# Activity Set 4. Trainer Guide

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1 Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development

2 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 NGSSS 7.G.2.2 Catching Volume In this activity, participants will create threedimensional models for various solid figures and compute the volumes of those shapes. MATERIALS Transparency/Page: Rectangular Prism Transparency/Page: Triangular Prism calculators (1 per pair of participants) VOCABULARY volume rectangular prism triangular prism TIME: 10 minutes 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. Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 1

3 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 INTRODUCE Rectangular prism Explain to participants that they are going to find the volume of some solid shapes. Ask participants to explain how to find the volume of a cube. (height width depth) Display Transparency: Rectangular Prism. 6 mm 6 mm 14 mm Ask participants how the formula that they just described applies to this shape. (The volume can be computed in the same way.) GEOMETRY AND MEASUREMENT OF SOLID FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: Rectangular Prism Triangular prism Point out that a cube is a special case of rectangular prism, where the depth is equal to the height and width. Ask participants to work in pairs to find the volume of the prism by using their calculators. ( = 504 cubic mm) DISCUSS AND DO 3.46 mm 9 mm Display Transparency: Triangular Prism and have participants take out their matching pages. 4 mm GEOMETRY AND MEASUREMENT OF SOLID FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: Triangular Prism Have participant pairs find the volume of the triangular prism. Give participants 2 3 minutes to complete their calculations. Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 2

4 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 CONCLUDE Have a volunteer write the problem and solution on the displayed transparency. [ 1 2 (4 3.46) 9 = cubic mm] Remind participants that while the two shapes that they 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 different surface area. The volumes of such shapes have formulas that are generated using calculus. Suggest to participants that students sometimes have trouble distinguishing between surface area and volume. Suggest that nets and models can provide a way to help students distinguish between the two. End of Catching Volume Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 3

5 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 Top-Down View In this activity, participants will create two-dimensional representations of three-dimensional shapes. Materials Transparency/Page: Top, Side, and Front Transparency/Page: Top, Side, and Front Answer Key Transparency/Page: Drawing Two-Dimensional Views Transparency/Page: Drawing Two-Dimensional Views Answer Key blank transparencies scrap paper rainbow cubes (60 per pair of participants) set of wooden solid shapes pencils Vocabulary model Time: 30 minutes TEACHING TIP: Prepare and distribute, in advance of the activity, 1 bag of 60 rainbow cubes to each pair of participants. Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 4

6 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 INTRODUCE Suggest to participants that students work in higher mathematics often requires an ability to visualize mathematics concepts in three dimensions, and that often they must do this from two-dimensional pictures. Suggest further that experience creating two-dimensional views of three-dimensional shapes helps students build a foundation for these later activities. Explain to participants that they now are going to practice such an activity, drawing two-dimensional views of three-dimensional figures. Display a blank transparency and pass out to participants scrap paper. Hold up, from your set of solid shapes, a cone. Ask a volunteer participant to sketch a top, a front, and a side view of the shape on the top half of the transparency while the others sketch them on their scrap paper. Sample sketches are shown below. Top Front Side Use the overhead projector to display the shadow of the shape from the various angles, if participants have trouble visualizing the shape. Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 5

7 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 Hold up, from your set of solid shapes, a half-sphere. Ask a volunteer participant to sketch a top, a front, and a side view of the shape on the bottom half of the transparency while the others sketch them on their scrap paper. Sample sketches are shown below. Top Front Side Discuss with participants the fact that the front and side views of these shapes look the same because the shapes are symmetrically curved around a central axis. Ask participants if they can name some shapes for which this would not be true. Appropriate responses include any solid shape that is skewed off a central axis. Use, after participants have suggested several objects, a rectangular prism to demonstrate the different views of a non-right prism. Draw a picture of a skewed cone on the transparency and point out how the views differ from the examples the participants sketched. Include the axes on the views to emphasize the skew. Sample drawings are shown below. Apex of the cone Top Front Side Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 6

8 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 DISCUSS AND DO Use rainbow cubes to build three-dimensional models of the three shapes described by the drawings below. Is there only one way to build each model? A. Top, Side, and Front front side top Explain to participants that they now will look at various two-dimensional views and try to reconstruct the three-dimensional objects from which the pictures were created. B. front side top Display Transparency: Top, Side, and Front and have participants take out their matching pages. C. front side top Explain to participants that their task will be to use their rainbow cubes to construct each shape pictured in three dimensions. GEOMETRY AND MEASUREMENT OF SOLID FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: Top, Side, and Front Assign half the group to work on the shapes in the order A, B, C. Assign the other half to work on the shapes in the order A, C, B. Direct participants to save the model they create last for discussion purposes. Give participants minutes to complete the activity. conclude Call the groups together. Have a volunteer pair build model B in the front of the room where everyone can watch. Repeat for model C. Refer to Transparency: Top, Side, and Front Answer Key to resolve any questions. Ask if everyone built the model the same way. There are alternate solutions, including those listed below. In model B, the top two cubes in the left rear column can be removed. In model C, the top two cubes in the front middle columns can be removed. Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 7

