Comparing Quantitiesmathincontext.eb.com/content/books/students/Comparing...4 Comparing Quantities...

Comparing QuantitiesAlgebra

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Mathematics in Context is a comprehensive curriculum for the middle grades.It was developed in 1991 through 1997 in collaboration with theWisconsin Centerfor Education Research, School of Education, University ofWisconsin-Madison andthe Freudenthal Institute at the University of Utrecht, The Netherlands, with thesupport of the National Science Foundation Grant No. 9054928.

The revision of the curriculum was carried out in 2003 through 2005, with thesupport of the National Science Foundation Grant No. ESI 0137414.

National Science FoundationOpinions expressed are those of the authorsand not necessarily those of the Foundation.

© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, thethistle logo, Mathematics in Context, and the Mathematics in Context logo areregistered trademarks of Encyclopædia Britannica, Inc.

All rights reserved.

No part of this work may be reproduced or utilized in any form or by any means,electronic or mechanical, including photocopying, recording or by any informationstorage or retrieval system, without permission in writing from the publisher.

International Standard Book Number 978-1-59339-917-7

Printed in the United States of America

1 2 3 4 5 C 13 12 11 10 09

Kindt, M., Abels, M., Dekker, T., Meyer, M. R., Pligge, M. A., & Burrill, G. (2010).Comparing quantities. InWisconsin Center for Education Research &Freudenthal Institute (Eds.), Mathematics in context. Chicago: EncyclopædiaBritannica, Inc.

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The Mathematics in Context Development TeamDevelopment 1991–1997

The initial version of Comparing Quantities was developed by Martin Kindt and Mieke Abels. It wasadapted for use in American schools by Margaret R. Meyer, and Margaret A. Pligge.

Wisconsin Center for Education Freudenthal Institute StaffResearch Staff

Thomas A. Romberg Joan Daniels Pedro Jan de LangeDirector Assistant to the Director Director

Gail Burrill Margaret R. Meyer Els Feijs Martin van ReeuwijkCoordinator Coordinator Coordinator Coordinator

Project Staff

Jonathan Brendefur Sherian Foster Mieke Abels Jansie NiehausLaura Brinker James A, Middleton Nina Boswinkel Nanda QuerelleJames Browne Jasmina Milinkovic Frans van Galen Anton RoodhardtJack Burrill Margaret A. Pligge Koeno Gravemeijer Leen StreeflandRose Byrd Mary C. Shafer Marja van den Heuvel-PanhuizenPeter Christiansen Julia A. Shew Jan Auke de Jong Adri TreffersBarbara Clarke Aaron N. Simon Vincent Jonker Monica WijersDoug Clarke Marvin Smith Ronald Keijzer Astrid de WildBeth R. Cole Stephanie Z. Smith Martin KindtFae Dremock Mary S. SpenceMary 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 StaffResearch Staff

Thomas A. Romberg David C. Webb Jan de Lange Truus DekkerDirector Coordinator Director Coordinator

Gail Burrill Margaret A. Pligge Mieke Abels Monica WijersEditorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator

Project Staff

Sarah Ailts Margaret R. Meyer Arthur Bakker Nathalie KuijpersBeth R. Cole Anne Park Peter Boon Huub Nilwik Erin Hazlett Bryna Rappaport Els Feijs Sonia PalhaTeri Hedges Kathleen A. Steele Dédé de Haan Nanda QuerelleKaren Hoiberg Ana C. Stephens Martin Kindt Martin van ReeuwijkCarrie Johnson Candace UlmerJean Krusi Jill VettrusElaine McGrath

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Cover photo credits: (left to right) © PhotoDisc/Getty Images;© Corbis; © Getty Images

Illustrations1 Holly Cooper-Olds; 2 (top), 3 © Encyclopædia Britannica, Inc.;23, 29 (left) Holly Cooper-Olds

Photographs4 (counter clockwise) PhotoDisc/Getty Images; © Stockbyte;© Ingram Publishing; © Corbis; © PhotoDisc/Getty Images;6, 7Victoria Smith/HRW; 10 SamDudgeon/HRW Photo;16©Corbis; 21© Stockbyte/HRW; 23 PhotoDisc/Getty Images;25 (left column top to bottom) © Corbis; PhotoDisc/Getty Images;© Corbis; 28Victoria Smith/HRW; 30 PhotoDisc/Getty Images

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Contents

Contents v

Letter to the Student vi

Section A Compare and ExchangeBartering 1Farmer’s Market 2Thirst Quencher 2Tug-of-War 3Summary 4Check Your Work 4

Section B Looking at CombinationsThe School Store 6Workroom Cabinets 10Puzzles 13Summary 14Check Your Work 14

Section C Finding PricesPrice Combinations 16Summary 20Check Your Work 20

Section D Notebook NotationChickens 22Mario’s Restaurant 23Chickens Revisited 24Sandwich World 25Summary 26Check Your Work 26

Section E EquationsThe School Store Revisited 28Hats and Sunglasses 29Return to Mario’s 30Tickets 31Summary 32Check Your Work 32

Additional Practice 34

Answers to Check Your Work 39

$50.00

TACOORDER

1

2

3

4

5

6

7

SALAD DRINK TOTAL

3.00

4 4

412 8.00

2—

—

1

11.00

$$$

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vi Comparing Quantities

Dear Student,

Welcome to Comparing Quantities.

