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RETEACH Workbook Visit The Learning Site! www.harcourtschool.com Title_CR_NLG5.indd 1 HSP Grade 5 6/18/07 5:19:59 PM 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. Permission is hereby granted to individuals using the corresponding student’s textbook or kit as the major vehicle for regular classroom instruction to photocopy entire pages from this publication in classroom quantities for instructional use and not for resale. Requests for information on other matters regarding duplication of this work should be addressed to School Permissions and Copyrights, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. Fax: 407-345-2418. HARCOURT and the Harcourt Logo are trademarks of Harcourt, Inc., registered in the United States of America and/or other jurisdictions. Printed in the United States of America ISBN 13: 978-0-15-356801-5 ISBN 10: 0-15-356801-1 If you have received these materials as examination copies free of charge, Harcourt School Publishers retains title to the materials and they may not be resold. Resale of examination copies is strictly prohibited and is illegal. Possession of this publication in print format does not entitle users to convert this publication, or any portion of it, into electronic format. 1 2 3 4 5 6 7 8 9 10 018 16 15 14 13 12 11 10 09 08 07 NL_SEGr5.indd template1 6/12/07 4:08:16 PM UNIT 1: USE WHOLE NUMBERS Chapter 1: Place Value, Addition, and Subtraction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Place Value Through Millions ........... RW1 Understand Billions ........................... RW2 Compare and Order Whole Numbers............................................. RW3 Round Whole Numbers .................... RW4 Estimate Sums and Differences ........................................ RW5 Add and Subtract Whole Numbers............................................. RW6 Problem Solving Workshop Strategy: Work Backward ................. RW7 Chapter 4: Expressions and Equations 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Write Expressions ............................ RW24 Evaluate Expressions ....................... RW25 Properties......................................... RW26 Mental Math: Use the Properties......................................... RW27 Write Equations............................... RW28 Solve Equations ............................... RW29 Functions.......................................... RW30 Inequalities ...................................... RW31 Problem Solving Workshop Strategy: Predict and Test ............... RW32 UNIT 2: USE DECIMALS Chapter 2: Multiply Whole Numbers 2.1 2.2 2.3 2.4 2.5 2.6 Mental Math: Patterns in Multiples ............................................ RW8 Estimate Products .............................. RW9 Multiply by 1-Digit Numbers .......... RW10 Multiply by Multi-Digit Numbers........................................... RW11 Problem Solving Workshop Strategy: Find a Pattern .................. RW12 Choose a Method ............................ RW13 Chapter 3: Divide by 1- and 2-Digit Divisors 3.1 Estimate with 1-Digit Divisors ............................................. RW14 3.2 Divide by 1-Digit Divisors................ RW15 3.3 Problem Solving Workshop Skill: Interpret the Remainder ........ RW16 3.4 Zeros in Division .............................. RW17 3.5 Algebra: Patterns in Division ............................................ RW18 3.6 Estimate with 2-Digit Divisors ............................................. RW19 3.7 Divide by 2-Digit Divisors................ RW20 3.8 Correcting Quotients ...................... RW21 3.9 Practice Division .............................. RW22 3.10 Problem Solving Workshop Skill: Relevant or Irrelevant Information .................... RW23 G5-NL TOC_TEnew.indd 1 Chapter 5: Understand Decimals 5.1 5.2 5.3 5.4 Decimal Place Value ........................ RW33 Equivalent Decimals ........................ RW34 Compare and Order Decimals ........ RW35 Problem Solving Workshop Skill: Draw Conclusions ................... RW36 Chapter 6: Add and Subtract Decimals 6.1 6.2 6.3 6.4 6.5 Round Decimals ............................... RW37 Add and Subtract Decimals ............ RW38 Estimate Sums and Decimals ......... RW39 Choose a Method ............................ RW40 Problem Solving Workshop Skill: Estimate or Find Exact Answer....... RW41 Chapter 7: Multiply Decimals 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Model Multiplication by a Whole Number ............................. RW42 Algebra: Patterns in Decimal Factors and Products ....................... RW43 Record Multiplication by a Whole Number ............................. RW44 Model Multiplication by a Decimal ......................................... RW45 Estimate Products ............................ RW46 Practice Decimal Multiplication ..... RW47 Problem Solving Workshop Skill: Multistep Problems ................ RW48 6/29/07 11:18:09 AM Chapter 8: Divide Decimals by Whole Numbers 8.1 8.2 8.3 8.4 Decimal Division .............................. RW49 Estimate Quotients.......................... RW50 Divide Decimals by Whole Numbers........................................... RW51 Problem Solving Workshop Skill: Evaluate Answers for Reasonableness ............................... RW52 UNIT 3: DATA AND GRAPHING Chapter 9: Data and Statistics 9.1 9.2 9.3 9.4 9.5 Collect and Organize Data ............. RW53 Mean, Median, and Mode .............. RW54 Compare Data ................................. RW55 Analyze Graphs ............................... RW56 Problem Solving Workshop Strategy: Draw a Diagram .............. RW57 Chapter 10: Make Graphs 10.1 Make Bar Graphs and Pictographs ...................................... RW58 10.2 Make Histograms ................... .........RW59 10.3 Algebra: Graph Ordered Pairs ........ RW60 10.4 Make Line Graphs ........................... RW61 10.5 Make Circle Graphs ......................... RW62 10.6 Problem Solving Workshop Strategy: Make a Graph.................. RW63 10.7 Choose the Appropriate Graph...... RW64 UNIT 4: NUMBER THEORY AND FRACTION CONCEPTS Chapter 11: Number Theory 11.1 Multiples and the Least Common Multiple ........................... RW65 11.2 Divisibility ........................................ RW66 11.3 Factors and Greatest Common Factor ............................... RW67 11.4 Prime and Composite Numbers...... RW68 11.5 Problem Solving Workshop Strategy: Make an Organized List ................................. RW69 11.6 Introduction to Exponents ............. RW70 11.7 Exponents and Square Numbers .... RW71 11.8 Prime Factorization ......................... RW72 G5-NL TOC_TEnew.indd 2 Chapter 12: Fraction Concepts 12.1 12.2 12.3 12.4 12.5 Understand Fractions ...................... RW73 Equivalent Fractions ........................ RW74 Simplest Form .................................. RW75 Understand Mixed Numbers .......... RW76 Compare and Order Fractions and Mixed Numbers........................ RW77 12.6 Problem Solving Workshop Strategy: Make a Model ................. RW78 12.7 Relate Fractions and Decimals........ RW79 UNIT 5: FRACTION OPERATIONS Chapter 13: Add and Subtract Fractions 13.1 Add and Subtract Like Fractions .... RW80 13.2 Model Addition of Unlike Fractions........................................... RW81 13.3 Model Subtraction of Unlike Fractions........................................... RW82 13.4 Estimate Sums and Differences ...... RW83 13.5 Use Common Denominators .......... RW84 13.6 Problem Solving Workshop Strategy: Compare Strategies......... RW85 13.7 Choose a Method ............................ RW86 Chapter 14: Add and Subtract Mixed Numbers 14.1 Model Addition of Mixed Numbers........................................... RW87 14.2 Model Subtraction of Mixed Numbers........................................... RW88 14.3 Record Addition and Subtraction .. RW89 14.4 Subtraction with Renaming ........... RW90 14.5 Practice Addition and Subtraction . RW91 14.6 Problem Solving Workshop Strategy: Use Logical Reasoning .... RW92 Chapter 15: Multiply and Divide Fractions 15.1 Model Multiplication of Fractions . RW93 15.2 Record Multiplication of Fractions RW94 15.3 Multiply Fractions and Whole Numbers........................................... RW95 15.4 Multiply with Mixed Numbers ....... RW96 15.5 Model Fraction Division .................. RW97 15.6 Divide Whole Numbers by Fractions...................................... RW98 15.7 Divide Fractions ............................... RW99 15.8 Problem Solving Workshop Skill: Choose the Operation.......... RW100 6/29/07 11:18:24 AM UNIT 6: RATIO, PERCENT, AND PROBABILITY Chapter 16: Ratios and Percent 16.1 Understand and Express Ratios .... RW101 16.2 Algebra: Equivalent Ratios and Proportions ........................................RW102 16.3 Ratios and Rates ............................ RW103 16.4 Understand Maps and Scales........ RW104 16.5 Problem Solving Workshop Strategy: Make a Table ................. RW105 16.6 Understand Percent ...................... RW106 16.7 Fractions, Decimals, and Percents RW107 16.8 Find Percent of a Number..............RW108 Chapter 17: Probability 17.1 17.2 17.3 17.4 Outcomes and Probability ............ RW109 Probability Experiments ................ RW110 Probability and Predictions .......... RW111 Problem Solving Workshop Strategy: Make an Organized List ............................... RW112 17.5 Tree Diagrams................................ RW113 17.6 Combinations and Arrangements................................ RW114 UNIT 7: GEOMETRY AND ALGEBRA Chapter 18: Geometric Figures 18.1 18.2 18.3 18.4 Points, Lines, and Angles .............. RW115 Measure and Draw Angles ........... RW116 Polygons......................................... RW117 Problem Solving Workshop Skill: Identify Relationships .......... RW118 18.5 Circles ............................................. RW119 18.6 Congruent and Similar Figures..... RW120 18.7 Symmetry ....................................... RW121 Chapter 19: Plane and Solid Figures 19.1 19.2 19.3 19.4 19.5 Classify Triangles ........................... RW122 Classify Quadrilaterals................... RW123 Draw Plane Figures ....................... RW124 Solid Figures .................................. RW125 Problem Solving Workshop Strategy: Compare Strategies....... RW126 19.6 Nets for Solid Figures .................... RW127 19.7 Draw Solid Figures from Different Views ............................. RW128 G5-NL TOC_TEnew.indd 3 Chapter 20: Patterns 20.1 20.2 20.3 20.4 20.5 Transformations ............................ RW129 Tessellations ................................... RW130 Create a Geometric Pattern.......... RW131 Numeric Patterns ........................... RW132 Problem Solving Workshop Strategy: Find a Pattern ................................ RW133 Chapter 21: Integers and the Coordinate Plane 21.1 Algebra: Graph Relationships ...... RW134 21.2 Algebra: Equations and Functions........................................ RW135 21.3 Problem Solving Workshop Strategy: Write an Equation ......................... RW136 21.4 Understand Integers ..................... RW137 21.5 Compare and Order Integers ....... RW138 21.6 Algebra: Graph Integers on the Coordinate Plane .................... RW139 UNIT 8: MEASUREMENT Chapter 22: Customary and Metric Measurements 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 Customary Length ......................... RW140 Metric Length ................................ RW141 Change Linear Units...................... RW142 Customary Capacity and Weight .................................... RW143 Metric Capacity and Mass ............. RW144 Problem Solving Workshop Skill: Estimate or Actual Measurement ................................ RW145 Elapsed Time.................................. RW146 Temperature .................................. RW147 Chapter 23: Perimeter 23.1 Estimate and Measure Perimeter ....................................... RW148 23.2 Find Perimeter ............................... RW149 23.3 Algebra: Perimeter Formulas ........................................ RW150 23.4 Problem Solving Workshop Skill: Make Generalizations ................... RW151 23.5 Circumference ............................... RW152 6/29/07 11:18:42 AM Chapter 24: Area and Volume 24.1 Estimate Area ............................... RW153 24.2 Algebra: Area of Squares and Rectangles ............................. RW154 24.3 Algebra: Relate Perimeter and Area ....................................... RW155 24.4 Algebra: Area of Triangles .......... RW156 24.5 Algebra: Area of Parallelograms ............................. RW157 24.6 Problem Solving Workshop Strategy: Solve a Simpler Problem ........................................ RW158 24.7 Surface Area ................................. RW159 24.8 Algebra: Estimate and Find Volume ................................. RW160 24.9 Relate Perimeter, Area, and Volume .................................. RW161 24.10 Problem Solving Workshop Strategy: Compare Strategies ..... RW162 G5-NL TOC_TEnew.indd 4 6/29/07 11:19:04 AM 0 Ones 1 2 Hundreds Hundreds 5 6 8 7 8 Ones 5. 617,008,235 4. 6. 3. 48,227,304 38,507 6/12/07 10:34:40 AM © Harcourt • Grade 5 Reteach 8. 60,000,000 1 30,000 1 9,000 1 20 1 4 RW1 153,709 347,254,901 Write each number in two other forms. 7. 803,154 2. 234,621,889 1. Write the value of the underlined digit. To find the value of a underlined digit, multiply the digit by its place value. In 301,256,878 the digit 3 is equal to 3 3 100,000,000 5 300,000,000. Word Form: Write the word name for the numbers in each period followed by the name of each period and a comma. three hundred one million, two hundred fifty-six thousand, eight hundred seventy-eight Expanded Form: Multiply each digit by its place value to write expanded form. 300,000,000 1 1,000,000 1 200,000 1 50,000 1 6,000 1 800 1 70 1 8 Standard Form: 301,256,878 Each period is separated by a comma. Tens 3 Tens Ones Period 8 3 5 Hundreds Hundreds 7 4 2 9 1 0 Hundreds Ones Period 5 0 809,237,228,771 433,173,983,021 6. 2. MXENL08AWK5X_RT_CH01_L2.indd 1 7. 3. 51,906,200,141 25,283,998,060 RW2 6,000 1 500 1 20 © Harcourt • Grade 5 Reteach 8. 5,000,000,000 1 200,000,000 1 800,000 1 621,389,007,718 275,487,601,035 Write each number in two other forms. 7. 3,209,003,812 5. 1. Write the value of the underlined digit. To find the value of an underlined digit, multiply the digit by its place value. In 38,752,491,050 the digit 4 is equal to 4 3 100,000 5 400,000. Word Form: Write the word name for the numbers in each period followed by the name of each period and a comma. thirty-eight billion, seven hundred fifty-two million, four hundred ninety-one thousand, fifty 30,000,000,000 1 8,000,000,000 1 700,000,000 1 50,000,000 1 2,000,000 1 400,000 1 90,000 1 1,000 1 50 Expanded Form: Multiply each digit by its place value to write expanded form. Each period is separated by a comma. Standard Form: 38,752,491,050 Tens Hundreds Period Ones Millions Period Hundreds Billions Period Tens Hundreds Period Ones Understand Billions Ones Millions Period Hundreds You can use a place-value chart to read larger numbers. The four periods shown in the place-value chart below are the ones, hundreds, millions, and billlions. Tens You can use a place-value chart to help you read and write whole numbers and find the value of a digit. A period is a group of three digits. The three periods shown in the chart below are ones, hundreds, and millions. Ones Place Value Through Millions Tens Name Tens Name Ones Grade5.indd 1 MXENL08AWK5X_RT_CH01_L1.indd 1 RW1-RW2 6/18/07 5:50:51 PM 6/12/07 10:35:37 AM 0 0 8 8 2 625,100 962,338 4. 525,100 7. 962,338 8. 18,181 5. 670,430 2. 130,870 . 18,818 640,470 130,870 9. 72,345,995 6. 13,275,104 3. 5,266,918 14. 324,060; 326,040 13. 3,541,320; 3,541,230 RW3 11. 270,908; 270,608 10. 3,218; 3,208 72,345,795 13,276,819 5,264,613 4 1 Ones 6/12/07 10:36:16 AM © Harcourt • Grade 5 Reteach 15. 12,452,671; 12,543,671 12. 8,306,722; 8,360,272 Name the greatest place-value position where the digits differ. Name the greater number. 2,815 1. 2,518 Compare. Write ,, ., or 5 for each Since 0 ten thousand is less than 1 ten thousand, then, 2,306,821 , 2,310,084 0,1 1 6 Compare the digits in the ten thousands place. 3 2 0 Ones Period 353 Tens Compare the digits in the hundred thousands place. 3 Ones 252 Hundreds 2 Tens Thousands Period Ones Millions Period Hundreds Compare the digits in the one millions place. Hundreds Compare 2,306,821 and 2,310,084. Write ,, ., or 5. A place-value chart can help you compare whole numbers. Compare and Order Whole Numbers Name Tens Grade5.indd 2 MXENL08AWK5X_RT_CH01_L3.indd 1 RW3-RW4 6/18/07 5:51:08 PM 10. 67,704,257 6. 35,118,247 MXENL08AWK5X_RT_CH01_L4.indd 1 17. 345,591 to 346,000 16. 216,593 to 200,000 RW4 14. 736,147 to 740,000 13. 52,398 to 52,000 12. 487,293,618 8. 849,207,284 4. 355,264,319 © Harcourt • Grade 5 Reteach 18. 3,517,004 to 4,000,000 15. 6,234,581 to 6,234,600 11. 517,218,137 7. 341,618,915 3. 83,445,182 Name the place to which each number was rounded. 9. 888,999,211 5. 6,024 2. 35,211 13,000,000 12,694,022 The 2 increases by 1 to a 3. Round each number to the place of the underlined digit. 1. 136,237,015 12,694,022 6 . 5, so you round up. So, 12,694,022 rounded to the nearest million is 13,000,000. Step 3: Rewrite all digits to the right of the underlined digit as zeros. Is the digit to the right 5 or greater? Increase by 1. Think: Is the digit to the right less than 5? The underlined digit stays the same. Step 1: Look at the digit to the right of the underlined digit. Round to the place value of the underlined digit. 12,694,022 You can round whole numbers by using the rounding rules. Round Whole Numbers Name 6/12/07 10:37:11 AM 237,150 2 __ 529,617 1__ 163,582 294,322 4. 72,543 29,583 1 __ 1 5. $63,895 10. 8. 47,738 78,905 1 __ 223,873 221,559 __ 11. 56,108 42,336 1 __ 2 RW5 1,825 2 __ 12. $8,423 9. 37,228 1 __ Estimate by using compatible numbers. 3. 1. 2. 773,645 95,223 1 103,229 1 __ 1 613,886 13. 2 135,710 1 __ 263,776 6. 2__ 173,509 925,461 14. 7. 6/12/07 10:41:00 AM © Harcourt • Grade 5 Reteach 125,318 2 __ 554,903 132.881 2 __ 745,556 You can estimate to find an answer that is close to the exact answer. Use compatible numbers to estimate. Compatible numbers are easy to compute mentally. Estimate using compatible numbers. 103,883 1 71,852 Think: 104 1 72 is easy to add mentally, so 103,883 104,000 104,000 1 72,000 are good compatible 1 71,000 1 72,000 numbers to use for an estimate. 176,000 Estimate by rounding. Add and Subtract Whole Numbers You can use a place-value chart to help you add or subtract. Estimate Sums and Differences millions 1 1 3 1 2 thousands 4 5 9 1 1 0 hundreds 1 9 5 3 4 5 9 5,382 241,393 1 __ 259,562 487,018 1 8,723 __ 6. 2. MXENL08AWK5X_RT_CH01_L6.indd A 9. 18,275 + 5,225 + 3,093 5. 1. 7. 3. 306,657 2 227,242 __ 319,007 182,322 2 __ RW6 10. 2,705,243 – 1,192,013 2 2,119,625 ___ 4,678,128 1 29,218 __ 33,617 Estimate. Then find each sum or difference. 1 617,634 __ 603,438 1 647,273 __ © Harcourt • Grade 5 Reteach 11. 500,601– 74,581 8. 4. 1,114,184 is close to the estimate of 1,000,000, so the answer is reasonable. Regroup 11 hundred thousands as 1 millions and 1 hundred thousands. 129,336 Add the thousands. Regroup 14 thousands as 1 ten thousands and 4 thousands. Then ad the hundreds. Then add then tens. First add the ones. Regroup 14 ones as 1 tens and 4 ones. Start adding from right to left. Regroup as needed. Estimate: 800,000 1 300,000 5 1,100,000. Then add the ten thousands. Regroup 11 ten thousands as 1 hundred thousands and 1 ten thousand. Last add the hundred thousands. + 1 8 hundred thousands 1 ten thousands 7 tens 6/12/07 10:48:37 AM Inverse operations are operations that undo each other. The inverse relationship allows you to check addition by using subtraction and to check subtraction by using addition. Estimate. Then find the sum of 789,039 1 325,155 When you add, you find the sum of two or more numbers. When you subtract, you find the difference of two numbers. Name Name ones Grade5.indd 3 MXENL08AWK5X_RT_CH01_L5.indd 1 RW5-RW6 6/18/07 5:51:32 PM Grade5.indd 4 Write your answer in a complete sentence. 6/12/07 10:51:10 AM © Harcourt • Grade 5 Reteach students signed up for summer sports increased by 635. From 2007 to 2008 the number increased by 224. In 2008, 1,783 students were signed up for summer sports. How many students were signed up in 2006? 7. From 2006 to 2007, the number of RW7 supplies to decorate the school. They also spent $85 to print banners to put above the bleachers. They now have $183 left in their budget. How much money did they start with in their budget? 6. The school pep club spent $326 on Work backward to solve. Check 5. Is there another strategy you could use to solve the problem? 4. worked backward to solve the problem. Solve 3. Solve the problem. Use the space below to show how you Plan 2. What strategy can you use to solve the problem? Read to Understand 1. What does the problem ask you to find? On Saturdays, Samantha has stretching class for 45 minutes and ballet for an hour and a half. After a 30-minute break, she has jazz class for 1 hour, which is over at 1:45 P.M. At what time does she begin? Mental Math: Patterns in Multiples Problem Solving Workshop Strategy: Work Backward 5 80 3 1 hundreds 5 80 hundreds 80 3 100 2 3 80 10. 3 3 600 MXENL08AWK5X_RT_CH02_L01.indd 1 26. 20 3 5,000 25. 8 3 300 24. 7 3 6,000 RW8 21. 800 3 80 16. 100 3 10 20. 5 3 200 15. 50 3 5,000 27. 800 3 400 22. 900 3 100 17. 7,000 3 40 12. 20 3 700 © Harcourt • Grade 5 Reteach 28. 70 3 100 23. 300 3 500 18. 9,000 3 20 13. 9 3 4,000 7 3 100 5 2 3 900 5 11. 60 3 50 7 3 10 5 6. 7 3 1 5 2 3 90 5 19. 80 3 6,000 14. 10 3 60 9. Find the product. 4 3 500 5 4 3 50 5 5. 2 3 9 5 6 3 200 5 9 3 400 5 5 3 300 5 4. 4 3 5 5 6 3 20 5 3. 6 3 2 5 basic fact times 100 basic fact times 10 9 3 40 5 2. 9 3 4 5 5 8,000 5 800 5 80 basic fact 5 3 30 5 1. 5 3 3 5 Find the missing numbers. 5 8 3 1 hundreds 8 3 100 5 80 hundreds 5 8 3 1 tens 8 3 10 5 8 tens You can use basic multiplication facts and patterns to find the product when you multiply by a multiple of 10. Find the product. 80 3 100 5 8 ones 58 831 5 8 3 1 ones Name Name MXENL08AWK5X_RT_CH01_L7.indd A RW7-RW8 7/19/07 5:55:16 PM 6/12/07 11:11:35 AM Grade5.indd 5 5 5 10. 44 3 260 11. 489 3 706 Estimate the product. 5. 456 3 76 6. 79 3 61 3 3. 63 3 43 3 1. 52 3 31 RW9 12. 3,485 3 59 7. 53 3 1,299 Round each factor and estimate the product. 30 3 700 5 21,000 32 3 723 13. 45 3 914 8. 26 3 725 3 4. 512 3 49 3 2. 731 3 47 STEP 2: Use basic multiplication facts and patterns to find the product of the rounded factors. 723 is closer to 700 than 800. 