Okay, so ‘infinity’ might be a bit of a stretch but I’m talking about the latest low-threshold, high-ceiling task to become my favourite puzzle^{1}.

Louise Hodgson shared this activity at the Mathematics Association of Tasmania conference at the weekend. The learning intention, as might be voiced to students, was:

*“There are patterns in the hundreds chart and the patterns can help us answer questions such as which number is 10 more or 10 less than another number.”*

Louise used the puzzle to illustrate how to teach maths in K-2 through problem-solving, focusing on the ‘teacher moves’. Her session was fabulous; it was thought-provoking, illuminating, and so much fun. Louise pitched this activity at K-2, but while I listened to the discussion, one eye was always on the puzzle — so many more questions were flashing through my mind.

Here it is. I’m not going to divulge actual solutions, but I will talk around the problem.

*What might the numbers be on the L-shaped piece if I know that one of the numbers is 65?*

Our table (two K-2 teachers, me, and a farmer turned facilitator) started by discussing the problem. We soon realised that we needed clarification on how many numbers are on the L-shape. After specialising for a bit, we decided to count the number of different possible orientations of the L-shape. That’s where the fun began. Louise had different groups share out their thinking. In the middle of Jamie’s explanation, our group had a revelation — were we only working out possible numbers written on the face-down side? Or were we considering possibilities for either side? Whoa! Which is it? A fabulous discussion ensued around the room as a result of what Jamie shared.

In the process of working on the problem, the following questions arose — not all intended for K-2.

- How many possibilities are there?
- Are any possibilities repeated?
- Given the pieces already there, are any of the possibilities not able to be placed on the hundreds chart?
- What are the limits of the L-shape? That is, what is the smallest number (or smallest ‘column/row’) that we can reach? What is the largest number (or largest ‘column/row’) that we can reach?
- In a 5×5 grid centred on 65, are there any numbers we can’t reach?
- Of the numbers we can reach, what is the frequency of each on all the possible L-shapes? Can we work this out without counting?
- What if instead of an L-shape we use a different shape? How do the answers to these questions change? For example, if the shape is symmetric, are there fewer possibilities?
- What if we extend to a 10×10×10 grid? And what does that even mean?

The consolidating task was as follows. What new questions does this generate for you?

*“The numbers 62 and 84 are on a jigsaw piece shaped like a letter of the alphabet. What might that piece look like?”*

This workshop was more evidence that delightful problems can come from any grade level. So much potential. So much fun.

[1] They’re all our favourites, right? Just ask Fawn.

Hi Amie!

So interesting and like you said, so much potential! With the “simple” question “What might the numbers be on the L-shaped piece if I know that one of the numbers is 65?” I could see students playing around with the shape as 65 could be any of the boxes with the L-shape meaning that each box could be a different numbers- in other words, it offers possibilities, free play and exploration and lots of thinking. Good stuff.

I’ll definately share this to the teachers I work with- you don’t happen to have this post already translated to French would you? 🙂 If I get feedback, I’ll share it out.

Merci!

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Pierre: thanks for the comment. I’d love to hear (as I’m sure would Louise) your experience with this activity.

Not already in French — my French extends to bon jour and merci! 🙂

– amie

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What a great activity that mixes number sense and spatial reasoning! I can imagine using it at a highschool level too!

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