This pdf covers comparing and creating geometric solids. In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces. Stereometry deals with the measurements of volumes of various solid figures, including pyramids, prisms, and other polyhedrons; cylinders; cones; truncated cones, and balls bounded by spheres.
1. Objective To review the properties of common geometric solids. 1 Teaching the Lesson materials Key Activities Math Journal 2, pp. 289 and 290 Students review common geometric solids—including prisms, pyramids, cylinders, cones, Student Reference Book, p. 101 and spheres—and investigate their properties. Study Link 11 1
Students construct rectangular prisms using straws and twist-ties. models of geometric solids (See Planning Ahead in Lesson 11 1.) Key Concepts and Skills straws and twist-ties • Identify parallel and intersecting line segments and parallel planes. [Geometry Goal 1] Class Data Pad (optional) • Describe, compare, and classify plane and solid figures. [Geometry Goal 2] • Identify congruent faces. [Geometry Goal 2] See Advance Preparation • Construct a rectangular prism. [Geometry Goal 2] Key Vocabulary geometric solid • rectangular prism • cylinder • triangular prism • cone • sphere • square pyramid • face • congruent • curved surface • edge • vertex (vertices) • cube • flat surface Ongoing Assessment: Recognizing Student Achievement Use journal page 290. [Geometry Goal 2] 2 Ongoing Learning & Practice materials Students determine how many of each U.S. coin are needed to make a 1-ounce weight. Math Journal 2, pp. 291 and 292 Students practice and maintain skills through Math Boxes and Study Link activities. Study Link Master (Math Masters, p. 328) balance or scale capable of measuring 1 ounce pennies, nickels, dimes, and quarters 3 Differentiation Options materials ENRICHMENT EXTRA PRACTICE ELL SUPPORT Teaching Aid Masters (Math Masters, pp. 389 and 390) Students explore the Students use a Venn Students create a Word Wall models of geometric solids relationships among the diagram to compare of geometry vocabulary. number of vertices, faces, geometric solids. computer with Internet access and edges of polyhedrons. Additional Information Advance Preparation For Part 1, construct a cube with 16 twist-ties and 12 straws, all the Technology same length. In four boxes, place enough twist-ties and full-size, 1 -size, and 3 -size straws Assessment Management System 2 4 that each pair of students can have 16 twist-ties and 8 straws of each length. Journal page 290, Problem 8 See the iTLG. See the Web site on page 859. 854 Unit 11 3-D Shapes, Weight, Volume, and Capacity
2. Getting Started Mental Math and Reflexes Math Message Write large numbers on the board, and have volunteers read Complete journal page 289. them aloud. Suggestions: 7,540,312 43,290,517 1,206,598,346 2,560,371 831,247,906 2,165,307,498 Study Link 11 1 Follow-Up 16,436,280 372,815,206 172,039,598,563 Ask small groups of students to compare Ask questions like the following: answers and to pose and solve the problems • What is the value of the digit x? they created. • Which digit is in the millions place? 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS ACTIVITY (Math Journal 2, p. 289) Display models of the six geometric solids—rectangular prism, cylinder, triangular prism, cone, sphere, and square pyramid—shown on journal page 289. Begin with the rectangular prism. To support English language learners, attach a tag and label it. Hold it up and ask the class to share examples of rectangular prisms in the classroom. You may wish to keep a list on the board or the Class Data Pad. Repeat this procedure for the remaining solids. When all of the solids have been discussed, ask students the following questions: Student Page ● Which solids were easy to find? Probably the rectangular prism, Date Time cylinder, and sphere LESSON 11 2 Geometric Solids Geometric shapes like these 3-dimensional ones are also called geometric solids. ● Which were hard to find? Probably the pyramid, triangular 101 102 prism, and cone ● Rectangular Cylinder Triangular Cone Sphere Square Why do you think some solids are more common than others? Prism Prism Pyramid Probably because they are easier to make or are more useful for Look around the classroom. Try to find examples of the geometric solids pictured above. Draw a picture of each. Then write its name (for example: book).Answers vary. storing things Example of rectangular prism: Example of cylinder: Example of triangular prism: Rectangular Cylinder Triangular Cone Sphere Square Name of object: Name of object: Name of object: Prism Prism Pyramid Geometric solids Example of cone: Example of sphere: Example of square pyramid: Name of object: Name of object: Name of object: 289 Math Journal 2, p. 289 Lesson 11 2 855
