Made to Measure - Universiteit Utrecht · 2016-04-13 · Blackline Masters Letter to the Family 56 Student Activity Sheets 57 Teaching Transparency 58 C D. Overview Made to Measureand

Made toMeasureGeometry andMeasurement

TEACHER’S GUIDE

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

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

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

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

All rights reserved.

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

International Standard Book Number 978-1-59339-945-0

Printed in the United States of America

1 2 3 4 5 13 12 11 10 09

de Lange, J.,Wijers, M., Dekker, T., Simon, A. N., Shafer, M. C., & Pligge, M. A.(2010). Made to measure. InWisconsin Center for Education Research &Freudenthal Institute (Eds.), Mathematics in context. Chicago: EncyclopædiaBritannica, Inc.

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

The Mathematics in Context Development TeamDevelopment 1991–1997

The initial version of Made to Measure was developed by Jan de Lange and Monica Wijers. It was adapted for use in American schools by Aaron N. Simon and Mary C. Shafer.

Wisconsin Center for Education Freudenthal Institute StaffResearch Staff

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

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

Project Staff

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

Revision 2003–2005

The revised version of Made to Measure was developed by Truus Dekker. It was adapted for use in American Schools by Margaret A. Pligge.

Wisconsin Center for Education Freudenthal Institute StaffResearch Staff

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

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

Project Staff

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

Cover photo credits: (left) © Getty Images; (middle) © Kaz Chiba/PhotoDisc/Getty Images; (right) © PhotoDisc/Getty Images

Illustrationsx Map from the Road Atlas © 1994 by Rand McNally; (right) © EncyclopædiaBritannica, Inc.; xii (bottom right) Holly Cooper-Olds; xviii (middle, bottomright, bottom middle) Christine McCabe/© Encyclopædia Britannica, Inc.;2, 14, 22 Holly Cooper-Olds; 23, 24 Christine McCabe/© EncyclopædiaBritannica, Inc.; 28 Holly Cooper-Olds; 31 Christine McCabe/© EncyclopædiaBritannica, Inc.; 32 Holly Cooper-Olds; 35 Christine McCabe/© EncyclopædiaBritannica, Inc.; 37, 38 Holly Cooper-Olds

Photographsx Historic Urban Plans, Inc.; xiii Courtesy of Michigan State UniversityMuseum; xvii © Corbis; 1 (counter clockwise) © PhotoDisc/Getty Images;© PhotoDisc/ Getty Images; © Ingram Publishing; © PhotoDisc/GettyImages; Sam Dudgeon/ HRW Photo; © Corbis; 10 Victoria Smith/HRW;17 Victoria Smith/HRW; 34 © EB Inc.

Made to Measure v

Contents

OverviewNCTM Principles and Standards for School Mathematics viiMath in the Unit viiiGeometry and Measurement Strand: An Overview xStudent Assessment in Mathematics in Context xivGoals and Assessment xviMaterials Preparation xviii

Student Material and Teaching NotesStudent Book Table of ContentsLetter to the Student

Section LengthsSection Overview 1AIntroduction: Selecting Strategies for Measurement 1Historical Measures: Comparing Systems of Measurement 2Comparing Systems: Converting Between Metric and Customary

Systems of Measurement 5Feet and Shoes: Converting from One Unit to Another 6Body Length and Fathom: Understanding Relationships Among Units 8Other Measures for Length: Comparing Processes of Measurement 9Summary 10Check Your Work 11

Section AreasSection Overview 12AA Body’s Surface Area: Understanding Surface Area 12Squares: Using Data to Estimate Surface Area 13Skinning a Square: Comparing Strategies for Determining Surface Area 14Hands and Body: Using Two-Dimensional Representations of

Three-Dimensional Objects to Solve Problems About Surface Area 15Surface Area by Formula: Using Geometric Models to Solve

Problems About Surface Area 17Height, Weight, and Area: Using Graphical Representations to

Solve Problems 18Early Areas: Converting from One Unit of Area to Another 19Summary 20Check Your Work 21

B

A

vi Made to Measure Contents

Contents

Section VolumesSection Overview 22AThe Volume of Your Heart: Estimating the Volume of a Solid 22Solids: Using Two-Dimensional Representations of Three-Dimensional

Objects to Solve Problems About Volume 22Liquids: Estimating the Volume of a Liquid; Using Graphical

Representations to Understand Relationships Among Units 26Measuring the Volume of Your Hand: Comparing Strategies for

Determining Volume 26The Volume of Your Body: Using Visualization to Solve Problems

About Volume 27Other Measures for Volume: Understanding Relationships Among

Units of Volume 28Summary 30Check Your Work 30

Section AnglesSection Overview 32AFurniture: Measuring and Understanding Relationships Among Arcs,

Lengths, and Angles 32Arcs of Movement: Measuring Arcs 32Summary 38Check Your Work 38

Additional Practice 40

Assessment and SolutionsAssessment Overview 43AQuiz 1 44Quiz 2 46Unit Test 48Quiz 1 Solutions 50Quiz 2 Solutions 51Unit Test Solutions 52

Glossary 54

Blackline MastersLetter to the Family 56Student Activity Sheets 57Teaching Transparency 58

C

D

Overview

Made to Measure andthe NCTM Principlesand Standards forSchool Mathematics forGrades 6–8The process standards of Problem Solving, Reasoning and Proof,Communication, Connections, and Representation are addressedacross all Mathematics in Context units.

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

Measurement

In grades 6–8 all students should:

• understand both metric and customary systems of measurement;

• understand relationships among units and convert from one unitto another within the same system;

• understand, select, and use units of appropriate size and type tomeasure angles, perimeter, area, surface area, and volume;

• use common benchmarks to select appropriate methods forestimating measurements;

• select and apply techniques and tools to accurately find length,area, volume, and angle measures to appropriate levels ofprecision; and

• develop strategies to determine the surface area and volume ofselected prisms, pyramids, and cylinders.

Geometry

In grades 6–8 all students should:

• draw geometric objects with specific properties, such as sidelengths or angle measures;

• use two-dimensional representations of three-dimensional objectsto visualize and solve problems such as those involving surfacearea and volume; and

• recognize and apply geometric ideas and relationships in areasoutside the mathematics classroom, such as art, science, andeveryday life.

Overview Made to Measure vii

Overview

Prior Knowledge

This unit assumes that students have anunderstanding of the following:

• the difference between length, area, and volume;

• how to use rulers to measure lengths in metricand customary units;

• how to find the areas of square and rectangularshapes in square units;

• units of measure used for liquid volume (liter,milliliter, gallon, and so on); and

• how to measure and draw angles using acompass card or protractor.

Throughout Made to Measure, students work withmetric and customary units to make actualmeasurements for length, area, and volume. Theybegin by studying historic units of measure basedon body parts, such as foot, pace, and fathom(length of outstretched arms). They relate theseunits to their own bodies and discuss the accuracyand usefulness of this type of measurement.

By comparing historic units of measurement tostandardized ones, students understand theimportance of the use of standard measurementunits. The metric system is based on powers of ten,which facilitates the process of convertingbetween different measurement units. Forexample:

• 1 meter (m) � 10 decimeters (dm)

• 1 dm � 10 centimeters (cm)

• 1 cm � 10 millimeters (mm)

For area the conversions are 1 m2 � 100 dm2, 1dm2

� 100 cm2, and so on. Finally, for volume theseconversions are used: 1 m3 � 1000 dm3; 1dm3 �1000 cm3, and so on. The liquid volume of 1 liter isthe same as 1dm3. Students need to decide whichunit of measurement is appropriate when solvingproblems.

The use of instruments like centimeter or inchrulers, centimeter measuring tapes, and compasscards or protractors is reviewed. Students use theirown measurements in activities about length, area,and volume to find estimation rules that makemental computation easier. Throughout the unit,they are encouraged to find their own points ofreference for conversions from one measurementsystem to the other, or within systems, such as:

• a gallon of milk is about 4 liters;

• a yard is a little less than one meter;

• a length of 4 inches is about the same as 10 cm;

• a football field is about the size of an acre.

Math in the Unit

viii Made to Measure Overview

Overview

When solving problems, students need to decidewhether an estimated answer, using estimationrules for conversion, is sufficient or if an exactanswer is needed. If an exact answer is needed,they must decide how many decimals areappropriate in the realistic situation they aredealing with.

The design, use, and critique of mathematicalmodels to represent a real world situation play animportant part in this unit. It is not possible to“measure” a person’s body’s surface area, so avariety of mathematical models is designed torepresent the human body. One mathematicalmodel is the use of two equal cylinders, each withthe height of the person and with a circumferencethat equals the person’s thigh measure. Studentslearn why mathematical models are used, tounderstand the advantages and disadvantages ofeach model and to find their limitations.

Students estimate and compare the volume of avariety of objects using their own points ofreference. They also use the formula volume �area of base × height and explain why this formulacan be used for certain objects, like boxes, but notfor others, like cones. By the end of the unit,students use angle measures for the angles theycan make with their wrists and investigate howfurniture and computers are designed to fit theangles of the body and other recommendationsfrom ergonomics.

When students have finished the unit they can:

• measure length, area, and volume using metricand customary units of measure. Students can:

• understand how historic units are related tobody measures;

• understand the importance of the use ofstandardized measurement units;

• convert units from one measurement systemto the other and within systems;

• find their own points of reference when usingestimation rules to make mental computationseasier;

• choose appropriate units of measurement in asituation; and

• decide whether an exact measure or anestimated one is appropriate in a situation.

• use and critique mathematical models torepresent an irregular shape. Students can:

• use and compare a variety of models to findthe body’s surface area, and

• use a cylinder filled with water to find thevolume of a hand.

• use the formula volume � area of base × heightto find the volume of some objects.

• use tools like centimeter rulers or inch rulersand compass cards and protractors. Studentscan:

• measure and draw angles using appropriateunits, and

• use a nomogram to relate a body’s surface areato a person’s height and weight.

• solve problems in a variety of situationsinvolving length, area and volume. Students can:

• investigate the relationship between footlength and shoe size;

• investigate the relationship between fathomand height; and

• use geometric models to solve problems likethe angle between back rest and seat of a chair.

Overview Made to Measure ix

Overview

x Made to Measure Overview

In the MiC units, measurement concepts and skillsare not treated as a separate strand. Many measure-ment topics are closely related to what studentslearn in geometry. The geometry and measurementunits contain topics such as similarity, congruency,perimeter, area, and volume. The identification ofand application with a variety of shapes, bothtwo-dimensional and three dimensional, is alsoaddressed.

The developmental principles behind geometryin Mathematics in Context are drawn from HansFreudenthal’s idea of “grasping space.” Throughoutthe strand, ideas of geometry and measurementare explored. Geometry includes movement andspace—not just the study of shapes. The majorgoals for this strand are to develop students’ abilityto describe what is seen from different perspectivesand to use principles of orientation and navigationto find their way from one place to another.

The emphasis on spatial sense is related to howmost people actually use geometry. The develop-ment of students’ spatial sense allows them to solveproblems in the real world, such as identifying acar’s blind spots, figuring out how much materialto buy for a project, deciding whether a roof orramp is too steep, and finding the height or lengthof something that cannot be measured directly,such as a tree or a building.

Mathematical Content

In Mathematics in Context, geometry is firmlyanchored in the physical world. The problemcontexts involve space and action, and studentsrepresent these physical relationships mathe-matically.

Throughout the curriculum, students discoverrelationships between shapes and develop theability to explain and use geometry in the realworld. By the end of the curriculum, studentswork more formally with such geometric conceptsas parallelism, congruence, and similarity, and usetraditional methods of notation as well.

Geometry and Measurement Strand:An Overview

SanJose

PacificOcean

Half MoonBay

Hayward

Golden Gate Bridge

1

SanFrancisco

680

92

10 miles5

10 16 kilometers5

0

Airports

S

N

W E

NE

NW

SW

SE

0 10 2030

40

5060

7080

90100

110120

130

140150

160170180190200210

220

230

240

250

26027

028

029

030

031

0

320330

340 350

101

280

San Francisco Bay

San FranciscoOakland BayBridge

OaklandInternationalAirport

84

Overview

Overview Made to Measure xi

Organization of the Geometry andMeasurement Strand

Visualization and representation is a pervasivetheme in the Geometry strand and is developedin all of the Geometry and Measurement strandunits. The units are organized into two substrands:Orientation and Navigation, and Shape andConstruction. The development of measurementskills and concepts overlaps these two substrandsand is also integrated throughout otherMathematics in Context units in Number, Algebra,and Data Analysis.

Orientation and Navigation

The Orientation and Navigation substrand is intro-duced in Figuring All the Angles, in which studentsare introduced to the cardinal, or compass,directions and deal with the problems that arisewhen people in different positions describe alocation with directions. Students use maps andcompass headings to identify the positions ofairplanes. They look at angles as turns, or changesin direction, as well as the track made by a sled inthe snow. They discern different types of anglesand learn formal notations and terms: vertex, �A,and so on. The rule for the sum of the angles in atriangle is informally introduced. To find anglemeasurements, students use instruments such as aprotractor and compass card.

123

Pathways through theGeometry and Measurement Strand

(Arrows indicate prerequisite units.)

Level 2

Level 1

Level 3

Figuring Allthe Angles

Reallotment

Made toMeasure

Packages andPolygons

Triangles andBeyond

30o

60o

90o

120o

150o

180o210o

240o

270o

300o

330o

N

30 mi.30 mi.30 mi

10 mi.10 mi.

20 mi.20 mi.

10 mi

20 mi

It’s All theSame

Looking atan Angle

Overview

xii Made to Measure Overview

In Looking at an Angle, the last unit in the Geometrystrand, the tangent ratio is informally introduced.The steepness of a vision line, the sun’s rays, a ladder,and the flight path of a hang glider can all be mod-eled by a right triangle. Considering the glide ratioof hang gliders leads to formalization of the tangentratio. Two other ratios between the sides of a righttriangle are introduced, the sine and the cosine.This leads to formalization of the use of thePythagorean theorem and its converse.

Shape and Construction

Reallotment is the first unit in the Shape andConstruction substrand. Students measure andcalculate the perimeters and areas of quadrilaterals,circles, triangles, and irregular polygons. Studentslearn and use relations between units of measure-ment within the Customary System and the MetricSystem.

Solids are introduced in Packages and Polygons.Students compare polyhedra with their respectivenets, use bar models to understand the conceptof rigidity, and use Euler’s formula to formallyinvestigate the relationships among the numbersof faces, vertices, and edges of polyhedra.

In Triangles and Beyond,students develop a moreformal understanding ofthe properties of triangles,which they use to constructtriangles. The concepts ofparallel lines, congruence,and transformation areintroduced, and studentsinvestigate the propertiesof parallel lines and paral-lelograms. A preformalintroduction to thePythagorean theoremis presented.

After studying this unit,students should be ableto recognize and classifytriangles and quadrilaterals.In the unit It’s All the Same,students develop an under-standing of congruency,

similarity, and the properties of similar trianglesand then use these ideas to solve problems. Theirwork with similarity and parallelism leads them tomake generalizations about the angles formedwhen a transversal intersects parallel lines, and thePythagorean theorem is formalized.

Does Euler’s formula workfor a five-sided tower?Explain your answer.

If a triangle has aright angle, then the squareon the longest side has the

same area as the othertwo combined.

Overview

Overview Made to Measure xiii

Measurement

The concept of a measurement system, standard-ized units, and their application overlaps the sub-strands of Orientation and Navigation, and Shapeand Construction. Furthermore, the developmentand application of measurement skills is integratedthroughout units in the Number, Algebra, andData Analysis strands, through topics such as useof ratio and proportion, finding and applying scalefactors, and solving problems involving rates (forinstance, distance-velocity-time relationships).

In Mathematics in Context, the Metric System isused not only as a measurement system, but alsoas a model to promote understanding of decimalnumbers.

The unit Made to Measure is a thematic measure-ment unit where students work with standard andnon-standard units to understand the systems andprocesses of measurement. They begin by studyinghistoric units of measure such as foot, pace, andfathom (the length of outstretched arms). Studentsuse their own measurements in activities aboutlength, area, volume, and angle and then examinewhy standardized units are necessary for each.

The relationships between measurement units areembedded in the number unit, Models You CanCount On, where students explore conversionsbetween measures of length within the MetricSystem. The measurement of area in both metricand Customary Systems is explicitly addressed inthe unit Reallotment. Students also learn somesimple relationships between metric and customarymeasurement units, such as 1 kilogram is about2.2 pounds, and other general conversion rules tosupport estimations across different measurementsystems. In Reallotment, Made to Measure, andPackages and Polygons, the concepts of volumeand surface area are developed. Strategies thatwere applied to find area measurements inReallotment are usedto derive formulas forfinding the volume ofa cylinder, pyramid,and cone.

Visualization andRepresentation

Visualization and representation is a componentof every Geometry unit. In Mathematics in Context,this theme refers to exploring figures from differentperspectives and then communicating about theirappearance or characteristics.

In Reallotment, students use visualizations andrepresentations to find the areas of geometricfigures. They decide how to reshape geometricfigures and group smaller units into larger, easy-to-count units. They also visualize and representthe results for changing the dimensions of a solid.In the unit It’s Allthe Same, studentsvisualize triangles tosolve problems.

Packages

Pencils

1

15

� 2 � 2

2

30

4

60

1 cm2

Overview

xiv Made to Measure Overview

Level IIIanalysis

Level IIconnections

Leve

ls o

f Rea

sonin

g

Questions P

osedDom

ains of Mathem

atics

Level Ireproduction

algebra

geometry

number

statistics &

probability

X

O

easy

difficult

Student Assessment inMathematics in ContextAs recommended by the NCTM Principles and Standards for SchoolMathematics and research on student learning, classroom assessmentshould be based on evidence drawn from several sources. An assessmentplan for a Mathematics in Context unit may draw from the followingoverlapping sources:

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

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

• products —Look for clarity and quality of thought in students’solutions to problems completed in class, homework, extensions,projects, quizzes, and tests.

Assessment Pyramid

When designing a comprehensive assessment program, the assessmenttasks used should be distributed across the following three dimensions:mathematics content, levels of reasoning, and difficulty level. TheAssessment Pyramid, based on Jan de Lange’s theory of assessment,is a model used to suggest how items should be distributed acrossthese three dimensions. Over time, assessment questions should“fill” the pyramid.

Overview

Overview Made to Measure xv

Levels of Reasoning

Level I questions typically address:

• recall of facts and definitions and

• use of technical skills, tools, and standardalgorithms.

As shown in the pyramid, Level I questions are notnecessarily easy. For example, Level I questions mayinvolve complicated computation problems. Ingeneral, Level I questions assess basic knowledgeand procedures that may have been emphasizedduring instruction. The format for this type ofquestion is usually short answer, fill-in, or multiplechoice. On a quiz or test, Level I questions closelyresemble questions that are regularly found in agiven unit substituted with different numbersand/or contexts.

Level II questions require students to:

• integrate information;

• decide which mathematical models or tools touse for a given situation; and

• solve unfamiliar problems in a context, basedon the mathematical content of the unit.

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

Level III questions require students to:

• make their own assumptions to solve open-endedproblems;

• analyze, interpret, synthesize, reflect; and

• develop one’s own strategies or mathematicalmodels.

Level III questions are always open-ended problems.Often, more than one answer is possible and thereis a wide variation in reasoning and explanations.There are limitations to the type of Level III prob-lems that students can be reasonably expected torespond to on time-restricted tests.

The instructional decisions a teacher makes as heor she progresses through a unit may influence thelevel of reasoning required to solve problems. If amethod of problem solving required to solve aLevel III problem is repeatedly emphasized duringinstruction, the level of reasoning required to solvea Level II or III problem may be reduced to recallknowledge, or Level I reasoning. A student who doesnot master a specific algorithm during a unit butsolves a problem correctly using his or her owninvented strategy may demonstrate higher-levelreasoning than a student who memorizes andapplies an algorithm.

The “volume” represented by each level of theAssessment Pyramid serves as a guideline for thedistribution of problems and use of score pointsover the three reasoning levels.

These assessment design principles are usedthroughout Mathematics in Context. The Goalsand Assessment charts that highlight ongoingassessment opportunities—on pages xvi and xviiof each Teacher’s Guide—are organized accordingto levels of reasoning.

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

Assessment Pyramid

10, 11

Use geometric models tosolve problems.

Overview

xvi Made to Measure Overview

Goals and AssessmentIn the Mathematics in Context curriculum, unit goals organized accordingto levels of reasoning described in the Assessment Pyramid on page xiv,relate to the strand goals and the NCTM Principles and Standards forSchool Mathematics. The Mathematics in Context curriculum is designedto help students demonstrate their understanding of mathematics in

each of the categories listed below. Ongoingassessment opportunities are also indicated ontheir respective pages throughout the Teacher’sGuide by an Assessment Pyramid icon.

It is important to note that the attainment ofgoals in one category is not a prerequisite to theattainment of those in another category. In fact,students should progress simultaneously toward

several goals in different categories. The Goals and Assessment table isdesigned to support preparation of an assessment plan.

Level I:

Conceptual

and Procedural

Knowledge

Ongoing Unit

Goal Assessment Opportunities Assessment Opportunities

1. Estimate lengths, Section B p. 15, #7b Quiz 1 #2eareas, volumes, and Section C p. 23, #3a Quiz 2 #2bangles using appropriate p. 26, #14 Test #1abcdeunits.

2. Find (compute, Section A p. 4, #10abcd Quiz 1 #1abdmeasure, and draw) p. 8, #23 Quiz 2 #2a, 3alengths, areas, volumes, Section B p. 19, #15 Test #5bc, 6and angles using Section D p. 33, #3abstandard andnonstandard units.

3. Use simple Section A p. 4, #14ab Quiz 1 #2abcdrelationships between p. 5, #15 Quiz 2 #1abcand within measurementsystems.

Overview

Overview Made to Measure xvii

Level II:

Reasoning,

Communicating,

Thinking,

and Making

Connections

Level III:

Modeling,

Generalizing,

and Non-Routine

Problem Solving

Ongoing Unit


4. Recognize that Section B p. 21, Quiz 1 #1cedifferent units of For Further Reflection Test #4measurement anddifferent measurementsystems exist.

5. Develop and use a Section A p. 9, #26c Quiz 1 #3abframe of reference for Section C p. 29, #19ab Quiz 2 #1bcmeasurement to solve Section D p. 39, Test #2ab, 4problems. For Further Reflection

6. Understand and use Section C p. 23, #4b Quiz 2 #3bthe relationships p. 27, #15b Test #5cbetween length, area, p. 31,volume, and angle. For Further Reflection

Ongoing Unit


7. Use geometric models Section B p. 17, #10, 11 Test #3to solve problems. Section D p. 35, #6ab

8. Develop and use Section C p. 25, #11, 12ab Quiz 2 #1dformulas to estimate and Test #5acalculate length, area,and volume.

