I taught arithmetic and geometric sequences today.

I’ve been concerned lately about how much students can engage with questions or small tasks that I give during notetaking because they are so absorbed in simply copying down the notes.

So today I tried separating the notes into two parts. In the first part, I told them not to write anything down and to just think about my questions so they can focus on the new information.

I threw some sequences up on the board and asked “What comes next?”

Most of the students across all of my blocks were able to answer quickly. They could catch on if the sequence was arithmetic or geometric even though we hadn’t discussed those terms yet. (So they explained the pattern but didn’t name it.) I mixed in a few where there was no easily discernible pattern or at least a non-arithmetic/non-geometric pattern. My 1st and 3rd blocks were responding really well so we had some fun with the Fibonacci and the Look-And-Say Sequence. I wanted to emphasize with them that there may not always be a pattern to the sequence or if there is, it might not be one of the two that we were studying.

I closed the examples by giving definitions of sequence, term, arithmetic, and geometric. Then I asked the following summary questions:

- Can you always find a pattern to a sequence?
- If there is a pattern, will it always be geometric or arithmetic?
- Explain how an arithmetic sequence is like a linear function.
- Explain how a geometric sequence is like an exponential function.

They did well on the first two, but struggled with the second two. Although I did have some great answers on the last two from a student who doesn’t always volunteer answers, but is brilliant whenever she does. (She always asks great questions too.)

Then after the summary questions we moved to the writing portion and I noticed an improvement in note-taking and engagement. Because we had primed the information a little bit the students weren’t trying to both absorb and write and it seemed to help. They still had a lot of questions while working on the in-class practice, but I would call this note experiment a success. And of course I figure this out on the last day I plan to give notes. (Today was the last lesson of the last unit. We are just reviewing for the final exam from now on.)

Two things I want to do differently next time:

- Not give away the formulas for finding the nth term of a geometric or arithmetic sequence. I’ll just give the students the first few terms and ask for really “far down” terms over and over again and see if any of the students can come up with the shortcut. Several of them already did before I gave them the formulas today so I know they are capable of doing it.
- Make the connection to linear and exponential functions more explicit. My lesson on sequences was totally un-integrated and stand alone. The only connection to linear and exponential growth was made by me explicitly talking to the class.

Maybe my daughter will be born now that I have finished teaching all the lessons that I am supposed to teach. It is just test review (and maybe some fun with statistics, paper airplane, and model rocket labs) for the rest of the year.

Thanks for reading.