In this unit, you will compare quantities such as prices, weights, and widths.

You will learn about trading andexchanging things in order to develop strategies to solve problems involving combinations of items and prices.

Combination charts and the notebook notation will help you find solutions.

In the end, you will have learned important ideas about algebra andseveral new ways to solve problems. You will see how pictures canhelp 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,

TThhee MMaatthheemmaattiiccss iinn CCoonntteexxtt DDeevveellooppmmeenntt TTeeaamm

$50.00

$50.00

Number of Erasers

Combination Chart

Nu

mb

er

of

Pen

cils

0 1 2 3

0

1

2

3

15 40 65

55

0 25

TACOORDER

1

2

3

4

5

6

7

SALAD DRINK TOTAL

3.00

4 4

412 8.00

2—

—

1

11.00

$$$

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A long time ago money did not exist. People lived in smallcommunities, grew their own crops, and raised animals such ascattle and sheep. What did they do if they needed something theydidn’t produce themselves? They traded something they producedfor the things their neighbors produced. This method of exchangeis 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, orexchange, them for bags of corn.

Section A: Compare and Exchange 1

ACompare and Exchange

Bartering

First he meets Aaron, who says, “I only tradesalt for chickens. I will give you one bag of saltfor 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 forthree bags of salt.”Paulo thinks, “That doesn’t help meeither.”

Then he meets Ranee. She will trade six chickensfor a goat, and she says, “My sister, Nina, iswilling to give you six bags of salt for everysheep you have.”

Paulo is getting confused. His family wants him to go home with bagsof corn, not with goats or sheep or chickens or salt.

1. Show what Paulo can do.

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2. How many bananas do you need to balance the third scale?Explain your reasoning.

3. How many carrots do you need to balance the third scale?Explain your reasoning.

4. How many cups of liquid can you pour from one big bottle? Explain your reasoning.

2 Comparing Quantities

Compare and ExchangeA

Farmer’s Market

Thirst Quencher

6 � �

�

4 � �

10 bananas 2 pineapples 1 pineapple 2 bananas 1 apple1 apple

6 carrots 1 ear of corn 1 ear of corn 2 peppers 1 pepper1 pepper

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Four oxen are as strong as five horses.

An elephant is as strong as one ox and two horses.

5. Which animals will win the tug-of-war below? Give a reason foryour prediction.


ACompare and Exchange

Tug-of-War

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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 producefor other things they need. Delia has some fish that she caught, andshe wants to trade them for other food. She hears that she can tradefish for melons, but she wants more than just melons. So she decidesto 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.

A

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1. Rewrite or draw pictures to represent the information so that it iseasier to use.

2. Use the information to write two more statements aboutexchanging apples, melons, corn, fish, and bread.

3. Delia says, “I can trade 10 fish for 10 apples.” Is this true? Explain.

4. Can Delia trade three fish for one loaf of bread? Explain why orwhy not.

5. Explain how Delia can trade her fish for ears of corn.

Explain how to use exchanging to solve a problem.

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Monica and Martin are responsible forthe school store. The store is open allday for students to buy supplies.Unfortunately, Monica and Martin can’tbe in the store all day to take students’money, so they use an honor system.Pencils and erasers are available forstudents to purchase on the honorsystem. Students leave exact changein a small locked box to pay for theirpurchases. Erasers cost 25¢ each, andpencils cost 15¢ each.

1. One day Monica and Martin find$1.10 in the locked box. How manypencils and how many erasershave been purchased?

2. On another day there is $1.50 inthe locked box. Monica and Martincannot decide what has been purchased. Why?

3. Find another amount of moneythat would make it impossible toknow what has been purchased.

BLooking at Combinations

The School Store

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Section B: Looking at Combinations 7

Monica wants to make finding the total price of pencils and eraserseasier, so she makes two price lists: one for different numbers oferasers and one for different numbers of pencils.

4. Copy and complete the price lists for the erasers and the pencils.

Erasers Price

0 $0.00

1 $0.25

2 $0.50

3 $0.75

4 $1.00

5 $1.25

6

7

Pencils Price

0 $0.00

1 $0.15

2 $0.30

3

4

5

6

7

One day the box has $1.05 in it.

5. Show how Monica can use her liststo determine how many pencilsand erasers have been bought.

Monica and Martin aren’t satisfied.Although they now have these twolists, they still have to do many calculations. They are trying to think of a way to get all the prices for all thecombinations of pencils and erasers in one chart.

6. Reflect What suggestions can youmake for combining the two lists?Discuss your ideas with your class.

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Looking at CombinationsB

Number of Erasers

Combination Chart

Nu

mb

ero

fP

enci

ls

0 1 2 3

0

1

2

3

15 40

0 25

Number of Erasers

Costs of Combinations (in cents)

Nu

mb

ero

fP

enci

ls

0 1 2 3

0

1

2

3

15 40

0 25

Monica and Martin come up with theidea of a combination chart. Here yousee part of their chart.

7. a. What does the 40 in the chartrepresent?

b. Howmany combinations oferasers and pencils canthis chart show?

