Comparing Quantities Algebra Mathematics in Context is a comprehensive curriculum for the middle grades It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No 9054928 The revision of the curriculum was carried out in 2003 through 2005, with the support of the National Science Foundation Grant No ESI 0137414 National Science Foundation Opinions expressed are those of the authors and not necessarily those of the Foundation Kindt, M., Abels, M., Dekker, T., Meyer, M R., Pligge M A., & Burrill, G (2006) Comparing Quantities In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopỉdia Britannica, Inc Copyright © 2006 Encyclopỉdia Britannica, Inc All rights reserved Printed in the United States of America This work is protected under current U.S copyright laws, and the performance, display, and other applicable uses of it are governed by those laws Any uses not in conformity with the U.S copyright statute are prohibited without our express written permission, including but not limited to duplication, adaptation, and transmission by television or other devices or processes For more information regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610 ISBN 0-03039627-1 073 09 08 07 06 05 The Mathematics in Context Development Team Development 1991–1997 The initial version of Comparing Quantities was developed by Martin Kindt and Mieke Abels It was adapted for use in American schools by Margaret R Meyer, and Margaret A Pligge Wisconsin Center for Education Freudenthal Institute Staff Research Staff Thomas A Romberg Joan Daniels Pedro Jan de Lange Director Assistant to the Director Director Gail Burrill Margaret R Meyer Els Feijs Martin van Reeuwijk Coordinator Coordinator Coordinator Coordinator Sherian Foster James A, Middleton Jasmina Milinkovic Margaret A Pligge Mary C Shafer Julia A Shew Aaron N Simon Marvin Smith Stephanie Z Smith Mary S Spence Mieke Abels Nina Boswinkel Frans van Galen Koeno Gravemeijer Marja van den Heuvel-Panhuizen Jan Auke de Jong Vincent Jonker Ronald Keijzer Martin Kindt Jansie Niehaus Nanda Querelle Anton Roodhardt Leen Streefland Adri Treffers Monica Wijers Astrid de Wild Project Staff Jonathan Brendefur Laura Brinker James Browne Jack Burrill Rose Byrd Peter Christiansen Barbara Clarke Doug Clarke Beth R Cole Fae Dremock Mary Ann Fix Revision 2003–2005 The revised version of Comparing Quantities was developed by Mieke Abels and Truus Dekker It was adapted for use in American schools by Gail Burrill Wisconsin Center for Education Freudenthal Institute Staff Research Staff Thomas A Romberg David C Webb Jan de Lange Truus Dekker Director Coordinator Director Coordinator Gail Burrill Margaret A Pligge Mieke Abels Monica Wijers Editorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator Margaret R Meyer Anne Park Bryna Rappaport Kathleen A Steele Ana C Stephens Candace Ulmer Jill Vettrus Arthur Bakker Peter Boon Els Feijs Dédé de Haan Martin Kindt Nathalie Kuijpers Huub Nilwik Sonia Palha Nanda Querelle Martin van Reeuwijk Project Staff Sarah Ailts Beth R Cole Erin Hazlett Teri Hedges Karen Hoiberg Carrie Johnson Jean Krusi Elaine McGrath (c) 2006 Encyclopædia Britannica, Inc Mathematics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopædia Britannica, Inc Cover photo credits: (left to right) © PhotoDisc/Getty Images; © Corbis; © Getty Images Illustrations Holly Cooper-Olds; (top), © Encyclopỉdia Britannica, Inc.; 23, 29 (left) Holly Cooper-Olds Photographs (counter clockwise) PhotoDisc/Getty Images; © Stockbyte; © Ingram Publishing; © Corbis; © PhotoDisc/Getty Images; 6, Victoria Smith/HRW; 10 Sam Dudgeon/HRW Photo; 16 © Corbis; 21 © Stockbyte/HRW; 23 PhotoDisc/Getty Images; 25 (left column top to bottom) © Corbis; PhotoDisc/Getty Images; © Corbis; 28 Victoria Smith/HRW; 30 PhotoDisc/Getty Images Contents Letter to the Student Section A Section B Section C Section D Section E Compare and Exchange Bartering Farmer’s Market Thirst Quencher Tug-of-War Summary Check Your Work vi 2 4 Looking at Combinations The School Store Workroom Cabinets Puzzles Summary Check Your Work 10 13 14 14 Finding Prices Price Combinations Summary Check Your Work 16 20 20 $50.00 Notebook Notation Chickens Mario’s Restaurant Chickens Revisited