9 GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity Set 4 Ask participants why this might be so. (Each of these shapes is a composite of other shapes, and the pictures did not provide enough information about each of the pieces.) Drawing Two-Dimensional Views Drawing Three-Dimensional Figures Draw top, front, and side views of each figure top front side GEOMETRY AND MEASUREMENT OF SOLID FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: Drawing Two-Dimensional Views Drawing Two-Dimensional Views Answer Key Drawing Three-Dimensional Figures Draw top, front, and side views of each figure top front side Suggest to participants that they have now seen the importance of various details in moving between two- and three-dimensional representations of objects, for example, the importance of axes and the importance of the number of views for each element of a shape. Display Transparency: Drawing Two-Dimensional Views and have participants take out their matching pages. Give participants 5 7 minutes to complete the drawings of the requested views. Have volunteers draw their answers on the displayed transparency. Refer to Transparency: Drawing Two-Dimensional Views Answer Key to resolve any questions. Display Transparency: Drawing Two-Dimensional Views Answer Key when all the answers have been resolved. Point out the dotted lines representing the axes on the last two shapes and emphasize how they can be used to help define the views. Suggest to participants that they can sketch such lines on shapes they need to reconstruct. 4. GEOMETRY AND MEASUREMENT OF SOLID FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development End of Top-Down View Transparency: Drawing Two-Dimensional Views Answer Key Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 8

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11 Rectangular Prism 6 mm 14 mm 6 mm GEOMETRY AND MEASUREMENT OF OF SOLID FIGURES Activity FIGURES AcTIvITY Set SET 4 Copyright by 2002 the by McGraw-Hill the McGraw-Hill Companies McGraw-Hill Professional Professional Development Development Mid_SGe_04_PM

12 Triangular Prism 9 mm 3.46 mm 4 mm GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity FIGURES AcTIvITY Set SET 4 Copyright by 2002 the by McGraw-Hill the McGraw-Hill Companies McGraw-Hill Professional Professional Development Development Mid_SGe_04_PM

13 Top, Side, and Front Use rainbow cubes to build three-dimensional models of the three shapes described by the drawings below. Is there only one way to build each model? A. front side top B. front side top C. front side top GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity FIGURES AcTIvITY Set SET 4 Copyright by 2002 the by McGraw-Hill the McGraw-Hill Companies McGraw-Hill Professional Professional Development Development Mid_SGe_04_PM

14 Top, Side, and Front Answer Key Use rainbow cubes to build three-dimensional models of the three shapes described by the drawings below. Is there only one way to build each model? 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 OF SOLID FIGURES Activity FIGURES AcTIvITY Set SET 4 Copyright by 2002 the by McGraw-Hill the McGraw-Hill Companies McGraw-Hill Professional Professional Development Development Mid_SGe_04_PM

15 Drawing Two-Dimensional Views Drawing Three-Dimensional Figures Draw top, front, and side views of each figure. 1. top front side GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity FIGURES AcTIvITY Set SET 4 Copyright by 2002 the by McGraw-Hill the McGraw-Hill Companies McGraw-Hill Professional Professional Development Development Mid_SGe_04_PM

16 Drawing Two-Dimensional Views Answer Key Drawing Three-Dimensional Figures Draw top, front, and side views of each figure. 1. top front side GEOMETRY AND MEASUREMENT OF SOLID FIGURES Activity FIGURES AcTIvITY Set SET 4 Copyright by 2002 the by McGraw-Hill the McGraw-Hill Companies McGraw-Hill Professional Professional Development Development Mid_SGe_04_PM

17 Glossary Geometry and Measurement of Solid Figures area base The number of square units in a region. 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 cube cylinder edge face hexahedron inscribed circle net octahedron 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. A rectangular prism that has six square faces. A three-dimensional figure that has two congruent, parallel, circular bases connected by a cylindrical surface. A line segment formed by the intersection of two faces of a solid figure. A flat side of a solid shape. A six-sided polyhedron; a regular hexahedron is a cube. A circle is inscribed within a polygon if every side of the polygon is tangent to the circle. A printed, two-dimensional pattern that can be cut and folded to make a solid shape. An eight-sided polyhedron. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Mid_SGe_04_PM

18 Glossary (continued) polyhedron (polyhedra, pl.) prism pyramid rectangular prism solid sphere surface area tetrahedron triangular prism A closed, three-dimensional object with a surface made up of polygons. A polyhedron that has parallelogram sides and two parallel, congruent bases; includes right prisms and oblique prisms. A polyhedron that has a polygon base and faces that are triangles. A prism with parallel bases that are congruent rectangles. A figure that has depth in addition to width and height. The set of all points in three-dimensional space that are equidistant from a specified point. The total area of the surface of a solid, including all faces and bases. A pyramid with a triangular base. 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. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Mid_SGe_04_PM

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