Proportional Reasoning Developed and Published by AIMS Education Foundation This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating Mathematics and Science) began in 1981 with a grant from the National Science Foundation. The non-profit AIMS Education Foundation publishes hands-on instructional materials that build conceptual understanding. The foundation also sponsors a national program of professional development through which educators may gain expertise in teaching math and science. Copyright 2000, 2003, 2009 by the AIMS Education Foundation All rights reserved. No part of this book or associated digital media may be reproduced or transmitted in any form or by any means except as noted below. A person purchasing this AIMS publication is hereby granted permission to make unlimited copies of any portion of it (or the files on the accompanying disc), provided these copies will be used only in his or her own classroom. Sharing the materials or making copies for additional classrooms or schools or for other individuals is a violation of AIMS copyright. For a workshop or conference session, presenters may make one copy of any portion of a purchased activity for each participant, with a limit of five activities or up to one-third of a book, whichever is less. All copies must bear the AIMS Education Foundation copyright information. Modifications to AIMS pages (e.g., separating page elements for use on an interactive white board) are permitted only for use within the classroom for which the pages were purchased, or by presenters at conferences or workshops. Interactive white board files may not be uploaded to any third-party website or otherwise distributed. AIMS artwork and content may not be used on non-aims materials. Digital distribution rights may be purchased for users who wish to place AIMS materials on secure servers for school- or district-wide use. Contact us or visit the AIMS website for complete details. AIMS Education Foundation 1595 S. Chestnut Ave., Fresno, CA 93702-4706 888.733.2467 aimsedu.org ISBN 978-1-881431-78-7 Printed in the United States of America TM PROPORTIONAL REASONING 2009 AIMS EDUCATION FOUNDATION
Proportional Reasoning Table of Contents Chinese Proverb... 3 Section Introductions... 5 National Documents... 7 Proportional Reasoning... 9 Helping Students Gain Understanding Through Investigations... 19 Foundations Fraction Equivalence with Pattern Blocks...29 Proportional Practice...33 Paper Clip Chains...57 Percent Bands... 61 Making Cents of Percents...71 Fraction Dominoes/Decimal and Percent Dominoes...77 Patterns in Equivalent Fractions...81 Recognizing and Building Proportional Relationships...85 The Parade of Triplets...91 Similarity Rectangle Ratios...95 The Shadow Knows...101 Mirror Ricochet...105 Enlargement Rubber Band Enlargements...109 Picturing Projections... 113 Overhead Enlargements... 117 Shrink Art...123 Growing Designs...129 Doin Dots...135 Proportional Reasoning in the Middle School...141 Scaling Playing at Math...145 Toy Soldiers Take the Court...151 Measuring Models...155 Making Movie Props...159 I Spy...163 Picturing Proportions...167 Rates Pulse Rates...177 Reading Rates...183 Time Trials...189 Making Good Lessons Better...193 Drop It!...199 Mixing Measures... 205 Human Proportion Head Hunters... 209 Tailor Made...215 Is Anyone Normal?...219 Are You Ideal?... 225 This Is So Typical... 233 AIMS Activities for Proportional Reasoning...245 PROPORTIONAL REASONING 1 2009 AIMS EDUCATION FOUNDATION
Topic Scaling Key Questions How much will this plastic shrink when it is heated? Will it shrink the same amount in all di rec tions? Focus Students will explore the characteristics of poly sty rene plas tic as it is heated. They will draw pictures on the plastic be fore heating it. After heating the plastic they will com pare the sizes of the pictures before and after to see if the shrinking was consistent in all di rec tions. Guiding Documents Project 2061 Benchmarks Mathematics is helpful in almost every kind of hu man endeavor from laying bricks to pre scrib ing medicine or drawing a face. In particular, mathematics has contributed to progress in sci ence and tech nol o gy for thou sands of years and still continues to do so. The global environment is af fect ed by national pol i cies and practices relating to energy use, waste dis pos al, ecological management, manufacturing, and population. NRC Standards Use appropriate tools and tech niques to gath er, an a lyze, and interpret data. Develop descriptions, ex pla na tions, pre dic tions, and models using ev i dence. Use mathematics in all aspects of sci en tific in qui ry. NCTM Standards 2000* Understand and use ratios and proportions to represent quantitative relationships Solve problems involving scale factors, using ratio and proportion Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects Model and solve contextualized problems using various representations, such as graphs, tables, and equations Math Measurement Proportional reasoning scaling percent Geometry similarity Science Environmental science recycling Integrated Processes Observing Collecting