This week, my 9th grade Prob/Stat class has been working through the Binomial Theorem. With many rich patterns and connections to explore, the unit begins with the class jigsaw, where students examine seemingly disconnected math concepts, and discover unexpected relationships. This year we have been using Wolfram|Alpha to validate our work, and discover more patterns.
Sharing the many patterns in Pascal’s Triangle is a real boost for me. This year, few students had seen the triangle before (which is a shame, as they are 9th graders!), so this was all new to them. If math is the science of patterns, then this is the center of the universe…would love to build a fun elective around it. Here are my 5 favorite Pascal’s Triangle ideas to share:
THE HOCKEY STICK THEOREM:
Start at any of the 1’s on the outside, slide your finger along the diagonal, going deeper into the triangle. Stop at any point, and instead of continuing along the diagonal, divert to the connected number in the next row. This number will be equal to the sum of the numbers along your traced path.
Here, 462 = 1 + 5 + 15 + 35 + 70 + 126 + 210
Can you verify this one?
And think about the proofs we can develop together. Move proofs from geometry and into combinatorics.
COOL DIAGONAL PATTERNS
Many students quickly identify what is happening along the diagonals:
- Along the sides, there are 1’s
- Along the next diagonal, we have consecutive numbers (the counting numbers)
But is that all? Nope…more cool stuff!
- The next diagonal reveals the TRIANGULAR NUMBERS: 1, 3, 6, 10, 15… (those numbers which make cute equilateral triangles if we make them with dots.
- Next, we have the TETRAHEDRAL NUMBERS: 1, 4, 10, 20, 35… (making cool pyramids with marbles!)
- And we can even locate the FIBONACCI NUMBERS, if we hunt enough!
Take your triangle, and color in all the multiples of 2. You’ll find some cool patterns emerge. How about multiples of 3?
This fun applet colors up to 256 rows. Try some prime numbers for wild, unexpected results. The coloring party continues when you look at coloring remainders.
COIN FLIPPING FUN
All of the entries in any row tell us the number of possible combinations in a binomial experiment. While coin flipping is not all that sexy, I have used these facts to play Plinko with my classes, which I have used in STEM talks before. Today in class, students created their own graphs of flipping results, using randint on their graphing calculators.
And…Galton Boards!!! Which I told my students I would offer 1,000,000 bonus points if they built for me.
IT RELATES TO GEOMETRY AND CHAOS
A few days before this unit, I had an animated gif of Sierpinski’s Triangle projected on my board as students walked in. Many students identified it from their Geometry class, and most were enthralled by the seemingly infinite self-similarity. So when a few days later I go rambling on about Pascal, dive into coloring patterns, and reveal the first 265 rows of colored even numbers……wait…what’s this?????
It’s our old friend Sierpinski out to play. And if you haven’t before, develop this cool visual by playing the Chaos Game on the rich Cut The Knot site.