Fun for All the Family

I'm always on the look out for math problems that can be used with people of any age. These puzzles are usually easy to explain but not so easy (or obvious) to solve. Here is one that I recently created and posted on our board's Math Twitter account (@DCDSBMath) that has proved popular with kids and adults alike.

To clarify: you can use any number of pieces to make a rectangle so right from the get go it has multiple entry points.
I gave this task to our principals and Math Lead teachers this week and as they worked through it, I was struck by the energy in the room. First two pieces were put together to make rectangles

then three:

and then the question was asked, "Is it possible to use all six pieces?" I put on my best enigmatic smile and said, "Perhaps!" With no immediate solution obvious, these adults had to persist at trying different arrangements.  Snippets of conversations from each table showed some common ground to the thinking going on:
"Are we allowed to flip the shapes?"
"How big must the rectangle be?"
"Either 6 by 4 or 3 by 8."
Then, around the room, shouts of delight went up and high fives abounded as different solutions were found.

When I made this puzzle, I did so knowing that this solution could yield a second solution.

As we moved from table to table airplaying the solutions, I was pleasantly surprised to see some I had not thought about. These two are a nice variation of each other:

As are these two:

But this one really toasted my crumpet:

Which of course begs the question: Can we find all the solutions? I'll leave that up to the reader to figure out.
A question arose during the session as to how to make this task more accessible to students who might have a visual-processing LD. One possible accommodation might be to colour-code the pieces and provide a template for them like this:

What I like about this problem is that not only is it great for developing spatial reasoning, it is one that can be attempted by adults and children alike: it is a problem that would be great for families to do together! And if you like problems such as this, then you will also love games such as Blokus and Kanoodle.
Or for those of you of a certain age, Tetris!

More Than One Way to Crack an Egg

Last week I posted this photo onto my Twitter account and it received a lot of interest:

I often am asked to help out with the initial Math assessment for students who have come from different countries into our board. These were two solutions from two different students, both from Vietnam. The first thing that struck me was that these methods were completely new to me:

Many folk agreed. Others still wondered why these methods were chosen. I agreed with Matt Dunbar that my preferred method would be this: 

My mantra has always been: 'First, isolate the variable'. Yet this is not what these students did. Other folk wondered, in the case of the student on the right, what the student was thinking:

So this week, I happened to see this student again and asked him if we would be happy to do some more questions for me. Fortunately, he was happy to help! Here's the first question I gave:

OK, so this tells me that his very first solution wasn't a one off: this 'first get-a-common-denominator' approach is a go-to strategy for him. I next gave him this question:

So now I see he is also paying attention to the numerator of the fraction in the first line. But still, he uses the 'first get-a-common-denominator' approach. I was now curious as to what he would do with a simpler equation:

So in this case, he does isolate the variable first: verrrry interrrresting! But now I want to see what he does when there is more than one denominator:

At this point, I am grinning like a Cheshire cat. What a neat way of solving this! When I asked him if everyone learns this method in his Vietnamese school, he replied that they did. So I showed him how I would have solved these (isolate first, then get rid of the denominators) and it was nice to see him smile and nod and say "Cool!"

I love it when I see something new like this. It reminds me that what we take for 'standard' here might not be standard everywhere. That doesn't mean to say that I will change the way I do these types of questions myself (I still like 'my' way!) but knowing that there is more than one way to crack an egg will help me as a teacher help students who might not get 'my' way.