This week’s lesson with the girls began with some basic combinatorics. I told them that I needed their help in solving a ‘real life’ problem. This seemed to catch their attention a bit quicker than usual.
Problem: On some days I have to pick up Katie on my way home from work. There are two routes that I can take from work to Katie’s school and two routes from Katie’s school to our house. In how many ways can I get home from work when I have to pick up Katie?
Katie: There are two ways to get from my school to our house?
And then before I have a chance to answer Katie realizes what the two ways are and gets really excited by this.
The problem turned out to be quite easy for the girls. They looked at the picture briefly and both came up with the correct answer. I then moved on to a different variation on the theme. I gave them three shapes and two colors and asked them to draw all the possible combinations of the two. Once again, they both came up with the correct number of objects without too much difficulty, but couldn’t explain to me how they knew that they got all of them. Also, because they had already solved the problem, they weren’t too interested in listening to my explanation either. At some point during the process Katie tried to draw a triangle with different colored sides. I told her that this was not allowed, but in retrospect I think that was not the right thing to do. I should have seen where she would take it and how many solutions she’d get. In general, I want to encourage creative and non-standard thinking.
The last problem I gave them was a little trickier. They had to figure out how many total chess games would be played if each one of us played with every other person exactly once. Without making any lists or doing any computations, Katie immediately volunteered six as the answer. I told them that they should try to list all of the matches that would be played. Katie took a long time writing out all of the names, but eventually had the three pairs written down. I could tell that she was briefly surprised by the fact that there were not six of them, but by this point Varya was on to something else and their attentions could only be recovered by moving on to something else entirely.