Cmp2 Geometry Storyline

The Mathematics of the Geometry Strand

Shapes and Designs – Grade 6 Overview

Summary of Investigation

Mathematics Background

• Polygon

• Sorting Shapes

• Symmetries of Shapes

• Tessellations

• Angle Measures

• Angles of a Polygon

• Angles and Parallel Lines

• Sums of the Angles of Polygons

• Interior Angles of Regular Polygons

• Exterior Angles of Regular Polygons

• Exploring Side Lengths of Polygons

Content Connections

Covering and Surrounding – Grade 6 Overview



• The Measurement Process

• Measuring Perimeter and Area

• Area and Perimeter of Rectangles

• Parentheses and Order of Operations

• Area of Triangles

• Area of Parallelograms

• Area and Circumference of Circles

• Estimating Perimeter and Areas of Irregular Figures

• Accuracy and Error

• Relationships Between Shapes and Size – Maximum and Minimum

• Fixed Area

• Fixed Perimeter

Content Connections


Stretching and Shrinking – Grade 7

Overview



• Similarity

• Creating Similar Figures

• Relationship of Area and Perimeter in Similar Figures

• Similarity of Rectangles

• Similarity Transformations and Congruence

• Comparing Area in Two Similar Figures Using Rep‐Tiles

• Equivalent Ratios

• Similarity of Triangles

• Angle‐Angle‐Angle Similarity for Triangles

• Solving Problems Using Similar Figures

Content Connections

Filling and Wrapping – Grade 7

Overview



• Rectangular Prisms

• Cylinders

• Relationship Between Surface Area and Fixed Volume

• Cones, Spheres, and Rectangular Pyramids

• Relationships Between Surface Area and Fixed Volume

• Effects of Changing Attributes – Similar Prisms

• Measurement

Content Connections


Looking for Pythagoras – Grade 8 Overview



• Finding Area and Distance

• Square Roots

• Using Squares to Find Lengths of Segments

• Developing and Using the Pythagorean Theorem

• A Proof of the Pythagorean Theorem

• Using the Pythagorean Theorem

• The Converse of the Pythagorean Theorem

• Special Right Triangles

• Rational and Irrational Numbers

• Converting and Repeating Decimals to Fractions

• Proof that √2 Is Irrational • Square Root Versus Decimal Approximation

• Number Systems

Content Connections

Kaleidoscopes, Hubcaps, and Mirrors – Grade 8 Overview



• Types of Symmetry

• Making Symmetric Designs

• Using Tools to Investigate Symmetries

o Transparent Reflection

o Hinged Mirrors

o Finding perpendicular bisectors

• Symmetric Transformations

• Congruent Figures

• Reasoning From Symmetry and Congruence

• Coordinate Rules for Symmetry Transformations

• Combining Transformations

Content Connections

OverviewShapes and Designs is the first unit in thegeometry strand. It develops students’ ability torecognize, display, analyze, measure, and reasonabout the shapes and visual patterns that areimportant features of our world. It builds onstudents’ elementary school exposure to simpleshapes, as they begin analyzing the properties thatmake certain shapes special. The unit focuses onpolygons and on the side and angle relationshipsof regular and irregular polygons (circles and othercurves are explored in later units).

In the Student Edition, the introduction developsthe broad theme of the unit: out of all the shapeswe use as basic components in buildings and art,some simple figures occur again and again becauseof properties that make them attractive and useful.The goal of Shapes and Designs is to have studentsdiscover and analyze many of the key properties ofpolygonal shapes that make them useful andattractive. As students become observant of themultitude of shapes that surround them and awareof the reasons that shapes are used for specificpurposes, they will be amazed by the visualpleasure and practical insights their new knowledgeprovides. We suspect that teachers will share thiseye-opening experience, finding new signs ofbeauty and structural significance in the thingsthey see everyday.

The approach to geometry in this unit issomewhat unique. First, the primary focus of theunit is on recognition of properties of shapes thathave important practical and aestheticimplications, not on simple classification andnaming of figures. While some attention is given tonaming familiar figures, each investigation focuseson particular key properties of figures and theimportance of those properties in applications. Forexample, students are periodically asked toidentify differences between squares, rectanglesthat are not squares, and parallelograms that arenot rectangles. We use a few special names fortypes of quadrilaterals (square, rectangle, andparallelogram) and triangles (isosceles, equilateral,and scalene). We frequently ask students to findand describe places where they see polygons ofparticular types and to puzzle over why thoseparticular shapes are used.

Summary of Investigations

Bees and PolygonsStudents sort polygons by common properties anddevelop rotation and reflection symmetries of ashape. Students also explore which shapes will tilea plane.

Polygons and AnglesThis investigation introduces three basic ways ofthinking about angles (turn, wedge, or rays) andthe ideas behind angle measurement. It givesstudents practice in estimating anglemeasurements based on a right angle. An angleruler is introduced, allowing more precisemeasurements. Students look at the consequencesof making measurement errors. They also explorethe angles formed by a transversal and parallellines. This will help students to better understandparallelograms.

Polygon Properties and TilingInvestigation 3 develops angle sums of polygons.Students begin by measuring the angles of regularpolygons and looking for patterns among the anglesums. As the number of sides of a polygon increasesby one, the sum increases by 1808. The teacherdemonstrates the sum of the angles of a triangleand the sum of the angles of a quadrilateral.

In this investigation, geometric arguments areused to help students confirm their conjecturesabout angle sums. Students see that the sum of theangles of a polygon is based on the sum of theangles of a triangle. That is, a polygon is subdividedinto triangles. Tiling is revisited and students usetheir knowledge about the measures of interiorangles and angle sums of regular polygons to explainwhy some regular polygons tile and others do not.

Two other ideas are also explored: (1) in apolygon the exterior angle and its correspondinginterior angle sum to 1808 and (2) all the exteriorangles of any polygon sum to 3608.

Investigation 33

Investigation 22

Investigation 11

Unit Introduction 3

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Back to Table of ContentsShapes and Designs Grade 6

Building PolygonsInvestigation 4 looks at the condition on sidelengths needed to form a triangle and then aquadrilateral. The last problem is a game usinggeoboards in which students change a quadrilateralby moving one or more vertices to form a newquadrilateral that fits a new set of criteria.

Mathematics BackgroundThe development in Shapes and Designs is basedon the van Hiele theory of geometry learning: webegin with recognition of shapes, then move toclassification of shapes, and then to analysis ofproperties of those shapes. The overalldevelopment progresses from tactile and visualexperiences to more general and abstractreasoning. We assume students have had priorexposure to the basic shapes and their names.

PolygonsThere are several different, but equivalent,definitions of polygons. Each of them is a bitsophisticated. Students at this level need informalexperiences with the concept of polygons beforethey see a more formal definition. At the beginningof this unit we say, “A polygon is a collection of linesegments put together in a special way.” Studentslook at two sets of shapes—polygons and non-polygons. After studying these two sets, andanswering the question, “What properties does ashape need to be a polygon?,” students develop abasic understanding of polygons that will continueto grow throughout the unit.

An important distinction to keep in mind inthis unit and other geometric units, is that apolygon consists of only the line segments (orsides) that make up the polygon. These linesegments enclose a region (of the plane). Thisregion is sometimes called the interior of thepolygon or polygonal region. The points in theinterior are not part of the polygon, and thepoints on the sides of the polygon are not part ofthe interior. We can also talk about the exteriorregion of a polygon—this is the set of points thatare neither on the polygon nor in the interior ofthe polygon. The distinctions that hold forpolygons and polygonal regions also hold for anyclosed plane figure, including circles.

In another sixth grade unit, Covering andSurrounding, the primary focus is on perimeter ofthe polygon and area of the polygonal region.Technically, when we talk about area we shouldsay area of the rectangular region or triangularregion, etc., but it has become common practice tosay area of a rectangle. It is understood that this isthe area of the interior of the rectangle. Thedistinction is important to note so that students donot go away with unintentional misconceptionsfrom the work or discussion in class. These ideaswill be addressed more directly in the studentmaterials of Covering and Surrounding.

Sorting ShapesBefore exploring tessellations, students spend alittle time on developing understanding ofpolygons. Students use the Shapes Set (a set ofplastic polygons) to sort the shapes into categoriesor subsets with specific properties that they havein common. Some may sort by regular and non-regular polygons. A regular polygon is a polygonwhose side lengths are all equal and whoseinterior angle measures are all equal. Studentslook at specific quadrilaterals and form anintuitive understanding of parallelograms andnon-parallelograms from their prior knowledge ofsquares and rectangles. They also sort triangles bythe length of their sides—connecting with priorknowledge of scalene, isosceles, and equilateraltriangles. Students use shapes to investigate whichshapes will tile a surface. This is a beginningattempt to answer the question of why the beesuse a hexagon as a shape of their honeycombs.For those shapes that do tile, students notice thatthe sides of the polygon must match and that theinterior angles of a polygon must fit exactlyaround a point in the plane.

Investigation 44

4 Shapes and Designs


Unit Introduction 5

Symmetries of ShapesStudents explore the symmetries of shapes, such asreflection (or line) symmetry and rotation (orturn) symmetry. Reflection symmetry is also calledmirror symmetry, since the half of the figure onone side of the line looks like it is being reflectedin a mirror. Rotation symmetry is also called turnsymmetry, because you can turn the figure aroundits center point and produce the same image.

A polygon with reflection symmetry has twohalves that are mirror images of each other. If thepolygon is folded over the line of symmetry, thetwo halves of the polygon match exactly.

All shapes have “trivial” rotation symmetry inthe sense that they can be rotated 3608 and lookthe same as before the rotation. When wedetermine whether or not a shape “has” rotationsymmetry we check for rotation symmetry forangles less than 3608. However the convention isthat once we determine that a shape has rotationsymmetry, when counting the rotation symmetrieswe include that “trivial” rotation as well.

For example, the shape at the right has two rotation symmetries: 1808

and 3608. This convention worksnicely because we can say that asquare has four rotation symmetries,a regular pentagon has five rotationsymmetries, and a regular hexagonhas six rotation symmetries.

A polygon with rotation symmetry can beturned around its center point less than a full turnand still look the same at certain angles of rotation.

Throughout the unit students are encouragedto look for symmetries of a shape.

TessellationsThe first big question presented in Shapes andDesigns, to motivate analysis of polygons, is theproblem of tiling or tessellating a flat surface. Thekey is that, among the regular polygons (polygonswith all edges the same length and all angles thesame measure), only equilateral triangles, squares,and regular hexagons will tile a plane.

triangles

There are other combinations of figures thatcan be used to tile a plane. Three are given below:

squares and octagons

triangles andpentagons

squares and triangles

Rotation Symmetry

Reflection Symmetry

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squares

hexagons


05

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centerline

center line

Some teachers use this opportunity to explainto students a short hand notation for describingthe shapes and combinations of shapes used totessellate. For example, to describe the tiling ofsquares one would write 4, 4, 4, 4 and for thetriangles 3, 3, 3, 3, 3, 3. To describe thecombination of shapes presented, you could write4, 3, 3, 3, 4. The notation identifies the shape by itsnumber of sides. It also tells the number of shapes,and the order in which the shapes surround apoint. You could suggest that the class look forinteresting tiling patterns in their homes or inschool. Have them make a sketch of any designsthey find.

For regular polygons to tile a plane, the anglemeasure of an interior angle must be a factor of360. So an equilateral triangle (608 angles), asquare (908 angles), and a regular hexagon (1208 angles) are the only three regular polygonsthat can tile a plane. Copies of each of these willfit exactly around a point in a plane.

There are eight combinations of regularpolygons that will tile so that each vertex hasexactly the same pattern of polygons. These aresometimes called semi-regular or Archimedeantessellations. (note the numbers in parenthesesrefer to the polygon by side number—8 means aregular octagon, 6 means a regular hexagon, etc.—and the order they appear around a vertex of thetiling):

2 octagons and 1 square (8-8-4) 1 square, 1 hexagon, and 1 dodecagon (4-6-12)4 triangles and 1 hexagon (3-3-3-3-6)3 triangles and 2 squares (4-3-4-3-3)1 triangle, 2 squares and 1 hexagon (4-3-4-6)1 triangle and 2 dodecagons (3-12-12)3 triangles and 2 squares (4-3-3-3-4)2 triangles and 2 hexagons (3-6-3-6)

See page 72 for pictures of these arrangements.Note that there are two arrangements with

triangles and squares, but depending on thearrangement they produce different tile patterns,so order is important.

In addition, any triangle or quadrilateral willtile a plane as in these examples:

When one understands the importantproperties of simple polygons, one can create anabundance of aesthetically appealing tilingpatterns, complete with artistic embellishments inthe style of artist M. C. Escher. However, it is thediscovery of what important properties of thefigures make the tiling possible, not the tilingquestion itself, that is one of the foci of the unit.

Angle MeasuresStudents explore angles in depth. The shape of apolygon is linked to the measures of angles formedwhere its sides meet. (The sides or edges of anangle are also called rays. The vertex of an angle isthe point where the two rays meet or intersect.)

Work is done to relate angles to right angles,focusing on developing students’ estimation skillswith angles. They use simple factors and multiplesof 908 turns to develop estimations. That is, theyuse 308, 458, 608, 908, 1208, 1808, 2708, and 3608 asbenchmarks to estimate angle size. Skills dealingwith these benchmarks are further developed in agame called Four-In-a-Row.

The need for more precision requires techniquesfor measuring angles. A new measuring tool, thegoniometer or angle ruler, a tool used in themedical field for measuring angle of motion orthe flexibility in body joints, such as knees isintroduced.

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Unit Introduction 7

This diagram illustrates why another methodfor measuring angles with the angle ruler, calledthe gripping method, gives the same results asplacing the rivet over the vertex of the anglebeing measured. The overlap of the sides of theruler forms a rhombus as you separate them.

