Introduction: "More Than One Nice Thing"

Students stand in a triangle formation to play ``More Than One Nice Thing'': Student at apex tells one nice thing to the person to the left behind him and to the person to the right behind him. Each person passes the same number of nice things he got down to each of his the neighbors behind him.

Be thinking about patterns you can conjecture as we explore what just happened. Questions:

1.

Who got the most Nice Things? the fewest?

2.

Did the person next to you get more or fewer Nice Things?

3.

Are you on the left or right side?

Write down the number of nice things everyone got on the board. Let's consider other ways to get the same results. Each person got the sum of the number of Nice Things as the two people in front of her. Try ``tracing'' the Nice Things. Each nice thing you got was because someone before you got a Nice thing from someone else. Trace the paths to the source of your nice things. Conclude that the number of Nice things you got was equal to the number of possible paths from the top of the triangle to you.

Experiment with developing a formula for the number of Nice Things a person in the mth spot in nth row would get. Remember that it's the same as the number of paths through the triangle to that spot. Is there another was to describe these numbers? Yes! Try to think of one. (Give time to talk.)

More viewpoints. Try skewing the triangle from equilateral to a right triangle. So, to get to the (m,n) spot in the triangle, you have to go down (m-1) times and to the right (n-1) times. so, you have to choose (n-1) right-shifts from (m-1) available spots. (Question: is their a better way to number the rows and columns? Yes: start from 0).

``Choose''. So, the number of Nice Things equals the number of paths equal the the number of ways to choose a subset. (Write a set on the board; label the triangle with the members of the set, show how the path describes the subset.)

Work through examples and questions. Tomorrow: Plan for analyzing the Triangle Homework: Make your own copy of the triangle, as far as you want. Whoever goes the farthest will get a prize. Make another triangle with a different rule from Pascal (Instead of adding the two above, dream up something else.)