If you extend this chart, as shownbelow, you can showmorecombinations.

Use the combination chart on Student Activity Sheet 1 to solve thefollowing problems.

8. Fill in the white squares with the prices of the combinations.

9. Circle the price of two erasers and three pencils.

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Number of Erasers

Nu

mb

er

of

Pen

cils

Number of Erasers

Nu

mb

er

of

Pen

cils

Number of Erasers

Nu

mb

er

of

Pen

cil

s

a b

Use the number patterns in your completed combination chart onStudent Activity Sheet 1 to answer problems 10–16.

10. a. Where do you find the answer to problem 1 ($1.10) in thechart?

b. How many erasers and how many pencils can be bought for $1.10?

11. a. Reflect What happens to the numbers in the chart as you move along one of the arrows shown in the diagram?

b. Reflect Does the answer vary according to which arrow you choose? Explain your reasoning.

12. What does moving along an arrow mean in terms of the numbersof pencils and erasers purchased?

13. a. Mark on your chart a move from one square to another thatrepresents the exchange of one pencil for one eraser.

b. How much does the price change from one square to another?

14. a. Mark on your chart a move from one square to another thatrepresents the exchange of one eraser for two pencils.

b. How much does the price change for this move?

15. Describe the move shown in charts a and b below in terms of theexchange of erasers and pencils.

16. There are many other moves and patterns in the chart. Find atleast two other patterns. Use different color pencils to mark them on your chart. Describe each pattern you find.

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Anna and Dale are going to remodel a workroom. They want to putnew cabinets along one wall of the room. They start by measuring the room and drawing this diagram.

Anna and Dale find out that the cabinets come in two different widths:45 centimeters (cm) and 60 cm.

17. How many of each cabinet do Anna and Dale need in order forthe cabinets to fit exactly along the wall that measures 315 cm? Try to find more than one possibility.



Workroom Cabinets

Win

do

w

Door

315 c

m

330 cm

60 cm45 cm

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Anna and Dale wonder how they can design cabinetsfor the longer wall.

The cabinet store has a convenient chart. The chartmakes it easy to find out how many 60-cm and 45-cmcabinets are needed for different wall lengths.

18. Explain how Anna and Dale can use the chart tofind the number of cabinets they need for thelonger wall in the workroom.



Win

do

w

Door

315 c

m330 cm

Number of Long Cabinets

Lengths of Combinations (in cm)

Nu

mb

er

of

Sh

ort

Cab

inets

270

315

360

405

450

495

330

375

420

465

510

555

390

435

480

525

570

450

495

540

585

510

555

5706

7

8

9

10

11

0

1

2

3

4

5

0

45

90

135

180

225

0

60

105

150

195

240

285

120

165

210

255

300

345

180

225

270

315

360

405

240

285

330

375

420

465

300

345

390

435

480

525

2 3 4 5 6 7 8

360

405

450

495

540

585

420

465

510

555

480

525

570

1

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19. Can the cabinet store provide cabinets to fit a wall that is exactly 4 meters (m) long? Explain your answer.

If cabinets don’t fit exactly, the cabinet store sells a strip to fill the gap. Most customers want the strip to be as small as possible.

20. What size strip is necessary for cabinets along a 4-m wall?

The chart has been completed to only 585 cm because longer rows ofcabinets are not purchased often. However, one day an order comesin for cabinets to fit a wall exactly 6 m long. One possible way to fillthis order is 10 cabinets of 60 cm each.

21. Reflect What are other possibilities for a cabinet arrangementthat will fit a 6-m wall? Note that although you do not see 600 inthe chart, you can still use the chart to find the answer. How?



0

45

90

135

180

225

270

315

60

105

150

195

240

285

330

375

120

165

210

255

300

345

390

435

180

225

270

315

360

405

450

495

240

285

330

375

420

465

510

555

300

345

390

435

480

525

570

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

360

405

450

495

540

585

420

465

510

555

Number of Long Cabinets

Lengths of Combinations (in cm)

Nu

mb

er

of

Sh

ort

Cab

inets

On the left is a part of the cabinetcombination chart.

22. What is special about themove shown by the arrow?

23. If you start in another squarein this chart and you makethe same move, what do younotice? How can you explainthis?

Strip

Wall

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24. Complete the puzzles on Student Activity Sheet 2.



Puzzles

0 5

18

0

27

37

0

24

20

0

35

55

a b

c d

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A combination chart can help youcompare quantities. A combinationchart gives a quick view of manycombinations.

Discovering patterns within combination charts can make your work easier by allowing you to discover patterns andextend the chart in any direction.

Charts can be used to solve many problems, as you studied in “The School Store” and“Workroom Cabinets.” In this chartthe arrow represents the exchangeof one pencil for one eraser.

Number of Erasers

Combination Chart

Nu

mb

er

of

Pen

cil

s

0 1 2 3

0

1

2

3

15 40 65

55

0 25

0

0

1 2

2

3

5

4 5

7

6 7

9

8

12

9

15

10

01234

4

56

6

10

78

Number of

Loop-D-Loop Rides

Numbers of Tickets

Nu

mb

er

of

Wh

irly

bir

d R

ides

This year the school fair hastwo rides. The Loop-D-Loopcosts five tickets, and theWhirlybird costs two tickets.

