Saw this tweet from Dan Anderson today:
I didn’t have the best math talks with the boys this morning and was sort of bummed out about it all day. Because the problem Dan posted has a really surprising and fun math connection in the solution I thought that going through it would help us end the day on a better note than we how we started it.
Slightly unfortunately we were a little pressed for time, but it was still a nice little project.
We started by just talking about Dan’s problem and about Pascal’s triangle in general by taking an initial look to see if we can find any pattern in the number of odd numbers in each row. The boys noticed a few interesting patterns.
We started this next section by talking in a little more detail about some of the patterns. My younger son noticed that there were a couple of patterns relating to the rows whose number was a power of 2.
The boys used the ideas they found here to take a guess at the number of odd numbers in rows 2048 and 2047, but weren’t sure how to get back to row 2015, yet.
This next section is where I’m a little bummed that we were pressed for time. I wanted them to brain storm about other things that were related to powers of 2, but they got a little stuck.
I used the idea from the picture in Dan’s tweet to write Pascal’s triangle mod 2. Writing out the triangle that way made my older son think about binary. They also were surprised to see something that looked like Sierpinski’s triangle emerging on the board!
From there we wrote out the row numbers in binary and looked for a connection to the number of odd numbers in each row. The surprise in this problem is that there is indeed a connection!
With the clock ticking down to when they had to get out the door, we went to Wolfram Alpha to see what 2015 was in binary. From there they boys guessed that that the number of odd numbers in row 2015 of Pascal’s triangle would be 1024. Wolfram Alpha confirmed this conjecture.
So, though I wish I would have had more time, this was still a fun little project. I’m always trying to help the boys see fun connections in math. Here I really had to show them the connection rather than letting them discover it, but that’s going to happen every now and then. Happy to end the day with this fun project.
4 thoughts on “Talking through Dan Anderson’s mod 2 Pascal’s Triangle”
I don’t quite understand your comment about being pressed for time. You homeschool, right? Doesn’t that mean you can leave open investigations like this, to be discussed again another day?
We really take advantage of the fact that we aren’t stuck on a fixed curriculum schedule nor face a standardized testing regime.
There is the the danger of being too unstructured, but we tend to split into explorations which are pretty open-ended and skill practice which has a clear scope (time, content).
Part of the spirit is the idea (not my original!): don’t do anything for a child that they can do for themselves, don’t say anything the children can say themselves.
I didn’t have a great morning with either kids and didn’t want to let the day end on that low note.
I got home from work at 5:45 and they needed to be out the door for their karate class at 6:15. Despite the time constraint I wanted to try to squeeze in this fun little project.