K to 2, to infinity

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 puzzle¹.

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.

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