Overview

The following items are the necessary materialsand resources to be used by the teacher andstudents throughout the unit. For further details,see the Section Overviews and the Materialssection of the Hints and Comments column oneach teacher page. Note: Some contexts andproblems can be enhanced through the use ofoptional materials. These optional materials arelisted in the Hints and Comments column.

Student Resources

Quantities listed are per student.

• Letter to the Family

• Student Activity Sheet 1

Teacher Resources

Quantities listed are per class, unless otherwiseindicated.

• 5-meter piece of rope

• Dictionary or encyclopedia

• Transparency Master 1

Student Materials

Quantities listed are per student, unless otherwiseindicated.

• Blank paper

• Centimeter cubes (made by the students)

• Centimeter graph paper (two sheets)

• Centimeter or inch rulers

• Centimeter or inch tape measures

• Compass card or protractor

• Graph paper (one sheet)

• Meter stick (one per group)

• Metric measuring cup or beaker (one per group)

• Newspapers (several per group)

• Orange juice carton, ketchup bottle, mayonnaisejar, and small carton of milk (one of each pergroup of students)

• Paper clips

• Tissue boxes (one per group of students)

• Yardstick (one per group)

xviii Made to Measure Overview

Materials Preparation

10 20 30 40 50 60 70 80 901 2 4 6 8 12 14 16 18 22 24 26 28 32 34 36 38 42 44 46 48 52 54 56 58 62 64 66 68 72 74 76 78 82 84 86 88 92 94 96 98 993 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

10 20 301 2 4 6 8 12 14 16 18 22 24 26 283 5 7 9 11 13 15 17 19 21 23 25 27 29

1 2 3 4 5 6 7 8 9 10 11 12

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SE

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S

StudentMaterial

andTeaching

Notes

StudentMaterial

andTeaching

Notes

Teachers Matter

v Made to Measure Teachers Matter

ContentsLetter to the Student vi

Section A LengthsIntroduction 1Historical Measures 2Feet and Shoes 6Body Length and Fathom 8Other Measures for Length 9Summary 10Check Your Work 11

Section B AreasA Body’s Surface Area 12Squares 13Hands and Body 15Surface Area by Formula 17Height, Weight, and Area 18Early Areas 19Summary 20Check Your Work 21

Section C VolumesThe Volume of Your Heart 22Solids 22Liquids 26The Volume of Your Body 27Other Measures for Volume 28Summary 30Check Your Work 30

Section D AnglesFurniture 32Summary 38Check Your Work 38

Additional Practice 40

Answers to Check Your Work 43

Teachers Matter

Teachers Matter Made to Measure vT

Dear Student,

Welcome to the Mathematics in Context unit Made to Measure. Thisunit is all about measuring: measuring your feet, your thumb, yourhands, and the angle made by your arm and your wrist.You will investigate how measuring units evolved.You will further investigate measurements for length,area, and volume. You might beamazed by what you can measure!

You will find that mathematics plays animportant role in measurement. Every timeyou measure something, you might ask yourself:

• Will every person measuring this item get the same measurementthat I did?

• Do all of these things have the same measurement?

• What other units of measure can I use?

• Are there other ways to measure these things?

Whenever you make a measurement in this unit,picture how big—or small or steep or short—that measurement is. When you can do thiswith all of the measurements in this unit,you are well on your way to becominga mathematician!

Sincerely,

The Mathematics in Context Development Team

1A Made to Measure Teachers Matter

Teachers MatterA

Section Focus

In this section, the use of the metric and customary systems to measure length,which students have studied in previous grades, is reviewed. Students relate thelength of different parts of the body to familiar objects and choose appropriatemeasurement units for a variety of situations. Historic units to measure lengthare compared to standardized ones so students understand the importance ofstandard measurement units.

Pacing and Planning

Day 1: Introduction Student pages 1–3

INTRODUCTION Problems 1 and 2 Relate the length of different parts of thebody to familiar objects.

CLASSWORK Problems 3–6 Choose an appropriate non-standardmeasurement unit to measure length.

HOMEWORK Problems 7–9 Investigate historical measurement unitsby averaging the length of body parts.

Day 2: Historical Measures (Continued) Student pages 4 and 5

INTRODUCTION Problems 10–12 Investigate historical measurement unitsby averaging the length of body parts.

CLASSWORK Problems 13 and 14 Compare the customary and metricmeasurement systems.

ACTIVITY Activity, page 5 Investigate the relationship betweencustomary and metric measurementunits and find conversion rules.

HOMEWORK Problem 15 List relationships between the customaryand metric measurement systems.

Day 3: Feet and Shoes Student pages 6–8

INTRODUCTION Problems 16–18 Investigate the relationship between shoesize and foot length by organizing theinformation into a table and graph.

CLASSWORK Problems 19–21 Find the relationship between foot lengthand shoe size and investigate differenthistorical standards for measuring a mile.

HOMEWORK Problems 22 and 23 Investigate the relationship between aperson’s height and his or her fathom.

Teachers Matter Section A: Lengths 1B

Teachers Matter A

Materials

Student Resources


• Letter to the Family

Teachers Resources

Quantities listed are per class.

• 5-meter piece of rope

Student Materials

Quantities listed are per pair of students, unlessotherwise noted.

• Centimeter or inch rulers

• Inch or centimeter tape measure

• Meter stick (one per group)

• Paper clips

• Yardstick (one per group)

* See Hints and Comments for optional materials.

Learning Lines

Number Sense

Students are encouraged to use and develop theirmeasurement sense by finding personal points ofreference having to do with length. Some examplesare:

• The distance from my house toour school is about 3 miles.

• When walking at a comfortablespeed, I move about 2 milesper hour.

• An inch is about the same asthe width of my thumb.

Students learn to decide when touse an exact measure and whenan estimate is more appropriate.

Measurement

Students use inch rulers or centimeter tapemeasures to measure the length of an object. Theyread rulers to the nearest tenth and understand themeaning of metric measurements for length. Forexample, the length 2.7 cm is 2 cm and 7 mm.Students investigate the use of historic measure-ments for length, such as a hand span, a thumb, ora fathom, and how they are related.

Problem Solving

A variety of problem-solving situations areprovided in this section. Students investigate therelationship between foot length and shoe sizeand between fathom and height of a person.

At the End of This Section: LearningOutcomes

By the end of this section, students are able to:

• measure length using metric, customary, andnon-standard units of measure;

• understand the importance of using standardmeasurement units;

• convert units of length within systems andbetween systems;

• choose an appropriate unit of measurement fora given situation; and

• decide whether to use an exact measure or anestimate.

Day 4: Other Measures for Length Student pages 9–11

INTRODUCTION Review homework. Review homework from Day 3.

CLASSWORK Problems 25 and 26 Investigate the use of travel times tomeasure length.

HOMEWORK Check Your Work Student self-assessment: IdentifyFor Further Reflection appropriate units of measurement and

estimate lengths and distances.

Additional Resources: Additional Practice, Section A, Student Book page 40

1 Made to Measure

Notes

Begin this unit with a dis-cussion about measure-ment. Ask students whatthey know about measure-ment and what they wouldlike to learn. This is a topicthat students have exper-ience with in many aspectsof their daily life.

2 Be sure to have studentsshare their responses tothis problem with the class.

LengthsA

Extension

The Romans invented the mile as a unit to measure distances. The wordmile comes from millia passuum, which means “1,000 double paces.” Havestudents find more information about historic measurements.

Hands-On Learning

Have students struggling with problem number 1 open a door, and thenask, What body parts would you need to know if you were to build a door?This can be repeated with sitting at a school desk and putting on a pair ofsocks.

Reaching All Learners

People who design the objects you use everyday have thought a lot about how big or howsmall those objects should be. Knowing thesizes of people’s arms, legs, and hands can bevery useful when designing furniture, clothes,toys, windows, doors, and many other items.

ALengths

Introduction

1. Which body measures would be useful to know if you weredesigning the following items?

a. doors d. pants

b. school desks e. baby cribs

c. shoes f. stairs

2. For what other objects would you need to know body measures?

Hints and CommentsOverviewStudents think about body measures that might beuseful to know when designing furniture, clothes,and other items.

Comments About the Problems

2. Explain to students that specific measurementsfor certain items are required by law to ensuresafety. For example, the spaces between thesidebars of baby cribs must be small enough sothat a baby’s head cannot fit through them. Toysare sometimes required to have warning labels ifthey contain small parts that might be swallowedby young children. Specific measurements, suchas optimal chair heights or distances from theeyes to computer terminals, are important in theworkplace.

Solutions and Samples

1. Answers will vary. Sample responses:

a. doors: height and width of person; size of aperson’s hand (for door knob size)

b. school desks: sitting height; length from kneedown; width of person

c. shoes: length and width of feet

d. pants: length from waist to ankle; hip size;waist size; inseam or leg length

e. baby cribs: height of children; width of head

f. stairs: length of foot

2. Sample responses:

• seats in cars

• seats in theaters

• hallways

• handgrips on bicycles or exercise equipment

• railings for stairs

• height of bicycle seats

Section A: Lengths 1T

2 Made to Measure

Notes3 Have students work ingroups for this problem.Not all students will befamiliar with all the termsused.

LengthsA

Extension

Have students do research about historical nonstandard units of measure.Isaac Asimov’s book Realm of Measure is a good resource. You might alsohave students investigate the history of terms, such as fathom or mile. Thefathom is currently used to measure water depth. Have students write shortreports and share their findings in class.


You probably discovered that the lengths and heights of different bodyparts are important for designing many common objects. At one time,all measurements for length were related to the human body. Someof these units of measure include the thumb, hand span, foot, yard,pace, and fathom.

3. Match each of the units of measure listed in the paragraph abovewith its drawing below.

LengthsA

Historical Measures

a.

c.

e.f.

b.

d.


Hints and CommentsOverviewStudents learn the names of some historical units ofmeasure for length that are related to the human bodyand match each measure with its drawing.

Planning

You may want students to work on this problem insmall groups or pairs.

Note that the thumb’s width is the measurement thatis now standardized as an inch.


3. a. The distance from the nose to the fingertips isa common way to estimate the length of fabricmaterial or thread.

c. The thumb is a measurement unit that refers tothe width of the thumb rather than its length.

f. The pace is a measurement unit that refers tothe distance from the back heel of one foot tothe front toe of the other foot. The pacerepresents the distance covered by a typicalstep of a person while walking.


3. a. yard

b. hand span

c. thumb

d. foot

e. fathom

f. pace

3 Made to Measure

Notes

4b Note that studentsshould take the measure-ment using a nonstandardunit from their own bodies,such as their own thumb,and not the units aspictured on page 2 of theStudent Book.

7 Some students mayneed to review what themean is.

LengthsA

Intervention

You may wish to discuss problem 5b with struggling students. Possiblediscussion topics could include the difficulty in measuring some objectsusing body parts as measurement units, the rationale behind their choicesof measurement units, and the difference in various body measures fromone individual to the next.

Extension

As an extension of problem 6, ask, How can the disadvantages of using bodyparts to make measurements be resolved? (Answers may vary. Some studentsmight suggest that the class should decide on a single foot or hand spanthat everyone could use, such as the average foot or hand span of thestudents in the class.)


A

4. a. Which of the units of measure fromproblem 3 would you use to find thelength of the nail shown here?

b. How long is this nail in the units of measure you chose foryour answer to a ?

5. a. Measure the length of your desk by using one or more of theunits of measure from problem 3.

b. List the class results in a table. Did everyone find the samelength? Why do you think this happened?

6. Reflect Name one advantage and one disadvantage of usingyour body to make measurements.

Lengths

In Scotland during the Middle Ages, a unit of measurecalled the Scottish thumb was used. A Scottish thumbis the mean of the thumb widths of three men: a largeman, an average-sized man, and a small man.

7. Why were three different-sized men used todetermine the Scottish thumb?

In 1616, the Germans decided to create a unit of measure called themean foot. To do this, they cut a piece of rope that was as long as thefeet of 16 men.

8. a. How do you think the rope was used to find the length of themean foot?

b. Which measurement is closer to the average person’smeasurement: the German mean foot or the Scottishthumb? Explain your answer.

9. With the help of 16 classmates, find the length of the mean foot inyour class by using the method described above.


Hints and CommentsMaterialscentimeter or inch rulers (one per student);5-meter piece of rope (one per class)

OverviewStudents choose appropriate nonstandard measurementunits from problem 3 to measure the lengths of a nailand their desks. They then take and compare the classresults. They also learn the origin and meaning of aunit of measure called the Scottish thumb and readabout how the Germans used 16 men to determine thelength of the mean foot. Students find the mean foot of16 classmates using the same method.

About the MathematicsProblems 5b and 6 give students an opportunity totalk about using appropriate measurement units, thevariability in measurement when using nonstandardunits, and the possible need for standardizing units.The concept of using a representative sample is impliedin the methods used by both the Scots and the Germansto establish the two nonstandard units presented here.In the unit Dealing with Data, students learned that arepresentative sample is a sample that has charac-teristics similar to the population and in roughly thesame proportion as that of the population.

PlanningMeasurements made using nonstandard units mayresult in answers that are not whole numbers. Thereare two methods that students might use to refinesuch measurements:

• use a combination of different measurement unitsin measuring one item, such as hand spans andthumbs, or

• use fractional parts of the measurement unitchosen. For example, the length of the desk mightbe 3 1__

2 hand spans.


6. Possible answers may arise from the classdiscussion in problem 5b.

See more Hints and Comments on page 59.


4. a. The thumb is the most appropriate unit tomeasure the length of the nail.

b. The nail is about 2 thumbs long.

5. a. Answers will vary, depending on the lengths ofindividual desks. Accept any reasonable lengthsmeasured in hand spans, feet, or acombination of hand spans and thumbs.

b. Students will probably have different answerseven if they used the same measures becauseof the variation in the sizes of body parts andthe level of precision used by differentstudents.


One advantage of using your body to makemeasurements is that you don’t need any specialtools, such as rulers.

One disadvantage of using your body to makemeasurements is that since different students’body parts are different sizes, your answers willnot be the same.

7. The Scots wanted to find an average width for thethumb unit of measurement, so they used thethumbs of three different-sized men.

8. a. Since there were 16 men, the mean foot wasdetermined by folding the rope into 16 pieces(folding it in half four times).

b. Answers will vary. Some students might saythat the mean foot is closer to the averageperson’s measurement since this unit wasdetermined using the feet of 16 people, whilethe Scottish thumb was determined using thethumbs of only three people.

9. Answers will vary. The length of the mean footshould be expressed in centimeters (cm) or inches(in.). In one pilot class, the total length of the feetof 16 students was about 379 cm, and the lengthof the mean foot was about 24 cm.

4 Made to Measure

Notes10. Some students mayneed to be reminded thatthe markings on rulers donot always begin at the farleft edge of the ruler. Sincedifferent methods can beused to measure one’space, students’ answersmay vary widely. Youmight wish to discuss thisin class.

13 and 14 Be sure todiscuss students’ responsesto evaluate their presentunderstanding of variouscustomary and metricunits and the relationshipsbetween the units withineach system.

LengthsA

Assessment Pyramid

10abcd, 14ab

Measure lengths usingstandard units.

Use simple relationshipsbetween measurementsystems.

Accommodation

Students may benefit from a pre-made table for use with problem 10.

Some students may wish to use a calculator for Problem 12.

Vocabulary Building

The terms metric system, foot, and customary system are introduced on thispage. You may wish to have students set up a vocabulary section in theirnotebooks and add these new terms to it.

English Language Learners

Ask English language learners to explain the measurement system they aremost familiar with. This could make an interesting whole-class discussion.


10. Measure the following in centimeters (cm). List the results in a table.

a. your thumb width c. your foot length

b. your hand span d. your pace

11. Use the table you made in problem 10 to answer the questionsabout relationships between measures.

a. How many thumb widths are in one hand span?

b. How many thumb widths are in one foot?

c. How many feet are in one pace?

You may add other units of measure and the relationships betweenthem to your list.

12. What is the size of the “typical” thumb for your classmates?Explain your answer.

Think about what it must have been like when everyone used his orher own thumbs for measuring. Today, of course, everyone hasstandard systems of measurement.

A few countries, including the United States, still use the foot as a unitof measure, but the length of a foot no longer refers to the length ofeach person’s foot. A standard has been officially established for thelength of a foot. Most countries use the metric system, which wasadopted in France in 1795.

13. a. The foot is a part of the Imperial, or English, system ofmeasurement. In the United States, we call this thecustomary system. List some other units of measurefor length that are part of the customary system.

b. In your notebook, write as many relationships between theunits of measure in the customary system as you can.

14. a. List some units of measure for length that are part of the metricsystem.

b. Write as many relationships between these units of measure asyou can.

Since the United States officially uses the metric system, it is importantfor you to have a sense of how the metric and the customary systemsrelate. The next activity will help you find some simple relationshipsbetween the metric and customary systems.

LengthsA


Hints and CommentsMaterialscentimeter rulers (one per student)

Overview

Students measure their thumb width and othernonstandard measurement units in centimeters (cm)and make a table showing the results. They then usethe table to describe relationships between thevarious nonstandard units.

About the Mathematics

The metric system is based on powers of 10, whichfacilitates the process of converting between differentmetric measurement units.For example, 10 millimeters � 1 centimeter;10 centimeters � 1 decimeter, and so on.

Planning

Some students may need more practice in workingwith metric units. You may need to give them anoverview of the units that are used most often as wellas their abbreviations. Note that abbreviations ofmeasurement are not followed by a dot. Commonmetric measurements are:

kilometer, km one mile is about 1.5 km;1 km � 1000 m

meter, m one meter is a little overone yard, 1 m � 10 dm

decimeter, dm 1 dm � 10 cm and aboutfour inches

centimeter, cm one inch is about 2.5 cmmillimeter, mm 1 cm � 10 mm, as shown on

a centimeter ruler



a. 2 cm

b. 19.5 cm

c. 24 cm

d. 58 cm

11. Answers will vary, depending on students’measurements for problem 10. Sample responses(based on sample answers in problem 10):

a. Almost 10 thumbs are equal to a hand span.

b. Twelve thumbs are equal to one foot length.

c. About 2 1__2 foot lengths are equal to one pace.

12. Some students might talk about the thumb widthmeasurement that occurred most often (themode), while other students may refer to theaverage thumb width of the class (the mean).Others may mention the middle value of the data(the median).

13. a. Common units of measure for length from theImperial measurement system are the inch,foot, yard, and mile.

b. Sample responses:

12 inches � 1 foot; 36 inches � 1 yard;3 feet �1 yard;

5,280 feet � 1 mile; 1,760 yards � 1 mile

14. a. Common units of measure for length from themetric measurement system are the millimeter,centimeter, decimeter, meter, and kilometer.


10 millimeters � 1 centimeter; 10 centimeters �1 decimeter;

100 centimeters � 1 meter; 1,000 meters �1 kilometer

5 Made to Measure

Notes

Activity

Be clear about how youwant students to recordthe information.

Prior to the activity, youmight hold up a length ofribbon or string and askstudents to estimate thelength. (About 1.6 meterswould probably be a goodlength.) Then invite twostudents to come up andmeasure the length inmeters. Ask, How would thelength of the ribbon changeif I measured it with ayardstick? Would there bemore or fewer yardscompared to meters?

LengthsA

Assessment Pyramid

15

Use simple relationshipsbetween measurementsystems.

Bringing Math Home

Have students discuss with their families and/or do additional research oncustomary and metric measurement units with which they are unfamiliar.

Intervention

If students struggle with the activity:

Say, This … measures 1.5 feet. How would this dimension change if I were tomeasure it in inches? Would it be more or less than 1.5 inches? (The smallerthe measuring unit, the greater the measurement number.)


Meters and Yards● Use a meter stick to measure your classroom. Predict how the

room dimensions would change if you measured with ayardstick.

● Use a yardstick to measure your classroom. Compare yourprediction to your actual measurement.

● Find a conversion rule for meters and yards you can use whendoing mental calculations.

Centimeters and Inches● Use a centimeter ruler to measure a paper clip. Predict how the

measurements would change if you measured using a rulerwith inches.

● Use an inch ruler to measure the paper clip. Compare yourprediction to your actual measurement.

● Find a conversion rule for centimeters and inches you can usewhen doing mental calculations.

Kilometers and MilesSince athletes compete internationally, all distances are in meters(m) or kilometers (km). Today, many U.S. high school cross-countryteams run 5-km races. Did you know that five kilometers is aboutthree miles?

● Investigate how your school measures the running events. Howlong is the running track that your school uses?

● Name a location that is about one mile away from your school.Would a location that is about one kilometer away from yourschool be closer or farther?

15. Write as many relationships between units in the customarymeasurement system and the metric system as you can.

Comparing Systems


Hints and CommentsMaterialspaper clips (one per student);centimeter and inch rulers (one of each per student);meter stick and yardstick (one per group of students)

Overview

Students investigate the relationship betweencustomary and metric measurement units. They alsofind conversion rules that can be used to makeestimations and mental calculations easier.


In the activity, students get a sense of how the metricand customary systems relate. Students measure theirclassroom (m) and a paper clip (cm) in metric unitsand then make predictions about how the dimensionsmight change if they were to measure in yards andinches. They measure in customary units andcompare their predictions. Finally, they generate arule for changing from one unit to another.

Planning

You may want to give some common reference pointsfor students to use when discussing metric measure-ments. For example, a door is about 2 meters high,and a thumb is about 2 centimeters wide. Moreimportant is that students identify their own pointsof reference when using measurements: the distancefrom home to school is about 3 miles or 4.5 kilometers,my height is about 1.6 m, and so on.

Solutions and SamplesActivity

Sample conversion rule for meters and yards:

• A yard is about 90 cm.

• A yard is a little less than one meter (or about10% less).

• One meter is about 1.1 yard.

• A meter is a little larger than a yard.

Sample conversion rule for centimeters and inches:

One inch is about 2.5 centimeters.

There are about 4 inches in 10 cm.

15. Answers will vary, depending on students’knowledge of the relationships between the unitsin both measurement systems. Sample responses:

1 meter (m) is about 3 feet; 1 inch is about 2.5 cm,1 km is about 2__

3 mile.

5 km is about 3 miles

6 Made to Measure

Notes

16c Be sure that studentsunderstand how toconstruct the graph andlabel its axes correctly. Ifall students’ shoes arelabeled within the sameshoe-size system, it islikely that students will seea relationship betweenshoe size and foot length.You may wish to discussany outliers on the graph.Are they true exceptions orare they measurementerrors?

