Two Labs to Introduce Rational Functions in Algebra 2

 

All the possible sizes for a "Cube-ie"

As our class was working through the Unit 3 topics for Pre-AP Algebra 2, we arrived at rational functions. Rational functions in the Pre-AP curriculum have two key learning objectives with several corresponding Essential Knowledge statements. Rational functions are one of the topics that need to be teacher developed. As I was thinking through LO 3.2.5 Construct a representation of a rational function, I decided that I wanted my students to discover rational functions in a hands-on way. I put together two lab exercises for students to do.

The first exploration involved measuring the length of time it takes a "cubie" to somersault 50 cm along a meter stick. A "cubie" is a segmented worm with anywhere from 1 cube body segment to 10 cube body segments (domain restriction). All we needed were a meter stick, several linking cubes, and a timer. 

One of the group's "cubie" somersaulting its way to 50 cm.

I borrowed the second exploration from an AP Environmental Lab on solar insolation. Students held a flash light above graph paper at different angles, counted how many squares were illuminated, and then divided the pre-measured brightness of the flashlight by the number of squares illuminated. This gave a measure of brightness per square.  We determined the brightness of the flashlight using the free Arduino SJ app on my school iPad. Phones should be able to do this too.

One thing I will do differently next year is to use actual graph paper instead of my big cling whiteboard graph paper because the squares were too big to see much of a difference as the angle of the flashlight changed. I had hoped that using the bigger squares wouldn't leave them counting tons of squares, but it turns out that regular graph paper does work better for this. 

Here's the lab I gave to students.

Desmos Activities for Precalculus Polynomial and Rational Functions

Made by Desmos.com

We came so close to finishing the first quarter face to face, but had to transition to remote learning for a week and a half starting on the last day of the quarter. Fortunately, I had been preparing my classes for this from the beginning of the year. They have been scanning and submitting all of their work in Google Classroom and we have been using Desmos on a regular basis. I would use Desmos wether we were face to face or remote, but I love Desmos even more when we have to go to remote learning. 

I found some great activity builders on Desmos for this unit that I didn't have last year. I even went to a virtual math seminar on using the computational layer in Desmos to add special elements to Desmos Activity Builder this past weekend and tried my hand at it for an activity we did this week. I must confess I made a couple of mistakes that my students helped me to realize and fix as we did it. It's not fancy, but it gave my students the group practice I wanted them to have with graphing rational functions with slant asymptotes. 

Here's the link to my whole collection: Precalculus Unit 3

Here are the individual activities:

Polynomial Equation Challenges

Constructing Polynomials

Polygraph: Polynomials (My students LOVE polygraphs!)

Polygraph: Rational Functions

What's Your End Behavior

Sketching Polynomials...without Desmos

Polynomials and Roots

Horizontal and Slant Asymptotes (I love the use of "which one doesn't belong" in this activity.)

Graphing Rational Functions (This is my creation...not fancy, but it accomplished what I was looking for.)