In a rhombus, opposite angles are equal. Thismeans that the rhombus angle at the rivet and theopposite angle are equal. The angle opposite therivet in the rhombus is also equal to the anglebetween the sides since they are vertical angles(i.e., angles formed by two intersecting lines). So,when you place a shape between the arms of theruler, the angle at the rivet has the same measureas the angle between the arms.

We use students’ intuitive knowledge of a right angle to define a one-degree angle as of a right angle (a 908 angle). Two important aspects of angles come into play in Problem 2.4 whenstudents investigate the results of a measurementerror that was made in the fatal flight of AmeliaEarhart. The issue on the measure of an angle notbeing dependent of the lengths of the sides of therays is a very important one. Students tend tohave a hard time holding onto what is beingmeasured when we measure an angle. Two thingscan cause confusion: the length of the rays and thedistance between them. When we measure angles,we are measuring the “opening” or turn betweenthe edges of the angle. The lengths of the twoedges (rays) that form the angle do not affect themeasure of the angle.

Angles of a PolygonConsistent with formal mathematical definitions,we refer to adjacent sides and adjacent angles of apolygon.

Some students may be familiar with anotherdefinition of adjacent angles for two angles thatare not part of a polygon:

With these students, you may prefer to use thealternate term consecutive angles when referringto adjacent angles in a polygon.

Angles and Parallel LinesStudents explore some interesting patterns amongthe angles created when a line cuts two or moreparallel lines. Below is a pair of parallel lines thatare cut by a third line. The line that intersects the parallel lines is called a transversal. As thetransversal intersects the parallel lines, it createsseveral angles.

Angles a and e, angles b and f, angles c and g,and angles d and h are called correspondingangles. Angles d and e and angles c and f arealternate interior angles. Parallel lines cut by atransversal make equal corresponding angles andequal alternate interior angles. Two lines thatintersect make two pairs of equal angles. In thediagram, angles b and c are equal, as are angles a

a b

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e f

adjacent angles not in a polygon

adjacentsides adjacent

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and d, e, and h, and f and g. These pairs of anglesare called vertical angles. Angles b and d aresupplementary angles. Their sum is 1808. At thispoint names are not stressed—only therelationship among angles. Students discover theserelationships and others by looking at patterns.

Parallelograms are defined early in the unit asquadrilaterals with opposite sides of the samelength and opposite angles of the same measure.A parallelogram can also be defined as aquadrilateral with opposite sides parallel. Parallellines help explain these and other special featuresof parallelograms such as that the sum of themeasures of two adjacent angles is 1808.

Students use these facts about angles formedby parallel lines and a transversal in anapplication problem to show that the sum of theangles of a triangle is 1808. In the picture below, ifline l is parallel to line BC, then angles 1 and 4 areequal and angles 3 and 5 are equal. Since the sumof angles 1, 2, and 3 is 180, then by substitutionthe sum of angles 2, 4, and 5 is equal to 1808.

Note that the sum of the angles of a triangle andother polygons is first looked at experimentally, asdiscussed in the next section.

Sums of the Angles of PolygonsStep 1: Finding the angle sum of a triangle.

Students use their knowledge of 1808 and 3608 toexperiment with the interior angles of a polygon.Students use three copies of a triangle. Each vertexin the triangle is numbered 1, 2, or 3. Students canarrange the three angles of the triangle around apoint. The three angles form a 1808 angle. Thisworks for any triangle.

A similar experiment is conducted for anyquadrilateral. The four angles of a quadrilateral

will form a 3608 angle or wrap around a pointexactly once. A similar pattern holds forpentagons. The interior angles will form a 5408

angle or wrap around a point one and one-halfcomplete turns. For a hexagon it is 7208 or twocomplete turns.

Also, to find the sum of the interior angles of apolygon with sides greater than three, the polygoncan be subdivided into triangles. The sum of theinterior angles of a polygon is determined by thenumber of non-overlapping triangles in thepolygon. See Step 2 for details.

Step 2: Using triangles to find the angle sum ofa polygon.

Method 1—subdividing polygons into trianglesusing diagonals

Some students will use the number of trianglesto determine the sum of the angles for otherpolygons. For example, they may notice that bysubdividing a polygon into triangles they will findthat for pentagons there are three triangles and forhexagons there are four triangles. See a subdividedquadrilateral and pentagon below.

To reason about the angle sum in a polygon,you can triangulate the polygon: start at anyvertex, and draw all the possible diagonals fromthat vertex. You can triangulate a square intotwo triangles, a pentagon into three triangles, ahexagon into four triangles, and so on. Each timethe number of sides increases by one, the numberof triangles increases by one, thus making apattern: 3 sides give 1 triangle, 4 sides give2 triangles, 5 sides give 3 triangles, 6 sides give4 triangles, and so on.

We can use symbols to state a rule for thispattern. If we let N represent the number of sidesin a polygon, then (N - 2) represents the numberof triangles we get by triangulating the polygon. Ifwe multiply by 1808 for each triangle, we have theformula: (N - 2) 3 180° = the angle sum in anN-sided polygon. Note that this is true for bothregular and irregular polygons.

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Unit Introduction 9

Method 2—subdividing polygons into trianglesusing a point in the interior of the polygon

Another method that students may use is todraw all the line segments from a point within apolygon to each vertex. This method subdivides thepolygon into N triangles. In a quadrilateral, fourtriangles are formed. The number of triangles is thesame as the number of vertices or sides of thequadrilateral. In the pentagon, five triangles are

formed. Again, the number of triangles is equal tothe number of sides or vertices of the pentagon.

The sum of the angles of the four triangles inthe quadrilateral is 1808 3 4. But this sumincludes 3608 around the central point. Therefore,to find the sum of the interior angles of thequadrilateral, 3608 must be subtracted from thesum of the angles of the four triangles. The sum ofthe interior angles of the quadrilateral is 1808 3 (4) - 3608 = 3608.

The sum of the angles of the five trianglesformed in a pentagon is 1808 3 5. But this sum alsoincludes 3608 around the central point. So, to findthe sum of the interior angles of a pentagon, 3608

must be subtracted from the sum of the angles ofthe five triangles. The sum of the interior angles ofthe pentagon is 1808 3 (5) - 3608 = 540°.

We notice that the sum of the interior angles ofa quadrilateral or pentagon is 1808 times thenumber of sides minus two. For the quadrilateralthe sum is 1808 (4 - 2) and for a pentagon thesum is 1808 (5 - 2).

This method works for any polygon. For apolygon with N sides, the sum of its interior anglesis: 1808(N) - 3608 = 1808(N - 2).

The data from the methods above are arrangedin a table, and students form the followinggeneralizations. (Figure 1)

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Figure 1

Regular Polygon

Triangle

Square

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

N sides

Number of Sides

3

4

5

6

7

8

9

10

N

Angle Sum

180

2(180) � 360

3(180) � 540

4(180) � 720

5(180) � 900

6(180) � 1,080

7(180) � 1,260

8(180) � 1,440

(N �2)(180)

Measure of Interior Angle

180 � 3 � 60

360 � 4 � 90

540 � 5 � 108

720 � 6 � 120

900 � 7 � 128.6

1,080 � 8 � 135

1,260 � 9 � 140

1,440 � 10 � 144

[(N �2)(180)] � N


Interior Angles of Regular PolygonsIf a polygon is regular, we can find the number ofdegrees in one of the angles by dividing the sumby the number of angles.

= the number of degrees in any

angle of a regular N-sided polygon.Students also notice that as the number of sides

of a regular polygon increases, the measure foreach interior angle also increases—it approaches1808, which occurs as the shape of the polygonapproaches that of a circle.

Exterior Angles of Regular PolygonsIn a regular polygon of N sides the sum of theinterior angles is (N - 2)1808. The measure of

each angle is . So the measure of each

corresponding exterior angle is

1808 - .

The sum of N exterior angles

= N

= 1808N - (N - 2)1808

= 1808N - 1808N + 3608

= 3608

Students arrive at this generalization bylooking for patterns.

Exploring Side Lengths of PolygonsStudents use polystrips to build triangles andquadrilaterals with given side lengths.

Students begin by considering the question ofwhether any three lengths will make a triangle.Students find that the sum of two side lengths of atriangle must be greater than the third side length.If the two side lengths equal the third side length,then the two smaller sides collapse or fit exactlyon the third side and no triangle is formed. If thesum is smaller, then the two short sides collapseonto the third side, but do not fit exactly.

If a triangle is possible, then there is exactlyone triangle that can be built. This explains therigidity of triangles and the extensive use of

triangular shapes in bracing in buildings: oncethree edges have been fitted together, they form astable figure.

This side length result for triangles is called theTriangle Inequality Theorem. For example, if theside lengths are a, b, and c, then the sum of anytwo sides is greater than the third:

a + b . c.b + c . a.c + a . b.

To make a quadrilateral, the sum of three sidelengths must be greater than the fourth sidelength. If a quadrilateral can be built, differentcombinations of the side lengths will producedifferent-shaped quadrilaterals. Also by pushingon the vertex of a quadrilateral, the shape willchange, thus producing different shapes with thesame arrangement of side lengths.

A quadrilateral can be distorted into manyother quadrilateral shapes, and can undergo atotal collapse much more easily than a triangle.However, when a diagonal is inserted to form twotriangles, the quadrilateral becomes rigid. Threeexamples of quadrilaterals that have correspondingsides with the same length are below.

equal lengths

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(N 2 2) 3 180°N



Unit Introduction 11

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Big Idea Prior Work Future Work

Finding area and perimeter of 2-D figures (Covering andSurrounding)

Exploring measurement ofline segments (elementaryschool)

Developingunderstanding andtechniques formeasuring angles

Exploring symmetry informally by looking at shapes ofdata sets (Data Distributions) Identifying symmetry in 3-D cubic figures (Ruins of Montarek ©2004); connectingsymmetry to isometries (Kaleidoscopes, Hubcaps, andMirrors)

Exploring symmetryinformally by looking atshapes of data sets (DataDistributions)

Exploring symmetriesin squares,rectangles,parallelograms, andequilateral triangles

Understanding area as the exact number of squareunits needed to cover a 2-D figure (Covering andSurrounding); subdividing figures into similar figures(Stretching and Shrinking); connecting tessellations toisometries (Kaleidoscopes, Hubcaps, and Mirrors)

Exploring how 2-D shapesfit together (elementaryschool)

Creating tilings withpolygons anddetermining theproperties of shapesthat can be used totile a surface

Learning important properties of rectangles, triangles,and parallelograms (Covering and Surrounding);studying properties of 3-D cube figures (Ruins ofMontarek ©2004); enlarging, shrinking, and distorting 2-D shapes (Stretching and Shrinking); learningproperties of 3-D figures (Filling and Wrapping);learning and applying the Pythagorean Theorem(Looking for Pythagoras)

Developing classificationskills through classifyingintegers (e.g., even, odd,abundant, deficient) anddata (e.g., categorical ornumerical) (Prime Time);developing shaperecognition skills(elementary school)

Learning importantproperties ofpolygons that relateto the angles andsides of polygons

Studying properties of 3-D cube figures (Ruins ofMontarek ©2004); exploring similarity of 2-D figures(Stretching and Shrinking); finding surface area andvolume of 3-D figures (Filling and Wrapping)

Developing mathematicalreasoning by analyzingintegers and data (PrimeTime); developing shaperecognition skills(elementary school)

Understanding partsof polygons and howparts of polygons arerelated


OverviewThroughout history we find records of theimportance of measurement. In fact, in the earlydevelopment of mathematics, geometry wassynonymous with measurement. Today we aresurrounded by increasingly complex measures suchas information-access rates, signal strength, andmemory capacity.

The overarching goal of this unit is to helpstudents begin to understand what it means tomeasure. Students study two kinds ofmeasurements: perimeter and area. Since studentsoften have misconceptions about the effects of eachof these measures on the other, it is critical tostudy them together and to probe theirrelationships. The problems in this unit arestructured so that students can build deepunderstanding of what it means to measure areaand what it means to measure perimeter. In theprocess, they develop strategies for measuringperimeter and area of both rectangular andnonrectangular shapes. As they discuss theirstrategies, students are supported in formulatingrules for finding area and perimeter of rectangles,triangles, parallelograms, and circles.

The name of this unit indicates the theme thatbinds the investigations together: covering (area)and surrounding (perimeter). A sub-theme runningthrough the unit focuses on questions of what isthe greatest and what is the least, since the notionsof maximum and minimum are importantthroughout mathematics. You will recognizeconnections throughout the Covering andSurrounding unit to all the units preceding it in thegrade 6 curriculum. The connections to factors andmultiples and to fractions are especially strong.


Designing Bumper CarsThis investigation introduces students to area andperimeter by asking them to create floor plans forbumper-car rides that are made from 1-meter-square floor tiles and 1-meter-long rail sections.

The floor tiles and rail sections allow students touse counting as a way to find the area andperimeter of the plans.

This investigation builds experience withanalyzing what it means to measure area andperimeter and develops efficient strategies forfinding area and perimeter of rectangles. Inaddition, students should begin to understand thedifference between area and perimeter. They shouldbe aware that shapes with the same area may nothave the same perimeter. Similarly, shapes with thesame perimeter may not have the same area. By theend of the investigation, students should be able towrite rules for finding area and perimeter of arectangle and be able to explain why these work.

Changing Area, Changing PerimeterIn this investigation, students explore fixed areaand fixed perimeter problems. These problems aresometimes referred to as maximum and minimumproblems. Holding one variable constant to studyhow another variable changes is a powerfulmathematical tool used to analyze a wide varietyof problems. It also helps strengthen students’understanding of area and perimeter and how theyare related.

Investigation 22

Investigation 11

Unit Opener 3

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Back to Table of ContentsCovering and Surrounding Grade 6

Measuring TrianglesIn this investigation, students deepen theirunderstanding of area and perimeter by findingthe areas and perimeters of triangles. The primarygoal is for students to develop, understand, anduse the formula for finding the area of a triangle.Students also learn to identify the base and heighton a triangle.

