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.
Anyway, one more of Ed’s awesome tasks (I’m leaving out several) was this:
There’s so much here. I don’t want to give it away because I had so much fun working with this question. Explore it. It’s awesome, I promise. This is also a great example of pseudocontext — my camera cut off the top, which says “Maria makes tables with square tops. She sticks tiles to the top of each table”. Really? I think this problem is perplexing without any fake reason that we should be interested in it beyond our own curiosity. Just my opinion.
Anyway, I totally buried the lede on this one. Ed had great materials for us to work on, but he was especially good at getting us to engage with the math and with each other. These tasks are great for learning and for assessment, but their true value was clear in the way Ed got us talking about the patterns and structure we found in each task, and we moved through so many in such a short time I was excited to incorporate these into my class on Monday.