Slide 1 / 311 Slide 2 / 311 Geometry 3D Geometry 2015-10-28 www.njctl.org Slide 3 / 311 Slide 4 / 311 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. Intro to 3-D Solids Click on the topic to go to that section Views & Drawings of 3-D Solids MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. Surface Area of a Prism MP3: Construct viable arguments and critique the reasoning of Surface Area of a Cylinder others. MP4: Model with mathematics. Surface Area of a Pyramid MP5: Use appropriate tools strategically. Surface Area of a Cone MP6: Attend to precision. MP7: Look for & make use of structure. Volume of a Prism MP8: Look for & express regularity in repeated reasoning. Volume of a Cylinder Additional questions are included on the slides using the "Math Volume of a Pyramid Practice" Pull-tabs (e.g. a blank one is shown to the right on Volume of a Cone this slide) with a reference to the standards used. Surface Area & Volume of Spheres If questions already exist on a slide, then the specific MPs that Cavaleri's Principle the questions address are listed in the Pull-tab. Similar Solids PARCC Sample Questions Slide 4 (Answer) / 311 Slide 5 / 311 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of Math Practice others. MP4: Model with mathematics. Intro to 3-Dimensional Solids MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math [This object is a pull tab] Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. Return to Table of If questions already exist on a slide, then the specific MPs that Contents the questions address are listed in the Pull-tab.
Slide 6 / 311 Slide 7 / 311 Intro to 3-D Solids Intro to 3-D Solids 2-dimensional drawings use only the x and y axes 3-dimensional drawings include the x, y and z-axis. X The z-axis is the third dimension. Y The third dimension is the height of the figure Z Length h t d w i Y width X height Length height w i d t h Length X X X Y Y Y Slide 8 / 311 Slide 9 / 311 Intro to 3-D Solids Intro to 3-D Solids Y Z Y Z r x X height height height X X Y Y X Y X Y Slide 10 / 311 Slide 11 / 311 Intro to 3-D Solids Intro to 3-D Solids A Polyhedron (pl. Polyhedra) is a solid that is bounded by To give a figure more of a 3-dimensional look, lines that polygons, called faces. An edge is the line segment formed by the are not visible from the angle the figure is being viewed intersection of 2 faces. A vertex is a point where 3 or more edges are drawn as dashed line segments. These are called meet hidden lines. Vertex Z Edge height height Face X Y
Slide 12 / 311 Slide 13 / 311 Intro to 3-D Solids Intro to 3-D Solids The 3-Dimensional Figures discussed in The 3-Dimensional Figures discussed in this unit are: this unit are: Cylinders Prisms Cones: Spheres: . C Pyramids Slide 14 / 311 Slide 15 / 311 Right Vs. Oblique Right Vs. Oblique In Oblique Prisms & Cylinders, the bases are not aligned In Right Prisms & Cylinders, the bases are aligned directly above one another. The edges are not directly above one another. The edges are perpendicular perpendicular with the bases. with both bases. Right Right Slide 16 / 311 Slide 17 / 311 Intro to 3-D Solids Right Vs. Oblique Prisms have 2 congruent polygonal bases. In Right Pyramids & Cones, the vertex is aligned directly The sides of a base are called base edges. above the center of the base. The segments connecting corresponding vertices are lateral edges. A In Oblique Pyramids & Cones, the vertex is not aligned B directly above the center of the base. C Oblique Right X Y In this diagram: Z There are 6 vertices: A, B, C, X, Y, & Z There are 2 bases: ABC & XYZ. There are 6 base edges: AB, BC, AC, XY, YZ, & XZ. Right Oblique There are 3 lateral edges: AX, BY, & CZ. This prism has a total of 9 edges.
