Chord Star in the Classroom

A million thanks to Zoya Voskoboynikov and her two sections of “regular” calculus at Lick for letting me come litter their otherwise pure math class with actual data. Of course, it was after the AP exam, and these are last-quarter seniors, so my being there didn’t interfere with any learning they needed to get through.

It worked great. It had what I most wanted: the aha experience of arriving at the destination by another route. Fortunately (and unsurprisingly), none of these successful math students remembered the theorem from the geometry class they took as frosh.

What we did

1. I set up the problem and had them predict, informally, what the function would look like. The main purpose of this is to orient students to what we’ll be measuring and to the idea that if you measure two quantities, you can see their relationship in a graph. Continue reading Chord Star in the Classroom

Beating the Modeling Drum

Hoping desperately it’s not also a dead horse…

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: Continue reading Beating the Modeling Drum

Suppose you find some big curved thing out in the world. Some things are curved more tightly than others. But how much more? How can we put a number on how tightly curved something is?

One way is to figure out the radius of curvature. The smaller the radius, the tighter the curve. (Would you tell students this at the beginning? Of course not. But I can’t describe how this can work without giving things away. So consider this a report on my own investigation.)

Let’s apply what we learned two posts ago. To review, we found out that if you pick a point inside a circle, and run a chord through it, the point divides the chord into two segments. The lengths of those segments are inversely proportional, that is, their product is a constant—it’s the same no matter which chord you pick.

Then, last time, we saw how that product varies with the point’s distance from the center.

Let’s see how we can use this to measure radii of curves out in the world. The cool thing is that we can do this remotely. Unlike most radii in school geometry, we can figure out the radius of curvature without ever finding the center of the circle.

The picture above is a hint. If that’s enough for you, don’t read further! Go do it! Continue reading Chord Star 3: Remote Radii

Chord Star: Another Geometry-Function-Modeling Thing

Last time I wrote about a super-simple geometry situation and how we could turn it into an activity that connected it to linear functions. What does it take to turn something from geometry into a function? This is an interesting question; in my explorations here I’ve found it helpful to look for relationships. And what I mean by that is, where do you have two quantities (in geometry, often distances, but it could be angles or areas or…) where one varies when you change the other.

So one strategy is, think of some theorem or principle, and see if you can find the relationship. To that end, remember teaching geometry and that cool theorem where if you have two chords that cross, the products are the same? That’s where this comes from. Oddly, it took a while to figure out what to plot against what to get a revealing function, but here we go.

Make a circle. Pick a point not near the center, but not too close to the circle itself. Draw a chord through that point. Measure the two segments. Call them $L_1$ and $L_2$. Or even x and y. Record the data. Continue reading Chord Star: Another Geometry-Function-Modeling Thing

Isosceles EGADs: Functions, Geometry, and Modeling

In trying to come up with more activities for EGADs (Exploring (or maybe Enriching) Geometry and Algebra though Data), the following dropped into my lap. Because it’s so simple and so interesting, I’d better write it down…

Everybody get a sheet of paper and draw an isosceles triangle. Try to make your triangle big enough to kinda fill the page, but also try to make it different from those around you. Make your triangle pretty carefully, but don’t measure and don’t use a straightedge.

Individuals can do this too, but I’m writing this as if it’s a class activity. The idea is to get a wide variety of shapes. It is not vital that these just be sketched, but (a) I think that makes the data more interesting, (b) it opens the possibility to drawing more carefully later, and (c) it’s much faster.

Measure the base angles and the vertex angle, and write them on the page.

If you need to introduce vocabulary, do it here. By the way, we don’t assume that these students know that the base angles should be the same. Also, we all know that measuring angles is hard, right?

We’re going to plot the measurements from the whole class. So write your angle measurements on the board.

You may need to help organize this. Will we plot both base angles? Up to you. If so, consider having each kid make two entries in the T-table or whatever.

Now make a graph. Put vertex angle on the horizontal axis and base angle on the vertical. Think about the range of values before you make your axes!

You may want to discuss what goes on which axis. Without having done this with kids, I bet most of us think of the vertex angle as the independent variable and base angle as the dependent. I, at least, think of the vertex angle as the defining angle in an isosceles triangle. This also has the happy consequence of requiring a change of axes in order to get the coolest version of the formula.

At any rate, the graph should look linear. Address outliers (probably due to bad measurement).

