grade four sampler

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G R A D E F O U R S A M P L E R

TH

I R D E D I T I O

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© 2008 Third Edition SampleLessons and OverviewThe purpose of this sampler is toprovide you with a comprehensiveoverview of the Math TrailblazersGrade 4 curriculum.

Foundations ofMath Trailblazers

2

Table of ContentsFoundations of Math Trailblazers............................................2Welcome ...............................................................................3Teacher Materials ..................................................................4Teacher Resources ................................................................5Student Components .............................................................6Kit Materials ..........................................................................7Unit Planning .........................................................................8Grade 4 Units and Lessons..................................................14Lesson Overview .................................................................16Student Pages .....................................................................27Assessment.........................................................................30Connections to Science .......................................................34Connections to Language Arts .............................................35Family Support ....................................................................36Math Facts ..........................................................................38A Curriculum for All .............................................................39What’s New in the Third Edition?......................................Back

Trailblazing MathMath Trailblazers is a complete, research-based, NationalScience Foundation-funded, K–5 mathematics program integrating math, science, and language arts. It embodies theNational Council for Teachers of Mathematics’ (NCTM) Principlesand Standards for School Mathematics and is based on theideas that mathematics is best learned by solving problems in real-world contexts and that a curriculum should balanceconceptual understanding and procedural skill.

Standards-Based MathematicsThis curriculum is modeled upon NCTM Standards and followsthe guidelines set by the NCTM’s Principles and Standards for School Mathematics. Through this curriculum, studentsbecome powerful mathematical thinkers by learning how andwhen to apply the lessons presented in Math Trailblazers.

Higher Expectations and EquityUsing approaches backed by solid research, Math Trailblazersengages students who have a wide range of abilities withchallenging content. Still, the program has the flexibility neededto reach the special-needs child. This edition contains newresources for teachers and administrators as they work withbilingual, special education, and talented and gifted students.

Connections to Science and Language ArtsConnecting math with other subjects greatly increases studentinterest and retention. Math Trailblazers takes advantage ofchildren’s natural curiosity about the world around them byusing science as a context to engage them in mathematicallybased investigations. Math Trailblazers also has many ties toLanguage Arts. Students use rich language as they exploremathematics. They become stronger communicators whenthey record solutions, discuss strategies, and record theirfindings in journals. Some lessons engage students throughthe use of trade books or through mathematical adventuresand historical stories.

The connections to science date back to the mid-1980s whena scientist and mathematician began collaborating to developa way for children to learn math and science that closelymatched the way they worked in their labs and classrooms.They assembled a team of math educators, classroom teachers, special education teachers, as well as scientistsand mathematicians to expand this work. This team, calledthe Teaching Integrated Mathematics and Science (TIMS)Project, continues to study the effects of, develop supportmaterials for, and revise Math Trailblazers.

A Balanced ApproachMath Trailblazers offers flexibility in teaching practices,lessons, and assessments to create a careful balancebetween conceptual understanding and procedural skills.Students will acquire problem-solving skills, learn to compute accurately, and develop fluency in the basic facts.

Collaborative WorkScientists, mathematicians, and most others who solve complexproblems in business and industry have always worked ingroups. In Math Trailblazers, a careful balance of individual and group work takes place. Social skills like sharing and cooperation increase. Through the discourse in small groups,students improve their math and communication skills.

A Problem-Solving ProgramMath Trailblazers adheres to the principle that mathematics is best learned through activeinvolvement in solving authentic problems. This strong emphasis on problem solving is thedefining characteristic of Math Trailblazers. Students become powerful mathematical thinkers as they develop concepts, skills, and procedures.

Welcome

3

®

Making Math Meaningful for Over 20 YearsWe are proud to introduce the third edition of MathTrailblazers. The third edition of Math Trailblazersrepresents more than 20 years of research andcollaboration with educators from around the country.

Highlights of the New Edition Include:• Teacher-friendly full-color format for the Unit

Resource Guide• Identified opportunities for differentiation• Reorganization of the Unit Resource Guide File• Administrator Handbook

Research BaseIn developing Math Trailblazers, the authors drew uponresearch findings from a wide variety of sources, as wellas from the broad experiences of the authoring group.With each edition, the authors consider the latest inresearch into mathematics learning, as well as theexperiences of teachers in districts across the country.For a comprehensive description of the research basefor Math Trailblazers, see the Foundations in MathTrailblazers section in the Teacher Implementation Guideor for a summary, visit www.mathtrailblazers.com.

NSF Support and Leadership from NCTM

The National Science Foundation (NSF)awarded a grant to the TIMS Project todevelop a comprehensive, elementarymathematics curriculum that wouldalign with the National Council of

Teachers of Mathematics’ (NCTM) Curriculum andEvaluation Standards for School Mathematics (NCTM,1989). The first edition of Math Trailblazers is the result of this work.

The second edition was revised to align with the currentNCTM recommendations as outlined in the Principles andStandards for School Mathematics (NCTM, 2000). Thisthird edition has an easier-to-use format, with additionalcontent for teachers and administrators.

Continuing Research and Results The TIMS Project continues to do research into theefficacy of Math Trailblazers. This research demonstratesthe advantages of using Math Trailblazers with students ina variety of settings. Districts around the country continueto report the success of using Math Trailblazers with theirstudents. Information on this research and reports fromdistricts using Math Trailblazers can be found in theResearch Base and Student Achievement brochureavailable from Kendall/Hunt by calling 800-542-6657,or visit www.mathtrailblazers.com.

Partners in Your SuccessAt Kendall/Hunt, we understand that the successfulimplementation of any curriculum develops over time.We are dedicated to working with you to design andimplement a long-range support program that fits yourneeds. We also offer a number of additional areas ofongoing support, including the Math Trailblazers ListServ,the Math Trailblazers Voice online newsletter, and the newsupport Web site at www.mymathtrailblazers.com.

Teacher Materials

Teacher Materials

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Unit Resource Guide FileThe Unit Resource Guide File organizes theteacher guides by quarter and fits on a shelffor easy accessibility. It includes:• All 16 Unit Resource Guides for Grade 4• Teacher Implementation Guide• Teacher Portfolio

Unit Resource GuidesA comprehensive guide to teachinglessons that includes:• Unit planner• How-to guide for each lesson• Assessment exercises• Pacing suggestions• Differentiation information• Background information, outline,

and planning tools• Letters home (English and

Spanish)

Teacher Portfolio• A portable Unit Resource Guide

organization tool• Includes program and grade-

level overview with professionaldevelopment details

Teacher Implementation Guide• Meeting individual needs

(ELL, Special Needs, TAG)• Unit overviews• Scope and sequence, and

standards• Philosophy behind math facts

and whole-number operations• Assessment overview• TIMS Tutors—an in-depth explo-

ration of mathematical concepts• Manipulatives, literature, and

software lists

5

Teacher Resources

www.mymathtrailblazers.com Online support for users, including teachers, administrators, and family.Includes lesson modeling videos, homework help, and much more!

Facts Resource GuideThis is a compilation of all the math facts andmaterials for each grade.• Select Daily Practice and Problem exercises• Fact quizzes and games • Math facts lessons• Math facts philosophy• Printable blackline and transparency masters

Administrator HandbookAdministrators can access information on how tosupport the implementation of Math Trailblazersin their school district.

Teacher’s Guide to Practice and Assessment CDAvailable for Grades 3, 4, and 5 and includes existingassessments from the curriculum and a supplementalstudent page builder. These are particularly helpful forteachers who wish to individualize instruction withextra practice or homework, or to prepare students forstandardized testing. Correlated to Math Trailblazersunits and national standards.

Teacher Resource CDThis CD contains all the resources you need, including:

• Daily Practice and Problems (DPP) with and without teacher notes• Observational Assessment Record pages (MS Word and PDF format)• Individual Assessment Record sheets (MS Word and PDF format) • Scope and Sequence• Teacher and Student Rubrics• Printable Blackline and Transparency Masters• Math Facts Calendar and other masters from Facts Resource Guide

FREEdemo

available!

www.mymathtrailblazers.com Online support for users, including teachers, administrators, and family.Includes lesson modeling videos, homework help, and much more!

Student Components

6

Student GuideThe Student Guide delivers the core materials for students includ-ing activities, labs, and games. Students often complete tablesand graphs or use data to solve problems. Key vocabulary termsare presented in boldface type.

Discovery Assignment BookThe Discovery Assignment Book is brimming with student materi-als—activities, flash cards, tables, graphs, and homework—thatcomplement materials in the Student Guide. Space is included forstudents to write in their answers, and pages can be torn out andhanded in to the teacher.

Adventure BookThe Adventure Book is a collection of original stories that engagestudents by showing them how mathematics and science areused in the world outside the classroom. These stories have avariety of themes, ranging from the history of mathematics tofantasies about kid detectives solving mysteries using length and volume.

Adventure Book on Audio CDThe Adventure Book on audio CD gives all students access to these enjoyable,informative stories.

All student

components

are available

in English and

Spanish

Kit Materials

7

Meaningful ManipulativesMath Trailblazers uses a greater and richer variety ofmanipulatives than any other program:• A variety of science and mathematics manipulatives are used in each grade level• The Complete Kit offers enough materials for up to 30 students per grade level• Kits contain both teacher and student materials, including overhead manipulatives

For more information on manipulative kits, callKendall/Hunt Publishing at 1-800-542-6657

Grade 4 Complete Kit Materials List

Quantity Description10 Balances Balance, Junior2 Packages Balls, Super / Package of 610 Balls Balls, Tennis3 Packages Base 10 Block (decimeter cube) / Package of 1015 Sets Base 10 Set12 Packages Connecting Cubes (PopCubes) / Package of 1002 Packages Connecting Cubes (unit), cm, 1 gm / Package of 5002 Sets Eyedroppers, Plastic / 1 Dozen Per Set3 Sets Fraction Pattern Blocks 1/2 cm1 Set Geometric Solids (3-D shapes)15 Cylinders Graduated Cylinder, 100 ml15 Cylinders Graduated Cylinder, 250 ml1 Package Marbles, Large, 3/4 in. / Package of 18015 Packages Marbles, Standard, 5/8 in. / Package of 2530 Metersticks Metersticks, Wood3 Packages Number Cubes (dot dice), 5/8 in., White / Package of 41 Set Overhead Base 10 Set1 Set Overhead Color Tiles, 1 in., 4-Color / Package of 481 Set Overhead Fraction Pattern Blocks1 Package Overhead Pattern Blocks / Package of 494 Packages Pattern Blocks, Plastic, 1/2 cm / Package of 25030 Protractors Protractor10 Ramps Ramp, Incline, Fiberboard 10 in. x 20 in. x 1/8 in., 50 cm30 Rulers Rulers / Transparent, cm, 12 in. Safe-T6 Packages Spinners / Package of 52 Packages Square Inch Color Tiles / Package of 40010 Sets Hexagram Standard Mass 27 / Set6 Totes Rough Tote, 10 gal. (to use as storage)

Unit Planning

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Unit Resource Guide — OutlineEach Unit Resource Guide begins with an Outline that indicates the subject of each lesson, supplies needed, andhow much time each lesson requires. The Outline also includes helpful information such as Pacing Suggestions,Assessment Indicators, related trade books and software, and differentiation opportunities.

