The hockey stick theorem and some fun geometry in Pascal’s triangle.

Yesterday’s project discussed this problem:

How many 3 digit numbers have strictly decreasing digits?

Not the best write up by me, sorry, but the approach that the kids took was interesting. They first tried case by case counting and found that the total was 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 = 120. This sum is a special case of the “hockey stick” theorem in combinatorics. That connection wasn’t part of yesterday’s project, so I thought it would be fun to discuss it today.

The first two parts of our discussion were about the sum they found and how that sum relates to Pascal’s triangle. The boys come up with some really interesting mathematical language in this part of the project – that was fun to hear:

 

 

For the final two pieces of our project we try to connect the hockey stick theorem with a little geometry – in this case triangles and pyramids.

 

 

So, although this project was on the difficult side for the boys, they still had lots of great ways to think through ideas in this project. The “hockey stick” is a really neat pattern in Pascal’s triangle, and I think the connection to geometry helps you see some of the easier cases. Fun way to spend our last morning math session before heading out on vacation!