Middle School Mathematics Assessments · at The University of Texas at Austin ... Middle School Mathematics Assessments: ... the questions included with the assessments to guide learning

Middle SchoolMathematics Assessments

The Charles A. Dana Centerat The University of Texas at Austin

With funding fromthe Texas Education Agency andthe National Science Foundation

Proportional Reasoning

iii

About the Charles A. Dana Center’s Work in Mathematics and Science

The Charles A. Dana Center at The University of Texas at Austin works to support education leaders and policymakers in strengthening Texas education. As a research unit of UT Austin’s College of Natural Sciences, the Dana Center maintains a special emphasis on mathematics and science education. We offer professional development institutes and produce research-based mathematics and science resources for educators to use in helping all students achieve academic success. For more information, visit the Dana Center website at www.utdanacenter.org.

The development of this work was supported in part by the Texas Education Agency, the National Science Foundation under cooperative agreement #ESR-9712001, and the Charles A. Dana Center at The University of Texas at Austin. Any opinions, fi ndings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily refl ect the views of the Texas Education Agency, the National Science Foundation, or The University of Texas at Austin.

Permission is given to any person, group, or organization to copy and distribute this publication, Middle School Mathematics Assessments: Proportional Reasoning, for noncommercial educational purposes only, so long as the appropriate credit is given. This permission is granted by the Charles A. Dana Center, a unit of the College of Natural Sciences at The University of Texas at Austin.

First printing February 2004

2004, The University of Texas at Austin. All rights reserved.

iv

Authors

Brenda DeBorde, Independent Mathematics ConsultantMary Alice Hatchett, Independent Mathematics ConsultantGena Silvas Lopez, Corpus Christi Independent School DistrictVicki Veigel Massey, Austin Independent School DistrictMaggie Myers, Charles A. Dana CenterPatricia A. Rossman, Austin Independent School DistrictTeresa Salas, Corpus Christi Independent School DistrictCarmen Whitman, Charles A. Dana Center

Charles A. Dana Center Production Team

Carmen Whitman, EditorDiane McGowan, Co-EditorBonnie McNemar, Co-EditorSusan Hudson Hull, Co-EditorAmy Dolejs, Production EditorPhil Swann, DesignerGeoff Potter, Graphics Assistant

Content EditorEva Gates, Independent Mathematics Consultant

Advisory TeamAmy L. Anding, Bryan Independent School DistrictCatherine Banks, Texas Woman’s UniversityLisa C. Brown, Austin Independent School DistrictShirl Chapman, Region 7 Education Service CenterSusan B. Empson, University of Texas at AustinYvonne Grant, Michigan State UniversityLynn Granzin, Region 15 Education Service CenterJulia Hankin, Austin Independent School DistrictSusan Hudson Hull, Charles A. Dana Center Ted H. Hull, Charles A. Dana CenterCarl Juenke, Charles A. Dana CenterGarland Linkenhoger, McAllen Independent School DistrictLaurie Mathis, Charles A. Dana CenterDiane McGowan, Charles A. Dana CenterTerri McLaughlin, Spring Branch Independent School DistrictBonnie McNemar, Charles A. Dana CenterBarbara J. Montalto, Texas Education AgencyLois G. Moseley, Mathematics EducatorMaggie Myers, Charles A. Dana Center

v

Rebecca Ontiveros, Region 19 Education Service CenterErika Pierce, Charles A. Dana CenterJoyce Schaeffer, North Lamar Independent School DistrictCathy L. Seeley, University of Texas at AustinCaren Sorrells, Birdville Independent School DistrictVanessa Spiegel, Port Neches-Grove Independent School DistrictPam Summers, Lubbock Independent School DistrictFrances M. Thompson, Texas Woman’s UniversityEmma Treviño, Charles A. Dana CenterLuAnn Weynand, North East Independent School District

With Special Thanks To:Brian Sargent, Charles A. Dana CenterHee-Joon Kim, Charles A. Dana CenterMountain Creek Neighborhood Kids, Pfl ugerville, TexasMartin Middle School Students, Corpus Christi, TexasBaker Middle School Students, Corpus Christi, TexasBedichek Middle School Students, Austin, Texas

vi

TEKS and TAKS Resources

The mathematics Texas Essential Knowledge and Skills (TEKS) were developed by the state of Texas to clarify what all students should know and be able to do in mathematics in kindergarten through grade 12. Districts are required to provide instruction that is aligned with the mathematics TEKS, which were adopted by the State Board of Education in 1997 and implemented statewide in 1998. The mathematics TEKS also form the objectives and student expectations for the mathematics portion of the Texas Assessment of Knowledge and Skills (TAKS) for grades 3 through 10 and for the Grade 11 Exit Level assessment.

