Algebra One Explicit Rules for Arithmetic Sequences

About 4 years ago we started to adopt the common core curriculum. At the high school level we started with Algebra One. It was a pretty crazy year but we got through it. One of the crazy learning expectations was that students would be able to create an explicit rule for both Arithmetic and Geometric sequences. It made me laugh out loud to see how we were supposed to do this with 8th & 9th graders. All the materials I read made it sound like students would intuitively develop the formulas just from examining sequences and recognizing regular behavior. hahahhahha. right. The first year I even tried to develop the formulas the usual way (that I've done for years in PreCalculus)

Nope, didn't stick. My students just did their best memorizing the formula.

Well finally I thought about - what do students already know? Yes in 8th grade they did learn about rate of change, slope and the slope intercept equation. And arithmetic sequences are linear. So I decided to use what they already know to develop explicit rules for arithmetic sequences

We started by graphing the sequence (took some time for students to figure out how to do this since they were only single number values and not ordered pairs).

Then we discussed how it appears to be linear. I did ask them should we connect the points to actually make a line (why isn't this continuous?). Then we talked about how we could write an equation for this line. They remembered rate of change really well and could tell me what it was from the graph. They were a bit puzzled at first on how to find the y-intercept. But they then used the slope to get to the y-axis. We did a bunch of these graphically. They were really liking it and were very good at finding the explicit rule.

But then I told them they should be able to do this without graphing - how can we do that? Students discussed this amongst themselves and came up with a method of doing this. Basically the sequence is "built" by adding the common difference. So if we do the opposite of the common difference we can get to the 0th term. Pretty cool. About half the class really got this. So those kids did really well on the homework. With some more examples for students to work through the next day I think they've all got it now.

Much better than that standard arithmetic explicit rule formula.