We had a lot of fun today playing with Pascal's Triangle, invented by Blaise Pascal, a French Mathematician in 1653. First I started out giving them the following math problem to solve.
Imagine you're buying an ice cream cone. If there are 5 flavors, how many possible combinations are there? How many ways of having no flavors? How many ways of having 1 flavor? How many ways of having 2 flavors? How many ways of having 3 flavors? How many ways of having 4 flavors? And, how many ways of having 5 flavors?
This was a good word problem for my 10-year old to solve. While he was working it out, I cut out squares of colored paper and glued them on another sheet of paper to form a pyramid or triangle. Once he finished solving the problem, I read to him some about Pascal and then I had him fill in all the outside squares on our triangle with the numeral 1. I then asked him to fill in the rest of the squares by adding together the numerals in the two squares that joined above each square. We only had seven rows, including the top 1, before we ran out of room. If I do this again, I might get a larger piece of paper to work from.
Now I told him to count down five rows, not counting the top 1, and look at the numbers that ran across this row. Did he see any similarities to the answers he had for the problem he had worked on earlier?
They, of course, matched.
We counted down five rows because the possible flavors we had to work with in the problem was five. Pascal's triangle can be used to find the answers to any of such type problems with ease.
I then asked him to add up the numbers in each row.
What pattern do you see?
He began saying the numbers out loud, "2, 4, 8...they double each time!"
I then read to him about something called a Galton board. The board is made with nails arranged like Pascal's triangle. Marbles are then poured through it. The probability of an marble ended up in a particular column is easy to work out by looking at the numbers in Pascal's triangle. The final pattern, too, is in the shape of a bell curve.
|Galton board, photo from Wikipedia|
There are so many fascinating patterns to be found in Pascal's Triangle. What patterns do you see?