The Home Plate Area Mystery A major-league home plate. Dimensions in inches. From Math World. Click the image to go there.

The illustration shows the dimensions, in inches, of a major league home plate, according to the official rules of baseball.

What’s the area of the plate?

Another way to present this question is to note that major-league bases are 15 inches square—and wonder which is bigger.

In either case, the problem of figuring out the area of this pentagon involves taking shapes apart or sticking them together. This skill, of dissecting or composing shapes, is important for students; they need to visualize the easy shapes that are inside (or outside) the hard ones. It even appears in the core standards, obliquely, in 7.G.6. It’s also wonderful that there are different ways to do it.

In this case, it’s not too hard. Students who know how to find the area of a right triangle can be successful without much teacher intervention.

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

Chord Star 3: Remote Radii 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 2: Choosing different points Last 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: 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.

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.

Modeling a Spiral, and enjoying Desmos At a recent meeting, I got to tell people about an old, non-finished book, EGADs (Enriching Geometry and Algebra through Data). The idea of the book is that there are geometrical constructions that have relationships under them—usually a relationship about length—that you can model using a symbolic formula.

Like that spiral. How does the length of the “spokes” of this spiral depend on the spoke number?

This post has two purposes:

1. To get you to try the spiral example.
2. To show how you can use the Desmos graphing calculator to do the graphing and calculation.

The draft of the book (link above) is free for now, but it occurred to me that you could do at least one activity (integrates trig, geometry, data, exponential functions) easily using Desmos’s cool new technology. Read on!

The activity: Spiral 20

Here’s what you do:

1. Download the PDF with the one-page handout. Print it out.
2. Measure the long legs of the triangles as suggested on the handout.
3. Go to the Desmos graphing calculator.
4. Enter the data.
5. Enter a function in that top panel. Try to match the data!
• If you leave parameters in the formula, it will ask if you want sliders. I love sliders. You will too.
• You can type values in, even if you have sliders.
• You can change the range of sliders by clicking the limits at the slider ends.
6. Play with the sliders and all the other features of this site.

Here is a graph of mine from Desmos, with only a little of the data: 