Three-Dimensional Figures 4

Problem 1A scenario is used whichprompts students to tape arectangle to a pencil and rotatethe pencil.PROBLEM 1Rectangular SpinnersYou and a classmate are starting a summer business, making spinning toys for smallchildren that do not require batteries and use various geometric shapes.They identify the threedimensional solid imageassociated with this rotationas a cylinder and relate thedimension s of the rectangle tothe dimensions of the imageof the solid. This activity isrepeated with a circle, anda triangle.Previously, you learned about rotations on a coordinate plane. You can also perform rotationsin three-dimensional space.1. You and your classmate begin by exploring rectangles.a. Draw a rectangle on an index card.b. Cut out the rectangle and tape it along the center to apencil below the eraser as shown.c. Hold on to the eraser with your thumb and index finger suchthat the pencil is resting on its tip. Rotate the rectangle byholding on to the eraser and spinning the pencil. You can getthe same effect by putting the lower portion of the pencilbetween both palms of your hands and rolling the pencil bymoving your hands back and forth.d. As the rectangle rotates about the pencil, the image of a three-dimensional solid isformed. Which of these solids most closely resembles the image formed by therotating rectangle?Groupingt Ask students to read theinformation. Discuss asa class.t Have students completeQuestions 1 with a partner.Then have students sharetheir responses as a class.4Figure 1Figure 2Figure 3Figure 4The image formed by the rotating rectangle closely resembles Figure 2.Guiding Questionsfor Share Phase,Question 1t If the rectangle was turnede. Name the solid formed by rotating the rectangle about the pencil.The solid formed by rotating the rectangle about the pencil is a cylinder.lengthwise and then taped,how would that affect theimage of the solid associatedwith the rotation?f. Relate the dimensions of the rectangle to the dimensions of this solid.The width of the rectangle is the radius of the cylinder. The length of therectangle is the height of the cylinder.in the middle, but tapedalong a side, how would thataffect the image of the solidassociated with the rotation?t If the rectangle was taped in a diagonal fashion to the pencil, how would thataffect the image of the solid associated with the rotation?t Will the image associated with this rotation always be a cylinder?338Chapter 4 Three-Dimensional Figures Carnegie Learningt If the rectangle wasn’t taped

GroupingHave students completeQuestion 2 with a partner.Then have students share theirresponses as a class.Guiding Questionsfor Share Phase,Question 2t If the circle was turned 90degrees and then taped, howwould that affect the imageof the solid associated withthe rotation?2. You and your classmate explore circles next.a. Draw a circle on an index card.b. Cut out the circle and tape it along the center to a pencil belowthe eraser as shown.c. Hold on to the eraser with your thumb and index finger suchthat the pencil is resting on its tip. Rotate the circle by holdingon to the eraser and spinning the pencil. You can get the sameeffect by putting the lower portion of the pencil between bothpalms of your hands and rolling the pencil by moving your handsback and forth.Remember, a circle is the set of all points that are equal distance from the center.A disc is the set of all points on the circle and in the interior of the circle.d. As the disc rotates about the pencil, the image of a three-dimensional solid isformed. Which of these solids most closely resembles the image formed by therotating disc?t If the circle wasn’t taped inthe middle, but taped slightlyto the right or to the left ofthe middle, how would thataffect the image of the solidassociated with the rotation?t If the circle was tapedalong its circumference tothe pencil, how would thataffect the image of the solidassociated with the rotation?Figure 1Figure 2Figure 3Figure 4The image formed by the rotating disc closely resembles Figure 4.4e. Name the solid formed by rotating the circle about the pencil.The solid formed by rotating the circle about the pencil is a sphere.t Will the image associatedwith this rotation always bea sphere?f. Relate the dimensions of the disc to the dimensions of this solid. Carnegie LearningThe radius of the disc is also the radius of the sphere.4.1 Rotating Two-Dimensional Figures through Space339

