Comparing Quantities

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Comparing Quantities

Algebra

TEACHER’S GUIDE

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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 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.

© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, the thistle logo, Mathematics in Context, and the Mathematics in Context logo are registered trademarks of Encyclopædia Britannica, Inc.

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 information storage or retrieval system, without permission in writing from the publisher.

International Standard Book Number 978-1-59339-927-6 Printed in the United States of America

1 2 3 4 5 13 12 11 10 09

Kindt, M., Abels, M., Dekker, T., Meyer, M. R., Pligge, M. A., & Burrill, G. (2010).

Comparing quantities. In Wisconsin Center for Education Research &

Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclopædia Britannica, Inc.

The Teacher’s Guide for this unit was prepared by David C. Webb, Teri Hedges, and Mieke Abels.

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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

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 Heuvel-Panhuizen

Peter Christiansen Julia A. Shew Jan Auke de Jong Adri Treffers

Barbara Clarke Aaron N. Simon Vincent Jonker Monica Wijers

Doug Clarke Marvin Smith Ronald Keijzer Astrid de Wild

Beth R. Cole Stephanie Z. Smith Martin Kindt

Fae Dremock Mary S. Spence

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

Beth R. Cole Anne Park Peter Boon Huub Nilwik

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

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

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

23, 29 (left) Holly Cooper-Olds Photographs

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

6, 7 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

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Comparing Quantities v

Overview

NCTM Principles and Standards for School Mathematics vii

Math in the Unit viii

Algebra Strand: An Overview x

Student Assessment in Mathematics in Context xiv

Goals and Assessment xvi

Materials Preparation xviii

Student Material and Teaching Notes

Student Book Table of Contents Letter to the Student

Section Compare and Exchange

Section Overview 1A

Bartering: Developing Strategies to Solve Problems 1

Farmer’s Market: Understanding Equivalence 2

Thirst Quencher: Using Equivalence to Solve Problems 2

Tug-of-War: Modeling Equivalence 3

Summary 4

Check Your Work 4

Section Looking at Combinations

Section Overview 6A

The School Store: Investigating Combinations Using Decimals 6 Workroom Cabinets: Using Tables to Investigate Combinations 10 Puzzles: Constructing Tables; Developing Strategies to Solve Problems 13

Summary 14

Check Your Work 14

Section Finding Prices

Section Overview 16A

Price Combinations: Applying Strategies for Solving Problems;

Calculating Using Decimals 16

Summary 20

Check Your Work 20

C B A

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vi Comparing Quantities Contents

Section Notebook Notation

Chickens: Using Equivalence to Solve Problems 22 Mario’s Restaurant: Applying Strategies for Solving Problems 23 Chickens Revisited: Using Symbols to Create Expressions 24 Sandwich World: Determining Patterns; Applying Strategies

for Solving Problems 25

Summary 26

Check Your Work 26

Section Equations

The School Store Revisited: Modeling and Using Equations 28

Hats and Sunglasses: Writing Equations 29

Return to Mario’s: Applying Equations 30

Tickets: Using Equations to Solve Problems 31

Summary 32

Check Your Work 32

Additional Practice

³⁴

Assessment and Solutions

Assessment Overview 39A

Quiz 1 40

Quiz 2 42

Unit Test 44

Quiz 1 Solutions 46

Quiz 2 Solutions 48

Unit Test Solutions 50

Glossary

⁵⁴

Blackline Masters

Letter to the Family 56

Student Activity Sheets 57

D

E

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Overview

Comparing Quantities

and the NCTM Principles and Standards for

School Mathematics for Grades 6–8

The process standards of Problem Solving, Reasoning and Proof, Communication, Connections, and Representation are addressed across all Mathematics in Context units.

In addition, this unit specifically addresses the following PSSM content standards and expectations:

Algebra

In grades 6–8 all students should:

•

represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;

•

relate and compare different forms of representation for a relationship;

•

develop an initial conceptual understanding of different uses of variables;

•

use symbolic algebra to represent situations and to solve problems;

•

recognize and generate equivalent forms for simple algebraic expressions; and

•

model and solve contextualized problems using various representations, such as graphs, tables, and equations.

Measurement

In grades 6–8 all students should:

•

understand the metric system of measurement.

Overview Comparing Quantities vii

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Overview

Prior Knowledge

This unit assumes that students can use symbols (words, pictures, letters) and mathematical expressions informally.

Students must be able to add and subtract three- digit numbers and multiply and divide two-digit numbers. They must be able to add and subtract fractions and decimals and multiply and divide simple fractions and decimals. Experience with reading tables is helpful.

Math in the Unit

Comparing Quantities is the second Algebra unit and the first unit in the Restrictions substrand. The unit introduces students to several informal strategies for solving systems of equations. At the end of the unit, students revisit these problem scenarios more formally as they use variables and formal equations to represent and solve problems.

