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Reteach Workbook PUPIL EDITION G ra d e 5 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 0-15-320813-9 1 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 student pages from this publication in classroom quantities for instructional use and not for resale. CONTENTS 4.5 Problem Solving Skill: Estimate or Find Exact Answer . . . . . . . . . . . . . . 20 Unit 1: USE WHOLE NUMBERS AND DECIMALS Chapter 1: Place Value of Whole Numbers 1.1 Understand Place Value . . . . . . . . . 1.2 Millions and Billions . . . . . . . . . . . . 1.3 Compare Numbers . . . . . . . . . . . . . 1.4 Order Numbers . . . . . . . . . . . . . . . . 1.5 Problem Solving Skill: Use a Table . . . . . . . . . . . . . . . . . . . . Chapter 2: Place Value of Decimals 2.1 Tenths and Hundredths . . . . . . . . . 2.2 Thousandths and Ten-Thousandths . . . . . . . . . . . . . . . 2.3 Equivalent Decimals . . . . . . . . . . . . 2.4 Compare and Order Decimals . . . 2.5 Problem Solving Skill: Draw Conclusions . . . . . . . . . . . . . . Chapter 3: Add and Subtract Whole Numbers 3.1 Round Whole Numbers . . . . . . . . . 3.2 Estimate Sums and Differences . . 3.3 Add and Subtract Whole Numbers . . . . . . . . . . . . . . . . . . . . . . 3.4 Choose a Method . . . . . . . . . . . . . . 3.5 Problem Solving Strategy: Use Logical Reasoning . . . . . . . . . . Unit 2: ALGEBRA, DATA, AND GRAPHING 1 2 3 4 5 Chapter 5: Algebra: Use Addition 5.1 Expressions and Variables . . . . . . . 5.2 Write Equations . . . . . . . . . . . . . . . . 5.3 Solve Equations . . . . . . . . . . . . . . . . 5.4 Use Addition Properties . . . . . . . . . 5.5 Problem Solving Skill: Use a Formula . . . . . . . . . . . . . . . . . 21 22 23 24 25 6 7 8 9 10 11 12 13 14 15 Chapter 4: Add and Subtract Decimals 4.1 Round Decimals . . . . . . . . . . . . . . . . 16 4.2 Estimate Decimal Sums and Differences . . . . . . . . . . . . . . . . . . . . 17 4.3 Add and Subtract Decimals . . . . . . 18 4.4 Zeros in Subtraction . . . . . . . . . . . . 19 Chapter 6: Algebra: Use Multiplication 6.1 Write and Evaluate Expressions . . 26 6.2 Order of Operations . . . . . . . . . . . . 27 6.3 Functions . . . . . . . . . . . . . . . . . . . . . . 28 6.4 Problem Solving Strategy: Write an Equation . . . . . . . . . . . . . . 29 6.5 Use Multiplication Properties . . . . 30 6.6 The Distributive Property . . . . . . . 31 Chapter 7: Analyze Data and Graphs 7.1 Collect and Organize Data . . . . . . 7.2 Find the Mean . . . . . . . . . . . . . . . . . 7.3 Find the Median and Mode . . . . . . 7.4 Problem Solving Strategy: Make a Graph . . . . . . . . . . . . . . . . . . 7.5 Analyze Graphs . . . . . . . . . . . . . . . . 32 33 34 35 36 Chapter 8: Make Graphs 8.1 Choose a Reasonable Scale . . . . . . 37 8.2 Problem Solving Strategy: Make a Graph . . . . . . . . . . . . . . . . . . 38 8.3 Graph Ordered Pairs . . . . . . . . . . . 39 8.4 Make Line Graphs . . . . . . . . . . . . . . 40 8.5 Histograms . . . . . . . . . . . . . . . . . . . . 41 8.6 Choose the Appropriate Graph . . 42 Unit 3: MULTIPLY WHOLE NUMBERS AND DECIMALS Chapter 9: Multiply Whole Numbers 9.1 Estimation: Patterns in Multiples . . . . . . . . . . . . . . . . . . . . . 9.2 Multiply by 1-Digit Numbers . . . . 9.3 Multiply by 2-Digit Numbers . . . . 9.4 Choose a Method . . . . . . . . . . . . . . 9.5 Problem Solving Skill: Evaluate Answers for Reasonableness . . . . Chapter 10: Multiply Decimals 10.1 Multiply Decimals and Whole Numbers . . . . . . . . . . . . . . . . . . . . . . 10.2 Algebra: Patterns in Decimal Factors and Products . . . . . . . . . . . 10.3 Model Decimal Multiplication . . . 10.4 Place the Decimal Point . . . . . . . . 10.5 Zeros in the Product . . . . . . . . . . . 10.6 Problem Solving Skill: Make Decisions . . . . . . . . . . . . . . . . 43 44 45 46 47 48 49 50 51 52 53 Unit 4: DIVIDE WHOLE NUMBERS AND DECIMALS Chapter 11: Divide by 1-Digit Divisors 11.1 Estimate Quotients . . . . . . . . . . . . 11.2 Divide 3-Digit Dividends . . . . . . . . 11.3 Zeros in Division . . . . . . . . . . . . . . . 11.4 Choose a Method . . . . . . . . . . . . . . 11.5 Algebra: Expressions and Equations . . . . . . . . . . . . . . . . . . . . . 11.6 Problem Solving Skill: Interpret the Remainder . . . . . . . . 12.3 12.4 12.5 12.6 Divide by 2-Digit Divisors . . . . . . . Correcting Quotients . . . . . . . . . . Practice Division . . . . . . . . . . . . . . . Problem Solving Strategy: Predict and Test . . . . . . . . . . . . . . . Chapter 13: Divide Decimals by Whole Numbers 13.1 Algebra: Patterns in Decimal Division . . . . . . . . . . . . . . . . . . . . . . . 13.2 Decimal Division . . . . . . . . . . . . . . . 13.3 Divide Decimals by Whole Numbers . . . . . . . . . . . . . . . . . . . . . . 13.4 Problem Solving Strategy: Compare Strategies—Work Backward or Draw a Diagram . . . . 13.5 Divide to Change a Fraction to a Decimal . . . . . . . . . . . . . . . . . . 62 63 64 65 66 67 68 69 70 Chapter 14: Divide Decimals by Decimals 14.1 Algebra: Patterns in Decimal Division . . . . . . . . . . . . . . . . . . . . . . . 71 14.2 Divide with Decimals . . . . . . . . . . . 72 14.3 Decimal Division . . . . . . . . . . . . . . . 73 14.4 Problem Solving Skill: Choose the Operation . . . . . . . . . . 74 Unit 5: NUMBER THEORY AND FRACTIONS 54 55 56 57 58 59 Chapter 12: Divide by 2-Digit Divisors 12.1 Algebra: Patterns in Division . . . . . 60 12.2 Estimate Quotients . . . . . . . . . . . . 61 Chapter 15: Number Theory 15.1 Divisibility . . . . . . . . . . . . . . . . . . . . 15.2 Multiples and Least Common Multiples . . . . . . . . . . . . . . . . . . . . . 15.3 Greatest Common Factor . . . . . . . 15.4 Problem Solving Skill: Identify Relationships . . . . . . . . . . . . . . . . . . 15.5 Prime and Composite Numbers . . 15.6 Introduction to Exponents . . . . . . 15.7 Evaluate Expressions with Exponents . . . . . . . . . . . . . . . . . . . . 15.8 Exponents and Prime Factors . . . . 75 76 77 78 79 80 81 82 Chapter 16: Fraction Concepts 16.1 Relate Decimals to Fractions . . . 16.2 Equivalent Fractions . . . . . . . . . . . 16.3 Compare and Order Fractions . . 16.4 Simplest Form . . . . . . . . . . . . . . . . 16.5 Understand Mixed Numbers . . . . 16.6 Problem Solving Strategy: Make a Model . . . . . . . . . . . . . . . . 83 84 85 86 87 88 Unit 6: OPERATIONS WITH FRACTIONS Chapter 17: Add and Subtract Fractions 17.1 Add and Subtract Like Fractions . . . . . . . . . . . . . . . . . . . . .89 17.2 Add Unlike Fractions . . . . . . . . . .90 17.3 Subtract Unlike Fractions . . . . . . .91 17.4 Estimate Sums and Differences . . . . . . . . . . . . . . . . . . .92 17.5 Use Least Common Denominators . . . . . . . . . . . . . . . .93 17.6 Add and Subtract Unlike Fractions . . . . . . . . . . . . . . . . . . . . .94 17.7 Problem Solving Strategy: Work Backward . . . . . . . . . . . . . . .95 19.3 Multiply Fractions and Mixed Numbers . . . . . . . . . . . . . . . . . . . . .103 19.4 Multiply with Mixed Numbers . . . . . . . . . . . . . . . . . . . . .104 19.5 Problem Solving Skill: Sequence and Prioritize Information . . . . . .105 Chapter 20: Divide Fractions 20.1 Divide Fractions . . . . . . . . . . . . . . .106 20.2 Reciprocals . . . . . . . . . . . . . . . . . . .107 20.3 Divide Whole Numbers by Fractions . . . . . . . . . . . . . . . . . .108 20.4 Divide Fractions . . . . . . . . . . . . . . .109 20.5 Problem Solving Strategy: Solve a Simpler Problem . . . . . . .110 Unit 7: ALGEBRA AND GEOMETRY Chapter 21: Algebra: Integers 21.1 Integers . . . . . . . . . . . . . . . . . . . . . .111 21.2 Compare and Order Integers . . .112 21.3 Add Integers . . . . . . . . . . . . . . . . . .113 21.4 Subtract Integers . . . . . . . . . . . . . .114 21.5 Subtract Integers . . . . . . . . . . . . . .115 21.6 Problem Solving Strategy: Draw a Diagram . . . . . . . . . . . . . . .116 Chapter 18: Add and Subtract Mixed Numbers 18.1 Add Mixed Numbers . . . . . . . . . . .96 18.2 Subtract Mixed Numbers . . . . . . .97 18.3 Subtraction with Renaming . . . . .98 18.4 Practice with Mixed Numbers . . . . . . . . . . . . . . . . . . . . .99 18.5 Problem Solving Skill: Multistep Problems . . . . . . . . . . . .100 Chapter 22: Geometry and the Coordinate Plane 22.1 Graph Relationships . . . . . . . . . . .117 22.2 Graph Integers on the Coordinate Plane . . . . . . . . . . . . .118 22.3 Use an Equation to Graph . . . . . .119 22.4 Problem Solving Skill: Relevant or Irrelevant Information . . . . . .120 Chapter 19: Multiply Fractions 19.1 Multiply Fractions and Whole Numbers . . . . . . . . . . . . . . . . . . . . .101 19.2 Multiply a Fraction by a Fraction . . . . . . . . . . . . . . . . . . . . . .102 Chapter 23: Plane Figures 23.1 Lines and Angles . . . . . . . . . . . . . .121 23.2 Measure and Draw Angles . . . . . .122 23.3 Angles and Polygons . . . . . . . . . . .123 23.4 Circles . . . . . . . . . . . . . . . . . . . . . . .124 23.5 Congruent and Similar Figures . .125 23.6 Symmetric Figures . . . . . . . . . . . .126 23.7 Problem Solving Strategy: Find a Pattern . . . . . . . . . . . . . . . . .127 Chapter 24: Classify Plane and Solid Figures 24.1 Triangles . . . . . . . . . . . . . . . . . . . . .128 24.2 Quadrilaterals . . . . . . . . . . . . . . . .129 24.3 Algebra: Transformations . . . . . . .130 24.4 Solid Figures . . . . . . . . . . . . . . . . . .131 24.5 Draw Solid Figures from Different Views . . . . . . . . . . . . . . .132 24.6 Problem Solving Skill: Make Generalizations . . . . . . . . .133 Unit 8: MEASUREMENT Chapter 25: Customary and Metric Systems 25.1 Customary Length . . . . . . . . . . . .134 25.2 Metric Length . . . . . . . . . . . . . . . .135 25.3 Change Linear Units . . . . . . . . . . .136 25.4 Customary Capacity and Weight . . . . . . . . . . . . . . . . . . .137 25.5 Metric Capacity and Mass . . . . . .138 25.6 Time . . . . . . . . . . . . . . . . . . . . . . . . .139 25.7 Problem Solving Strategy: Make a Table . . . . . . . . . . . . . . . . .140 Chapter 26: Perimeter and Area 26.1 Perimeter . . . . . . . . . . . . . . . . . . . .141 26.2 Algebra: Circumference . . . . . . . .142 26.3 Algebra: Area of Squares and Rectangles . . . . . . . . . . . . . . . . . . .143 26.4 Relate Perimeter and Area . . . . .144 26.5 Algebra: Area of Triangles . . . . . .145 26.6 Algebra: Area of Parallelograms . . . . . . . . . . . . . . . .146 26.7 Area of Irregular Figures . . . . . . .147 26.8 Problem Solving Strategy: Solve a Simpler Problem . . . . . . .148 Chapter 27: Surface Area and Volume 27.1 Nets for Solid Figures . . . . . . . . . .149 27.2 Surface Area . . . . . . . . . . . . . . . . . .150 27.3 Algebra: Volume . . . . . . . . . . . . . .151 27.4 Measure Perimeter, Area, and Volume . . . . . . . . . . . . . . . . . .152 27.5 Problem Solving Skill: Use a Formula . . . . . . . . . . . . . . . .153 Unit 9: RATIO, PERCENT, AND PROBABILITY Chapter 28: Ratio 28.1 Understand Ratios . . . . . . . . . . . .154 28.2 Express Ratios . . . . . . . . . . . . . . . .155 28.3 Ratios and Proportions . . . . . . . .156 28.4 Scale Drawings . . . . . . . . . . . . . . . .157 28.5 Problem Solving Skill: Too Much/Too Little Information . . .158 Chapter 29: Percent 29.1 Understand Percent . . . . . . . . . . .159 29.2 Relate Decimals and Percents . . . . . . . . . . . . . . . . . . . . .160 29.3 Relate Fractions, Decimals, and Percents . . . . . . . . . . . . . . . . . .161 29.4 Find a Percent of a Number . . . .162 29.5 Mental Math: Percent of a Number . . . . . . . . . . . . . . . . . . . .163 29.6 Problem Solving Strategy: Make a Graph . . . . . . . . . . . . . . . .164 29.7 Compare Data Sets . . . . . . . . . . . .165 Chapter 30: Probability 30.1 Probability Experiments . . . . . . . .166 30.2 Outcomes . . . . . . . . . . . . . . . . . . . .167 30.3 Probability Expressed as a Fraction . . . . . . . . . . . . . . . . . . . .168 30.4 Compare Probabilities . . . . . . . . .169 30.5 Problem Solving Strategy: Make an Organized List . . . . . . . .170 LESSON 1.1 Name 0 9 0 0 1 3 9 1 , , , , , , 0 0 0 5 5 Ones 3 Hundreds Ones Ones Tens Hundreds Thousands The number 391,568 may be easier to read and write if you use a place-value chart. Tens Understand Place Value 0 0 0 0 6 6 0 0 0 0 0 8 Standard form: 391,568 Expanded form: 300,000 90,000 1,000 500 60 8 Word form: Three hundred ninety-one thousand, five hundred sixty-eight Write the number in the place-value chart. Then write the number in expanded form. 1. 716,583 2. 78,056 Thousands © Harcourt H T Ones O H T Thousands O H T Ones O H T O Use the place-value chart to help you write the value of the bold-faced digit. 3. 58,346 4. 723,308 5. 468,005 6. 420,822 Reteach RW1 LESSON 1.2 Name Millions and Billions You can use a place-value chart to help you read and write greater numbers such as 721,306,984. Millions H 7 T 0 2 Thousands O 0 0 1 , , , H 0 0 0 3 T 0 0 0 0 0 Ones O 0 0 0 0 0 6 , , , , , , H 0 0 0 0 0 0 9 T 0 0 0 0 0 0 0 8 O 0 0 0 0 0 0 0 0 4 seven hundred twenty-one million, three hundred six thousand, nine hundred eighty-four Standard form: 721,306,984 Expanded form: 700,000,000 20,000,000 1,000,000 300,000 6,000 900 80 4 Word form: seven hundred twenty-one million, three hundred six thousand, nine hundred eighty-four Write the number in word form. 1. 2,267,025,142 2. 702,326,500 3. 600,000,000 50,000,000 9,000,000 800,000 40,000 3,000 700 1 4. thirty-five billion, eight hundred six million, four hundred eighty-six thousand, two hundred twenty-six RW2 Reteach © Harcourt Write the number in standard form. LESSON 1.3 Name Compare Numbers You can use a place-value chart to compare numbers. Thousands Ones Compare the digits from left to right. H T O H T O 2 8 9 8 6 5 ← First number: 289,865 2 8 9 7 6 5 ← Second number: 289,765 ↑ ↑ ↑ ↑ same same same 8 > 7 So, 289,865 > 289,765. Complete the place-value chart. Write <, >, or for each 1. Thousands H T O 375,841 3. T T 5. O T H T H T 4. 47,206,385 T O O H H T T O 677,860 Thousands 476,935 H Ones 677,860 Ones O 986,496 Thousands O Thousands 367,841 Millions H O Ones 467,935 © Harcourt H Thousands H 2. Ones . H T O 986,495 Ones O H T O 47,083,219 Reteach RW3 LESSON 1.4 Name Order Numbers You can use a place-value chart to order numbers. Compare the digits from left to right. Since 4 > 2, 342,198 is the greatest Thousands Ones number. H T O H T O 3 2 2 6 7 8 Continue to compare with the remaining two numbers. 3 4 2 1 9 8 Since 6 > 5, 322,678 > 322,501. 3 2 2 5 0 1 same 4 > 2 same 6 > 5 So, 342,198 > 322,678 > 322,501. Use the place-value chart to order the numbers. 1. 144,421; 144,321; 145,221 Thousands H T 2. 532,124; 58,124; 532,876 Ones O H T Thousands O H T Ones O H T O 3. 456,342,523; 456,342,876; 494,123,563 Millions T O H T Ones O H T O © Harcourt H Thousands RW4 Reteach LESSON 1.5 Name Problem Solving Skill Use a Table Tables help organize data so you can make comparisons. Suppose you want to compare the sizes of four planets. You could make the following table. THE PLANETS Name Diameter (miles) Mercury 3,030 Venus 7,517 Earth 7,921 Mars 4,222 Mercury Venus Earth Mars • Look at the diameters. Compare the digits from left to right. • The smallest planet is Mercury. The largest planet is Earth. Use the tables to answer the questions. 1. This table shows the sales for a popular music store chain. Which type of music had the greatest sales amount? the least sales amount? MUSIC SALES Music Type Sales (in dollars) Alternative Rock 1,345,850 Classical 548,290 Country 1,930,000 Light Rock © Harcourt 2. This table shows the areas of some of the world’s oceans. Which of these oceans has the greatest area? the least area? 425,830 OCEANS Name Area (square miles) Indian 31,507,000 North Pacific 32,225,000 South Pacific 25,298,000 North Atlantic 18,059,000 South Atlantic 14,426,000 Reteach RW5 LESSON 2.1 Name Tenths and Hundredths Money can be used to model decimals. A dollar represents one whole, or $1.00. one 1.0 1 The whole is divided into 10 equal parts. One dime is 110 of a dollar, or $0.10. one tenth 0.1 1 10 The whole is divided into 100 equal parts. 1 of a One penny is 100 dollar, or $0.01. one hundredth 0.01 1 100 Write as a decimal. 1. 2. 3. 4. 1 dollar, 2 dimes, and 5. 3 dollars and 6 dimes 6. 7 dollars, 5 dimes, 9 pennies and 7 pennies Write as a decimal and a fraction. 7. 4 dimes and 6 pennies 8. 2 dollars and 7 dimes 9. 3 dollars, 6 dimes, © Harcourt and 5 pennies 10. five and six tenths 11. five hundredths 12. four and three tenths 13. one and eighty- 14. nine and seventeen 15. two and eight tenths three hundredths RW6 Reteach hundredths LESSON 2.2 Name Thousandths and Ten-Thousandths A place-value chart can help you find the value of each digit in a decimal. Ones Decimal: Read: Write: 2 two 2.0 Tenths • Hundredths Thousandths Ten-Thousandths 3 6 5 three tenths six hundredths five thousandths 0.3 0.06 0.005 1 one ten-thousandth 0.0001 In Standard Form: 2.3651 In Expanded Form: 2.0 0.3 0.06 0.005 0.0001 In Word Form: two and three thousand, six hundred fifty-one ten-thousandths Record each decimal in the place-value chart. Write each decimal in expanded form and word form. 1. 1.5138 Ones Tenths Hundredths Thousandths Ten-Thousandths Tenths Hundredths Thousandths Ten-Thousandths Tenths Hundredths Thousandths Ten-Thousandths • 2. 4.973 Ones © Harcourt • 3. 7.0458 Ones • Reteach RW7 LESSON 2.3 Name Equivalent Decimals Equivalent decimals are different names for the same number or amount. 2 tenths 20 hundredths 0.20 0.2 In the place-value chart, both numbers have a 2 in the tenths place. Ones • Tenths 0 • 2 0 • 2 Hundredths 0 → → 2 tenths 20 hundredths The zero to the right of the 2 does not change the value of the decimal. So, they are equivalent. Write the numbers in the place-value chart. Then write equivalent or not equivalent to describe each pair of decimals. 1. 2.5 and 2.50 Ones • 2. 0.73 and 0.703 Tenths Ones •Tenths Hundredths Thousandths Hundredths • • • • Write the two decimals that are equivalent. 3.050 3.500 4. 1.110 1.1 1.11 5. 0.180 0.0180 0.018 Write an equivalent decimal for each number. 7. 0.05 9. 2.875 RW8 Reteach 8. 2.100 10. 0.040 6. 7.77 7.707 7.770 © Harcourt 3. 3.05 LESSON 2.4 Name Compare and Order Decimals Hundredths Thousandths 7 4 1 7 4 2 • • → 6 Equal Equal Equal numbers numbers numbers of of ones of tenths hundredths → 6 Tenths → Ones → You can use a place-value chart to compare 6.741 and 6.742. 21 So, 6.742 6.741. Write the numbers in the place-value chart. Then write , , or in each . 1. 2.45 O • 2.54 T H • • • T O 72.658 • T 7. 6.275 H Th 4. 564.876 H Th H T 564.786 O• T • • • • Write , , or in each 5. 3.21 6.230 O• T • 3. 72.648 © Harcourt 2. 6.23 3.210 6.257 H Th . 6. 721.460 72.146 8. 468.036 468.136 Order from least to greatest. 9. 16.54, 16.56, 16.55 10. 3.400, 3.004, 3.040 Reteach RW9 LESSON 2.5 Name Problem Solving Skill HOURS MICHAEL EXERCISED Draw Conclusions Michael exercises at 4:00 P.M. daily unless he is sick. The table shows the number of hours Michael exercised last week. Can the conclusion be drawn from the information given? Write yes or no. Explain your choice. Michael usually eats dinner at 5:30. Day Hours Monday 1.8 Tuesday 1.5 Wednesday 0 Thursday 2.2 Friday 1.6 Michael was sick on Wednesday. Can the conclusion be drawn from the information given? Write yes or no. Explain your choice. During a kickball game between two gym classes, the final score was 25 to 22. Each team had 15 players. 1. There were more boys on the 2. There was a winning team. © Harcourt winning team than on the losing team. 3. Each player kicked a homerun. RW10 Reteach 4. More than 40 points were scored in the game. LESSON 3.1 Name Round Whole Numbers You can round whole numbers by using the rounding rules. Step 1: UNDERLINE the digit in the place to which you want to round. Step 2: COMPARE the digit to the right of the underlined digit to 5. Round Down: If the digit to the right is less than 5, the underlined digit stays the same. Round Up: If the digit to the right is 5 or greater, increase the underlined digit by 1. Step 3: CHANGE all digits to the right of the underlined digit to zeros. A. Round 43,658 to the nearest hundred. UNDERLINE. 43,658 COMPARE. 5 5 Round Up. CHANGE. 43,700 B. Round 9,309,587 to the nearest million. UNDERLINE. 9,309,587 COMPARE. 3 5 Round Down. CHANGE. 9,000,000 Round each number to the place of the bold-faced digit. 1. 38,761 2. 719,432 UNDERLINE 38,761 COMPARE UNDERLINE 719,432 5 COMPARE Round Round CHANGE © Harcourt 5 CHANGE Round 2,409,485 to the place named. 