723 rounds to 700. 32 is closer to 30 than 40. 32 rounds to 30. 6/12/07 11:29:02 AM © Harcourt • Grade 5 Reteach 14. 38 3 4,118 9. 71 3 $9.58 5 5 Record the 4 in the ones place. Write the 2 above the tens place. 4 4 3 3 464 7 42 3 3 MXENL08AWK5X_RT_CH02_L03.indd 1 6. 1. 7. 2. 3 764 8 65 3 4 2 5 2 T 4 4 6 O 8. 3. 3 604 5 RW10 1,208 9 3 Record the 2 in the tens place. Write the 2 above the hundreds place. 9. 4. 3 532 6 3 2 5 2 H 2 5 2 T 4 4 6 O 10. 5. 3 3 © Harcourt • Grade 5 Reteach 4,365 6 745 3 Record the 2 in the hundreds place and the other 2 in the thousands place. 6/12/07 11:31:06 AM 4 3 5 hundreds 5 20 hundreds 20 hundreds 1 2 hundred 5 22 hundreds 2 T Step 3: Multiply the hundreds. 3,045 8 3 4 3 5 tens 5 20 tens 20 tens 1 2 ten 5 22 tens 5 H 5 T 2 6 O 2 T 4 3 6 ones 5 24 ones 24 ones 5 2 ten 4 ones 5 H Step 2: Multiply the tens. Estimate. Then find the product. 3 T Step 1: Multiply the ones. Estimate: 600 3 9 5 2,400 You can use rounding numbers and use multiplication facts to estimate products. Remember: If the digit to the right of the greatest place is 0– 4, round down. If the digit is 5– 9, round up. Multiply by 1-Digit Numbers You can use a place-value chart to help you multiply by 1-digit numbers.1 Estimate. Then find the product 556 3 4 Estimate Products Estimate the product. 32 3 723 STEP 1: Round both factors to the greatest place. Name Name MXENL08AWK5X_RT_CH02_L02.indd 1 RW9-RW10 7/19/07 5:55:35 PM Grade5.indd 6 1 6 3 8 5 8 6 T 6 H Think: 7 3 688 5 4,816 4 Th 6 7 8 O 4 Tt 287 3 38 6. 11. 237 3 16 5 253 3 17 1. 7. 2. 392 3 81 439 3 56 5 6 2 8 3 8 1 6 8 4 5 6 T H 466 3 29 $324 3 45 RW11 12. 407 3 28 5 8. 3. 9. 4. 0 6 7 8 O Think: 60 3 688 5 41,280 1 4 Th Step 2: Multiply the tens. Estimate. Then find the product. Tt Step 1: Multiply the ones. 688 3 67 Estimate: 700 3 70 5 49,000 Estimate. Then find the product. 6 4 0 2 9 8 1 6 3 8 8 6 5 4 5 6 T H O 6 0 6 7 8 10. 5. 6/29/07 3:09:20 PM © Harcourt • Grade 5 Reteach 189 3 86 805 3 62 13. 683 3 53 5 507 3 54 576 3 43 Think: 46,096 is close to the estimate of 49,000. So the answer is reasonable. 1 4 Th 4 Tt Step 3: Add the partial products. MXENL08AWK5X_RT_CH02_L05.indd 1 11 3 35 5 385, 11 3 42 5 462, and 11 3 63 5 693, what is the middle digit in the product of 11 and 81? 7. If 11 3 23 5 253, 11 3 33 5 363, Find a pattern to solve. 6. How can you check your answer? Check numbers? © Harcourt • Grade 5 Reteach 8. What is the sum of the first 8 odd RW12 5. Write your answer in a complete sentence. 3. Solve the problem. Describe the strategy you used. Solve 2. How can finding a pattern help you solve the problem? Plan 1. Write the question as a fill-in-the-blank sentence. Read to Understand A large art museum is planning a show of famous oil paintings. The show will last for 4 days. If 38,888 people attend the show each day, how many people will attend the show in all? Problem Solving Workshop Strategy: Find a Pattern Multiply by Multi-Digit Numbers You can use a place-value chart and regrouping to help you multiply by 2-digit numbers. Name Name MXENL08AWK5X_RT_CH02_L04.indd 1 RW11-RW12 7/19/07 5:55:52 PM 6/12/07 11:33:25 AM Grade5.indd 7 11. 7 3 2,394 5 6. 450 3 18 3 7. 384 3 258 8. RW13 12. 54 3 37 3 19 5 9,000 30 3 6/12/07 11:36:31 AM © Harcourt • Grade 5 Reteach 8 9 10. 702 3 5. 143 13. 40 3 400 3 10 5 423 3 12 9. Find the product. Choose mental math, paper and pencil, or a calculator. 2. 453 3. 5,000 4. 285 1. 305 3 24 3 627 3 123 3 30 Multiply. 3 3 75 3 216 This is a good problem to solve using a calculator because it requires two calculations and uses greater numbers. This is a good problem to solve using paper and pencil since a 3-digit number is multiplied by a 2-digit number. You can use a calculator to solve problem with greater numbers or problems with more than one step. 3,432 429 8 3 __ 27 You can use paper and pencil to solve problems when mental math is too difficult. Multiply. 42 3 60 42 3 60 5 (40 1 2) 3 60 5 (40 3 60) 1 (2 3 60) 5 2,400 1 120 5 2,520 6qw 216 12 18 24 30 180 4 6 5 30 216 4 6 MXENL08AWK5X_RT_CH03_L1.indd 1 10. 499 4 7 5 7. 265 4 4 5 5qw 314 3qw 252 Estimate the quotient. So, 180 4 6 is about 30 4. 42 48 157 2qw 546 6qw RW14 11. 345 4 6 5 8. 344 4 8 5 5. 2. So, 216 4 6 is about 40 240 4 6 5 40 216 4 6 8qw 289 254 4qw 54 12. 189 4 8 5 9. 372 4 5 5 6. 3. 18 and 24 are both close to 21. You can use either number, or both numbers to estimate the quotient. Step 2: Estimate using compatible numbers. 1. 36 Find multiples that are close to the first 2 digits of the dividend. 6 Step 1: Think of the multiples of 6: Estimate the quotient. Compatible numbers are numbers that are easy to work with mentally. In division, one compatible number divides evenly into the other. Think of the multiples of a number to help you find compatible numbers. You can use mental math, paper and pencil, or a calculator to find a product. This is a good problem for mental math because you can use the Distributive Property to compute mentally. Estimate with 1-Digit Divisors Choose a Method You can use mental math to solve problems with numbers that are easy to compute. Name Name MXENL08AWK5X_RT_CH02_L06.indd 1 RW13-RW14 7/19/07 5:56:10 PM Reteach © Harcourt • Grade 5 6/12/07 11:39:38 AM Grade5.indd 8 22 Divide. 4qw Multiply. 4 3 5 5 20 Subtract. 22 2 20 5 2 Compare. 2 , 4 8qw 136 3qw 741 Divide. 1. 7qw 297 7qw 456 2. 5qw 8,126 RW15 8,659 4qw 3. 6qw 5,238 256 4qw 1,027 2 8 22 2 20 27 2 24 3 4. 6/12/07 11:44:03 AM © Harcourt • Grade 5 Reteach 4qw 9,449 4,973 7qw 27 Divide. 4qw Multiply. 4 3 6 5 24 Subtract. 27 2 24 5 3 Compare. 3 , 4 Write the remainder. Step 4: Bring down the 7 ones. Then divide the 27 ones. Name the position of the first digit of the quotient. Then find the first digit. 25 4qw 1,027 2 8 22 2 20 27 Step 3: Bring down the 2 tens. Then divide the 22 tens. I will only use the remainder B C MXENL08AWK5X_RT_CH03_L3.indd 1 © Harcourt • Grade 5 Reteach 8 miles per day along a 125-mile long trail. How many days will Jessie and her family hike exactly 8 miles? 7. Hannah and her family want to hike RW16 troop. They will canoe a total of 75 miles, and want to travel 8 miles each day. How many days will they need to travel the entire distance? 6. Harry goes on a canoe trip with his scout Solve the following problems and then tell how you would interpret the remainder. 5. How can you check to see if your answer is reasonable? 4. How many minivans are needed for the field trip to the park? D I will only use the quotient I will add 1 to the quotient I will use the quotient and write the remainder as a fraction A 3. How would you interpret the remainder? Circle your answer then explain. 9qw 75 2. What is the quotient? What is the remainder, if there is one? 1. Write the question as a fill-in-the-blank sentence. A total of 75 fifth-grade students are going on a field trip to a local park. The school is providing minivans to take the students to the park. If each minivan holds 9 students, how many minivans are needed? Problem Solving Workshop Skill: Interpret the Remainder Divide by 1-Digit Divisors You can use estimation to help you place the first digit in the quotient. Then, you can follow steps to divide. Name the position of the first digit of the quotient. Then find the first digit. Divide. 4qw 1,027 Step 1: Step 2: Use estimation to place the first digit. Divide the 10 hundreds. 10 Divide. 4qw 2 1,027 4 4 4qw 1,027 Multiply. 4 3 2 5 8 2 8 800 4 4 5 200 Subtract. 10 2 8 5 2 2 So, the first digit, 2, is in the hundreds place. Compare. 2 , 4 Name Name MXENL08AWK5X_RT_CH03_L2.indd 1 RW15-RW16 6/18/07 5:52:29 PM 6/12/07 11:45:07 AM Grade5.indd 9 9. 8qw 330 13. 846 4 7 RW17 8. 7qw 843 6/12/07 11:47:53 AM © Harcourt • Grade 5 Reteach 14. 5,420 4 5 1,423 10. 7qw 5. 3qw 6,024 1 0 2 Multiply: 7 3 2 = 14 714 7qw 27 Subtract: 14 – 14 = 0 014 214 Compare: 0 < 7 0 4. 5qw 5,412 1 < 7, so there are not enough tens to divide. Write 0 in the tens place of the quotient. 3. 3qw 927 12. 3,291 4 3 7. 4qw 8,126 807 6. 2qw 11. 605 4 3 2. 3qw 624 10 7qw 714 27 _____ 01 1. 8qw 872 Divide. So, 714 4 7 5 102. Compare: 0 < 7 Subtract: 7 – 7 = 0 Multiply: 7 3 1 = 7 Divide the ones. Bring down the 1 ten. Divide the tens. Divide the hundreds. 7 7qw 714 20 _____ 0 Step 4: Step 3: Step 2: So, the first digit is in the hundreds place. 700 4 7 = 100 714 4 7 Step 1: Use estimation to place the first digit. MXENL08AWK5X_RT_CH03_L5.indd 1 17. 8,000 4 10 13. 6,300 4 7 9. 210 4 3 5. 500 4 50 1. 40 4 2 18. 1,400 4 7 14. 6,000 4 2 10. $300 4 10 6. 120 4 40 2. 160 4 8 RW18 19. $2,400 4 30 15. 3,000 4 30 11. 630 4 90 7. 480 4 6 3. $270 4 90 Use basic facts and patterns to find the quotient. 36,000 4 60= 600 Step 3: Divide. Use the basic fact. 36,000 4 60 Step 2: There are zeros in the dividend and the divisor. Cancel out one zero each. The basic fact is 36 4 6 = 6. 36,000 4 60 Step 1: Find the basic fact. Use basic fact patterns to find the quotient. 36,000 4 60. You can use basic facts and patterns to find quotients. 714 Divide. 7qw Algebra: Patterns in Division Zeros in Division Name When you have zeros in division, you treat them the same way you would treat any other digit when dividing. Name MXENL08AWK5X_RT_CH03_L4.indd 1 RW17-RW18 6/18/07 5:53:21 PM © Harcourt • Grade 5 Reteach 20. 5,600 4 8 16. $4,500 4 50 12. 540 4 60 8. 560 4 70 4. 420 4 6 6/12/07 11:50:04 AM Grade5.indd 10 157 42qw 622 11. 12qw Estimate the quotient. 6. 409 4 63 1. 73q 268 12. RW19 34qw 293 7. 478 4 19 2. 13. 738 81qw 8. 7,145 3. Write two sets of compatible numbers for each. Then give two possible estimates. So, 8 and 9 are reasonable estimates of the quotient. or 500 4 50 5 9 427 4 49 400 4 50 5 8 427 4 49 Step 3: Divide the compatible numbers to find the estimates. 427 is between 400 and 500 Step 2: Find two numbers close to the dividend that are compatible with the rounded divisor. Step 1: Round the divisor to the nearest ten. 49 rounds to 50. Reteach 6/12/07 11:51:44 AM © Harcourt • Grade 5 2,369 Divide. 53qw Estimate the quotient. 343 54qw You can use estimation to help you place the first digit in the quotient. Then, you can follow steps to divide. Compatible numbers are numbers that are easy to work with mentally. In division, one compatible number divides evenly into the other. Think of the multiples of a number to help you find compatible numbers. MXENL08AWK5X_RT_CH03_L7.indd 1 6,413 4. 43qw 612 1. 52qw Divide. Check your answer. So, 2,369 4 53 5 44 r37. 44 r37 53qw 2,369 2 212 249 2 212 37 RW20 4,684 5. 27qw 917 2. 63qw Step 3: Bring down the 9 ones. Then divide the 249 ones. 4 53qw 2,369 2 212 24 Step 2: Divide 236. 40 2,000 50qw 1,597 6. 89qw 608 3. 24qw © Harcourt • Grade 5 Reteach 6/12/07 11:52:11 AM Write the remainder to the right of the whole number part of the quotient. Compare: 37 , 53 Subtract: 249 2 212 5 37 Multiply: 53 3 4 5 212 Think: Multiply: 53 3 4 5 212 Subtract: 236 2 212 5 24 Compare: 24 , 53 Think: So, the first digit is in the tens place. Step 1: Use estimation to place the first digit. Remember to use compatible numbers to estimate. Divide by 2-Digit Divisors Estimate with 2-Digit Divisors 49qw 427 Name Name MXENL08AWK5X_RT_CH03_L6.indd 1 RW19-RW20 6/18/07 5:53:31 PM Grade5.indd 11 20 400 6. 16qw 845 Divide. 1. 58qw 1,325 20 7. 24qw 217 6 2. 37qw 241 80 RW21 8. 37qw 4,819 3. 29qw 2,276 Write low, high, or just right for each estimate. So, for 16qw 416 the estimated quotient, 20, is too low. 96 is much greater than 16. 96 > 16 20 416 16qw 2 32 96 2 0 96 9. 71qw 488 10 4. 82qw 910 Then divide using your compatible dividend and divisor. 20 20qw 400 2 40 00 Next, use the estimated quotient to check the degree of accuracy. 16 416 Write too high, too low, or just right for the estimate below. 20 16qw 416 First, use compatible numbers for the dividend and divisor. 6/12/07 11:57:09 AM © Harcourt • Grade 5 Reteach 10. 43qw 9,189 60 5. 63qw 3,784 Write 1 as the remainder. Compare: 1 < 4 Subtract: 33 – 32 = 1 Multiply: 8 3 4 = 32 Compare: 3 < 4 Subtract: 35 – 32 = 3 Multiply: 8 3 4 = 32 6qw 115 MXENL08AWK5X_RT_CH03_L9.indd 1 6. 219 4 7 1. 7. 935 4 4 2. 9qw 326 RW22 8. 6,121 4 5 3. 7qw 2,198 Divide. Multiply to check your answer. So, 353 4 4 5 88 r1 88 r1 4qw 353 2 32 33 2 32 1 Step 3: Divide the ones. 8 4qw 353 2____ 32 3 Step 2: Divide the tens. So, the first digit is in the tens place. 350 4 4 = 8 353 4 4 Step 1: Use estimation to place the first digit. 353 . Multiply to check your answer. Divide 4qw 3,504 9qw 9. 9,217 4 7 4. 3,167 6qw © Harcourt • Grade 5 Reteach 10. 8,032 4 4 5. When you divide, it helps to remember that division is an operation that tells the number of equal groups, or the number in each equal group. Practice Division Correcting Quotients Estimates can help you identify the first digit in the quotient, but sometimes you will need to correct the quotient. Name Name MXENL08AWK5X_RT_CH03_L8.indd 1 RW21-RW22 6/18/07 5:53:40 PM 6/12/07 11:59:23 AM Grade5.indd 12 7 1 (72 4 9) RW23 elderly neighbor with chores. Her goal is to earn $1,300. She saves $25 per week of her earnings. She spends $10 at the shopping mall with her friends. Her brother, Germaine, is saving $15 per week to buy an MP3 player. How many weeks must Shonda save $25 to reach her goal? 6/12/07 12:02:24 PM © Harcourt • Grade 5 Reteach 1,890 miles. He paid an average of $2.57 for gas and his car got 15 miles to the gallon. On average, how many miles did Mr. Greene drive each day? 2. 52 more than 24 3. 6 plus the quotient of 56 and 7 MXENL08AWK5X_RT_CH04_L1.indd 1 He grew out of 1 pair and bought some more jeans. 4. Kelly has 4 pairs of jeans. © Harcourt • Grade 5 Reteach more markers than crayons. He then hiked for 20 more minutes. RW24 6. Quinn has three times 5. Adam hiked for a while. Write an algebraic expression. Tell what the variable represents. week and 17 miles the next week. Write a numerical expression. Tell what the expression represents. 1. Hank ran 14 miles one Tell which information is relevant and irrelevant to solve the problem. Then solve. 6. Over a 45-day period, Mr. Greene drove Think: It is a good idea to use a variable that helps you remember what it represents. In this case, n = books. n19 let n 5 the number of books Dee had. 19 Then, “given 9 more books” So, “Dee had some books. She was given 9 more books.” can be represented as variable n First, “some books” Dee had some books. She was given 9 more books. Write an algebraic expression. Tell what the variable represents. An algebraic expression is an expression with at least one variable. A variable is a letter or symbol that stands for one or more numbers. So, 7 plus the quotient of 72 divided by 9 represents a sum. Use clue words to help you write Write a numerical expression. Tell what the expression represents. expressions. For example: more, 7 plus the quotient of 72 divided by 9 sum, added, and plus indicate First, “7 plus” 71 addition. Then, “the quotient of 72 divided by 9” 72 4 9 A numerical expression has only number and operation signs. 5. Shonda earns $35 per week helping her 4. What was the price for each student ticket? information in the space on the right to write an equation to solve? 3. How could you use the relevant A total of 48 fifth graders and 4 teachers went on a field trip to the. The total cost for the students’ tickets was $576. The total cost for the teachers’ tickets was $60. What was the price for each student ticket? cross out irrelevant information? What would you circle, if you were told to circle relevant information? 2. What would you cross out in the problem below, if you were told to 1. How would you write the problem as a fill in the blank sentence? A total of 48 fifth graders and 4 teachers went on a field trip to the museum. The total cost for the students’ tickets was $576. The total cost for the teachers’ tickets was $60. What was the price for each student ticket? Write Expressions Problem Solving Workshop Skill: Relevant or Irrelevant Information An expression has numbers, operation signs, and sometimes variables. An expression does not have an equal sign. Name Name MXENL08AWK5X_RT_CH03_L10.indd 1 RW23-RW24 6/18/07 5:53:50 PM 6/12/07 12:09:43 PM Grade5.indd 13 10. 38 2 (18 4 6) 9. 3 3 (54 4 9) 11. 64 2 (24 4 3) 7. (20 2 13) 3 5 3. (75 1 5) 2 8 2 12. (13 2 7) 3 4 8. (21 1 42) 2 6 4. (16 3 2) 2 10 3418 8 1 8 5 16 (6 4 3) 3 4 1 8 14. (16 3 w) 2 5 if w 5 4 18. (22 1 k) 3 5 if k 5 3 13. 2 1 (15 4 n) if n 5 3 17. (8 1 4) 3 n if n 5 2 RW25 if z 5 185 19. z 1 (8 3 1) if r 5 7 15. (r 1 9) 1 6 Reteach 6/12/07 12:13:34 PM © Harcourt • Grade 5 6 if m 5 30 __ 1 14 20. m if x 5 5 16. (96 4 12) 5 x Evaluate the algebraic expression for the given value of the variable. 6. (81 1 11) 2 8 2. 33 2 9 1 14 5. 6 3 (63 4 7) 1. 15 1 6 1 3 Evaluate each expression. So, (6 4 3) 3 4 1 8 5 16. • Perform the operation in parenthesis first: • Multiply or divide, from left to right: • Add or subtract, from left to right: Remember to follow the order of operations. (6 4 3) 3 4 1 8 Evaluate the expression. Commutative Property Zero Property of Multiplication MXENL08AWK5X_RT_CH04_L3.indd 1 4. 12 3 (6 3 8) 5 (12 3 n) 3 8 1. 27 3 (n 3 8) 5 (27 3 9) 3 8 RW26 © Harcourt • Grade 5 Reteach 3. 6 3 n 5 85 3 6 5. (4 1 n) 1 3 5 4 1 (7 1 3) 6. n 3 120 5 0 2. 61 1 33 5 33 1 n Find the value of n. Identify the property used. So, this follows the Identity Property of Multiplication, n 5 1. Notice that the product and the factor other than n are the same. 15 3 0 5 0 The product of any number and zero is zero. Find the value of n. Identify the property used. 81 3 n 5 81 4 3 11 5 11 3 4 8325238 If the order of the addends or the factors is changed, the sum or product stays the same. (7 3 4) 3 5 5 7 3 (4 3 5) (9 1 3) 1 2 5 9 1 (3 1 2) 51055 33156 The way addends are grouped on factors are grouped does not change the sum or product. Associative Property The sum of zero and any number equals that number. The product of one and any number equals that number. Identity Property Expressions using addition and multiplication follow certain properties. Properties Evaluate Expressions To evaluate an expression, or to find the value of an expression, you have to perform each operation separately to determine the answer. Name Name MXENL08AWK5X_RT_CH04_L2.indd 1 RW25-RW26 6/18/07 5:54:01 PM 6/12/07 12:18:21 PM Grade5.indd 14 RW27 14. 68 1 81 1 42 13. 73 3 3 8. 3 3 360 7. 3 3 9 3 2 11. 37 3 5 5. 21 1 39 1 38 4. 7 3 5 3 3 10. 7 3 3 3 9 2. 4 3 27 1. 27 1 26 1 33 Use properties and mental math to find the value. So, 5 3 29 5 145. 5 145 5 100 1 45 5 (5 3 20) 1 (5 3 9) 5 3 29 5 5 3 (20 1 9) 15. 12 3 4 3 5 Reteach 6/12/07 12:19:17 PM © Harcourt • Grade 5 12. 43 1 (47 1 46) 9. 62 1 28 1 17 6. 6 3 45 3. (24 1 19) 1 16 Use the Distributive Property and mental math to work out 5 3 29. 5 3 29 Use properties and mental math to find the value. You can use addition or subtraction to break apart a factor. The Distributive Property states that you can break apart a factor to multiply. Let k stand for the number of keys Omar has. Choose a variable. Step 2 MXENL08AWK5X_RT_CH04_L5.indd 1 © Harcourt • Grade 5 Reteach seedlings and some cucumber seedlings. She planted 31 seedlings altogether. How many cucumber seedlings did Mrs. Greene plant? 4. Mrs. Greene planted 18 tomato After he bought the jacket he had $18 left. How much did the jacket cost? 2. Jerrod saved $85 to buy a new jacket. RW28 test. He studied 3 times longer for his science test than for his spelling test. How long did Ryan study for his spelling test? 3. Ryan studied 45 minutes for his science muffins. Beth’s family ate some. Now there are 16 muffins left. How many muffins did Beth’s family eat? 1. Beth’s mother made 24 blueberry k25=7 Write an equation. Step 3 Write an equation for each. Tell what the variable represents. So, the equation is k2557 Let k 5 number of keys Omar has. A number decreased by 5 is 7. Write a representative sentence. Step 1 Tia has 5 fewer keys than Omar. If Tia has 7 keys, how many does Omar have? Write an equation. Tell what the variable represents. An equation is a number sentence that shows that two quantities are equal. Like an expression, an equation has numbers, operation signs, and sometimes variables. An equation is different from an expression because an equation does have an equal sign. Write Equations Mental Math: Use the Properties You can use properties and mental math to help you solve problems. Name Name MXENL08AWK5X_RT_CH04_L4.indd 1 RW27-RW28 6/18/07 5:54:13 PM 6/12/07 1:03:40 PM Grade5.indd 15 A function is a relationship between two quantities. One quantity depends upon the other. You can show a function using a function table. To solve an equation, you find a value for the variable that makes the equation true. That value is the solution. 15 4 3 5 5 Try 3 Yes No Yes 6. 7 3 v 5 42 5. f 1 17 5 20 7. 48 4 k 5 8 3. 5 3 u 5 60 10. 32 5 p 1 9 14. 18 5 a 1 2 9. 45 5 5 3 t 13. 24 5 6 3 z RW29 15. x 2 11 5 23 11. 35 2 n 5 14 Use mental math to solve each equation. Check your solution. 2. 12 4 e 5 4 1. t 2 5 5 1 Which of the numbers 3, 6, or 12 is the solution of the equation? So, for 15 4 r 5 5, r 5 3. 3 3 5 5 15 Check: Use an inverse operation to check your work. 15 4 5 5 3 Try 5 Test some possibilities by replacing r with your predictions. Think: 15 divided by what number equals 5? 15 4 r 5 5 6/12/07 1:04:58 PM © Harcourt • Grade 5 Reteach 16. 18 4 v 5 6 12. 77 4 y 5 11 8. 41 2 g 5 29 4. 6 1 p 5 12 5 30 6 36 42 8 48 0 y 1 MXENL08AWK5X_RT_CH04_L7.indd 1 k j 3. k 5 7j 1 1 0 x 1. y 5 4x 3 1 5 2 Complete each function table. 7 3 64 9 4 0 b a 2 4. b 5 6a 2 8 n m 2. n 5 5m 1 4 RW30 So, when x 5 42 in the function table, y 5 7. Rule: Divide by 6 Equation: x 4 6 5 y Replace: 42 4 6 5 y 75y Think of a rule. Write it as an equation. Look at the pattern. Outputs Inputs 16 4 3 6 29 5 Write an equation to represent each function. Then complete the table. Functions Solve Equations Use mental math to solve the following equation. Check your solution. Name Name MXENL08AWK5X_RT_CH04_L6.indd 1 RW29-RW30 7/19/07 6:04:07 PM 8 8 © Harcourt • Grade 5 Reteach 10 10 6/15/07 3:14:17 PM Grade5.indd 16 The capacity of a large aquarium is 12 gallons more than the capacity of a small aquarium. The aquariums hold a total of 52 gallons of water. What is the capacity of each aquarium? Read to Understand 1. What are you asked to find? An inequality is a number sentence that shows that two amounts are not equal. 925<6 or 4 < 6 is true 10 2 5 < 6 or 5 < 6 is true 6. x 1 9 , 18 7. x 1 5 . 13 8. x . 15 2 6 13. x 2 14 . 3 9. x . 16 14. x 1 5 . 11 10. x , 11 1 7 RW31 15. x 1 3 , 22 11. x . 24 2 8 6/12/07 1:22:47 PM © Harcourt • Grade 5 Reteach 16. x 2 9 , 8 12. x 1 1 , 18 Which of the numbers 16, 17, and 18 are solutions of each inequality? 5. x 2 4 , 6 Which of the numbers 8, 9, and 10 are solutions of each inequality? 