Plan and Organize with Confidence

2 URG • Grade 4 • Unit 2 • Outline

OutlineGeometric Investigations: A Baseline Assessment Unit

Major Concept Focus• perimeter

• area

• length

• portfolios and collection folders

• width

• TIMS Laboratory Method

• Student Rubric: Telling

• acute, obtuse, and right angles

• subtraction facts review

• point graphs

• estimating angle size

• communicating problem-solving strategies

This unit has two major goals: to develop basic geometry concepts and to pro-vide opportunities to gather information about students’ mathematical abilitiesand attitudes at the beginning of the school year. In the first lesson, students

review perimeter and area. In the second lesson, Perimeter vs. Length, students use the TIMSLaboratory Method to investigate the relationship between the length and width of rectanglesand their perimeters. Using the Student Rubric: Telling as a guide, students then communicatetheir conclusions.

To solve a problem, Helipads for Antopolis, they must apply area and perimeter concepts andthen communicate their problem-solving strategies. Each student starts a collection folder andplaces work in a portfolio to establish a baseline for documenting growth throughout the year.Students also explore angles in this unit. They identify acute, right, and obtuse angles by explor-ing the amount of turning in an angle. The DPP for this unit reviews the subtraction facts.

EstimatedClass Sessions

15-16

Unit Summary

Pacing Suggestions

• Lesson 2 Perimeter vs. Length is a laboratory investigation in which students must collect, organize, graph,and analyze data using data tables and line graphs. If these skills are new to your students, use the maximumnumber of class sessions to complete the lab.

• Lesson 6 Angles and Lesson 7 Angles in Pattern Blocks introduce angles. Students estimate angle measuresand identify acute, obtuse, and right angles. Students practice these skills in the Daily Practice and Problemsand Home Practice in succeeding units before they measure angles using protractors in Unit 9. Use the recommended number of class sessions for Lessons 6 and 7.

Use the following Assessment Indicators and theObservational Assessment Record that follows theBackground section in this unit to assess studentson key ideas.

A1. Can students use patterns in data tables andgraphs to make and test conjectures?

A2. Can students collect, organize, graph, andanalyze data?

A3. Can students make and interpret pointgraphs?

A4. Can students find the perimeter of polygons?

A5. Can students find the area of polygons?

A6. Can students estimate the size of an angle?

A7. Can students identify acute, obtuse, andright angles?

A8. Can students communicate solutionstrategies?

A9. Do students demonstrate fluency with thesubtraction facts?

Assessment Indicators

These helpful hints assist in schedulingand highlight lessons that review contentand extend mathematical themes to assistin differentiation.

Pacing Suggestions

Provides a description of the unit’s content,including a focus on the systematicapproach to learning the math facts.

Unit Summary and MajorConcept Focus

Assist the teacher in assessing student growthin both conceptual understanding and skills.


• 1 piece of stringabout 39 incheslong per studentpair

• 10 square-inchtiles per student

• 1 ruler per student,optional

• overhead square-inch tiles, optional

• 60 square-inchtiles per studentgroup

• rulers

Lesson 1Investigating

Perimeter andArea

URG Pages 35–49SG Pages 28–30

DAB Pages 15–16

DPP A–DHP Part 3

Lesson 2Perimeter vs.

Length


DPP E–JHP Parts 1–2

Lesson 3Letter to

Myrna

URG Pages 69–80

DPP K–N

Copies/TransparenciesSuppliesLesson Information

Unit Planner

ActivityStudents use string to explore the perimeter ofshapes. They also investigate the area andperimeter of shapes in the imaginary town ofAntopolis using square-inch tiles.

Homework1. Assign the Homework section of Investigating

Perimeter and Area Activity Pages.2. Assign Part 3 of the Home Practice for

additional practice finding area and perimeter.

Assessment1. Use the Student Rubric: Telling to guide

students as they communicate the problem-solving process.

2. Use the Observational Assessment Record todocument students’ abilities to find perimeterand area.

Assessment LabStudents help Myrna the ant plan rectangular run-ways made of square-inch tiles for the Antopolisairport. They apply the TIMS Laboratory Methodto find the relationship between length andperimeter for rectangles of fixed width.

Math FactsDPP items G, H, and I provide practice withaddition and subtraction.

Homework1. Assign the two parts of the Homework section

of the Perimeter vs. Length Lab Pages ondifferent nights.

2. You may assign Parts 1 and 2 of the HomePractice at any time.

Assessment1. Evaluate students’ work on the lab following

the suggestions in the Assessment section ofthe Teacher Implementation Guide.

2. Students should save their work fromPerimeter vs. Length Lab Pages to put in theirportfolios. (See Lesson 5.)

3. Use DPP Task J to assess area and perimeter.

Assessment ActivityStudents write a letter to an ant named Myrnaexplaining the results of their investigation inLesson 2.

Math FactsDPP Bit M is a diagnostic test for the subtractionfacts.

HomeworkHave students revise their letters for homework.

• 1 copy of Two-column DataTable URG Page 45 per student,optional

• 1 copy of Square-Inch GridPaper URG Page 46 per student

• 1 transparency of Square-InchGrid Paper URG Page 46,optional

• 1 transparency or poster ofStudent Rubric: Telling TIG,Assessment section

• 1 copy of ObservationalAssessment Record URG Pages 11–12 to be usedthroughout this unit

• 1 copy of Centimeter GraphPaper URG Page 64 per student

• 1 copy of Three-column DataTable URG Page 65 per student

• 1 transparency of Three-columnData Table URG Page 65

• 1 transparency of CentimeterGraph Paper URG Page 64

• 1 copy of Subtraction Facts:Count-Ups URG Page 31 perstudent

• 1 copy of A Letter to MyrnaURG Page 80 per student

• 1 transparency or poster ofStudent Rubric: Telling TIG,Assessment section

• 1 copy of TIMSMultidimensional Rubric TIG,Assessment section


KEY: SG = Student Guide, DAB = Discovery Assignment Book, AB = Adventure Book,URG = Unit Resource Guide, DPP = Daily Practice and Problems, HP = Home Practice(found in Discovery Assignment Book), and TIG = Teacher Implementation Guide.


2


2


3-4

Unit Planning

9

Indicates all the manipulatives,supplies, and copying or trans-parencies needed for each lesson.

Supplies and Copies/Transparencies

Outlines lesson type and content.

Lesson Information

Lesson 4Helipads for

Antopolis


DPP O–RHP Part 6

Lesson 5Portfolios


DPP S–T

Lesson 6Angles


DPP U–ZHP Parts 4–5

Copies/TransparenciesSuppliesLesson Information

• 1 copy of Subtraction Facts:Count-Backs URG Page 32 perstudent

• 1 copy of Subtraction Facts:Using a Ten URG Page 33 perstudent

• 1 copy of Square-Inch GridPaper URG Page 46 per student, optional

• transparencies of Square-InchGrid Paper URG Page 46

• 1 copy of Subtraction Facts:Doubles and Others URG Page 34 per student

• 2 copies of Angle Circles URGPage 108 per student group of4, copied on heavy stock paperin contrasting colors

Assessment1. Use the students’ work A Letter to Myrna to

assess their current abilities to communicatetheir solution strategies in writing.

2. DPP Bit M is a diagnostic test for thesubtraction facts.

ActivityStudents design helipads for the Antopolisairport. Each helipad must be a rectangle with aperimeter of 24 inches. Students find the designwith the biggest area.

Math FactsDPP Bits O and Q are diagnostic tests for thesubtraction facts.

Homework1. Ask students to explore helipads with

different perimeters. (optional)2. Assign Part 6 of the Home Practice.

Assessment1. You can use Questions 4–7 to assess problem-

solving and communication skills.2. Use the Observational Assessment Record to

document students’ progress in communicat-ing problem-solving strategies.

Assessment ActivityStudents organize collection folders and portfo-lios, choose one or two pieces for the portfolio,and begin a table of contents.

Math FactsDPP Bit S is the fourth diagnostic test for thesubtraction facts.

HomeworkAssign Questions 1–2 as homework.

Assessment1. Evaluate portfolios using criteria established

in class.2. DPP Bit S is the fourth diagnostic test for the

subtraction facts.

ActivityAngles are introduced. Students identify acute,obtuse, and right angles.

Homework1. Assign the Homework section on the Angles

Activity Pages in the Student Guide.2. Assign Parts 4 and 5 of the Home Practice.

AssessmentUse the Observational Assessment Record todocument students’ abilities to identify right,acute, and obtuse angles with their angle circles.

• 50 square-inchtiles per studentgroup

• 24-inch piece ofwire or string perstudent group

• 2 file folders perstudent

• rack for hangingfiles or a file box

• 1 pair of scissorsper student and2 pairs withblades of differentlengths for theteacher

• 2 rulers

• 1 calculator perstudent

• 2 metersticks

All blackline masters including assessment, transparency, and DPP masters are also on the Teacher Resource CD.

URG • Grade 4 • Unit 2 • Outline 5


2


1


3 (Continued)

Indicates opportunities for review, practice,and extension of concepts.

Homework

Highlights the ongoing math fact programthrough review, practice, and assessmentopportunities found in the Daily Practiceand Problems and lessons.

Math Facts

Indicates opportunities to help informinstruction and monitor student progress.

Assessment

Unit Planning

10

URG • Grade 4 • Unit 2 • Outline 7

ConnectionsA current list of literature and software connections is available at www.mathtrailblazers.com. You can also find information on connections in the Teacher Implementation Guide Literature List and Software List sections.

Literature ConnectionsSuggested Titles

• Burns, Marilyn. Spaghetti and Meatballs for All! Illustrated by Debbie Tilley. Scholastic Press Inc., New York, 1997.

• Neuschwander, Cindy. Sir Cumference and the Great Knight of Angleland. Illustrated by WayneGeehan. Charlesbridge Publishing, Watertown, MA, 2001.

• Neuschwander, Cindy. Sir Cumference and the Isle of Immeter. Charlesbridge Publishing, Watertown,MA, 2006.

Software Connections• Carmen Sandiego’s Math Detective provides practice with math facts, estimation, ordering numbers,

and word problems.

• The Factory Deluxe promotes spatial reasoning and practice finding area.

• Graph Master allows students to collect data and create their own graphs.

• Ice Cream Truck develops problem solving, money skills, and arithmetic operations.

• Kid Pix allows students to create their own illustrations.

• MicroWorlds EX is a drawing program that helps students develop spatial reasoning and an understand-ing of coordinates while making shapes.

• Math Arena is a collection of math activities that reinforces many math concepts.

• Math Mysteries: Whole Numbers is a series of structured word problems dealing with whole numbers.

• Mighty Math Number Heroes poses short answer questions about fractions, number operations,polygons, and probability.

• Number Facts Fire Zapper provides practice in number facts in an arcade-like game.

• Number Sense and Problem Solving: Puzzle Tanks develops logical thinking while practicing math facts.