3. hundred thousands 4. hundreds UNDERLINE 2,409,485 COMPARE 5 UNDERLINE 2,409,485 COMPARE Round CHANGE 5 Round CHANGE Reteach RW11 LESSON 3.2 Name Estimate Sums and Differences You can estimate sums and differences by rounding the numbers in the problem before performing the operation. One way to round is to round to the greatest place-value position. For example: A. Estimate the sum by rounding. 3,709,525 567,802 → → B. Estimate the difference by rounding. 4,000,000 600,000 4,600,000 539,014 205,918 The sum is about 4,600,000. → → 500,000 200,000 300,000 The difference is about 300,000. Estimate by rounding. 3. 5. 7. 9. 473,542 207,958 → → 8,619,724 3,970,685 → → 724,581 219,067 → → 521,739 659,931 → → 4,516,361 3,497,205 → → RW12 Reteach 2. 4. 6. 8. 10. 741,356 157,900 → → 5,101,118 496,007 → → 192,837 445,672 → → 911,011 187,408 → → 6,212,345 3,493,968 → → © Harcourt 1. LESSON 3.3 Name Add and Subtract Whole Numbers You can add or subtract to find an exact answer. Estimates will help you determine if you have a reasonable answer. Suppose you have saved 3,857 pennies. Then your mom gives you 2,234 more pennies to help you buy a present for a friend. How many pennies do you have altogether? First, estimate. Then, add to find the exact answer. 1 3,857 2,234 → → 4,000 2,000 6,000 The answer should be close to 6,000. 1 3 8 5 7 2 2 3 4 6 0 9 1 6,091 is close to the estimate, so the answer is reasonable. You have 6,091 pennies. Estimate. Then find the exact sum or difference. 1. 3. © Harcourt 5. 5 4 9 2 4 0 7 8 → → 2 9 5 3 6 → 1 0 8 1 9 → 1 3 7 6 4 3 2 → → 2. 7 9 0 6 4 2 3 4 → → 4. 6 8 4 4 4 7 3 9 → → 6. 3 6 7 4 8 → 1 4 2 4 7 → Reteach RW13 LESSON 3.4 Name Choose a Method You add and subtract greater numbers the same way you add and subtract smaller numbers. It may become difficult to keep place values aligned when adding and subtracting greater numbers. Commas help you to line up the numbers. For example, find the sum of 6,716,678 and 5,014,209. • Line up the addends along the commas. • Add to find the exact answer. 1 1 6,716,678 5,014,209 11,730,887 → → 7,000,000 5,000,000 12,000,000 • Estimate the sum to see if your answer is reasonable. 11,730,887 is close to the estimate of 12,000,000, so the answer is reasonable. Find the sum or difference. Estimate to check. 1. 8,432,790 3,876,339 → → 3. 9,010,776 4,573,932 → → 2. 4,918,471 1,839,220 → → 4. 3,825,449 4,361,749 → → Copy the problem. Use commas to help you line up the numbers. Find the sum or difference. Estimate to check. → → RW14 Reteach 6. 7,927,881 4,618,532 → → © Harcourt 5. 6,654,148 4,732,387 LESSON 3.5 Name Problem Solving Strategy Use Logical Reasoning A table can help you with logical reasoning. Elizabeth, Alan, Calvin, and Marie each ordered a different ice cream flavor. The flavor choices were vanilla, peach, chocolate, and strawberry. Neither Alan nor Marie ordered vanilla. Calvin had a brown ice cream stain on his t-shirt. Marie is allergic to strawberries. Which flavor ice cream did each person order? • Calvin had a brown stain on vanilla his t-shirt. Put a yes in the chocolate column for Calvin Elizabeth Yes and a no in each empty box in Alan No that row and in that column. Calvin • Marie is allergic to strawberries and she did Marie not order vanilla. Put a no in those boxes. Put a yes in the remaining box, peach, and a no in the remaining boxes in that column. peach chocolate strawberry No No No No No Yes No No Yes No No Yes No No • Alan did not order vanilla. Put a no in that box. That leaves strawberry. • So, Elizabeth ordered vanilla. Put a yes in that box. Use logical reasoning and the table to solve. © Harcourt 1. Rishawn, Julie, Kevin, and LaTia each have a different favorite subject. Julie likes to use paint and chalk. LaTia enjoys using numbers. Science is not Kevin’s favorite subject. What is each student’s favorite subject? art math music science Rishawn Julie Kevin LaTia Reteach RW15 LESSON 4.1 Name Round Decimals The same rules you learned for rounding whole numbers can be used to round decimals. Step 1: Underline the digit in the place to which you want to round. Step 2: Compare the digit at the right of the underlined digit to 5. Round Down: If the digit at the right is less than 5, the underlined digit stays the same. Round Up: If the digit at the right is 5 or more, increase the underlined digit by 1. Step 3: Rewrite all digits to the right of the underlined digit as zeros. An equivalent decimal can be written by leaving off trailing zeros. A. Round 5.6431 to the nearest B. Round 0.8287 to the nearest hundredth. thousandth. Underline. 5.6431 Compare. 35 Rewrite. 5.6400 or 5.64 Round down. Underline. 0.8287 Compare. 75 Rewrite. 0.8290 or 0.829 Round up. 2. Round 82.64751 to the nearest 1. Round 4.1872 to the place of the bold-faced digit. thousandth. Underline. 4.1872 Underline. 82.6475 Compare. 5 Round . Compare. 5 Round . Rewrite. Rewrite. Round each number to the place of the bold-faced digit. 4. 9.0287 5. 108.108 6. 26.3199 © Harcourt 3. 7.325 Round 12.8405 to the place named. 7. hundredths RW16 Reteach 8. ones 9. tenths 10. thousandths LESSON 4.2 Name Estimate Decimal Sums and Differences Jonas earned $25.87. Kevin earned $20.94. About how much did they earn in all? About how much more did Jonas earn than Kevin? You can estimate decimal sums and differences by rounding the amounts to the nearest whole number and then adding or subtracting. A. Estimate the sum by rounding. B. Estimate the difference by rounding. $25.87 → $26 → 20.94 21 $47 $26 $25.87 → 20.94 → 21 $5 They earned about $47. Jonas earned about $5 more than Kevin. Estimate the sum or difference by rounding to the nearest whole number or dollar. 1. $63.98 → 5.29 → 2. 9.684 → 2.395 → 3. 25.39 → 17.71 → Estimate the sum or difference to the nearest tenth. 4. 8.604 → 6.71 → 5. 26.4572 → 11.3518 → → 8.592 → 6. 56.8 Estimate the sum or difference. © Harcourt 7. 8.453 → 1.21 → 7.05 0.63 10. → → 8. 11. 8.25 → 0.385 → 5.128 → 1.56 → 9. 12. 9.52 → 1.29 → 2.31 → 4.804 → Reteach RW17 LESSON 4.3 Name Add and Subtract Decimals To add or subtract decimals, line up the decimal points in the problem. Finding an estimate first will help you determine if your answer is reasonable. First, estimate. → 18.948 5.765 → Then, subtract to find the exact answer. 19 6 13 1 8 1 5 3 . . . 8 9 14 4 8 7 1 6 8 5 3 13.183 is close to the estimate, so the answer is reasonable. The answer should be close to 13. Estimate. Then find the exact sum or difference. 1. 1 4 . . 8 5 . . 5 5 8 → 3 → 1 8 3 5. 7 2 6 → 3 → 1 6 4 . . . . 3. 5 2 → 7 3 → 6 2 7 6. 3 2 → → 8 6 3 . . . 3 1 8 9 → 8 → 5 → . . 2 9 8 → 6 → © Harcourt 4. 2. 7. 5.86 8.79 n RW18 Reteach 8. 14.09 2.87 n LESSON 4.4 Name Zeros in Subtraction Find 1.34 1.256. • To subtract decimal numbers, line up the numbers along the decimal points. 1.34 . 1.256 . 1.340 . 1.256 . • Add zeros so both numbers have the same number of decimal places. • Subtract. 2 1310 1.4 30 1.2 5 6 0.0 84 • Place a decimal point in the answer, below the decimal points in the problem. So, 1.34 1.256 0.084. Find the difference. 1. 2.7 1.5 2. 3.94 2.6 3. 4.75 2.56 4. 6.8 3.9 6. 3.5 2.8 7. 4.4 1.65 8. 7.643 3.4 9. 13. 9.41 6.527 14. © Harcourt 11. 4.2 2.83 12. 5.6 3.58 5. 5.1 3.08 11.904 8.626 10. 16.24 9.1 14.5 8.872 15. 35.4 15.567 16. 3.84 2.68 n 17. 2.7 0.312 n 18. 4.1 3.3 n 19. 6.57 1.898 n 20. 5.2 2.623 n 21. 7.42 3.416 n Reteach RW19 LESSON 4.5 Name Problem-Solving Skill Estimate or Find Exact Answer Both estimation and exact answers are useful when shopping. Estimations help you determine if you have enough money. Exact answers help you determine if you received the correct change. Suppose you have $5.00, and want to buy 5 drinks for $0.85 each. Do you have enough money? How much change will you receive? Exact Answer $0.85 Estimation $0.85 → $1.00 $0.85 → $1.00 $0.85 $0.85 → $1.00 $0.85 $0.85 → $0.85 → $1.00 $1.00 $5.00 So, you have enough money. $5.00 4.25 $0.75 change $0.85 $0.85 $4.25 So, you should receive $0.75 change. Write an estimate of the total amount. Then solve. a magazine for $3.25, a small pizza for $3.89, and two drinks for $1.15 each. How much change will Pat receive? 3. Jenny has $10.00. She wants to buy 6 pounds of apples costing $0.75 per pound and a bag of oranges costing $1.45. What is Jenny’s exact cost? How much change will she receive? RW20 Reteach 2. Paula has $45. She wants to pur- chase CDs costing $12.99, $14.99, $9.99, and $11.99. Does Paula have enough money? If so, how much change will she receive? 4. Erin has $50.00. She wants to buy a purse for $17.99, gloves for $10.98, and a sweater for $19.95. What is her exact cost? How much change will Erin receive? © Harcourt 1. Pat has $10.00. He wants to buy LESSON 5.1 Name Expressions and Variables An expression has numbers and operation signs. It does not have an equal sign. Use these words to help you write expressions. Addition: more, sum, plus, added, gave Subtraction: less, minus, loss, difference, spent, left John had 12 marbles. He won 7 more. Mary had $10. She spent $3. Translate this into an expression. Translate this into an expression. Clue Word: more 12 7 Clue Word: spent 10 3 An expression may have a variable. A variable is a letter or symbol that can stand for a number. Peter caught 2 fish in the morning. In the afternoon, he caught some more. Susan had 4 sharpened pencils. Then she broke the point off of some of them. Translate this into an expression. Translate this into an expression. Clue Word: more 2n Clue Word: left 4n Write the clues. Then write an expression using n for the unknown number. Explain what the variable represents. 1. The temperature dropped © Harcourt 7 degrees and then went up 4 degrees. Clue Words: 3. Steven wrote 8 pages for homework. The dog ate some of them. Clue Words: 2. When the train stopped, 5 people boarded and 2 got off. Clue Words: 4. Gabriel collected 7 stones. John gave some more stones. Clue Words: Reteach RW21 LESSON 5.2 Name Write Equations An equation is a number sentence that uses the equal sign to show that two amounts are equal. You can use variables to stand for numbers you do not know. Peter had 10 books. After his birthday party, he had 16 books. How many books did he receive for his birthday? books he has plus books received total books 10 books books received total books 10 n 16 Write an equation with a variable. Explain what the variable represents. 2. Mary Beth loves chocolate chip were 22 students in the class. How many more cups did he need to serve punch to all his classmates? cookies. Her mother took a sheet of 12 out of the oven. Mary Beth ate some. Now there are 8 left. How many did she eat? cups he had cups he needed total cups for punch total cookies number eaten number left 3. Jennifer had spent $32 for a new jacket. She had $12 left. How much did she have originally? original amount amount spent amount left RW22 Reteach 4. Monica had a collection of stickers. She bought 7 and had a total of 21. How many did she originally have? number in collection number gained total amount © Harcourt 1. Joseph had 7 paper cups. There LESSON 5.3 Name Solve Equations When you solve an equation, you find the value of the variable that makes the equation true. In an equation, the amounts on both sides of the equal sign have the same value. It is like a balanced scale. n 6 10 To solve, ask, “How many counters would I need to add to the left side of the scale to make it balanced?” Use mental math to find the missing addend. The solution equation will be n 6 10. Think: what number plus 6 equals 10? n4 Check your solution. Replace n with 4. n 6 10 4 6 10 10 10 Use mental math to solve. Check your solution. 1. n 5 15 2. n 6 6 3. n 10 20 Solve the equation. Check your solution. 5. n 8 12 6. 25 n 22 7. n 10 6 7 8. 22 n 7 18 9. 14 8 n 13 © Harcourt 4. 15 n 22 Reteach RW23 LESSON 5.4 Name Use Addition Properties You can use the properties of addition to help you solve problems. The Associative Property states that you may group addends differently without changing the value of the sum. 7 (8 4) (7 8) 4 7 12 15 4 19 19 The Commutative Property states that addends may be added in any order without changing the value of the sum. The Zero Property states that you may add zero to any number without changing the value of the number. 6556 11 11 505 Name the addition property used in each equation. 1. 223 0 223 2. (5 6) 3 5 (6 3) 3. 56.4 10 10 56.4 Find the value of n. Identify the addition property used. 4. 200 n 100 200 5. 78 (5 n) (78 5) 7 6. 4 n 7 4 7. 0 88 n Algebra: Name the addition property used in each equation. 10. w 0 w RW24 Reteach 9. p (q r) (p q) r 11. d f f d © Harcourt 8. g h h g LESSON 5.5 Name Problem Solving Skill Use a Formula To find the perimeter of a figure, you add the lengths of its sides. Remember that perimeter measures the distance around a figure. You can use a formula to find the perimeter. Use a different letter for each side of the figure. 10 10 Pabc P 10 10 12 A triangle needs 3 letters. P 32 12 Find the perimeter of the following figures. 1. 6 2. 5 5 5 8 12 12 3. 15 20 4. 15 22 15 12 34 © Harcourt Use a formula to solve. 5. Jeff wants to build a rectangular fence in his yard for his dog. The yard is 35 feet by 40 feet. How much fencing must Jeff buy? 6. Draw a square with each side measuring 8 units and find the perimeter. Reteach RW25 LESSON 6.1 Name Write and Evaluate Expressions You can write and evaluate expressions to model different situations. Ms. Hartwick has 6 rows of students in her classroom. She has the same number of students in each row. To model how many students are in Ms. Hartwick’s class, you can write an expression. 6 rows times number of students in each row → → → 6 n If there are 5 students in each row, how many students are in Ms. Hartwick’s class altogether? Replace the variable n in the expression with 5 to find how many students are in Ms. Hartwick’s class altogether. Evaluate the expression if n 5. 65 Replace n with 5. → 6n → 30 So, there are 30 students in Ms. Hartwick’s class altogether. Write an expression. If you use a variable, tell what it represents. 1. Beth runs 4 days a boxes of canned dog food. Each box had 9 cans of dog food. 3. Marcus has 7 shelves of CDs. Each shelf holds the same number of CDs. © Harcourt week. She runs the same number of miles each day. 2. Caitlin bought 12 Evaluate each expression. 4. n 2 if n 12 RW26 Reteach 5. 9 n if n 7 6. 22 (n 3) if n 6 LESSON 6.2 Name Order of Operations When an expression has more than one operation, you evaluate it using the order of operations. The order of operations is a set of rules that tells you which operation to do first. Evaluate 18 (4 6) 2. Step 1 18 (4 6) 2 Operate inside parentheses . 4 6 24 Step 2 Step 3 18 24 2 Multiply and divide from left to right. 24 2 12 18 12 Add and Subtract from left to right. 18 12 30 So, 18 (4 6) 2 30 30 Complete to evaluate the expression. 1. 10 (7 4) 8 2. 15 5 9 4 8 10 94 8 4 © Harcourt Evaluate the expression. 3. 14 (5 2) 2 4. 2 8 (16 4) 5. 5 7 24 8 6. 4 (55 11) 6 7. 29 (6 3) 2 8. (27 9) 8 7 9. 8 6 7 2 12. 19 7 (12 6) 10. 30 (10 10) 13 11. 6 7 4 5 13. 7 48 (7 5) 14. 27 3 (1 5) Reteach RW27 LESSON 6.3 Name Functions When one quantity depends on another quantity, the relationship between the quantities is called a function. Paintbrushes cost $4 each. How much will 5 paintbrushes cost? You can write an equation to represent the function. Number of dollars 4 the number of paintbrushes d 4 p d 4 5 d 20 You can also use a function table to show the number of dollars different numbers of paintbrushes cost. paintbrushes, p 1 2 3 4 5 dollars, d 4 8 12 16 20 So, 5 paint brushes will cost $20. Complete the function table. 1. b 9c c 2 2. s 7t 4 6 8 10 6 7 8 9 10 s b 3. h 6j 4 j t 8 6 4. f 6 5g 4 2 0 h 0 5 10 15 20 f 5. d 3a 2 a 12 10 g 6. n 15 2m 4 8 6 4 d m 3 5 7 9 11 n 7. y 8x 7 for x 3, 4, 5 9. y 6x 15 for x 6, 7, 8 RW28 Reteach 8. y 100 4x for x 5, 10, 20 10. y 49 3x for x 8, 9, 10 © Harcourt Use the function. Find the output, y for each input, x. LESSON 6.4 Name Problem Solving Strategy Write an Equation You can write an equation to help you solve a problem. Felicity and Alex were in charge of parking cars in the small parking lot at the State Fair. The lot was filled with 72 cars in all by noon of the first day. The cars were organized into 9 equal rows of cars. How many cars were in each row? Write an equation to find the number of cars parked in each row. Think 9 times what number equals 72. So, each row had 8 cars. total cars rows number of cars in lot of cars in each row 72 9 c 9 8 72 8 c Write and solve an equation for each problem. Explain what the variable represents. 1. Jacob has to stack boxes in the grocer’s storage room. The room is 96 inches high. Each box is 12 inches high. How many boxes can Jacob stack on top of each other? © Harcourt 3. Chelsea has to line up 48 chairs in 6 equal rows. How many chairs should she put in each row? 2. The shelves that the grocer stacks the canned goods on are 30 inches high. The grocer stacked the cans 5 high. How tall is each can? 4. Troy made a striped blanket for his bed. The blanket was 54 inches wide with 9 equal stripes. How wide was each stripe? Reteach RW29 LESSON 6.5 Name Use Multiplication Properties You can use mental math and the properties of multiplication to solve problems. Property of Multiplication Example Explanation Commutative Property 42n4 4224 n2 You can multiply numbers in any order. The product is always the same. Associative Property (3 n) 5 3 (4 5) (3 4) 5 3 (4 5) n4 You can group factors differently. The product is always the same. Property of One n15 515 n5 When one of the factors is 1, the product equals the other number. Zero Property 4n0 400 n0 When one factor is 0, the product is 0. 1. n 3 0 2. n 3 3 2 3. 4 (2 5) (n 2) 5 4. 1 n 8 5. (n 3) 2 5 (3 2) 6. 6 7 7 n 7. (7 3) n 7 (3 2) 8. 8 2 n 8 9. 3 n 3 RW30 Reteach © Harcourt Solve the equation. Identify the property used. LESSON 6.6 Name The Distributive Property You can use the Distributive Property to break apart numbers to make them easier to multiply. To find 20 13, you can break apart 13. ← Break apart. 20 13 20 (10 3) (20 10) (20 3) ← Multiply. ← Add. (200) (60) 260 Use the Distributive Property to restate each expression. Find the product. 2. 20 18 1. 20 12 Break apart. 20 ( Multiply. Add. 20 20 200 ) Multiply. 30 30 Add. ) © Harcourt 20 Break apart. 12 ( Multiply. 12 12 Add. 5. 30 26 ) 6. 25 17 Break apart. 30 ( Add. ) 4. 12 45 Break apart. 30 ( Multiply. 20 Add. 3. 30 16 Multiply. Break apart. 20 ( 30 30 ) Break apart. 25 ( Multiply. Add. 25 25 Reteach ) RW31 LESSON 7.1 Name Collect and Organize Data The tally table shows how many fifth grade students rode the bus during the first four weeks of school. How can you find the total number of students in the fifth grade that rode a bus to school? Week 1 2 3 The information in the tally table can be made easier to read and understand by using a frequency table. The frequency for each week tells how many fifth grade students rode a bus that week. The cumulative frequency column shows a running total of the number of students who rode a bus. Step 1 Count the tally marks for each week. Place the total for each week in the column labeled Frequency on the frequency table. Step 2 For each new line of data, write the sum of the frequencies in the Cumulative Frequency column. The last number in the Cumulative Frequency column will tell you the total number of fifth graders that rode a bus. Fifth Grade Riders 4 FIFTH GRADERS RIDING A BUS Week Frequency (Number of Students) Cumulative Frequency 1 7 7 2 6 7 6 13 3 4 How many fifth graders rode a bus? Suppose 2 more fifth graders rode a bus in Week 3. In addition, 7 new fifth graders enrolled in school. 4 of the new students are walkers and 3 rode a bus in Week 2. Use this information to complete a new frequency table. What is the new total number of fifth graders riding a bus? What is the new range? FIFTH GRADERS RIDING A BUS Week 1 2 3 4 RW32 Reteach Frequency (Number of Students) Cumulative Frequency © Harcourt The range is the difference between the greatest and the least numbers in a set of data. Greatest Number Least Number Range Use the frequency table to find the range of the number of fifth graders who rode a bus. Show your work. LESSON 7.2 Name Find the Mean Tom has taken three tests. He wants to know his average score for the three tests. The type of average Tom is looking for is called the mean. Tom’s Test Scores Test Score Step 1 1 2 3 80 70 90 Step 2 Add the three test scores together. Divide the sum by the number of tests. 80 70 90 240 240 3 80 So, Tom’s mean test score is 80. Write an addition sentence for the sum of each set of numbers. 