Some photos of the “Duplicator Lab” activity at the end:

Level Onestudents build the solution. It’s important to realize that they see their work as challenging. Keeping track of the recurring duplication is a big task, and they feel pleased when they arrive at duplication event 10. They seem to recognize that they’re working at full horsepower!

Level Two students begin writing a table of the solution, as soon as they can visualize how the duplication events occur.

Level Three students find the solution non-visually. Because they see patterns quickly, they proceed directly to the abstract recording of the pattern they saw: each term is the sum of the term directly preceding it and the term two steps before that. Because they self-identify as speedy in math , they get pleasure from carrying the calculation into 5- and 6-digit sums… We neither encourage nor discourage this algorithmic flight of fancy. We accept their need to dazzle!