Try this: using these four digits—1, 2, 8, 9—make 100 different equations. The answers to the equations must be between 1 and 100, with no two equations having the same answer. Each must use all four digits, while duplicating none of them. So this one works:
But this one doesn’t:
To get a result of 31, you might do something like this:
You can use roots and exponents, as in this solution for 32, remembering you need a 2 to make a squared power:
To complete the challenge you’ll need factorials, negative integers, fractions, lots of creative thinking and, as Pooh would say, a good deal of pencil, as in this solution for 3:
Ok, go.
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In April this year, as every year since 2019, that’s the challenge that has been put to the Grade 7s. Each student is given their own unique numbers, and over the six weeks of this challenge, those four numbers become ingrained in their brains. In the first week they have a goal of 15 equations, the next, another 15, etc. “To get your first one is really easy, you just add your four numbers together,” says STEM teacher Ruth McArthur. “So that sense of accomplishment happens early. But then as you get further on, you have to struggle.”
Toward the end, it’s strikingly difficult to do. Given the investment up to that point, it’s equally difficult to let go of. “That’s the thing that we love about it,” says McArthur. “It’s the kids pushing themselves to explore more things mathematically.” They are personally committed to their numbers, and the challenge of completing the task. “We encourage kids to not just do it on their own,” but rather “to help each other out, to go home and talk to their families, because we believe that the more you’re talking about math and thinking about math, the better it is.” She says,” I know grandparents, siblings, parents have sat down and thought about the four numbers in different ways.”
It’s not a way of looking at math that the students have done in the past, or, for that matter, most people ever do. The rules provide a set of guardrails in the way that writing a haiku or a sonnet does: while the criteria are clear, everything in between feels creative, and no two results—i.e. a full set of 100 equations—are exactly the same. The students use whatever they know as if drawing on a set of tools.
To keep them within the guardrails, and to maintain the momentum, students are required to submit their equations each Friday for checking. The process stresses the importance of breaking down a larger challenge into smaller goals, getting feedback along the way, and seeking support when needed. The teachers check to make sure they are showing their work, that they are able to communicate what they are doing, and that they meet all the criteria. Says McArthur, “if I can’t see their calculations, if they jump steps, if they don’t have the appropriate numbers at their start, or if they’ve had a zero—they can’t bring in a zero anywhere—then those equations don’t count.” So they reset, and move ahead from that point. As they get toward the end, each new equation added to the tally can feel like a win.
What teachers find is that students often gain a sense of discovery, which in math can be a very new experience for them, one that can bring a sense of pride and ownership. If they learn a new way to use their numbers, they will then go and show somebody else how to use them in the same way, such as adding exponents to exponents. “It challenges students at their respective levels while also building a collective excitement, interest, and enthusiasm in mathematics,” says middle school STEM teacher Braden Kenney. “They really do become each other’s teachers, sharing their own strategies and new mathematical concepts they’ve discovered.”
At the deadline, which this year fell on May 19, they came together as a community for the “Push Day,” helping those students who had solved 90 or more to work through their final equations. It becomes a whole-school event, where senior school students, teachers, and staff members join in the challenge.
Working in small teams, calculators out and brains on, they puzzle away at their final numbers. “I love the excitement they get when they are near the end and they're helping each other try and solve the problem,” says teacher Kate Hubbs. She recalls a student who found a quiet spot to focus on a problem while others, all high school students, continued working as a group. “He then quietly waved us over to say he got it, and he got to show them his solution.” She says, “that was the highlight of this year,” and it was no doubt a highlight for the student as well.
For many of the senior school students, it’s a full-circle moment, as many of them did this exact exercise when they themselves were in Grade 7. They had their own four unique numbers (some even remember what they were) and recall the senior students who helped them work through their final equations. As they do that with the Grade 7s now, they share strategies and new math concepts they have learned in high school.
In the final moments, they divide out into separate rooms according to a team roster. If they have 90 or more equations signed off, then the whole community comes together to try to get them to 100. Together they complete a challenge that, if we’re being honest, most adults would struggle with, even with a good deal of pencil.
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How would you do? To see one way of completing the 1 to 100 challenge with 1, 2, 8, and 9, click here.
This challenge is adapted from the activity adapted from Peter Liljedahl by Kimberley Krasevich in “TE Number Sense and Numeration” with Carly Ziniuk at The Bishop Strachan School. Peter Liljedahl is an Associate Professor of Mathematics Education in the Faculty of Education and co-director of the David Wheeler Institute for Research in Mathematics Education at Simon Fraser University.