Draw the line you think best approximates the data. Find its equation.

Be ready to present your data and line, and explain as much as you can about the line. In particular, why does it have that slope and intercept?

In the spirit of SERP “Poster problems” this could be a poster-plus-gallery-walk event.

…and why somebody might try to convince you they are.

It’s even in the Core Standards. This is taken out of context—but not very far:

A model can be very simple, such as writing total cost as a product of unit price and number bought… (Common Core, p 72)

Seriously?

Okay, I could make a case for it, but I won’t.

I’m becoming more convinced that the real hallmark of modeling is simplification  (see this post for more).

Modeling is not simply using math on real-world problems, though that is a Good Thing; you can model to help with pure math as well. And I bet we could find good real-world problems that don’t involve modeling.

But back to simplification. The key element (I believe this afternoon anyway) is taking something and using math in a way that makes it simpler, less complicated than the thing itself. We model to make things tractable. We can handle the model even when the thing it represents it too complicated. If it’s a good model, it captures the essence of what we’re looking at; and exactly what that means may depend on the specific context.

• We might model a hexnut as a hexagonal prism with a cylindrical hole, and use that geometrical model to find a volume. We avoid the threads and the easing on the corners: they’re too complicated—but we hope our model captures the essence of the hexnut.
• We might model some messy data as a line or a curve. We can’t make a reasonable prediction from the mess, but we can with a function: just plug in a value and calculate.
• We might explore the behavior of a system of linked differential equations by creating a numerical model, a system of difference equations we can evaluate on a computer. It’s conceptually simpler (for the computer at least), so we sacrifice some precision for tractability.
• We might even take all the complexity of Americans and do a Census. When we do, we create a data model: the structure for the information we will collect. We have only approximately captured the people’s information (this is the Census, right, not the NSA). We hope our data has the essence that we need to know—but there is a huge amount of detail that we have ignored. Like the threads, like the deviations in the scatter plot, like the inaccuracies in the numerical model.

What does this have to do with word problems?

Suppose we ask, if Eduardo buys four cans of orange juice for \$2.49 a can, how much does he pay altogether?

There is math here, no question. We can argue whether it’s real life.

But it doesn’t involve simplification. All of the information is present. There is no model, and no need for one.

Modeling Digression: Coin Weights

Too much philosophy, let’s get some data!

Part of the motivation for the recent posts on modeling is that I’m writing a paper for a friend. It takes off from the hexnut weight data and cubic model that appear in The (still unpublished, sigh) Model Shop. More on that soon, because I’ve found some interesting features in hexnut data. But first, another friend sent me a link to this post on doghousediaries I now share with you. Check it out.

Because I had just been doing hexnuts, I immediately thought about coins as an alternative: how do you suppose the weight of coins (Physics people: I will be reporting weight in grams. We both know that’s mass, not weight. We can handle that, right?) is related to their size?

I mean, hexnuts look more or less geometrically similar (they’re not, but that’s the other story), so you’d think their mass is cubic in the linear size: a nut that fits a half-inch bolt ought to be eight times the weight of one that fits a quarter-inch bolt. And that model fits pretty well.

So as teachers, we look for situations that stretch that understanding. EGADs has you cut out cardboard squares or circles and weigh them; since the difference in size is two-dimensional, the weight-size relationship is quadratic.

But what about coins? They look kind of two-dimensional, but the big ones tend to be thicker. So are they cubic or quadratic or something in between? Well. Whenever I come back from some other country, I usually have leftover coinage, which gets saved in that jar in the top drawer. And I have a good scale and (I know, I’m a measurement geek) a decent micrometer. So it’s an empirical question.

What’s Modeling Good For?

What’s the purpose of mathematical modeling? The easy answer is something like, to understand the real world. When I look more deeply, however, I see distinct reasons to model—and to model in the classroom. I hope that trying to define these will help me clarify my thinking and shed light on some of the worries I have about how modeling might be portrayed.

(So this is the third in a series on modeling. We began with some definitions, then proceeded to look at “genres” of modeling.)

Let’s look at a few purposes and try to distinguish them. To save the casual reader time, I’ll talk about prediction, finding parameter values, and finding insight. I think the last is the most subtle and the one most likely to be missed or misused by future developers.

Maybe I’ll post more about each of these in detail later, but for now I’ll move quickly and not give extended examples.