Experience First

The structure of today’s activity is slightly different than our usual format. Instead of having students work in groups through a sequence of questions, we had each group work on a whiteboard (or other non-permanent vertical surface) to explore the rate of change of a growing rectangle, using a variety of problem solving techniques. In part 1, students think about a rectangle whose width is constant but whose length is increasing by 3 inches every second. Students should realize that after every second the area increases by 21, not 3 square inches. This is an essential understanding before students move onto part 2. As you monitor groups, have groups verbalize why the instantaneous rate of change of the area is 21 square inches per second. Continue to feign confusion until each group member is able to provide a convincing justification.

 

In part 2, students explore what happens when the height and the width are increasing. Expect multiple solution paths. To prepare for today’s consolidation and debrief, spend time anticipating student solutions and planning on how you will connect them and what questions you will ask to assess and advance students using that particular method. (We use Smith and Stein’s 5 Practices for Orchestrating Productive Discussions approach to guide us). We predicted that students would make a table, draw a visual, or expand the area expression and then take a derivative. For students using the analytical approach, challenge them to think about where the 41 comes from and where the 12t comes from. Students should be able to make sense of both of these numbers using a visual.

 

It is also worth asking students why the original area of the rectangle (70 square inches) is nowhere to be found in the derivative equation. This helps students solidify their understanding of why the derivative of a constant is 0, namely because the 70 does not contribute to the change in the size of the rectangle--it is a constant!

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Formalize Later

When debriefing the lesson, spend sufficient time discussing why in part a, the area increases by 21 square inches. In part 2, we want to help students see why we might be multiplying the rate of change of the length by the current width (7+2t) and the rate of change of the width by the current length (10+3t). This is the essence of the product rule. Since the width is not always 7, we are adding more than just a 3x7 rectangle every time. The same can be said for the length. Show algebraically that the product rule and their original method (expanding then taking a derivative) create equivalent results. Ask students why the product rule might be useful or alternatively, why expanding the product might not always be the best strategy or even a possible strategy (What if the two functions were x^3 and sin x?)

 

Help students see that each passing second produces two new rectangles of size 10x2 and 3x7 (a total of 41 square units) AND 12t square units represented by the missing bottom corner of the rectangle. The 12t can also be seen in a table, as the constant second difference of the areas is 12.

 

One common question is why the bottom right rectangle is not 6 (since 3x2=6). Similarly, when looking at the table, the areas change by 6, then by 18, then by 30, etc. (not 12, 24, 36…) This goes a bit beyond where students are in a Precalculus course, but there is a distinction between the change in the area of the rectangle over one second and the instantaneous rate of change of the area at a single instant. One way I’ve found to explain this to students is to say that the instantaneous rate of change is a description of how the area is changing at any particular instant, not the amount it changes (since nothing happens at one instant). This helps explain why we see a 12t, and not a 6t, since we are describing what happens to the areas in the lower right hand corner (increasing by 12 more square units every second), not talking about the actual area added in the lower right corner.