Sandwich World Summary Check Your Work 22 23 24 25 26 26 Equations The School Store Revisited Hats and Sunglasses Return to Mario’s Tickets Summary Check Your Work 28 29 30 31 32 32 ORDER TACO SALAD DRINK TOTAL Additional Practice 34 Answers to Check Your Work 39 — — 4 $ 3.00 $ 8.00 $ 11.00 Contents v Dear Student, Welcome to Comparing Quantities In this unit, you will compare quantities such as prices, weights, and widths You will learn about trading and exchanging things in order to develop strategies to solve problems involving combinations of items and prices $50.00 $50.00 Combination charts and the notebook notation will help you find solutions Combination Chart Number of Pencils ORDER TACO SALAD DRINK TOTAL 55 15 40 — 4 $ 3.00 $ 8.00 $ 11.00 65 25 — Number of Erasers In the end, you will have learned important ideas about algebra and several new ways to solve problems You will see how pictures can help you think about a problem, how to use number patterns, and will develop some general ways to solve what are called “systems of equations” in math Sincerely, The Mathematics in Context Development Team vi Comparing Quantities A Compare and Exchange Bartering A long time ago money did not exist People lived in small communities, grew their own crops, and raised animals such as cattle and sheep What did they if they needed something they didn’t produce themselves? They traded something they produced for the things their neighbors produced This method of exchange is called bartering Paulo lives with his family in a small village His family needs corn He is going to the market with two sheep and one goat to barter, or exchange, them for bags of corn First he meets Aaron, who says, “I only trade salt for chickens I will give you one bag of salt for every two chickens.” “But I don’t have any chickens,” thinks Paulo, “so I can’t trade with Aaron.” Later he meets Sarkis, who tells him, “I will give you two bags of corn for three bags of salt.” Paulo thinks, “That doesn’t help me either.” Then he meets Ranee She will trade six chickens for a goat, and she says, “My sister, Nina, is willing to give you six bags of salt for every sheep you have.” Paulo is getting confused His family wants him to go home with bags of corn, not with goats or sheep or chickens or salt Show what Paulo can Section A: Compare and Exchange A Compare and Exchange Farmer’s Market How many bananas you need to balance the third scale? Explain your reasoning 10 bananas pineapples pineapple bananas apple apple How many carrots you need to balance the third scale? Explain your reasoning carrots ear of corn pepper ear of corn peppers pepper Thirst Quencher How many cups of liquid can you pour from one big bottle? Explain your reasoning ؋ ؍ ؍ ؋ Comparing Quantities ؍ Compare and Exchange A Tug-of-War Four oxen are as strong as five horses An elephant is as strong as one ox and two horses Which animals will win the tug-of-war below? Give a reason for your prediction Section A: Compare and Exchange A Compare and Exchange These problems could be solved using fair exchange In this section, problems were given in words and pictures You used words, pictures, and symbols to explain your work Delia lives in a community where people trade goods they produce for other things they need Delia has some fish that she caught, and she wants to trade them for other food She hears that she can trade fish for melons, but she wants more than just melons So she decides to see what else is available This is what she hears: • • • • For five fish, you can get two melons For four apples, you can get one loaf of bread For one melon, you can get one ear of corn and two apples For 10 apples, you can get four melons Comparing Quantities For each of the following puzzles, find the number that goes in the circle and explain your strategy a b 51 10 18 27 Section C Finding Prices Three T-shirts and four caps are advertised for $96 Two T-shirts and five caps cost $99 How much does a single T-shirt cost? How much does a cap cost? Show your work $ 96 $ 99 Additional Practice 35 Additional Practice Three tall candles and five short candles cost $7.75 Two tall candles and two short candles cost $3.50 $7.75 $3.50 Margarita used a