Comparing quantities grade 6

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..

Comparing Quantities

Algebra

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: Encyclopædia Britannica, Inc.

Copyright © 2006 Encyclopædia Britannica, Inc.

All rights reserved.

Printed in the United States of America.

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ISBN 0-03039627-1

1 2 3 4 5 6 073 09 08 07 06 05

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

Project Staff

Jonathan Brendefur Sherian Foster Mieke Abels Jansie Niehaus Laura Brinker James A, Middleton Nina Boswinkel Nanda Querelle James Browne Jasmina Milinkovic Frans van Galen Anton Roodhardt Jack Burrill Margaret A Pligge Koeno Gravemeijer Leen Streefland Rose Byrd Mary C Shafer Marja van den Adri Treffers

Peter Christiansen Julia A Shew Heuvel-Panhuizen Monica Wijers

Barbara Clarke Aaron N Simon Jan Auke de Jong Astrid de Wild

Doug Clarke Marvin Smith Vincent Jonker

Beth R Cole Stephanie Z Smith Ronald Keijzer

Fae Dremock Mary S Spence Martin Kindt

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

Project Staff

Sarah Ailts Margaret R Meyer Arthur Bakker Nathalie Kuijpers

Erin Hazlett Bryna Rappaport Els Feijs Sonia Palha

Teri Hedges Kathleen A Steele Dédé de Haan Nanda Querelle Karen Hoiberg Ana C Stephens Martin Kindt Martin van Reeuwijk Carrie Johnson Candace Ulmer

Jean Krusi Jill Vettrus

Elaine McGrath

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

© Corbis; © Getty Images

Illustrations

1 Holly Cooper-Olds; 2 (top), 3 © Encyclopædia Britannica, Inc.;

23, 29 (left) Holly Cooper-Olds

Photographs

4 (counter clockwise) PhotoDisc/Getty Images; © Stockbyte;

© Ingram Publishing; © Corbis; © PhotoDisc/Getty Images;

6, 7 Victoria Smith/HRW; 10 Sam Dudgeon/HRW Photo;

16 © Corbis; 21 © Stockbyte/HRW; 23 PhotoDisc/Getty Images;

Letter to the Student vi

Section A Compare and Exchange

Section B Looking at Combinations

Section C Finding Prices

Section D Notebook Notation

The School Store Revisited 28

1 2 3 4 5 6 7

SALAD DRINK TOTAL

3.00

4 441

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

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

1 2 3

15 40 65 55

0 25

TACO ORDER

1 2 3 4 5 6 7

SALAD DRINK TOTAL

3.00

4 1

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 do if they needed something theydidn’t produce themselves? They traded something they produced for the things their neighbors produced This method of exchange

is called bartering.

Compare and Exchange Bartering

Paulo lives with his family

in a small village His familyneeds corn He is going tothe market with two sheepand one goat to barter, orexchange, them for bags

of corn

First he meets Aaron, who says, “I only trade salt forchickens I will give you one bag of salt for every twochickens.”

“But I don’t have any chickens,” thinks Paulo, “so Ican’t trade with Aaron.”

Later he meets Sarkis, who tells him, “I will give youtwo bags of corn for three bags of salt.”

Paulo thinks, “That doesn’t help me either.”

Then he meets Ranee Shewill trade six chickens for agoat, and she says, “Mysister, Nina, is willing togive you six bags of salt forevery 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

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

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 for

your prediction

Tug-of-War

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 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

exchanging 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 or

why not

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

Explain how to use exchanging to solve a problem

Monica and Martin are responsible forthe school store The store is open allday for students to buy supplies.

Unfortunately, Monica and Martin can’t

be 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 change

in 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 in

the locked box Monica and Martincannot decide what has been purchased Why?

3 Find another amount of money

that would make it impossible toknow what has been purchased

Looking at Combinations The School Store

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.

One day the box has $1.05 in it

5 Show how Monica can use her lists

to 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

15 40

0 25

Monica and Martin come up with the idea of a combination chart Here you see part of their chart

7 a What does the 40 in the chart

represent?

b How many combinations of

erasers and pencils can Monica and Martin show

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

9 Circle the price of two erasers and three pencils.

Use the number patternsin your completed combination charton

Student Activity Sheet 1 to answer problems 10–16.

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

chart?

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 numbers

of pencils and erasers purchased?

13 a Mark on your chart a move from one square to another that

represents 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 that

represents 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 the

exchange of erasers and pencils

16 There are many other moves and patterns in the chart Find at

least two other patterns Use different color pencils to mark them on your chart Describe each pattern you find

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 for

the cabinets to fit exactly along the wall that measures 315 cm? Try to find more than one possibility

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 to

find the number of cabinets they need for thelonger wall in the workroom

Door

330 cm

Number of Long Cabinets

Lengths of Combinations (in cm)

270315360405450495

330375420465510555

390435480525570

450495540585

510

555

5706

7891011

012345

04590135180225

060105150195240285

120165210255300345

180225270315360405

240285330375420465

300345390435480525

360405450495540585

420465510555

480525570

1

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 comes

in 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?

120165210255300345390435

180225270315360405450495

240285330375420465510555

300345390435480525570

420465510555

Number of Long Cabinets

Lengths of Combinations (in cm)

On the left is a part of the cabinetcombination chart

22 What is special about the

move shown by the arrow?

23 If you start in another square

in this chart and you makethe same move, what do younotice? How can you explainthis?

Strip Wall

24 Complete the puzzles on Student Activity Sheet 2.

0

24 20

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 exchange

of one pencil for one eraser

1 2 3 4

4

5 6

6

10

7 8

Number of Loop-D-Loop Rides

1 In your notebook, copy the

combination chart thatshows how many ticketsare needed for differentcombinations of these tworides Complete the chart

as necessary to solve theword problems

3 Janus has 19 tickets How can she use these tickets for both rides

so 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 and

vertical pattern? Why or why not? What about a pattern on the

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 found

in the chart

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

of fair exchange

Finding 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

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

Sean bought two T-shirts and one sweatshirt for a total of $30 When

he 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?

Denise spent $7 to buythree clipboards and

10 pencils

$80.00

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 the

prices of one cap and one umbrella Is this the same answer youfound for problem 4 on page 17?

9 Study the two pictures of sunglasses and shorts Use one of the

extra charts on Student Activity Sheet 4 to make a combination

chart 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)

$50.00

$50.00

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.

$4.35

You can use different strategies to solve shopping problems

If you can find a pattern in a picture, you can use the fair exchange

method 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 pattern

in 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 buy

candles The candles are available

in different combinations of sizes

a Without calculating prices,

determine which is more expensive, the short or the tallcandle

b What is the difference in price

between one short and one tallcandle?

c Draw a new picture that shows

another combination of shortand tall candles Write the price

of the combination

d What is the price of a single

short candle?

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 solve

problems

$5.80

$10.20