### Purpose

To explore characteristics of shapes by making and using tangram sets; to discover how the tangram pieces are related to one another; and to determine how many different combinations of the triangles, squares, and parallelograms in tangram sets can make a given shape.

### Context

By using tangram shapes, children learn the relationships between shapes, for instance, that two identical right isosceles triangles fit together to form a square. Additionally, children learn that three basic shapes—triangles, squares, and parallelograms, each composed of one or more small right triangles—can fit together to form many other shapes and figures.

One line of research on how people learn emphasizes the helpfulness of making multiple representations of the same idea and translating from one to another. When a student can begin to represent a relationship in tables, graphs, symbols, and words, one can be confident that the student has really grasped its meaning.

Students should begin to use all these features in describing and designing things and increase substantially the number of geometric shapes and concepts they are familiar with. For more information on the development of geometrical thinking, see the related research for this Benchmark section. (*Benchmarks for Science Literacy*, p. 352.)

Experiences in exploring and understanding relationships of shapes to each other are important prerequisites for advanced mathematics. The opportunity to explore shapes will introduce vocabulary and geometric concepts that are part of mathematics and everyday problem solving, such as storing articles in tight spaces or building a model car.

### Planning Ahead

Each student will need a set of tangrams for this activity. See Activity 1 (in the Development) for instructions on how to make your own tangram sets.

### Motivation

We see shapes around us all the time. Ask students to think about shapes by asking the following questions:

- What's the simplest shape (closed figure) you can make just with straight lines?
- What's this shape called? (A triangle.)
- What are its characteristics?
- Can you construct one that has sides that are 10, 20, and 30 cm long, respectively? Why or why not?
- Does this lead to any further characteristics of your shape or to modification of those you already have?

If you have time, you can extend this discussion to four-sided figures enclosed by straight lines. Or, you can go right on to introducing the following activities, that involve combining different three- and four-sided shapes to get more complicated shapes.

### Development

**Activity 1: Constructing Tangrams** (optional)

While you can use plastic tangram sets, having students make and cut out their own set of tangrams using heavy, colored cardboard can help them understand the relationship of the shapes. See Constructing Your Own Set of Tangrams on Tom Scavo's Tangrams, in which students start with a rectangular piece of paper, folding and cutting to create their own set of tangrams.

**Activity 2: How Do Tangrams Work? **

In this activity, students will try to create a specified shape using different combinations of their tangram pieces and then fill out a data sheet to show which shapes they used.

For detailed instructions and student-ready worksheets, use the How Do Tangrams Work? student sheet for this lesson.

Each worksheet displays a shape and includes a data table. Students should try to make the shapes using as many different combinations of the tangram pieces as they can.

To guide students through the activity, ask:

- Is there another way, different from the one you just found, to make the shape? If so, fill in the second column of the data table.
- How are the columns different? How are they the same?
- How can you be sure that the new way to make the shape is different form the old?
- Encourage students to find three, four, or even more ways to create the shape.

Or, you can devise a shape of your own by combining the tangram pieces to make a shape and tracing the outline of the shape. Using shapes that are drawn to scale and having students fill these in may be more appropriate for younger students. More advanced students should enjoy the challenge of trying to recreate a shape that is smaller or larger than their tangram pieces.

**Activity 3: How are the shapes related? **

Begin this activity by discussing shapes and revisiting some of the concepts discussed in Activity 2. Have students think about and define some of the characteristics of other shapes, such as squares, parallelograms, and rectangles. Now that students have explored shapes using tangrams, they should be able to further refine their thinking about how these shapes are related and how they are different.

Then, reflecting back on the different combinations they used in Activity 3, ask students to analyze what they did using the guiding questions below. Ask:

- What process(es) did you use to figure out the shapes?
- What did you discover about triangles? About the other shapes?
- What did you discover about the relationship of the tangram pieces to one another?
- Which has the larger area, the square or the parallelogram? Explain your answer.
- Which has the larger area, the square or the medium triangle? Explain your answer.
- What do your results tell you about the relationships among the various tangram pieces? Can one shape be made from another (or others)?
- Is there a basic shape that could be used to make them all?
- How can you prove your answers? Try it!

### Assessment

Individual or class portfolios of examples of patterns and relationships collected over time could be used as the raw material for reflecting on how mathematics defines a pattern or relationship so that it transcends and is more powerful than individual instances of it.

As students conduct these activities on shapes using tangrams, have them keep a folder of images and ideas from a variety of contexts that provide examples of patterns and relationships. Their portfolio should include not only pictures and images, but also their reflections as well as answers to the questions in the activities.

### Extensions

Can a geometric figure, for example, a parallelogram, be used over and over again to cover a flat surface completely? Try it with the various tangram pieces. Can just any shape be used to do this? (Disregard the rough edges that may be formed.)

Shapes that can be simply slid along (in any direction) and/or rotated to cover a surface are said to tessellate the surface. To investigate tessellations and other shape explorations, you can begin with books such as *The Sneaky Square* & *113 Other Math Activities for Kids.*

Try some of the other activities on Tom Scavo's Tangrams. You can also refer to the Tangram Resources listed on these pages for ideas for further explorations. Or, visit Enchanted Minds Tangrams, another source for a tangram Java applet.

Tantalizing Tangrams, a unit from the Mathematics TEKS Toolkit, includes a tutorial, interactive activities, and teacher ideas for more tangram activities.