Concept: Development of geometric thinking through exploration. Why this problem? Your Task. This problem: Getting Ready

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1 Concept: Development of geometric thinking through exploration Your Task Getting Ready Part A 1. Read the background for the activity. (See page 3.) 2. Go to the NCTM Tangrams Electronic Example. ( eexamples/chap4/4.4/index.htm#applet) 3. To choose a puzzle, you should click one of the eight gray shape buttons beneath the work area. Why this problem? This problem: Provides the learners with experience finding the area of a figure by direct comparison and lays the foundation for work with square units of area; Provides the learners with experience composing and decomposing shapes. 1. Read How to Use the Interactive Figure (link above the workspace) for a description of how to use the tool. 2. Do two of the puzzles. a. Choose a picture and use all seven pieces to fill in the outline. b. If necessary, click the Show Hint button to display an outline of the shapes. 3. For each shape, answer the following questions: a. Can you fill the outlines in another way? b. How many different ways are there to fill in this shape? c. What do you do when you cannot figure out a puzzle? d. Can some tangram pieces be substituted for others? 4. Answer the next two teacher-reflection questions in your Learning Log. a. How would you use computer manipulatives in your teaching to help all children learn the mathematics? b. What additional problems would you create to focus your students' attention on the relationships among the tangram pieces? Page 1 of 6

2 Part B (Optional) 1. Click the Tangram Challenges icon (located at top right or bottom right of the NCTM Tangrams Electronic Example). 2. Scroll down to the Task area and do the first four tangram challenges. a. Make a square using only one tangram piece. b. Make a square using two tangram pieces. c. Make a square using three tangram pieces. d. Make a square using four tangram pieces. 3. Is it possible to solve all these problems? Use the definition of a square to help you find the answer. 4. Try the additional tangram challenges with the virtual tangrams. Make at least two of the following four figures using all seven tangram pieces: o A trapezoid; o A rectangle that is not a square; o A parallelogram that is not a square; o A triangle. * Answers on page 4 Page 2 of 6

3 Tangram Puzzles: Activity Background By Pamela Buffington, Ph.D. It is critical that students have experiences with geometric shapes before using formal labels, descriptions, and properties. As the reading in Session 1 stated, "teaching experiments with area measure have revealed that second graders could develop a comprehensive understanding of area measure when they began solving problems involving partitioning and redistributing areas without measuring." (Kilpatrick, J., Swafford, J., & Findell, B. (Eds.), p. 283). The use of tangrams and mosaic puzzles provides opportunities for students to explore the relationship between geometric figures and puzzle pieces of different shapes and sizes. They can also experiment with the concept of area without using the formal "unit squares" or counting the "square units." In this activity you will visit the NCTM Illuminations site and use the "Tangram Puzzle" electronic examples. In Part A of the activity, you will be asked to use the tangram online tool to complete several puzzles. This tool provides multiple representations that have the same area. Researcher and mathematician Pierre M. van Hiele states that "placing pieces to fill in the space in puzzles also provides experiences with area. By direct comparison, students can show that some pieces take up more space than others." (p. 315). In Part B of the activity, you will be asked to use the tangram online tool to discover the relationships between and among the pieces by comparing the pieces directly, combining the pieces, and recombining the pieces. For example, the first figure above is half of the second figure, as is shown by directly comparing or combining in the third figure. In addition, van Hiele states that "these kinds of experiences with area lay a foundation for later work with square units of area and the discovery of ways to find the area of various shapes --- for example, why the area of a right triangle is one-half the area of a rectangle --- and how the enlargement of a shape, for instance, by doubling the lengths of its sides, affects its area. (p. 315) Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press. Van Hiele, P. M. (1999). Developing Geometric Thinking through Activities That Begin with Play. Teaching Children Mathematics, 5(6), Page 3 of 6

4 Answers Part A 2. Here is a sample solution for the third figure in the top row, which looks like the letter E. 3. Using the same example figure above, here are the answers to the four questions listed: a. Can you fill the outlines in another way? Yes. The figure has a mirror plane of symmetry along the horizontal axis. There are at least two possible configurations of the pieces for this puzzle. Solution given by hint: Mirror image: b. How many different ways are there to fill in this shape? There are two ways to fill in this shape, as shown above. c. What do you do when you cannot figure out a puzzle? If you cannot figure out a puzzle, it might be helpful to think about what kinds of shapes can be made by putting more than one piece together. For instance, think about which different shapes could be used to fill in a square. Practicing with other tangrams may also be helpful, since similar strategies can be used to fill the spaces across figures. Page 4 of 6

5 d. Can some tangram pieces be substituted for others? Some tangram pieces can be substituted for others if they have equivalent areas and can make the same shapes. This will often require rotating or flipping pieces. Part B (Optional) 2. Scroll down to the Task area and do the first four tangram challenges. a. Make a square using only one tangram piece. b. Make a square using two tangram pieces. c. Make a square using three tangram pieces. d. Make a square using four tangram pieces. Here are two possible solutions: 3. Is it possible to solve all these problems? Use the definition of a square to help you find the answer. Yes, see above. 4. Try the additional tangram challenges with the virtual tangrams. Make at least two of the following four figures using all seven tangram pieces: o A trapezoid; Page 5 of 6

6 o A rectangle that is not a square; o A parallelogram that is not a square; o A triangle. Page 6 of 6