Let me encourage you to go to your hardware store and get some hexnuts. You won’t regret it. Now let’s see if I can write a post about it in under, like, four hours.
(Also, get a micrometer on eBay and a sweet 0.1 gram food scale. They’re about $15 now.)
I have been focusing on using functions to model data plotted on a Cartesian plane, so let’s start there. Suppose you go to the hardware store and buy hexnuts in different sizes. Now you weigh them. How will the size of the nut be related to the weight?
A super-advanced, from-the-hip answer we’d like high-schoolers to give is, “probably more or less cubic, but we should check.” The more-or-less cubic part (which less-experienced high-schoolers will not offer) comes from several assumptions we make, which it would be great to force advanced students to acknowledge, namely, the hexnuts are geometrically similar, and they’re made from the same material, so they’ll have the same density.
Most students, however, won’t have that instant insight. So we begin by asking them to predict. They will draw graphs that increase, which is a good start, and many student graphs will curve upwards. Great! Then when they measure, students see something like the graph. (Here is a Desmos document with the data.)
They will look at this graph, and many will say it looks like a parabola. Which it does. Kinda. But if you put a quadratic of the form on the graph, you will never get it to fit well, no matter what value of k you choose.
We’re doing this quickly, so let’s skip ahead to say that when you try , things go much better (though they aren’t perfect). Then you can explore why you think it “goes like” x cubed. You might get there by discussing the quarter-inch nut and the half-inch nut, and pass out samples, and ask, why isn’t the half-inch twice the weight of the quarter-inch. And so forth.
Now the discussion can go in two interesting and different directions. One goes to the wonderful snook data where you can do a similar exploration about fish, and (especially if you do a log transform) you discover that the best exponent for these fish is about 3.24 rather than 3.00. And you gotta wonder how that could be.
But we’re not going there. Instead we start with a great question that astute readers must have asked, but we have avoided until now: what do we mean by the size of a hexnut? The answer is, the diameter of the bolt that it fits. Right? A quarter-inch nut is the one that fits a quarter-inch bolt. It’s bigger (duh) than a quarter inch across.
A Geometrical Model
So let’s consider modeling the geometry of the situation. A hexnut is a hexagonal prism, right? And it has a circular hole.
We can measure more than the weight. Suppose we measure the distance across the faces (f) and the thickness (t) as well. Then the area of the relevant hexagon is , and the resulting volume of the solid—taking the hole into account—is
So we can model the mass with , where is the density of the material. The next graph, from Fathom, shows this calculated model mass against the measured mass, along with and a residual plot. We have already slid the rho slider to a decent value.
I have positioned the slider so the first few points are flat. When I do that, they are not centered at zero. Also, the larger nuts deviate from the line more and more. This is an indication that there are systematic effects that our model doesn’t account for. I leave it to you to think about what those might be!
The value of rho? For this graph, 8.22 grams per cc, which is not bad (just a little high) for the density of steel.
And the modeling point? If you agree with my earlier assertion that simplification and abstraction are the hallmarks of modeling, then this is modeling even though we didn’t make a fancy function. We assumed that the nut is truly shaped like a hexagonal prism with a cylindrical hole cut in it. That’s patently false, but it’s a completely decent approximation, and much easier to deal with than the ugly, messy truth. What’s more, we can use the ways reality deviates from the simple, abstract model to decide whether our model values might be a little high or low.
Here are the data from that graph. Lengths are inches, mass is in grams:
boltSize flat thick mass 0.25 0.4298 0.2148 3.09 0.3125 0.4951 0.2651 4.72 0.375 0.5555 0.3236 6.73 0.4375 0.6807 0.3802 12.77 0.5 0.7407 0.4253 15.85 0.625 0.9223 0.5450 31.06 0.75 1.0920 0.6337 48.8
There is a lot more to say, but hey, it’s almost four hours so it’s way time to stop.