. 70 4 280 2. 2,000 2 4,000 3. 600 8 4,800 Tell how many zeros will be in the product. 4. 300 60 5. 5,000 300 6. 40 800 8. 800 400 9. 500 800 Find the product. 30 300 9, 32 , 40 10. 900 3,000 11. 12,000 12. 9,000 30,000 13. 14. 400 3,000 15. 1,200 RW58 Reteach , 900 60 500 1,100 © Harcourt 7. LESSON 11.2 Name Multiply by Multiples of 10 The multiples of 10 are 10, 20, 30, 40, and so on. When one of the factors in a multiplication problem is a multiple of 10, you can find the product by solving a simpler problem and writing a zero. Example: 15 30 Solve the problem 3 15, then write one zero. 3 15 45 so, 30 15 450. Write a simpler problem to be solved. 1. 22 40 2. 52 60 Find the product. 3. 12 40 4. ? 5. ? 82 40 ? Simpler Problem: Simpler Problem: Simpler Problem: 4 12 6 29 4 82 Write 1 zero so, Write 1 zero so, Write 1 zero so, 40 12 56 30 60 29 . 7. 75 40 40 82 . 8. . 26 80 © Harcourt 6. 29 60 Reteach RW59 LESSON 11.3 Name Estimate Products You can round numbers and use basic facts to estimate products. For 2-digit numbers: For 3-digit numbers: If the ones digit is 0–4, the digit in the tens place stays the same. If the tens digit is 0–4, the digit in the hundreds place stays the same. If the ones digit is 5–9, the digit in the tens place increases by 1. If the tens digit is 5–9, the digit in the hundreds place increases by 1. For example: Round 41–44 to 40. Round 45–49 to 50. For example: Round 700–749 to 700. Round 750–799 to 800. 50 45 → 42 → 40 2,000 749 → 700 44 → 40 28,000 1. 4. 41 → 36 → 160 → 41 → 2. 157 → 57 → 3. 125 → 25 → 5. 187 → 72 → 6. 236 → 45 → 7. 349 → 74 → 8. 456 → 56 → 9. 568 → 27 → 10. 638 → 16 → 11. 774 → 55 → 12. 836 → 43 → 13. 719 → 85 → 14. 468 → 68 → 15. 229 → 54 → RW60 Reteach © Harcourt Round each factor and estimate the product. LESSON 11.4 Name Model Multiplication A model can be used to simplify multiplication. This is a model for 12 26. Multiply to find the number of squares in each rectangle. Then add the partial products. 6 20 10 10 2 26 12 2 20 40 10 6 60 10 20 + 200 2 312 6 20 When you add the number of squares on each rectangle in the grid, you find that the product of 12 and 26 is 312. Multiply and solve by using the model and writing partial products. 1. 26 14 46 6 20 10 10 4 20 10 6 4 4 6 20 10 20 © Harcourt 2. 23 17 20 73 3 10 10 7 20 10 3 7 7 20 3 10 20 Reteach RW61 LESSON 11.5 Name Problem Solving Strategy Solve a Simpler Problem Solve. 60 217 = ? , or 217 60 • You can find the product by breaking the three-digit number apart. • Multiply, using the simpler numbers. First, multiply 60 times 7. 60 7 420 partial products Second, multiply 60 times 10. 60 10 600 Third, multiply 60 times 200. 60 200 12,000 Fourth, add the partial products to find the product. 420 600 12,000 13,020 Find each product by using partial products. Show the four steps you used. 1. 241 6 2. 237 30 3. 675 20 5. 295 70 6. 327 50 616 6 40 240 6 200 1,200 6 240 1,200 1,446 555 40 © Harcourt 4. RW62 Reteach LESSON 12.1 Name Multiply by 2-Digit Numbers Solve 12 15 by using the partial-products method. Step 1 Step 2 Step 3 Step 4 15 12 15 12 15 12 15 12 Partial Products 1 5 1 2 25 1 0 Step 5 2 10 2 0 Add partial products together. 10 5 5 0 10 10 1 0 0 10 20 50 100 180 product → 1 8 0 The product of 12 and 15 is 180. Multiply by using the partial-products method. © Harcourt 1. 3 1 1 7 2. 4 6 2 8 3. 7 9 5 6 71 86 69 7 30 8 40 6 70 10 1 20 6 50 9 10 30 20 40 50 70 product → product → product → 4. 82 25 5. 63 47 6. 92 34 Reteach RW63 LESSON 12.2 Name More About Multiplying by 2-Digit Numbers Find the product 28 136. • Estimate by rounding. Since 30 100 is 3,000, your answer should be close to 3,000. • Multiply. 136 28 1,088 ← 136 8 2,720 ← 136 20 3,808 Remember: 136 8 is the same as • Compare the product to your estimate. Since 3,808 is close to your estimate of 3,000, it is a reasonable product. 48 86 240 8 30 800 8 100 1,088 Estimate and multiply. 1 2 6 126 → 100 45 → 50 5,000 2. 4 5 276 → 38 → 6 3 0 3. 737 → 47 → RW64 Reteach 5, 0 4 0 5, 6 7 0 7 3 7 4 7 2 7 6 3 8 4. 545 → 24 → 5 4 5 2 4 © Harcourt 1. LESSON 12.3 Name Multiply Greater Numbers When multiplying greater numbers, it is more important than ever to work neatly. Using a grid may help to keep your regroupings from getting mixed up. 3,846 27 5 3 4 3, 8 4 6 ← Use one row for the first regroupings. 2 7 2 6, 9 2 2 1 1 5 3 4 3, 8 4 6 ← Use another row for the second regroupings. 2 7 2 6, 9 2 2 7 6, 9 2 0 1 0 3, 8 4 2 Multiply. © Harcourt 1. 2. 6, 5 5 1 8 4 4, 3 7 5 4 7 Reteach RW65 LESSON 12.4 Name Practice Multiplication Betty is selling her CD collection. She sold 12 CDs at $14.95 each. How much money did she receive for her CDs? • What do you need to do? Multiply 12 $14.95. • First, estimate by rounding. Since 12 $14.95 is close to 10 $15.00, your answer should be close to $150.00. • Multiply. 14.95 12 2990 ← 1495 2 14950 ← 1495 10 $179.40 ← Find the sum and place the decimal point. • Write the answer in dollars and cents. • Compare the product with your estimate. Since $179.40 is close to your estimate of $150.00, it is a reasonable product. 1. 3. 5. 7. $7.98 51 → $8.00 50 $400.00 $80.10 → 53 913 44 → 645 83 → RW66 → 391 28 → → 391 28 $80.10 4. 2,356 → 45 53 → 2356 45 → 913 6. 5,469 → 32 44 → 5469 32 → 645 83 $86.32 → 61 → $86.32 61 Reteach 2. → $7.98 51 798 39900 $406.98 8. © Harcourt Estimate and multiply. LESSON 12.5 Name Problem Solving Skill Multistep Problems To solve a problem, you need to make several decisions. One way to approach a problem is to organize the data, select the operation needed, and then solve. The fourth-grade class sold treats at a flea market. On Friday, they sold 144 chocolate chip cookies and 36 crispy treats. On Saturday, they sold 136 chocolate chip cookies and 46 crispy treats. Each chocolate chip cookie sells for $0.25, and each crispy treat sells for $0.50. How much money did they make? Break the problem down into steps. 1. Find the total number of chocolate chip cookies sold. Operation: addition Equation: 144 136 2. Find the total number of crispy treats sold. Operation: addition Equation: 36 46 3. Find the amount of money earned from each type of cookie. Operation: Equations: 280 $0.25 82 $0.50 4. Solve the problem. Operation: addition © Harcourt Equation: Organize the information. Select the operations. Solve. 1. Molly is the star forward on her basketball team. In Saturday’s game she scored eleven 2-point baskets and four 3-point baskets. How many points did Molly score? 2. Each egg carton at the dairy holds 144 eggs. There are 4 cartons on the truck and 34 cartons in the warehouse. How many eggs are there altogether? Reteach RW67 LESSON 13.1 Name Divide with Remainders What is 23 5? The model below shows one way to think about this division problem. Think: If you divide 23 into 5 equal groups, how many are in each group? 23 5 There are 5 groups of 4, with 3 left over. 4 r3 3 52 20 3 23 5 4 r3 The remainder is the amount left over when a number cannot be divided evenly. You can multiply and subtract to find the remainder. Complete 1–5 to find the remainder. 6r 1. 53 2 30 4r 2. 73 1 28 9r 3. 43 9 6r 31 9 4. 6r 5. 96 2 6. 74 3 RW68 Reteach 7. 51 9 8. 32 2 9. 21 7 © Harcourt Divide. LESSON 13.2 Name Model Division Divide 92 into 4 equal groups. Write 92 4. Step 1 Step 2 Show 92 as 9 tens and 2 ones. 2 Begin by dividing the 2 9 tens. Place an equal 49 8 number of tens into 1 each group. 2 tens in each group 8 tens used 1 ten left Draw 4 circles to show 4 groups. Step 3 Regroup the 1 ten left into 10 ones. Now there are 12 ones. 2 2 bring 49 8↓ ones 12 down Step 4 Divide the 12 ones. Place an equal number of ones into each group. 23 3 ones in 2 each group 49 8 12 12 12 ones used 0 0 ones left So, 92 4 23. Use base-ten blocks to model each problem. Record the numbers as you complete each step. 24 8 2. 39 6 3. 45 2 4. 23 5 5. 56 5 © Harcourt 1. Reteach RW69 LESSON 13.3 Name Division Procedures What is 47 3? Remember these steps: Divide Multiply Subtract 1 7 34 3 1 Compare Bring down 1 34 313 431 13 Divide the tens. Multiply. Subtract. Compare. 1 34 7 3↓ 17 Bring down? 15 7 34 3 17 15 2 Yes. Bring the 7 down to make 17. 5 31 7 3 5 15 17 15 2 23 Divide the ones. Multiply. Subtract. Compare. 15 r2 7 34 3 17 15 2 Bring down? Nothing is left to bring down. If the number left over is less than the divisor, record it as the remainder. 1. 56 5 RW70 Reteach 2. 35 2 3. 25 1 4. 49 1 © Harcourt Solve. Use the D, M, S, C, B steps shown above. LESSON 13.4 Name Problem Solving Strategy Predict and Test 1. There are 87 marbles. After the marbles are divided evenly into bags, there are 2 marbles left over. There are fewer than 10 bags. How many bags are there? How many marbles are in each bag? Predict: Test: Remember, there must be 2 marbles remaining. 3 Bags? 29 38 7 6 27 27 0 No remainder 4 Bags? 21 r3 7 48 8 7 4 3 5 Bags? Remainder of 3 Remainder of 2 You found the answer! 17 r2 7 58 5 37 35 2 There are 5 bags with 17 marbles in each bag, and 2 marbles are left over. 2. There are 82 cookies. When the cookies are divided evenly on plates, there are 4 cookies left over. There are fewer than 10 plates. How many plates are there? How many cookies are there on each plate? HINT: Predict numbers between 0 and 9. You may need to predict more than three times. © Harcourt Predict: Plates? Plates? Plates? Test: Remember, there must be 4 cookies remaining. Reteach RW71 LESSON 13.5 Name Mental Math: Division Patterns You can find how many fives are in 2,000 by following these steps. • Rewrite each problem as a multiplication fact. 20 5 4 5 4 20 5 4 0 20 0 • Circle all the numbers in the basic fact to see how many zeros to write in the quotient. 200 5 40 2,000 5 ? 5 4 00 20 00 So, 2,000 5 400. • Write the quotient. If there is a zero in the basic fact, the quotient has one fewer zero than the dividend. Otherwise the quotient has the same number of zeros as the dividend. You can estimate a quotient by using numbers that are easy to divide. Use a basic division fact and patterns to find each quotient. 1. 4. 45 9 2. 21 3 3. 81 9 450 9 210 3 810 9 4,500 9 2,100 3 8,100 9 36 6 5. 72 8 6. 12 4 360 6 720 8 120 4 3,600 6 7,200 8 1,200 4 Basic Fact: RW72 Reteach Basic Fact: Basic Fact: © Harcourt Divide mentally. Write the basic division fact and the quotient. 7. 480 6 8. 1,500 3 9. 36,000 ÷ 6 LESSON 13.6 Name Estimate Quotients Compatible numbers are numbers that are close to the actual numbers and can be divided evenly. They can help you estimate a quotient. Estimate. 43 9 5 Step 1 Round the dividend to a number that can be divided evenly by the divisor. Step 2 Rewrite the division problem with the compatible numbers, and solve. The number 395 can be rounded to 360. This number can be divided evenly by 4. The number 395 can also be rounded to 400. This number can also be divided evenly by 4. 