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.