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.
I recently had the opportunity to work with a calculus class on curve sketching and how derivatives affect the shape of a graph. The classroom teacher and I brainstormed some ideas about how we might infuse some hands on activity into the lesson. We decided to try an activity called Functions on the Floor. I originally saw an outline of this activity in a presentation from Liana Dawson called Hands-On Calculus Activities.
Warm up - Functions and Their Derivatives
We started the class out with a Demos activity called Functions and Their Derivatives. We had students work in pairs on this activity. In the first part of the activity, students are presented with the graphs of three functions and they have to decide which is the original function, which is the derivative and which is the second derivative. In the second part of the activity, students create their own challenge and then get an opportunity to try out the challenges created by other groups. I really like the collaboration and discussions creating by working on these challenges.
Functions on the Floor Instructions
After the warm up, we transitioned to the Functions on the Floor activity. We used masking tape to create several coordinate systems on the floor with the x- and y-axis labeled from -3 to 3. At each of these stations was a list of information about a continuous function. Students used a small rope to create a function on the axis that satisfied all of the conditions listed. They then drew their function into a Desmos activity I had prepared. Using the drawings in the Desmos activity we could monitor students activity and plan for our review of the functions at the end of class. We heard some really constructive conversations taking place. The Google slides for the stations can be found here.
Sharing our Work
I thought that this activity was a nice way to incorporate both hands-on physical problem solving while still leveraging some of the power of online tools like Desmos. I learned a few lessons doing this activity and the classroom teacher and I had a productive discussion after the lesson. We talked about how the lesson went and where we saw areas for improvement. We both agreed that eight stations was more than needed as it took some time for the discussions in the student groups to come up with a reasonable graph. We thought that some of the stations could have fewer constraints to consider as well. Also, I think next time I would use something besides rope. I had pretty inexpensive rope and it wasn't as pliable as I would have liked. I think a thick piece of yarn might have worked just a well. If you give this activity a try, let me know how it goes.
Nova Scotia Mathematics Curriculum Outcomes
Calculus 12 B15 - Demonstrate an understanding of critical points and absolute extreme values of a function
Calculus 12 B16 - Find the intervals on which a function is increasing or decreasing
Calculus 12 C5 - Apply the First and Second Derivative Tests to determine the local extreme values of a function
Calculus 12 C6 - Determine the concavity of a function and locate the points of inflection by analyzing the second derivative