Today I tried this one.
My grade 10 students quickly realized that the pattern was not linear but a number of them still wanted to represent the pattern with a linear relation. They struggled for a bit and then we talked about how we could use an area model to get something like this.
They realized quickly that the height was just the step number and that the widths were growing linearly. We found an expression for the width in terms of the step number, then multiplied by the height to find the number of helmets. This was a great way to combine linear and quadratic relations.
I had asked students to find the an equation that represents the pattern and to find the number of helmets in the 43rd step. The best part about this pattern was that before most students had even started working on it, one of my students, K., who struggles to write stuff down was madly working away on his calculator. He was clearly working on the specific case rather than the general case. After I had given some time for everyone to work I brought the class together and asked for some ideas. K. was the first to raise his hand. His solution was essentially "multiply 43 by 43 and add 43". He explained why he thought this was the solution but it was clearly over many of my students' heads. I left it out there and took other suggestions. We worked at coming up with the solution algebraically and came up with #helmets = 2n2+n, where n is the step number. As we finished I noticed the similarity between K.'s solution and this one. K.'s solution was # helmets = n2+n. I looked at the image again, knowing the algebraic solution and trying to visualize how K.'s solution fit in. As I looked I saw this.