1. x . 8 2. x , 10 3. x , 12 2 3 4. x . 6 1 1 So, 8, 9, and 10 are solutions to the inequatity x 2 5 < 6. 825<6 or 3 < 6 is true Replace x with each of the numbers that are possible solutions. ( 1 2 12) 2 12) 5 5 52 52 MXENL08AWK5X_RT_CH04_L9.indd 1 and bettas cost $5 each. Andy spent $19 at the store. How many of each type of fish did Andy buy? 6. Catfish cost $3 each at the pet store RW32 © Harcourt • Grade 5 Reteach tetras in her aquarium. The product of the numbers of each type of fish is 63. If Charlotte has more goldfish than tetras, how many of each type does she have? 7. Charlotte has a total of 16 goldfish and How can you use the equation c + (c + 12) = 52 to check your answer? Predict and test to solve. 5. ( 1 Small Aquarium What is the capacity of each aquarium? Check 4. Large Aquarium Solve 3. What might 2 of your predictions be? Test these in the equations below. Plan 2. What do you know about the capacities of the two aquariums? Problem Solving Workshop Strategy: Predict and Test Inequalities Which of the numbers 8, 9, and 10 are solutions of the inequality, x – 5 < 6? Name Name MXENL08AWK5X_RT_CH04_L8.indd 1 RW31-RW32 6/18/07 5:54:32 PM 6/12/07 1:24:26 PM Grade5.indd 17 3. 2 2 2 2 Hundredths Thousandths 5. 4. 7. 3.01 8. 3.01 9. Find the value of the underlined digit. 2. 1. RW33 9.814 10. 54.236 Write the decimal shown by the shaded part of each model. So, the decimal represented by the model is 0.08. Tenths Ones There are 100 squares in a hundredths model. In this model 8 squares are shaded. Use a place value chart to help write the decimal. 6. 11. 54.236 6/12/07 1:30:05 PM © Harcourt • Grade 5 Reteach Equivalent decimals are different name for the same number or amount. You can use decimal squares to find equivalent decimals. You can use models to write decimals. This model shows 420 thousandths, or 0.420. There are 420 shaded parts. The model shows 1,000 equal parts. Each part represents 1 thousandth, or 0.001 of the model. Now divide each of the 100 parts into 10 equal parts. The model at the right shows what each small square would look like. 4.87 and 4.870 0.23 and 0.230 5. 2. 9.87 and 9.78 0.51 and 0.500 0.830 0.803 0.83 MXENL08AWK5X_RT_CH05_L2.indd template1 7. 8. RW34 0.93 0.093 0.930 Write the two decimals that are equivalent. 4. 1. 9. 6. 3. 1.007 1.070 1.07 © Harcourt • Grade 5 Reteach 1.11 and 1.111 0.680 and 0.68 Write equivalent or not equivalent to describe each pair of decimals. Each model shows the same amount shaded. So, 0.42 and 0.420 are equivalent. This model shows 42 hundredths, or 0.42. 42 of the parts are shaded. The model shows 100 equal parts. Each part is 1 hundredth of the model. 0.42 and 0.420. Write equivilant or not equivilant to describe the pair at decimals below. Equivalent Decimals Decimal Place Value Write the decimal shown by the shaded part of the model below. Name Name MXENL08AWK5X_RT_CH05_L1.indd 1 RW33-RW34 6/18/07 5:55:16 PM 6/12/07 1:33:01 PM Grade5.indd 18 8.106 0.6 8.16 0.603 4. 7. 11. 14. 0.614, 0.641, 0.64 3.08, 3.801, 3.8 13. 8. 5. 2. 10. Order from least to greatest. 9.9 9.39 1. Compare. Write <, >, or = for each So, 0.28 > 0.208. 7.8 0.89 0.30 RW35 0.159, 0.154, 0.14 1.576, 1.765, 1.567 0.78 0.69 0.308 . 4.71 1.83 7.245 7.254 4.071 1.833 6/12/07 1:36:16 PM © Harcourt • Grade 5 Reteach 15. 7, 6.99, 6.099, 7.001 12. 3.971, 3.9, 4, 3.901 9. 6. 3. Analyze Complete the table. RW36 © Harcourt • Grade 5 Reteach Julia has three plants. One plant receives 1 tablespoon of plant food each month and grows to a height of 7 cm. A second plant receives 3 tablespoons of plant food each month and grows to a height of 8.5 cm. A third plant is not given any plant food and grows to a height of 5.75 cm. What conclusion can you draw about how the amount of food given affects plant growth? MXENL08AWK5X_RT_CH05_L4.indd template1 5. Conclusion What conclusion can Mr. Hall draw about the number of weeks students spent on their plant experiments? Solve by drawing a conclusion. 4. How much time did the least number of students spend on the experiment? How much time did the greatest number of students spent on the experiment? 3. students spent 4 to 6 weeks. students spent 2 to 4 weeks. Since 8 hundredths is greater than 0 hundredths, 0.28 is greater than 0.208. students spent less than 2 weeks. 8>0 How would you complete the statements below using the details from the problem? 2=2 2. Compare the digits in the hundredths place. 0 Compare the digits in the tenths place. 8 What does the problem ask you to find? 0=0 2 0 Hundredths Thousandths Compare the digits in the ones place. Tenths Ones Use a place value chart. 1. Mr. Hall discovers that 5 of his students spent less than 2 weeks on their plant experiment, 16 students spent 2 to 4 weeks, and 3 students spent 4 to 6 weeks. What conclusion can Mr. Hall draw about the number of weeks students spent on their plant experiment? A place-value chart can help you compare decimals. You may need to add zeros so you can compare the same number of digits in each decimal. Compare the digits from left to right. . Problem Solving Workshop Skill: Draw Conclusions Compare and Order Decimals Compare 0.28 and 0.208. Write <, >, or = for the Name Name MXENL08AWK5X_RT_CH05_L3.indd template1 RW35-RW36 6/18/07 5:58:35 PM 6/12/07 1:41:29 PM Grade5.indd 19 ROUNDING RULES: • If the digit to the right is less than 5, the underlined digit stays the same. Rewrite: 0.130 or 0.13 Step 2: Rewrite all digits to the right of the underlined digit as zeros. An equivalent decimal can be written by leaving off the trailing zeros. 6. 11.323 10. 20.595 5. 12.63 9. 0.964 11. 6.89 7. 4.289 3. 108.108 13. 12.35 to 12.4 RW37 14. 0.428 to 0.43 Name the place to which each number was rounded. 2. 9.028 1. 7.325 Round each number to the place of the underlined digit. Compare: 4 < 5 4 is less than 5, so the digit stays the same. Step 1: Compare the digit to the right of the underlined digit to 5 using the rounding rules. hundredths place 15. 9.462 to 9.46 Reteach 6/12/07 1:46:43 PM © Harcourt • Grade 5 12. 32.514 8. 7.547 4. 26.199 • If the digit to the right is greater than or equal to 5, the underlined digit increases by 1. $ 1 3. 0 4 2 $ 0. 9 5 $ 1 2. 0 9 $13 $1 $12 0. 4 5 1 0. 7 1. 5 8 1 4. 5 3 MXENL08AWK5X_RT_CH06_L2.indd 1 9. 3.5 1 2.89 1 0.4 5. 1. 1 4 6. 8 $1 8. 5 2 1 $3. 7 3 11. 7. 3. RW38 10. $15 2 $1.27 6. 2. Find the sum or difference. 21.05 2 2.65 2. 9 2 0. 6 3 6. 3 9 2. 1 8 1 7. 8 5 So, $13.04 2 $0.95 5 $12.09 is close to the estimate, $12. So the answer is reasonable. $13.04 2 $0.95 2 $0.95 __ 12 19 14 You can use lined notebook paper to add and subtract decimals. You can use the same rules you learned for rounding whole numbers to round decimals. Subtract. $13.04 Add and Subtract Decimals Round Decimals Round 0.134 to the place of the underlined digit. Name Name MXENL08AWK5X_RT_CH06_L1.indd 1 RW37-RW38 6/18/07 5:58:44 PM $2 1. 40 2 $1. 33 0. 3 2 1 2 0. 1 2 3 © Harcourt • Grade 5 Reteach 12. $7.00 2 $1.05 8. 4. 6/12/07 2:10:37 PM Grade5.indd 20 5.4 3.4 3.4 + 0.8 0.82 0.305 0.78 0.8 0.5 2 0.3 6.17 2 3.5 51.234 2 28.4 11. 0.427 + 0.711 6. 1. 7. 2. 1.73 1.4 1 3.17 8. 3. 3.28 2 0.86 7.6 2 2.15 0.78 2 0.305 RW39 9. 4. 10. 5. 6/12/07 2:41:00 PM © Harcourt • Grade 5 Reteach $23.07 2 $ 7.83 2.083 0.56 1 0.41 13. 40.512 + 30.399 15.27 1 41.8 0.443 1 0.207 • greater than or equal to 5, round up. If the digit to the right of the place you are rounding to is: • less than 5, round down. Remember the rounding rules: 12. 61.05 – 18.63 $29.38 1 $42.75 Estimate by rounding. So, the difference is about 0.5. 2 Estimate the difference by rounding. 1 1.2 1.247 Round and add to estimate. First, decide what place to round to. Since two of the addends have digits to the hundredths, tenths would be a good place to round to. 8 . 4 4 5 _ 5 . 8 71.4 1__ 11.5 1 73.9 1 4.37 MXENL08AWK5X_RT_CH06_L4.indd 1 5. 1. 6. 2. 90.4 1 88.76 127.35 1 928.527 3.3 1_ 5.6 1 RW40 7. 1__ 2.25 1 3. 10 10 1 7 13 1 38.445 2 25.86 12.585 2 . 5 • Use a calculator for difficult numbers or very large numbers. $ 48.60 1 32.81 __ $ 15.79 0.4 • Use paper and a pencil for larger numbers. 1.2 2 0.8 _ Choose a method. Find the sum or difference. So, 3 2 25.86 __ Choose a method. Find the difference. 2 Estimate the sum by rounding. These are very large numbers, so use a calculator. There is more than one way to find the sums and differences of whole numbers and decimals. You can use mental math, a calculator, or paper and pencil. 38.445 You can estimate to find an answer that is close to the exact answer. You can use the rounding rules to help you estimate sums or differences. 8. 4. 8 • Use mental math for problems with fewer digits or rounded numbers. Choose a Method Estimate Sums and Differences 1.247 0.82 1 3.4 Name Name MXENL08AWK5X_RT_CH06_L3.indd 1 RW39-RW40 6/18/07 5:58:56 PM © Harcourt • Grade 5 Reteach 14.219 1.793 1 15.881 1 0.364 1 1__ 1.558 5 38.445 6/12/07 2:45:39 PM Grade5.indd 21 How long does her last throw need to be for her to advance to the final round? 4. 5. A waitress charged Hannah $6.75 for lunch. Hannah wants to tip the waitress about 20% of $6.75. How much should she leave? RW41 6/12/07 2:48:05 PM © Harcourt • Grade 5 Reteach car. Each tire costs $110.60. What is the total cost? 6. Lance is purchasing 4 new tires for his Tell whether you need an estimate or an exact answer. Then solve the problem. Show how you solve the problem in the space below. Will you estimate or find an exact answer to solve this problem? Explain your choice. 2. 3. What are you asked you to find? 1. In a baseball-throwing contest, a score of 50 meters or more is needed to advance to the final round. Jenna’s first two throws were 16.64 meters and 15.33 meters. How long does her last throw need to be for her to advance to the final round? Model Multiplication by a Whole Number Problem Solving Workshop Skill: Estimate or Find Exact Answer MXENL08AWK5X_RT_CH07_L1.indd 1 7. 4 3 0.12 4. 8 3 0.05 1. 9 3 0.23 Find the product. So, 2 3 0.96 5 1.92 RW42 8. 0.09 3 6 5. 2 3 0.84 2. 7 3 0.25 Step 1 Shade 0.96 two times using a different color each time. Find the product. 2 3 0.96 9. 3 3 0.32 6. 6 3 0.52 3. 4 3 0.71 © Harcourt • Grade 5 Reteach Step 2 Count how many squares are shaded. There are 192 hundredths squares or 1 whole and 92 hundredths. Place the decimal after the whole number. You can use the hundredths models, to show multiplication of decimals and whole numbers. Name Name MXENL08AWK5X_RT_CH06_L5.indd 1 RW41-RW42 7/19/07 6:04:26 PM 6/21/07 12:22:15 PM Grade5.indd 22 6.37 3 1,000 5 0.185 3 1,000 5 $3.75 3 1,000 5 0.008 3 1,000 5 $0.25 3 10,000 5 1.313 3 10,000 5 3.94 3 1,000 5 3.94 3 10,000 5 2.002 3 1,000 5 2.002 3 10,000 5 RW43 3.94 3 100 5 2.002 3 100 5 11. 3.94 3 10 5 $0.25 3 1,000 5 1.313 3 1,000 5 10. 2.002 3 10 5 $0.25 3 100 5 1.313 3 100 5 8. $0.25 3 10 5 $3.75 3 100 5 0.008 3 100 5 7. 1.313 3 10 5 $3.75 3 10 5 0.008 3 10 5 5. $3.75 3 1 5 6.37 3 100 5 0.185 3 100 5 4. 0.008 3 1 5 6.37 3 10 5 2. 6.37 3 1 5 0.185 3 10 5 1. 0.185 3 1 5 Use patterns to find the product. 0.005 3 1 0.005 3 10 0.005 3 100 0.005 3 1,000 Answer 6/13/07 1:52:33 PM © Harcourt • Grade 5 0.05 3 10,000 5 0.05 3 1,000 5 0.05 3 100 5 12. 0.05 3 10 5 0.6 3 10,000 5 0.6 3 1,000 5 0.6 3 100 5 9. 0.6 3 10 5 89.36 3 1,000 5 89.36 3 100 5 89.36 3 10 5 6. 89.36 3 1 5 9.999 3 1,000 5 9.999 3 100 5 9.999 3 10 5 Reteach 0.005 0.05 0.5 5 3. 9.999 3 1 5 Number of Zeros in Number of Places to Whole Number Move Decimal Point 0 0 1 1 2 2 3 3 Use the patterns to find the product. When multiplying a decimal by 10, 100, 1,000 or 10,000, first count the number of zeros in the whole number. Then move the decimal point one place to the right in your answer for every zero that you counted. Record Multiplication by a Whole Number Algebra: Patterns in Decimal Factors and Products MXENL08AWK5X_RT_CH07_L3.indd 1 32 21. 237.89 3 32 16. 25.68 11. 0.04 3 86 6. 6.31 3 53 1. 0.933 3 6 3 39 22. 77.42 37 17. 159.46 12. 33.33 3 72 7. 8.492 3 10 8. 0.688 3 2 3. 32.96 3 44 RW44 3 70 23. 3.043 3 44 18. 621.3 3 54 24. 0.333 32 19. 736.07 14. 1.917 3 41 9. 121.3 3 5 4. 379.4 3 5 25. © Harcourt • Grade 5 Reteach 39.83 3 66 3 51 20. 800.9 15. 5.585 3 28 10. 9.57 3 34 5. 0.007 3 26 Start at the right end of your answer and count to the left the same number of places as are in the decimal factor. There are two digits to the right of the decimal point. 13. 290.6 3 6 126.88 1 1 5 4 4.88 3 26 _ 2928 1 9760 __ 2. 5.27 3 18 Find and record the product. Write the problem vertically and multiply like whole numbers. Find and record the product. 4.88 3 26 When given a horizontal multiplication problem, rewrite the problem vertically. Multiply and put the decimal point in the answer the same number of places to the right as the decimal point in the problem. Name Name MXENL08AWK5X_RT_CH07_L2.indd 1 RW43-RW44 6/18/07 5:59:21 PM 6/13/07 2:08:08 PM Grade5.indd 23 7. 4. 1. 0.8 3 0.3 0.4 3 0.2 0.6 3 0.8 8. 5. 2. RW45 0.7 3 0.8 0.3 3 0.9 0.5 3 0.5 Use the model to find the product. So, 1.6 3 0.5 5 0.80 9. 6. 3. 1.4 3 0.1 1.6 3 0.8 1.1 3 0.7 First, use a color pencil to shade 16 columns. Next, use a different color pencil to shade 5 rows. Then, count the squares in the area in which the color shading overlaps. Make a model to find the product of 1.6 3 0.5 Reteach 6/13/07 2:57:12 PM © Harcourt • Grade 5 8 2 3 4 _ 2.15 3 3.92 __ MXENL08AWK5X_RT_CH07_L5.indd 1 RW46 11. 300.59 3 0.3 8. 6.6 3 8.2 7. $0.39 3 291 10. 487.66 3 2.12 5. 29.3 3 0.31 2. $26.83 3 11 4. 6.71 3 4.22 1. 4.25 3 7.82 Estimate the product. 12 3 3 4 _ 6 Round both numbers up. Get the range Round both numbers down. 2.15 2 3 3.92 3 3 __ _ Does 8 fall between 6 and 12? Yes, so this estimate is reasonable. Round 2.15 3 3.92 __ Estimate the product. 2.15 3 3.92 12. 0.409 3 1.47 9. 7.6 3 9.217 6. $8.54 3 9 3. 3.3 3 9.4 © Harcourt • Grade 5 Reteach Your orginal estimated answer should fall between the other two estimates in the range. When estimating to find the product of two decimals, use a range to determine if your answer is reasonable. Estimate Products Model Multiplication by a Decimal When you use models to multiply decimals, remember that each square in the hundredths grid represents 0.01. So, 60 squares represent 60. 0.1 represented in the hundredths model is the entire first column or 10 squares. One full grid represents the whole number, one. Name Name MXENL08AWK5X_RT_CH07_L4.indd 1 RW45-RW46 7/19/07 6:04:46 PM 6/13/07 2:59:37 PM Grade5.indd 24 .4 0.41 8. 3.43 3 9.3 RW47 15. 0.08 3 2 12.4 3 8.5 14. $6.95 3 5.3 7. 0.5 3 0.7 12. 0.381 3 14 7.9 3 6.6 3. 11. 194.6 3 0.2 6. Estimate. Then find the product. 1. 72 2. 0.9 3 1.3 3 0.4 You know your actual answer should be close to 0.24. So, the actual answer is 0.2296. 2,296 Multiply as whole numbers. 41 3 56 __ 0.41 3 0.56 __ 9. 4. 3 3 5 .24 10. 5. 16. 1.11 3 1.1 13. $4.50 3 9.5 60.2 3 2.6 3.6 3 0.8 .6 0.56 Now estimate the factors. 4.1 3 5.3 6/13/07 3:01:50 PM © Harcourt • Grade 5 Reteach $5.94 3 0.07 A fishing boat caught a total of 4,012 lobsters in one season, selling the lobsters to fish markets for $0.05 each. How much money did the boat earn for the season? When you mulitply decimals, first mulitiply the factors as whole numbers. Use estimation to figure out where to place the decimal point in your answer. You can do this by estimating your factors. Multiply your estimated factors and place the decimal point in the answer. Your estimate should show that your actual answer is reasonable. the season? 4. How much money did the boat earn for MXENL08AWK5X_RT_CH07_L7.indd 1 RW48 © Harcourt • Grade 5 Reteach Describe the steps required to solve. Then solve the problem. 6. The U.S. population in 1990 was 248.7 American ate 92.4 lb of red meat that million people. The 2000 population was year. They ate 1.21 times that amount in 1.13 times that. The 2010 population is 2003. In 1940, Americans ate an average projected to be 1.1 times the 2000 of 12.3 lb of poultry for the year, and 5.79 population. How many people, to the times that in 2003. How many pounds of nearest million, are projected to live in poultry and red meat did the average the United States in 2010? American eat per year? 3 $0.05 __ 4,012 5. FAST FACT in 1940, the average 3. Multiply. 2. What step or steps are needed to solve problems? 1. What are you asked to find? Problem Solving Workshop Skill: Multistep Problems Practice Decimal Multiplication Estimate. Then find the product. Name Name MXENL08AWK5X_RT_CH07_L6.indd 1 RW47-RW48 6/18/07 5:59:41 PM 6/13/07 3:05:51 PM Grade5.indd 25 There are four groups of 7 squares. Since we are using hundredths models, the answer must be in hundredths. Now divide the squares into four groups of the same size. 5. $7.32 4 6 8. 8.56 4 4 12. 4.88 4 8 4. 6.03 4 9 7. 0.94 4 2 11. $9.99 4 3 RW49 2. 1.95 4 3 1. 0.36 4 6 Use decimal models or play money to model the quotient. Record your answer. So, 0.28 4 4 5 0.07. Show 0.28 using a decimal model. 13. 5.53 4 7 9. $9.18 4 9 6. 7.63 4 7 3. $9.75 4 5 Reteach 6/13/07 3:25:11 PM © Harcourt • Grade 5 8. 91.7 4 18 5. 6.4 4 9 2. 46.8 4 5 MXENL08AWK5X_RT_CH8_L02.indd 1 10. 47.8 4 59 RW50 11. 8.91 4 27 Find two estimates for the quotient. 7. 65.8 4 22 4. 5.3 4 8 1. 23.7 4 9 Estimate the quotient. So, an appropriate estimate is 0.04. • 3.4 and 80 are closer to 3.409 and 83. Which compatible numbers are closer to the original problem? • 3.6 and 60 are compatible. 3.6 4 60 = 0.03 • 3.2 and 80 are compatible. 3.2 4 80 = 0.04 Think of some compatible numbers near 3.409 and 83: 3.409 4 83 12. 7.42 4 35 9. 45.43 4 36 6. 0.312 4 5 3. 94.2 4 6 When dividing a decimal by a two-digit number, use compatible numbers for the dividend and divisor. Compatible numbers are numbers that are easy to divide. Just as you did with whole numbers, you can use decimal models to divide decimals into groups. Estimate the quotient. Estimate Quotients Decimals Division Use decimal models to model the quotient. Record your answer. 0.28 4 4 Name Name MXENL08AWK5X_RT_CH8_L01.indd 1 RW49-RW50 7/19/07 6:03:49 PM Reteach © Harcourt • Grade 5 6/13/07 3:41:46 PM Grade5.indd 26 5 3.15 046 84 38.64 10. 14. 13. 82 459.2 6. 9. 5 34.15 Find the quotient. 5. 77 $24.64 2 85.12 $092 58 $53.36 RW51 15. 11. 7. 4 1.74 17 $99.28 0063 64 4.032 Copy the quotient and correctly place the decimal point. 0285 876 35 1. 2. 3. 6 1.71 9 78.84 47 164.5 So, 3.15 4 5 5 0.63 Step 3. Divide. Step 2. Place a zero in the ones place above the dividend if the division is greatest then the divdend. Step 1. On the quotient, put the decimal point above the decimal point of the dividend. Find the quotient. . 5 3.15 16. 12. 8. 4. Reteach 6/13/07 3:45:11 PM © Harcourt • Grade 5 61 1.159 38 78.66 0008 7 0.056 $241 39 $93.99 0.63 5 3.15 30 15 0. 5 3.15 + 0.57 1.05 0.97 MXENL08AWK5X_RT_CH08_L04.indd 1 RW52 her average speed was faster than Danielle’s total average speed. Susie drove 96.4 mph, 88.5 mph, and 83.9 mph. Is Susie’s answer reasonable? Explain. 6. For the first three laps of the race, Susie says average speed of 19.18 mph. Brent says Buddy was faster by 9.18 mph. Whose answer is reasonable? 5. Buddy says he was faster than Brent by an USE DATA For 5-6, use the table. 4. Whose answer is reasonable? Explain. 45 73.25 mph 89.47 mph Susie Brent © Harcourt • Grade 5 Reteach 50 50 43 99.10 mph Rico 21 98.65 mph 91.43 mph Number of Laps Finished Average Speed per Lap Middleville Auto Race Danielle Buddy Driver 3. Which answer, Brad’s or Brittany’s, is closer to your estimate? rounding them to whole numbers? 2. How can you estimate the three decimal parties of fruit, 1. What are you asked to find? Britney bought 0.97 kilogram of apples, 1.05 kilograms of bananas, and 0.57 kilogram of oranges. Britney says she bought 25.9 kilograms of fruit. Brad says that Britney bought 2.59 kilograms of fruit. Whose answer is reasonable? Problem Solving Workshop Skill: Evaluate Answers for Reasonableness Divide Decimals by Whole Numbers When dividing decimals, placing the decimal point in your answer is an important step to finding the correct value. Name Name MXENL08AWK5X_RT_CH08_L03.indd 1 RW51-RW52 6/18/07 6:00:06 PM 6/13/07 3:52:16 PM Grade5.indd 27 people who live in California. 4. a random sample of 200 RW53 adults who live in Los Angeles. 5. a random sample of 200 6/14/07 7:17:23 AM © Harcourt • Grade 5 Reteach people who live in Los Angeles. 6. a random sample of 200 A radio station wants to find out the favorite type of music of people that live in Los Angeles, California. Tell whether each sample represents the population. If it does not, explain. A playground maker wants to find out if children in grades 4–6 like their new playground equipment. Tell whether each sample represents the population. If it does not, explain. 2. a random sample of 3. a random sample of 1. a random sample of 400 400 teachers children in grades 4–6 boys in grades 4–6 So, the survey does not represent the population because the sample includes children at only one school. • The survey does involve children ages 10–14. • But, it is a survey of only one school. Does the random sample fairly represent the population? A fruit juice company wants to survey children ages 10–14. Tell whether the sample below represents the population. If it does not, explain. A random sample of 100 children at one school. The median is 8. (8 1 8) 4 2 5 8 Step 2: Find the middle number. Since there is an even number of data, the median is between the two middle numbers. 6. 21, 24, 22, 24, 21, 31, 25 8. 43, 53, 63, 53 5. $25, $36, $28, $27 7. 350, 378, 350, 252, 275 9.8, 10.2, 11.6, MXENL08AWK5X_RT_CH09_L2.indd 1 15. 10.5, 12.5, 13. 11. $64, $48, $40, 12. 75, 62, , 10, 13; mean: 11 RW54 ; mean: 73 © Harcourt • Grade 5 Reteach , $7; mean: $10.60 , 45, 32; mean: 33 16. $12, $8, $17, , 9.7; mean: 10.3 14. 21, ; mean: $51 Use the given mean to find the missing number in each data set. 10. 1.3, 1.55, 2.75, 1.3, 2.6 4. 164, 215, 174, 174, 193 3. 7.8, 9.4, 10.6, 7.8, 7.8, 9.4 9. 873, 954, 896, 941 2. 641, 874, 614, 755 Sometimes there is only one mode or no mode. There are two modes 5 and 8. Step 2: Find the number that occurs most often. 1. 21, 15, 17, 21, 16 Find the mean, median, and mode for each set of data. The mean is 13. 78 4 6 5 13 Step 2: Divide the sum by the number of addends. 5, 5, 8, 8, 14, 38 Step 1: Order the data from least to greatest. Step 1: Order the data from least to greatest. Step 1: Add to find the sum. 8 1 8 1 14 1 5 1 38 1 5 5 78 The mode is the number that occurs most often in a set of data. The median is the middle number when a set of data is arranged in order. Find the mean, median, and mode for the data set: 8, 8, 14, 5, 38, 5 A survey is a way to gather information about a group. When you are gathering information about a group, • the whole group is called the population, • the people surveyed are called the sample. The mean is the average of a set of data. Mean, Median, and Mode Collect and Organize Data A sample must fairly represent the population. In a random sample, everyone in the population has an equal chance of being surveyed. Name Name MXENL08AWK5X_RT_CH09_L1.indd 1 RW53-RW54 6/18/07 6:00:15 PM 6/14/07 7:18:43 AM 10 24 68 52 65 52 31 68 12 69 53 72 Range: 72 2 12 5 60 2. 