The mathematics TEKS can be downloaded in printable format, free of charge, from the Texas Education Agency website (www.tea.state.tx.us/teks). Bound versions of the mathematics and science TEKS are available for a fee from the Charles A. Dana Center at The University of Texas at Austin (www.utdanacenter.org).

Resources for implementing the mathematics TEKS, including professional development opportunities, are available through the Texas Education Agency and the Charles A. Dana Center, formerly the state-designated Mathematics Center for Educator Development. Online resources can be found in the Mathematics TEKS Toolkit at www.mathtekstoolkit.org.

Additional products and services that may be of interest are available from the Dana Center at www.utdanacenter.org. These include the following:

• TEKS, TAAS, and TAKS: What’s Tested at Grades 3–8? charts

• Mathematics Abridged TEKS charts

• Mathematics TEKS “Big Picture” posters

• Mathematics Standards in the Classroom; Resources for Grades 3–5

• Mathematics Standards in the Classroom; Resources for Grades 6–8

• Algebra I Assessments and the corresponding professional development

• Geometry Assessments and the corresponding professional development

• Algebra II Assessments and the corresponding professional development

• TEXTEAMS professional development mathematics institutes

• TEKS for Leaders professional development modules for principals and other administrators

vii

CONTENTS

Introduction xi

Middle School Mathematics Assessment Solution Guide xiv

Mathematics TEKS Alignment xv

Mathematics Grade 6 TAKS Alignment xxi

Mathematics Grade 7 TAKS Alignment xxii

Mathematics Grade 8 TAKS Alignment xxiii

Chapter 1: Number, Operation andQuantitative Reasoning 1

Overarching Fun Park Party Grade 6 3 Fun Park Party Grade 7 13 Fun Park Saturday Grade 8 17

Grade 6 Homecooking Chili 23 Spring Sensations 27

Grade 7 Bargain Shopping 31 Rose Garden Plan 37

Grade 8 Talk, Talk, Talk 43 Half-Life Happening 48

Chapter 2: Patterns, Relationships, andAlgebraic Thinking 55

Overarching At Home in Space Grade 6 57 Solar Cells for Science Grade 7 61 City in Space Grade 8 67

Grade 6 Community Clean-Up 75 Towering Pizzas 85

viii

Grade 7 South Texas Natives 95 Working Smarter 99

Grade 8 Fast Food Workout 103 Global Warming: Texas-Size 109

Chapter 3: Geometry and Spatial Reasoning 115

Overarching By the Sea Grade 6 117 By the Sea Grade 7 123 By the Sea Grade 8 135

Grade 6 Sweet Trip to the Candy Shop 147 The Round Table 155

Grade 7 Sorting Rectangles 159 Mighty Mascot! 163

Grade 8 Javier Builds a Model 169 Student Council President 175

Chapter 4: Measurement 185

Overarching Lights at the Marleys’ and Farleys’ Grade 6 187 Gardens at the Marleys’ and Farleys’ Grade 7 193 Storage Boxes at the Marleys’ and Farleys’ Grade 8 201

Grade 6 Extravaganza 211 Matchmaker 221

Grade 7 Bug Juice 235 Photographic Memories 241

ix

Grade 8 Liberty Enlightening the World 251 In the Rafters 257

Chapter 5: Probability and Statistics 263

Overarching Perplexing Polygons Grade 6 265 Perplexing Polygons Grade 7 270 Perplexing Polygons Grade 8 279

Grade 6 Science Quiz 287

Grade 7 Big Money Prizes 293

Grade 8 Five Friends 297

Chapter 6: Underlying Processes andMathematical Tools 305

Overarching What’s in Your Wallet? Grade 6 307 What’s in Your Wallet? Grade 7 315 What’s in Your Wallet? Grade 8 323

Grade 6 Gone Fishin’ 331 Secret Recipe 343

Grade 7 Rx 353 It’s a Weighty Matter 361

Grade 8 Java Joe’s 369 How Green is Green? 379

x

xi

1 National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics.

Reston, VA: National Council of Teachers of Mathematics.

Introduction

The Dana Center developed Middle School Mathematics Assessments: Proportional Reasoning as a resource for teachers to use to provide ongoing assessment integrated with middle school mathematics instruction.

Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) states: “Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.” 1 Further, NCTM (1995) identifi ed the following six standards to guide classroom assessment: 2

Standard 1: Assessment should refl ect the mathematics that all students needto know and be able to do.

Standard 2: Assessment should enhance mathematics learning.

Standard 3: Assessment should promote equity.

Standard 4: Assessment should be an open process.

Standard 5: Assessment should promote valid inferences about mathematics learning.

Standard 6: Assessment should be a coherent process.

Implementing these assessment standards may require signifi cant changes in how teachers view and use assessment in the mathematics classroom. Teachers should assess frequently to monitor individual performance and guide instruction.

What is Middle School Mathematics Assessments:Proportional Reasoning?

Middle School Mathematics Assessments: Proportional Reasoning contains problems that refl ect what all students need to know and be able to do in sixth-, seventh-, and eighth-grade mathematics. The resource focuses on exploring proportionality through the content and process strands. These assessments may be formative, summative, or ongoing. The problems focus on students’ conceptual understanding as well as their procedural knowledge. The tasks require more than right or wrong answers; they focus on how students are thinking about a situation.

Why focus on proportional reasoning?

The “big idea” of proportionality, including proportional relationships and proportional reasoning, is central to an understanding of middle and high school mathematics, and

2 National Council of Teachers of Mathematics. 1995. Assessment Standards for School Mathematics. Reston, VA:

National Council of Teachers of Mathematics.

xii

provides a critical foundation for formal algebra study. The concept of proportionality operates as a single thread running through and connecting each of the content and process strands of middle school mathematics.

Proportionality involves recognizing quantities that are related by multiplication. Numbers, tables, graphs, words, and equations are used to think about the quantities and their relationships. Fluency with proportionality develops through problem solving and reasoning in many areas, including ratio and proportion, percent, similarity, scaling, linear equations, slope, and probability.

What is the purpose of Middle School Mathematics Assessments: Proportional Reasoning?

The purpose of these assessments is to make clear to teachers, students, and parents what is being taught and learned about proportionality throughout middle school mathematics. Teachers should use evidence of student insight, student misconceptions, and student problem-solving strategies to guide their instruction. Teachers may also use the questions included with the assessments to guide learning and to assess student understanding. The use of these assessments should help teachers enhance student learning and provide them with a source of evidence on which they may base their instructional decisions.

What is the format of Middle School Mathematics Assessments: Proportional Reasoning?

This book contains 51 problems divided by chapter according to their TEKS strand. The problems address how proportionality can be used to show understanding of the TEKS.

The problems have been divided into six categories:

Number, Operation, and Quantitative Reasoning

Patterns, Relationships, and Algebraic Thinking

Geometry and Spatial Reasoning

Measurement

Probability and Statistics

Underlying Processes and Mathematical Tools

Each problem

• includes a mathematics task,• is aligned with the Grades 6, 7, and 8 mathematics Texas Essential Knowledge and

Skills (TEKS) student expectations,• is aligned with the Texas Assessment of Knowledge and Skills (TAKS) objectives,• includes “scaffolding” questions that the teacher may use to help the student to

analyze the problem,

xiii

• provides a sample solution,* and• includes extension questions to bring out additional mathematical concepts in a

summative discussion of solutions to the problem.

*The sample solution is only one way that a problem may be approached and is not necessarily the “best” solution. For many of the problems there are other approaches that will also provide a correct analysis of the problem. The authors have attempted to illustrate a variety of methods in the different problem solutions. Several of the problems include samples of anonymous student work.

Following this introduction are alignments of all the problems to the TEKS and to the grade-level TAKS objectives.

What is the solution guide?

The solution guide is a problem-solving checklist that may be used to understand what is necessary for a complete problem solution. When assigning the problem, the teacher will give the students the solution guide and will indicate which of the criteria should be considered in the problem analysis. In most problems all of the criteria are important, but initially the teacher may want to focus on only two or three criteria. On the page before a student work sample in this book, comments on some of the criteria that are evident from the student’s solution are given. The professional development experience described below will help the teacher use this tool in the classroom and will also help guide the teacher to use other assessment evaluation tools.

TEXTEAMS Practice-Based Professional Development—Middle School Mathematics Assessments: Proportional Reasoning

The Dana Center has developed a three-day TEXTEAMS institute in which participants experience selected assessments, examine the assessments for alignment with the TEKS and TAKS, analyze student work to evaluate student understanding, consider methods for evaluating student work, view a video of students working on the assessments, develop strategies for classroom implementation, and consider how the assessments support the TAKS. Teachers should contact their local school district or regional service center to determine when this institute is offered.

xiv

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