LengthsA

ExtensionSince shoe-size systems differ for boys and girls, for problem 16, you maywant to have the boys make a table and graph of their data and the girls dothe same.

Hands-On LearningStudents might compare the standard thumb (one inch) to their ownthumbs. The widths of students’ thumbs will probably measure less thanone inch, unless their thumbs are “squashed” flat. Students should use anEnglish ruler to measure their feet since they need to find the length inthumbs (inches).

Vocabulary BuildingHave students add the term standard thumb to the vocabulary section oftheir notebooks.


In problem 10 c, you measured the length of your foot in centimeters.The length of your foot is different from your shoe size.

16. a. Do you think that there is a relationship between shoe size andfoot length? Explain your answer.

b. Make a table that lists the foot length (in cm) and correspondingshoe size for each student in your class.

c. Graph the results. Put foot length in centimeters on thehorizontal axis and shoe size on the vertical axis. What doesyour graph tell you about the relationship between foot lengthand shoe size?

Just as countries use different systems of measurement, they alsohave different systems for determining shoe sizes. For some shoes,you can find at least three different sizes:

● European size—usually a number between 33 and 47● U.K. (United Kingdom) size—usually a number between 1 and 15● U.S. size—usually a number between 1 and 15 (slightly larger

than U.K. sizes)

A Lengths

Feet and Shoes

The U.K. system of shoe sizes began in the seventhcentury.

Shoe sizes were measured with a standard thumb

(now called an inch).

17. How many standard thumbs (or inches) long isyour foot?

To get a more accurate measurement, the U.K. introduced a smallerunit of measure, the stitch. Three stitches are in one standard thumb.

18. How many stitches long is your foot?

In the U.K. system, shoe sizes are based on the number of stitches.The first 25 stitches are not counted in adult shoe sizes. Size 1 is,therefore, really 26 stitches, or 8 2��3 inches (in.).


Hints and CommentsMaterialscentimeter or inch rulers (one per student)

Overview

Students investigate the relationship between shoesize and foot length by organizing the informationinto a table that lists the foot lengths andcorresponding shoe sizes for each student anddisplaying the results in a graph. They also learnabout three different measurement systems fordetermining shoe sizes.


16. a. Yes, there is a relationship between shoe sizeand foot length. Sample explanation:

If there were no relationship between footlength and shoe size, there would be no way forpeople to buy shoes that fit them withouttrying on many pairs of shoes.

b. Sample table:

c. Graphs will vary, depending on students’ datafor part b. Sample graph based on sample tablein part b:

The graph shows that there is a relationshipbetween shoe size and foot length. As footlength increases, shoe size increases. Thepoints on students’ graphs should roughly|form a straight line or a long oval.

10

9

8

7

6

5

20 25 30

sho

e si

ze

foot length (cm)

Comparison of Shoe Size and Foot Length

17. Answers will vary, depending on the lengths ofstudents’ feet. Sample response:

If your foot length is about 24 1__2 cm, then it is

about 9 2__3 inches.

18. Answers will vary, depending on students’responses for problem 17. Answers should beabout three times the answer to problem 17.

Length of Foot

(in cm)Shoe Size

Length of Foot

(in cm)Shoe Size

24

24

22

22

25

25

23.5

25

24

7.5

7.5

7

6.5

8.5

8

6.5

8

7

25

21

22

22

21

22

24

25

25

8

5.5

5.5

5

5

6

8

9.5

9.5

7 Made to Measure

Notes

20 Discuss this problemwith students. Have themshare their responses andlisten to each other’s ideas.

21 If comparing height toshoe length will offend anyof your students, you mayprovide data to discussinstead.

LengthsA

Intervention

You might suggest that students add two columns in the table for problem19b to include the U.S. shoe sizes for both men and women. Be sure thatstudents use adult shoe sizes here rather than children’s shoe sizes.

Parent Involvement

Have students measure the foot lengths of family members in stitches orinches to determine their U.K. shoe size using the table they made inproblem 19a or applying the rule they formulated in problem 19c. They canalso compare the U.K. shoe sizes they found for family members with theirU.S. shoe sizes.


A

19. a. Copy the table into your notebook and continue it to shoe size 8.

b. Use your answer to problem 18 to find your U.K. shoe size inthe table you made in part a. How does your U.K. shoe sizecompare with your U.S. shoe size?

c. Formulate a rule that helps you find someone’s foot length ininches if you know his or her U.K. shoe size. Write the rule inarrow language.

? ?U.K. shoe size ⎯⎯→ ……… ⎯⎯→ foot length(in stitches) (in inches)

20. Reflect After you finish problem 19, look back at your answers toproblem 16. Would you change your answers now? Whyor why not?

In problem 10 you measured your pace and your foot length.

21. Make a table of the pace and foot-length measurements of all ofthe students in your class. Use the results to determine whetherthe two measures are related. For example, you can find out if theperson with the biggest foot also has the longest pace or if theperson with the shortest foot has the shortest pace. Drawing agraph may be helpful.

Lengths


Hints and CommentsOverviewStudents make a table of foot lengths and U.K. shoesizes, and formulate a rule to find someone’s footlength if they know the U.K. shoe size. They alsocompare U.K. shoe sizes to U.S. shoe sizes.

Planning

For some students you may need to review the use ofarrow strings. Extra practice can be found in the unitNumber Tools. Students are introduced to using arrowlanguage to express rules or formulas in the unitExpressions and Formulas.


20. This problem asks students to find a relationshipbetween U.K. shoe sizes and foot length in inches.In problems 16b and 16c, students made a tableand graph to investigate the relationship betweenfoot length and their own shoe size.

21. This problem challenges students to find therelationship between pace and foot length.More advanced students may use sophisticatedmathematical arguments.


19. a.

Note: Three stitches is about one inch.

b. In general, U.S. shoe sizes are larger than U.K.shoe sizes. Using the foot length of 29 stitches(9 2__

3 inches), the U.K. shoe size is 4.

c. U.K. size ⎯� 25⎯→ in stitches ⎯� 3⎯→ foot lengthin inches

20. If students indicated in their answer for problem16c that there is no relationship between footlength and shoe size, they should see the need tochange their answer after completing problem 19.

21. Tables and graphs will vary. Students’ responsesshould indicate that there is probably a relation-ship between foot length and pace. Somestudents may draw a scatter plot, as shown.

In the graph, the relationship between pace lengthand foot length is about 3:1. (The length of the paceis about three times the length of the foot.) Sincethe length of the pace is also related to the length ofthe legs, the relationship above is no more than ageneral rule.

26

27

28

29

30

9

9

10

1

2

3

4

5

31 10 6

32 10 7

33 11 8

8

9

Foot Length

(in stitches)

Foot Length

(in inches)

U. K.

Shoe Size

2

3

2

3

2

3

1

3

1

3

0.5

Foot Length (in cm)

Pace L

en

gth

(in

m)

21 22 23 24 25 26 27 28

1

Comparison of Pace Length and Foot Length

8 Made to Measure

Notes

22 Have students usecentimeter or inch rulersto measure the lengthbetween the fingertipsof the horizontallyoutstretched arms of theperson in the drawing andcompare that length withthe person’s height. Sincethe person more or lessfits into a square, moststudents will see that aperson’s height is approx-imately equal to his or herfathom.

LengthsA

Assessment Pyramid

23

Measure lengths usingstandard units.

ExtensionThe following activity is taken from the book Mathematics, Measurementand Me by Doug M. Clarke. Read the following description aloud tostudents: People tend to be divided into three categories: tall doors (wheretheir height is greater than their arm span); wide doors (where their height isless than their arm span); and square doors (where their height is very closeto their arm span). Ask, Are you a tall door, a wide door, or a square door?

Vocabulary BuildingHave students add the term fathom to the vocabulary section of theirnotebooks.

English Language LearnersEnglish language learners may benefit from some additional discussion ofthe term fathom.


The fathom is another unit of measure associated with the humanbody. You can measure a person’s fathom by having him or her standtall and extend his or her arms out from both sides, horizontal to theground. The fathom originated as the distance from the middlefingertip of one hand, to the middle fingertip of the other hand.

The picture is based on a famous drawing by Leonardo da Vinci. Thegirl more or less fits in a square.

22. Based on the picture on this page, what is the relationshipbetween a person’s height and his or her fathom?

23. Measure your height and your fathom to decide how preciselyyou would fit into a square.

LengthsA

Body Length and Fathom


Hints and CommentsMaterialscentimeter or inch rulers (one per student); inch orcentimeter tape measures (one per student)

Overview

Students learn the meaning of another measurementunit associated with the human body known as afathom. They investigate the relationship between aperson’s height and his or her fathom and measuretheir own heights and fathoms to decide howprecisely they would fit into a square.

Planning

If measuring students’ own height and fathom is aproblem, you might use the heights of five different7th grade students as shown on page 13 of the StudentBook and ask students what the measurements of eachstudent’s fathom could be to replace problem 23.


23. Students may measure their heights and fathomsusing a metric or customary tape measure.

This is a nice relationship to graph. The horizontalnature of the fathom and vertical nature of theheight transfers well to the graph. Besides thesquare, people lie along the center line, y � x.

Did You Know?

Leonardo da Vinci was an Italian artist and scientistwho lived from 1452–1519. The Mona Lisa was one ofhis famous paintings. He was also an accomplishedsculptor and architect. His scientific endeavorsincluded studying the anatomy of the human bodyand investigating flight.


22. A person’s height is approximately equal to his orher arm span (or fathom).

23. Most students will respond that they would fit oralmost fit in a square: their arm span (or fathom)will most likely be the same length as their height.For some students the “square” will be more like arectangle.

9 Made to Measure

Notes

25 You might discusshow distance is figured inmountainous areas, suchas by travel time or as-the-crow-flies distance. Actualdistances in these areasare difficult to measure.The distances between twopoints are not often givenas precise road distances.

LengthsA

Assessment Pyramid

26c

Develop and use a frame ofreference for measurementto solve problems.

Extension

Have students ask several people how far they live from home to somelocation, such as their place of work or a shopping mall. Students can notewhether people respond using units of distance or time and share theirfindings with the class.

Writing Opportunity

Have students write a paragraph about the Empire of Mali, which flourishedin western Africa from about 1240 to 1500 A.D., and the contributions ofMansa Musa, who reigned as emperor from 1312 to 1337 A.D. The empirestretched from the northern coast of modern Senegal along the AtlanticOcean to modern Nigeria and from the rain forests in the south to oases inthe Sahara Desert.


A

There are many other ways to measure length. In Papua, New Guinea,for example, a local unit of distance is “a day’s travel.”

24. Why might a day’s travel make sense as a unit of distance?

In mountainous regions of New Guinea, walking distances areexpressed in hours, not in kilometers or miles. Puli, a citizen of NewGuinea, says, “It will take us two hours to cover the distance from thevillage to the lake in the mountains, but we save time on the returntrip. The distance back will only take five quarters of an hour.”

25. a. Why do you think there are two travel times?

b. Is the distance in kilometers different for the two directions?Explain.

In the 14th century, the biggest trading empire in Africa was theEmpire of Mali. Mansa Musa was one of its emperors. Sheik Uthmaned-Dukkali, a learned Egyptian who lived in Mali for 35 years, declaredthat Mali was “four months of travel long and three months wide.”

Lengths

Other Measures for Length

26. a. Use the map to estimate the length and width of the Empireof Mali in both customary and metric units of measure.(Note: 500 miles (mi) equals 800 km.)

b. What is the distance in miles or kilometers of “one month oftravel”?

c. Based on your answer to part b, how do you think people inMali traveled in the 14th century?

Source: Data from Basil Davidson,African Kingdoms(New York: Time Incorporated, 1966).

SCALE

km0 1,000 2,000

Mali

Mali (1200 A.D. – 1500 A.D.)


Hints and CommentsMaterialsworld map (optional, one per class)

Overview

Students learn about other ways to measure length,such as the distance a person can travel in one day.Students reason about how such a unit of distancemakes sense. They also compare this distance unitwith distance measured in kilometers. Students use amap scale and information about the size of theEmpire of Mali expressed in travel time to investigatethe distance in kilometers that people could havetraveled in one month.


The concept of using distance units other than lengthis not new for most students. In the unit Figuring Allthe Angles, students are introduced to the differencebetween as-the-crow-flies distance and “taxicabdistance.”

Planning

You may want to locate Papua, New Guinea, an islandin the Pacific Ocean north of Australia, on a world mapand discuss its mountainous terrain. You might alsowant to locate Africa on a world map to examine thecurrent boundaries of the country of Mali today. Youcould compare the relative size of the Empire (and/orcountry) of Mali with that of the United States.


26. c. This question is intended to help students get asense of how fast people traversed Malicarrying cargo many years ago. Students withsome previous mountain hiking experiencemight recall that locations are often referred toby how many days of travel they are from thebase of the mountain.


24. Sample response:

If a person is traveling in the mountainous areasof Papua, New Guinea, the time it takes to travelacross the terrain on narrow and winding roadswould be more useful than knowing the distancebetween two given points. A person might need todetermine when to leave home in order to arriveat another point at a certain time.

25. a. Since it requires more time to walk up themountain than it does to walk down themountain, there are two different travel times.

b. The distance remains the same if the samepath is used on both trips.

26. a. The Empire of Mali measures about 2,250 km(length) by 1,200 km (width), or 1406 miles by750 miles.

b. The distance is about 400 km or 250 miles permonth. Sample explanation:

1,200 km in 3 months is 1,200 � 3 � 400 km permonth. 800 km � 500 mi, so 400 km � 250 miper month.

c. Answers will vary. The people of Mali couldhave traveled a distance of 18 km per day onfoot, which is a 4–5-hour trip, depending onthe type of terrain.

10 Made to Measure

Notes

Students often think theycan skip the Summarysince there are noproblems attached

Some suggested strategiesto overcome this:

• Read the Summaryaloud as a class. Askvolunteers to read aparagraph.

• Ask students tosummarize eachparagraph.

• Ask students how theSummary might behelpful.

LengthsA

Extension

Have students select more objects or distances and choose appropriateunits of measurement to determine their lengths.


Lengths

In this section, you learned many different ways to estimate andmeasure length. In the past, people used thumbs, feet, and arms tomeasure length, but each person often measured the same distancedifferently. Some people use units of time to measure distances. Forexample, Cedric takes one hour to hike around the nature trail.

Today, two standard systems of measurement exist. Most countriesof the world use the metric system, in which length is measured incentimeters (cm), meters (m), and kilometers (km).

1 kilometer = 1000 meters1 meter = 100 centimeters

1 centimeter = 10 millimeters

A few countries use the customary, or Imperial, system, in whichlength is measured in inches (in.), feet (ft), yards (yd), and miles (mi).

1 mile = 5,280 feet1 yard = 3 feet1 foot = 12 inches

Here are some relationships between both measuring systems. Youmay need to convert from one system to another.

1 mile is about 1.5 km (to be exact: 1 mi = 1.6 km)1 yard is a little less than 1 m (to be exact: 1 yd = 0.9144 m)1 foot is a little more than 30 cm (to be exact: 1 ft = 30.48 cm)1 inch is about 2.5 cm (1 in. = 2.54 cm)

A


Hints and CommentsOverviewStudents read and discuss the Summary that reviewsthe nonstandard measurement units associated withthe human body.

11 Made to Measure

Notes

1 When working on thisproblem, encouragestudents to measure astack of pennies and thendivide the total height ofthe stack by the number ofpennies to determine thethickness of one penny.

For Further ReflectionReflective questions aremeant to summarize anddiscuss important concepts.

LengthsA

Assessment Pyramid

4

1, 2, FFR

3

Assesses Section A Goals

Parent Involvement

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


1. Which unit of measurement would you use to measure thefollowing lengths?

a. the distance from your home to school

b. the length of your classroom

c. the thickness of a penny

2. List some distances expressed in units of time rather than units oflength. Explain why using time is appropriate in each case.

3. Most rulers have markings for both the customary and metricsystems.

a. Another common metric unit is a decimeter (dm);1 dm = 10 cm. Use a ruler to draw exactly one dm.

b. Underneath your decimeter, use a ruler to draw 1 in.

c. Estimate about how many inches are in 1 dm.

d. Rewrite the list of metric measuring units in the Summary toinclude a decimeter.

Neville, who lives in Denmark, wrote to his friend in Texas. “Today myfather and I went for a very long walk, about 16 km! I was very tiredwhen we got home!”

4. Estimate how many miles Neville walked.

Make a list of your own reference points that have to do with length.For example, the distance from my house to our school is aboutthree miles.


Hints and CommentsMaterialsmetric or English tape measures (optional, one perstudent);inch or centimeter rulers (optional, one per student);pennies (optional, 10 per student)

Overview

Students use the Check Your Work problems as self-assessment. The answers to these problems are alsoprovided on page 43 of the Student Book.

After students complete Section A, you may assign ashomework appropriate activities from the AdditionalPractice section, located on page 40 of the StudentBook.

Solutions and SamplesAnswers to Check Your Work

1. Here are some possible units of measurements.

a. Miles, kilometers, paces, minutes.

b. Meters, decimeters, or feet. Many classroomsare about 8 m, or about 25 feet long.

c. Millimeters. A penny has a thickness of about1.5 mm.

Check with a classmate if your answers do not matchany of these.

2. Here are some examples, but yours may be different.

• Distance from home to school. It is importantto know how much time it takes to go to schoolin order to be on time for school.

• Walking in the mountains. In the mountainsyou may want to be back before dark, so ithelps to know how long it will take. In general,walking speed is about 4 km or 3 miles perhour but much slower when you have to climbuphill or walk through sand.

3. a. Here is 1 dm.

b. Here is 1 inch.

c. A good estimate is about 4 in.; it is actually alittle less than 4 in.

d. 1 km � 1,000 m

1 m � 10 dm or 100 cm

1 dm � 10 cm

Note that some measurements are not in the list:

one hectometer (hm), one decameter (dam), and

one millimeter (mm). The complete list follows:

1 km � 10 hm 1 m � 10 dm

1 hm � 10 dam 1 dm � 10 cm

1 dam � 10 m 1 cm � 10 mm

4. About 10 miles. Using the information fromproblem 26:

500 miles is about 800 km, so 8 km is about5 miles and 16 km is about 10 miles.

For Further Reflection

Here is one sample list, but students’ lists will bedifferent.

• The height of a door is about 2 m.

• My height is 153 cm or 1.53 m.

• My pace is about 70 cm.

• One inch is about 2.5 cm.

• One mile is about 1.5 km (or 1.6 km).

Students continue to add to this list while they workthrough the unit.


Teachers MatterB

Section Focus

In Section B, the focus is on area and related units of measurement. Metricunits for length in the metric system are, for example, km, dm, m, and cm.The units for area are km2, dm2, m2, and cm2. In the customary system,units for length are inch (in.), mile (mi), or feet (ft), and the units for areaare sq in., sq mi, and sq ft. The concept of surface area is reviewed.

Pacing and Planning

Day 5: A Body’s Surface Area Student pages 12–14

INTRODUCTION Problems 1–3 Investigate various methods to estimatethe surface area of the human body.

ACTIVITY Activity, page 14 Make a square fathom out of newspaper toestimate the surface area of the body.

HOMEWORK Problems 4 and 5 Determine the length and width of arectangle with a given area.

Day 6: Hands and Body Student pages 15 and 16


CLASSWORK Problems 6–8a Measure the area of a handprint usinggraph paper and use the value to calculatethe surface area of the body.

HOMEWORK Problem 8b Use the rule of nines to calculate thesurface area of the body.

Day 7: Surface Area by Formula Student pages 17–19

ACTIVITY Review homework. Review homework from Day 6.

CLASSWORK Problems 9–12a Use the formula for the surface area of acylinder to estimate the surface area ofthe body.

HOMEWORK Problems 12bc and 13 Use a chart called a nomogram to determinethe surface area of the body.

Teachers Matter Section B: Areas 12B

Teachers Matter B

MaterialsStudent Resources

No resources required

Teachers Resources

No resources required.

Student Materials

Quantities listed are per student, unless otherwisenoted.

• Centimeter graph paper

• Centimeter rulers or tape measures

• Newspapers (several per group)


Learning Lines

Number Sense

Students are encouraged to use and develop theirmeasurement sense with areaby finding personal points ofreference. Some examples are:

• An acre is about the samearea as that of a football field.

• On average, the body surfacearea of an adult is about twosquare meters (m2).

• The surface area of my mathbook is about 600 cm2.

Measurement

Students learn how, initially, area was measuredusing units that originated from the human body,like square thumbs and square fathoms. Theyselect and use appropriate measurement unitsfor area using both metric and customary unitsof measurement.

Other Representations

Students use a special chart named a nomogramto relate a person’s body area to his or her weightand height.


By the end of this section, students are able to:

• calculate the area of a rectangle or a squareusing appropriate units of measurement;

• calculate the area of a cylinder used as a modelto represent the human body;

• compare and critique different methods ofcalculating the surface area of an irregularlyshaped object;

• model real objects using regular geometric shapes;

• recognize historical units of measurement forarea such as the acre; and

• use a nomogram to investigate the relationshipbetween a person’s height, weight, and body’ssurface area.

Day 8: Early Areas Student pages 19–21


CLASSWORK Problems 14 and 15 Use a square furlong to determine thedimensions and the area of one acre andcompare the results to the results to thearea of a football field.

ASSESSMENT Check Your Work Student self-assessment: Use formulasFor Further Reflection and rules to solve area problems.

Day 9: Summary


ASSESSMENT Quiz 1 Assesses Section A and B Goals

Additional Resources: Additional Practice, Section B, Student Book page 41

12 Made to Measure

Notes

Throughout this section,students may use themeasurements of onestudent they think issimilar to them in order toavoid students feelinguncomfortable measuringthemselves.

For ease in comparing, allskin surface areameasurements in thissection should be recordedin square centimeters.

Note: 5 centimeters is equalto 2 inches.

AreasB

Intervention

Some students may need help understanding that to find the surface areaof Ray’s body they will need to multiply his height by his shoulder widthand then by 2.

Vocabulary Building

Have students add the term surface area to the vocabulary section of theirstudent notebooks. Students should give examples that they can refer back to.


Students sometimes skip the reading to get to the numbered problems.Read the text on this page with English language learners in order to clearlyset the context for the problem.


A body’s surface area is the amount ofskin that covers a body. Sometimes it isimportant to know the surface area of aperson’s body. For example, health careworkers estimate the surface area of aburned patient to decide how muchliquid the patient needs to replace lostfluids.

