We just did a three-post sequence about “Chord Stars,” finishing up with how we could use insights from data to find radii of curvature remotely, that is, without ever finding the center of the circle. There’s a lot to discuss about that process; this post is part of that discussion.
In particular, it’s an interesting example of modeling. Quite a while ago I was worrying about the definition of modeling, not simply to get it “right”—many people model in different ways—but rather to try to identify things that we were pretty sure demonstrated modeling. Part of my anxiety, as the Core Standards lumber into classrooms, is that people will carelessly define modeling as “real-world” (or something equally weak) and we will lose a great opportunity to improve math education.
I often think of modeling in terms of using functions to model data. That’s partly because some of the coolest, most wonderful math experiences I’ve had have revolved around finding a function that was a good approximation to data. The process of measurement, improving those measurements, finding a suitable function, getting insight about the function as I wrangled it, and getting insight into the situation and the data from the function—all that together is an intoxicating cocktail of mathy-worldy wonderfulness.
But it’s not all there is to modeling, so I want to pause to point out another modeling genre (one of the ones I listed in this old post) that just appeared in Chord Star 3, namely, modeling real-world stuff with geometrical objects.
In fact, here are a curb with tools, and the relevant part of a Sketchpad sketch:
They clearly resemble each other, but I want to make two observations:
- The moment you go into the real world and try to measure something like the length of a chord, all kinds of trouble arises. The curb is not in fact perpendicular to the roadway. The edge is not sharp. Where exactly do you measure from? I love this stuff. I want students to struggle with issues like these and develop intuition about what matters.
- If you just look at the “geometry” picture above, you can’t really tell it’s part of a circle. Could it be a parabola? Sure. So in a geometry text, you most often see the illustration here. You get the whole circle, and the chord is not nearly as close to the circle itself. And it’s not obvious that the two diagrams are really the same.
But one we see it, the full circle with the chord is an elegant model. It’s a simplification of reality that helps us organize our thinking, see relationships, and make calculations. When we get good, the whole-circle picture helps us generalize. We can see that ab = cd and that therefore, even if we can’t see the center, if we know a, b, and c, we can calculate d. And furthermore (we realize after making several calculations), d will be so close to c + d that we might as well treat it as the diameter and make life easy.
This kind of model helps us explain and apply the functions we get when we just plot the numbers and fit functions—like the plot at right from the first Chord Star post.
Looking at that plot, and seeing where we went with it, we can see that one goal of modeling with functions is to put the function out of a job. After all, when we went to the curb, we did not plot the function. By then, we had already used the function to learn that the products were the same. We used that next-level discovery to calculate the radius of curvature.
Which leads to two concluding observations:
- Taking the time to measure and find relationships using graphs and functions can be an important step in between the real world and the most elegant abstraction—in this case, the circle with the diameter and chord. Often, we can’t see that great diagram in the real world—it’s the wrong part of the diagram, or the wrong scale. You know that problem with the guy at the top of the mast and you have to calculate how far he can see given the radius of the Earth? I bet it would be easier if kids had a chance to draw and measure and fit first. They’d try lots of triangles and see how they are related. It wouldn’t be a “gotcha” question where you had to see how to apply this product property (or, better in this case, Pythagoras) in order to solve it.
- Measuring the geometrical drawing on paper, and finding the relationships among the measurements, is useful because it lessens but does not eliminate challenges in measurement that you meet in the real world. (And that’s the whole point of EGADs.)