90 43 6 0 100 44 0 0 So, one estimate of the quotient is 90. A second estimate is 100. Write two pairs of compatible numbers for each. Give two possible estimates. 1. 74 6 0 2. 53 7 0 3. 64 4 3 5. 76 ,5 2 1 6. 93 1 5 __________________ 83 ,0 6 2 © Harcourt 4. Reteach RW73 LESSON 14.1 Name Place the First Digit When you begin a division problem, you must decide where to place the first digit of the quotient. 2 371 6↓ 11 Example B hundreds tens ones tens ones Example A Divide the tens. Multiply. 326 Subtract. 761 Compare. 13 Bring down? Yes. Bring down 1 to make 11. 3 4, so ask yourself 4 3 9 2 how many 4’s are in 39. Divide the ones. 339 2 3 r2 Multiply. Subtract. 11 9 2 37 1 Compare. 23 6↓ Bring down? Nothing is left. 1 1 If there is a 9 remainder, 2 record it. 98 4 3 92 36↓ 0 32 32 0 Divide the tens. Multiply. Subtract. Compare. Bring down. Divide the ones. Multiply. Subtract. Compare. Nothing is left. Write an x where the first digit in the quotient should be placed. 1. 53 4 – 3 0 4 2. 4 37 – 6 ↓ 1 4 3. 1 2 9 5 1 8 ↓ 1 5 4. 45 3 9 4 1 3 5. 58 3 – 5 ↓ 3 3 6. 32 9 RW74 Reteach 7. 24 7 8. 7 1 9 5 0 0 0 0 0 0 0 0 0 9. 5 7 5 0 0 4 0 0 0 © Harcourt Divide. LESSON 14.2 Name Divide 3-Digit Numbers These examples show how to divide a three-digit number. Example A Example B 1 9 352 3↓ 22 Divide the hundreds. Multiply. Subtract. Compare. Bring down? 615 9 There are not enough hundreds to divide. 17 2 9 35 3 22 21↓ 19 Divide the tens. Multiply. Subtract. Compare. Bring down? 2 9 615 12↓ 39 176 r1 Divide the ones. 2 9 35 Multiply. 3 Subtract. 22 Compare. 21 Bring down? If there is a 19 remainder, 18 record it. 1 Divide the tens. Multiply. Subtract. Compare. Bring down? Divide the ones. 26 r3 Multiply. 59 61 Subtract. 12 Compare. 39 Bring down? If there is a 36 remainder, 3 record it. Write an x where the first digit in the quotient should be placed. 1. 53 4 9 – 3 0 ↓ 4 9 2. 37 4 5 – 6 ↓ 1 4 3. 21 9 8 – 1 8 ↓ 1 8 4. 45 3 2 – 4 ↓ 1 3 © Harcourt Divide. 5. 53 4 9 6. 37 4 5 7. 21 9 8 8. 45 3 2 Reteach RW75 LESSON 14.3 Name Zeros in Division In some division problems, you need to place a zero in the tens place or the ones place of the quotient. Zero in ones place Zero in tens place H T O 2 0 8 8 2 43 8 0 3 0 3 2 3 2 0 428 Write 2 in quotient. 43 400 Write 0 in quotient. 43 2 4 8 32 Write 8 in quotient. 48 326 Write 2 in quotient. 31 5 3 5 15 Write 5 in quotient. 32 300 Write 0 in quotient. H T O 2 5 0 r2 5 2 37 6 1 5 1 5 0 2 0 2 37 Divide and check. 36 1 4 2. 4. 7 9 1 3 5. 44 2 8 38 4 0 3. 6. 28 1 9 56 5 2 © Harcourt 1. RW76 Reteach LESSON 14.4 Name Divide Greater Numbers To divide greater numbers, repeat the division pattern until there is nothing left to bring down and the difference is less than the divisor. T H T O 3 2, 5 6 9 0000 1. 2. 3. 4. 5. 6. 7. 8. 9. Divide the hundreds. Multiply and subtract. Bring down the 6 tens. Divide the tens. Multiply and subtract. Bring down the 9 ones. Divide the ones. Multiply and subtract. There is nothing left to bring down. 1 3. The remainder is 1. Write divide, multiply and subtract, or bring down to show the next step. 8 7 2 1. 2. 5 4, 2 8 5 6 4, 3 3 7 4 0 4 2 2 8 1 3 © Harcourt Divide. 3. 41 ,8 7 3 4. 7$ 4 5 .9 2 5. 23 ,4 9 8 6. 6$ 3 5 .4 6 7. 87 ,8 9 3 8. 39 ,8 2 2 Reteach RW77 LESSON 14.5 Name Problem Solving Skill Interpret the Remainder You can use a remainder in different ways. Look for word clues in problems to help you decide what to do with the remainder. Look at the list of clues below. Use it to help solve the problems. a. If the problem asks how much is left over, use the remainder for your answer. b. If the problem asks how many groups are needed to divide a number into groups, increase the quotient by 1. c. If the problem asks how many groups can be made from a larger number, drop the remainder. Example: Thomas has to divide his 24 classmates into groups of 5. How many students will be left over? 24 5 4 r4 4 students will be left over. Example: How many tables will be needed to seat the 24 classmates at tables of 5? 24 5 4 r4 5 tables will be needed (4 tables will not be enough.) Example: How many 10-person spelling bee teams can be made with 24 classmates? 24 10 2 r4 There will be 2 teams. (There are not enough classmates to make a third team of 10.) 1. Mark is cutting ribbon to wrap a gift. He has a spool of 45 inches of ribbon. How many 8-inch lengths of ribbon can he make? 2. Scott needs 27 sheets of paper for an art project. Paper comes in packages of 20 sheets. How many packages of paper should Scott buy? 3. Anne is making bead bracelets. Each bracelet takes 5 beads. Anne has 37 beads. How many beads will she have left over? 4. Alex is giving 6 brownies to each teacher. She has 38 brownies. How many teachers can she give brownies to? RW78 Reteach © Harcourt Look for the clues in these problems. Solve. Then write a, b, or c from above to tell how you interpreted the remainder. LESSON 14.6 Name Find the Mean The mean is the number found by dividing the sum of a set of numbers by the number of addends. To find the mean: Step 1 Add the numbers. Step 2 Count the number of addends. Step 3 Divide the sum by the number of addends. Examples: Find the mean of 21, 14, 26, and 47. Find the mean of 35, 47, 28, 29, and 31. Step 1 21 14 26 47 108 Step 1 35 47 28 29 31 170 Step 2 There are 4 addends. Step 2 There are 5 addends. Step 3 108 4 27 Step 3 170 5 34 The mean is 27. The mean is 34. Find the mean of each set of numbers. 1. 5, 6, and 10 2. Step 1 5 6 10 $1.89; $2.85; $3.24 Step 1 Step 2 There are Step 3 addends. 3 The mean is Step 2 There are addends. Step 3 . The mean is . © Harcourt Find the mean. 3. 4, 5, 5, 8, 8 4. 19, 29, 12, 48 5. $3,112; $2,018; $4,527 6. 32, 71, 89, 64 7. 151, 109, 133 8. $1.19, $0.78, $0.61 Reteach RW79 LESSON 15.1 Name Division Patterns to Estimate In the problem 594 18 33, 594 is the dividend, 18 is the divisor, and 33 is the quotient. To estimate the quotient, follow Steps 1–3. Step 1 Step 2 Step 3 Round the divisor to the nearest tens place. Round the dividend to the nearest multiple of the new divisor. Divide, using the rounded numbers. 18 → 20 595 → 600 600 20 30 Write the numbers you would use to estimate the quotient. Then write the estimate. 1. 36 11 n 2. 78 13 n 3. 56 19 n Write the numbers you would use to estimate the quotient. Then write the basic fact that helps you find the quotient. The first one has been done for you. 4. 446 92 n 5. 52 11 n 6. 162 79 n 8. 272 28 n 9. 235 56 n 450 90 5 45 9 5 7. 76 24 n 10. Dividend Divisor 40 20 20 11. 12. 4,000 RW80 Reteach 20 Quotient 13. 20 Dividend Divisor 90 30 30 14. 15. 9,000 30 Quotient 30 © Harcourt Complete the tables. LESSON 15.2 Name Model Division You can divide by using models. What is 35 12? Step 1 Step 2 Step 3 Count out 35 objects. Make as many groups of 12 as you can. Record your work. Count 2 r11 the number 5 123 of groups 24 and how 11 many are left over. © Harcourt Make a model to divide. 1. 154 2 2. 174 9 3. 251 0 3 4. 216 5 5. 113 5 6. 311 2 8 7. 169 3 8. 195 9 9. 248 4 Reteach RW81 LESSON 15.3 Name Division Procedures What is 231 19? Follow Steps 1–4 when you divide a 3-digit number by a 2-digit number. Step 1 Step 2 How many groups of 19 are in 2? Zero groups. Write an x in the hundreds place. How many Multiply. groups of 19 are Subtract 19 and in 23? One group. bring down the 1. Write a 1 in the tens place. x 3 1 192 Step 3 Step 4 How many groups of 19 are in 41? Two groups. Write a 2 in the ones place. Subtract. x1 3 1 192 19 41 x1 3 1 192 x12 3 1 192 19 41 38 3 Compare the remainder, 3, with the divisor, 19. The remainder is less than the divisor. Write the remainder as part of the quotient. So, 231 19 12 r3. Divide. 118 0 6 2. 121 8 5 3. 151 2 2 4. 161 1 3 6. 131 8 3 7. 185 8 1 8. 221 1 9 © Harcourt 1. 5. 174 1 4 RW82 Reteach LESSON 15.4 Name Correcting Quotients Divide. 321 38 n 383 2 1 When you are dividing, sometimes the first digit you write in the quotient is too high or too low. Too High 9 383 2 1 342 Since 342 > 321, the quotient 9 is too high. Use a lesser number. Too Low Just Right 7 383 2 1 266 55 Since the remainder, 55, is greater than the divisor, 38, the quotient 7 is too low. Use a greater number. 8 2 1 383 304 17 Since the remainder, 17, is less than the divisor, 38, the quotient 8 is just right. © Harcourt Write too high, too low, or just right for each estimate. Explain. 6 3 6 1. 181 108 28 7 0 3 2. 161 112 8 3 9 3. 171 136 3 5 1 9 4. 231 115 4 6 3 6 5. 322 192 44 7 7 0 6. 281 196 Divide. 7. 433 6 5 8. 343 3 8 9. 523 7 9 10. 684 5 4 Reteach RW83 LESSON 15.5 Name Problem Solving Skill: Choose the Operation Add Joining groups of different sizes Erin read a biography of Thomas Jefferson that was 312 pages long. Next she read a fiction book which was 289 pages long. How many pages did she read altogether? 312 289 601 Subtract Taking away or comparing groups How many pages longer was the biography than the fiction book? 312 289 23 Multiply Joining equal-sized groups Finding how many in each group or how many equal-sized groups 23 pages longer Erin thinks it takes about 3 minutes to read one page of a book. How many minutes should it take her to read the biography? 3 312 936 Divide 601 pages 936 minutes Erin wants to read the fiction book in 17 sittings. How many pages should she read at each sitting? 289 17 17 17 pages 1. There are 12 players on Coach Walker’s soccer team, 14 players on Coach Lee’s team, and 24 adults. How many people are there? 2. If each of the players at the soccer game is wearing 2 shin guards, how many shin guards are at the game? 3. Coach Walker has brought 3 balls to the soccer game. Before the game, he wants equal groups of his players to warm up with the balls. How many players should be in each group? 4. Each quarter of the soccer game lasts 12 minutes. If 9 minutes have gone by, how many minutes remain in the quarter? RW84 Reteach © Harcourt Solve. Name the operation you use. Write add, subtract, multiply, or divide. LESSON 16.1 Name Factors and Multiples Use a multiplication table to find the multiples of a given number. A multiple is the product of a given whole number and another whole number. Look at the row or column for a given number. All the numbers in that row or column are multiples of that number. Example: Look at the row or the column for 6. Some multiples of 6 are 6, 12, 18, 24, , 66, and , , 48, 54, , . 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 1 1 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 You can also use the table to find factors of a given number. To find factors of 36, search the table for 36. When you see 36, look at the row and column numbers to find the factors. 12 Some factors of 36 are 3, 4, 6, 9, and 12 24 36 48 60 72 84 96 108 120 132 144 . Use the multiplication table to find factors for each product. 1. 45 2. 24 3. 20 6. 