OverviewOverviewUnit 8 revolves around perimeter, area, and scale drawings. It beginswith a review of perimeter and area concepts previously introduced inEveryday Mathematics, and extends knowledge by developing formulasas mathematical models for the areas of rectangles, parallelograms, andtriangles. Unit 8 also explores applications of area with the help of scaledrawings. Unit 8 has three main areas of focus:

◆ To review perimeter and area concepts,

◆ To develop formulas as mathematical models for the areas ofrectangles, parallelograms, and triangles, and

◆ To explore applications of area with scale drawings.

642 Unit 8 Perimeter and Area

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Lesson Objective Page

Contents

8◆1 Kitchen Layouts and Perimeter 658To provide experience measuring and adding distances; finding the median and other landmarks of a set of measurements; and finding the perimeters of triangles.

8◆2 Scale Drawings 664To provide practice measuring distance to the nearest foot; and to provide experience creating a scale drawing on a grid using measurements and a given scale.

8◆3 Area 670To review basic area concepts; to provide practice estimating the area of a polygon by counting unit squares and using a scale drawing to find area.

8◆4 What Is the Area of My Skin? 675To demonstrate how to estimate the area of a surface having a curved boundary; and to provide practice converting from one square unit to another.

8◆5 Formula for the Area of a Rectangle 681To guide the development and use of a formula for the area of a rectangle.

8◆6 Formula for the Area of a Parallelogram 687To review the properties of parallelograms; and to guide the development and use of a formula for the area of a parallelogram.

8◆7 Formula for the Area of a Triangle 693To guide the development and use of a formula for the area of a triangle.

8◆8 Geographical Area Measurements 699To discuss how geographical areas are measured; and to provide practice using division to compare two quantities with like units.

8◆9 Progress Check 8 704To assess students’ progress on mathematical content through the end of Unit 8.

644 Unit 8 Perimeter and Area

To provide experience measuring andadding distances; finding the medianand other landmarks of a set ofmeasurements; and finding theperimeters of triangles.

To provide practice measuring distanceto the nearest foot; and to provideexperience creating a scale drawing on a grid using measurements and agiven scale.

To review basic area concepts; toprovide practice estimating the area of apolygon by counting unit squares andusing a scale drawing to find area.

To demonstrate how to estimate the areaof a surface having a curved boundary;and to provide practice converting fromone square unit to another.

To guide the development and use of aformula for the area of a rectangle.

To review the properties ofparallelograms; and to guide thedevelopment and use of a formula forthe area of a parallelogram.

To guide the development and use of aformula for the area of a triangle.

To discuss how geographical areas aremeasured; and to provide practice usingdivision to compare two quantities withlike units.

Lesson Objectives Links to the Past Links to the Future

Learning In Perspective

Grades 1 and 2: Define and calculate minimum,maximum, middle value (median), and mode by listing or tallying data. Grade 3: Define andcalculate mean (average) and range. Use lists, tally charts, line plots, and graphs to determine landmarks.

Grade 4: Use a map scale to estimate distances.Grade 3: Use a map scale to estimate the directdistance between two places.

Grade 2: Tile surfaces. Estimate areas usingcentimeter and inch grids; find areas of geoboardrectangles. Grade 3: Estimate areas of surfaces andmeasure with 1-foot or 1-yard squares. Find areasof rectangles by counting squares.

Grade 3: Refer to tables of equivalent units of measure.

Grade 2: Explore area and area units. Grade 3: Tilerectangles with pattern blocks; review the meaningof area and square units using models of squarefeet and square yards; discuss how to find the area of a room to be carpeted; use a geoboard to actout calculating areas of rectangles.

Grade 2: Explore area and area units. Grade 3: Tilerectangles with pattern blocks; review the meaningof area and square units using models of squarefeet and square yards; discuss how to find the area of a room to be carpeted; use a geoboard to actout calculating areas of rectangles.

Grade 2: Explore area and area units. Grade 3: Tilerectangles with pattern blocks; review the meaningof area and square units using models of squarefeet and square yards; discuss how to find the area of a room to be carpeted; use a geoboard to actout calculating areas of rectangles.

