Students had to figure out how many layers of toothpicks would be in the entire structure once Dan ran out of toothpicks.
This might have been a bit too easy. One student simply drew a diagram with 250 little lines, each one representing a toothpick. I was a little worried at first that many students would do this. I wanted them to think about and use patterns. So I came up with a sequel where drawing would be the last thing you'd want to do....more on that later.
Later I decided that I was really glad that a student tried this method. Good problems are supposed to have multiple solution paths and it really falls on me to show how these different solutions are similar/different.
Other students did something similar, but first divided the number of toothpicks in the container by the number of toothpicks in a triangle. They then drew 83 triangles.
And finally, some looked for patterns: there is 1 triangle in the first layer, 3 triangles in the first two layers, 6 triangles in the first three layers, 10 triangles in the first four layers, and so on. They actually saw this pattern a week ago when they tried to figure out how many cups were in my son's cup pyramid. Sadly, no one made the connection.
Knowing that this problem would be so easy, I came up with a sequel that I knew would keep them busy. How many toothpicks would it take to make the largest triangle I could fit in the classroom?
To be continued.