3. Student Page Geometric Solids Geometry and Constructions Reviewing Vocabulary for WHOLE-CLASS DISCUSSION Polygons and circles are flat, 2-dimensional figures. The surfaces they enclose take up a certain amount of area, but they do not have any thickness and do not take up any volume. face Geometric Solids Three-dimensional shapes have length, width, and thickness. They take up volume. Boxes, chairs, and balls are all examples. face (Math Journal 2, p. 289; Student Reference Book, p. 101) face A geometric solid is the surface or surfaces that surround a 3-dimensional shape. The surfaces of a geometric solid may be cube flat or curved or both. A flat surface of a solid is called a face. A curved surface of a solid does not have any special name. Use the display models of the six geometric solids and Student A cube has 6 square faces that are the same size. Three of the cube’s faces cannot be seen in the figure at the right. face Reference Book, page 101 to review vocabulary associated with A cylinder has 3 surfaces. The flat top and flat bottom are faces that are formed by circles. A curved surface connects the curved geometric solids. To support English language learners, discuss top and bottom faces. A food can is a good model of a cylinder. A cone has 2 surfaces. The flat bottom is a face that is formed surface face the meaning of each term. Pose questions like the following: by a circle. A curved surface is connected to the bottom face and cylinder comes to a point. An ice cream cone is a good model of a cone. ● Which of these geometric solids has 6 faces? Rectangular However, keep in mind that a cone is closed; it has a “lid.” The edges of a geometric solid are the line segments or curves where surfaces meet. A corner of a geometric solid is called a prism curved vertex (plural vertices). A vertex is usually a point at which surface ● edges meet, but the vertex of a cone is an isolated corner. It is completely separated from the edge of the cone. face Which solids have congruent faces? Rectangular prism, cone vertex edges vertices vertices edge no vertices cylinder, triangular prism, square pyramid edges edge ● Which solids have a curved surface? Sphere, cone, and edge A sphere has one curved surface but no edges and no vertices. sphere no edges, no vertices cylinder A basketball or globe is a good model of a sphere. Check Your Understanding ● Which has the most edges? Rectangular prism 1. a. How are cylinders and cones alike? b. How do they differ? 2. a. How are spheres and cones alike? b. How do they differ? Check your answers on page 343. ● Which two have the fewest vertices (corners)? Cylinder and Student Reference Book, p. 101 sphere What is the singular form of the word vertices? vertex ● Which has two faces and one curved surface? cylinder Have students look around the classroom and point out the faces, edges, and vertices of objects that have shapes similar to those in the display. Links to the Future Encourage students to use the geometry vocabulary, but do not expect them to be precise at this time. Student Page Modeling Geometric Solids PARTNER ACTIVITY Date Time (Math Journal 2, p. 290) LESSON 11 2 Modeling a Rectangular Prism 101 102 After you construct a rectangular prism vertices Show the class the cube you constructed out of straws. (See with straws and twist-ties, answer the questions below. edges Advance Preparation.) Point out that it shows only the edges of the faces. It is a “frame” for the geometric solid; the flat surfaces faces of the cube must be imagined. 1. How many faces does your rectangular prism have? 6 face(s) 2. How many of these faces are formed by rectangles? 6 face(s) 3. How many of these faces are formed by squares? 0, or 2 face(s) 4. Pick one of the faces. How many other faces are parallel to it? 1 face(s) 5. How many edges does your rectangular prism have? 12 edge(s) 6. Pick an edge. How many other edges are parallel to it? 3 edge(s) 7. How many vertices does your rectangular prism have? 8 vertices 8. Write T (true) or F (false) for each of the following statements about the rectangular prism you made. Then write one true statement and one false statement of your own. a. T It has no curved surfaces. b. F All of the edges are parallel. c. T All of the faces are polygons. d. F All of the faces are congruent. e. True Answers vary. f. False Answers vary. 290 Math Journal 2, p. 290 856 Unit 11 3-D Shapes, Weight, Volume, and Capacity
4. Ask students what geometric solid this construction represents. Cube, or rectangular prism Demonstrate how the vertices are put together. Cube made out of straws and twist-ties Distribute straws and twist-ties. (See Advance Preparation.) Have partners work together to make a rectangular prism. One way is to start with a rectangle and build up. Have the straw cube, as well as other models of rectangular prisms, available for inspection. When their rectangular prism is finished, partners should complete journal page 290. NOTE Problem 3 on journal page 290 asks students to identify the number of square faces in the rectangular prisms they have made. Depending on the straw sizes used, the prisms will have either 0 or 2 square faces. Students cannot construct cubes (which have 6 square faces) because they have only 8 straws of each length per partnership. Journal Ongoing Assessment: Recognizing Student Achievement page 290 Problem 8
Use journal page 290, Problem 8 to assess students’ ability to describe a rectangular prism. Students are making adequate progress if they are able to Student Page correctly identify the given statements as true or false and write their own true Date Time and false statements. Some students may write statements that involve LESSON comparisons among geometric solids. 11 2 Making a 1-Ounce Weight 140 1. Estimate how many of each coin you think it will take to make a 1-ounce weight. Then use a [Geometry Goal 2] balance or scale to determine exactly how many of each coin are needed. Coin Estimated Number Actual Number of Coins of Coins penny Answers vary. 11 or 12 nickel 5 or 6 dime 12 or 13 quarter 5 2 Ongoing Learning & Practice 2. Describe how you estimated how many of each coin it might take to make a 1-ounce weight. Sample answer: I know there are about 28 g in 1 ounce, and a nickel weighs 5 g. Therefore, 6 nickels should weigh 30 g (5 6 30), so 6 nickels equal about 1 ounce. I estimate that 1 pennies weigh 2 as much as nickels, so it should take twice as Making a 1-Ounce Weight SMALL-GROUP ACTIVITY many pennies, which is 12. Dimes are smaller than pennies, so I guessed 15 dimes. Quarters are heavier than nickels, so I guessed 4. (Math Journal 2, p. 291) Try This 3. About what fraction of an ounce does each coin weigh? Sample answers: 1 1 1 1
