INTERACTIVE B O O K WRITING AND SIMPLIFYING EXPRESSIONS 57 011 AIMS Education Foundation
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 01 by the AIMS Education Foundation Solving Equations by Working Backwards 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 978-1-60519-071-6 Printed in the United States of America AIMS Education Foundation 1595 S. Chestnut Ave., Fresno, CA 9370 888.73467 aimsedu.org WORKING BACKWARDS 3 01 AIMS Education Foundation
Solving Equations by Working Backwards Table of Contents Table of Contents Welcome to the AIMS Essential Math Series! BIG IDEA: Equivalent expressions can be made by combining like terms and applying the commutative and distributive properties. Lesson Two: Loading Dock : Multiplying and Dividing Lesson One: Loading Dock : Adding and Subtracting Investigation Loading Dock : Adding and Subtracting...9 Keeping inventory of pallets, packs, and units on a loading dock reinforces the understanding of the commutative property and combining like terms. Comics Loading Dock : Adding and Subtracting...13 Clarify the process of combining like terms and the meanings of term, coefficient, and variable. Investigation Loading Dock : Multiplying and Dividing...19 Keeping inventory of pallets, packs, and units as multiple loads are shipped and received reinforces understanding of the distributive property in a meaningful way. Practice Loading Dock : Multiplying and Dividing...5 Comics Loading Dock : Multiplying and Dividing...7 Relate the simplifying of expressions using the distributive property to multiplication with numbers. WORKING BACKWARDS 5 01 AIMS Education Foundation
BIG IDEA: Equations can be thought of as a coded description of a series of actions leading to an outcome. Undoing each event using the inverse operation reveals the unknown condition represented by a variable. Lesson Three: Making Pirate Codes Investigation Making Pirate Codes...9 Translating a series of instructions into code provides an understanding of the meaning of an equation. Comics Making Pirate Codes...35 Develop the concept that an equation can use symbols to describe a process. Lesson Four: Solving Pirate Codes Investigation Solving Pirate Codes...39 Finding a treasure by undoing a sequenced code develops the concept of solving equations using inverse operations. Comics Solving Pirate Codes...43 Demonstrate that equations can be undone by working backwards in the sequence they were developed. Practice Solving Pirate Codes...46 Lesson Five: Deciphering Pirate Codes Investigation Deciphering Pirate Codes...49 Undoing using inverse operations with a variety of forms of equations develops confidence in working with algebraic equations. Practice Deciphering Pirate Codes...5 Comics Deciphering Pirate Codes...55 Reinforce the order in which the equations are solved when working backwards. Interactive Game Pirate Codes...57 Practice in solving equations in an engaging format. Lesson Six: Tracking Number Tricks Investigation Tracking Number Tricks...59 Solving a set of number tricks provides practice in writing, simplifying, and solving equations. The amazement and excitement of the tricks is evidence of the power of algebra. Practice Tracking Number Tricks...63 Comics Tracking Number Tricks...65 Clarify how algebraic symbols and using inverse operations can solve problems efficiently. Problem Solving: Solving a Trick...69 Students apply algebra to a trick they play on others. WORKING BACKWARDS 6 01 AIMS Education Foundation
BIG IDEA: Acting out or visualizing a problem can help in understanding a situation so it can be translated into an equation. As an equation, the problem can be solved by undoing the operations. Lesson Seven: Drawing Solutions Glossary...91 Common Core Standards and Materials...93 The Story of Solving Equations with Inverse Operations...95 Using Comics to Teach Math...99 Using Animations to Teach Math...101 The AIMS Model of Learning...103 Investigation Drawing Solutions...71 Acting out several jumping situations is an opportunity to recognize that visualizing a situation helps in translating and solving a problem. Comics Drawing Solutions...78 Relate the numeric, visual, and algebraic solutions of equations. Lesson Eight: Picturing Problems Problem Solving: Picturing Problems...81 Making a drawing helps make sense of problems so they can be translated, simplified, and solved in algebraic form. Comics Picturing Problems...84 Explain a problem-solving process and how visual sketches can improve understanding of substitution of equivalent expressions. Assessment...87 Students apply and practice solving equations in both familiar and novel settings. Interactive PDF