Pascal’s triangle and some fun counting for kids

Last night I asked my younger son what he’d like to learn about in today’s Family Math and his answer was Pascal’s triangle.  Since we just started our little summer project on counting and probability, this was a timely suggestion.

We started with a simple review of how you create the triangle and also talked about some simple patterns.  This video went a little over 5 minutes because my younger son noticed an interesting pattern with prime numbers that took an extra minute to explain:

The next part of today’s talk was relating the numbers in Pascal’s triangle to ways we can pick groups of objects.  We illustrated our groups with snap cubes.  After a little introduction to ways to choose groups from sets of two and three objects, we show that the main identity in Pascal’s triangle – that two adjacent numbers in a row add together to get the numbers in the next row – can be understood in terms of selecting groups.   I’m not sure how clear the explanation was, but I hope it made sense:

The next step was to show one way that picking groups of blocks can help us understand why the rows in Pascal’s triangle always add up to be a power of 2.   This fact is a little easier to understand that the example in the last video (as long as you know binary).  In retrospect, I should have done this identity first.

Finally, we wrap up back at the whiteboard talking about why we see powers of 11 in Pascal’s triangle.  We actually did an entire Family Math about this fact a while ago:

https://mikesmathpage.wordpress.com/2014/01/12/pascals-triangle-and-powers-of-11/

I was pretty happy that my son remembered the powers of 11 in the first video, so I was really happy to be able to do go over this idea again.  We revisited why those powers show up (which involves a short discussion of polynomials) and then use the same idea to compute a few other similar computations.  (and ugh!  sorry we went off of the bottom of the screen at the end 😦  )

All in all a fun morning talking about Pascal’s triangle.  It is always fun to revisit old topics and dive a little deeper than we did previously!