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Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTab 5: GeometryTable of ContentsMaster Materials List5-iiPicture This!Handout 1: Part I DiagramsHandout 2: Drawings in Two-DimensionsHandout 3: Isometric DrawingsTransparency 1Transparency 2Transparency 3Transparency 4Transparency 5Transparency 6Transparency 7Transparency 85-15-45-65-95-115-125-135-145-155-165-175-18Texas “T” ActivityHandout 1: Part IHandout 2: Part ITransparency 1: Part II Activity ATransparency 2: Part II Activity ATransparency 3: Part II Activity BTransparency 4: Part II Activity BTransparency 5: Part II Activity CTransparency 6: Part II Activity CHandout 3: Part III5-195-225-235-245-255-265-275-285-295-30Tab 5: Geometry: Table of Contents5-i

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTab 5: GeometryMaster Materials ListThree-dimensional cube modelGrid paperSnap cubesRulersCalculatorMarkersPicture This!: Transparencies and handoutsTexas “T” Activity: Transparencies and handoutsThe following materials are not in the notebook. They can be accessed on the CDthrough the links below.Isometric dot paper (1 per participant) and transparencies (1 per group)Tab 5: Geometry: Master Materials List5-ii

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityActivity:Picture This!TEKS:This activity supports teacher content knowledge needed for:(G.6) Dimensionality and the geometry of location. The studentanalyzes the relationship between three-dimensional geometric figuresand related two-dimensional representations and uses theserepresentations to solve problems.The student is expected to:(C) use orthographic and isometric views of three-dimensionalgeometric figures to represent and construct threedimensional geometric figures and solve problems.Overview:This activity encourages participants to explore and draw orthographicand isometric views. It explores some non-technical aspects oforthographic drawings and the relationship between isometric andorthographic drawings.Materials:Isometric dot paperGrid paperSnap cubesRulersTransparencies: 1-8 (pages 5-11 – 5-18)Isometric dot paper (1 per participant) and transparencies (1 per group)Three-dimensional cube modelHandout 1 (pages 5-4 – 5-5)Handout 2 (pages 5-6 – 5-8)Handout 3 (pages 5-9 – tricorthographicperspectivevanishing pointGrouping:Each participant should complete these activities. Participants maywork in groups of 3-4 so that they can help one another.Time:1 to 1 ½ hoursPicture This!5-1

Mathematics TEKS Refinement 2006 – 9-12Lesson:ProceduresPart I:1. Distribute Handout 1, Part 1 Diagrams,(pages 5-4 – 5-5) to participants and atleast one model of a cube to eachgroup. Have participants determinewhich of the diagrams could represent acube.2. Provide participants with Handout 2,Drawings in Two-Dimensions, (pages5-6 – 5-8) terms with definitions.Discuss and define perspective views,vanishing point, axonometric views andorthographic views.Have participants decide in their groupswhich type of drawing each diagramfrom Handout 1 represents. Groupsshould be prepared to share theirchoices with the whole group.Part II:3. Distribute two pre-made snap cubeexamples, a transparency of isometricdot paper (see Materials List for a link tothis), and transparency pens to eachgroup; also provide a sheet of isometricdot paper for each participant.Have each participant sketch theisometric view of the given example,and then have the group sketch theview onto their transparency. Eachgroup will share its result.4. Give participants a sheet of grid paperand have them sketch the threeorthographic views (front, side and top).Tarleton State UniversityNotesParticipants should be in groups fordiscussion; allow time for discovery that allof these diagrams represent a cube.Ask the participants why the diagrams ofthe cube in Part I look so different.The diagrams show the cube fromdifferent angles and perspectives.(See discussion of vocabulary below)Transparencies 1-6 (pages 5-11 – 5-16)for each diagram are provided.Be sure to bring the participants backtogether for a whole class discussion.The trainer can make two of the givensample shapes, or make two simpleshapes of his/her own.Optional: To save prep time, have theparticipants build the models shown onTransparency 7 (page 5-17).Volunteers may share their sketches withthe class.Give each group a new set of pre-made Trainer can choose any of the remaining(more challenging) examples and repeat sample shapes, or create their own.the process.Picture This!5-2

