My intro physics kids fear math—Dan Meyer’s angry wolverine metaphor describes their view of the subject perfectly. This partly because even though they’re better at math than 95% of kids their age, they’re at a hyper competitive private school, and have found themselves tracked into ‘regular math’, which means they only took algebra in 8th grade. But my main goal in physics is to help my students to see how science changes the way you see the world. In particular, your pattern seeking mind begins to realize that the patterns we see all around us can be described mathematically, and we can use those patterns to make predictions, search for causes and learn subtle, profound and beautiful things about the world around us. But no one wants to play with the wolverine.

So yesterday, despite still feeling linearizing a square root relationship with intro physics freshmen, might not be the best way to go, I jumped head first into trying to make this exciting, picking up where we left off a few days ago with the cool picture we had before.

When you set up pendulums by equal intervals of time, you see the lengths don't grown linearly.

Most kids were starting to see this as neat. So we said this was a graph—just like what they’d see in a textbook, or on a calculator. The only strange thing about it is that it’s rotated from how we normally view graphs of period vs length.

A picture is a graph, and a graph is a picture.

But if we just rotate it 90°, this graph looks just like the ones we’ve made from our data.

Picture with a twist

Most of them know that the graph of a parabola is $y=x^2$ (I said they were good at math), and so I have them graph it, and then literally turn their calculators until it looks like our “graph”. So everyone turns their calculators on its side, and they see that the shape matches.

Now, we say the $y$ axis is now horizontal, and I ask what is being plotted along the horizontal: length $L$.

We then see that $x$ must correspond to period ($T$) and so if we rewrite $y=x^2$ we get

$L=T^2$

Then I ask, don’t we want a formula for period, not length? How do we solve for period? Simple, they say—square root

$T=\sqrt{L}$

And there we go. From there we take a quick trip to google spreadsheets and make the computer do all the hard work calculating the roots of 20 numbers for us in under a second, plot the data, and see a beautiful direct relationship.

linearization of period-length relationship.

Here we print off copies of this graph and students have to find the slope by hand. Most of the time, I really wish gdocs would add a simple “find trendline” command, but when students are first learning this, I actually really love that it doesn’t and then have to find the line of best fit by hand.

Finally, we come back to one last puzzle. Our equation is something like

$T=0.2\sqrt{L}$

But if you look closely at the units, you see that seconds must somehow be equal to $\textrm{centimeters}^{1/2}$ which makes no sense. Then somebody realizes we forgot the units on the slope, so we calculate that, and what do you know? They turn out to be $\frac{\textrm{s}}{\sqrt{\textrm{cm}}}$, just what we need to make everything work out. And so, we put it all together plotting our data with our model, and presto:

functions describe nature!

So we just described nature with an equaiton, and I think maybe slightly less than have the kids really started to say “cool.” For the rest, I put together a set of jing videos to on our blog to show them how to follow the steps again.

Tomorrow (or today, since I’m posting late): we’ll break this model with a meter stick.