Slide 18 / 311 Slide 19 / 311 Intro to 3-D Solids 1 Choose all of the base edges. The polygons that make up the surface of the figure are AB A called faces. The bases are a type of face and are parallel and congruent to each other. The lateral B C B DE edges are the sides of the lateral faces. A D FS C F E A In this diagram: B CP D There are 2 bases: ABC & XYZ. N P FA E C Q There are 3 lateral faces: AXBY, M CD R BYCZ, & CZAX. F S X Y G NP This prism has a total of 5 faces. BC Z H DQ I Slide 19 (Answer) / 311 Slide 20 / 311 2 Choose all of the lateral edges. Choose all of the base edges. 1 AB A B C AB A A CD B D B C DE B A F E D ER C FS C F E BN N D P Answer D CP A, B, E, F, G, H N P Q M E DQ FA E R S Q M QR F CD S R F MS G NP G AM H [This object is a pull tab] BC H CP I I DQ Slide 20 (Answer) / 311 Slide 21 / 311 2 Choose all of the lateral edges. 3 Chooses all of the bases. AB A B C B C AFSM A A A B CD D D FERS B F F E E ER C C EDQR N N BN P P D Answer D ABCDEF Q Q M M DQ C, D, E, H, I E S R S R CDQP E QR F BCPN F G MS MNPQRS G AM H [This object is a pull tab] H ABNM I CP
Slide 21 (Answer) / 311 Slide 22 / 311 3 Chooses all of the bases. 4 Chooses all of the lateral faces. B C B C A AFSM A AFSM A A D D F FERS FERS E B B F E EDQR EDQR C C N P Answer N P D, G D ABCDEF D ABCDEF Q M Q M R S R S CDQP CDQP E E F BCPN F BCPN MNPQRS MNPQRS G G [This object is a pull tab] ABNM ABNM H H Slide 22 (Answer) / 311 Slide 23 / 311 4 Chooses all of the lateral faces. 5 Chooses all of the faces. B C B C A D A A AFSM A AFSM D F E F E FERS FERS B B N P C EDQR C EDQR N P Answer Q M ABCDEF ABCDEF D Q D M A, B, C, E, F, H R S R S CDQP CDQP E E F BCPN F BCPN G MNPQRS G MNPQRS [This object is a pull tab] H ABNM H ABNM Slide 23 (Answer) / 311 Slide 24 / 311 Intro to 3-D Solids 5 Chooses all of the faces. B C A pyramid has 1 base with vertices and the lateral A D edges go to a single vertex. AFSM A F E A FERS B N P All of the choices Answer C EDQR This pyramid has: are faces Q 6 lateral edges, M D ABCDEF S R 6 base edges, 12 edges (total) CDQP E 7 vertices P N BCPN F M Q [This object is a pull tab] MNPQRS G S R H ABNM
Slide 25 / 311 Slide 26 / 311 Intro to 3-D Solids 6 Choose all of the base edges. V A pyramid has faces that are polygons: VN A 1 base and triangles that are the lateral faces. KN B A VL C N This pyramid has: K M D LM 6 lateral faces, 1 base, VM E L 7 faces (total) N P VK F M Q G KL S R H NM Slide 26 (Answer) / 311 Slide 27 / 311 6 Choose all of the base edges. 7 Choose all of the lateral edges. V V A VN A VN KN KN B B VL Answer VL N C C K M N B, D, G, H K M LM LM D D L VM VM E E L F VK F VK [This object is a pull tab] G KL G KL H NM H NM Slide 27 (Answer) / 311 Slide 28 / 311 7 Choose all of the lateral edges. 8 How many edges does the pyramid have? V V VN A KN B Answer A, C, E, F N C VL K M N LM K M D L VM E L VK F [This object is a pull tab] G KL H NM
Slide 28 (Answer) / 311 Slide 29 / 311 8 How many edges does the pyramid have? 9 Choose all of the lateral faces. V V KNV A NMV B N KLMN C K M N Answer K M D VML 8 L KLV E L [This object is a pull tab] Slide 29 (Answer) / 311 Slide 30 / 311 10 Choose all of the bases. 9 Choose all of the lateral faces. V V A KNV A KNV NMV NMV B B N KLMN KLMN N C C K M K M Answer A, B, D, E VML VML D D L L KLV KLV E E [This object is a pull tab] Slide 30 (Answer) / 311 Slide 31 / 311 10 Choose all of the bases. 11 How many faces does the pyramid have? V KNV V A NMV B Answer N C KLMN K M C N VML D K M L KLV E L [This object is a pull tab]
Slide 31 (Answer) / 311 Slide 32 / 311 Intro to 3-D Solids 11 How many faces does the pyramid have? A cylinder has 2 bases which are congruent circles. The V lateral face is a rectangle wrapped around the circles. A . Answer A & B 5 N K M are the bases of the cylinder. . L B [This object is a pull tab] A cylinder can also be formed by rotating a rectangle about an axis. Click for sample animation Slide 33 / 311 Slide 34 / 311 Intro to 3-D Solids Intro to 3-D Solids A sphere is a 3-dimensional circle in that every point on the sphere is A cone, like a pyramid, has one base which is a circle. the same distance from the center. . N N is the base of the cone. . C V V is the vertex of the cone. A cone can also be formed by rotating a right triangle about Similar to a circle, a sphere is named by its center one of its legs. point. Sphere C is the solid shown above. Click for sample animation Slide 35 / 311 Slide 35 (Answer) / 311 12 Which solids have 2 bases? 12 Which solids have 2 bases? Prism Prism A A Pyramid Pyramid B B Answer C Cylinder C Cylinder A & C Cone Cone D D Sphere Sphere E E [This object is a pull tab]
Recommend
More recommend
