# NCTM – How Can “10 Minute Tasks” Change My Classroom

First really stellar talk of the week at NCTM was Ed Nolan’s How Can “10 Minute Tasks” Change My Classroom.

I was next to Jen Campbell and we had a great time exploring patterns and algebraic structure during Ed’s talk. It was full of great low-floor questions — simple presentation, easy to conjecture, but truly stretched for miles. I particularly loved this question, which we explored for awhile.

Ed did a great job getting us thinking about different ways of representing the pattern. Then, we looked at 6 different samples of student work and explored some really fascinating misconceptions along the way. My favorite way of looking at it was.
1 row, two dots = 1/2
2 rows, 6 dots = 2/6 = 1/3
3 rows, 12 dots = 3/12 = 1/4.
This pattern continues, and reveals a really fascinating aspect of the quadratic structure of this pattern.
While Jen and I were playing with the numbers, Ed threw up a slide noting that the differences increased at a constant rate, or in other words the second difference was constant:
l     Step 1      Step 2       Step 3       Step 4
l         2                6               12            20
l                +4             +6              +8
l                        +2             +2
And I had an awesome a-ha moment when I realized that that corresponded with the fact that the second derivative is constant. Jen and I checked it for a cubic:
l         1             8              27               64              125
l                +7            +19           +37             +61
l                      +12            +18           +24
l                                +6             +6
Maybe I’m behind the curve here, but that really blew my mind.