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Exploring linearity

August 26, 2010
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Ever since I visited the Park School, and learned about its innovative math curriculum, I’ve been thinking a lot more about the mathematics in my classroom. In this particular case, the idea of linearity.

My students have figured out that the length of the pendulum is the only thing that affects its period, and after they roughly collected a few data points, they came to conclusions like this

For every 6 cm you shorten the length, the period changes by 0.2 s.

This statement is the hallmark of linearity, but the pendulum’s period does not vary linearly with its length. How to show this? Just the day before, I tried to walk the group that made this statement through the linearization process: graphing their data, seeing that it curves, then getting them to try various plots to find two things that are directly related (like T and \sqrt{L} ), and they did this, but since they made the statement above the next day they clearly didn’t “get it.”

To explore this idea further, we graphed Dan Meyer’s grocery cart data, and talked about how making a graph can really make you famous. From this, I think the students saw the true meaning of the slope—for every item you add to your cart, it takes 3 more seconds on average to check out.

That made a nice segue into the previous statement about the length of a pendulum and its period, which quickly led us to discover we probably needed some better data. So I decided to film 5 pendulums of varying length, since much of the problem is just taking good data for the period of very short pendulums, and we spent few minutes in class just measuring the period of the 5cm pendulum video as a group, and then watching the sequel, the 10cm pendulum in the group and measuring that period. From there, I posted the statement again about how the length affects the pendulum, and asked them if it was fair to call this a linear relationship. When students saw my line was predicting a 0 length pendulum to have a non-zero period, I think they were beginning to get it. Too bad the bell rang.

So for homework, I assigned students to individually make pendulums with different periods. Today, we’ll hang them from the bulletin board, arranged by period, and see that the strings actually form some sort of non-linear pattern.

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