Constructing Geometric Solids


 Robyn Anissa Lamb
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1 Constructing Geometric Solids Objectives To provide practice identifying geometric solids given their properties; and to guide the construction of polyhedrons. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Identify parallel and intersecting line segments and parallel faces. [Geometry Goal ] Describe, compare, and classify plane and solid figures. [Geometry Goal 2] Identify congruent faces. [Geometry Goal 2] Construct polyhedrons; sketch twodimensional representations of polyhedrons. [Geometry Goal 2] Key Activities Students practice identifying geometric solids by solving riddles about their properties. Students construct polyhedrons with straws and twistties. Students explore ways to draw a cube. Key Vocabulary pyramid prism polyhedron regular polyhedron triangular pyramid tetrahedron dodecahedron Materials Math Journal 2, pp Student Reference Book, p. 02 Study Link 2 Math Masters, pp. 452 and 453 (optional) models of geometric solids (See Planning Ahead in Lesson.) straws and twistties (See Lesson 2.) blank paper straightedge transparent tape slate set of polyhedral dice (optional) dictionary (optional) Plotting Book Heights Math Journal 2, pp. 295A and 295B Students plot book heights on a line plot. Math Boxes 3 Math Journal 2, p. 296 Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 3a. [Number and Numeration Goal 3] Study Link 3 Math Masters, p. 329 Students practice and maintain skills through Study Link activities. READINESS Sorting Geometric Solids 30 objects of various shapes Students sort common objects by their properties. ENRICHMENT Creating Cube Nets Math Masters, pp. 39 and 444 scissors Students find all possible cube nets. EXTRA PRACTICE Taking a 50Facts Test Math Masters, pp. 4 and 44; p. 46 (optional) pen or colored pencil Students take a 50facts test. They use a line graph to record individual and optional class scores. Advance Preparation For Part, you need the geometric solids from Lesson 2, plus a triangular pyramid and a cube. Use everyday objects or make them from Math Masters, pages 452 and 453. For the optional Readiness activity in Part 3, use cans, egg cartons, party hats, paper cups, tubes, and boxes. Teacher s Reference Manual, Grades 4 6 pp Unit 3D Shapes, Weight, Volume, and Capacity
2 Getting Started Mental Math and Reflexes Pose problems involving the multiplication of a fraction by a whole number. Have students find each product. Suggestions: 4 _ 2 = 2 24 _ = 3 30 _ 6 = 5 5 3_ 5 = 3 9 2_ 3 = 6 6 3_ 4 = 2 5 _ 2 = 2 _ 2 4 _ 3 = 4 2_ 3 0 3_ 4 = 7 2_ 4, or 7 _ 2 Math Message Open your Student Reference Book to page 02. Solve the following riddle: I have the same number of faces as vertices. What am I? Study Link 2 FollowUp Have partners compare answers. Ask: How many pairs of parallel faces does the rectangular prism have? 3 How many pairs of parallel faces does the tetrahedron have? 0 Teaching the Lesson Math Message FollowUp (Student Reference Book, p. 02) WHOLECLASS DISCUSSION All of the pyramids shown at the top of page 02 of the Student Reference Book have the same number of faces as vertices. Pyramids are named for the shape of their base. All of the remaining faces are triangles that meet at a vertex. Pose another riddle: I have 6 faces. All of my faces are rectangles. What am I? Rectangular prism Prisms are named for the shape of their two parallel bases. Emphasize that a polyhedron (plural polyhedrons or polyhedra) is a geometric solid whose surfaces are all formed by polygons. A polyhedron does not have any curved surfaces. Display the six geometric solids from Lesson 2 and the triangular pyramid and cube. Ask: Which of these solids are NOT polyhedrons? Cylinder, sphere, cone A polyhedron is a regular polyhedron if each face is formed by a regular polygon; the faces all have the same size and shape; and each vertex looks exactly the same as every other vertex. Ask: Which of these solids are regular polyhedrons? Cube and triangular pyramid, or tetrahedron NOTE Triangular pyramid and tetrahedron are two names for the same geometric solid. When people refer to a tetrahedron, they often mean a regular tetrahedron. Not all tetrahedrons (triangular pyramids) are