1. 3, 5, 4, 1, 7 2. 20, 15, 10 3. 22, 26, 28, 32 Write how many numbers are listed in each set of numbers. 4. 3, 5, 4, 1, 7 5. 20, 15, 10 6. 22, 26, 28, 32 Write a division sentence to find the mean for each set of numbers. 7. 3, 5, 4, 1, 7 8. 20, 15, 10 9. 22, 26, 28, 32 © Harcourt 10. One month later, Tom took 5 more tests. His scores were 80, 70, 90, 90, and 100. What is the mean of these test scores? Show your work. Find the mean for each set of data. 11. 9, 11, 13, 13, 9 12. 33, 28, 35, 33, 26 13. 105, 112, 133, 118, 102 Reteach RW33 LESSON 7.3 Name Find the Median and Mode Sam takes tests to see how many words he can type in a minute. The data in the table show his first 7 tests. Number of Words Typed in a Minute Test 1 2 3 4 5 6 7 Score 22 16 18 14 16 34 20 You can find Sam’s median score and the mode of the data. Step 1 Step 2 Step 3 List the scores from least to greatest. To find the median score, cross off a number from each end until there is only one number left in the middle. Find the score that occurred most often. — 16, — 18, 20, — — — 16, 14, 22, — 34 Sometimes there is more than one mode or no mode. 14, 16, 16, 18, 20, 22, 34 The number 18 is the median score. Sam scored 16 twice. The number 16 is the mode. Arrange the numbers from least to greatest. Circle the median number. 16, 17, 19 2. 24, 32, 28, 45, 19, 23, 16, 51, 32 3. 103, 98, 105, 101, 99 Arrange the numbers from least to greatest. Find the median and the mode. 4. 9, 7, 5, 11, 11 5. 14, 12, 12 6. 3, 7, 2, 9, 6, 5, 3, 1, 3 median: median: median: mode: mode: mode: RW34 Reteach © Harcourt 1. 13, 12, 11, 11, 9, 8, LESSON 7.4 Name Problem Solving Strategy Make a Graph Mr. Schwartz recorded the number of newspapers he sold in his store every day of the week for two weeks. Newspapers sales were 60, 65, 66, 71, 71, 72, 74, 75, 76, 77, 79, 80, 81, and 83. Is the number sold usually in the 60’s, 70’s, or 80’s? You can make a stem-and-leaf plot to organize the data by place value. Stem Make a column of the tens digits of the data, listing them in order from least to greatest. These are the stems. Leaves 6 7 8 Beside each tens digit, record the ones digits of the data, in order from least to greatest. These are the leaves. Stem Leaves 6 0 5 6 7 1 1 2 4 5 6 7 9 8 0 1 3 The stem-and-leaf plot shows the greatest number of leaves are on the 7 stem. So, the number of newspapers sold is usually in the 70’s. Make a graph to solve. 1. Lynnette’s golf scores are 72, 74, 74, 78, © Harcourt 80, 82, 83, 87, 88, and 91. Does she usually score in the 70’s, 80’s, or 90’s? 2. The coach of the Tigers recorded the number of parents that attended each home baseball game. Parents’ attendance was 16, 17, 23, 24, 29, 30, 33, 36, 36, and 38. Is parents’ attendance usually in the 10’s, 20’s, or 30’s? Reteach RW35 LESSON 7.5 Name Analyze Graphs Books Read Number of Books Graphs help you to draw conclusions, answer questions, and make predictions about the data. Study the following graphs to answer the questions. 1. A bar graph is useful when comparing data by groups. 10 8 6 4 2 0 John Sally Student Which student read the most books? the least? Lori 2. Line graphs are helpful to see how data changes over a period of time. What happened to the temperature as the week passed? Temperature (degrees F) Daily Temperatures 80 70 60 50 40 30 20 10 0 Mon Tues Wed Day Thurs Fri Pet Store Population Cats 3. A circle graph shows how parts of data Birds relate to each other and to the whole. Dogs About one half of the animals in the pet store are what type of animal? Types of Books in Mr. Williams’ Class 4. A pictograph displays countable data with symbols or pictures. Pictographs have a key to show how many each picture represents. How many books does Mr. Williams have in his class? Fantasy Mystery Biography Poetry Key: RW36 Reteach = 4 books © Harcourt Reptiles LESSON 8.1 Name Choose a Reasonable Scale Henry kept track of how much mail his family received in one week. Mail Received in a Week Mon Tue Wed Thu Fri Day He put the data in a table. Number He wants to put the data in a line graph. of Pieces 8 10 He must select a scale. A scale is the series of numbers placed at fixed distances. The difference between one number and the next on the scale is called the interval. 6 4 2 The scale must include the numbers 2 through 10. It must include a number less than the least data and a number greater than the greatest data. Look at four ways Henry can display the mail data. 11 10 9 8 7 6 12 5 10 4 8 12 3 6 9 15 2 4 6 10 1 2 3 5 0 M T W Th F 0 By 1's M T W Th F 0 By 2's M T W Th F 0 M By 3's T W Th F By 5's Henry selects a scale with intervals of 2. From the box, choose the most reasonable interval for each set of data. List the numbers needed in the scale. 1. 5, 15, 20, 25, 10, 18 Interval 2. 50, 125, 100, 150, 100, 20 a. By 25’s © Harcourt b. By 20’s c. By 10’s 3. 8, 12, 10, 20, 10, 30 4. 20, 101, 40, 59, 115 d. By 5’s Reteach RW37 LESSON 8.2 Name Problem Solving Strategy: Make a Graph Then he planned how to display the data using a bar graph. The interval skipped from 0 to 450, so Sid used a zigzag line to show a break in the scale. He finds the range is 170. He chooses the interval of 50. Using the graph, Sid predicts that next year’s population will be about 650 students. Milton Elementary School Population 1997 1998 1999 2000 2001 Year Number of 450 Students 520 560 580 620 Milton Elementary School Population Number of Students The school population has changed over the last five years. Sid wants to use this data to predict next year’s school population. He organized the data into a table. 650 600 550 500 450 0 1997 1998 1999 Year 2000 2001 Make a graph to solve. 1. Mr. Struther surveyed some students FIELD TRIP IDEAS Locations Zoo Museum Aquarium Number of Students 40 30 50 2. Attendance at the zoo was organized into a table. What graph or plot would best display the data? Make a graph or plot. 3. A baseball team kept track of the number of parents at the baseball games. The team organized the data into a table. What graph or plot would best display the data? Make a graph or plot. RW38 Reteach ATTENDANCE AT THE ZOO Month April May June July Number of People 640 620 680 600 © Harcourt to find ideas for a field trip. He organized the data into a table. What graph or plot should he use to display the data? Make a graph or plot. PARENTS AT BASEBALL GAMES Game 1 2 3 4 5 6 7 Number of Parents 23 12 24 17 29 16 21 LESSON 8.3 Name Graph Ordered Pairs Points on a coordinate grid can be given a unique name in the same way each house on a street has a unique number. Houses on a street follow an order so people can tell them apart and points also follow an order. 8 O 7 L 6 H 5 W I 4 The order of the numbers in an ordered pair is always expressed the same way. The first number in an ordered pair tells how far to move horizontally from the origin. The second number tells how far to move vertically. D C 3 N 2 0 E A 1 1 2 3 4 5 6 (3, 4) Move 3 horizontally; Move 4 vertically 7 8 Name the ordered pair for each point. 1. E 2. H 3. O 4. C 5. A 6. D 7. N 8. I 9. W 10. L Graph and label the following points on a coordinate grid. 11. M 15. S (6, 2) (6, 0) 12. N 16. A 20. H (0, 5) (2, 5) (2, 6) 13. P 17. V 21. T (3, 4) (4, 1) (1, 7) 14. R 18. G 22. Y (1, 0) (3, 7) (6, 3) © Harcourt 19. B (5, 7) Reteach RW39 LESSON 8.4 Name Make Line Graphs The table shows how much money the XYZ Toy Company made for the last five years. The company wants to display the data in a line graph. XYZ Toy Company Year 1995 1996 1997 1998 1999 Sales in Millions ($) 120 60 80 140 160 The greatest number in the table is 160 million. The least number is 60 million. The difference between the greatest and the least number in the set of data is the range. 160,000,000 60,000,000 100,000,000 XYZ Toy Company There is a break in the scale from 0 to 60 and the interval is 20. 160 Sales in Millions ($) The vertical axis is labeled with the amounts of money; the horizontal axis is labeled with the years. 180 140 120 100 80 60 0 1995 1996 1997 1998 1999 For each set of data, write a subtraction sentence to find the range. 1. 12, 18, 9, 13, 7 2. 20, 100, 40, 60, 35 3. 1, 9, 6, 5, 3 4. 5. On-Line Hours Used Month Oct Nov Dec Hours Used 30 RW40 Reteach 60 Jan 100 125 Jelly Bean Sales Days Boxes Sold 1 2 3 4 5 100 50 150 250 200 © Harcourt Make a line graph for each set of data. LESSON 8.5 Name Histograms Histograms are a type of bar graph. The bars in a histogram are related and follow an order. They show the number of times the data occur within intervals. The bars in a bar graph are not related to one another. To decide whether to make a histogram or bar graph, you need to decide whether the data fall within intervals. Every 30 minutes, the popcorn popper records the number of boxes sold in the movie theater. Here is the information. 5:00 5:30 28 26 6:00 6:30 7:00 7:30 8:00 33 45 56 86 85 8:30 9:00 57 25 9:30 10:00 10:30 11:00 35 48 32 21 Find the range for the set of data. What interval could you use to make the histogram? Select an interval to divide the data equally. Make a frequency table with the intervals and record the number of boxes of popcorn sold during these time periods. Use the frequency table to make the histogram. Label the scale for the number of boxes sold and title the graph. Graph the frequency for each interval. Remember that the bars in a histogram are side-by-side. Decide which graph would better represent the data below, a bar graph or histogram. Then make the graph. © Harcourt MONEY SPENT ON LUNCH Amount of Money Number of Students $1.25 4 $1.75 3 $2.00 5 $2.50 4 $2.75 3 $3.00 1 Reteach RW41 LESSON 8.6 Name Choose the Appropriate Graph Michael’s Test Scores To display data, it is important to select the most appropriate graph or plot. Test 1 2 Line Plot Stem-and-Leaf Plot A line plot is used to record data as they are collected. x x x x x A stem-and-leaf plot is used to organize data by place value. 87 92 95 4 5 Score 87 75 92 95 100 Two different graphs and two different plots for displaying data are shown. 75 3 stem leaf 7 8 9 10 Key: 7 5 a score 100 Bar Graph Line Graph A bar graph is used to compare facts about groups. A line graph shows change over time. Michael's Test Scores Michael's Test Scores 120 120 100 100 80 80 60 60 Score Score 5 7 2 5 0 of 75 40 20 40 20 0 1 2 3 Test 4 0 5 1 2 3 4 5 Test Michael chose the line graph because it shows his test scores over time. Write the best type of graph or plot for the data. 6 major cities. 3. Display the daily growth of a sunflower in inches. RW42 Reteach 2. Record the letter grades (A–F) for a class of 30 students. 4. Record the temperature every hour for 24 hours. © Harcourt 1. Compare the population of LESSON 9.1 Name Estimation: Patterns in Multiples You can round numbers and use basic facts to estimate products. Count the number of zeros in your rounded numbers. They will appear to the right of your basic fact in your estimate. For 2-digit numbers: For 3-digit numbers: If the ones digit is 0–4, round down. If the tens digit is 0–4, round down. If the ones digit is 5–9, round up. If the tens digit is 5–9, round up. For example: Round 41–44 to 40. Round 45–49 to 50. For example: Round 700–749 to 700. Round 750–799 to 800. 45 → 50 42 → 40 2,000 700 749 → 44 → 40 28,000 2 zeros 3 zeros Round each factor and estimate the product. 1. 4. © Harcourt 7. 10. 13. 141 → 36 → 160 → 41 → 349 → 74 → 638 → 16 → 719 → 85 → 2. 5. 8. 11. 14. 157 → 57 → 187 → 72 → 456 → 56 → 774 → 55 → 468 → 68 → 3. 6. 9. 12. 15. 125 → 25 → 236 → 45 → 568 → 27 → 836 → 43 → 229 → 54 → Reteach RW43 LESSON 9.2 Name Multiply by 1-Digit Numbers Multiply the ones. Multiply the tens. Multiply the hundreds. 143 3 29 143 3 429 143 3 9 Sometimes you need to regroup. Step 1 Multiply the ones. 3 3 ones 9 ones 143 3 9 Step 2 Multiply the tens. 3 4 tens 12 tens Write the 2. Regroup the 10 tens as 1 hundred. 143 3 29 1 1 Step 3 Multiply the hundreds. 3 1 hundred 3 hundreds 143 Now add the regrouped hundred. 3 3 hundreds 1 hundred 4 hundreds So, 3 143 429. 429 Tell which place-value positions must be regrouped. Find the product. 451 2 2. 328 3 3. 715 5 4. 1,458 5. 2,473 6. 6,925 7. 3,562 8. 20,317 9. 13,234 6 7 RW44 Reteach 2 4 4 3 © Harcourt 1. LESSON 9.3 Name Multiply by 2-Digit Numbers You can multiply by two-digit numbers by breaking apart one of the factors. To find 21 14, you can break apart 14 into 1 ten 4 ones. 10 4 Step 1 Multiply by the ones. 21 4 84 Step 3 Add the products. 21 14 84 ← 4 21 210 ← 10 21 294 21 Step 2 Multiply by the tens. 21 10 210 So, 21 14 294. Complete to find the product. 1. 13 12 3. 30 17 5. 40 19 © Harcourt 7. 37 15 2. ← ← 4. ← ← ← ← ← ← 45 15 ← ← 8. ← ← 28 14 6. ← ← 22 15 28 16 ← ← Reteach RW45 LESSON 9.4 Name Choose a Method You can multiply three-digit numbers by breaking apart one of the factors. To find 312 143, break apart 143 into 1 hundred 4 tens 3 ones. Step 1 Step 2 Step 3 Step 4 Multiply by the ones. Multiply by the tens. Multiply by the hundreds. Add the products. 312 3 936 312 40 12,480 312 100 31,200 312 143 936 ← 3 312 12,480 ← 40 312 31,200 ← 100 312 44,616 So, 312 143 44,616. Complete to find the product. 3. 5. 423 146 ← ← ← 354 246 ← ← ← 672 334 2. 4. 6. ← ← ← RW46 Reteach 231 123 ← ← ← 438 253 ← ← ← 596 254 ← ← ← © Harcourt 1. LESSON 9.5 Name Problem Solving Skill Evaluate Answers for Reasonableness You can use estimation to check if an answer is reasonable. Use your knowledge of patterns in multiples to help you with large numbers. At the garden center, there were 174 rows of flowers. Each row contained 86 flowers. Estimate first. Then solve the problem and compare it to your estimate to see if it is reasonable. Estimate 174 86 1,044 13,920 14,964 200 90 18,000 Your estimate was more than your exact amount because you used greater numbers. How would your estimate compare to your exact answer if you rounded both factors down? Choose the most reasonable answer without solving. 1. Joel’s dad sold each of his paint- materials to create each of his paintings. About how much does it cost him to create the 26 paintings he sold? A $7,000 B $12,000 F $600 G $1,500 C $20,000 D $60,000 3. A small airport has 21,795 © Harcourt 2. Joel’s dad pays $157 for the ings at an art show for $750. He sold 26 at the show. About how much money did he get? H $5,000 J $60,000 4. The average length of important passengers each year. About how many passengers will they have altogether in 8 years? rivers in the world is 2,142 miles. If we measured 18 of these rivers, about how many miles would we measure? A 2,000 B 20,000 F 4,000 miles G 20,000 miles C 200,000 D 2,000,000 H 40,000 miles J 400,000 miles Reteach RW47 LESSON 10.1 Name Multiply Decimals and Whole Numbers To multiply a whole number and a decimal, modeling with money can be helpful. To multiply 3 0.2, follow these steps. Step 1 Write 0.2 as 0.20. You can add a zero at the end of a decimal without changing the value. Draw that amount of money. Step 2 Draw 3 groups of coins of $0.20. The 2 dimes equal $0.20. You could also draw 4 nickels or 20 pennies. Count the total amount. $0.20 $0.20 $0.20 $0.60 So, 3 0.2 0.60, or 0.6. Draw the coins that equal the decimal amount. Use the fewest coins possible. 2. 0.30 3. 0.16 4. 0.52 5. 0.80 6. 0.24 © Harcourt 1. 0.28 Make a money model to find each product. 7. 4 0.15 8. 3 0.1 9. 2 0.21 10. 4 0.01 11. 3 0.06 12. 2 0.78 13. 3 0.32 14. 4 0.12 15. 2 0.53 RW48 Reteach LESSON 10.2 Name Algebra: Patterns in Decimal Factors and Products You can use patterns to place the decimal point in a product. Factors Product 2 1 2 212 ← no decimal places in factors 2 0.1 0.2 2 0.1 0.2 ← one decimal place in factors 2 0.01 0.02 2 0.01 0.02 ← two decimal places in factors The number of decimal places in the factors equals the number of decimal places in the product. Complete the tables. 1. 4. © Harcourt 7. 10. 2 3 2. 2 4 3. 3 3 2 0.3 2 0.4 3 0.3 2 0.03 2 0.04 3 0.03 3 5 5. 3 6 6. 3 7 3 0.5 3 0.6 3 0.7 3 0.05 3 0.06 3 0.07 2 8 8. 4 5 9. 6 7 2 0.8 4 0.5 6 0.7 2 0.08 4 0.05 6 0.07 15 1 11. 28 1 12. 32 1 15 0.1 28 0.1 32 0.1 15 0.01 28 0.01 32 0.01 Reteach RW49 LESSON 10.3 Name Model Decimal Multiplication To multiply 0.3 0.2, a 10-by-10 model will help. Step 1: Draw diagonal lines through the bottom 3 rows. Step 2: Draw diagonal lines through 2 columns. The 3 rows represent 0.3. The 2 columns represent 0.2. Step 3: The overlapping squares that have an x in them show the product of 0.3 0.2. The 6 squares with x’s represent 0.06. The product of 0.3 0.2 is 0.06. Write a number sentence for each drawing. 1. 2. 3. 4. Make a model for each and find the product. 6. 0.2 0.8 7. 0.5 0.9 8. 0.7 0.5 © Harcourt 5. 0.1 0.5 RW50 Reteach LESSON 10.4 Name Place the Decimal Point How many decimal places are in the product of 0.21 and 0.03? Step 1 Step 2 Add the number of decimal places from each factor. Multiply the numbers just like whole numbers. To have 4 decimal places, you have to add 2 zeros before the 63. 0.21 0.03 ? 2 places 2 places 4 places 0.21 0.03 0._ _ _ _ 0.21 0.03 63 000 0.0 0 6 3 Write how many decimal places are in each number. 1. 0.105 2. 0.0006 3. 0.008 Write how many decimal places are in each product. Then write the product. The first one has been done for you. 4. 0.3 0.5 5. 0.6 0.03 6. 0.002 0.8 8. 3 0.4 9. 0.7 0.2 0. _ _ 0.15 7. 0.24 0.01 Find each product. 11. 0.06 1.8 12. 7 0.08 © Harcourt 10. 0.5 0.03 Reteach RW51 LESSON 10.5 Name Zeros in the Product Be careful when multiplying by decimals to include all of the decimal places in the product. Example: Find 0.013 0.6. Step 1 Step 2 Step 3 Find the number of decimal places the product should have. Multiply. Place the decimal point. 0.013 has three decimal places and 0.6 has one decimal place. The product should have 3 1 4 decimal places. 0.013 0.6 78 The product should have 4 decimal places. There are two digits, so write 2 zeros in the product and place the decimal point. 0.013 0.6 0.0078 1. Find 0.03 0.4. Step 1: Find the number of decimal places the product should have. Step 2: Multiply Step 3: Place the decimal point. 0.03 0.4 0.03 0.4 Step 2: 0.047 0.07 Step 3: 0.047 0.07 Step 2: Step 3: 2. Find 0.047 0.07. Step 1: Step 1: 4. Find 0.054 0.007. RW52 Reteach 5. Find 0.0942 0.7. © Harcourt 3. Find 0.0732 0.8. LESSON 10.6 Name Problem Solving Skill Make Decisions We make decisions every day. There are often many things to consider. Use the questions below to guide you through making decisions. Your neighbors have invited you to go with them on Saturday. Julia’s family is going to the museum and to a movie. Karl’s family is going on a bakery tour and to a football game. You must decide which invitation to accept. 1. If the museum visit will cost $3.00 and the movie will cost $4.75, how much will the trip with Julia’s family cost? 2. If a football ticket costs $14.50 and the bakery tour is free, how much will the trip with Karl’s family cost? 3. If you had to make your decision based on total cost, which trip would you choose? Why? 4. Julia’s family will start their trip at 8:30 A.M. Breakfast will take 45 minutes. They plan to stay at the museum for 2 hours. Lunch will take 45 minutes, and the movie will last 2 hours and 30 minutes. When will the trip with Julia’s family end? 5. The bakery tour will take 1 hour and 30 minutes. Lunch will take 30 minutes. The football game will take 3 hours and 30 minutes, and dinner with Karl’s family will take 1 hour. If this trip starts at 11:00 A.M.,when will it end? 6. If you had to make your decision based on the total time of the trip, the start © Harcourt time, or the end time of the trip, which invitation would you accept? Why? 7. If you had to make your decision based on the activities you like better, which invitation would you accept? Why? Reteach RW53 LESSON 11.1 Name Estimate Quotients Compatible numbers are numbers that are easy to compute mentally. One compatible number divides evenly into the other. Think of number factors to help you find compatible numbers. What is 85 5 4 ? Step 1 Think: What are the multiples of 8? 8 16 24 32 40 48 56 64 Which multiple is closest to 55? 56 is close to 55. 8 and 560 are compatible numbers. Step 2 Divide. 560 8 70 A good estimate for 554 8 is 70. 1. 32 5 2 2. 65 4 6 3. 41 5 4 9 2 4. 91 2 9 5. 71 6 5 6. 42 4 4 7. 83 8 0 8. 54 9 7 9. 24 8 7 10. 32 ,5 5 8 11. 56 ,0 9 7 12. 65 RW54 Reteach © Harcourt Follow the steps above to estimate each quotient. LESSON 11.2 Name Divide 3-Digit Dividends Bryan has 522 coins. He divides them among 3 jars. How many coins are in each jar? Divide. 