Section 2.12 Sensemaking: Covariational Reasoning
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.
Sensemaking Strategy 2.12.1. Covariational Reasoning.
Covariational Reasoning, or
Covariation, involves examining the relationship between two (or more) quantities to see how they are changing together
1 . The objective is to state
how the quantities are related and
why that relationship makes sense.
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 physically
2 and what the equation actually says. Several examples are provided below.
Example 2.12.2. Covariational Sensemaking: Area of a Circle.
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 consider 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!
Example 2.12.3. Covariational Sensemaking: Displacement.
Consider the equation \(\Delta x = v\Delta t\text{,}\) rewritten from the definition of average velocity. This equation relates \(\Delta x\) to two other quantities: velocity \(v\) and time \(\Delta t\text{.}\) We can investigate each relationship with \(\Delta x\) separately by holding the other quantity constant.
Physically, velocity is a measure of how fast something is traveling. In general, a faster object with travel a greater distance in the same amount of time than a slower object: you might think about a race or a car chase. On the other hand, two objects with the same speed can travel different distances if they travel for different times, and the more time spent traveling, the farther something will travel. The equation agrees with both because \(\Delta x\) is the product of \(v\) and \(\Delta t\text{.}\) If either \(v\) or \(\Delta t\) had been divided instead, we would have seen a different relationship.
Exercises Activities: Practice Sensemaking
The statements below are incorrect. For each statement, use a sensemaking strategy to determine why it is incorrect, such as the one introduced in this section or those in
Section 2.3.
1.
Your friend has built a rocket and reports that it has a top speed of \(v=700 \mathrm{~m/s^2}\text{.}\)
2.
The Wizard of Oz claims the volume of his balloon \(V\) is related to the temperature \(T\) by a constant \(\alpha\text{:}\) \(V = \frac{\alpha}{T}\text{.}\)
3.
A car dealer claims that a certain new car has a mass of \(4000 \mathrm{~g}\text{.}\)
4.
A manufacturer claims that an office space with length \(x\) and width \(y\) can fit the following number of their cubicles: \(N=5\frac{y}{x}\text{.}\)
"co" meaning together and "variation" meaning changing
often, drawing a diagram or a graph can help your physical reasoning
