Arithmetic and geometric sequences was another unit where this was my first time teaching this when I was the one introducing this to students. There were parts of this unit that I really liked, but there are definitely things I want to improve for next year. Looking back on the unit, I’m surprised at how many students struggled with arithmetic sequences after having just spent so much time on linear functions.
I started day 1 with this warm-up:
After that we did Notice/Wonder using the following image. Next time, do I keep the sequences color coded but have them random on the page rather than sorted with the line down the middle?
We then took notes on arithmetic sequences, common differences, geometric sequences, and common ratios. I used some of Sarah Hagan’s notes found here and made others similar to hers after reading this post. I also used her half sheet as practice. The link to my notes is at the bottom of this post.
The warm-up for day 2 was this Which one Doesn’t Belong?
On day 2, we talked about writing the rule for the arithmetic sequences. We made the connection to common difference and the slope of a line as well as the zero term and the y-intercept.
Sara Van Der Werf’s “Sarified” standards have become my Bible for my state standards. I LOVE how she has combined the vocab allowed along with the examples given for each benchmark. However, I’ll admit I was a little bit perplexed by the standards and examples for arithmetic sequences, so if you’re a MN teacher reading this and can clarify for me, please do!
Here’s how the standards define an arithmetic sequence:
f(x) = mx + b, where x = 0, 1, 2, 3, …
But then in this example, unless I’m completely off track here -which could absolutely be the case, -1 would have to be the 1st term based on the answers given.
So after debating and overthinking, I decided to give students some examples where the first number given was the zero term and others where the first number given was the first term. I also told them in the directions which term the first number given was. Was this the best solution? I don’t know, but it’s what I did. If you have thoughts on what I could do next time, please share!
Next up was writing rules for geometric sequences. I didn’t want to just give the formula to students and have them use it, but I couldn’t think of anything I felt was great. Last minute, I came up with the following and again had students notice and wonder things about it. Was it the best thing ever? No, but my students were able to come up with the formula for geometric sequences on their own after looking at this image.
Again, we took notes on geometric sequences and did a couple examples together as a class.
When students graphed the sequences, we also looked at the graph on Desmos and talked about how because we see a small portion of the graph, it may look linear, but when we look at more of the graph we see that it isn’t linear.
When students were working with arithmetic and geometric sequences separately, they did fine, but the wheels started falling off for some students when the sequences were mixed up. Students mixed up where the zero term went and where the common difference/common ratio went. I was frustrated that my students were frustrated because, to me, this wasn’t really that much new stuff. Arithmetic sequences were like linear functions which we had just spent weeks working with. After school that day I put on my “non-math teacher brain” and tried to look at the formulas from my students’ perspective.
Arithmetic: f(x) = mx + b (m = common difference; b = zero term)
Geometric: f(x) = ab^x (a = zero term; b = common ratio)
When I wrote the two formulas out like that, I understood why students were frustrated. Although there’s a “b” in both formulas, it doesn’t represent the same thing in both. In arithmetic sequences it’s the zero term, but in geometric sequences it’s the common ratio. I could see and understand why they felt the “rules” were changing on them. Once I understood the confusion, I was able to address this better when working with students. When students were confused, I pointed out that in both formulas the common ratio/common difference is the number “with the x“. I don’t know that’s the best way to go about helping students who struggle with that though.
If anyone else has students who struggle with this and have found ways to introduce this so that students don’t get mixed up with this or have ways of explaining this so students understand, please share!
Here is the link to the notes I used in this unit as well as a few worksheets.
Much of the format in the notes is stuff I’ve borrowed or modified from Sarah Hagen. I don’t do true interactive notebooks (INB). I upload the notes as a pdf to Google Drive for my students at the start of a unit, and they use an app called Good Reader on their iPads so they can write or type in the notes. Here is Sarah’s page of INB resources.