When people begin to move from arithmetic to algebra, they start thinking about properties of operations -- specifically on "what undoes what." Function machines are used both to build algorithms and to provide experience in thinking about how to "undo," or create inverses of, those algorithms.
We will also focus on the concept of uniqueness, looking at non-numerical examples to illustrate the uniqueness of functions. A key idea in the study of functions is that inputs must give unique outputs, but outputs may not have unique inputs. If the latter is the case, these functions cannot be undone.
By the end of the session, we will see that the same function can be expressed by different rules or algorithms, something that we have seen informally in Sessions 1 and 2.
Groups: Discuss any questions about the homework. Look at the descriptions of the rules found for Problem H1, and, if it has not already been done, write the rule using symbols. Discuss the notation in x2 + 1, especially reviewing the meaning of the exponent. This session contains a brief problem on the function y = x4, and later, Session 7 contains extensive work with exponential functions, so it's good to preview some of these ideas.
Another interesting question on the homework is how we can know that 102 wouldn't appear in the "output" column of the table (as long as the "inputs" remained integers). One explanation: The input of 10 gives 101, and the input of 11 gives 122. Since the function seems to be only increasing, it has skipped 102. Since each number in the table is one more than a perfect square, another good explanation would be that 101 is not a perfect square, so 102 cannot appear in the table.
Groups: Consider sharing solutions and representations for the frog in the well problem, as well.