522 3 n Step 1 Since 5 hundreds can be divided by 3, the first digit is in the hundreds place. Divide. 35 Multiply. 3 1 Subtract. 5 3 Compare. 2 3 Step 2 2 Bring down the tens. Divide. 32 Multiply. 3 7 Subtract. 22 21 Compare. 1 3 Step 3 2 Bring down the ones. Divide. 31 Multiply. 3 4 Subtract. 12 12 Compare. 0 3 Since n 174, each jar contains 174 coins. 1 35 2 2 3 2 17 35 2 2 3 22 21 1 174 35 2 2 3 22 21 12 12 0 © Harcourt Follow the steps above to find each quotient. 1. 39 2 8 2. 71 4 9 3. 58 4 5 4. 48 9 2 5. 63 9 9 6. 38 7 3 7. 97 6 5 8. 59 3 4 Reteach RW55 LESSON 11.3 Name Zeros in Division There are 618 pencils in the supply room. They are to be divided evenly among 6 classes. How many pencils will each class receive? You will use division to find the answer. 618 6 n Step 1 1 66 1 8 6 0 Since 6 hundreds can be divided by 6, the first digit will be in the hundreds place. Divide. 10 66 1 8 6 01 0 1 Step 2 Bring down the tens. Divide the 1 ten. Since 6 >1, write 0 in the quotient. Step 3 103 66 1 8 6 01 0 18 18 0 Bring down the ones. Divide. So, each class will receive 103 pencils. Multiply. 616 Subtract. 660 Compare. 06 Multiply. 600 Subtract. 101 Compare. 16 Multiply. 6 3 18 Subtract. 18 18 0 Compare. 06 1. 39 2 7 2. 88 7 2 3. 55 4 2 4. 66 0 8 2 4 5. 36 0 7 6. 28 2 6 7. 48 4 3 8. 78 RW56 Reteach © Harcourt Follow the steps above to find each quotient. LESSON 11.4 Name Choose a Method Divide 42,574 by 7. Divide 7 into 42 to get 6. Multiply 6 by 7 to get 42. Subtract 42 from 42 to get 0. Bring down the 5 to get 05. 6,082 74 2 ,5 7 4 Divide 7 into 5 to get 0; Multiply 0 by 7 to get 0. 42 ↓ Subtract 0 from 05 to get 5. Bring down the 7 to get 57. 05 0↓ Divide 7 into 57 to get 8. Multiply 8 by 7 to get 56. 57 Subtract 56 from 57 to get 1. Bring down the 4 to get 14. 56↓ Divide 7 into 14 to get 2. Multiply 2 by 7 to get 14. 14 Subtract 14 from 14 to get 0. 14 0 Answer = 6,082 © Harcourt Follow the steps above to find each quotient. 1. 94 5 ,0 3 5 2. 59 ,0 8 5 3. 41 6 ,0 8 7 4. 57 0 ,8 6 1 5. 68 5 6 ,4 1 2 6. 51 8 ,0 0 5 7. 42 0 0 ,0 8 8 8. 57 ,5 5 5 9. 56 5 4 ,3 2 1 10. 52 1 ,0 7 6 11. 33 5 6 ,7 8 9 12. 36 7 ,5 3 0 14. 74 ,3 2 6 15. 81 ,9 9 9 16. 36 4 5 ,1 2 3 13. 63 ,7 9 1 Reteach RW57 LESSON 11.5 Name Algebra: Expressions and Equations An expression combines numbers or variables with operations. Problem: twenty-four divided by a number Expression: 24 n The value of the expression depends on the number, n. If the number is 2, the value is 12. If the number is 8, the value is 3. An equation is a number sentence that uses an equal sign to show that two amounts are equal. Equation: 24 n 6 Problem: Twenty-four divided by a number is six. To solve the equation, think: 24 divided by what number equals 6? You can predict and test to solve. Predict: 3 Test: 24 3 8; too high Predict: 4 Test: 24 4 6; correct Evaluate the expression for n. 1. 48 n n 2, 6, 12 2. n 10 n 100, 60, 70 3. n 12 n 12, 36, 72 4. 18 n n 2, 6, 18 Determine which value is a solution for the given equation. n 5, 6, or 7 9. 350 n 50 n 5, 6, or 7 6. 195 n 65 n 2, 3, or 4 10. 200 n 5 n 30, 40, or 50 7. n 6 8 8. n 5 50 11. n 6 12 12. n 6 14 n 36, 42, or 48 n 72, 76, or 78 n 200, 250, or 300 n 80, 82, or 84 © Harcourt 5. 49 n 7 Solve each equation. Then, check the solution. 13. 45 n 5 14. 100 n 10 15. n 6 7 16. n 12 9 17. 65 n 13 18. 120 n 30 19. n 3 31 20. n 4 21 RW58 Reteach LESSON 11.6 Name Problem Solving Skill Interpret the Remainder When there is a remainder in a division problem, you need to look at the question to see what is being asked. You may drop the remainder, or round the quotient to the next greater whole number, or you may use the remainder as a fractional part of your answer. Andy made punch with 48 ounces of apple juice, 36 ounces of grape juice, and 60 ounces of lemon soda. How many 5-ounce servings did he make? 48 36 60 144 oz 28 r4 51 4 4 10 44 40 4 There are 4 ounces left over. That is not enough for another 5-ounce serving. Drop the remainder. So, Andy made 28 five-ounce servings. Solve. Explain how you interpreted the remainder. 1. Mia bought 10 feet of wire for a © Harcourt science project. She divided the wire equally into 3 pieces. How long was each piece of wire? 3. A total of 175 players signed up for a baseball league. There are 9 teams in the league. If the players are divided among the teams, what is the greatest number of players on any team? 2. A total of 325 people will be attend- ing a sports banquet. There will be 8 people seated at each table. How many tables will be needed? 4. Jennie baked 132 cookies. She wants to divide them evenly among her 7 friends. How many cookies will she give to each friend? Reteach RW59 LESSON 12.1 Name Algebra: Patterns in Division Rick has a 1,313-page book. If he reads 14 pages a day, about how long will it take him to finish reading the book? You divide to find the answer. 1,313 14 You can estimate to find the number of days it will take Rick to read the book. Estimate: 1,313 rounds down to 1,000. 14 rounds down to 10. 1,000 10 There are zeros in the dividend and in the divisor. Cancel out one zero in each. 1,000 10 100 So, it will take Rick about 100 days to read the book. You can check this estimate by multiplying. Multiply the divisor by the quotient. 10 100 1,000 Find each quotient. Cancel out the zeros if appropriate. Write a multiplication sentence to check. The first one is done for you. 2. 560 70 3. 720 80 4. 2,100 70 5. 480 60 6. 2,500 50 7. 36,000 90 8. 24,000 40 9. 5,600 80 1. 1,500 30 50 RW60 Reteach © Harcourt 30 50 1,500 LESSON 12.2 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. 421 ,5 7 4 Step 1 Round the divisor. The number 42 rounds to 40. It can also be rounded to 50. Step 2 Round the dividend. The number 1,574 can be rounded up to 1,600 or rounded down to 1,500. Step 3 Rewrite the division problem with the compatible numbers, and solve. 40 401 ,6 0 0 30 501 ,5 0 0 So, one estimate of the quotient is 40. A second estimate is 30. © Harcourt Write two pairs of compatible numbers for each. Give two possible estimates. 1. 483 ,3 6 7 2. 764 ,1 1 7 3. 378 4 7 4. 542 ,4 3 8 5. 684 ,8 3 1 6. 732 6 ,9 7 0 Reteach RW61 LESSON 12.3 Name Divide by 2-Digit Divisors A total of 6,501 people attended the local theater. A movie was shown 20 times during a 5-day period. The same number of people attended each showing except for the first. How many people attended each showing? Step 1 Step 2 Decide where to place the ,5 0 1 206 first digit in the quotient. Are there enough thousands? ■ No, 6 20. Are there 206 ,5 0 1 enough hundreds? Yes, 65 20. The first digit goes in the hundreds place. 5 Divide the hundreds. 206 Write the 3 in the hundreds 3 place. 206 ,5 0 1 Multiply. 20 3 – 60 Subtract. 65 60 Compare. 5 20 5 Step 3 Step 4 0 Divide the tens. 205 Write the 2 in the tens place. Multiply. 20 2 Subtract. 50 40 Compare. 10 20 32 206 ,5 0 1 –6 0 50 – 40 10 0 1 Divide the ones. 201 Write the 5 in the ones place. Multiply. 20 5 Subtract. 101 100 Compare. 1 20 325 r1 206 ,5 0 1 6 0 50 40 101 100 1 So, 325 people attended each showing of the movie, with 1 more person, or 326 people, attending the first showing. ,2 1 9 1. 526 ,0 1 7 2. 819 ,0 0 8 3. 246 4. 179 2 ,4 1 8 ,8 5 0 5. 326 6. 41 8 7 ,4 0 9 RW62 Reteach © Harcourt Follow the steps above to find each quotient. LESSON 12.4 Name Correcting Quotients Maria collects postcards. She has 389 postcards in her collection. The cards are organized in albums that hold 48 postcards each. How many albums has Maria used? Divide. 389 48 Step 1 Write two pairs of compatible numbers, and estimate the answer. 9 403 6 0 8 0 0 504 Step 2 Use one of your estimates. Step 3 Divide. Since 5 48, the estimate is just right. The divisor, 48, is closer to 50. Use 8 as the first digit in the quotient. 8 r5 483 8 9 384 5 So, Maria has 8 full albums and 1 album with only 5 postcards in it. Use the steps above to find each quotient. 2. 319 7 3 5 3. 482 7 5 4. 749 5. 625 5 7 6. 272 9 2 7. 525 0 9 6 8 8. 857 © Harcourt 1. 196 7 9. 755 ,3 8 7 10. 49 8 ,3 7 2 11. 65 4 ,7 160 ,5 34 12. 54 59 Reteach RW63 LESSON 12.5 Name Practice Division Ron’s Record Shop received a shipment of 756 tapes. The tapes were packaged in 28 cartons. Each carton held the same number of tapes. How many tapes were in each carton? 756 28 Step 1 Decide where to place the first digit. Are there enough hundreds? No, 7 28. Place the first digit in the tens place. Step 2 ■ 5 6 287 2 287 5 6 56 19 Divide the 75 tens. Multiply. 28 2 Subtract. 75 56 Compare. 19 28 Step 3 Divide the 196 ones. Multiply. 28 7 Subtract. 196 196 So, each carton held 27 tapes. 27 5 6 287 56 196 196 0 You can use multiplication to check the answer. Multiply the divisor by the quotient. Add any remainder. 28 27 756 The answer checks. 1. 172 5 5 2. 263 9 6 3. 334 5 8 4. 497 2 1 5. 45 6 ,0 0 4 6. 39 7 2 ,1 1 8 7. 15 4 9 7 3 66 ,5 5 8. 54 RW64 Reteach © Harcourt Follow the steps above to find each quotient. Check by multiplying. LESSON 12.6 Name Problem Solving Strategy: Predict and Test Rhea has 253 stickers. She has them stored in equal groups in containers and has started a new container with 3 stickers in it. How many containers of stickers does she have? How many stickers are in each container? Step 1 Subtract the 3 stickers in the new container from the 253 total number of stickers. 253 3 250 There are 250 can be divided by 5. Step 2 Use predict and test to find the number of equal groups in 250. The number ends with 0, so 250 can be divided by 5. Step 3 Divide. 250 5 50 Check. 50 5 250 250 3 253 ✔ The answer checks. So, Rhea has 5 containers of stickers with 50 stickers in each container. There are 3 stickers in the new container. Predict and test to solve. © Harcourt 1. James has 467 bookmarks in his collection. He has them stored in equal groups in boxes. He then starts a new box with 5 bookmarks in it. How many boxes of bookmarks does he have? How many bookmarks are in each box? 2. Nora baked 156 brownies. She is putting them into packages with an equal number of brownies in each. She eats 2 brownies. How many packages does she make? How many brownies are in each package? Reteach RW65 LESSON 13.1 Name Algebra: Patterns in Decimal Division Kara is dividing $3 equally into 5 boxes. How much money should go into each box? $3 5 ? Using a pattern can help you find the exact answer. Write similar number sentences with zeros added to the dividends. The decimal point shifts one place to the left each time. 3,000 5 600.0 300 5 60.0 30 5 6.0 3 5 0.6 So, each box gets 0.6, or $0.60. Complete each number sentence. Look for a pattern. 1. 3,000 6 500 6 50 2. 4,500 5 900 3. 6,400 5 90 8 800 4. 2,800 8 80 7 400 280 7 30 6 45 5 64 8 28 7 36 4.5 5 6.4 8 2.8 7 Use a pattern to write the quotients. 8 6. 600 4 7. 800 5 40 8 60 4 80 5 48 64 85 8. 1,400 7 9. 13,000 5 10. 2,700 9 140 7 1,300 5 270 9 14 7 130 5 27 9 1.4 7 13 5 2.7 9 RW66 Reteach © Harcourt 5. 400 LESSON 13.2 Name Decimal Division You can use a centimeter ruler to help you divide 1.4 by 2. cm 1 2 1 stands for 1 centimeter. Each space stands for 110 , or 0.1, cm. Step 1 Step 2 Find 1.4 centimeters on the ruler. Count the number of spaces. Divide the number of spaces by 2. 14 2 7 1.4 cm cm Count over 7 spaces. 1 The seventh space is 0.7 cm. So 1.4 2 0.7. 2 0.7 cm There are 14 spaces. cm 1 2 Use the centimeter ruler to find the quotient. 1. 2.4 6 cm 1 2 2. 2.5 5 3 cm © Harcourt 4. 2.4 4 cm 1 2 1 2 2 3 5. 2.1 7 = 3 cm 7. 3.9 3 cm 1 3. 1.8 3 1 2 4 cm 1 2 1 2 3 6. 1.6 8 = 3 8. 3.3 3 3 cm cm 1 2 3 9. 0.8 8 = 3 cm 1 2 Reteach 3 RW67 LESSON 13.3 Name Divide Decimals by Whole Numbers You can use a centimeter ruler to help you divide 3.6 by 2. cm 1 stands for 1 centimeter. Each space stands for 110 or 0.1 cm. 1 2 3 4 Step 1 Step 2 Find 3.6 centimeters on the ruler. There are 3 whole centimeters. 1 23 .6 Divide them into 2 equal groups. There is 1 centimeter 2.0 1.6 in each group with 1.6 centimeters left over. 3.6 cm cm 1 2 3 4 1 cm 1 cm 1.6 cm Step 3 Count the spaces for the remaining 1.6 cm. There are 16 spaces. Divide them into 2 groups. There are 2 groups of 8 spaces. Each group is 0.8 centimeter. 1 cm 1 cm cm 1 2 There are 2 equal groups of 1.8 centimeters. 0.8 cm 0.8 cm 1.8 cm 1 2 3 4 Step 4 1 cm cm 3 4 1.8 cm 0.8 cm 1 cm 0.8 cm So, 3.6 2 = 1.8 Use the ruler to find the quotient. 1 cm 2. 22 .8 2 3 4 4. 44 .0 cm RW68 cm 1 3. 23 .2 2 3 4 2 Reteach 3 4 cm 1 1 2 3 4 1 2 3 4 6. 43 .6 5. 21 .2 1 cm 2 3 4 cm © Harcourt 1. 33 .6 LESSON 13.4 Name Problem Solving Strategy Compare Strategies Problem Since the school year began, Jill has grown 0.75 inches. Now she measures 58.5 inches. What did she measure when the year began? What strategy can you use? Work backward: What information can you use to find out how tall Jill was when the year began? You can start by using the information at the end. Now she is 58.5 inches. Then use the fact that she grew 0.75 inch. Work backward to find out how tall she was at the beginnning of the year. 58.5 current height 0.75 height she grew 57.75 inches height at beginning of school year. Work backward You can also use predict and test. Predict: She was 57 inches. Test: 57 0.75 57.75; too low Predict again, using a higher number. Predict and Test Predict: She was 57.75 inches. Test: 57.75 0.75 58.5 inches Solve and write the problem solving strategy you used: work backward or predict and test. 1. Anthony started with his favorite 2. Forty-seven baseball players need a ride to the play-off game. Each car has seat belts for 4 players and can make 2 trips. How many cars will be needed? © Harcourt number. Then he subtracted 7 from it. He multiplied this difference by 3 and then added 5. Finally he divided this number by 11. His end result was 1. What was Anthony’s favorite number? 3. The sum of 2 numbers is 40 and their difference is 2. What are the two numbers? 4. The school spent $438.75 to buy art supplies and gym supplies. The total cost of the art supplies was $230.60. How much was spent on the gym supplies? Reteach RW69 LESSON 13.5 Name Divide to Change a Fraction to a Decimal Fractions can be written as decimals by dividing the numerator by the denominator. numerator → denominatorn u m e ra to r denominator 3 To write as a decimal, divide 3 by 5. 5 0.6 3 53 .0 ← numerator 5 ↑ denominator 3.0 0 Write as a decimal. 3 50 2. 8 10 7. 6. 4 10 3. 16 100 4. 15 20 8. 4 5 9. 1 8 13. 63 100 12. 3 8 17. 3 15 18. 14 25 22. 47 50 23. 11. 16. 21. RW70 Reteach 3 4 5. 42 50 10. 4 8 15. 4 100 20. 7 8 25. 1 4 14. 721 1,000 19. 8 1,000 24. 20 40 10 25 6 25 5 8 30 1,000 © Harcourt 1. LESSON 14.1 Name Algebra: Patterns in Decimal Division Helen has $2.00, and she is putting $0.50 into each box. How many boxes can she fill? $2.00 0.50 ? 20 5 4 Using a pattern can help you find the exact answer. 2.0 0.5 4 Notice that moving the decimal point one place to the left for the divisor and the dividend will give the same answer each time. 0.2 0.05 4 So, She can fill 4 boxes with $0.50 in each box. Complete each number sentence. Look for patterns. 1. 240 6 40 24.0 40 2.4 0.06 2. 320 8 40 32.0 3. 490 7 70 40 3.2 0.08 49.0 4. 540 9 60 70 4.9 0.07 54.0 60 5.4 0.09 Use patterns to write the quotients. 5. 36 6 7. 54 6 3.6 0.6 4.2 0.7 5.4 0.6 0.36 0.06 0.42 0.07 0.54 0.06 8. 32 4 © Harcourt 6. 42 7 9. 56 8 10. 63 7 3.2 0.4 5.6 0.8 6.3 0.7 0.32 0.04 0.56 0.08 0.63 0.07 11. 28 4 12. 24 6 13. 48 6 2.8 0.4 2.4 0.6 4.8 0.6 0.28 0.04 0.24 0.06 0.48 0.06 Reteach RW 71 LESSON 14.2 Name Divide with Decimals Use the following rules to divide a decimal by another decimal: 1.) Move the decimal point in the divisor as far right as possible. 2.) Move the decimal point in the dividend to the right the same number of places as you did in the divisor. 3.) Put the decimal point in the quotient directly above the new decimal point in the dividend. 4.) Divide the numbers to obtain the quotient. Example: Move the decimal point 2 places to the right. quotient 6. 0.070 .4 2 7.0 4 2 . Divide the numbers to obtain the quotient. Find the quotient. Check by multiplying. 2. 6.4 0.8 3. 0.21 0.07 4. 0.50 0.25 5. 3.9 1.3 6. 0.96 0.24 7. 2.4 0.4 8. 0.49 0.07 © Harcourt 1. 3.2 0.4 RW 72 Reteach LESSON 14.3 Name Decimal Division When dividing a decimal by another decimal, you must change the divisor to a whole number by multiplying the divisor by 10 or 100. Whatever number you use to multiply the divisor, you must also use to multiply the dividend. Your divisor is a whole number and your dividend is larger by the same amount. 1 8 7. Divide 52.36 by 0.28. 0.2 8 5 2 3 6. 2 8 0 0. Write the division problem on graph paper. 2 4 30. 2 2 4 0. Move the decimal point in the divisor by multiplying by 100. 1 9 6. 0.28 becomes 28.0. 1 9 6. Move the decimal point in the dividend by multiplying it by 0 0 0. 100. 52.36 becomes 5236.0. Divide as if you were working with whole numbers. Find the quotient. Check by multiplying. Check: 2. 5.24 .5 2 4 Check: 3. 0.80 .8 9 6 Check: 4. 0.562 4 .4 7 2 Check: 5. 24.78 0.3 Check: 6. $39.00 $0.52 Check: 7. 9.144 0.36 Check: 8. $1.84 $0.04 Check: © Harcourt 1. 0.59 .2 5 Reteach RW73 LESSON 14.4 Name Problem Solving Skill Choose the Operation Type of Animal Ricardo and his two friends raise small animals. Ricardo buys rabbits, hamsters, mice, and gerbils. If Ricardo and his friends each take an equal number of animals, how many animals will each person get? Number Rabbits 15 Hamsters 27 Mice 36 Gerbils 9 There are 15 rabbits for 3 people. Should you multiply? or 15 3 45 Should you divide? 15 3 5 Which answer makes more sense? Since they bought only 15 rabbits, 5 rabbits each makes the most sense. You should divide. For Problems 1–6, use the table to solve each problem. Name the operation you used. 1. Ricardo and his two friends purchase animal food. They share what they buy equally. What is Ricardo’s share of the rabbit food? Type of Food Rabbit Food Amount (in pounds) 186.3 Hamster Food 53.1 Mouse Food 26.9 Gerbil Food 12.6 4. Ricardo spent $37.17 buying 2. Ricardo buys the same amount of gerbil food each month for 5 months. How much gerbil food does Ricardo buy? hamster food. What was the cost per pound for the hamster food? 5. How much animal food do 3. Ricardo pays $1.25 per pound for a month’s worth of gerbil food. How much does the gerbil food cost in all? RW 74 Reteach 6. What is Ricardo’s share of the hamster food? © Harcourt Ricardo and his friends buy in all? LESSON 15.1 Name Divisibility The rules for divisibility by 3 and 9 are special. They depend on finding the sum of the digits. • A number is divisible by 3 if the sum of the digits of the number is divisible by 3. • A number is divisible by 9 if the sum of the digits of the number is divisible by 9. 1. Decide if 615 is divisible by 3. a. What is the sum of the digits 6, 1, and 5? b. Is 12 divisible by 3? c. Is 615 divisible by 3? 2. Decide if 615 is divisible by 9. a. What is the sum of the digits 6, 1, and 5? b. Is 12 divisible by 9? c. Is 615 divisible by 9? © Harcourt Tell if each number is divisible by 3 or 9. 3. 90 4. 315 5. 390 6. 405 7. 75 8. 4,770 9. 320 10. 3,705 11. 801 12. 408 13. 117 14. 490 15. 81 16. 906 17. 432 18. 235 19. 123 20. 684 21. 963 22. 91 Reteach RW75 LESSON 15.2 Name Multiples and Least Common Multiples Sam and Mary love to count. Sam counts by 3’s and Mary counts by 4’s. Sam Mary 3, 6, 9, 12 , 15, 18, 21, 24, 27, 30, 33, . . . 4, 8, 12 , 16, 20, 24 , 28, 32, 36, 40, . . . Sam and Mary both say the numbers 12 and 24. These numbers are called the common multiples of 3 and 4. The first common multiple is 12, so it is called the least common multiple of 3 and 4. List the first 6 multiples of the number. 1. 2 2. 5 3. 6 4. 7 5. 8 6. 9 7. 10 8. 11 9. 12 Find the first 2 common multiples of each pair of numbers. 11. 4 and 8 12. 6 and 8 13. 4 and 12 © Harcourt 10. 2 and 5 Find the least common multiple of each pair of numbers. 14. 3 and 8 RW76 Reteach 15. 6 and 9 16. 5 and 8 17. 3 and 7 LESSON 15.3 Name Greatest Common Factor You can find the greatest common factor of two numbers. It is the greatest factor that the two numbers have in common. Find the greatest common factor of 9 and 15. Step 1 Step 2 Step 3 List all the factors of each number. Note the common factors. Which factor is greater? 9: 1, 3, 9 The common factors of 9 and 15 are 1 and 3. 3 is greater than 1. 15: 1, 3, 5, 15 So, the greatest common factor of 9 and 15 is 3. Use the factors given to find the greatest common factor (GCF) for each pair of numbers. 1. 10: 1, 2, 5, 10 2. 25: 1, 5, 25 21: 1, 3, 7, 21 GCF 3. 18: 1, 2, 3, 6, 9, 18 GCF 28: 1, 2, 4, 7, 14, 28 4. 