Mathematics TEKS Refinement 2006 – 9-12ProceduresPart III:6. Distribute Handout 3, IsometricDrawings, (pages 5-9 – 5-10). Haveparticipants practice creating isometricdrawings from orthographic viewswithout the aid of three-dimensionalmodels.Bring the participants back together fora whole class discussion and to analyzetheir drawings.Questions to ask: “What difference do you see betweenthe orthographic drawings of diagram1 and diagram 2?” “How can the orthographic drawingsbe changed to include additionalinformation differentiating diagram 1and diagram 2?”Tarleton State UniversityNotesOn Handout 3, what is the differencebetween the orthographic drawings ofdiagram 1 and diagram 2? (see page 510)They are the same!There are easy solutions to this dilemma.Try adding segments in the drawing tooutline the cube faces seen in each view.Indicate different distances from theviewer to different parts of the orthogonaldrawings by using different line weights.Faces nearest the viewer can be outlinedwith dark segments. Faces one cubefarther away can be a medium weight andfaces two cubes farther away can be alight or dashed segment. Alternatively,add segments in the top view to outline thecube faces and number the squares toindicate the number of cubes in the“stack.” (See Transparency 8 (page 5-18)for solutions) These drawings are called“mats.”Show a Transparency 8 after discussion.Extensions:Picture This!After students have learned to draw orthographic and isometric figures,they can investigate surface area and volume. Initially, they can usesnap cube models, but as visualization skills improve, there should beless dependence on models and more interpretation from the diagrams.5-3

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Part 1 DiagramsWhich of these two-dimensional pictures could be used to represent a cube?Picture This!Handout 1-15-4

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Part 1 Diagram SolutionsWhich of these two-dimensional pictures could be used to represent a thographicIsometricPicture This!Handout 1-25-5

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Drawings in Two-DimensionsUse the definitions below to determine which of these descriptions best fits each of thesketches given in Part I.Perspective View: The technique of portraying solid objects and spatial relationships ona flat surface. There are numerous methods of depicting an object, depending uponthe purpose of the drawing.Vanishing Point: In an artistic perspective drawing, receding parallel lines (lines that runaway from the viewer) converge at a vanishing point on the horizon line. Thismaintains a realistic appearance of the object depicted, even as the vantage pointchanges.Orthographic Drawing: Ortho means “straight” and the views in orthographic drawingsshow the faces of a solid as though you are looking at them “head-on” from the top,front or side. Orthographic drawings are used in engineering drawings to convey allthe necessary information of how to make the part to the manufacturing department.The line of sight is perpendicular to the surfaces of the object. Two conventions areused in technical drawings. These are first angle and third angle, which differ only inposition of the plan (top, front, and side views). They are derived from the method ofprojection used to transfer two-dimensional views onto an imaginary, transparentbox surrounding the object being drawn. This is not a distinction the state of Texashas seen fit to recognize in the TEKS.appears astopPicture This!frontright sideHandout 2-15-6

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityA common use of a “top” orthographicview is the floor plan of a house or otherbuilding.Axonometric drawing: A two-dimensional drawing showing three dimensions of anobject. The vantage point is not perpendicular to a surface of the object beingdrawn. Axonometric means, “to measure along the axes.” There are threetypes: isometric, dimetric, and trimetric.Isometric Drawing: A two-dimensional drawing thatshows three sides of an object in one view. Thevantage point is 45 to the side and above theobject being viewed. The resulting angle betweenany two axes appears to be 120 . Isometricmeans, “one measure.” In this view, 90 anglesappear to be 120 or 60 .120 120 120 Video games, such as SimCity 2000,frequently use isometric drawingsCourtesy Electronic Arts, Inc.Picture This!Handout 2-25-7