1. In your notebook, copy thecombination chart thatshows how many ticketsare needed for differentcombinations of these tworides. Complete the chartas necessary to solve theword problems.

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2. How many tickets are needed for two rides on the Loop-D-Loopand three rides on the Whirlybird?

3. Janus has 19 tickets. How can she use these tickets for both ridesso that she has no leftover tickets?

4. a. On your combination chart, mark a move from one square to another that represents the exchange of one ride on the Whirlybird for two rides on the Loop-D-Loop.

b. How much does the number of tickets as described in 4a,change as you move from one square to another?

5. Use the combination chart on Student Activity Sheet 3.

a. Write a story problem that uses the combination chart.

b. Label the bottom and left side of your chart. Give the chart a title and include the units.

c. What do the circled numbers represent in your story problem?

Do you think combination charts will always have a horizontal andvertical pattern? Why or why not? What about a pattern on the diagonal?

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

0 2 4 6 8 10

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So far you have studied two strategiesfor solving problems that involvecombinations of items. The first strategy, exchanging, applied to theproblems about trading food at thebeginning of the unit. The secondstrategy was to make a combinationchart and use number patterns foundin the chart.

In this section, you will apply thestrategy of exchanging to solve problems involving the method of fair exchange.

CFinding Prices

Price Combinations

$50.00

$50.00

Use the drawings below to answer problems 1–3.

1. Without knowing the price of a pair of sunglasses or a pair of shorts, can you determine which item is more expensive? Explain.

2. How many pairs of shorts can you buy for $50?

3. What is the price of one pair of sunglasses? Explain your reasoning.

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4. What is the price of one umbrella? One cap?

Sean bought two T-shirts and one sweatshirt for a total of $30. Whenhe got home, he regretted his purchase. He decided to exchange oneT-shirt for an additional sweatshirt.

Sean made the exchange, but he had to pay $6 more because thesweatshirt is more expensive than the T-shirt.

5. What is the price of each item? Explain your reasoning.

Denise wants to trade Josh two pencils for a clipboard.

6. Is the trade a fair exchange? If not, who has to pay the difference,and how much is it?

7. What is the price of a pencil? What is the price of a clipboard?Explain your reasoning.

Section C: Finding Prices 17

CFinding Prices

$80.00

$76.00

$8.00

$7.00

Josh spent $8 to buyfour clipboards andeight pencils.

Denise spent $7 to buythree clipboards and 10 pencils.

$80.00

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You can use a chart to solve some of these shopping problems.

This combination chart represents the problem of the caps and theumbrellas (page 17).

8. Complete this chart on Student Activity Sheet 4. Then find theprices of one cap and one umbrella. Is this the same answer youfound for problem 4 on page 17?


Finding PricesC

9. Study the two pictures of sunglasses and shorts. Use one of theextra charts on Student Activity Sheet 4 to make a combinationchart for these items. Label your chart. What is the price of onepair of sunglasses? One pair of shorts?

At Doug’s Discount Store, all CDs are one price; all DVDs are another price.

David buys three CDs and two DVDs for $67.Joyce buys two CDs and four DVDs for $90.

10. What is the price of one CD? One DVD? You may use any strategy.

0 1 2 3 4 5

80

0

1

2

3

4

5

76

Number of Caps

Costs of Combinations

(in dollars)

Nu

mb

er

of

Um

bre

lla

s

$50.00

$50.00

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On a visit to Quinn’s Quantities, Rashard finds the prices for variouscombinations of peanuts and raisins.

11. What does Rashard pay for a mixture of 5 cups of peanuts and 2 cups of raisins? You may use any strategy.

12. Reflect Create your own shopping problem. Solve the problemyourself, and then ask someone else to solve it. Have the personexplain to you how he or she found the solution.

In solving shopping problems, you have used exchanging and combination charts. Joe studied the problem below and used a different strategy.

Follow Joe’s strategy to see how he found the price of each candle.

13. Explain Joe’s reasoning.


CFinding Prices

$7.30

$3.40

Joe

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

$1.70

$5.10

$2.20

$1.10

$0.60

• A mixture of 3 cups of peanutsand 2 cups of raisins costs $3.30.

• A mixture of 4 cups of peanutsand 3 cups of raisins costs $4.55.

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$4.20

$4.35


Finding PricesC

You can use different strategies to solve shopping problems.

If you can find a pattern in a picture, you can use the fair exchangemethod. To do so, continue exchanging until a single item is left soyou can find its price. If not, combining information may help youfind the price of a single item.

Another strategy is to make a combination chart and look for a patternin the prices. Use the pattern to find the price of a single item. Youmay also use the fair exchange method with a combination chart.

1. Felicia and Kenji want to buycandles. The candles are availablein different combinations of sizes.

a. Without calculating prices,determine which is more expensive, the short or the tallcandle.

b. What is the difference in pricebetween one short and one tallcandle?

c. Draw a new picture that showsanother combination of shortand tall candles. Write the priceof the combination.

d. What is the price of a singleshort candle?

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2. Roberto bought two drinks and two bagels for $6.60.

Anne bought four drinks and three bagels for $11.70.

Use a combination chart to find the cost of a single drink.

