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## The Heater II: Instructor's Guide

### Main Ideas

- All derivatives require knowing “with respect to what”.

### Students' Task

*Estimated Time: 15–30 minutes*

Students work in groups to determine derivatives using a contour diagram showing temperature as a function of both distance and time in

### Prerequisite Knowledge

- None

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student

### Activity: Introduction

This activity naturally follows the Heater activity, although it is not necessary that these activities be done at or near the same time.

A good introduction to the activity is to give students a SWBQ asking what the derivatives of $x^2$, $y^2$, $a^2$, $e^2$, and $\pi^2$ are, carefully without mentioning with respect to what. It doesn't take long for students to get the idea…

### Activity: Student Conversations

The prompt deliberately fails to ask which derivative is being sought. Students will quickly catch on to this omission, with or without haveing done the SWBQ above. As groups figure this out, ask them to compute both derivatives, with respect to time and with respect to distance.

### Activity: Wrap-up

The need to specify with respect to what should be obvious to all by the end of the activity, requiring little if any further discussion. A discussion of the techniques used to “compute” derivatives using contour diagrams may be useful, possibly involving a discussion of “thick” derivatives. (Add reference.)

### Extensions

A discussion of determining derivatives from real-world data, given numerically or obtained via measurement, possibly involving a discussion of “thick” derivatives.

The computational part of this activity can itself be regarded as an extension, as it is also quite reasonable to stop the activity at the point where students realize the need to distinguish between the derivatives with respect to time and distance. In this case, it's probably best to skip the introductory SWBQ.