What learning math sometimes looks like: counting arrangements

A problem from an old MOEMS test gave my son a lot of trouble this morning. It was one of those times where you start looking at a problem one way and it is so hard to move away from that viewpoint.

It did make for a nice project this morning, though. Here’s the problem:

Sara places four books on a shelf. The blue book must be somewhere to the left of the green book. The red book must be somewhere to the left of the yellow book. In how many different orders can Sara place the books?

We started talking about the problem using snap cubes – one idea he has here is that the arrangements will come in pairs. You have to be careful with that idea, though, because you might already accidentally have a pair.

He also tries some counting ideas, but we also need to be a little careful to make sure the arrangements we are counting satisfy the conditions of the problem.


Next we try to find a systematic way of counting the arrangements. The first thing we try is to find all of the arrangements with the yellow book in the first slot.

After a long conversation about that case, he had an easier time understanding the cases where the green cube is in the first slot.


To wrap up the conversation this morning I tried having him look at other ways to divide the 6 cases into two groups of 3. He noticed that we could look at the blue and red cubes instead.


So, a tough problem for my son. Hopefully this conversation helped him see a few ways that looking at patterns help you count. His original focus was on finding the number of different possibilities for each slot, but that’s a tough way to approach this problem. An alternate approach that we didn’t cover today involves picking two slots out of the four for the green and blue books – I’ll leave that approach until the next time we talk about this problem.

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