1-2 Drawings, Nets, And Other Models

1-2Drawings, Nets, andOther Models1-21. Plan1What You’ll Learn212345 To draw nets for three-1.dimensional figuresIsometric DrawingOrthographic DrawingFoundation DrawingIdentifying Solids From NetsDrawing a NetTo make a foundationdrawing, as in Example 3Math BackgroundGeometric principles are thebasis of drawing techniques.For example, isometric dot paperis a pattern of the vertices ofequilateral triangles and caneasily be used to draw equilateraltriangles and hexagons thattesselate a plane.AB C. . . And Why1Lesson 1-1Draw the next figure in each sequence. See margin.orthographic drawingsExamplesGO for HelpCheck Skills You’ll Need To make isometric andATo make isometric andorthographic drawingsTo draw nets for threedimensional figuresAObjectivesC DD E2.OXH OXNew Vocabulary isometric drawing orthographic drawing foundation drawing netDrawing Isometric and Orthographic ViewsYou will study both two-dimensional and three-dimensional figures in geometry.A drawing on a piece of paper is a two-dimensional object. It has length and width.Your textbook is a three-dimensional object. It has length, width, and height.Representing a three-dimensional object on a twodimensional surface requires special techniques.Vocabulary TipIn Greek, isos means“equal” and metronmeans “measure.”In an isometric drawing,all 3-D measurementsare scaled equally.You can make an isometric drawing on isometric dotpaper to show three sides of a figure from a corner view.The simple drawing of a refrigerator at the right is anisometric drawing.More Math Background: p. 2CLesson Planning andResources11. Answers mayvary. Sample:EXAMPLEIsometric DrawingMake an isometric drawing of the cube structure at the left.See p. 2E for a list of theresources that support this lesson.Isometric drawing:PowerPointBell Ringer PracticeFrontCheck Skills You’ll NeedStep 1For intervention, direct students to:Finding and Using a PatternQuick CheckLesson 1-1Extra Skills, Word Problems,Proof Practice, Ch. 1E B1 On isometric dot paper,make an isometric drawingof the cube structure.See back of book.H1010 cm10 cmChapter 1 Tools of Geometry10 cm47 cmSpecial Needs10 cm10Step 3 A1, 2.Step 2htRigBelow LevelL1Have students reproduce the net in Example 4 usingscissors and graph paper.learning style: tactileL2Provide cubes for students to examine as they workthrough Examples 1–3.learning style: tactile

2. TeachAn orthographic drawing is another way to show a three-dimensional figure.It shows a top view, front view, and right-side view.2EXAMPLEGuided InstructionOrthographic DrawingMake an orthographic drawing from the isometric drawing at the left.Isometric drawing:2Orthographic drawing:nthtRig3TopDashed linesshow hiddenedges.FrontQuick CheckCareersArchitects use orthographicdrawings for floor plans. Ask: Afloor plan of your school buildingwould show which view? topSolid linesshow visibleedges.FroEXAMPLEEXAMPLEError PreventionBecause a foundation drawingshows the base, students maythink there is a bottom view.Remind them that the top viewshows the base.RightPowerPoint2 Make an orthographic drawing from this isometric drawing.See back of book.Additional Examples1 Make an isometric drawingof the cube structure below.FrontA foundation drawing showsthe base of a structure and theheight of each part. A foundationdrawing of the Sears Tower isshown at the right.EXAMPLE498965109 10989654989The Sears Toweris made up ofnine sections.The numbers tellhow many storiestall each section is.Foundation DrawingFroIsometric drawing:2 Make an orthographic drawingfrom the isometric drawing inAdditional Example 1.Connection3The foundation drawingshows four heights in thenine sections of the SearsTower in Chicago, Illinois.221FrontQuick CheckhtRig1FrontFront3 a. How many cubes would you use to make the structure in Example 3? 9 cubesb. Critical Thinking Which drawing did you use to answer part (a), the foundationdrawing or the isometric drawing? Explain.Answers may vary. Sample: The foundation drawing; you can justadd the five numbers.Lesson 1-2 Drawings, Nets, and Other ModelsAdvanced LearnersHave students research the Platonic Solids orArchimedean Solids and draw their nets. Students canthen make models of these solids.learning style: verbalHave each student bring an empty cereal or othertype of box to cut into a net. Clarify that more thanone net or pattern can be drawn for a given solid.learning style: tactileTopRight3 Create a foundation drawingfor the isometric drawing inAdditional Example 1.11English Language Learners ELLL4RiRightReal-WorldFoundation drawing:tghntMake a foundation drawing for the isometric drawing at the left.21Right3htRig1Front11