Students use a balance or scale to determine how many of each 1 penny 12 oz 1 nickel 6 oz 1 dime 13 oz 1 quarter 5 oz available type of U.S. coin are needed to make a 1-ounce weight. Explain how you found your answers. Sample answer: I used the number of coins that equal 1 oz Students can display the results of their experiments in the Gram as the denominator. For example, because 12 pennies equal 1 1 oz, 1 penny equals 12 oz. & Ounce Museum. 291 Math Journal 2, p. 291 Lesson 11 2 857
5. Student Page Math Boxes 11 2 Date Time
INDEPENDENT LESSON 11 2 Math Boxes ACTIVITY 1. The object below has the shape of a 2. Draw the figure after it is rotated (Math Journal 2, p. 292) 1 geometric solid. What is the name of the clockwise 4-turn. solid? Circle the best answer. A. rectangular prism Mixed Practice Math Boxes in this lesson are linked B. cone with Math Boxes in Lessons 11-4 and 11-6. The skills C. cylinder D. square pyramid VOLUME 0 2 4 6 8 10 in Problems 5 and 6 preview Unit 12 content. Writing/Reasoning Have students write a response to the 101 102 106 107 3. Write a number model to estimate the 4. Insert , , or to make a true number answer. Then correctly place the decimal point. sentence. following: For Problem 6, how would you determine the 1 . a. 0.97 4 3 8 8 a. 12 19 number of calories in 32 bagels? Sample answer: Multiply
Number model: 144 b. 44
26 the number of calories in 1 bagel by 3; 230 * 3 690. Then divide c. 64 0.43 . b. 1 8 7 74.8 4 1 d. 2 4
the number of calories in 1 bagel by 2; 230 / 2 115. Add the 80 4 20 8 quotient to the number of calories in 3 bagels; 115 690 805. Number model: e. 0.28 0.37 6 60 5. Round each number to the nearest tenth. 6. A cinnamon raisin bagel has about 230 calories. How many calories are a. 2.34 2.3 in one dozen bagels? b. 0.68 c. 14.35 0.7 14.4 About 2,760 calories Study Link 11 2 INDEPENDENT ACTIVITY d. 1.62 1.6 (Math Masters, p. 328) e. 5.99 6.0 182 183 47 Home Connection Students identify geometric solids 292 Math Journal 2, p. 292 represented by various objects. They also identify the vertices and the number of edges in two geometric solids. Study Link Master Name Date Time STUDY LINK 11 2 Solids 1. The pictures below show objects that are shaped approximately like geometric solids. Identify each object as one of the following: cylinder, 101 102 cone, sphere, triangular prism, square pyramid, or rectangular prism. a. b. c. Type: square Type: cone Type: sphere pyramid d. e. f. Type: cylinder Type: rectangular Type: triangular prism prism 2. Mark Xs on the vertices of the 3. How many edges does the rectangular prism. tetrahedron have? 6 edges Practice 4. Circle the numbers that are multiples of 7. 132 7,000 63 560 834 91 5. Circle the numbers that are multiples of 12. 24 120 38 600 100 75 Math Masters, p. 328 858 Unit 11 3-D Shapes, Weight, Volume, and Capacity
6. Teaching Aid Master Name Date Time 3 Differentiation Options Venn Diagram INDEPENDENT ENRICHMENT ACTIVITY Exploring Euler’s 15–30 Min Polyhedral Formula (Math Masters, p. 389) Technology Link To apply students’ ability to describe solid figures, have them explore the relationships among vertices, edges, and faces of polyhedrons at http://nlvm.usu.edu/en/nav/ frames_asid_128_g_2_t_3.html?openinstructions. On an Exit Slip, ask students to record their observations and draw a conclusion. Sample answer: The sum of the numbers of faces and vertices is 2 more than the number of edges: E 2 F V. Euler’s polyhedral formula states that the number of vertices minus the number of edges plus the number of faces is always Math Masters, p. 390 equal to 2. VEF2 NOTE The Web site is part of the National Library of Virtual Manipulatives for Interactive Mathematics developed by Utah State University. See http://nlvm.usu.edu. INDEPENDENT EXTRA PRACTICE ACTIVITY Comparing Geometric Solids 5–15 Min (Math Masters, p. 390) To practice comparing the attributes of solid figures, have students choose two geometric solids and use them to complete the Venn diagram on Math Masters, page 390. SMALL-GROUP ELL SUPPORT ACTIVITY Creating a Word Wall 30+ Min To provide language support for geometry vocabulary, have students illustrate definitions of key geometric terms for a classroom display. NOTE It might appear that there is an overwhelming number of geometry terms to know and understand. Each of these terms should be discussed in the context of solving problems and should relate to students’ experiences. Teaching these terms in isolation or reducing them to a vocabulary list that needs to be memorized will not produce successful results in most cases. Students should have opportunities to work with each of these terms, build models of them, write them, and discuss them. Lesson 11 2
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