5 Use the multiplication table to write 4 multiples for each number. 4 5. 9 © Harcourt 4. Write as many factors as you can for each product. 7. 18 8. 36 Reteach RW85 LESSON 16.2 Name Factor Numbers An array is an arrangement of objects in rows and columns. Arrays can show how factors and products are related. Use arrays to break down 56 into factors. STEP 1 Outline a rectangle that has 56 squares. Use your basic facts to help decide the length and width of the rectangle. What two factors have a product of 56? STEP 2 Split the rectangle into two equal parts to find other factors. These arrays show ( ) 56 STEP 3 Can you split your array into different equal parts to find more factors? These arrays show ( ) 56 Write an equation for the arrays shown. 1. 2. Show two ways to break apart the model. 4. © Harcourt 3. Write two ways to break down the number. 5. 42 RW86 6. Reteach 36 LESSON 16.3 Name Prime and Composite Numbers A prime number has exactly two factors, 1 and the number itself. A composite number has more than two factors. The number 1 is a special number because it is neither prime nor composite. It has only 1 factor, 1. Make arrays to find the factors. Write prime or composite for each number. 8 2. 23 3. 35 4. 9 5. 24 6. 11 © Harcourt 1. Reteach RW87 LESSON 16.4 Name Find Prime Factors You can find the prime factors of a number by using a factor tree. Begin by writing the number. 80 2 40 Use what you know about multiplying to write the number as a product. If one factor is prime, underline it. You can not break that factor down any further. 2 2 20 2245 22225 Break down the factors that are not underlined. Continue until all factors are prime. They should all be underlined at the bottom of your tree. So, 80 2 2 2 2 5. Use factor trees to write each as a product of prime factors. 30 2. 45 3. 50 4. 64 © Harcourt 1. RW88 Reteach LESSON 16.5 Name Problem Solving Strategy Find a Pattern Carrie has finished part of her square quilt. How many patches of each color will be in the final quilt? Look for patterns to help you make a plan for solving. Since this is a square quilt and there are 9 patches along the top, you know there must be 9 rows. B R B R B R RWB WB R RWB WB R RWB WB R B Blue Look closely at the quilt. There is a pattern of 3 blues, 3 reds, and 3 whites in each row. Multiply each color by the 9 rows. So, there are 27 blue, 27 red, and 27 white patches. Blue RW WB RW WB RW WB R Red Red B R B R B R RW WB RW WB RW WB W White White 3 9 27 3 9 27 3 9 27 Fill in the missing pieces of the pattern. © Harcourt 1. 2. 128, 64, 32, 16, 3. 4, 6, 8, 9, 10, 12, 14, 15, , , 4. Reteach RW89 LESSON 17.1 Name Read and Write Fractions A fraction is a number that names a part of a whole. Read: two thirds two out of three two divided by three Read: one half one out of two one divided by two parts shaded 2 total parts 3 parts shaded number of equal parts 1 2 Write a fraction for each model. 1. parts shaded total parts 3. parts shaded total parts 5. 2. parts shaded total parts 4. parts with stars number of equal parts parts with cats number of equal parts 6. part with baseballs number of equal parts Write two ways to read each fraction. 3 4 8. 1 2 2 3 9. © Harcourt 7. RW90 Reteach LESSON 17.2 Name Equivalent Fractions 1 1 4 Fractions that name the same amount are called 1 8 1 1 6 1 8 1 1 6 1 1 6 1 1 6 equivalent fractions. 14, 28, and 146 are different names for the same number. So, 14 28 146 , which makes them equivalent fractions. Find the equivalent fraction. 1. 2. 2 3 3. 1 2 6 8 4. 3 5 3 4 Color the correct number of parts to show an equivalent fraction. Then write the equivalent fraction. 5. 6. © Harcourt 6 9 7. 2 4 8. 1 3 1 4 Reteach RW91 LESSON 17.3 Name Equivalent Fractions You can use different fractions to name the same amount. Fractions that name the same amount are called equivalent fractions. You can find equivalent fractions in three ways. Use a number line. 0 1 1 2 0 2 4 1 4 3 4 1 You can see that 12 24, so they are equivalent fractions. Multiply both the numerator and the denominator by the same number. Divide both the numerator and the denominator by the same number. 1 13 3 3 33 9 6 6÷2 3 8 8÷2 4 The fraction 13 names the same amount as 39, so they are equivalent fractions. The fractions 68 and 34 name the same amount, so they are equivalent fractions. Use the number lines to find out if the fractions are equivalent. Write yes or no. 1 4 3 12 8 12 3 4 1. 2. 0 0 1 2 12 12 3 4 2 4 1 4 4 3 12 12 5 12 7 6 12 12 8 12 1 10 11 12 9 12 12 1 2 3 25 35 4 5 43 53 3 8 32 82 4. 1 7 14 74 6. 3. 5. Divide both the numerator and the denominator to find the simplest form of each fraction. 7 28 77 28 7 16 24 16 8 24 8 12 16 12 4 16 4 8. 10 15 10 5 15 5 10. RW92 Reteach 7. 9. © Harcourt Multiply both the numerator and the denominator to name an equivalent fraction. LESSON 17.4 Name Compare and Order Fractions You can compare fractions that have the same denominators. 2 5 Compare 25 and 35. Compare the shaded areas in the fraction models. 2 3, so 25 35. 3 5 You can also compare fractions that have different denominators. Compare 23 and 12. Compare the shaded areas in the fraction models. Since 23 has a larger shaded area, 23 12. 2 3 1 2 Compare the fraction models. Write , , or in the 1. 2. 4 6 5 6 4. © Harcourt 3. 3 4 6 8 5. 2 3 2 4 7. 1 3 2 3 3 5 2 4 3 6 4 1 0 5 8 6. 5 8 6 8 8. 3 9 . 9. 3 4 3 5 Reteach RW93 LESSON 17.5 Name Problem Solving Strategy Make a Model Beth made a snack mix for the class party. She used 12 lb of peanuts, 23 lb of pretzels, and 14 lb of dried fruit. List the ingredients in order from greatest to least amount. You can make a model to help solve the problem. Use fraction bars to model the fractions. Line up the fraction bars so you can tell which bar has the most shaded and which bar has the least shaded. 1 2 1 2 lb of peanuts 2 3 lb of pretzels 1 3 1 4 lb of dried fruit 1 4 1 3 The bar for pretzels has the most shaded and the bar for dried fruit has the least shaded. So, the ingredients in order from greatest to least are pretzels, peanuts, and dried fruit. Make a model to solve. The amusement park has a colorful sidewalk. Starting with the first square, every third square is orange, and every fifth square is green. Draw a model of the first 12 squares of the sidewalk in the space below. What fraction of the first 12 squares are either orange or green? 2. Patti made punch from 12 gallon of grape juice, 23 gallon of kiwi juice, and 58 gallon of papaya juice. Draw a model for each in the space below. Then list the ingredients in order from greatest to least. 1 2 1 3 grape juice 1 3 1 1 1 1 1 8 8 8 8 8 RW94 Reteach kiwi juice papaya juice © Harcourt 1. LESSON 17.6 Name Mixed Numbers A mixed number is made up of a whole number and a fraction. These models represent a mixed number. 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 11 3 3 There are 3 whole figures shaded. 3 shaded. The last figure is 2 3 figures are shaded. So, 32 31 of the figures are shaded. Or, you can say 1 © Harcourt Write a fraction and a mixed number for each picture. 1. 2. 3. 4. Rename each fraction as a mixed number. You may wish to draw a picture. 3 2 6. 13 5 7. 13 3 9. 26 6 10. 5. 8. 22 4 18 8 Reteach RW95 LESSON 18.1 Name Add Like Fractions When you add like fractions, add only the numerators. 2 1 5 5 ? + 2 parts shaded 5 parts = 1 part shaded 5 parts 3 parts shaded 5 parts 2 1 → Add the numerators. 3 → 21 → 5 5 → Write the denominator. → 5 → 5 Find the sum. Show how you added the numerators. 2 3 23 5 6 6 6 6 2. 1 4 8 8 4. 2 1 4 4 6. 4 5 10 10 8. 4 6 12 12 10. 3 2 55 12. 3. 5. 7. 9. 11. RW96 Reteach 3 3 7 7 2 2 5 5 1 1 3 3 4 1 99 2 1 8 8 5 6 88 © Harcourt 1. LESSON 18.2 Name Subtract Like Fractions When you subtract like fractions, subtract only the numerators. 2 6 8 8 ? X X 6 parts shaded 8 pa rt s 2 parts shaded 8 pa rt s 4 parts shaded 8 pa rt s 1 6 2 → Subtract the numerators. → 6 2 → 4 , or 8 8 → Write the denominator. → 8 → 8 2 Find the difference. Show how you subtracted the numerators. 5 3 53 2, or 1 3 6 6 6 6 2. 7 4 8 8 4. 2 1 4 4 6. 7 5 10 1 0 8. 10 6 12 12 10. 9 3 10 1 0 12. 1. 3. 5. © Harcourt 7. 9. 11. 6 3 7 7 4 2 5 5 2 1 3 3 4 1 9 9 2 1 88 9 1 12 1 2 Reteach RW97 LESSON 18.3 Name Add and Subtract Mixed Numbers When adding mixed numbers, add the fractions first. Then add the whole numbers. 43 Add. 72 6 6 Step 1 Add the fractions. Step 2 Add the whole numbers. 6 72 6 43 6 72 6 43 5 6 6 115 Step 3 . The sum is115 6 43 115 . So, 72 6 6 6 When subtracting mixed numbers, subtract the fractions first. Then subtract the whole numbers. 4 16 Subtract. 85 6 Step 1 Subtract the fractions. Step 2 Subtract the whole numbers. 85 6 85 6 4 16 4 16 1 6 76 Step 3 The difference 1 is 76. 5 4 1 So, 86 16 76 1 Find the sum or difference. 9 52 2. 9 24 5. 5 84 6. Reteach 5130 4. 2160 7 93 7. 72 7 8 63 52 8 RW98 3. 6 71 31 5 9. 6 83 8 13 9 45 8. 31 9 0 1 81 12 12 4 62 31 4 3 10. 8 97 11. 4 32 4 71 © Harcourt 1. LESSON 18.4 Name Problem Solving Skill Choose the Operation In deciding if you should add or subtract to solve a problem, think about the answer. Should the answer be greater than the numbers in the problem? If so, then consider adding. If not, then consider subtracting. Eldrick and Erin wrote their ages as mixed numbers. Eldrick is 10172 years old and Erin is 9112 years old. How much older than Erin is Eldrick? Think: Should the answer be more than or less than 10172? less than Solve: 10172 1 912 1162 or 112 years Read the problem. Answer the Think question. Then solve. 1. © Harcourt 2. Devon needs 2 pieces of lumber to build a frame for his treehouse. One should be 1212 feet long and the other should be 312 feet long. How long are the 2 pieces of lumber together? Think: Should the answer be greater than or less than 1212? Alicia is knitting a scarf that will be 523 feet long. So far she has knitted 313 feet. How much more does she have to knit? Think: Should the answer be greater than or less than 523? Solve: Solve: Reteach RW99 LESSON 18.5 Name Add Unlike Fractions Yesterday, Jill walked 14 mile, and Bob walked far did they walk in all? To answer this, you add 1 4 1 3 mile. How 13. The fractions 14 an 13 are unlike fractions because the denominators are not the same. To add unlike fractions, you can use the following steps: Step 1 Step 2 Use fraction bars to model 14 13. Find the like fraction Write an equation for bars that are equivalent the sum shown by the to 14 13 in length. like bars. Step 3 1 1 4 1 3 1 1 1 4 1 4 1 3 1 3 1 1 1 1 1 1 1 12 12 12 12 12 12 12 ? 7 1 1 4 3 12 1. 1 1 6 4 2. 2 2 5 10 3. 1 5 3 6 4. 2 1 8 4 5. 1 1 5 2 6. 1 1 3 4 7. 4 1 6 4 8. 1 1 3 2 9. 3 3 5 10 10. 1 2 4 6 3 3 8 4 12. 1 2 6 3 11. RW100 Reteach © Harcourt Use fraction bars to find the sum. LESSON 18.6 Name Subtract Unlike Fractions Subtract. 