# 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!

Anna

• 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 (nrich.maths.org) 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 http://nrich.maths.org/6606 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. This has to be one of my all time favourites – along with the locker problem. 🙂

3. Hello! I could have sworn I’ve been to this website before but after going
through a few of the posts I realized it’s new to me.

Regardless, I’m definitely delighted I came across it and I’ll
be bookmarking it and checking back regularly!