Mathematics Instructional Materials SAS#.1 (one per pair of students) SAS#.2 (one per pair of students) TIS#.1 (transparency) TIS#.2 (transparency) TIS#.3 (Journal prompt) Isometric Dot Paper Isometric Line Paper Pop Cubes, Connecting Cubes, LinkerCubes, Multilink Cubes or Centimeter Connecting Cubes Student journal Program Overview Students create three-dimensional shapes shown in drawings using cubes that connect. As they learn to interpret the two-dimensional drawing, they explore spatial relationships of their cube creations including volume and surface area. Students learn to draw a two-dimensional figure from a three-dimensional object. NCTM Standards Addressed Standard: Geometry Analyze characteristics and properties of twoand three-dimensional geometric shapes and develop mathematical arguments about geometric relationships: Grades 3 5 identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes; classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids; investigate, describe, and reason about the results of subdividing, combining, and transforming shapes; make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions. Grades -8 precisely describe, classify, and understand relationships among types of two- and threedimensional objects using their defining properties. Use visualization, spatial reasoning, and geometric modeling to solve problems Grades 3 5 build and draw geometric objects; identify and build a three-dimensional object from two-dimensional representations of that object; identify and build a two-dimensional representation of a three-dimensional object. Grades -8 use two-dimensional representations of threedimensional objects to visualize and solve problems such as those involving surface area and volume. Standard: Measurement Apply appropriate techniques, tools, and formulas to determine measurements Grades 3-5 select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles; develop strategies to determine the surface areas and volumes of rectangular solids. Grades -8 develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders. Instructional programs from prekindergarten through grade 12 should enable all students to Standard: Problem Solving build new mathematical knowledge through problem solving. Standard: Reasoning and Proof make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs. 1
Standard: Communication communicate their mathematical thinking coherently and clearly to peers, teachers, and others; use the language of mathematics to express mathematical ideas precisely. Standard: Connections recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole. California Standards Addressed Measurement & Geometry Grade 5 2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures: 2.3 Visualize and draw two- and threedimensional objects made from rectangular solids. Mathematical Reasoning Grade 5 3.0 Students move beyond a particular problem by generalizing to other situations: 3.3 Develop generalizations of the result obtained and apply them in other circumstances. Grade 3.3 Develop generalizations of the result obtained and the strategies and apply them in new problem situations. Program Outline Program Viewing Welcome Focus Activity #1 DLI CT S Activity #2 DLI CT S Activity #3 DLI CT S Activity #4 DLI Introduction and Teaser Students practice spatial visualization activities. Build It Introduces and demonstrates the activity YOUR TIME Facilitates activity Work in groups to build shapes on worksheet Draw It Introduces and models the task YOUR TIME Reinforces and clarifies directions. Facilitates by observing groups Builds a short train of cubes and uses the isometric paper to make drawings of the different views Make a Prediction Displays a diagram, students predict how many cubes it took to build YOUR TIME Facilitates discussion and has students share strategies and predictions. Make and discuss predictions Drawing Pictures of 3-D Shapes Demonstrates the activity, for use as a post viewing activity YOUR TIME 2
CT S Journal Writing Makes sure all students understand task, facilitates activity Builds shapes with a few cubes and draws it on paper What professions might need good spatial skills? KEY DLI: Distance Learning Instructor CT: Classroom Teacher S: Students Activity #1: Build It Instructions for Program Viewing Description Students work with partners using Pop Cubes to build figures on SAS#.1 Advance Preparation Have Pop Cubes ready for distribution, approximately 20 per student Materials Pop Cubes, LinkerCubes, Centimeter Connecting Cubes, or MultiLink Cubes SAS#.1 Lesson Implementation The Distance Learning Instructor introduces the activity. Each student in the group selects a drawing from SAS#.1 to build with the cubes. They check each other s building for accuracy. Activity #2: Draw It Description Students build a short train of Pop Cubes (2-3) and draw three or four different views on the isometric dot-paper. Advance Preparation Prepare a transparency of the isometric paper to use to help clarify instructions, draw a few more examples, or so students can display and share their solutions. Materials Isometric paper, duplicated for students (at least 2 per person) Pop Cubes Lesson Implementation The Distance Learning Instructor demonstrates how to use the isometric paper and gives the students an opportunity to practice before the YOUR TIME segment. Students create a train using 2 or 3 Pop Cubes, then draw as many views of their train on the isometric paper. They shade in their drawings for clarity. 3