GroupingHave students completeQuestion 3 with a partner.Then have students share theirresponses as a class.Guiding Questionsfor Share Phase,Question 3t If the triangle was turned3. You and your classmate finish by exploring triangles.a. Draw a triangle on an index card.b. Cut out the triangle and tape it lengthwise along the center toa pencil below the eraser as shown.c. Hold on to the eraser with your thumb and index finger such thatthe pencil is resting on its tip. Rotate the triangle by holding on tothe eraser and spinning the pencil. You can get the same effect byputting the lower portion of the pencil between both palms of yourhands and rolling the pencil by moving your hands back and forth.d. As the triangle rotates about the pencil, the image of a three-dimensional solid isformed. Which of these solids most closely resembles the image formed by therotating triangle?sideways and then taped,how would that affect theimage of the solid associatedwith the rotation?t If the triangle wasn’t tapedin the middle, but tapedalong a side, how would thataffect the image of the solidassociated with the rotation?t If the triangle was tapedupside down to the pencil,how would that affect theimage of the solid associatedwith the rotation?4t Will the image associatedwith this rotation always bea cone?Figure 1Figure 2Figure 3Figure 4The image formed by the rotating triangle closely resembles Figure 3.e. Name the solid formed by rotating the triangle about the pencil.The solid formed by rotating the triangle about the pencil is a cone.f. Relate the dimensions of the triangle to the dimensions of this solid.The base of the triangle is the diameter of the cone. The height of the triangle isthe height of the cone. Carnegie LearningBe prepared to share your solutions and methods.340Chapter 4 Three-Dimensional Figures

Check for Students’ UnderstandingAssociate a word in the first column to a word in the second column. Explain your rTriangle—ConeThe image of a cone can be visualized by rotating a triangle on a pencil.Rectangle—CylinderThe image of a cylinder can be visualized by rotating a rectangle on a pencil.Circle—SphereThe image of a sphere can be visualized by rotating a circle on a pencil. Carnegie Learning44.1 Rotating Two-Dimensional Figures through Space340A

Carnegie Learning4340BChapter 4Three-Dimensional Figures

Cakes and Pancakes4.2Translating and StackingTwo-Dimensional FiguresLEARNING GOALSIn this lesson, you will:t Apply translations to two-dimensional planefigures to create three-dimensional solids.t Describe three-dimensional solids formedby translations of plane figuresthrough space.t Build three-dimensional solids by stackingcongruent or similar two-dimensionalplane figures.ESSENTIAL IDEASt Rigid motion is used in the process of Carnegie Learningredrawing two-dimensional plane figures asthree-dimensional solids.t Models of three-dimensional solids areformed using translations of plane figuresthrough space.t Models of two-dimensional plane figuresare stacked to create models of threedimensional solids.KEY TERMSt isometric papert right triangular prismt oblique triangular prismt right rectangular prismt oblique rectangular prismt right cylindert oblique cylinderTEXAS ESSENTIAL KNOWLEDGEAND SKILLS FOR MATHEMATICS(11) Two-dimensional and three-dimensionalfigures. The student uses the process skillsin the application of formulas to determinemeasures of two- and three-dimensional figures.The student is expected to:(C) apply the formulas for the total and lateralsurface area of three-dimensional figures,including prisms, pyramids, cones,cylinders, spheres, and composite figures,to solve problems using appropriate unitsof measure(D) apply the formulas for the volume ofthree-dimensional figures, includingprisms, pyramids, cones, cylinders,spheres, and composite figures, to solveproblems using appropriate unitsof measure341A

OverviewModels of two-dimensional figures are translated in space using isometric dot paper. Students analyzethe three-dimensional solid images associated with the translations. Technically, the translation of asingle point or collection of points changes the location of the point or collection of points. Applied to theactivities in this lesson, it is not an actual solid that results from translating the plane figure, rather animage to the eye that is associated with this movement. The activity that includes stacking the twodimensional models should provide opportunities for these types of discussions as well. Both right andoblique prisms and cylinders are used in this lesson. Carnegie Learning4341BChapter 4Three-Dimensional Figures

Warm Up1. Translate the triangle on the coordinate plane in a horizontal direction.y8642028 26 24 222468x222426282. Describe the translation of each vertex.Answers will vary.Each vertex was moved to the right 5 units.43. Translate the triangle on the coordinate plane in a vertical direction.y8642028 26 24 222468x Carnegie Learning222426284.2 Translating and Stacking Two-Dimensional Figures341C

4. Describe the translation of each vertex.Answers will vary.Each vertex was moved to the up 3 units.5. Translate the triangle on the coordinate plane in a diagonal direction.y8642028 26 24 222468x2224262846. Describe the translation of each vertex.Answers will vary.Each vertex was moved to the right 5 units and up 3 units.7. What do you suppose is the difference between translating a triangle on a coordinate plane, as youhave done in the previous questions, and translating a triangle through space?When a figure is translated on a coordinate plane, the vertices are moved in two directions, leftor right, and up or down. When a figure is translated through space, space is three-dimensionalso the vertices move in three directions, left or right, up or down, and backward or forward.341DChapter 4Three-Dimensional Figures Carnegie LearningAnswers will vary.

