Monday, March 11, 2019

Discovering Average Value in AP Calculus

I saw this tweet ages ago and it's been screenshot and sitting on my desktop for almost a year.
Before teaching average value this year, I say down to recreate it as best I could.  A few people asked me for it, so I shared my file below, but all credit goes to Nicole Lalanne for this one!


Also, if I could do it again, I'd use split peas or something else that is flat. Even my always responsible and mature AP kids couldn't keep popcorn kernels from rolling all over the classroom at some points (despite their ingenius pencil trick seen below).



Let me know if you make any modifications to it! I'd love to know any and all tweaks we can make! 

Rectilinear Motion & Balloon Rockets in Calculus

We are just wrapping up rectilinear motion in my non-AP Calculus class and I wanted a lab for kids to make the theoretical connections to the motion of real life objects. I also wanted them to get a feel for some rudimentary mathematical modeling. I didn't want them getting to the test feeling like they could ROCKET to an A with their knowledge and then having this kind of result:


I decided to write a performance assessment that would have students model a scenario and analyze the fit of their model based on their actual observations. I'm counting it as 15% of their applications of derivatives unit assessment. 


We work in 3 phases:
  • Rocket Construction
  • Basic Rocket Motion
  • Advanced Rocket Motion

Basic construction is easy and cost me $3 in materials from the dollar store. A few groups struggled to get their rockets to move, based on how they attached their balloons. Most just needed to readjust so the air shot directly back out of the opening instead of up or down. 

For both phases- the basic and the ridiculous- students used Logger Pro to perform video analysis of the motion. They did a great job and the directions seemed to be clear to them. These were modified from directions to a Related Rates project I stole from Sam Shah, which he stole from someone else. Sam's version of the project is here and the original Logger Pro directions for that project are here. (Also I've used that project in my class before and it's fun if you're looking for a related rates adventure in modeling). 

What I love about Logger Pro is how easily students can look at best fit models for various functions. They were able to easily analyze the correlation and make decisions on which model was most accurate. 




I limited their options, as seen in the table below, based on which they could actually differentiate based on their limited algebra and calculus knowledge. 


Students then used their models and their knowledge of calculus to answer these....
Simple Motion:

1. At t=2, what direction is your particle moving? Use calculus to support your answer.

2. Use calculus to prove whether the particle is speeding up or slowing down at t=4 seconds. Explain how you know.

3. Based on your model, after how many seconds should the particle be at rest? (Use your calculator to solve, if necessary)

4. What should the position of the particle be when it comes to a stop?

5. How closely did the mathematical model represent your rocket’s path? Give evidence to support your argument.

6. How could we make another trial run more accurate?

Advanced Motion: 

1. When should your rocket be at rest?

2. On what intervals should your rocket be moving right? Left? Use a sign chart to make your conclusions and explain how you know.

3. On what intervals is your particle accelerating in a positive direction? Negative direction? Use a sign chart to make your conclusions and explain how you know.

4. On what intervals should your rocket be speeding up? Slowing down? How do you know?

5. How closely did the mathematical model represent your rocket’s path? Give evidence to support your argument.

So far, so good! If you try using this with your students, please send me any tweaks you make! I'm presenting this to some of my colleagues this week and can tell you there's lot so applications you can make of this lab to quadratics (at any level), parametric equations, and more! It's a fun way to bring math to life! 

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!