Variables

Variables occur mainly in the role of unknowns.

In the beginning of the unit, the variables are presented in pictures. The pictures of the object symbolize a quantity tied to that object. Students operate with pictures in an algebra sense. At the end of the unit, letters are used to represent the quantities. In between, students can choose their own form of representation: pictures, words, symbols, or letters.

Equations, Systems of Equations

The unit focuses on solving sets of equations and therefore on equivalence of equations. Making equivalent equations is an important part of solving an equation. Students develop different strategies to solve picture equations, story equations, and formally described equations. In the unit Graphing Equations, students will solve systems of equations graphically and by

substitution.

Strategies

The focus is on informal methods to solve sets of equations; no algorithms are being introduced.

Fair Exchange

Within the context of bartering, students are introduced to the concept of substitution, by the method of fair exchange.

Throughout this unit, students will use these exchange situations to develop more sophisticated strategies for solving equation-like problems. The combination charts in Sections B and C and the notebook notation in Section D are based on the exchange strategies introduced in this section. In formal algebra, the exchange process is called substitution and the equation-like problems are referred to as a system of equations.

Combination Chart

The principle of fair exchange is visualized in the combination chart. Students identify and use the number patterns in these charts to solve problems.

Math in the Unit

viii Comparing Quantities Overview

•

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.

Number of Erasers Combination Chart

Number of Pencils

0 1 2 3

15 40 65

55

0 25

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Overview

Combine and Reason

Students’ work with problems involving

combinations of items is extended as they explore problems about shopping. Given two “picture equations” of different quantities of two items and their combined price, students find the price of a single item by combining and reasoning.

Notebook Notation:

The notebook notation strategy is based on a matrix with coefficients. This strategy also allows solving sets of equations with more than two variables.

When students have finished the unit they will:

•

be able to organize information from problem situations;

•

have some understanding of the concepts of variable and equation;

•

start using symbolic language; and

• Students use pictures, words, or symbols to describe one’s own solution process.

• Students start manipulating with symbols, like A A A 3A.

•

informally solve systems of equations.

• Students use their informal knowledge of bartering and exchanging to solve problems.

• Students are able to produce equivalent equations using pictures, words, symbols, or letters.

• Students are able to follow the solution process of someone else.

• Students find that a guess-and-check strategy is often inefficient.

• Students use different representations like pictures, notebooks, charts, and equations and choose strategies appropriate for the problem situation.

• Students interpret a mathematical solution in terms of the problem situation.

• Students develop their reasoning skills.

Overview Comparing Quantities ix

$109

$101

TACO ORDER

1 2 3 4 5 6 7

SALAD DRINK TOTAL 3.00

4 4

1 4

2 8.00

— 2

— 1

11.00

$

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Overview

xx Comparing Quantities Overview

Mathematical Content

The Algebra strand in Mathematics in Context emphasizes algebra as a language used to study relationships among quantities. Students learn to describe these relationships with a variety of representations and to make connections among these representations. The goal is for students to understand the use of algebra as a tool to solve problems that arise in the real world or in the world of mathematics, where symbolic representations can be temporarily freed of meaning to bring a deeper understanding of the problem. Students move from preformal to formal strategies to solve problems, learning to make reasonable choices about which algebraic representation, if any, to use.

The goals of the units within the algebra strand are aligned with NCTM’s Principles and Standards for School Mathematics.

Algebra Tools and Other Resources

The Algebra Tools Workbook provides materials for additional practice and further exploration of algebraic concepts that can be used in conjunction with units in the Algebra strand or independently from individual units. The use of a graphing calculator is optional in the student books. The Teacher’s Guides provide additional questions if graphing calculators are used.

Organization of the Algebra Strand

The theme of change and relationships encompasses every unit in the Algebra strand. The strand is organized into three substrands: Patterns and Regularities, Restrictions, and Graphing. Note that units within a substrand are also connected to units in other substrands.

Algebra Strand:

An Overview

1 2 3

Patterns and Figures

Algebra Rules!

Building Formulas

Ups and Downs Operations

Graphing Equations Expressions

and Formulas

Level 1

Level 2

Level 3

Algebra Tools Workbook

Pathways through the Algebra Strand

(Arrows indicate prerequisite units.)

Comparing Quantities

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Overview

Overview Comparing Quantities xi

In Building Formulas, students explore direct and recursive formulas (formulas in which the current term is used to calculate the next term) to describe patterns. By looking at the repetition of a basic pattern, students are informally introduced to the distributive property. In Patterns and Figures, students continue to use and formalize the ideas of direct and recursive formulas and work formally with algebraic expressions, such as 2(n + 1).

In a recursive (or NEXT-CURRENT) formula, the next number or term in a sequence is found by performing an operation on the current term according to a formula. For many of the sequences in this unit, the next term is a result of adding or subtracting a fixed number from the current term of the sequence. Operations with linear expressions are connected to “Number Strips,” or arithmetic sequences.