15 12 20 15 9 17 13 16 5 8 3 2 3 2 4 1 7 0 6 9 2 3 A: Songs Students Heard 13 11 A: Weights of Boxes 4 1 RW55 18 13 21 15 14 16 B: Weights of Boxes 5 10 7 5 2 7 0 9 37 68 9 6 15 6/14/07 7:21:00 AM © Harcourt • Grade 5 Reteach 52 22 18 B: Songs Students Heard 11 22 Compare the mean, median, and range of the data sets. 1. 12 52 Median: (52 1 52) 4 2 5 52 The median for data set B is the same as the median for data set A. Median: (52 1 52) 4 2 5 52 The range for data set A is greater than the range for data set B. Range: 79 2 10 5 69 68 60 Mean: 543 4 12 5 45.25 32 15 The mean for data set A is greater than the mean for data set B. Mean: 573 4 12 5 47.75 43 79 A: Pages Students Read 31 Duke 27 23 Women 25 26 33 Tennessee Connecticut Michigan State School 14 Men 30 Key: Poetry Biography Mystery Fantasy = four books Types of Books in Mr. Li’s Class A pictograph displays countable data using pictures and symbols. Pictographs have a key to show how many each picture or symbol stands for. Birds, 20 wins? MXENL08AWK5X_RT_CH09_L4.indd 1 RW56 basketball had? 4. What if football had 20 wins. How would this amount be shown on the pictograph? 2. How many wins did baseball have? 3. How many more wins did soccer have than Tues. Wed. Day Thurs. Fri. Key: Hockey Basketball Baseball Soccer = 8 wins © Harcourt • Grade 5 Reteach Sport Team Wins The temparure increases steadily. . Which sport had this number of 1. Twenty-eight wins would be shown as Mon. Daily Temperatures How can you describe the trend in temperature from Wednesday to Friday? 100 80 60 40 20 0 A line graph shows how data changes over a period of time. USE DATA For 1–4, use the pictograph at the right. What kinds of animals does the pet store have the same number of? cats and dogs Dogs, 10 Cats, 10 Reptiles, 5 Pet Store Population A circle graph shows how parts of data are related to each other and to the whole. Which team had the most wins? How many fantasy books are in Mr. Li’s Duke: 27 1 31 5 58 Connecticut: 23 1 25 5 48 class? Tennessee: 14 1 30 5 44 Michigan: 26 1 33 5 59 The key shows each symbol stands for 4 books. A half symbol stands for 2 books. (5 3 4) 1 2 5 22 fantasy books So, Michigan had the most wins. 40 35 30 25 20 15 10 5 0 NCAA Basketball Wins 2004–2005 A bar graph uses bars to display countable data. A bar graph is useful when comparing data by groups. Graphs can help you draw conclusions, answer questions, and make predictions about data. You can compare data sets using the mean, range, and median. B: Pages Students Read Analyze Graphs Compare Data Compare the mean, median, and range of the data sets. Name Name Number of Wins Grade5.indd 28 MXENL08AWK5X_RT_CH09_L3.indd 1 RW55-RW56 6/18/07 6:00:28 PM 6/14/07 8:13:20 AM 3 5 people used the Internet and 4 people used an atlas. Two of these people used the Internet and an atlas. How many people used the Internet or an atlas during the research period? 6. During a one-hour research period, 8 Draw a Venn diagram to solve. RW57 Check 5. What is one way you could check your solution. 4. How would you write your answer as a complete sentence. 2 U.S. Presidents First Ladies Solve 3. Solve the problem using the Venn diagram. Plan 2. How can drawing a diagram help you solve the problem? 1. How would you write the question as a fill-in-the blank sentence? Read to Understand Five students wrote a report about U.S. Presidents, 8 wrote a report about U.S. first ladies, and 3 wrote a report about both U.S. Presidents and first ladies. How many students wrote reports? Reteach 6/14/07 8:13:40 AM © Harcourt • Grade 5 13 11 Boots MXENL08AWK5X_RT_CH10_L01.indd 1 6 19 Stoves Backpacks /VNCFS4PME (FBS Tents May Camping Sales 1. Make a bar graph for the data set. • Add a key for Week 1 and Week 2. • Use one color for Week 1 and a different color for Week 2. • Add the second set of data using the key for Week 2. 56 48 40 32 24 16 8 0 24 16 8 0 40 32 56 48 Name Emma Name 5JOB 8FFL &NNB Biking Record Tina 8FFL .BSDVT Marcus Week 1 Biking Record Philip 1IJMJQ © Harcourt • Grade 5 Reteach from exercise 1 and this data: June Sales: tents, 20; stoves, 9; boots, 11; backpacks, 15. RW58 Describe how to make a double-bar graph of the data in the table and this data: Week 2: Marcus, 56 miles; Tina, 44 miles; Emma, 32 miles; Phillip, 48 miles. A double-bar graph is used to compare similar kinds of data. • Draw a bar to show 40 miles for Tina • Draw a bar to show miles for Phillip Use the data at the right to complete the bar graph. Week 1 Biking Record Name Distance (in miles) Macus 48 Tina 40 Emma 28 Phillip 52 2. Make a double-bar graph for the data set Make Bar Graphs and Pictographs Problem Solving Workshop Strategy: Draw a Diagram Pictographs use pictures or symbols to display or organize data. Bar graphs display data using vertical or horizontal bars. Name Name Distance (in miles) Distance (in miles) Grade5.indd 29 MXENL08AWK5X_RT_CH09_L5.indd 1 RW57-RW58 7/19/07 5:56:28 PM 6/15/07 1:16:13 PM 2 8 7 6 5 4 3 2 1 0 1-3 List the intervals. 1. Use 10 inches for each interval. Height (in inches) of Students 52 48 47 41 54 60 42 46 39 57 44 49 46 47 61 43 63 49 56 52 51 60 54 42 62 USE DATA For 1–2, use the table. RW59 10-12 4 10-12 6 Height (in Inches) of Students 6/15/07 1:17:11 PM © Harcourt • Grade 5 Reteach 2. Make a histogram of the data. 4-6 7-9 Ages (in years) 6 11 Frequency Table 1-3 4-6 7-9 3 8 3 Ages of Students Taking Swimming Lessons Interval Frequency 5 6 12 12 10 3 6 11 4 7 3 • So, 8 children ages 4-6 take swimming lessons the most. • Label each bar with the frequency table interval and the axis. • Make each bar the same width. • Draw the bars touching, but not overlapping. • Use the frequency table to find how long to make each bar. • Make the histogram by first choosing an appropriate scale for the vertical axis. Label the axis. • Make a frequency table to determine the number of times each age appears in the data. • A reasonable interval would be 3 years, starting at 1. • The data in the table shows ages from 2 to 12. • To make a histogram, you need to find a reasonable interval. 8 4 6 11 9 6. F (6, 2) 4. D (5, 5) 2. B (1, 9) 12. M 11. L MXENL08AWK5X_RT_CH10_L03.indd 1 10. K 8. H 9. J 7. G Use the coordinate grid. Write an ordered pair for each point. 5. E (9, 3) 3. C (3, 7) 1. A (1, 6) Graph and label the following points on a coordinate grid. (x , y) The x-coordinate is the 1st number in the ordered pair. The y-coordinate is the 2nd number in an ordered pair. (10, 4) RW60 • Start at 0. • Since x comes before y in the alphabet, remember to move across the x-axis first. • Move across the x-axis to the number 10. • Then, go straight up to the line labeled 4 on the y axis. • Draw a dot at the intersection of the two lines, and label the ordered pair (10, 4). An ordered pair contains two numbers, x and y. To graph an ordered pair: Graph and label (10, 4) on the coordinate grid. 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 1 K 1 2 2 A coordinate grid is like a sheet of graph paper with two borders, the x-axis and y-axis. The x-axis is the horizontal line at the bottom of the grid. The y-axis is the vertical line on the left side of the grid. An ordered pair is the location of a point on the grid. A histogram shows the number of times, or frequency, an event or data item occurs. Ages of Children Taking Swimming Lessons Algebra: Graph Ordered Pairs Make Histograms Swimming Lessons Make a histogram of the data. How many children ages 4-6 take swimming lessons? Name Name ZBYJT ZBYJT Grade5.indd 30 MXENL08AWK5X_RT_CH10_L02.indd 1 RW59-RW60 7/19/07 5:56:44 PM 3 G J 3 4 4 5 L 8 M 9 10 9 10 11 8 © Harcourt • Grade 5 Reteach H 6 7 YBYJT 6 7 YBYJT 5 6/15/07 1:19:20 PM (10, 4) Age Height 4 41 5 43 Jim’s Check-Up Heights 3 39 Jim’s Height (inches) 6 46 7 48 1 3 54 2 3 .POUI 4 5 4 62 5 70 Reteach July 66 6/15/07 1:21:12 PM © Harcourt • Grade 5 Sacramento, CA Average Lows San Diego, CA Average Lows Month Mar. April May June Temperature (oF). 53 53 60 62 R61 2. 2 44 Tupelo, MS Average Highs 1 40 se ar ch ar se Re ch Research Re ail E-m E-m ail Tom’s Weekly Computer Usage Activity Hours E-mail 3 Games 1 Research 6 MXENL08AWK5X_RT_CH10_L05.indd 1 1. Use the data to make a circle graph. 2. RW62 • Shade and label one section each as guitar, violin, and drums. • Shade and label two of the sections piano. • The circle should be divided into five equal sections. • Use this number to divide the circle into equal sections. • There are 10 students who take music lessons. • To make a circle graph, you need to determine the total number of parts of the data. arch MXENL08AWK5X_RT_CH10_L04.indd 1 1. Make a line graph for each the data set. • Connect the points to show the average temperature rising from month 1 to month 5. • Use the data to find where to graph each point. • Write the months along the horizontal axis using even spacing. • The scale should go from 08 to 808 using intervals of 208. • The greatest temperature in the table is 708. Month o Temperature (in F) P • To make a line graph, you need to determine the greatest number on the vertical scale. 5FNQFSBUVSFJO ' Use the data to make a circle graph. ail Average Monthly Temperature in Tupelo, MS A circle graph shows how parts of the data are related to the whole and to each other. A line graph shows how a set of data changes over time. E-m Make a line graph of the data in the table. Make Circle Graphs Make Line Graphs Name violin drums Cat Cat Pet Fish Dogs Cats Rabbits 3 4 2 1 Fish Reteach © Harcourt • Grade 5 Dog Number of Votes Favorite Pets in CJ’s Class guitar piano Music Lessons Students 4 2 2 2 piano Instrument Piano Guitar Violin Drums b Rese Ra Dog Name D bit g RW61-RW62 Do Grade5.indd 31 og Fis h Fish s me Ga Research Rese arch 7/19/07 5:57:02 PM 7/11/07 4:42:48 PM Grade5.indd 32 Leaves Did the basketball team score least often in the 50’s, 60’s or 70’s? Stem 6. RW63 The number of people eating lunch at a diner for the past 12 days was 41, 17, 28, 12, 37, 44, 32, 26, 18, 24, 36, and 25. Should the diner be more prepared to serve customer numbers in the 10’s, 20’s, 30’s or 40’s range? Make a graph to solve. Check 5. What other graph might you have used to solve this problem? 4. Solve 3. How would you fill out the stem–and–leaf plot for this problem in the space at the right? Plan 2. How can you organize the numbers to better answer the question? Read to Understand 1. What are you asked to find? Hampton High’s basketball team had the following scores this season 63, 67, 73, 55, 61, 53, 60, 63, 52, 61, and 64. Did the basketball team score least often in the 50s, 60s, or 70s? Reteach 6/15/07 1:26:33 PM © Harcourt • Grade 5 Choose the Appropriate Graph Problem Solving Workshop Strategy: Make a Graph Stem Leaves A circle graph shows how parts of the data are related to the whole and to each other. 4 3 MXENL08AWK5X_RT_CH10_L07.indd 1 season. hours. © Harcourt • Grade 5 Reteach 4. The temperature every hour for 24 inches. 2. The daily growth of a sunflower in RW64 3. The scores of a basketball team for one 6 major cities. 1. The populations of males and females in Choose the best type of graph or plot for the data. Explain your choice. 5 6/15/07 1:28:12 PM A stem-and-leaf plot is used to organize data by place value. Stem-and-leaf Plot A line graph shows change over time. Circle Graph Line Graph A line plot is used to record data as it is collected. Double-bar Graph A double-bar graph is used to compare two sets of data. Line Plot A circle graph will best display how Jen spends the hours in one day because it will show each hour spent in relation to the whole. 4 3 1 2 Bar Graph A bar graph is used to compare data by category. How Jen spends the hours of the day. Choose the best type of graph or plot for the data. Explain your choice. Bar and double-bar graphs, line graphs, line plots and stem-and-leaf plots organize numerical data. Name Name MXENL08AWK5X_RT_CH10_L06.indd 1 RW63 RW64 7/19/07 5:57:20 PM Grade5.indd 33 9 3 3 = 27 9 3 8 = 72 9 3 4 = 36 9 3 9 = 81 9 3 5 = 45 9 3 10 = 90 2. 10 3. 6 and 7 RW65 4. 4, 5, and 10 Write the least common multiple for each set of numbers. 1. 6 List the first eight multiples of each number. So, 24 is the least common multiple. Multiples of 12: 12 24 36 48 60 72 84 96 ... 8 16 24 32 40 48 56 64 ... Multiples of 8: 8 12 16 20 24 28 32 ... 4 Multiples of 4: Write the least common multiple for: 4, 8, and 12. 6/14/07 8:18:00 AM © Harcourt • Grade 5 Reteach The least common multiple, or (LCM), of two or more numbers is the least number that is a multiple of all of the numbers. When a number is a multiple of two or more numbers it is a common multiple. So, the first 10 multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, and 90. 9 3 2 = 18 9 3 7 = 63 931=9 9 3 6 = 54 To find the first eight multiples, multiply 9 by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. 5, 6, 7, 5 51617155 23 5, 6 7 5 51617155 23 5, 6 7 5 MXENL08AWK5X_RT_CH11_L02.indd 1 3. 6,420 1. 489 RW66 4. 8,703 2. 364 Test each number to determine whether it is divisible by 2, 3, 5, 6, 9, or 10. So, 5,675 is divisible by 5. It is not divisible by 2, 3, 6, 9, or 10. A number is divisible by 10 if the last digit is 0. 5,675 is not divisible by 10. A number is divisible by 9 if the sum of its digits is divisible by 9. 5,675 is not divisible by 9. A number is divisible by 6 if is divisible by 2 AND by 3. 5,675 is not divisible by 3, so it is not divisible by 6. A number is divisible by 5 if the last digit is 0 or 5. 5,675 is divisible by 5. A number is divisible by 3 if the sum of its digits is divisible by 3. 5,675 is not divisible by 3. A number is divisible by 2 if the last number is an even number. 5,675 is not divisible by 2. Test the following number to see if it is divisible by 2, 3, 5, 6, 9, or 10. 5,675 24 is divisible by 4. 24 is not divisible by 5. A number is divisible by another number if the quotient is a whole number and the remainder is 0. A multiple of a number is the product of that number and any other number. 24 4 4 = 6 24 4 5 = 4 R4 Divisibility Multiples and the Least Common Multiple List the first 10 multiples of 9. Name Name MXENL08AWK5X_RT_CH11_L01.indd 1 RW65-RW66 7/19/07 5:57:40 PM © Harcourt • Grade 5 Reteach not a 0 sum not divisible by 9 5 sum not divisible by 3 7/11/07 4:47:59 PM odd numbe Grade5.indd 34 Reteach A composite number is a number that has more than two factors. List the factors of 21. 4 8 10 5 40 Factors of 40 2. 27 3. 16 6. 35, 28 7. 18, 24 9. 20, 35 10. 18, 42 RW67 11. 30, 35 Write the greatest common factor for each set of numbers. 5. 21, 30 Write the common factors for each set of numbers. 1. 14 List the factors of each number. 12. 49, 35 8. 45, 15 4. 32 Identify the greatest common factor in the overlap section, which in this case is 8. So, the greatest common factor of 16 and 40 is 8. 12 16 Factors of 16 Use a Venn diagram to find the common factors. Write the greatest common factor for 16 and 40. The greatest common factor, or GCF, is the greatest factor that two or more numbers have in common. A common factor is a factor that two or more numbers share. So, the factors of 21 are 1,3,7, and 21. 3 3 7 = 21 6/14/07 8:26:59 AM © Harcourt • Grade 5 A prime number has exactly two factors, 1 and itself. A factor is a number multiplied by another number to find a product. 438 32 3 1 834 MXENL08AWK5X_RT_CH11_L04.indd 1 7. 67 4. 38 1. 15 8. 44 5. 45 2. 19 RW68 9. 31 6. 24 3. 11 Write prime or composite. You may use counters or draw arrays. So, the number 32 is composite. Since 32 can already be represented by more than 2 arrays there is no reason to keep going. Draw arrays to represent 32. • Set an array of 1 row of 32. Label it. “1 3 32” • Set an array with 8 rows of 4. Label it “8 3 4”. • Set an array of 32 rows at 1. Label it. “32 3 1” • Set an array with 4 rows of 8. with Label it. “4 3 8” Write prime or composite 32. You may use couners or draw arrays. 32 You can make arrays to find if a number is prime or composite. An array is an arrangement of objects in rows and columns. A number with exactly two arrays is prime. A number with more than two arrays is composite. The number 1 is neither prime nor composite. Prime and Composite Numbers Factors and Greatest Common Factor 1 3 21 = 21 Name Name MXENL08AWK5X_RT_CH11_L03.indd 1 RW67-RW68 7/19/07 5:57:57 PM © Harcourt • Grade 5 Reteach 1 3 32 6/14/07 8:41:41 AM Grade5.indd 35 . Introduction to Exponents Problem Solving Workshop Strategy: Make an Organized List 6. ; red and ; green and Look back at the problem. Does the answer make sense? Explain. RW69 Anna can have a sandwich on white, pumpernickel, or wheat bread. She can have it grilled, baked, or cold. How many sandwich choices does Anna have? Make an organized list to solve. 5. ; white and How many ways can Toni and Hector combine two colors? white and red and white; red and Color Combinations How can you complete the organized list below to help solve the problem? Do not list any combination more than once. For example, red and white is the same as white and red. Check 4. 3. Solve Plan 2. How can an organized list help you solve the problem? Reteach 6/14/07 8:48:11 AM © Harcourt • Grade 5 100 1,000 What is the value of 10 3 10? What is the value of 100 3 10? MXENL08AWK5X_RT_CH11_L06.indd 1 5. 10 4 Find the value. 6. 10 9 3. 10 3 10 3 10 3 10 3 10 3 10 3 10 1. 10 3 10 3 10 3 10 3 10 7. 10 6 8. 10 8 © Harcourt • Grade 5 Reteach 4. 10 3 10 3 10 3 10 3 10 3 10 3 10 RW70 Write in exponent form. The find the value. 2. 10 3 10 103. So, 10 3 10 3 10 written in exponent form is So, the value of 103 5 1,000. 3 times How many times is the base being used as a factor? What is the base? 10 10 3 10 3 10 How would you write the question as a fill-in-the blank sentence? Read to Understand 1. Write in exponent form. Then find the value. Toni and Hector are weaving potholders using the colors red, white, green, and blue. If they use only two colors for each potholder, how many ways can they combine two colors? The exponent is a number that tells how many times another number, the base, is used as a factor. Name Name MXENL08AWK5X_RT_CH11_L05.indd 1 RW69-RW70 7/19/07 5:58:15 PM 6/14/07 8:59:46 AM Grade5.indd 36 base 4. 27 Find the value. 1. 5 3 5 3 5 3 5 3 5 3 5 3 5 5. 123 RW71 2. 8 3 8 3 8 6. 192 6/14/07 9:29:08 AM © Harcourt • Grade 5 Reteach 3. 2 3 2 3 2 3 2 3 2 Exponent form: 63 Words: the third power of six or six cubed Write in exponent form and then write in words. Exponent form: 42 Words: the second power or four or four squared Write 4 3 4 and 6 3 6 3 6 in exponent form and in words. There are two ways to write the word form for an exponent of 2 or 3. A perfect square or a square number is the product of a number and itself. A square number can be represented with the exponent 2. 36 in words is “the sixth power of three.” The exponent 6 means “the sixth power.” Write 36 in words. A base with an exponent can be written in words. 3 3 3 3 3 3 3 3 3 3 3= 36 The base is repeated 6 times, so 6 is the exponent. 3 is the repeated factor, so 3 is the base. exponent A factor tree is a diagram that shows the prime factorization of a composite number. In the last lesson, the base was always 10. The base does not always have to be 10, though. 36 Prime factorization is a way to show a composite number as the product of prime factors. You already know that an exponent is a number that tells how many times the base is used as a factor. 3 3 2 3 3 3 2. 20 5. 3 3 5 3 3 3 5 MXENL08AWK5X_RT_CH11_L08.indd 1 7. 52 3 7 RW72 8. 2 3 3 3 5 3 2 3 7 3 3 Find the number for each prime factorization. 4. 2 3 3 3 5 3 5 Rewrite the prime factorization using exponents. 1. 42 Find the prime factorization. You may use a factor tree. 2 3 2 3 3 = 22 3 3 Rewrite 2 3 2 3 3 using exponents. You can use exponents in prime factorization for factors that appear two or more times. 9. 23 3 3 3 112 © Harcourt • Grade 5 Reteach 6. 2 3 5 3 3 3 5 3 2 3 5 3. 90 Write each composite number as a product of two factors. Write the number as a product of two factors. Write the number being factored at the top. So, the prime factorization of 12 is 2 3 2 3 3. 2 4 12 Find the prime factorization of 12. You may use a factor tree. Prime Factorization Exponents and Square Numbers Write 3 3 3 3 3 3 3 3 3 3 3 in exponent form. Name Name MXENL08AWK5X_RT_CH11_L07.indd 1 RW71-RW72 7/19/07 5:58:33 PM 6/14/07 9:32:17 AM Grade5.indd 37 6. 5. 7. 3. 12. 9. 0 0 A B 1 1 13. 10. 0 0 RW73 D E Write a fraction to name the point on the number line. 2. 1 1 14. 11. 0 0 8. 4. F Write a fraction for the shaded part. Write a fraction for the unshaded part. 1. 3 1 1 6/14/07 9:42:40 AM © Harcourt • Grade 5 Reteach C 4 3 3. What is the fraction? __ 2. Unshaded parts? 1. Parts in the group? 4 1. How many parts make up the group? 4 2. How many parts are shaded? 1 1 3. What is the fraction? __ 4 Unshaded Part: Shaded Part: Equivalent fractions are fractions that name the same number or amount. 1 are equivalent fractions because they are equal in value. __ and __ For example, 3 6 2 2 Write an equivalent fraction for __ . 5 A fraction describes the number of parts of a whole or a group. __ shows that 5 parts make a whole, and 4 parts are being used. For example, 4 5 Numerator 4 __ 5 Denominator __ , which have 1 as the numerator, are called unit fractions. Fractions such as 1 7 Write a fraction for the shaded part. Write a fraction for the unshaded part. MXENL08AWK5X_RT_CH12_L02.indd 1 8 2 21. __ 10 4 16. ___ 8 3 11. __ 3 __ 6. 2 10 8 22. ___ 9 __ 17. 5 7 __ 12. 4 6 __ 7. 4 Write an equivalent fraction. 1 __ 1. __ 2. 1 2 6 RW74 11 3 23. ___ 12 6 18. ___ 6 3 13. __ 9 __ 8. 3 5 __ 3. 2 12 2 24. ___ 10 7 19. ___ 8 6 14. __ 4 2 9. __ 7 __ 4. 2 6 6 25. __ 5 4 20. __ 12 1 15. ___ 7 5 10. __ 4 __ 5. 3 © Harcourt • Grade 5 Reteach “What you do to the top (numerator) you must do to do the bottom (denominator).” Equivalent Fraction Rule: Equivalent Fractions Understand Fractions __ : To write an equivalent fraction for 2 5 1. Choose a number to multiply with. Let’s use 2. 2. Multiply the top (numerator) by 2. 232=4 3. Multiply the bottom (denominator) by 2. 5 3 2 = 10 4 . __ is 4 over 10, or ___ So, an equivalent fraction for 2 10 5 Name Name MXENL08AWK5X_RT_CH12_L01.indd 1 RW73-RW74 7/19/07 5:58:50 PM 6/14/07 9:48:39 AM Grade5.indd 38 90 20 ___ 35 7. 21 ___ 9 2. 6 __ RW75 27 18. 18 ___ 9 30 17. 