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityZDimetric drawing: A two-dimensional drawingsimilar to an isometric drawing, but with onlytwo of the resulting angles between the axeshaving the same apparent measure. Dimetricmeans, “two measures.”130 130 100 XYTrimetric drawing: Again, A two-dimensionaldrawing similar to an isometric drawing, butwith none of the resulting angles betweenthe axes having the same apparentmeasure. Trimetric means, “threemeasures.”Picture This!Handout 2-35-8

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Isometric DrawingsMake orthographic drawings of each diagram. You should include drawings from thetop, front, and right side.Diagram 1Picture This!Diagram 2Handout 3-15-9

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Isometric Drawings Solutionstopfrontright sidetopfrontright sideThe orthogonal drawings are the same!There are easy solutions to this dilemma. Try adding segments in the drawing to outlinethe cube faces seen in each view. Indicate different distances from the viewer todifferent parts of the orthogonal drawings by using different line weights. Faces nearestthe viewer can be outlined with dark segments. Faces one cube farther away can be amedium weight and faces two cubes farther away can be a light or dashed segment.Alternatively, add segments in the top view to outline the cube faces and number thesquares to indicate the number of cubes in the “stack.”Picture This!Handout 3-25-10

Mathematics TEKS Refinement 2006 – 9-12Picture This!Tarleton State UniversityTransparency 15-11

Mathematics TEKS Refinement 2006 – 9-12Picture This!Tarleton State UniversityTransparency 25-12

Mathematics TEKS Refinement 2006 – 9-12Picture This!Tarleton State UniversityTransparency 35-13

Mathematics TEKS Refinement 2006 – 9-12Picture This!Tarleton State UniversityTransparency 45-14

Mathematics TEKS Refinement 2006 – 9-12Picture This!Tarleton State UniversityTransparency 55-15

Mathematics TEKS Refinement 2006 – 9-12Picture This!Tarleton State UniversityTransparency 65-16

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Materials: snap cube shapes, grid paper, isometric paper, and rulersPrepare orthographic and isometric drawings of each shape.Sample shapes:Picture This!Transparency 75-17

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPICTURE THIS!Isometric Drawingstopfrontright sidetopfrontright sideMatsPicture This!Transparency 85-18

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityActivity:Texas “T” ActivityTEKS:This activity supports teacher content knowledge needed for:(G.2) Geometric Structure. The student analyzes geometricrelationships in order to make and verify conjectures.The student is expected to:(A) use constructions to explore attributes of geometric figuresand to make conjectures about geometric relationships.(G.3) Geometric Structure. The student applies logical reasoning tojustify and prove mathematical statements.The student is expected to:(D) use inductive reasoning to formulate a conjecture.(G.5) Geometric Patterns. The student uses a variety ofrepresentations to describe geometric relationships and solveproblems.The student is expected to:(A) use numeric and geometric patterns to develop algebraicexpressions representing geometric properties.(B) use numeric and geometric patterns to make generalizationsabout geometric properties, including properties of polygons,ratios in similar figures and solids, and angle relationships inpolygons and circles.(G.6) Dimensionality and the geometry of location. The studentanalyzes the relationship between three-dimensional geometric figuresand related two-dimensional representations and uses theserepresentations to solve problems.The student is expected to:(B) use nets to represent and construct three-dimensionalgeometric figures.(G.11) Similarity and the geometry of shape. The student appliesthe concepts of similarity to justify properties of figures and solveproblems.The student is expected to:(A) use and extend similarity properties and transformations toexplore and justify conjectures about geometric figures.(B) use ratios to solve problems involving similar figures.(D) describe the effect on perimeter, area, and volume when oneor more dimensions of a figure are changed and apply thisidea in solving problems.Texas “T” Activity5-19