35: 1, 5, 7, 35 21: 1, 3, 7, 21 49: 1, 7, 49 GCF GCF List the factors of each number. Write the greatest common factor (GCF) for each pair of numbers. The first one is done for you. © Harcourt 5. 8 1, 2, 4, 8 12 1, 2, 3, 4, 6, 12 GCF 7. 9 6. 6 24 4 GCF 8. 4 27 14 GCF GCF Reteach RW77 LESSON 15.4 Name Problem Solving Skill Identify Relationships Identifying relationships can help you solve some word problems. There is a relationship between the product of two numbers and the product of their least common multiple (LCM) and greatest common factor (GCF). Example: • Find the relationship between the product of 6 and 9, and the product of their LCM and GCF. The LCM of 6 and 9 is 18. The GCF of 6 and 9 is 3. 6 9 54 LCM GCF 18 3 54 So, the product of two numbers is equal to the product of their LCM and GCF. Use the relationship between the given numbers to complete the table. First Number Second Number Product of Numbers 6 15 90 8 4 8 Product of LCM and GCF 32 56 12 36 Use the relationships between the given numbers to solve. GCF of 4 and another number is 36. What is the other number? 3. The product of the LCM and GCF of two numbers is 55, and neither of the numbers is 1. What are the two numbers? RW78 Reteach 2. The product of two numbers is 98. The GCF of the two numbers is 7. What is their LCM? 4. The product of two numbers is 320. The GCF of the two numbers is 4, and one of the numbers is 16. What is the other number? © Harcourt 1. The product of the LCM and LESSON 15.5 Name Prime and Composite Numbers You can use squares to see if a number is prime or composite. A prime number has exactly two factors, 1 and the number itself. A composite number has more than two factors. Is the number 5 prime or composite? Is the number 8 prime or composite? 51 81 15 Only 2 arrangements of squares are possible (5 1, 1 5). The number 5 has exactly two factors, so it is a prime number. 18 42 24 More than 2 arrangements of squares are possible (8 1, 1 8, 4 2, 2 4). The number 8 has more than two factors, so it is a composite number. Draw squares to see if each number is prime or composite. Write prime or composite. 1. 7 2. 6 Write the possible arrangements of squares for each number. Then write prime or composite. The first one is done for you. © Harcourt 3. 4 1 4, 4 1, 2 2; composite 4. 9 5. 10 6. 11 7. 12 8. 13 Reteach RW79 LESSON 15.6 Name Introduction to Exponents Exponents are also called “powers.” 10 10 102 102 10 to the power of 2 10 10 10 103 103 10 to the power of 3 Show 10 to the power of 8 in four different ways. Exponent Form 108 Expanded Form Standard Form Word Form 108 10 10 10 108 100,000,000 One hundred 10 10 10 million 10 10 Write in expanded form. 1. 100 2. 10,000 3. 1,000 4. 100,000 5. 106 6. 103 7. 105 8. 107 Write in exponent form. 10. 10,000 11. 10,000,000 12. 1,000 14. 10 to the 15. 10 to the 16. 10 to the © Harcourt 9. 100,000 13. 10 to the power of 3 RW80 Reteach power of 7 power of 9 power of 10 LESSON 15.7 Name Evaluate Expressions with Exponents What number does 65 represent? 6 is called the base. The 5 is called the exponent. The exponent tells you how many times the base is used as a factor. 65 6 6 6 6 6 = 7,776 43 4 4 4 64 4 is the base and 3 is the exponent. Write the base and the exponent. 1. 46 2. 64 3. 918 4. 57 Base Base Base Base Exponent Exponent Exponent Exponent Write the equal factors. 5. 99 7. 39 8. 126 10. 810 11. 1111 12. 244 14. 64 15. 93 16. 52 © Harcourt 9. 142 6. 67 Find the value. 13. 46 Reteach RW81 LESSON 15.8 Name Exponents and Prime Factors You can think about prime factorization as a series of division problems. Begin with the number you need to factor: 48 What is the least possible prime number that divides 48? 2 Keep dividing by prime divisors until you get 1 as a quotient. 1. Divide 2 into 48. 24 24 8 2. Is the quotient 1? No. 12 22 4 Repeat the process. 3. Is the quotient 1? No. Repeat the process. 4. Is the quotient 1? No. Repeat the process. 5. Is the quotient 1? Yes. 6 21 2 3 26 1 33 Stop. Write the prime divisors as factors of 48. 48 2 2 2 2 3 Use what you know about exponents to write the factors. Write the prime factorization of the number. Use exponents when possible. 1. 12 2. 24 3. 28 4. 45 5. 36 6. 125 7. 256 8. 81 RW82 Reteach © Harcourt 48 24 3 LESSON 16.1 Name Relate Decimals to Fractions You can write a fraction or a decimal to tell what part is shaded. Model Fraction Decimal O 4 shaded parts 100 parts T Read H four hundredths 0 •. O 25 shaded parts 100 parts 0 •. 0 4 T H twenty-five hundredths 2 5 Complete the table. Model 1. Fraction Decimal O Read T H T H T H T H T H shaded parts parts •. 2. O •. 3. O © Harcourt •. 4. O •. 5. O •. Reteach RW83 LESSON 16.2 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 are equal, 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. 0 3 4 2 4 1 4 2. 2 4 5 3 8 32 82 4. 1 7 14 74 6. 3. 5. 2 3 25 35 4 5 43 53 Divide both the numerator and the denominator to name an equivalent 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. RW84 Reteach 7. 9. 7 8 10 © Harcourt 1 Multiply both the numerator and the denominator to name an equivalent fraction. LESSON 16.3 Name Compare and Order Fractions The three fractions 23, 34, and 26 are arguing about who is the largest. You can settle the argument by finding a common multiple for the denominators. Step 1 Step 2 Step 3 Find the product of all three denominators. Rename each fraction so that 72 is the denominator. 2 24 48 3 24 72 Compare the numerators. Put them in order from least to greatest. 24 48 54 72 72 72 3 4 6 72 72 is a common multiple. Use it for the denominator. 3 18 54 = 4 18 72 2 12 24 = 6 12 72 2 2 3 6 3 4 So, 34 is the largest fraction. Find the product of the denominators. 2 3 5 5 4 7 1. , , 2 1 1 9 3 2 2. , , 1 1 1 2 5 8 3. , , Rename the fractions by using a common denominator. 2 3 5 5 4 7 © Harcourt 4. , , 2 1 1 9 3 2 5. , , 1 1 1 2 5 8 6. , , Compare and order from least to greatest. 2 3 5 5 4 7 7. , , 2 1 1 9 3 2 8. , , 1 1 1 2 5 8 9. , , Reteach RW85 LESSON 16.4 Name Simplest Form You can use fraction bars to find the simplest form of a fraction. Find the simplest form for 132. Step 1 Model 132 with fraction bars. Step 2 Line up other fraction bars to find all the equivalent fractions for 132. You can see that 28 and 14 are equivalent fractions for 132. Step 3 The equivalent fraction that has the largest fraction bar possible is in the simplest form. 1 12 1 12 1 12 3 12 1 12 1 12 1 12 3 12 1 8 1 8 1 4 2 8 1 4 So, 14 is the simplest form of 132. Use the fraction bar outlines below to model each fraction and equivalent fractions. Divide the outline into equal parts or keep it whole. Write the fraction in its simplest form. 9 12 1. Simplest form 4 12 Simplest form RW86 Reteach © Harcourt 2. LESSON 16.5 Name Understand Mixed Numbers John drank 234 cartons of milk with his lunch. Whole Number The number 234 is a mixed number. A mixed number is made up of a whole number and a fraction. → → 234 Fraction 3 4 2 cartons carton In the mixed number 234, the whole number 2 represents two whole cartons of milk. In the mixed number 234, the fraction 3 represents a part of another carton. 4 You can divide all three cartons into 4 equal parts to show how many fourths John drank. 1 1 4 There are 11 shaded parts. Each part is 14 carton. cartons So, John drank 141, or 234, cartons of milk. Write both a fraction and a mixed number for each figure. 2. 3. 4. 5. 6. © Harcourt 1. Reteach RW87 LESSON 16.6 Name Problem Solving Strategy Make a Model Trisha spent 34 hour on math homework, 38 hour on science, and 12 hour on language arts. Which homework did she spend the most time on? You can make a model to solve this problem. Step 1 For each fraction, draw a box. Shade the box to show the fraction. Step 2 Find the LCD, and divide each box into that many equal parts. 3 4 3 8 1 2 6 8 3 8 4 8 The LCD is 8. 6 Step 3 Compare the numerators. is the 8 greatest fraction. So, Trisha spent the most time on math homework. Make a model to solve. he used flour in three different recipes. The amounts were 34 cup, 1 2 cup, and 4 cup. What was the 4 least amount called for? pound ground 3. Nick bought 2 3 beef, 1112 pound ground turkey, and 34 pound ground veal. Which meat did he buy the most of? RW88 Reteach of a mile from her 2. Karen walked 5 6 house to a friend’s house. Joe walked 172 of a mile to his friend’s house. Who walked a greater distance? of the 4. In the store display, 2 5 T-shirts were yellow and 14 were blue. Were there more yellow or blue T-shirts? © Harcourt 1. Joe loves to cook. Last weekend LESSON 17.1 Name Add and Subtract Like Fractions The denominators must be the same when adding or subtracting fractions. Add 26 16. Step 1 Step 2 Step 3 Are the denominators the same? Yes. Add the numerators. The denominator stays the same. 2 ← 2 sixths 6 1 ← 1 sixth 6 3 sixths Write the sum over the denominator. Write it in simplest form. 2 6 1 6 3 1 6 2 2 6 1 6 So, 26 16 12. To subtract like fractions, subtract the numerators. Remember, the denominator stays the same. Then write the difference over the denominator. Find the sum or difference. Write the answer in simplest form. 1 5 2 5 2. 8 9 7 9 5. 2 6 3 6 8. 6 8 1 8 11. 1. © Harcourt 4. 7. 10. 3 7 2 7 3. 4 9 2 9 7 8 1 8 6. 1 8 3 8 9. 4 6 1 6 12. 9 12 5 12 6 10 3 10 7 14 4 14 Reteach RW89 LESSON 17.2 Name Add Unlike Fractions Use fraction bars to add fractions. 1 1 Show with fraction bars. 3 6 Now, find like fraction bars that 1 1 fit exactly under the sum . 3 6 So, three sixth bars fit under the sum. 3 1 equals in simplest form. 6 2 1 1 1 So, . 3 6 2 1 1 3 1 6 ←→ 1 1 3 1 6 1 6 1 6 1 6 Use fraction bars to find the sum. 2. 1 1 2 1 8 3. 1 8 1 8 1 1 10 10 5. 1 1 10 10 1 10 1 5 1 2 3 5 3 10 11. RW90 Reteach 1 5 1 5 1 2 6. 1 3 7. 1 1 1 1 1 12 12 12 12 12 1 1 5 1 1 6 1 3 4. 1 1 1 10 10 1 1 1 3 1 4 2 3 3 4 9. 1 4 5 6 13. 8. 12. 1 4 1 3 2 6 1 4 10. 1 3 1 2 3 8 3 4 14. 1 6 2 3 © Harcourt 1. LESSON 17.3 Name Subtract Unlike Fractions 1 Use fraction bars to subtract fractions. 5 1 Show with fraction bars. 6 3 Now, find like fraction bars that fit 5 1 exactly under the difference . 6 3 So, three sixth bars fit under the difference. 3 1 equals in simplest form. 6 2 5 1 1 So, . 6 3 2 1 6 1 6 1 6 1 3 1 6 1 6 ? (difference) 1 1 6 1 6 1 6 1 6 1 3 1 6 ? 1 6 1 6 1 6 Use fraction bars to find the difference. 1. 2. 1 1 1 1 1 1 1 1 1 1 1 1 1 12 12 12 12 12 12 12 12 12 12 12 1 3 1 1 1 10 10 10 ? 3. 1 6 1 6 1 6 1 2 1 8 © Harcourt 1 3 3 4 1 2 7. 1 6 1 8 1 8 1 8 1 5 1 5 1 1 1 10 10 10 6 8 1 4 8. 1 8 1 1 5 ? 1 8 ? 6. 1 3 1 6 1 8 1 4 1 1 6 1 ? 5. ? 4. 1 1 6 1 2 1 5 ? 2 3 1 2 9. Reteach RW91 LESSON 17.4 Name Estimate Sums and Differences You can round fractions to 0, to 12, or to 1 to estimate sums and differences. Estimate the sum 35 89. Step 1 Step 2 Step 3 Find 53 on the number line. Is it closest to 0, 12, or 1? The fraction 35 is closest to 12. 1 5 0 5 2 5 3 5 1 2 0 0 9 8 9 Find on the number line. Is it closest to 0, 12, or 1? The fraction 89 is closest to 1. 1 9 4 5 2 9 3 9 4 9 1 5 9 6 9 7 9 To estimate the sum 35 89, add the two rounded numbers. 1 2 8 9 9 9 1 1 2 0 5 5 1 112 So, 35 89 is about 112. Use the number lines to estimate whether each fraction is closest to 0, to or to 1. Then find the sum or difference. The first one is done for you. 1 , 2 0 6 1 6 2 6 3 6 1 2 0 1. 4 6 1 2 1 8 0 4 6 5 6 6 6 0 8 1 0 2. 2 6 1 8 2 8 3 8 4 8 5 8 6 8 7 8 1 1 2 7 8 8 8 3. 5 6 3 8 4. 4 6 © Harcourt 1 2 3 8 RW92 Reteach 5. 7 8 5 6 6. 1 6 7 8 LESSON 17.5 Name Use Least Common Denominators Tim and Barbara shared a pizza. Tim ate 13 of the pizza, and Barbara ate 49. How much of the pizza did they eat in all? You add 13 49 to answer this question. Use the least common denominator to add the unlike fractions. The least common denominator is the least common multiple of the denominators. multiples of 3: 3, 6, 9, 12 Step 1: Find the least common multiple of the denominators. multiples of 9: 9, 18, 27 1 3 4 4 = , and = 3 9 9 9 Step 2: Rename each fraction, using the least common denominator. Step 3: Add the like fractions. Step 4: This sum is already in simplest form. 3 4 7 + = 9 9 9 7 = 9 So, Barbara and Tim ate 79 of the pizza. To subtract unlike fractions, follow steps 1 and 2. Then subtract and simplify. Find the least common denominator. 1 3 1 5 © Harcourt 1. 2 5 1 2 3 6 2. 3 4 7 9 3. 5 6 4. Find the sum or difference. Write the answer in simplest form. 5 6 3 4 6. 5 6 7 12 9. 5. 8. 7 12 1 4 1 2 9 12 6 10 3 5 7. 3 4 1 3 10. Reteach RW93 LESSON 17.6 Name Add and Subtract Unlike Fractions When you add or subtract two fractions with unlike denominators, you need to make the denominators the same. Find the least common denominator (LCD), and change the fractions to like fractions with that denominator. Add. 23 14 n Step 1 Step 2 Step 3 Find the multiples of both denominators to determine the LCM. Use the LCD to make like fractions. Multiply the numerator and denominator by the same number. 8 2 24 3 3 4 12 Add the fractions. 8 12 3 12 11 12 So, n 11 . 12 3 3, 6, 9, 12, . . . 4 4, 8, 12, 16, . . . The LCM of 3 and 4 is 12. So, the LCD of 23 and 14 is 12. 3 1 13 4 4 3 12 So, the sum of 2 11 3 1 . This answer 4 12 is in simplest form. To subtract fractions with unlike denominators, follow these 3 steps. However, in Step 3, subtract the fractions and write the answer in simplest form. Write like fractions. Then find the sum or difference. Write the answer in simplest form. 3 1 3 4 9 Simplest form: RW94 Reteach 1 2. 2 2 5 1 2 2 5 Simplest form: 3 3. 9 1 6 3 9 1 6 Simplest form: © Harcourt 1 1. LESSON 17.7 Name Problem Solving Strategy Work Backward The students in Jason’s class started measuring their heights at the beginning of January. By March 1, Jason had grown 3 3 4 inch. In February, Jason grew 8 inch. How much did he grow in January? You can solve the problem by working backward. Start with the amount he had grown by March 1, and subtract the amount he grew in February. 3 3 Find by using the LCD method. 4 8 The LCD of 4 and 8 is 8. Change each fraction into eighths, and subtract the numerators. 1 3 2 6 4 2 8 1 3 1 3 8 1 8 6 3 3 8 8 8 So, Jason grew 38 inch in January. Work backward to solve. 1. Paula is in Jason’s class. By March 1, she had grown 78 inch. In February, she grew 14 inch. © Harcourt How much did she grow in January? 3. Harry started the day by trading 5 of his comic books for 7 of Jenny’s. Next, he bought 8 at the store. Then he gave Tom 9 comic books. Harry came home with 12 comic books. How many did Harry start the day with? 2. Sid is in Jason’s class. By April 1, he had grown 1156 inch. In March, he grew 18 inch, and in February, he grew 38 inch. How much did he grow in January? 4. Wesley started with his favorite number. Then he subtracted 7 from it. He multiplied this difference by 3 and then added 5. Finally, he divided this number by 11. His end result was 1. What is Wesley’s favorite number? Reteach RW95 LESSON 18.1 Name Add Mixed Numbers Fred and Gregg are going to put up a tent. They need two pieces of rope to secure the tent. One has to be 314 feet long and the other 212 feet long. How much rope do they need? To find the answer, you must add 314 212. You can add mixed numbers by following these steps. Step 1 Add the whole numbers. 3 2 5 Step 2 Find the LCD. Write equivalent fractions. Add the fractions. multiples of 4: 4, 8, 12 multiples of 2: 2, 4, 6 11 1 4 1 4 12 2 22 4 1 1 4 2 1 2 3 4 4 4 Step 3 Add the sum of the whole numbers to the sum of the fractions. Write the answer in simplest form if needed. 3 3 5 5 4 4 1 1 3 So, 3 2 5. 4 2 4 1. 5. 5 3 8 1 2 8 2. 1 7 2 1 2 3 RW96 6. Reteach 1 6 3 1 2 12 3 4 5 1 2 10 3. 7. 1 4 4 1 2 4 1 4 2 3 3 8 4. 8. 3 5 7 3 1 7 3 3 4 1 2 8 © Harcourt Find the sum in simplest form. LESSON 18.2 Name Subtract Mixed Numbers 1 Sonia cut out a pattern for a new skirt from the 32 yards of material she bought. The pattern used 213 yards. How much material was left? 1 You can answer the question by subtracting, 32 213. To subtract mixed numbers, follow these steps. Step 1 Step 2 Find the LCD of the fractions by listing the multiples of each number. Change the fractions into like fractions with 6 as the denominator. Multiples of 2: 2, 4, 6, 8, Multiples of 3: 3, 6, 9, 12, 1 3 3 2 3 6 10 15 1 2 2 3 2 6 Since 6 is the first common multiple, it is the least common multiple. Step 3 Step 4 Subtract the fractions. Subtract the whole numbers. 1 3 3 3 2 6 1 2 2 2 3 6 1 6 1 3 3 3 2 6 1 2 2 2 3 6 1 1 6 1 So, Sonia has 1 yards left. 6 Subtract. Write the answer in simplest form. 4 8 4 5 10 1 1 1 1 10 10 © Harcourt 1. 4 4. 1 4 8 8 3 12 1 3 1 1 4 12 2. 5. 2 4 6 6 3 6 1 1 4 4 6 6 7 7 2 2 8 8 1 4 1 1 2 8 3. 6. 3 9 7 7 4 12 5 5 4 4 12 12 7 7 6 6 9 9 2 6 4 4 3 9 Reteach RW97 LESSON 18.3 Name Subtraction with Renaming Wayne had 414 feet of rope. He gave 223 feet to his friend. How much rope did he have left? You can answer the question by subtracting, 414 – 223. To subtract mixed numbers, follow these steps. Step 1 Step 2 Find the LCD. Change each fraction into a fraction with the denominator 12. 4, 8, 12, Multiples of 3: 3, 6, 9, 16 12 So, 12 is the LCD. 13 3 4 3 12 24 8 3 4 12 Step 3 Step 4 Replace the unlike fractions with the like fractions. Rename 1 whole from 4 to subtract the fractions. Rename the 1 as 1122. 3 1 4 4 12 4 8 2 2 2 12 3 3 15 4 3 12 12 8 8 2 2 12 12 Step 5 Subtract the fractions. 3 15 4 3 12 12 8 8 2 2 12 12 7 12 Step 6 Subtract the whole numbers. 3 15 4 3 12 12 8 8 2 2 12 12 7 1 12 7 So, Wayne has 1 feet left. 12 3 12 15 12 12 12 Find the difference in simplest form. 1. 1 5 4 1 2 RW98 2. Reteach 1 6 8 1 2 4 3. 3 5 10 3 1 5 4. 1 4 6 2 2 3 © Harcourt Multiples of 4: LESSON 18.4 Name Practice with Mixed Numbers Larry made 256 pounds of baked ziti. He and his brother ate 113 pounds. How much was left over? Use fraction bars to find the answer. Subtract. 256 113 11 1 1 Estimate: about 112 pound 1 1 1 1 1 6 6 6 6 6 256 113 136 or 112 pounds Add or subtract. Write the answer in simplest form. Estimate to check. 1. 5. 2. 3. 6. 7. 5 1 12 1 2 6 13 3 15 1 2 5 4 9 5 2 2 3 3 2 8 7 4 8 4. 2 6 3 10 1 12 3 5 4 7 3 8 8. 2 4 5 1 1 3 1 7 12 1 2 6 Algebra Find the value of n. 1 2 © Harcourt 9. 3 n 5 6 7 1 7 11. 4 n 2 1 8 1 2 10. n 4 6 1 6 1 3 12. n 11 15 Reteach RW99 LESSON 18.5 Name Problem Solving Skill: Multistep Problems Hank bought a piece of wood that was 8 feet long. He used 114 feet for a shelf in his room and 214 feet for a shelf in his sister’s room. Then he made a box using another 314 feet. How much of the wood does he have left? You can solve the problem by doing more than one operation. First add the 114 feet for his shelf, the 214 feet for his sister’s shelf, and the 314 feet for his box. 1 1 4 1 2 4 1 3 4 3 6 Then you would subtract the total 4 amount of 634 feet from the 8 feet he bought. 8 634 114 1 So, Hank has 1 feet of wood left. 4 Solve. made four projects. The first one used 312 feet, the second one used 241 feet, the third one used 234 feet, and the fourth one used 141 feet. How much wood did he have left? 3. On Monday Charley drove 32 miles, on Tuesday 58 miles, on Wednesday 88 miles, and on Thursday 94 miles. His total for five days was 335 miles. How far did he drive on Friday? RW100 Reteach 2. Nancy read every day for five days. She read 8 pages on Monday, 12 pages on Tuesday, 25 pages on Thursday, and 40 pages on Friday. If she read a total of 156 pages, how many pages did she read on Wednesday? 4. Lacy was serving pizza at a party. She gave the first person 18 of the pizza, the second person 38, and the third person 14 of the pizza. How much of the pizza is left? © Harcourt 1. Ralph bought 12 feet of wood. He LESSON 19.1 Name Multiply Fractions and Whole Numbers Hector had 12 baseball cards. He gave 23 of them to his friend Ned. How many baseball cards did he give to Ned? You can answer the question by multiplying 23 12. To multiply a fraction and a whole number you can use a model: Step 1 Draw 12 rectangles to show the cards. Step 2 The denominator of the fraction 23 is 3. This means there are 3 equal parts, so divide the rectangles into 3 equal groups. Step 3 The numerator of the fraction 2 3 is 2. This means there are 2 parts given, so shade 2 of the groups. Step 4 Count the shaded rectangles, or cards. There are 8 cards. 2 So, 12 8. 3 Write the number sentence each model represents. © Harcourt 1. 2. 3. Draw a picture to help you multiply. Find the product. 4 9 4. 27 1 6 5. 12 3 5 6. 20 Reteach RW101 LESSON 19.2 Name Multiply a Fraction by a Fraction 3 3 Multiply. 