Today in calculus I wanted to check if students really understood what they were doing when they were finding the instantaneous rate of change. (We haven’t learned the word derivative yet, but this is the formal definition of the derivative.)
So I handed out this worked out problem.
And I had them next to each of the letters write a note answering the following individually (not as a group):
A: write what the expression represents graphically and conceptually
B: write what the notation 
C: write what mathematical simplification is happening, and why were are allowed to do that
D: write what the reasoning is behind why were are allowed to make this mathematical move
E: explain what this number (-1) means, both conceptually and graphically
It was a great activity. I had them do it individually, but I should have had students (after completing it) discuss in groups before we went to the whole group context. Next time…
Anyway, the answers I was looking for (written more drawn out):
A: the expressions represents the average rate of change between two points, one fixed, and the other one defined in relation to that first point. The average rate of change is the constant rate the function would have to go at to start at one point and end up at the second. Graphically, it is the slope of the secant line going through those two points.
B: the
C: we see that 
D: as we bring h closer and closer to 0, we see that 
E: the -1 represents the instantaneous rate of change of 
I loved doing this because if a student were able to properly answer each of the questions, they really truly understand what is going on.
Continuous Everywhere but Differentiable Nowhere 
This is a great activity for assessing understanding. Getting students to explain their answers really exposes whether they are stuck on procedures or actually understand the concepts.
Reblogged this on Math Education Concepts.
This is awesome. I feel like I should do activities like this more often with my students. Seems like a great way to assess whether or not they know exactly what is going on. I think activities like this are also a great way to de-emphasize “getting the right answer,” which seems to be the only thing many students care about.
Hi Jeff! Thanks. I am now at a point in my classroom where my kids get tested on both things — the procedure, but also the understanding of the procedure.
Sam
Wow, wow, wow. Seeing this, daughter brings to a higher register your prowess as a math teacher. Mesmerized at how colloquial and down to earth how you communicate your process. Formulas are pretty, aren’t they. Make we want to turn a problem in to a wall art poem.
Very nice. Would be interested in what some of the responses looked like.
A question I like a lot is to give a few expression that look like the limit definition and ask which will work, which won’t work, and graphically what each is measuring.
f(3 + 2h) – f(3) /2h ; f(3 + h) – f(3 – h) / 2h ; f(3) – f(h) / h f(x) – f(3) /x-3 etc.
(with limit notation of h -> 0 or x -> 3)