• Shape Up! is a geometric program that contains five sets of shapes that students can manipulate andexplore.

• Ten Tricky Tiles provides practice with number facts through engaging puzzles.

• TinkerPlots allows students to record, compare, and analyze data in tables and graphs.


Teaching All Math Trailblazers Students

Math Trailblazers lessons are designed for students with a wide range of abilities. The lessons are flexible anddo not require significant adaptation for diverse learning styles or academic levels. However, when needed,lessons can be tailored to allow students to engage their abilities to the greatest extent possible while buildingknowledge and skills.

To assist you in meeting the needs of all students in your classroom, this section contains information aboutsome of the features in the curriculum that allow all students access to mathematics. For additional informa-tion, see the Teaching the Math Trailblazers Student: Meeting Individual Needs section in the TeacherImplementation Guide.

Differentiation Opportunitiesin this Unit

Games

Use games to promote or extend understanding ofmath concepts and to practice skills with childrenwho need more practice.

• DPP Item N Digits Game from Lesson 3Letter to Myrna

• DPP Item P Digits Game from Lesson 4Helipads for Antopolis

Laboratory Experiments

Laboratory experiments enable students to solveproblems using a variety of representationsincluding pictures, tables, graphs, and symbols.Teachers can assign or adapt parts of the analysisaccording to the student’s ability. The followinglesson is a lab:

• Lesson 2 Perimeter vs. Length

Journal Prompts

Journal prompts provide opportunities for studentsto explain and reflect on mathematical problems.They can help both students who need practiceexplaining their ideas and students who benefitfrom answering higher order questions. Students

with various learning styles can express them-selves using pictures, words, and sentences.Teachers can alter journal prompts to suit students’ability levels. The following lesson contains ajournal prompt:


DPP Challenges

DPP Challenges are items from the Daily Practiceand Problems that usually take more than fifteenminutes to complete. These problems are morethought-provoking and can be used to stretch stu-dents’ problem-solving skills. The following lessonhas DPP Challenges in it:

• DPP Challenges V and X from Lesson 6Angles

Extensions

Use extensions to enrich lessons. Many extensionsprovide opportunities to further involve or chal-lenge students of all abilities. Take a moment toreview the extensions prior to beginning thisunit. Some extensions may require additionalpreparation and planning. The following lessonscontain extensions:


• Lesson 4 Helipads for Antopolis

• Lesson 5 Portfolios

Identifies ways to help challenge and support students with a wide rangeof abilities or diverse learning styles.

Teaching All MathTrailblazers Students

Suggests literature and software toenhance and extend lessons. Indicatesessential trade book titles around whichactivities are constructed.

Connections

2. Transfer appropriate documentation from theUnit 2 Observational Assessment Record tostudents’ Individual Assessment Record Sheets.

2

Preparing for Upcoming LessonsPlace pattern blocks, base-ten pieces, connecting cubes, and square-inch tiles in a learning center for studentsto explore.

Offers a “heads-up” look atfuture lessons to aid in planning.

Preparing for UpcomingLessons

Unit Planning

11

Seeing the Big PictureUnit Resource Guide — BackgroundThis section of the Unit Resource Guide explains what students will learn and places the material in the larger context of the curriculum.

This unit is a collection of activities that will pro-vide you with baseline information about students’mathematical knowledge. All these activities areboth useful as assessments and worthwhile aslearning experiences.

GeometryThe unit has two major goals. The first is to reviewbasic geometric ideas and explore patterns throughgeometric investigations. As students investigaterelationships between length, area, and perimeter,they will collect and organize data, create andinterpret graphs, and make and check predictions.Students also begin their study of angles. Studentsfirst explore angles dynamically as an amount ofturning. They then explore angles in shapes. Theemphasis is on developing an intuitive sense aboutthe size of an angle, rather than finding a precisemeasurement. For this reason, protractors are notintroduced until Unit 9. Throughout, students willreason mathematically and communicate aboutmathematics.

Baseline AssessmentThe second goal is to provide opportunities togather information about students’ mathematicalabilities and attitudes at the beginning of theschool year to document their progress throughoutthe coming months. Each student is valued for theknowledge and skills he or she brings to theclassroom and each student is expected to makesignificant gains during the year. Students willbegin collecting materials for their portfolios.Each student will add to his or her portfoliothroughout the year as a record of progress. Seethe TIMS Tutor: Portfolios for information on theuse of assessment portfolios. See the Assessmentsection in the Teacher Implementation Guide formore information about the overall assessmentprogram for fourth grade.

As part of the assessment process, students arereintroduced to the Student Rubric: Telling. Theyuse the rubric as a guide as they talk and write aboutthe problems they solve. Later units will introducethe other two rubrics, Solving and Knowing. Allthree rubrics will help them reflect upon mathemat-ics as they begin their work and as they revise it.Samples of student writing scored using theteacher’s version of the rubric are included in theUnit Resource Guide so you can better evaluateyour students’ work. See the Assessment section ofthe Teacher Implementation Guide for a full discus-sion of the TIMS Multidimensional Rubric.

Resources The following books about assessment were writ-ten with classroom teachers in mind:

• Mathematical Sciences Education Board.Measuring Up: Prototypes for MathematicsAssessment. National Academy Press,Washington, DC, 1993.

• Payne, J.N. (Ed.). Mathematics for the YoungChild. National Council of Teachers ofMathematics, Reston, VA, 1990.

• Pellegrino, J.W., N. Chudowsky, and R. Glaser(Eds.). Knowing What Students Know: TheScience and Design of Educational Assessment.National Research Council, National AcademyPress, Washington, DC, 2001.

• Principles and Standards for SchoolMathematics. National Council of Teachers ofMathematics, Reston, VA, 2000.

• Stenmark, J.K. (Ed.). Mathematics Assessment:Myths, Models, Good Questions, and PracticalSuggestions. National Council of Teachers ofMathematics, Reston, VA, 1991.

• Webb, N.L. (Ed.). Assessment in theMathematics Classroom. National Council ofTeachers of Mathematics, Reston, VA, 1993.

BackgroundGeometric Investigations: A Baseline Assessment Unit

URG • Grade 4 • Unit 2 • Background 9

Provides in-depth information aboutthe mathematics contained in theunit, including other resources inMath Trailblazers that supplementknowledge on a variety of topics.

Background

Unit Planning

12

Assessing Student GrowthUnit Resource Guide — Observational Assessment RecordThe Observational Assessment Record offers another avenue for noting students’ progresson a range of concepts and skills.

12 URG • Grade 4 • Unit 2 • Observational Assessment Record

Name A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Comments

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

Observational Assessment Record

Can students use patterns in data tables and graphs to make and test conjectures?

Can students collect, organize, graph, and analyze data?

Can students make and interpret point graphs?

Can students find the perimeter of polygons?

Can students find the area of polygons?

Can students estimate the size of an angle?

Can students identify acute, obtuse, and right angles?

Can students communicate solution strategies?

Do students demonstrate fluency with the subtraction facts?

A10

A9

A8

A7

A6

A5

A4

A3

A2

A1

URG • Grade 4 • Unit 2 • Observational Assessment Record 11


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Also available on the Teacher Resource CD in PDF and MS Word format.

A list for teachers to assess keyskills, behaviors, and knowledge.Identifies what to look for asstudents are observed.


Unit Planning

13

A Continuing ReviewUnit Resource Guide — Daily Practice and ProblemsThe Daily Practice and Problems (DPP) are short exercises that provide an ongoing review, practice, and assessmentof math concepts, skills, and facts. Each lesson includes suggestions on which Daily Practice and Problems to use.

URG • Grade 4 • Unit 2 • Daily Practice and Problems 13

Daily Practice and ProblemsGeometric Investigations:A Baseline Assessment Unit

Number SenseC, D, H, I, K, N, P,R, T, V, X, AA, CC

Math FactsG–I, M,

O, Q, S, AA

Computation F, G, K, N, P, R, U,

W, CC

MoneyR, U

TimeA, B, E, T,

W, Y, Z

MeasurementC, D, J, BB, DD

GeometryJ, L, Y, Z,BB, DD

DataX

57

The Daily Practice and Problems (DPP), found atthe beginning of each unit, is a set of short exer-cises that:

• provides distributed practice in computationand a structure for systematic study of thebasic math facts;

• develops concepts and skills such as numbersense, mental math, telling time, and work-ing with money throughout the year; and

• reviews topics from earlier units, presentingconcepts in new contexts and linking ideasfrom unit to unit.

There are three types of items: Bits, Tasks, andChallenges. Bits are short and should take no morethan five or ten minutes to complete. They oftenprovide practice with a skill or basic math facts.Tasks take ten or fifteen minutes to complete.Challenges usually take longer than fifteen

minutes to complete and the problems are morethought-provoking. They can be used to stretchstudents’ problem-solving skills. Tasks andChallenges may be appropriate for homework orassessment.

Refer to the Daily Practice and Problems and Home Practice Guide in the TeacherImplementation Guide for further information on the DPP. This guide includes information onhow and when to use the DPP.

Practice and Assessment of the Subtraction Facts

By the end of second grade, students using MathTrailblazers are expected to demonstrate fluencywith all the subtraction facts. In Grade 3, studentsreview the subtraction facts and fluency is con-firmed. The DPP for this unit provides a quick

A DPP Menu for Unit 2

Two Daily Practice and Problems (DPP) items are included for each class session listed in theUnit Outline. A scope and sequence chart for the DPP is in the Teacher Implementation Guide.

Icons in the Teacher Notes column designate the subject matter of each DPP item. The firstitem in each class session is always a Bit and the second is either a Task or Challenge. Eachitem falls into one or more of the categories listed below. A menu of the DPP items for Unit 2follows.

Student Questions Teacher Notes

Magic Squares or Not?

In a magic square, the sums of the rows,columns, and diagonals are all the same.

Which of the following are magic squares?

1. 2.

TIMS Task1. This is a magic square. All rows,

columns, and diagonals sumto 30.

2. This is not a magic square. Thesum of the first row is 75, butthe sum of the last row is 65.

3. This is a magic square. All rows,columns, and diagonals sumto 36.

Birthday Party

John’s birthday party will start at 11:00 A.M.and end at 2:30 P.M.

1. How long will John’s party last?

2. It takes Dave 20 minutes to walk toJohn’s house. What time should heleave home to get to the party on time?

TIMS Bit1. 3 �

21

� hours

2. 10:40 A.M.

9 8 13

14 10 6

40 5 30

15 25 35


Units

1. Frank measured his hand length. Heforgot to include the unit in his measurement. He just wrote “15.” Do you think he measured in centimeters or inches?

2. Which is longer, 15 cm or 15 inches?

3. Find an object that is about 15 centimeters long.

4. Find an object that is about 15 inches long.

TIMS Bit1. centimeters

2. 15 inches

3. a pen

4. width of a TV; width of a smallwindow

Could Be or Crazy?

Decide whether each measurement is a “could be” or “crazy” measurement.

