Table of Contents Appendix E-examples Resources

Developing Geometry Understandings and Spatial Skills through Puzzlelike Problems with Tangrams: Tangram Challenges Tangram Puzzles Tangram Challenges

Describing figures and visualizing what they look like when they are transformed through rotations or flips or are put together or taken apart in different ways are important aspects of geometry in the lower grades. This two-part tangram example demonstrates the potential for high-quality experiences provided by computer "shape" environments for students as they learn concepts described in the Geometry Standard. Problem-solving tasks that involve physical manipulatives as well as virtual manipulatives afford many students an entry into mathematics that they might not otherwise experience. In Tangram Puzzles, students can choose a picture and use all seven pieces to fill in the outline. In this second part, Tangram Challenges, students can use tangram pieces to form given polygons.

Task

Is it possible to complete all these tasks? Try these tangram challenges with the virtual tangrams:

Make a square using only one tangram piece. Make a square using two tangram pieces. Make a square using three tangram pieces. Make a square using four tangram pieces.

Make a square using five tangram pieces. Make a square using six tangram pieces. Make a square using all seven tangram pieces.

Which of these figures can you make using all seven tangram pieces?

• A trapezoid
• A rectangle that is not a square
• A parallelogram that is not a square
• A triangle
  

[How to Use the Interactive Figure]

[Stand-alone applet]

Working with Tangram Challenges in the Classroom

Many students will find these tasks very interesting but challenging. Young students are learning about position in space, new vocabulary, and properties of figures all at the same time. The computer tangrams may help them become more aware of the properties of figures and the processes they use in manipulating shapes because they must plan the moves they need to make. Teachers can encourage students to become more deliberate in their planning by having them work with a partner to talk about the actions they need to take. For example, students have to figure out explicitly how to place the tangram pieces in relation to one another in activities such as these, where there are no outlines. The built-in rotation and flip tools are also a good way to help students see the transformation motions.

These tangram challenges may be made easier by giving students outlines to use at their desks so they can experiment with fitting the seven tangram pieces into the outlines.

Assessment through Observations and Conversations

Activities such as Tangram Puzzles and Tangram Challenges can serve as vehicles for assessing students' thinking. In observing and talking with students, teachers might consider questions such as these:

• How easily do students manipulate the shapes?
• What mathematical vocabulary are students using as they talk with one another?
• Do students recognize congruence and relationships among combinations of shapes?
• Do students appear to use what they have learned in earlier tasks in solving new problems?

Take Time to Reflect

How might work with computer manipulatives facilitate the learning of many children with special needs? What additional challenges might teachers create that would focus students' attention on the relationships among the tangram pieces?

Tangram Puzzles

Tangram Challenges

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