Geometry and Measurement of Solid Figures

Activity Set 4

Trainer Guide

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TG

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

NGSSS 7.G.2.2Catching volume

In this activity, participants will create three-dimensional models for various solid figures and compute the volumes of those shapes.

MATERIALS

• Transparency/Page:RectangularPrism• Transparency/Page:TriangularPrism• calculators(1perpairofparticipants)

VOCABULARY

• volume• rectangularprism• triangularprism

TIME: 10minutes

TEAChIng TIp: For an advanced group of participants, skip the rectangular prism (Introduction) and do the triangular prism (Discuss and Do) as a whole group.

INTRODUCE

•Explaintoparticipantsthattheyaregoingtofindthevolume of some solid shapes.

•Askparticipantstoexplainhowtofindthevolumeofacube.(height•width•depth)

•DisplayTransparency:RectangularPrism.

•Askparticipantshowtheformulathattheyjustdescribedappliestothisshape.(Thevolumecanbecomputedinthesameway.)

•Pointoutthatacubeisaspecialcaseofrectangularprism, where the depth is equal to the height and width.

•Askparticipantstoworkinpairstofindthevolumeoftheprismbyusingtheircalculators.(6•6•14=504cubicmm)

DISCUSS AND DO

•DisplayTransparency:TriangularPrism and have participantstakeouttheirmatchingpages.

•Haveparticipantpairsfindthevolumeofthe triangular prism.

•Giveparticipants2–3minutestocompletetheir calculations.

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TG

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Rectangular prism

14 mm

6 mm

6 mm

Transparency: Rectangular Prism

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Triangular prism

4 mm

3.46 mm

9 mm

Transparency: Triangular Prism

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TG

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

CONCLUDE

•Haveavolunteerwritetheproblemandsolutiononthe displayed transparency. [ 12 (4•3.46)•9=62.28cubicmm]

•Remindparticipantsthatwhilethetwoshapesthatthey have used during this activity have volumes that can be computed by “counting” layers, some shapes such as pyramids have volumes that cannot be computed in this manner because each “layer” has a differentsurfacearea.Thevolumesofsuchshapeshave formulas that are generated using calculus.

•Suggesttoparticipantsthatstudentssometimeshavetrouble distinguishing between surface area and volume.

•Suggestthatnetsandmodelscanprovideawaytohelp students distinguish between the two.

End of Catching volume

Top-Down view

In this activity, participants will create two-dimensional representations of three-dimensional shapes.

MATERIALS

• Transparency/Page:Top,Side,andFront• Transparency/Page:Top,Side,andFrontAnswerKey• Transparency/Page:DrawingTwo-DimensionalViews• Transparency/Page:DrawingTwo-DimensionalViews

AnswerKey• blanktransparencies• scrappaper• rainbowcubes(60perpairofparticipants)• setofwoodensolidshapes• pencils

VOCABULARY

• model

TIME:30minutes

TEAChIng TIp: Prepare and distribute, in advance of the activity, 1 bag of 60 rainbow cubes to each pair of participants.

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TG

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

INTRODUCE

•Suggesttoparticipantsthatstudents’workinhighermathematics often requires an ability to visualize mathematics concepts in three dimensions, and that often they must do this from two-dimensional pictures.

•Suggestfurtherthatexperiencecreating two-dimensional views of three-dimensional shapes helps students build a foundation for these later activities.

•Explaintoparticipantsthattheynowaregoingtopractice such an activity, drawing two-dimensional views of three-dimensional figures.

•Displayablanktransparencyandpassouttoparticipants scrap paper.

•Holdup,fromyoursetofsolidshapes,acone.

•Askavolunteerparticipanttosketchatop,afront,and a side view of the shape on the top half of the transparencywhiletheotherssketchthemontheirscrappaper.Samplesketchesareshownbelow.

•Usetheoverheadprojectortodisplaytheshadowofthe shape from the various angles, if participants have trouble visualizing the shape.

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TG

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

Top Front Side

•Holdup,fromyoursetofsolidshapes,ahalf-sphere.

•Askavolunteerparticipanttosketchatop,afront,and a side view of the shape on the bottom half of thetransparencywhiletheotherssketchthemontheirscrappaper.Samplesketchesareshownbelow.

•Discusswithparticipantsthefactthatthefrontandsideviewsoftheseshapeslookthesamebecausetheshapesaresymmetricallycurvedaroundacentralaxis.

•Askparticipantsiftheycannamesomeshapes forwhichthiswouldnotbetrue.Appropriateresponsesincludeanysolidshapethatisskewed offacentralaxis.

•Use,afterparticipantshavesuggestedseveralobjects,a rectangular prism to demonstrate the different views of a non-right prism.

•Drawapictureofaskewedconeonthetransparencyandpointouthowtheviewsdifferfromtheexamplestheparticipantssketched.Includetheaxesontheviewstoemphasizetheskew.Sampledrawingsareshown below.

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

Top Front Side

Top Front Side

Apex of the cone

DISCUSS AND DO

•Explaintoparticipantsthattheynowwilllookatvarious two-dimensional views and try to reconstruct thethree-dimensionalobjectsfromwhichthepictureswere created.

•DisplayTransparency:Top,Side,andFrontand have participantstakeouttheirmatchingpages.

•Explaintoparticipantsthattheirtaskwillbetousetheir rainbow cubes to construct each shape pictured in three dimensions.

•AssignhalfthegrouptoworkontheshapesintheorderA,B,C.AssigntheotherhalftoworkontheshapesintheorderA,C,B.

•Directparticipantstosavethemodeltheycreatelastfor discussion purposes.

