Geometry

Expanding Cylinders

In class, and on Twitter, I posed a question that led to lots of great conversation.

To increase the volume of a cylinder, is it better to increase the radius or the height? Great conversation starter today.#math #mathchat

— Patrick Honner (@MrHonner) January 9, 2015

There are many reasons I love this particular question. It’s familiar, accessible, and usually counter-intuitive. And it bridges algebra and geometry in a very natural way.

A reasonable response is to argue that an increase in radius will be better. The volume of a cylinder is $V(r,h) = \pi r^2 h$ , so an increase in radius seems to have a squared effect on the volume, while the effect of an increase in height is only linear.

If you imagine the grey cylinder shown below, this argument seems to make sense. Increasing the height a little bit adds a small blue disk of volume to the top, but increasing the radius a little bit adds a large blue shell. The additional volume of the shell clearly appears to be more than that of the disk.

However, this argument is a lot less convincing if you imagine a different cylinder.

It’s not obvious which additional volume here is larger, which suggests some further thinking is in order. At this point, some multidimensional extreme-case thinking usually leads to an appropriate conclusion: Namely, that the answer depends on the dimensions of the cylinder.

This problem is my standard introduction to partial derivatives. It creates great context for computing and comparing

$V_r = 2\pi r h$ and $V_h = \pi r^2$

But its versatility is another reason I like this problem so much. Geometry and Calculus students can both engage in this problem in a meaningful way, using the tools available to them to analyze the situation. And it always results in great conversations!

By MrHonner, 11 yearsJanuary 11, 2015 ago

Math Photo: Riemann Shadows

The shadows fall like approximating rectangles under a sine curve. This brings to mind a basic approach in computing the area of curved regions: approximating the curved region with a series of flat regions, whose areas are easy to compute.

The angle of the sun makes the shadows more like approximating parallelograms, though. So we’d probably need a change-of-coordinates to complete our calculation!

By MrHonner, 11 yearsJanuary 4, 2015 ago

Geometry Resources Technology

Circumcircles in Desmos

I’m presenting on Desmos at today’s AMAPS meeting in New York City, and preparing my talk was an object lesson in how wonderful this technology is.

Part of my presentation demonstrates simple ways that Desmos can be a part of every high school math class: Algebra, Geometry, Trigonometry, Pre-Calculus, and Calculus. While Geogebra is generally more suitable for demonstrating and exploring geometry, Desmos certainly can be useful in that course, so I wanted to show something relevant and interesting as part of my talk. I thought, “Why not compute the circumcircle for an arbitrary triangle?”

While all the pieces of the mathematical puzzle were there for me, figuring out how to put them together in Desmos was a fun, frustrating, and worthwhile challenge. I had to play around with the basic concepts associated with perpendicular bisectors and think creatively about some mathematical problems and equations. I even ended up using the new regression feature in Desmos in a clever way!

I often get caught up in little challenges like this, and this is why Desmos is so wonderful: it provides us a mathematical makerspace. It invites us to play around, to create, to engineer, to build. And all of this happens through using the language and concepts of mathematics.

You can see my circumcircle demonstration here, and you can find more of my work in Desmos here.

By MrHonner, 11 yearsDecember 10, 2014 ago

Geometry Photography

Math Photo: Dendroconcentricity

I’ve always loved tree rings, in part because the word dendrochronology fascinated me a child. This lovely cross-section evokes so many mathematical ideas: concentricity, network theory, and polar representations, to name a few.

By MrHonner, 11 yearsDecember 7, 2014 ago

Appreciation Geometry Photography