I wanted to be able to transfer that understanding to functions in table form. It seems to be a similar kind of thinking, but every year I get a significant fraction of the class who would come to this conclusion whenever they work on their own:
I introduced the word "term" and we began with simple questions like:
What is the term in the sequence above the arrow? How did you find it?
Is there a shorter way that you could find it?
What would be the 20th term in the sequence? How did you find it?
I was amazed at how quickly the students picked up on the idea, and were able to do their own reasoning and explaining about finding missing terms. I wanted them to be able to reason backwards as well, and to think about gaps in the sequence:
Here is some student thinking as we were beginning to write functions for the sequences:
Below are some of the worksheets I used this year, due for some reworking this summer.
Questions I still have:
1) Does it matter if the indices (0,1,2,3,4,5,..) are above the sequence or below it? Does one or the other help the transition to vertical tables?
2) How can I connect the students' understanding here to the continuity of a function, and the line on a graph?
3) How can I better relate this way of looking at a table (as a sequence with indices) to looking at a table as a set of points?