Effects of Changing Lengths 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 010 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. ISBN 978-1-6019-0-1 AIMS Education Foundation 19 S. Chestnut Ave., Fresno, CA 970 888.7.67 aimsedu.org Printed in the United States of America EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation
Effects of Changing Lengths BIG IDEA: Table of Contents Welcome to the AIMS Essential Math Series! The perimeter of scaled figures changes by a factor the same as the scale factor while the area changes by the square of the scale factor. Lesson One: Sim Growth 1 Investigation Sim Growth... 9 Students generalize the relationship of scale factor to perimeter and area by cutting out and measuring similar polygons. Comic Sim Growth... 1 Emphasizes that the perimeter grows by the same factor as a side, while area grows by a factor that is the length factor times itself. Animation Sim Growth... 17 A triangle changes incrementally against a background grid, providing a dynamic source of data. Lesson Two: Patio Pavers Investigation Patio Pavers... 19 Using interlocking cubes to simulate pavers, students investigate the changes in in relationship to different-sized patios. Practice extends understanding to scaled pictures. Comic Patio Pavers... Extends the idea that the factor of change (scale factor) in the length changes the perimeter by the same factor and the area by the scale factor times the scale factor, or the scale factor squared. Lesson Three: Copy Cat Investigation Copy Cat... Comparing enlarged and reduced pictures reinforces the concept of area growth and its relationship to scale factor. While seeing the effects of stretching in two dimensions, students are introduced to fractional scale factors. EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation
Comic Copy Cat... 7 Reinforces the idea of area growth as the square of the scale factor and clarifi es the idea of scale factor as a fraction. Lesson Four: Skee Ball Investigation Skee Ball... 9 Comparing the area of the different-sized circles of a Skee Ball game extends the generalizations of to all twodimensional shapes. Comic Skee Ball... 1 Looking at a Skee Ball game provides an application for the growth of area by relating it to the reduction in points. Lesson Five: Grid Growth 6 Investigation Grid Growth... Scaled images made on a coordinate grid provide the opportunity to see that area grows as a square of the scale factor even on irregular shapes. Animation Zooms... 6 Dynamically stretching an image allows students to play with scaled enlargements and distortions. BIG IDEA: Comic Grid Growth... 7 Presents the new concepts needed for this activity: coordinate graphing, representing scale factor as a percent, enlarging a picture by multiplying the coordinates by the scale factor. Practice... 9 Problem-solving situations allow the opportunity to practice and apply what has been learned. The surface area of scaled solids grows by the square of the scale factor while the volume changes by the cube of the scale factor. Lesson Six: Bigger Boxes 7 Investigation Bigger Boxes... 1 Building scaled boxes out of cubes extends the growth relationship to the surface area and volume of three-dimensional solids. EFFECTS OF CHANGING LENGTHS 6 010 AIMS Education Foundation
Comic Bigger Boxes... Reviews how a change in the scale factor relates to growth for a three-dimensional solid. Perimeter increases by the scale factor. Surface area (two-dimensional) grows by the square of the scale factor. Volume (three-dimensional) grows by the cube of the scale factor. Animation Pyramid Comparison... 6 The dimensions of length, surface area, and volume of two scaled pyramids are physically compared. Lesson Seven: Growing Skins and Guts 8 9 Investigation Growing Skins and Guts... 7 Comparing the skin (surface area) and guts (volume and mass) of scaled fi gures made out of cubes extends the concepts of growth to irregular shapes. Video How Much Bigger?... 6 Students compare scale models of a mint box and a soda can to see that the relationships of dimensions extends to all three-dimensional shapes. Comic Growing Skins and Guts... 7 Reviews the growth concepts applied in the investigation while validating the reality of the simulation to growth or size comparisons of humans. Practice: Selling Growth... 9 Problem-solving situations allow the opportunity to practice and apply what has been learned. 10 Assessment... 6 Two assessments provide real-world contexts for applying the effects of changing lengths. Glossary... 67 National Standards and Materials... 69 The Story of Effects of Changing Lengths... 71 Using Comics to Teach Math... 7 Using Animations to Teach Math... 77 The AIMS Model of Learning... 79 EFFECTS OF CHANGING LENGTHS 7 010 AIMS Education Foundation
