Struggling through surface area of cylinders

The problem set I gave my students included surface areas of rectangular prisms; triangular, square, and hexagonal pyramids; and cylinders. The going was slow finding the surface are for all of these. Part of the issue was that I asked for the lateral surface area and the total surface area.

For pyramids, the lateral surface area was easy to describe, since it was all the sides other than the base upon which the pyramid stood. This then caused some minor confusion for cylinders, especially, as students put all areas other than the bottom into lateral surface area. I had to explain that we don't include the top in the lateral area either.

With this understanding, students worked through most of the pyramid and rectangular prisms. The hexagonal prism still posed problems. As I drew diagonals and divided the figure into triangles, students could see that they could find the area of the hexagonal base by finding the area of each triangle. We revisited the interior angle sums for polygons and used this to determine the interior angle size of a regular hexagon. Some students saw that the diagonal would bisect the angle so each triangle had half of this angle measurement (θ). With some prodding, they could see that the height (h) of the triangle bisected the hexagon side (s). Working with the base angle, students then had to revisit their trigonometric ratios to find that tan(θ) = h / (s/2).

For cylinders, students just had trouble making connections. I try to use an analogy of wrapping paper around the cylinder. It's a easy, hands-on way for students to visualize the lateral surface area of the cylinder. This helped, but students were not comfortable with the whole idea. This presented the biggest struggle of the day.

A couple of questions dealt with reasoning and problem solving. For example, given the surface area of a cube, what is the cube's side length? One question was, for a given cylinder, would doubling the height or doubling the radius result in more surface area? Most of the students felt doubling the height would produce more surface area. For this, and as part of our brain break, I took students outside and had them stand in a tight circle. I had students kneel or squat down and said this represented the original cylinder. Standing up is equivalent to doubling the height. Next, I paced off the diameter of the circle we formed. I told students that we would step back to simulate doubling the radius. Looking at the much larger circle, I asked which cylinder had the larger surface area. Students could see doubling the radius results in a bigger surface area.

I had another set of problems that concerned volume that I gave to students for homework. We'll discuss these problems and work through a few additional surface area and volume problems next class.