Teacher Note Isometric Paper Suggest to the students that they use the dot that matches the corners of the cube. They may need to move their train or building so that the corners of the cubes are positioned to match the dots on the paper. Have students use shading to enhance their drawings. Drawings can be from different views, looking down, looking up, to the right, or left (see examples below): Activity #3: Make a Prediction Description The Distance Learning Instructor displays a drawing of a building, the Classroom Teacher facilitates the activity. Students predict how many cubes it took to build. They make predictions. Students then use Pop Cubes to make the building to verify their prediction. Students begin working on SAS#.2. Advance Preparation Duplicate SAS#.2 (one per pair) Materials Pop Cubes SAS#.2 (one per pair) Lesson Implementation The Distance Learning Instructor displays a drawing of a building. Students predict the number of cubes necessary to build it. Students then build it, to verify their prediction. Students call-in with their prediction and the actual number of cubes it took. Students begin working on SAS#.2, they will need to complete after the program. Teacher Notes -- Answer Key SAS#.2 How Many Cubes? 1) Actual: 7 cubes 2) Actual: 8 cubes 3) Actual: 8 cubes Activity #4: Drawing Pictures of 3-D Shapes Description Students create buildings using Pop Cubes. Advance Preparation None 4
Materials Pop Cubes Isometric paper (one per student) Lesson Implementation The Distance Learning Instructor describes the activity. Students create a building then draw this building on the isometric paper. Students send this drawing to another student to see if it can be built correctly. Students do this as a post-viewing activity. Post-Viewing Activities Being able to draw in perspective or interpret a drawing is an important skill in our daily lives. Interpreting maps, giving directions, building an object from written plans, are a variety of tasks that need spatial awareness. Complete Activity #3 and Activity #4. Draw pictures of 3-D shapes. Students create buildings and draw them on isometric paper. Students send their drawings to another student to see if it can be built correctly. Quick Takes Make transparencies of different shapes (TIS#.1), flash them on the overhead for a few seconds, and have students build them. Display this shape again for a few more seconds for students to make adjustments to their building. When students have had enough time, display it again for students to revise and correct their creation. Students share their strategies for maintaining and organizing their mental picture. Skeleton Tower Display TIS #.2 and have students share predictions how many cubes it takes to build. Allow time for students to share their strategies for their prediction and then allow them to build it to verify their prediction. Resources Software to Practice Spatial Visualization Activities: Building Perspective, by Sunburst Students need to predict how buildings are arranged on a grid by viewing from the front, back, left and right. Spatial problem solving and logical reasoning are involved to come up with a solution. Books Seeing Solids and Silhouettes, 3-D Geometry, TERC, Dale Seymour Publications. The Super Source, Snap Cubes, ETA/Cuisenaire Company of America. Credits Skeleton Tower from the Shell Centre for Mathematical Education. England: University of Nottingham, 1984. 5
SAS#.1 Build It Name Date
SAS#.2 Draw It Name Date How Many Cubes? How many cubes does it take to make each building? Predict. Then build with cubes to check. 1) Prediction: cubes Actual: cubes 2) Prediction: cubes Actual: cubes 3) Prediction: cubes Actual: cubes 7
TIS#.1 Quick Take 8
TIS#.2 Possible Assessment: Skeleton Tower (i) (ii) (iii) (iv) How many cubes are needed to build this tower? How many cubes are needed to build a tower like this, but 12 cubes high? Explain how you worked out your answer to part (ii). How would you calculate the number of cubes needed for a tower n cubes high? Shell Centre for Mathematical Education, University of Nottingham, 1984. 9
(i) Skeleton Tower Rubric Showing an understanding of the problem by dealing correctly with a simple case. Answer: 2 marks for a correct answer (with or without working) Part mark: Give 1 mark if a correct method is used but there is an arithmetical error. (ii) Showing a systematic attack in the extension to a more difficult case. Answer: 27 4 marks if a correct method is used and the correct answer is obtained. Part marks: Give 3 marks if a correct method is used but the work contains an arithmetical error or shows a misunderstanding (e.g. 13 cubes in the centre column) Give 2 marks if a correct method is used but the work contains two arithmetical errors/misunderstandings. Give 1 mark if the candidate has made some progress but the work contains more than two arithmetical errors/misunderstandings. (iii) Describing the methods used. 2 marks for a correct, clear, complete description of what has been done providing more than one step is involved. Part mark: Give 1 mark if the description is incomplete or unclear but apparently correct. (iv) Formulating a general rule verbally or algebraically. 2 marks for a correct, clear, complete description of method. Accept "number of cubes=n (2n-1)" or equivalent to 2 marks. Ignore any errors in algebra if the description is otherwise correct, clear and complete. Part mark: Give 1 mark if the description is incomplete or unclear but shows that the candidate has some idea how to obtain the result for any given value of n. Shell Centre for Mathematical Education, University of Nottingham, 1984. 10
Isometric Dot Paper 11
Isometric Line Paper 12
TIS#.3 Journal Writing Journal Writing What professions might need good spatial skills? 13