Calculus for Geometry Students

In Geometry, we’re beginning a chapter on areas of polygons, and the first lesson is area of a rectangle. Pretty exciting, huh? My students are mostly ninth-graders, with a few tenth-graders, and I thought they might enjoy seeing how the area of a rectangle is used to estimate the area under a curve, i.e. a Riemann Sum.

I used a Geogebra activity created by Alex Kasantsidis to demonstrate a simple Riemann sum. We discussed how the sum of the rectangles can either overestimate or underestimate the area under the parabola, and how we can get a better approximation of the area by increasing the number of rectangles used.


Then, I had my students work through an activity (you can download it here) to estimate the area under the curve y = 12 – x^2 for x = -1 to x = 3 using eight rectangles. After averaging the left-hand and right-hand sums, they came up with 38.5. The actual area is 38.66…, so with only eight rectangles they achieved very good results!

What my students enjoyed even more, though, was the satisfaction of learning calculus-level mathematics. Hopefully, this activity allayed some of the apprehension they might have when they hear the word “calculus”.



Math Is Everywhere (cont.)

Being a math teacher is both a blessing and a curse (to paraphrase Adrian Monk). I’ve just returned from an overnight trip to Decatur, AL with my cross country team (where our varsity crushed all competition with an incredible team score of 20, and our JV earned third with an excellent 69 points). As I got off the elevator last night, I noticed the carpet in the hallway had a very interesting pattern. I’d be willing to bet the carpet company has an undercover mathematician working in its art department! Parts of it looked like Riemann Sums:

Carpet1 Carpet2

and the overall pattern looked like a sum of trigonometric functions:

Carpet3 Carpet4

I’ve been playing around on desmos, trying to match it (and not being very successful). Does anyone have any suggestions?