Main ideas

- Previewing the Math Bits theme of Variables and Representations
- Teaching differentials, small changes, and zapping with d

## Lecture (30 minutes)

Ask students to make a sketch of f = 7x^{2} on the big white board.

What are the variables of interest? One idea here is to note that x is not the only variable here: f is also a variable from a physics perspective.

What representations do we have for this relationship? The symbolic equation is one, and the graph is another.

Introduce the differential quantities df and dx as small changes in f and x, respectively. Ask students to add df and dx to their graphs.

Then, ask students how df and dx are related to each other. Students should be able to articulate that this is a derivative.

Is the relationship the same if we choose a different point on the graph (a different initial x)?

Then relate df and dx using the symbolic representation: df = 14xdx.

- The following two steps can be replaced by other activities on the relevant Hour page.
- Give the students f = 5x
^{2}y^{3}. What are the variables? What representations do we have (if time, hand out surfaces and have students make sketches similar to the above)? What does a differential relationship look like? For this last one, use the generic form df = A dx + B dy, and talk about the fact that this is another representation. - Do some specific examples of using the zapping with d strategy (see Zapping with d).