I've been experimenting with something and I think it might be working! So, here it is. I am going to describe briefly how I approach linear reasoning with my 7th-graders and then open it up for feedback.
1. I draw a visual pattern with dots.
2. I guide the kids to look for how many dots are added at each stage.
3. The kids visually group dots into n groups of k, where n matches the stage # and k matches the stepwise increase. They'll notice that at each stage, there is a certain number of "leftover" dots.
4. Kids start to generate expressions like 1(3) + 1 to represent "1 group of 3, plus 1 extra dot." Then 2(3) + 1 for the next stage, then 3(3) + 1 for the next stage...
5. Eventually, kids can figure out that for stage 1000, the number of dots involved would be 1000(3) + 1. (This inductive thinking, it turns out, takes a couple of days to develop and for them to feel comfortable with...)
6. You then coax them into figuring out that for any stage n, the formula is n(3) + 1. It helps to keep pointing in turn at the formula and back at the concrete numerical examples 1(3) + 1, 2(3) + 1, etc. to emphasize that n is just a placeholder.
7. When kids start to gain confidence in their formula, ask them to start making predictions for small stages n=5 or n=6 and then verify their results by actually continuing the visual patterns.
8. After doing this a bunch of times, they then try to do this with numbers instead of visual dots.
The goal is that they should be able to look at a rough "table" that looks like this (with no actual dots drawn):
Stage 1: 16 dots
Stage 2: 20 dots
Stage 3: 24 dots
...and figure out that the algebraic expression for predicting the size at stage n is n(4) + 12, and then also understand that that formula matches 1(4) + 12, 2(4) + 12, 3(4) + 12, etc... One advantage I have already noticed about this method of teaching linear formulas is that it continuously reinforces the idea that the formula is merely an abstraction (or generalization) of an already-existing pattern.
9. Then, when they're comfortable with the entire process, repeat with negative slopes. Instead of circling groups of added dots, they start circling spaces of taken away dots.
A kid should be able to look at this (after a bit of guidance) and write the equation y = 16 - x(2) for dots at stage x.
What do you think? Do you think this is another rote method of linear patterns, or do you think this could actually increase student understanding of linear equations? Where do you think I should go from here, to transition their understanding into a formal knowledge of linear patterns within the coordinate plane?
PS. You are welcome to download all my lessons thus far (and planned ahead a bit, even!) here. Teaching 5 grades at once is super fun!!