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityOverview:This activity encourages participants to explore patterns in height,surface area, and volume of similar figures when one or moredimensions are changed.Materials:Snap cubesCalculatorTransparencies 1-6 (pages 5-24 – 5-29)MarkersHandout 1 (page 5-22)Handout 2 (page 5-23)Handout 3 (pages 5-30 – 5-34)Grouping:Groups of 3-4Time:1 ½ hoursLesson:ProceduresPart I:1. Distribute Handout 1, Part I, (page 5-22) of theactivity to each participant. Distributeapproximately 250 snap cubes and a calculatorto each group.Each participant builds his/her own “T” #1 andcompletes handout 1 of Part I of the activity.As a group, participants build additional “Ts” andcomplete Handout 2 of Part I (page 5-23).2.NotesThis activity will help participantsto discover the relationshipbetween linear ratios, arearatios, and volume ratios whensimilar three-dimensional objectsare built.Participants should work alone tobuild the first “T” and answer thequestions, then discuss theiranswers in their groups.Participants should then discusstheir answers to page 1 beforebeginning to work as a group tocomplete page 2.Bring the class together for a whole groupdiscussion of the findings for Part I.Questions to ask: “Did anyone build all 4 ‘Ts’? If not, why not?” “Were you able to complete the table withoutbuilding all the models?” “Did you notice any patterns as you completedthe table?” “How did you describe the patterns in terms ofn?”Texas “T” Activity5-20

Mathematics TEKS Refinement 2006 – 9-12ProceduresPart II:3. Using a jigsaw cooperative learning procedure,assign each group Activity A, Activity B, orActivity C [Transparencies 1 & 2 (pages 5-24 –5-25), Transparencies 3 & 4 (pages 5-26 – 527), or Transparencies 5 & 6 (pages 5-28 – 529)] and provide each group with theappropriate transparencies for its assignedactivity.Tarleton State UniversityNotesSelect one group per activity toactually present its results. Askfor discussion from other groupswho completed the same activity.Each group should select a reporter to completethe transparency for the assigned activity, andbe prepared to present to the entire group.Part III:4. Distribute Handout 3, Part III, (pages 5-30 – 534), “One and Two-Dimensional Change,” toeach participant. Have participants work in theirgroups to complete Part III. Then bring theclass back together to report their findings anddiscuss them as a whole group.Assign each group a piece ofPart III to report out. Selectgroups that did not report out forthe previous parts in step 3.Questions to ask: “How did what you discovered in Activities A,B, and C relate to what you did in Part III?” “When is the change in dimension(s) linear?Quadratic?”5.Lead participants to begin a discussion aboutthe “Think About It” questions in Part III.Texas “T” ActivityAfter the group determines that achange in the radius of a canwould not be a linear change involume, ask, “What wouldhappen if we made a onedimensional change in asphere?”5-21

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part IUsing snap cubes with the edge of one cube representing 1 unit of length, build the first“T” like the one shown in the illustration. The “T” is 4 units tall and 3 units wide.Use the “T” to complete the table below:N “T”NUMBERHEIGHT OF “T”TOTAL SURFACEAREAVOLUME“T” #1What do you think will happen to the height of the “T” when the dimensions of each partare doubled? tripled?What do you think will happen to the surface area of the “T” when the dimensions ofeach part are doubled? tripled?What do you think will happen to the volume of the “T” when the dimensions of eachpart are doubled? tripled?Texas “T” ActivityHandout 15-22

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part IIn your group, build at least the next two “Ts”, where n represents the height of the “T.”To build each “T”, take the dimensions of “T” #1 and multiply by n (n 2, n 3, n 4).N “T”NUMBERHEIGHT OF “T”TOTAL SURFACEAREAVOLUME“T” #2“T” #3“T” #4“T” NTexas “T” ActivityHandout 25-23

