## Parabolic Telescopes

Last week as I was going through a whirlwind tour of conics to wrap up my Math Analysis class, I wanted to illustrate some real-life examples. Rather than just saying “these things are parabolas, these things are ellipses [etc]” I wanted to have the kids DO something with conics.

I’m not sure I succeeded, but I did come up with one activity that, with further work, could be a nice problem for future Advanced Algebra / Analysis classes.

I had a nice high quality picture of the Very Large Array in New Mexico, since my sister and I visited there on a cross-country drive. I wondered if it was possible to find an equation that would actually model the parabolic shape of the telescopes. So I popped the picture into Geogebra and constructed a parabola via the locus tool (which I just recently learned how to use).

The stated goal of the assignment is “find an equation for the telescope.” I told them not to worry about the rotation. We could handle that later. (we didn’t handle it, but perhaps next time, if this comes after matrices we could multiply by a rotation matrix… find the angle by inverse trig based on the slope of the dish)

But I think if I had used Geogebra with the kids more this year, they might have the capability to construct the locus themselves instead of me giving them focus / directrix. Basically, if they did what I did, I feel they’d get a good understanding of what a parabola is from a locus standpoint.

I didn’t put this on the mathlet, but the diameter of one dish is 85 feet. Two ways to handle that: (1) rescale it so the geogebra numbers match it. (2) have the kids find the scaling factor (put points on the locus, measure the diameter using a segment). I like option 2, but that does require them to be well versed in Geogebra (goal: verse my kids in Geogebra next year!)

Well, I appreciate any comments / critique.

Hey Scott, I followed your e-mail here and I sympathize. I drive by Stanford’s satellite dishes on Highway 280, now and then, and marvel both a) at their size and b) how I simply can’t figure out a good approach vector for a classroom activity.

The question has to be visceral. It has to have multiple entry points so that learners all across the spectrum can buy-in. I think you sense that “what is the equation of the parabola?” is kind of limp.

But that’s the best I have also. Keep us posted if you figure anything out.

Dan Meyer

I strongly recommend familiarizing your students with geogebra, especially if you can get them each access to a computer now and then (or all the time). The locus tool might be a goal set too high – I haven’t thought about how to teach it to students yet – but they will certainly be happy learning the basics of variables, sliders, and equations. The kids in my class can fit a parabola to a picture in no time flat. Give them the power to move parabolas before giving them this problem with the dishes and I suspect you’ll have more luck with engagement levels.

If I were a kid, I would find it much more interesting to think about what the focus means in a physical parabola – ie. the incoming rays of info all bounce off of the dish and focus at that one spot, where the stellite dish has its antennae. And that’s just a plain reflection problem (albeit on a curved surface, so you may need to teach the kids how to estimate the direction of tangent lines?).

Is that possible to bring into this lesson? What if the kids had to build their own satellite models as a project? Is that do-able?

Definitely a physical model would be very fun to do. The Exploratorium in San Francisco has very nice parabolic sound dishes, which allow you to whisper a conversation with someone in the opposite dish 100 yards away, across a crowded and loud warehouse. Move your head out of the focus, and you can’t hear them anymore.

I think the lesson as I presented it a couple years ago suffered in that it did not have a clear focus, pun intended. As Dan pointed out, why should we care about the equation of that dish? On the other hand, in the precalculus class, thats what the topic was: conic sections and their equations. I may be revisiting this topic later on this year in an Algebra 2 course.