25 ___ 8 16. __ 18 13. ___ 5 12. 3 __ 4 9 45 8. 30 ___ 12 3. ___ 3 11. __ 2 Write each fraction in simplest form. 6. 8 1. 4 __ Name the GCF of the numerator and denominator. 10 is 2. The GCF of the numerator and denominator for ___ 14 • Divde the numerator and denominator by the GCF 2. __ 10 4255 _______ 14 4 2 7 10 in simplest for is 5 __ . So, ___ 14 7 2. What is the greatest common factor? 2 49 28 19. ___ 12 14. ___ 4 7 4 9. __ 16 4. ___ 8 18 13 20. 13 ___ 24 15. ___ 16 10 10. ___ 5 5. ___ 6 Another way to find the greatest common factor (GCF) of the numerator and denominator is to list the factors of the lesser number and eliminate the ones that aren’t common. Reteach 6/14/07 9:54:09 AM © Harcourt • Grade 5 A mixed number is made up of a whole number and a fraction. A fraction greater than 1 is sometimes called an improper fraction. A fraction is in simplest form when the numerator and denominator have 1 as their only common factor. 10 Write ___ 14 in simplest form. 1. List the factors of 10 and 14. 10: 1, 2, 5, 10 14: 1, 2, 7, 14 Understand Mixed Numbers Simplest Form 2 1 20 5 22 __ 42 5 Step 2. Add Step 3. Remember to keep the same denominator, 5. 5 3 4 5 20. 2 4 __ 5 MXENL08AWK5X_RT_CH12_L04.indd 1 10 11. 17 ___ 6 6. ___ 19 4 12. 7. 4 6 __ 5 3 3 __ 8 2 2. ___ 11 13. 8. RW76 7 4 ___ 12 2 5 __ 5 5 3. ___ 12 3 14. ___ 23 9 5 1 __ 6 9. ___ 29 4. 15. Reteach © Harcourt • Grade 5 1 7 __ 8 7 1 2 __ 3 10. ___ 18 5. Write each mixed number as a fraction. Write each fraction as a mixed number. 1. 7 __ 3 2 ___ 10 Step 1. Multiply 22. 2 written as a fraction is ___ So, the mixed number 4 __ 5 5 2 as a fraction. Write 4 __ 5 2 To change 4 __ to a 5 fraction, multiply the denominator by the whole number, then add the numerator to the sum. The denominator stays the same for the answer. 23 3 ___ So, the fraction ___ 10 written as a mixed number is 2 10. 23 ___ Write the fraction 10 as a mixed number. 23 ___ Step 1. Divide 10 To change 10 to a mixed 23 . number, divide the Step 2. See where the following go: numerator over the 2r3 whole number denominator. 23 10 remainder 220 divisor 3 Name Name MXENL08AWK5X_RT_CH12_L03.indd 1 RW75-RW76 7/19/07 5:59:08 PM 6/14/07 10:15:00 AM Grade5.indd 39 1 1 4 1 __ 3 5 __ 7 9 7 ___ 12 4 18 10. 5 ___ 7 24 9. 7 3 __ 7 6. ___ 15 3 __ 5 5. 5 __ 12 2. ___ 7 4 __ 9 1. 5 __ 9 Compare. Write <, >, or = for each . 11. 4 4 __ 7 5 7. __ 4 2 3. 1 __ RW77 __ 54 9 7 ___ 12 5 ___ 12 7 7 7 7 5 ___ 5 ___ 5 ___ 5 ___ 21 21 21 21 3 3 3 5 ___ 9 9 3 ______ 5 ___ 5 __ 7 733 21 21 Then, compare. 9. 7 ,53 7 , 5 ___ __ because 5 ___ 5 ___ 7 21 21 21 6 4 ___ 11 10 ___ 11 5 __ 6 __ 22 3 6 5 12. __ 8. 8 4. __ 2 6/14/07 10:24:26 AM © Harcourt • Grade 5 Reteach 13 ___ 16 __ 25 6 1 __ 4 12 12 12 12 12 12 12 12 12 Next, rename the fractions using the common denominator. Multiples of 7: 7, 14, 21 Multiples of 21: 21, 42, 63 7 Compare. Write ,, ., or 5 for 5 ___ 21 First, list the multiples of denominators 4 1 __ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 4 1 __ The two rows are of the same length. Or you can look for the common denominators. 9 . __ 5 1 ___ So, 1 3 4 12 Use fraction bars to compare. What is the order of the four people? 2 3 Bernice threw the shotput 10 5_9 feet. Terry threw the shotput 10 4_7 ft, and Carla threw the shotput 10 2_5 ft. Who threw the shotput the longest distance? Who threw for the shortest distance? MXENL08AWK5X_RT_CH12_L06.indd 1 6. Make a model to solve. RW78 Check 5. What other strategy could you use to solve the problem? 4. 1 Paul Solve 3. How can you use the strategy to solve the problem? Fill out the model. Plan 2. What strategy can you use to solve the problem? Read to Understand 1. What are you asked to find? Amber, Marcus, Paul, and Shelly line up to make their jumps. Shelly is not first. Amber has at least two people ahead of her. Paul is third. Give the order of the four. 4 © Harcourt • Grade 5 Reteach Problem Solving Workshop Strategy: Make a Model Compare and Order Fractions and Mixed Numbers Fractions and mixed numbers can be compared by using fraction bars. 9 Compare. Write <, >, or = for 1 _3_ 1 ___ 4 12 . Name Name MXENL08AWK5X_RT_CH12_L05.indd 1 RW77-RW78 7/19/07 5:59:23 PM 6/14/07 10:29:39 AM Grade5.indd 40 7. 1.7 6. 0.004 9. 5.58 8. 3.92 9 17. 3 ____ 1000 100 20 437 16. _____ 4 1 12. ___ __ 11. 1 1000 RW79 667 18. 1 _____ 8 3 13. __ 4 Reteach Step 2 ( (' ) (' * (' + (' , (' (' . (' / (' 0 (' ( ' ( (' ) (' * (' + (' , (' (' . (' / (' 0 (' ( ' ( (' ( * ) * ( ' ( * ) * ( 6 6 10 RW80 10 3 1 ___ 5 2. ___ Find the sum or difference. Write it in simplest form. 115 __ 1. __ (' . (' / (' 0 (' ( 5 5 423 __ 3. __ ' ( * ) * ( 6/15/07 1:30:01 PM © Harcourt • Grade 5 Reteach __ . The subtracted fraction is 1 3 1. Jump back 1 to __ 3 , (' Step 3 + (' 2. The first fraction is __ 32 . Start at 0. Jump 2 to __ 3 * (' Step 2 2 1 1 1 You land at 1 third, so __ 2 __ = __ . __ is written in simplest form. 3 3 3 3 ' The denominators are 3. Divide a number line into 3 equal parts. Step 1 ) (' 1 The second addend is ___ 10 . 7 ___ and jump Start at 10 1 more tenth. Step 3 8 8 54 8 , so ___ 7 1 ___ 1 5 ___ __ . . Written in simplest form, ___ You land at ___ 10 10 5 10 10 10 You can also use a number line to subtract fractions. 1 2 2 __ Find the difference, write it in simplest form. __ 3 3 ' 7 . The denominator of each The first addend is ___ 10 fraction is 10. Divide a number Start at 0 and jump 7. 7 . line into 10 equal parts. This brings you to ___ 10 Step 1 6/14/07 10:33:37 AM MXENL08AWK5X_RT_CH13_L01.indd 1 © Harcourt • Grade 5 3 20. 5 __ 7 19. 9 ___ 35 7 15. __ 8 10. 4.35 5. 0.365 22 14. ___ 25 4. 0.45 3. 0.78 Write each fraction or mixed number as a decimal. 2. 0.32 1. 0.8 Write each decimal as a fraction or mixed number in simplest form. 51 . So, 0.255 written as a fraction is ____ 200 Simplify. 255 4 5 1,000 4 5 Step 2. 51 5 ____ 200 0.255 has three digits to the right of the decimal point, so the denominator will have three zeros. 255 0.255 5 ______ 1,000 You can use a number line to add fractions. 1 7 1 ___ Find the sum, write it in simplest form. ___ 10 10 You can represent a fraction as a decimal and vice versa. For 1. example, 0.5 is the decimal equivalent to __ 2 Write the decimal 0.255 as a fraction or mixed number in simplest form. Step 1. Add and Subtract Like Fractions Relate Fractions and Decimals To write a decimal as a fraction, remove the decimal point and place the number over the appropriate power of 10. Name Name MXENL08AWK5X_RT_CH12_L07.indd 1 RW79-RW80 7/19/07 5:59:40 PM RW81-RW82 Grade5.indd 41 7/19/07 5:59:59 PM 12 12 12 3 1 ___ 5 2 5 ___ ___ Add the fractions with like denominators. __ bar for Trade each 1 4 1 bars. three ___ 12 3 1 ___ 1 2 5 ___ 2 __ 1 ___ 4 12 12 12 12 12 1 12 1 12 1 12 1 12 1 12 1 12 MXENL08AWK5X_RT_CH13_L02.indd 1 RW81 Find each sum using fraction bars. Write it in simplest form. 3 3 5 1 ___ 2 7 __ 1 1 __ __ 1 __ __ 4 4 12 1. 3 2. 8 3. 6 4 5 . 2 5 ___ 1 1 ___ So, __ Step 3 Step 2 1 12 Reteach © Harcourt • Grade 5 10 5 ___ 2 10 1 ___ 5 10 4 ___ Subtract the fractions with like denominators. 5 2 ___ 1 1 5 ___ 1 __ 2 ___ 2 10 10 10 __ bar for Trade the 1 2 1 bars. five ___ 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 MXENL08AWK5X_RT_CH13_L03.indd 6/15/07 1:30:55 PM 1 3 4 221 __ 1. __ 8 2 1 __ 2 __ 2. 7 RW82 6 3 __ 2 1 __ 3. 5 Use fraction bars to find the difference. Write it in simplest form. 1 5 ___ 4 . Written in simplest form, ___ 4 5 __ 2. 1 2 ___ So, __ 2 10 10 10 5 Step 3 Step 2 1 2 1 __ 2 ___ Subtract. 1 2 10 Step 1 __ fraction bar to Use a 1 2 model the first fraction. 2 1 1 ___ Write the sum using fraction bars. Write it in simplest form. __ 4 12 1 4 You can use fraction bars to help you subtract fractions with unlike denominators. Trace fraction bars of fractions with unlike denominators for equivalent bars of fractions with like denominators. You can use fraction bars to help you add fractions with unlike denominators. Trade fractions bars of fractions with unlike denominators for equivalent bars of fractions with like denominators. __ bar and two Use a 1 4 1 bars to model fractions ___ 12 with unlike denominators. Model Subtraction of Unlike Fractions Model Addition of Unlike Fractions Step 1 Name Name Reteach 6/15/07 1:31:13 PM © Harcourt • Grade 5 RW83-RW84 Grade5.indd 42 7/19/07 6:00:15 PM 2 4 __ 6 6 4 __ 1 1 ( - , - ( - 8 7 5. __ + - 3 __ 8 ( ) * - 2. __ 2 8 ) - 1 __ MXENL08AWK5X_RT_CH13_L04.indd 1 4. 1. ' ' - Estimate each sum or difference. 6 6 9 add the two rounded numbers. 411 __ , To estimate the sum __ __ on the number line. Find 1 9 1 ,or 1? Is it closest to 0, __ 2 __ is closest to 0. The fraction 1 9 4 on the number line. Find __ 6 1 ,or 1? Is it closest to 0, __ 2 4 1. __ The fraction is closest to __ 6 2 4 1 __ 1 __ 6 9 1. 4 1 __ 1 is about __ So, __ 6 9 2 Step 3 Step 2 Step 1 Estimate the sum. 5 __ 6 7 __ 8 ( - ) 0 * 0 ) - + 0 ( ) ' ( ) * - ( ) , 0 6. 0 + - . 0 , - / 0 6 1 __ 6 1 2 ( ( 0 0 7 __ 8 3 __ 8 ( , - . / 0 / / / / / 3. 5 __ ' ( ) * + / / / / / __ 1051 2 RW83 2 1 2 1 __ ' ' ( 0 0 ' ' - Reteach © Harcourt • Grade 5 3 12 12 12 2 10 10 10 MXENL08AWK5X_RT_CH13_L05.indd 6/15/07 1:31:39 PM 1 7 1 __ __ 5. 8 2 4 3 2 __ __ 6. 4 2 3 RW84 9 4 ___ __ 7. 10 2 5 Find the sum or difference. Write the answer in simplest form. 1 2 1 1 3 1 __ __ __ __ 1. __ 1 __ 2. 2 1 5 3. 4 1 6 5 3 10 10 12 5 8. 8 __ 2 5 __ 9 6 3 1 __ __ 4. 5 1 4 6/15/07 1:32:08 PM © Harcourt • Grade 5 Reteach Stop when you find 2 fractions with like denominators. 4 9 2 ___ 5 5 ___ 9 2 __ 1 becomes ___ 4 . Written in simplest form, ___ 4 5 __ 2. ___ 9 2 __ 1 Example 2: Subtract ___ 10 2 9 , 18 9 . ___ ___, 27 ___, 36 ___ Step 1 Write equivalent fractions for ___ 10 10 20 30 40 1 1. 5 2, 3 4, ___ __, __ __, __ Step 2 Write equivalent fractions for __ 2 2 4 6 8 10 Step 3 Rewrite the problem using the equivalent fractions. Then subtract. 12 5 1 __ 5 1 ___ 9 . Written in simplest form, ___ 9 5 __ 3. 1 becomes ___ 4 5 ___ ___ To add or subtract unlike fractions, you need to rename them as like fractions. You can do this by making a list of equivalent fractions. When you find two fractions with the same denominator, they are like fractions. 5 1 __ 1 Example 1: Add. ___ 12 3 5 , 10 5 . ___ ___, 15 ___, 20 ___ Step 1 Write equivalent fractions for ___ Stop when you find 12 12 24 36 48 1 1 3 2 4 __ __ __ __ ___ 2 fractions with , , , Step 2 Write equivalent fractions for . 3 3 6 9 12 like denominators. Step 3 Rewrite the problem using the equivalent fractions. Then add. Use Common Denominators Estimate Sums and Differences 1 , or to 1 estimate sums and differences. You can round fractions to 0, to __ Name Name RW85-RW86 Grade5.indd 43 7/19/07 6:00:31 PM What information will you use? MXENL08AWK5X_RT_CH13_L06.indd 1 the morning. He spends 15 minutes getting dressed, 15 minutes eating breakfast, 5 minutes brushing his teeth, and 20 minutes riding to school. What time should he get out of bed? 7. Jeff needs to be at school at 8:00 in Compare strategies to solve. RW85 © Harcourt • Grade 5 Reteach 1 are __ are 6th-graders, __ 8th-graders. If 3 8 __ 4 are 8th-graders, 7th-graders, and 1 6 then what fraction are 5th-graders? 8. Central School has 5th-, 6th-, 7th-, and How much money did David have before his purchases? Check 6. How can you check your answer? 5. Solve 4. You can write an equation. Let n 5 the amount of money David had before his purchase. Solve n 5 3.99 1 1.24 1 4.55 1 1.57. Plan 3. What are two different strategies you could use to solve this problem? 2. Read to Understand 1. What are you asked to find? David bought some supplies for the science project. He spent $3.99 for poster board, $1.24 for a glue stick, and $4.55 for color pencils. If David had $1.57 when he left the store, how much money did he have before his purchases? Choose a Method Problem Solving Workshop Strategy: Compare Strategies X X paper and pencil X calculator MXENL08AWK5X_RT_CH13_L07.indd 6/15/07 1:32:47 PM 1 9 6 1. __ 512 __ 12 RW86 12 2. ___ 5 7 2 ___ Choose a method. Find the sum or difference. Write it in simplest form. 4 8 3. 3 __ 2 3 __ The chart states that if one denominator is a multiple of the other, use paper and pencil. 2 1 ___ 1 1 ___ 9 . 7 5 ___ 7 becomes ___ Rewrite using equivalent fractions: __ 5 10 10 10 10 9 . 7 5 ___ __ 1 ___ So, 1 5 10 10 5 is a multiple of 10. Choose a method. Find the sum. Write it in simplest form. 7 1 1 ___ __ 5 10 1 1 ___ 7 5 10 The fractions do not have like denominators: __ 5 10 Is one denominator a multiple of the other? Yes Denominators are not alike and are not multiples of one another. One denominator is a multiple of the other. Denominators are alike. mental math You can use this chart to decide whether to use mental math, paper and pencil, or a calculator to add or subtract fractions. Name Name Reteach 6/15/07 1:34:37 PM © Harcourt • Grade 5 Grade5.indd 44 1 1 1__ 4 4. 10 4 3 __ 1 2 10 2 _ 3 3 _ 12 4 1. 5 1 __ 5. 11 5 8 3 _ 1 _ 4 RW87 5 _ 8 1 _ 14 2 2. 2 Use fraction bars to find the sum. Write it in simplest form. 12 12 12 6. 2 1 _ 14 2 5 _ 6 1 _ 13 6 3. 3 1 STEP 3 Count like bars to find the sum. 1 bars is 2___ 7 . Two 1 bars and seven ___ 12 12 7. __ 1 11 __ 5 2___ So, 11 4 3 12 1 1__ 3 1 __ 1 1 __ __ 1 12 12 12 12 1 1__ 4 1 1__ 3 1 __ 1 __ 1 __ 1 __ 1. 1 and 1__ STEP 2 Draw like fraction bars to model 1__ 4 3 1 _ 3 1 _ 4 STEP 1 Draw a picture to show the addends. 1 __ 1 11 __ 11 4 3 5 1 _ Reteach 6/13/07 9:31:29 AM © Harcourt • Grade 5 1 4 _ _ 52 . 2 8 1 3 _ _ 2 1 2 8 3 1 1 _ _ _ 21 51 . 2 8 8 4 3 1 _ _ _ 21 51 . 8 8 8 2 3 _ __ 2 1 10 5 MXENL08AWK5X_RT_CH14_L02.indd 1 4. 4 5. 5 RW88 1 5 _ _ 22 6 4 Use fraction bars, or draw a picture to find the difference. Write it in simplest form. 1 1 4 1 _ _ _ __ 1. 6 2 4 2. 3 2 2 3 6 5 10 So, 2 2 3. Cross out bars to subtract 1__ 3. STEP 3 Subtract 1__ 8 8 Count the bars that are left. 2 1 bars. STEP 2 Rename the bars with __ 4 2 Fraction bars can help you subtract mixed numbers. Fraction bars can help you add mixed numbers. Draw a picture to find the difference. Write the answer in simplests form. 1 STEP 1 Draw a picture to show 2 _ . 2 Model Subtraction of Mixed Numbers Model Addition of Mixed Numbers Add. Write the answer in simplest form. Name Name MXENL08AWK5X_RT_CH14_L01.indd 1 RW87-RW88 6/18/07 6:03:40 PM 6. 2 3. 6 5 2 _ __ 21 3 12 1 3 _ _ 24 4 8 4 _ 8 1 _ 2 1 _ 2 © Harcourt • Grade 5 Reteach 2 2 2 6/13/07 9:41:07 AM Grade5.indd 45 4 13 9 __ __ __ 11 5 . 12 12 12 3 1 1 _ _ __ . 1 1 57 4 3 12 1 __ . 12 1. 2 1 2 _ _ 14 9 6 2. 10 3 5 _ _ 15 6 4 RW89 3. 11 5 7 _ _ 29 8 6 Find the sum or difference. Write it in simplest form. So, 5 Write the answer in simplest form 7 13 13 __ __ 1656 . 12 12 4 9 __ __ 11 56 12 12 Add the sums: 5 Step 4 Add the whole numbers. 5 Step 3 Add the fractions. 4. 18 Reteach 6/13/07 9:46:00 AM © Harcourt • Grade 5 1 3 _ _ 2 14 5 2 3 3 1 _ _ _ 2 1 5 . 2 4 4 6 3 1 3 3 _ _ _ _ _ 2 1 or 1 2 1 5 . 2 4 4 4 4 5. 5 1 3 _ _ 2245 3 4. 6 MXENL08AWK5X_RT_CH14_L04.indd 1 2. 5 5 1 _ _ 21 5 8 5 1. 3 6. 4 5 1 _ _ 22 5 6 6 RW90 3. 4 3 7 __ __ 2 3 10 5 10 Use fraction bars to find the difference. Write it in simplest form. So, 2 Subtract, 2 4 _ Rename one whole bar as . 4 1 3 1 _2 _ _ _ and . Rename as . 2 4 2 4 1 1 _ _ using two whole bars and one bar. 2 2 3 1 _ _ 2 1 2 4 Think of the LCD for Model 2 2 3 1 _ _ 21 5 5 5 5 1 _ _ 21 5 3 6 When you subtract mixed numbers you may need to rename the whole numbers. When you add or subtract mixed numbers, you may need to rename the fractions as fractions with a common denominator. 3 1 _ _ Find the sum. Write the answer in simplest form. 5 1 1 . 4 3 3 1 _ _ Step 1 Model 5 and 1 . 4 3 1 9 4 3 3 1 _ __ _ __ Step 2 The LCD for _ and _ is twelfths, so rename 5 as 5 and 1 as 1 . 4 12 3 12 4 3 Subtraction with Renaming Record Addition and Subtraction Use fraction bars to find the difference. Write it in simplest form. Name Name MXENL08AWK5X_RT_CH14_L03.indd 1 RW89-RW90 7/19/07 6:00:50 PM Reteach © Harcourt • Grade 5 6/13/07 9:48:11 AM Grade5.indd 46 STEP 2 STEP 3 STEP 4 35 5 7 15 3573 ____ ___ 7__ 9 9 35 45 39 5 15 ___ 9 __ 5 15 1 ____ 151 5 5 39 45 STEP 2 Write equivalent fractions using the LCD. Estimate. 45 9 15___ 5 ___ 739 45 ___ 2 715 45 __ 45 54 14___ 1. ___ 1 8 11 15 __ 2 4 __ 9 2. 6 _ 5 2 8__ 3 12__ 8 RW91 3. 2 _ __ 1 91 2 2__ 7 4. 6/13/07 9:54:14 AM © Harcourt • Grade 5 Reteach 7 2 7 ___ 12 __ 1 15 __ 3 13 43 5 7___ ______ 5 7 39 4543 15 STEP 3 Rename so you can subtract. Subtract the whole numbers. Subtract the fractions. Write the answer in simplest form. 15 2 7 5 8 ___ 1114 15 Add the whole Write the answer numbers. Add in simplest form. the fractions. 10 4 5 11___ 4 ___ 5 10 1 15 ___ 1 ___ 1019 2___ 15 15 15 15 15 9 __ So, 22 1 8___ 3 15 __ 3 19 1 8__ 10___ 5 15 _ Estimate. Find the sum or difference. Write it in simplest form. 5: 5,10,15, ..., 45 9: 9,18,27, ..., 45 STEP 1 Find the LCD. Subtract. 358 333 9 8__ 5 8___ 5 533 15 358 __ 2 7__ 151 5 9 Find the LCD. Write equivalent fractions using the LCD. 10 2 3 5 5 2___ 2 __ 2 3: 3, 6, 9, 12, 15 ... 3 5 2 3 3 5 15 5: 5, 10, 15, 20 ... STEP 1 4 7 ft farther than Raul not the shortest hit; 1 __ 8 Sonya MXENL08AWK5X_RT_CH14_L06.indd 1 © Harcourt • Grade 5 Reteach grade watch baseball and football. If 132 students watch football and 119 watch baseball, how many students watch both baseball and football? 7. In October, 200 students in Fiona’s RW92 down the total distances they ran last week. The farthest distance ran was 5 miles farther 1 miles. Deidre ran 8 __ 19 __ 8 8 3 __ than Amy and 3 miles farther than 8 17 miles less than Brad. Clarence ran 6 ___ 20 Brad. Who ran the shortest distance, and what was that distance? 6. Amy, Brad, Clarence, and Deidre wrote Use logical reasoning to solve. 5. How did the strategy help you solve the problem? Check 4. Who hit the ball the farthest, and what distance was the ball hit? not the shortest hit __ ft farther than Raul; 3 1 __ ft less than Stanley not the shortest hit; 71 2 8 Lenny and Sonya hit the ball farther. Distance Clues Stanley Raul Lenny Player Solve 3 feet. 3. Complete the table. Remember that the shortest hit was 100 __ 2. How can you organize the data to solve the problem? Plan Read to Understand 1. How could you write the question as a fill-in-the blank sentence. Lenny, Raul, Stanley, and Sonya wrote down their longest hits during batting practice. The 3 feet. Lenny hit the ball 7 __ 1 feet farther than Raul, but 3 __ 1 shortest of the 4 hits was 100 __ 4 2 8 __ feet farther than Raul did. Of Stanley, Raul, feet less than Stanley. Sonya hit the ball 1 7 8 Sonya, and Lenny, who hit the ball the farthest, and what distance was the ball hit? Problem Solving Workshop Strategy: Use Logical Reasoning Practice Addition and Subtraction Estimate. Then write the sum or difference in simplest form. 3 __ 22 3 Estimate. 1 9 Add. _ __ 1 83 12 5 Name Name MXENL08AWK5X_RT_CH14_L05.indd 1 RW91-RW92 7/19/07 6:01:08 PM 6/15/07 1:35:42 PM Grade5.indd 47 Y 3 6 __ __ 3 5 9. 2 5 Find the product. 1 3 __ 25 5. __ 5 3 1. 4 5 5 4 __ 3 3 __ 5 10. 1 3 3 __ 25 6. __ 2. 8 5 4 3 __ 3 2 __ 5 11. 3 5 3 __ 45 7. __ 3. RW93 Write the product each model represents. • The area of the blue shaded part is 1 3 1, or 1 square units. __ . • The fraction of the part that is shaded blue is 1 4 1 1 1 5 __ __ 3 __ So, 2 4. 2 Use the denominator of the first factor, 2, to make rows. 1 of the yellow column blue. Shade __ 2 • The area of the rectangle is 2 3 2, or 4 square units. Use the denominator of the second factor, 2, to make columns. Shade the left column yellow. 1 3 __ 1 __ 2 2 3 5 8 3 6/13/07 11:33:00 AM © Harcourt • Grade 5 Reteach __ 3 2 __ 5 12. 7 35 1 3 __ 8. __ 4. B Y You can use an area model to record multiplication of fractions. You can use grid paper to model multiplication of fractions. You will use the grid to make and shade a rectangle. 5 MXENL08AWK5X_RT_CH15_L02.indd 1 3 10 7. 1 __ 3 2 __ 5 3 6 4. 2 9 5 __ 3 ___ 5 1. __ 3 3 5 __ 5 10 8 5 __ RW94 12 2 5 3 __ 3 8 5 5 9 3 ___ 8. ___ 5. 9 3 5 2. 4 __ 3 __ Find the product. Write in simplest form. 6 5 ______ 6 4 6 5 __ 1 __ 3 3 __ 5 ___ So, in simplest form, 2 3 4 12 12 4 6 2 3 ___ 6 2 __ __ So, 3 3 4 5 12 . area of rectangle shaded twice ___ ____________________________ 5 6 12 area of big rectangle Use the two areas to write a fraction. • The rectangle has an area of 12 square units. 2 7 8 9 5 4 5 1 3 __ 9. __ 3 6. __ 4 5 2 3 __ 6 5 1 3 __ 3. __ • The area of the rectangle that is shaded twice is 6 square units. Then use the denominator of the first factor, 3, to make 2 of the rows. Shade __ 3 shaded columns. Use the denominator of the second factor, 4, to make columns. __ of the columns. Shade 3 4 2 __ 3 3 __ 3 4 Find the product. Write the answer in simplest form. Record Multiplication of Fractions Model Multiplication of Fractions Find the product: Name Name MXENL08AWK5X_RT_CH15_L01.indd 1 RW93-RW94 6/18/07 6:04:09 PM Reteach © Harcourt • Grade 5 6/13/07 10:01:07 AM Grade5.indd 48 1365 7. __ 3 4 __ 5 4. 2 3 1 4 __ 5 1. 5 3 3 Find the product. 40 5 8 4 5 ___ So, 10 3 ___ 5 5 8. 5. 2. Step 4 Count the shaded parts. There are 40 shaded parts. RW95 1 3 10 5 __ 4 2355 __ 3 2345 __ 5 4 Step 3 The numerator of the fraction __ 5 is 4. So shade 4 parts of each rectangle. __ is 5. Step 2 The denominator of the fraction 4 5 This means there are 5 equal parts. So divide the rectangles into 5 equal parts. Step 1 Draw 10 rectangles. 9. 6. 3. 2 1 __ 335 __ 5 235 6 __ 5 931 5 Reteach 6/13/07 10:02:45 AM © Harcourt • Grade 5 You can use paper and pencil to multiply mixed numbers. 2 1 3 2 __ Find the product. 1 __ 2 3 You can use a model to multiply a fraction and a whole number. 131 __ 1 __ 3 2 4 3 2 __ 1 1 __ 5 5 1 __ 3 3 __ 21 5 2 MXENL08AWK5X_RT_CH15_L04.indd 1 7. 4. 1. Find the product. 2 1 So, 1 __ 3 2 __ 5 4. 2 3 5 6 RW96 3 1 __ 3 1 __ 8. 3 5 4 3 __ 3 2 2 __ 8 3 2 __ 3 1 __ 5. 1 3 2. 2 8 3 3 1 __ 3 3 __ 9. 1 5 3 2322 __ 6. 2 __ 8 3. 2 5 __ 3 1 1 __ 54 24 ___ Step 3 Write the product in simplest form 6 3 8 5 24 __ 3 __ ___ 2 3 6 © Harcourt • Grade 5 Reteach (3 3 2) 1 2 8 2 5 ___________ 2__ 5 __ 3 3 3 3 1) 1 1 3 __ 5 (2 ___________ 5 __ 11 2 2 2 Step 2 Multiply the fractions. Step 1 Write each mixed number as a fraction. First multiply the denominator by the whole number, then add the numerator. Multiply with Mixed Numbers Multiply Fractions and Whole Numbers Find the product. __ 10 3 4 5 Name Name MXENL08AWK5X_RT_CH15_L03.indd 1 RW95-RW96 6/18/07 6:04:19 PM 6/13/07 10:07:42 AM Grade5.indd 49 2 10 10 10 10 10 10 10 10 10 10 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 ___ 1 1 ___ ___ 5. 1. 3 1 6. 3 4 __ 12 __ 141 5 4 1 1 4 ___ 2. __ 141 __ __ 2 4 Use fraction bars to find the quotient. 1 5 2. __ 4 ___ So, 1 5 10 5 1. There are 2 tenths in __ 2 5 __ 7. 