Body surface area is also importantwhen caring for babies. Did you knowthat babies cool down faster thanadults do? A person’s body cools downby sweating in relation to body surfacearea but warms up in relation to bodymass. A baby’s skin area is very large inrelation to his or her weight, so a babycools down much faster than an adult.Babies can feel uncomfortably coldeven when adults feel warm. So whenyou take care of a baby, don’t forget tomonitor his or her skin temperature.

BAreas

A Body’s Surface Area

1. How might you measure your body’s surface area or amount ofskin?

2. a. Estimate Ray’s body surface area in square centimeters. Ray is157 cm tall, and his shoulder-to-shoulder width is 46 cm. Hereis one square centimeter (cm2) you can use as a reference.

Section B: Areas 12T

Hints and CommentsOverviewIn this section, students investigate various ways toestimate the surface area of their bodies. On this page,students think about ways in which they mightmeasure their surface area and estimate their surfacearea in square centimeters (cm2).


1. and 2.By estimating their body’s surface area, studentslearn the meaning of surface area and thechallenges of finding the surface areas of irregularshapes. The emphasis in these two problems is onstudents’ thinking about possible methods withwhich to estimate their surface area.


1. Sample responses:I could use a rectangular sheet of paper toestimate the area of one side of my body anddouble the answer to include the other side.I could wrap my body in newspaper and estimatethe area of the sheets used.I could estimate the number of sheets of paper ofa particular size, such as self-stick notes or 8 1__

2 by11 sheets, that would cover my body and estimatethe total area of the sheets used.

2. a. Sample response:Ray’s surface area is about 14,000 cm2.Strategies will vary. Here is a sample strategyusing the rectangle method:Height: 157 cm tall (which is about 5 ft 2 in.)Shoulder to shoulder width: 46 cm (which isabout 18 in.)Surface area � 157 � 46 � 2, or about 14,000 cm2

13 Made to Measure

Notes

Be sure that studentsrecord the results in theirSurface Area EstimationTables. Tell students it isimportant they choose thesame student name fromthe table throughout thesection. Compare theresults found by eachgroup of students.

AreasB

Accommodation

Create a table like the one on this page for students who may have difficultycreating their own.


There is a large amount of text on this page. Have English language learnersread the text aloud and check for understanding.


As you go through this section, you will estimate a person’s bodysurface area in many different ways. Here are measurement data forfive different 7th graders.

b. Estimate the body surface area of the 7th grader that is closestto your own height. Copy this table into your notebook to keeptrack of your estimations. Enter your method and estimate forthe 7th grader you chose.

Every time you make a new estimate, record your answer in your table.

BAreas

7th Grader Age (yrs) Height (cm) Shoulder Width (cm)

Joyce 12 135 38

Deon 13 147 46

Nora 12 151 43

Emmanuel 13 165 52

Luther 12 178 50

Surface Area Estimation Table

7th Grader Estimation Method Estimate (cm2)

SquaresIn Section A, you investigated how wellyour body fits inside a square. Of course, noteveryone’s square is the same size. For somepeople, it resembles a rectangle rather thana square.

You can use the area of a body’s square(or rectangle) to make another estimate of aperson’s surface area. It might surprise youthat three-fifths of the area of a body’s squareis a good estimate of a body’s surface area.

3 Explain why 3��5 of a body’s square is agood estimate of a person’s surface area.


Hints and CommentsOverviewStudents start making a Surface Area Estimation Tablewhere they compare a variety of methods to estimate aperson’s body surface area. The method introduced onthis page is to use the diagram shown on page 8 andtake 3__

5 of the area of the square or rectangle that abody fits into.


This section provides an introduction into mathe-matical modeling; since it is not possible to measurea person’s surface area in an exact way, mathematicalmodels that represent the body are needed. Studentsinformally assess these models by comparing theestimates found when using the different models.


2. b. Note that students only need to choose one7th grader from the list.

3. Sample answer:

If you “roll” your body over the square, it rollsabout 3__

5 of the way before you get back to whereyou started. The front and back of your body eachtakes up about 1__

5 of the square, with your twosides together taking up another 1__

5 .



Joyce 135 � 38 � 2 � 10,260 10,000

Deon 147 � 46 � 2 � 13,524 13,500

Nora 151 � 43 � 2 � 12,986 13,000

Emmanuel 165 � 52 � 2 � 17,160 17,000

Luther 178 � 50 � 2 � 17,800 17,800

14 Made to Measure

Notes

Be sure that students recordthe results found whiledoing the Activity in theirSurface Area EstimationTables. Tell students it isimportant that they choosethe same student namefrom the table throughoutthe section.

5 You might want todiscuss the range ofheights for average adultmales in centimeters

AreasB

Intervention

You might suggest that students use tape to secure the newspaper sections.Some students will cut and fit the newspaper sections in ways that fit betterthan methods used by other students. After students have approximated orcalculated the area of their squares in square centimeters, they might usevarious methods to find three-fifths of the squares’ area. Some studentsmay divide the total area of the square by five and multiply that answer bythree to find three-fifths of the area.


You will need a newspaper for this activity.

● Make a life-size body square out of newspaper. Use the datafrom one of the 7th graders on the previous page. You willprobably need to tape some newspaper pages together tocreate the entire body square.

● Shade and measure 3��5 of the square. Use square centimetersto calculate the area of the shaded part of your square. This isa new estimate for your 7th grader’s body surface area. Recordthe result in your Surface Area Estimation Table.

● How close do you think your estimate is? Cut up the shadedpart of your square and try to piece together the front side ofyour 7th grader. Report how well your “skin” would coveryour person. Save this person’s skin. You will need it later inthe section.

Skinning a Square

There are other ways to find your body’s surfacearea. Timm Ulrichs, a German artist, did manyartistic experiments that were quite mathemati-cal. Mr. Ulrichs found his own surface area usingsmall sticky squares. He took thousands of littlesticky squares of foil, each exactly the size of onecm2, and stuck them on his body until he wascompletely covered.

Since each piece of foil was 1 cm2, Ulrichs wasable to find the total surface area of his body. Heplaced all of the foil squares on graph paper inthe shape of a large rectangle. He counted 18,360squares and concluded that his body’s surfacearea was 18,360 cm2.

1 cm2

4. Find some possible dimensions for the length and width of TimmUlrichs’s rectangle. You don’t have to be exact.

5. a. Suppose Timm Ulrichs’s height is 180 cm. What is the area ofa square with this height?

b. Is Timm Ulrichs’s body surface area of 18,360 cm2 equal to3��5 of the square? Explain your answer.


Hints and CommentsMaterialsnewspapers (several per group);centimeter rulers or tape measures (one per group);scissors, optional (one pair per student);tape, optional (one roll per group)

Overview

Students further investigate the estimation methodusing 3__

5 of a person’s body square. They make a life-size version of this square (or rectangle) based on themeasurements of the 7th grader shown in the table onpage 13 and shade 3__

5 of it to estimate the surface area.Students also read about an estimation method usedby the German artist Timm Ulrichs to approximatehis body’s surface area. They figure out possibledimensions of a rectangle that has the same area asUlrichs.


The relationships between length, width, and area areused in different ways:

The area � length � width and area � width � lengthformula is used to determine a possible length, giventhe area of a rectangle.


4. You might suggest that students round the givensurface area to 18,000 cm2 to make it easier forthem to find a possible length and width for therectangle. To help students develop a frame ofreference for this measurement, you mightremind them that the dimensions 180 cm � 100cm and 200 cm � 90 cm are close to 2 meters �1 meter, the common dimensions of a door.

Solutions and SamplesActivity

Note that students only need to choose one 7th graderfrom the list. The areas in the list are calculated;students’ estimate may vary from the calculations!

4. Answers will vary. Sample responses that result inan area of about 18,000 cm2:180 cm � 100 cm200 cm � 90 cm150 cm � 120 cm300 cm � 60 cm250 cm � 70 cm (area of 17,500 cm2)

5. a. 180 cm � 180 cm � 32,400 cm2.

b. 3__5 of 32,400 cm2 is 19,440 cm2, so the area of the

square is not equal to the body surface area of18,360 cm2, but close to it.



Joyce 135 � 38 � 2 � 10,260 10,000

135 � 135 � 18,225

1__5

of 18,225 is 3,645

3__5

of 18,225 is 3 � 3,645 � 10,935 11,000

Deon 147 � 46 � 2 � 13,524 13,500

147 � 147 � 21,609

1__5

of 21,609 is 4,322

3__5

of 21,609 is 12,965 13,000

Nora 151 � 43 � 2 � 12,986 13,000

151 � 151 � 22,801

1__5

of 22,801 is 4,560

3__5

of 22,801 is 13,680 13,700

Emmanuel 165 � 52 � 2 � 17,160 17,000

165 � 165 � 27,225

1__5

of 27,225 is 5,445

3__5

of 27,225 is 16,335 16,300

Luther 178 � 50 � 2 � 17,800 17,800

178 � 178 � 31,684

1__5

of 31,684 is 6,337

3__5

of 31,684 is 19,010 19,000

15 Made to Measure

Notes

7 Be sure that students usecentimeter graph paper totrace their hand to make iteasier to estimate the areaby simply counting thecentimeter squares coveredby their handprint.

Note: Handprint estimatesmight be about 80% of thearea of the rectangle thatencloses the hand. SurfaceArea Estimate is about 100handprints.

AreasB

Assessment Pyramid

7b

Estimate areas usingappropriate units.

Accommodation

Students having difficulty tracing their own hand could ask a classmate orteacher to trace it for them.


BAreas

Hands and Body6. Trace your hand on cm-graph paper

as shown. Estimate the area of yourhandprint in square centimeters.

Here is the height and width data for hand prints of the 7th graders.

You can use the area of your handprint to estimate your body’ssurface area. Legend has it that it takes about 100 handprints tocover the body.

7. a. Use the handprint dimensions to sketch a handprint of oneperson.

b. Estimate the area of the handprint and the person’s bodysurface area.

c. Does the handprint method give the same estimate of thebody’s surface area as Timm Ulrichs’s sticky squares method?

d. Record your new estimate in your Surface Area EstimationTable.

7th Grader Age (yrs) Hand Height (cm) Hand Width (cm)

Joyce 12 14 9

Deon 13 15 10

Nora 12 17 11

Emmanuel 13 18 14

Luther 12 21 13



6. Answers will vary, depending on the size ofstudents’ hands.

Sample response: 115 cm2.

7. a. The dimensions of the handprints studentsdrew should match the hand height and widthof the 7th grader they chose from the list.

b. Estimates will vary. Note that students only needto estimate the area of the handprint and theperson’s body surface area for one 7th grader.

c. No, the estimates are different. The handprintmethod often results in a smaller surface areaestimate than a response using the rectanglemethod.

d. Students should add their results to the tableshown on the previous page.

Hints and CommentsMaterialscentimeter graph paper (one sheet per student);centimeter tape measures or rulers (one per student)

Overview

Students trace their hand on centimeter graph paperand estimate the area of their handprint to determinetheir body surface area using the rule of thumb thattheir surface area is about the same area covered bythat of 100 handprints. This is yet anothermathematical model used to estimate a person’ssurface area.

Hand RectangleEstimate Handprint

Hand Hand Area of Body

7th Age Height Width Handprint Estimate

Grader (yrs) (cm) (cm) (cm2) (cm2)

Joyce 12 14 9 100 10,000

Deon 13 15 10 120 12,000

Nora 12 17 11 �150 15,000

Emmanuel 13 18 14 �202 20,200

Luther 12 21 13 �218 21,800

16 Made to Measure

Notes

Be sure that studentsrecord the results in theirSurface Area EstimationTables. Remind students itis important they choosethe same student namefrom the table throughoutthe section.

AreasB

Intervention

Make sure that students understand the “rule of nines” before they startworking on problem 8. You might ask students to compare the result of the“rectangle method” of estimating surface area used on Student Book page13 with that of the “handprint method” used here. Ask, Which method gaveyou a surface area that was smaller? (The handprint method) What mightbe some reasons that the two methods resulted in two different answers?(The handprint method is a more precise measurement strategy.)


Burns can be very serious. The seriousness of a burn depends onhow much of the body has been burned. To estimate the extent of apatient’s burns, health care workers use the “rule of nines.” This ruledivides the body into eleven sections, each of which accounts for 9%of the total surface area, as shown in the picture.

8. a. Think of a way to measure as precisely as possible one ofthe “rule of nines” sections of the person you created in theactivity on page 14. Measure the area of this section in squarecentimeters.

b. Use the result to calculate your person’s body surface area.Record your new estimate in your Surface Area EstimationTable.

AreasB

Each Upper Leg 9%(Includes Front and Back)

Each Lower Leg 9%(Includes Front and Back)

Head and Neck 9%

Each Arm 9%

Abdomen 9%

Chest Front 9%

Abdomen Back 9%

Chest Back 9%


Hints and CommentsOverviewStudents relate the body’s surface area to estimationsof the extent of a patient’s burns.


Once again this is a model that is used in daily life toestimate one’s surface area.


8. Students may use different methods with varyingaccuracy to measure each of the 11 sections of thebody. Some sections are easier to find the area ofthan others are. For example, it is easier tomeasure the surface area of an arm or leg thanthat of the head and neck. Some students maynote that the percents add up to 99% rather than100% (11 � 9% � 99%). The difference of onepercent can be ignored since the rule of nines isan estimation strategy.


8. a. Sample response:

The dimensions of the abdomen of my“person” are about 28 cm by 40 cm. I multipliedto find its surface area: 28 � 40 � 1,120 cm2.

b. Response using the sample dimensions in part a:

Using the rule of nines, I multiplied the surfacearea of the abdomen by 11 to find the totalbody surface area: 11 � 1,120 � 12,320 cm2.

Students should add their results to the tableshown on page 13.

17 Made to Measure

Notes

Be sure that studentsrecord the results in theirSurface Area EstimationTables. Tell students it isimportant they choose thesame student name fromthe table throughout thesection.

AreasB

Assessment Pyramid

10,11

Use geometric models tosolve problems.

Extension

Have students use the following formula to estimate the surface area oftheir bodies: surface area � hip circumference � height and compare it tothe formula based on the “two cylinder method.”

You might ask students to explain why or for which group of persons thisformula may give a more accurate approximation of the body’s surface thanthe “two cylinder method” used above. (Since the hip circumferenceencompasses the outer circumference of the body, it results in a moreaccurate answer. For people with a larger waistline and thin legs, thisformula would not work.)


BAreas

The formula is based on the formula for the surface area of twocylinders.

10. Explain how you can model the body using two cylinders ofequal height.

11. Reflect Would the formula work for babies? Explain why orwhy not.

Surface Area by Formula

There are both simple and complex formulas for finding the surfacearea of a person’s body. One simple way is to multiply your height byyour thigh circumference (the length around your thigh) and doublethe answer. Here is a formula for this method.

height � thigh circumference � 2 � body surface area(in cm) (in cm) (in cm2)

9. Use the new formula and the measurement data for one7th grader to estimate the person’s body surface area. Writeyour new estimate in your Surface Area Estimation Table.

7th Grader Age (yr) Height (in cm) Thigh Circumference (in cm)

Joyce 12 135 36

Deon 13 147 48

Nora 12 151 45

Emmanuel 13 165 54

Luther 12 178 48


Hints and CommentsOverviewStudents use the measurements of height and thighcircumference of their chosen 7th grader and find thesurface area using a given formula. They explain howthe body’s surface area can be seen as equal to that ofthe two cylinders in the formula.


Students first learn to calculate the surface area ofcylinders in the unit Reallotment. An easy way tovisualize or explain the surface area of a cylinder is tocut the cylinder and make it into a flat rectangle, thedimensions of which would be the person’s heightand the circumference of their thigh. Themathematical model of the human body used here isthat of two identical cylinders placed next to eachother. In problem 11, students are informallyintroduced to the concept of domain of a function. Ofcourse, the formal term is not used.


10. Encourage students to think whether this is anaccurate way to measure body surface area.


9. Students should add the results of one of the7th graders to their own Surface Area EstimationTable.

10. Since the height is used, the two cylinders are astall as a person with a circumference of the size ofone thigh. For a human body, the two cylindersencompass the legs and are as tall as the body.Since there are two legs, doubling the answergives an estimate of the body’s surface area.

11. No, the formula will probably not give a goodestimation of a baby’s surface area, since thethigh’s circumference of a baby is much smallerthan the torso and head.



Joyce 135 � 36 � 2 � 9,720 9,700

Deon 147 � 48 � 2 � 14,112 14,000

Nora 151 � 45 � 2 � 13,590 13,600

Emmanuel 165 � 54 � 2 � 17,820 17,800

Luther 178 � 48 � 2 � 17,080 17,100

18 Made to Measure

Notes

12b Create a table forstudents who would findthis time-consuming.

12c Students shouldcontinue to use the same7th grader as in previousproblems.

AreasB

Intervention

You may want to discuss the different methods that were used forestimating a person’s body surface area in class. Have students findmathematical arguments to decide whether one method gives a betterestimate in their view than the other one.

Vocabulary Building

Have students add the term nomogram to the vocabulary section of theirnotebooks. Ask them what the advantage of using this type of nomogram isas compared to the other methods of estimation.


The surface area of a person’s bodyrelates to his or her height and weight.

Unfortunately, there is no easy formulato find out exactly how these threemeasures are related.

Instead, healthcare workers use aspecial chart called a nomogram.

First they find the patient’s height(in centimeters) in the left-hand column.

Next, they find the patient’s weight(in kilograms) in the far right handcolumn.

They connect these two points witha line and read the estimate for thepatient’s surface area (in squarecentimeters) from the middle column.The estimate is where the line crossesthe middle scale.

AreasB

Height, Weight, and Area

12. a. A line for one patient has already been drawn on thisnomogram. What information about the patient is indi-cated by the nomogram?

Here is the weight data for the 7th graders.

b. To use the nomogram, the weight must be in kilograms.Use this ratio table to calculate the kilogram weight ofeach 7th grader.

c. Without drawing in your book, use the nomogram and astraightedge to estimate one 7th grader’s body surfacearea. Write your new estimate in your Surface AreaEstimation Table.

7th Grader Weight

(in lb)

Joyce 84

Deon 96

Nora 102

Emmanuel 135

Luther 125

Kilograms 1

Pounds 2.2

Adapted with permission from Arithmetic Teacher, © May 1989 by the National Councilof Teachers of Mathematics. All rights reserved.


Hints and CommentsMaterialsstraightedges or rulers (one per student)

Overview

Students use a nomogram chart and informationabout a person’s height and weight to determine thebody surface areas in yet another way.


12. a. The patient is 180 cm tall, weighs 70 kilograms(kg), and has a body surface area of about19,000 cm2.

b. Sample ratio tables, based on the weight ofeach 7th grader:

c. Students should add the results of one of the7th graders to their own Surface AreaEstimation Table.

Sample estimates in cm2, using the nomogram:

Joyce 38 kg, 135 cm 11,800 cm2

Deon 44 kg, 147 cm 13,400 cm2

Nora 46.5 kg, 151 cm 14,000 cm2

Emmanuel 61 kg, 165 cm 16,800 cm2

Luther 57 kg, 178 cm 17,200 cm2

Nora

kilograms (kg) 1 0.5 3 10 50 47 46.5

pounds 2.2 1.1 6.6 22 110 103.4 102.3

Emmanuel

kilograms (kg) 1 2 10 50 60 61

pounds 2.2 4.4 22 110 132 134.2

Joyce

kilograms (kg) 1 2 10 40 38

pounds 2.2 4.4 22 88 83.6

Deon


pounds 2.2 8.8 22 88 96.8

Luther


pounds 2.2 6.6 22 132 125.4

19 Made to Measure

Notes

You may want to add a fewmore reference pointsabout area; for example,have students find objectsin the classroom the sizeof 1 m2 and have themmake a drawing of 1 dm2

(10 cm � 10 cm) with arow of 10 cm2 squares init for their list.

AreasB

Assessment Pyramid

15

Compute areas usingstandard units.

ExtensionHave students measure the dimensions and calculate the area of the plot ofland on which their house or apartment is built. Many such land plots areone-fourth acre in area. Have students try to imagine the areas of theirproperty compared to that of a football field. Students might also comparethe areas of the school parking lot or of a soccer field to the area of afootball field and acre.

InterventionMore practice can be found in the unit Number Tools.

English Language LearnersThe large amount of text on this page may be a deterrent for Englishlanguage learners. Read through the text with them to be sure theyunderstand the context being developed on this page.


BAreas

On average, the surface area of an adult’s body is about two squaremeters (m2).

One square meter is exactly 10,000 cm2 (100 cm � 100 cm).

13. Does the patient from problem 12a have an average body surfacearea? What about the 7th grader you have been working with?

In Section A, you learned how body measures were first used asmeasuring units for lengths. It wasn’t until later that measuring unitsbecame standard. Initially, area was measured using measuring unitsoriginating from the human body. Square feet, square thumbs, squarehand span, and square fathoms, were used to measure area.

The English used the units rood and acre.

Area units in the metric system relate to units of length; squarecentimeters (cm2), square meters (m2), and square kilometers (km2) toname a few. Some Americans measure area with square inches (in.2),square feet (ft2), and square miles (mi2). To measure land area, theyuse an ancient unit called the acre. One acre is 43,560 ft2 ( 1��640 mi2 or4047 m2). Unlike the other measuring units for area, the acre does notuse the word “square” in its name. Parcels of land are often irregularin shape.

Suppose each of these pieces of land measures one acre.

square rectangle

14. a. What are the dimensions (in feet) of each acre?

b. Make a scale drawing of a parcel of land that measures10 acres.

Early Areas

The playing area of a football field is100 yards long by 50 yards wide.

15. Use a calculation to show that afootball field is about one acre.


Hints and CommentsOverviewStudents learn about historic area measurement unitsbased on the human body, such as square fathomsand square thumbs. They explore the area unit acre(a square unit) and determine how many acres are ina football field.


Common units of area measurement in the metricsystem are:

square kilometer, km2

1 km � 1000 m

1 km2 � 1000 m � 1000 m � 1,000,000 m2 square meter

Planning

The reference point of “one football field is about oneacre” is one of the relationships between units ofmeasurement useful for students’ understanding.


14. The purpose of this problem is to help studentsget a sense of the size of an acre. They relate theunit acre to the dimensions and area of a footballfield in order to develop a frame of reference formeasurement.


13. Yes, the patient has a body’s surface area of 19,000square centimeters (cm2), which is close enoughto the average of 20,000 cm2 two square meters.

Students will find a body surface area smaller than2 m2, because 7th graders are not adults, so theycould have less than an average body surface area.

14. a. Square: 209 ft � 209 ft.