TIMS Task1. Could be: 80 inches is about the

length of two metersticks.Fourth graders are between1 d 2 t t ll

Daily Practice and ProblemsStudents may solve the items individually, in groups, or as a class. The items may also be assigned for homework. The DPPs are also available on the Teacher Resource CD.


Passing Time

Answer the following questions in hours andminutes. How much time has gone by from:

1. 8:00 A.M. to 11:00 A.M.?

2. 11:00 A.M. to 3:00 P.M.?

3. 9:45 A.M. to 10:00 A.M.?

TIMS Bit1. 3 hours

2. 4 hours

3. 15 minutes

4. 15 minutes

The DPP review topics from earlier units, presentingconcepts in new contexts and linking ideas fromunit to unit. They provide a structure for systematicstudy of the basic math facts, practice in computa-tion, and work with other math concepts.

Daily Practice and Problems Menu

These DPP Icons denote themath category for each DPPitem, including math facts andcomputation for planning.

DPP Icons

The Daily Practice and Problems are available on the Teacher Resource CD, wherethey can be sorted by type and category. Theyare also available on www.mymathtrailblazers.com.

The Teacher Notes section offersideas for a balance betweenconceptual understanding andprocedural skills, solution strate-gies, and possible answers tothe student questions.

Teacher Notes

Grade 4 Lessons

14

UNIT 1: Data About UsLesson 1: Getting to Know Room 204Lesson 2: Getting to Know Room 204 a Little BetterLesson 3: An Average ActivityLesson 4: The Four ServantsLesson 5: Arm Span vs. HeightLesson 6: Solving Problems About Room 204

UNIT 2: Geometric Investigations:A Baseline Assessment Unit

Lesson 1: Investigating Perimeter and AreaLesson 2: Perimeter vs. LengthLesson 3: Letter to MyrnaLesson 4: Helipads for AntopolisLesson 5: PortfoliosLesson 6: AnglesLesson 7: Angles in Pattern Blocks

UNIT 3: Numbers and Number Operations

Lesson 1: Multiplying and Dividing with 5s and 10sLesson 2: Roman NumeralsLesson 3: Place ValueLesson 4: The TIMS Candy CompanyLesson 5: Addition and SubtractionLesson 6: What’s Below Zero?Lesson 7: At the Hardware Store

UNIT 4: Products and FactorsLesson 1: Multiplication and RectanglesLesson 2: FactorsLesson 3: Floor TilerLesson 4: Prime FactorsLesson 5: Product BingoLesson 6: Multiplying to Solve Problems

UNIT 5: Using Data to PredictLesson 1: Predictions from GraphsLesson 2: Another Average ActivityLesson 3: The Meaning of the MeanLesson 4: Bouncing BallLesson 5: Two Heads Are Better Than OneLesson 6: Professor Peabody Invents a BallLesson 7: Speeds at the Indianapolis 500

UNIT 6: Place Value Patterns Lesson 1: NewswireLesson 2: DoublesLesson 3: Big Base-Ten PiecesLesson 4: News Number LineLesson 5: Close EnoughLesson 6: Using EstimationLesson 7: 9 to 5 War

UNIT 7: Patterns in MultiplicationLesson 1: Order of OperationsLesson 2: Divisibility RulesLesson 3: Oh, No! My Calculator Is BrokenLesson 4: Multiplying by 10sLesson 5: MultiplicationLesson 6: EstimationLesson 7: Multiplying Round NumbersLesson 8: A Camping Trip

UNIT 8: Measuring Up: An Assessment Unit

Lesson 1: VolumeLesson 2: Fill It FirstLesson 3: Volume vs. NumberLesson 4: Review ProblemsLesson 5: Hour WalkLesson 6: Midyear TestLesson 7: Midyear Experiment and Portfolio ReviewLesson 8: Facts I Know: Multiplication and

Division Facts

Grade-Level Listing of Lessons by Unit:

Grade 4 Lessons

15

UNIT 9: Shapes and Solids Lesson 1: LinesLesson 2: What’s Your Angle?Lesson 3: SymmetryLesson 4: Journey to FlatopiaLesson 5: PrismsLesson 6: Finding the Volume of a PrismLesson 7: Building an OctahedronLesson 8: Constructing a Prism

UNIT 10: Using Decimals Lesson 1: m, dm, cm, mmLesson 2: TenthsLesson 3: HundredthsLesson 4: Downhill RacerLesson 5: Decimal HexLesson 6: Alberto in TenthsLand

UNIT 11: MultiplicationLesson 1: Modeling MultiplicationLesson 2: More MultiplicationLesson 3: Compact MultiplicationLesson 4: All-Partials RevisitedLesson 5: More Compact MultiplicationLesson 6: Phil and Howard’s Excellent Egyptian

AdventureLesson 7: Visiting Egypt

UNIT 12: Exploring FractionsLesson 1: Fraction StripsLesson 2: Adding and Subtracting with Fraction StripsLesson 3: Comparing FractionsLesson 4: Frabble Game and Bubble SortLesson 5: Equivalent FractionsLesson 6: Pattern Block FractionsLesson 7: Solving Problems with Pattern BlocksLesson 8: Fraction PuzzlesLesson 9: Midterm Test

UNIT 13: Division Lesson 1: TV SurveyLesson 2: DivisionLesson 3: More DivisionLesson 4: Solving Problems Using Multiplication

and DivisionLesson 5: Plant Growth

UNIT 14: Chancy Predictions: An Introduction to Probability

Lesson 1: Chance DiscussionsLesson 2: Bean Counter’s GameLesson 3: Rolling a Number CubeLesson 4: From Number Cubes to SpinnersLesson 5: Exploring SpinnersLesson 6: Make Your Own SpinnersLesson 7: Probe Quest

UNIT 15: Using Patterns Lesson 1: Plant Growth ConclusionLesson 2: In the Shade of the Old Meranpi TreeLesson 3: Planet GzorpLesson 4: Function MachinesLesson 5: Taste of TIMSLesson 6: Patterns and Problems

UNIT 16: Assessing Our Learning Lesson 1: Experiment ReviewLesson 2: Problems and PracticeLesson 3: Area vs. LengthLesson 4: The Many-Eyed DragonflyLesson 5: End-of-Year TestLesson 6: Portfolios

Math FactsDPP Bits O and Q are diagnostic tests for the subtraction facts.

Homework1. Ask students to explore helipads with different perimeters. (optional)2. Assign Part 6 of the Home Practice.

Assessment1. Use Questions 4–7 to assess problem-solving and communication skills.2. Use the Observational Assessment Record to document students’ progress in communicating problem-

solving strategies.

Key Content• Using area and perimeter to solve problems.

• Developing mathematical arguments to supporta statement.

• Connecting mathematics to real-world problems.

Key Vocabulary• helipad

• rectangle

• square

URG • Grade 4 • Unit 2 • Lesson 4 81

Helipads for Antopolis

Students design helipads for the Antopolis Airport. After designing severalhelipads—rectangles with a perimeter of 24 inches—they find which helipad hasthe maximum area. They must also explain how they know they have found the

maximum-area design.


2

Lesson Overview

Lesson Overview

16

Everything You Need to TeachUnit Resource Guide — LessonIncludes detailed instruction on what’s involved in teaching each lesson. Try this lesson with your students.

Alerts teacher to homework assignments.

Homework

Lists important mathematical ideas studentswill encounter in the lesson. Key Content isaligned by the NCTM PSSM in the MathTrailblazers Scope and Sequence located inthe Teacher Implementation Guide.

Key Content

Provides opportunities to assess yourstudents’ progress and guide instruction.

Assessment

Identifies the mathfacts in the lesson.

Math Facts

Lesson Overview

17

Communicating Solutions to Open-Response Problems

Students will be introduced to the other two studentrubrics in later units. They will have many opportu-

nities to write about the problems they solve and toimprove their communication skills. See Units 5, 6,8, 10, 12, and 20.

Communicating Solutions to Open-Response Problems

In third grade, students used the three student rubrics(Knowing, Solving, and Telling) as guides for writing

about their problem-solving strategies. See the fol-lowing lessons in Grade 3: Lesson 6 in Unit 2,Lesson 5 in Unit 5, Lesson 2 in Unit 7, Lesson 3 inUnit 10, and Lesson 4 in Unit 20 for samples of stu-dent work using the rubric.

Before This Unit

After This Unit

Curriculum Sequence

70 URG • Grade 4 • Unit 2 • Lesson 3

Not all lessons include a Curriculum Sequence since some of the same topics are being covered, as in the case of this lesson.The example shown is from the lesson prior to this one.

Highlights the content coherence found in MathTrailblazers. This aids in planning and instruction byhelping teachers know where students have beenand where they are headed.

Curriculum Sequence

Lesson Overview

18


Student BookHelipads for Antopolis (Student Guide Pages 37–38)

Daily Practice and Problems and Home Practice DPP items O–R (Unit Resource Guide Pages 24–25)Home Practice Part 6 (Discovery Assignment Book Page 14)

Note: Classrooms whose pacing differs significantly from the suggested pacing of the units should use theMath Facts Calendar in Section 4 of the Facts Resource Guide to ensure students receive the complete mathfacts program.

Assessment ToolsObservational Assessment Record (Unit Resource Guide Pages 11–12)

Supplies and Copies

All blackline masters including assessment, transparency, and DPP masters are also on the Teacher Resource CD.

Student Teacher

Copies

• 1 copy of Subtraction Facts: Count-Backs per student(Unit Resource Guide Page 32)

• 1 copy of Subtraction Facts: Using a Ten per student(Unit Resource Guide Page 33)

• 1 copy of Square-Inch Grid Paper per student, optional(Unit Resource Guide Page 46)

Materials List

Supplies for Each Student Group

• 50 square-inch tiles

• 24-inch piece of wire or string

Supplies

Copies/Transparencies

• transparencies of Square-Inch Grid Paper(Unit Resource Guide Page 46)

Helps locate daily exercises.

Daily Practice and Problems

Suggested ways to assess student growththroughout this lesson and unit.

Assessment Tools

Indicate manipulatives, supplies,and copying or transparenciesneeded for each lesson.

Supplies and Copies/Transparencies

Lesson Overview

19


O. Bit: Subtraction Facts Test:Count-Backs (URG p. 24)

Take the subtraction diagnostic test SubtractionFacts: Count-Backs. Your teacher will suggestsome additional activities if you need morepractice.

57 Q. Bit: Subtraction Facts Test:

Using a Ten (URG p. 25)

Take the diagnostic test Subtraction Facts: Using aTen. Your teacher will suggest some additionalactivities if you need more practice.

57

R. Task: Sharing Money (URG p. 25)

Four children found $7.00 at the bus stop. Theydecide to share it equally. How much should eachchild get?

Suggestions for using the DPPs are on pages 86–87.


P. Task: Play Digits Game Again (URG p. 24)

Draw boxes like these on your paper.

As your teacher or classmate chooses thedigits, place them in the boxes. Try to find thelargest difference. Remember each digit will beread only once. Once you place a digit, it cannot bemoved.

–

Summary of all student questionsfrom the Daily Practice and Problemsfor this lesson.


Short items providing quick review orfocused practice of a specific topic or skill.

