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!