4 3 5 10 To subtract unlike fractions, you can follow these steps. Step 1 Step 2 Place four 15 fraction bars under a 1 whole bar. Then place three 110 fraction bars under the four 15 bars. Compare the bars. Find like fraction bars that fit exactly under the difference 45 130 . 1 1 1 5 1 5 1 5 1 5 1 5 1 5 1 1 1 10 10 10 1 1 1 10 10 10 1 5 1 5 ? 1 1 1 1 1 10 10 10 10 10 So, 4 5 130 5 10, or 1 2 . © Harcourt Use fraction bars to find the difference. Write the answer in simplest form. 1. 3 1 4 2 2. 5 1 8 4 3. 2 1 3 2 4. 7 2 10 5 5. 6 1 8 2 6. 1 1 1 12 3 7. 2 1 3 6 8. 3 1 4 2 9. 9 1 10 2 10. 3 3 4 8 1 1 2 6 12. 1 3 2 10 11. Reteach RW101 LESSON 19.1 Name Relate Fractions and Decimals You can write a fraction or a decimal to tell what part is shaded. Model Fraction Decimal O 4 shaded parts 10 parts 0 T . O 25 shaded parts 100 parts 0 . H four tenths 4 T •. Read H twenty-five hundredths 2 5 Complete the table. Model Fraction Decimal O 1. shaded parts parts Read T H T H T H T H T H . • O 2. . • O 3. . O 4. © Harcourt • . • O 5. . •.• RW102 Reteach LESSON 19.2 Name Decimals Greater Than 1 6110 A mixed number is made up of a whole number and a fraction. Decimal Number 6.1 A decimal greater than 1 is made up of a whole number and a decimal. $500.75 Whole Number Two whole blocks and three tenths of a block are shaded. Write: 2130, or 2.3 Write a mixed number and a decimal greater than 1 for each model. 1. 2. 3. 4. Write each mixed number as a decimal. 5. 3140 6. 23 9 10 0 7. 4140 8. 9 3 100 © Harcourt Write each decimal as a mixed number. 9. 5.6 10. 7.02 11. 3.78 12. 6.09 13. 2.8 14. 6.07 15. 8.02 16. 9.4 Reteach RW103 LESSON 19.3 Name Equivalent Decimals Equivalent decimals are different names for the same amount. 0.3 Ones = 0.30 Tenths 0 . 3 0 . 3 The decimal models show that 0.3 and 0.30 are equivalent decimals. The same amount is shaded in each model. Hundredths 0 The place-value chart also shows that 0.3 = 0.30. Both numbers have the digit 3 in the tenths place. Shade an equivalent amount in each pair of models. Write the decimal for each shaded amount. 1. 2. 3. 4. 6. RW104 Reteach © Harcourt 5. LESSON 19.4 Name Compare and Order Decimals Write these numbers in order from greatest to least: 0.4, 0.35, 0.37, 0.05. Step 1 Write the numbers in a place-value chart. Step 2 Write a zero in the empty hundredths place. Step 3 Compare the digits in each place, beginning at the left. Ones Tenths Hundredths 0 . 4 0 0 . 3 5 0 . 3 7 0 . 0 5 Greatest 0.40, 0.37, 0.35, 0.05 Least Compare. Then write the greater number. 1. 0.2 0.3 2. 2.25 2.27 3. 0.36 0.34 4. 3.5 3.9 5. 0.37 0.3 6. 0.02 0.10 7. 0.38 0.40 8. 1.04 1.2 Compare. Write , , or in the . 9. 10. © Harcourt 0.2 0.19 11. 0.50 0.39 12. 0.2 13. 0.34 0.39 14. 2.1 2.09 0.25 0.47 15. 0.24 0.34 16. 0.04 0.4 0.41 Reteach RW105 LESSON 19.5 Name Problem Solving Strategy Use Logical Reasoning Laverne, Jody, Pearl, Leroy, and Cal are each performing in a variety show. Laverne will perform after Leroy. Pearl will perform third. Jody will perform before Pearl. Cal will perform before Jody. In what order will they perform? Use notecards to help you solve the problem. Write each person’s name on a slip of paper. When you know a person’s position, place the slip of paper with his or her name in that position. rne Lave Jody Pearl Leroy Cal Pearl will perform third. ? ? Pearl ? ? Jody will perform before Pearl. Cal will perform before Jody. Cal Jody Pearl ? ? Leroy Laverne Laverne will perform after Leroy. Cal Jody Pearl So, the order is: Cal, Jody, Pearl, Leroy, Laverne. 1. Vicky has 5 classes before lunch each day. She has math before science. She has Spanish before math. She has English right before lunch. She has history after science. In what order does Vicky have classes in the morning? 2. There are 6 children in the Ingraham family. Denise is older than Steven, but younger than Eliot. Kayla is younger than all her brothers and sisters except for Kirk. Bernie is younger than Steven. Write the names of the Ingraham children in order from oldest to youngest. RW106 Reteach © Harcourt Use notecards and logical reasoning. LESSON 19.6 Name Relate Mixed Numbers and Decimals Change 23 to a decimal. 5 Change 4.70 to a mixed number. 3 6 0.6 5 10 Read 4.70 as “4 and 70 hundredths.” So, 23 2.6 5 7 0 4.70 4 100 Write the mixed number in simplest form. So, 4.70 47. 10 Write an equivalent decimal for each mixed number. 1. 32 3 3. 5 10 2. 23 4 3. 43 10 Write an equivalent mixed number for each decimal. 4. 3.6 3 and 6. 4.90 tenths 3 10 3 5. 2.8 7. 3.75 Write an equivalent mixed number or decimal. 8. 63 5 9. 4.20 10. 1 2 10 2 0 8 100 12. 31 2 13. 7.75 14. 2 3 5 15. 4.6 16. 7.9 © Harcourt 11. Reteach RW107 LESSON 20.1 Name Round Decimals Round 2.53 to the nearest whole number. Step 1 Underline the digit in the ones place. 2 .53 Step 2 Look at the digit to the right. If the digit is 5 or greater, the digit in the rounding place is increased by 1. → 2 .53 So, round 2 to 3. Step 3 Rewrite with zeros in all places to the right. 3.00 Use similar steps to round 1.26 to the nearest tenth. → 1. 2 6 Round 2 to 3. 1.30 Complete the chart. 1. 3.25 2. 14.85 3. 21.51 4. 65.28 5. 75.31 6. 80.09 7. 114.72 8. 151.38 RW108 Reteach Round to the nearest whole number. © Harcourt Round to the nearest tenth. LESSON 20.2 Name Estimate Sums and Differences You can estimate the sum of $25.36 and $14.84 by rounding to the nearest dollar and then adding. $2 5. 0 0 $2 5. 3 6 → $1 5. 0 0 $1 4. 8 4 $4 0. 0 0 You can estimate the difference between $38.63 and $12.49 by rounding to the nearest dollar and then subtracting. $3 8. 6 3 → $3 9. 0 0 $1 2. 4 9 $1 2. 0 0 $2 7. 0 0 Estimate the sum or difference. © Harcourt 1. $3.51 → $1.21 $4. 0 0 $1. 0 0 2. 15. 84 22. 31 3. $9.51 $6.22 4. 85.43 19.84 5. $8.62 $4.25 6. $19.65 $ 8.42 7. $78.23 $48.61 8. 69.58 10.71 9. 50.28 49.72 10. 9.54 34.19 Reteach RW109 LESSON 20.3 Name Add Decimals How can you add 1.65 to 0.37? Remember to keep digits lined up by place value and their decimal points. Step 1 Step 2 Identify the place value of each digit. 1.65 0.37 1 is in the ones place. 0 is in the ones place. 6 is in the tenths place. 3 is in the tenths place. 5 is in the hundredths place. 7 is in the hundredths place. Line up each digit Step 3 Add, starting from the right. by its place value. Regroup if you need to. O T H . 0 •. 3 •. 1 •6 O T H . 0 •. 3 2 •. 0 7 1 5 7 11 •161 5 2 So, 1.65 0.37 equals 2.02. Rewrite each problem in the place-value grid. Find the sum. 0.5 O T 2. 0.9 0.6 H O •. T 6. 1.4 H •. •. RW110 Reteach •. O 7. T •. O 8. 29.09 15.8 T T H •. •. O T H •. •. H •. 21.34 40.82 H T •. •. T •. 1.46 0.17 H •. •. T 4. •. 1.5 O •. O •. 1.2 O H •. •. 5. 0.6 1.67 0.58 •. •. T 3. •. •. © Harcourt 1. 0.8 LESSON 20.4 Name Subtract Decimals How can you subtract 0.58 from 1.72? Remember to keep digits lined up by place value and their decimal points. Step 1 Step 2 Identify the place value of each digit. 1.72 0.58 1 is in the ones place. 0 is in the ones place. 7 is in the tenths place. 5 is in the tenths place. 2 is in the hundredths place. 8 is in the hundredths place. Line up each digit Step 3 Subtract, starting from the by its place value. right. Regroup if you need to. O T . 0 •. 5 •. 1 •7 H O 2 1 •67 1 . 0 •. 5 1 •. 1 8 T H 2 11 12 8 1 4 So, 1.72 0.58 equals 1.14. Rewrite each problem in the place-value grid. Find the difference. 0.7 0.1 1. O 2. 0.9 T H O •. •. T O T •. •. 0.51 O T 4. 1.91 H •. 18.9 5.2 T O H T •. 7. 52.61 32.84 H T O •. •. •. H •. •. 6. T •. •. 0.85 O •. •. H •. 31.7 24.5 © Harcourt T 3. 1.72 •. •. 5. 0.5 T H •. •. •. Reteach RW111 LESSON 20.5 Name Add and Subtract Decimals How can you add 15.3 and 2.45? Step 1 Write the decimals, keeping the decimal points in a line. T O . 2 •. 4 •. 1 5•3 Step 2 Write an equivalent decimal with a zero in the hundredths place so that both numbers have the same number of digits after the decimal point. T O 5 T H . 2 •. 4 7 •. 7 1 5•3 0 1 Step 3 T H 5 5 Add as you would with whole numbers. Follow the same steps when you subtract. Rewrite each problem in the place-value grid. Add or subtract. 1.34 0.5 O T 2. 21.67 4.2 T H O •. T 5. O H •. •. RW112 O •. Reteach •. 6. 25.2 8.71 T H T •. •. •. T H •. 7 1.57 •. T •. 1.8 0.52 O T H •. •. 4. 5 24.89 •. •. 3. O T H •. •. •. © Harcourt 1. LESSON 20.6 Name Problem Solving Skill Evaluate Reasonableness of Answers Carrie walks 2.5 miles to Sue’s house. Then she and Sue walk 1.6 miles to Joan’s house. All three girls then walk 3.1 miles to Carrie’s house. How many miles did Carrie walk? Complete the chart. Find the actual distance and the estimated distance. Distance in miles between: Actual Estimate Carrie’s and Sue’s house 2.5 3.0 Sue’s and Joan’s house 1.6 2.0 Joan’s and Carrie’s house 3.1 3.0 Total 7.2 8.0 Compare the actual distance to the estimated distance. 7.2 miles is close to the estimate of 8 miles. So the answer is reasonable. Carrie walked 7.2 miles. For 1–4, use the information above. Circle the more reasonable answer. 1. Sue decided to walk home from Carrie’s house. How far did she walk in all? a. 2.5 miles © Harcourt 3. 5. b. 7.2 miles How much closer does Carrie live to Sue than to Joan? a. 0.6 mile Joan decided to walk home from Carrie’s house. How far did she walk in all? a. 6.2 miles 4. b. 5.6 miles Carrie walked to Sue’s house and back home 3 times last week. About how far did she walk in all? a. 18 miles 2. How much closer does Joan live to Sue than to Carrie? a. 4.7 miles 6. b. 3.1 miles b. 1.5 miles How much closer does Sue live to Joan than to Carrie? a. 0.9 miles b. 4.1 miles b. 9 miles Reteach RW113 LESSON 21.1 Name Choose the Appropriate Unit The customary system of measurement uses inches, feet, yards, and miles to measure length, width, distance, and height. You can use the following sentence to help you remember the order of the sizes of these measurements. In Fourth grade You Measure. Inch Foot Yard Mile Use inches to measure small items, such as a pen or your finger. Use feet for greater measurements, such as heights. Use yards to measure distances, such as the length of a football field. Use miles to measure great distances, such as the distance between cities. Name the greater measurement. 1. 234 mi or 234 yd 4. 100 yd or 100 in. 2. 1 in. or 1 ft 3. 45 ft or 45 yd 5. 80 in. or 80 mi 6. 17 ft or 17 mi Choose the most reasonable unit of measure. Write in., ft, yd, or mi. The width of your hand is about 3 8. . 9. 11. 10. . The length of one lap around a running track is 220 The width of your desk is 2 12. . 234 mi or 234 yd notebook paper is 11 RW114 Reteach 14. 1 in. or 1 ft . The length of a piece of Circle the greater measurement. 13. . to Boston is about 200 The length of your classroom is 25 The distance from New York City 15. 