Rectangle: 52.5 ft � 832 ft.

b. Here are two scale drawing of 10 acres; one is arectangle and the other reallots this rectangle.

1 in. represents 209 ft.


A football field measures 300 ft by 150 ft, so it hasa total area of 45,000 ft2 (300 � 150 � 45,000).Since an acre has an area of 43,560 ft2, a footballfield takes up about the same space as one acre.

418 ft

1,042 ft

20 Made to Measure

Notes

Read and discuss theSummary with students.You might want them tolook through this sectiononce more to review thedifferent area estimationmethods. Students may usetheir Surface AreaEstimation Tables to helpthem compare the variousmethods.

AreasB

Parent Involvement

Have students discuss the Summary problem with their parents, showingthem examples in the section.


Areas

Surface area is important for figuring out the amount of materialneeded to cover something. You use body surface area to determinethe amount of fluid needed by burn victims.

B

Some metric units for measuring surfacearea are square centimeters and squaremeters. One square meter is the same as10,000 square centimeters.

Some customary units for measuringsurface area are square inches andsquare feet. One square foot is thesame as 144 square inches.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 23 24

25 26 27 28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 48

49 50 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81 82 83 84

85 86 87 88 89 90 91 92 93 94 95 96

97 98 99 100 101 102 103 104 105 106 107 108

109 110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131 132

133 134 135 136 137 138 139 140 141 142 143 144

An acre is an area unit for measuring land (1 acre = 43,560 ft2).

In this section, you learned about several different methods to find aperson’s body’s surface area.

• 3��5 of the person’s square • about 100 handprints

• height � thigh circumference � 2 • a nomogram scale


Hints and CommentsOverviewStudents read and discuss the Summary, whichreviews the main estimation methods used to find aperson’s body surface area.

21 Made to Measure

Notes

Be sure to discuss CheckYour Work with yourstudents so they under-stand when to givethemselves credit for ananswer that is differentfrom the one at the backof the book.

For Further ReflectionReflective question aremeant to summarize anddiscuss important concepts.

AreasB

Assessment Pyramid

3a

1c, 3b FFR

1ab, 2

Assesses Section B Goals

Parent Involvement

Have parents review the problems with their child to relate the Check YourWork problems with the problems in the section.


Nicola is 1.58 m (or 158 cm) tall. She uses the method of 3��5 of asquare to find her body’s surface area.

1. a. Calculate Nicola’s body’s surface area.

b. Nicola’s answer to question 1a was 14,978.4 cm2. Why would15,000 cm2 be a better estimate of her body’s surface area?

c. Is Nicola’s surface area less than, more than, or equal to 2 m2?

2. Trace your foot on graph paper and estimate the area of yourfootprint in square centimeters.

You have used this formula for a person’s surface area.

height � thigh circumference � 2 = body surface area(in cm) (in cm) (in cm2)

This formula is based on the mathematical model of two cylinders.

There is another simple formula, based on the mathematical model ofone cylinder.

height � hip circumference � body surface area(in cm) (in cm) (in cm2)

3. a. Use the new formula to find the body surface area of an adultwho is 1.85 m tall and has a hip circumference of 105 cm.

b. Is the body surface area you found about average?

Write a paragraph or two comparing the different methods you usedfor estimating a person’s body surface area. Which of these methodsdo you think is most accurate? Why?


Hints and CommentsOverviewStudents use the Check Your Work problems as self-assessment. The answers to these problems are alsoprovided on page 44 of the Student Book.

After students complete Section B, you may assignappropriate activities as homework from theAdditional Practice section, located on Student Bookpage 41.


This problem informally assesses students’ ability torecognize that different units of measurement anddifferent measurement systems exist, to recognize theimportance of standard measurement units, and touse a variety of mathematical models to calculate andestimate area. It also assesses their ability to developand use a frame of reference for measurement tosolve problems and their ability to select and useappropriate measurement units.


1. a. About 15,000 cm2.

Rounding Nicola’s height up to 160 cm, Nicola’spersonal square has an area of 25,600 cm2. One-fifthis 5,120 cm2 and three-fifth is 15,360 cm2. Adjustingfor rounding, a good estimate is 15,000 cm2.

b. Having a decimal in the answer is not appropriatehere. This method is only an estimate, so15,000 cm2 is a more reasonable for estimating.

c. Nicola’s surface area is less than two squaremeters.

Recall that 1 m2 � 10,000 cm2, since 1 m2 is a100 cm by 100 cm square and 100 cm � 100 cm �10,000 cm2; 2 m2 is double that amount, or20,000 cm2. Two square meters is well overNicola’s surface area of 15,000 cm2.

2. Estimates will likely be between 130–280 cm2.Exchange your grid with a classmate to verifyyour estimate.

3. a. Using the new formula, the surface area isabout 19,425 cm2. Using hip circumference of105 cm and a height of 185 cm, you calculate185 cm � 105 cm � 19,425 cm2.

b. This is about average since the body surfacearea for an adult is about 2 m2 and19,425 cm2 � 1.9 m2.


Discuss with a classmate which method you prefer andwhy. Here is an overview of the various methods usedin this section to estimate your body’s surface area:

• Make an estimation if you know the size of 1 cm2.Not very accurate.

• Skinning a square. Make a life-size version of yourbody’s square and find 3__

5 of this square as anestimate for your body’s surface area. More accuratethan the first estimate but still very rough.

• Use sticky squares of 1 cm2 each to cover yourbody and count them to find your body’s surfacearea. Rather accurate but a lot of work!

• Trace handprints on cm-graph paper and use therule that 100 handprints cover the body. Not veryaccurate.

• Use the mathematical model of two equal-sizedcylinders to represent the surface area. Easy to dobut not really accurate.

• Use a nomogram relating height, weight, andbody surface area. Easy to do; you usemeasurements. You cannot know how accuratethis method is because you do not know how thenomogram was made.


Teachers MatterC

Section Focus

Students investigate different methods for estimating, calculating, and determiningthe volume of different objects, containers, or spaces using metric and customaryunits of measurement.

Students make a cubic centimeter (cm3) from a net and investigate how many areneeded to fill a cubic decimeter (dm3). They decide which unit of measurement isappropriate when measuring the volume of different objects.

Pacing and Planning

Day 10: The Volume of Your Heart Student pages 22–24

INTRODUCTION Problem 1 Discuss measurement units for volumeand use two closed fists to estimate thevolume of a human heart.

CLASSWORK Problems 2–7ab Construct a centimeter cube, use centimetercubes to measure volume, and develop arule for calculating the volume of a box.

HOMEWORK Problems 7cd Determine the dimensions of differentboxes that can contain 60 cubes.

Day 11: Solids (Continued) Student pages 24–26

INTRODUCTION Problems 8 and 9 Estimate the volume of a classroom usingappropriate measurement units.

CLASSWORK Problems 10–13 Investigate objects whose volume cannotbe calculated using the rule for calculatingthe volume of a box.

HOMEWORK Problem 14 Use the volumes of familiar objects toestimate the volumes of other objects.

Day 12: Measuring the Volume of Your Hand Student pages 26–28


ACTIVITY Activity, page 26 Measure the volume of a hand in millilitersby submersing a hand in a beaker of water.

CLASSWORK Problem 15 Make a table listing the two estimates ofhand volume and make a graph thatdisplays the relationship.

HOMEWORK Problems 16 and 17 Estimate the volume of the body using ablock model.

Teachers Matter Section C: Volumes 22B

Teachers Matter C

Day 13: Other Measures for Volume Student pages 28–31

INTRODUCTION Problem 18 Compare a cubic cubit with a cubic meter.

CLASSWORK Problems 19 and 20 Solve problems involving cubic fathomsand cords of firewood.

ASSESSMENT Check Your Work Student self-assessment: Solve problemsFor Further Reflection involving volume.

Day 14: Summary


ASSESSMENT Quiz 2 Assesses Section A–C Goals

Additional Resources: Additional Practice, Section C, Student Book pages 41 and 42

Materials

Student Resources


Teachers Resources


Student Materials

Quantities listed are per student, unless otherwisenoted.

• Centimeter cubes (made by students on page 22)

• Centimeter graph paper (two sheets)

• Centimeter rulers

• Metric measuring cups or beakers (one per group)

• Orange juice carton, ketchup bottle, mayonnaisejar, small carton of milk (one of each per groupof students)

• Tissue box (one per group)


Learning Lines

Number Sense

Students are encouraged to enhance their senseof volume by finding personal points of reference.Some examples include:

• About 1,200 centimeter cubes fill a tissue box.

• The volume of my hand is about 200 cm3.

• One gallon of milk is about 4 liters.

• A can of water is about 1__3 liter.

Measurement

Students estimate and compare the volumeof a variety of objects using their own points ofreference. They investigate the formula, volume �area of base � height and explain why it works forsome objects, like a box or a cylinder, but not forothers, like a pyramid or a cone. The need forconsistent measuring units is emphasized. Studentsalso learn about historic units of measurementrelated to body sizes, such as the cubic cubit anda full cord.


By the end of this section, students can:

• measure volume in two ways, using liquid unitsof measure such as liters, and solid units ofmeasure such as cubic centimeters (cm3);

• calculate the volume of a box using the formula,volume � area of base � height, and are able toexplain in which situations this formula will notwork; and

• estimate and compare the volume of a variety ofobjects using appropriate units of measurement.

22 Made to Measure

Notes

1 and 2 You may wantto start this section byworking on these problemsas a whole-class activity.Problem 1 is criticalbecause it is the first timethat students are asked tothink about volume. Thiswould be a good place todiscuss the different waysthat students measurevolume. Some studentsmay use liquid measure-ments, some may usesolid measurements, andsome may use analogiesto describe the size of theheart.

VolumesC

Accommodation

You may wish to make a cubic decimeter so that students can place theircubic centimeters for problem 2b in it to get a better grasp of the volume inquestion.

Intervention

You might ask students whether they know of any relationships betweenthe different measures they come up with. It is not important for studentsto “look up” relationships, however. It is more important for students to geta general sense of the sizes of various units of volume, both liquid andsolid. This is a major focus of this section.


In the last two sections, you measuredlengths and surface area. You can alsomeasure, estimate, and calculate volume.Knowing the volume of a three-dimensionalobject is useful when you want to describehow much space it takes up.

Your heart is about the size of your two fistsclenched together.

1. Estimate the volume of your heart.

CVolumes

The Volume of Your Heart

SolidsYou can use cubes to completely fill up a solid, a namefor a 3-dimensional shape. To make a centimeter cube(cm3), draw a figure like the one shown here. The figureis called the net of a cube.

2. a. Use centimeter graph paper to draw a net of acentimeter cube. Cut it out and fold along thedotted lines. Tape the tabs to make one cubiccentimeter (cm3).

b. Does your class have enough centimeter cubesto fill up one cubic decimeter (dm3)?(Remember: 10 cm � 1 dm.)

c. How many centimeter cubes do you need tocompletely fill one dm3?

1 cm

1 cm

Section C: Volumes 22T

Hints and CommentsMaterialscentimeter graph paper (one sheet per student);cubic decimeter made from a net, optional(one per class)

Overview

Students read the introduction to this section. Theylearn that their heart is about the size of their two fistsclenched together, and they are asked to estimate itsvolume. Students make a cubic centimeter from a netand investigate how many are needed to fill a cubicdecimeter. They think about the appropriateness ofmeasuring objects in different metric units.


1. Students may come up with nonstandard units ofmeasure or standard units of measure, such ascubic centimeters (cm3), milliliters (ml), cubicinches, and so on.


1. Answers will vary. Sample responses:

• 1–2 cups

• a pint

• about the volume of a grapefruit

2. a. Check to see that students correctly made amodel of a cubic centimeter.

b. No. It would take 1,000 cubic centimeters to filla cubic decimeter (10 cm � 10 cm � 10 cm �1,000 cm3). Cubes made by one class would noteven fill a single layer.

c. It takes 1,000 cubic centimeters to fill a cubicdecimeter.

23 Made to Measure

Notes

4a Students do not haveto calculate the volumesof the objects they name.They are merely asked todescribe which objects areappropriately measured inthe given units.

VolumesC

Assessment Pyramid

4b

3a

Understand relationshipsbetween areas.

Estimate volumes usingappropriate units.

Hands-On Learning

You may wish to have centimeter place value cubes on hand so strugglingstudents can actually fill the box.

Vocabulary Building

Make sure that students understand that the term net refers to a flat (two-dimensional) pattern that can be cut and folded into a three-dimensionalshape like the one pictured on this page. This term with illustrations can beadded to the vocabulary section of student notebooks.


Maria wants to find the volume of the box shown here.She started filling the box with the centimeter cubes.

Maria says, “I can easily find the volume of the box.Since the box measures 8 cm by 7 cm, I can fit 56 cubeson the bottom layer.”

5. Explain what else Maria has to do to find thevolume of the box.

6. Find the volume of this container.

7 cm 8 cm

4 cm

Find an empty tissue box.

3. a. Estimate how many centimeter cubes you would need to fillup the tissue box.

b. Cut off the top. How many centimeter cubes do you need tocompletely fill it? Explain how you found your answer.

If 32 centimeter cubes completely fill a box, then the volume of the boxis 32 cm3. For larger objects, you would use larger measuring units.For example, you would measure the volume of your classroom incubic decimeters or cubic meters (in the metric system) or in cubicfeet or cubic yards (in the customary system).

4. a. For what other kinds of objects would you measure the volumein cubic centimeters? Cubic decimeters? Cubic meters?

b. Write some statements about how cubic centimeters, cubicdecimeters, and cubic meters are related.

CVolumes

5 cm

Area ofthe base:100 cm2


Hints and CommentsMaterialstissue boxes (one per group of students);centimeter cubes made by students

Overview

Students start investigating a formula to measurethe volume of a solid. They think about theappropriateness of measuring objects in differentmetric units.


For rectangular boxes, the volume may be calculatedusing the formula: volume � base area � height.This formula may also be used for other objects, likecylinders and prisms. Note that it cannot be used forpyramids or conical objects.

Another formula that may be used is volume � length� width � height.

In the metric system, volume is measured in cubiccentimeters (cm3), cubic decimeters (dm3), and cubicmeters (m3).

1 m � 10 dm

1 m3 � 10 dm � 10 dm � 10 dm � 1000 dm3

1 dm � 10 cm

1 dm3 � 10 cm � 10 cm � 10 cm � 1000 cm3

A cubic decimeter, or 1,000 cm3, is the same as oneliter, which is used for liquids.

Note that understanding the concept of volume andbeing able to identify appropriate measurement unitsis more important than being able to apply one of theformulas mentioned.


3. Students should realize that they do not needto fill the box with cubic centimeters. They maymeasure the dimensions of the box in cm (usingcubes) and use a formula for volume. Or they canuse the cubic centimeters they made to find outhow many fit along the length and the width ofthe box (multiplying these numbers will give thenumber of cubic centimeters of one layer) andthen again using a cubic centimeter to find outhow many layers will fit into the box.


3. a. Students’ estimates will vary. Discuss answersin class.

b. Sample response:

My tissue box measures 5 cm � 11 cm � 23 cm.The bottom layer requires 253 cubes, andwith five layers, there would be 253 � 5 or1,265 cubes.

4. a. Answers will vary. Sample responses:

• cm3: small boxes, bags of flour or sugar

• dm3: large boxes, cans, milk cartons

• m3: houses, grain, coal

b. Sample response:

• 1 dm3 � 1,000 cm3.

Since there are 10 cm in a dm, there are10 � 10 � 10 � 1,000 cm3 in a dm3.

• 1 m3 � 1,000 dm3.

Since there are 10 dm in 1 m, the dimensionsof one cubic meter are 10 dm � 10 dm �10 dm. So 1 m3 � 1,000 dm3.

• 1 m3 � 1,000,000 cm3.

Since there are 1,000 cm3 in a cubic decimeter,there are 1,000,000 cm3 in a cubic meter.

5. Maria has to multiply the 56 cubes in the bottomlayer by four, which is the total number of layers(4 � 56 � 224 cm3).

6. 500 cm3. Sample strategy:

The area of the base is 100 cm2. This means thereare 100 cubes that make up the bottom layer. Thebox will hold five layers of 100 cubes, so itsvolume is 500 cm3.

24 Made to Measure

Notes

7a You may want to makea model of a cheese cubewith edges 2 cm. Somestudents may need thisvisual aid to solve theproblems on this page.

VolumesC

Intervention

Students struggling with problem 9b may need you to help them calculatethe area of one wall. Then ask, How can you use this information tocalculate the volume of the classroom? What additional information will youneed to complete the problem?


The 1, 2, 3 Cheese Factory makes cheese cubes. Theywrap each cube so that the cheese cubes look likenumber cubes. The edges of the cubes have a lengthof 2 cm.

7. a. How many of the one-cubic centimeter cubeswould you need to fill the space taken by onecheese cube? (Hint: It may help to make adrawing.)

The 1, 2, 3 Cheese Factory packages the cheese cubesin a large plastic container, also the shape of a cube.

b. Would 100 cheese cubes exactly fill up thisplastic cube? Explain why or why not.

VolumesC

Maria designed a 10-cm tall container with base dimensions 6 cm by4 cm. She says, “To find the volume, I figured out the area of the baseof the container and multiplied by 10 cm.”

8. a. Use Maria’s method to find the volume of her container.

b. How many cheese cubes would fit in her container?

9. Suppose that you want to determine the volume of yourclassroom.

a. Is it possible to use centimeter cubes (like the one you madeon page 22) to find the volume of the classroom?

b. Can you use the area of one wall to find the volume of yourclassroom? Find the volume of your classroom. Be sure to usethe right unit measurements.

The 1, 2, 3 Cheese Factory is designing a new plasticcontainer that is 10 cm tall and holds exactly 60cheese cubes.

c. What is the volume of this new plasticcontainer?

d. List some possible dimensions for this newdesign. (Hint: It may help to make a drawing.)


Hints and CommentsMaterialscentimeter rulers (one per student)

Overview

Within the context of cheese cubes sold by a cheesefactory, students further investigate and solveproblems about volume. They make an estimation ofthe volume of their classroom using appropriatemeasurement units.

Planning

Students could work on the problems in small groupsor pairs. Another possibility is to assign theseproblems as homework to informally assess students’abilities to solve problems involving volume.


7. b. This problem relates to the use of cubicnumbers—a common misconception amongstudents is that 100 is a cubic number.


7. a. You would need eight 1-cm cubes to fill up onecheese cube.

Sample explanation:

The dimensions of the cheese cube are 2 cm � 2cm � 2 cm. You can fit four 1-cm cubes on thebottom, two along each edge. The height of thecheese cube is 2 cm, so two layers of 1-cm cubesfill the space. In total, there are eight 1-cm cubes(2 � 4 cubes).

b. No, you cannot fill the plastic cube with exactly100 cheese cubes.

Here are two sample explanations:

• Suppose you have three cheese cubes alongeach edge. The base would hold 9 cheesecubes (3 � 3) There would be 3 layers ofcheeses, so in total there are 27 cheese cubes(3 � 9).

With four cheese cubes along each edge,there would be 64 cheese cubes in total(4 � 4 � 4).

With five cheese cubes along each edge,there would be 125 (5 � 5 � 5). This is morethan 100 cheese cubes.

• Since the container must be a cube, you canonly fill it with these amounts of cheese cubes:1, 8, 27, 64, 125, 216,… none of which isexactly 100 cheese cubes.

c. The volume is 480 cm3. Each cheese cube has avolume of 8 cm3. Since there are 60 cheesecubes, the volume is 60 � 8 cm3 or 480 cm3.

d. Here are some possible dimensions:2 cm � 24 cm � 10 cm (5 layers of 1 � 12)4 cm � 12 cm � 10 cm (5 layers of 2 � 6)6 cm � 8 cm � 10 cm (5 layers of 3 � 4)

Check that student’s answers have a height of10 cm; the remaining dimensions both need tobe multiples of 2 and have a product of 48 (thevolume is 480 cm3).

Sample explanation:

Each cheese cube is 2 cm tall. Since mycontainer is 10 cm tall, I can stack 5 layers ofcheese (10 cm � 2 cm). Each layer will have12 cheese cubes

(60 cheese cubes � 5 layers). I can make a3 � 4 arrangement with 12 cheese cubes.The dimensions of each layer would be6 cm � 8 cm (3 cheese cubes � 2 cm and4 cheese cubes � 2 cm).

8. a. Area of the base is 24 cm2 (6 cm � 4 cm). And24 cm2 � 10 cm is 240 cm3, which is the volume.

b. 30 cheese cubes can fill up Maria’s container.

Sample strategies:

Since the height is 10 cm, there are again 5 layers.Each layer is 6 cm � 4 cm, so 3 � 2 cheese cubesfit in each layer (6 cm � 2 cm and 4 cm � 2 cm).That makes 6 cheese cubes on each of the5 layers for a total of 30 cheese cubes.

I noticed that this volume is exactly half thevolume of the container holding 60 cheese cubes.So there must be half as many: 30 cheese cubes.

9. a. Students should notice that while it would bepossible in theory to fill the classroom with cmcubes and count them, in practice it would beimpossible to do so.

b. Yes, the area of the wall times the length of thefloor will give the volume. Students may usemeters or feet in measuring, so the volume willbe in cubic meters or cubic feet.

25 Made to Measure

Notes

11 When working on thisproblem, be sure thatstudents understand whenthe given volume formulais applicable.

VolumesC

Assessment Pyramid

11, 12

Develop formulas toestimate and calculatevolume.

Hands-On Learning

Have students draw or cut out pictures of shapes for which the formula willwork and shapes for which the formula will not work.

Intervention

You might ask students whether the volume changes when a stack of paperis “slanted.” Students are not expected to use the formula A � πr 2 to findthe base area of a soda can.


10. Is it possible to use centimeter cubes (like the one you made onpage 22) to find the volume of a soda can? Explain your reasoning.

Expressed as a formula, this was Maria’s strategy.

volume � area of the Base � height

11. Can this formula be used to find the volume of a stack of paper?A soda can?

The formula does not work forall three-dimensional objects.For example, you cannot find thevolume of the pyramid by usingthis formula.

12. a. Give an example for which the formula does work and anotherexample for which the formula does not work.

b. Reflect Why does the formula give the correct answer forsome objects but not for others?

You can use the area of your handprint (Section B, problem 7) toestimate the volume of your hand.

13. a. Besides the area of your handprint, what other measure(s) doyou need to estimate the volume of your hand?

b. Estimate the volume of your hand.

CVolumes

hh


Hints and CommentsOverviewStudents investigate the formula volume � area of thebase � height and explain why it works for someobjects but not for others. They also find a newestimate for the volume of their hands.


