Tuesday, February 12, 2019

Teaching Derivatives of Inverse Functions - The Easy Way!

I told someone last week that I'd send them a summary of how I teach derivatives of inverse functions, so I figured I would post it here too in case it could also help others.  My first year teaching AB this felt like a small formula that kids would need to have memorized for the potential of 1(or maybe 2) questions on the AP exam- ugh. I had multiple teachers my first year tell me just to skip it....it could be left out and no one would be the wiser. I really struggled with that idea, so I stumbled my way through it.

I was lucky enough to have another teacher share their strategy for teaching derivatives of inverses with me and I've never looked back. My kids rock at it now and it makes sense to them- not another formula to memorize. We are building on the idea of inverses from previous courses....that if (a,b) is a point on f we know (b, a) must be a point on the inverse of f.
I've also linked to a discovery activity where students see that derivatives of inverses are reciprocals by looking at linear functions. It's simple, but lets them do a bit more of the heavy lifting.

Happy deriving! 

Sunday, February 10, 2019

Polynomial Symmetry vs. Polynomial End Behavior

Let me dramatically reenact any given pre-calculus, algebra, or calculus class to you from my past almost decade of teaching....

Scenario #1:

Student: "This function is even."

Me: (Secretly lighting up inside because they understood the lesson!) "Why do you say that?"

Student: "Because both ends go up!"

**Face Palm**

Scenario #2: 

Student: "The function has an odd degree."

Me: (Secretly lighting up inside because they understood the lesson!) "Why do you say that?"

Student: "Because it's symmetric about the origin!"

**Face Palm"

Does this sound familiar to anyone else? 

I am fighting the good fight this year and preemptively planning an activity for my A2 students to work on to get them thinking about the differences between symmetry and end behavior. 

It starts with students identifying both the symmetry and degree of given polynomials:

Then, asks them to agree or disagree with some statements about symmetry and degree:

Then, finally, asks them to draw their own pictures to fit constraints. 

Will report back on how this goes. If you have any tricks to help students keep these straight, please share! 

Thursday, February 7, 2019

Optimization & Rewriting with Constraints

One of the joys of teaching a Non-AP Calculus class is that I can spend some time digging into a topic that's glossed over far too quickly in AP Calculus: Optimization. It's one of the most intuitive topics for students- they understand the frustrations of trial and error when a better mathematical process exists and the applications of trying to find the most or the least something.

I am anticipating that the algebra behind optimization will be the real issue for my students, so I created this scaffolded assignment to get them thinking about using constraints to write equations in terms of one variable. It builds from being very apparent about which formulas to use and how to solve them to being completely open ended- the way they will actually appear in a Calculus questions. Note that this would be appropriate for an Algebra 2 or a Pre-Calculus class as they study optimization from non-calculus perspective as well! Even an advanced Algebra I class would be able to tackle many of these! 

I thought about it in 3 phases...

Phase 1- Working with the Algebra
In each of these questions, the equations are given. Students need to isolate variables and use substitution to rewrite. 

Phase 2- Basic Modeling
In each of these questions, a basic scenario is given that can be translated easily into an equation. Constraints are labelled as such. Students need to generate their own equations, then use substitution to rewrite. 

Phase 3- Removing the Scaffolding
These questions progressively remove the scaffolding, ending in a problem that is worded in the way an optimization problem would be. Students should be able to rely on their experience form the previous questions to solve these. 

Feel free to adapt and let me know your thoughts! Happy Optimizing!