TIMS Bits

More difficult or longer problems whichsometimes ask students to use previouslylearned concepts in new contexts.

TIMS Tasks

Opportunities for students to extend theirmath skills.

TIMS Challenges

Identifies subject matter covered in each Daily Practiceand Problem.

Daily Practice andProblems Icons

Lesson Overview

20


Teaching the ActivityBegin this activity by giving each group of two orthree students one piece of wire or string, 24 incheslong. Have each student group measure the wire andthen make a rectangle with it. If string or wire is notavailable, ask student groups to draw a rectanglewith a perimeter of 24 inches on a sheet of Square-Inch Grid Paper.

Then, compare rectangles. Ask each group thefollowing:

• Is the perimeter 24 inches?

• What is the length and what is the width?

• What is the area?

Students should realize that even though the perime-ter of each rectangle is 24 inches, the rectangles havedifferent shapes and areas.

When creating and comparing the rectangles, somestudents may not know that a square is a rectangle.Knowing that a square is a rectangle is necessary tosolve the problem to be explored later in this lesson.Ask students to define a rectangle and then explainwhether a square fits that definition. See the ContentNote for definitions of a rectangle and square.

For this activity, 24-inch twist-ties work well. They are sometimesavailable in bulk in garden stores.

Squares and Rectangles in Everyday Language and inMathematics. A rectangle is defined as a four-sided figure withfour right angles. If all four sides of a rectangle are the samelength, then it is a square. A square is a kind of rectangle.

In talking about vocabulary with students, distinguish mathematicallanguage from everyday language. Some everyday terms are alsoused in mathematics and have precise definitions that differsignificantly from the meaning familiar to students. This can beconfusing. For example, in everyday English, squares andrectangles sometimes are considered to be two different shapes.Both have four sides and four right angles, but the square is theone with four equal sides and the rectangle is the one whoselength differs from its width. When students hear that a square is arectangle because it satisfies the definition of rectangle, they maydistrust the mathematical definition because it differs from theirprior understanding. Avoid this confusion by addressing the issueand discussing the distinction between mathematical and everydayusage.

Get the most out of each lesson with these practicalsuggestions and maximize learning opportunitiesfor reaching all students. Includes hints regardingfacilitating student collaboration.

TIMS Tip

Specific mathematical content informationfor the teacher relating to lesson activities.

Content Note

Describes what students do for each activity.Includes discussion prompts, information aboutstudent page questions, and lesson instructions.

Teaching the Activity

Lesson Overview

21


Myrna Myrmidon’s Aunt Pennylikes to fly helicopters. Whenshe flew to Ladybug Airport, she landed her helicopter ona helipad. The helipad atLadybug Airport is 4 inches long and 2 inches wide.

1. Ladybug Airport is building a new helipad. They are also buying newperimeter lights. The lights are attached to a wire that goes around theentire helipad. How long does the wire need to be to fit around this helipad?

2. What is the area of this helipad?

While the construction crew at Ladybug Airport waited for the lights to bedelivered, they discussed the new helipad. They knew that the perimeter of thehelipad could not be changed since the wire had already been ordered. However,they could change the helipad’s area. The head construction worker remindedthem that the helipad needed to be a rectangle made with square-inch tiles.

3. Help the construction crew.

A. Are there other rectangles besides the one shown above that have the same perimeter as the current helipad at Ladybug Airport? Use square-inch tiles to help you.

B. What is the area of eachrectangle you found?

C. Which rectangle would you recommend for the new helipad? Why?


Helipads for Antopolis SG • Grade 4 • Unit 2 • Lesson 4 37

Student Guide - page 37 (Answers on p. 89)

Aunt Penny wants a helipad included in the Antopolis airport. Myrna agrees, buttells her aunt that only 24 inches of wire (for perimeter lights) can be allowed forthe helipad. Myrna also says that the helipad must be a rectangle built withsquare-inch tiles.

4. Using square-inch tiles, find all the possible helipads with a perimeter of 24 inches. Sketch each helipad on a piece of paper, showing the lengthand width. Be sure your helipads are rectangles.

5. What is the area of each of your helipads?

6. Penny wants the helipad to be as big as possible. Design a helipad with aperimeter of 24 inches with the largest possible area.

7. Explain why you think your helipad has the largest possible area.

SG • Grade 4 • Unit 2 • Lesson 4 Helipads for Antopolis38

Student Guide - page 38 (Answers on p. 89)

Turn to the Helipads for Antopolis Activity Pages inthe Student Guide. Read and discuss Questions 1–3together as a class. Questions 1 and 2 ask studentsto find the area and perimeter of the 4-inch by 2-inchhelipad shown on the Student Guide page. The areais 8 square inches and the perimeter is 12 inches.

Encourage students to use the square-inch tiles tofind other rectangles with a perimeter of 12 inches(Question 3A). To begin the exploration, use square-inch tiles on the overhead to create a rectangle thatdoes not have a perimeter of 12 inches (such as a 2-inch by 3-inch rectangle). Ask a student to come tothe overhead and find the perimeter.

Then have the students find other rectangles with aperimeter of 12 inches. Using square-inch tiles, thereare two possibilities: 1 inch by 5 inches and 3 inchesby 3 inches. The former has an area of 5 squareinches and the latter has an area of 9 square inches(Question 3B). Discuss Question 3C and accept allanswers. Question 3C encourages students to thinkabout the reasonableness of their mathematicalanswer in the context of the problem’s specificneeds. Here, there is more than one mathematicallycorrect answer. Therefore, the students must thinkthrough why one helipad shape may work better thananother and give reasons for their choices. This ques-tion will also prepare them for their explanation inQuestion 7.

Students should now be ready to answerQuestions 4–7. Read the paragraph that precedesQuestion 4 and briefly discuss what the studentsneed to do. Review the Student Rubric: Tellingbefore students begin to record their solutions andsolution paths.

For Question 4 students will look for rectangles thathave a perimeter of 24 inches. Some students mayfind the piece of wire distributed at the beginning ofclass helpful in testing the perimeter of each of therectangles they create. Other students may prefercounting out the perimeter of each rectangle or usingother strategies (e.g., finding the length and widthand then doubling).

Since we are restricted to square-inch tiles, all lengthsand widths must be whole numbers. There are six possible rectangles: 11-inches by 1-inch, 10-inches by2-inches, 9-inches by 3-inches, 8-inches by 4-inches,7-inches by 5-inches, and 6-inches by 6-inches.

Lesson Overview

22


Encourage students to explore their data and look forpatterns to be sure they found all the possible heli-pads. Some patterns that students may recognize intheir data are described here.

• Students may recognize that the more compactthe rectangle, the greater the area. For example,an 8-inch by 4-inch rectangle with an area of32 square inches is more compact than an 11-inch by 1-inch rectangle with an area ofonly 11 square inches. This may spur them tofind an even more compact rectangle.

• Students may notice that the sum of the lengthand width in each of their rectangles is 12 inches(one-half of the perimeter). Knowing that thissum is 12 inches in every case may help themknow when they have found all the rectangles.

• Students may notice that in each succeeding rec-tangle, the length decreases one inch as the widthincreases one inch.

Question 5 asks students to record the area of eachof their helipads. Remind them to use the unit ofmeasure, square inch, when labeling their helipads.Question 6 asks students to compare their helipadsto find the rectangle with the largest possible areaand a perimeter of 24 inches. The 6-inch by 6-inchrectangle has the greatest area—36 square inches.

In Question 7, students are asked to explain whythey think the rectangle they chose in Question 6 hasthe biggest possible area. Ask students how theyknow they found all the possible rectangles and howthey can be sure that there is not one that has a largerarea. If some students did not find the rectangle with the largest area (the 6-inch by 6-inch square),encourage them to look back at their data. Ask themto explain any patterns they see and guide them to the more compact helipad. While answeringQuestion 7, encourage students to use the StudentRubric: Telling. Suggest that aside from words, theycan use pictures, data tables, or symbols to showtheir solutions and strategies.

Math FactsDPP Bits O and Q are the next two diagnostic testsfor the subtraction facts. They feature the counting-back strategy and the using-tens strategy.

Maximum Area and Rectangles. When areas of differentrectangles that have the same perimeter are compared, the squarewill always have the maximum area. If the problem states thelengths and widths must be whole numbers and there is no squarewith the given perimeter, then the rectangle that is closest to thedimensions of a square will have the maximum area. Weencourage you to allow students to explore rectangles, area, andperimeter, so they may possibly discover this on their own.

Highlights math facts for review, practice, or assessment.

Math Facts

Lesson Overview

23


Homework and Practice

• You can modify this activity by changing therequired perimeter. For example, ask students toexplore helipads with perimeters of 18 inches. Or ask students to explore helipads that have aperimeter of their own choosing. If assigned forhomework, students will need to take homesquare-inch tiles or Square-Inch Grid Paper.

• DPP Task P is a version of the Digits Game inwhich students work with two-digit subtraction.DPP Task R provides practice in dividing a sumof money.

• Assign Part 6 of the Home Practice for home-work. Do not assign Part 5 until Lesson 6 iscomplete.

Answers for Part 6 of the Home Practice are in the AnswerKey at the end of this lesson and at the end of this unit.

Assessment

• Questions 4–7 can help you assess how well yourstudents can solve a problem and communicate asolution. Score Question 7 using the Tellingdimension of the TIMS Multidimensional Rubric.

To assist you in scoring your students’ work, hereare questions specific to this task.1. Is the response complete and clear? Did the

student describe his or her approach to find-ing the rectangles with perimeter of24 inches? Did the student explain his or herapproach for determining if all the possiblehelipads were found?

2. Did the student tell why his or her chosenhelipad has the maximum area possible?

3. Did the student use pictures or data tables tosupport the argument?

4. Did the student use the terms length, width,perimeter, and area correctly?

5. Did the student use the appropriate units ofmeasurement when reporting the length,width, perimeter, and area of the rectangles?

• Use the Observational Assessment Record torecord students’ abilities to communicate theirproblem-solving strategies.

ExtensionCan you use square-inch tiles to make a helipad thatis a rectangle with a perimeter of 9 inches? Can arectangle made from square-inch tiles have an odd-numbered perimeter? Explain.

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Shopping with GrandmotherSolve the following problems. Show how you solved each one using picturesor words.

1. Ming went to the mall with his grandmother. They left the house at 8:00 A.M.

They returned home 6 hours later. When did they get home?

2. The bus fare to the mall was 85¢ for his grandmother and 75¢ for Ming.What was the total bus fare to and from the mall? (Remember, they need tocome home too!)

3. The mall has two floors. The first floor has 62 stores. The second floor has48 stores. How many stores are in the mall?

4. The mall newspaper claims that nearly half the stores are participating in afall sale. About how many stores are participating in the fall sale?

5. Ming and his grandmother shopped in 12 stores. How many stores did theynot visit?

6. On the way home, they stopped in the grocery store. Ming went to the delicounter to buy lunch meat. He took a number from the counter that gave his turn in line. The girl behind the counter was waiting on Number 54. AfterNumber 54, nine more people needed to be served before it was Ming’sturn. What number did Ming have?