SimGrowth How do the length and area Students compare the change in length and area of scaled sets of similar polygons to the polygon. The generalization that lengths grow by the same factor as the scale and the area grows as a square of the scale factor is evident. tig Investigation The focus of this investigation is growth, but it provides a natural way to review similarity. SimGrowth Paralgram Growth Use your group's four paralgrams to make comparisons. Two times Three times Materials Markers Scissors The group assigns each student the scale factor polygon they are to make. EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation Four times Five times Draw and cut out paralgrams with sides two, three, four, and five times longer then the. Divide students into groups of four and give each student scissors, marker, and student page. Original Paralgram SimGrowth EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation Draw and cut out triangles with sides two, three, four, and five times longer then the. Original Triangle Students should use the marker to plan the outline of the polygon they are making before they cut it out. SimGrowth SimGrowth How many times longer is each side than the paralgram? Triangle Growth Use your group's four triangles to make comparisons. How many times higher is this paralgram than the paralgram? How many paralgrams does it take to cover this paralgram? change in height compare to the change in length? What relationship is there between the length and height growth and the area growth? 9 16 They are all the same. If you double one length, you double all lengths. All lengths grow at the same factor as the scale and the area grows as a square of the scale factor. this should be a piece of cake! before students write their summaries, Discuss the two concluding questions to check that they recognize the big ideas. How many times longer is each side than the triangle? How many times higher is this triangle than the triangle? How many triangles does it take to cover this triangle? Two times Three times 9 Four times 16 Five times group members share their polygons and collaboratively complete the record pages. EFFECTS OF CHANGING LENGTHS Comics Animation 010 AIMS Education Foundation change in height compare to the change in length? They are all the same. If you double one length, you double all lengths. What relationship is there between the length and height growth and the area growth? All lengths grow at the same factor as the scale and the area grows as a square of the scale factor. no problemo! Emphasize that the perimeter grows by the same factor as a side, while area grows at a factor that is the length factor times itself. EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation A triangle changes incrementally against a background grid, providing a dynamic source of data. EFFECTS OF CHANGING LENGTHS 9 010 AIMS Education Foundation
SimGrowth Paralgram Growth Use your group's four paralgrams to make comparisons. Two times Three times Four times Five times How many times longer is each side than the paralgram? How many times higher is this paralgram than the paralgram? How many paralgrams does it take to cover this paralgram? change in height compare to the change in length? this should be a piece of cake! What relationship is there between the length and height growth and the area growth? EFFECTS OF CHANGING LENGTHS 10 010 AIMS Education Foundation
Draw and cut out paralgrams with sides two, three, four, and five times longer than the. SimGrowth Original Paralgram EFFECTS OF CHANGING LENGTHS 11 010 AIMS Education Foundation
SimGrowth Triangle Growth Use your group's four triangles to make comparisons. Two times Three times Four times Five times How many times longer is each side than the triangle? How many times higher is this triangle than the triangle? How many triangles does it take to cover this triangle? change in height compare to the change in length? no problemo! What relationship is there between the length and height growth and the area growth? EFFECTS OF CHANGING LENGTHS 1 010 AIMS Education Foundation
Draw and cut out triangles with sides two, three, four, and five times longer than the. Original Triangle SimGrowth EFFECTS OF CHANGING LENGTHS 1 010 AIMS Education Foundation
KEEP GOING KEEP GOING
