I have recently had a rather unusual opportunity. I’ve talked a bit over the last few months about my consulting work producing online videos for a flipped classroom; I’ve been working busily on the Geometry unit.
Now the last section of this unit is on three-dimensional geometry, and I’ve been given pretty free reign as to what to cover in this 20-lectures series of 5-7 minute videos. And given my interest in polyhedra (which I could focus on exclusively with no shortage of things to discuss!), I felt I had a good start.
But the challenge was also to cover some traditional topics (cones, cylinders, spheres, etc.) — as well as more advanced topics — while not using mathematics beyond what I’ve used in the first several sections of the Geometry unit.
There is, of course, no “correct” answer to this problem. But I thought I’d share how I’d approach this series of lectures, since geometry is such a passion of mine — and I know it is for many readers as well. The process of reforming high school geometry courses is now well underway; I hope to contribute to this discussion with today’s post.
Where to start? Cones and cylinders — a very traditional beginning. But I thought I’d start with surface areas. Now for cylinders, this is pretty straightforward. It’s not much more difficult for cones, but the approach is less obvious than for cylinders.
Earlier in the unit, we derived the formula for the area of a sector of a circle, so finding the lateral surface area of a cone is a nice opportunity to revisit this topic. And of course, finding the lateral surface area of a cylinder involves just finding the area of a rectangle.
Now what do both of these problems have in common? Their solution implies that cones and cylinders are flat. In other words, we reduce what is apparently a three-dimensional problem (the surface area of a three-dimensional object) to a two-dimensional problem.
This is in sharp contrast to finding the surface area of a sphere — you can’t flatten out a sphere. In fact, the entire science of cartography has evolved specifically in response to this inability.
So this is a nice chance to introduce a little differential geometry! And no, I don’t really intend to go into differential geometry in any detail — but why not take just a minute in a lecture involving spheres to comment on why the formulas for the surface areas of cones and cylinders are fairly easy to derive, and why — at this level — we’re just given the formula for the surface area of a sphere.
I try to mention such ideas as frequently as I can — pointing out contrasts and connections which go beyond the usual presentation. Sure, it may be lost on many or most students, but it just may provide that small spark for another.
I think such comments also get at the idea that mathematics is not a series of problems with answers at the back of the book…on the face of it, there is no apparent reason for a student to think that finding the surface area of a cone would be simpler than finding the surface area of a sphere. This discussion gets them thinking.
Next, I’m planning to discuss Archimedes’ inscription of a sphere in a cylinder (which involves the relative volumes). This is a bit more straightforward, and it’s a nice way to bring in a little history.
I also plan to look at inscribing a sphere in a right circular cone whose slant height is the same as the diameter of the base, so that we can look at a two-dimensional cross-section to solve the problem. In particular, this revisits the topic of incircles of triangles in a natural way — I find it more difficult to motivate why you’d want to find an incircle when looking at a strictly two-dimensional problem.
Now on to calculus! Yes, calculus. One great mystery for students is the presence of “1/3” in so many volume formulas. There is always the glib response — the “3” is for “three” dimensions, like the “2” in “1/2 bh” is for “two” dimensions.
When deriving these formulas using integration, this is actually exactly a fairly solid explanation. But for high school students who have yet to take calculus?
It is easy to approximate the volume of a right circular cone by stacking thin circular disks on top of each other. If we let the disks get thinner and take more and more of them, we find the volume of the cone as limit of these approximations. All you need is the sum
I plan to prove that
and then prove (or perhaps just suggest — I’m not sure yet) the formula for the sum of squares.
I think a fairly informal approach could be successful here. But I do think such discussions are necessary — in calculus, I’ve routinely asked students why certain formulas they remember are true, and they struggle. As a simple example, students can rarely tell me why the hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg.
When teachers just give students formulas and ask them to plug numbers in to get answers to oversimplified word problems, of course there is a sense of mystery/confusion — where did these formulas come from? I’m hoping that this discussion suggests that there is a lot more to mathematics than just a bunch of formulas to memorize.
As usual, I realize I have much more to say on this topic than I had originally supposed…I’ve only discussed up to the fifth lecture so far! Since I have not had extensive experience teaching more traditional topics at the high school, it has been an interesting challenge to tackle the usual geometry topics in a way that grabs students’ attention. It’s a challenge I enjoy, and of course I’ll have much more to say about it next week….