[Desmos Design] Algebra Is Power, Not Punishment

This is the first of several posts where I’ll use the activities Desmos created last quarter to illustrate our design principles.

One of those principles is:

Create an intellectual need for new mathematical skills.

Nowhere is that principle more necessary, in our view, than in the instruction of algebraic expressions. Three of my least favorite words in the English language are “write an expression” because they so often mean we’re asking the student to do the difficult work of variable manipulation without experiencing any of the fruit of that work.

In both of the questions below, students are likely to experience the work of writing an expression as punishment, not power.

Given the width of the lawnmower (W) and the length of the rope (L), write an expression for the pole radius (R) that will make the lawnmower cut the lawn in a perfect spiral.

Given the width of the pool in tiles (n), write an expression for the number of tiles that will fit around the pool border.

We recognize that one reason variables give us power is that they let us complete lots of versions of the same task quickly and reliably. So in our version of both of the above problems, we asked students first to work numerically, both to acclimate them to the task, but especially to establish the feeling that, “Okay, doing a lot of these could get tedious.”

And then we use their expression to power ten pool borders.

And ten lawnmowers.

Those activities are Pool Border Problem and Lawnmower Math. In Picture Perfect, for another example of this principle, we give students the option of either a) filling in a table with 24 rows, or b) writing an algebraic expression once.

In each case, students are more likely to see algebra as power than punishment.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I am confused a bit as to the lawnmower question. Power implies purpose… there is no real purpose to the example. It seems to be a variation of the goat on a rope problem, which is more real. I wonder if we do not have real problems, how will we find power.

    • Hi Joe, thanks for the comment. I’m afraid I don’t understand how a goat on a rope is more (or less) real than a lawnmower on a pin. Maybe it would help you to know that people actually do this to mow their lawns.

      Me, I don’t find “real” to be precise enough of a term to be a useful critique in these situations. My view is that “real” is a matter of what students do with a context in their minds as much as where that context exists in the physical world.

    • I think the power of these examples is less about the applicability to a life situation, and more about allowing students to create something meaningful from mathematics. That is I want to say it is more about the ability for students to guide mathematics rather than the mathematics guiding them.

  2. I tested the lawnmower math activity with 2 kids (7yo and 9yo), so this is just anecdotal, but the activity did a great job accomplishing the objective of creating a need for an expression.

    I’d note that these are urban kids with almost no experience of lawnmowing. While that context didn’t connect with their personal experience, the video introduction made sense to them why someone might use this mowing strategy and why missed patches would be a problem. They were less clear why it would matter if the loops overlapped, but that was easy to address with wasted fuel.

    Second, the animations are compelling. This was really the key hook that interested them in the activity.

    Third, it was very hard to use the scrollbar to get the right diameter for the pole. This was frustrating, almost enough to put them off the activity. When they got the chance to enter an expression, it was a relief. When they got to see the animation apply their expression several times, it was a delight.

  3. Thanks for the feedback, Joshua.

    Third, it was very hard to use the scrollbar to get the right diameter for the pole. This was frustrating, almost enough to put them off the activity.

    Can you expand on this? What were they frustrated about? What do you think would help?

    • For estimate #3, they had trouble getting the size of the pole within the tolerance using the bar. To be honest, I think this is a feature of the activity, as long as the teacher/guide knows to push through. This experience underscored the value of using numbers to specify measurements and expressions for calculating those values.

      If you felt a need to address this specific issue, one alternative would be to provide an input box where the students could type in a numerical value for the radius of the pole.

  4. Alison Gilbert

    April 3, 2017 - 8:31 pm -

    Hi Dan! I just needed to point out here that there is an error in the Desmos student application… the first problem of a 3 x 3 pool seems to be depicted by a 5 X 5 grid…. hopefully we can get that fixed because I really want to use the tutorial in my class! Thank you for all that you do, Alison

    • Thanks for your question, Alison. In that problem, the pool is 3×3. The tiles that surround the pool are 5×5. Does that make sense?