A Calculus Rant (with stats at the end)

Let’s look at a simple optimization problem. Bear with me, because the point is not the problem itself, but in what we have to know and do in order to solve it. Here we go:

Suppose you have a string 12 cm long. You form it into the shape of a rectangle. What shape gives you the maximum area?

Traditionally, how do we expect students to solve this in a calculus class? Here is one of several approaches, in excruciating detail: Continue reading A Calculus Rant (with stats at the end)

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Chord Star in the Classroom

ChordStarOverview
from this post

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

RemoteRadiusModel2Hoping 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:

ImageImage

They clearly resemble each other, but I want to make two observations: Continue reading Beating the Modeling Drum

Chord Star 3: Remote Radii

remoteRadiusTeaserSuppose 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 2: Choosing different points

ChordStar2ThumbLast time we saw how you could make a “chord star” by picking a point inside a circle and drawing chords through that point. Then we measured the two lengths of the partial chords (let’s call them L_1 and L_2) and plotted them against one another. We got a rectangular hyperbola, suggesting (or confirming if we remembered the geometry) that L_1 L_2 = k , some constant.

But we asked, “what effect does your choice of point have on the graph and the data?” So of course we’ll take an empirical approach and try it. If you have a classroom full of students, and they used the same-sized circle and picked their own points, you could immediately compare the points they chose to the functions they generated. Or you could do it as an individual. The photo shows what this might look like, and here is a detail:

ChordStar2_Drawing_detail

Now we’ll put the data in a table, but this time,

  • In addition to L1 and L2, we’ll record R, the distance from the center to the point. It may not be obvious to students at first that all points the same distance from the center (or the edge) will give the same data, but I’ll assume we get that.
  • We’ll double the data by recording the data in the reverse order as well. It makes the graph look better.

Here’s the graph, coded by distance (in cm) of the point from the center.

Image Continue reading Chord Star 2: Choosing different points

Chord Star: Another Geometry-Function-Modeling Thing

ImageLast 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

ImageIn 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.

Continue reading Isosceles EGADs: Functions, Geometry, and Modeling