45 ft or 45 yd . © Harcourt 7. LESSON 21.2 Name Measure Fractional Parts MATERIALS scissors, paper, inch ruler You can make the following paper model to help identify the marks on a ruler. • Cut a long strip of printer paper. Label the left edge 0 and the right edge 1. • Fold the strip in half four times. • Unfold the strip of paper. You should now have 16 equal sections and 15 folds. Count over 8 folds from 0 and draw a line on that fold. Label this line 12. • Starting again at 0, count over 4 folds and 12 folds. Draw lines on the folds. Label the first one 14 and the second one 34. 1 2 0 0 1 4 1 2 1 1 3 4 1 1 Use a ruler to measure the length to the nearest 4 inch and to the nearest 8 inch. 1. 2. 3. 4. Draw a chain of 3 large paper clips. Make each paper clip 134 inches long. 6. What is the length of your paper clip chain measured to the nearest 14 inch? © Harcourt 5. Reteach RW115 LESSON 21.3 Name Algebra: Change Linear Units To change units, think which measurement is larger. Then think about how the two units relate. 2 yd 24 in. ft ft Yards are larger than feet. There are 3 feet in every yard. To find the number of feet in 2 yards, multiply. Inches are smaller than feet. One foot is 12 inches. To find the number of feet for 24 inches, divide. 236 24 12 2 So, 2 yd 6 ft. Larger Unit to Smaller Unit So, 24 in. 2 ft. Smaller Unit to Larger Unit Number of Miles 1,760 Yards Number of Inches 12 Feet Number of Miles 5,280 Feet Number of Inches 36 Yards Number of Yards 3 Number of Yards 36 Number of Feet 12 1. Feet Inches Inches Number of Feet 3 Yards Number of Yards 1,760 Number of Feet 5,280 Miles Miles List the linear units of measurement from least to greatest. Think about how to change the unit. Write either larger unit to smaller unit or smaller unit to larger unit. Write multiply or divide to tell how to change the units. 9 feet to 3 yards 3. 2 miles to 10,560 feet 4. 36 inches to 1 yard © Harcourt 2. Change the unit. 5. 3 yd RW116 Reteach ft 6. 72 in. ft 7. 3 mi ft 8. 300 ft yd LESSON 21.4 Name Capacity To change units of capacity, think which unit is larger. Then think about how the two units relate. Smaller Unit Larger Unit 2 cups 1 pint 2 pints 1 quart 4 quarts 1 gallon When changing from larger units to smaller units, multiply. When changing from smaller units to larger units, divide. 2 gal 6c qt 248 So, 2 gallons equals 8 quarts. pt 623 So, 6 cups equals 3 pints. Find the equivalent measurement by drawing a picture. 12 c pt 2. 4 qt pt 3. 2 gal c © Harcourt 1. Reteach RW117 LESSON 21.5 Name Weight Lightest Unit Ounce (oz) Heaviest Unit 1 Pound (lb) 16 oz 1 Ton (T) 2,000 lb To change units of weight, think which unit is heavier. Then think about how the two units relate. 3 lb 32 oz oz lb Ounces are lighter than pounds. One pound is 16 ounces. Multiply 3 by 16. Pounds are heavier than ounces. There are 16 ounces in a pound. Divide 32 by 16. 3 16 48 32 16 2 So, 3 pounds equals 48 ounces. So, 32 ounces equals 2 pounds. Think about how to change the unit. Write either lighter unit to heavier unit or heavier unit to lighter unit. Write multiply or divide to tell how to change the units. 1. 2 tons to 4,000 pounds 2. 64 ounces to 4 pounds 3. 1 pound to 16 ounces Circle the more reasonable measurement. 5. 300 T or 300 lb 6. 4 oz or 4 lb © Harcourt 4. 12 oz or 12 lb Write the equivalent measurement. 7. 3 lb RW118 Reteach oz 8. 7 T lb 9. 50,000 lb T LESSON 21.6 Name Problem Solving Strategy Compare Strategies You can use many different strategies to solve a problem. Sometimes one strategy is more helpful or easier to understand than another. Charlotte is making a household cleaner. She mixes together 3 gallons of water and one quart of ammonia. She is going to pour the cleaner into some quart-sized bottles. How many bottles can she fill? You can use the strategies draw a picture and make a table to solve this problem. Draw a Picture Make a Table } water ammonia gallons 1 2 3 quarts 4 8 12 The picture shows that the cleaner will be made with 13 quarts of liquid. The table shows that 3 gallons 12 quarts So, Charlotte can fill 13 bottles. 12 quarts water 1 quart ammonia 13 total quarts. So, Charlotte can fill 13 bottles. Choose a strategy to solve. © Harcourt 1. Sheri is making a household cleaner. She mixes 3 pints of bleach and 2 quarts of water. How many one-pint squeeze bottles can she fill? 2. Bethany is putting a border around the trees in her yard. Each tree needs 12 feet of border. She buys 20 yards of border. How many trees can she surround? Reteach RW119 LESSON 22.1 Name Linear Measure The centimeter, decimeter, and meter are units of length. 1 cm The width of a large paper clip is about 1 centimeter (cm). 1 dm 1m The width of a computer disk is about 1 decimeter (dm). The width of a door is about 1 meter (m). Use the references above to help you decide if the measurement is close to a cm, dm, or m. 1. the thickness of a calculator 2. the width of a small paperback book 3. the height of a counter 5. the thickness of a workbook 4. 6. the length of a large dog the height of a big apple 7. width of a softball a. 1 cm b. 1 dm c. 1m 8. width of a basketball a. 25 cm b. 25 dm c. 25 m 9. height of an average ceiling a. 3 cm b. 3 dm c. 3m length of a bed a. 200 cm b. 200 dm c. 200 m 10. RW120 Reteach © Harcourt Choose the most reasonable unit of measure. Write a, b, or c. LESSON 22.2 Name Algebra: Change Linear Units You have to multiply or divide when you change units. Multiply when you change from a larger unit to a smaller unit. 3 meters ? dm 3 meters ? cm Larger Unit 1 Meter 3 m 10 30 dm 3 m 100 300 cm Smaller Unit 100 Centimeters 10 Decimeters 10 10 Divide when you change from a smaller unit to a larger unit. 300 cm ? dm 300 cm ? m Smaller Unit 100 Centimeters 300 cm 10 30 dm 300 cm 100 3 m 10 Decimeters Larger Unit 1 Meter 10 10 Circle the larger unit. 1. meter or decimeter 2. centimeter or decimeter 3. meter or centimeter Would you multiply by 10 or by 100 to change the larger units to the smaller units? Write 10 or 100. 4. 11 m ? cm 5. 20 dm ? cm 6. 8 m ? dm Would you divide by 10 or by 100 to change the smaller units to the larger units? Write 10 or 100. 7. 180 cm ? dm 8. 200 cm ? m 9. 800 dm ? m © Harcourt Complete the table. Meters 10. 9 13. 17 Centimeters 900 100 11. 12. Decimeters 1,000 170 120 1,200 Reteach RW121 LESSON 22.3 Name Capacity You can measure capacity, or how much liquid a container can hold when it is filled, by using the units milliliter (mL) or liter (L). A box the size of a base-ten ones unit holds 1 milliliter (mL). 1,000 boxes the size of a ones unit hold a liter (L). A small springwater bottle also holds 1,000 mL, or 1 L. Use the pictures above to help you choose the equivalent measurement. Circle a, b, or c. 3 liters ? milliliters a. 2. 3,000 mL 50 L b. c. 5,000 mL 4,000 mL c. 400 mL 10 mL c. 100 mL 3,000 mL c. 30 mL 200 mL c. 2,000 mL 5L 4 mL b. 1,000 mL b. 3 L ? milliliters a. 6. c. 1 L ? milliliters a. 5. 300 mL 4 liters ? milliliters a. 4. b. 5 liters ? milliliters a. 3. 30 mL 300 mL b. 2 L ? milliliters a. 20 mL b. © Harcourt 1. Write the equivalent measurement. 7. 4L mL 8. 1L mL 9. 8L mL 10. 2L mL 11. 5L mL 12. 3L mL RW122 Reteach LESSON 22.4 Name Mass You can measure mass with the units grams (g) and kilograms (kg). A base-ten ones unit has a mass of about1 gram (g). 100 100 100 100 100 100 100 100 100 100 If you put 1,000 ones units in a bag, the bag would have a mass of 1 kilogram (kg), or 1,000 grams. Or you can think of a large container of oatmeal. It has a mass of about 1 kg, or 1,000 g. Circle each object that has a mass greater than 1 kg. 1. an airplane 2. a strand of hair 3. a pencil 4. a bed 5. a full backpack 6. a computer disk Choose the more reasonable unit of measure. Write g or kg. 7. a shirt 9. a refrigerator 10. a big bag of pet food a baby 12. a pair of glasses 11. 8. a photograph Circle the more reasonable unit of measure. spool of thread a. 10 g b. 10 kg 14. car tire a. 20 g b. 20 kg 15. bag of flour a. 5g b. 5 kg 16. hammer a. 2g b. 2 kg 17. scarf a. 25 g b. 25 kg © Harcourt 13. Reteach RW123 LESSON 22.5 Name Problem Solving Strategy Draw a Diagram Row1 Row 2 Row 3 Row 4 Megan is planting rows of watermelon plants. The plants have to be spaced 2 meters apart. Her garden is 8 meters by 8 meters. She will start her first row 1 meter in from the edge of the garden. How many rows of watermelon plants can she plant? • To solve, use grid paper to draw a diagram of the garden. Have the length of a square represent 1 meter. • Since each square represents 1 meter, start the first row 1 space from the edge. Draw the next row 2 meters or 2 spaces over. Continue drawing rows until no more can fit. =1 meter =1 meter So, Megan can plant 4 rows. Draw a diagram to solve. Megan’s watermelon plants need to be spaced 2 meters away from every other watermelon plant. She will plant the first plant in each row 1 meter from the edge of the garden. How many watermelon plants can Megan fit in her 8 meter by 8 meter garden? 2. Tom wants to plant a watermelon garden like Megan’s. He has a square area that is 12 meters by 12 meters. How many watermelon plants can he plant? 3. If Megan and Tom plant marigold plants every 1 meter around the perimeter of their gardens starting at a corner, how many marigold plants will they plant? © Harcourt 1. RW124 Reteach LESSON 23.1 Name Temperature: Fahrenheit You can find the temperature by reading a thermometer. A thermometer is like a vertical number line. The scale of a thermometer is usually marked in intervals of 5° or 10°. As it gets warmer, the alcohol in the thermometer expands, and we say the temperature rises. As it gets colder, the alcohol in the thermometer contracts, and we say the temperature falls. Most household thermometers measure temperatures from about 20°F to 120°F. This thermometer reads 80°F. 3. 112°F Use skip-counting to find the change in temperature. 20°F to 40°F 4. 5. 30°F to 75°F 6. 0°F to 40°F © Harcourt Fill in the thermometer to show the given temperature. 1. 60°F 2. 30°F Estimate the temperature of each of the following. 7. a cup of hot cocoa 8. a hot summer day 9. a warm bath Reteach RW125 LESSON 23.2 Name Temperature: Celsius The customary unit for measuring temperature is degrees Fahrenheit (°F). The metric unit for measuring temperature is degrees Celsius (°C). Water freezes at 0°C. Water boils at 100°C. Room temperature is about 20°C. The greater the temperature, the warmer it is. So, 80°C is warmer than 72°C because 80 > 72. Temperature can be compared by using subtraction. So, 80°C is 8° warmer than 72°C. 80° – 72° 8° Compare the temperatures. Write or to show which is warmer. 1. 90° 70° 50° 30° 10° 10° 90° 70° 50° 30° 10° 10° 80° 60° 40° 20° 0° 80° 60° 40° 20° 0° 2. 90° 70° 50° 30° 10° 10° °C 50°C 90° 70° 50° 30° 10° 10° 80° 60° 40° 20° 0° 80° 60° 40° 20° 0° 3. 90° 70° 50° 30° 10° 10° 10°C 80° 60° 40° 20° 0° °C °C 40°C 90° 70° 50° 30° 10° 10° 80° 60° 40° 20° 0° 30°C 80°C 70°C Write the temperatures. Write or to show which is warmer. Find the change in temperature. 