Cakes and Pancakes4.2Translating and StackingTwo-Dimensional FiguresLEARNING GOALSKEY TERMSIn this lesson, you will:t Apply translations to two-dimensional planefigures to create three-dimensional solids.t Describe three-dimensional solids formedby translations of plane figuresthrough space.t Build three-dimensional solids by stackingcongruent or similar two-dimensionalplane figures.t isometric papert right triangular prismt oblique triangular prismt right rectangular prismt oblique rectangular prismt right cylindert oblique cylinderYou may never have heard of isometric projection before, but you have probablyseen something like it many times when playing video games.4Isometric projection is used to give the environment in a video game a threedimensional effect by rotating the visuals and by drawing items on the screen usingangles of perspective.One of the first uses of isometric graphics was in the video game Q*bert, released in1982. The game involved an isometric pyramid of cubes. The main character, Q*bert,starts the game at the top of the pyramid and moves diagonally from cube to cube,causing them to change color. Each level is cleared when all of the cubes change color.Of course, Q*bert is chased by several enemies. Carnegie LearningWhile it may seem simple now, it was extremely popular at the time. Q*bert had hisown line of toys, and even his own animated television show!3414.2 Translating and Stacking Two-Dimensional Figures341

Problem 1A scenario in which studentsuse isometric paper to createthe images of three-dimensionalsolids on two-dimensionalpaper is the focus of thisproblem. The three-dimensionalsolids highlighted in thisproblem are both right andoblique; triangular prisms,rectangular prisms, andcylinders. Students concludethat the lateral faces of prismsare parallelograms. Rigidmotion helps students visualizehow models of solids canbe formed from models ofplane figures.PROBLEM 1These Figures Take the CakeYou can translate a two-dimensional figure through space to create amodel of a three-dimensional figure.1. Suppose you and your classmate want to design a cakewith triangular bases. You can imagine that the bottomtriangular base is translated straight up to create the toptriangular base.Recall thata translation is atransformation that“slides” each point of afigure the same distancein the same direction.a. What is the shape of each lateral face of this polyhedronformed by this translation?Each lateral face is a rectangle.Groupingb. What is the name of the solid formed by this translation?Have students completeQuestions 1 through 4 with apartner. Then have studentsshare their responses asa class.4The solid formed by this translation is a triangular prism.A two-dimensional representation of a triangular prism can be obtained by translating atriangle in two dimensions and connecting corresponding vertices. You can use isometricpaper, or dot paper, to create a two-dimensional representation of a three-dimensionalfigure. Engineers often use isometric drawings to show three-dimensional diagrams on“two-dimensional” paper.Guiding Questionsfor Share Phase,Questions 1 through 4t Are the sides or lateral facest What is a prism?t How are prisms named?t What is a triangular prism?t What are some properties ofa triangular prism?t How many units and inwhat direction did youtranslate the triangle on theisometric paper?t When you connected the corresponding vertices, do the sides appear to beparallel to each other?t Do the lateral faces appear to be parallelograms? How do you know?t Are you translating the triangle through space in a direction that isperpendicular to the plane containing the triangle? How do you know?t What is the difference between an oblique triangular prism and a righttriangular prism?342Chapter 4 Three-Dimensional Figures Carnegie Learningformed by parallel lines?How do you know?