Measuring ParallelogramsIn this investigation, students deepen theirunderstanding of area and perimeter by findingthe areas and perimeters of parallelograms. Theprimary goal is for students to develop,understand, and use procedures for finding areasand perimeters of parallelograms.

Measuring Irregular Shapes and CirclesThe first problem in the investigation looks atcounting techniques for estimating areas andperimeters of non-regular shapes. These shapesoften cannot be covered with whole square units.The counting techniques used to estimate areasand perimeters of non-regular shapes arepowerful yet concrete. They let students see thepower of formulas to solve some cases and theneed to understand the concepts underlying areaand perimeter to solve others. Appropriately, thisproblem pushes students to think about two ideasthat are central to this unit: covering andsurrounding.

The last three problems in this investigationhelp students discover how diameter and radius ofa circle are related to its circumference and area.These three problems lay the foundation forstudents to develop an understanding of thenumber p (pi), which represents the number ofdiameters needed to surround a circle and thenumber of radius squares (r 2) needed to cover acircle. Some students will have already seen theexpressions C = pd (or C = 2pr) and A = pr 2 butdo not understand what they mean.

Mathematics Background While this unit does not explicitly focus on themore global aspects of what it means to measure,it does lay the groundwork for teachers to raiseissues that help students begin to see relationshipsand characteristics of all measurements.

The Measurement ProcessThe measurement process involves several keyelements.

• A phenomenon or object is chosen, and anattribute that can be measured is identified.This could involve such disparate properties asheight, mass, time, temperature, and capacity.

• An appropriate unit of measurement isselected. The unit depends on the kind ofmeasure to be made and the degree ofprecision needed for the measure. Units ofmeasurement include centimeters, angstroms,degrees, minutes, volts, and decibels.Instruments for measuring include rulers,calipers, scales, watches, ammeters, springs,and weights.

• The unit is used repeatedly to “match” theattribute of the phenomenon or object in anappropriate way. This matching might beaccomplished, for example, by “covering,”“reaching the end of,” “surrounding,” or“filling” the object.

• The number of units is determined. Thenumber of units is the measure of theproperty of the phenomenon or object.

Measuring Perimeter and AreaCovering and Surrounding highlights two importantkinds of measures (perimeter and area) that dependon different units and measurement processes.Counting is a natural and appropriate way forstudents to find area and perimeter, becausemeasurement is counting. When we measure, weare counting the number of measurement unitsneeded to “match” an attribute of an object.

Measuring perimeter requires linear units.Measuring area requires square units. When findingthe perimeter of a figure, students will often saythey counted the number of squares along a sideto find the length. Students need to be aware thatperimeter is a linear measure. To measure the

Investigation 55

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4 Covering and Surrounding



perimeter you count (measure) the number ofunit lengths that form the border of the figure.

In the figure at the left below, the 12 squaretiles border a 4 by 4 square. This is not theperimeter. Instead, the perimeter comprises16 unit lengths, shown in the figure below.

A strong emphasis on formulas that precedesunderstanding the methods may contribute to thisconfusion. While students can become adept atplugging numbers into formulas, they often have ahard time remembering which formula does what.This may be because they have an incompletefundamental understanding of what themeasurement is about and how the formulacaptures their more informal, intuitive computations.

Many students think that area and perimeterare related in that one determines the other. Theymay think that all rectangles of a given area havethe same perimeter or that all rectangles of agiven perimeter have the same area. Alternatively,students may not see any distinction between areaand perimeter, giving area answers for perimeterproblems or vice versa. The investigations inCovering and Surrounding help students realizefor themselves the inaccuracy of such notions andhelp them to analyze the distinctions between thetwo measures.

In this unit, students work with tiles, transparentgrids, grid paper, string, rulers, and other devicesof their choice to develop a dynamic sense ofcovering and surrounding to find area andperimeter. Once students have an understandingof area and perimeter, they are ready to developrules or formulas for finding area and perimeterin certain situations. This should be encouraged,but not forced too early. Some students need thehelp of a more hands-on approach to measuringfor quite a while. The payoff for allowing studentsthe time and opportunity to develop levels ofabstraction with which they are comfortable isthat, through these explorations, they make senseof perimeter and area in a lasting way.

Area and Perimeter of RectanglesIn Investigation 1, students first explore area andperimeter of non-rectangular shapes. Afterbuilding shapes with square tiles and computingthe perimeter and area by counting the units, thestudents investigate rectangles displayed on agrid. Again they find that they can find the areaby counting the number of squares enclosed bythe rectangle and the perimeter by counting thenumber of linear units surrounding the rectangle.Students may have found that, once you havecounted the grid squares in one row, you couldmultiply by the number of rows to find the totalnumber of squares in the rectangle. In otherwords, you can find the area of a rectangle bymultiplying the length by the width.

For example, in this rectangle there are 5 squares in the first row and 7 rows in all. Thearea of the rectangle is 5 3 7 = 35 square units orin general, O 3 w.

Similarly, the perimeter is 2 (7 + 5) or 2 3 7 + 2 3 5 or in general, 2 (O + w) or 2O + 2w.

Parentheses and Order of OperationsThe two equivalent forms for the perimeter of arectangle provide an opportunity to discuss therole of parentheses in expressions and the orderof operations. Parentheses indicate that thenumbers in them need to be operated on first. Inthe perimeter formula, perimeter = (length +

width) 3 2, the parentheses indicate that you needto add the length and width before you multiplyby two. Students who are using a scientificcalculator, or one that follows order of operations,need to be aware that they must enter the stringcorrectly for it to give the correct answer. Forexample, for a rectangle with a length of 4 and awidth of 3, if you enter the string of numbers andoperations as 4 + 3 3 2, most calculators willautomatically follow the order of operations andmultiply 3 3 2 before adding the 4. If you want to

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add 4 + 3 first, either key in 4 + 3 and press equalto get the sum before keying in to multiply by 2,or use the parentheses keys and key in the following: 4 3 2 to find theperimeter.

Area of TrianglesIn Investigation 3, students use what they havelearned from finding the area and perimeter of arectangle to find the area and perimeter of atriangle. Essentially any triangle can be thought ofas half of a rectangle. If we surround a trianglewith a rectangle in a particular way, two smalltriangles are formed, both of which are inside therectangle and outside the triangle. These aretriangles 1 and 4 in the diagram below.

Referencing the grade 6 unit Shapes andDesigns, students notice that the two trianglescreated by drawing in the height of the triangle(triangles 2 and 3 in the diagram) are congruentto the other two triangles in the rectangle (that is,triangles 1 and 2 are congruent, as are triangles 3and 4 in the diagram).

area of triangle 1 +



area of triangle 4 = area of rectangleSince the areas of triangle 1 and triangle 2, as wellas triangle 3 and triangle 4, are the same:

2(area of triangle 2) +2(area of triangle 3) = area of rectangle

Hence the area of the original triangle is b 3 h

where b is the base of the triangle (or the length of the corresponding rectangle) and h is the heightof the triangle (or the width of the rectangle).

Obtuse triangles are reoriented because everyobtuse triangle has one orientation where the

smallest upright rectangle does not have an area equal to twice the triangle. In the firstarrangement below, the rectangle and the obtusetriangle have the same base and height. In thesecond orientation, the base of the obtuse triangleis shorter than the base of the enclosing rectangle.(In Problem 3.2, students find that the area of atriangle is the same regardless of which side ischosen for the base.)

The formula for area of a triangle still holds forobtuse triangles, regardless of orientation. However,the approach modeled in Problem 3.1 does notdemonstrate why this is so. Here is a proof of whythe formula works for obtuse triangles:

The following obtuse triangle has a base b anda height h. It is embedded in a rectangle. Notethat the bottom side of the rectangle is made upof two parts, the base of the obtuse triangle andthe base of a right triangle, which is x. From thepreceding demonstrations, the area of a righttriangle is known. So the length or base of therectangle is b + x and its height is h.

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Area of ParallelogramsThe rule for area of a parallelogram is developedfrom students’ experience finding the area oftriangles. This may be a different approach thanthe one you have used to develop a rule for areaof a parallelogram. It is not uncommon to see therule for area of a parallelogram developed out ofthe rule for area of a rectangle. When studentsinformally explore ways to find the area ofparallelograms, as they will in Problem 4.2, theyoften offer that they can cut the parallelogramand rearrange it into a rectangle. They can thenuse their rule for area of a rectangle to find thearea of a parallelogram. This approach can beillustrated with this diagram:

While this method works for most parallel-ograms, and it should be accepted when studentsoffer it as an informal strategy, it will not workwith parallelograms oriented like this one:

It is not possible to make one vertical cut andrearrange the pieces to form a rectangle. (Of course,one can reorient the parallelogram so that thelonger side is the base. Then the rearrangementworks.) But, every parallelogram can be dividedinto two congruent triangles by drawing onediagonal on the parallelogram, as shown:

In doing so, the length of the base and height ofthe parallelogram is the same length as the baseand height of the triangle. From this students cansee that they find the area of the parallelogram bymultiplying the base and height, without dividing

by two, as they did when finding the area of atriangle. The area of a parallelogram is2 3 ( b 3 h), or just b 3 h.

The height of a parallelogram is the perpen-dicular distance from the base to the side parallelto the base. As is the case with triangles, theheight of a parallelogram depends on the side thatis chosen for the base.

Area and Circumference of CirclesIn Investigation 5, the area of a circle is developedby finding the number of squares, whose sidelengths equal the radius, that cover the circle.In the diagram, the circle is enclosed in a square.Two perpendicular diameters are drawn whichmakes four squares whose areas are r 2. The areaof the circle is less than 4r 2. Then by either findingthe number of radius squares that cover the circleor by counting the area outside the circle, butinside the larger square, the area of the circle isapproximately 3 radius squares [p(r 3 r) or pr 2].

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The circumference is found by counting thenumber of diameter lengths needed to surroundthe circle. The number is about 3. Thecircumference of a circle is pd.

Estimating Perimeter and Area of Irregular FiguresFinding an exact measurement for area and perimeter is not always possible or even necessary.However, there are methods that allow for moreprecise estimates. In the last investigation of theunit before finding the area and circumference of acircle, students will look at irregular shapes, usingthe context of lakes and shorelines, to explore waysto find reasonable estimates for perimeter and area. When estimating perimeter, students use astring to wrap around the picture of the lake andthen measure the length of the string with theappropriate linear measure. A measure of length isalways approximate. Once the unit of length hasbeen selected, it can be subdivided to make moreaccurate measurements because there is less roomfor error in the approximation.

For example, suppose you approximate thelength of the segment measured with two scalesbelow.

Using the upper measurement scale, you find the length is approximately 4. Using the lower measurement scale, you approximate the length

as .To measure the surface area of the lake,

students select a corresponding square unit tocover the surface with. The number of unitsneeded to cover the lake is the area. Just as withlength, using smaller units for measuring areagives a more precise measurement.

In the following pictures, the object is measuredfirst with centimeter squares and then with half-centimeter squares.

From the pictures you can see more of the tree iscovered with whole units of half-centimeter squares,giving a better approximation for the answer.

Accuracy and ErrorMeasuring objects and comparing data from thewhole class is an excellent way to help studentsbegin to see that all measures are approximate.One can fine-tune measurements to get a degreeof precision in a particular situation, but nomatter how precise the instruments with which ameasurement is made, error will always exist.

The investigations in Covering and Surroundingprimarily involve whole-number situations asstudents begin to develop methods for finding areaand perimeter. But they will also encounter manysituations with rational numbers. In addition,students are likely to need fractions or decimalswhen measuring real objects. Students, especiallythose uncomfortable with fractions or decimals, maytry to round all measurements to whole numbers.

You will need to encourage them to use fractionsor decimals so that their measurements are moreaccurate.

Measurement gives rise to proportionalreasoning through measurement conversions, asencountered with the units in the tree exampleabove.

The familiarity of the situation can helpstudents make sense of the relationships. Becausethey will be working with drawings that represent

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real objects, students will encounter problems ofscale. We have tried to keep these problemsmanageable by carefully selecting scales thatmake for easy transitions from the model to thereal object. The investigations have a mix ofmetric and standard measures.

Students will often be asked to estimate or tocompare measurements. Comparing and estimatingare important skills used in many kinds ofquantitative situations, and they will help studentsdevelop skills in knowing whether an estimate isreasonable and appropriate, how to makeestimates, and how to compare measurements inmeaningful ways.

The Relationship Between Shape andSize—Maximum and MinimumIt is important for students to explore howmeasurements are affected when an attribute suchas area or perimeter is held constant. For example,some students hold the misconception that if you know the area of a shape, you can find theperimeter. Yet, for any given area you can makemany different shapes with that area, all of whichcould have varying perimeters. For example, both a1-by-9 and a 3-by-3 rectangle have an area of9 square units. The 1-by-9 rectangle is long and thin, and it will have a larger perimeter than the3-by-3 rectangle that is square in shape. The same is true when perimeter is held constant. For a givenperimeter, there are many different areas that canbe designed. Of the set of rectangles with a fixedarea, the rectangle that is most shaped like a squarehas the least perimeter. With no restrictionsspecified, there is no rectangle that has a greatestperimeter because we always can divide the width

in half and double the length, 2O 3 = Ow, to

make the perimeter greater. Where O . w,

2(2O 1 ) . 2O 1 2w. Because 4O 1 w . 2O 1 2w,

the perimeter is greater than the perimeter of theoriginal rectangle with dimensions O and w. If werestrict the dimensions to whole numbers, it wouldbe the rectangle that is the longest.

If the area is 9 square units, then the1-by-9 rectangle has the greatest perimeter. But ifthe dimensions are not restricted to whole numbers,

then a -by-18 rectangle has a greater perimeter

and a -by-36 rectangle has an even greater perimeter.

