This semester I'm teaching Calculus I and Linear Algebra. In each class, I've had a moment of discovery in the past week or so.

**Calculus: Derivatives from Graphs **

In calculus, I work with them on what the derivative graph of a function would look like, given just the graph of the function. So if the graph of f is this ...

... then what would f' look like? The activity (with 8 different graphs) went as it usually does.

- Step 1: Find where the slope is 0, and give f' a value of 0 at that x.
- Step 2: Where the slopes of f are positive, highlight positive values for f' (and similarly for negative slopes). (Actually, the highlighting was new. I usually just draw dotted lines.)

- Step 3: Draw a curve that connects it all.

We had an absolute value curve and discussed where the derivative is undefined. (Which I marked with vertical red lines.)

And then we got to this one ...

I said that *w'* looked like this ...

And **a student asked** how I knew the lines were straight. Hmm, *do* I know that? "I'm not sure. Let's see..."

I thought about the curve given for *w* and said it looked like a bunch of parabola shapes (which I know have straight line derivatives), ... or like the absolute value of sine. I decided this was a fascinating question, and put both on desmos.

The red is y = |sin(π/2*x)|, and the blue is y = -(x+1) ^{ 2 } + 1 and y = -(x-1) ^{ 2 } + 1. To me, it looks like the original graph of *w* could be either one. But the derivative is the straight line segments only if *w* came from parabolas. If it came from a sine wave, then the derivative is curved (coming as it does from cosine). Using orange for the derivative of the sine graph and purple for the derivative of the parabolas graph, I got this in desmos...

Very different look to the derivatives, even though the original *w* could have been either of the original functions I put onto desmos. Fascinating!

**Linear Algebra: Pivots vs Free Variables**

We are using some fabulous activities from the Inquiry-Oriented Linear Algebra project, along with our textbook, *Linear Algebra and Its Applications*, by David Lay (we're using the 4th edition). We had just done part 3 of the Magic Carpet project the day before, and I was summarizing. We were talking about the span of a set of 3 vectors in ℝ^{3}, and saw that the span made a plane through the origin. This was because there were 2 pivot columns. And then **a student asked**, "But don't we use the number of free variables to decide whether we have a line or a plane?"

To me that felt like a very deep question for a student to be asking this early in the semester. I said I'd answer the next day, since we were almost out of time. The next day I said, "We looked at the pivots because we were asking about span, which is all the linear combinations of the column vectors. Until we started considering span, we more typically asked about all the solutions to a set of equations, which is a different sort of question. For that, we look at how many free variables to determine if all our solutions create a line or a plane (or something more)."

I have never had a student ask a question like this, and was quite intrigued. I told them we'd explore somewhat similar questions in our 3rd unit (chapter 4 of Lay), when we will explore column space and null space. Once again, I was fascinated.

I've been teaching for over 30 years. I know calculus I inside and out. I've taught linear algebra often enough to feel like I'm a pretty solid expert on the basics. (I'd love to have more expertise on where this class might lead them.) Even so, I learn new things each semester. Even teaching beginning algebra, I have repeatedly seen it from a new perspective when prodded by some unique question a student was asking.

Yay for student questions.