3. The prices of drinks and bagels have changed.

a. Use any strategy to find the new cost of a drink.

b. How much is a single bagel now?

Write several sentences describing the differences between using the method of fair exchange and using combination charts to solveproblems.

$5.80

$10.20

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Three chickens weighed themselves in different combinations.

1. What should the scale read in the fourth picture?

2. Show how to find out how many kilograms (kg) each chickenweighs.

DNotebook Notation

Chickens

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Section D: Notebook Notation 23

DNotebook Notation

Mario’s Restaurant

3. Some of the orders do not have total prices indicated. What are the prices of these orders?

4. Make up two new orders and write them in your notebook. Fill in the prices of these orders.

5. What is the price of each item?

Mario runs a Mexican restaurant, and he is very busy. He moves fromone table to another, writing down all the orders. You can see belowhow he writes the orders on his order pad.

TACOORDER SALAD DRINK TOTAL1

2

3

4

5

6

7

8

9

10

104

4

2

2

832

2

2

2

--

--

1

3

1

1

93

3

1

1

--

--

$$$

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The way Mario wrote the orders in his notebook gave him a goodoverview of many combinations. Such notation can also be applied to other problems. If you apply Mario’s notebook notation

to the chicken problems, you might come up with this chart.


Notebook NotationD

Chickens Revisited

Number of Each Size of Chicken

SM L

Weight

(in kg)

00

0

1

11

1

11

10.6

8.5

6.1

S is the weight of the small chicken.

M is the weight of the medium chicken.

L is the weight of the large chicken.

6. How can you find the total weight of the three chickens by usingnotebook notation?

7. Make new combinations until you find the weight of each chicken.

18.CQ.SB.0509.eg.qxd 05/14/2005 11:04 Page 24

Here are some orders that were served at Sandwich World today. You can write these orders in notebook notation.


DNotebook Notation

Sandwich World

ApplesOrder Milk Sandwich Total

3.40$$$

011 1

4.20102 1

2.80113

4

5

6

7

8

9

10

0

$3.40

$4.20

$2.80

8. In your own notebook,make new combinationsuntil you can determinethe price of each item.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 25


Notebook NotationD

In this section, you explored notebook notation as a good way to getan overview of the information contained in a problem. You can makenew combinations in a notebook by:

• adding rows;

• finding the difference between rows; and

• doubling or halving rows; and so on.

The new combinations you create can help you find solutions to newproblems.

You can write these combinations of fruits in notebook notation.

1. In your own notebook, make new combinations until you find the price of each item.

$1.10

$1.20

$1.30

Price of Combinations

Apple Banana Pear Price

0 1 1 1.30$

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 26


2. Study the following notebook showing lunch orders at Mario’s restaurant.

a. Find the cost of one salad. Explain how you got your answer.

b. How can you find the cost of one drink? One taco?

3. Can you solve problem 2 by using a combination chart? Why or why not?

TACOORDER

1

2

3

4

5

6

7

SALAD DRINK TOTAL

3.00

4 4

412 8.00

2—

—

1

11.00

$$$

Write a description to tell an adult in your family about all of the waysyou have learned so far in this unit to solve problems. Show him orher examples of what you have learned.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 27


The prices of pencils anderasers have changed, soMartin and Monica have tomake a new price chart.Student Activity Sheet 5

contains a combinationchart for you to complete.

EEquations

The School Store Revisited

0 1 2 3 4 5

0

1

2

3

4

5

6

7

6 7

Number of Erasers

Prices of Combinations (in cents)

Nu

mb

er

of

Pen

cils

130

2. What number belongs in the emptycircle? Write an equation representingthis situation.

3. Use Student Activity Sheet 5 to writethe information from the two equations in notebook notation.

4. Monica tells you that 1E � 2P � 75.Express this information in both the combination chart and the notebooknotation on Student Activity Sheet 5.

5. Find the new price for a single eraser and a single pencil. You may use eithernotebook notation or the combinationchart.

Information about the number in the picture above can be expressedin a formula. This formula is also called an equation.

2E � 3P � 130

1. a. What does the letter E in this equation represent? What doesthe letter P represent?

b. Describe in words the meaning of this equation.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 28

Section E: Equations 29

EEquations

6. Each of these pictures can be replaced by an equation. Write thetwo equations, using the symbol H for the price of a hat and thesymbol S for the price of a pair of sunglasses.

7. Write an equation that shows the total price of one hat and fourpairs of sunglasses.

8. What is the price of one pair of sunglasses? What is the price ofone hat? Show how you found these prices.

Hats and Sunglasses

$109

Here is another chart that represents the pricesof combinations of two items.

9. Write an equation in which the price of oneitem is A and the price of the other item isBfor each of the circled numbers. You shouldhave five different equations.

10. Make up a price for each item so that priceA is higher than price B. Use your prices tocomplete the chart in your notebook.

11. Reflect Do you think all of the students inyour class have the same numbers in theircharts? Explain why or why not.

12. Now you can make many equations basedon the information in your chart. Writethree equations.

0 1 2 3 4 5

0

1

2

3

4

5

Number of First Item

(Price A)

Prices of Combinations

(in dollars)

Nu

mb

er

of

Seco

nd

Ite

m

(Pri

ce B

)

8

16

24

32

40

0

$101

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 29

Some prices in Mario’s restaurant have changed.You now have to pay $6.50 if you order one taco,two salads, and one drink. You pay $11.50 for onetaco, four salads, and three drinks. For $4.50, youcan buy one taco and two drinks.