Instructions The digital format of this book, whether downloaded or on disc, is an interactive PDF. Interactive links display as colored text or icons. Explore the links to enrich the learning experience with videos, animations, and Flash applications. Using a projector or interactive whiteboard allows the materials to display for the class and supplements the teacher s instruction. A green link on a student page plays a video explaining the key question and focus of the activity. Blue links explain the procedure and suggestions for implementation. The red link explains and summarizes the learning gained in the activity. Solutions can be toggled on and off by clicking on the record tables. The link for the comics displays them in slideshow format. Right clicking on an interactive window allows the option of making the window full screen. For Mac users without right click functions, use control click. WORKING BACKWARDS 7 01 AIMS Education Foundation
In a context of a loading dock, a running total of pallets (p ), packs (p), and units are recorded to keep track of what is on the dock as shipments are shipped and received. This provides the opportunity to combine the same kind of items or like terms. tig Investigation WORKING BACKWARDS 13 01 AIMS Education Foundation Materials Scissors Loading Dock How can a code be used to keep track of what is on a loading dock as shipments are shipped and received?. stores receive gizmos in Packs are formed by gizmos cubes being bound together in a long string. The shipping supervisor is really only concerned with how many packs are sent or received. So, packs is shortened to (p). Copy a materials page for each group of students and have them cut pieces out. They are used in multiple activities, so you might laminate them and keep sets in envelopes or small plastic bags. These boxes are recorded as Orientation the ever-popular gizmo is individually wrapped in a cubic box. Since gizmos are the only things shipped, the shipping supervisor simply records the number of gizmos for units shipped. Those would be recorded as 4P. WORKING BACKWARDS 01 AIMS Education Foundation MOS GI MOS GI MOS GI MOS GI MOS GI MOS GI MOS GI GI GI MOS MOS GI GI MOS MOS GI GI MOS MOS GI GI MOS MOS GI GI MOS MOS GI GI MOS MOS Loading Dock Materials. for larger shipments, packs are bound together in pallets. Pallets are formed as square flats that are one pack long the dimensions of and one pack wide. a pallet are P X P or P. the supervisor refers to pallets as pack squared., Inc. shipping. Those pallets on the forklift would be recorded as 3p. + 3P + 4P + 5 WORKING BACKWARDS 9 01 AIMS Education Foundation
Have students view the video or read the loading dock orientation before beginning the activity. WORKING BACKWARDS 15 01 AIMS Education Foundation tig Investigation Loading Dock 6. Shipping/Receiving Invoice Act as the shipping supervisor. Use your model units, packs, and pallets to act out loads being shipped or received. Then use symbols to make an accounting of what has been done on the loading dock. 7. 8. 9. 10., Inc. a. Received three pallets, eight packs, and 1 units. a. Received five pallets, five packs, and five units. b. Shipped six units, two pallets, and eight b. Shipped five units, four packs, and three pallets. a. Received four packs and ten units. b. Shipped five units and four a. Received six pallets, four packs, and six units. a. Received six pallets and nine b. Shipped four packs, five units, and five pallets. b. Shipped four pallets and eight Shipping/Receiving 3p + 8p + 1 6 + p + 8p 1p + 6 5 + 4p 5p + 5p + 5 5 + 4p + 3p p + p 4p + 10 6p + 4p + 6 4p + 5 + 5p 6p + 9p 4p + 8p WORKING BACKWARDS 14 01 AIMS Education Foundation Loading Dock 5 1p + 1 p + 1p Shipping/Receiving Invoice +. +, Inc. + + + How can a code be used to keep track of what is on a loading dock as shipments are shipped and received? a. Received four packs and 10 units. b. Received five units and six a. Received three pallets, four packs, and five units. a. Received four pallets, two packs, and three units. a. Received a pallet, five packs, and seven units. a. Received four pallets and five Shipping/Receiving Act as the shipping supervisor. Use your model units, packs, and pallets to act out loads being shipped or received. Then use symbols to make an accounting of what has been done on the loading dock. b. Received four units, two packs, and one pallet. b. Received four units and three b. Received two pallets, three units, and two b. Received nine packs and eight units. 4p + 10 5 + 6p 10p + 15 3p + 4p + 5 4 + p + 1p 4p + 6p + 9 4p + p + 3 4 + 3p 1p + 5p + 7 p + 3 + p 4p + 5p 9p + 8 Comics 4p + 5p + 7 3p + 7p + 10 4p + 14p + 8 What do you need to keep in mind as you TAKE inventory of what is on the loading dock? Use the materials to model one or two problems. Check that students can model and record actions. Using the materials, students work in small groups to model and record what is happening. Discuss what students have learned before they write the summary response. Clarify the process of combining like terms and the meanings of term, coefficient, and variable. WORKING BACKWARDS 10 01 AIMS Education Foundation
