6th-grade students. 12-year-olds. Given this problem today: Find the rule for the pattern.

Their answer:

Amazing! Such good thinking today. This is the day we work for all year long when it comes together so beautifully, and the students reach what I initially may think they cannot, and then I get blown away by what they can do when given the opportunity.

Here’s how it came together today. My only wish is that I had taken pictures of student work. We had so much to do today, and this warm-up was quickly pushed aside as we started the lesson. I asked one student at the beginning of the second period, but he’d already tossed the scratch paper he’d worked on. But maybe it wouldn’t have shown much, the conversation was around the pattern and that discussion built on the responses other students gave.

I presented the visual and then, just like the other times we’ve done this throughout the year, asked students to draw the next stage, fill out a table, and then see if they could find a rule that would match the pattern they saw.

Draw next.

Complete the table.

Working a visual pattern today was last minute switch, so I hadn’t picked a pattern out in advance and randomly selected a page at www.visualpatterns.org and grabbed the first one to catch my eye. I started working on the problem at the same time as my students and as I solved my thought was, “This is too much for them, we won’t get to the rule today, so we’ll spend time talking about how we see the pattern growing.”

Anticipating the need for students to speak about the part of the task they would be successful at and build from there, I had them turn and talk to a neighbor about how they saw the pattern growing. “Leave the numbers for now and just discuss how you see the pattern growing.” After a minute or two, the discussion died down, and I asked them to compare their tables. The conversation buzz was short, and so we started discussing.

Students saw the pattern growing in a doubling pattern. “First sorta downward, then to the right, then down.” Mass agreement. Positioning a ruler across the lines of reflection, I asked the students to give an angle measure, “45º”, quickly followed by 90º and 180º. This also fit their doubling observations. We talked about what the next image would look like. Most of the class reflected it over a vertical line, while a handful reflected it over a horizontal line. No one offered up a third alternative.

As we start to fill out the chart, as student pipes up with answers in prime factor form as he saw the pattern that way, so we fill in the chart thusly:

I ask about the pattern and if anyone has an idea about the rule. The doubling pattern is noted a few times. I mention that the rule is something more than sixth graders are expected to work with, but they do know a notation for multiplying the same number over and over.

And the conversation tumbled on, quicker than I could have imagined.

Student A: “What if we used it like an exponent, but it’s always one less than the number, so n-1.”

In my head: Oh, my stars, **they used a variable expression without me prompting that.** Last visual pattern they would not use the variable in the expression even with me steering the conversation there constantly, instead, they wanted to define each new step recursively and felt just fine about working out 42 steps to find step 43.

Me: “How would that look?”

Student B: “It would be like 3 times exponent of n-1.”

Me: Draws on board and then this as student clarifies her meaning to be 3 as the base with an exponent of n-1.

Happy murmuring agreement in the class.

Me: “Great, let’s check it.”

Didn’t work.

Me: Not wanting to discourage her great thinking even though it didn’t result in the right answer at first, “Okay, so that didn’t work, but you’re actually really close. Let’s think.” And I paused. I’m so proud of myself now that I paused! Becuase a half a breath later, another student spoke.

Student C: “What if we did 3 times 2 to the n-1?”

Me: Write it on the board, “Yes, that’s exactly what it is!” Now, I wish I had just calmly said, let’s check it, but I didn’t.

Another student gave us their answer for the 10th step, her method was to continuously double, and there was a general agreement for this method. We used a calculator to find the 43 step, and I told them the answer in scientific notation, which they aren’t used to, and they insisted on seeing the number written out on the board. It’s impressive. A short discussion about bacteria, exponential growth and the actual numbers of bacteria that we all have on us all the time ensued and I was reminded of a book I have that I should pull out for the warm-up tomorrow. I can’t remember the title, but it has students figure out things like, how many eyelash mites can fit on an average eyelash?