2 4 1 RW97 Step 3 Count the number of shaded tenths. Step 2 Divide the parts into tenths. Step 1 Draw one whole rectangle and shade one fifth of it. 141 __ 3. __ 4 10 __ 8. 3 4 1 2 1 __ 4 ___ 4. 1 6/13/07 10:08:28 AM © Harcourt • Grade 5 Reteach __ 447 8 Divide. MXENL08AWK5X_RT_CH15_L06.indd 1 5 3 __ 6. 4 4 2 __ 5. 4 4 2 5 3 2. 3 4 __ 4 3 1. 1 4 __ 7 4 __ 8 8 5 ___ 32 3 __ 7 7 8 __ 8. 1 4 5 6 7. 3 4 __ 7 1 4. 4 4 __ 2 © Harcourt • Grade 5 Reteach 32 4 ___ 54__ 7 7 1 4 __ 4 __ 3 8 __ 1 7 1 4 __ 2 3. 6 4 __ 3 Think: The reciprocal 8. __ is __ of 7 8 7 RW98 Find the quotient. Write it in simplest form. __ 5 4 4 __ . So, 4 4 7 7 8 Step 4 Rewrite product as a mixed fraction, if needed. Step 3 Multiply Step 2 Use the reciprocal of the divisor to write a multiplication problem. 4. Think: Write 4 as __ 1 Find the quotient. Write it in simplest form. You can use pictures to model division of fractions. Step 1 Rewrite 4 as a fraction. Divide Whole Numbers by Fractions Model Fraction Division 1 1 __ 4 ___ 5 10 Name Name MXENL08AWK5X_RT_CH15_L05.indd 1 RW97-RW98 6/18/07 6:04:27 PM 6/13/07 10:10:27 AM Grade5.indd 50 3 8 __ 3 1 4 __ 3 __ 3 8 __ 5 4 14 1 4 8 4 __ 4 3 __ 3. 2 1 9 6 545 __ 4. __ 5. 2. RW99 Divide. Write the answer in simplest form. 1. Write a division sentence for each model. Write a division sentence for 3 41 __ 4 ___ 10 5 Step 2 Write the reciprocal of the divisor. To write the reciprocal, switch the numerator and denominator of the divisor. 80 5 8 8 __ 4 3 __ 5 ___ __ So, 3 1 . 3 8 9 9 We can use a model to write a division sentence. 143 __ 3 __ 3 8 The divisor is always the second number. Step 1 Locate the divisor. Step 3 Write the multiplication problem. 6. 6/13/07 10:11:20 AM © Harcourt • Grade 5 Reteach 80 __ 5 ___ 38 3 9 5 __ 4 ___ 32 7 14 3 10 ___ reciprocal and change 4 to __ 3. The mixed number 3 1 3 10 __ written as a fraction is 3 143 __ 3 __ 3 8 Replace the divisor with its How can you check to see if your answer is reasonable? How many fish are goldfish? What equation can you write to solve? What operation can you use to solve the problem? Explain your choice. What are you asked to find? Twelve students in Danita’s cooking class play a musical __ of the entire instrument. This is 3 4 class. How many students are in Danita’s cooking class? MXENL08AWK5X_RT_CH15_L08.indd 1 6. RW100 7. © Harcourt • Grade 5 Reteach 1 Lance spends __ 30 of each year 12 studying on weekends and ___ 60 of each year studying on weekdays. What fraction of the year does Lance study? Tell which operation you would use to solve the problem. Then solve. 5. 4. 3. 2. 1. 2 are goldfish. Tanya has 10 fish in her aquarium. Of these fish, __ 5 How many fish are goldfish? Problem Solving Workshop Skill: Choose the Operation Divide Fractions We can rewrite all division problems as multiplication problems. 3. 1 4 __ Write the answer in simplest form. Divide 3 __ 3 8 Write the division problem as a multiplication problem. Name Name MXENL08AWK5X_RT_CH15_L07.indd 1 RW99-RW100 6/18/07 6:04:37 PM 6/13/07 10:11:43 AM Grade5.indd 51 Red 3 blue stripes 8 stripes in all Red 5. triangles to hearts 4. hearts to all shapes RW101 2. all shapes to triangles 1. circles to triangles Write each ratio in three ways. Then name the type of ratio. __ The ratio can be written: 3 to 8, 3:8, or 3 8 . The type of ratio is: part to whole 3 stripes What is the ratio of blue stripes to all stripes? The ratio can be written as: 5 to 3 5:3 5 __ 3 The type of ratio is: part to part. Red 5 red stripes Count the stripes of each color. Blue 6/13/07 10:29:26 AM © Harcourt • Grade 5 Reteach 6. circles to all shapes 3. circles to hearts Red Blue (632) 12 6 12 3 3. 5:25 13 MXENL08AWK5X_RT_CH16_L2.indd 1 8 __ and 10 ___ 5. 5 7 21 12 _ and ___ 6. 4 RW102 40 8 7 ___ and _ 7. 35 Tell whether the ratios form a proportion. Write yes or no. Write two equivalent ratios for each ratio. Use multiplication and division. ___ 1. 15 2. 12 to 16 20 3 8 5 __ 254 2 are proportions. __ __ and ___ A proportion is an equation that shows that 2 ratios are equal. 6 (642) 3 So, two equivalent ratios for 4 to 6 are 8 to 12 and 2:3. 2 = 2 to 3 4 = (442) __ _____ = __ Use division to write an equivalent fraction. Divide the numerator and the denominator by the same number. 6 8 = 8 to 12 4 = (432) __ ______ = ___ Use multiplication to write an equivalent fraction. Multiply the numerator and the denominator by the same number. 4 4 to 6 becomes __ 6 42 6 © Harcourt • Grade 5 Reteach 5 28 8. ___ and __ 4. 9 to 18 Write to equivalent ratios for 4 to 6. Use multiplication and division. Write the ratio as a fraction with the first number being the numerator and the second number being the denominator. Write the ratio in three ways. Then name the type of ratio. Blue Equivalent ratios make the same comparison. You can use pictures to model equivalent ratios. A ratio is the comparison of two quantities. Ratios can compare part to part, part to whole, or whole to part. Red Algebra: Equivalent Ratios and Proportions Understand and Express Ratios Red stripes:blue stripes. Name Name MXENL08AWK5X_RT_CH16_L1.indd 1 RW101-RW102 7/19/07 6:05:05 PM 6/13/07 10:31:01 AM Grade5.indd 52 hours miles ______ The denominator is 14. Divide the numerator and denominator by 14. 14 770 ____ 2. 65 miles in 5 hours RW103 8. 624 miles in 13 hours 7. 24 paintings in 12 days $42 for 6 hours of work 5. 4. 72 dog toys in 12 boxes 6/13/07 10:38:37 AM © Harcourt • Grade 5 Reteach 9. $1.14 for 6 ears of corn 6. 27 plants in 3 rows 3. $2.70 for 6 fruit bars 55 5 55 770 4 14 5 ___ _________ 14 4 14 1 Unit Rate Write each ratio in fraction form. Then find the unit rate. 1. 231 miles in 3 hours Write each ratio in fraction form. The unit rate, 55, is the number of miles traveled per hour. 14 770 ____ Rate Write the rate as fraction Write the ratio in fraction form. Then find the unit rate. 770 miles in 14 hours A unit rate is a rate that when written as a fraction has a 1 as its denominator. 3 3 1 3.6 Map Distance (cm) Actual Distance (km) 1 7 720 200 MXENL08AWK5X_RT_CH16_L4.indd A 2. 6.5 in. 3. 1.25 in. RW104 4. 3 in. © Harcourt • Grade 5 Reteach 5. 4.5 in. 28 200 200 3 actual distance 1:200 1 inch 0 1 inch 5 200 miles * Georgetown COLORADO scale The map distance is given. Find the actual distance. The scale is 1 in. = 300 mi. 1. Complete the ratio table. So, the actual distance is 720 miles. 3 distance on map 2 * Northglen Write ratios for the map scale and the actual distance. The map distance is given. Find the actual distance of 3.6 in. The scale is 1 in. 5 200 mi. A map scale is the ratio that compares the | distance on a map to the actual distance. A scale drawing is a reduced or enlarged drawing whose shape is the same as an actual object and whose size is determined by the scale. Understand Maps and Scales Ratios and Rates A rate is a ratio that compares two quantities having different units of measure. Name Name MXENL08AWK5X_RT_CH16_L3.indd A RW103-RW104 7/11/07 5:32:39 PM 7/11/07 4:50:11 PM Grade5.indd 53 200 2 Multiply. 2.5 3 77 5 192.5 77 80 2.5 grouped by age: 21–30, 31–40, 41–50, and 51+. The ages of the students are 25, 68, 33, 61, 46, 59, 46, 29, 35, 28, 21, 51, 47, and 79. Make a table to show how many students are in each age group. 6. In aerobics class, the students are RW105 Check 5. How can you check to see if your answer is reasonable? Explain. 4. How could you write your answer in a complete sentence. Weight on Venus Weight on Earth 1 Divide. 200 4 80 5 3. Make and complete the table to solve the problem. Solve Plan 2. How can making a table help you answer the question? 1. How could you write the question as a fill-in-the-blank sentence? 6/13/07 10:55:21 AM © Harcourt • Grade 5 Reteach Percent is the fraction part of a number to 100. Fractions and decimals can be written as percents. The symbol for percent is %. This symbol means “per hundred.” Tania weighs 80 pounds on Earth and 77 pounds on Venus. Tania’s father weighs 200 pounds. How much would Tania’s father weigh on Venus? 2. 3. MXENL08AWK5X_RT_CH16_L6.indd A 4. 5. RW106 6. Write a decimal and a percent to represent the shaded part. 1. Write a fraction and a percent to represent the shaded part. Decimal: 0.54. Divide. 54 4100 5 0.54 54 Fraction: ____ 100 Percent: 54% 54 of the hundred squares are shaded. 60 Decimal: ____ 100 Write a decimal and a percent to represent the shaded part. Percent: 60% 60 squares are shaded. shaded 5 ____ 60 . Write the fraction as parts: _______ 100 whole 60 ____ Fraction: 100 The grid is divided into 100 squares. Write a fraction and a percent to represent the shaded part. Understand Percent Problem Solving Workshop Strategy: Make a Table Read to Understand Name Name MXENL08AWK5X_RT_CH16_L5.indd A RW105-RW106 6/18/07 6:05:09 PM Reteach © Harcourt • Grade 5 6/13/07 10:56:01 AM Grade5.indd 54 5. 76% 4. 32% 8. 0.45 11. 0.10 8 10. ___ 20 5 3 7. __ RW107 Write each fraction or decimal as a percent. 2. 35% 1. 80% 12. 0.62 10 9 9. ___ 6. 51% 3. 40% Write each percent as a decimal and as a fraction in simplest form. 6/13/07 10:56:32 AM © Harcourt • Grade 5 Reteach To write a fraction in simplest form, divide the numerator and the denominator by the same number. Keep doing this until 1 is the only common factor. 98 4 2 5 49 98 5 ________ ____ ___ 100 100 4 2 50 So, 98% as a decimal 5 0.98 49 98% as a fraction 5 ___ 50 98 • 98% also means ___ 100 • 98% means ninety-eight hundredths, or 0.98. Write 98% as a decimal and as a fraction in simplest form. 20% 30% 40% 50% 60% 70% 2 3 4 20% 5 6 7 MXENL08AWK5X_RT_CH16_L8.indd 1 RW108 7. 25% of 20 5. 40% of 35 6. 50% of 26 3. 50% of 12 8 80% Find the percent of each number. 2. 90% of 20 1. 40% of 80 So, 25% of 24 is 6. Change the color of the counters in 1 of the 4 groups. Count the counters whose color you changed. 1 _ Since 25% 5 , separate the counters into 4 4 equal groups. Show 24 counters. What is 25% of 24? 1 2 So, 90% of 10 is 9. Count by 1s to find 90% of 10. 10% Flip over 90% of the counters. 1 Each counter also represents 1, since 10 3 1 5 10. 10% What is 90% of 10? Show 10 counters. Each counter represents 10%. You can use two-color counters to model percent of a number. Find Percent of a Number Fractions, Decimals, and Percents Percents can be written as decimals or as fractions and vice versa. Name Name MXENL08AWK5X_RT_CH16_L7.indd A RW107-RW108 6/18/07 6:05:19 PM 10 100% 8. 75% of 28 4. 75% of 32 9 90% Reteach © Harcourt • Grade 5 6/13/07 10:57:42 AM RW109 RW110 Grade5.indd 55 6/18/07 6:05:30 PM R B 6 equal sections. 3 2 4 sections MXENL08AWK5X_RT_CH17_L1.indd 1 1. red 2. blue G RW109 Y © Harcourt • Grade 5 Reteach 3. a color other than red Use the spinner to find the probability of spinning each event. • Place the number of favorable outcomes in the numerator of the probability formula: __ P(color other than blue) 5 4 6 4 1 • Next, count the number of favorable outcomes number of favorable outcomes P(color other than blue) 5 ____________________________ 6 • Then, place the number of equally likely outcomes in the probability formula: • First, find the number of equally likely outcomes Use the spinner to write the probability of spinning a color other than blue. B Probability is the likelihood that an event will happen. A prediction is a reasonable guess about the possible outcome, or result, of a probability experiment. An event is the set of one or more outcomes in a probability experiment. The sample space of an event is the set of all possible outcomes. If the sample space is divided into equal parts, each part is equally likely to be selected. The theoretical probability can be expressed in words, as a ratio, or as a fraction: number of favorable outcomes . P(event) 5 _______________________________ number of equally likely outcomes The probability of an event can range from impossible to certain, or from 0 to 1. Y Probability Experiments Outcomes and Probability Total Number of Pulls 12 Red 4 Blue Marble Experiment 7 3 10 Blue Green Total Red Tally Tile Experiment Color MXENL08AWK5X_RT_CH17_L2.indd 6/15/07 1:37:43 PM 1 RW110 predict that Sarah would pull a blue tile or green tile more often if she pulled tiles from the bag 60 more times? Explain. 3. Based on the experimental probabilities, would you 6/15/07 1:38:17 PM © Harcourt • Grade 5 Reteach 2. What is the experimental probability of Sarah pulling a blue tile? a red tile? a green tile? and put the tile back in the bag 20 times. Predict how many times out of 40 pulls that Sarah would pull a red tile from the bag. 1. Sarah pulled a tile from a bag, recorded its color, For 1–3, use the table. So, you can predict that a red or blue marble will be pulled 32 times out of 50. 16 3 2 5 32 Multiply by 2 since 25 goes into 50 2 times. 9 Green 25 Add the times a red marble was pulled to how many times a blue marble was pulled. 12 1 45 16 Count the total number of pulls/tally marks. Predict the number of times out of 50 pulls that Dylan would get a marble that is either red or blue. A box contains 6 black marbles and 3 white marbles. Toby shakes the box and pulls a marble without looking. He records the color marble on a tally table, replaces the marble, and repeats the experiment. An experimental probability of an event is the ratio of the number of times an event occurs to the total number of times the activity is performed. Name Name Grade5.indd 56 14 coin tosses pulls; 30 more pulls 1. 12 blue marbles in 20 RW111 cube tosses; 45 more tosses 2. 8 evens in 12 number more spins 6/15/07 1:38:34 PM © Harcourt • Grade 5 Reteach 3. 4 purple in 24 spins; 60 Express the experimental probability of as a fraction in simplest form. Then predict the outcomes of future trials. • Place the number of favorable outcomes in the numerator of the probability formula and simplify: 6 53 __ P(color other than blue) 5 ___ 14 7 • Finally, to predict the outcome of future trials, multiply the probability by the number of future trials. 3 3 35 ___ 5 15 P(color other than blue) 5 __ 7 1 3 __ So, 6 heads in 14 coin tosses is in simplest form. 7 You can expect 15 heads in 35 coin tosses. • Then, place the number of trials as a denominator in the probability formula: number of times an event occurs P(heads) 5 ______________________________ 14 • Next, count the number of times an event occurs 4 sections • First, find the number of trials Use the spinner to write the probability of spinning a color other than blue. Express the experimental probability of 6 heads in 14 coin tosses as a fraction in simplest form. Then predict the outcome of 35 more tosses. MXENL08AWK5X_RT_CH17_L4.indd 1 cereal, fruit, yogurt, egg © Harcourt • Grade 5 Reteach chooses 3 items each morning instead of two. How many different breakfasts can he make from Joshua’s list. 8. Suppose Joshua’s brother Steven RW112 Joshua make if he choose exactly two different items from this list each morning? 7. How many different breakfasts can Make a list to solve. 6. How do you know your answer is correct? Check 5. What is the answer to the question? 4. How can you use the strategy to solve the problem? Solve 3. What strategy can you use to solve the problem? Plan 2. What are you asked to find? 1. What math details can you identify? Read to Undertand Nia, Sam, and Toni are waiting in line to be seated at Kaye’s Restaurant. How many different ways can they line up? Problem Solving Workshop Strategy: Make an Organized List Probability and Predictions Experimental probability is a number taken from an actual situation. To find experimental probability of an event, write the ratio of the number of times the event occurs to to total number of trials. of times an event occurs. ______________________________ Experimental probability 5 number total number of trials Name Name MXENL08AWK5X_RT_CH17_L3.indd 1 RW111-RW112 6/18/07 6:05:42 PM 6/15/07 1:38:54 PM Grade5.indd 57 A combination is a selection of different items in which the order is not important. An example of a combination is a table place setting that consists of a cup, saucer, plate, and bowl. Listing them in that order or as bowl, plate, cup, and saucer does not change the combination of dishes. To find all possible outcomes of an event, you can use either of the following: 3 Another event has n possible outcomes 5 Total outcomes of both events occurring together Spinner 2 red blue yellow red blue yellow red blue yellow Outcomes red, red red, blue red, yellow blue, red blue, blue blue, yellow yellow, red yellow, blue yellow, yellow green, blue, or red shirt, and white or tan shorts 1. choosing outfits with a RW113 tossing a number cube labeled 1 to 6. 2. tossing a coin and 6/15/07 1:39:10 PM © Harcourt • Grade 5 Reteach labeled 1 to 6 and spinning a spinner labeled 1 to 4. 3. tossing a number cube Draw a tree diagram or use the Fundamental Counting Principle to find the total number of outcomes. So, there are 9 possible outcomes. yellow blue red Spinner 1 First, Make a column for each event. Next, list all possible Outcomes for the 1st event and the second event. Draw a tree diagram. Draw a tree diagram or use the Fundamental Counting Principle to find the total number of outcomes for using two spinners, both with 3 equal sections colored red, blue, and yellow. One event has m possible outcomes • Fundamental Counting Principle. In Fundamental Counting Principle, the total possible outcomes of two events can be stated as a multiplication formula. Topping cheese pepperoni cheese pepperoni MXENL08AWK5X_RT_CH17_L6.indd 1 letters T, E, A. 1. ways to arrange the RW114 A, T so they form a real word 2. ways to arrange letters C, © Harcourt • Grade 5 Reteach and Martin can stand in a ticket line 3. ways that Amy, Greg, Lisa Make a list or draw a tree diagram to find the total possible choices. Crust thin thin thick thick So, there are four possible combinations. Step 3 Count the number of pizza choices. Step 2 List all possible outcomes each event. A table can help you. Make sure each topping is listed with each type of crust. The pizza is the same if you order the crust either before or after the topping. So, this is a combination. Order does not matter. Step 1 Decide if it is a combination or an arrangement. pizzas with choices of thick or thin crust and either cheese or pepperoni topping Make a list or draw a tree diagram to find the total number of possibilities. An arrangement is a group of items that are ordered. An example of an arrangement is assigned seats in a math class. Only one person is assigned any particular seat. Combinations and Arrangements Tree Diagrams • Tree diagram. A tree diagram lists all possible choices or outcomes. Name Name MXENL08AWK5X_RT_CH17_L5.indd 1 RW113-RW114 6/18/07 6:05:54 PM 6/15/07 1:41:35 PM A Z X 6 7 4 8 9 10 5 6/13/07 10:59:06 AM © Harcourt • Grade 5 MXENL08AWK5X_RT_CH18_L02.indd 1 2. /KXN RW116 3. /MXN 4. /KXL J K N M Reteach X Z © Harcourt • Grade 5 L obtuse angle 5 6. 4 13 50 0 1. /LXN 3 12 0 60 right angle 2 110 70 5. 2 Y 100 80 Estimate the measure of each angle. Then use a protractor to find the measure. Z T 1 90 acute angle X Y mm cm 80 70 100 110 4. T U W U 1 60 20 50 0 1 13 W Step 3 Measure /UXY. So, /UXY is 105°. intersecting lines Reteach X Y Step 2 Place the center point of the protractor _ on the vertex, point › X, so that UX passes through 0˚. T U W Point Step 1 Think: _› _› X is the vertex. Rays XY and XU from the sides of an angle. 3. RW115 G Z perpendicular lines E Y 180 0 MXENL08AWK5X_RT_CH18_L01.indd 1 RW115-RW116 2. C F X W 180° 170 10 parallel lines H D T U 135° Use a protractor to measure /UXY. A protractor is a tool for measuring angles. The measure of /UXY is about 90˚. Compare /UXY to the benchmark 90˚. Think: /UXY appears to be greater than 45˚. /UXY appears to be less than 135˚. 90° 160 20 1. B 45° Estimate the measure of /UXY. 0° You can use benchmarks to estimate angle measures. Measure and Draw Angles 0 30 Draw the following on the grid. Label your drawings. Straight angle 5 180° Obtuse angle . 90° Acute angle , 90° Right angle 5 90° ‹_› Look for lines that cross at exactly one point. ‹_› AD and BC cross at exactly one point. ‹_› ‹_› So, AD and BC are intersecting lines. Use the figure. Name an example of intersecting lines. Parallel lines are lines in a plane that are always the same distance apart. Perpendicular lines are two lines that intersect to form right angles. Intersecting lines are lines that cross at exactly one point. A line segment is part of a line between two endpoints and all of the points between them. Use both endpoints to name a line segment. A line is a straight path in a plane. It has no endpoints and can be named by any two points on the line. A plane is an endless flat surface. A point is an exact location. In geometry, objects have special names. Points, Lines, and Angles Name 4 14 0 0 30 15 0 Name 15 20 160 0 40 0 10 180 170 Grade5.indd 58 14 7/19/07 6:01:28 PM 7/11/07 5:39:19 PM Grade5.indd 59 4 sides and angles 3 sides and angles 5 sides and angles 6 sides and angles Regular Polygons Pentagon Hexagon 8 sides and angles Octagon 10 sides and angles Decagon 4 sides and angles 5 sides and angles 6 sides and angles 1. 2. RW117 3. Name each polygon and tell if it is regular or not regular. The polygon has 10 sides. It is a decagon. Measure the sides to see if they are each the same length. The sides are not each the same length. So, the figure is a pentagon that is not regular. Octagon 4. 8 sides and angles Name the polygon below and tell if it is regular or not regular. 3 sides and angles Triangle Polygons that Are Not Regular Quadrilateral Pentagon Hexagon 6/13/07 11:06:22 AM © Harcourt • Grade 5 Reteach 10 sides and angles Decagon A polygon that has sides and angles that are not the same measure is not regular, but it is still named by the number of sides and angles. Quadrilateral Triangle MXENL08AWK5X_RT_CH18_L04.indd 1 page, predict how many lines will be drawn for 7-, 8-, and 9-sided figures if you connect their vertices. 6 RW118 © Harcourt • Grade 5 Reteach with 1-inch sides is 6 inches. The distance around a regular hexagon with 2-inch sides is 12 inches, and the distance around a regular hexagon with 3-inch sides is 18 inches. What is the distance around a regular hexagon with 6-inch sides? 7. The distance around a regular hexagon 5 2 +3 5 4 6. Using the information from the top of the Identify the relationship. Then solve. Number of Sides Number of Lines Connecting Vertices out the missing parts of the table? 3. Using the relationships already demonstrated by the table, how would you fill How many lines did you draw? 2. Connect the vertices of the hexagon at the right. 