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part IIActivity A: Comparing Height in Similar FiguresUsing the completed tables from Part I, fill in the ratios for the following table. Recordfractions in lowest terms. Use the table data to answer the questions below.RATIO OF FIGURESRATIO OF HEIGHT“T” #1:”T” #2“T” #2:”T” #3“T” #1:”T” #3“T” #3:”T” #4“T” #1:”T” #4“T” #2:”T” #51. Was your prediction in Part I about the height of the “Ts” accurate? Explain youranswer.2. What pattern do you observe about the simplified ratios of the heights?3. Using what you observed about the patterns, can you determine the ratio of theheight of “T” #20 to the height of “T” #12?4. If the heights of the “Ts” were in the ratio of 100/40, which “Ts” would you becomparing? Explain your answer.5. Determine which “T” has a height of 60 units. Explain your answer.Texas “T” ActivityTransparency 15-24

Mathematics TEKS Refinement 2006 – 9-12Tarleton State University6. What is the ratio of the height of “T” #48 to “T” #16?7. If you know the height of “T” #1, how can you find the height of “T” #N?Based upon your results, if the ratio of two similar figures is 4/5, the ratio of the heightswould be .Write a rule based on your results:If the ratio of the heights in two similar figures is m/n, the ratio of their heights would be.Texas “T” ActivityTransparency 25-25

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part IIActivity B: Comparing Height and Surface Area in Similar FiguresUsing the completed table from Part I, fill in the ratios for the following table. Recordfractions in lowest terms. Use the table data to answer the questions below.RATIO OF FIGURESRATIO OF HEIGHTRATIO OF SURFACE AREA“T” #1:”T” #2“T” #2:”T” #3“T” #1:”T” #3“T” #3:”T” #4“T” #1:”T” #4“T” #2:”T” #51. Was your prediction in Part I about the surface areas of the “Ts” accurate?Explain your answer.2. What pattern do you observe about the simplified ratios of the heights andsurface areas?3. Using what you observed about the patterns, can you determine the ratio of thesurface area of “T” #20 to the surface area of “T” #12?4. If the surface area of the “Ts” were in the ratio of 256/49, which “Ts” would yoube comparing? Explain your answer.5. Determine which “T” has a surface area of 3,744 square units. Explain youranswer.Texas “T” ActivityTransparency 35-26

Mathematics TEKS Refinement 2006 – 9-12Tarleton State University6. What is the ratio of the surface area of “T” #50 to “T” #15?7. If you know the surface area of “T” #1, how can you find the surface area of “T”#N?Based upon your results, if the ratio of two similar figures is 4/5, the ratio of the heightswould be and the ratio of the surface area would be.Write a rule based on your results:If the ratio of the heights in two similar figures is m/n, then the ratio of their surface areawould be .Texas “T” ActivityTransparency 45-27

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part IIActivity C: Comparing Total Surface Area and Volume in Similar FiguresUsing the completed table from Part I, fill in the ratios for the following table. Recordfractions in lowest terms. Use the table data to answer the questions below.RATIO OF FIGURESRATIO OFHEIGHTRATIO OFSURFACE AREARATIO OFVOLUME“T” #1:”T” #2“T” #2:”T” #3“T” #1:”T” #3“T” #3:”T” #4“T” #1:”T” #4“T” #2:”T” #51. Was your prediction in Part I about the volumes of the “Ts” accurate? Explainyour answer.2. What pattern do you observe about the simplified ratios of the heights, surfaceareas and volumes?3. Using what you observed about the patterns, can you determine the ratio of thevolume of “T” #20 to the height of “T” #12?4. If the volume of the “Ts” were in the ratio of 1000/512, which “Ts” would you becomparing? Explain your answer.Texas “T” ActivityTransparency 55-28

Mathematics TEKS Refinement 2006 – 9-12Tarleton State University5. Determine which “T” has a volume of 34,992 cubic units? Explain your answer.6. What is the ratio of the volume of “T” #50 to “T” #15?7. If you know the volume of “T” #1, how can you find the volume of “T” #N?8. What happens to the volume if all the dimensions of the “T” are reduced by onehalf? Explain your answer.Based upon your results, if the ratio of the heights of two similar figures is 4/5, the ratioof the surface area would be , and the ratio of the volume would be.Write a rule based on your results:If the ratio of the heights in two similar figures is m/n, the ratio of their surface areawould be and the ratio of their volume would be .Texas “T” ActivityTransparency 65-29