7. When Ming got home, he and his grandmother played a basketball game onthe computer. His grandmother won the game! Her team scored 17 morepoints than Ming’s team. If Ming’s team scored 74 points, how many pointsdid his grandmother’s team score?

PART 6

Name Date

DAB • Grade 4 • Unit 2 GEOMETRIC INVESTIGATIONS: A BASELINE ASSESSMENT UNIT14

Discovery Assignment Book - page 14 (Answers on p. 90)

Suggested assignments,review, and practice.

Homework andPractice

Assessment

Activities or suggestionsfor further exploration on the same topic.Provides an opportunityfor differentiation.

Extension

Evaluates your students’skills and understandingof concepts. Includesmultiple assessmentideas to improve thevalidity of judgmentsabout student learning.

Lesson Overview

24


2


Math Facts and Daily Practice and ProblemsDPP Bits O and Q are diagnostic tests for the subtraction facts. DPP Task P provides practice in double digitsubtraction. DPP Task R provides practice in division.

Teaching the Activity 1. Each student group measures a 24-inch piece of wire or string and makes a rectangle with it, or draws

a rectangle with a perimeter of 24 inches.

2. Groups compare their rectangles.

3. Students should note that the rectangles have different shapes and areas.

4. Read and discuss Questions 1–2 on the Helipads for Antopolis Activity Pages in the Student Guide.

5. Students use square-inch tiles to find rectangles with a perimeter of 12 inches. (Question 3)

6. Read the paragraph that precedes Question 4 and briefly discuss what the students need to do. Remindstudents to use the Student Rubric: Telling as a reference as they record their solutions and strategies.

7. Students look for rectangles that have a perimeter of 24 inches using the square-inch tiles. They recordthe area of each rectangle they find. (Questions 4–5)

8. Students explore their data and look for patterns to be sure they have found all the possible helipads.

9. Students compare their helipads to find the rectangle with the largest possible area and a perimeter of24 inches. (Question 6)

10. Students explain why they think the rectangle they chose has the largest possible area. (Question 7)

Homework1. Ask students to explore helipads with different perimeters. (optional)

2. Assign Part 6 of the Home Practice.

Assessment1. Use Questions 4–7 to assess problem-solving and communication skills.

2. Use the Observational Assessment Record to document students’ progress in communicating problem-solving strategies.

ExtensionAsk:

Can you use square-inch tiles to make a rectangular helipad with a perimeter of 9 inches?

Can a rectangular helipad made from square-inch tiles have an odd-numbered perimeter?

Answer Key is on pages 89–90.

Notes:

At a Glance

Summarizes suggested steps for teachingwhile highlighting math facts practicetechniques, homework, and assessment.Serves as a quick review for the teacherwho has taught the lesson previously orplanned for it.

At a Glance

Lesson Overview

25

URG • Grade 4 • Unit 2 • Lesson 4 • Answer Key 89

Answer Key • Lesson 4: Helipads for Antopolis











Student Guide - page 37







Student Guide - page 38

Student Guide (p. 37)


1. 12 inches

2. 8 square inches

3. A. Yes, 1 � 5 inches and 3 � 3 inches

B. 5 square inches and 9 square inches

C. Answers will vary. Students may think that asquare helipad is better for helicopters thana rectangular helipad. Accept all answers.

Student Guide (p. 38)

4.–5.

6. 6 inch � 6 inch rectangle

7. Answers will vary. The 6-inch square gives usthe largest area—36 square inches. SeeContent Note in the Lesson Guide.*

L = 11 inches, W = 1 inch, A = 11 square inches

L = 10 inches, W = 2 inches, A = 20 square inches





*Answers and/or discussion are included in the Lesson Guide.

A red bar at the top of each page identifiesthe answer key that follows each lesson.Answers appear next to reduced pages.

Answer Keys

Lesson Overview

26

90 URG • Grade 4 • Unit 2 • Lesson 4 • Answer Key

*Answers for all the Home Practice in the Discovery Assignment Book are at the end of the unit.

Answer Key • Lesson 4: Helipads for Antopolis

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PART 6

Name Date


Discovery Assignment Book - page 14

Discovery Assignment Book (p. 14)

Home Practice*

Part 6. Shopping with Grandmother

1. 2 P.M.

2. $3.20

3. 110 stores

4. Answers will vary. 55 stores is exactly half.Accept estimates from 50 to 60 stores.

5. 98 stores

6. 64

7. 91 points

Student Pages

27











All student

components are

available in English

and Spanish

Student Pages

28







Student Pages

29

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PART 6

Name Date


Go to www.mathtrailblazers.com to download the remaining studentpages for this sampler or to download the entire sampler!

Assessment

30

A Balanced Assessment That’s Both

Observational AssessmentsObservational assessments in Math Trailblazers are designedto allow teachers to gather information about students asthey are learning.

Assessment IndicatorsA list for teachers to assess key skills, behaviors, and knowledge to look for as you observe students.

Observational Assessment RecordUse this form to record informal observations of students’ progress. Found in the Unit Resource Guide.

Individual Assessment Record SheetRecord observations on individual students. Includes allunits for the year, allowing teachers to keep all theirobservations for each child in one place. Found in theTeacher Implementation Guide.

Let’s look at the three major assessment goals in Math Trailblazers:• Help teachers explore student thinking to best guide instruction.• Clearly communicate the goals of the curriculum to students and parents.• Inform students and parents of progress in meeting curriculum goals and possible paths for further efforts.

Assessment in Math Trailblazers reflects the breadth and balance of the curriculum. Numerous opportunitiesfor both formal and informal assessment of student learning are integrated into the program.

Many assessment activities are incorporated into daily lessons, while others are in formal assessment units.Activities include a mix of short, medium-length, and extended activities that are hands-on investigations orpaper-and-pencil tasks.

In all cases, assessment activities are designed to be worthwhile educational experiences that elicit morethan just an answer.

Observational Assessment Record

Can students use patterns in data tables and graphs to make and test conjectures?

Can students collect, organize, graph, and analyze data?

Can students make and interpret point graphs?

Can students find the perimeter of polygons?

Can students find the area of polygons?

Can students estimate the size of an angle?

Can students identify acute, obtuse, and right angles?

Can students communicate solution strategies?

Do students demonstrate fluency with the subtraction facts?

A10

A9

A8

A7

A6

A5

A4

A3

A2

A1


1.

2.

3.

Individual Assessment Record SheetName

Unit 1: Data About Us � Date and Comments:

A1. Can students identify categorical and numerical variables?

A2. Can students find the median of a data set?

A3. Can students make and interpret bar graphs?

A4. Can students make and interpret point graphs?

A5. Can students use patterns in data tables and graphs to make predictions?

A6. Can students measure length in inches?

A7. Do students demonstrate fluency with the addition facts?

A8.

Unit 2: Geometric Investigations: � Date and Comments:A Baseline Assessment Unit

A1. Can students use patterns in data tables and graphs to make and test conjectures?

A2. Can students collect, organize, graph, and analyze data?

A3. Can students make and interpret point graphs?

A4. Can students find the perimeter of polygons?

A5. Can students find the area of polygons?

A6. Can students estimate the size of an angle?

A7. Can students identify acute, obtuse, and right angles?

A8. Can students communicate solution strategies?

A9. Do students demonstrate fluency with the subtraction facts?

*Formative and Summative AssessmentThe comprehensive assessment program built into MathTrailblazers offers opportunities for both formative andsummative looks at student learning. Formative assess-ments provide “snapshots” of students at various timesover the course of the program in order to monitorongoing progress. They are useful in making decisionsthat inform instruction, helping move the student towardlearning outcomes. Observational assessment is one ofthe many components of formative assessment.

Math Trailblazers also contains opportunities for sum-mative assessments, which are more comprehensive.The periodic end-of-year tests in Grades 1–5, quarterlyassessments in Grades 3–5, as well as the math factquizzes in Grades 2–5 are examples of summativeassessments because they are cumulative in nature,often reaching back several units for content.

Assessment

31

Written AssessmentsWritten assessments in Math Trailblazers aredesigned to provide multiple opportunities for stu-dents to show what they know.

Assessment IdeasUse suggestions offered by the Lesson Guide toassess Math Trailblazers lesson content. Essentialtools include Daily Practice and Problems andHomework.

Math JournalsGather information about students’ understandingof specific concepts and their ability to communi-cate this understanding in writing and expresstheir attitude about math.

Assessment Activity PagesChart progress of student groups via these short“paper-and-pencil” assessments, or use theseactivity pages as quizzes to assess each studentindividually.

Assessment LabsThere are 3 or 4 assessment labs provided inGrades 2–5. These extended investigations helpstudents apply the mathematics they are learningin a new context. Use them also to gauge students’ ability to work together.


The following samples of student work have been scored using these criteria.

Scoring Keenya’s work

Figure 9: Keenya’s work

Assessment Lessons

Formative and Summative*

Name Date

If you walked steadily for an hour, about how many steps would you take?

A. 500 B. 1000 C. 5000 D. 10,000 E. 50,000 F. 100,000

Make an estimate without walking for one hour. Explain how you madeyour estimate. Show all your work.

Hour Walk

Assessment LessonsDevelop and document student progress withthese specially designed lessons.

Assessment UnitsAssess student progress on many tasks throughthese grade-specific units and lessons.

PortfoliosStudents’ written work and teachers’ anecdotalrecords provide a complete picture of progress.

Assessment

32

Fact Self-AssessmentEmpower students to record their progress withcustom “Facts I Know” charts.

Facts QuizConduct periodic quizzes with materials from theDaily Practice and Problems section.

RubricsThe TIMS Rubrics for aspects of mathematicsunderstanding—Solving Problems, UnderstandingMathematical Content, and Communication—assist teachers in informing students and parentsof performance.

Student RubricsStudent Rubrics make the expectations for performance clear to students.

SG • Grade 4 • Unit 8 • Lesson 8 Facts I Know234

Picturing Fact Families1. The picture below represents the following problem: If a rectangle has

a total of 20 squares in 4 rows, how many squares are in each row?

What division sentence describes this problem?

2. The picture below represents the following problem: If a rectangle has a total of 20 squares in 5 rows, how many squares are in each row?

A. What division sentence describes this problem?

B. These two division sentences are members of the same fact family.What are the other number sentences in this same fact family?

3. Solve the given fact. Then name other facts in the same fact family.

A. 9 × 7 = ? B. 6 × 4 = ? C. 7 × 8 = ?

Division Facts and Triangle Flash Cards4. The directions that follow tell you how to use your Triangle Flash Cards to

practice the division facts. Work with a partner. Use your Triangle FlashCards: 5s and 10s.

Facts I Know: Multiplication and Division Facts

Name Date

Division Facts I Know• Circle the facts you know well.• Keep this table and use it to help you divide.• As you learn more facts, you may circle them too.

0 0

81 2 3 4 5 6 7

00 0 0 0 0

166 8 10 12 14

21 246 9 12 15 18

9

27

0

182

3

0

0

0

0

4

0

1

2

3

0 92 7 83 4 5 61×

10

30

0

20

10

TIMS Multidimensional RubricSolving

Identifies the elements of theproblem and their relationships toone another.