and recording data Interpreting data Applying data Materials Clear polystyrene (resin code 6) Permanent colored markers Colored pencils Fine sandpaper Rulers Scissors Single hole punch Oven or toaster oven Cookie sheet Hot pads Background Information Plastics, also called resins, have different prop er ties. There are six resins which are used most com mon ly in pack ag ing. These six resins have dif fer ent per for mance char ac ter is tics, so handlers of re cy cled plas tics are chal lenged to avoid the con tam i na tion that is the re sult of resin types mix ing. In 1988, The Society of the Plas tics Industry, Inc. in tro duced a res in cod ing system to be placed on the bot tom of con tain ers to help facilitate the sep a ra tion and pro cess ing of plas tics. The six pack ag ing resins were each as signed a num ber 1-6. The num ber is found inside a tri an gle of chas ing arrows. The type of plastic used in this activity is poly sty rene. Its resin code is 6 often with the resin ab bre vi a tion, PS, be neath the triangle. Poly sty rene is a clear, rig id, and brittle plas tic. It is used to make cups, tableware, food packages, and even furniture. When air is add ed, mak ing poly sty rene foam, it can be used for hot bev er age cups and pack ag ing pea nuts. One of the characteristics of polystyrene is that it has memory. Memory means that when it is heated it will re form back into its original size and shape. Flat polystyrene will shrink consistently when heat ed. This makes it an interesting material on which to draw pic tures and watch them shrink as they are heated. While the length and width of the plastic is reduced, the thick ness increases. PROPORTIONAL REASONING 123 2009 AIMS EDUCATION FOUNDATION
Management 1. Gather clear polystyrene. The clear disposable food containers at take-out restaurants (salads) and gro cery stores (baked goods) are often poly sty rene. Well be fore the activity have students start gathering ap pro pri ate containers. Have them check that the plas tics they gath er have a resin code of 6. The con tain ers can also be pur chased at a restaurant supply store. 2. This activity works well if it is broken up into two pe ri ods. One to draw and make the shrink art, the sec ond to de ter mine if it has shrunk proportionally. 3. Two alternatives for student interpretation are provided. The first has stu dents de ter mine if the scale factor is con sis tent for all di men sions and provides an ap pli ca tion for proportional reasoning. The second alternative has students com pare many corresponding dimensions and interpret their relationship graphically to en cour age al ge bra ic think ing along with pro por tion al reasoning. Be fore beginning the activity determine which al ter na tive is most ap pro pri ate for the class. 4. Before beginning the activity, the teacher needs to try out the materials and become familiar with the pro ce dure. If a toaster oven is being used, check that it will work; some ovens do not have enough circulation to evenly heat the plastic. A heat gun may be used in place of an oven. 5. Because the thickness of the plastic increases as it melts, it is most easily cut and holes punched before it is heat ed. The brittle nature of the plastic re quires that sharp scis sors be used in cutting. To pre vent sharp edg es and cor ners, the scissors should be held per pen dic u lar to the plastic when cutting it out. 6. The colored designs will adhere to the plastic bet ter if the plastic is roughed up with fine sand pa per be fore trac ing. The designs should be shad ed in. The colors will become more intense as the plas tic shrinks. 7. When the plastic pieces are initially put into the 350 oven, they will curl in upon themselves and look like they are ruined. However, when thor ough ly heated, they will be shrunk and lying flat. They should then be re moved from the oven and allowed to cool on a flat sur face. 8. Although designed as a math investigation, this ac tiv i ty makes a nice craft project for making hol i day or na ments or key chains. If the shapes are cut out before heating they can be strung or hung as jew el ry. 9. Students should be familiar with finding percents from ratios before beginning this activity. Procedure 1. Show the students a teacher-made sample of shrink art. Have them discuss how big they think the sam ple was before it was heated. Discuss the Key Ques tions. 2. Have the students cut out a flat rectangular piece of clear polystyrene from what they collected. 3. Using the first student sheet, tell students to draw a de sign they want to go on their plas tic within the out line provided. 4. Direct the students to transfer the design onto the plas tic by placing the plastic on top of the outline and trac ing the design with permanent markers or colored pen cils. The design should be shaded in. Emphasize that the mea sure ment dots need to be traced onto the plas tic also. If the art is to be hung, punch a hole in the plas tic be fore heating it. 5. After placing the plastic on a cookie sheet, put it in a 350 oven for several minutes. When the plastic piec es are flat, take them out and al low time for them to cool. Alternative 1 1. Using their orig i nal drawing and the shrink art, di rect stu dents to measure and record the be fore and af ter di men sions to the nearest millimeter on the second stu dent sheet. In struct them to use the measuring dots as points of ref er ence for find ing the width, height, and diagonal lengths. 