EquationsE

Return to Mario’s

13. Write an equation that corresponds to each of the orders above.

14. By combining the orders, you can make new equations. Whatequation do you get when you add the last two orders?

15. Make up two other equations by combining orders.

16. Show how you can combine equations to get the equation 1S � 1D � $2.50.

17. Find the new price for each of the three items.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 30

This afternoon a new animated movie is playing at the movie theater.Many adults and children are waiting in line to buy their tickets.

18. How much will the ticket seller in the third picture charge?

19. How much will you pay if you go to this theater alone?


EEquations

Two adults andtwo children.

Twentydollars,please.

One adult andthree children.

Seventeendollars, sir.

Three adults andfive children.

. . . .

Tickets

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 31


EquationsE

Many problems compare quantities such as prices, weights, and widths.

One way to describe these problems is by using equations. For example, study the picture of the umbrellas and cap.

If you let U represent the price of oneumbrella and C represent the price of one cap, the equation is 2U � 1C � $80.

These problems can also be solved with combination charts if thereare only two different items. When there are more than two items,you can use notebook notation to find the solution.

1. At a flower shop Joel paid $10 for three irises and four daisies.Althea paid $9 for two irises and five daisies.

a. Write equations representing this information.

b. Write an equation to show the price of one iris and six daisies.

c. Find the cost of one iris and the cost of one daisy.

0 1 2 3 4 5

0

1

2

3

4

5

Number of Erasers

Price of Combinations

Nu

mb

er

of

Pen

cils

Notebook Notation

TACOORDER SALAD DRINK TOTAL

$ 80.00

18.CQ.SB.0913.qxd 11/19/2005 21:42 Page 32


2. At a movie theater, tickets for three adults, two seniors, and twochildren cost $35. Tickets for one senior and two children cost$12.50. Tickets for one adult, one senior, and two children cost $18.50.

a. Write three equations representing the ticket information. UseA to represent the price of an adult’s ticket, S to represent theprice of a senior’s ticket, and C to represent the price of achild’s ticket.

b. Write two additional equations by combining your first threeequations.

c. Explain how you can combine equations to get the equation2A � 1S � $16.50.

d. Explain how you can combine equations to get the equation A � $6.

e. What is the cost of each ticket?

3. In the following equations, the numbers 96 and 27 can representlengths, weights, prices, or whatever you wish.

4L � 3M � 96

L � M � 27

a. Write a story to fit these equations.

b. Find the value of L and the value of M.

4. In the following equations, find the value of C and the value of K.Imagining a story to fit the equations may help you solve for thevalues.

5C � 4K � 50

4C � 5K � 58

Refer back to Quinn's Quantities on page 19. Write an equation thatrepresents the price of the two mixtures. Tell which is easier for you touse, the problem posed in words or represented by equations andexplain why.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 33

Susan and her friends like to collect and trade basketball cards. Todayafter school, Susan made these trades:

• two Tigers for three Lions• three Cougars for four Tigers• one Cougar for one Tiger and two Bears• four Panthers for two Cougars

1. Use the information to think of two card trades that would be fair.

2. James offers Susan six Lions for three Cougars. Should Susanmake this trade? Why or why not?

3. James then offers one Panther for two Tigers. Should Susanmake this trade? Why or why not?

4. Susan has five Cougars. How many Bears can she get for herCougars?


Additional Practice

Section Compare and ExchangeA

Section Looking at CombinationsB

Number of Large Canoes

Numbers of People on Canoe Trip

Nu

mb

er

of

Sm

all C

an

oe

s

0 1 2 3 4 5 6 7 8 9 10

0123456789

101112131415

0

2

3

5

7

8

9

4

6

6

A Girl Scout troop wants to rent canoes for agroup of 25 people. Both small and large canoesare available. Each small canoe holds two people,and each large canoe carries three people.

Use the combination chart on Student Activity

Sheet 6 to solve the problems. You do not needto complete the entire chart.

1. What combinations of small and large canoeswill accommodate exactly 25 people? Find allthe possibilities.

2. One person broke her leg a week before thetrip and is unable to go on the canoe trip.Name one possibility for a combination ofcanoes 24 people can rent.

3. Explain why the chart starts with (0, 0) andnot with (1, 1).

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 34


4. For each of the following puzzles, find the number that goes inthe circle and explain your strategy.

a. b.

0

10 18

0

27

51

Section Finding PricesC

$ 99

1. Three T-shirts and four caps are advertised for $96. Two T-shirtsand five caps cost $99. How much does a single T-shirt cost? How much does a cap cost? Show your work.

$ 96

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 35

Three tall candles and five short candles cost $7.75. Two tall candlesand two short candles cost $3.50.

Margarita used a combination chart to find the prices of short and tall candles.

2. a. Use Margarita’s chart to show how she might solve theproblem.

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.