10. The intent of this question is to help studentsrealize the limitations of using “strict” cubiccentimeters as measuring tools and to see thatsometimes, cubic centimeters need to be divided.

12. This problem formalizes the volume formula(volume � base area � height) and the concept ofusing consistent measuring units.

13. b. To determine the depth of their hand, studentsmay decide to measure the thickness in severalestablished places and then find the average ofthose measurements.


10. Answers can vary; check for justification. Here aretwo samples.

• No, because cubes cannot fill the entire cansince the can is not a block.

• Yes, you can cut up the cubes into pieces andmake them fit inside the can.

Or students may suggest ways of estimatingwith cubes.

11. You can use the formula for a stack of paper. Thearea of the base tells how many cubes you needfor the bottom layer, and the height tells howmany layers you need. You can use the formulafor a soda can, but since the Base is a circle, youneed the formula for the area of the circle;

area of a circle � π � r2, where r represents theradius of the circle-shaped base of the can.

12. a. Examples will vary. The volume formula worksfor prisms or cylinders, like a bar of soap, fishtank, cylindrical candle, or book. The volumewill not work for cones or pyramids, like acone-shaped carton of milk, a pyramid orconical candle, or bottle of glue.

b. The formula only works when all the slicesmatch the footprint of the Base. The area of theBase tells how many cubes you need for thebottom layer, and the height tells how manylayers you need. This works only if all of thelayers have the same area. The slices or “layers”in the object must all be the same. If the objectis a pyramid or a cone, an irregular shape, thisis not the case.

13. a. The thickness of the hand.

b. Estimates will vary. Students may note that thethickness of the hand varies from the fingertipsto the wrist. Using the estimate of the surfacearea of the hand from Section B (115 cm2), thevolume of the hand is about 130 cm3.

26 Made to Measure

Notes

Have students read theintroductory text. Thendiscuss with the class howliquids are measureddifferently from solids.

VolumesC

Assessment Pyramid

14

Estimate volumes usingappropriate units.

Extension

You may wish to bring to class a beaker that shows milliliters to measurethe volumes of containers. Fill some empty jars with water, have studentsestimate the volume of each jar in liters, and then pour the contents intothe beaker to determine the actual volume.

Parent Involvement

Students may investigate the volumes (in liters) of containers that they findat home and then write reports on their findings.


The units used to measure liquids are different from the units usedto measure solids. Pints, quarts, gallons, liters, and milliliters are alltypical measures of liquid volume.

VolumesC

Measuring the Volume of Your HandFor this activity, you will need a can or beaker that measures inmilliliters. It should be big enough so that you can put yourhand in it.

Pour water into the can or beaker so that it is about half full.Measure and record the level of the water. Then, put a rubberband on your wrist, and put your entire hand in the water upto the bottom edge of the rubber band. You may want to makea fist. Be sure that your whole hand is underwater.

Measure and record the new water level. The difference betweenthe old level and the new level is the volume of your hand inmilliliters (ml).

Liquids

Here are some bottles whose contents are measuredwith units of liquid volume.

If you are estimating volumes, it may be helpful toknow that:

• a regular soda can contains about 1��3 liter.

• a cubic decimeter contains exactly one literof liquid.

14. Estimate the volume (in liters) of the following objects.

a. an orange juice container c. a small cup of juice

b. a large glass of water d. a gallon of milk

liter 1 liter 2 liters12


Hints and CommentsMaterials1__2-liter, 1-liter, and 2-liter bottles (optional, one of

each per class);orange juice carton, ketchup bottle, mayonnaise jar,and small carton of milk (one of each per group ofstudents);metric measuring cup or beaker (one per group)

Overview

Students learn that measurement units for liquids aredifferent from units used to measure solids. Theyestimate the volumes, in liters, of food packages. Theyinvestigate a method to measure the volume of theirhand by immersing their hand in water in ameasuring cup or beaker. They investigate therelationship between hand volume in cubiccentimeters and hand volume in milliliters.

Planning

Students may want to add their own reference pointsregarding volume, like “a regular soda can containsabout 1__

3 liter,” to their list. You may want to bring inactual objects and have students estimate theirvolume.

Students may do the activity in small groups. Ask thescience teacher at your school for beakers that arelarge enough for students to immerse their handsinto (1,000-ml beakers will work). If large beakers areunavailable, 2-pound coffee cans will work. Variouswater levels can be determined and marked on theinside of the can by using a graduated cylinder andwaterproof marking pen. Ask students whether theythink the answer would change if they made theirhand into a fist.


14. Students may use either the actual objects or thereference measures to estimate volumes. Theymay have to convert pints and gallons to liters,or they may estimate the volume in liters.



a. an orange juice container: about 2 liters

b. a large glass of water: about 1__3 liter

c. a small cup of juice: about 1__4 liter

d. a gallon of milk: about 4 liters

27 Made to Measure

Notes15 By this point in the unit,students should realizethat 1 milliliter � 1 cubiccentimeter and that thereis a relationship betweenliquid size and solid size,although they are measuredin units that have differentnames. You may want toremind students that onecubic decimeter is thesame as one liter.

VolumesC

Assessment Pyramid

15b

Understand volumerelationships.

Extension

Weight is not a topic in this unit, but you may also want to mention tostudents that 1 liter of water (which equals 1 cubic decimeter) weighs onekilogram (and, therefore, 1 milliliter of water weighs 1 gram).

Intervention

Make the table in problem 15 as a whole-class activity, and have studentswork in small groups to discover the relationship between the twomeasurements. Note that one milliliter is 1_____

1000 of a liter. Some students mayneed help creating the axes for the graph.


You now have two estimates of the volume of your hand: one in cubiccentimeters (from problem 13) and the other in milliliters. How arethese two measurements related?

15. a. Make a table listing the two estimates of hand volumes forten students.

b. Find a relationship between the two measurements—milliliters and cubic centimeters. Drawing a graph mightbe helpful.

CVolumes

Cubic Centimeters

0 50 100 150 200 250

50

Millilite

rs

100

150

200

250

The Volume of Your BodyYou could find the volume of your body in the same way that youfound the volume of your hand, using liquid measures, but youwould probably need a bathtub to do it!

Another way is to use solid units of measureand to think of your body as being made upof cubes.

16. Estimate your body’s volume by modelingyour body with cubes as shown on theleft. You will need to decide on a good unitof measure for the cubes.

To estimate the volume of your body, you canalso use a block to model your body, as shownon the left.


Hints and CommentsMaterialsgraph paper (one sheet per student)

Overview

Students used the measurements of the volume of thehand of the students in their class to find a relationshipbetween cubic centimeters and milliliters. They startinvestigating the volume of a person’s body and findappropriate measurement units.


15. a. Sample table:

b. The volume in cm3 is the same as the volumein milliliters (ml) since 1 ml of water occupies1 cm3 of space. The points on the graph shouldshow a strong linear relationship. (In fact, thegraph should be roughly a line with slope 1 andy-intercept 0.) If differences exist, they may bedue to errors in measurement. The volume willprobably be about 200 cm3.

Sample graph:

16. Estimates will vary depending on the size of aperson. Here is a strategy for someone 140 cm tall.

There are 38 cubes in the model of the person onStudent Book page 27. The model is 14 cubes talland 1 cube wide. Since I am about 140 cm tall(14 dm), I can model my body’s volume by makingeach edge of the cube to be 1 dm. The volume of1 cube is 1 dm3. Using exactly 38 cubes, my body’svolume is 38 dm3, assuming I am also about 1 dmthick.

Name of

Student

Volume of

the Hand

(in cm3)

Volume of

the Hand

(in ml)

Cubic Centimeters

Relationship Between Milliliters

and Cubic Centimeters

0 50 100 150 200 250

50

Millilite

rs

100

150

200

250

28 Made to Measure

Notes

17 Students may needhelp understanding thatthey are being asked tofind the width and depthof a block 16 cm tall anda volume of 67 dm3.

VolumesC

Vocabulary Building

Some students may not be familiar with the term mass. While technicallyit is a measure of an object’s resistance to acceleration, it is probablysufficient to tell students that mass is the “stuff” that they are made of.Mass is the scientific term for what we know as “weight.”

Example: If a student is 157 cm tall, then because the model is 14 blockshigh, each block would be 157 � 14 or about 11 cm high or 11 cm on a side.Each cube would then have a volume of 11 � 11 � 11 � 1,331 cm3. Thereare 38 cubes in the model shown, so the volume of the shape would be38 � 1,331 � 50,578 cm3, which is the same as 50.5 dm3.


VolumesC

Other Measures for Volume

The table above contains some information about Mindy when shewas 10, 13, and 16 years old.

17. Use a block to model Mindy’s volume when she was 13 years old.Find the width and depth measurements of the block.

Data for Mindy at Three Ages

Ages (in years) 10 13 16

Height (in dm) 14 16 17

Mass (in kg) 31 52 65

Surface Area (in dm2) 110 150 175

Volume (in dm3) 41 67 79

In the past, units of measure for volume were sometimes related tobody sizes. It was more common, however, to use measurement toolslike the cup as standards for measuring volume.

The ancient Egyptians had the first unit of measure for length—thecubit. The cubit is the distance from a person’s elbow to the tip ofthe extended middle finger.


Hints and CommentsOverviewStudents estimate their body volume by modelingtheir body with cubes. They also find dimensions ofa single block that can be used to model the givenvolume of a 13-year-old girl. Students investigatehistorical measures of volume related to the body.


Often in mathematics, problems that are morecomplicated are solved by looking at simpler cases,or situations that are “idealized” in some way. In thesituation on Student Book page 27, a body modeledwith cubes or rectangular blocks simplifies theproblem of finding the volume of a person’s body.


17. The width is most likely 4 dm and depth is 1 dm(height is 16 dm). Other answers are possibleprovided the area of the Base is about 4 dm2 andvolume is 67 dm3.

Sample strategy:

The area of the Base of the block would be whatis left over after dividing Mindy’s volume by herheight (67 dm3 � 16 dm). The area of the Base ofthe block is about 4 dm2. One probable depthmeasurement is 1 dm, so the width must be 4 dm.The block that models Mindy’s volume at age 13is 1 dm � 4 dm � 16 dm.

29 Made to Measure

Notes18 You may want to tellstudents to first find outhow many cubits are in1 meter before determiningthe volume of one cubiccubit in cubit meters.

19 You may wantstudents to outline thebase of a cubic fathom onthe ground to see how bigit would be (once theyfigure in the height).

VolumesC

Assessment Pyramid

19ab

Develop and use a frame ofreference for measurementto solve problems.

Vocabulary Building

Encourage students to draw sketches indicating the dimensions of a fullcord and a face cord in the vocabulary section of their notebooks. Thesketches may show them that it is not necessary to calculate both volumesto compare them since only the depths are different.


The Egyptians used a standard length called the royal cubit, which is52.4 cm, or 20.62 in.

In the ancient Egyptian system, the volume unit of measure is thecubic cubit.

18. a. Use your two arms to get a rough idea of how big a cubiccubit is.

b. About how many “royal” cubic cubits are there in a cubicmeter?

The cubic fathom is a unit of measure used in the 1800s in Europeto measure volumes of firewood. (Remember that a fathom is thedistance between your two outstretched arms.)

19. a. Estimate the amount of firewood that can fit into your cubicfathom.

b. What else can you measure with a cubic fathom? With a cubiccubit?

In the United States, firewood is measured in cords. A face cord is 4 fthigh, 8 ft long, and 16 in. deep. A face cord usually fits in the back of apickup truck. A full cord is 4 ft high, 8 ft long, and 4 ft deep.

20. How much more firewood is in a full cord than in a face cord?

CVolumes


Hints and CommentsOverviewStudents compare the measures cubic cubit and cubicfathom to cubic meters. Then they compare the volumeof a face cord of firewood with that of a full cord.


Doubling the dimensions (length, width, and height)of an object does not result in a volume that is twiceas big: the volume is enlarged by a factor of eight.


18. a. Students need not calculate an answer; theyare merely getting an idea of size. However,they may want to make a sketch and indicatedimensions in centimeters.

19. a. Students’ responses may be in logs or involume measures. If students are able to findone type of measure easily, they might be askedto find the other.


18. a. 50 cm3. Arm positions may vary. One possibilityis to bend one arm so that the forearm ishorizontal and parallel to the chest. The otherarm should be bent so that the fingers arepointing up and the elbow is in contact withthe fingers of the other hand.

b. There are about 8 cubic cubits in 1 cubic meter.

One edge of a cubic meter is 100 cm. Since1 cubit � 50 cm, then 2 cubits � 100 cm.All the edges of a cubic meter are 2 cubits long.The volume is 8 cubic cubits (2 cubits � 2 cubits� 2 cubits).

19. a. Answers will vary depending on a person’sheight and the size of the firewood. About147 logs will fit into a person’s cubic fathomthat is 1.5 m tall.

Sample strategy:

My height is 1.50 m, so my cubic fathom is3.38 m3 (1.5 m � 1.5 m � 1.5 m).

I assume that firewood is about 0.5 m long andabout 0.2 m wide and 0.2 m high.

Stacking the firewood in my cubic fathom willbe 3 logs deep, 7 logs wide, and 7 logs high.

So together there would be 147 logs (3 � 7 � 7).


• cubic fathom: space in a car, bus, hallway, orclassroom

• cubic cubit: a laundry basket, boxes,cupboards, or a bushel basket

20. There is three times more wood in a full cord thanin a face cord.

Here are two strategies.

• The face cord is 16 inches deep, while thefull cord is 48 inches deep, so the full cordhas three times as much wood. The otherdimensions are the same.

• Finding the volume directly:

Face cord: 4 ft � 8 ft � 11__3 ft � 42 2__

3 ft3.

Full cord: 4 ft � 8 ft � 4 ft � 128 ft3.

Since the ratio of the volumes is 1:3, there isthree times as much wood in the full cord.

30 Made to Measure

Notes

After having a student readthe Summary aloud, youmay wish to have them goback through the sectionand find problems thatsupport the conceptstaught. This will encouragestudents to actively use theSummary section as astudy tool.

VolumesC

Assessment Pyramid

1, 2

Assesses Section C Goals

Parent Involvement

Have students discuss the Summary with their parents, showing themexamples from the section.


Volumes

Volume is important for measuring the sizes of such objects as apackage, a drink, or even your hand.

In this section, you measured volume in two ways—with liquid unitsof measure (such as liters) and with solid units of measure (such ascubic centimeters).

You also measured the following:

• the volume of an object when you wanted to know how muchspace it takes up or how much it holds.

• the surface area or area of an object when you wanted to knowwhat it takes to cover it.

• the length of an object when you wanted to know how long orhow tall it is.

1. When would it be useful to know the volume of an object?

2. Which unit measurement would you use for the followingobjects? (Choose from liters, cubic centimeters, and cubicmeters.)

a. a bottle of water

b. the volume of water in a swimming pool

c. the volume of a package containing math books

C


Hints and CommentsOverviewStudents read the section Summary. They use theCheck Your Work problems as self-assessment.The answers to these problems are also providedon page 44 of the Student Book.

Check Your Work problems assess students’ ability toestimate lengths, areas, and volumes visually usingappropriate units; to find (compute, measure, anddraw) lengths, areas, and volumes using standard andnonstandard units; and to recognize that differentunits of measurement and different measurementsystems exist. It also assesses their ability to use therelationships between length, area, and volume; touse formulas and rules of thumb to estimate andcalculate length, area, and volume; and to select anduse appropriate measurement units.


1. Here is one response. Yours may be different.

You might need to know an object’s volume forpacking, shipping, or pricing.

2. a. a bottle of water: 1 liter.

b. the volume of the water in a swimming pool:900 m3.

c. the volume of a package of math books;3,000 cm3.

31 Made to Measure

Notes

Be sure to discuss CheckYour Work with students sothey understand when togive themselves credit foran answer that is differentfrom the one at the back ofthe book.


Reflective questions aremeant to summarize anddiscuss important concepts.

VolumesC

Assessment Pyramid

3a, FFR

3b

Assesses Section C Goals

Parent Involvement

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


In a supermarket, you can buy small packages of fruit drinks indifferent sizes.

3. a. Could both packages show: Contains 0.2 liters? Why orwhy not?

b. Three packages of Frisca are packaged together and soldfor $1.98. What is the price of one liter of Frisca?

Linda knows that one liter equals “one cubic something.” She doesn’tremember whether it is 1 cm3, 1 dm3, or 1 m3. Explain to Linda whichunit of measure equals one liter.


Hints and CommentsOverviewStudents use the Check Your Work problems as self-assessment. The answers to these problems are alsoprovided on page 45 of the Student Book.

Planning

After students complete Section A, you may assign ashomework appropriate activities from the AdditionalPractice section, located on Student Book pages 41and 42.

Check Your Work problems assess students’ ability toestimate lengths, areas, and volumes visually usingappropriate units; to find (compute, measure, anddraw) lengths, areas, and volumes using standard andnonstandard units; and to recognize that differentunits of measurement and different measurementsystems exist. It also assesses their ability to use therelationships between length, area, and volume; touse formulas and rules of thumb to estimate andcalculate length, area, and volume; and to select anduse appropriate measurement units.


This problem assesses students’ ability to use simplerelationships between and within measurementsystems.


3. a. Yes, both packages can show, “contains 0.2 liters.”One liter is the same as 1,000 cm3 (1 dm3), so0.2 liters is about 200 cm3. The shorter packagehas a volume of 202.5 cm3 (7.5 cm � 4.5 cm �6 cm). The taller package has a volume of216 cm3 (4.5 cm � 4 cm � 12 cm). The volumeof both packages needs to be a little more than200 cm3, so the juice will not spill out of thecontainer too easily.

b. One liter costs $3.30. Sample explanation:

Three packages contain 0.6 liters (3 � 0.2 liters).Use a ratio table to find the price of one liter.


A liter is equal to 1 dm3. Here is one sample explanationfor Linda.

Linda, look at our cubic centimeter. It is way toosmall to hold one liter. Here, let’s get the meterstick. Imagine a cubic meter. That is too large.Since 10 cm = 1 dm, a cubic decimeter is just theright size to hold exactly 1 liter.

Volume (in liters) 0.6 0.3 0.1 1.0

Price (in dollars) 1.98 0.99 0.33 3.30


Teachers MatterD

Section Focus

In this section, students explore the angles that they can make withtheir wrists and ankles and investigate how furniture and computersare designed to fit the angles of the body. The use of a compass cardor protractor is reviewed. Students who have used the unit FiguringAll the Angles will be familiar with the use of a compass card.

Pacing and Planning

Day 15: Furniture Student pages 32 and 33

INTRODUCTION Problems 1 and 2 Discuss the angles that students can makewith their arms, ankles, wrists, and headsand the importance of these angles forfurniture design.

ACTIVITY Activity, pages 32 and 33 Perform an experiment to determine theProblems 3 and 4 different range of motion for the right and

left hand.

Day 16: Furniture (Continued) Student pages 34–39

INTRODUCTION Problem 5 Draw a split computer keyboard at a25-degree angle.

CLASSWORK Problems 6–10 Make a diagram of furniture that fitscertain angle requirements and measureangles in a diagram.

HOMEWORK Check Your Work Student self-assessment: Use body angleFor Further Reflection measurements to reason about the design

of man-made objects.

Additional Resources: Additional Practice, Section D, Student Book page 42

Teachers Matter Section D: Angles 32B

Teachers Matter D

Materials

Student Resources


• Student Activity Sheet 1

Teachers Resources

Quantities listed are per class.

• Dictionary or encyclopedia

Student Materials


• 8 1__2 ′′ × 11′′ paper, (two sheets per student)

• Compass card from Transparency Master 1 orprotractor


Learning Lines

Measurement

Students measure and draw angles usingappropriate units. The compass card is a tool usedthroughout Mathematics in Context to measureangles and navigate. In this section, students usethese 360-degree protractors to measure angles.Students should be given the opportunity toinvestigate the use of this measurement tool sothat they can learn to use it appropriately and candistinguish the differences between this tool andthe protractor.

Problem Solving

Students discuss the layout of a traditionalkeyboard and why the design does not fit thenatural angles of the wrist. Students then makediagrams of furniture to fit certain ergonomicrequirements.


By the end of this section, students can:

• measure and draw angles using a compass cardor protractor and

• use geometric models to solve problems.10° to 15°

32 Made to Measure

Notes

2 You may want to ask onestudent to look up theword ergonomics and thenshare the definition withthe class.

AnglesD

Parent Involvement

Ergonomics does not need to be restricted to the workplace. Students maylook around at home to see for which objects it is important to keepcomfortable designs in mind and talk about this issue with their families.They may look at chairs, countertops, tables, doors, bathtubs, toilets, sinks,and so on. Have them focus on the height and placement of objects, payingparticular attention to angles.


English language learners may need some help with the context ofergonomics. It may help to give them a variety of examples.


When designing furniture, cars, computers, and other items, it issometimes necessary to know the angles that a person can make withhis or her arms, ankles, wrists, and head.

1. For what objects might it be important to keep angles in mind?

Ergonomics is important for designing work environments. Designersuse ergonomics to determine the placement of office equipment andthe size and dimensions of furniture that will create safe, efficientworking environments for people. Designers also consider ergonomicswhen creating new buses and trains for commuters, for example, byplacing controls close to the driver.

2. Find the dictionary definition of ergonomics.

DAngles

Furniture

Arcs of Movement

The first drawing shows the “arc of movement” of someone’s lefthand. The picture shows how far to the left and right the wrist canbend.

Section D: Angles 32T

Hints and CommentsMaterialsdictionary or encyclopedia (one per class)

Overview

Students learn the meaning of the word ergonomics,how it involves learning about people’s range ofmovement, and how it impacts the design of suchthings as chairs and computer keyboards.

Planning

Problem 1 may be answered in small groups orindividually; problem 2 can be done as a class. Inaddition, you may want to have students consultother reference books. Information in encyclopediasmay be found under systems engineering. Anothergood reference is Fitting the Task to the Man, byEtienne Grandjean.


1. After investigating an uncomfortable chair, it isvery likely that students will mention all sorts offurniture. Encourage them to think of otherobjects as well such as musical instruments,vehicles, and so on.



• a computer keyboard

• a car (the positions of the steering wheel, thebrake pedal, and so on)

• a piano (or other musical instrument)

• a chair

• a bike

2. Ergonomics is the study of people and their workenvironments. Some of the goals of ergonomicsare the following:

• to reduce stress on a person’s body caused byinteraction with and use of objects;

• to design equipment and machines for safeand efficient use;

• to construct work stations to ensure correctbody posture; and

• to adjust lighting, air-conditioning, and noiselevels for optimal working conditions.