Students learn that they can combine sequences by addition and subtraction. In Patterns and Figures, students also encounter or revisit other mathematical topics such as rectangular and triangular numbers. This unit broadens their mathematical experience and makes connections between algebra and geometry.

Patterns and Regularities

In the Patterns and Regularities substrand, students explore and represent patterns to develop an understanding of formulas, equations, and expres- sions. The first unit, Expressions and Formulas, uses arrow language and arithmetic trees to represent situations. With these tools, students create and use word formulas that are the precursors to algebraic equations. The problem below shows how students use arrow language to write and solve equations with a single unknown.

The students use an arrow string to find the height of a stack of cups.

number

⎯¹⎯→ ⎯⎯⎯³⎯→ ⎯⎯⎯¹⁵⎯→ height

of cups of stack

a. How tall is a stack of ten of these cups?

b. Explain what each of the numbers in the arrow string represents.

c. These cups need to be stored in a space 50 cm high. How many of these cups can be placed in a stack? Explain how you found your answer.

As problems and calculations become more complicated, students adapt arrow language to include multiplication and division. When dealing with all four arithmetic operations, students learn about the order of operations and use another new tool — arithmetic trees — to help them organize their work and prioritize their calculations. Finally, students begin to generalize their calculations for specific problems using word formulas.

frame height saddle

height

saddle height (in cm) inseam (in cm) 1.08 frame height (in cm) inseam (in cm) 0.66 2

4 9 14 19

24

29 34

4 5n

1 3 5 7 9 11 13

rim

hold

base

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Overview

xxii Comparing Quantities Overview

Within such contexts as bartering, students are introduced to the concept of substitution (exchange) and are encouraged to use symbols to represent problem scenarios. Adding and subtracting relationships graphically and multiplying the values of a graph by a number help students develop a sense of operations with expressions.

To solve problems about the combined costs of varying quantities of such items as pencils and erasers, students use charts to identify possible combinations. They also identify and use the number patterns in these charts to solve problems.

Students’ work with problems involving combinations of items is extended as they explore problems about shopping. Given two “picture equations”

of different quantities of two items and their combined price, students find the price of a single item. Next, they informally solve problems involving three equations and three variables within the context of a restaurant and the food ordered by people at different tables.

This context also informally introduces matrices. At the end of the unit, students revisit these problem scenarios more formally as they use variables and formal equations to represent and solve problems.

0 1 2 3 4 5

80

0 1 2 3 4 5

76

Number of Caps Costs of Combinations

(in dollars)

Number of Umbrellas

TACO ORDER

1 2 3 4 5 6 7 8 9 10

SALAD DRINK TOTAL

$ 10 4

4 2

2

$ 8 3 2 2 2 2

—

— 1 3 1 1

$9 3

3 11

—

$80.00

$76.00

In the unit Graphing Equations, linear equations are solved in an informal and preformal way. The last unit, Algebra Rules!, integrates and formalizes the content of algebra substrands. In this unit, a variety of methods to solve linear equations is used in a formal way.

Connections to other strands are also formalized.

For example, area models of algebraic expressions are used to highlight relationships between symbolic representations and the geometry and measurement

strands. In Algebra Rules!, students also work with quadratic expressions.

The Patterns and Regularities substrand includes a unit that is closely connected to the Number strand, Operations. In this unit, students build on their informal understanding of positive and negative numbers and use these numbers in addition, subtraction, and multiplication. Division of negative numbers is addressed in Revisiting Numbers and in Algebra Rules!

Restrictions

In the Restrictions substrand, the range of possible solutions to the problems is restricted because the mathematical descriptions of the problem contexts require at least two equations. In Comparing Quantities, students

explore informal methods for solving systems of equations through nonroutine, yet realistic, problem situations such as running a school store, renting canoes,

and ordering in a restaurant.

a

a a²

b ab

ab b

b²

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Overview

Overview Comparing Quantities xiii

In Graphing Equations, students move from locating points using compass directions and bearings to using graphs and algebraic manipulation to find the point of intersection of two lines.

Students may use graphing calculators to support their work as they move from studying slope to using slope to write equations for lines. Visualizing frogs jumping toward or away from a path helps students develop formal algebraic methods for solving a system of linear equations. In Algebra Rules!, the relationship between the point of inter- section of two lines (A and B) and the x-intercept of the difference between those two lines (A – B) is explored. Students also find that parallel lines relate to a system of equations that have no solution.

Graphing

The Graphing substrand, which builds on students’

experience with graphs in previous number and statistics units, begins with Expressions and Formulas where students relate formulas to graphs and read information from a graph.

Operations, which is in the Patterns and Regularities substrand, is also related to the Graphing substrand since it formally introduces the coordinate system.