2. Translate each triangle to create a second triangle. Use dashed line segments toconnect the corresponding vertices.a. Translate this triangle in a diagonal direction.b. Translate this right triangle in a diagonal direction.c. Translate this triangle vertically.d. Translate this triangle horizontally.43. What do you notice about the relationship among the line segments connecting thevertices in each of your drawings? Carnegie LearningThe line segments appear to be parallel to each other, and they are congruent.4.2 Translating and Stacking Two-Dimensional Figures343

GroupingHave students completeQuestions 5 through 9 with apartner. Then have studentsshare their responses asa class.When you translate a triangle through space in a direction that is perpendicular to the planecontaining the triangle, the solid formed is a right triangular prism. The triangular prismcake that you and your classmate created in Question 1 is an example of a right triangularprism. When you translate a triangle through space in a direction that is not perpendicular tothe plane containing the triangle, the solid formed is an oblique triangular prism.An example of an oblique triangular prism is shown.Guiding Questionsfor Share Phase,Questions 5 through 9t Are the sides or lateral facesformed by parallel lines?How do you know?4. What is the shape of each lateral face of an oblique triangular prism?Each lateral face is a parallelogram.t What is the differencebetween a triangular prismand a rectangular prism?5. Suppose you and your classmate want to design a cake with rectangular bases. Youcan imagine that the bottom rectangular base is translated straight up to create the toprectangular base.t What do a triangular prismand rectangular prism havein common?t What are some properties ofa rectangular prism?t How is isometric dot paper4different than Cartesiangraph paper?a. What is the shape of each lateral face of the solid figure formed by this translation?t How does isometricEach lateral face would be a rectangle.dot paper help visualizethree dimensions?t How many units and inwhat direction did youtranslate the rectangle on theisometric paper?b. What is the name of the solid formed by this translation?The solid formed by this translation is a rectangular prism.corresponding vertices,do the sides appear to beparallel to each other?t Do the lateral faces appear tobe parallelograms? How doyou know?t Are you translating the rectangle through space in a direction that isperpendicular to the plane containing the rectangle? How do you know?t What is the difference between an oblique rectangular prism and a rightrectangular prism?344Chapter 4 Three-Dimensional Figures Carnegie Learningt When you connected the

A two-dimensional representation of a rectangular prism can be obtained by translating arectangle in two dimensions and connecting corresponding vertices.6. Draw a rectangle and translate it in a diagonal direction to create a second rectangle.Use dashed line segments to connect the corresponding vertices.7. Analyze your drawing.a. What do you notice about the relationship among the line segments connecting thevertices in the drawing?The line segments appear to be parallel to each other, and they are congruent.b. What is the name of a rectangular prism that has all congruent sides?A rectangular prism with all congruent sides is a cube.c. What two-dimensional figure would you translate to create arectangular prism with all congruent sides?To create a rectangular prism with all congruent sides,I would translate a square.4What othershapes can Itranslate to createthree-dimensionalfigures? Carnegie Learningd. Sketch an example of a rectangular prism with allcongruent sides.4.2 Translating and Stacking Two-Dimensional Figures345

When you translate a rectangle through space in a direction that is perpendicular to theplane containing the rectangle, the solid formed is a right rectangular prism. Therectangular prism cake that you and your classmate created in Question 8 is an example of aright rectangular prism. When you translate a rectangle through space in a direction that isnot perpendicular to the plane containing the rectangle, the solid formed is an obliquerectangular prism.8. What shape would each lateral face of an oblique rectangular prism be?Each lateral face would be a parallelogram.9. Sketch an oblique rectangular prism.Grouping10. Suppose you and your classmate want to design a cake with circular bases. You canimagine that the bottom circular base, a disc, is translated straight up to create the topcircular base.Have students completeQuestions 10 through 14 witha partner. Then have studentsshare their responses asa class.4Guiding Questionsfor Share Phase,Questions 10through 14t Are the sides or lateral facesa. What shape would the lateral face of the solid figure formed by this translation be?The lateral face would be a rectangle.formed by parallel lines?How do you know?b. What is the name of the solid formed by this translation?The solid formed by this translation is a cylinder.between a prism anda cylinder?t What do a prism and cylinderhave in common?t What are some properties ofa cylinder?t When you connected thecorresponding tops andbottoms of the cylinder,do the sides appear to beparallel to each other?346t Do the lateral faces appear to be parallelograms? How do you know?t Are you translating a circle through space in a direction that is perpendicular tothe plane containing the circle? How do you know?t What is the difference between an oblique cylinder

Surface Area 11.A 11.C 1 Students review surface area and lateral surface area and develop formulas for determining the total and lateral surface areas of various solid "gures. X X 4.7 Using Surface Area and Volume Formulas 10.A 10.B 11.C 11.D 1 This lesson provides student with the opportunity to solve