33 Made to Measure

Notes

3a Students may needhelp placing their compasscards/protractors correctly.

AnglesD

Assessment Pyramid

3ab

Measure angles usingstandard units.

Extension

You may wish to have students investigate whether the angles of theirdominant hand are bigger than the angles of their other hand or whetherpeople with bigger hands make bigger angles.


You can measure the arc of movement of your own hands.

• Draw a small x on the bottom of a sheet of paper.

• Lay your forearm flat, with your left hand and wrist on the sheetof paper. Put the middle of your wrist over the small x.

• Put a mark on the paper at the top of your middle finger.

• With your arm and hand flat on the table, bend your wrist to theright as far as you can without moving your arm. Put anothermark on the paper at the top of your middle finger.

• Bend your wrist to the left as far as you can without movingyour arm. Mark the location of the top of your middle finger.

• Draw lines from each of the three finger marks to the small x(the middle of your wrist).

3. a. Use your compass card or protractor to measure the anglesthat your left hand can bend to the right and to the left.

b. What do you think the angles will be for your right hand?

Measure the arc of movement of your right hand following the sameinstructions on page 32 and above.

4. a. Do you think the results of problem 3 will vary for differentstudents in your class?

b. Why might someone be interested in studying the motion ofthe wrist joint?

DAngles

When typing on a computer keyboard, your handsshould rest on the keys. In order to reach all of thekeys, your hands must bend sideways. If you do a lotof typing, the position in which you hold your handson the keyboard can cause physical discomfort inyour hands, wrists, or forearms.


Hints and CommentsMaterialssheet of paper (one per student);compass card or protractor (one per student).(Note that a Transparency Master page is available inthis Teacher’s Guide to provide for compass cards.)

Overview

Students draw the arc of movement of their left handand measure the angle formed. They measure theangles that describe the range of movement of theirleft hands using the drawing they made on theprevious page. They then measure the range for theirright hands. They compare the right-hand anglemeasurements for all students in the class, and thinkabout why the motion of the wrist joint might beimportant in ergonomics.

Planning

Students may work on the Activity individually or ingroups. It is important that the forearm stays fixed onthe table. Students may mark their wrist with an x aswell to make sure that it stays on the x on the sheet ofpaper.

The three lines students draw on their paper shouldbe equally long, since all three represent the length oftheir middle finger.

Students may work individually on problem 3.Problem 4a is a whole-class activity. Students maywork on problem 4b individually.

For some students you may need to review howangles are measured.


3. Students must decide how to place their compasscards or protractors on their drawings.

4. a. Students might represent the data in ahistogram or use one-number statistics (mean,median, or mode) to summarize the data.


3. a. The measures will be about 33° on the thumbside and 45° on the little finger side.

b. The right-hand angles will mirror the left-handangles. The smaller angle will occur on thethumb side.

4. a. Different students will have different results.In one pilot class, the mean angle measure onthe thumb side was 32°, and on the little fingerside, it was 40°. The range of angle measures onthe thumb side was 24° to 40°. The range on thelittle finger side was 30° to 50°.

b. Students should mention that wrist movementdata is important to designers of objects likecomputer keyboards.

�B

� � 33�B � 45�

34 Made to Measure

Notes

5 Before students beginwork on this problem,make sure they understandwhat they are expected todo. Have students thenshare their work in class.

5b To answer this problem,students may use theirdata on the hand arcs fromStudent Book page 33.

AnglesD

Accommodation

If available, some students may need to look at a split keyboard to be ableto make the drawing for problem 5a. A photograph should also work.

Vocabulary Building

A "line of sight" is an an imaginary straight line from a person’s eye to anobject. In a later Geometry unit, Looking at an Angle, the phrase “visionline” is also used. To help students relate this concept to more familiarcontexts, ask students how the “line of sight” might apply to games likeHide and Seek, Capture the Flag, or Laser Tag.


Studies on ergonomic keyboard design have found that most peopleprefer a split keyboard. A split keyboard is divided into two parts: theleft part ends with the keys T-G-B, and the right part starts with keysY-H-N.

On a split keyboard, the two parts make an angle of about 25°, that is,the lines drawn through T-G-B and Y-H-N make an angle of 25°. Thedistance between the two parts—or between the keys G and H—should be about 95 mm.

5. a. Make a drawing of a split keyboard. You do not have to drawall of the keys, only an outline of the two parts.

b. What is one advantage to using a split keyboard?

Another important angle in design involves your line of sight. If youlook straight ahead, your line of sight follows a horizontal line. Yournormal line of sight is typically 10° to 15° below the horizontal.

In the best ergonomic computer design, the screen should not belower than the sight line of 15° below the horizontal.

AnglesD

10° to 15°


Hints and CommentsMaterialscompass cards from Transparency Master 1 orprotractors (one per student)

Overview

Students read a description of a split keyboard andthen draw one. They think about the advantages ofthis kind of keyboard. They also draw lines of sightthat make a given angle, which is important for theergonomic design and placement of computerscreens.


5. a. Students might make a scale drawing of a splitkeyboard, although it is possible to draw themiddle of the keyboard to full scale.


5. a. Drawings will vary in their level of detail, butthe angle and distance measurements givenshould be represented in the sketch. Samplestudent drawing:

Students can draw one line, measure thedistance (95 millimeters), plot a point at thatdistance, and then use a protractor to find theline through the point that makes an angle of25° with the first line.

b. Answers will vary. Basically, the wrists do notneed to bend as much with a split keyboard,which makes typing more comfortable.


Hints and CommentsMaterialsStudent Activity Sheet 1 (one per student);compass cards from Transparency Master orprotractors (one per student);straightedge or ruler (one per student)

Overview

Students use what they have learned on the previouspages to solve a real life problem regarding designrecommendations for computers. They startinvestigating design recommendations for easy chairs.


The mathematical concept of sight lines is furtheraddressed in the unit Looking at an Angle.

Planning

The problem on this page may be used to informallyassess students’ ability to draw vision lines and tomeasure angles.


6. The purpose of this question is to have studentsdraw the angle (by first drawing the line toindicate the horizontal plane).


6. a. Sample drawing:

b. Ellen’s computer screen is in the right position;it fits the requirements.



Overview

Students use what they have learned on the previouspages to solve a real life problem regarding designrecommendations for computers. They startinvestigating design recommendations for easy chairs.


The mathematical concept of sight lines is furtheraddressed in the unit Looking at an Angle.

Planning

The problem on this page may be used to informallyassess students’ ability to draw vision lines and tomeasure angles.


6. The purpose of this question is to have studentsdraw the angle (by first drawing the line toindicate the horizontal plane).


6. a. Sample drawing:

b. Ellen’s computer screen is in the right position;it fits the requirements.

36 Made to Measure

Notes

7 Students may want totrace the outlines of bothchairs and draw ahorizontal line beforemeasuring the anglesinvolved with a compasscard or protractor.

9a Again students maytrace the outline of the seatand the backrest and drawa horizontal line beforemeasuring angles.

AnglesD

Parent Involvement

Students might use the given recommendations to judge easy chairs athome. They can rate the comfort of these chairs or ask family members torate them. Students can then write reports about their findings.


Here are two different easy chairs.

7. Check to see if these chairs fit the recommendations on page 35.

8. Based on these recommendations, design your own easy chair.Draw a side view of your chair on an appropriate scale.

Shown here is the side viewof an office chair.

9. a. Use Student Activity Sheet 1 to measure the angles the seatand the backrest make to the horizontal plane. Also measurethe angle of the backrest to the seat. Draw extra lines if youwant to.

b. Compare your findings to the recommendations above foreasy chairs. What conclusion can you make?

AnglesD



Overview

Students learn about recommendations for the designof an easy chair and then judge two easy chairs andan office chair based on these recommendations.They also design their own easy chair.

Planning

Have students work on problems 7 and 8 in smallgroups. Students may work on problem 9 individually.It may be assigned as homework or used forassessment. Discuss students’ answers to problems7–9 in class.


Remind students that when they measure the angle ofthe backrest to horizontal, they must measure fromthe same zero line they used to measure the seatangle (although the zero line will change from chairto chair).

Some students may notice that the recommendedminimums for the seat to horizontal and for thebackrest to the seat do not add up to the minimumfor the backrest to horizontal. The same holds true forthe maximums. Each angle was evidently testedseparately for comfort.

9. b. Office chairs are a bit more upright than easychairs.


7. Neither chair fits all recommendations.

The chair pictured on the left has a seat that istilted backwards about 9° from the horizontalplane, which is too little. The angle between theseat and the backrest for the left chair is about103°, which is not quite enough. To the horizontalplane, the backrest is tilted 9° � 103° � 112°. Thisis acceptable. The left chair, therefore, fits one ofthe three recommendations.

For the right chair, the seat is tilted backwardsabout 19° from the horizontal plane, which isacceptable. The angle between the seat and thebackrest is about 103°, which is not quite enough.To the horizontal plane, the backrest is tilted19° � 103° � 122°. This is acceptable. Theright chair, therefore, fits two of the threerecommendations.

8. Designs will vary. Students’ chairs should fit therecommendations and be drawn to scale.

9. a. The seat is about 5° to the horizontal plane,while the backrest is about 110° to thehorizontal. The angle of the backrest to theseat is about 105° (the difference betweenthe two angles).

b. Answers will vary, but students should notethat the seat barely fits the recommendations,which is not surprising since an office chair isnot an easy chair.

37 Made to Measure

Notes

10a You may wantstudents to make a sketchshowing what is describedin the text. Such a drawingwill help them to see howto complete part b. Askstudents to explain whythe 145° is fixed for everyperson. (Because the armis blocked by the body.)

AnglesD

Intervention

Have students draw a huge compass card on the floor (with refinement upto 10°) and have one student stand on it with the shoulder right above thecenter, outstretch the arm, and use a plumb line attached to the hand tomeasure the angle between the side position and the back position.


The following text is taken from a book on ergonomics, Fitting theTask to the Man, by Etienne Grandjean.

“The arm can rotate through an angle of 250° about itsaxis…of which a half-circle (180°) lies in front of the body,and a further 70° or thereabouts, backwards.”

10. a. Read the text carefully. Write in you own words what the textmeans.

b. Check to see if your arm can make an angle of 70° backward.First, think of a way to measure this angle.

Challenged by the Egyptians!

It is sometimes said that measurement is the oldest appearance ofmathematics. The name geometry literally means “measuring theearth.” Several pieces of papyrus have been found showing thatover 5,000 years ago the Egyptians measured the area of the fieldsafter the river Nile flooded. This may have been used to calculatethe amount of taxes people had to pay. Many calculations and

measurements were necessary tobuild the pyramids in Egypt.

The Egyptians also knew abstract factsabout area and volume. The so-calledMoscow Papyrus, illustrated here,shows how the volume of a truncatedpyramid was calculated.

This is what the truncated pyramidlooks like. The base is a 4 � 4 squareand the top is a 2 � 2 square.The height is 6. Could you find thevolume of this truncated pyramid?The answer is 58. Show how youfound the answer.

DAngles

4

4

6

22

Math History


Hints and CommentsMaterialscompass cards from Transparency Master or protractors(one per group);string, optional (one piece about 1.5 m long per group);pens, optional (one per group);drawing chalk or a sheet of paper of about 2m2,optional (one per group)

Overview

Students read the text taken from a book on ergo-nomics, which focuses on the angle of rotation of aperson’s arm. They check the statement in the text bymeasuring the rotation angle of their arm.


10. This problem is an informal assessment ofstudents’ ability to estimate angles using appro-priate units; to find (compute, measure, anddraw) angles; and to use geometric models tosolve problems.

b. Students should think of a good way to measurethis angle. It works best to have them carry outthe measuring in groups. You may ask studentsto write a short report about their measuringmethod and their findings, including sketchesthat show what they measured and how.


10. a. Explanations will vary. The text refers torotating the arm in the horizontal plane,moving it extended from the front of the bodyto the back. It can rotate through 180° in frontof the body and another 70° behind it.

b. Student data will vary. A compass card orprotractor can be positioned so that the centeris at the joint of the arm and the torso. Theangle measure can then be read after a studenthas moved his or her arm. Two people areneeded to do this. Each student can lay his orher arm on a table on a large sheet of paper.Another student can draw lines starting at theshoulder for the positions of the arm and thenmeasure the angle.

180�

70�

38 Made to Measure

Notes

After having a student readthe Summary aloud, youmay wish to have them goback through the sectionand find problems thatsupport the conceptstaught. This will encouragestudents to actively use theSummary section as astudy tool.

AnglesD

Assessment Pyramid

1b

1a

Assesses Section D Goals

Parent Involvement

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


Angles

In this section, you investigated the movement for both the right andleft hand. You also explored how the placement of a computer screenand the angles between the seat and the backrest of an easy chair canaffect your comfort.

D

Frank is a small boy sitting behind a computer desk. The horizontal sightline from his eye to the screen ends at the midpoint of the screen.

1. a. Measure the angle between the two sight lines shown in thedrawing.

b. Is the computer screen positioned according to the designrecommendations?


Hints and CommentsOverviewStudents read the section Summary. They use theCheck Your Work problem as self-assessment. Theanswers to these problems is also provided onStudent Book page 45.


1. a. The angle measures about 25°.

b. The angle below the horizontal sight line is halfof 25° or 12.5°. The angle requirements aresatisfied, but the screen is too high!

39 Made to Measure

Notes

Be sure to discuss CheckYour Work with students sothey understand when togive themselves credit foran answer that is differentfrom the one at the back ofthe book.


Reflective questions aremeant to summarize anddiscuss important concepts.

AnglesD

Assessment Pyramid

2b, FFR

2a

Assesses Section D Goals

Extension

Have students complete their list of conversion formulas and then use theirresponses as a peer assessment opportunity where they exchange papers toverify the reasonableness of the formulas.


(“In the best ergonomic computer design, the screen should not belower than the sight line of 15° below the horizontal.”)

2. a. Spread your fingers to find the maximum angle between twoof your fingers. Make a drawing to show your work.

b. Do you think all students in your class found approximatelythe same answer to problem 2a? Why or why not?

Complete the list of personal reference points you started in Section A.Write formulas you can use to convert commonly used measuringunits. Adapt each formula to make it easy to do the conversionsmentally.

Customary to Metric Metric to Customary

miles ➝ kilometers kilometers ➝ milesinches ➝ centimeters centimeters ➝ inchesfeet ➝ centimeters centimeters ➝ feetpounds ➝ kilograms kilograms ➝ pounds


Hints and CommentsOverviewStudents use the Check Your Work problem as self-assessment. The answer to this problem is alsoprovided on page 45 of the Student Book.

Planning

After students complete Section D, you may assign ashomework appropriate activities from the AdditionalPractice section, located on Student Book page 42.


2. a. Your angle probably measures between 30°and 40°.

b. No, probably not. Some students have a widerangle between their fingers than others.Compare your results with those of classmates.


Students complete their list with all personalreference points they found throughout the unit.They compare their lists to those from otherstudents in their class.

Here are formulas for making conversions betweencommon units for each system.

Customary to Metric Metric to Customary

miles ⎯� 1.61⎯⎯→ kilometers kilometers ⎯� 0.62⎯⎯→ miles(multiply by 5 and (multiply by 6 anddivide by 3) divide by 10)

inches ⎯� 2.54⎯⎯→ centimeters centimeters ⎯� 0.39⎯⎯→ inches(multiply by 5 and (multiply by 4 anddivide by 2) divide by 10)

feet ⎯� 30.48⎯⎯→ centimeters centimeters ⎯� 0.033⎯⎯→ feet(multiply by 30) (multiply by 3 and divide

by 100)

pounds ⎯� 0.45⎯⎯→ kilograms kilograms ⎯� 2.2⎯⎯→ pounds(divide by 2) (multiply by 2; and add

1 kg for every 5 pounds)

Additional Practice SolutionsStudent Page

40 Made to Measure Additional Practice Solutions Student Page

1. List some body measures that would be useful to know if youwere designing the following items.

a. telephones

b. children’s beds

c. kitchen cabinets

2. List all of the units of (length) measure that you know thatare related to the human body. Explain the meaning of eachmeasure.

3. Which unit of measurement would you use to measure thefollowing. (Note: Use the metric as well as the customarysystem.)

a. the height of a door

b. the length of a city block

c. the length of a post-it-note

Here is the rule, expressed as an arrow string, to find someone’s footlength in inches if you know his or her U.K. shoe size:

U.K. size ⎯⎯�25⎯→ ______ ⎯⎯�3⎯→ foot length (in inches)

4. a. Matthew wears size 5 in the U.K. system. What length (ininches) is his foot?

b. Sondra measured her foot length, 9.5 in. Which shoe size(U.K. system) do you advise her to choose?

5. Which would you use to express the following distances—lengthor time? Explain.

a. the distance from your home to school

b. the distance from your city to New York City

c. the distance from start to finish of a hiking trail

Additional Practice

Section LengthsA

Section A. Lengths


a. the size of a person’s head, especially fromthe ear to the chin; the size of a person’shand

b. the height of a child; the length of a child’slegs; the width of a child’s body

c. the height of an average adult person; thelength of his or her arms


thumb, hand span, foot, yard, pace, fathom,and cubit

Sample definitions:

Thumb: the width of an average man’s thumb;

Hand span: the width of an outstretched handfrom the tip of the last finger to the tip of thethumb;

Foot: the length of an average man’s foot;

Yard: the length of a man’s reach from his faceto the tip of his finger;

Pace: the distance from the back heel of onefoot to the front toe of the opposite foot afterone step has been taken;

Fathom: the distance from the middle fingertipof one hand to the middle fingertip of the otherhand of a person holding his or her arms fullyextended sideways;

Cubit: the distance from a person’s elbow tothe tip of the extended middle finger.

3. a. feet or meters

b. pace or part of a mile, meters

c. thumb, inches, centimeters

4. a. Matthew’s foot length is 10 inches:

5 ⎯� 25⎯→ 30 ⎯� 3⎯→ 10

b. Accept sizes 3, 3.5, and 4. Make a reversearrow string:

3.5 ←� 25⎯⎯ 28.5 ←� 3⎯⎯ 9.5


a. I would express the distance from home toschool in time units because I know it takesme 10 minutes to walk to school, but I’mnot sure what the actual distance is.

b. I would want to know the distance frommy city to New York City in miles so I couldget an idea about how far away it is frommy city.

c. I would express the distance from start tofinish of a hiking trail using time units,especially if it is a hiking trail in themountains. The actual trail distance may beshort, but if the terrain is rocky or steep, itmight take a long time to hike it.

Additional Practice Solutions

Additional Practice Solutions Made to Measure 40T



12 cm

Area of

the base:

80 cm26 cm

4 cm

15 cm

1. Estimate the surface area of the following objects.

a. a basketball b. a book c. a cereal box

2. For which object in problem 1 was it easiest to find the surfacearea? Why?

3. How many square centimeters are in one square meter?

The units for length in the metric system relate to each other.

kilometers ⎯�10⎯→ hectometers ⎯�10⎯→ decameters ⎯�10⎯→ meters

⎯�10⎯→ decimeters ⎯�10⎯→ centimeters ⎯�10⎯→ millimeters

4. Make a similar arrow string showing the relationship betweenunits for area in the metric system.

5. Enrique lives on a 15-acre farm.

a. About how many football fields would cover his farm?

b. What are possible dimensions for Enrique’s farm?

Section AreasB

Section VolumesC

1. List all of the units of measure for volume that you know andexplain how they relate to each other.

2. Find the volume in cubic centimeters of the following.

a. b.

Section B. Areas

1. Estimates will vary. Sample estimates:

a. 1,800 cm2. Strategies will vary. Samplestrategy:

I wrapped a basketball in newspaper andmade sure that no newspaper overlapped.Then I removed the newspaper andrearranged the pieces into the shape of arectangle. I measured the length and widthof the rectangle with a centimeter ruler andmultiplied to find the area.

b. 580 cm2. Sample strategy:

I measured the length, width, and depth ofthe book:

length: 18 cm; width: 11 cm; depth: 3 cm.

Then I estimated the area of the front of thebook and doubled it:

11 cm � 18 cm is about 10 � 20 � 200 cm2

200 cm2 � 2 � 400 cm2

Then I estimated the area of the book’sspine and doubled it:

3 cm � 18 cm is about 3 � 20 � 60 cm2

60 cm2 � 2 � 120 cm2

Then I estimated the area of the top of thebook and doubled it:

3 cm � 11 cm is about 3 � 10 � 30 cm2

30 cm2 � 2 � 60 cm2

So I estimated the total surface area of thebook to be:

400 cm2 � 120 cm2 � 60 cm2 � 580 cm2

c. 1,656 cm2 (for a cereal box with dimensions6 cm � 18 cm � 30 cm). Students may usethe same strategy as illustrated in part b.

2. Sample response:

It was easy to estimate the surface area of thecereal box and the book since I could measurethe dimensions of each object using a ruler. Icould not estimate the dimensions of the ballsince a ball has a curved surface.

3. 10,000 cm2 are in one m2. Along each side of asquare meter, 100 cm fit. 100 � 100 � 10,000.

4. The metric system has base 10 for lengths.

The metric system has base 100 for areameasurements.

square kilometer ⎯� 100⎯⎯→ square hectometer

⎯� 100⎯⎯→ square decameter ⎯� 100⎯⎯→

square meter ⎯� 100⎯⎯→ square decimeter

⎯� 100⎯⎯→ square centimeter ⎯� 100⎯⎯→

square millimeter

5. a. A little less than 15 football fields.

Sample strategy:

One football field is 45,000 sq. ft.(300 ft � 150 ft � 45,000 sq. ft). One acre isthe same as 43,560 sq. ft. So I can use thearea of a football field as a point of referenceto know how large an acre is.

b. Sample response:

600 feet by 1,090 ft. Fifteen acres is15 � 43,560 � 653,400 sq. ft. I used aguess-and-check strategy with my calculatoruntil I found two numbers that I multipliedtogether to make a product of 654,000, whichis close enough to 653,400.

Section C. Volumes


One liter equals 1 dm3, or 1,000 cm3.

One gallon is about 4 liters, or 4 quarts,or 4 dm3.

One quart equals about 1 liter.

Two pints equal 1 quart.

2. a. 960 cm3. Sample strategy:

I used the volume formula:volume � area of the Base � height:

� 80 cm2 � 12 cm

� 960 cm3

b. 360 cm3. Sample strategy:

I used the volume formula:volume � length � width � height:

� 4 cm � 15 cm � 6 cm

� 360 cm3





Additional Practice

One liter is one cubic decimeter (dm3).

3. a. What are possible dimensions (in inches) of a package with avolume of approximately one liter.

b. Use your answer for a to fill in the sentence.