In Ups and Downs, students use equations and graphs to investigate properties of graphs corresponding to a variety of relationships: linear, quadratic, and exponential growth as well as graphs that are periodic.

In Graphing Equations, students explore the equation of a line in slope and y-intercept form.

They continuously formalize their knowledge and adopt conventional formal vocabulary and notation, such as origin, quadrant, and x-axis, as well as the ordered pairs notation (x, y). In this unit, students use the slope-intercept form of the equation of a line, y = mx + b. Students may use graphing calcu- lators to support their work as they move from studying slope to using slope to write equations for lines. Students should now be able to recognize linearity from a graph, a table, and a formula and know the connections between those representations. In the last unit in the Algebra strand, Algebra Rules!, these concepts are formalized and the x-intercept is introduced. Adding and subtracting relationships graphically and multiplying the values of a graph by a number help students develop a sense of operations with expressions.

April 20

Water Level (in cm)

Sea Level +80 +60 +40 +20 0 -20 -40 -60 -80 -100 Time

A.M.

1 3 5 7 9 11 1 3 5 7 9 11 P.M.

A +B A

B –20

–10

–100 0 10 20 30 50 60

10 20 30 40 50 60 70

y

x A

A –B B

40

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Overview

xiv Comparing Quantities Overview

Level III

analysis

Level II

connections

Levels of Reasoning

Questions Posed Domains of Mathematics

Level I

reproduction

algebra

geometry

number

statistics &

probability

X O

easy

difficult

Student Assessment in Mathematics in Context

As recommended by the NCTM Principles and Standards for School Mathematics and research on student learning, classroom assessment should be based on evidence drawn from several sources. An assessment plan for a Mathematics in Context unit may draw from the following overlapping sources:

•

observation—As students work individually or in groups, watch for evidence of their understanding of the mathematics.

•

interactive responses —Listen closely to how students respond to your questions and to the responses of other students.

•

products —Look for clarity and quality of thought in students’

solutions to problems completed in class, homework, extensions, projects, quizzes, and tests.

Assessment Pyramid

When designing a comprehensive assessment program, the assessment tasks used should be distributed across the following three dimensions:

mathematics content, levels of reasoning, and difficulty level. The Assessment Pyramid, based on Jan de Lange’s theory of assessment, is a model used to suggest how items should be distributed across these three dimensions. Over time, assessment questions should

“fill” the pyramid.

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Overview

Overview Comparing Quantities xv

Levels of Reasoning

Level I questions typically address:

•

recall of facts and definitions and

•

use of technical skills, tools, and standard algorithms.

As shown in the pyramid, Level I questions are not necessarily easy. For example, Level I questions may involve complicated computation problems. In general, Level I questions assess basic knowledge and procedures that may have been emphasized during instruction. The format for this type of question is usually short answer, fill-in, or multiple choice. On a quiz or test, Level I questions closely resemble questions that are regularly found in a given unit substituted with different numbers and/or contexts.

Level II questions require students to:

•

integrate information;

•

decide which mathematical models or tools to use for a given situation; and

•

solve unfamiliar problems in a context, based on the mathematical content of the unit.

Level II questions are typically written to elicit short or extended responses. Students choose their own strategies, use a variety of mathematical models, and explain how they solved a problem.

Level III questions require students to:

•

make their own assumptions to solve open-ended problems;

•

analyze, interpret, synthesize, reflect; and

•

develop one’s own strategies or mathematical models.

Level III questions are always open-ended problems.

Often, more than one answer is possible and there is a wide variation in reasoning and explanations.

There are limitations to the type of Level III problems that students can be reasonably expected to respond to on time-restricted tests.

The instructional decisions a teacher makes as he or she progresses through a unit may influence the level of reasoning required to solve problems. If a method of problem solving required to solve a Level III problem is repeatedly emphasized during instruction, the level of reasoning required to solve a Level II or III problem may be reduced to recall knowledge, or Level I reasoning. A student who does not master a specific algorithm during a unit but solves a problem correctly using his or her own invented strategy may demonstrate higher-level reasoning than a student who memorizes and applies an algorithm.

The “volume” represented by each level of the Assessment Pyramid serves as a guideline for the distribution of problems and use of score points over the three reasoning levels.

These assessment design principles are used throughout Mathematics in Context. The Goals and Assessment charts that highlight ongoing assessment opportunities — on pages xvi and xvii of each Teacher’s Guide — are organized according to levels of reasoning.

In the Lesson Notes section of the Teacher’s Guide, ongoing assessment opportunities are also shown in the Assessment Pyramid icon located at the bottom of the Notes column.

Assessment Pyramid

6, 7

Develop reasoning skills to solve equation-like problems.