combination chart to find the prices of short and tall candles a Use Margarita’s chart to show how she might solve the problem Margarita’s Chart Prices of Combinations (in dollars) Number of Tall Candles 7.75 3.50 0 Number of Short Candles b Margarita wrote the first combination as 3T ؉5S ؍$7.75 What does the letter T represent? The letter S? c Write a similar statement for the second combination 36 Comparing Quantities Additional Practice Section D Notebook Notation Some of today’s orders at Fish King are shown in the notebook a In your own notebook, list at least three new combinations b What is the price of each item at Fish King? ORDER DRINK FRIES FISH TOTAL 1 $ 8.80 $ 3.60 $ 7.40 — For a total cost of $18.40, Gideon went on the Whirling Wheel four times, in the Haunted House two times, and on the Roller Coaster four times For a total cost of $18, Louisa went on the Whirling Wheel five times and on the Roller Coaster five times Bryce likes only the Roller Coaster, and he rode it 10 times! He spent one dollar less than Louisa What is the price of each attraction? Solve the problem using notebook notation Show all of your calculations Create a problem of your own, using notebook notation Show a detailed solution to your problem Additional Practice 37 Additional Practice Section E Equations Five large rowing boats and two small boats can hold 36 people Two large rowing boats and one small one can hold 15 people a Write two equations representing the information Use the letters L and S b What the letters L and S in your equations represent? c How many people can one large boat hold if it is full? Show your work A mixture of cups of almonds and cups of peanuts costs $9.20 A mixture of cup of almonds and cups of peanuts costs $5.20 a Write two equations representing the information Use the letters A and P b What the letters A and P in your equations represent? c What is the price for a mixture of cups of almonds and cups of peanuts? Show your work Imagine a story for the system of equations below 2A ؉ 4C ؍27 3A ؉ 1C ؍23 a What the letters or variables in this system of equations represent in your story? b Choose any strategy to find the value of A and the value of C Kevin invented a story that is represented by this system of equations 5P ؉ 3K ؍8 10P ؉ 6K ؍16 Can Kevin find the value for P and the value for K? Explain why or why not 38 Comparing Quantities Section A Compare and Exchange You may sketch pictures, similar to the work below Or you may write words If so, be sure to check your numbers five fish four apples one melon 10 apples for two melons for one loaf of bread for one ear of corn and two apples for four melons You should have two correct statements If your statement does not appear here, discuss it with a classmate to see if they agree with you Sample responses: eight apples one melon two melons eight ears of corn two ears of corn one fish two ears of corn four fish for two loaves of bread for five ears of corn for five apples for one loaf of bread for one apple for one apple for one fish for one loaf of bread Yes, Delia’s statement is true Remember: you need to provide an explanation! Sample explanations: • In problem 2, I found that one fish trades for one apple, so 10 fish trade for 10 apples • Since you can trade five fish for two melons, you can trade 10 fish for four melons You can trade four melons for 10 apples from the original information, so you can trade 10 fish for 10 apples Answers to Check Your Work 39 Answers to Check Your Work No, this statement is not true Remember: you have to give an explanation! Sample explanations: • I found in problem that four fish can be traded for one loaf of bread, so three fish are not enough to get one loaf of bread • I found in problem that one fish can be traded for one apple, so three fish will be worth only three apples Because four apples are the same as one loaf of bread, three fish are not enough You can have several different solutions and still be correct Check your solution with another student You may make an assumption about the number of fish Delia has Sample responses: • If she has five fish, she can trade for two melons Then she can get two ears of corn and four apples, because one melon