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FIGURE IT out Series FIGURE IT OUT Book 4 Thinking Like a Math Problem Solver Sandra R. Cohen CURRICULUM ASSOCIATES®, Inc. Table of Contents Page For the Student...............................................................1 Act It Out........................................................................2 Guess and Check ............................................................4 Draw a Picture................................................................6 Make a List......................................................................8 Mixed Practice ..............................................................10 Look for a Pattern .........................................................12 Solve a Simpler Problem ................................................14 Work Backward.............................................................16 Use Logical Reasoning ..................................................18 Mixed Practice ..............................................................20 Review: Act It Out.........................................................22 Review: Guess and Check .............................................22 Review: Draw a Picture .................................................23 Review: Make a List.......................................................24 Review: Look for a Pattern ............................................25 Review: Solve a Simpler Problem ...................................26 Review: Work Backward................................................26 Review: Use Logical Reasoning .....................................27 Final Review .................................................................28 ISBN 0-7609-2374-4 ©2003, 1995—Curriculum Associates, Inc. North Billerica, MA 01862 No part of this book may be reproduced by any means without written permission from the publisher. All Rights Reserved. Printed in USA. 15 14 13 12 11 10 987654321 For the Student Figure It Out is a booklet that teaches problem-solving skills. In each lesson you will learn a strategy that you can use to solve problems. Your teacher will ask you questions to help you use the strategy to solve the first problem in each lesson. Your teacher will also guide you through the second problem in each lesson, giving you some things to think about and ways to help you find the solution. The last two problems in each lesson are for you to do on your own. These problems give you practice using the strategy you have just learned. By the time you have completed this booklet, you will have learned eight strategies to use when solving problems. The strategies will be useful in school and in your everyday life. Hopefully, you will find that calculators and computers are useful for computation, but human beings are needed to solve problems. When You Use This Booklet • Read each problem carefully before you begin to solve it. • Think about the questions that follow the first two problems in each lesson. They will help you to understand the problems and find the solutions. • Use the blank space on the page to work through problems. You can write anywhere in this booklet if it will help you to solve a problem. • Once you have solved a problem, check your solution to be sure it makes sense. Some problems have more than one correct answer. • Write the solution to a problem on the line that follows the problem. This Figure It Out booklet was prepared for students by Sandra R. Cohen 1 Act It Out 1. Kate, Pete, Joan, Amos, and Paul are the 5 finalists in the Main Street School Spelling Bee. Before the final round of the bee, each student must shake hands with every other student. What is the total number of handshakes? Questions a. Estimate the answer. __________ How did you make your estimate? _________________________________________________________________ b. When 2 students shake hands with each other, how many handshakes is that? __________________________________________________________ c. Suppose Kate is shaking hands with Pete. Could you also say that Pete is shaking hands with Kate? ____________________________________ d. If 1 student shakes hands with 4 others, how many handshakes is that? _____________________________________________________________ Apply the Strategy Work with 4 other students to act out the problem. Pretend you are the finalists. Each person should shake hands with the others. Record the number of times you shake hands. Solution____________________________________________________________ e. How can you be sure you recorded all the possible handshakes? _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 2 2. Which is worth more: 30 centimeters of quarters placed side by side or 60 centimeters of dimes placed side by side? About how much more? RT RT RT BE Y BE Y E Y I I IB L L S O S O S O L TE F A TE F A TE F A TA M TA M TA M S E S E S E D R D R D R E I E I E I C C C T T T A A I A I I N N E E N IN GOD W IN GOD W GOD WE NUM NUM NUM U U IN U US U US U US U TRUST TRUST RUST RIB RIB RIB T E PLU E PLU E PLU 1995 1995 95 O E O E O E 19 N E D I M N E D I M N E D I M | |||||||||||||||||||| 0 1234567891011121314151617181920 Centimeters Think about: • Which do you think is worth more? Why? • Which coin is larger? • About how much longer is the row of dimes than the row of quarters? • About how many quarters will measure 10 centimeters? • About how many dimes will measure 10 centimeters? Work with other students. Use real or play money. Act out the problem to find the solution. If you use play money, it should be the same size as real money. Solution ____________________________________________ On Your Own Work with other students. Act out the problems to solve them. 3. Miguel has 3 times as many video games as Paula. Both children have an even number of games. Together they have between 20 and 30 games. How many games does each child have? ____________________________________________________ 4. There were 12 boys who lived near the Bethpage Golf Course. As a team, the boys found and sold golf balls. Each odd- numbered boy found the same number of balls as his number. Each even-numbered boy sold 1 ball. So, Boy 1 found 1 ball, Boy 3 found 3 balls, Boy 5 found 5 balls, and so on. Boys 2, 4, 6, 8, 10, and 12 each sold 1 ball. When the 12 boys were finished, how many golf balls were not sold? ____________________________________________________ 3 Book TEACHER GUIDE 4 FIGURE IT OUT Thinking Like a Math Problem Solver Sandra R. Cohen CURRICULUM ASSOCIATES®, Inc. Introduction The Teacher Guide Problem solving is the process of applying acquired The teacher guide consists of procedures for teaching knowledge to different situations. It is the basic skill of the strategy lessons and guidance for presenting the mathematics and an integral part of the mathematics Mixed Practice and Review pages. The teacher curriculum at all levels of instruction. guide also contains two blackline masters that can be duplicated and distributed to students for use in solving Figure It Out is a series of booklets designed to teach the problems. strategies for solving mathematical problems. As students work through a booklet, they learn to read Using the Teacher Guide problems carefully, to think about the content of Suggestions for instruction are provided throughout the problems, and to use what they know about numbers teacher guide. These include questions to ask students, and mathematics to decide how to find solutions. teaching tips, and diagrams and tables for student use. The teacher guide also provides answers to questions The problems included in each booklet are considered and solutions to problems posed in the student book. to be non-routine problems. Their scope extends beyond Some problems have more than one correct answer. All that of routine problems, or those which students can possible answers are listed. It is recommended that the solve by merely reading and identifying the necessary teacher read through the teaching notes for each mathematical operation. Each problem in Figure It lesson before presenting the lesson to students. Out has some unique quality that requires students to think carefully about how to solve it. Many are problems Teaching Strategy Lessons that students can relate to real life. Though the first problem in each strategy lesson is intended to be primarily teacher-directed, and the The most exciting aspect of teaching mathematics is second is meant to be more self-guided, the teacher the discoveries students make as they work through can approach the two problems in a similar manner. problems. Guide them with questions, encourage the The teacher should read the problems aloud or have a use of manipulatives, and be sure to give students time student volunteer read the problems. He or she should and space to discover. also read each of the Questions and Think about The Student Book questions aloud and lead the class in a discussion about The student book consists of lessons that teach eight the students’ answers. During questioning, the teacher different strategies that can be used to solve nonroutine should encourage students to explain how they arrived problems. Each lesson opens with a problem followed at their answers. Explanations should be requested for by Questions designed to help students think about correct and incorrect student responses. From students’ the problem and how to solve it. After the Questions, answers, the teacher will see the wide variety
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