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Overview

xvi Comparing Quantities Overview

Goals and Assessment

In the Mathematics in Context curriculum, unit goals, organized according to levels of reasoning described in the Assessment Pyramid on page xiv, relate to the strand goals and the NCTM Principles and Standards for School Mathematics. The Mathematics in Context curriculum is designed to help students demonstrate their

understanding of mathematics in each of the categories listed below. Ongoing assessment opportunities are also indicated on their respective pages throughout the teacher guide by an Assessment Pyramid icon. It is important to note that the attainment of goals in one category is not a prerequisite to attaining those in another category. In fact, students should progress simultaneously toward several goals in different categories. The Goals and Assessment table is designed to support preparation of an assessment plan.

Level I:

Conceptual and Procedural Knowledge

Ongoing Unit

Goal Assessment Opportunities Assessment Opportunities 1. Use pictures, words, or Section A p. 2, #4 Quiz 1 #1

symbols to solve problems. p. 3, #5 Test #1a, 2

2. Use exchanging, Section C p. 18, #10 Quiz 1 #1 substituting, and other Section D p. 25, #8 Quiz 2 #1ab, 2 strategies to solve problems. Section E p. 31, #18 Test #1c, 2 3. Interpret and use Section B p. 11, #18 Quiz 1 #2, 3abc

combination charts, p. 13, #24 Quiz 2 #1b, 2

notebook notation, and Section D p. 25, #8 Test #1bc, 3, 4 equations.

4. Develop an understanding Section E p. 28, #1ab Test #4 of equation and variable. p. 29, #12

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Overview

Overview Comparing Quantities xvii

Level II:

Reasoning, Communicating, Thinking,

and Making Connections

Level III:

Modeling, Generalizing, and Non-Routine Problem Solving

Goal Assessment Opportunities Assessment Opportunities 5. Describe one’s own Section C p. 19, #13 Quiz 1 #2, 3c

solution process and follow Test #2, 5

the solution process of someone else.

6. Discover, investigate, and Section B p. 9, #16 Test #1c

extend patterns. p. 11, #18

Section C p. 19, #13

7. Organize information from Section E p. 28, #4 Test #3, 5 problem situations using p. 30, #13

combination charts, p. 31, #19

notebook notation, and equations.

8. Interpret a mathematical Section B p. 12, #21 Quiz 1 #3ac solution in terms of the Section C p. 19, #13

problem situation.

9. Develop reasoning skills Section B p. 15, Quiz 2 #1b to solve equation-like For Further Reflection

problems. Section C p. 17, #6, 7

Goal Assessment Opportunities Assessment Opportunities 10. Recognize similarities in Section C p. 19, #12

solution strategies. p. 21,

For Further Reflection

11. Recognize the advantages Section D p. 27, Test #2, 5 and disadvantages of For Further Reflection

various solution strategies Section E p. 33, and choose strategies For Further Reflection appropriate for the problem

situation.

12. Informally solve problems Section C p. 19, #12 Test #4 that involve systems of Section D p. 25, #8

equations. Section E p. 28, #5

p. 30, #17 p. 31, #19

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Overview

The following items are the necessary materials and resources to be used by the teacher and students throughout the unit. For further details, see the Section Overviews and the Materials section at the top of the Hints and Comments column on each teacher page. Note: Some contexts and problems can be enhanced through the use of optional materials. These optional materials are listed in the corresponding Hints and Comments column.

Student Resources

Quantities listed are per student.

•

Letter to the Family

•

Student Activity Sheets 1–5

Teacher Resources

No resources required

Student Materials

Quantities listed are per pair of students.

•

Colored pencils, one box

xviii Comparing Quantities Overview

Materials Preparation

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Student

Material

Teaching and

Notes

Student

Material

Teaching and

Notes

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Teachers Matter

v Comparing Quantities Teachers Matter

Teachers Matter

Teachers Matter vT

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.

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

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, T

The Mathematics in Context Development Team

$50.00

Number of Erasers Combination Chart

Number of Pencils

0 1 2 3

15 40 65

55

0 25

TACO ORDER

1 2 3 4 5 6 7

SALAD DRINK TOTAL 3.00 41 44

2 8.00

— 2

— 1

11.00

$

$$

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1A Comparing Quantities Teachers Matter

Teachers Matter A

Section Focus

The focus of this section is that students exchange goods and quantities to solve problems involving bartering, balances, and simple equations. In this way, they informally solve problems involving systems of equations.

Pacing and Planning

Day 1: Bartering Student pages 1 and 2

INTRODUCTION Problem 1 Exchange goods in a bartering situation.

CLASSWORK Problems 2 and 3 Exchange quantities to balance a scale.

HOMEWORK Problem 4 Use exchange strategies to determine

quantities that satisfy simple situations.

Day 2: Tug-of-War Student pages 3 and 4

INTRODUCTION Review homework. Review homework from Day 1.

CLASSWORK Problem 5 Use exchange strategies to determine

which team of animals will win a tug of war.

HOMEWORK Check Your Work and Student self-assessment: Use exchange

For Further Reflection relationships in a trading situation to solve problems.