Guided Instruction21Nets for Three-Dimensional FiguresPowerPointA net is a two-dimensional pattern that you can fold to form a threedimensional figure. A net shows all of the surfaces of a figure in one view.Additional Examples4 Is the pattern a net for a cube?If so, name two letters that will beon opposite faces.yes; A andAC, B andE, D and FBDEF4Identifying Solids From NetsEXAMPLEMultiple Choice The net at the left shows all the surfaces of a three-dimensionalfigure. Which figure can you fold from the net?C5 Draw a net for the figure witha square base and four isoscelestriangular faces. Label the netwith its dimensions.10The black triangle is shown on the largest side of the figure instead of the smallestin choices C and D. Those choices cannot be correct. The black triangle will be atthe same end as the two diagonal lines when the net is folded into a box. Choice Bis correct.cm145AABB8 cmm0cDCCDEEATest-Taking TipBCDEThe number of surfaceson a solid must matchthe number of regionsin its net.1Package designers can use nets to help design containers.5Drawing a NetEXAMPLEPackaging Draw a net for the graham cracker box. Label the net withits dimensions.Resources Daily Notetaking Guide 1-2 L3 Daily Notetaking Guide 1-2—L1Adapted Instruction14 cmGRAHAMCRACKERS20 cmClosure7 cm20 cm7 cm14 cmExplain how isometric, orthographic, and foundation drawingsare alike and how they aredifferent. Sample: All showthree-dimensional figures ona two-dimensional surface;isometric drawings show threefaces, orthographic drawingsshow the outlines of three views,and foundation drawings usethe top view of orthographicdrawings to show the heightof each part of the figure.Quick Check5 Draw a net for the solid shown. Label the net with its dimensions. See margin.10 cm10 cm7 cm1212FEDCBEDCBA3EDCBA2EDCBA4 The net at the right folds into the cube shownbeside it. Draw the cube and show which letterswill be on its front and top. E, CD8 cmQuick CheckChapter 1 Tools of Geometry4 cm

EXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.3. PracticePractice and Problem SolvingAssignment GuideAPractice by ExampleExample 1GO forHelpOn isometric dot paper, make an isometric drawing of each cube structure.1–3. See margin.1.2.3.(page 10)1 A B 1-10, 17-22, 28-302 A B11-16, 23-27, 31-32C Challenge33-34Test PrepMixed ReviewExamples 2, 3(page 11)For each figure, make (a) an orthographic drawing, and(b) a foundation drawing.4–6. See back of book.4.5.Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 4, 14, 18, 22, 26.6.htFroRigntExercises 1–3 If possible, providehtRigFront35-3839-45Frontwooden cubes so that studentscan model each cube structure.ghtRiHow many cubes would you use to make each of the following?Example 47. the structure in Exercise 4 68. the structure in Exercise 5 69. the structure in Exercise 6 810. a model of the Sears Tower onpage 11 713GPS Guided Problem SolvingL2ReteachingL1Adapted PracticeMatch each three-dimensional figure with its net.PracticeName(page 12)C11.A12.L3L4EnrichmentB13.ClassL3DatePractice 1-2Points, Lines, and PlanesRefer to the diagram at the right for Exercises 1–15.* )1. Name AB in another way.2. Give two other names for plane Q.3. Why is EBD not an acceptable name for plane Q?AQAre the following sets of points collinear?* )* )A.B.C.DE4. AB and C5. B and F6. EB and A7. F and plane QAre the following sets of points coplanar?* )* )F9. DB and FC11. AE and DC* )12. F, A, B, and C13. F, A, B, and D14. plane Q and EC15. FB and BD* )* )C* )* )8. E, B, and F10. AC and ED* )B* )Find the intersection of the following lines and planes in the figureat the right.* )* )16. GK and LGHG17. planes GLM and LPNLK18. planes GHPN and KJPMJ19. planes HJN and GKLNP Pearson Education, Inc. All rights reserved.* )* )20. KP and plane KJN21. KM and plane GHLRefer to the diagram at the right.Example 5(page 12)22. Name plane P in another way.Draw a net for each figure. Label the net with its dimensions. 14–16. Seeback of book.14.15.16.7mD23. Name plane Q in another way.24. What is the intersection of planes P and Q?25. Are A and C collinear?CBAQ26. Are D, A, B, and C coplanar?27. Are D and C collinear?* )P* )28. What is the intersection of AB and DC ?2 in.2 in.4 in.30 mm10 m12 mm17. Answers may vary.Sample:a.6mApply Your Skills17. a. Open-Ended Make an isometric drawing of a structure that can be builtusing 8 cubes. a–b. See margin.b. Make an orthographic drawing of this structure.c. Make a foundation drawing for this structure. See left.Lesson 1-2 Drawings, Nets, and Other Models1.2.FronthtRig* )30. Are AB and plane Q coplanar?31. Are B and C collinear?8m36 mmB29. Are planes P and Q RightTop13

4. Assess & ReteachFor each foundation drawing, make (a) an isometric drawing on dot paper, and(b) an orthographic drawing. 18–20. See back of book.2119.1FrontUse the figure below for Exercises1–3.232120.Front132321Right3RightLesson Quiz3Right18.GPSPowerPointFrontRead the comic strip and complete Exercises 31 and 32.FrohtntRig1. Make an isometric drawing ofthe cube structure.Fro21. orthographic topview22. Answers may vary.Sample: You maywant a bird’s-eyeview for a touristmap showinglocations ofattractions.htntRigVisualization Think about how each net can be folded to form a cube. What is thecolor of the face that will be opposite the red face?bluegreenorangepurple23.24.25.26.2. Make an orthographicdrawing.FrontTopRight3. Is the pattern a net for a cube?If so, name two letters that willbe on opposite faces. no27b. Answers may vary.Sample: the netshown in Exercise23; it is easy to cutand fold.FA21. What type of drawing that you’ve studied in this lesson is a “bird’s-eye view”?See left.22. Writing Photographs of the Washington Monument are typically not takenfrom a bird’s-eye view. Describe a situation in which you would want a photoshowing a bird’s-eye view. See left.BC27. There are eleven different nets for a cube. Four of them are shown above.a. Draw as many of the other seven as you can. (Hint: Two nets are the same ifyou can rotate or flip one to match the other.) See margin.b. Writing If you were going to make 100 cubes for a mobile, which of theeleven nets would you use? Explain why. See left.DMake an orthographic drawing for each isometric drawing.Eyes; A and C, B and D,E and F28.GOHomework HelpVisit: PHSchool.comWeb Code: aue-01024. Draw a net for the figure.29.30.nlineFronthtRigtFro28–30. See back of book.nthRightFrontRig31. Draw a net for a cylinder. (Hint: The net needs to show three regions: twocircles and a rectangle.) See margin.Sample:32. There are eight different nets for a pyramid with a square base. Draw as manyof them as you can. See margin.14Chapter 1 Tools of Geometry31. Answers mayvary. Sample:27. Answers may vary. Sample:1432.