Pascal’s and Sierpinski’s triangle

We’ve really been enjoying “The Math Book” by Clifford Pickover (sorry, I don’t know Latex well enough to embed the \alpha and \beta into the book title).  We started reading it last weekend and did a project on the Prince Rupert problem:

During the week I’ve been having the kids read a section of their choice and write a little one page report.  They’ve written on the Menger sponge, the Klein bottle, the Hilbert hotel, slide rules, the 15 puzzle, and Pascal’s triangle.  The short (one page, mostly) sections in the book allow the kids to read and the write about interesting math, so these short projects have been a lot of fun.
Yesterday my youngest son wrote about Pascal’s triangle, and my older son had an interesting comment on the pictures in the book – why was there a picture of Sierpinski’s triangle in the section about Pascal’s triangle?    Good question, and one that we attempted to tackle this morning in our weekend Family Math series.

The first step was a short talk about the basics of Pascal’s triangle.  It is a nice little arithmetic review for younger kids, and there are so many fun identities hiding in the triangle that you could talk about Pascal’s triangle many times without worrying about running out of material.  In fact, just this week Alexander Bogomolny at Cut the Knot posted this neat set of identities that I don’t remember ever seeing previously:

For now, the infinite series math is a little over our heads, but there is still plenty of interesting math for kids in Pascal’s triangle:

After this short little discussion of Pascal’s triangle and how it worked, I showed them how you could simplify the triangle and get something that starts to look like Sierpinski’s triangle:

Always fun to play around with Pascal’s triangle, and if you are looking for a book that can help kids see some really fun math, get your hands on Pickover’s new math book as fast as you can!