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part IIIOne and Two-Dimensional ChangeThus far we have examined the effect on surface area and volume of a figure if alldimensions are changed by the same factor. What do you suppose will happen tosurface area and volume if only one dimension is changed?For this exercise, let’s use a simpler figure.l lengthw widthh heightVolume of a rectangular prism: V lwhSurface area of a rectangular prism: SA 2(lw) 2(wh) 2(hl)Volume:A. Change in one dimension:Given a rectangular prism with a length of 2, width of 2 and a height of 1;1. Without changing the length and width, change the height by a factor of n andcomplete the following table.nVolume1423456n2. Enter the data into lists in your calculator as L1 and L2 and plot the data.Texas “T” ActivityHandout 3-15-30

Mathematics TEKS Refinement 2006 – 9-12Tarleton State University3. Write in words the pattern you observe in the volumes.4. Write an algebraic function for the pattern.5. Without changing the length and height, change the width by a factor of n andcomplete the following table.nVolume1423456n6. Write in words the pattern you observe in the volumes.7. Write an algebraic function for the pattern.8. Why is the pattern the same as #3?Texas “T” ActivityHandout 3-25-31

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityB. Change in two dimensionsGiven a rectangular prism with a length of 2, width of 2 and a height of 1;1. Without changing the length, change the height and width by a factor of n andcomplete the following table.nVolume1234546n2. Enter the data into lists in your calculator as L1 and L3 and plot the data.3. Write in words the pattern you observe in the volumes.4. Write an algebraic function for the pattern.5. Write a function to predict the volume of our figure if all three dimension changeby factor of n.Texas “T” ActivityHandout 3-35-32

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversitySurface Area:A. Change in one dimension:Given a rectangular prism with a length of 2, width of 2 and a height of 1;1. Without changing the length and width, change the height by a factor of n andcomplete the following table.nSurfaceArea123456n2. Enter the data into lists in your calculator as L1 and L2 and plot the data.3. Write in words the pattern you observe in the surface area.4. Write an algebraic function for the pattern.Texas “T” ActivityHandout 3-45-33

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityB. Change in two dimensions:Given a rectangular prism with a length of 2, width of 2 and a height of 1;1. Without changing the length, change the height and width by a factor of n andcomplete the following table.nSurfaceArea123456n2. Enter the data into lists in your calculator as L1 and L2 and plot the data.3. Write in words the pattern you observe in the volumes.4. Write an algebraic function for the pattern.Think about it:Does a one dimensional change in any 3-dimensional figure create a linear change involume? What if the figure were a can?Texas “T” ActivityHandout 3-55-34

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityTEXAS “T”: Part I SOLUTIONSHEIGHT OF “T”TOTALSURFACEAREA“T” #14266“T” #28104 (26*22)48 (6*23)“T” #312234 (26*32)162 (6*33)“T” #416416 (26*42)384 (6*43)“T” N4N26N26N3N “T”NUMBERTexas “T” ActivityVOLUME5-35

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPart IIActivity A: Comparing Height in Similar FiguresRATIO OF FIGURESRATIO OF HEIGHT“T” #1:”T” #24/8 1/2“T” #2:”T” #38/12 2/3“T” #1:”T” #34/12 1/3“T” #3:”T” #412/16 3/4“T” #1:”T” #44/16 1/4“T” #2:”T” #58/20 2/51. Was your prediction about the height of the “Ts” accurate? Explain youranswer. Participants should predict that the height doubles when thedimensions are doubled and triples when the dimensions are tripled.2. What pattern do you observe about the simplified ratios of the heights?The simplified ratio is the same as the ratio of the “Ts”. For example, “T” 2:”T”3 8/12 2/33. Using what you observed about the patterns, can you determine the ratio ofthe height of “T” 20 to the height of “T” 12? The ratio is 20/12.4. If the heights of the “Ts” were in the ratio of 100/40, which “Ts” would yoube comparing? Explain your answer. You would be comparing “T” 25 to “T”10. 100/4 25 and 40/4 10.5. Determine which “T” has a height of 60 units. Explain your answer. “T” 15has a height of 60 since 60 divided by 4 equals 15.6. What is the ratio of the perimeters of “T” 48 to “T” 16? The ratio is 48/16which simplifies to 3/1.Texas “T” Activity5-36

