On the end of year assessment we did a few weeks ago, I noticed that many students were not sure how to approach one of the problems. It involved finding all the different ways/orders that three children could sit on a sled. I also noticed that some students got a few different ways and then really didn't know how to go over their work to know whether or not they had found all the possibilities.

I thought we would try another problem together and thought of the classic book, A Three Hat Day. I've used this book in many classes over the years and think it's a great one to engage students in a problem that really leads to thinking through a problem-solving process. I asked the students to consider how many ways R.R. Pottle the Third could wear the his three hats (in a different order). I also discussed the importance of "proof" in mathematics and asked them to consider how they could prove that they had found all the possible ways to wear the hats.

This student loves to draw so I was not surprised when she drew each set of hats. For her, it actually didn't take too long but I do remind the students not to spend too much time on the "art" aspect of representing how they solved a problem.

This student started to draw all the sets of hats but quickly realized that this was going to take too long and switched over to coloured circles to represent the hats instead.

This student used a chart and number coding to systematically create all the different possibilities.

This student used letter coding (and provided a key!) to create the six ways that R.R. Pottle could wear his hats.

This student also used letter coding and a key to solve this problem.

The students were also asked to create a similar problem. Most of them, like this student, modeled their problem on the three hat problem.

As students finished solving and recording how they solved the problem, I asked small groups of students who had solved the problem in different ways to get together and share how they solved the problem with each other.

Although the days are slowly slipping away, I might try and fit in one more problem like this before the end of the year and specifically ask students to solve it in at least one different way than they did for this one.

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