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In this activity, you will work on both visualization and communicating mathematically.

The National Council of Teachers of Mathematics writes:

*Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment…. Because mathematics is so often conveyed in symbols, oral and written communication about mathematical ideas is not always recognized as an important part of mathematics education.*

*Principles and Standards of School Mathematics,* p. 59

Reston, VA: National Council of Teachers of Mathematics, 2000

The goal in this activity will be to use mathematical language to express your ideas and to understand the ideas of others. In the following Activity, you will look at the effectiveness of various types of descriptions of geometric designs, ranging from descriptions that use informal language to those whose language is precise and mathematical.

This activity also works best when done in groups. Go to **Note 3** for suggestions for doing the “Building from Directions” activity with a group.

If you are working alone, consider asking a friend or colleague to work with you. Otherwise, print the following final designs (PDF). Put the final designs aside without looking at them. Then begin with the first design description. Follow the instructions to build the described design with pattern blocks, or draw it on a piece of paper. When you are finished, compare your drawing with the final design for this description. Repeat this activity with the second and third designs.

The design looks like a bird with

- a hexagon body;
- a square for the head;
- triangles for the beak and tail; and
- triangles for the feet.

- Start with a hexagon.
- On each of the three topmost sides of the hexagon, attach a triangle.
- On the bottom side of the hexagon, attach a square.
- Below the square, attach two more triangles with their vertices touching.

- Start with a hexagon. Position it so that it has two horizontal sides.
- On each of the non-horizontal sides, attach a triangle so that the side of the triangle exactly matches the side of the hexagon.
- On the top side of the hexagon, attach a triangle so that the side of the triangle exactly matches the side of the hexagon.
- Take a square and place it above the top triangle. It should be placed so that the vertex of the triangle is at the midpoint of a side of the square and the sides of the square are horizontal and vertical.
- On each of the two top vertices of the square, attach a triangle. Place each triangle so that the vertex of the square is at the midpoint of the side of the triangle and the side of the triangle makes 45° angles with the two sides of the square it touches.

When you were building from the given descriptions, what pieces were clear, and what pieces were unclear? What elements of the descriptions made it possible for you to picture (or not to picture) what was described and recreate it?

How closely did your designs match the target? Describe any differences between them and why you think they occurred.

*Principles and Standards of School Mathematics* (Reston, VA: National Council of Teachers of Mathematics, 2000), 59. Reproduced with permission from the publisher. Copyright © 2000 by the National Council of Teachers of Mathematics. All rights reserved.

**Note 3**

If you are working in a group, first, have everyone take about five minutes to explore the pattern blocks. Then divide the group into pairs and have the partners sit with each other. Each partner will work with the same set of pattern blocks. Each pair should use a divider, such as a propped-up binder, to prevent the partners from seeing each other’s work. Then follow these steps:

- Partner 1 will build a design with his pattern blocks. He should build a design flat on his desk, not a tower or building.
- Partner 1 will then describe his design to Partner 2. He may use words only — no gestures, drawings, or other visual cues. Partner 2 may ask for clarification, but should try not to ask too many questions. The goal is for Partner 1 to describe the design completely.
- Partner 2 will build the design described by Partner 1. When the design is built, they will lift the divider to compare their designs.
- The partners should switch roles.

After the partners on each team have had a chance to describe their designs to each other, discuss the questions below and jot down answers to share with the whole group. During the discussion, make a list of terms that were used, ideas where a term was needed and not known, discrepancies between the designs and the target, and what might have been said to eliminate those differences.

**Consider These Questions**

- When you described your design for your partner, what was difficult to describe? What was easy to describe?
- Jot down some of the mathematical terms you used in your descriptions. Were there terms you didn’t know or couldn’t think of, but that you felt the need to use?
- When you were building from your partner’s description, what pieces were clear, and what pieces were unclear? Were you able to picture what your partner described and recreate it?
- How closely did your designs match the target? Describe any differences between them and why you think they occurred.

Answers will vary. Here are some examples of what may not have been clear from the given descriptions. In Description 1, it is not clear what is meant by “looks like a bird.” Also, the description doesn’t tell you how to place each shape in relation to the others. In Description 2, the hexagon could be oriented in different ways (top and bottom sides horizontal or left and right sides horizontal). It is also not clear how to attach the triangles and squares to the hexagon or the two triangles to the square. Description 3 removes these ambiguities.

Answers will vary.