1. What are you asked to do? Connect the vertices within a square, and a regular pentagon. A square and a regular pentagon are shown at the right. Count the lines within each figure. How many lines would you draw within a regular hexagon? Problem Solving Workshop Skill: Identify Relationships Polygons A polygon is a closed plane figure formed by three or more line segments. Polygons are named by the number of sides and angles. A regular polygon has sides that are all the same length and angles that are all the same measure. Name Name MXENL08AWK5X_RT_CH18_L03.indd 1 RW117-RW118 6/18/07 6:06:12 PM 6/13/07 11:08:19 AM Grade5.indd 60 The length of a diameter is always twice the length of a radius. N M B 13.5 cm 4. Name a radius. V H RW119 Complete 5–6. Then use a compass to draw each circle. Draw and label the measurements. 5. radius 5 1.5 cm 6. radius 5 diameter 5 diameter 5 2.5 cm 3. Name a chord. Use data for 1–4, use the circle at the right. 1. Name the circle. 2. Name a diameter. So, the diameter is twice this length or 7 cm. V F X You know that the radius is 3.5 cm. Use a compass set at 3.5 cm to draw this circle around a center point. Label your center point, point V. Complete. Then use a compass to draw the circle. Label the measurements. Radius: 3.5 cm Diameter: A chord has its endpoints on the circle. NO and MP are chords. • A diameter of a circle passes through the center and has it endpoints on the circle. MP is a diameter. A radius of a circle connects the center of the circle with any point on the circle. BL is a radius. BM and BP are also radii. O P L Reteach 6/13/07 11:16:06 AM © Harcourt • Grade 5 E R T S W V 2. 6. GH MXENL08AWK5X_RT_CH18_L06.indd 1 5. EF 7. /G 3. RW120 8. /M Identify the corresponding side or angle. 1. G E L J 4. Write whether the two figures appear to be congruent, similar, or neither. /R corresponds to /U /S corresponds to /V So, /T corresponds to /W. Figures RST and UVW are congruent. Identify the angle that corresponds to /T So, the two figures are neither. Think, do the figures appear to be the same size? • No Think, do the figures appear to be the same shape? • No The figure on the left appears to have greater height than the figure on the right. Tell whether the two figures at the right appear to be congruent, similar, or neither. Figures can be congruent, similar, both, or neither. Congruent figures have the same size and the same shape. Similar figures have the same shape, but may or may not be the same size. Corresponding angles and corresponding sides are in the same related position in different angles. A circle is round. All of its points are the same distance from its center. A circle is named by its center point. Look at circle B. Congruent and Similar Figures Circles Circle B Name Name MXENL08AWK5X_RT_CH18_L05.indd 1 RW119-RW120 6/18/07 6:06:23 PM F K © Harcourt • Grade 5 Reteach M H U 6/13/07 11:16:30 AM Grade5.indd 61 À 6. 2. 7. 3. 9. 10. RW121 11. Draw lines of symmetry. Tell whether each figure has rotational symmetry. Write yes or no. 5. 1. Tell whether the parts on each side of the line match. Is the line a line of symmetry? So, the line on this figure is a line of symmetry. Fold the cut-out figure along the dashed line. Use a piece of paper to trace the figure and line. Cut out the figure and line of you piece of paper. Tell whether the parts on each side of the line match. Is the line a line of symmetry? 12. 8. 4. Reteach 6/13/07 11:30:48 AM © Harcourt • Grade 5 2 cm 15 cm 14 cm 5 in. 2. 5 in. 5 in. MXENL08AWK5X_RT_CH19_L01.indd 1 4. 5. RW122 Classify each triangle. Write acute, right, or obtuse. 9 cm 1. 6. 3. Classify each triangle. Write isosceles, scalene, or equilateral. It is an acute triangle. It has three angles, each less than 90°. Classify the triangle. Write acute, right, or obtuse. It is a scalene triangle. It has no equal side lengths. Classify the triangle. Write isosceles, scalene, or equilateral. 10 cm 10 cm 8 cm 7 cm right triangle: One angle is right and the other two angles are acute. © Harcourt • Grade 5 Reteach 4 cm obtuse triangle: One angle is obtuse and the other two angles are acute. isosceles triangle: Two sides are the same length. scalene triangle: All sides are of different lengths. Use the corner of a paper to classify angles. acute triangle: All three angles are acute. equilateral triangle: All sides are the same length. You can classify triangles either according to their sides or according to their angles. If you fold the picture of the light bulb in half along the dashed line, the two parts will match exactly. The light bulb has line symmetry. Use a ruler to compare side lengths. Classify Triangles Symmetry A figure has rotational symmetry if it can be rotated less than 360˚ around a center point and still match the orginal figure. Name Name MXENL08AWK5X_RT_CH18_L07.indd 1 RW121-RW122 6/18/07 6:06:34 PM 6/6/07 11:38:42 AM Grade5.indd 62 trapezoid quadrilateral one pair of parallel sides You can draw a plane figure based on a description. You can use this chart to classify quadrilaterals. rhombus parallelogram 4 congruent sides 2. 3. RW123 4. Describe each quadrilateral using parallel, perpendicular, and congruent. 1. Classify each figure in as many ways as possible. Write quadrilateral, parallelogram, square, rectangle, rhombus, or trapezoid. 1 in. 1 in. 1 in. 1 in. 1 in. 1 in. © Harcourt • Grade 5 Reteach pair of angles measuring 60°, the other pair measuring 120° 2. quadrilateral: four 3-centimeter sides; 1 RW124 side between the angles that measures 4 centimeters 1. triangle: 2 angles that measure 30°; one Use a protractor and a ruler to draw each figure. Classify each figure. It has 4 congruent sides and 4 right angles, so the figure is a square. 1 in. 1 in. Use a protractor to make sure the other 3 angles are right angles. Draw two more 1 in. lines to connect your figure. 1 in. Step 4 Step 3 1 in. Use a protractor to make sure the angle formed by the two lines you drew is a right angle. Draw: 1 in. Step 2 Step 1 1 in. A quadrilateral with 4 right angles; 4 congruent sides each measuring 1 inch. 6/13/07 9:44:51 AM MXENL08AWK5X_RT_CH19_L03.indd 1 © Harcourt • Grade 5 Reteach It has 4 sides so it is a quadrilateral. Its opposite sides are parallel and congruent so it is a parallelgram. It has 4 right angles with opposite sides that are parallel and congruent. However, not all of its sides are congruent, so it is a rectangle. Classify the figure in as many ways as possible. Write quadrilaterals, parallelogram, square, rectangle, rhombus, or trapezoid. square rhombus rectangle rectangle parallelogram 4 right angles parallelogram quadrilateral opposite sides are parallel opposite sides congruent Draw the figure on a coordinate plane. Classify the figure. Draw Plane Figures Classify Quadrilaterals quadrilateral 4 sides Name Name MXENL08AWK5X_RT_CH19_L02.indd 1 RW123-RW124 7/19/07 6:01:49 PM 7/11/07 4:37:23 PM Grade5.indd 63 The one base is a hexagon. hexagonal pyramid hexagon prism pentagonal prism shape of a ball and has no base The two bases are circles. The two bases are hexagon. The two bases are pentagons. All faces are squares. 1. 2. RW125 3. 4. Classify each solid figure. Write prism, pyramid, cone, cylinder, or sphere. So, the figure is a pyramid. Reteach 6/13/07 10:44:31 AM © Harcourt • Grade 5 MXENL08AWK5X_RT_CH19_L05.indd 1 © Harcourt • Grade 5 Reteach prism. How many faces does the prism have? How many vertices does the prism have? 7. Mackenzie used 15 straws to make a RW126 does it have? How many vertices does it have? 6. A pyramid has 9 faces. how many edges Solve by comparing strategies. Check 5. How can you check your answer? The figure has only one base. 4. Write your answer in a complete sentence. 3. Solve the problem. Describe the strategy used. Solve 2. What are two strategies you can use to solve the problem? Plan 1. Write the quesiton as a fill-in-the-blank sentence. Read to Understand The figure is a polygon. Classify the solid figure. Write prism, pyramid, cone, cylinder, or sphere. sphere A solid figure with curves is not a polyhedron. cone The one base cylinder is a circle. The one base is a pentagon. pentagonal pyramid The one base is a square. square pyramid cube rectangular prism The one base is a rectangle. rectangular pyramid All faces are rectangles. A prism has two congruent and parallel polygons as bases. triangular The two prism bases are triangles. A pyramid is a polyhedron with only one polygon base. triangular All faces are pyramid triangles. Mackenzie used 12 straws as edges to build a pyramid. How many sides did the base of her pyramid have? How many vertices did her pyramid have? Problem Solving Workshop Strategy: Compare Strategies Solid Figures You can identify a polyhedron by the shape of its faces. They are solid figures with faces that are polygons. Name Name MXENL08AWK5X_RT_CH19_L04.indd 1 RW125-RW126 7/19/07 6:02:07 PM 6/6/07 11:42:29 AM Grade5.indd 64 c. 2. b. 1. a. Match each solid figure with its net. So, the solid figure matches net b. RW127 c. 3. Net b has 3 rectangular and 2 triangular faces. Nets a and b each have 5 faces. Look for the net with 5 faces. It has 5 faces; 3 rectangular and 2 triangular. The figure is a triangular prism. Name the solid figure for the net. d. 4. d. b. Reteach 6/6/07 11:43:00 AM © Harcourt • Grade 5 shape of base shape of base with a point in the center circle top view top top MXENL08AWK5X_RT_CH19_L07.indd 1 3. 1. front front side side RW128 4. 2. top top top circle triangle rectangle front view Identify the solid figure that has the given views. So the figure is a square pyramid. The top view is a square. All inside segments meet at a point. The front and side views are triangles, so the figure is a pyramid or cone. Identify the solid figure that has the given views. sphere pyramid or cone prism or cylinder side © Harcourt • Grade 5 Reteach side side circle triangle rectangle side view front front front You can use this chart to help you identify solids from different views. You can identify solid figures by their nets. A net can be cut out, folded, and taped together to form a polyhedron. a. Draw Solid Figures from Different Views Nets for Solid Figures Match the solid figure with its net. Name Name MXENL08AWK5X_RT_CH19_L06.indd 1 RW127-RW128 6/18/07 6:07:05 PM 6/6/07 11:43:56 AM Grade5.indd 65 Can you pick it up and turn it over? Yes, so it is a reflection. • 2. 5. 1. 4. RW129 6. 3. 6/6/07 11:30:42 AM © Harcourt • Grade 5 Reteach Can you turn it about a point? No, so it is not a rotation. • Name each transformation. Write translation, reflection, or rotation. So, the transformation is a reflection. Can you push it along a straight line? No, so it is not a translation. • Ask how the figure changes position. Name the transformation. Write translation, reflection, or rotation. A rotation is a turn. An example is the way the hands on an analog clock move. A reflection is a transformation that flips the figure over a line. An example is turning a page in a book. MXENL08AWK5X_RT_CH20_L02.indd 1 4. 1. 5. 2. and RW130 6. 3. © Harcourt • Grade 5 Reteach 6/13/07 10:49:49 AM If you try to cover a surface with closed figures, an you have gaps or overlaps, then you have not made a tessellation. Predict whether the figure or figures will or will not tessellate. Write yes or no. So, yes, the figure will tessellate. The pair of figures do not tessellate. Example – Not a Tessellation A tessellation is a design that uses closed figures. In a tessellation, one or more figures are put together to cover a surface without any gaps or overlaps. A transformation is a change in position of a figure that does not change its size or shape. Example – Tessellation Predict whether the figure will tesselate. Test your prediction. Tessellations Transformations A translation is a transformation that slides the figure along a straight line. An example is a sled sliding downhill. Name Name MXENL08AWK5X_RT_CH20_L1.indd 1 RW129-RW130 7/19/07 6:02:38 PM Grade5.indd 66 3. Rotation A reflection is a flip over a line. It must be picked up for this transformation. 1. 2. RW131 Tell how each pattern might have been created. This pattern might have been created by translating the figure to the right. The triangles are all facing the same way. Each triangle can slide to the next position without being picked up. Describe how the pattern might have been created. This pattern might have been created by reflecting the figure over a vertical line. This figure must be picked up and flipped to form this pattern. It is not a translation because the figure turns in two different ways. Describe how the pattern might have been created. A translation is a slide that occurs without turning or picking the figure up. 189 4 63 5 3 5. 3888, 648, 108, 18, ... 8. 828, 818, 808, 798, ... 4. 101, 120, 139, 158, ... 7. 7, 35, 63, 91, ... RW132 2. 313, 307, 301, 295, ... 63 4 21 5 3 1. 15, 45, 135, 405, ... Identify the rule of each pattern. Step 3 Write the rule. So the rule is divide by 3. • Are the quotients the same? Yes, so the rule is division. 567 4 189 5 3 Then, test division, the other remaining operation. © Harcourt • Grade 5 Reteach 9. 12, 48, 192, 768, ... 6. 612, 598, 584, 570, ... 3. 240, 120, 60, 30, ... 567 2 378 5 189 189 2 378 fi 63 • Are the differences the same as the pattern in the problem? No. So the rule subtract 378 doesn’t work. Step 2 Test subtraction, one of the remaining operations. Think of a number to subtract from 567 to get 189. • Does the pattern decrease? Yes. So the rule is either subtraction or division. 6/13/07 10:57:48 AM MXENL08AWK5X_RT_CH20_L4.indd 1 © Harcourt • Grade 5 Reteach A rotation is a turn about a point. Step 1 See if you can eliminate two operations • Does the pattern increase? No. So eliminate addition and multiplication. 567, 189, 63, 21, ... Identify the rule for the pattern. You can find a rule for a number pattern and use it to extend the pattern. You can use reflections, rotations, or translations to create a pattern. Reflection Numeric Patterns Create a Geometric Pattern Translation Name Name MXENL08AWK5X_RT_CH20_L03.indd 1 RW131-RW132 7/19/07 6:02:56 PM 6/6/07 11:34:21 AM ? ? What are the next two figures in his pattern? RW133 6/6/07 11:36:57 AM © Harcourt • Grade 5 Reteach What will the next three figures in Amelia’s design be? Look for a pattern to solve. 7. Amelia uses parallelograms to decorate a 6. Nate uses a design made of figures to picture frame. Her design so far is shown decorate a ceramic bowl he made in below. pottery class. His design so far looks like this: What colors will the next two squares be? Check 5. How did finding the pattern help you solve the problem? 4. Answer the question. Then draw a line under the part that repeats. 3. Color the first 10 squares to show Suri’s pattern so far. Solve a sequence? 2. What kind of pattern does Suri use, a repeating pattern or Plan Read to Understand 1. Write the question as a fill-in-the blank sentence. Suri is painting a design that is a pattern of squares. She paints the squares black, white, black, gray, black, white, black, gray. What color could the next two squares be? Algebra: Graph Relationships Problem Solving Workshop Strategy: Find a Pattern 10 5 Number of sides, y 15 3 20 4 6 3 Number of Vertices, y 9 3 3 18 RW134 2 1 2 1 Number of cubes, y Number of Triangles, x 12 6 Number of faces, x MXENL08AWK5X_RT_CH21_L1.indd 1 2. 1. Write the ordered pairs. Then graph them. For this relationship, the y-coordinate is always 5 times the x-coordinate. You can graph the ordered pairs for the relationship shown in the table. Each column can be written as an ordered pair (x,y): 1 pentagon, 5 sides: (1,5) 2 pentagons, 10 sides: (2,10) 3 pentagons, 15 sides: (3,15) 4 pentagons, 20 sides: (4,20) 2 1 Number of pentagons, x A table can show the relationship. Write the ordered pairs and then graph them. 12 4 0 2 4 6 y 2 You can graph ordered pairs to show relationships between two amounts. For example, you can show the relationship between a number of pentagons and the number of sides. Name Name 4 Number of Sides Grade5.indd 67 MXENL08AWK5X_RT_CH20_L05.indd 1 RW133-RW134 7/11/07 5:33:43 PM 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 16 18 20 22 y 6 y 2 8 4 6 8 4 10 6 12 14 x x x Reteach 10 18 10 © Harcourt • Grade 5 8 16 Number of Pentagons 2 (1,5) (2,10) (3,15) (4,20) 7/11/07 4:35:02 PM Grade5.indd 68 0 y 1 4 2 8 +8 3 12 4 4 16 3 5 2 4 1 3 output, y 2. input, x 2 10 1 5 15 3 4 25 5 RW135 3. +6 x x 4 16 Plan 2. What strategy can you use to solve the problem? 0 +2 +4 +6 0 +4 +8 +2 +4 output, y input, x 1 0 4 1 +6 2 +8 Reteach 13 6/13/07 11:02:01 AM © Harcourt • Grade 5 10 3 MXENL08AWK5X_RT_CH21_L3.indd 1 lunches. How much does he spend for lunches in one year? 6. Richard spends $150 each month on Write an equation to solve. sense for the problem. RW136 © Harcourt • Grade 5 Reteach paycheck. She has $360 in her savings account. How much will she have in the acount 12 weeks from now? 7. Mary saves $20 each week from her 5. Look back at the problem. Explain why your answer makes Check 5. How many more miles does Nancy need to travel? 4. Solve the equation for n. 84 5 Solve +4 3 12 3. Write an equation. Let n represent the number of miles Nancy still must travel. +2 2 8 +16 y 1 4 1. What information is given? Read to Understand +12 +18 +8 0 0 4 3 4 5 16 3 3 4 5 12 x y 23458 13454 x y Find the rule to complete the function table. Then write an equation. output, y input, x graph the ordered pair. Rule: Subtract 3. Equation: y 5 x 2 1. Complete the function table. Then Graph the ordered pairs. Use the function table to write the ordered pairs. (0,0), (1,4), (2,8), (3,12), (4,16) Equation: 4x Rule: multiply by 4. • Use the rule to write an equation. So, the value of y when x 5 0 is 0. 03450 • Complete the function table. Use the rule to find the value of y when x 5 0. • Find a rule. Each value of y is 4 times the value of x. So, the rule is multiply 4. Find a rule and complete the function table. Then write an equation. Nancy has traveled 39 miles on a train to visit her grandmother. The total distance of the trip is 84 miles. How many more miles will Nancy travel to reach her grandmother’s house. Problem Solving Workshop Strategy: Write an Equation Algebra: Equations and Functions You can write an equation and make a function table to show the relationship between two amounts. Name Name MXENL08AWK5X_RT_CH21_L2.indd 1 RW135-RW136 7/19/07 6:05:23 PM 6/13/07 11:18:17 AM Grade5.indd 69 -6 -4 -2 -8 -6 -4 -2 0 +2 +4 +6 +8 +10 2 2, 1, and 5. 0 +2 +4 +6 +8 +10 No profit or loss is at $0, and profit is a positive word. -8 -6 -4 -2 0 +2 +4 +6 +8 +10 2. a gain of 12 yards RW137 3. 7 degrees below freezing Write an integer to represent each situation. -10 Identify the integers graphed on the number line. So, the integer for a profit of $800 is 1800, or just 800. 1. positive integers 6/13/07 11:30:55 AM © Harcourt • Grade 5 Reteach 4. 8 floors up Some words used with Some words used with negative integers positive integers loss gain or profit decrease increase behind ahead backward forward below above down up under over withdrawal deposit to the left to the right Write an integer that describes a profit of $800. So, the interger that represents 79 degrees below 0 is 279. “Below” is a word used with negative integers. Use the chart at the right to see which category “below” falls under. Notice how the word “below” is in the situation. Write an integer to represent 79 degrees below 0. So, the integers are • The point that is 2 units left of 0 represents 22. •The point that is one place to the right of 0 represents 1. • The point that is 5 units right of 0 represents 5. -10 Identify the integers graphed on the number line. negative integers -8 -8 -6 -4 -2 0 +2 +4 -8 -6 -4 -2 0 -8 -6 -4 -2 0 +2 +4 -8 -6 -4 -2 0 +2 +4 +6 +6 +6 +6 +8 +8 +8 +8 +10 +10 +10 +10 1 1 6 2 7 1 2 1 2 1 6 1 MXENL08AWK5X_RT_CH21_L5.indd 1 7. 4. 1. Compare. Write ,, ., or 5. 8. 5. 2. 1 2 2 5 2 9 6 10 8 RW138 2 2 2 9. 6. 3. 2 2 1 3 2 4 2 1 3 6 0 Two integers are equal only when they have the same sign and same number. -10 +4 Since 24 is to the left of 19, 24,19 Compare 24 and 19. Use ,, ., or 5. -10 +2 Since 14 is to the left of 19, 14,19 Compare 14 and 19. Use ,, ., or 5. -10 Since 14 is to the right of 28, 14.28 Compare 14 and 28. Use ,, ., or 5. -10 Since 24 is right of 28, 24.28 Compare 24 and 28. Use ,, ., or 5. You can use a number line to compare and order integers by graphing the integers on a number line. Integers are the set of whole numbers and their opposites. -10 Compare and Order Integers Understand Integers Integers can be graphed on a number line. Positive integers are always greater than 0. Numbers with no sign in front of them are positive integers. Negative integers are always less than 0. The absolute value of an integer is its distance from 0 on a number line. Name Name MXENL08AWK5X_RT_CH21_L4.indd 1 RW137-RW138 6/18/07 6:07:57 PM Reteach © Harcourt • Grade 5 6/13/07 11:38:57 AM Grade5.indd 70 to 23 and to 12 To graph (23, 12), you would start at (0,) and the go U H 6. 9. E Q 7. 8. Z 5. D 4. N 3. S 2. RW139 For 2–9, identify the ordered pair for each point. 1. So, the ordered pair (4, 24) names the location of point F. • Point F is 4 units below the x-axis, so the y-coordinate is 24. • Point F is 4 units right of the y-axis, so the x-coordinate is 4. What ordered pair names the location of point F? So, the ordered pair (21, 3) names the location of point C. • Point C is 3 units above the x-axis, so the y-coordinate is 3. • Point C is 1 unit left of the y-axis, so the x-coordinate is 21. What ordered pair names the location of point P on the coordinate plane? The y-coordinates above the x-axis are positive integers. The y-coordinates below the x-axis are negative integers. +5 +4 C + 3 +2 +1 E N -4 Q E -2 Z D +2 H y- axis -4 -2 0 +U2 +4 y- axis S x -axis x -axis 6/13/07 11:50:21 AM © Harcourt • Grade 5 Reteach +4 B -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 -1 A -2 -3 F -4 -5 The x-coordinates to the right of the y-axis are positive integers. The x-coordinates to the left of the y-axis are negative integers. The origin is the place where the x- and y-axis intersect. The horizontal line is the x-axis. The vertical line is the y-axis. Just as a coordinate grid is formed by two perpendicular rays, a coordinate plane is formed by two intersecting and number perpendicular lines. Customary Length Algebra: Graph Integers on the Coordinate Plane 2 MXENL08AWK5X_RT_CH22_L1.indd 1 7. RW140 8. __ inches 5. 3 feet or 34 1 1 inch. Measure the length to the nearest __ 8 8 7 inches or 6 inches 4. 5 __ 2 1 2. nearest __ inch. Tell which measurement is more precise. 1. nearest inch 4 © Harcourt • Grade 5 Reteach 1 inches 6. 