Nrich – Factors and Multiples Puzzle

Nothing gets me more excited about teaching mathematics than a task that can engage my lower level students while simultaneously challenge my high achieving students. The Factors and Multiples Puzzle from Nrich did just that. (Thanks to @drrajshah for posting this on twitter.)

I’m glad I used this in multiple classes because if nothing else, it gave students the opportunity to learn about triangular numbers! What a testament to the fact that we don’t allow students to explore with numbers nearly enough: I’ll bet only one student out of 60 had any idea what triangular numbers were.  A fantastic, interesting set of numbers, arithmetically and visually, was unbeknownst to 99% of my students.

My math recovery students were intrigued by the puzzle portion of it. In fact, I have one student in particular who is not particularly motivated by much . He’s a ‘too cool for school’ kind of kid, and he’ll tell you as much. When I bust out a puzzle, he’s all in. And when I say ‘all in,’ I mean 100%, until he solves it. It’s pretty awesome stuff to have been able to catch his attention and see how cleverly he thinks through things. Amazing.

I also gave this task to a group of advanced students. An interesting strategy these students developed was to grab a whiteboard to work out some patterns in groups of numbers.  I loved walking around and hearing their strategies.  As some groups finished, they started walking around and giving tips (not answers) to other groups.  It was wonderful.

One of my particularly eager students taped his together uniquely.  I appreciated his humor.  🙂



  1. Hello Megan!

    I’m a pre-service teacher working towards my secondary math certification, and I came across this post after Dan Meyer recommended it to me. I’ve been grappling with the question of how to engage students in mathematics, especially when the material seems so disconnected from their lives. I always hear students asking, “When will I EVER use this?” And I want to be able to give them a valid response. Lots of math textbooks provide these pre-packaged “real-world” problems that are supposed to help students connect mathematics to their everyday life. Perhaps the idea behind that is… if math problems were less abstract, more concrete and grounded in a real-world context, the students would care to solve the problems and feel like it’s actually worth solving. It’s one way to try to engage students in mathematics. But Dan proposes the idea that it doesn’t matter whether a problem uses “real-world” or “fake-world” objects, but instead, it’s more important to consider how the problem puzzles and intrigues the student. After reading your post, I’m wondering how your assignment caused some of the most disinterested students to engage 100%, especially given that the topic was pretty abstract and didn’t include a “real-world” context. I think it’s fantastic that your lower-level students were so engaged, and I would love to find ways to help my students engage as well. So what made a puzzle, like yours for example, so engaging? I’d like to suggest a few thoughts on this: The puzzle has an appropriate challenge level. The directions are clear and easy to follow, and there’s a likelihood that it can be solved within the time constraint given, giving the student a reason to try to attempt the problem and persist through it. Also, the “game”-like quality of a puzzle might engage students who typically enjoy games of that nature. Something I’m wondering is the students’ motivation to work through the puzzle. It seems that the students’ motivation is less about learning the math, but more about the satisfaction of solving the puzzle. If that’s the case, the math may still seem distant and disconnected to their lives, which in the long run, could resurface their frustration and disengagement in learning math. My questions would be: What do you think made your puzzle so engaging for the students? And are there any ways you’ve engaged students with a puzzle that also helped them to see how math was relevant to their lives? Is there a way to both engage them and also help students find some personal value in the mathematics they’re learning? It’d be great to hear your thoughts on this. Thanks for sharing your ideas and experiences – this was a great post!


    • Thank you for such a thoughtful, intriguing comment. I hope I can help answer some of your questions.
      I have found that a good “puzzling” math problem can be very engaging to all levels of students. There are a few things I look for in a good puzzling task:
      1. Multiple entry points or a low barrier to entry.
      -in this case, all of my students were able to start somewhere because the directions were clear and concise.
      2. It engages at multiple levels
      -after explaining triangular numbers, the faster kids were off, determined to solve this in the allotted time (about half an hour). Some of the struggling kids were able to pair up and share their ideas. I’m able to also assist those still struggling to remember what a factor is. Some of the extensions (which I did not utilize with this particular class) include coming up with their own categories and fitting as many numbers as they can.
      3. Students who do not arrive at a full solution still can feel some level of success.
      -this is tough to do sometimes. In this task though, the directions ask them to fit as many numbers as they can. This allows for students who cannot fit all 25 into the grid to still feel as though they had some success because they were able to get 20 or even 24 numbers. Then they could perhaps think of their own number(s) to fill in the slots that they couldn’t finish.

      Sometimes, yes, I struggle with the overall engagement being about the “puzzle” rather than the math behind the puzzle. In this case, however, the students are able to see how different sets of numbers (evens, odd, factors, multiples, square numbers, triangular numbers) relate and overlap. The math doesn’t get totally lost in the puzzle solving. In fact, the puzzle is unsolvable without it. I’m not sure one of them knew what a triangular number was before this task and they now know that 3 is triangular and prime. Perhaps more obviously, they connected that evens and odds needed to be on the same side of the grid (similarly with less than and greater than 20). As a result, they also discovered that there are no numbers that are both square numbers and prime.
      As for finding tasks that do this sort of thing (engage kids in a puzzle like feel as well as keep the math connected to the task), the nrich website ( is truly masterful at this. There are so many tasks and problems on that website that offer this puzzle like feel while keeping the math connected to it. One in particular that I think would also work this way is Dicey Operations This task has more of a “game” feel than a puzzle, but the same principle needs to be true: The motivation for the students to engage might be the game feel of the task, but the math IS the game, not a sidekick to the game.
      I hope this is helpful. Thanks for reading my blog post. I really appreciate the important questions you pose.

  2. Pingback: dy/dan » Blog Archive » [Fake World] Limited Theories of Engagement

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