Grade 3: Discuss tools used to measure distances;explore division as equal sharing. Solve divisionproblems by direct modeling, arrays, and othermethods.

Grades 5 and 6: Applications and maintenance.

Grade 5: Use a map scale to estimate actualdistances. Grade 6: Use a scale to calculate actual size from a scale drawing.

Grade 5: Find areas of rectangles on grids. Use a rectangle method for finding areas of triangles and parallelograms.

Grades 5 and 6: Applications and maintenance.

Grade 4: Use formulas to find volumes (Units 11and 12). Grade 5: Develop and apply formulasfor areas of triangles and parallelograms. Identify personal references for metric andcustomary units of area. Grade 6: Use formulasto find perimeter, circumference, and area.

Grade 4: Use formulas to find volumes (Units 11and 12). Grade 5: Develop and apply formulasfor areas of triangles and parallelograms. Identify personal references for metric andcustomary units of area. Grade 6: Use formulasto find perimeter, circumference, and area.

Grade 4: Use formulas to find volumes (Units 11and 12). Grade 5: Develop and apply formulasfor areas of triangles and parallelograms. Identify personal references for metric andcustomary units of area. Grade 6: Use formulasto find perimeter, circumference, and area.

Grades 5 and 6: Applications and maintenance.

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Create a tally chart. Data and Chance Goal 1Find the minimum, maximum, mode, and median of a data set; use landmarks to draw conclusions. Data and Chance Goal 2Measure distances in feet and inches. Measurement and Reference Frames Goal 1Calculate the perimeter of a triangle. Measurement and Reference Frames Goal 2Add mixed units; convert between feet and inches. Measurement and Reference Frames Goal 3

Find the median of a data set. Data and Chance Goal 2Make a rough floor plan of the classroom. Measurement and Reference Frames Goal 1

and Operations and Computation Goal 7Make a scale drawing of the classroom. Measurement and Reference Frames Goal 1

and Operations and Computation Goal 7Measure to the nearest foot. Measurement and Reference Frames Goal 1

Use a scale drawing to estimate the area of the classroom. Operations and Computation Goal 7 and Measurement and Reference Frames Goal 2

Find the areas of polygons by counting squares and partial squares. Measurement and Reference Frames Goal 2Identify polygons. Geometry Goal 2

Use the terms estimate and guess. Operations and Computation Goal 6Use an estimate to judge the reasonableness of a solution. Operations and Computation Goal 6Count squares and partial squares or use a formula to estimate area. Measurement and Reference Frames Goal 2Convert between square inches and square feet. Measurement and Reference Frames Goal 3

Rename fractions as decimals. Number and Numeration Goal 5Count unit squares or use a formula to find the area of a rectangle. Measurement and Reference Frames Goal 2Use patterns in a table to develop a formula for the area of a rectangle. Patterns, Functions, and Algebra Goal 1Apply the Distributive Property of Multiplication over Addition. Patterns, Functions, and Algebra Goal 4

Find the area of a rectangle. Measurement and Reference Frames Goal 2Develop a formula for calculating the area of a parallelogram. Measurement and Reference Frames Goal 2Calculate perimeter. Measurement and Reference Frames Goal 2Identify perpendicular line segments and right angles. Geometry Goal 1Describe properties of parallelograms. Geometry Goal 2

Find the areas of rectangles and parallelograms. Measurement and Reference Frames Goal 2Develop a formula for calculating the area of a triangle. Measurement and Reference Frames Goal 2Identify perpendicular line segments and right angles. Geometry Goal 1Describe properties of and types of triangles. Geometry Goal 2Evaluate numeric expressions containing parentheses. Patterns, Functions, and Algebra Goal 3

Use division to compare two quantities with like units. Operations and Computation Goal 4Use “times as many” language to compare area measurements. Operations and Computation Goal 4Estimate and compare area measurements. Measurement and Reference Frames Goal 2

Key Concepts and Skills Grade 4 Goals*

* For a detailed listing of all Grade 4 Goals, see the Appendix.

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Key Concepts and Skills

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Ongoing Learning and Practice

Math BoxesMath Boxes are paired across lessons as shown in the brackets below.This makes them useful as assessment tools. Math Boxes also previewcontent of the next unit.

Ongoing Learning and Practice

8◆1 Fraction Match Naming equivalent fractionsNumber and Numeration Goal 5

8◆3 Fraction Top-It Comparing and ordering fractionsNumber and Numeration Goal 6

8◆6 Fraction Of Finding fractions of collectionsNumber and Numeration Goal 2

8◆7 Rugs and Fences Calculating the area and perimeter of a polygonMeasurement and Reference Frames Goal 2

8◆8 Grab Bag Calculating the probabilities of eventsData and Chance Goal 4

Lesson Game Skill Practiced

See the Differentiation Handbook for ways to adapt games to meet students’ needs.

646 Unit 8 Perimeter and Area

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Home Communication Study Links provide homework and home communication.

Home Connection Handbook provides more ideas to communicateeffectively with parents.

Unit 8 Family Letter provides families with an overview, Do-AnytimeActivities, Building Skills Through Games, and a list of vocabulary.

Practice through Games Games are an essential component of practice in the Everyday Mathematicsprogram. Games offer skills practice and promote strategic thinking.

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Mixed practice [8◆1, 8◆3], [8◆2, 8◆4], [8◆5, 8◆7], [8◆6, 8◆8]

Mixed practice with multiple choice 8◆1, 8◆2, 8◆4, 8◆7

Mixed practice with writing/reasoning opportunity 8◆1, 8◆3, 8◆4, 8◆7, 8◆8

Encourage students to use a variety of strategies to solve problems and toexplain those strategies. Strategies that students might use in this unit:

◆ Drawing and using a picture ◆ Using computation◆ Using estimation ◆ Using a formula◆ Making a model ◆ Making a graph

Lesson Activity

See Chapter 18 in the Teacher’s Reference Manual for more information about problem solving.

8◆1 Analyze and rate the efficiency of a kitchen.

8◆2 Make a classroom floor plan.

8◆3 Estimate areas of polygons by counting squares.

8◆4 Estimate the area of your skin.

8◆5–8◆7 Develop formulas for area.

8◆8 Compare country areas.

MOST CLASSROOMS

F E B R U A R Y M A R C H A P R I L

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Problem SolvingProblem Solving

Content Standards: 1 Number and Operations, 2 Algebra, 3 Geometry, 4 Measurement, 5 Data Analysis and ProbabilityProcess Standards: 6 Problem Solving, 7 Reasoning and Proof, 8 Communication, 9 Connections, 10 Representation

Planning Tips

Lessons thatteach throughproblem solving,not just aboutproblem solving

PacingPacing depends on a number of factors, such as students’ individual needsand how long your school has been using Everyday Mathematics. At thebeginning of Unit 8, review your Content by Strand Poster to help you seta monthly pace.

NCTM StandardsUnit 8Lessons

NCTMStandards

8 ◆1 8 ◆2 8 ◆3 8 ◆4 8 ◆5 8 ◆6 8 ◆7 8 ◆8 8 ◆9

3–5,6–10

3, 4,6–10

3–5,6–10

4, 6–10

2–4,6–10

2–4,6–10

2–4,6–10

1, 4,6–10 6–10

648 Unit 8 Perimeter and Area

Balanced Assessment

8◆1 Rename fractions with denominators of 10 and 100 as decimals and percents. [Number and Numeration Goal 5]

8◆2 Use data to create a line graph.[Data and Chance Goal 1]

8◆3 Count whole squares and half squares to find the area of a polygon. [Measurement and Reference Frames Goal 2]

8◆4 Count unit squares and fractions of unit squares to estimate the area of an irregular figure.[Measurement and Reference Frames Goal 2]

8◆5 Calculate the perimeter of a figure when the length of one side is missing.[Measurement and Reference Frames Goal 2]

8◆6 Solve fraction addition and subtraction problems. [Operations and Computation Goal 5]

8◆7 Describe a strategy for finding and comparing the areas of a square and a polygon. [Measurement and Reference Frames Goal 2]

8◆8 Calculate and express the probability of an event as a fraction.[Data and Chance Goal 4]

Lesson Content Assessed

Use the Assessment

Management System

to collect and analyze dataabout students’ progressthroughout the year.

Ongoing Assessment

Recognizing Student AchievementOpportunities to assess students’ progress toward Grade 4 Goals:

Informing InstructionTo anticipate common student errors and to highlight problem-solving strategies:

Lesson 8◆2 Use equal floor lengths in making a rough classroom floor plan

Lesson 8◆3 Use strategies to find area

Lesson 8◆5 Use different multiplication strategies for finding the area of a triangle

Lesson 8◆6 Understand the difference between parallelogram and rectangle perimeter measurement

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Rename tenths and hundredths as decimals.[Number and Numeration Goal 5]

Order fractions. [Number and Numeration Goal 6]

Use manipulatives, mental arithmetic, and calculators toadd and subtract fractions.[Operations and Computation Goal 5]

Use scaling to model multiplication and division.[Operations and Computation Goal 7]

Predict the outcomes of experiments; express theprobability of an event as a fraction.[Data and Chance Goal 4]

Measure length to the nearest centimeter.[Measurement and Reference Frames Goal 1]

Describe and use strategies to measure the perimetersof polygons.[Measurement and Reference Frames Goal 2]

Describe and use strategies to find the areas ofpolygons. [Measurement and Reference Frames Goal 2]

Describe and compare plane figures using appropriategeometric terms. [Geometry Goal 2]

CONTENT ASSESSED Self Oral/Slate Written Open Response

ASSESSMENT ITEMS

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Periodic Assessment8◆9 Progress Check 8

Portfolio OpportunitiesOpportunities to gather samples of students’ mathematical writings, drawings, and creations to add balance to the assessment process:

◆ Determining expected results, Lesson 8◆1◆ Investigating pattern-block perimeters, Lesson 8◆1◆ Making a scale drawing of a bedroom, Lesson 8◆2◆ Naming the greater fraction, Lesson 8◆3◆ Using a geoboard to find polygons, Lesson 8◆3◆ Constructing parallelograms and perpendicular line segments, Lesson 8◆6◆ Writing probability questions, Lesson 8◆7◆ Finding the area and perimeter of a nonregular hexagon, Lesson 8◆7◆ Comparing the areas of shapes, Lesson 8◆9

Assessment HandbookUnit 8 Assessment Support

◆ Grade 4 Goals, pp. 37–50 ◆ Unit 8 Open Response◆ Unit 8 Assessment Overview, pp. 110–117 • Detailed rubric, p. 114

• Sample student responses, pp. 115–117Unit 8 Assessment Masters

◆ Unit 8 Self Assessment, p. 189◆ Unit 8 Written Assessment, pp. 190–192◆ Unit 8 Open Response, pp. 193 and 194◆ Unit 8 Class Checklist, pp. 276, 277, and 303◆ Unit 8 Individual Profile of Progress, pp. 274,

275, and 302

◆ Exit Slip, p. 311◆ Math Logs, pp. 306–308◆ Other Student Assessment Forms, pp. 304,

305, 309, and 310

Daily Lesson Support

Differentiated Instruction

ENGLISH LANGUAGE LEARNERS

READINESS ENRICHMENT

EXTRA PRACTICE8◆2 Listing everyday and mathematical

meanings of the word scale8◆3 Building a Math Word Bank8◆5 Building a Math Word Bank

8◆1 Constructing rectangles and squares of a given perimeter on a geoboard

8◆2 Using a “foot-long foot” to measureobjects

8◆3 Identifying squares on a geoboard8◆4 Determining the area of an irregular

region8◆5 Building a rectangle and finding its

area8◆8 Exploring area comparison using a

concrete model

8◆1 Using pattern blocks to make polygons8◆2 Making scale drawings8◆3 Using geoboards to find polygons 8◆5 Finding the areas of rectangles 8◆5 Finding rectangles with various areas

that all have the same perimeter 8◆6 Constructing figures 8◆6 Combining two-dimensional shapes 8◆7 Making area comparisons 8◆7 Finding the area of a nonregular hexagon8◆8 Determining gravitational pull 8◆8 Exploring similar figures

8◆3 Calculating areas of polygons 8◆4 Estimating areas of irregular regions 8◆7 Playing Rugs and Fences

5-Minute Math 8◆1 Solving perimeterproblems

Adjusting the Activity8◆1 Thinking in terms of the partial-sums

algorithm; using a yardstick to renamemeasurements

8◆3 Estimating the area of the classroomfloor in square yards

8◆4 Modeling the conversion from squareinches to square feet ELL

8◆4 Determining latitude and longitude8◆5 Labeling the base and height of each

rectangle ELL8◆7 Drawing the height of a triangle with

an obtuse base angle ELL8◆8 Using the constant feature on

calculators to repeat divisionA U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

650 Unit 8 Perimeter and Area

Cross-Curricular LinkIndustrial Arts LinkLesson 8◆1 Students learn about kitchen work triangles.

ScienceLesson 8◆8 Students determine the gravitational pull of each planet relative to Earth.

Differentiation HandbookSee the Differentiation Handbook for materials on Unit 8.

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Social StudiesLesson 8◆4 Students visit a second country inSouth America on the World Tour.

Lesson 8◆8 Students compare areas of countriesin South America to Brazil’s area.

Unit 8 Vocabularyareabaseequilateral triangleformulaheightisosceles trianglelengthperimeterperpendicularrough floor planscalescale drawingsquare unitstime-and-motion studyvariablewidthwork triangle

Grade 3

Grade 4

Grade 5

3◆4 3◆6,3◆7

3◆6–3◆8

3◆7

8◆1 8◆2 8◆3 8◆4 8◆5 8◆6 8◆7 8◆8

9◆4–9◆6

9◆5 9◆4 9◆6 9◆6 9◆7

Professional Development

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Teacher’s Reference Manual LinksSection

14.1.1

14.3

15.4.2

14.4.1

14.4.1

Topic

Measurement and Estimation

Length

Map and Model Scales

Discrete and Continuous Modelsof Area

Discrete and Continuous Modelsof Area

Lesson

8◆1

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8◆4

14.4.2

13.4.2

14.4.2

13.4.2

14.4.2

13.4.2

14.10.2

Section

Area Formulas

Polygons

Area Formulas

Polygons

Area Formulas

Polygons

Area in Geography

Topic

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Lesson

Language SupportEveryday Mathematics provides lesson-specific suggestions to help allstudents, including non-native English speakers, to acquire, process, andexpress mathematical ideas.

Connecting Math and LiteracyActual Size, by Steve Jenkins, Houghton Mifflin, 2004Castle, by David Macaulay, Houghton Mifflin, 1982

Student Reference Bookpp. 114, 117, 118, 133, 243–245, 247, 249, 260, 261, 286, 287,

and 295

Multiage Classroom ◆ Companion LessonsCompanion Lessons from Grades 3 and 5 can help you meet instructionalneeds of a multiage classroom. The full Scope and Sequence can be foundin the Appendix.

Materials

Lesson Masters Manipulative Kit Items Other Items

* Denotes optional materials

Technology Assessment Management System, Unit 8iTLG, Unit 8

Study Link 7◆11 geoboard and rubber bands scissors; tape; straightedge; yardstick*;Study Link Master, p. 247 Geometry Template ruler; Fraction Match CardsTeaching Masters, pp. 248 and 249 pattern blocksTeaching Aid Master, p. 437 slate

Study Link 8◆1 per group: 1 tape measure straightedge; tape; ruler; scissorsTeaching Aid Masters, pp. 388 or slate

389, 413, 414, 416, and 443*transparency of Math Masters,

p. 443*Study Link Master, p. 250Teaching Masters, pp. 251–253

Study Link 8◆2 Geometry Template unit squares; scissors; tape; Study Link Master, p. 254 geoboard and rubber bands straightedge; 1 deck of 32 Fraction CardsGame Master, p. 506 slateTeaching Aid Master, p. 437

Study Link 8◆3 calculator scissors; tape; objects with flat irregular Teaching Aid Masters, pp. 385, 388 surfaces; masking tape; square

or 389, 419*, 420*, and 444 stick-on notesStudy Link Master, p. 255

Study Link 8◆4 1 six-sided die 24-inch string loop; tapeStudy Link Master, p. 256 36 square pattern blocksTeaching Masters, pp. 257–259 calculatorTeaching Aid Master, p. 444 slate

