I have been using patterns like the ones on Fawn Nguyen's site to help students explore functions. I've learned a lot from the geometric grouping strategies that Fawn and her students use, and I've had some great conversations with my own students this year. The students can see the pattern growing, can work with it in a way that makes sense to them, and they learn to express those relationships using algebra. They are a great way to learn about the role functions play, and where equations come from.
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 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:
 | And it tended to be a very sticky misconception. I always ended up feeling like I was just telling them what to do, and they were not seeing any reason behind it. I realized that when I look at this table, I see a sequence in the y values that continues in both directions, and I can imagine other values that are part of that sequence and how to get them. That seemed to be something that the students didn't see or know how to reason with. |
So I started to formulate some questions about sequences, asking them to make predictions about specific terms in sequences, and discussing strategies that were useful and why.
I introduced the word "term" and we began with simple questions like:
I introduced the word "term" and we began with simple questions like:
What is the next term in the sequence?
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:
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 my recording of some student thinking:
We extended the concept by numbering the terms, introducing larger gaps, and gradually moving to a table format. All along the way, I also included sequences that were decreasing as well, and ones that involved negative numbers:
In discussing how to find the larger numbers, and searching for more generalized methods, we were able to talk about how the number of the term (I was not always strict with using the word "index") was related to the value of the term. They didn't come up with the idea of a zeroth term on their own, but when I did introduce it, it made a lot of sense, and they immediately saw the value.
Here is some student thinking as we were beginning to write functions for the sequences:
Here is some student thinking as we were beginning to write functions for the sequences:
While we were exploring sequences, I also introduced in parallel the idea that a function could generate a sequence:
In the end, of course there were still students who got confused when we were working with vertical tables and graphing points, but the sequences were the go-to tool for discussions at that point, and brought a lot of "aha"s much more quickly than in previous years. I will definitely be keeping sequences as part of my math talks next year.
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?
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?
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| 02_06_w6d_7_sequences_4.docx |
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