Fixed AreaSuppose the area of a rectangle is 24 square unitsand its dimensions are restricted to whole numbers.The rectangle with the least perimeter is the4-by-6 rectangle. In the set of real numbers, therectangle with the least perimeter is the square

whose side lengths are . This will be explored in the Looking for Pythagoras unit in eighth gradeafter they have been introduced to square rootsand irrational numbers.

If we allow our shape to be something otherthan a rectangle (and, in context, if we hadflexible walls!), then the best design is a circlewith radius of approximately 2.76 meters. Thecircumference (perimeter) of this circle isapproximately 17.37 meters.

When you are working with circles inInvestigation 5 you might want to return to thisproblem and explore what happens to theperimeter when you use a circle with area of 24 square units instead of a rectangle.

Fixed PerimeterSuppose the perimeter of a rectangle is 24 unitsand its dimensions are restricted to whole numbers.The rectangle that has the greatest area is the 6-by-6 rectangle. The 1-by-11 rectangle has theleast area. If the dimensions are any real numbers,then there is no smallest area. For example,

a -by-11 rectangle has an area of 3 = =

5 square units, which is smaller than 11 squareunits. This process could continue infinitely. Wecould have a perimeter of 24 units with side

lengths by 11 . The area would be 3 = ,

which is 2 square units. The relationship

between length and area has the shape of aparabola. This relationship will be revisited in thealgebra unit, Frogs, Fleas, and Painted Cubes.

In Investigation 2, and other places throughoutthe unit, students will have an opportunity toexplore the relationship between shape and size.They will consider situations where area is heldconstant and perimeter varies as well as situationswhere perimeter is held constant and area varies.These investigations help develop understandingof area and perimeter. The relationship betweensize and shape will be revisited when studentsstudy volume and surface area in the seventh-gradeunit Filling and Wrapping. This work also providesa foundation for future studies in calculus.

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Developing and applying algorithms for performing decimal calculations (Bits and Pieces III); developingstrategies and algorithms for finding the surface areaand volume of prisms, cones, and spheres (Filling andWrapping); developing the Pythagorean Theorem andother equations to model algebraic and geometric patterns (Thinking With Mathematical Models; Lookingfor Pythagoras; Growing, Growing, Growing; Frogs,Fleas, and Painted Cubes; Say It With Symbols; TheShapes of Algebra)

Performing operations withwhole numbers and findingfactor pairs of whole numbers (Prime Time);subdividing and comparingshapes (Shapes and Designs);collecting data and lookingfor and generalizing patterns(Prime Time, Shapes andDesigns)

Developing strategiesand algorithms forfinding the perimeterand area of rectangles, triangles,parallelograms, andcircles

Studying the relationship between the dimensions andvolume of a prism (Filling and Wrapping)

Effects of side lengths onshapes of polygons (Shapesand Designs)

Studying the relationship betweenperimeter and areain rectangles

Finding the area and side lengths of shapes on a coordinate grid (Looking for Pythagoras)

Performing operations withrational numbers; estimatingsums of rational numbers(Prime Time, Bits and PiecesI, Bits and Pieces II)

Developing strategiesfor finding theperimeter and areaof irregular two-dimensional shapes

Studying two-dimensional (surface area, square units)and three-dimensional (volume, cubic units) measures offigures (Filling and Wrapping)

Side lengths of polygons(Shapes and Designs)

Interpreting perimeteras the number of (linear) units neededto surround a two-dimensional shape

Studying relationships between three-dimensional models and two-dimensional representations of themodels (Ruins of Montarek ©2004); comparing areas oftwo-dimensional shapes to test for similarity (Stretchingand Shrinking); finding surface area and volume ofthree-dimensional figures (Filling and Wrapping)

Making tessellations (Shapesand Designs)

Interpreting area asthe number of squareunits needed to covera two-dimensionalshape



OverviewKnowledge of similarity is important to thedevelopment of children’s understanding of thegeometry in their environment. In their immediateenvironment and in their studies of natural andsocial sciences, students frequently encounterphenomena that require familiarity with the ideasof enlargement, scale factors, area growth, indirectmeasurement, and other similarity-relatedconcepts.

Similarity is an instance of proportionality. Forexample, if you increase the size of a diagram by50%, then distances in the enlarged diagram areproportional to distances in the original diagram.Specifically, every distance in the enlargement is aconstant multiple (1.5) of the correspondingdistance in the original. It is generally understoodthat understanding proportional reasoning is animportant stage in cognitive development.

Students in the middle grades often experiencedifficulty with ideas of scale. They confuse addingsituations with multiplying situations. Situationsrequiring comparison by addition or subtractioncome first in students’ experience with mathematicsand often dominate their thinking about anycomparison situation, even those in which scale isthe fundamental issue. For example, whenconsidering the dimensions of a rectangle thatbegan as 3 units by 5 units and was enlarged to asimilar rectangle with a short side of 6 units, manystudents will say the long side is now 8 units ratherthan 10 units. They add 3 units to the 5 units ratherthan multiply the 5 units by 2, the scale factor. Thesestudents may struggle to build a useful conceptionthat will help them distinguish between situationsthat call for addition and those that aremultiplicative (calling for scaling up or down).

The problems in this unit are designed to helpstudents begin to accumulate the knowledge andexperiences necessary to make these kinds ofdistinctions and to reason about scaling ingeometry situations. The next unit, Comparing and Scaling, continues to develop these ideas innumerical, rather than geometric, contexts.


Enlarging and Reducing ShapesSimilarity is introduced at an informal level.Students use their intuition about enlargementsand reductions to answer questions. Students makedrawings of similar figures using a pair of rubberbands. Then, they compare side lengths, anglemeasures, perimeters, and areas of the original andenlarged figures.

Similar FiguresStudents build a good working definition of similar in mathematical terms. They begin to seeconnections between geometry and algebra.Using the coordinate system, they draw severalgeometric figures. Some of the figures are similarto one another and others are not.

They explore algebraic rules that cause imagesto change size and to move about the coordinateplane. They also compare angle measures andlengths of corresponding sides informally as theyinvestigate transformations. Students find that fortwo figures to be similar corresponding anglesmust be congruent and corresponding sides mustgrow or shrink by the same factor.

Similar PolygonsStudents deepen their understanding of what itmeans for two figures to be similar. In addition,they explore the relationship between the areas ofsimilar figures. The idea that area does not grow atthe same rate as side length when a figure isenlarged is difficult for students to grasp. Throughexperiments with rep-tiles (shapes where copiesare put together to make larger, similar figures),students explore the relationship between theareas of two similar figures. They also discoverhow triangles are special. These experiences helpthem build mental images to support theirevolving ideas about the relationship betweenscale factor and area.

Investigation 33

Investigation 22

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Back to Table of ContentsStretching and Shrinking Grade 7

Similarity and RatiosStudents use equivalent ratios to test if figures aresimilar. They compare ratios of the sides withinrectangles (length to width of one rectangle andlength to width of the other). Students learn thatfor non-rectangular shapes such as triangles, youneed information about angle measures as well.They learn that between two similar figures, youcan find the length of missing sides using eitherratios or scale factors.

Using Similar Triangles and RectanglesStudents apply their knowledge about similarityof triangles to real-world problems. They use theshadow and mirror methods to find the height ofa tall object. They compare their data to decidewhich method gives more consistent results. Theyalso use similar triangles to find the distanceacross a physical feature, such as a river. In eachproblem, they find that triangles are similar iftheir corresponding angles are equal. In the ACE,they use their knowledge about similar rectanglesto make similar rectangles and to find missingmeasurements.

Mathematics Background The activities in the beginning of the unit elicitstudents’ first notions about similarity as twofigures with the same shape. Students may havedifficulty with the concept of similarity because ofthe way the word is used in everyday language—family members are “similar” and houses are“similar.” The unit begins by having studentsinformally explore what it means for twogeometric figures to be similar. They createsimilar figures using rubber bands. Early on, theybegin to see that some attributes of similar figuresare the same while others are not. For example,corresponding angle measures appear to be thesame, but corresponding side lengths aredifferent—yet these differences are predictable.

Students’ experiences with photocopiersenlarging or shrinking pictures provide anotherfamiliar context to begin the exploration ofsimilar figures.

Through the activities in Stretching andShrinking, students will grow to understand thatthe everyday use of a word and its mathematicaluse may be different. For us to determinedefinitively whether two figures are similar,similarity must have a precise mathematicaldefinition.

SimilarityTwo figures are similar if:

• the measures of their corresponding angles areequal

• the lengths of their corresponding sides increaseby the same factor, called the scale factor.

The two Figures A and B below are similar.

The corresponding angle measures are equal.The side lengths from Figure A to Figure B growby a factor of 1.5. Thus the scale factor fromFigure A to Figure B is 1.5. (Figure A stretches, oris enlarged.) You can also say the scale factor

from Figure B to Figure A is , or . (Figure B

shrinks, or is reduced.)

Creating Similar FiguresThe rubber-band stretcher introduced inInvestigation 1 is a tool for physically producing asimilarity transformation. It does not give preciseresults, but it is an effective way to introduce

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Investigation 55

Investigation 44

4 Stretching and Shrinking



students to similarity transformations. Moreprecision is gained in transformations usingalgebraic rules that specify how coordinateschange. See Problem 1.2 in the Student Editionfor instructions.

In this unit, students make figures on a coordinatesystem and use algebraic rules to transform theminto similar figures. For example, if the coordinatesof a figure are multiplied by 2, the algebraictransformation is from (x, y) to (2x, 2y). In general,if the coordinates of a figure are (x, y), algebraicrules of the form (nx + a, ny + b) will transform itinto a similar figure with a scale factor of n.Thesealgebraic rules are called similarity transformations,which are not introduced as vocabulary in this unit.In the preceding figures, Figure B has beentransformed from Figure A by the rule (1.5x, 1.5y).

Relationship of Area and Perimeter inSimilar FiguresThe perimeters of one rectangle A and rectangle Bbelow are related by a scale factor of 2. The areaincreases by the square of the scale factor, or 4.This can be seen by dividing rectangle B into fourrectangles (labeled A) congruent to rectangle A.

Similarity of Rectangles Since all of the angles in rectangles are rightangles, you need only check the ratios of thelengths of corresponding sides. For example,rectangles C and D are similar, but neither issimilar to rectangle E.

The scale factor from rectangle C to rectangleD is 2 because the length of each side of rectangleC multiplied by 2 gives the length of thecorresponding side of rectangle D. The scale

factor from rectangle D to rectangle C is

because the length of each side of rectangle D

multiplied by gives the length of the

corresponding side of rectangle C. Rectangle E isnot similar to rectangle C, because the lengths ofcorresponding sides do not increase by the samefactor.

Similarity Transformations and CongruenceIn general, algebraic rules of the form (nx, ny) arecalled similarity transformations, because they willtransform a figure in the plane into a similarfigure in the plane. If the figure described by therule, (x, y), is compared to the figure described bythe rule, (nx, ny), n is the scale factor from theoriginal figure to the image. The scale factor fromFigure A (x, y) to Figure B (2x, 2y) is 2. The scalefactor from Figure A (x, y) to Figure C (3x, 3y) is3. This is a special case where n = 1 in Figure A.If we compare two figures created by rules when n ≠ 1 in both figures, then the scale factor is not n.An example is Figure B (2x, 2y) and Figure C (3x, 3y). They are similar to each other. But the

scale factor from B to C is .

Note that similarity is transitive. If Figure A issimilar to Figure B and Figure B is similar toFigure C, then Figure A is similar to Figure C.

In Problem 2.2, the students will see that addingto x and/or y moves the figure around on the grid,but does not affect its size. This means that a moregeneral form of similarity transformations of thissort is (nx + a, ny + b). Rules of this form, wherethe coefficient of both x and y is 1 [such as (x + 3,y - 2)], move the figure around, but the figure

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stays exactly the same shape and size (it iscongruent to the original).

Congruent is a term from the sixth-grade unitShapes and Designs. Note that the scale factorbetween two congruent figures is 1. Therefore,congruent figures are also similar.

There are other transformations in the planethat preserve congruence, such as flips and turns.These are studied in the eighth-grade unitKaleidoscopes, Hubcaps, and Mirrors.

Comparing Area in Two Similar FiguresUsing Rep-TilesIt is generally surprising to students that if youapply a scale factor of 2 to a figure, the areabecomes 4 times as large. One approach is to havestudents calculate the area of a figure and that ofits image and compare the results. In the first twoinvestigations, area is explored informally. InInvestigation 3, we use rep-tiles to demonstratethat when you apply a scale factor of 2, it requiresfour copies of the original figure. In this case, youare really measuring area using the original figureas the unit, rather than square inches or squarecentimeters. If congruent copies of a shape can beput together to make a larger, similar shape, theoriginal shape is called a rep-tile. It takes fourcongruent triangles to create a larger similartriangle with a scale factor of 2 or nine congruenttriangles to create a larger similar triangle with ascale factor of 3. The large triangle below is madefrom four congruent copies of the smaller triangle.The scale factor from the original triangle to thelarger triangle is 2. From the diagram it is fairlyeasy to see that corresponding angles have equalmeasures.

The following examples are also rep-tiles with ascale factor of 2 from the smaller shape to thelarger shape.

A misconception that can arise is the idea thattiling is related to similarity. Figures that tile maynot make a larger, similar figure. In addition, anyfigure can be transformed into a larger or smallerimage, regardless of whether the figure can tilethe plane. Rep-tiles are special because they makearea comparisons easy.

Equivalent RatiosIn similar figures, there are several equivalentratios. Some are formed by comparing lengthswithin a figure. Others are formed by comparinglengths between two figures. For the rectangles

below, the ratio of length to width is or 1.6– for

rectangle P and or 1.6– for rectangle R.

