Second Thinking Problem

September 6, 2012

Yesterday, I said:

I'll start by having groups try to describe the characteristics of each type of cube (for example, the cubes with 3 faces covered are in the corners). Depending on how well that goes, I might challenge them with with an nxnxn cube, or give them another problem to work on.

Well it didn't turn out how I had hoped (understatement 🙂 ). Students had difficulty putting into words a description of which cubes would have 0 faces covered, 1 face covered, 2, etc. Since this was required to go further, I didn't see any alternative to directly teaching this and giving them sample descriptions.

After I did that I asked them to use those descriptions to determine the number of cubes of each category for a 4x4x4 cube--yes we did this yesterday, but I was hoping they could use these descriptions rather than brute-forcing the solutions. Well that was very optimistic of me! Not only did they have lots of difficulty, but they were disengaging; I think probably because they had had enough of this problem. (I didn't even get to trying to generalize--way too optimistic!)

So I switched to another problem:

On a table there are 1001 pennies lined up in a row. I then come along and replace every second coin with a nickel. After this, I replace every third coin with a dime. Finally, I replace every fourth coin with a quarter. After all this, how much money is on the table?

As with yesterday, initially there was a lot of standing around with blank faces. But within 5 minutes everyone was at least looking at a whiteboard, and greater than 50% were actively engaged with their groups.

Here's one of the most advanced group's work:

Four things stand out from my two classes: