Notes 
Algebraic Strategies >>> Golden Apples
The activity talks about a prince who has picked a basketful of golden apples. On his way home, he is stopped by three different trolls, each of them demanding payment of onehalf of the apples the prince currently has, plus two more. They are told that when the prince finally gets home he only has two apples left, and are asked to calculate how many apples the prince had when he started. (Note: the correct answer to this initial problem is 44 apples.)
I told the students that they had 20 minutes to look at the problem and the followup questions and then we would get together during the last 10 minutes of class to discuss the activity. I also told them that I wanted them to try to come up with a general formula or algorithm that they could use to find the starting number of apples if given the ending number of apples. I wanted them to think of this problem as one that could be solved by coming up with a function for which if the input were 2, the output would be 44.
When we got together as a class during the last ten minutes to discuss any patterns they discovered, they all seemed to have solved this problem quite easily. Looking at their work on the papers that I collected, they came up with the correct answers to all of the three problems they were asked to do. As far as the function that they were to come up with, about threefourths of them came up the function f(x) = 8x + 28 where x represented the number of apples that were left when the prince got home.
Several other students came up with their answers by starting off with x representing the number of apples at the start, and then said that after the third troll took his apples, the remaining number of apples would be represented by the expression (x/8  7/2). They then solved the problem by setting this expression equal to the number of apples left and then solved for x.
Both of the methods worked, and I think it was good for the students to see that this problem could be solved both by doing and undoing. Finding input from output is something they should probably spend more time working on, since we work with that quite a bit when finding derivatives and then antiderivatives in calculus.
Overall, I was pleased with this activity and will probably try it again. I also plan to try it on my applied trigonometry students to see how they do. We have spent quite a bit of time lately talking about functions, so this should be a good activity for them.