Additional Practice

$7.75

$3.50

Prices of Combinations (in dollars)

Margarita’s Chart

Nu

mb

er

of

Tall

Can

dle

s

Number of Short Candles

5

4

3

2

1

0

0 1 2 3 4 5

3.50

7.75

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 36


Additional Practice

Section Notebook NotationD

Some of today’s orders at Fish King are shown in the notebook.

1. a. In your own notebook, list at least three new combinations.

b. What is the price of each item at Fish King?

For a total cost of $18.40, Gideon went on the Whirling Wheel fourtimes, 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 timesand on the Roller Coaster five times.

Bryce likes only the Roller Coaster, and he rode it 10 times! He spentone dollar less than Louisa.

2. What is the price of each attraction? Solve the problem usingnotebook notation. Show all of your calculations.

3. Create a problem of your own, using notebook notation. Show adetailed solution to your problem.

DRINKORDER

1

2

3

4

5

6

7

FRIES FISH TOTAL

$ 8.801

1

2

$ 3.60

2

—1

3 $ 7.401

1

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 37

1. 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 theletters L and S.

b. What do the letters L and S in your equations represent?

c. How many people can one large boat hold if it is full? Showyour work.

2. A mixture of 3 cups of almonds and 2 cups of peanuts costs$9.20. A mixture of 1 cup of almonds and 2 cups of peanuts costs $5.20.

a. Write two equations representing the information. Use theletters A and P.

b. What do the letters A and P in your equations represent?

c. What is the price for a mixture of 2 cups of almonds and 3cups of peanuts? Show your work.

3. Imagine a story for the system of equations below.

2A � 4C � 27

3A � 1C � 23

a. What do the letters or variables in this system of equationsrepresent in your story?

b. Choose any strategy to find the value of A and the value of C.

4. 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 orwhy not.


Additional Practice

Section EquationsE

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 38

1. You may sketch pictures, similar to the work below.

Or you may write words. If so, be sure to check your numbers.

five fish for two melonsfour apples for one loaf of breadone melon for one ear of corn and two apples10 apples for four melons

2. You should have two correct statements. If your statement doesnot appear here, discuss it with a classmate to see if they agreewith you.

Sample responses: eight apples for two loaves of breadone melon for five ears of corntwo melons for five appleseight ears of corn for one loaf of breadtwo ears of corn for one appleone fish for one appletwo ears of corn for one fishfour fish for one loaf of bread

3. Yes, Delia’s statement is true. Remember: you need to provide anexplanation!

Sample explanations:

• In problem 2, I found that one fish trades for one apple, so 10fish 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.


Section Compare and ExchangeA

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 39


Answers to Check Your Work

4. No, this statement is not true. Remember: you have to give anexplanation!

Sample explanations:

• I found in problem 2 that four fish can be traded for one loaf ofbread, so three fish are not enough to get one loaf of bread.

• I found in problem 2 that one fish can be traded for one apple,so three fish will be worth only three apples. Because fourapples are the same as one loaf of bread, three fish are notenough.

5. You can have several different solutions and still be correct.Check your solution with another student. You may make anassumption about the number of fish Delia has.

Sample responses:

• If she has five fish, she can trade for two melons. Then she canget two ears of corn and four apples, because one melon isworth one ear of corn and two apples. I know from problem 2that one apple is worth two ears of corn. So if she wants morecorn, she can trade four apples for eight ears of corn. Delia willthen have traded 10 ears of corn in total for five fish.

This means that one fish is worth two ears of corn. So for eachfish Delia has, she can get two ears of corn.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 40

1. You might have filled out the chart in a different way.

2. 16 tickets

Different strategies are possible:

• In the chart, you can see that for two Loop-D-Loop rides youneed 10 tickets, and for three Whirlybird rides you need sixtickets. So altogether you need 16 tickets.

• You can draw arrows that go up one square and to the rightone square, like on the chart above. This move adds seventickets, and 7 � 9 � 16.

3. Janus can go on three Loop-D-Loop rides and two Whirlybirdrides 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 theother combinations are for either too many or too few tickets.



Section Looking at CombinationsB

00

1 2

2

35

4 5

7

6 7

9

8

12 17 22

14 19 2416

915 20

1001234

4

56

6

10

78

Number of

Loop-D-Loop Rides

Numbers of Tickets

Nu

mb

er

of

Wh

irly

bir

d R

ides

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 41



4. a. Different charts are possible. You should draw an arrow thatgoes down one square and to the right two squares, like onthe chart below.

b. The number of tickets increases by eight.

0 1 2 3 4 5

0

1

2

3

4

5

Number of Motorcycles

Number of People

Nu

mb

er

of

Min

ibu

ses

12

22

32

42

52 54

24

34

44

36

46

56

48

58 60

0 2 4

14

6

16

26

8

18

28

38

10

20

30

40

50

10

20

30

40

50

Numbers of Tickets

0

0

1 2

2

3

5

4 5

7

6 7

9

11

13

15

17

19

8

12

14

16

18

20

9

15 20

22

24

26

25

27

29

30

32

34

35

37

39

40

42

45

47

50

17

19

21

23

25

10

4

6

8

10

12

14

16

10

Number of

Loop-D-Loop Rides

Nu

mb

er

of

Wh

irly

bir

d R

ides

0

1

2

3

4

5

6

7

8

5. Discuss and check your answers to problem 5 with a classmate.

One example of a story:

a. A motorcycle holds two people, and a minibus holds 10 people.

b.

c. The circled entry 16 stands for the number of people travelingon three motorcycles and one minibus. The circled entry 40stands for the number of people traveling in four minibuses.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 42



Section Finding PricesC

1. a. You can have different explanations that are correct. Twoexamples are:

• In both pictures there are five candles, but the price is higher in the second picture. Since there are more short candles inthe picture on the right, they must be more expensive.