4. 90° 70° 50° 30° 10° 10° 90° 70° 50° 30° 10° 10° 80° 60° 40° 20° 0° °C RW126 Reteach 80° 60° 40° 20° 0° 5. 90° 70° 50° 30° 10° 10° 90° 70° 50° 30° 10° 10° 80° 60° 40° 20° 0° °C 80° 60° 40° 20° 0° 6. 90° 70° 50° 30° 10° 10° 90° 70° 50° 30° 10° 10° 80° 60° 40° 20° 0° °C 80° 60° 40° 20° 0° LESSON 23.3 Name Negative Numbers A number line can help you to compare two numbers. The number that is to the right is the greater number. Compare 5 and 4. Write a number sentence using , , or . -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Since 4 is to the right of 5, 4 5 or, 5 4. Locate each pair of numbers on the number line by drawing dots. Then write a number sentence using or to compare them. 1. 1 and 3 2. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 3. 2 and 7 4 and 2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 4. 1 and 3 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Draw dots for each number. Then order the numbers from least to greatest. 5. 5, 7, 2 6. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 7. 4, 2, 1, 0 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0, 3, 4, 1 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 8. 4, 4, 2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Reteach RW127 LESSON 23.4 Name Problem Solving Skill Make Generalizations Thomas and Tina are planning a trip. A travel agent has given them some information about two locations. Temperature (in °F) Average Monthly Temperature Month Thomas and Tina can make some generalizations about the weather in each location. A generalization is a broad statement made after seeing some specific information. Thomas makes the generalization that Skitown is a colder location than Beachville. Use the graph to answer the questions. Number of Points Ted’s and Al’s Bowling Scores Ted 1. What was Ted’s bowling score in Week 3? 2. What was Al’s bowling score in Week 1? 3. What generalizations can you make about Ted’s and Al’s bowling abilities? 4. What generalization can you make about Al’s bowling scores over the four-week period? RW128 Reteach © Harcourt Al LESSON 24.1 Name Use a Coordinate Grid You can find the point (3,4) on the grid. Step 2 Go straight across to number 3. Step 3 Go straight up to the line labeled 4. The path to (3,4) is traced on the grid. Always go straight across first, and then go straight up. y-axis Step 1 Start at 0. 8 7 6 5 4 3 2 1 (3,4) 0 1 2 3 4 5 6 7 8 x-axis Remember: Across begins with an A, and A comes first in the alphabet. Complete. 1. The numbers in an represent the number of spaces you move across and up to locate a point on the grid. Trace the path from zero to each of the following points on the grid. Plot each point. (3,4) 3. (5,6) 4. (2,3) 5. (6,7) 6. (4,5) 7. (7,8) 10 9 8 7 6 5 4 3 2 1 y-axis 2. 0 1 2 3 4 5 6 7 8 9 10 x-axis Plot each ordered pair on the grid. Connect the points to make a figure. (2,3), (4,1), (7,1), (9,3), (7,5), (4,5) 9. Name the figure you drew. 10 9 8 7 6 5 4 3 2 1 y-axis © Harcourt 8. 0 1 2 3 4 5 6 7 8 9 10 x-axis Reteach RW129 LESSON 24.2 Name Length on the Coordinate Grid You can find the distance between two points on a coordinate grid by counting units or by using subtraction. When finding the horizontal distance between two points, look at the x-coordinates. When finding the vertical distance between two points, look at the y-coordinates. What is the horizontal distance from point (2,3) to (9,3)? y-axis Step 1 Graph the ordered pairs (2,3) and (9,3) on the coordinate plane to the right. Connect the two points. Step 2 Count the units between the two points or subtract the x-coordinates. 9 2 7 So, the horizontal distance between the two points is 7 units. 10 9 8 7 6 5 4 3 2 1 (2, 3) 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 x-axis Find the length of each line segment. 5 4 3 2 1 RW130 10 9 8 7 6 5 4 3 2 1 (4, 8) (4, 3) (5, 6) (5, 2) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x-axis x-axis 10 9 8 7 6 5 4 3 2 1 4. 10 9 8 7 6 5 4 3 2 1 y-axis y-axis 3. 2. y-axis 10 9 8 7 6 5 4 3 2 1 y-axis 1. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x-axis x-axis Reteach (9, 3) LESSON 24.3 Name Use an Equation An equation may have 2 variables. The equation y 5x can be written y 5 x. You can think of this equation as a rule that says: To find the value of y, you must multiply x by 5. If the value of x is 3, the value of y 5 3 or 15. If you need to find several values of y, you can use a function table to organize the data. You choose any number as the value of x. This is called the input. Then you use the equation as a rule to find the output. The output is the value of y. Example: y 5x 2 You choose 1 as the value of x. You use the rule to find y 7. You choose 3 as the value of x. You use the rule to find y 17. You choose 6 as the value of x. You use the rule to find y 32. Find the value of y when x 4. 1. y 6x 3. y 7x 10 Input x Rule y 5x 2 Output y 1 y (5 1) 2 7 3 y (5 3) 2 17 6 y (5 6) 2 32 2. y 9x 3 4. y 3x 8 Use the equation to complete each function table. 5. y 4x 6. y 3x 41 Input x 1 2 3 Rule y 4x Output y Input x 4 5 6 Rule Output Reteach RW131 LESSON 24.4 Name Graph an Equation To graph an equation on a coordinate grid, you first make a function table to find values of x and y. Then you write the values of x and y as ordered pairs and graph the pairs on the grid. Finally, you draw a line to connect the points. Graph the equation y 2x 1. x y 1 1 2 3 3 5 4 7 5 9 y-axis Make a function table. Write the values of x and y as ordered pairs. (1,1), (2,3), (3,5), (4,7), (5,9) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x-axis Graph the ordered pairs on the coordinate grid. Draw a line to connect the points. Graph the equation y x 3. 1. Complete the function table. x y 1 2 3 4 5 Write the values of x and y as ordered pairs. 3. Graph the ordered pairs on the coordinate grid. Draw a line to connect the points. y-axis 2. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x-axis Graph the equation y 6 x. Complete the function table. 1 2 3 4 5 5. Write the values of x and y as ordered pairs. 6. Graph the ordered pairs on the coordinate grid. Draw a line to connect the points. RW132 Reteach 10 9 8 7 6 5 4 3 2 1 0 © Harcourt x y y-axis 4. 1 2 3 4 5 6 7 8 9 10 x-axis LESSON 24.5 Name Problem Solving Skill Identify Relationships You can identify relationships between x and y by looking at a function table and writing a rule for the values of x and y. Look at the function table below and identify the relationship between the input x and the output y. Input, x Output, y 1 3 2 6 3 9 4 12 5 15 6 18 7 21 8 24 9 27 10 30 Rule: Multiply x by 3 to get y. You can also identify relationships by looking at a coordinate grid. y-axis Look at the grid below. What is the relationship between the x values and the y values? Step 1 Write the ordered pairs in a table. Input, x Output, y 1 3 2 4 3 5 4 6 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x-axis Rule: Add 2 to the value of x to get y. 1. Describe the relationship between x and y. 2. Use the graph to complete the function table. Then describe the relationship between x and y. Input, x Output, y 1 2 Input, x Output, y 3 4 1 6 2 12 3 18 y-axis © Harcourt Step 2 Write a rule. 4 24 5 30 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x-axis Reteach RW133 LESSON 25.1 Name Lines, Rays, and Angles F G • A line is straight. It continues in both directions. It does not end. • line FG A A ray is part of a line and has one endpoint. You may think of the sun’s rays, which start at the sun (endpoint) and go on forever into space. B • • ray AB An angle is formed when two rays have the same endpoint. E • There are three types of angles: • D angle C • C A right angle forms a square corner. An acute angle is an angle with a measure less than a right angle. An obtuse angle is an angle with a measure greater than a right angle. Obtuse begins with an O. Think of an Open angle. HINT: • • • right angle acute angle obtuse angle Write the name of each figure. 1. E F • 2. Y Z • • 3. B 4. • • • C • D What kind of angle is each? Write acute, right, or obtuse. • 6. 7. • © Harcourt • 5. 8. 9. • • RW134 10. Reteach • LESSON 25.2 Name Line Relationships • D F • Lines that intersect to form 4 right angles are perpendicular lines. Lines AB and CD are perpendicular lines. G • •E • A C • Lines that never intersect are parallel lines. Lines GH, and LM, and RS are parallel lines. D • • B • All lines that cross each other are intersecting lines. Lines DE and FG intersect. H • M S• G •L • R • • For 1–3, draw perpendicular lines. 2. • • N • • B Y • • M • • A 3. • 1. Z • For 4–5, use the following map. way © Harcourt eS t. St. chuset Lak Massa Foster Tufts S Adams S t. t. d Broa ts Ave . 4. Name the streets parallel to Adams Street. 5. Name the streets perpendicular to Massachusetts Avenue. Reteach RW135 LESSON 25.3 Name Congruent Figures and Motion Congruent figures have the same size and shape. They can be in different positions and still be congruent. Three ways to move a figure are flip, slide, and turn. • You can slide a figure in any direction. The figure looks exactly the same but is in a new position. • You can flip a figure over an imaginary line. The flipped figure is a mirror image. • • You can turn a figure around a point. The point holds the corner of the figure while the rest of the figure rotates around the point. Tell how each figure was moved. Write slide, flip, or turn. 1. 2. 3. 4. Use each original figure to draw figures to show a slide, flip, and turn. Original Slide Flip Turn © Harcourt 5. 6. RW136 Reteach LESSON 25.4 Name Symmetric Figures A figure has rotational symmetry if it can be turned about a central point and still look the same in at least two different positions. This figure has rotational symmetry. Original Position Figure Matches Turning Figure Matches Turning Trace and cut out the figure below. Fold it along the dotted line. When you unfold the figure, you can see that one side is a reflection of the other. When a figure can be folded about a line so that its two parts are identical, the figure has line symmetry. Trace the following figures and turn them to determine whether each figure has rotational symmetry. Write yes or no. 1. 2. 3. © Harcourt Is the dotted line a line of symmetry? Trace and cut out each figure. Fold to check. Write yes or no. 4. 5. 6. Reteach RW137 Name LESSON 25.5 Problem Solving Strategy Make a Model You can make a model to solve some problems. Read the following problem. 1. Underline what you are asked to find. 2. What strategy can you use to solve the problem? 3. How can you use the strategy to solve the problem? 4. Solve the problem with the grid at the right. Make a model and solve. 