Additional Resources: Algebra Tools; pages 3, 45, and 46. Additional Practice, Section A, Student Book page 34

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Teachers Matter Section A: Compare and Exchange 1A

Teachers Matter A

Materials

Student Resources

Quantities listed are per student.

•

Letter to the Family Teachers Resources No resources required Student Materials No materials required

* See Hints and Comments for optional materials.

Learning Lines

Fair Exchange

In this section, students use their informal understanding of bartering and exchanging items to solve equation-like problems. Within the context of bartering, students are introduced to the concept of substitution, by the method of fair exchange.

Throughout this unit, students will use these exchange situations to develop more sophisticated strategies for solving equation-like problems.

In formal algebra, the exchange process is called substitution.

Equations

In this section, students begin to solve equation-like problems. In formal algebra, these equation-like problems are referred to as a system of equations.

For instance, students are shown two combinations of fruits that balance a scale, and they must exchange, add, and subtract to find what will balance with a single piece of fruit. Exchange or balance situations like this represent equivalent quantities.

The balance represents an equal sign. The total weight of the items on the left side equals the total weight of the items on the right side.

Variable

In this section, the variables are presented in pictures. Students are encouraged to use words or symbols to record and explain the process used to solve the problems.The problems in this section involve the number of items rather than the monetary value of the items. In later sections, students find the cost or weight of each item and the cost or weight of a combination of items.

At the End of the Section:

Learning Outcomes

Students use their informal knowledge of bartering and exchanging to solve problems. Students develop their reasoning skills. They informally solve systems of equations. Students use pictures, words, or symbols to describe one’s own solution process, and they are able to follow the solution process of someone else.

10 bananas 2 pineapples 1 pineapple 2 bananas 1 apple

1 apple

6 carrots 1 corn 1 pepper

1 corn 2 peppers 1 pepper

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Compare and Exchange A

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 they didn’t produce themselves? They traded something they produced for the things their neighbors produced. This method of exchange is calledbartering.

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.

A Compare and Exchange

Bartering

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.

1. Show what Paulo can do.

Reaching All Learners

Act It Out

Have students play the roles of Paulo, Aaron, Sarkis, Ranee, and Nina. The pictures on page 1 can be enlarged, cut out, and handed to the students.

Have students act out the bartering problem.

Hands-On Learning

It may help students if you have labeled cards or cards with pictures on them so students can physically make the trades at their desks.

Vocabulary Building

Discuss the term bartering. Have students begin a vocabulary section in their notebook and add this term to it.

1 Comparing Quantities

Notes

1Point out to students that although this problem has a lot of text and may sound confusing, students KNOW that they can find the solution. (See Act It Out and Hands-On Learning in the Reaching All Learners section below.)

You may need to simplify the text in this problem for English Language Learners.

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Section A: Compare and Exchange 1T

Hints and Comments

Materials

pictures of sheep, goats, salt, chickens, and corn, optional

Overview

Students experiment with different exchange strategies involving pictures, diagrams, and lists to solve a bartering problem.

About the Mathematics

The pictured items are mathematically spoken variables. They each represent a certain value;

however, their values stay unknown. The information given is how each item is related to another item.

This makes it possible to exchange items for other items, which is a start of the development of the concept of substitution. In the Solutions column, the tree diagram in Strategy 3 for problem 1 is similar to a method used in a previous Algebra unit, Expressions and Formulas.

Comments About the Solutions

1. Allow students to struggle with this exchange problem. Encourage them to develop their own methods for solving it. The exchange from salt to corn may be the most difficult for students since the ratio of salt to corn is 3:2.

Writing Opportunity

Ask students to write a journal entry describing a personal experience in which they traded or bartered to get something they wanted. You may use this writing opportunity to point out that bartering is still useful in modern society.

Did You Know?

Bartering means trading goods and services rather than paying money for goods and services. Buying goods with money is more convenient than trading because money is easily carried and can be divided more easily than, say, one bull. Unlike foodstuffs, money will not spoil.

Solutions and Samples

1. Strategies may vary. Sample strategies:

• Some students may use logical reasoning.

Paulo should trade the 2 sheep for 12 bags of salt and the goat for 6 chickens. Then he should trade the 6 chickens for 3 bags of salt. Paulo will have a total of 15 bags of salt. He should trade the 15 bags of salt for 10 bags of corn and go home.

• Some students may use pictures.

• Some students may use a tree diagram.

• Some students may organize the solution according to the exchange for each merchant.

1st: Nina 2 sheep➞ 12 bags of salt 2nd: Ranee 1 goat➞ 6 chickens 3rd: Aaron 6 chickens➞ 3 bags of salt 4th: Sarkis 15 bags of salt➞ 10 bags of corn

Goat

Chickens (6)

Salt (3)

Corn (10) Sheep (2)

Salt (12)

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Compare and Exchange A

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?