Study Link 8◆5 straws and twist ties scissors; tape; index card or other Teaching Masters, pp. 260, 263 compass square-corner object; counters*;

and 264 centimeter ruler straightedge; Fraction Of CardsStudy Link Masters, pp. 261 and 262Game Master, p. 479Teaching Aid Master, p. 437

Study Link 8◆6 slate scissors; tape; index card or otherTeaching Masters, pp. 265, 267 centimeter ruler square-corner object

and 268transparency of Math Masters,

p. 403*Study Link Master, p. 266Game Masters, pp. 498–502

Study Link 8◆7 2 six-sided dice world map or globe; Grab Bag Cards; Study Link Master, p. 269 calculator scissorsGame Master, p. 485 slateTeaching Masters, pp. 270–273 centimeter ruler

Study Link 8◆8 slate scissorsAssessment Masters, pp. 189–194 centimeter rulerStudy Link Masters, pp. 274–277

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652 Unit 8 Perimeter and Area

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The discussion below highlights the major content ideas presented in Unit 8 and helps establish instructional priorities.

Perimeter and Area (Lessons 8◆1 and following)Unit 8 begins with a review of perimeter (Lesson 8-1). Students use themeasurements they made for Study Link 7◆11 (Math Masters, pages 235and 236) to evaluate the arrangement of appliances in their home kitchens.Then, in Lesson 8-2, they make scale drawings of their classroom. Thisactivity serves several purposes:

◆ It provides practice in measuring lengths and in using a scale to makea scale drawing. Students use this skill again to make scale drawingsof their bedrooms and their bedroom furniture in an optional activityin Lesson 8-2. They use their scale drawings to evaluate variousarrangements of their bedroom furniture.

◆ Students use their scale drawings of the classroom floor to find thearea of the floor in Lesson 8-3.

Length and perimeter (or circumference) are measures of distance along alinear path. Area is a measure of a finite amount of surface. This surfacemay be “flat” (for example, the interior of a rectangle), or it may be “curved”(for example, the surface of a cylinder or cone).

It is important to note that, like other measures, area always includes both a number and a unit. Units of area are typically square units based on linearunits, such as a square yard, a square meter, and a square mile. Note thatthere are several units of area in which the word square does not appear; forexample, an acre of land (now �6

140� of a square mile) is said to have been

based, a long time ago, on the amount of land a farmer could plow in oneday. In the metric system, the hectare is used to measure land areas.

Mathematical Background

In most schoolbooks, the definition of area is based on the idea of “tiling,” orcovering a surface with identical unit squares, without gaps or overlaps, andthen counting those units.

If the surface is bounded by a rectangle, it is natural to arrange the tiles inan array, and to multiply the number of tiles per row by the number of rows.The usual formulas, A � l � w or A � b � h, are then easily linked to arraymultiplication: area is equal to the number of square unit tiles in one row(equal to the length of the base in some linear unit) times the number ofrows (equal to the width, or height, in that same linear unit). For othersurfaces, defined by regular or irregular boundaries, tiling with square unitscan be thought of as (or actually done by) laying a grid of appropriate squareunits on the region and counting, estimating, or calculating how manysquares it takes to cover that region.

Another, more dynamic conception of area has proven to be useful in some applications and in more advancedmathematics courses. Imagine running a one-foot-wide paintroller on the floor of a rectangular room along one wall. Forevery foot the roller travels, 1 square foot of the floor ispainted. Now suppose the room is 20 feet wide and the rolleris the width of the room (a 20-foot wide roller). Then, forevery foot the roller travels along the length of the floor, 20 square feet of floor are painted. When the roller reachesthe other side of the floor, the entire floor will have beenpainted. If you think of the floor as the interior of a rectangle,then the area of the rectangle is obtained not by countingsquares (a discrete conception), but by sweeping the width ofthe rectangle across the interior of the rectangle, parallel toits base (a continuous conception). The area is simply theproduct of the length of the base and the width of therectangle. In the classroom, this can be shown by rubbing the long part of a piece of chalk on the board to mark arectangular surface—the farther you sweep it along, thebigger the rectangle and the greater the area.