5. Mike wants to make a larger picture of the figure at the right. Use the grid to help Mike make a larger picture. RW138 Reteach © Harcourt Hannah wants to make a poster for her room. She wants to make a larger picture of the geometric design at the right. How can she make a model to help her? LESSON 26.1 Name Perimeter of Polygons A polygon is a closed plane figure with straight sides. Polygons are named by the number of sides or number of angles. Here are some polygons. TRIANGLE QUADRILATERAL PENTAGON HEXAGON OCTAGON 3 sides 3 angles 4 sides 4 angles 5 sides 5 angles 6 sides 6 angles 8 sides 8 angles 2 cm The perimeter is the distance around a polygon. Add the lengths of all the sides to find the perimeter. P 4 cm 6 cm 2 cm 5 cm P 17 cm 5 cm 4 cm So, the perimeter is 17 cm. 6 cm Find the perimeter. Name the polygon. 1. 2. 2 cm 2 cm 3. 2 cm 2 cm 2 cm 2 cm 4 cm 3 cm 4 cm 2 cm 2 cm 4 cm 4. 1 cm 5. 1 in. 4 in. 3 cm 6. 5 in. 7 in. 8 in. © Harcourt 3 in. 7 in. 3 in. 5 in. 6 in. 5 in. 3 in. 1 in. 12 in. 10 in. 7 in. Reteach RW139 LESSON 26.2 Name Estimate and Find Perimeter Perimeter is the distance around a figure. You already know that you can find the perimeter by adding the lengths of the sides of a figure. 7 in. 3 in. 2 in. 3 in. 1 in. 1 3 2 7 3 10 = 26 The perimeter is 26 inches. 10 in. Rectangles and squares have two pairs of sides that are the same length. You can find the perimeter of a rectangle by multiplying and adding. For squares, multiply the length of one side by 4 to find the perimeter. 10 in. 9 ft 3 ft 3 ft 10 in. 10 in. 9 ft (2 3) (2 9) = 6 18 = 24 ft 10 in. 4 10 = 40 in. Find the perimeter. 1. 2. 3 cm 3 cm 3cm 5m 3m 3 cm 3. 3m 4. 4 ft 4 ft 3 in. 4 in. 4 ft 5m 3 in. 4 ft 5. 6. 3m 3m 7. 9 ft 9 ft 9 ft 3 yd 5 yd Reteach 9 ft 8. 10 in. 3 in. 2 yd 3in. 8 yd 3m RW140 4 in. 10 in. LESSON 26.3 Name Estimate and Find Area Area is the number of square units needed to cover a flat surface. You can find the area of each figure on the geoboard by counting the number of square units. 1 2 3 4 5 6 1 unit 2 1 unit 2 1 The area is 6 square units. The area is 2 square units. 1 2 3 4 5 6 7 8 The area is 8 square units. You can find the area of a rectangle by using multiplication. You multiply the length times the width of a rectangle to find the area. Write your answer in square units, such as sq ft or sq yd. l w = A length width = area 7 ft 3 ft = 21 sq ft 7 ft 3 ft Find the area. 1. 2. 5. 3. 6. 6 in. 3 in. 4. 7. 4 yd 3m 8m 4 yd Reteach RW141 LESSON 26.4 Name Relate Area and Perimeter Perimeter is the distance around a figure. Perimeter is measured in linear units. Area is the number of square units needed to cover a flat surface. Area is measured in square units. Figures with the same area can have different perimeters. Area 20 sq units Perimeter 22 units Area 20 sq units Perimeter 24 units Figures with the same perimeter can have different areas. Area 25 sq units Perimeter 28 units Area 16 sq units Perimeter 28 units Find the area and perimeter of each figure. 1. 2. 3. Find the area of each figure. Then draw another figure that has the same area but a different perimeter. 5. © Harcourt 4. area RW142 Reteach area LESSON 26.5 Name Relate Formulas and Rules You can use formulas, or rules, to find unknown lengths when you know the area or perimeter and one or more sides. Formula: Perimeter a b c EXAMPLE A EXAMPLE B Find the unknown length. The perimeter is 22 inches. 6 in. c b 7 in. Pabc 22 a 6 7 22 a 13 22 13 a 9a The length is 9 inches. Find the unknown length. The perimeter is 94 meters. 22 m a 34 m c b a Pabc 94 a 22 34 94 a 56 94 56 a 38 a The length is 38 meters. Formula: Area l w EXAMPLE C EXAMPLE D Find the unknown width. The area is 64 square yards. 16 yd Alw 64 16 w 64 16 w 4w The width is 4 yards. Find the unknown length. The area is 44 square miles. 4 mi w Alw 44 l 4 44 4 w 11 w The length is 11 miles. l Find the unknown length in each triangle. 1. 2. 5 ft 16 cm ? 3. 4 ft 19 cm Perimeter 47 cm © Harcourt ? ? 3 mi 3 mi Perimeter 12 ft Perimeter 9 mi Complete for each rectangle. 4. Area 144 sq. m Length 12 m Width m 5. Area 84 sq. in. Width 6 in. Length 6. in. Area 360 sq. cm Length 15 cm Length Reteach cm RW143 LESSON 26.6 Name Problem Solving Strategy Find a Pattern 4 cm 3 cm You can find patterns to solve long problems quickly. Read the following problem. Kyle has several different computer chips. Some of his computer chips are twice as long as others. He wants to know how the area changes as the length of the computer chips are doubled. How do the areas change? 1. Underline what you are asked to find. 2. What strategy can you use to solve the problem? 3. Complete. Width Length Computer chip A 2 2 Computer chip C 3 4 Computer chip B 2 4 Computer chip D 3 8 Width Length Computer chip E 4 4 Computer chip G 6 4 Computer chip F 4 8 Computer chip H 6 8 Width Length Area Width Length Area Area Use find a pattern and solve. © Harcourt 4. Area 8 cm 3 cm RW144 Reteach LESSON 27.1 Name Faces, Edges, and Vertices Polygons are two-dimensional figures because they have length and width. Solid figures are three-dimensional figures because they have length, width, and height. vertex edge face You can identify solid figures by the plane figures that are their faces, and by the number of faces, edges, and vertices they have. Cube Rectangular Prism Triangular Prism Triangular Pyramid Square Pyramid Some solid figures have curved surfaces. Cylinder Cone Sphere © Harcourt Which solid figure do you see in each? 1. 2. 3. 4. 5. 6. Reteach RW145 LESSON 27.2 Name Patterns for Solid Figures You can make a three-dimensional figure from a two-dimensional pattern called a net. Look at the net at the right. Can you see how it forms the three-dimensional rectangular prism? Take another look at the net, and count the faces. 4 Since the three-dimensional rectangular prism has 6 faces, the net must also have 6 faces. 1 2 3 5 6 Draw a line from the three-dimensional figure to its net. 2. 3. 4. 5. 6. 7. 8. © Harcourt 1. RW146 Reteach LESSON 27.3 Name Estimate and Find Volume of Prisms Cubic unit The measure of the space that a solid figure occupies is called volume. Volume is measured in cubic units. You can use what you know about unit cubes to estimate volume. These are two ways to find the volume of this rectangular prism. One way: You can count the number of cubic units. Another way: You can find the length, width, and height of the rectangular prism in cubes. Then multiply to find the volume in cubic units. Step 2 Step 3 Step 4 Find the length. Count the number of cubes in one row. Find the width. Count the number of rows in one layer. Find the height. Count the number of layers in the prism. Multiply length width height to find the volume. ↔ Step 1 2 4 → The length is 4 cubes. 2 → The width is 2 cubes. The height is 2 cubes. 4 cubes 2 cubes 2 cubes 16 cubes. So, the volume is 16 cubic units. Find the volume. 2. 3. © Harcourt 1. 4. What is the volume of a solid figure that is 4 cubes by 7 cubes, by 9 cubes? Reteach RW147 LESSON 27.4 Name Problem Solving Skill: Too Much/Too Little Information The cost of a box of 400 cotton swabs is $4.35. Each box 10cm has a height of 4 cm, a length of 18 cm, and a width of 4cm 10 cm. What is the volume of the box of cotton swabs? 18cm Step 1 Step 2 What does the problem ask you to find? What information do you need to solve the problem? Step 3 Step 4 Decide if there is too much or not enough information. Explain. Solve the problem, if possible. 1. There are 12 boxes of paper in a case. Each case is 6 feet in length and 4 feet in height. What is the volume of each case? 2. There are 24 tops in a package. Each package has a height of 5 inches, a width of 6 inches, and a length of 10 inches. What is the volume of one package? 3. Ms. Maguire has a cardboard box that is 1 meter wide and 4 meters long. What is the volume of Ms. Maguire’s cardboard box? 4. There are 3 people in the pool. The pool is 10 feet deep, 50 feet long, and 20 feet wide. What is the volume of the pool? RW148 Reteach © Harcourt Decide if the problem has too much information or too little information. Then solve the problem, if possible. LESSON 28.1 Name Turns and Degrees The unit used to measure an angle is a degree (°). There are 360° in a circle. The circle at the right shows and XB . The angle between them is 20°. two rays, XA A X B Turning XB around the circle results in angles of different sizes. A complete turn around the circle is 360°. Angle measurements can be related to a complete turn (360°). 3 4 a turn (270°) 1 2 1 4 a turn (180°) a turn (90°) 1 4 1 2 3 4 Tell whether the rays on the circle show a , , or turn. 1. 2. 3. © Harcourt Tell whether the rays have been turned 90°, 180°, 270°, or 360°. 4. 5. 6. 7. 8. 9. Reteach RW149 LESSON 28.2 Name Measure Angles Use a protractor to measure the size of the opening of an angle. The scale on a protractor is marked from 0° to 180°. Measure angle ABC with a protractor. Step 1 Step 2 Step 3 Place the center of the protractor on the vertex of the angle. Line up the center point and the 0° mark on the protractor with one ray of the angle. Read the measure of the angle where the other ray passes through the scale of the protractor. A 0 12 80 70 60 50 20 C 13 0 0 12 80 70 Write angle measure in (°). 60 50 14 40 0 20 180 170 160 30 vertex 10 13 0 B C 0 0 12 110 100 90 A 80 70 60 50 14 40 0 B 90 0 10 180 170 160 30 15 0 40 A 100 110 15 0 0 90 14 0 13 100 110 20 180 170 160 30 15 0 ray 10 0 B The measure of ABC 60°. Use a protractor to measure the angle between the two rays. Remember to line up the center mark and the 0° mark on the protractor with one of the rays. 1. 2. 3. A C B A B ABC A B C ABC 4. C ABC 5. 6. A A A B ABC RW150 Reteach C B ABC C B ABC C C LESSON 28.3 Name Circles A circle is named by its center point. All the points on the circle are the same distance from the center point. A chord is a line segment that has its endpoints on the circle. •F Chord EF E C • • diameter CD A •D • radius AB A radius of a circle is a line segment from the center to any point on the circle. B• Circle A A diameter of a circle is a chord that passes through the center. For 1–4, use Circle L and a centimeter ruler. 1. 2. The center of the circle is point . Name each radius of the circle shown. Each radius measures cm. M • L N • •O • Circle L 3. A diameter of the circle is line segment . The diameter measures A diameter is also a cm. . 4. Name three points on the circle. 5. Draw a circle. Label the center point R. Draw a radius RS. Draw a diameter TV. S • R Measure the diameter. Reteach • • V • Measure the radius. T RW151 LESSON 28.4 Name Circumference The measure of the distance around a circle is its circumference. Since a circle is not a polygon, its circumference cannot be measured easily with a ruler. The circumference of a circle is about 3 times the diameter. If you know the diameter of the circle, you can estimate its circumference. 