4 5 To multiply fractions you can use a rectangle model. Follow these guidelines: • Draw a rectangle, and divide the rectangle into 5 equal columns. This is for the denominator of 35. • Shade 3 of the columns. This is for the numerator of 53. • Divide the rectangle into 4 equal rows. This is for the denominator of 43. ;;; ;; ;; • Shade 3 of the rows with diagonal lines. This is for the numerator of 43. • Count how many pieces the rectangle is divided into. There are 20 pieces. This is the new denominator. • Count how many pieces have overlapping lines and shading. There are 9. This is the new numerator. 3 3 9 So, . 4 5 20 1 3 5 6 2. 2 5 1 3 5. 1 4 5 6 8. 3 5 3 5 11. 1. 4. 7. 10. RW102 Reteach 5 8 3 4 3. 1 4 3 8 1 2 7 8 6. 5 6 3 4 2 3 1 4 9. 2 7 3 4 4 5 1 2 12. 5 9 1 2 © Harcourt Divide and shade a rectangle model to find the product. LESSON 19.3 Name Multiply Fractions and Mixed Numbers Multiply. 23 214 You can find the product by using the Distributive Property. The Distributive Property allows you to break apart numbers to multiply. To multiply a fraction and a mixed number, break apart the mixed number. 2 1 2 1 2 2 Break apart the mixed number. 3 4 3 4 2 2 1 Multiply each part. 2 3 3 4 4 2 3 12 16 2 Find the LCD and rename the fractions. 12 12 18 1 Add the products. Simplify the sum. 1 12 2 2 1 1 So, 2 1. 3 4 2 Multiply. Write the answer in simplest form. 1 3 1 5 2. 2 1 6 2 3 4. 2 1 3 1 2 6. 4 3 8 1 4 8. 2 1. 3 © Harcourt 3. 3 5. 3 7. 1 1 2 3 4 1 4 5 6 1 8 1 4 4 5 1 2 Reteach RW103 LESSON 19.4 Name Multiply with Mixed Numbers 2 3 1 2 Multiply. 1 1 To multiply two mixed numbers, follow the same steps you use to multiply a fraction and a mixed number. Step 1 Step 2 Step 3 Write each mixed number as a fraction. Multiply the fractions, Write the product as a mixed number in simplest form. (3 1) 2 3 2 3 5 3 1 (2 1) 1 2 1 2 3 2 1 5 3 15 32 6 or cancel the 3 in the numerator and denominator. 1 5 3 5 32 2 15 3 1 2 2 6 6 2 or 5 1 2 2 2 1 Multiply. Write the answer in simplest form. 1 2 1 5 2. 1 1 1 2 1 4 4. 1 3 3. 1 1 1 2 3 5 5. 6 1 1 5 1 2 7. 1 1 RW104 Reteach 1 3 3 4 2 3 1 2 1 2 2 3 6. 1 1 1 2 3 5 8. 2 1 © Harcourt 1. 2 1 LESSON 19.5 Name Problem Solving Skill Sequence and Prioritize Information The Perez family planned an evening event that includes a snack, dinner, dessert, and game time. 1 There are 6 hours planned for the evening. of the evening’s time 3 1 will be devoted to dinner. of the time will be spent on having a 6 snack. How many hours will be spent on playing games and dessert? Sequencing the information may help you solve this problem. Start with events for which you have some information. Event Time 1 3 1 6 hours total 1 hour for snacks 6 6 hours total 2 hours for dinner Dinner Snack Now subtract the snack and dinner time to find how much time can be devoted to games and dessert. Event Games and Dessert Time Total time spent on dinner and snacks 3 hours 6 hours total 3 hours for dinner and snacks 3 hours left for games and dessert Sequence the information by starting with the information you know. Then solve the problem. © Harcourt 1. John drives a total of 350 miles a day. He makes 3 stops. He drives 150 miles to his first stop. From the second stop to the third stop, he drives 75 miles. How many miles does he drive from the first stop to the second stop? 2. Mary spent $45.00 altogether at the store. She bought some food for $32.75 and some school supplies. How much did she spend on school supplies? Reteach RW105 LESSON 20.1 Name Divide Fractions You can use pictures to model division of fractions. 4 51 1 3 5 10 Step 1: Draw 5 whole circles and Step 1: Draw one whole rectangle shade all 5. and shade four fifths of it. Step 2: Divide each circle into thirds. 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 Step 3: Count the number of shaded thirds. There are 15 thirds in 5. So, 5 1 15. 3 Step 2: Divide the rectangle into tenths. 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 Step 3: Count the number of shaded tenths. There are 8 tenths in 4 . So, 4 1 8. 5 5 10 Draw a model for the division problem and find the quotient. 2 3 1 9 2. 2 3 4 1 8 4. 3 1 2 1 8 6. 3. 5. RW106 Reteach 1 5 1 4 1 3 1 6 © Harcourt 1. LESSON 20.2 Name Reciprocals Reciprocals are two fractions that have a product of 1. Fractions: The reciprocal of 3 is 8 . 8 3 3 8 2 4 1 8 3 24 To find the reciprocal of a fraction, switch the numerator and denominator. Whole Numbers: 77 . 1 The reciprocal of 7 is 1 . 1 7 To find the reciprocal of a whole number, first write it as a fraction. Then switch the numerator and denominator. Mixed Numbers: 7 52 1 3 3 7 is 3. The reciprocal of 1 3 17 To find the reciprocal of a mixed number, first write it as a fraction. Then switch the numerator and denominator. Are the two numbers reciprocals? Write yes or no. 1 9 1. and 19 5 13 3 10 10 3 3. 1 and 1 10 1 10 7. 2 and 2. and 3 5 5. and 2 6. and 3 5 8 5 4. 5 and 1 5 1 4 4 9 8. and 7 12 12 7 Write the reciprocal of each number. © Harcourt 5 12 1 7 10. 11. 6 2 11 15. 11 16. 1 9. 14. 3 8 5 9 12. 3 1 2 17. 6 5 13. 18. 100 Reteach RW107 LESSON 20.3 Name Divide Whole Numbers by Fractions Beth is working on a science project. She needs pieces of wire that are 2 yd long for the project. She bought a piece of 3 wire that is 6 yd long at the hardware store. How many 2 pieces can she cut from this 6-yd piece? 3 6 Step 1: Write a division sentence to Think: Write 6 as 6 2 . 1 3 1 find this amount. 6 Think: The reciprocal 3 Step 2: Use the reciprocal of 1 2 the divisor to write a of 2 is 3 . 3 2 multiplication problem. 18 6 3 9 1 2 2 Step 3: Multiply. So, Beth can cut 9 pieces of wire. Use the reciprocal to write a multiplication problem. Solve the problem. Write the answer in simplest form. 2. 5 1 2 3. 10 4 5 6. 8 3 4 7. 18 5. 12 4 5 3 4 10. 16 3 8 14. 9 9. 6 13. 9 RW108 Reteach 1 5 2 3 4. 27 3 5 3 8 8. 7 4 5 6 7 12. 2 3 20 16. 20 11. 9 15. 6 3 10 4 5 © Harcourt 1 8 8 3 24 1 1 1. 3 LESSON 20.4 Name Divide Fractions Connie is working on a craft project. She needs 3 -yd pieces 8 3 of ribbon for the project. She bought a -yd piece of ribbon 4 at the craft store. How many 3 -yd pieces can she cut from 3 -yd piece? 8 4 3 Step 1: Write a division sentence to 3 4 8 find this amount. 3 Think: The reciprocal 8 Step 2: Use the reciprocal of 4 3 the divisor to write a of 3 is 8 . 8 3 multiplication problem. 3 8 2 4 2 Step 3: Multiply. 4 3 1 2 So, Connie can cut 2 pieces of ribbon. Use the reciprocal to write a multiplication problem. Solve the problem. Write the answer in simplest form. 3 8 1. 24 5 6 1 3 © Harcourt 5. 1 4 1 2 1 3 1 4 2 3 3. 4 5 5 8 3 4 7. 6 1 4 1 3 10. 1 2 3 4 1 4 14. 1 2 3 4 5 6. 9. 2 13. 5 9 2. 5 12 5 8 4. 1 15 8. 1 7 1 3 1 2 12. 1 1 5 6 1 3 16. 1 11. 15. 1 3 2 3 1 2 1 3 Reteach RW109 LESSON 20.5 Name Problem Solving Strategy Solve a Simpler Problem The bank gave Jim a loan of $4,000. This is 1 of the amount 8 they gave him last year. How much did the bank loan Jim last year? You can solve a more difficult problem by first solving a simpler one. Step 1: If you can, change the numbers so that they are easier to work with. Let 4 represent 4,000. Step 2: Write the problem, using the new number. 41 4 8 8 1 1 Step 3: Solve the problem, using the new number. 4 8 32 1 1 Step 4: Adjust the answer, using the original number. Multiply the answer by 1,000 to adjust. Think: The reciprocal of 1 is 8 . 8 1 So, 32 1,000 32,000. So, the bank loaned Jim $32,000 last year. Use a simpler problem to solve. Then adjust your answer. bike. This was 2 of his 3 savings. How much money was in his savings? 2. The distance from Barbara’s house to Raymond’s house is 3,200 miles. You can travel 3 of 4 the distance by highway. How many miles cannot be traveled by highway? RW110 Reteach © Harcourt 1. Charles spent $600 on a new LESSON 21.1 Name Integers Integers are whole numbers and their opposites. The positive integers are to the right of 0. The negative integers are to the left of 0. 0 is neither positive nor negative. negative integers positive integers -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 neither positive nor negative How far from 0 is 5 and in what direction? How far from 0 is 5 and in what direction? 5 and 5 are opposites. They are the same distance from 0 on a number line, but in opposite directions. Some other opposites are 1 and 1, 6 and 6. The distance a number is from 0 is referred to as its absolute value. 5 and 5 are both 5 units from 0. So, 5 and 5 both equal 5. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 How far from 0 is 8? 8 units The symbol means absolute value. 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 So, 8 8 because it is 8 units from 0. Name the integer which describes each situation below. 1. an increase in price 2. 10 minutes before © Harcourt of $30.00 3. 4 feet above ground school starts Write the opposite of each integer. 4. 6 5. 28 6. 1,489 7. 2,000 Name each integer’s absolute value. 8. 34 9. 30 10. 235 Reteach RW111 LESSON 21.2 Name Compare and Order Integers Integers increase as you move right on a number line and decrease as you move left. Compare 7 to 8. Use , or . -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 The numbers increase as you move right, and 7 is to the right of 8. So, 7 8. Order 5, 5, 3, and 7 from least to greatest. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 Look at the number line. Since the numbers increase as you move right on the number line, the order from least to greatest is 5, 3, 5, 7. Name the integer that is 1 less than 8. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 1 less means a decrease, and decreasing amounts move left on a number line. So, 1 to the left of 8 is 9. Compare. Write , , or for each 1. 6 6 2. 7 2 . 3. 3 1 4. 10 8 Order each set of integers from greatest to least. 6. 2, 5, 6, 4 7. 10, 10, 1, 1 8. 4, 2, 5, 1 © Harcourt 5. 9, 1, 0, 4 Name the integer that is one less than the given integer. 9. 6 10. 4 11. 0 12. 21 13. 25 Name the integer that is one more than the given integer. 14. 10 RW112 15. 4 Reteach 16. 3 17. 32 18. 1 LESSON 21.3 Name Add Integers This pan balance “weighs” positive and negative numbers. Negative numbers go on the left of the balance and positive numbers go on the right. 8 2 1 3 7 11 Find 11 8. Find 2 7. Find 1 3. The scale will tip to the left side, because it is 3 “heavier.” The scale will tip to the right side, because it is 5 “heavier.” Both 1 and 3 go on the left side. The scale will tip to the left side, because it is 4 “heavier.” Find how much “heavier” the lower side is. 1. 2. 3. 2 7 4. 9 10 5. 101 8 6 101 6. 2 3 5 9 Solve. © Harcourt 7. 7 4 11. 9 3 8. 10 4 12. 9 1 9. 6 0 13. 5 3 2 10. 5 4 14. 3 5 Reasoning Without adding, tell whether the sum will be negative, positive, or zero. 15. 18 25 16. 9 20 17. 427 427 18. 75 19 Reteach RW113 LESSON 21.4 Name Subtract Integers You can use drawings to subtract integers. Use circles with signs to represent positive integers and circles with signs to represent negative integers. Find 8 6. Step 1 Step 2 Step 3 First, make a drawing of 8. To subtract 6, take away 6 of the circles. The number of circles left represents the difference. So, 8 6 2. Find 5 3. Step 1 Step 2 Step 3 Step 4 First, make a drawing of 5. You cannot subtract 3 until you add positive circles. To subtract 3, take away 3 of the circles. The number of circles left represents the difference. Add both the positive number and its opposite; 3 and 3. So, 5 3 8. 1. 5 3 2. 6 2 3. 7 3 4. 9 2 5. 8 1 6. 10 4 7. 4 4 8. 7 6 11. 7 6 12. 9 3 RW114 Reteach 10. 8 2 Solve. 9. 5 4 © Harcourt LESSON 21.5 Name Subtract Integers When you subtract a positive integer from a negative integer, you add the opposite. Replace the negative integer with its opposite, and change the operation to addition. Example: Find 6 2. Step 1 Find the opposite of the number being subtracted. The opposite of 2 is 2. Step 2 Add the opposite. Change the operation to addition and replace the number being subtracted with its opposite. 6 2 6 2 Step 3 Add the negative numbers. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 So, 6 2 8. Solve. 1. 5 3 2. 4 7 3. 2 8 4. 7 1 5. 2 1 6. 4 6 7. 7 3 8. 0 8 9. 9 2 10. 5 5 11. 2 4 12. 5 1 © Harcourt Algebra Complete. 13. 2 6 2 14. 7 7 7 15. 3 9 3 16. 5 5 5 Compare. Write , , or in each 2 6 . 17. 5 6 18. 2 8 19. 3 7 20. 5 2 3 7 5 7 6 1 Reteach RW115 LESSON 21.6 Name Problem Solving Strategy Draw a Diagram Draw a diagram to solve. PROBLEM: Erik and his friends are practicing scuba diving in a 20-foot-deep, 30-foot-long pool for class. First, they had to go down 15 feet. Then they had to go down 2 more feet to practice clearing the water out of their masks. Then they went up 9 feet and back down 10 feet. At what depth are they now? When solving problems involving integers, first look for key words to determine the positive and negative numbers. 0 1 A few key words are listed. Step 1 down 15 feet (15) Step 2 down 2 more feet (2) Step 3 up 9 feet (9) Step 4 down 5 feet (5) Step 4 The key words in the problem are up () and down (). Use the key words to step out the problem. Step 3 positive () up above raise Step 2 negative () down below drop Step 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Making or using a diagram will help you visualize the steps to solve the problem. Look at the diagram. Draw a diagram to solve. 1. The next day a new set of divers were practicing in the pool. They began by diving 19 feet. Then they rose 9 feet, went back down 7 feet and up 5 feet. Where are they now? RW116 Reteach 2. Jan went swimming. She dove 15 feet, came up 8 feet, went down 1 foot, and came back up 5 more feet. Where is she now? © Harcourt The last step shows that Erik and his friends will go from 8 feet down 5 more (5) feet. The students are now at 13 feet below the surface, or 13 feet. LESSON 22.1 Name Graph Relationships Number of dollars, x 1 2 3 4 Rolls of Pennies Bill put his collection of pennies in Number of rolls of pennies, y 2 4 6 8 $0.50 rolls. Every two rolls held $1. y He made a table to show the 9 relationship between number of 8 dollars and number of rolls of pennies. Bill wrote the data as ordered pairs: (1,2), (2,4), (3,6), and (4,8). Then he graphed the points and drew a line to connect them. The ordered pair (2,4) means that Bill has $2 if he has 4 rolls of pennies. 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 x Numbers of Dollars Write the ordered pairs. Then graph the ordered pairs. 1. Input, x 1 2 3 4 2. Output, y 4 8 12 16 3. Input, x 2 4 6 8 © Harcourt Output, y 3 5 7 9 Input, x 8 10 12 14 Output, y 4 5 6 7 4. Input, x 8 7 6 5 Output, y 6 5 4 3 5. In the problem with Bill’s pennies, 6. In the problem with Bill’s pennies, what does the ordered pair (3,6) mean? what would be the next ordered pair? 7. How did you decide the answer for problem 6? Reteach RW117 LESSON 22.2 Name Graph Integers on the Coordinate Plane A coordinate plane is For (2,4), move 2 units For (2,4), move 2 units formed by a horizontal left on the x-axis and 4 right on the x-axis and 4 units up on the y-axis. units up on the y-axis. number line (x-axis) and a vertical number line y (y-axis), which intersect. The point at +5 +4 (+2,+4) which the two lines intersect is named (-2,+4) + 3 by the ordered pair (0,0) and is called +2 the origin. The numbers in the ordered +1 x pair are called coordinates. - - - - + + + + + To plot ordered pairs on a coordinate plane, begin at the origin. Positive numbers are to the right and above (0,0). Negative numbers are to the left and below (0,0). 5 4 3 2 10 -1 -2 -3 -4 (-2,-4) -5 For (2,4), move 2 units left on the x-axis and 4 units down on the y-axis. 1 2 3 4 5 (+2,-4) For (2,4), move 2 units right on the x-axis and 4 units down on the y-axis. Write the ordered pair described. Then plot and label the point on the coordinate plane. 1. Start at the origin. Move right 5 units 2. Start at the origin. Move left 4 units and up 1 unit. 3. Start at the origin. Move right 2 units and down 3 units. 4. Start at the origin. Move left 1 unit D +5 +4 +3 +2 +1 y x C -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 -1 F -2 A -3 -4 -5 B and down 3 units. Identify the ordered pair for each point on the coordinate plane above. 5. Point A 6. Point B 7. Point C 8. Point D 9. Point E 10. Point F RW118 Reteach E © Harcourt and up 3 units. LESSON 22.3 Name Use an Equation to Graph Sally earns $3 for each hour that she baby-sits. She can graph how much she earns according to the number of hours she baby-sits. Sally can show this relationship with x and y values in a function table. Hours, x 1 2 3 4 5 Earns, y $3 $6 $9 $12 $15 This table shows the amount of money that Sally can earn for the hours she baby-sits. There is a pattern. Find the rule and write the equation that shows the relationship between x and y. Equation: y 3x Rule: Multiply x by 3. Sally knows that for any number of hours, she earns three times that number. She can find out exactly how much she will make by substituting the number of hours (x) and multiplying it by 3 to get y, how much she earns. Use a rule to complete the table. Then write the equation. 1. © Harcourt 3. Table, x 1 2 3 4 Table legs, y 4 8 12 Input, x 2 3 4 Output, y 4 5 6 7 2. 4. Fingers, x 5 10 15 Hands, y 1 2 3 child, x 1 2 3 eyes, y 2 4 6 20 4 Use each equation to make a table, write 4 ordered pairs, and then make a graph. 5. y x 4 6. y x 2 7. y x 4 x x x y y y Reteach RW119 LESSON 22.4 Name Problem Solving Skill Relevant or Irrelevant Information Jonathan gave a map to his visiting cousin so she would be able to find the places she needs. She is looking for the Pet Store and knows that its x-coordinate is the same as the Food Store’s x-coordinate. The Snack Shop is north of the Cinema. Jonathan said that the Pet Store is 4 blocks south of the Toy Store. Can you help her find the Pet Store? Snack Shop +5 +4 +3 +2 +1 y Toy Store x C -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 -1 -2 Food Store -3 Cinema 4 -5 Step 1 Decide what you are trying to find. the coordinates of the Pet Store Step 2 Read each fact and decide whether it is relevant or irrelevant to solving the problem. • The Pet Store has the the same x-coordinate as the Food Store. relevant • The Snack Shop is north of the Cinema. irrelevant • The Pet Store is 4 blocks south of the Toy Store. relevant Step 3 Use the relevant information to solve the problem. • The Food Store’s x-coordinate is 3. • The Toy Store’s y-coordinate is 5. • So the Pet Store is at (3,1). 1. A group of 72 students visited the science center. One third of them visited © Harcourt the planetarium. One half of that number went to the weather exhibit. The remaining students visited the electricity exhibit. Most of the students liked the science center. How many students saw the electricity exhibit? 2. Which information is relevant to this problem? RW120 Reteach LESSON 23.1 Name Lines and Angles In geometry, objects have special names. You can make lines by connecting any two points. Lines go on forever. You show this by putting arrows at the ends of the line. You can make line segments by joining two points. Line segments do not go on forever. They do not have arrows at the ends. A D C B D A A line segment is part of a line. It is the shortest distance between two points on a line. line segment CD, or C D Lines that cross at one point are intersecting. C B D BC and AD go on A line is a straight path in a plane. It has no ends. It can be named by any two points on the line. line AB, or AB, and line BC, or BC cross each CD and AD other at point D. and AD intersect AB to form right angles. C B A forever, and they will never cross. Lines that intersect to form four right angles are perpendicular. Lines in a plane that never intersect and are the same distance from each other are parallel. © Harcourt Draw and label each object. 1. lines AB and CD 2. line segment KL 3. line FG 5. lines NO and QR 6. lines HI and JK parallel to each other 4. lines EF and GH intersecting at point A perpendicular to each other parallel to each other Reteach RW121 LESSON 23.2 Name Measure and Draw Angles You can use a protractor to measure the angle at the right. A protractor is a tool for measuring the size of the opening of an angle. The unit used to measure an angle is a degree. D F E A protractor has a center point at the bottom where two lines form right angles. To the right of this is the 0° mark. To the left is the 180° mark. 10 01 0 100 90 80 7 0 12 60 50 10 20 180 170 1 60 15 30 180˚ 0 14 40 0 13 0˚ 0 center point Step 1 10 01 0 100 90 80 7 0 12 60 50 14 40 0 13 0 15 F 0 180 170 1 60 20 E 30 D 10 Place the protractor on the angle so that the center point lines up with the vertex and the horizontal line on the protractor lines up with ray EF. Step 2 10 01 0 100 D 14 0 15 60 50 20 F 0 E 10 180 170 1 60 80 7 0 30 Read the number of degrees the pencil is pointing to. 90 12 0 13 40 To measure the angle, place a pencil on top of the other ray of the angle. So, the measure of DEF is 50°. Use a protractor to measure and classify the angle. 