Review of optimization

I'm planning to use the following sort-in-order activity with a group that will review calculus and especially struggled with optimization.
For each group - 2 to 3 students - I will make horizontal cuts across the paper, so that each step, why, and example will be on one piece of paper, and then shuffle them.  The kids are then instructed to put the pieces in order of how they would solve an optimization problem that will be different for each group of students. They are also asked to provide an explanation for why each step is done and the example solution problem they are assigned.



I wonder if this task would also work as an introductory activity to optimization, as in: make sense of this, given whatever conceptual understanding you have of derivatives.

Any feedback on this?

Apparently, this is how to teach my kids vector equations of lines

Vector equations of lines usually hit my students like a nasty shock, after weeks of soft and cushy work with vector operations. This year, I tried a constructive approach, using this silly and in my opinion boring and too structured investigation:



The plan was that students should use this geogebra activity together with the written instructions. That immediately failed, since school computers did not have updated geogebra. So they used mini-whiteboards instead, and it worked well. And when I say it worked well, I mean it worked amazing. I have no idea why, because seriously that investigation isn't exactly a masterpiece of pedagogy, but this group of students caught on to both the activity and the conclusions. Above all, even the weakest students in class could arrive at, explain and use vector equations of lines.

I am SO back

Wow, take a break for 18 months and when you're back, things are the same (whiteboards) and different (each kid with their own laptop, and oooh, check out Desmos). So what did I do? Completely screw up (in a let-this-be-a-warning-to-you-all kinda way) one of my first lessons with a smart, nice group of seniors.

I've been debating this with myself. Should my first blog post after an extended break be about a teaching mistake? Doesn't everyone want to get inspiration for good teaching, rather than bad? Nah. We all need to learn from each other's mistakes, and we should discuss them rather than pretend we never make any.

So, with that in mind...

The aim of the lesson was to introduce students to derivatives of trig, exponential, and logarithmic functions.
I decided to use Desmos for an investigative activity (technology! and inquiry-based math!).
I created a graph with instructions, and shared with my students.
In case you're too lazy to click that link and check out the instructions, here's a summary:

1. Check out the graph of f(x)= sin(x). What do you think the derivative will look like?
2. Ask desmos to graph derivative. Huh. What IS that function do you think? Graph your guess to see if it is correct. If it's correct, write down your conclusion. If not, try again. 
3. Repeat with cos(x), tan(x), e^x, and ln(x)

Once students got the hang of the instructions, they engaged very actively with the task and were more or less successful with guessing the derivative functions. At the end of 30 minutes or so we could summarize the findings. Everyone was happy. It felt like an awesome lesson. 

But it was crap. I realized it a while later, out of nowhere. There was nothing to trigger conceptual understanding, no variety of thinking and problem-solving. Instead of consulting desmos to ask for derivatives, students would have been better served by using their formula booklets, which is what they'll have on exam,

I'm writing this lesson up as a warning example of the seductive nature of technology and inquiry-based teaching. Both have their place, of course, but only as tools for obtaining specific goals relevant to learning. 

In this case, what I should have done, is to build on student understanding of the derivative as a gradient function by assigning them to groups, one per function, and having each group work old-school with paper and ruler to draw tangents, estimate gradients, and plot the derivative function based on these values. Then they could have tried to identify the formula of that function. How do you beat paper and ruler for an activity like this? You don't.