One of my favorite start-of-the-year activities is to have students produce a growth rate graph from the CDC growth chart for children. Here is a link to a student-facing version of the activity, which you are free to copy and adapt for use in your class1. And here is a quick-start guide to using this activity in class.
This activity has students compute slopes using both difference quotients and tangent lines. It asks them to abstract the difference quotient (in spreadsheet notation), so that it can be computed quickly for hundreds of data points. It gets them warmed up to the idea of viewing slope as a function, illustrating a situation where graphing slope as its own function is quite illuminating. It helps them see how a function relates to its derivative and prepares them for interpreting derivatives in the future, both in applied contexts and in curve sketching.
The graphs below2 give a quick sense of how this activity turns out. Students start with the graph on the left (data from the CDC), and end up producing the graph on the right. Features that are barely visible on the left become obvious on the right, for example the incredibly rapid growth of 2-year-olds and the growth spurt around age 12. Truly, the growth rate graph gives a much richer sense of how human growth varies throughout childhood.
The right edge of the growth rate graph also illustrates how a function can continue to increase even if its derivative decreases. Every year, I say to my students after they do this: “the rate of growth graph drops off pretty fast after age 12. Does that mean the person is shrinking?” And they immediately say, “no, it means that they’re growing more slowly.” In this context, students see that this makes sense: the declining growth rate corresponds to a flattening slope, not to a decreasing height. Later in the year, I remind them of this every time someone gets confused about the distinction between decreasing f’ and decreasing f.
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This is one of several numerical examples I like to do at the start of the year. By doing some examples with real data, students get to see how informative derivatives and integrals can truly be: sometimes things jump out in a graph of f’, even though they are barely visible in the graph of f. In addition, doing some numerical calculus really drives home to students how accumulation connects to area and how rate of change connects to slope. Engaging students with these questions before adding limits into the mix lets them get a feel for many of the core ideas of calculus in a setting that is as simple as possible.
Doing some numerical examples also ties some of the big ideas of calculus to students’ intuition about rate of change and accumulation in everyday life. For example, the conversation about f “increasing at a decreasing rate” makes a whole lot of common sense in this growth rate activity, it a way it doesn’t when f and f’ have no concrete meaning. This is a great example to refer back to when introducing increasing/decreasing, concavity, and curve sketching.
- Within the terms of the creative commons license in the document. ↩︎
- The graphs shown here (taken from my book) are based on the growth chart for girls. The graphs for boys (used in the activity) are slightly different. And comparing the two is actually quite interesting. These graphs were initially designed using the Desmos Graphing Calculator, built by Desmos Studio PBC. They have been adapted and modified using Adobe Illustrator. ↩︎