2. 3. 4. 5. 6. © Harcourt 1. RW122 Reteach LESSON 23.3 Name Angles and Polygons How can you remember polygons and their angles? One way is to learn the meanings of the words that describe each shape. Remembering other words that use the same roots can also help you remember the figures. triangle tri- means 3 tricycle— a 3-wheeled bicycle quadrilateral quad- means 4 quadruplets — 4 babies born at once to the same mother pentagon pent- means 5 the Pentagon — a 5-sided building in Washington, D.C. hexagon hex- means 6 hex sign — Pennsylvania Dutch art that uses 6-sided figures drawn inside a circle octagon oct- means 8 octopus — an animal with 8 legs October — used to be the 8th month on the calendar polygon poly- means many polyhedra — a term used for the many-sided shapes of crystals © Harcourt Name each polygon. 1. 2. 3. 4. 5. 6. 7. 8. Many quadrilateral shapes have their own meanings as well. Look in the dictionary and find these meanings. Find another word that can help you remember the meaning. Then draw the shape. 9. trapezoid 10. parallelogram 11. rectangle 12. rhombus Reteach RW123 LESSON 23.4 Name Circles You need a centimeter ruler and a compass to construct a circle with a radius of 2 cm. A radius is a line segment that connects the center with a point on the circle. A diameter is a chord that passes through the center of a circle. • Draw a point at the center of the circle. • Start at the point. Use a centimeter ruler to draw a line segment 2 cm long. This is the radius. • Place the point of the compass on the center point. Place the pencil point on the other end of the radius. • Hold the compass point still. Turn the compass around on the point to make a complete circle. Use a centimeter ruler and a compass to construct each circle. 2. radius 1.5 cm 3. radius 2 cm 4. diameter 1.0 cm 5. diameter 2.4 cm 6. diameter 5.0 cm © Harcourt 1. radius 1 cm RW124 Reteach LESSON 23.5 Name Congruent and Similar Figures Two figures are similar if they have the same shape. Two figures are congruent if their matching sides and angles are the same. To determine if triangles ABC and DEF are congruent: • Measure the sizes of the matching angles to see if they are equal. • Measure the lengths of the matching sides to see if they are equal. B 8 cm E 37 90 A 10 cm 8 cm 37 90 53 C 6 cm D Lengths of Sides AB = DE, BC = EF, and AC = DF 10 cm 53 6 cm F Angles A = D, B = E, and C = F The matching sides are equal, and the matching angles are equal. So, the two triangles are congruent. Find one pair of similar figures and four pairs of congruent figures. A. B. 9 in. 90 C. 9 in. 90 90 90 90 90 90 E. 9 in. © Harcourt 90 F. 70 90 70 110 120 8 in. 110 60 120 90 90 90 8 in. 70 60 9 in. I. 90 9 in. 9 in. 70 H. G. 70 8 in. 110 110 8 in. 110 9 in. 70 110 70 9 in. 8 in. 110 70 8 in. 90 9 in. 70 8 in. 110 90 9 in. 9 in. 90 D. 9 in. 110 8 in. J. 110 70 Reteach RW125 LESSON 23.6 Name Symmetric Figures A figure has line symmetry when it can be folded on a line so that its two parts match. The two halves of the pentagon are congruent. Trace the pentagon. Fold it in half along the dotted line. The left half is congruent to the right half. A figure can have more than one line of symmetry. Find all the lines of symmetry for the pentagon. The pentagon has five lines of symmetry in all. Draw the lines of symmetry. How many lines of symmetry does each figure have? 2. 3. 4. 5. 6. © Harcourt 1. RW126 Reteach LESSON 23.7 Name Problem Solving Strategy Find a Pattern Leonardo Fibonacci was one of the most talented mathematicians of the Middle Ages. One of his hobbies was studying number patterns. One of his most famous patterns is shown below. What is the next term in the pattern? 1, 1, 2, 3, 5, 8 Step 1 What does the problem ask? It asks what the next term in the number pattern is. Step 2 Find a pattern. The next term is the sum of the two previous terms. 1 1 2, 1 2 3, 2 3 5, 3 5 8 Step 3 Use this information to solve the problem. 5 8 13. 13 is the next term in the pattern. © Harcourt Find a pattern to solve. 1. What is the next shape in this pattern? 2. When Fred’s number is 1, Ann’s number is 3. When Fred’s number is 2, Ann’s number is 5. If Fred’s is 6, what is Ann’s number? 3. Write a rule for the pattern described in Problem 2. 4. Alex read 45 pages on Sunday, 90 pages on Monday and 135 pages on Tuesday. If he continues this pattern, how many pages will he read on Friday? Reteach RW127 LESSON 24.1 Name Triangles Triangles are polygons with 3 sides and 3 angles. One method of classifying triangles is by the lengths of their sides. To classify a triangle using this method, you need to know the lengths of its sides. 7 in. 5 in. 3 congruent sides = equilateral triangle 4 in. 2 congruent sides = isosceles triangle 0 congruent sides = scalene triangle Each side is a different length, so this is a scalene triangle. List the number of congruent sides. Then name each triangle. Write isosceles, scalene, or equilateral. 1. 2. 12 m 3. 15 m 8 in. 12 m 4. 5. 10 cm 6 in. 5 in. 12 m 15 m 15 m 6. 9m 13 cm 10 cm 9m 9m 3 cm 11 cm 6 cm 8. 5 ft 7 ft 5 ft RW 128 Reteach 10 m 9. 6m 4m 10 m 8m 4m © Harcourt 7. LESSON 24.2 Name Quadrilaterals Polygons that have 4 sides and 4 angles are quadrilaterals. Quadrilaterals can be classified by looking at the number of parallel sides, the lengths of their sides, and the measures of their angles. Trapezoid Parallelogram Rectangle Rhombus Square 2 pairs Number of parallel sides 1 pair 2 pairs 2 pairs 2 pairs Number of congruent sides 0 pair 2 pairs 2 pairs all 4 sides all 4 sides Number of congruent angles 0 pair 2 pairs all 4 angles 2 pairs all 4 angles Examples To classify the quadrilateral at the right, identify the following characteristics. 6m 140˚ 40˚ Number of parallel sides: 2 pairs 6m Number of congruent sides: all 4 sides 6m 40˚ 140˚ 6m So, the figure is a rhombus. Number of congruent angles: 2 pairs Classify each quadrilateral. Write quadrilateral, trapezoid, parallelogram, rectangle, rhombus, or square. 1. 2. 8m 3m 6m 3. 9m 6m 12 m 3m 6m 9m 2m 7m © Harcourt 8m 4. 5. 5m 6m 4m 6. 6m 6m 6m 2m 2m 2m 2m 8m 6m Reteach RW 129 LESSON 24.3 Name Algebra: Transformations When you move a figure, it is called a rigid transformation. A translation is one type of transformation. When you translate, or slide, a figure on a coordinate plane, the coordinates change. The figure may move up or down, left or right, or both. Here are three examples of translations. 3 spaces to the right 2 spaces up y y y 10 9 8 7 6 5 4 3 2 1 0 3 spaces to the right and 2 spaces up (2,4) (2,1) (4,4) (5,4) (4,1) (5,1) 10 9 8 7 6 5 4 3 2 1 (7,4) (7,1) x 0 1 2 3 4 5 6 7 8 9 10 New ordered pairs: (2,1) to (5,1), (4,1) to (7,1), (2,4) to (5,4), (4,4) to (7,4) (2,6) 10 9 8 7 6 5 4 3 2 1 (4,6) (2,4) (4,4) (4,3) (2,3) (2,1) x (4,1) 0 1 2 3 4 5 6 7 8 9 10 (5,6) (2,4) (4,4) (5,3) (2,1) (7,6) (7,3) x (4,1) 1 2 3 4 5 6 7 8 9 10 New ordered pairs: (2,1) to (5,3), (4,1) to (7,3), (2,4) to (5,6), (4,4) to (7,6) New ordered pairs: (2,1) to (2,3), (4,1) to (4,3), (2,4) to (2,6), (4,4) to (4,6) Translate each figure. Draw the new figure with its coordinates. Name the new ordered pairs. 1. Translate the figure 2. Translate the figure 3. Translate the figure 5 spaces to the right and 4 spaces up. 3 spaces to the right and 4 spaces down. 4 spaces to the left and 4 spaces down. 0 1 2 3 4 5 6 7 8 9 10 RW 130 Reteach y x 10 9 8 7 6 5 4 3 2 1 0 y (2,9)(3,9) (3,6) (2,5) (4,6) (5,5)(6,5) (4,5) x 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 0 (8,8) (7,6) (10,8) © Harcourt y 10 9 8 7 6 (1,5) (2,5) 5 4 3 2 1 (1,1)(2,1) (9,6) (5,2) x 1 2 3 4 5 6 7 8 9 10 LESSON 24.4 Name Solid Figures A prism is a solid figure that has two congruent faces called bases. A prism is named by the polygons that form its bases. The prism at the right is a hexagonal prism. base • The faces of this solid figure are rectangles. face • The bases of this solid figure are hexagons. A pyramid is a solid figure with one base that is a polygon and three or more faces that are triangles with a common vertex. A pyramid is named by the polygon that forms its base. vertex This is a hexagonal pyramid. face • The faces of this solid figure are triangles. base • The base of this solid figure is a hexagon. Classify the solid figure. Then write the number of faces, vertices, and edges. 1. 2. 3. Write the name of the solid figure. © Harcourt 4. I have a base with 5 equal sides. My faces are 5 triangles. 5. All 6 of my faces are squares. 6. I have 2 congruent pentagons for bases. I have 5 rectangular faces. Reteach RW 131 LESSON 24.5 Name Draw Solid Figures from Different Views A solid figure looks different when it is viewed from different positions. Top Look at the solid figure at the right. • There are 2 cubes in the top layer. • There are 4 cubes in the middle layer. • There are 6 cubes in the bottom layer. This is a drawing of the figure viewed from the top. This is a drawing of the figure viewed from the side. Side Front This is a drawing of the figure viewed from the front. For 1–6, use the figure on the right. Tell how many cubes are in each row. Top 1. top layer 2. middle layer Front 3. bottom layer Draw the figure from different views. 5. side view 6. front view © Harcourt 4. top view Side RW 132 Reteach LESSON 24.6 Name Problem Solving Skill Make Generalizations When you generalize, you make a statement that is true about a whole group of similar situations. Read the following problem. Jesse is going camping. He will take with him: a box of cereal, paper towels, a flashlight, a soccer ball, and a tent shaped like a teepee. What polyhedrons will Jesse take on his camping trip? 1. Use what you know about each object to make a generalization. You can create a chart: OBJECT GENERALIZATION cereal Cereal usually comes in rectangular boxes. paper towels Paper towels come in a roll, which is a cylinder. a flashlight a soccer ball tent 2. Sort the shapes. 3. Solve the problem. Which shapes are polyhedrons? Explain. 4. Describe the strategy you used. © Harcourt Use what you know about each object to make a generalization. Then solve. 5. Annie takes a book, 2 cans of fruit juice, and a wedge pillow to the beach. What solid figures does she have? How many of these are polyhedrons? Reteach RW 133 LESSON 25.1 Name Customary Length You can measure more precisely by using smaller units of measure. Measured to the nearest inch: 2 in. 1 1 Measured to the nearest inch: 2 in. 4 4 1 2 1 3 Measured to the nearest inch: 2 in. 16 16 inches 1 So, the measurement to the nearest inch is most precise. 16 For 1–5, use a customary ruler to measure your textbook. 1. to the nearest inch: 1 2 1 3. to the nearest inch: 4 1 4. to the nearest inch: 8 1 5. to the nearest inch: 16 2. to the nearest inch: height width height width height width height width height width 6. Which is the most precise measure? least precise measure? 1 Measure each line segment to the nearest inch. 16 © Harcourt 7. 8. 9. 10. 11. RW 134 Reteach LESSON 25.2 Name Metric Length Use your fingers to help you estimate metric length. 1 2 3 4 5 6 7 8 9 10 Compare the width of each of your fingers to 1 centimeter. Is one of your fingers about 1 cm wide? Use your fingers to help estimate the length of each object. Then use a ruler to measure to the nearest centimeter and millimeter. 1. 2. 3. 4. 5. 6. © Harcourt Use a ruler to draw a line segment of the given length. 7. 6 cm 3 mm 9. 9 cm 8. 3 cm 8 mm 10. 52 mm Reteach RW 135 LESSON 25.3 Name Change Linear Units Use a mental image to help you decide whether to multiply or divide when changing linear units. 6 yd ft 48 in. ft 3ft 3ft 3ft 3ft 3ft 3ft Since each yard has 3 feet, multiply 6 by 3 to find the number of feet in 6 yards. 12 12 1ft 1ft 12 12 1ft 1ft Since each foot has 12 inches, divide 48 by 12 to find the number of feet in 48 inches. Use a mental image to help you change the units. 1. 3 ft 4. 36 ft 7. 30 yd 2. 12 ft yd 5. 80 mm ft 8. 7 ft in. in. cm in. 3. 15 km 6. 36 ft in. 9. 2 mi yd m Complete. 3 ft 2 ft in. 11. 3 ft 2 ft 12. 3 km 9 m 2 km ft 2 ft 3 km 9 m 2 km m in. 7 cm 8 mm 6 cm mm 13. km 9 m 2 km m9m 2 km m 8 mi 30 yd 7 mi yd © Harcourt 10. Find the sum or difference. 14. 2 ft 3 in. 4 ft 10 in. RW 136 Reteach 15. 2 ft 1 in. 9 in. 8 m 4 cm 5 m 80 cm 16. 5 m 13 cm 1 m 5 cm 17. LESSON 25.4 Name Customary Capacity and Weight You can change units of weight with multiplication or division. Change larger units to smaller units by using multiplication. Change smaller units to larger units by using division. 3 lb oz 48 oz lb Ounces are smaller than pounds, so divide. 48 16 3 → → Pounds are larger than ounces, so multiply. 3 16 48 (16 oz in 1 lb) (16 oz in 1 lb) So, 3 lb 48 oz. So, 48 oz 3 lb. Write multiply or divide. 1. When I change pounds to tons, I Customary Units for Measuring Weight . 16 ounces (oz) 1 pound (lb) 2,000 pounds 1 ton (T) 2. When I change ounces to pounds, I . 3. When I change tons to pounds, I 4. When I change pounds to ounces, . I . Multiply to solve. 5. 6 lb oz 6. 15 lb oz 7. 4 T lb 8. 1T oz © Harcourt Divide to solve. 9. 20,000 lb T 10. 128 oz lb 80 oz lb 12. 14,000 lb T 11. Multiply or divide to solve. 13. 96 oz lb 14. 5T lb 16. 12,000 lb 15. 20 lb oz T Reteach RW 137 LESSON 25.5 Name Metric Capacity and Mass Use the conversion table to determine whether to multiply or divide to change metric units. Metric Units of Capacity and Mass 1,000 mL 1 L 250 mL 1 metric cup 4 metric cups 1 L 1,000 liters 1 kL 1,000 mg 1 g 1,000 g 1 kg Change the unit. 5 liters metric cups Think: 1 liter 4 metric cups Multiply by 4 to change liters to metric cups. So, 5 liters 20 metric cups. Change the unit. 1. 750 mL metric cups 250 mL metric cups 2. 8.5 L mL 1 L by to change mL to metric cups by change L to mL 750 mL 8.5 L 3. 5,000 g kg metric cups 4. 3 kL 6. 3,000 mg to mL L g © Harcourt 5. 7 L metric cups mL RW 138 Reteach LESSON 25.6 Name Time You can calculate elapsed time by using a clock. Think of a clock as a circular number line. Count the hours by ones and then count the minutes by fives. Mark worked from 9 A.M. to 5:15 P.M. How many hours did Mark work? 5 min 3 hr 2 hr 1 hr Start 4 hr 10 min 5 hr 11 12 1 15 min 2 10 9 3 6 hr 4 8 7 6 5 7 hr Count the hours and then count the minutes. So, Mark worked 8 hours 15 minutes. 8 hr End Use the clocks to determine the missing information. 1. Begin End 11 12 1 2 10 9 3 4 8 7 6 5 A.M. Begin End 11 12 1 2 10 9 3 4 8 7 6 5 11 12 1 2 10 9 3 4 8 7 6 5 11 12 1 2 10 9 3 4 8 7 6 5 P.M. P.M. A.M. Elapsed time: © Harcourt 3. 2. Elapsed time: 16 hr 10 min Begin End 11 12 1 2 10 9 3 4 8 7 6 5 A.M. 4. Begin End 11 12 1 2 10 9 3 4 8 7 6 5 11 12 1 2 10 9 3 4 8 7 6 5 11 12 1 2 10 9 3 4 8 7 6 5 P.M. P.M. A.M. Elapsed time: 9 hr 45 min Elapsed time: Reteach RW 139 LESSON 25.7 Name Problem Solving Strategy Make a Table You can make a table to show elapsed time. Tara and Erin need to catch a school bus at 7:45 A.M. Before they catch their bus, they need 15 minutes to shower, 15 minutes to dress, 20 minutes to eat, and 10 minutes to walk to the bus. For what time should Tara and Erin set their alarm? A table can help organize the information. Work backward from the final time. The starting time of one activity becomes the ending time of the previous activity. Activity Start Time End Time Elapsed Time Shower 6:45 A.M. 7:00 15 min Dress 7:00 7:15 15 min Eat 7:15 7:35 20 min Walk to bus 7:35 7:45 A.M. 10 min So, Tara and Erin must set their alarm for 6:45 A.M. Make a table to solve. Chenoa is planning a hike. He will hike for 40 minutes, eat for 20 minutes, hike for 30 minutes, rest for 10 minutes, and hike for 40 minutes. He wants to end his hike at 1:30 P.M. At what time should Chenoa start his hike? Activity Start Time End Time Elapsed Time 12:50 1:30 P.M. 40 min Hike © Harcourt Eat Hike Rest Hike RW 140 Reteach LESSON 26.1 Name Perimeter Since opposite sides of a rectangle are equal, you can use a formula to find the perimeter of a rectangle. Since the sides of a regular polygon are equal, you can use a formula to find the perimeter of a regular polygon. 10 yd 4 cm 5 yd 4 cm 4 cm Perimeter (P) (2 l) (2 w) P (2 10) (2 5) P 20 10 P 30 Perimeter (P) (number of sides) s P34 P 12 So, the perimeter of the triangle is 12 cm. So, the perimeter of the rectangle is 30 yd. Find the perimeter of each polygon. 7 ft 4 cm 1. 2. 7 ft 2 cm length P width ) (2 P (2 © Harcourt side P P Perimeter is P Perimeter is ) . 3 in. 4. . 3. length 6 in. width 3m side P P Perimeter is ) (2 ) P P Perimeter is . P (2 . Reteach RW141 LESSON 26.2 Name Algebra: Circumference The distance around a circular object is called its circumference. A chord that passes through the center of a circle is a diameter. If you know the diameter of a circle, you can find the circumference. Remember. . . The relationship of the diameter to the circumference of a circle, C d, is about 3.14 and is called pi. Circumference diameter 3.14, or C d 3.14 means “is approximately equal to”. Find the circumference of this circle. Diameter 4 4 cm diameter Circumference diameter 3.14 4 3.14 12.56 cir cu m ference The circumference is approximately equal to 12.56 cm. The diameter of each circle is given. Multiply the diameter times 3.14 to find the circumference. 2. 5 cm 4. Reteach 3 cm 6 cm 5. 8.2 cm RW142 3. © Harcourt 1. 6. 7.25 cm 4.3 cm LESSON 26.3 Name Algebra: Area of Squares and Rectangles You can use a formula to find the area of a rectangle. You can use a formula to find the area of a square. 6 cm 3.1 in. 2 cm 3.1 in. Area (A) l w A62 A 12 Area (A) s s A 3.1 3.1 A 9.61 So, the area of the rectangle is 12 cm2. So, the area of the square is 9.61 in.2 Find the area of these squares and rectangles. 1. 5 ft 3m 2. 2m 5 ft side A length (l) width (w) A A . Area is A 2.5 yd 3. . Area is © Harcourt 2 yd 4. 8.4 m 8.4 m length (l) side width (w) A A A A Area is . Area is . Reteach RW143 LESSON 26.4 Name Relate Perimeter and Area Rectangles with the same perimeter can have different areas. Look at the rectangles below. Each rectangle has a perimeter of 24 cm, but their areas are different. Remember. . . Area (A) length (l) width (w) 1 cm 2 cm 3 cm 4 cm Rectangle A: 1 cm 11 cm Area 11 cm2 Rectangle B: 2 cm 10 cm Area 20 cm2 C D 6 cm 9 cm 10 cm 11 cm B 8 cm 6 cm A Rectangle C: Rectangle D: 3 cm 9 cm 4 cm 8 cm 2 Area 27 cm Area 32 cm2 E Rectangle E: 6 cm 6 cm Area 36 cm2 Rectangle E is the rectangle with the greatest area, 36 cm2. © Harcourt Use the grid to draw rectangles for the given perimeter. Name the length and width of the rectangle with the greatest area. 1. Perimeter 12 cm RW144 Reteach 2. Perimeter 28 cm LESSON 26.5 Name Algebra: Area of Triangles Use what you know about the area of a rectangle to find the area of a triangle. • Area of a rectangle equals length width. (A l w) 6 cm • The area of a triangle is half the area of a rectangle with the 1 same base and height. (A b h) 2 1 Area (A) b h 2 1 A 3 6 2 1 A 18 9 2 base (b) 3 cm height (h) 6 cm Area is 9 cm2. 3 cm Find the area of these triangles. 2. 5 cm 10 in. 1. 6 cm 5 in. base (b) base (b) height (h) height (h) 1 2 A A 1 2 A Area is A . Area is . 4. © Harcourt 4 mi 2 yd 3. 9 yd 4 mi base (b) base (b) height (h) height (h) 1 2 A A 1 2 A Area is . A Area is Reteach . RW145 LESSON 26.6 Name Algebra: Area of Parallelograms Use what you know about the area of a rectangle to find the area of a parallelogram. • Area of a rectangle equals length width. (A l w) • The area of a parallelogram is equal to the area of a rectangle with the same base (length) and height (width). (A b h) You can use a formula to find the area of a parallelogram. h = 4 cm Area (A) b h b = 5 cm A54 base (b) 5 cm height (h) 4 cm A 20 Area is 20 cm2. Find the area of these parallelograms. 2. 6 yd 2m 1. 8m 5 yd base (b) base (b) height (h) height (h) A Area is . 1 ft A A Area is . 4. 3 ft © Harcourt 5m 3. A 10 m base (b) base (b) height (h) height (h) A A Area is RW146 Reteach . A A Area is . LESSON 26.7 Name Area of Irregular Figures Count whole and half square units to find the area of an irregular figure on grid paper. Count: 8 whole squares 12 half squares Divide: 12 2 6 Add: 8 6 14 The area of the figure is 14 square units. When an irregular figure on grid paper has partial squares that cannot be counted exactly, use the averaging method to estimate the area. Count: 8 8 Divide: 8 Add: 8 whole squares partial squares 24 4 12 The area of the figure is about 12 square units. Find the area. Each square is 1 cm2. 