9 in. Estimate the circumference of the circle. diameter 9 inches The circumference is about 3 9 in., or about 27 in. Estimate each circumference. 2. 5. 6. 13 ft 12 in. 7. 8. 9. 11. 52 in. RW152 Reteach 9 cm 6m 22 yd 10. 16 mi 11 m 2 cm 4. 3. 8 cm 12. 63 ft 51 mi © Harcourt 1. LESSON 28.5 Name Classify Triangles Triangles can be classified according to the lengths of their sides. 6 in. 5 in. 5 in. 5 in. 5 in. 5 in. 5 in. 7 in. 6 in. A triangle with 3 sides congruent is an equilateral triangle. A triangle with 2 sides congruent is an isosceles triangle. A triangle with no sides congruent is a scalene triangle. Classify each triangle by the lengths of its sides. Write isosceles, scalene, or equilateral. 1. 2. 6 in. 3. 8 ft 6 in. 10 ft 6 in. 4. 3 mi 3 mi 5. 6. 1 ft 1 ft 4 cm 6 cm 3 cm 1 ft 7 cm © Harcourt 3 mi 8 ft 4 cm 4 cm 11 yd 7. 8. 9. 5 cm 4 cm 10 yd 4 in. 10 yd 2 in. 3 in. 6 cm Reteach RW153 LESSON 28.6 Name Classify Quadrilaterals A quadrilateral has four sides and four angles. Quadrilaterals are grouped by their angles, by the length of their sides, and by any parallel sides. The table will help you compare different quadrilaterals. Grouping Side Lengths Parallel Sides Angles General Quadrilaterals 4 different lengths 0 pairs parallel sides angles vary Trapezoids lengths can vary 1 pair parallel sides angles vary Parallelograms opposite sides congruent 2 pairs parallel sides 2 acute and 2 obtuse, or 4 right Rectangles opposite sides congruent 2 pairs parallel sides 4 right Rhombuses 4 sides congruent 2 pairs parallel sides 2 acute, 2 obtuse, or 4 right Squares 4 sides congruent 2 pairs parallel sides 4 right Match the drawing with as many terms as possible. 1. 2. 3. 4. 5. 6. rhombus D. rectangle B. trapezoid E. parallelogram C. A. square F. general quadrilateral 7. How are rhombuses and squares alike? 8. How are rhombuses and squares different? RW154 Reteach © Harcourt For 7–8, use the table above. LESSON 28.7 Name Problem Solving Strategy Draw a Diagram Ms. Mellot wants her students to sort the figures shown below according to those that have line symmetry and those that have rotational symmetry. How should the students sort the figures? Figure A Figure B Figure C Figure D Figure E 1. What you are asked to do? 2. What information will you use? 3. What strategy can you use to solve the problem? 4. What kind of diagram can you make? Figure F A Venn diagram uses circles to show relationships among different sets of things. © Harcourt Use these 2 circles to make a Venn diagram. Label one circle Line Symmetry and the other Rotational Symmetry. Then sort the figures by writing the letters A–F in the correct circles. Line Symmetry Rotational Symmetry A, B, D C, E, F Reteach RW155 LESSON 29.1 Name Record Outcomes When you toss one coin, there are 2 possible ways the coin can land: heads or tails. When you toss two coins, there are 4 possible ways the coins can land. POSSIBLE WAYS FOR TWO COINS TO LAND First Coin Second Coin There is 1 way to get 2 heads. → heads heads → heads tails → tails heads There is 1 way to get 2 tails. → tails tails There are 2 ways to get 1 head and 1 tail. When you spin the pointer on this spinner once, there are 3 possible numbers you can get: 2, 3, or 4. Complete the table to show all the possible sums you can get if you spin the pointer twice and add the numbers. 2 3 4 POSSIBLE SUMS FOR TWO SPINS Second Spin Sum 1. 2 2 22 2. 2 3 23 3. 2 4. 3 5. 3 6. 3 7. 4 8. 4 9. 4 10. 2 Look at the sums. Which sum is found most often? RW156 Reteach © Harcourt First Spin LESSON 29.2 Name Tree Diagrams Making a tree diagram can help you determine the possible outcomes for a probability experiment. Sharon has a coin and a spinner divided into two sections: red and yellow. She will toss the coin and spin the pointer on the spinner. What are the possible outcomes? How many are there? Spinner red yellow Coin heads tails heads tails Outcomes red and heads red and tails yellow and heads yellow and tails So, there are 4 possible outcomes. Complete the tree diagram to solve. 1. Jerome is conducting a probability experiment with a coin and a bag of marbles. He has 3 marbles in the bag: 1 red, 1 purple, and 1 brown. He will replace the marble after each turn. How many possible outcomes are there for this experiment? What are they? Marbles Coin Outcomes heads red and heads red tails purple © Harcourt brown There are 2. possible outcomes. Sarah has 10¢. How many different outcomes of coins could she have? List the outcomes. Reteach RW157 LESSON 29.3 Name Problem Solving Strategy Make an Organized List You can use a table to help you make an organized list of all the possible outcomes of an experiment. Juanita is making airplane reservations. She can choose a window seat, a middle seat, or an aisle seat. She can choose regular or vegetarian meals. In how many different ways can she plan her flight? Make a table to help you see all the possible outcomes. AIRPLANE RESERVATIONS SEAT MEAL window regular window vegetarian middle regular middle vegetarian aisle regular aisle vegetarian 1. Name the choices Juanita has. 2. How many outcomes are there? 3. Charisse is choosing new eyeglasses. She can choose metal or plastic frames, and can have clear or tinted lenses. List all the possible choices. RW158 Reteach 4. Scott is redecorating his room. He can either paint or wallpaper, and have carpeting, tile, or rugs. In how many different ways can he decorate his room? © Harcourt Make an organized list to solve. LESSON 29.4 Name Predict Outcomes of Experiments Look at the spinner. 0 30 Is spinning a number with 2 as a digit a likely or an unlikely event? 6 24 To find out if an event is likely or unlikely, list all the numbers you could spin. 18 13 • If the event describes half or more than half of the numbers, it is likely. • If the event describes fewer than half of the numbers, it is unlikely. Numbers you could spin: Does it have 2 as a digit? 0 no 6 no 13 no 18 no 24 yes 30 no The event describes fewer than half of the numbers. So, it is unlikely you will spin a number with 2 as a digit. For Problems 1–4, use the spinner above. Decide whether the event describes each number. Write yes or no. © Harcourt 0 1. a number greater than zero 2. a 2-digit number 3. a number less than 10 4. a number divisible by 3 6 13 18 24 30 For Problems 5–8 use the spinner above. Tell whether each event is likely or unlikely. 5. a number greater than zero 6. a 2-digit number 7. a number less than 10 8. a number divisible by 3 Reteach RW159 LESSON 30.1 Name Probability as a Fraction Probability can be described by a number written as a fraction. Probability the number of favorable outcomes total possible outcomes Using the spinner at the right, what is the probability of the pointer stopping on a circle? • List all the 4 outcomes possible outcomes. • Identify the outcomes 1 circle that are circles. • Record the probability as a fraction. number of circles Probability of circle 1 4 total number of shapes Use the spinner at the right. Circle the favorable outcomes. Then write the probability as a fraction. The first one has been done for you. What is the probability of the pointer stopping on a ball? 2. What is the probability of the pointer stopping on something that is not a ball? 3. What is the probability of the pointer stopping on a football? 4. What is the probability of the pointer stopping on a ball or a baseball bat? RW160 Reteach 3 5 © Harcourt 1. LESSON 30.2 Name More About Probability A box contains 4 black marbles and 4 white marbles. Experiment: Shake the box, and pull a marble. Record the color; then replace the marble. There are two possible outcomes for this experiment. • The marble is black. • The marble is white. Tom conducts the experiment 16 times. He predicts that the outcomes will be 8 black and 8 white marbles. He records the actual results in a table. MARBLE EXPERIMENT Possible outcomes Predicted frequency Black White 8 8 Actual frequency 1. What is the mathematical probability of drawing a white marble? 2. Based on the experiment, what is the probability of drawing a white marble? © Harcourt 3. How does this compare to the mathematical probability? 4. If Tom were to pull a marble 20 times, how many white marbles would he predict if he used mathematical probability? Reteach RW161 LESSON 30.3 Name Test for Fairness Fairness in a game means that one player is as likely to win as another. To see if a game is fair, compare the probabilities of winning and not winning. A game is fair if the probabilities are equal. Game Description 12 8 • One player tries to spin even numbers. • One player tries to spin odd numbers. Is the game fair if this spinner is used? probability of spinning an even number 3 5 2 probability of spinning an odd number 5 The probabilities are not equal, so the game is not fair. 11 10 9 Use the spinner above. Write the probabilities. Write fair or not fair to tell if the game is fair. Game Description • One player wins if the pointer stops on 9 or 12. • One player wins if the pointer stops on 8 or 12. 1. The probability of spinning 9 or 12 is . 2. The probability of spinning 8 or 12 is . 3. The game is . Write the letter of the spinner that makes the game fair. 4. Game Description B C 5. Game Description • One player tries to spin a rectangle. • One player tries to spin a triangle. • One player tries to spin a shape with 4 or fewer sides. • One player tries to spin a shape with 4 or more sides. Spinner Spinner RW162 Reteach © Harcourt A LESSON 30.4 Name Problem Solving Skill Draw Conclusions Daryl, Tat, Judy, and Nora have written their first initials in a spinner with 8 equal sections to play a game. Read each clue to help you write the letters on the spinner. ANALYZE CONCLUDE The probability of the pointer stopping on a D is 38. 3 out of 8 sections have a D in them. The sections with a T take up 14 of the spinner. 1 4 of 8 is 2, so 2 sections have a T in them. There are the same number of sections with J as with T. sections have J in them. Each remaining section has an N. sections have N in them. Use what you concluded to write the letters on the spinner. 1. Is the spinner fair? Explain. © Harcourt ge purple oran ge d re d re purple gre en blue 2. or an blue Kelsey made a spinner with 10 equal sections. She colored 2 opposite sections blue. She colored the section to the left of each blue section orange. She made 120 of the spinner green and 120 of the spinner red. The rest of the sections she colored purple. Did Kelsey make a fair spinner? gree n Read the problem. Draw conclusions from each fact and then color the spinner to solve. Use crayons. How many sections are each color? blue orange red purple green Write is or is not to complete the sentence. 3. The spinner fair. Reteach RW163