Working through surface area and volume problems

Here in Colorado. we use PARCC for our state mandated assessment. A colleague had put together a table of PARCC approved formulas.

I made copies of this table and told students to paste or tape these into their notes. To be sure the notation was clear, I explained and illustrated the meaning of b versus the meaning of B in the formulas. Students seemed to understand that B was the area of the base in a 3-dimensional object while b was, essentially, the width in a 2 dimensional object. I asked students to read through the formulas in case there were any other questions about the notation. One student checked to make sure that d referenced the diameter of a circle.

With that, I asked students to work through the surface area and volume problems they were given last class. These problems weren't really worked on as the students were confused and the substitute teacher was not helpful, although they said he was a nice guy.

I walked around the room and checked with students as to how they were doing and to hear about their thinking. There were some common points of misunderstanding, such as a 3-d representation of a square pyramid not looking square even though the labeling said it was a square pyramid.

There were also issues with finding the base area of a regular hexagonal pyramid. For this, I showed how the base could be divided into triangles by connecting the polygon's center point to each vertex. The triangle sides were angle bisectors of the interior angles. I reminded students that we had looked at the sum of interior angles for polygons. Using this information, the base length, and the fact that the triangle height was a perpendicular bisector of the hexagon's side, to establish a right triangle in which we knew one side length, needed a second side length, and knew all the angles measurements of the right triangle. This leads us right back to using trigonometric ratios to find the height of the triangle. This process works for any regular polygon, the only things that change are the size of the interior angles, the number of triangles that are formed, and whatever the given side length is.

I continued to walk around and check on student progress. While there continued to be a bit of uncertainty and lack of confidence in reading diagrams, for the most part, the class seemed to grasp the general ideas of lateral surface area, total surface area, and finding base areas.

I asked students to complete their work on these problems before next class. We'll look at results and move onto another set of practice problems. I may introduce some additional variations from the problems, for example, finding the height of the pyramids they worked with and then using that information to calculate volumes.