is worth one ear of corn and two apples I know from problem that one apple is worth two ears of corn So if she wants more corn, she can trade four apples for eight ears of corn Delia will then have traded 10 ears of corn in total for five fish This means that one fish is worth two ears of corn So for each fish Delia has, she can get two ears of corn 40 Comparing Quantities Answers to Check Your Work Looking at Combinations You might have filled out the chart in a different way Numbers of Tickets Number of Whirlybird Rides Section B 6 16 14 19 24 12 17 22 10 15 20 10 Number of Loop-D-Loop Rides 16 tickets Different strategies are possible: • In the chart, you can see that for two Loop-D-Loop rides you need 10 tickets, and for three Whirlybird rides you need six tickets So altogether you need 16 tickets • You can draw arrows that go up one square and to the right one square, like on the chart above This move adds seven tickets, and ؉ ؍16 Janus can go on three Loop-D-Loop rides and two Whirlybird rides or one Loop-D-Loop ride and seven Whirlybird rides If you keep filling out the chart, each entry is either greater or less than 19 except for those two combinations So all of the other combinations are for either too many or too few tickets Answers to Check Your Work 41 Answers to Check Your Work a Different charts are possible You should draw an arrow that goes down one square and to the right two squares, like on the chart below b The number of tickets increases by eight Numbers of Tickets 16 Number of Whirlybird Rides 14 19 12 17 10 15 20 25 13 18 23 11 16 21 26 14 19 24 29 34 39 12 17 22 27 32 37 42 47 0 10 15 20 25 30 35 40 45 50 10 Number of Loop-D-Loop Rides Discuss and check your answers to problem with a classmate One example of a story: a A motorcycle holds two people, and a minibus holds 10 people b Number of Minibuses Number of People 50 52 54 40 42 44 46 48 50 30 32 34 36 38 40 20 22 24 26 28 30 10 12 14 16 18 20 56 58 60 10 Number of Motorcycles c The circled entry 16 stands for the number of people traveling on three motorcycles and one minibus The circled entry 40 stands for the number of people traveling in four minibuses 42 Comparing Quantities Answers to Check Your Work Section C Finding Prices a You can have different explanations that are correct Two examples are: • In both pictures there are five candles, but the price is higher in the second picture Since there are more short candles in the picture on the right, they must be more expensive • When one tall candle is replaced by one short candle, the price increases $0.15 The short candles are more expensive than the tall candles b The short candles are $0.15 more expensive than the tall candles c Compare your answer with your classmates There are several possible combinations You can add all the candles and prices to get one combination: $8.55 Some other examples you get when you exchange candles are below and on the next page $3.75 $3.90 Answers to Check Your Work 43 Answers to Check Your Work $4.05 –15¢ $4.20 ؉15¢ $4.35 ؉15¢ $4.50 d One short candle costs $0.90 Different strategies are possible Discuss your strategy with a classmate An example of one strategy follows: Exchange each tall candle for a short candle (See pictures in answer c.) When you have five short candles, the total price is $4.50 $4.50 ، ؍$0.90 44 Comparing Quantities Answers to Check Your Work Costs of Combinations (in dollars) 7.20 8.70 10.20 11.70 10 ؊1.50 ؊5 1.80 3.30 1.50 10 6.60 ؊5 Number of Drinks One strategy is to subtract the price of one bagel and two drinks to find a difference of $5.10 on the diagonal and repeat this to get $1.50 for one bagel Another strategy is that if the entry for the (2,2) cell is $6.60, then the entry for the (1,1) cell is $3.30 So going up the diagonal by moving over one and up one (an increase of one drink and one bagel), the next diagonal cell would be $9.90 and the next to the right of $11.70 would be $13.20 This makes the cost of one bagel $1.50, which can be used to go back to the cost of drinks and no bagels Once you know that four drinks cost $7.20, you can divide to find the cost of one drink You may have filled out other parts of the chart You not need to fill out the