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

Compare and Exchange

A

Farmer’s Market

Thirst Quencher

6

4

10 bananas 2 pineapples 1 pineapple 2 bananas 1 apple 1 apple

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

Reaching All Learners

Actually pouring water from one container into another may help some students solve problem 4.

Extension

You may have students create their own balance problems and show their solutions. Ask students to use at least three different objects in each problem. Have them explain their reasoning.

Assessment Pyramid

4

Use pictures, words, or symbols to solve problems.

Notes

2–4Encourage students to use whatever method they prefer. Some students may feel more comfortable using pictures. Others may use letters or other symbols. Look for clarity in their explanations.

4Students struggling with this problem may need you to point out that if 6c 3 small bottles then 1 small bottle 2 cups.

This way they will have a multiple of 8 to work with.

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Hints and Comments

Materials

pictures of sheep, goats, salt, chickens and corn, optional;

a variety of sizes of cups and bottles, optional

Overview

Students solve problems that involve balance and measurement by using the concept of exchanging.

About the Mathematics

The balance represents an equal sign. The total weight of the items on the left side equals the total weight of the items on the right side. Students will continue to develop their understanding of the concept of variable. They may even start to use letters as a symbol.

As shown in the Solutions column, some students may assign a numerical value to the items to help them solve the problem.

Comments About the Solutions

4. If students are having difficulty, ask What are you trying to find out? (The number of cups in one large bottle.) and From the picture, what do you know about the large bottle? (It holds four medium bottles.) Encourage students to use that information to determine the number of cups in the large bottle.

Other students may be familiar with the relationships among the quantities:

• 2 cups 1 pint

• 2 pints 1 quart

• 4 cups 1 quart

• 4 quarts 1 gallon

Solutions and Samples

2. Three bananas. Sample strategies:

• Some students may draw pictures.

• Some students may use words in equations.

10 bananas 2 pineapples, so 5 bananas 1 pineapple.

1 pineapple 2 bananas 1 apple.

5 bananas 2 bananas 1 apple, so 3 bananas 1 apple.

• Some students may assign an arbitrary amount for the weight of each item.

From the scale on Student Book page 2, if each banana weighs 1 kilogram, then each

pineapple must weigh 5 kilograms. So the middle scale must have 5 kilograms on the left and 2 kilograms and one apple on the right.

Then one apple must weigh 3 kilograms, which is the same weight as three bananas.

3. Two carrots. Explanations will vary. Sample explanation:

1 corn 2 peppers.

6 carrots 1 corn 1 pepper, so 6 carrots 2 peppers 1 pepper.

6 carrots 3 peppers, so 1 pepper 2 carrots.

4. 16 cups. Explanations will vary. Students may use methods similar to those in problem 2.

Sample explanation:

Using the information in line 2, begin by substituting two small bottles for the medium bottle in line 1 and proceed as shown below.

6 cups 3 small bottles 2 cups 1 small bottle 4 cups 1 medium bottle, so 4 medium bottles 16 cups 1 large bottle

so,

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Compare and Exchange A

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.

Compare and Exchange A

Tug-of-War

Reaching All Learners

Intervention

You might provide each group with a blank transparency so they can use the overhead projector to show the strategy used to solve this exchange problem to the class.

Students may continue to need cutout animal shapes to explain their solutions.

Advanced Learners

Have students find the amount in horses for 1 elephant and 3 horses or ask them to find how much stronger 1 elephant is than 4 oxen.

Assessment Pyramid

5

Use pictures, words, or symbols to solve problems.

Notes

5Students don’t need to find an exact amount to solve this problem. Expect them to be able to tell you that 1 elephant and 3 horses are stronger than 4 oxen.

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Hints and Comments

Materials

blank transparencies, optional (one per group of students);

overhead projector, optional (one per class);

copies of animal shapes, optional (five horses, four oxen, one elephant per group of students)

Overview

Students solve a more complex exchange problem involving the strength of horses, elephants, and oxen.

About the Mathematics

When letters are used to solve exchange problems, each letter represents an actual object. Later in the unit, letters will be used to represent the value or quantity of each object. Make sure that students are aware of the meaning of the letters as they progress through the problems.

These exchange problems provide a new context for the use of symbolic expressions and an opportunity to introduce students to basic computation with symbols, such as the fact that 3 A, or 3A, stands for A A A.

Comments About the Solutions

5. You can discuss the advantages and disadvantages of each strategy shown in the Solutions column, but do not label one strategy as better than another. If students use letters to solve the problem, you may discuss this as a handy shortcut. Some students may feel more comfortable using pictures or words than letters.

The fourth strategy works because the point values are chosen so that all the equalities in the problem hold. Other point values will work as long as the ratio of horses to oxen to elephants is 4:5:13.

Solutions and Samples

5. The elephant and three horses will win the tug-of- war. Strategies will vary. Sample strategies:

• Some students may draw pictures and use substitution.