7 inches or 7 __ 8 1 3. nearest __ inch. Estimate the length of the caterpillar in inches. Then measure the length to the nearest inch. 5 in. mark. The crayon is almost exactly on the 2__ 8 1 inch, the crayon is 2__ 5 inches long. So, to the nearest __ 8 8 __ inch using a ruler. Now measure the crayon to the nearest 1 8 Estimate: About 3 inches long Estimate the length of the crayon in inches. Then measure the length to the nearest 1_8 inch. When you measure with precision, you use the smallest unit possible. Measuring to the nearest 1_2 inch is more precise than measuring to the nearest inch. Measuring to the nearest 1_4 inch is even more precise. Name Name MXENL08AWK5X_RT_CH21_L6.indd 1 RW139-RW140 6/18/07 6:08:07 PM 6/13/07 11:57:52 AM Grade5.indd 71 1 cm 1 cm 5 10 mm CM kilometer (km) 1 km 5 1,000 m 1 m 5 100 cm 1 m 5 1,000 cm 5 6 2. to the nearest millimeter. inches smaller 15 in. 10 ft ft 4. 3 ft 1 5 ft 5 in. 4 in. Find the sum or difference. 1. 33 yd 5 Change the unit. 5. RW142 5 yd 1 8 yd 2. 500 cm 5 Rename, because 15 in. > 1 ft. 15 in. 5 1 ft 3 in. 10 ft 15 in. = 10 ft 1 1 ft 3 in. 5 11 ft 3 in. So, 6 ft 7 in. 1 4 ft 8 in. 5 11 ft 3 in. 7 in. 8 in. 6 ft 1 4 ft 4 ft. 2 ft. 2 ft. 8 yd 2 2 yd 6 yd m 6. 10 ft 2 3 ft 3. 13 cm 5 Reteach © Harcourt • Grade 5 4 in. 11 in. mm So, 9 yd 1 ft 2 2 yd 2 ft 5 6 yd 2 ft Subtract. Subtract. 9 yd 1 ft 2 2 yd 2 ft There are not enough feet to subtract 2 feet from 1 foot, so rename. 9 yd 1 ft 5 8 yd 4 ft Add feet. Add inches. 2 ft. 2 ft. 100 m 5 1 m, so 800 m 5 cm Metric Units of Length 10 mm 5 1 100 cm 5 1 m 1,000 m 5 1 km 12in. 5 1 ft 3 ft 5 1 yd 5,280 ft 5 1 mi 1,760 yd 5 1 mi Customary Units of Length Add. 6 ft 7 in. 1 4 ft 8 in. So, 800 centimeters 5 8 m. m larger Divide. 800 4 100 5 8 Think: cm smaller STEP 3 STEP 2 Multiply. 3 3 12 5 36 STEP 3 Decide: Multiply or Divide m 1 ft 5 12 in., so 3 ft 5 in Think: STEP 2 inches STEP 1 Change the unit: 800 cm 5 So, 3 feet 5 36 inches. feet larger Decide: Multiply or Divide STEP 1 Change the unit: 3 feet 5 Divide to change from smaller to larger linear units 6/13/07 12:01:10 PM MXENL08AWK5X_RT_CH22_L3.indd 1 © Harcourt • Grade 5 Reteach 6. width of the North American Continent 5. height of a skyscraper RW141 4. distance from Austin to Phoenix 3. width of a paperclip wire Write the appropriate metric unit for measuring each. 1. to the nearest centimeter. Estimate the length of the string in centimeters. Then measure the length These 2 cities are in different states, so there is a considerable amount of distance between them. So, the most appropriate unit for measuring the distance from Boston to Dallas is the kilometer. Write the appropriate metric unit for measuring the distance from Boston to Dallas. So, to the nearest centimeter, the paperclip is 4 cm long. The paperclip is exactly 4 centimeters long. Notice where the paperclip ends on the ruler. 4 Align the left side of the paperclip with the zero mark on the ruler. This is already shown in the picture. 3 STEP 2 2 STEP 1 cm 1 How long is the paperclip to the nearest centimeter? You can use a centimeter ruler to measure length. 1 mm CM meter (m) Some common metric units of length are: centimeter (cm) Change Linear Units Multiply to change from larger to smaller linear units. Metric Length millimeter (mm) Name Name MXENL08AWK5X_RT_CH22_L2.indd 1 RW141-RW142 7/11/07 5:34:28 PM 7/11/07 4:38:55 PM Grade5.indd 72 oz. 1 lb 5 16 oz, __ lb 5 oz so 21 2 qt oz gal 1. 14 pt = 4. 7.5 lb = 7. 20 qt 5 Change the unit. 8. 3 T 5 RW143 5. 9 gal = 2. 12,000 lb = 1 pounds is equal to 40 ounces. So, 2__ 2 ounces smaller Think: Decide: Multiply or Divide pounds larger STEP 2 STEP 1 __ lb 5 Change the unit: 21 2 lb qt T 6/13/07 12:16:47 PM © Harcourt • Grade 5 Reteach pt qt fl oz 2,000 lb 5 1 T 16 oz 5 1 lb 9. 16 fl oz 5 6. 16 c = 3. 7 c = Multiply. __ 3 16 5 21 2 8 5 ___ 16 5 40 __ 23 1 STEP 3 Customary Units of Weight 4 qt = 1 gal 2 c = 1 pt 2 pt = 1 qt So, there are 2 qt in 8 c. 4 c 5 1 qt, so 8 c 5 oz 8 fl oz = 1 c 4 c = 1 qt 8 c 4 4 5 2 qt Think: Decide: Multiply or Divide cups quarts so divide to change units smaller larger Divide. STEP 2 STEP 1 kilograms larger 9,000 g 5 1 kg, so 9,000 g 5 kg Think: STEP 2 milligrams smaller 1 g = 1,000 mg, so 10 g = mg Think: STEP 2 mg. MXENL08AWK5X_RT_CH22_L5.indd 1 7. 20 metric cups 5 4. 10,000 mL 5 1. 6,000 L 5 Change the unit. kL L L RW144 8. 9 g 5 5. 4 kg 5 2. 350 mg 5 So, there are 10,000 milligrams in 10 grams. grams larger Decide: Multiply or Divide STEP 1 Change the unit: 10 g 5 So, there are 9 kilograms in 9,000 grams. grams smaller Decide: Multiply or Divide STEP 1 Change the unit: 9,000 g 5 Change the unit: 8 c 5 STEP 3 Divide to change from smaller to larger units. Divide to change from smaller to larger units. kg Multiply to change from larger to smaller units. Multiply to change from larger to smaller units. Customary Units of Capacity Customary Capacity and Mass Customary Capacity and Weight qt. Name Name MXENL08AWK5X_RT_CH22_L4.indd 1 RW143-RW144 6/18/07 6:08:24 PM mg g g 9. 2,500 ml 5 6. 4 L 5 metric cu Reteach © Harcourt • Grade 5 kL 6/13/07 1:07:46 PM metric cups 1,000 g = 1 kg 1,000 mg = 1 g Metric Units of Mass 1,000 liters = 1 kL 4 metric cups = 1 L 250 mL = 1 metric cup 1,000 mL = 1 L Metric Units of Capacity 3. 1,750 mL 5 10 3 1,000 510,000 mg Multiply. STEP 3 Divide. 4,000 ______ 1,000 5 9 kg STEP 3 Grade5.indd 73 329-cm-long piece of corduroy. He needs to use 146-cm-long pieces for two pillows and an 83-cm-long piece for the third. Does Colin have enough corduroy? 7. Colin wants to make pillows from a RW145 6/6/07 11:16:38 AM © Harcourt • Grade 5 Reteach costumes for the school play. She doesn’t want any left-over fabric. She needs 1.2 meters for three costumes and 1.9 meters for two costumes. How much fabric should Julie buy? 8. Julie needs to buy fabric to make Tell whether you need an estimate or an actual measurement. Then solve. 6. How can you check your answer? 5. How far did Jennifer jog last week? 4. What equations can you use to solve the problem? 3. What operations will you use to solve the problem? measurement? Why? 2. Can you estimate, or do you need to find an actual 1. How could you write the question as a fill-in-the blank sentence. Jennifer went jogging last week. She jogged 2.65 kilometers on Tuesday and twice as far on Wednesday. How far did Jennifer jog last week? Problem Solving Workshop Skill: Estimate or Actual Measurement Name MXENL08AWK5X_RT_CH22_L6.indd 1 RW145-RW146 6/18/07 6:08:35 PM Count the number on minutes in 5 minute intervals. 11 12 1 2 10 9 3 4 8 7 6 5 3 T 4 W Jan. 5 R 6 F 7 S 3 F 4 S Count days. Step 2 March 2 to March 5 is 3 days. January 12 to March 5 is 7 weeks. MXENL08AWK5X_RT_CH22_L7.indd 1 End: 9:52 P.M. End: RW146 Elapsed time: 3 hr 29 min 3. Start: Elapsed time: 2 hr 5 min 2. Start: 4:35 P.M. Write the time for each. June 30. How many weeks away is Larry’s trip? 1. On June 2, Larry planned a whitewater rafting trip for 6 M 7 T 8 1 W March 3 F 4 S 9 10 11 2 R 5 M 6 T 7 W June 3 S 9 10 2 1 8 F R 25 26 27 28 39 30 18 19 20 21 22 23 24 11 12 13 14 15 16 17 4 S 26 27 28 29 30 31 19 20 21 22 23 24 25 12 13 14 15 16 17 18 5 S 5 minute interval: ending on the 6 11 12 1 2 10 9 3 4 8 7 6 5 © Harcourt • Grade 5 Reteach End: June 25, 5:30 P.M. Elapsed time: 4. Start: June 21, 3:20 P.M. So, the elapsed time from January 12 to March 5 is 7 weeks and 3 days. Count weeks. Step 1 26 27 28 9 10 11 2 R 19 20 21 22 23 24 25 8 1 W Feb. 12 13 14 15 16 17 18 7 T 29 30 31 6 M 22 23 24 25 26 27 28 5 S 15 16 17 18 19 20 21 9 10 11 12 13 14 2 1 8 M S What is elapsed time from January 12 to March 5? You can use a calendar to find elapsed time in weeks and days. So, the elapsed time is 2 days 1 hr, and 30 min. Count hours until you reach 2 P.M. 11 12 1 2 10 9 3 4 8 7 6 5 Count the hours and minutes. Step 2 May 5 – May 3 5 5 2 3 5 2 days Count the days. Start: May 3, 1 P.M. End: May 5, 2:30 P.M. Step 1 Write the elapsed time. You can use a clock to find elapsed time in hours and minutes. Elapsed Time Name 6/13/07 1:18:11 PM Grade5.indd 74 212 °F 32 49 7. -20 ºF to 45 ºF 6. 111 ºF to 77 ºF 8. 8 ºF to 103 ºF 4. 75 ºC to 39 ºC RW147 3. -35 ºC to 50 ºC 2. 7 ºC to 61 ºC Use the thermometer to find the change in temperature. By how many degrees Fahrenheit did the temperature change? 105 °C –45 –35 –25 –15 –5 5 15 25 35 45 55 65 75 85 95 95ºC 6/13/07 1:21:26 PM © Harcourt • Grade 5 Reteach 9. 69 ºF to -13 ºF 5. 52 ºC to -44 ºC 1. The temperature is 49 ºF. Seven hours later, the temperature is 39 ºF. Use the thermometer to solve. So, the change in temperature from -25 ºC to 10 ºC is 35 ºC. • Equation: 25 1 10 5 35 • Change: Increase • Think: -25 ºC is lower than 10 ºC Find the change in temperature from -25 ºC to 10 ºC. So, the change in temperature from 102 ºF to 95 ºF is 7 ºF. • Equation: Subtraction 102 2 95 5 7 • Change: Decrease • Think: 102 ºF is higher than 95 ºF Find 102 ºF to 95 ºF Use the thermometer to calculate changes in temperature. Temperatures below zero are written as negative numbers. For example, 9 degrees below zero is written -9 ºC or -9 ºF. The thermometer at the right measures degrees Fahrenheit and degrees Celsius. 2 3 4 5 6 7 8 9 10 MXENL08AWK5X_RT_CH23_L01.indd 1 3. 1. RW148 4. 2. Find the perimeter of each polygon in centimeters. The string is about 13 centimeters long, so the perimeter of the polygon is about 13 centimeters. centimeters 1 Step 3 Lay the string in a straight line and measure its length with a centimeter ruler. Step 2 Cut the string where it meets itself. Step 1 Lay a piece of string around the figure. Find the perimeter of the polygon in centimeters. 11 12 13 The perimeter of a figure is the distance around the figure. You can use a piece of string to estimate the perimeter of a figure. Estimate and Measure Perimeter Temperature Degrees Fahrenheit (°F) are the customary units for measuring temperature. Degrees Celsius (°C) are the metric units for measuring temperature. Name Name MXENL08AWK5X_RT_CH22_L8.indd 1 RW147-RW148 7/19/07 6:03:13 PM 14 © Harcourt • Grade 5 Reteach 15 6/13/07 1:23:49 PM Grade5.indd 75 4. 1. 5 in. 7 in. 4 cm 4 in. 2 cm 3 in. 4 in. 5. 2. 7 ft 6 ft RW149 7 ft 4 ft 3 ft 5 ft 6. 3. 6/13/07 1:28:39 PM © Harcourt • Grade 5 Reteach 3m 7 cm So, the perimeter of the polygon is 24 ft. So, the perimeter of the polygon is 18 in. Find the perimeter of each polygon. 8 1 4 1 8 1 4 5 24 4 ft Each of the 3 sides are 6 in. long. Multiply. 6 3 3 5 18 6 in. 8 ft 6 ft 6 in. 8 ft 6 ft MXENL08AWK5X_RT_CH23_L03.indd 1 4. 1. 3 in. 5. 2. 3 yd 8 yd 5 yd RW150 4.5 m Find the perimeter of each polygon by using a formula. So, the perimeter of the square is 36 in. P 5 36 P5439 Perimeter (P) 5 (number of sides) 3 S So, the perimeter of the pentagon is 40 cm. P 5 40 P5538 Perimeter (P) 5 (number of sides) 3 S Find the perimeter of each polygon by using a formula. Since the sides of a regular polygon are equal, you can use a formula to find the perimeter of a regular polygon. Find the perimeter of each polygon. Since opposite sides of a parallelogram are equal, you can use addition to find the perimeter of a parallelogram. Algebra: Perimeter Formulas Find Perimeter Since the sides of a regular polygon are equal, you can use multiplication to find the perimeter of a regular polygon. Name Name MXENL08AWK5X_RT_CH23_L02.indd 1 RW149-RW150 7/19/07 6:03:31 PM 2 yd 6. 3 in. 5.7 cm 3. 3.6 cm 10 in. 3 in. © Harcourt • Grade 5 Reteach 5.7 cm 3.6 cm 10 in. 9 in. 8 cm 6/13/07 1:35:08 PM Grade5.indd 76 shape of a regular hexagon. The perimeter is 84 inches. What is the length of each side of the lid? 6. The lid of a designer box is in the Make generizations to solve. 5. How can you check your answer? Check RW151 6/6/07 10:45:15 AM © Harcourt • Grade 5 Reteach rectangular shaped pen in her backyard. The width of the pen is 15 feet. What is the length of the pen? 7. Jen has 96 feet of fencing to make a 4. What is the length of one side of the base of the other tissue box? 3. Which formula would you use to solve the problem? Solve 2. How does knowing that the two tissue boxes are congruent help you? MXENL08AWK5X_RT_CH23_L05.indd 1 4. a radius of 9 ft 1. a diameter of 4 m RW152 5. a diameter of 20 cm 2. a radius of 10 in. 1 in. 6. a radius of 2 yd © Harcourt • Grade 5 Reteach 3. a diameter of 10 in. To the nearest tenth, find the circumference of a circle that has So, the circumference of the circle is about 6.28 in. C = 6.28 C = 3.14 3 2 C=p3d Use the formula now that you have the length of the diameter Plan To the nearest hundredth, find the circumference of a circle that has a radius of 1 in. If you know the diameter, d, of the circle, you can use the formula C = p 3 d to find the circumference, C. The distance across a circle is called the diameter. Half the distance across a circle is called the radius. The distance around, or circumference, of a circle is p (about 3.14) mulitplied by the distance across the circle. Circumference Name The radius is half of the diameter so multiply the measurement at the radius by 2. 1 3 2 5 2 in. 1. What are you asked to find? Read to Understand Two tissue boxes are congruent cubes. If the perimeter of the base of one tissue box is 16 in., what is the length of one side of the base of the other tissue box? Problem Solving Workshop Skill: Make Generalizations Name MXENL08AWK5X_RT_CH23_L04.indd A RW151-RW152 6/18/07 6:09:02 PM 6/6/07 10:45:40 AM Grade5.indd 77 4 1 18 5 22 Add the values from Steps 2 and 3. Step 4 2. 5. 1. 4. RW153 6. 3. Estimate the area of the shaded figure. Each square on the grid is 1 cm2. So, the estimated area of the shaded figure is about 22 in.2 Total number of full squares: 18 Count the number of full squares, including those missing only a tiny corner. 84254 Divide the total by 2. Step 2 Step 3 Total number of partial squares: 8 Count the number of partial squares. Skip squares with only a tiny corner. Step 1 2 Each square on the grid is 1 in. Estimate the area of the shaded figure. Reteach 6/13/07 1:38:22 PM © Harcourt • Grade 5 12.4 km 9.6 km 8 cm MXENL08AWK5X_RT_CH24_L2.indd 1 1. 8 cm Find the area of each figure. 2. So, the area of the figure is 211.36 km2. 153.76 1 57.6 5 211.36 RW154 4 yd 10 yd 3. 5 in. © Harcourt • Grade 5 Reteach 12 in. 5 in. 5 in. 6 km Area of a polygon. To find the total area, add the area of the square to the area of the rectangle. A 5 9.6 3 6 or A 5 57.6 To find the area of the rectangle, use the formula A 5 1 3 w. Area of a rectangle. The other part of the figure is a 6 km-by-9.6 km rectangle. A 5 12.42 or A 5 153.76 To find the area of the square, use the formula A 5 S2. 12.4 km You can find the area of a square or rectangle by using formulas. Find the area of the figure at the right. The area of a figure is the number of square units needed to cover it. Area of a square. One part of the figure is a 12.4 km-by-12.4 km square. Algebra: Area of Squares and Rectangles Estimate Area You can use centimeter grid paper to estimate the area of a figure. Name Name MXENL08AWK5X_RT_CH24_L1.indd 1 RW153-RW154 6/18/07 6:09:14 PM 6/13/07 1:39:00 PM Grade5.indd 78 Make as many different rectangles as possible. Find the area of each. Complete a table showing possibilities. Stop when you realize you have found the greatest area. Divide the perimeter by 2. The length and width of a rectangle whose perimeter is 44 have a sum of 22. 1. 20 in. 2. 38 km RW155 3. 32 cm 4. 26 in. Area 21 40 57 72 85 96 105 112 117 120 121 120 117 6/6/07 1:45:38 PM © Harcourt • Grade 5 Reteach 5. 8 yd You can use factors of a given area to find the length and width of rectangles. Length Width Perimeter 1 21 44 2 20 44 3 19 44 4 18 44 5 17 44 6 16 44 7 15 44 8 14 44 9 13 44 10 12 44 11 11 44 12 10 44 13 9 44 For the given perimeter, find the length and width of the rectangle with the greatest area. Use whole numbers only. For a given perimeter, the square has the greatest area. So , the rectangle with the greatest area for a rectangle with a perimeter of 44 yards is 11 yd 3 11 yd. Step 2 Step 1 2 8 in. 10.5 in. MXENL08AWK5X_RT_CH24_L4.indd 1 1. 2. 7m Find the area of each triangle. So the area is 17.5 m . Find the area of the triangle. RW156 7 ft 5m __ ft 51 8 3. __ bh A51 2 __ 3 7 3 5 A51 2 A 5 17.5 If b is the base and h is the height, you can use the formula 1 bh to find the area, A, of a triangle. A 5 __ 2 How does the area of the triangle relate to the area of the rectangle? Repeat steps 1 through 4 with the same-size rectangle, but place the dot in Step 3 on a different side. • What does that say about the area of the triangle? Fold the rectangle along two sides of the triangle. • Does it cover the triangle without overlapping? © Harcourt • Grade 5 Reteach 5.6 m 7.6 m Place a dot on one side of the rectangle. Use a ruler to draw a line from the dot to each opposite corner. You have made a triangle. Step 3 Step 4 Cut out the rectangle. Step 2 Draw a rectangle on grid paper. Make each side greater than 10 units. Find and record the area of the rectangle. You can use grid paper to find a relationship between the areas of triangles and rectangles. You can use grid paper to relate perimeter and area. Step 1 Algebra: Area of Triangles Algebra: Relate Perimeter and Area For the perimeter 44 yards, find the length and width of the rectangle with the greatest area. Use whole numbers only. Name Name MXENL08AWK5X_RT_CH24_L3.indd 1 RW155-RW156 6/18/07 6:09:25 PM 6/13/07 1:43:06 PM Grade5.indd 79 1. 2. Find the area of each parallelogram. A 5 15 units2 A5335 A 5 bh So, the area of the parallelogram is: RW157 The formula for the area of a parallelogram is base 3 height Count the grid squares to find the area of the parallelogram. The base is 3 units and the height is 5 units. Step 4 Cut out the triangle on the bottom and move it to the top of the parallelogram to form a rectangle. Step 3 Draw a line segment at the bottom row to form a right triangle. Step 2 Draw a parallelogram on grid paper and cut it out. Step 1 Write the base and height of the parallelogram. Then find its area in square units. You can use grid paper and the base and height of a parallelogram to find its area. Algebra: Area of Parallelograms Name MXENL08AWK5X_RT_CH24_L5.indd 1 RW157-RW158 7/20/07 11:49:04 AM 3. 3 b h 5 Reteach 6/6/07 1:47:08 PM © Harcourt • Grade 5 MXENL08AWK5X_RT_CH24_L6.indd 1 2m 8m 5m 7m 6. Find the area of the figure. RW158 © Harcourt • Grade 5 Reteach long and one of its sides is 8 cm while its perimeter is 25 cm, then what is the length of its other 3 sides combined? 7. If the area of a rectangle is 36 cm Use the solve a simpler problem strategy to solve. 5. What is another way to solve this problem? Check sun catcher? 4. Multiply your answer from question 3 by 6. How many small squares are in Fran’s How many small squares are in this 1st row? 3. The first row of the sun catcher is shown. Solve 2. How can pictures help you break the problem into simpler parts? Plan 1. What are you asked to find? Read to Understand Fran’s sun catcher has 6 rows of 5 squares. Each square has 3 rows of 3 small squares. How many small squares are in Fran’s sun catcher? Problem Solving Workshop Strategy: Solve a Simpler Problem Name 7/20/07 11:44:26 AM Grade5.indd 80 1. RW159 2. Use the net to find the surface area of each prism in square units. So, the surface area of the square pyramid is 48 square units or 48 units2. Find the area of the square A5s3s A5434 A 5 16 Find the area of one triangle A 5 1_2 3 b 3 h A 5 1_2 3 4 3 4 A58 Add the areas of the square and the 4 triangles; 16 1 8 1 8 1 8 1 8 E C D Reteach 6/13/07 1:44:20 PM © Harcourt • Grade 5 Volume is measured in cubes, or in cubic units. Use the net to find the surface area of the figure in square units. B Area is measured in squares, or in square units. Another way to find the surface area is to use a net. 5 in 5 in MXENL08AWK5X_RT_CH24_L8.indd 1 1. 4 in 8 cm 2. RW160 10 cm 8 cm 3. Find the volume of each rectangular prism in cubic centimeters. So, the volume of the prism is 27 cubic feet, or 27 ft3. The formula for volume is Volume 5 length 3 width 3 height V 5 3ft 3 3ft 3 3ft V 5 27ft3 The volume will tell you how much space can fit inside the box, the cube, or any other figure. Find the volume of the rectangular prism. 3 ft Just as you used squares to find the area of a rectangle, you can use cubes to find the volume of a rectangular prism. You can find the surface area, the total area of the surface of a solid figure, by adding the area of each face. A Algebra: Estimate and Find Volume Surface Area Each face of a square is one square unit. Name Name MXENL08AWK5X_RT_CH24_L7.indd 1 RW159-RW160 6/18/07 6:11:10 PM 3.1 m 3 ft Reteach 4.8 m © Harcourt • Grade 5 2.5 m 3 ft 6/13/07 1:48:05 PM Grade5.indd 81 5 in2 ft 3 ft 3 ft 5 ft 4 ft 1. area of this parallelogram 5 cm RW161 3 cm triangular prism 2. volume of this Write the units you would use for measuring each. 7 cm Use cubic units or cm3, yd3, in3, mi3, m3, or km3. So, the unit for volume 5 ft3 3 5 3 ft 3 3 ft 3 7 ft 5 63 ft3 Volume 5 l 3 w 3 h Volume Use the unit or cm., ft, yd, mi, in, or k. So, unit for perimeter 5 m 5 5 m 1 5 m 1 5 m 1 5 m 1 5 m 1 5 m 5 30 m Perimeter 5 side 1 side 1 side1 side 1 side 1 side Perimeter Use unit square or cm2, ft2, yd2, mi2, m2, or km2. So, unit for area 5 in2 in 3 in 5 8 in 3 4 in 5 32 in2 Area A513w 4 in. 5m 3 ft 3 ft 7 ft 8 in pentagon 6/13/07 1:53:06 PM © Harcourt • Grade 5 Reteach 3. perimeter of this regular 8 in. Write the units you could use for measuring the area of this triangle Volume is the amount of space a solid figure takes up. Area is the space on a flat surface of a flat figure. MXENL08AWK5X_RT_CH24_L10.indd 1 Reteach © Harcourt • Grade 5 Its floor is a 10 ft-by-10 ft square. What is its height? 6. The volume of an elevator is 1200 ft3. RW162 pizza. What is the most number of pieces possible? [Hint: Pieces do not have to be equal in size or shape.] 5. Three straight slices are made across a Make a model or write an equation to solve. 4. How can you check that the missing length is 12 feet? Check Which is the better method? Explain. Another way to solve the problem is to Write an Equation and solve it. Since A 5 l 3 w, 156 5 13 3 w. a 13 3 1 rectangle. Then see if it has 156 squares. If it does not, draw a 13 3 2 and then a 13 3 3 rectangle. Continue until you have 156 squares inside the rectangle. 3. One way to solve is to use Make a Model. The length is 13 feet, so use grid paper to draw Solve 2. Name two different strategies you can use to solve this problem. Plan 1. What is the shape and size of the kitchen floor? Read to Understand Tashi is repairing tiles on a kitchen floor. The floor is in 2 the shape of a rectangle. The area of the floor is 156 ft . The length is 13 feet. What is the width of the floor? Problem Solving Workshop Strategy: Compare Strategies Relate Perimeter, Area, and Volume Perimeter is the distance around a flat figure. Name Name MXENL08AWK5X_RT_CH24_L9.indd 1 RW161-RW162 6/18/07 6:11:35 PM 6/6/07 1:50:52 PM