regular, however. Geometry and Constructions Polyhedrons A polyhedron is a geometric solid whose surfaces are all formed by polygons. These surfaces are the faces of the polyhedron. A polyhedron does not have any curved surfaces. Pyramids and prisms are two important kinds of polyhedrons. Polyhedrons That Are Pyramids The shaded face of each pyramid above is called the base of the pyramid. The shape of the base is used to name the pyramid. For example, the base of a square pyramid has a square shape. The faces of a pyramid that are not the base are all shaped like triangles and meet at the same vertex. The two shaded faces of each prism above are called the bases of the prism. The bases of a prism are the same size and shape. They are parallel. All other faces join the bases and are shaped like parallelograms. The shape of the bases of a prism is used to name the prism. For example, the bases of a pentagonal prism have the shape of a pentagon. Many polyhedrons are not pyramids or prisms. Some are illustrated below. A rhombicuboctahedron has 26 faces. Eighteen of them are squares and are triangles. triangular pyramid square pyramid pentagonal pyramid hexagonal pyramid Polyhedrons That Are Prisms triangular prism rectangular prism pentagonal prism hexagonal prism Polyhedrons That Are NOT Pyramids or Prisms Student Reference Book, p. 02 Lesson 3 6
3 Adjusting the Activity Distribute a set of polyhedral dice. Ask students to find the name of each die and determine the die that is not a regular polyhedron. The names of the dice are as follows: tetrahedron, octahedron, decahedron, dodecahedron, icosahedron. The decahedron is not a regular polygon because its faces are not regular polygons. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Solving Geometry Riddles PROBLEM SOLVING WHOLECLASS 3 Construction of Polyhedrons Polyhedrons are geometric solids with flat surfaces formed by polygons. For each problem below Decide what the polyhedron should look like. Use straws and twistties to model the polyhedron. Answer the questions about the polyhedron. Look at page 02 of the Student Reference Book if you need help with the name.. I am a polyhedron. I have 5 faces. Four of my faces are formed by triangles. One of my faces is a square. a. After you make me, draw a picture of me in the space to the right. b. What am I? c. How many corners (vertices) do I have? d. What shape is my base? 2. I am a polyhedron. I have 4 faces. All of my faces are formed by equilateral triangles. All of my faces are the same size. a. After you make me, draw a picture of me in the space to the right. b. What am I? c. How many corners (vertices) do I have? d. What shape is my base? Math Journal 2, p Square pyramid 5 Square Triangular pyramid, or regular tetrahedron 4 Triangle Tell students that in this lesson they will explore the properties of polyhedrons by solving more riddles like the one in the Math Message. Pose additional riddles for students to solve. You might read one clue at a time and have students guess each time. When all clues have been given, ask a student to come up and display the correct solid, or a picture of it in the Student Reference Book, and name it. Ask the student to explain why it is that particular solid. Language Arts Link Consider having students look up the word parts tetra, hedron, poly, octa, deca, and dodeca in the dictionary to give them a better understanding of the origins and meanings of geometric terms. Riddle I am a geometric solid. I have six faces. All of my faces are squares. What am I? cube Riddle 2 I am a geometric solid. I have two surfaces. My base is formed by a circle. I come to a point at the top. What am I? cone Riddle 3 I am a polyhedron. I have the fewest number of faces of all the polyhedrons. All of my faces are triangular. I come to a point at the top. What am I? Triangular pyramid, or tetrahedron Riddle 4 I am a polyhedron. My faces are pentagons. I am useful for calendars. My picture is on page 03 of the Student Reference Book. What am I? dodecahedron 62 Unit 3D Shapes, Weight, Volume, and Capacity