You can also look at the ratios of correspondingsides across two figures. In this situation it iswidth-to-width and length-to-length. The ratios

are and , respectively. These ratios are equivalent, and are also equivalent to 2, the scalefactor. This second kind of ratio is not formallydiscussed in the unit, but students have used itinformally when they divide corresponding sidelengths between two similar figures to find thescale factor. This ratio also appears in an ACE.

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Unit Introduction 7

The perimeter grows by a factor of 2 and the areagrows by a scale factor of 2 3 2, or 4.

Similarity of TrianglesFor polygons other than triangles, you must make sure that the lengths of corresponding sides increase by the same scale factor and thatcorresponding angle measures are equal whenconsidering similarity.

In Shapes and Designs, students explored animportant property of triangles—angles determinea triangle’s shape. The property leads to theAngle-Angle-Angle Similarity Theorem forTriangles:

If the measures of corresponding angles intwo triangles are equal, then the two trianglesare similar.

For triangles you only have to check the angles to determine whether two triangles are similar.However, this fact about triangles is only hinted atin the unit. At this stage of their development ofunderstanding of similarity, it is best if studentsoperate with the general definition that applies toall polygons:

Corresponding angle measures are equal andcorresponding side lengths grow by the samescale factor. The next section explains thistheorem.

Angle-Angle-Angle Similarity for TrianglesThe line that connects the midpoints of twoopposite sides of a triangle is parallel to the thirdside and its length is equal to half the length ofthe opposite side. The parallel lines create equalcorresponding angles.

Parallel lines also cut transversals into segmentswhose ratios are equal. In the following figure,segment DE is parallel to segment CB, so theratios of the lengths of the segments ADto DC and the lengthsof the segments AE toEB are equal.

That is, = .

These facts can be used to show that if thecorresponding anglesof one triangle arecongruent to the

corresponding angles of another triangle then thetwo triangles are similar. This is known as theAngle-Angle-Angle Similarity Theorem. This istrue only for triangles. Also, because you knowthat the angles of a triangle add to 180° you needonly check two angles of a triangle in order toverify similarity.

This unit presents an alternative test forsimilarity. If the corresponding angle measures areequal, then instead of checking the ratio betweencorresponding sides (the scale factor), you couldcheck the ratios of sides within each figure. Given

the two figures below, if = and = ,

then the figures are similar.

Solving Problems Using Similar FiguresEquivalent ratios can be used to solve interestingproblems. For example, shadows can be thought ofas sides of similar triangles because the sunlighthits the objects at the same angle. A building ofunknown height and a meter stick, both of whichare casting shadows, are shown below. To find theheight of the building, you can use the scale factorbetween the lengths of the shadows. Since goingfrom 0.25 to 10 involves a scale factor of 40,multiply the height of the meter stick by 40 toobtain the height of the building, 40 3 1 m =

40 m. You could also think of this as = .

Finding the value of x that makes the ratiosequivalent gives you the height of the building.

1 m

0.25 m10 m

Not drawn to scale.x

10.25

x10

F

H G M L

K

LMKM

GHFH

KLLM

FGGH

AEEB

ADDC

A

E

B

C

D

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Exploring ratios and proportional relationships(Comparing and Scaling); developing the concept ofslope (Moving Straight Ahead)

Exploring properties of two-dimensional shapes;finding areas, perimeters,and side lengths of shapes(Shapes and Designs;Covering and Surrounding)

Applying propertiesof similar figures

Scaling and comparing figures and quantities(Comparing and Scaling); using slope to solve problemsinvolving linear relationships (Moving Straight Ahead)

Using factors and multiples(Prime Time); measuringtwo-dimensional figures(Covering and Surrounding);using ratios in fraction form(Bits and Pieces I; Bits andPieces II); using maps(Variables and Patterns)

Analyzing scalefactors betweenfigures; analyzingratios within twofigures; applyingscale factors to solvetwo-dimensionalgeometric problems

Finding the equation of a line (Moving Straight Ahead);expressing linear relationships with symbols;determining whether linear expressions are equivalent(Say It With Symbols); writing directions for isometries intwo dimensions (Kaleidoscopes, Hubcaps, and Mirrors)

Constructing two-dimensional shapes(Shapes and Designs); usingsymbols to communicateoperations (Variables andPatterns); exploringsymmetries of a figure(Shapes and Designs)

Describing andproducingtransformations ofplane figures

Analyzing how two-dimensional shapes are affected by different isometries; generating isometrictransformations (Kaleidoscopes, Hubcaps, and Mirrors)

Developing and applyingconcepts of vertex, angle,angle measure, side, andside length (Shapes andDesigns; Covering andSurrounding)

Identifying thecorresponding partsof similar figures

Scaling quantities, objects, and shapes up and down(Comparing and Scaling; Filling and Wrapping)

Finding angle measures,lengths, and areas of planegeometric figures (Shapesand Designs; Covering andSurrounding)

Enlarging andshrinking planefigures



Overview

In Filling and Wrapping, students explore thesurface areas and volumes of rectangular prismsand cylinders in depth. They look informally athow changing the scale of a box affects its surfacearea and volume. They also informally investigateother solids—including cones, spheres, and squarepyramids—to develop volume relationships.


Building BoxesStudents are introduced to the ideas of volumeand surface area through the concepts of wrappingand filling, building on their knowledge of areaand perimeter of two-dimensional figures from theCovering and Surrounding unit. Rectangularprisms are described by their dimensions: length,width, and height.

Designing Rectangular BoxesStudents continue their exploration of surface areaand investigate its relationship to volume. Theterms surface area and volume are introduced asvocabulary. By thinking about filling boxes inlayers, students develop the formula for volume ofa rectangular prism. The volume of a box is thenumber of blocks in the bottom layer multipliedby the number of layers—the area of the basetimes the height of the prism. (This strategy holdsfor all prisms.)

Prisms and CylindersStudents compare the volumes and surface areasof a variety of prisms with regular bases and acommon height. Students build prisms by foldingseveral sheets of congruent rectangular paper intothe shapes of triangular, rectangular, and

hexagonal prisms. They find the volume of anyrectangular prism by determining how many unitcubes would fill the prism. They observe that thevolume of a prism increases as the number oflateral sides increases. The volume of a cylinder isthe area of the base (the number of unit cubes inthe bottom layer) of the cylinder multiplied by itsheight (the number of layers). Surface area isinformally looked at as the sum of the area of thebases (circles) and lateral side (rectangle).

Prisms and cylinders come together as studentsdesign a rectangular box with the same volume asa given cylinder. They find the surface area of thebox is greater than the surface area of the cylinder.

Cones, Spheres, and PyramidsStudents compare the volumes of cones, cylinders,and spheres in an application. Students determinehow many times the volume of the cone or spherewill fill the cylinder and then look for relationshipsamong the three volumes. (Finding the surfaceareas of cones and spheres is not considered in thisunit.) Students also compare the volume of asquare pyramid to a cube.

Scaling BoxesStudents study the effects of changing thedimensions or the volume of a rectangular prism inthe context of designing compost containers. Theyexplore two central ideas: how to double thevolume of a rectangular prism and examine howother measures change as a result, and the effectsof applying scale factors to the dimensions ofrectangular prisms. What effect does doubling(tripling, quadrupling, etc.) each dimension of arectangular container have on its volume andsurface area? Students apply their knowledge ofsimilarity and scale factors to explore therelationships between a model of a cruise ship andthe actual cruise ship. This last problem connectsmany of the ideas discussed in this unit.

Investigation 55

Investigation 44

Investigation 33

Investigation 22

Investigation 11

Unit Introduction 3

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Back to Table of ContentsFilling and Wrapping Grade 7


Rectangular PrismsStudents begin the unit by exploring the surfacearea of a rectangular box. The strategy for findingthe surface area of a box is to determine the totalarea needed to wrap the container. Studentscreate nets that can be folded into boxes. The areaof the net becomes the surface area of the box.

This provides a visual representation of surfacearea as a two-dimensional measure of athree-dimensional object.

In the student edition, a rectangular prism isdefined as a three-dimensional shape with sixrectangular faces. Technically, this defines a rightrectangular prism. An oblique rectangular prismalso has opposite sides that are rectangles, but atleast two opposite sides must be nonrectangularparallelograms.

In this unit we only discuss right rectangularprisms. There is one ACE question on obliquerectangular prisms. The following figures are rightrectangular prisms drawn on isometric dot paper.

Isometric drawings are another useful2-dimensional representation of 3-dimensionalobjects. You may wish to have isometric dot paperavailable throughout the unit for students to usein making sketches of the 3-dimensional prismthey create.

The strategy for finding the volume of arectangular box is to count the number of layersof unit cubes it takes to fill the container. Thenumber of unit cubes in a layer is equal to thearea of the base—one unit cube sits on eachsquare unit in the base. The volume (the totalnumber of unit cubes) of a rectangular prism isthe area of its base (the number of unit cubes inthe first layer) multiplied by its height (the totalnumber of layers).

The same layering strategy is used to generalizethe method for finding the volume of any prism.The volume of any prism is the area of its basemultiplied by its height.

RectangularPrism

TriangularPrism

HexagonalPrism

one layer five layersfill the box

Obliquerectangular

prism

4 Filling and Wrapping



Unit Introduction 5

Students also informally compare the volume oftwo rectangular boxes by filling one box with riceor sand and then pouring the sand into the otherrectangular box.

CylindersThe surface area and volume of a cylinder aredeveloped in a similar way. Like a prism, acylinder has two identical faces (circles). Also likea prism, a cylinder has a lateral surface thatflattens to a rectangle. Students will notice all ofthese similarities. If they need language for therectangle that is part of the net of both cylindersand prisms, feel free to introduce the term lateralsurface. The term is not included in the studentmaterials. Students cut and fold a net to form acylinder. In the process, they find that the surfacearea of the cylinder is the area of the rectanglethat forms the lateral surface plus the areas of thetwo circular ends.

Cylinders can be thought of as circular prisms. Inthis case, it is easy to extend the techniques formeasuring prisms to techniques for measuringcylinders. The volume of a cylinder is developed asthe number of unit cubes in one layer (the area ofthe circular base) multiplied by the number oflayers (the height) needed to fill the cylinder.Because the edge of the circular base intersectsthe unit cubes, students will have to estimate thenumber of cubes in the bottom layer.

From the Covering and Surrounding unit,students know the formula for the area of a circle.They can apply this formula to find the area of thebase of a cylinder. The area of the base ismultiplied by the height to find the volume.

The volume of a cylinder = pr2h. Studentsinvestigate rectangular prisms with polygonalbases. If the prisms have the same heights, then asthe number of sides of the polygonal baseincreases, the shape gets closer to a cylinder.

Cones, Spheres, and Rectangular PyramidsStudents conduct an experiment to demonstratethe relationships among the volumes of a cylinder,a cone, and a sphere. If all three have the sameradius and the same height (the height beingequal to two radii), then it takes three cones fullof sand to fill the cylinder, and one and a halfspheres full of sand to fill the cylinder.

These relationships may also be expressed asfollows:

volume of the cone = of the volume of the

cylinder or

volume of the sphere = of the volume of the

cylinder or or

For cones and spheres, only the volume is studied.Surface area in these two cases is not consideredhere because the reasoning needed would take ustoo far afield. Formulas for these are sometimesconsidered within the context of high schoolgeometry or calculus courses.

The volume of a square pyramid is found in asimilar way by comparing it to a square prism.This is easily generalized to finding the volume ofa rectangular pyramid. If the base of the pyramidis a polygon, then as the number of sides of thepolygon increases, the shape of the pyramid getscloser to a cone.

43pr3

23pr2(2r)

23

13pr2h

13

�

�

Estimate thenumber of unit

cubes in one layer.

And multiply by the number

of layers.

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Relationship Between Surface Area andFixed VolumeStudents also investigate the effects of a change indimension, surface area, or volume on the otherattributes of a three-dimensional object. Forexample, if 24 unit cubes are arranged in arectangular shape and packaged in a rectangularbox, which arrangement of the cubes will requirethe least (the most) packaging material? Byphysically arranging the blocks and determiningthe surface area of each arrangement, studentsdiscover that a column of 24 cubes requires themost packaging, and the arrangement that is themost like a cube (2 by 3 by 4) requires the leastamount of packaging. This is similar to ideasstudents have studied about plane figures: For afixed area, the rectangle that is most like a squarehas the least perimeter of any rectangle with thesame area. A similar relationship holds for a fixedsurface area. The rectangular prism that is themost like a cube will have the greatest volume fora fixed surface area.

In fact, it is not cube-ness that minimizessurface area, but sphere-ness. For a fixed volume,a sphere has the smallest surface area and,conversely for a fixed surface area, a sphere hasthe largest volume.

Effects of Changing Attributes—Similar PrismsThrough the context of designing an indoorcompost box, students explore the effects thatchanging a box’s dimension have on the volumeand surface area of the box. Given the dimensionsof a compost box known to decompose a halfpound of garbage per day, students investigatewhat size box would decompose one pound ofgarbage per day. They find that they need to

double only one dimension of a rectangular boxto double its volume.

Students also look at the effects of doubling allthree dimensions of a box. Making scale modelsof the original box and the new box helps studentsvisualize the effect of the scale factor. Doublingeach dimension of a rectangular prism increasesthe surface area by 2 3 2 = 4 times (a scalefactor of 22) and volume by 2 3 2 3 2 = 8 times(a scale factor of 23). The surfaces of the twoprisms are similar figures with a scale factor of 2from the small prism to the large prism. Thisexploration connects back to ideas in thesimilarity unit, Stretching and Shrinking.

When we describe a cylinder, we generally giveonly two dimensions: the height and the radius.The radius is constant in every direction, so weneed not give a “length radius” and a “widthradius.” Yet, when we change the radius, wechange both the length and the width of the base.That is, we change two dimensions, not just one.In the ACE for Investigation 5, studentsinvestigate similar cylinders. At that point,students may need to discuss whether a cylinderis 2- or 3-dimensional in this sense.

