Learning Introductory Physics with Activities

You have previously learned how to make sense of both units and numbers in Section 2.3. Perhaps the most powerful type of physics sensemaking is symbolic, since so much of physics is concerned with symbolic equations. The most general way to make sense of a symbolic equation is with covariational reasoning.

When you use covariational reasoning, continue to follow the sensemaking steps you saw in Figure 2.3.4. In particular, you want to know both what should be true physically² and what the equation actually says. As an example, consider the equation for the area of a circle in terms of its radius: \(A_{circle} = \pi r^2\text{.}\) Thinking physically first, you might think about all the various circles you have drawn or seen. You might even draw a few circles with different radii to see how they behave. Together, physical reasoning should tell you that increasing the radius of a circle will also lead to an increase in the area, since radius and area are both measures of the size of a circle. Examining the equation next, \(A_{circle}\) clearly depends on \(r\text{;}\) in fact, increasing \(r\) leads to an increase in \(A_{circle}\text{.}\) In this instance, the physical reasoning and the equation agree with each other. You can actually probably say a little more: the symbolic relationship is quadratic, so doubling \(r\) actually increases \(A_{circle}\) by a factor of four! This agrees with the physical behavior of a circle because changing the radius changes the size of the circle in all directions!