In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Brendan Gunnoe. His topic, from Geometry: finding the area of a circle.

**History: Squaring the circle**

The ancient Greeks and other groups at the time had a fascination with geometry. These cultures tended to like thinking in terms of simpler geometric shapes, such as circles, equilateral triangles and squares. One of the classic problems proposed by these ancient peoples was “Can you create a square with the same area as a circle with finitely many steps only using a compass and straightedge?”. This problem stood for thousands of years, stumping even the most brilliant of mathematicians that attempted to show it true. Eventually, in the year 1882, it was finally proven impossible because of a property of the number π. It’s not too hard to show that π isn’t an integer, nor is it rational. What was left to show is whether π was algebraic or transcendental. The proof from 1882 showed that π is in fact transcendental, proving that it cannot be made using the rules set out by the original question. If a number is algebraic, then it is a solution to a polynomial with rational coefficients.

**Curriculum: Using limit of triangular approximations to get the integral**

The teacher starts off class by drawing a circle with an inscribed triangle, another with a square, and so on until a hexagon is inscribed. The teacher then draws isosceles triangles that originate at the circles center and extend to the corners of the polygons. The teacher could ask questions like “What do you notice about the total area of the triangles and the area of the circle as we keep adding sides to the polygon?” and “What do you notice about the triangles we made and the little wedges of the circle, what’s the same and what’s different about them?”. Then the teacher could arrange both the triangles and wedges in an alternating up and down fashion, almost like two zippers, to line up the triangles and wedges. The teacher could ask “What’s the length of the top of the triangles? What about the tops of the wedges, what’s their length?”.

Finally, the teacher asks “What happens when we let the number of pieces gets REALLY big? What happens to difference between the area of the triangles and wedges? What about the tops of the triangles and the tops of the wedges?”. In the limit, the upper edge converges to half of the circumference of the circle and the height of the triangles converges to the radius of the circle. Using this line of thinking, the teacher guides the students into seeing how you can derive the equation for the area of a circle by using approximating it with triangles, and then looking at what happens in the limit.

**Application**

A telescope’s lens is what’s used to control how much light gets into the eye piece. Suppose you’re an astronomer and want to take a photo of the full Moon on a clear night, which gives off 0.25 lumens/s-m^{2}. Suppose your camera needs to get a total of at least 3 lumens to produce a good photo and 5 lumens to get an amazing photo. What’s the radius of a lens (in centimeters) that can take a good photo in 10 minutes? What’s the radius of a lens that can take an amazing photo in 10 minutes?

Now suppose you’re working with the Hubble space telescope in low Earth orbit trying to get photos of a nearby star system. The radius of the main telescope is 120cm and the star system you want to observe is giving off light at a rate of 10^{-5} lumens/s-m^{2}. How long will it take to get a good photo with Hubble? What about a great photo?