MeasurementAll measurements are approximations. In thework in this unit, this idea will become moreapparent than usual. Students’ calculations ofsurface area and volume will often involve anapproximation of the number p, and they willoften use a calculated amount as a value in asubsequent calculation. Be aware that althoughstudents’ answers will often differ. The answersmay reflect correct reasoning and correctmathematics.

6 Filling and Wrapping



Unit Introduction 7

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Enlarging, shrinking, and distorting 2-Dfigures (Stretching and Shrinking); scalingquantities up and down using ratios andproportions (Comparing and Scaling)

Studying the effects ofapplying scale factor to thedimensions of a prism to itsvolume and surface area

Using variables to represent avariety of relationshipsalgebraically (Moving StraightAhead; Frogs, Fleas, and PaintedCubes; Growing, Growing,Growing; Say It With Symbols)

Developing strategies and algorithms forfinding the perimeter and area ofrectangles, triangles, parallelograms, andcircles (Covering and Surrounding)

Developing strategies andalgorithms for finding thesurface area and volume ofprisms and cones, and thevolume of cones and spheres

Algebraically analyzing suchrelationships in geometricfigures (Frogs, Fleas, andPainted Cubes)

Studying the relationship betweenperimeter and area in rectangles (Coveringand Surrounding)

Studying the relationshipsamong the dimensions,surface area, and volume ofprisms and cylinders

Comparing areas and perimeters ofdifferent 2-dimensional figures (Coveringand Surrounding)

Developing strategies forfinding and comparingvolumes and surface areas of different 3-dimensionalfigures

Interpreting perimeter as the number oflinear units that surround a 2-dimensionalfigure (Covering and Surrounding);interpreting area as the number of squaresthat cover a 2-dimensional figure (Coveringand Surrounding)

Interpreting surface area asthe number of square unitsthat cover or wrap theexterior of a 3-dimensionalfigure

Interpreting area as the number of squaresthat cover a 2-dimensional figure (Coveringand Surrounding); making minimal andmaximal buildings (Ruins of Montarek © 2004); studying relationships between 3-D models and 2-D representations ofthe models (Ruins of Montarek © 2004)

Interpreting volume as thenumber of unit cubes that filla 3-dimensional figure



OverviewIn Looking for Pythagoras, students explore twoimportant ideas: the Pythagorean Theorem andsquare roots. They also review and makeconnections among the concepts of area, distance,and irrational numbers.

Students begin the unit by finding the distancebetween points on a coordinate grid. They learnthat the positive square root of a number is theside length of a square whose area is that number.Then, students discover the Pythagoreanrelationship through an exploration of squaresdrawn on the sides of a right triangle. In the lastinvestigation of the unit, students apply thePythagorean Theorem to a variety of problems.


Coordinate GridsStudents review coordinate grids as they analyze amap in which streets are laid out on a grid. Theymake the connection between the coordinates oftwo points and the driving distance between them.This sets the stage for finding the distance betweentwo points on a grid without measuring. Studentsinvestigate geometric figures on coordinate grids.Given two vertices, they find other vertices thatdefine a square, a non-square rectangle, a righttriangle, and a non-rectangular parallelogram.And, they calculate areas of several figures drawnon a dot grid.

Squaring OffStudents explore the relationship between the areaof a square drawn on a dot grid and the length ofits sides. This provides an introduction to theconcept of square root. They find the distancebetween two points by analyzing the line segmentbetween them: they draw a square using thesegment as one side, find the area of the square,and then find the positive square root of that area.

The Pythagorean TheoremStudents develop and explore the PythagoreanTheorem. They then investigate a geometric puzzlethat verifies the theorem, and they use thetheorem to find the distance between two pointson a grid. In the last problem, they explore andapply the converse of the Pythagorean Theorem.

Using the Pythagorean TheoremFor students to appreciate the mathematical powerof the Pythagorean Theorem, they need toencounter situations that can be illuminated by thetheorem. Students explore an interesting patternamong right triangles, apply the PythagoreanTheorem to find distances on a baseball diamond,investigate properties of 30-60-90 triangles, andfind missing lengths and angle measures of atriangle composed of smaller triangles.

Mathematics BackgroundStudents’ work in this unit develops an importantrelationship connecting geometry and algebra: thePythagorean Theorem. The presentation of ideasreflects the historical development of the conceptof irrational numbers. Early Greek mathematicianssearched for ratios of integers to represent sidelengths of squares with certain given areas such as2 square units. The square root of 2 is an irrationalnumber, which means that it cannot be written as aratio of two integers.

Finding Area and DistanceStudents find areas of plane figures drawn on dotgrids. This reviews some concepts developed in thegrade 6 unit Covering and Surrounding. Onecommon method for calculating the area of afigure is to subdivide it and add the areas of thecomponent shapes. A second common method isto enclose the shape in a rectangle and subtractthe areas of the shapes that lie outside the figure

Investigation 44

Investigation 33

Investigation 22

Investigation 11

Unit Introduction 3

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Back to Table of ContentsLooking for Pythagoras Grade 8

from the area of the rectangle. Below, the area ofthe shape is found with each method.

In Investigation 2, students draw squares withas many different areas as possible on a 5 dot-by-5 dot grid. There are eight possiblesquares, four “upright” and four “tilted.”

Square RootsIf the area of a square is known, its side length iseasy to determine: it is the number whose squareis the area. The fact that some of these lengthsare not whole numbers prompts the introduction

of the symbol. The lengths of the sides of the

preceding squares (in units) are 1, 2, 3, 4, , ,

, and . Because the grid is a centimeter grid, students can estimate the values of the squareroots by measuring these lengths with a ruler. Bymaking these ruler estimates and comparing themto estimates obtained by computing square rootson a calculator, students develop a sense of thesenumbers and begin to realize that they cannot beexpressed as terminating or repeating decimals.

Students also develop benchmarks for

estimating square roots. For example, isbetween 2 and 3 because 4 � 5 � 9, and since 5 is

closer to 4 than 9, we estimate that is closer to2 than 3. Students might try 2.25. But 2.252

= 5.06.

So, is between 2 and 2.25, but closer to 2.25.They might try 2.24 to get 2.242

= 5.0176, which is closer. This method can be continued until thedesired accuracy is obtained. Students alsoestimate square roots with a number line ruler,which helps them to develop a sense of the size of

the irrational numbers such as , , and .

One way to locate on the number line is asfollows:

The square below has an area of 2 square units.

The length of a side of this square is units.If we draw a number line as shown, and use acompass to mark off a segment with the samelength as a side of the square, we can see that thesegment is about 1.4 units long.

Using Squares to Find Lengths of SegmentsFinding the areas of squares leads students to amethod for finding the distance between two dots.The distance between two dots on a dot grid is thelength of the line segment connecting them. Tofind this length, students can draw a square with

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Subdivide to find the area:2 � 2 � 1 � 1 � 6

4 Looking for Pythagoras



Unit Introduction 5

the segment as one side. The distance between thetwo dots is the square root of the area of thesquare.

To use this method to find all the differentlengths of line segments that can be drawn on a 5 dot-by-5 dot grid, the grid must be extended tofit the squares associated with those lengths. Forexample, the bold line segment below is the sideof a square (shaded) with an area of 25 square

units, so the segment has length units, or 5 units.

To draw the square with the given side length,many students will use an “up and over” or “downand over” method to go from one point to thenext. For example, to get from the lower endpointof the segment above to the other endpoint, yougo up 4 units and right 3 units. These endpointsare two vertices of the square. To get the thirdvertex, go right 4 units and down 3. To get thefourth, go down 4 units and to the left 3. In thisway, they are developing intuition about thePythagorean Theorem.

Developing and Using the Pythagorean TheoremOnce students are comfortable with finding thelength of a segment by thinking of it as the side ofa square, they investigate the patterns among theareas of the three squares that can be drawn onthe sides of a right triangle.

The observation that the square on thehypotenuse has an area equal to the sum of theareas of the squares on the legs leads students tothe Pythagorean Theorem: If a and b are thelengths of the legs of a right triangle and c is thelength of the hypotenuse, then a2

+ b2= c2.

A theorem is a general mathematical statementthat has been proven true. Over 300 differentproofs have been given for the PythagoreanTheorem. It is regarded as one of the mostimportant developments in mathematics because itallows us to link ideas of number to ideas of space.

c2

a2 a2 � b2 � c2

b2

5 4

1

8 8

16

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A Proof of the Pythagorean TheoremStudents solve a puzzle that gives a geometricproof of the Pythagorean Theorem. The puzzlepieces consist of eight congruent right trianglesand three squares.

The side lengths of the squares are the lengthsof the three triangle sides.

To solve the puzzle, students must arrange thepieces to fit into two square puzzle frames.Students’ arrangements of the 11 shapes maydiffer slightly, but all arrangements lead to thesame conclusion.

One possible arrangement is shown below. Thesides’ lengths of the right triangle have beenlabeled a, b, and c.

Once the shapes are arranged, you can reasonas follows:

• The areas of the frames are equal. They aresquares with side lengths of a 1 b.

• Each frame contains four identical righttriangles. The other shapes are squares with areaa2, b2, and c2.

• If the four right triangles are removed fromeach frame, the area remaining in the twoframes must be equal. That is, the sum of theareas of the squares in one frame must equalthe area of the square in the other frame.

Geometrically, the diagram shows that if thelengths of the legs of a right triangle are a and b,and the length of the hypotenuse is c, then a2

+ b2= c2. You can make similar puzzle pieces

starting with any right triangle and then arrangethe shapes in the same way. Therefore, thisstatement is true for any right triangle.

In later courses, students may see this geometricargument presented algebraically. The sum of theareas of the two squares and the four triangles inthe left frame equals the sum of the areas of thesquare and the four triangles in the right frame:

a2+ b2

+ = c2+

a2+ b2

= c2

The Pythagorean Theorem has manyapplications that connect the concepts of linesegment lengths, squares, and right angles.

Using the Pythagorean Theorem to Find LengthsStudents use the Pythagorean Theorem to find thedistance between two dots on a dot grid. Thelength of a horizontal or vertical line segmentdrawn on a dot grid can be found by counting theunits directly. If the segment is not vertical orhorizontal, it is always possible to treat it as the

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Puzzle frames Puzzle pieces




Unit Introduction 7

hypotenuse of a right triangle with vertical andhorizontal legs. The length of the hypotenuse—and thus the distance between thedots—can then be found with the PythagoreanTheorem.

In high school, students will see the followingformula for finding the distance between twopoints, (x1, y1) and (x2, y2) in the plane:

d =

This is simply the Pythagorean Theorem wherea = x1 - x2 (the horizontal distance between twopoints), b = y1 - y2 (the vertical distancebetween two points), and c = d.

To find the length of line segment AB below,draw a right triangle with segment AB as thehypotenuse. Calculate the areas of the squares onthe legs of the triangle (4 square units each), addthese areas (8 square units, which is the area ofthe square drawn on the hypotenuse), and take

the square root. The length of AB is units.

The Converse of the Pythagorean TheoremThe converse of a statement of the form “If pthen q” is “If q then p.” The converse of thePythagorean Theorem states: If a, b, and c are thelengths of the sides of a triangle and a2

+ b2= c2,

then the triangle is a right triangle. The converse ofa true statement is not always true. However, theconverse of the Pythagorean Theorem is true andcan be used to show that a given triangle is a righttriangle. For example, if you know the side lengthsof a triangle are 6 in., 8 in., and 10 in., thenbecause 62

+ 82= 102, you can conclude that the

triangle is a right triangle.Students do not formally prove the converse of

the Pythagorean Theorem in this unit. Rather,they build triangles with a variety of different sidelengths and determine whether they are righttriangles. Based on their findings, they conjecturethat triangles whose side lengths satisfy a2

+ b2= c2 are right triangles.

Students are asked to explain why theirconjecture is true. One explanation is: “Supposewe know that Triangle 1 has sides a, b, and c, thatsatisfy the relationship a2

+ b2= c2. Suppose

Triangle 2 has sides a, b, and d and we know thatTriangle 2 is a right triangle with leg lengths of a and b. Then a2

+ b2= d2. From the first

statement we know that a2+ b2

= c2. Logically,this gives us that c2

= d2, and, therefore, c = d(because they must both be positive numbers).Now Triangle 1 and Triangle 2 have the samethree measures for their sides. In Shapes andDesigns, students learned that once you know allthree sides of a triangle, it is uniquely identified.They will investigate this idea more formally whenthey study congruence of triangles in Hubcaps,Kaleidoscopes, and Mirrors. So these two trianglesare identical, right-angled triangles. In other wordsit is impossible for a triangle whose sides fit therelationship a2

+ b2= c2 to not be a right-angled

triangle.An interesting byproduct of the converse of the

Pythagorean Theorem is the concept ofPythagorean triples, sets of numbers that satisfythe relationship a2

+ b2= c2. Students discover

that finding Pythagorean triples means findingtwo square numbers whose sum is also a squarenumber. Multiples of one triple will generatecountless others. For example, once you establishthat 3-4-5 is a Pythagorean triple, you know that6-8-10, 9-12-15, and so on, are also Pythagoreantriples.

Special Right TrianglesIn Investigation 4, students learn about 30-60-90triangles by starting with an equilateral triangle (a 60-60-60 triangle). They use the line ofsymmetry to show the reflection line forms twocongruent 30-60-90 triangles. For each of thesetriangles, they deduce that the leg opposite the 308 angle is half the length of the side of theoriginal triangle. They then use the PythagoreanTheorem to find the length of the other leg.

The Pythagorean Theorem can be used to showsome special relationships among side lengths of30-60-90 triangles (that is, triangles with 308, 608,and 908 angles).

