I recently attended Nat Banting's presentation at the Ontario Association for Mathematics Education annual conference in Ottawa (#OAME2019). He talked about "Creating Mathematical Possibility by Constraining Mathematical Possibility" (you can check out the slides from this presentation on his website). Last week I saw several educators sharing variations of a task that Nat presented on his blog called a 'menu task'.
In a menu task, students are given a list of specifications and are asked to create functions that satisfy these specifications. It would be a fairly straightforward task for students to create a different function for each single specification (e.g. create a function that has a positive y-intercept). Students are challenged by asking them to use as few functions as possible to satisfy all the specifications in a list (in whatever combinations they desire).
Amie Albrecht shared a Linear Relationships version of this menu task which inspired me to try creating one for high school calculus.
Calculus Functions Menu Task
Instead of jumping right into the menu, I wanted to make sure that students were familiar with the expectation so I decided to build up to the menu. I started by asking students to come up with one function to satisfy each specification. Then I used the same specifications but asked students to satisfy them with only two functions.
After getting used to the idea, I then continued with the Calculus functions menu task. The students have just started integration so this is mainly a review with just a bit of integration thrown in.
If you'd like to give this task a try with your students, here is a link to my google slides.
Here are three example functions that I came up with to satisfy these ten specifications (some specifications are satisfied by more than one function):
A,D,E,H) y = -(x+2)(x-1)(x-3)
C,G,H) y = (x+2)(x)(x-2)
A,B, F, I,J) y = -[(x+1)(x+1)(x-1)]/(x-1)
Students were able to come up with fewer functions for this task. Here is one student's two functions to satisfy the ten specifications.