4 Riddle 5 I am a polyhedron. I have two triangular bases. My other faces are rectangles. Sometimes I am used for keeping doors open. What am I? Triangular prism Riddle 6 I am a geometric solid. I have only one surface. My one surface is curved. I have no base. What am I? sphere 3 Knowing how to draw is a useful skill in mathematics. Here are a few ways to draw a cube. Try each way. Tape your best work at the bottom of page 295. A Basic Cube Draw a square. Draw another square that overlaps your first square. The second square should be the same size as the first. Connect the corners of your 2 squares as shown. This picture does not look much like a real cube. One problem is that the picture shows all 2 edges, even though not all the edges of a real cube can be seen at one time. Another problem is that it is hard to tell which face of the cube is in front. A Better Cube Begin with a square. Drawing a Cube Next, draw 3 parallel line segments going right and up from 3 corners of your square. The segments should all be the same length. Using Straws and TwistTies to Model Polyhedrons (Math Journal 2, p. 293) PARTNER Remind students how the straws and twistties were used to make frames for cubes and rectangular prisms in Lesson 2. Students should work with partners to construct polyhedrons and answer the riddles on journal page 293. Although students constructions might differ in size, the shapes should have the same properties. Drawing Cube Models (Math Journal 2, pp. 294 and 295) WHOLECLASS Have students follow the directions on journal pages 294 and 295 to draw cubes in three different ways. Tell students to practice on blank paper and then tape their best example of each method on the bottom of journal page 295. Ask students to share any other method they use to draw a cube. 2 Ongoing Learning & Practice Plotting Book Heights (Math Journal 2, pp. 295A and 295B) PROBLEM SOLVING PROBLEM SOLVING INDEPENDENT Students plot book heights in fractions of an inch on a line plot. Then they use the line plot to solve fraction and mixednumber addition and subtraction problems. Finally, connect the ends of the 3 line segments. This cube is better than before, but it shows only the edges and corners, not the faces. If you want, try shading your cube to make it look more realistic. Math Journal 2, p Designing a Bookcase Stephen wants to build a bookcase for his books. To help him design the bookcase, he measured the height of each of his books. He rounded each measurement to the nearest _ of an inch. His measurements are given below. Book Heights (to the nearest _ inch) 6 _ 2, 9 _ 4, 7 _, 7 _ 2,, 6 7_, 9 _ 4, 9 _ 4, 9 _ 4, 9 _ 4, 9 _ 4, _ 4,, _ 4, 3_, 6 _ 2, 7 _, 9, 6 7_, 9 3_, 6 7_, 7 _ 2,, _ 4, 9 _ 4, 6 7_, 6 7_, _ 4, _ 4, _ 4 Plot the data set on the line plot below. Number of Books Adjusting the Activity Book Heights ELL If possible, have several geometric solids available for students to handle as they work in their journals. AUDITORY KINESTHETIC TACTILE VISUAL X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Height (inches) Sample number models are given. 2 9 Use the completed plot to answer the questions below and on journal page 295B. Write a number model to show how you solved each problem a. What is the maximum book height? 9 3_ inches b. What is the minimum book height? 6 _ 2 inches c. What is the range of the data set? 2 7_ inches Number model: 9 3_ 6 _ = 2 2 7_ Math Journal 2, p. 295A Lesson 3 63
5 3 Designing a Bookcase continued a. What is the median of the data set? _ 4 inches Math Boxes 3 (Math Journal 2, p. 296) INDEPENDENT b. How much longer is the maximum height than the median height? _ inches Number model: 9 3_ _ 4 = _ 3. Suppose that Stephen wants to make the space between the shelves on his bookcase 7_ of an inch taller than his tallest book. a. How far apart should he make the shelves? 0 2_ inches apart Number model: 9 3_ + 7_ = 0 2_, or 0 _ 4 b. If the thickness of the wood he uses for the shelves is 5_ inch, what will be the total height of each shelf? (Hint: The total height is the thickness of the wood plus the distance between shelves.) 0 7_ inches Number model: 0 2_ + 5_ = 0 7_ 4. Stephen decides to make the bookshelf two shelves high. He will put all the books that are inches tall or shorter on the top shelf and all the books that are more than inches tall on the bottom shelf. a. What will be the difference in height between the tallest books on the top shelf and the shortest books on the top shelf? _ 2 inches Number model: 6 _ 2 = _ 2 b. What will be the difference in height between the tallest book on the bottom shelf and the shortest books on the bottom shelf? _ inches Number model: 9 3_ _ 4 = _ 5. Make up and solve your own problem about the book height data. Answers vary. Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson . The skill in Problem 5 previews Unit 2 content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 3a Use Math Boxes, Problem 3a to assess students ability to find multiples of whole numbers less than 0. Students are making adequate progress if they are able to name the first ten multiples of 6. Some students might be able to solve Problem 3b, which involves finding multiples of whole numbers greater than 0. [Number and Numeration Goal 3] Number model: Math Journal 2, p. 295B Study Link 3 (Math Masters, p. 329) INDEPENDENT Home Connection Students solve riddles about geometric solids. 3 Differentiation Options READINESS Sorting Geometric Solids SMALLGROUP 5 30 Min 3 Math Boxes. Draw the figure after it is translated to the right. 3. Name the first ten multiples of each number Find the solution of each open sentence. a. 7_  s = _ b. t + _ 4 = _ 2 c. 3_ 0  m = _ 5 d. 2_ + x = 3_ 4 a. 6 2,, 24, 30, 36, 42, 4, 54, 60 b. 6 72, 25, 344, 430, 56, 602, 6, 774, 60 s = t = m = x = 4_ 6 4 _ 0, or _ To investigate attributes of geometric solids, have students sort common household items into groups based on appearance and discuss how the objects in each group are the same and how they are different. Then have one student sort the items according to a different attribute and ask the other students to determine how they were sorted. Finally, have students sort the objects by use. Ask questions such as the following: Which containers would be easiest to pack together in a box? Why might containers for liquids and dry materials have different shapes? Why might the word container be a good description for most of the objects? 4. Insert parentheses to make each number sentence true. ( ) ( ) ( ) ( a = 250 b. 2.7 / = 3. c > 5 30 d = ) 5. Gum costs $0.0 per pack. What is the cost of a. 4 packs of gum? $3.20 b. 0 packs of gum? $.00 c. 6 packs of gum? $2.0 d. 33 packs of gum? $ Math Journal 2, p Unit 3D Shapes, Weight, Volume, and Capacity
6 ENRICHMENT Creating Cube Nets (Math Masters, pp. 39 and 444) SMALLGROUP 5 30 Min To apply students understanding of attributes of geometric solids, have them find all possible nets, or patterns of squares that can be folded to form a cube. (See margin.) Students should record their nets on inch grid paper (Math Masters, page 444). Students should eliminate any nets that are duplicates when reflected or rotated in their unfolded state. Eleven nets are possible: Name STUDY LINK 3 Geometry Riddles Answer the following riddles.. I am a geometric solid. I have two surfaces. One of my surfaces is formed by a circle. The other surface is curved. What am I? cone 3. I am a polyhedron. I am a prism. My two bases are hexagons. My other faces are rectangles. What am I? hexagonal prism Try This 5. Write your own geometry riddle. Answers vary. Study Link Master I am a geometric solid. I have one square base. I have four triangular faces. Some Egyptian pharaohs were buried in tombs shaped like me. What am I? square pyramid or rectangular pyramid 4. I am a polyhedron. All of my faces are the same. All of my faces are equilateral triangles. I have eight faces. What am I? octahedron Practice $0 6. $20 + $30 = 7. = $35 + ($35) $5. = $0 + ($25) 9. $0 + ($00) = $55 $70 $ $5 + ($40) =. $400 = $300 + ($00) Math Masters, p. 329 On an Exit Slip, have students describe common features of nets that will and will not result in cubes. Sample answer: Nets that will result in cubes have 6 squares and 4 sides. Nets that will not result in cubes have fewer than 6 squares, have 4 squares that share a single vertex, or have more than 4 squares in a single row. EXTRA PRACTICE Taking a 50Facts Test (Math Masters, pp. 4, 44, and 46) SMALLGROUP 5 5 Min Folding a net to form a cube See Lesson 34 for details regarding the administration of a 50facts test and the recording and graphing of individual and optional class results. Lesson 3 65
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