To gain fluency in the abstract form, use the practice page. If students require materials, allow their use. WORKING BACKWARDS 16 01 AIMS Education Foundation Loading Dock Write a coded description for each order. Practice Order A: Order B: Order C: Order D: Order E: 3p + 6p + 10 4p + 8p + 1 5p + 10p + 15 6p + 10p + 18 8p + 1p + 0 Describe in code what is shipped or received from the dock and what that equals on the dock. shipping Describe in code what is shipped or received from the dock and what that equals on the dock.. Receive Order A and Order B. 7p + 14p + Receive Order A and Order D. 9p + 16p + 8 Receive Order C and Order B. 9p + 18p + 7 Receive Order E and Order B. 1p + 0p + 3 Receive Order C and Order E. 13p + p + 35 6. 7. 8. 9. Receive Order A, Order B, and Order C. 1p + 4p + 37 Receive Orders A, B, D, E. 1p + 36p + 60 Receive Order E and ship Order D. p + p + Receive Order D and ship Order A. 3p + 4p + 8 10. Receive Order C and ship Order A. p + 4p + 5 WORKING BACKWARDS 11 01 AIMS Education Foundation
Materials WORKING BACKWARDS 1 01 AIMS Education Foundation
Orientation How can a code be used to keep track of what is on a loading dock as shipments are shipped and received?. These boxes are recorded as the ever-popular gizmo is individually wrapped in a cubic box. Since gizmos are the only things shipped, the shipping supervisor simply records the number of gizmos for units shipped. stores receive gizmos in Packs are formed by gizmos cubes being bound together in a long carton. The shipping supervisor is really only concerned with how many packs are sent or received. So, packs is shortened to (p). Those would be recorded as 4P.. for larger shipments, packs are bound together in pallets. Pallets are formed as square flats that are one pack long the dimensions of and one pack wide. a pallet are P X P or P. the supervisor refers to pallets as pack squared., Inc. shipping. Those pallets on the forklift would be recorded as 3p. + 3P + 4P + 5 WORKING BACKWARDS 13 01 AIMS Education Foundation
How can a code be used to keep track of what is on a loading dock as shipments are shipped and received? Shipping/Receiving, Inc. Shipping/Receiving Invoice a. Received four packs and 10 units. b. Received five units and six Act as the shipping supervisor. Use your model units, packs, and pallets to act out loads being shipped or received. Then use symbols to make an accounting of what has been done on the loading dock. + a. Received three pallets, four packs, and five units. b. Received four units, two packs, and one pallet.. + a. Received four pallets, two packs, and three units. b. Received four units and three + a. Received a pallet, five packs, and seven units. b. Received two pallets, three units, and two + a. Received four pallets and five + b. Received nine packs and eight units. What do you need to keep in mind as you TAKE inventory of what is on the loading dock? WORKING BACKWARDS 14 01 AIMS Education Foundation
Shipping/Receiving 6., Inc. Shipping/Receiving Invoice a. Received three pallets, eight packs, and 1 units. Act as the shipping supervisor. Use your model units, packs, and pallets to act out loads being shipped or received. Then use symbols to make an accounting of what has been done on the loading dock. b. Shipped six units, two pallets, and eight a. Received five pallets, five packs, and five units. b. Shipped five units, four packs, and three pallets. 7. 8. a. Received four packs and ten units. b. Shipped five units and four 9. a. Received six pallets, four packs, and six units. b. Shipped four packs, five units, and five pallets. 10. a. Received six pallets and nine b. Shipped four pallets and eight WORKING BACKWARDS 15 01 AIMS Education Foundation
Write a coded description for each order. Practice Order A: Order B: Order C: Order D: Order E: Describe in code what is shipped or received from the dock and what that equals on the dock. Receive Order A and Order B. 6. Receive Order A, Order B, and Order C. Describe in code what is shipped or received from the dock and what that equals on the dock.. Receive Order A and Order D. 7. Receive Orders A, B, D, E. Receive Order C and Order B. 8. Receive Order E and ship Order D. shipping Receive Order E and Order B. 9. Receive Order D and ship Order A. Receive Order C and Order E. 10. Receive Order C and ship Order A. WORKING BACKWARDS 16 01 AIMS Education Foundation
