Visualizing Functions Summary
Monday - Friday, June 29 - July 3, 2009
We've been looking at visual patterns, identifying functions that represent them, and then trying to find how each component of the function manifests itself within the picture. We've discovered that there are frequently a great many different functions that can represent the same pattern and that there are often different ways to "see" the same function within the picture.
We began by looking at three different variations of Regina's Logo. The goal was to write a function that could be used to predict how many squares or tiles would beused to create the figure of a given size. While the pattern itself was the same ineach variation of the logo, each variation began at different stages of the pattern, for example, variation 1 began with one picture at size 1, while variation 2 began with the same exact picture, but called it size 0 (and the same picture was size 2 in variation 3). The challenge for many of us existed not in finding a representative function, but in finding how that particular function manifested itself pictorially.
Next, we looked at a new visual pattern, but using several different approaches. First, the goal was to find a function that predicted how many squares or tiles would be used to create the figure of a given size. Then, we identified functions that predicted how many "sticks" or edges would be used. Then we briefly looked at functions that predicted just the perimeter, which was followed by creating functions that predicted how many "dots" or vertices would be used in the figure.
Once most of us were ready to move on to something different, we began looking at the numerical patterns that existed in a sort of modified multiplication table (multiplication table plus). We were first given a table that had some of the initial entries and had to find the pattern to fill it in. We wrote a linear combination to represent the contents of any cell and then discussed whether the table was closed under addition and multiplication. We were then given a table with a few random entries and had to determine the initial values so we could fill it in. We discussed various ways for successfully accomplishing this.
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.