Finally came the day we gathered all the coefficients and wrote them down row by row.
At first we wrote:
row 0 = 1
row 1 = 1 1
row 2 = 1 2 1
row 3 = 1 3 3 1
row 4 = 1 4 6 4 1
And then we noticed some patterns: 1s at beginning and end, next diagonal says 1, 2, 3, 4, ; there is symmetry to each line.
And we wondered: 1, 3, 6.....is the next number 9?
Today I asked if they could help fill in the 5th row. They got:
row 0 = 1
row 1 = 1 1
row 2 = 1 2 1
row 3 = 1 3 3 1
row 4 = 1 4 6 4 1
row 5 = 1 5 5 1 but they got stumped in the middle.
I changed it up (to look more like the triangle we all know and love) and said "Well, lots of times people write it like this:"
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 5 1
and suddenly a whole bunch of students shouted, "10 and 10!!", while others yelled back, "WHAT?? How did you get that??? Why??? WHAT?????"
Finally we started to put little black boxes over every odd number. This was going to take a long time figuring out what number was in next line, so we looked at adding two odd numbers, two even numbers, and an odd and even.
"Go forth and color, my cherubs!" And they did. And it was good.