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Suppose the hypotenuse of a 30-60-90 triangle has length c. The length of the side opposite the 30° angle

must be half this length, or .

Using the Pythagorean Theorem, the square of thelength of the longer leg is

c2- , or . So, its length

is , or .

Students also explore isosceles right triangles(45-45-90 triangles), and find that the length ofthe hypotenuse is always the length of one of the

legs times . If the length of each leg is a then,by the Pythagorean Theorem, the square of thelength of the hypotenuse must be a2

+ a2, or 2a2.Therefore, the length of the hypotenuse is

= .

Rational and Irrational NumbersWhen we examine patterns in the decimalrepresentations of fractions, or rational numbers,we find that the decimals either terminate

or repeat. For example, is equal to 0.2

(a terminating decimal) and is equal to

0.33333. . . (a repeating decimal).

Numbers such as , , and cannot beexpressed as repeating or terminating decimals.Students create line segments with these lengths.

For example, is the length of the hypotenuseof a right triangle whose legs have length 1. Theythen locate the lengths on a number line. Thisprocedure helps students to estimate the size ofthese irrational numbers.

Converting Repeating Decimals to FractionsBecause all repeating decimals are rationalnumbers, they can be represented as fractions.It is not always obvious, though, what fraction isequivalent to a given repeating decimal. Onemethod for converting a repeating decimal to a

fraction involves solving an equation. To convert12.312312. . . to a fraction, for example, call theunknown fraction N. Thus, N = 12.312312. . . .Multiply both sides of the equation by 1,000 (the power of 10 that moves a complete repeatinggroup to the left of the decimal point), whichgives 1,000N = 12,312.312312. . . . Then, subtractthe first equation from the second, which gives

999N = 12,300. Therefore, N = , or .

The decimal equivalents of fractions withdenominators of 9, 99, 999, and so on, displayinteresting patterns that can be used to writerepeating decimals as fractions. For example, alldecimals with a repeating part of one digit, suchas 0.111 . . . and 0.222 . . . , can be written as afraction with 9 in the denominator and the

repeated digit in the numerator, such as and .

Decimals with a repeating part of two digits, suchas 0.010101 . . . and 0.121212 . . . , can be written asa fraction with 99 in the denominator and the

repeated digits in the numerator, such as and .

Proof that �–2 Is Irrational

In high school, students may prove that is nota rational number. Its irrationality can be proved inan interesting way—a proof by contradiction. Theproof is given here for the teacher’s information.

Assume is rational. Then, there exist

positive integers p and q such that = . So,

= p. Squaring both sides gives 2q2= p2.

From the Prime Time unit students learned thatall square numbers have an odd number offactors. The reason is that factors of a numbercome in pairs. In a square number the factors inone of the pairs must be equal, which makes thenumber of factors for a square number odd. Thismeans that if p and q are positive integers, then p2 and q2 each have an odd number of factors.Since p2

= 2q2, p2 has the same number of factorsas 2q2. But 2q2 has an even number of factors (Thefactor 2 plus the odd number of factors of q2.) Thisis a contradiction. Therefore, p and q cannot exist

with these properties and must be irrational.

Square Root Versus Decimal ApproximationProblems involving the Pythagorean Theoremoften result in square roots that are irrationalnumbers. Students at this level are often reluctant

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Unit Introduction 9

to leave numbers in a square root form. For

example, rather than give an exact answer of ,they give a decimal approximation, such as 1.732.Some students are not comfortable thinking aboutsquare roots as numbers. Although it is importantto know the approximate size of an answer,especially in a practical problem, it is sometimesbetter to give an exact answer, and this oftenmeans using square root form. For example, in thestudy of 30-60-90 triangles,

Here, is much easier to remember than amulti-digit decimal approximation, and theexpression using the square root gives the exactresult. Similarly, in a right triangle, if thehypotenuse has a length of 9 units and one leg hasa length of 8 units, then the length of the other leg

is = units. This answer is exact,while the calculator answer, 4.123105626, is anapproximation. This is not to say that all answersshould be left in square root form—context needsto be considered. Heights of buildings are moreeasily comprehended in whole-number or decimalform, even if that form does not give the preciseanswer. Students should be encouraged to leave ananswer in square root form when there is nopractical reason to express it as a decimalapproximation. The hope is that all students willbecome comfortable with square roots as numbersin contexts where expressing an answer as a squareroot is appropriate. In this unit, we want students tohave a “sense” of square roots as numbers and

some idea of where they fit on the number line orbetween what two rational numbers they occur.

Number Systems New number systems are created when a problemarises that cannot be answered within the systemcurrently in use, or when inconsistencies arise thatcan be taken care of only by expanding thedomain of numbers in the system.

The historical “discoveries” of new numbersystems in response to needs are reflected in thenumber sets students use in grades K–12.Elementary students begin with the countingnumbers, also called natural numbers. Then, zerois added to the system to create the set of wholenumbers. Later, students learn that negativenumbers are needed to give meaning in certaincontexts, such as temperature. Now they have thenumber system called the integers.

In elementary and middle school, studentslearn about fractions and situations in whichfractions are useful, as in many division problems.Students’ number world has been expanded to theset of rational numbers.

In this unit, students encounter contexts inwhich the need for irrational numbers arises.Specifically, they need irrational numbers toexpress the exact lengths of tilted segments on agrid. The set of rational numbers and the set ofirrational numbers compose the set of realnumbers. The diagram in Figure 1 is one way torepresent these sets of numbers.

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Investigating symmetry (Kaleidoscopes, Hubcaps, andMirrors); finding the equation of a circle (Shapes ofAlgebra)

Finding slopes of lines andinvestigating parallel lines(Variables and Patterns;Moving Straight Ahead)

Understanding sloperelationships ofperpendicular andparallel lines

Solving quadratic equations (Growing, Growing,Growing; Frogs, Fleas, and Painted Cubes; Say It WithSymbols)

Representing fractions asdecimals and decimals asfractions (Bits and Pieces I,II, and III)

Understandingirrational numbers asnon-terminating,non-repeatingdecimals

Exploring sampling and approximations (Growing,Growing, Growing; Samples and Populations; Frogs,Fleas, and Painted Cubes)

Understanding fractions anddecimals (Bits and Pieces I, II,and III)

Investigating rationalnumbers written asdecimals

Solving geometric and algebraic problems (Growing,Growing, Growing; Frogs, Fleas, and Painted Cubes; SayIt With Symbols; Kaleidoscopes, Hubcaps, and Mirrors)

Solving problems ingeometric and algebraiccontexts (Shapes andDesigns; Moving StraightAhead; Thinking WithMathematical Models;Covering and Surrounding)

Using thePythagoreanTheorem to solveproblems

Formulating and using symbolic rules and the syntax formanipulating symbols (Frogs, Fleas, and Painted Cubes;Say It With Symbols; Shapes of Algebra)

Formulating, reading, andinterpreting symbolic rules(Variables and Patterns;Moving Straight Ahead;Thinking With MathematicalModels; Covering andSurrounding); working withthe triangle inequality(Shapes and Designs)

Understanding the PythagoreanTheorem and how itrelates the areas ofthe squares on thesides of a righttriangle

Looking for patterns in square numbers (Frogs, Fleas,and Painted Cubes); looking for patterns in exponents(Growing, Growing, Growing)

Applying the formula forarea of a square (Coveringand Surrounding)

Understandingsquare roots aslengths of sides ofsquares

Studying transformations and symmetries of planefigures (Kaleidoscopes, Hubcaps, and Mirrors)

Measuring areas of polygonsand irregular figures (Bitsand Pieces I; Covering andSurrounding) and surfaceareas of three-dimensionalshapes (Filling and Wrapping)

Finding areas offigures drawn on acoordinate grid withwhole-numbervertices

Finding midpoints of line segments (Kaleidoscopes,Hubcaps, and Mirrors)

Measuring lengths (Shapesand Designs; Covering andSurrounding); working withcoordinates (Variables andPatterns; Moving StraightAhead; Thinking WithMathematical Models)

Calculating thedistance betweentwo points in theplane



OverviewStudents often have an intuitive understanding ofsymmetry. They recognize that a design is symmetricif some part of it is repeated in a regular pattern.Though students begin recognizing symmetricfigures at an early age, the understanding neededto confirm symmetry and to construct figures withgiven symmetries requires greater sophistication.Kaleidoscopes, Hubcaps, and Mirrors, the lastgeometry and measurement unit in the ConnectedMathematics 2 curriculum, helps students to refinetheir knowledge of symmetry and use it to makemathematical arguments.

Symmetry is commonly described in terms oftransformations. Symmetry transformations, orrigid motions, include reflections, rotations, andtranslations. They produce congruent figures, asopposed to similar figures discussed in the grade 7unit Stretching and Shrinking. Similaritytransformations change the size of a figure whilepreserving its shape (unless the scale factor is 1 : 1).In contrast, symmetry transformations preserveboth angle measures and side lengths, resulting inan image that is congruent to the original figure.The purpose of this unit is to stimulate and sharpenstudents’ awareness of symmetry, congruence, theirconnections, and to begin to develop theirunderstanding of the underlying mathematics.

Students will explore congruence, symmetry, andtransformations in greater depth in futuremathematics classes.


Three Types of SymmetryStudents are introduced to reflection, rotation, andtranslation symmetry. They identify the symmetriesin several designs and make designs with givensymmetries. Students are also introduced to toolsand procedures for testing for symmetry andmaking symmetric figures. The goal is to heightensensitivity to various forms of symmetry and todevelop geometric techniques for testing anddrawing symmetric figures.

Symmetry TransformationsStudents are challenged to describe the motionsinvolved in constructing symmetric designs. Theyexplore the relationships between figures and theirimages under reflections, rotations, and translations.They use their findings to write precise rules forfinding images under each type of transformation.

Exploring CongruenceThis investigation emphasizes the connectionbetween symmetry, transformations, andcongruence. Students use their intuitions aboutwhat figures or parts of figures appear to becongruent. Then they use symmetry arguments toverify their intuition. The Side-Angle-Side,Angle-Side-Angle, and Side-Side-Side congruenceconditions emerge from this investigation. Parts ofthe problem will reveal to students thatcongruence of quadrilaterals requires moreinformation (in general 5 pieces of side and angleinformation suffice, but no less).

Applying Congruence and SymmetryStudents are given some opportunities to reasonwith congruence conditions from Investigation 3and symmetry arguments to establish congruence,and to use “corresponding parts of congruenttriangles” to draw conclusions about side andangle measurements not available in the originaldata. Students also are asked to reason informallyto conclusions about parallelograms and otherfigures. For example, if a quadrilateral has rotationsymmetry, the quadrilateral must be aparallelogram, or the diagonals of a parallelogrambisect each other, and so on. This experience givesstudents an informal introduction to arguments likethose they will use more extensively in high schoolgeometry, without formal axiomatic reasoning.

Investigation 44

Investigation 33

Investigation 22

Investigation 11

Unit Introduction 3

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Back to Table of ContentsKaleidoscopes, Hubcaps, and Mirrors Grade 8

Transforming CoordinatesStudents work with figures drawn on a coordinategrid. By writing computer commands for makingfigures and their images under varioustransformations, students develop rules forlocating the image of a general point (x, y) undera particular reflection, rotation, or translation, anduse these rules to locate images undercombinations of transformations.

Mathematics BackgroundIn this unit, students study symmetry, symmetrytransformations, and their connections tocongruence. They learn to recognize and makedesigns with symmetry, and they learn to describemathematically the transformations that lead tosymmetric designs. They explore the concept andconsequences of congruence of two figures bylooking for symmetry transformations that willmap one figure exactly onto the other.

Types of SymmetryIn the first investigation, students learn torecognize designs with symmetry and to identifylines of symmetry, centers and angles of rotation,and directions and distances of translations.

A design has reflection symmetry, also calledmirror symmetry, if a reflection in a line maps thefigure exactly onto itself. The letter A hasreflection symmetry because a reflection in thevertical line through the vertex will match eachpoint on the left half with a point on the right halfand vice versa. The vertical line is the line ofsymmetry for this design. The image, orcorresponding point, of each point on one side ofthe line of symmetry is the same distance from theline of symmetry as the original point.

A design has rotation symmetry if a rotationother than a full turn about a point maps thefigure onto itself. The design below has rotationsymmetry because a rotation of 120° or 240° aboutpoint P will match each point on each flag with acorresponding point on another flag. Point P isreferred to as the center of rotation. The angle ofrotation for this design is 120°, the smallest anglethrough which the design can be rotated to matchwith the original design. The image of each pointis the same distance from the center P; and theoriginal point, center P, and image point alwaysform a 120° angle.

A design has translation symmetry if atranslation, or slide, maps the figure onto itself. Thefigure below is part of a translation-symmetricdesign. If this design continued in both directions,a slide of 1 inch to the right or left would matcheach point of each flag in the design with acorresponding point on another flag. (Figure 1)Distances between corresponding points are allthe same.

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Investigation 55

4 Kaleidoscopes, Hubcaps, and Mirrors

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Unit Introduction 5

Making Symmetric DesignsOnce students learn to recognize symmetry ingiven designs, they make their own symmetricdesigns. Students may use reflecting devices,tracing paper, and angle rulers or protractors tohelp them construct such designs.

• A design with reflection symmetry can bemade by starting with a basic figure and thendrawing the reflection of the figure in a line.The original and its reflection image make adesign with reflection symmetry.

• A design with rotation symmetry can be madeby starting with a basic figure and making n � 1 copies of the figure, where each copy is rotated degrees about a center point starting from the previous copy. The originaland its n � 1 rotation images make a designthat has rotation symmetry.

• A figure with translation symmetry can bemade by making copies of a basic figure, sothat each copy is the same distance and samedirection from the previous copy. The figureand its translation images make a design withtranslation symmetry.

