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! 

Thursday, December 13, 2018

Implicit Differentiation Sorting Activity

If you're a calculus teacher, you know this struggle...

My non-AP calc students have truly blown me away with the ease at which they've taken to derivative rules this unit. I went into this unit with a relatively open idea of pacing- what my AP kids can do in a day might take 3-4 with these kiddos. But, they're killing it....when isolated, their calculus is beautiful.

Unfortunately, the calculus isn't going to be "isolated" as we move on in the course and the struggle became very, very real as we started implicit differentiation. With that in mind, I write this sorting activity to help scaffold students in their algebraic skills.

I have students identify the initial problem, then work on whiteboards to determine the next steps. They had great conversations and asked even better questions. Once this was finished, I saw a lot more independence as we solved implicit differentiation problems without the scaffolding....success!

Here's the original document if you want it!

Thursday, November 29, 2018

Did I Finally Figure Out How to Teach Recursion?

Recursion has always been a weird topic for me- one where I try all sorts of different things to try to get kids to tie the relatively basic conceptual idea to the funky notation that is sometimes associated with it. I've taught it at the middle school level (next=now blah blah blah), in Algebra 1, in Algebra 2, and in Pre-Calculus and I inevitably always have some kids who struggle make the transition from the idea to the notation and requirements. So, yet again, I tried tweaking this year. Here's what I did. 

I started by putting this slide on the board: 
Students had 30 seconds in groups to tell me the exact sequence about which I was thinking....this obviously caused some unrest. Some groups were prepared with random guesses, some got indignant that they didn't have enough information, some argued among themselves about which side to take. I let groups guess and every time I said "WRONG!" until someone finally pointed out that this was rigged. 

I let them know that I'd allow the class to ask 2 questions to try to get it right. They needed to prioritize those questions in their groups and then we continued the discussion. The 2 questions they asked:

1) What's the pattern?

We discussed- was this strong enough to give me the EXACT sequence I was thinking about? Nope! 

2) What's the first term?

That's it! We need to know where to start and how to proceed from there! Those 2 magical ingredients of a recursive formula were just generated by the class- score. 

Next, I challenged groups to complete this table together. They had great discussions, challenged each other, and the wait time to try to get them to ask questions at the end was excruciating because they really felt confident about it. 

Welp, guess they don't need me now. They just discovered how to write these without me ever teaching it. I'll be in the Math Office drinking my latte with my feet up on my desk. 

After some wrap up notes, we played a round of Quiz Quiz Trade- one of my favorite Kagan Strategies. Below are my rules:

And here's the actual document:

These were small tweaks to how I normally teach recursion, but they made a huge difference for my kiddos this year. The extra work we did with notation previously made a big difference as well (see my last blog post). I loved seeing the kids excited about a topic that has sometimes caused anguish for others in the past! 

Friday, November 16, 2018

Building Understanding of Sequence Notation

Sequences and series are never a particularly hard topic for my Algebra 2 kids- conceptually, at least. They get it. They've been looking for patterns since they were tiny little humans and it's a fun puzzle for many of them to do so in class. What they struggle with consistently is the notation we demand of them at this level of mathematics. Some mastered it in Algebra 1, but many just show up and pray they do it right this time or memorize what their teacher told them to do instead of trying to battle the sense-making involved to understand and be able to apply this knowledge. This activity is one I use to combat that and it builds to have students generate their own equation for recursive sequences. 

Phase I:
We start by doing a sorting activity, where students are given the cards at right and asked to order them. This is an easy enough task- definitely low floor for some of my weaker students. This is a springboard for discussing the "why" and asking questions like:
  • Which comes directly after n? How do you know?
  • Which comes directly before n? How do you know?
  • If you have n-3, which would come 2 after that? 
You can really extend it as far as you want to go here. 

Phase II:
From there, we look at the actual expressions we use for terms. 

 Using what we previously did, ordering these isn't usually much of a challenge. From there, students work in their small groups to extend:

Phase III:
Here is where the wrap up discussion as a class occurs and we begin to test our understanding of the notation. In small groups, we first just examine the differences between position in the sequence and actual value of the term:

Then, we start translating from words into notation. 

I've always heard it said that students are more receptive to an idea if they think it came from themselves or another student....this activity has been a huge help in clarifying misconceptions with the help of other students, not just a teacher re-explaining it the same way for the 400th time. 

If you have anything else that you love for teaching recursion, sequences, or series, please share!! 

Monday, November 5, 2018

Limit Definition of the Derivative Carousel Activity

I've been working on building up the limit definition of the derivative conceptually with my non-AP Calc class for the last week (see my recent post on what I've been doing so far ) and today we started putting it all together! 

To into this activity, I used an awesome warm up from Math Teacher Mambo (Thanks, Shireen, for sharing on the AP Calc Facebook group). As soon as she shares it publicly, I'll link it here. It's so, so good!

Future Home of Warm Up Activity Link

After completing the warm up, here's how the carousel worked:

1) Put students in to groups and have them start at a blank poster or piece of VNPS

2) Each group completes one step at their poster, then rotates. They will then check the next group's work, correct it, and then add the next step to the poster. Here were the directions:

  • Draw a blank axis- I told them just first quadrant- and a function of their choice
  • Sketch a secant line
  • Label x, x+h, and h
  • Label f(x) and f(x+h)
  • Write an expression for slope of the secant line
  • Transform expression into slope of the tangent line

Here were some of our results:

Awesome conversations ensued, including whether it would make sense for f(x) to equal f(x+h) and what that would mean for the secant line (Helllllloooooo, Rolle's Theorem!). Kids were explaining to each other, critiquing each other's work, and doing a lot of sense making among themselves! 

We'll see how this translates to retention beyond today on their next quiz, but I loved seeing the progress that they're making. I think many could explain it better than a few of my AP kids right now- a good "challenge accepted" moment for me to amp this conceptual understanding up more in AP, too.