In addition, for x and y larger than 0, they can create a three-dimensional plot of the surfaces on each side of the above equation. They can then rotate the plot and examine different views of the surfaces in order to see that they never intersect. Since Carolyn Gordon was looking for soundalike drums, she had to find two drums with the same area and perimeter that also sounded the same when they were played. Explain that Carolyn Gordon proved that her drums sounded the same without actually testing them out in real life and explain why a mathematical proof does not need to be constructive. Since the students will be worried about this, explain that some physicists created the drums and tested them in real life.
Students can cut one of the drums into pieces and try to fit the pieces onto the other drum.
They will discover that the drums do not have the same area, which contradicts the fact that they must in order to sound the same. Students can resolve the apparent conflict by investigating the accuracy of the models. Students can identify which drum resembles the drum that Carolyn is holding in Figure 1. Student can cut along their folds and then try to fit the pieces onto the other drum. Once they identify which piece does not fit properly, they can compare an uncut version of the drum to Carolyn's drum in Figure 1 in order to see that this piece was not drawn to scale.