Uses problem-solving strategieswhich are . . .

Organizes relevant information . . .

Relates the problem and solutionto previously encountered mathe-matics and makes connectionsthat are . . .

Persists in the problem solvingprocess . . .

Looks back to examine the rea-sonableness of the solution anddraws conclusions that are . . .

Level 4

All major elementsidentified

Systematic,complete, efficient,and possiblyelegant

Systematically andefficiently

At length, elegant,and meaningful

At length

Insightful andcomprehensive

Level 3

Most elementsidentified

Systematic andnearly complete,but not efficient

Systematically, withminor errors

Evident

Until a solution isreached

Correct

Level 2

Some, but shows little under-standing of relationships

Incomplete or unsystematic

Unsystematically

Brief or logicallyunsound

Briefly

Incorrect or logicallyunsound

Level 1

Few or none

Not evident orinappropriate

Not at all

Not evident

Not at all

Not present

Knowing

Understands the task’s mathe-matical concepts, their propertiesand applications . . .

Level 4

Completely

Level 3

Nearly completely

Level 2

Partially

Level 1

Not at all

Student Rubric: Telling

In My Best Work in Mathematics:

• I show all of the steps that I used to solve the

problem. I also tell what each number refers to

(such as 15 boys or 6 inches).

• I explain why I solved the problem the way I did so

that someone can see why my method makes sense.

• If I use tools like pictures, tables, graphs, or number

sentences, I explain how the tools I used fit the

problem.

• I use math words and symbols correctly. For example,

if I see “6 – 2,” I solve the problem “six minus two,”

not “two minus six.”

SG • Grade 3 • Appendix C 305

What does this

rubric tell you?

It helps me talk

and write about math

and solving problems!

KH4176_SG3_ppStudent Rubric: Solving

In My Best Work in Mathematics:• I read the problem carefully, make a good plan for

solving it, and then carry out that plan.• I use tools like graphs, pictures, tables, or numbersentences to help me.

• I use ideas I know from somewhere else to help mesolve a problem.

• I keep working on the problem until I find a goodsolution.

• I look back at my solution tom k

How does thisrubric help you?

It helps me planstrategies, find solutions, and check my work when I solve problems.

Student Rubric: Knowing

In My Best Work in Mathematics:

• I show that I understand the ideas in the problem.

• I show the same mathematical ideas in different ways.

I use pictures, tables, graphs, and sentences when

they fit the problem.

• I show that I can use tools and rules correctly.

• I show that I can use the mathematical facts that

apply to the problem.

What is a rubric?

It tells me how

to make sure I’ve done

my best work!

KH4176_SG3_Appendix_p3

03-306 5/24/07 10:06

AM Page 303

Assessment

Assessment OverviewThis section within the Teacher ImplementationGuide provides a unit-by-unit listing of where theassessments are located in each grade.

CustomizedAssessments The Teacher’s Guide to Practice and AssessmentCD creates customized assessments.

33

Teacher Resource CDThis CD contains all the resources you need, including:

• Daily Practice and Problems (with and without teacher notes)• Observational Assessment Record pages (MS Word and PDF format)• Individual Assessment Record sheets (MS Word and PDF format) • Scope and Sequence• Teacher and Student Rubrics• Printable Blackline and Transparency Masters• Math Facts Calendar and other masters from Facts Resource Guide

Assessment Component Component Description Location

Unit 1—Data About Us

Observational Assessment Record Assessment Record URGIndividual Assessment Record Sheet Assessment Record TIG

Lesson 3—An Average Activity Lesson URGDPP Item J—Variables and Values Assessment Item URGHome Practice Part 3—Finding the Median Assessment Page DAB

Lesson 4—The Four Servants Adventure Book URGDPP Item K—Addition Test: Doubles, 2s, 3s Assessment Item URG

Lesson 5—Arm Span vs. Height Lab URGDPP Item M—Addition Test: More Addition Facts Assessment Item URGDPP Item T—Favorite Sandwiches Assessment Item URGMore Arm Span vs. Height Data Assessment Page URG

Observational Assessment Record Assessment Record URGIndividual Assessment Record Sheet Assessment Record TIG

Lesson 1—Investigating Perimeter and Area Lesson URGStudent Rubric: Telling Student Rubric SG/TIG

Lesson 2—Perimeter vs. Length Assessment Lab URG, SG, DAB

DPP Item J—Area and Perimeter Assessment Item URG

Lesson 3—Letter to Myrna Assessment Lesson URG, SGDPP It M S bt ti F t T t C t U S bt ti F t A t It URG

Unit 2—Geometric Investigations: A Baseline Assessment Unit

Teacher’s Guide to Practiceand Assessment CDAvailable for Grades 3, 4, and 5 and includes existing assess-ments from the curriculum and a supplemental student pagebuilder. These are particularly helpful for teachers who wish toindividualize instruction with extra practice or homework, or toprepare students for standardized testing. Correlated to MathTrailblazers units and national standards.

FREEdemo

available!

Connections to ScienceThe TIMS® Laboratory Method is a simplified version ofthe scientific method used by scientists and serves as aneffective problem-solving tool.

In Grade 4, laboratory investigations appear in Units 1, 2,5, 8, 10, 13, 14, 15, and 16.

Investigations begin with discussions of experimentalsituations, variables, and procedures.

Next, students DRAW pictures in which they indicate theexperimental procedures and identify key variables.

Students then gather or COLLECT data and organize it indata tables. Next, they GRAPH their data.

The last phase or EXPLORE part of theexperiment is an in-depth analysis of theresults structured in a series of analyticalquestions.

Science

34

Finally, when their graph was finished, they analyzed and discussed their results.

Irma and Jerome chose toinvestigate the twovariables, hand length and height. Youalso will investigate twovariables that describeyour class. The armspan and height ofeach student in yourclass will be measured. Your job is to find out whether you can predict a fourth-grade student’sheight if you know his or her arm span.

Begin by drawing a picture of what you will do in the experiment. Then collectand organize data in a table. Next, make a graph of the data. Finally, explore thedata by looking for patterns.

Draw a picture of the setup for your experiment. Show the variables Arm Span(S) and Height (H) in your picture. Use Irma’s hand length and height picture tohelp you draw a picture of your Arm Span vs. Height experiment. Remember tolabel the variables.

1. A. Is arm span a categorical or a numerical variable?

B. Is height a categorical or a numerical variable? Explain how you know.

2. What is the same about all the people you measured for this experiment?

SG • Grade 4 • Unit 1 • Lesson 5 Arm Span vs. Height20

It looks like she’s in a hurry. Let’s

ask her if we can measure her hand. We can use our

graph to predict her height.

We never measured the

principal’s hand length and height.

Measure the arm span and height of each person in your group to the nearest inch. Record your group’s data in a data table like the one at the right. Discuss with your group what the letters S and H stand for.

Discuss any patterns you see in the data table.

• Graph your group’s data. Plot arm span on the horizontal axis and heighton the vertical axis. Scale your horizontal axis to at least 75 inches andthe vertical axis to at least 100 inches. Remember to label each axis.

• A class graph of Arm Span vs. Height will provide more data for you toanalyze. Plot one point, your own data, for arm span and height on the class graph.

Use your class data and graphs to help you answer the following questions.Include units with your answers. Be ready to share your answers with theentire class.

3. A. Describe your group’s graph. What do you notice about the points?

B. Describe the class graph. What do you notice about the points?

4. Compare your group’s graph and the class graph. How are they alike?How are they different?

Arm Span vs. Height SG • Grade 4 • Unit 1 • Lesson 5 21

Arm Span vs. Height Data Table

HHeight

(in inches)

SArm Span(in inches)

Name

Jerome graphed his family’s data as a point graph:

Jerome graphed hand lengthon the horizontal axis andheight on the vertical axis. To plot a point for his brotherPeter’s data, he first located hishand length, 12 cm, on thehorizontal axis. Then he foundhis height, 102 centimeters, onthe vertical axis.

• Make sure you see howJerome graphed hisbrothers’ and sisters’ data.

• Use the graph to find thehand length and height ofJerome’s parents.

Irma and Jerome measuredtheir classmates. They alsocollected data from otherfamilies, schoolchildren, andeven teachers. When theyfinished graphing the data,the graph represented HandLength vs. Height for theirneighborhood.

Arm Span vs. Height SG • Grade 4 • Unit 1 • Lesson 5 19

00

10

20

30

40

50

60

70

80

90

4 8 12 16 20 24 28 32 36

Jerome’s Family Graph

100

110

120

130

140

150

160

170

180

Peter

Jenny

Timothy

LHand Length (in centimeters)

HH

eigh

t (in

cen

timet

ers)

Abby

Mom

Jerome

Dad

The TIMS Laboratory Method

Irma and Jerome noticed that the Adventure Book story The Four Servants took

place in China. All the measurements were of Chinese people. They wondered if

the four servants would have found the same results if they had measured the

people in Irma and Jerome’s neighborhood.

Irma and Jerome used the TIMS Laboratory Method to help them solve problems

involving hand length and height in their neighborhood. The four servants used

this four-step method. First, Irma and Jerome drew a picture of the steps they

would follow in the experiment. Irma’s picture is shown below.

Arm Span vs. Height

I wonder if we

would get a line too? The

people in our neighborhood

are different.

Let’s try it! We can

start by collecting data for

our family members.

H4177_SG4_U01_002-02

5 6/25/07 2:42 PM

Page 17

Connections to Language Arts

Language Arts

35

Now we have to graph thedata and answer the questions.

We are supposed to predictthe bounce height when the drop

height is 60 cm. I get 33 cm.

I get 35 cm. Let’s try it out.

AB • Grade 4 • Unit 5 • Lesson 5 19

Two Heads Are Better Than One

OK. Ready.

It went to 34 cm.Good. We’re both close.

Now we have to predict howhigh the ball will bounce if we drop

it from 160 cm. We’ll have tomake the line longer.

That’sextrapolation.

AB • Grade 4 • Unit 5 • Lesson 520

Two Heads Are Better Than One

See video of examples of students collaborating anddiscussing mathematics in aMath Trailblazers classroom at www.mathtrailblazers.com.

Students increase mathematicalunderstanding and confidence bydiscussing and defending their ideas.

Writing and Discourse

Trade books launch and extend mathematical investigations.Original stories from the Adventure Books show concept applicationand relate concepts to science, history, and other subject areas.

Reading

Family Support

Supporting Family and StudentsOur engaging student materials, along with written and online support, create the foundationfor a successful partnership between educators, students, and their families.

36

Players

This is a game for two or more players.

Materials

• �12� sheet of Centimeter Grid Paper

• Spinners 1–4 and 1–10 Activity Page• a clear plastic spinner or a paper clip and pencil • a crayon or marker for each player

Rules

1. The first player spins twice so that he or she has two numbers. The playermay either spin one spinner twice or spin each spinner once.

Floor Tiler

4 1

3 2

11023

4789

The sketches below show the number of Chocos made by workers at the TIMSCandy Company. Write the amount of candy using numbers.