SOLVING EQUATIONS with INVERSE OPERATIONS what are like terms? ESSENTIAL MATH SERIES Loading Dock - what is meant by the variable part of a term? you can t put packs and pallets into the same pile! I mean, what would you get if you combined three pallets and five packs? eight palletacks?? hang on, I think I get it now. the reason we can t combine 5A and 7B is because we have 5 of one thing and 7 of a different thing. and the reason we can t combine 3p and 5P is because we have 3 of one thing and 5 of a different thing. so p and p are different things, just like pallets and packs are different things?. what is a coefficient? ms. cho, I m confused. combining terms used to seem so easy. count up the a s, count up the b s, and that was it. Yeah. these new problems have only one letter. sometimes we combine terms, sometimes we don t. why is that? well, let s start with this: how did you know to combine 5A with 6A? that s right, redmond. in math we say that 3p and 5p are unlike terms because they have different variable parts. but ms. cho, what s a variable part? good question, elora. class, what do you think the variable part might be in each of these terms? + (6a + 4b) (5a + 3b) 11a + 7b 11a 3p 7b 5p that s easy. they re like terms. right. and the same is true for 4B and 3B. now, why not also combine 11A with 7B? uh because okay. you got me. I just watch what you do and never add terms with different letters. you have some good observation skills, redmond. 11A and 7B are what we call unlike terms because they have different variables. I think the variable part in 11A must be a. and for 7B, it s just b. does that mean the variable part for 3p is p? and the variable part for 5p is p? ms. cho, if p is called the variable part of 3p, is 3 called the number part? Q1 excellent, elora. you re perfectly right! let s look at the loading dock activity and see if we can find a more complete answer to your question. consider this problem: (p + p) + (p + 3p) 3p + 5p well, in the activity we thought of p as one pallet, and p as two pallets. that s three pallets in all: 3p. that s correct, vanessa. but the difficult part of redmond s question is this: why not combine 3p and 5p? that s a pretty good name, but the number part has a more official name. we call it a coefficient, which means the number with. this diagram provides a nice summary of this important vocabulary. 3p the variable part (p ) tells us what we have why do we combine p and p? Coefficient Variable Part Q while the coefficient (3) tells us how many we have. WORKING BACKWARDS 17 01 AIMS Education Foundation
redmond, what was your original question again? why is it that sometimes we combine terms and sometimes we don t? we only combine terms if they have the same exact variable part. Loading Dock - EXTRA QUESTIONS class, what s the answer? Use the questions on this page to enhance your discussion of the comic. Find the panels with the black Q circles that correspond to the learning prompts below. Use them as stopping points to help your class understand the concepts being discussed. that s right. and by variable part we mean the letter and the exponent. well, the first one can t be combined because the terms have different variable parts ( versus ). so we can t combine terms unless they have the same letter and the same exponent? Q3 it s sounds like you ve got it, redmond! now, let s see if everyone understands. consider these three expressions. can any of them be combined? 7x + 7x 3y + 10y 5a + 5b but the second one can, since they have the same variable and the same exponent. Q1 Q3 do you understand the idea of like terms and unlike terms? what do you think the class will say? write down a pair of like terms. then write down a pair of unlike terms. be sure to label which pair is which. which of these three expressions contain like terms? explain how you know. 7x + 7x Q like terms have identical variable parts, though their coefficients can be different. identify the variable part and coefficient for each of these terms: 3a 4b 5c d 3 3y + 10y 13y 3y + 10y 5a + 5b the third one is tricky, ms. cho, using the same coefficient (5) and the same exponent (). but you can t fool me now! the variable parts are different because they use different letters, so the terms can t be combined. well done, class. it s essential that you know what to combine when simplifying, and it looks like you ve got that down. WORKING BACKWARDS 18 01 AIMS Education Foundation