• When one tall candle is replaced by one short candle, the priceincreases $0.15.

The short candles are more expensive than the tall candles.

b. The short candles are $0.15 more expensive than the tallcandles.

c. Compare your answer with your classmates. There are severalpossible combinations. You can add all the candles and pricesto get one combination:

Some other examples you get when you exchange candles are belowand on the next page.

$8.55

$3.90

$3.75

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 43

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 � 5 � $0.90



–15¢

�15¢

�15¢

$4.05

$4.20

$4.35

$4.50

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 44

2. One strategy is to subtract the price of onebagel and two drinks to find a differenceof $5.10 on the diagonal and repeat this toget $1.50 for one bagel. Another strategyis that if the entry for the (2,2) cell is $6.60,then the entry for the (1,1) cell is $3.30. Sogoing up the diagonal by moving overone and up one (an increase of one drinkand one bagel), the next diagonal cellwould be $9.90 and the next to the right of$11.70 would be $13.20. This makes thecost of one bagel $1.50, which can beused to go back to the cost of 4 drinks andno bagels. Once you know that four drinkscost $7.20, you can divide to find the costof one drink.

You may have filled out other parts of thechart. You do not need to fill out the wholechart to find the answer.

The cost of a drink is $1.80.



Costs of Combinations

(in dollars)

Nu

mb

er

of

Dri

nks

Number of Bagels

5

4

3

2

1

0

0 1 2 3 4

7.20

�1.50�

5.10

�5.

10

8.70

1.80 3.30

1.50

10.20

6.60

11.70

3. 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:

2 drinks � 2 � $1.40 � $5.80So two drinks must cost $3.00.So one drink costs $1.50.

b. The cost of a single bagel is $1.40.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 45

1. Discuss your solution with a classmate.

Different strategies are possible. For example:

Subtract any one of the first three rowsfrom row 4.

In this example, the price of one apple isfound by subtracting row 1 from row 4.

Answers:One apple costs $0.50.One banana costs $0.60.One pear costs $0.70.



Section Notebook NotationD

1

2

3

4

56

Apple Banana Pear Price

0 1 1 $1.30

1 1 0 $1.10

1 0 1 $1.202 2 2 $3.601 1 1 $1.80

1 0 0 $0.50

TacoOrder Salad Drink Total

$ 3.00--11 2

$ 8.00122 4

$ 11.004--3 4

$ 6.00--24 4

$ 2.001--5 --

6

7

�2

2. a. One salad costs $2. You may have doubled the firstorder and then subtracted this from the second orderto find the price of a salad, as shown.

b. One drink costs $0.75. One taco costs $1.50. Compareyour work with a classmate’s work.

Sample strategy:

From answer a, you know that a salad costs $2.00.In order 3, there were four salads: 4 � $2 � $8.00.The price of the order was $11.00, so four drinks cost $11.00 � $8.00 � $3.00.$3.00 � 4 � $0.75 is the price of one drink.

In order 1:1 taco � 2 drinks � $3.001 taco � 2 x $0.75 � $3.001 taco � $1.50 � $3.00So one taco costs $1.50.

3. No, a combination chart cannot be used to solve theproblem. A combination chart can be used only for acombination of two items.

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 46

1. a. 3I � 4D � $10

2I � 5D � $9

b. 1I � 6D � $8

c. An iris costs $2, and a daisy costs $1. You may have differentexplanations.

You may continue the pattern by removing one iris and addingone 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.

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

� 1A � 1S � 2C � $18.50

2A � 1S � $16.50

d. You can subtract the second equation from the third.

1A � 1S � 2C � $18.50

� 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 � $4.00Answers to Check Your Work 47


Section EquationsE

(

(

subtract

(

(

subtract

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 47

0 1 2

L

M

3

0

1

2

3

15 27

54

96

81

12

4

4 �15

�27

X 3�

L M

41

81

3

33

15—1

19627

3. a. Different stories are possible. Here is one example of a story.

Ronnie can read four library books and three magazines in 96hours. He can read one library book and one magazine in 27hours.

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 � 2715 � M � 27

M � 12

• Combination chart:

• Equations:



4L � 3M � 96

� 3 L � M � 27 3L � 3M � 81

1L � 15 15 � M � 27

M � 12

subtract

18.CQ.SB.0509.eg.qxd 05/13/2005 08:19 Page 48

4. Discuss your strategy with a classmate.

The combination chart shows one strategy.

9 C � 18

C � 2



0 1 2

K

C

3

0

1

2

3

4

58

4

5

5

50

6

42

7

34

8

26

9

18

�8

Since C � 2, then there are several waysof finding K so that: K � 10

18.CQ.SB.0913.qxd 11/19/2005 21:44 Page 49

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