Mathematics TEKS Refinement 2006 – 9-12Tarleton State University7. If you know the height of “T” 1, how can you find the height of “T” N? Ntimes the height of “T” 1Based upon your results, if the ratio of two similar figures is 4/5, the ratio of theheights would be 4/5 .Write a rule based on your results:If the ratio of two similar figures is m/n, the ratio of their heights would bem/n .Texas “T” Activity5-37

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPart IIIActivity B: Comparing Height and Surface Area in Similar FiguresRATIO OF HEIGHTRATIO OF SURFACEAREA“T” 1:”T” 24/8 1/226/104 1/4“T” 2:”T” 38/12 2/3104/234 4/9“T” 1:”T” 34/12 1/326/234 1/9“T” 3:”T” 412/16 3/4234/416 9/16“T” 1:”T” 44/16 1/426/416 1/16“T” 2:”T” 58/20 2/5104/650 4/25RATIO OF FIGURES1. Was your prediction about the surface areas of the “Ts” accurate? Explainyour answer. The participants should predict that the surface area will increaseby the original surface area times the square of the dimensional change (height).2. What pattern do you observe about the simplified ratios of the heights andsurface areas? The ratio of the surface area is equal to the ratio of the heightssquared.3. Using what you observed about the patterns, can you determine the ratio ofthe surface area of “T” 20 to the surface area of “T” 12? 400/144 25/94. If the surface area of the “Ts” were in the unsimplified ratio of 256/49,which “Ts” would you be comparing? Explain your answer. You would becomparing “T” 16 to “T” 7. You take the square root of 256 and 49 to determinethe answer.5. Determine which “T” has a surface area of 3,744 square units. Explain youranswer. 3,744 26 N2Divide by 26 and take the square root. “T” 12 has asurface are of 3,744.6. What is the ratio of the surface area of “T” 50 to “T” 15? 2500/225 100/9Texas “T” Activity5-38

Mathematics TEKS Refinement 2006 – 9-12Tarleton State University7. If you know the surface area of “T” 1, how can you find the surface area of“T” N? 26N2Based upon your results, if the ratio of two similar figures is 4/5, the ratio of theheights would be 4/5 and the ratio of the surface area would be16/25 .Write a rule based on your results:If the ratio of two similar figures is m/n, the ratio of their heights would bem/n and the ratio of their surface area would be m2/n2 .Texas “T” Activity5-39

Mathematics TEKS Refinement 2006 – 9-12Tarleton State UniversityPart IIActivity C: Comparing Total Surface Area and Volume of Similar FiguresRATIO OFFIGURESRATIO OFHEIGHTRATIO OFSURFACE AREARATIO OFVOLUME“T” 1:”T” 24/8 1/226/104 1/46/48 1/8“T” 2:”T” 38/12 2/3104/234 4/948/162 8/27“T” 1:”T” 34/12 1/326/234 1/96/162 1/27“T” 3:”T” 412/16 3/4234/416 9/16162/384 27/64“T” 1:”T” 44/16 1/426/416 1/166

Overview: This activity encourages participants to explore and draw orthographic and isometric views. It explores some non-technical aspects of orthographic drawings and the relationship between isometric and orthographic drawings. Materials: Isometric dot paper Grid paper Snap cubes Rulers Transparencies: 1-8 (pages 5-11 – 5-18)

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