By: Natalie Hobson, Sonoma State University
If you give calculus students graphs, they are going to draw tangent lines. As instructors we often encourage students to rely on tangent lines so heavily that discussions about rates of change become lessons about sliding lines along graphs, rather than about understanding the relationships that these graphs represent in the context of a given problem.
So how do we help students develop a deeper understanding of these relationships? Let’s consider an exercise that you might give your own students during a lesson on tangent lines and rates of change.
Exercise 1: The Growing Cone
The images in Figure 1 (left) depict a growing cone. The graph in Figure 1 (right) represents the relationship between the outer surface area and height of the growing cone. Describe the rate of change of the surface area with respect to the height of the cone as it grows.
A Common Response: The “Sliding Tangent Line”
A typical solution to the Growing Cone exercise involves drawing a collection of tangent lines along the graph in Figure 1 and exploring the steepness of these lines as the height increases (see Figure 2). One could observe that the tangent lines become steeper, thus the rate of change increases.
So, what’s so bad about this strategy? This issue is that students often use this technique successfully without interacting with the quantities of surface area and height. Furthermore, to determine an increase in “steepness” they do not need to measure or determine slope at all, they only need to observe visual properties of the line without actually doing mathematics. Students begin to rely on and practice such techniques exclusively. As a result they are not able to reason in situations where these strategies break down, which we will see in Exercises 2 and 3 below.
Moving Towards Deeper Understanding
What does a response for Exercise 1 with deeper understanding look like? At a beginning level, a student could describe that as the height of the cone increases, the surface area of the cone also increases. At a more sophisticated level, a student could describe that as the height increases in equal sized amounts, the surface area begins to increase by larger amounts. This means that the surface area increases at an increasing rate with respect to the height of the cone. As a result, the graph of the surface area and height would appear steeper for sections of the graph corresponding to points where the cone is taller.
Supporting students’ understanding of the quantitative relationships we have discussed above provides them with a deeper understanding of tangent lines and other tools in calculus. A student who is able to understand the quantitative relationship in Exercise 1, should be able to illustrate the changes in the surface area across equal changes in the height of the cone both in the picture of the cone (see Figure 3 (left)) and on the graph (see Figure 3 (right)). From here, students can more deeply connect slopes of secant lines on graphs with the changes in quantities of the growing cone as illustrated in Figure 3 (right) and revisit slopes of tangent lines as ratios of infinitesimal changes. A sophisticated understanding of tangent lines then should provide a student with the necessary tools to interpret a tangent line by comparing relative changes of the quantities the line represents.
Exercises to Elicit Deeper Understanding
The “sliding tangent line” strategy is exclusive to graphs in the Cartesian coordinate system with the independent variable on the horizontal axis and the dependent variable on the vertical axis. To help students develop understanding of the co-varying quantities represented in a graph, we can provide them exercises in which tricks like the “sliding tangent line” strategy don’t apply. For example, does this strategy work in Exercise 2?
Exercise 2: The Growing Cone Exercises Part II
Describe the rate of change of the height with respect to the surface area of the cone in the Growing Cone exercise.
A student who is able to quantitatively reason about the situation in Exercise 1 could respond to this question without needing to redraw the graph in Figure 1. The student could instead consider changes in the height of the cone that correspond to equal changes in the surface area of the cone. The cones and graph in Figure 4 illustrate the changes of height in the cone corresponding to an equal partition of the surface area. The changes in height decrease as the surface area increases in equal amounts. Thus, the student could conclude that the height of the cone increases at a decreasing rate with respect to surface area.
Graphs in Polar Coordinates
It could be argued that students could redraw the graph of height and surface area in the previous example so that the “sliding tangent” trick still applies. But what about in polar coordinates? Consider Exercise 3 which explores the graph of the cosine function in polar coordinates.
Exercise 3: Polar Coordinate Graph
Describe the rate of change of the radius with respect to the angle in the graph in Figure 5.
Standard tangent lines do not make sense in this problem. We could draw a line that is tangent to the graph as shown in Figure 6. The slope of this line represents the vertical change in relation to the horizontal change, not the change in radius with respect to change in angle.
We could, however, reason with the graph in Figure 5 in a similar way as we’ve previously seen by creating changes in angle and radius. In Figure 7, a collection of equally spaced angles has been marked with brown arcs and the corresponding changes in radius are drawn with orange segments. We can first observe that the radius decreases as the angle increases. We can then compare the changes in radius as the angle increases. The changes in radius get larger as the angle moves from 0o to 30o to 60o to 90o. A student could then conclude that as the angle increases from 0 to 90 degrees, the radius decreases at an increasing rate.
Teaching quantities in calculus
The focus in calculus should be the measurable attributes, or quantities, involved in rates of change, not on tangent lines drawn on a graph. The tricks and associations we teach students provide them quick ways to draw inferences but leave them unable to understand the implications of these inferences on the quantities. As a result, students come away from calculus knowing how to slide lines along graphs but not knowing how to make comparisons in changing quantities in their world. We can help students develop deeper understandings in calculus by asking questions that focus on the fundamental relationships between changing quantities and challenge students to think beyond memorized strategies.
Several of the ideas and examples here are inspired by the work of Advancing Reasoning, an NSF-funded research project whose mission is to support students’ and teachers’ mathematical thinking and learning by developing products that create transformative learning experiences. For more ideas on how to support students’ quantitative reasoning, see our project page at: https://sites.google.com/site/advancingreasoning/. All figures in this post were created in the GeoGebra application.