Using precise language is key to doing mathematics effectively. When students learn to apply terms accurately and to describe their own mathematical thinking clearly their understanding improves.

Let's look back at the Tile Patterns problem, introduced earlier in this session. We'll use it to focus on this key element of the communication standard, precise language.

As you explored that problem, you may have discovered that the more precise your description, the better your friend would be able to construct the pattern. Precise language may have aided in determining the functional notation, as well.

Here is one precise way you might have described the tile problem to your friend:

"The first term is made from two square tiles, one above the other."

"The second term uses five tiles. The bottom is a square formed by four tiles, two rows of two tiles, with the fifth tile forming a third row just above the left-hand side of the square."

"In the third term, there are ten tiles. Nine tiles make a three by three square, with the tenth tile on the top left of the square."

"So, to build the fourth term, you need seventeen tiles: sixteen to make a four by four square, with one tile on the top left."

"The n^th term would require n² tiles for the n by n square, and one more for the top (placed on the top left), n² + 1 tiles."

The process of describing how to build the term forces the listener and the speaker to concentrate on the relation between the number of the term and the value of the term. Such descriptions illustrate that understanding mathematics relies on communicating about representation. Finding a way to talk precisely about the visual and symbolic aspects of this problem helps complete the task.

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