I spent the weekend at the Park City Mathematics Institute’s Teacher Leadership Program outreach event in Denver. It’s pretty awesome spending two days hanging out with a small group of passionate teachers, doing math and talking about teaching.

I had a bunch of takeaways from the weekend, from some fun math we did to great conversations at dinner. One particularly interesting one was about modifying tasks. We spent three 75-minute sessions reflecting on using tasks to promote student discussion — identifying worthwhile tasks, modifying tasks to meet our needs, and implementing them to help students make connections and engage in meaningful mathematical discourse. One tension we ran into was that many of the geometric tasks we looked at seemed to lend themselves more to discussion; they were visual and offered multiple solution paths and a low floor to entry in a way that more algebraic tasks did not. One fun geometric task was to find quadrilaterals whose diagonals were perpendicular, given two points on the plane. But when we looked at a task involving exponent rules, the task didn’t seem to have as much potential, and the topic didn’t lend itself to discussion in the same way as thinking about perpendicular lines and properties of quadrilaterals did.

This all got me thinking about how to write some better exponent rules tasks. I took the approach of Peter Liljedahl, Norma Gordon, and many others who have been talking about #SmudgedMath on Twitter. Here are three tasks I came up with. For each, I would ask students, “Is this possible?”

I don’t think these tasks are groundbreaking, just that they’re a better way to spend five or ten minutes than many other approaches to exponent rules. There are two things I really like about these Smudged Math problems:

- Ambiguity. The question requires conversation to clarify and understand. Teachers over the weekend were curious if the smudges have to be the same number. I think that’s a great question to put back on students, and ask them if it is possible for each problem. I also really like using a smudge rather than a variable, because it creates a sense of mystery without the intimidation that comes from variables, and could be used to smudge something other than numbers in certain problems.
- Multiple entry points. Students could argue about whether it is possible if smudges represent the same number, whether it’s possible at all, finding and verifying individual examples, or trying to find a generalization that includes more variables — and even proving that generalization algebraically.

What I think I like best about Smudged Math problems is that they get me thinking differently about topics and tasks. Rather than looking at some topics as those boring units I need to power through, it adds another tool to my toolbox to engage students in discourse, push thinking to higher levels, and lower the floor while also raising the ceiling.