whole chart to find the answer Number of Bagels The cost of a drink is $1.80 a Check your strategy with a classmate The cost of a drink is $1.50 Sample strategy: Double the first picture: Four drinks and four bagels cost $11.60 Compare this with the second picture: Four drinks and three bagels cost $10.20 The difference on the left is one bagel The difference on the right is $1.40 So one bagel costs $1.40 To find the price of a drink, take the first picture: drinks ؉ ؋ $1.40 ؍$5.80 So two drinks must cost $3.00 So one drink costs $1.50 b The cost of a single bagel is $1.40 Answers to Check Your Work 45 Answers to Check Your Work Section D Notebook Notation Discuss your solution with a classmate Different strategies are possible For example: Apple Banana Pear Price 1 $1.30 1 $1.10 1 $1.20 2 $3.60 1 $1.80 0 $0.50 Subtract any one of the first three rows from row In this example, the price of one apple is found by subtracting row from row Answers: One apple costs $0.50 One banana costs $0.60 One pear costs $0.70 a One salad costs $2 You may have doubled the first order and then subtracted this from the second order to find the price of a salad, as shown b One drink costs $0.75 One taco costs $1.50 Compare your work with a classmate’s work Sample strategy: Order Taco Salad Drink Total 1 $ 3.00 2 $ 8.00 4 $ 11.00 $ 6.00 $ 2.00 ؋2 From answer a, you know that a salad costs $2.00 In order 3, there were four salads: ؋ $2 ؍$8.00 The price of the order was $11.00, so four drinks cost $11.00 ؊ $8.00 ؍$3.00 $3.00 ، ؍$0.75 is the price of one drink In order 1: taco ؉ drinks ؍$3.00 taco ؉ x $0.75 ؍$3.00 taco ؉ $1.50 ؍$3.00 So one taco costs $1.50 No, a combination chart cannot be used to solve the problem A combination chart can be used only for a combination of two items 46 Comparing Quantities Answers to Check Your Work Equations a 3I ؉ 4D ؍$10 2I ؉ 5D ؍$9 b 1I ؉ 6D ؍$8 c An iris costs $2, and a daisy costs $1 You may have different explanations You may continue the pattern by removing one iris and adding one daisy, and then the total cost goes down by one dollar So 7D ؍$7 One daisy costs $1 Now, 1I ؉ 6($1) ؍$8, so one iris costs $2 a 3A ؉ 2S ؉ 2C ؍$35.00 1S ؉ 2C ؍$12.50 1A ؉ 1S ؉ 2C ؍$18.50 b Different answers are possible Sample responses: 3A ؉ 3S ؉ 4C ؍$47.50 1A ؉ 2S ؉ 4C ؍$31.00 c You can subtract the third equation from the first 3A ؉ 2C ؉ 2C ؍$35.00 subtract ؊ 1A ؉ 1S ؉ 2C ؍$18.50 2A ؉ 1S ؍$16.50 d You can subtract the second equation from the third 1A ؉ 1S ؉ 2C ؍$18.50 subtract ؊ 1S ؉ 2C ؍$12.50 1A ؍$ 6.00 e An adult’s ticket costs $6.00 A senior’s ticket costs $4.50, and a child’s ticket costs $4.00 Strategies may vary Sample strategy: 2A ؉ 1S ؍$16.50 2($6.00) ؉ 1S ؍$16.50 1S ؍$4.50 $4.50 ؉ 2C ؍$12.50 2C ؍$8.00 C ؍$4.00 ( ( ( ( Section E Answers to Check Your Work 47 Answers to Check Your Work a Different stories are possible Here is one example of a story Ronnie can read four library books and three magazines in 96 hours He can read one library book and one magazine in 27 hours b L ؍15, M ؍12 Discuss your solution with a classmate Different strategies are possible Sample strategies: • Notebook notation: So L ؍15, and L ؉ M ؍27 15 ؉ M ؍27 M ؍12 ؊ • L Combination chart: 96 81 54 ؉15 15 27 ؉27 12 0 M • Equations: ؋3 48 Comparing Quantities 4L ؉ 3M ؍96 L ؉ M ؍27 3L ؉ 3M ؍81 1L ؍15 15 ؉ M ؍27 M ؍12 subtract L M 3 — 96 27 81 15 X3 Answers to Check Your Work Discuss your strategy with a classmate The combination chart shows one strategy 58 ؊8 50 42 K 34 26 18 0 C C ؍18 C؍2 Since C ؍2, then there are several ways of finding K so that: K ؍10 Answers to Check Your Work 49 ... 465 525 585 180 240 300 360 420 480 540 135 195 255 315 375 435 495 555 90 150 210 270 330 390 450 510 45 105 165 225 285 345 405 465 0 60 120 180 240 300 360 420 Number of Long Cabinets 12 Comparing. .. Minibuses Number of People 50 52 54 40 42 44 46 48 50 30 32 34 36 38 40 20 22 24 26 28 30 10 12 14 16 18 20 56 58 60 10 Number of Motorcycles c The circled entry 16 stands for the number of people traveling... circled numbers represent in your story problem? 50 52 54 56 58 60 40 42 44 46 48 50 30 32 34 36 38 40 20 22 24 26 28 30 10 12 14 16 18 20 10 Do you think combination charts will always have