One liter is approximately _______________ cubic inches.

4. Estimate the volume in liters of:

a. a bathtub

b. a quart of milk

c. a bottle of shampoo

Section AnglesD

1. Draw a side view of a chair so that the angle between the seatand the backrest is 115° and the angle between the seat and thehorizontal is 7°.

2. Below is a side view of a computer screen and a keyboard on adesk.

a. Measure the angle of the keyboard to the desk.

b. Measure the angle of the monitor to the horizontal plane ofthe desk.

3. a. Sample strategy:

I know that 1 dm is about the same lengthas 4 inches (1 inch � 2.54 cm). So 1 dm3 or1 l is the volume of a package of about4 � 4 � 4 � 64 cubic inches.

Sample dimensions: length 4 inches, width2 inches, height 8 inches (4 � 2 � 8 � 64).

b. One liter is approximately 64 cubic inches.

4. Estimates will vary. Sample estimates:

a. about 160 liters

b. about 1 liter

c. about 1__2 liter

Section D. Angles

1.

2. a. The angle is 8° .

b. The angle is 7° .



115�7�

Assessment Overview

Assessment Overview

43A Made to Measure Assessment Overview

Assessment Overview

Unit assessments in Mathematics in Context include two quizzes and aUnit Test. Quiz 1 is to be used anytime after students have completedSection B. Quiz 2 can be used after students have completed Section C.The Unit Test addresses most of the major goals of the unit. You canevaluate student responses to these assessments to determine whateach student knows about the content goals addressed in this unit.

Pacing

Each quiz is designed to take approximately 25 minutes to complete.The Unit Test is designed to be completed during a 45-minute classperiod. For more information on how to use these assessments, seethe Planning Assessment section on the next page.

Goals Assessment Opportunities Problem Levels

• Estimate lengths, areas, volumes, and Quiz 1 Problem 2eangles using appropriate units. Quiz 2 Problem 2b

Test Problems 1abcde

• Find (compute, measure, and draw) lengths, Quiz 1 Problems 1abd Iareas, volumes, and angles using standard Quiz 2 Problems 2a, 3aand nonstandard units. Test Problems 5bc, 6

• Use simple relationships between and within Quiz 1 Problems 2abcdmeasurement systems. Quiz 2 Problems 1abc

• Recognize that different units of Quiz 1 Problems 1cemeasurement and different measurement Test Problem 4systems exist.

• Develop and use a frame of reference for Quiz 1 Problems 3abmeasurement to solve problems. Quiz 2 Problems 1bc II

Test Problems 2ab, 4

• Understand and use the relationships Quiz 2 Problem 3bbetween length, area, volume, and angle. Test Problem 5c

• Use geometric models to solve problems. Test Problem 3

• Develop and use formulas to estimate and Quiz 2 Problem 1dIII

calculate length, area, and volume. Test Problem 5a

Assessment Overview

Assessment Overview Made to Measure 43

Assessment Overview


These assessment activities assess the majority of the goals forMade to Measure. Refer to the Goals and Assessment Opportunitiessection on the previous page for information regarding the goalsthat are assessed in each problem. Some of the problems that involvemultiple skills and processes address more than one unit goal. Toassess students’ ability to engage in non-routine problem solving(a Level III goal in the Assessment Pyramid), some problems assessstudents’ ability to use their skills and conceptual knowledge in newsituations. For example, in the surface area problem on the Unit Test(problem 3), students must reason about a geometric model for thehuman body to solve a new problem.

Planning Assessment

These assessments are designed for individual assessment; however,some problems can be done in pairs or small groups. It is importantthat students work individually if you want to evaluate each student’sunderstanding and abilities.

Make sure you allow enough time for students to complete theproblems. If students need more than one class session to completethe problems, it is suggested that they finish during the next math-ematics class, or you may assign select problems as a take-homeactivity. Students should be free to solve the problems their own way.However, student use of a calculator on these assessments is at theteacher’s discretion.

If individual students have difficulty with any particular problems,you may give the student the option of making a second attemptafter providing him or her a hint. You may also decide to use one ofthe optional problems or Extension activities not previously done inclass as additional assessments for students who need additional help.

Scoring

Solution and scoring guides are included for each quiz and the UnitTest. The method of scoring depends on the types of questions oneach assessment. A holistic scoring approach could also be used toevaluate an entire quiz.

Several problems require students to explain their reasoning or justifytheir answers. For these questions, the reasoning used by students insolving the problems as well as the correctness of the answers shouldbe considered in your scoring and grading scheme.

Student progress toward goals of the unit should be considered whenreviewing student work. Descriptive statements and specific feedbackare often more informative to students than a total score or grade.You might choose to record descriptive statements of select aspectsof student work as evidence of student progress toward specific goalsof the unit that you have identified as essential.

44 Made to Measure Quiz 1 Mathematics in Context

Made to Measure Quiz 1 Page 1 of 2

Name ____________________________________________ Date ______________________

Use additional paper as needed.

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1. a. Use a historical measurement unit related to the humanbody to find the length and width of the sheet of paper thisquiz is printed upon.

Length: ____________

Width: ____________

b. Explain why you chose the measurement unit you did.

c. Explain why it is not likely that everyone in your class gotthe same answer for part a.

d. Use a centimeter ruler to find the length and width of thesheet of paper this quiz is printed upon. Write yourmeasurements in whole centimeters.

Length: ____________

Width: ____________

e. Why should all of the students in this class get the sameanswers for part d?

2. Fill in the blanks in the statements below.

a. 1 m = _____ cm

b. 1 dm2 = _____ cm2

c. 10 cm is about the same size as _____ inches.

d. 1 mile is about the same length as _____ kilometers.

e. The surface area of the sheet of paper this quiz is printed onis _________ than one square decimeter.

(less/more)

Mathematics in Context Made to Measure Quiz 1 45

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Page 2 of 2 Made to Measure Quiz 1

3. In a survey, people were asked how far they commute dailyfrom home to their place of work. Below are some of theresponses.

• Jody said, “I commute about three miles to get to workevery day.”

• Carl said, “It takes me about 15 minutes to walk to my office.”

• Brenda said, “I live a little less than 8 km from my work.”

a. Who do you think lives the closest to his or her place ofwork, Jody, Carl, or Brenda?

b. Give mathematical reasons to support your reasoning.


Made to Measure Quiz 2 Page 1 of 2

Name ____________________________________________ Date ______________________


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1. If you know the height and weight of an adult, you can use this graphto find out if he or she is overweight, underweight, or has a healthyweight. Note: This graph can be used only for adults.

a. Simone is 176 cm tall and has a weight of 95 kilograms (kg). Jacqui isthe same height as Simone but weighs 55 kg. Using the approximation2.2 pounds = 1 kilogram, find Simone’s and Jacqui’s weight in pounds.

Simone: __________ pounds

Jacqui: __________ pounds

1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 1.92 1.96 2.0

1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 1.92 1.96 2.0

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Page 2 of 2 Made to Measure Quiz 2

b. If you were Simone’s and Jacqui’s doctor, what would you recommend to each of them?

c. Carey has a height of 1.65 m and her weight is 57 kg.According to this graph, should she worry about her weight?Explain why or why not.

d. Use the estimation rule, “A person’s body’s surface area isabout 3__

5 of the person’s square,” to estimate Carey’s body surface area. Use an appropriate unit of measurement for your answer.

2. A package contains napkins that are 25 cm � 25 cm.

a. Find the surface area of one napkin. Use an appropriate unit of measurement.

b. The text on the package shows, 97.5 sq in. for the surface area.Estimate the dimensions in inches for one napkin. Show yourwork.

3. Kathleen wants to use a cardboard container to send some booksby mail.

The dimensions of the container are 30 cm � 24 cm � 20 cm.

a. Find the volume of the box in cm3.

b. Kathleen needs to send almanacs by mail. Each almanac hasthe following dimensions.

• length: 21 cm

• width: 14 cm

• thickness: 5 cm

How many of these books will fit in the box? Show your work.

48 Made to Measure Unit Test Mathematics in Context

Made to Measure Unit Test Page 1 of 2

Name ____________________________________________ Date ______________________


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1. Name something that you would measure in:

a. centimeters (cm)

b. kilometers (km)

c. liters (l)

d. square meters (m2)

e. cubic centimeters (cm3)

2. In Europe, the maximum driving speed in a city often is 50 kmper hour.

a. Approximately how many miles per hour are you allowed todrive there?

b. Which estimation rule did you use to find your answer?

3. Ned says he estimated his body surface area to be about 8 m2.Explain to Ned how his estimate could not be correct.

4. Karl’s parents bought about half an acre of land to build a newhouse with a yard. Give two possible dimensions for this pieceof land. You may choose your own units of measurement.(Remember that one acre is about the size of a football field.)

Mathematics in Context Made to Measure Unit Test 49

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Page 2 of 2 Made to Measure Unit Test

5. The formula that you used to find the volume of a box is:

Volume = Area of the Base � Height

a. Explain why this formula cannot be used to find thevolume of a pyramid-shaped box.

b. Use the formula to find the volume of an aquarium withthe dimensions:

length 50 cm, width 40 cm, and height 25 cm.

c. Suppose Arno fills this aquarium with water until the levelof water is 4 cm below the top. Does it contain more orless than 40 liters? Show your work.

(Recall that 1 dm = 10 cm and 1 liter would fill a volume of 1 dm3.)

6. Easy Does It, a chair company, designs easy chairs so that theseat tilts 17 degrees down toward the back of the chair. Thechairs are designed this way so that a person sitting in thechair will not slide off the front.

Make an accurate sketch of the easy chair described using acompass card or protractor.

Made to Measure Quiz 1Solution and Scoring Guide


Possible student answer Problem levelSuggested numberof score points

1. a. Accept any reasonable answer. Students may 3 Iuse the hand span or thumb measurement units to determine the length and width of the sheet of paper.

b. Accept any reasonable answer. Sample student 1 Ianswer: I chose hand span and thumb because the other historical measurement units like foot and yard are too large to measure this sheet of paper.

c. Sample explanations: 1 II

• Students may have used different units.

• Every student has different-sized hands and thumbs.

• It is difficult to measure the dimensions accurately using historic measurement units.

d. The dimensions will probably be close to 2 I21 cm width and 28 cm length.

e. Sample student answers: 1 I

• With a centimeter ruler you can be very accurate.

• Everyone used the same type of ruler, so the size of the ruler and the marks for cm are equal for all students.

• Student answers may differ if answers were given in mm, but all of the answers for length and width should be close to the same size.

2. a. 1 m � 100 cm 5 I

b. 1 dm2 � 100 cm2

c. 4

d. 1.5 or 1.6

e. more

3. a. Carl lives the closest to work. 1 II

b. Sample reasoning: 1 II

• Jody probably travels 3 miles in about 5 minutes.

• Carl lives about 1 km, or less than 1 mile, from work.

• It would take Brenda about an hour and a half to walk to work.

Total score points 15

Made to Measure Quiz 2Solution and Scoring Guide


Possible student answer Problem levelSuggested numberof score points

1. a. Simone weighs about 209 pounds. 2 I

Jacqui weighs about 121 pounds.

b. Students should indicate that to be a healthier 2 I/IIweight, Jacqui needs to gain weight and Simone needs to lose weight. Sample student response:

If I was their doctor, I would tell Jacqui that she needs to gain about 5 kg to be in the healthy category. Simone needs to lose at least 15 kg.

c. Carey does not have to worry about her weight. 1 I/II

According to the graph, her weight is “healthy.”

d. Carey’s body’s surface area is about 1.6 m2. 3 IIISample student work:

1.65 m � 1.65 m � 2.7225 m2

3__5 of 2.7225 � 1.6335

The answer needs to be rounded off since an estimation rule was used.

2. a. 25 cm � 25 cm � 625 cm2 2 I

b. Accept answers for length and width of a little 2 Iless than 10 inches.

Sample student work:

97.5 is a little less than 100. If it was 100 exactly, the dimensions would be 10 in. � 10 in.

Note: The exact dimensions are 9 7__8 in. � 9 7__8 in.

3. a. The volume is 30 cm � 24 cm � 20 cm � 2 I14,400 cm3.

b. 8 books would fit in the box. 2 II

Sample student work: I assumed the base of the box measured 24 � 30 cm. Two books will fit, placed next to each other. Since the height is 20 cm, four layers of two books each can be used.

Note: If students divide the volume of the box by the volume of one book, the answer they find is 9 books. But a book cannot be cut to fill empty space.


(1 point for a correct estimation, 1 point for a correct

unit of measurement, 1 point for a

correct computation.)

(1 point for a correct answer,

1 point for a correct unit of measurement.)

(1 point for an acceptable answer,

1 point for a correct explanation.)

(1 point for a correct answer,

1 point for a correct explanation.)

Possible student answer

Made to Measure Unit TestSolution and Scoring Guide

52 Made to Measure Unit Test Mathematics in Context

Problem levelSuggested numberof score points

1. Different answers are possible. Accept any 5 Ireasonable answer.

Sample answers:

a. The height or width of a photo.

b. The distance from home to school.

c. The volume of a water bottle.

d. An amount of carpet.

e. The volume of a soup bowl.

2. a. Accept answers between 30 and 35 miles 1 IIper hour.

b. Sample estimation rule: One mile is about 1 II1.5 kilometers (km).

3. Sample explanation: 2 III

Even if Ned is very tall, his height is probably not over 2 meters. Suppose his shoulder width is 0.5 m, which would be a huge shoulder! The surface area ofhis sides would be less than 0.5 m � 2 m � 2 � 2m2.If Ned is 1 m wide, then his front and back together would be 4 m2. Unless Ned is over 2.5 m tall with a wrestler-type body, there is no way the rest of his body surface area would add up to 8 m2!

4. Using the football field estimate, student responses 2 IIshould equal about 2,500 square yards. Sample answer:

A football field is about one acre. The dimensions of a football field are about 100 yards � 50 yards. Half an acre could be 25 yards � 100 yards, or 50 yards � 50 yards.

Made to Measure Unit TestSolution and Scoring Guide

Mathematics in Context Made to Measure Unit Test 53

Possible student answer Suggested numberof score point

Problem level

5. a. Sample explanation:The area of the base will tell you how many cubes 2 IIIare needed for the bottom layer. However, the formula only works if all of the layers are equal in size. Since the layers of a pyramid get smaller andsmaller as you go from the base to the peak, this formula will not work.

b. volume � 50 cm � 40 cm � 25 cm � 50,000 cm3 2 I

c. There are more than 40 liters in the aquarium. 3 I/II

Sample student work:

50 cm � 40 cm � 21 cm � 42,000 cm3

1 dm3 � 1,000 cm3, so 42,000 cm3 � 42 dm3

42 dm3 is equal to 42 liters

or

1 dm3 � 1,000 cm3, so 50,000 cm3 � 50 dm3;50 dm3 is equal to 50 liters.The maximum amount of water that can be contained in the aquarium is 50 liters. The volume that is left at the top of the aquarium is50 cm � 40 cm � 4 cm � 8,000 cm3, or 8 liters. So Arno filled the aquarium with 42 liters.

6. Accept a correct drawing with an angle of 16°, 17°, 2 Ior 18° between the “seat” and the “horizontal.”


17˚

(1 point for a correct number,

1 point for correct units.)

(Award no points if students only answered“more,” without a correct

explanation.)

Glossary

54 Made to Measure Glossary

GlossaryThe Glossary defines all vocabulary words indicated inthis unit. It includes the mathematical terms thatmay be new to students, as well as words having todo with the contexts introduced in the unit. (Note:The Student Book has no Glossary. Instead, studentsare encouraged to construct their own definitions,based on their personal experiences with the unitactivities.)

The definitions below are specific for the use of theterms in this unit. The page numbers given are fromthe Student Books.

acre (p. 19) a unit of area used in the U.S. andEngland; equal to 160 square rods, 4,840 squareyards, or 4,047 square meters. One acre is about thesize of a football field.

cord (p. 29) a unit of wood cut for fuel equal to astack 4 feet × 4 feet × 8 feet, or 128 cubic feet

cubit (p. 28) an ancient unit of measure for length,equal to the distance from a person’s elbow to thetip of the extended middle finger

customary system (p.4) The measurementsystem used in the United States, in which length ismeasured in inches, feet, yards, and miles. It is alsocalled the Imperial system.

ergonomics (p. 32) the study of people and theirwork environments. One goal of ergonomics is todesign equipment and machines for safe andefficient use.

face cord (p. 29) a stack of firewood 4 feet × 4 feet× 16 inches, or 42 2__

3 cubic feet. A full cord is a stackof firewood 4 feet × 4 feet × 8 feet, or 128 cubic feet.

fathom (p. 8) the distance from the middle fingertip of one hand to the middle finger tip of the otherhand of a person holding his or her arms fullyextended sideways. The fathom is a historic unit oflength, now standardized to 6 feet, used especiallyfor measuring the depth of water.

foot (p. 4) the distance from the heel to the toe.This historic measurement is now standardized to12 inches.

hand span (p. 2) the distance from the tip of thethumb to the tip of the little finger when the fingersare spread out as far as possible

metric system (p. 4) the measurement systemused in most countries, based on powers of ten.Basic units are: length—the meter; mass—thegram; liquid volume—the liter.

mile (p. 6) the Roman mile was 1,478.7 meterslong. This historic measurement is nowstandardized to be 5,280 feet.

net (p. 22) a flat (two-dimensional) pattern to becut and folded into a three-dimensional shape

nomogram (p. 18) a graphic representation thatconsists of several lines marked to scale, likenumber lines. The lines are arranged in such a waythat by using a straightedge to connect knownvalues on tow lines, an unknown value on the thirdline can be read at the point of intersection.

rood (p. 19) a historic British unit of land areaequal to 1__

4 acre.

royal cubit (p. 29) an ancient Egyptian standardlength, which measures 52.4 cm, or 20.62 in.

Scottish thumb (p. 3) the mean distanceacross the thumbs of three men—a large man, anordinary-size man, and a small man. The Scottishthumb, or standard thumb, is now called an inch.

solid (p. 22) a geometric figure that has threedimensions, such as a cube, pyramid, or sphere

stitch (p. 6) one-third of a standard thumb

surface area (p. 12) the area of all of thesurface or material that covers an object

yard (p. 2) the distance from the nose to thefingertips when the arm is held out to the side.This historic unit of measurement is nowstandardized to 3 feet.

BlacklineMasters

BlacklineMasters

56 Made to Measure Letter to the Family

Letter to the Family

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Dear Family,

Your child is about to begin working on the Mathematics in Context unitMade to Measure. This unit is all about the measurement of lengths,areas, volumes, and angles.

In the unit, students measure the lengths of their feet, hands, and thumbsand look for relationships between them. They measure the area of afootball field and estimate the surface areas ofhands and legs. They measure volumes ofbottles, boxes, and stacks of firewood. Theyalso measure the angles made by arms andwrists.

To help your child, get him or her involved inmeasuring around the house. Ask your childto measure ingredients in recipes, fabric forsewing, the length of a hall for carpet, or thedimensions of the sheets needed for a bed.Compare your child’s height with theheights of other members of the family.

Discuss situations in which you mightestimate a measurement. Have your child estimate, for example, how manypennies are in a jar, how long it will take something to cook, or how long a log will burn in the fireplace.

By showing children how measurementis used daily, they will better appreciatewhat they are studying in school.

Sincerely,

The Mathematics inContext DevelopmentTeam

Made to Measure

Dear Student,

Welcome to the Mathematics in Context unit Made to Measure. This

unit is all about measuring: measuring your feet, your thumb, your

hands, and the angle made by your arm and your wrist.

You will investigate how measuring units evolved.

You will further investigate measurements for length,

area, and volume. You might be amazed by what you can measure!You will find that mathematics plays an

important role in measurement. Every time you measure something, you might ask yourself:• Will every person measuring this item get the same measurement

that I did?• Do all of these things have the same measurement?• What other units of measure can I use?• Are there other ways to measure these things?Whenever you make a measurement in this unit,

picture how big—or small or steep or short—that measurement is. When you can do this with all of the measurements in this unit, you are well on your way to becoming a mathematician!

Sincerely,

The Mathematics in Context Development Team

Student Activity Sheet 1 Made to Measure 57

Student Activity Sheet 1Use with Made to Measure,

pages 35 and 36.

Name ________________________________________

©EncyclopædiaBritannica,Inc.Thispagemaybereproducedforclassroomuse.

Ellen’s desk. The scale of the drawing is 1:10.

Name ________________________________________

58 Made to Measure Teaching Transparency

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Teaching Transparency

Made to Measure Section A: Lengths; page 33

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Section A: Lengths 59

A

9. If Dealing with Data was taught prior to this unit,you might discuss the importance of using arepresentative sample of 16 students. Aftercompleting this problem, students can investigatewhether one student’s foot is the same length asthe mean foot of the class.

Hints and Comments(continued from page 3T)


8. a. Notice that the Germans chose the length ofthe feet of 16 men to determine the mean foot.Since the number 16 is a power of two, a ropemade up of lengths that are a power of twocould be easily folded.

b. This question is intended to be open-ended toencourage students to discuss its implications.Some students may say that the German meanfoot is closer to the average person’s measure-ments since it is based on the foot lengths of16 men. Other students might prefer using theScottish thumb since it is easier to find.Students may also comment on the variabilityof the two nonstandard measures. It is likelythat there will be greater differences in thelengths of feet than in the widths of thumbs.

A

4. a. Which of the units of measure fromproblem 3 would you use to find thelength of the nail shown here?

b. How long is this nail in the units of measure you chose foryour answer to a ?

5. a. Measure the length of your desk by using one or more of theunits of measure from problem 3.

b. List the class results in a table. Did everyone find the samelength? Why do you think this happened?

6. Reflect Name one advantage and one disadvantage of usingyour body to make measurements.

Lengths

In Scotland during the Middle Ages, a unit of measurecalled the Scottish thumb was used. A Scottish thumbis the mean of the thumb widths of three men: a largeman, an average-sized man, and a small man.

7. Why were three different-sized men used todetermine the Scottish thumb?

In 1616, the Germans decided to create a unit of measure called themean foot. To do this, they cut a piece of rope that was as long as thefeet of 16 men.

8. a. How do you think the rope was used to find the length of themean foot?

b. Which measurement is closer to the average person’smeasurement: the German mean foot or the Scottishthumb? Explain your answer.

9. With the help of 16 classmates, find the length of the mean foot inyour class by using the method described above.

Lengths

Documents

Made to Measure - Universiteit Utrecht · 2016-04-13 · Blackline Masters Letter to the Family 56 Student Activity Sheets 57 Teaching Transparency 58 C D. Overview Made to Measureand