Substitute two horses and one ox for the elephant in the third picture.

Subtract one ox from each side.

From the first picture in the problem, five horses are as strong as four oxen.

So the elephant and the three horses will win.

• Some students may substitute five horses for the four oxen in the third picture. The result is one elephant and three horses versus five horses, or one elephant versus two horses. The second picture shows that one elephant is as strong as two horses and one ox, so the elephant and three horses will win.

• Some students may use substitution with symbols to represent the animals.

1st picture: O O O O H H H H H 2nd picture: E O H H

3rd picture: E H H H O O O O

Substitute E O H H into the 3rd picture:

O H H H H H O O O O

Substitute five horses (see first picture) for the four oxen on the right:

O H H H H H H H H H H

So the left side will win because you have five horses on each side with one ox on the left.

• Some students may assign points to each animal.

1 horse 4 points, then 1 ox 5 points (first picture), and 1 elephant 13 points (second picture).

Adding points in the third picture yields 25 points versus 20 points, indicating that the left side will win.

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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.

A

Reaching All Learners

Parent Involvement

Have parents review the section with their child to relate the Check Your Work problems to the problems from the section.

Vocabulary Building

The term fair exchange is introduced in the Summary on this page.

Students can add this term to the vocabulary section of their notebook.

Notes

Read the Summary

problem to your class. Have them give you examples of how words, pictures, and symbols are used to help them solve the problems in this section.

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Hints and Comments

Materials

play money kits, optional (one per group)

Overview

Students read the Summary, which reviews the main concepts covered in this section. They then use various strategies to solve additional exchange problems in a bartering context. Students use these Check Your Work problems as self-assessment. The answers to these problems are also provided in the Student Book.

Planning

The Check your Work problems may be assigned as homework.

After students complete Section A, you may assign for homework appropriate activities in the Additional Practice section, located on page 34 of the Student Book.

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Compare and Exchange A

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

2. Use the information to write two more statements about 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.

Reaching All Learners

Intervention

As a warm-up activity for the next few class sessions, you may apply the exchange or substitution principle to money using nickels, dimes, quarters, and dollars.

Extension

To see how well students understand exchanging, you may have them create their own problems. If they have trouble finding a context, suggest a bartering problem or one involving a balance, as in problems 2 and 3.

Assessment Pyramid

5, FFR 1, 2, 3, 4 Assesses Section A Goals

Notes

For Further Reflection Reflective questions are meant to summarize and discuss important concepts.

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Section A: Compare and Exchange 5T Because four apples is the same as one loaf of bread, three fish are not enough.

• I found in problem 2 that one fish can be traded for one apple, so three fish will be worth only three apples.

Because four apples are worth one loaf of bread, three fish are not enough.

5. 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 2 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.

Solutions and Samples

Answers to Check your Work

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 melons four apples for one loaf of bread

one melon for one ear of corn and two apples 10 apples for four melons

2. 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 for two loaves of bread one melon for five ears of corn two melons for five apples eight ears of corn for one loaf of bread two ears of corn for one apple one fish for one apple two ears of corn for one fish

four fish for one loaf of bread 3. 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.

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

Sample explanations:

• I found in problem 2 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 2 that one fish can be traded for one apple, so three fish will be worth only three apples.

Hints and Comments

Materials

play money kits, optional (one per group)

Overview

Students use these Check Your Work problems as self- assessment. The answers to these problems are provided in the Student Book.

Comments About the Solutions

5. This is an open question; the number of fish that Delia has to trade for corn is not stated. Some students may trade for some corn, while others may trade until they have only corn.

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6A Comparing Quantities Teachers Matter

Teachers Matter B

Section Focus

This section introduces combination charts to organize information about combinations of two items. Students explore exchange patterns in the combination charts and use the patterns to solve problems.

Pacing and Planning

Day 3: The School Store Student pages 6 and 7

INTRODUCTION Problems 1–3 Determine how many pencils and erasers

were purchased at the student store.

CLASSWORK Problems 4 and 5 Create a table that lists the costs of pencils and erasers and use the table to solve problems.

HOMEWORK Problem 6 Describe a method for combining the lists

of costs for pencils and erasers.

Day 4: The School Store (continued) Student pages 8 and 9

INTRODUCTION Problem 7 Interpret the meaning of the entries in a

combination chart.

CLASSWORK Problems 8–14 Describe number patterns and exchange

patterns in a combination chart.

HOMEWORK Problems 15 and 16 Describe exchange patterns on the

combination chart in terms of pencils and erasers.

Day 5: Workroom Cabinets Student pages 10 –13

INTRODUCTION Problem 17 Determine the number of cabinets that fit

along a wall.

CLASSWORK Problems 18–23 Investigate a combination chart for short

and long cabinets.

HOMEWORK Problem 24 Solve combination chart puzzles.