•Giveparticipants10–15minutestocompletetheactivity.

CONCLUDE

•Callthegroupstogether.

•HaveavolunteerpairbuildmodelBinthefrontofthe room where everyone can watch.

•RepeatformodelC.

•RefertoTransparency:Top,Side,andFrontAnswerKey to resolve any questions.

•Askifeveryonebuiltthemodelthesameway.Thereare alternate solutions, including those listed below.

◆ InmodelB,thetoptwocubesintheleftrearcolumn can be removed.

◆ InmodelC,thetoptwocubesinthefrontmiddlecolumns can be removed.

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Use rainbow cubes to build three-dimensional modelsof the three shapes described by the drawings below.

Is there only one way to build each model?

Top, Side, and Front

A.

B.

C.

front

front

front

side

side

side

top

top

top

Transparency: Top, Side, and Front

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_TG

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GEOMETRY AND MEASUREMENT OF SOLID FIGURESACTIvITY SET 4

•Askparticipantswhythismightbeso.(Eachoftheseshapes is a composite of other shapes, and the pictures did not provide enough information about eachofthepieces.)

•Suggesttoparticipantsthattheyhavenowseentheimportance of various details in moving between two- and three-dimensional representations of objects,forexample,theimportanceofaxesandtheimportance of the number of views for each element of a shape.

•DisplayTransparency:DrawingTwo-DimensionalViews andhaveparticipantstakeouttheirmatchingpages.

•Giveparticipants5–7minutestocompletethedrawings of the requested views.

•Havevolunteersdrawtheiranswersonthedisplayedtransparency.

•RefertoTransparency:DrawingTwo-DimensionalViewsAnswerKey to resolve any questions.

•DisplayTransparency:DrawingTwo-DimensionalViewsAnswerKey when all the answers have been resolved.

•Pointoutthedottedlinesrepresentingtheaxesonthe last two shapes and emphasize how they can be used to help define the views.

•Suggesttoparticipantsthattheycansketchsuchlineson shapes they need to reconstruct.

End of Top-Down view

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Drawing Three-Dimensional FiguresDraw top, front, and side views of each figure.

top front side

1.

2.

3.

4.

Drawing Two-Dimensional Views

Transparency: Drawing Two-Dimensional Views

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Drawing Three-Dimensional FiguresDraw top, front, and side views of each figure.

top front side

1.

2.

3.

4.

Drawing Two-Dimensional ViewsAnswer Key

Transparency: Drawing Two-Dimensional Views Answer Key

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional DevelopmentGEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Rectangular Prism

14 mm

6 mm

6 mm

All pages are live in the transparency file

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional DevelopmentGEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Triangular Prism

4 mm

3.46 mm

9 mm

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional DevelopmentGEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Use rainbow cubes to build three-dimensional modelsof the three shapes described by the drawings below.

Is there only one way to build each model?

Top, Side, and Front

A.

B.

C.

front

front

front

side

side

side

top

top

top

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional DevelopmentGEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Use rainbow cubes to build three-dimensional modelsof the three shapes described by the drawings below.

Is there only one way to build each model?

Top, Side, and FrontAnswer Key

A. There is only one way to build this model.

B. This is one of multiple ways to build this model.

C. This is one of multiple ways to build this model.

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional DevelopmentGEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Drawing Three-Dimensional FiguresDraw top, front, and side views of each figure.

top front side

1.

2.

3.

4.

Drawing Two-Dimensional Views

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional DevelopmentGEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 TRANS_MS_SG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Drawing Three-Dimensional FiguresDraw top, front, and side views of each figure.

top front side

1.

2.

3.

4.

Drawing Two-Dimensional ViewsAnswer Key

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

GlossaryGeometry and Measurement of Solid Figures

area The number of square units in a region.

base The bottom side or face of a solid shape.

circumscribed circle A circle is circumscribed around a polygon if every vertex of the polygon lies on the circle.

cone A solid that has a circular base with the surface formed by line segments joining every point on the circumference of the circle to a shared vertex.

cube A rectangular prism that has six square faces.

cylinder A three-dimensional figure that has two congruent, parallel, circular bases connected by a cylindrical surface.

edge A line segment formed by the intersection of two faces of a solid figure.

face A flat side of a solid shape.

hexahedron A six-sided polyhedron; a regular hexahedron is a cube.

inscribed circle A circle is inscribed within a polygon if every side of the polygon is tangent to the circle.

net A printed, two-dimensional pattern that can be cut and folded to make a solid shape.

octahedron An eight-sided polyhedron.

All glossary pages are live in Activity 1 BLM file

GEOMETRY AND MEASUREMENT OF SOLID FIGURES—AcTIvITY SET 4 Mid_SGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

polyhedron A closed, three-dimensional object with a (polyhedra, pl.) surface made up of polygons.

prism A polyhedron that has parallelogram sides and two parallel, congruent bases; includes right prisms and oblique prisms.

pyramid A polyhedron that has a polygon base and faces that are triangles.

rectangular prism A prism with parallel bases that are congruent rectangles.

solid A figure that has depth in addition to width and height.

sphere The set of all points in three-dimensional space that are equidistant from a specified point.

surface area The total area of the surface of a solid, including all faces and bases.

tetrahedron A pyramid with a triangular base.

triangular prism A prism with triangular bases.

vertex (vertices, pl.) A point common to two sides of an angle, two sides of a plane polygon, or three or more edges of a solid figure; the apex of a cone.

volume The cubic measure of the interior of the space of a three-dimensional figure.

Glossary (continued)