1. 2.

3. 4.

Dear Family Member:

Your child is reviewing place value—the idea that the value of a digit in a number

depends upon where it is placed. For example, the 2 in 329 stands for 2 tens but

the 2 in 7293 is 2 hundreds.

In class your child uses base-ten pieces to represent numbers. When the pieces are

not available, students are encouraged to draw pictures of the base-ten pieces. We

call these drawings of the base-ten pieces base-ten shorthand. To help your child

with homework Questions 1–11, you may wish to review the Base-Ten Shorthand

section on the previous pages.

Thank you for your help.

Families are encouragedto take part in gamesand activities.

Games

Includes notes to the family onassignments and explain how theycan help their student at home.

Homework Notes

Delivers core materials including activi-ties, labs, and games. Students completetables and graphs or use data to solveproblems. Key vocabulary terms are presented in boldface type.

Student Guide

This index provides page references for the Student Guide. Definitions or explanations of key terms can be found on the pages listed in bold.

Additionwith base-ten pieces, 78–80, 84–85multidigit, 174paper-and-pencil, 79–80, 84, 174

Angle, 41–56, 42, 243–250acute, 50–51, 54–55comparing, 43–44, 53drawing, 47, 54measuring, 243–250obtuse, 51, 54–55in pattern blocks, 55–56right, 48–49, 54–55in shapes, 45–46, 56vertex of, 243

Area, 28–38, 29counting square units, 433–436

Arm span, 20–23

Base (with exponents), 113Base-ten

division, 364–378multiplication, 298–303

Base-Ten Board, 68Base-ten pieces, 68–85

bit, 68flat, 68pack, 68skinny, 68

Base-Ten Recording Sheet, 68Base-ten shorthand, 71–85, 280–282, 285–287Best-fit line, 123–124

Categorical variable, 7, 20Certain event, 385

C

A

Index/Glossary

Grade 4

This Index/Glossary provides pagereferences for the Student Guide.Definitions or explanations of keyterms can be found on the pageslisted in bold.

Index/Glossary

Com

pany

Letter HomeGeometric Investigations: A Baseline Assessment Unit

Date:

Dear Family Member:

Geometry is an important part of the Math Trailblazers® curriculum because geometric knowledgeis useful in everyday situations. Many trades, such as carpentry and clothing design, are based ingeometry.

In this unit, your child will explore geometry and measurement, reasonmathematically, and communicate about mathematics. As your childinvestigates relationships between length, area, and perimeter, he orshe will collect and organize data, create and interpret graphs, andmake and check predictions. Your child will also look for angles in hisor her environment and in geometric shapes.

Your child also will review the subtraction facts.

Finally, your child will begin a math portfolio. Collecting completedassignments in a portfolio now and throughout the year will allow usto see how your child’s mathematical abilities develop in fourth grade.

As we explore mathematics concepts in the classroom, you can pro-vide mathematical opportunities at home. For example:

Acute

Obtuse

Com

pany

Designed to go out under theteacher’s signature, the Letter Homein each Unit Resource Guide explainswhat students will study in the unitand how their families can help.

Letter Home

Letter Home

also available in

Spanish

Family Support

37

Back to Index

Back to Index

Offers Access to Support for:• Homework help• TIMS Tutors—An in-depth exploration of the

mathematical concepts and ideas behindMath Trailblazers

• Glossary• Activities to extend learning• Frequently Asked Questions

Provides background information and suggestedapproaches and examples for families.

Online Homework Help

www.MyMathTrailblazers.com

Home PracticeTriangle Flash Cards: 9s

Study for the quiz on the multiplication facts for the nines. Take home your Triangle Flash Cards: 9s and your list of facts you need to study.

Here’s how to use the flash cards. Ask a family member to choose one flash cardat a time. He or she should cover the corner containing the highest number. Thisnumber will be the answer to a multiplication fact. Multiply the two uncoverednumbers.

Your teacher will tell you when the quiz on the 9s will be.

Mixed-Up Multiplication Tables

1. Complete the table. Then, describe any patterns you see.

PART 2

PART 1

Name Date

4

6

7

8

2 5 9 103×

18

Unit 6

Name Date

Predicting Prices

Look for patterns in each graph. If the points form a line, draw a best-fit lineon the graph.

1.

050545862667074788286909498

0 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08

Cos

t of 1

lb o

f Let

tuce

in C

ents

Cost of LettuceThe Discovery Assignment Book is brimmingwith student materials—activities, flashcards, tables, graphs, and homework—thatcomplement materials in the Student Guide.Space is included for students to write intheir answers, and pages can be torn outand handed in to the teacher.

Discovery Assignment Book

Use of StrategiesIn early grades the basic facts are approached as problems to be solved rather than facts to be memorized.In all grades we encourage the use of strategies to findfacts, so students become confident they can findanswers to problems that they do not immediatelyrecall. Research has shown that an overemphasis onmemorization and frequent administration of timedtests is counterproductive.

Distributed Review of the FactsStudents study small groups of facts that can be foundusing similar strategies. Students review these facts,one group at a time. Starting in Grade 2 students useflash cards to review facts in school and at home.

Ongoing Practice Math facts are distributed throughout the curriculum,especially in the Daily Practice and Problems, HomePractice, and games.

Multiyear Approach In Grades 1 and 2, Math Trailblazersemphasizes strategies that lead tofluency in addition and subtractionfacts. In Grade 3, students gain fluency with the multiplication factswhile reviewing the addition and subtraction facts. In Grade 4,students achieve fluency with thedivision facts and verify fluencywith the multiplication facts. InGrade 5, the multiplication anddivision facts are systematicallyreviewed and assessed.

Appropriate AssessmentTeachers observe and assess students’ knowledge ofthe facts as they work on activities, labs, and games as well as through written tests and quizzes. Periodicshort quizzes in the Daily Practice and Problems sections naturally follow the study of small groups of facts organized around specific strategies.

Facts Are Not Gatekeepers Students are not prevented from learning more complexmathematics because they do not perform well on facttests. Use of strategies, calculators, and other math tools(e.g. manipulatives, hundreds charts, and multiplicationtables) allow students to continue working on importantmathematical concepts while they are still learning the facts.

Math Facts

Math FactsA major goal of Math Trailblazers is to prepare students to compute accurately and flexibly in all situations.Acquisition of basic math facts and fluency with whole-number operations is covered extensively.

A Careful BalanceIn creating our program we sought a balance between developing strategies and providing practice. This approachis based on a large body of research and advocated by the NCTM’s Principles and Standards for School Mathematics(2000). This research indicates that the methods used in Math Trailblazers lead to more effective learning and better retention of math facts.

38

Mrs. Haddad noticed that 4 of theworkers at the TIMS Candy Companyeach made 32 Chocos in one hour. She wanted to find the total numberof Chocos made. Mrs. Haddadused base-ten pieces and wrote32 + 32 + 32 + 32. She said thiscould be an addition problem or amultiplication problem.

Mrs. Haddad used the all-partials method to solve the problem by multiplication. Since there are 4 groups of 2 bits, there are 8 bits total.

Since there are 4 groups of 3 skinnies, this makes 12 skinnies or 1 flat 2 skinnies and

MultiplicationBase-Ten Board

3 2× 4

8

Recording Sheet

Recording Sheet

Nicholas watched 347 minutes of television in 5 days. If he watched about thesame number of minutes each day, about how many minutes per day did he watch television?

1. Estimate how many minutes Nicholas watched television each day. Did hewatch more than 100 minutes each day?

2. Model the problem using the base-ten pieces or base-ten shorthand.

3. Will 5 divide into 347 evenly? Why or why not?

4. What is the average number of minutes Nicholas watched television eachday? Explain the remainder.

Keenya showed her method again for doing division.

More Division

How do you know what numbers

to pick?

You don’t always.Try to get a good estimate. The only

time you have to erase is if your estimate is too big. I try to choose easy numbers

at the beginning. There are many ways of getting the correct answer.

The Beautiful Blooms Garden Store sells many trays of flowers. A tray of flowers has 6 rows. There are 8 flowers in each row.

1. How many flowers are in each tray?

Ana and Grace went to the BeautifulBlooms Garden Store. They noticed thatthe trays of flowers were stored on ashelf. Ana counted 28 trays on a shelf.

Ana said, “I wonder how many flowers areon one of these shelves.”

Grace replied, “We can estimate thenumber of flowers on a shelf: 48 flowers is about 50 and 28 trays is about 30 trays. So what is 50 × 30?”

Multiplying Round Numbers

A Curriculum for All

39

Integrated DifferentiationScienceThe hands-on nature of scientific investigation allowschildren with different learning styles and interests toexplore the mathematics.

Language ArtsMath concepts are introduced and developed usingcontexts rich in language. Children can explain theirthinking and solutions to problems in a variety of ways,using both oral and written communication.

ManipulativesUsing manipulatives provides different representationsof math concepts to help children better understandand remember the mathematics.

Collaborative WorkStudents with different abilities and styles learn fromone another and gain confidence in their problem-solving abilities.

Balanced AssessmentMultiple forms of assessment provide students with avariety of ways to show progress to their teacher.

Curriculum SequenceConcepts are revisited in different contexts within thegrade and across all grades, giving students repeatedlearning opportunities.

Reaching All StudentsThrough High ExpectationsWith problem solving central to the design of Math Trailblazers, opportunities to differentiate instruction are builtinto the program in multiple ways. Lessons are designed with the flexibility to reach students with a wide range ofabilities and learning styles. Some opportunities to differentiate instruction are specifically identified in a specialsection of each unit and in tips. Other opportunities are integrated throughout the curriculum.

Identified DifferentiationTeaching All Math Trailblazers Students(Appears in the Unit Resource Guide Outline)Identifies games, challenges, extensions,laboratory experiments, and journal prompts.

TIMS Tips(Appear in lessons in the Unit Resource Guide)Provides management suggestions to promote learning for all students.

Daily Practice and Problems(Appears in the Unit Resource Guide)Provides practice and review of math concepts,skills, and facts.

For Grades 3–5, three different types of Daily Practice and Problems are included: Bits (review and practice),Tasks (grade level and current learning), and Challenges (deeper thinking). All aid in differentiating instruction.

Highlights of What’s New inMath Trailblazers Third Edition?

• Reorganized teacher-friendly Unit Resource Guide – New full-color format – Identified differentiation – Lessons contain all necessary BLM – New Unit Planner and revised Outline – Reduced student pages within Answer Keys

• Administrator Handbook • Teacher Implementation Guide includes new or expanded

sections on:– Meeting individual needs– The Math Trailblazers classroom– Working with parents– Language in the Math Trailblazers classroom

• Reorganization of the Unit Resource Guide File • And much more! Contact Kendall/Hunt for a complete

listing of all the changes to the third edition.

Catalog AvailableIn addition to our grade-level samplers, you may want to look at our program product catalog. It contains additional information on Math Trailblazers, including pricing information. Request them from your sales representative or download it at www.mathtrailblazers.com.

C H E C K U S O U T A T. . .

This program originallyreceived funding from theNational Science Foundation.

grade four sampler

Documents