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Patterns, Functions, and Algebra
Session 7 Part A Part B Part C Part D Homework
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Notes for Session 7, Part C

Note 9

This section focuses on figurate numbers as a visual context for introducing and exploring quadratic functions. The idea is to get a sense for another kind of function without worrying too much about the details. In working through these problems, some people might come up with closed-form rules to describe the figurate numbers, and others might come up with recursive or even doubly-recursive rules (using the fact that the second differences are constant). All of these approaches make it possible to work through the problems and see how these functions are different from both the linear and exponential functions studied previously.

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Note 10

Graph paper can be used to work through Problems C1-C9. If using the spreadsheet program to create the graphs, only use the data for the first six figures (or fill in the in-between entries in the table). Large jumps in inputs cause the spreadsheet graphs to distort and look less like parabolas. Since there are only a few data points for each, it might be just as easy to graph them by hand.

If you have trouble coming up with a rule for the triangular numbers, remember that the triangles you are building look a lot like the staircases built in a previous session. (In fact, the pattern is exactly the same.)

Groups: To wrap up this part, discuss how tables were filled in, particularly how missing inputs were found when given an output. Also compare rules. Some people will likely have come up with closed-form rules and others with rules like "add the previous output and the current input" (for triangular numbers). They might even have extended patterns like these:

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Note 11

Groups: Share thoughts on how the two graphs in this part were similar to each other and different from others they have seen. If possible, look at a picture of a parabola that includes negative inputs in order to clarify the difference in shape between a parabola and the exponential functions they saw earlier.

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