Students are being asked to develop twoseparate but related skills. The first is to recognizesymmetries within a given design. The second is to make designs with one or more specifiedsymmetries starting with an original figure (whichmay not, in itself, have any symmetries). Thus, it is important to give students experience both inanalyzing existing designs to identify theirsymmetries, and also to give them experienceusing symmetry transformations to make designsthat have symmetry.

Using Tools to Investigate SymmetriesThere are a number of tools available that canhelp students visualize and describe symmetries.This section includes information about thesetools and suggests some ways of using the tools.

Transparent Reflection When using a mirror to test for reflection symmetry,it is hard to check the details of a design. It may bedifficult to determine whether a reflected imagematches the design behind the mirror exactly,since that part of the design is not visible.

Transparent reflection tools, such as ImageReflectors allow the viewer to see a reflected

image while simultaneously looking at the rest ofthe object through the transparent plastic. Thishelps the user to match the reflected image withthe part of the design behind the plastic. When thetwo halves match, the line of symmetry can beidentified by drawing a line segment along thebottom edge of the plastic.

Hinged MirrorsUsing two mirrors, you can demonstrate thepatterns made by reflections in a hinged mirror.You might allow students to experiment with themirrors during this investigation.

1. Place the two mirrors at an angle facing eachother. Tape the mirrors together so that theycan be opened as if hinged.

2. Draw a dark line on a piece of paper.

3. Stand the mirrors on the line, positioned sothat an equilateral triangle is formed by thereflections in the mirrors with the part of theline between the mirrors as one side of thetriangle.

4. Reposition the mirrors by opening or closingthe hinge to show a square, a pentagon, andso on.

5. Look for the lines of symmetry in eachreflected image.

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6. Place an object or draw a design between theline on the paper and the two faces of themirrors; for example:

Now set the mirrors to show an equilateraltriangle, a square, a pentagon, and so on, andobserve the reflected images.

It is a nice extension to measure the anglebetween the mirrors for each polygon, make atable of the data, and generalize the angle neededto make an n-sided polygon by observing thepattern in the table. From the data below, it isevident that the angle of the mirrors necessary to

produce an n-sided polygon is .

Finding Perpendicular BisectorsTo find the perpendicular bisector of the linesegment connecting A to A9 using a straightedgeand compass, set the compass on point A to drawan arc that extends a bit beyond the midpoint ofAA9. Using that setting, draw an intersecting arcfrom point A9.

Draw the line connecting the points ofintersection of the arcs. We can show that this lineis the perpendicular bisector of . Note thatthe intersection points of the arcs, X and Y, areequidistant from A and A9, that is,

= = = . We have a rhombusand its diagonals. From their explorations studentsknow that the diagonal of a rhombus is an axis ofsymmetry; thus, is a line of symmetry. Because

is a line of symmetry we know that the distancefrom A to the line of symmetry is the same as thedistance from A9 to the line of symmetry, and theline of symmetry is perpendicular to . Thus

is also the perpendicular bisector of . (Analternative way of making a convincing argumentwould be to rely on showing there are fourcongruent triangles, which would allow us todeduce that = and &AZX = &A9ZX.Since these angles are supplementary adjacentangles they must be right angles.)

Students can find the midpoint of usingseveral informal methods:

• measuring to find the midpoint

• folding the line segment in half

• marking the length of the segment on a stripof paper and folding the strip of paper in half

• using a transparent reflection tool

After locating the midpoint, students can use asquare corner, an angle ruler, or a protractor todraw a perpendicular line through the midpoint.The idea of a perpendicular bisector is useful whenstudents try to find an unknown center of rotation.

Symmetry TransformationsThe concepts of symmetry are used as the

starting point for the study of symmetrytransformations, also called distance-preservingtransformations or rigid motions. Thesetransformations—reflections, rotations, andtranslations—relate points to image points so thatthe distance between any two original points isequal to the distance between their images. Theinformal language used to specify thesetransformations is slides, flips, and turns. Some

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Unit Introduction 7

children will have used this language and willhave had informal experiences with thesetransformations in the elementary grades.

In this unit, students examine figures and theirimages under reflections, rotations, andtranslations by measuring key distances andangles. They use their findings to determine howthey can specify a particular transformation sothat another person could perform it exactly.Students learn that a reflection can be specifiedby giving the line of reflection. They learn thatunder a reflection in a line l the point A and itsimage point A9 lie on a line that is perpendicularto the line of symmetry and are equidistant fromthe line of symmetry.

A rotation can be specified by giving the centerof rotation and the angle of the turn. In this unit,the direction of the rotation is assumed to becounterclockwise unless a clockwise turn isspecified. For example, a 72° rotation about apoint P is a counterclockwise turn of 72° with P asthe center of the rotation. Students learn that apoint B and its image point B9 are equidistantfrom the center of the rotation P. They see thatthe image of a point under a rotation travels onthe arc of a circle and that the set of circles onwhich the image points of the figure travel areconcentric circles with P as their center. They alsofind that the angles formed by the points on theoriginal figure and their corresponding rotation

images, such as &BPB9, all have measures equalto the angle of turn.

A translation can be specified by giving thelength and direction of the slide. This can be doneby drawing an arrow with the appropriate lengthand direction. Students find that if you draw thesegments connecting points to their images, forexample, CC9, the segments will be parallel and allthe same length. The length is equal to themagnitude of the translation.

This work helps students to realize that anytransformation of a figure is essentially atransformation of the entire plane. For every pointin a plane, a transformation locates an image point.And it is not uncommon to focus on the effect ofa transformation on a particular figure. This unitattempts to give mathematically precise descriptionsof transformations while accommodating students’natural instinct to visualize the figures moving.Thus, in many cases, students are asked to study afigure and its image without considering the effectof the transformation on other points. However,the moved figure is always referred to as the imageof the original, and the vertices of the image areoften labeled with primes or double primes toindicate that they are indeed different points.

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An interesting question is, “For whichtransformations are there points that remainfixed?” These are called fixed points. The image ofeach such point is simply the point itself. For areflection, the points on the line of reflection arefixed points. For a rotation the only fixed point isthe center of rotation. For a translation, all pointshave images with new locations, so there are nofixed points.

Congruent FiguresThe discussion of distance-preservingtransformations leads naturally to the idea ofcongruence. Two figures are congruent if theyhave the same size and shape. Intuitively, thismeans that you could move one figure exactly ontop of the other by a combination of symmetrytransformations (rigid motions). In the languageof transformations, two figures are congruent ifthere is a combination of distance-preservingtransformations (symmetry transformations) thatmaps one figure onto the other. Several problemsask students to explore this fundamentalrelationship among geometric figures.

The question of proving whether two figures arecongruent is explored informally from a differentdirection as well. A major question asked is whatminimum set of equal measures of correspondingsides and/or angles will guarantee that two trianglesare congruent. In the unit, students meet these ideasin an exploratory game situation where there is apayoff for finding the least set of measures neededto confirm congruence. It is likely that studentswill discover the angle congruence theorems thatare usually taught and proved in high schoolgeometry– Side-Side-Side, Side-Angle-Side, and Angle-Side-Angle. This engagement with theideas in an informal way will help make the moreproof oriented approach of high school geometrymore understandable. They should also find thatAngle-Angle-Angle and Side-Side-Angle do notguarantee congruence. Angle, angle, angleguarantees similarity, same shape, but not samesize. With Side-Side-Angle, in some cases thereare two possibilities, so you cannot know forcertain that you have congruence.

In a right triangle, with the right angle and anytwo corresponding sides given you can use thePythagorean Theorem to find the third side. Thisgives you two sides and the included angle, or

three sides—enough to know that two trianglesare congruent.

Reasoning From Symmetry and CongruenceSymmetry and congruence give us ways ofreasoning about figures that allow conclusions tobe drawn about relationships of line segments andangles within the figures.

For example, suppose that line is a line ofreflection symmetry for triangle ABC; themeasure of &CAM is 37°; the measure of = 6;and the measure of = 4.

As a consequence of the line symmetry, you cansay that

• C is the reflection of B.

• A is the reflection image of A.

• M is the reflection image of M.

• Segment CA is the reflection image of segmentBA, which means they have equal lengths.

• Segment CM is the reflection image of BM,so each has measure 3.

• is the reflection image of .

• So &AMC � &AMB; thus each anglemeaures 90°.

•&CAM � &BAM, so each angle measures 37°.

• So &C � &B, and each angle must measure 180° - (90° + 37°) = 53°.

In the grade 6 unit Shapes and Designs,students explored the angles made by atransversal cutting a pair of parallel lines. Forsome of the reasoning problems in this unit,students will probably need to use ideas ofvertical angles, supplemental angles, and alternateinterior angles from Shapes and Designs. For

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Unit Introduction 9

example, in the diagram below, L1 and L2 aregiven parallel lines and we know the measure oftwo angles. (We are not given that L3 and L4 areparallel.)

From this we can find the measures of angles a, b, c, e, f, g, and h. &b = 1208 because verticalangles are equal. &f = 1208 because alternateinterior angles are equal. &g = &f = 1208 becausevertical angles are equal. &a is supplementary to1208 and, therefore, is 608. This means that &c, &e,and &h are all 608 using vertical angles andalternate interior angles. We can then go on to saythat L3 and L4 must be parallel because we knowthat alternate interior angles are congruent, &eand the angle marked as 60°.

The relationships among parallel lines and theirrespective transversals can help especially whenreasoning about parallelograms. For example, inthe parallelogram shown below we know, bydefinition, that there are two pairs of parallel linesand transversals. This relationship results inseveral pairs of congruent angles.

This prior reasoning from parallel lines cut by atransversal combined with congruence andsymmetry culminate in Shapes and Designs,Investigation 4.3. Here students are asked to thinkabout what is given in a quadrilateral and to drawconclusions about whether or not the shape is aparallelogram.

Coordinate Rules for SymmetryTransformationsIn the final investigation of the unit, we look attransformations of figures on a coordinate plane.

Students also write rules for describingreflections of figures drawn on a coordinate grid.Such rules tell how to find the image of a generalpoint (x, y) under a reflection. For example, areflection in the y-axis matches (x, y) to (-x, y); areflection in the x-axis matches (x, y) to (x, -y);and a reflection in the line y = x matches (x, y) to (y, x).

A9 is the image of Aunder a reflection inthe y-axis.A0 is the image of Aunder a reflection inthe x-axis.A- is the image of Aunder a reflection inthe line y � x.

As with reflections, students learn to specifycertain rotations by giving rules for locating theimage of a general point (x, y). For example, arotation of 90° about the origin matches the point (x, y) to the image point (-y, x), and a rotation of180° about the origin matches (x, y) to (-x, -y).

A9 is the image of Aunder a 90° rotationabout the origin.A0 is the image of Aunder a 180°rotation about theorigin.

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A translation can also be specified by giving arule for locating the image of a general point (x, y).For example, a vertical translation of 3 units upmatches point (x, y) to (x, y + 3), and a horizontaltranslation of 3 units to the right matches (x, y) to (x + 3, y). A translation along an oblique line canbe specified by considering the vertical andhorizontal components of the slide. For example, atranslation in the direction of the line y = x, twounits right and two units up, matches (x, y) to (x + 2, y + 2). A translation of 2 units to the rightand 4 units down matches (x, y) to (x + 2, y - 4).

A9 is the image of A under a translation of 3 units up.A0 is the image of A under a translation of 3 units to the right.A- is the image of A under a translation of 2 units to the right and 4 units down.A-9 is the image of A under a translation inthe direction of the line y � x, 2 units rightand 2 units up.

Combining TransformationsIn very informal ways, students explorecombinations of transformations. In a fewinstances in the ACE extensions, students are askedto try to describe a single transformation that willgive the same result as a given combination. Forexample, reflecting a figure in a line and thenreflecting the image in a parallel line has the sameresult as translating the figure in a directionperpendicular to the reflection lines for a distanceequal to twice that between the lines.

Reflecting a figure in a line and then reflectingthe image in an intersecting line has the sameresult as rotating the original figure about theintersection point of the lines for an angle equalto twice that formed by the reflection lines. Noticethat reflecting the triangle ABC in line 1 and thenreflecting the image A9B9C9 in line 2 does NOTgive the same result as reflecting triangle ABC inline 2 first and then reflecting the image in line 1.

Students revisit this idea in the project whenthey explore combinations of transformations thatmap a geometric figure onto itself. In thisinstance, the figure used is an equilateral triangle.

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Using matrices to represent transformations (high school)Describing similaritytransformations in wordsand with coordinate rules(Stretching and Shrinking)

Describing rigidmotions in words andwith coordinate rules

Finding equations for circles and points on circles (The Shapes of Algebra)

Reasoning about congruence theorems in geometry(high school)

Performing and analyzingsimilarity transformations(Stretching and Shrinking)

Using symmetry andcongruence toreason about figures

Describing symmetry in graphs, such as graphs ofquadratic functions, periodic functions, and powerfunctions (high school)

Relating similaritytransformations to theconcept of similarity(Stretching and Shrinking)

Relating rigidmotions to theconcept of symmetry

Relating rigid motionsto the congruence offigures

Making inferences and predictions based on observationand proving predictions (high school)

Looking for regularity andusing patterns to makepredictions (all ConnectedMathematics units)

Looking for patternsthat can be used topredict attributes ofdesigns

Recognizing symmetry in graphs of functions (high school)

Applying the ideas of symmetry to other subjects, suchas graphic design and architecture (high school)

Recognizing and completingmirror reflections (Shapesand Designs)

Recognizing and completingdesigns with rotationsymmetry (Shapes andDesigns)

Rotating cube buildings(Ruins of Montarek, © 2004)

Recognizing, analyzing, andproducing tessellations(Shapes and Designs)

Recognizingsymmetry in designs

Determining thedesign element thathas been reflected,rotated, or translatedto produce a designwith symmetry

Making designs withreflection, rotation, ortranslation symmetries



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