1. Area 2. whole squares half squares 2 cm2 Area whole squares half squares 2 cm2 © Harcourt Estimate the area. Each square is 1 cm2. 3. whole squares partial squares 2 Area is about cm2 4. whole squares partial squares 2 Area is about cm2 Reteach RW147 LESSON 26.8 Name Problem Solving Strategy Solve a Simpler Problem Peter wants to paint a triangle with red paint on the playground. The height of the triangle will be 20 meters and the base 5 meters. Each container of red paint covers 10 square meters. How many containers of red paint will Peter need to paint his whole triangle? Step 1 What does the problem ask? It asks how many containers of paint Peter will need. Step 2 Find the area of the triangle. 1 2 Area (A) base (b) height (h) 1 2 A 5 20 A = 50 The area is 50 m2. Step 3 Identify the number of 10 m2 containers needed to cover 50 m2. Divide. 50 10 5 So, Peter needs 5 containers of paint to paint the triangle on the playground. Break these problems into simpler steps to solve. The living room is 12 feet long and 13 feet wide. The dining room is 15 feet long and 11 feet wide. How many square feet of carpet will be needed? RW148 Reteach 2. Tom is laying new sod in his yard. His front yard is 20 yd by 15 yd, and his backyard is 20 yd by 20 yd. Sod is sold by the square foot. How many square feet of sod does Tom need? © Harcourt 1. Frank’s house needs new carpet. LESSON 27.1 Name Nets for Solid Figures A net is a two-dimensional pattern for a three-dimensional prism or pyramid. Look at the net at the right. base • It has 1 triangular base. face • It has 3 triangular faces. Think about how you could fold it to make a solid figure. • It folds into a triangular pyramid. faces Look at the second net at the right. • It has 2 rectangular bases. • It has 4 rectangular faces. bases Think about how you could fold it to make a solid figure. • It folds into a rectangular prism. Match each solid figure with its net. 2. 3. © Harcourt 1. a. b. c. Reteach RW149 LESSON 27.2 Name Surface Area The surface area of a solid figure is the sum of the areas of its faces. To find the surface area of a box, add the areas of the 6 faces. Find the area of each face. Then add the areas. Face Area Top 5 4 20 cm2 Bottom 5 4 20 cm2 Left 4 7 28 cm2 Right 4 7 28 cm2 Front 5 7 35 cm2 Back 5 7 35 cm2 7 cm 4 cm 5 cm 2 Total Area 166 cm Use the tables to find the surface area of each box. 1. 2. 3 in. 3 ft 5 in. 2 ft 8 in. 9 ft Top Area 58 Face Top Bottom Bottom Left Left Right Right Front Front Back Back Total Area Total Area RW150 Reteach Area 92 © Harcourt Face LESSON 27.3 Name Algebra: Volume Volume is the amount of space a solid figure occupies or can hold. The formula for volume is: Volume length width height Look at the rectangular prism at the right. V l w h V (8 5) 6 6 cm V 40 6 240 So, the volume is 240 cm3. 8 cm 5 cm To find a missing dimension, use the formula for volume. Step 1 Substitute the known values in the formula. V l w h 200 (10 4) h Step 2 Multiply. 200 40 h Step 3 Use Mental Math. Think: 40 times what number equals 200? So, the height is 5 ft. 200 40 5 5h ? 10 ft 4 ft V 200 ft3 Find the volume. 1. l 12 yd, w 2 yd, h 6 yd 2. l 7 m, w 12 m, h2m V 3. l 11 cm, w 7 cm, h 3 cm V V Find the missing dimension. 4. length 6 in. 5. width 8 in. width 5 ft height Volume 120 ft3 © Harcourt height 9 ft Volume 270 ft3 Volume 150 m3 8. length 6 cm width height 12 cm Volume 216 cm3 length 10 ft width height 2 m Volume 240 in.3 length 4 ft 6. width 5 m height 7. length 9. length width 14 in. height 7 in. Volume 1,960 in.3 Reteach RW151 LESSON 27.4 Name Measure Perimeter, Area, and Volume Keywords can help you decide upon the appropriate unit of measure. • Use units to measure the length of or distance around an object. Keywords: around, length, height, distance, perimeter • Use square units to measure the area of an object. Keywords: cover, area, surface area • Use cubic units to measure the volume of an object. Keywords: volume, capacity, fill, space Substitute inches, meters, and so on for units when you are given specific measurements. Underline the keywords and tell the appropriate units to measure each. Write units, square units, or cubic units. 1. capacity of a mug 2. paper to cover a box 3. length of a room Underline the keyword and write the units you would use to measure each. 4. surface area of this cube 5. perimeter of this square 8 in. © Harcourt 4 ft 6. volume of this prism 7. area of this parallelogram 3 cm 2m 7 cm 8 cm 4m RW152 Reteach LESSON 27.5 Name Problem Solving Skill Use A Formula Paul wants to send some things to his brother at camp. He finds a box in his garage that is 12 inches long and 10 inches wide. It has a volume of 1,200 cubic inches. Paul wants to pack a container that is 14 inches high. Will the container fit in the box? To answer the question, you need to find the height of the box. Use the formula for volume. Vlwh • Substitute. 1,200 12 10 h • Multiply. 1,200 120 h • Use Mental Math. 1,200 120 10 The height is 10 inches. The answer is no, the container will not fit in the box. h 10 Use a formula and solve. 1. Rita wants to put a wallpaper © Harcourt border around her room. Her room is 11 ft by 13 ft. How many ft of border does she need to do the job? How many sq ft of carpet are required to carpet the room? 3. Mark needs to ship 500,000 cm3 cubic centimeters of peanuts. The shipping crate is 90 centimeters by 80 centimeters by 70 centimeters. Is it large enough to ship the peanuts? Explain. 2. Matthew needs to return a lamp that measures 15 inches by 12 inches by 8 inches. He has a box that is 16 inches long and 13 inches wide. It has a volume of 2,080 cubic inches. Is the box big enough for the lamp? 4. How many square feet of carpet do you need to cover a 12-foot by 15-foot room? how many square yards? Reteach RW153 LESSON 28.1 Name Understand Ratios You can use decimal models to help find ratios. Ratios compare two quantities. There are three types of ratios. Part to Whole Whole to Part Part to Part Shaded parts: 4 Total parts: 10 Shaded parts: 3 Total parts: 10 Shaded parts: 6 Unshaded parts: 7 So, the ratio of part to whole is 4 to 10. So, the ratio of whole to part is 10 to 6. So, the ratio of part to part is 3 to 7 or 7 to 3. Complete the ratios. 1. 2. shaded parts: shaded parts: total parts: unshaded parts: part to whole ratio part to part ratio: 4. part to whole ratio: part to whole ratio: whole to part ratio: whole to part ratio: part to part ratio: part to part ratio: Tell which type of ratio is expressed. 5 to 5 5. 10 to 5 RW154 Reteach © Harcourt 3. LESSON 28.2 Name Express Ratios You can write ratios in three ways. A part to whole ratio can be written: 6 to 10 6 10 6:10 A whole to part ratio can be written: 10 to 6 10 6 10:6 A part to part ratio can be written: 6 to 4 6 4 6:4 Write each ratio in three ways. Then name the type of ratio. 1. 4 red counters to 3 green counters 2. 12 pencils to 6 pens 3. 2 soccer balls out of 11 balls 4. 16 of 24 students are boys Write a, b, or c to show which ratio represents each comparison. 5. 3 red apples out of 8 apples 3 a 8 b 8:3 8 a c 3 to 11 7 7. 8 baseballs to 13 basketballs © Harcourt a 21:8 b 13 to 8 6. 7 boys to 8 girls 8 c 13 b 7:8 c 7 to 15 8. 1 month out of 12 months a 1:11 b 12 to 1 c 1 to 12 Write each ratio in two other ways. 9. 35 10. 11 to 13 11. 28 to 47 12. 14: 6 21 13. 7 14. 4 19 Reteach RW155 LESSON 28.3 Name Ratios and Proportions You can use pictures to show equivalent ratios. This shows the ratio 34. This shows an equivalent ratio, 68. You can also use fractions to show equivalent ratios. 3 6 12 15 18 9 4 8 12 16 20 24 Draw pictures to determine if the ratios are equivalent. Then write yes or no. 1. 12 and 36 2. 34 and 46 3. 23 and 35 4. 25 and 410 Write two fractions that are equivalent to each ratio. 5. 4 5 6. 9 2 11 12 8. 6 10 8 6 3 7 7. 9. 10. 11. 23 12. 3 to 4 13. 8 to 12 14. 57 15. 119 16. 16 to 4 17. 16 18. 2 to 10 19. 711 20. 26 RW156 Reteach © Harcourt Write two ratios that are equivalent to each ratio. LESSON 28.4 Name Scale Drawings You can use map scales and equivalent ratios to determine actual distances. Scale: 4mm 1 mi In this map of New Jersey, the scale is 4 mm 1 mi. So, the ratio of millimeters to miles is 4:1. Paterson Leonia The map distance from Paterson to Leonia is about 36 mm. What is the actual distance in miles? Montclair Use equivalent ratios. Orange 4 36 n 1 ? Newark 4 9 36 199 Since 4 9 36, you would multiply 1 9. So, the distance from Paterson to Leonia is 9 miles. Use the map scale above and equivalent ratios to find the actual distance. 1. The map distance from Paterson to Newark is 48 mm. What is the distance in miles? 2. The map distance from Montclair to Orange is 12 mm. What is the distance in miles? 3. The map distance from Paterson to Orange is 42 mm. © Harcourt What is the distance in miles? Use a map scale of 1 cm 15 mi and equivalent ratios to complete the table. 4. Watertown to Belmont 6 cm 5. Arlington to Bedford 4 cm 6. Belmont to Avon 11 cm 7. Franklin to Millis 3.5 cm Reteach RW157 LESSON 28.5 Name Problem Solving Skill Too Much/Too Little Information Sometimes you have too much or too little information to solve a problem. When you are given too much information, you must decide what information to use to solve the problem. When you are given too little information, you can’t solve the problem. Jared’s Fish Populations Read the table carefully. Look at the question and decide if you have too much or too little information. Fish to Types of Fish fish : catfish fish : rainbowfish fish : tetras What is Jared’s ratio of rainbowfish to tetras? What information you Know: 5:1 25:3 25:5 • You know Jared’s ratio of fish to rainbowfish is 25:3. • You know Jared’s ratio of fish to tetras is 25:5. What information you Don’t Need: • You don’t need the information on catfish. You have too much information, so you can solve the problem. Jared’s ratio of rainbowfish to tetras is 3:5. Use the table to complete each problem. 2. How many fish are there for does Jared have for every one rainbowfish? every one tetra? What you Know: What you Know: What you Don’t Need: What you Don’t Need: What you Need to Know: What you Need to Know: RW158 Reteach © Harcourt 1. How many red rainbowfish LESSON 29.1 Name Understand Percent You can represent part of the whole by using percents. Percent means “per hundred.” 100 percent is the whole. The 10 10 grid has 100 squares. Each square represents 1 percent. 33 squares are shaded. So, 33% of the squares are shaded. 67% of the squares are unshaded. 56 squares are shaded. So, 56% of the squares are shaded. 44% of the squares are unshaded. Write the percents for the shaded and unshaded squares. 1. 2. 3. Percent shaded Percent shaded Percent shaded Percent unshaded Percent unshaded Percent unshaded Shade the 10 10 grid to show the percent. 5. 69% 6. 82% © Harcourt 4. 34% Reteach RW159 LESSON 29.2 Name Relate Decimals and Percents Percents and decimals both represent a part of a whole, or of 100. You can use money to compare percents and decimals. Decimal: Quarter Dime Nickel Penny $0.25 $0.10 $0.05 $0.01 Read: twenty-five hundredths ten hundredths five hundredths one hundredth Ratio: 25 out of 100 Percent: 25% of a dollar 10 out of 100 5 out of 100 1 out of 100 10% of a dollar 5% of a dollar 1% of a dollar Write a decimal and a percent to describe each total amount. 2. 1 quarter, 1 dime, 1 penny decimal decimal percent percent 3. 3 quarters, 3 pennies 4. 8 dimes, 3 nickels, 2 pennies decimal decimal percent percent 5. 12 nickels, 4 pennies 6. 3 pennies decimal decimal percent percent Write the number as a decimal and as a percent. 7. forty-five hundredths 9. eighty-four hundredths RW160 Reteach 8. twenty-one hundredths 10. seventy-two hundredths © Harcourt 1. 1 quarter, 2 dimes LESSON 29.3 Name Relate Fractions, Decimals, and Percents Percents can be written as decimals, or as fractions with 100 as the denominator. • 45% means forty-five hundredths, or 0.45. 4. 5 • 45% also means 100 To write a fraction in simplest form, divide the numerator and the denominator by the same number. Keep doing this until 1 is the only common factor. 4 5 45 5 9 • 45% 100 100 5 20 So, 45% 0.45 9. 20 To write a fraction as a percent, write a fraction with the percent as the numerator and 100 as the denominator. 3 7 37%, or 0.37 • 100 25 7 5 75%, or 0.75 • 3 3 4 4 25 100 Complete. Write each as a decimal, a percent, and a fraction in simplest form. 35 100 1. 0.35 35% 2. 25% 100 100 25 3. 20% 100 100 4. 0.90 100 100 5. 0.16 100 100 6. 0.49 49% 100 © Harcourt 35 5 100 5 100 Complete. Write as a decimal and as a percent. 1 20 7. 11 100 100 9. 2 5 100 11. 100 , or , or , or 3 10 100 6 25 100 8. 10. 3 4 12. 100 , or , or , or Reteach RW161 LESSON 29.4 Name Find a Percent of a Number You can make a model to find a percent of a number. Find 40% of 30. Step 1 Step 2 Step 3 Use pieces of paper to represent 30. Separate the pieces of paper into 10 equal groups. Each group represents 10%. Separate 4 groups from the rest. These 4 groups represent 40%. Since each group has 3 pieces of paper, 4 groups have 12 pieces of paper. So, 40% of 30 equals 12. You can find a percent of a number by changing the percent to a decimal and multiplying. Step 1 Step 2 Change the percent to a decimal. Multiply the number by the decimal. 40% 0.40 0.40 30 12 © Harcourt So, 40% of 30 equals 12. Use a decimal to find the percent of the number. 1. 10% of 30 2. 20% of 50 3. 15% of 40 4. 20% of 60 5. 25% of 40 6. 3% of 18 RW162 Reteach LESSON 29.5 Name Mental Math: Percent of a Number You can use mental math to find a percent of a number. Find 30% of 50 chips Think: 30% 10% 10% 10% 30% of 50 (10% 50) (10% 50) (10% 50) ↓ ↓ ↓ (0.1 50) (0.1 50) (0.1 50) ↓ ↓ ↓ 5 5 5 15 So, 30% of 50 15. © Harcourt Use mental math to find the percent of each number. 1. 60% of 20 2. 80% of 30 3. 25% of 60 4. 15% of 40 5. 70% of 50 6. 45% of 20 7. 130% of 4,000 8. 20% of 150 9. 10% of 2,000 10. 70% of 80 11. 15% of 20 12. 200% of 10,000 13. 85% of 100 14. 60% of 500 15. 70% of 800 Reteach RW163 LESSON 29.6 Name Problem Solving Strategy FAVORITE MAGAZINES Make a Graph Magazine You can make a graph to display percent data. Step 1 Review your data. If your data show the relationship of parts to a whole, you can use a circle graph. Percent Sports Illustrated for Kids 20% National Geographic 30% Nickelodeon 10% Zoo Zillions 20% People 20% Step 2 Divide a circle into 10 equal sections. FAVORITE MAGAZINES Sports for Kids 20% Puzzle Power 20% Step 3 Label the number of sections that show each percent. 1 section represents 10%. Buy Smart 20% 2 sections represent 20%. There are three 20% sections. 3 sections represent 30%. Around the World 30% Media Talk 10% Step 4 Label the percents and title the circle graph. Use a 10-section circle and the data in the table to make a circle graph. FAVORITE VACATIONS Vacation Place 2. FAVORITE HOBBIES Hobby Percent Percent National Park 20% Painting 20% Beach 20% Collecting Stamps 10% Amusement Park 40% Making Models 30% Foreign Country 10% Collecting Stuffed Animals 30% Famous City 10% Other 10% RW164 Reteach © Harcourt 1. LESSON 29.7 Name Compare Data Sets You can find a percent of a number to compare the results of two or more sets of data. The circle graphs below show the results of surveys Joshua conducted. Joshua’s family There were 20 people surveyed. Joshua’s neighborhood There were 60 people surveyed. FAVORITE ICE-CREAM FLAVORS IN JOSHUA’S FAMILY FAVORITE ICE-CREAM FLAVORS IN JOSHUA’S NEIGHBORHOOD Mint Chip 10% Strawberry 10% 5% Other Chocolate 45% Vanilla 30% Strawberry Mint Chip 15% 15% 10% Other Chocolate Vanilla 40% 20% Joshua wants to know in which survey vanilla got more votes. Joshua’s family Find 30% of 20 people. Joshua’s neighborhood Find 20% of 60 people. Step 1 Step 2 Step 1 Step 2 Change the percent to a decimal. Multiply the total number of people by the decimal. Change the percent to a decimal. Multiply the total number of people by the decimal. 30% 0.30 0.30 20 6 20% 0.20 0.20 60 12 So, 6 people voted for vanilla. So, 12 people voted for vanilla. So, vanilla received more votes in Joshua’s neighborhood survey. For 1–2, use the circle graphs above. © Harcourt 1. In which survey did chocolate receive more votes? How many more votes? 2. If vanilla had received only 10% of the votes in Joshua’s neighborhood, in which survey would vanilla have received more votes? Explain. Reteach RW165 LESSON 30.1 Name Probability Experiments A box contains 6 black marbles and 2 white marbles. Experiment: Shake the box, and pull a marble. Record the color; then replace the marble. There are two possible events for this experiment. • The marble is black. • The marble is white. Tom conducts the experiment 16 times. He predicts that the events will be 12 black and 4 white marbles. He records the actual results in a table. MARBLE EXPERIMENT Events Predicted frequency Black White 12 4 Actual frequency 1. Is white or black a more likely event? Why? © Harcourt 2. What were the actual frequencies after 16 trials? 3. Explain why Tom predicted 12 black and 4 white marbles. RW166 Reteach LESSON 30.2 Name Outcomes Your cafeteria offers a choice of tuna, turkey, or veggie sandwiches. You can also choose between white and wheat bread. What are your possible choices? A tree diagram shows you all the possible choices. Breads white wheat Sandwiches Choices tuna tuna sandwich on white bread turkey turkey sandwich on white bread veggie veggie sandwich on white bread tuna tuna sandwich on wheat bread turkey turkey sandwich on wheat bread veggie veggie sandwich on wheat bread So, you have 6 possible choices. 1. Amanda must choose swimming a 25-meter, 50-meter, or 100-meter race. She can swim in either the freestyle or backstroke division. How many choices does Amanda have? Complete the tree diagram. Divisions © Harcourt freestyle Races Choices 25-meter 25-meter freestyle 50-meter backstroke 100-meter Amanda has choices. 2. Brian’s parents are buying a new car. They can choose a sedan or a minivan. Both cars are available in red or white. How many choices do they have? Reteach RW167 LESSON 30.3 Name Probability Expressed as a Fraction You can predict the probability, or chance, that an event will happen. Ben has a spinner with six sections. The possible outcomes are spinning blue, spinning red, spinning yellow, or spinning green. What is the probability of spinning blue? green blue yellow blue red red 2 number of blue sections Probability of spinning blue total number of sections 6 2 1 So, the probability of spinning blue is 6, or 3. For Problems 1–4, use spinner A. Give the probability of spinning each color. 1. blue 2. red A blue red green yellow 3. green yellow yellow blue red 4. yellow 5. 1 6. 2 7. 3 8. 4 9. 5 RW168 Reteach 1 3 2 3 4 5 14 1 2 B © Harcourt For Problems 5–8, use spinner B. Give the probability of spinning each number. LESSON 30.4 Name Compare Probabilities You can compare the probabilities of events to determine whether one event is more likely than another. Kim has a bag of marbles with 6 blue and 3 green marbles. Which color marble is she more likely to pull? First, find the probability of each event. number of blue marbles 6 Probability of blue total number of marbles 9 Probability of green number of green marbles 3 total number of marbles 9 Then, compare the probabilities. 6 9 3 9 So, Kim is more likely to pull a blue marble. For Problems 1–2, use the spinner. Find the probability of each event. Decide which event is more likely. red red 1. The pointer will land on blue; the pointer will land blue on red. green Probability of blue number of blue sections 2 total number of sections 8 Probability of red yellow green blue red More likely event: 2. The pointer will land on yellow; the pointer will land © Harcourt on green. Probability of Probability of More likely event: Reteach RW169 LESSON 30.5 Name Problem Solving Strategy Make an Organized List Making an organized list can help you determine the possible outcomes of 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 spinner. What are the possible outcomes? How many are there? Spinner Coin Outcomes heads red and heads red tails red and tails heads yellow and heads yellow tails yellow and tails So, there are 4 possible outcomes. Make an organized list to solve. 1. Jereme 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 © Harcourt purple brown There are possible outcomes. 2. Sarah has 10¢. How many different combinations of coins could she have? RW170 Reteach