Like many folk, I'm a big fan of number puzzles. I always look forward to the Kakuro in the Toronto Star every Sunday: sometimes this will take me 5 minutes to solve, sometimes 5 days. I also love KenKens and have been seeing these used in classrooms as a great way of practicing number bonds and problem solving at the same time (always a win-win situation in my view). Sudokus are also OK but I won't go out of my way to work on one of these. I recently came across Multiplication Squares on the nrich maths site. This one particular problem struck me as having lots of potential of allowing students to practice and to problem solve both with additive and multiplicative thinking. I quickly created a few of these and, as I often do, I tried these with my own children.
I started with a 2 by 2 multiplicative one for my daughter:
The idea is to fill in the cells so the top two multiply to give 40, the bottom two multiply to give 14, the two left multiply to give 56 and the two right multiply to give 10. As she did this, I could see her use multiplication and division to fill in the cells. She also had to think about which pairs of numbers multiplied to give 40: in my opinion this is a far richer question than the more closed 4×10. I then gave her a 3 by 3 one:
At this point, my son wanted a piece of the action, so I gave him a 2 by 2 additive thinking square:
When I created this, I had a 2 in the top left hand cell: he started with 5. And that's when it suddenly hit me: there are an infinite number of solutions to these! I began to love these even more.
My son, seeing his sister work on the 3 by 3 multiplication, then asked for one for himself. I duly obliged:
As he persevered on this, it was clear that there was lots of lovely decomposition and recomposition going on. Then this happened:
I tried these puzzles in a grade 3/4 class for the first time yesterday. We began with a 2 by 2 additive puzzle which they absolutely gobbled up. When they finished, it was really powerful to say, "Compare your solution to your neighbour's" and then see them get excited when they had done it in different ways.
I then wanted to see if they were ready for a 2 by 2 multiplicative puzzle so gave them this:
Now this is a class which has very recently begun learning about multiplication through the use of arrays and been recording their findings:
Without a doubt, they had to think more about this puzzle but the fact that they could think of 12 in different pairs of factors helped them.
* * *
As a teacher, it takes hardly any time at all to create these and I can structure them such that if I want to focus on a particular fact, I can include that as a possible solution. I can also add an extra level by including some restrictions to the solution. For example, my son gave me this one to try:"I already know the answer," he told me. I looked at it and said, "Is it all fives?"
"Yes, but your not allowed to use fives!"
So this got me thinking. I can create a whole new set of puzzles with restrictions. These will require more perseverance to solve (you're welcome). For example:
Call me bias, but I really like these and as I haven't seen this type of puzzle before, I get to choose what to call them.
So I have decided to call these puzzles: YOHAKU.
I have created a website with these on: http://www.yohaku.ca/ Who knows, maybe people will like doing these and Yohaku will become really popular. Maybe one day they will be as big as Sudoku and KenKen! I'd love to know what you think.