In June, a colleague invited me into his classroom to teach a Desmos modelling task — Predicting Movie Ticket Prices — in his Math 12 class. Students experienced exponential functions earlier in the course. We were curious about whether his students would apply what they knew about exponential functions to a task situated outside of an exponential functions unit — a task not having to do with textbook contexts of half-life, bacteria, or compound interest. They did. And they deepened their understanding of how change by a common *ratio* appears in exponential equations (vs. change by a common *difference* in linear equations). They did this within 45 minutes of a 75-minute class. So my colleague let me try out another, less sexy, task — one adapted from MARS. This task, like much of Math 12, is about naked functions; no real-world context here. Nat Banting’s closing keynote at #NWmath reminded me of it. Watch Nat’s talk; view his slides.

The original MARS task above is closed: two functions, one linear and one quadratic, each passing through four points. I wanted to open it up so I changed the prompt: **“A set of functions pass through the points shown. What could the equations for the functions be?”** Also, I removed one of the points — (5, 3) — to allow for different solutions of two functions. The thinking is that open questions encourage a variety of approaches. And then, from fifteen pairs of students:

I anticipated this. The points scream linear and quadratic. They are *sources of coherence*. I had lowered the floor but no Rileys entered *y* = 5, *y* = 7, *y* = 8, *y* = 9. The problem wasn’t problematic. I had raised the ceiling but no one wrestled with equations for sinusoidal or polynomial or radical or rational functions. The *freedom* within my open question didn’t bring about new and diverse ideas. To support creativity — *mathematical* creativity! — I had to introduce a *source of disruption*, a *constraint*: **“A set of nonlinear functions pass through the points shown. What could the equations for the functions be?”**

A student could have used the linear nature of absolute value functions to get around my nonlinear constraint — a bit of a Riley move? — but no one did.

Instead, some students picked up on the symmetry of two new possible parabolas:

Writing the equation of the second parabola — finding the parameters *a* and *q* — presented more of a problem.

Others bent the line; they saw the middle of its three points as the vertex of a cubic function that had been vertically stretched and reflected:

Some saw four compass points and wrote an equation of a circle. This led to a function vs. not a function conversation: “Does that count?” Others saw a sine function that passed through three of these four points. There were “close enough” solutions — great for Coin Capture but not quite passing through the given points:

I didn’t anticipate this. Students weren’t as constrained by “pass through” as I was. Also, they were motivated to capture the points using only two functions, as before.

With more time, I could have shifted constraints again: **“A set of functions pass through the points shown. What could the equations for the functions be? (P.S. The graph of at least one of them has an asymptote.)”** This would have triggered exponential and logarithmic or rational functions. (Even without introducing this constraint, we noticed at least one student playing with rational functions at the end of class.)

Above, there’s evidence to support Nat’s #NWmath conjecture: “Shifting constraints triggered new mathematical possibilities.” My (more) open question didn’t cut it. The student thinking — and conversations — that I had hoped for only emerged when freedom “sloshed against” constraints.

## Update

Edited & published in BCAMT’s *Vector* 62(2).

## One Reply to “A Function of Freedom and Constraints”