EFFECTS OF CHANGING LENGTHS 1 010 AIMS Education Foundation Ori g g ra m Ori g g ra m how does the perimeter of that parallelogram compare to the? well, the sides of the one you just drew are each double the. l l so, when you add up the lengths of the sides to find the perimeter, that turns out to be double, too. well, the length of each side is doubled. l l l the sides of the next bigger ones we did were each times as long. for the next one they were times as long, and for the biggest one, the sides were times as long as the. Ori gi gram I gave you a sheet that looked like this, right? you drew several different parallelograms around the parallelogram, each of which was an enlargement of the. then you cut them out. I ve drawn one of them here. 1 wait, how do we know the perimeter is double? we don t know the lengths of the sides! so, how could you add them up? let s look at the paralgram that I drew that has sides that are times as long as the sides of the. how did the sides of the enlarged parallelograms that you cut out compare to the?. to enlarge a triangle or paralgram, each side of the shape is multiplied by the same number. what is the name for that number?. how does the perimeter change when a paralgram is enlarged by multiplying each side by? how does the area change? this activity is about what happens to a parallelogram or a triangle as we enlarge it. Or ig g le n Tr ia ESSENTIAL MATH SERIES 1. how is a shape like a paralgram or a triangle enlarged? THINGS TO LOOK FOR: Sim Growth EFFECTS OF CHANGING LENGTHS P = 10 + 1 + 10 + 1 = 0 al Or igin ram log Paralle when you multiply by, the perimeter is times as big. okay, class, what happens to the perimeter if the parallelogram is enlarged so the sides are times as long? well, double is, and double is 6. mark, let s see if knowing the lengths of the sides of the might help you. 6 nal Ori gi g ra m llel o I get it. so, if you multiply each side by or or, then the perimeter will be that many times bigger. it s almost the same as when you double it, only this time the perimeter is times as big. P=+6++6 = 0 that means two sides are and two are 6. when you add them up, you get 0. i nal Or i g g ra m lo le l what if for the parallelogram we say the length of the base is and the side is? remember that the opposite sides of a parallelogram are the same length as well. P=6+9+6+9 = 0 Ori gi gra m okay, mark, is this perimeter double the perimeter of the? we know that if you multiply the length of each side by a number, then the perimeter is multiplied by that number, too. P = 8 + 1 + 8 + 1 = 0 g and, when you multiply each side by, the perimeter is times as big. yeah, I see that now. if you double the lengths of the sides, the length of the perimeter gets doubled, too. Ori g g ra m if we double the lengths of the sides of the paralgram, what are the lengths of the sides of the enlargement? but it doesn t matter what the lengths of the sides of the shape are, right? P=+++ = 10 then, like you said, elora, we can find the perimeter by adding up the lengths of all the sides.
1 0 0 1 11 11 10 10 9 8 7 6 1 0 0 number of pavers change as the dimensions of a patio grow? Using flats of blocks to represent patio pavers, students determine the amount of curbing required to surround a patio (perimeter) and the number of pavers required to cover the patio (area). By building patios of different dimensions, students recognize that as the curbing (perimeter) increases by a factor, the number of pavers (area) increases by the square of that factor. tig Investigation Materials Hex-a-Link Cubes (Item #19) Divide students into groups of four and give each group 100 cubes. number of pavers change as the dimensions of a patio grow? Pavers are flats of cement formed to look like an arrangement of bricks. Covering a patio with pavers takes less time because there are fewer pieces to lay. Make each size of patio by covering the area with two-by-three flats. Use patios to complete the table. Complete this chart by using two-by-three pavers: Patio Size (units) x x 6 6 x 9 8 x 1 when the pavers are laid out, curbing is used to surround the patio and keep the pavers in place. Curbing Needed (units) Patio Area (sq. units) PRACTICE BY MAKING PATIOS OF DIFFERENT SIZES AND SEE IF YOU CAN FIND AN EASY WAY TO DETERMINE HOW MUCH CURBING AND HOW MANY PAVERS NEED TO BE BOUGHT. 1 6 7 x Pavers Required 10 0 6 1 0 9 0 96 16 Each group assembles the cubes into twoby-three flats that represent pavers. 1. How many times longer is each patio than a single paver? x 6, times; 6 x 9, times; 8 x 1, times x, times; 6 x 6, times; 8 x 8, times x, times; 6 x 6, 9 times; 8 x 8, 16 times. How many times more curbing is needed for each patio than a single paver?. How many times more area does each patio have than a single paver?. For each patio, what is the relationship of the area number in question three to the curbing number in question two? The growth in area is the growth in perimeter squared. 6 times 6 times 6, or 6 squared, or 6 6 x 6 = 1 x 18 = 16 square units. How many times longer is a 1 x 18 patio than a single paver? 