Today, I thought I’d share a little more about things learned along the way with my curriculum consulting. As I mentioned before, I’m creating a series of online lectures for the Geometry unit. This past week, the section I was working on (and will still be working on into next week) is Circle Geometry.
As I also remarked earlier, I’m using the University of Chicago School Mathematics Project’s textbook on Geometry as a reference. In this text are many theorems about the measure of the angle between two intersecting lines in terms of the measures of the intercepted arcs.
This image is certainly familiar.
The question I had to consider was how to organize all these results in a coherent 5–7-minute lecture. It turns out that there was too much for just one lecture, so I did spread it out into two. But I still needed a flow.
Although the results were not new to me, I had never taught this topic before. My main experience teaching geometry at the high school level was designing and teaching a course on spherical trigonometry as it applies to studying polyhedra. So this gave me an opportunity to stand back and just think about putting it all together.
I was happy with what I came up with — an approach which could be classified under “combinatorial geometry.” I decided to pose the following question:
In other words, if you have two intersecting lines, and you draw a circle so that it intersects both lines, what configurations are possible?
Looked at in this way, there are just two considerations: whether the intersection of the lines is outside, on, or inside the circle, and whether the lines are secant or tangent lines.
It’s not difficult to make the enumeration, so I’ll just give it briefly here. There is only one configuration if the intersection of the lines lies inside the circle, since both lines must be secant lines.
When the intersection of the lines is on the circle, one of the lines may be tangent, although both cannot be since there is a unique tangent at any given point on a circle. And when the intersection of the lines is outside the circle, zero, one, or two of the lines may be tangent to the circle.
This enumeration allows for a systematic approach. If you’ve ever worked through find the angle measures, you know that starting with the arrangement in the upper right corner is the way to begin. I won’t go through all the details, but I will just indicate that the following figure is all you need:
This simple case is analyzed by considering as an angle exterior to The analysis of all the other cases builds from this.
I decided to include a discussion not found in the UCSMP text — continuity. Of course this is not a topic which can be rigorously discussed at say, the 10th-grade level. But why not give students an intuition of the idea?
Let’s start with a few diagrams.
This series illustrates the case that the intersection of the lines is outside the circle, and one of the lines is tangent. We look at this as the limiting case of a series of pairs of secant lines.
This argument depends upon the fact that the measurement of all arcs and angles varies continuously as moves around the circle. While, as mentioned, this cannot be addressed rigorously, it is a very intuitive argument. Moreover, there are many different software packages you could use to make an animation of this process, and display all the arc and angle measurements as moves around the circle.
There is no reason not to introduce this argument. In my pair of lectures, I used more traditional geometrical arguments as well. It doesn’t hurt students to be exposed to a wide range of proof ideas.
I summarize all of these results in the following graphic.
The measure of the angle indicated with the red dot is half the measure of the intercepted arc, or the sum/difference of the measures of the intercepts arcs, shown in red and blue. An arc in blue indicates its measure is to be subtracted rather than added. I was very happy with this graphic. I think that if a student followed the lecture, they could state every result just by looking at it.
This also proved to be a great segue into looking at the power of a point. I thought I’d begin with the figure in the upper left, proving the usual theorem using similar triangles.
And now for another continuity argument!
This is a nice way to see that the power of any point on the circle is 0. It is also a nice contrast to the theorems about the angle between the intersecting lines: when and eventually reach 0, you’re not able to conclude anything about a relationship between and
This means that there is no theorem relating lengths of segments for the two cases when the intersection of the lines lies on the circle. I use the following graphic to indicate this, with two cases grayed out when the power of the point offers no conclusion.
Now all of these results are the usual ones found in high school geometry textbooks; nothing new here. But for me, just having to step back and think about how to put them all together was a fun challenge.
Again, I am surprised at how much I’m learning even though I’m just putting together a few slides on elementary geometry. The process of writing these lectures is an engaging one, and I hope the students who will eventually watch them will benefit from a perspective not found in more traditional textbooks.