654 Unit 8 Perimeter and Area

Using tiling to demonstrate area.(discrete model)

Use painting to demonstrate area.(continuous model)

For most purposes, you will probably choose the traditional conception ofarea—counting or computing the number of square units required to cover a surface. However, the authors recommend that you also introduce thecontinuous conception, for it is easily extended to conceptions of volume. For example, students can think of the volume of a prism in terms of a prism that is gradually filled with water: The surface of the water is shapedlike the base of the prism; the higher the level of the water, the more space it occupies and the greater the volume. This leads to the formula for thevolume of a prism and a cylinder as the area of the base multiplied by theheight. This formula for volume works no matter what the shape of the base.

Students often confuse perimeter with area, perhaps because theyare not clear about the meaning of formulas they have beentaught by rote. Perimeter is a measure of length, or distance;area is a measure of surface. Perimeter can beillustrated with a trundle wheel that rollsalong the boundary of a surface; the wordperimeter contains the word rim. Areacan be illustrated with the sweep of apaint roller across a surface or a pieceof chalk across the board. Perimeter ismeasured in units of length. Area ismeasured in square units—the numberof unit squares needed to cover a surface.Occasionally remind your students ofthese basic differences.

For additional information regarding perimeterand area, see Section 14.4 of the Teacher’s Reference Manual.

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Use the trundle wheel todemonstrate perimeter.

Developing and Using Formulas(Lessons 8◆4–8◆8)In Lessons 8-5, 8-6, and 8-7 students develop and use formulas for findingthe area of rectangles, parallelograms, and triangles.

Lessons 8-4 and 8-8 deal with areas that are not calculated from formulas.In Lesson 8-4, students estimate the area of one side of their hands bytracing them onto a grid and counting squares and fractions of squares.Students then use these estimates to compute the area of the skin on theirentire bodies by applying a rule of thumb (area of skin is about 100 timesthe area of one side your hand). In Lesson 8-8, students learn about areameasurements of some of the geographical features they have encounteredon the World Tour. They use division and a calculator to compare areas ofSouth American countries.

656 Unit 8 Perimeter and Area

Brazil’s area: 3,300,000 square miles

Peru’s area: 500,000 square miles

Ecuador’s area: 110,000 square miles

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In this unit, the formula for the area of arectangle (area � length of base times width,or height) is assumed as a basic “axiom.” Theformula for the area of a parallelogram isdeveloped by cutting apart a parallelogramand rearranging the parts into a rectangularshape. Similarly, the formula for the area ofa triangle is developed by cutting trianglesand reassembling them to formparallelograms. Similar cut-and-paste

methods will be used later in Everyday Mathematics to develop formulas forthe areas of other two-dimensional shapes—always with the idea oftransforming a given shape into another shape of the same area, for which aformula is already known.

Once students learn a formula, they tend to use it without much thought asto its origin. It is important to remind students occasionally of why formulasmake sense. Such derivations have the advantage of demonstrating animportant mathematical process—one fact is taken as an “axiom” on whichthere is agreement, and other rules or relationships are developed from the axiom.

Most schoolbooks (and many standardized tests) express the formula forthe area of a rectangle as the product of the length times the width

(A � l � w). The formulas for the area of aparallelogram (A � b � h) and a triangle (A � �

12� � b � h) are expressed in terms of the

length of the base and the height. Studentswho have been taught only the formula A � l � w will often mistakenly multiply the lengths of two adjacent sides whencalculating the area of a parallelogram.Similar errors are made in finding theareas of triangles. While students should befamiliar with the formula A � l � w, point

out that since a rectangle is a special kind of parallelogram, it makessense to use the formula A � b � h for the area of rectangle.

An added advantage to using the base-times-height formula is that it isconsistent with the formulas used to find the volumes of many three-dimensional figures. For example, no matter what the shape of the base, thevolume of a prism is always equal to the area of a base multiplied by its height.

In most math textbooks, nearly every illustration in connection with the areaof a polygon shows the base as a horizontal segment on which the figure“sits.” But the base can be any side, and the height can be measured on aperpendicular to whichever side is designated as the base.

To learn more about developing and using formulas, see Section 14.4.2in the Teacher’s Reference Manual.

height

heightbase

base

Two ways of designating the base and height of a parallelogram.

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