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At the beginning of this session, we learned that finding, describing, explaining, and predicting with patterns are some of the most important skills in mathematics. We had a chance to practice these skills throughout the session as we explored several patterns.

We have used many different kinds of descriptions for patterns. Two kinds of descriptions, in particular, are very important:

A *recursive description* tells you how to get from one term to the next. In the table in Part A, for instance, some people might say, “Each output is 4 more than the previous one.” That’s a recursive description. In the pattern of Part D, if you said, “To get from staircase n to staircase n + 1, add n + 1 blocks,” you were using a recursive description.

Recursive descriptions are most useful when the instruction to get from one step to the next works no matter where you start in the pattern. But you need to know the output from one or more of the previous steps to find the next one.

A *closed-form description* tells you how to get from any input to any output, without needing to know any previous outputs. In the table in Part A, the instruction “take the input, multiply it by 4, and add 2” or “the output for n is 4n + 2” are closed-form descriptions. In the pattern of Part D, if you said, “Staircase n has n(n + 1) / 2 blocks,” you were using a closed-form description.

**Problem E1: Write and Reflect**

Give an example from your work so far that has been a recursive description of a pattern. Did you ever describe a pattern with a closed-form description? Can you describe the use of the variable in your description?

**Problem E2: Write and Reflect**

In your opinion, why is the study of patterns important, and how does it connect to algebraic thinking?

**Note 12**

Review the “Big Four” of patterns: finding, describing, explaining, and predicting with patterns. Do you think you have practiced all four of these skills? When? What was the most enjoyable? Most difficult? Do you practice these processes with your students?

Read about recursive and closed-form descriptions.

**Groups:** Discuss the difference between closed form and recursive form. Take five minutes to jot down answers to Problems E1 and E2, then spend some time sharing a few of the answers.

**Problem E1**

Recursive descriptions include things like “Take the previous number of blocks and add n” and “Take the previous number of toothpicks and add 4.” Closed-form descriptions include things like “The number of toothpicks is given by the formula T = 4n + 2” and “The number of blocks in the nth staircase is given by the formula n * (n + 1) / 2.”