6. Since a 1 x 18 patio is six times longer than a paver, how many pavers will you need to cover this patio? 7. If a single paver covers 6 square units, how much does a 1 x 18 unit patio cover? 8. If you know how many times longer a patio is than a paver, how do you determine how many times larger the area of the patio is? (growth factor of length) = area EFFECTS OF CHANGING LENGTHS 0 010 AIMS Education Foundation EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation 1 6 7 8 Comics Extend the idea that the factor of change (scale factor) in the length changes the perimeter by the same factor and the area by the scale factor times the scale factor, or the scale factor squared. EFFECTS OF CHANGING LENGTHS 19 010 AIMS Education Foundation
EFFECTS OF CHANGING LENGTHS 1 010 AIMS Education Foundation okay, I guess that makes sense, but what about the word factor? what does that mean? That s right. if the sides of the enlargement are twice as long as the, then the perimeter will be twice as long. now, class, what do you think happens to the area when the parallelogram is enlarged? very good, redmond. in this activity we ve used,,, and as scale factors. a factor is something you multiply, red. sometimes we call a toy car like that a scale model of the real car. well, let s see if we can make sense of that. I m sure you ve seen some toy cars that looked almost exactly like real ones. mr. david, is there a name for the number that you multiply times the lengths of the sides. = 6 like times is 6, right? Two and are called factors, those are the numbers you multiplied. that means it s the same shape and it has the same details, but it s a different size. yes, it has a name. it s called the scale factor. P Ori g g ra m e lo a ra ll when the scale factor is and we double the lengths of the sides of the paralgram, what happens to the area? oh, okay. I think I get it. so, it s called a scale factor because it s a number that is multiplied times each side of the parallelogram to make it bigger. so, scale is really about how much bigger or smaller one thing is compared to another thing. scale factor??? how come it s called that? P e lo a ra ll yeah, it s like you have of those parallelograms in a row and you have rows of them. when the scale factor is, each side of the parallelogram is multiplied by. what happens to the area? P e lo a ra ll look, if you just doubled the parallelogram in one direction, it would double the area. How many times greater is the area than the? 9 16 How many times greater is the perimeter than the? Each side of the is multiplied by that s right, mark. class, you can see that that s what we recorded in the bottom row of the chart. and then if you triple the height, it s like you triple the triple. that s times. so, the area is 9 times as much as the. the area is times as much. g g g g g g g g g g g g class, I have another question. when the scale factor is, you multiply the sides by and you multiply the area by times, right? g g g g when the scale factor is, you multiply the sides by. then you have parallelograms in a row times rows. that s 16 times as big. Pa r a l sides by area by = =9 = 16 = MULTIPLY in fact, when you multiply the sides by a number, you multiply the area by that number times itself. l l l l l l l l l l l l l l l l l l l l l l l l And when the lengths of the sides of the are multiplied by, the area is times equals times as big. if you triple the width, you triple the area. you have a row of three parallelograms. so, the enlargement is times or times as much as the. instead of doubling, you re tripling in each direction. but, when you double the in two directions, it s like you double the double.
Watch the entire video. Pause at this frame: SimGrowth Animation area of triangle change as the length of the base grows? How many times longer is the base than the unit (1) triangle s base? How many unit triangles does it take to cover this larger triangle? What is the relationship of the bases's growth to the area s growth? Continue playing the animation. Pause at this frame: How many times longer is the base than the unit (1) triangles s base? How many unit triangles does it take to cover this larger triangle? What is the relationship of the bases growth to the area s growth? Continue playing the animation. Pause at this frame: What fraction is the base of this triangle of the unit triangle s base? What fraction is the area of this triangle of the unit triangle s base? What is the relationship of the base s growth to the area s growth? Explain how the scale of change in a triangle's area relates to the change in the length of the base. EFFECTS OF CHANGING LENGTHS 17 010 AIMS Education Foundation
EFFECTS OF CHANGING LENGTHS 010 AIMS Education Foundation