2. Have students determine the scale factor as ratio, decimal equiv a lent, and per cent using their mea sure ments. 3. Direct stu dents to find the av er age per cent scale fac tor from their three sets of data. 4. Have students compare their averages with oth er stu dents and dis cuss how they could make shrink art of a spe cific size us ing their data. Alternative 2 1. Using their original drawing and the shrink art, have the students choose a variety of di men sions in their art. Di rect them to measure and record the cor re spond ing di men sions of both the drawing and the shrink art on the third student sheet. 2. Have students fi nd the decimal ratio de scrib ing the re la tion ship of the shrink art to the draw ing for all the dimensions. 3. Direct the students to make a scatter plot of the corresponding dimensions to see the relationship of the shrink art to the drawing. 4. Discuss how the line formed by the data relates to the relationship of the shrink art dimensions and the orig i nal art. 5. Using the data and/or the graph have the students devel op an equa tion that describes the relationship of the shrink art dimensions and the original art s di men sions. Discussion 1. How much did your plastic piece shrink? 2. How close did it come to shrinking the same amount in all directions? 3. What about the ratios communicates that the plastic shrinks con sis tent ly? [all ratios are very close to the same] 4. What about the graph communicates that the plastic shrinks con sis tent ly? [forms a line] 5. Did everyone s art shrink the same amount? Explain. 6. How could you predict the final size of art on a new piece of plas tic? PROPORTIONAL REASONING 124 2009 AIMS EDUCATION FOUNDATION
7. How can you use the scale factor or ratio to predict the final size of art on a new piece of plastic before it is heated? 8. How can you use the graph to predict the final size of art on a new piece of plastic before it is heated? 9. Write an equation that can be used to predict the final size of art on a new piece of plastic before it is heated? 10. How could this shrinking plastic be used? Extensions 1. Have students do a data analysis of the class s av er age scale factors to see if they can summarize the data. They might find mean, median, mode, and range to check the consistency of change. 2. Have students make a design the size they would like their art to be in final form. Have them use their scale factor to determine how big to make the de sign so it will be the desired size when shrunk. 3. Have students shrink different objects made of poly sty - rene (cups, bowls, flatware) to see them re mem ber the shape of the original form. * Reprinted with permission from Principles and Standards for School Mathematics, 2000 by the National Council of Teachers of Mathematics. All rights reserved. PROPORTIONAL REASONING 125 2009 AIMS EDUCATION FOUNDATION
Make a Drawing a. Draw a design or picture that will fill the frame below. b. Use colored pencils to shade in the draw ing the way you want it. Prepare a Plastic Sheet a. Cut out a square of polystyrene to cover the frame. b. Rough up one side of the plastic with sandpaper until it is frosted. Transfer the Drawing a. Place the plastic, frosted side up, on top of your design in the frame. b. With colored pencils trace your design onto the plastic. c. Be sure to trace the four black dots onto the plastic. Shrinking the Plastic a. Preheat the oven to 350 F. b. Place the plastic on a cookie sheet. c. Put the cookie sheet with the plas tic in the oven. d. Watch it. The plastic will curl up and then flatten out. If a corner should get stuck on itself, you may need to re move it from the oven and pull the corner away from the plastic. Return it to the oven. e. When the plastic has been flat for 30 seconds, remove it from the oven and cool it on a flat surface. PROPORTIONAL REASONING 126 2009 AIMS EDUCATION FOUNDATION
Using the measuring marks on your original draw ing and shrink art, mea sure the three di men sions below to the nearest mil li me ter. Then de ter mine the scale factor for each di men sion. Width Original Measurement: mm Shrink Art Measurement: mm Scale Factor Ratio Decimal Percent Equivalent Original Measurement: mm Shrink Art Measurement: mm Height Diagonal Scale Factor Ratio Decimal Percent Equivalent Original Measurement: mm Shrink Art Measurement: mm Scale Factor Ratio Decimal Percent Equivalent What was the average scale factor for your art? PROPORTIONAL REASONING 127 2009 AIMS EDUCATION FOUNDATION
Dimension Description Original (mm) Shrunk (mm) Ratio: shrunk original 100 Average Ratio: 90 80 Shrunken Dimension (mm) 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Original Dimension (mm) PROPORTIONAL REASONING 128 2009 AIMS EDUCATION FOUNDATION