Rates of Change & Slope

As school started back up again, in algebra 1 we started our unit on equations of lines by talking about rates of change. Last year I just jumped right into slope first, and verbally mentioned some applications (if I remember correctly), but this year I thought I'd give them a context first. I started with a worksheet on rates of change and had them work through the first 2 problems with not much prompting. Then we discussed their answers. I made sure they used units. I made sure we saw ALL ways of writing the numbers. My classes (everyone's?) seem unsure that 9/4 and 2.25 and 2 1/4 can all be correct answers. They still look at me like they don't QUITE believe I'm not pulling a fast one on them.

Then they did the next two, and we discussed, and so on. I made sure to mix up the units, I have some with negative rates of change on the back. After a few, if it didn't come up, I discussed and made them write their calculations on the side as (y2-y1)/(x2-x1) using the appropriate numbers. If it didn't come up, I made sure to investigate what would happen if you used different points or a different order.

In my block class, we had just enough time to do this and then to have a discussion about rates of change and slope and how they're the same (barring units).

I'm thinking this went well because on the second day, when we discussed rise/run and started graphs to y=mx+b equations, they were pretty good and quick on calculating slope. Also, when I stepped back and asked them to identify positive and negative slopes, they had a picture of water levels increasing or decreasing and were pretty accurate.

Oh, one funny thing. I like to give them memory tools to keep things straight, and on "rise over run", I said, well what goes on top? the y's, and y's rhymes with rise. Then I told them I couldn't think of a memory tool that links run with x's, so I put it to them. Maybe these have already been done before, but I forgot, so I really liked some of the ones my kids came up with:

you run from your ex
you run from T-rex
you run cross country (x)
you run on a flat surface, and x is on the horizontal axis.