We've been building up to this for a week. My students "discovered" how to find the slope of a line given two points.
Back up to early last week. We used Fawn's steepness worksheet to start exploring slope via a discussion of steepness of staircases. Initially, a lot of my students wanted to count the number of steps. This created some bizarre conclusions like "F is steeper than E." Given the opportunity to measure, the diagonal seemed to be the length of choice. With a little prodding, a few at least started measuring the angle of elevation. I brought the class together and asked how reasonable it seemed to say that F was the steepest (since it had the longest diagonal). Most kids agreed that this didn't make sense and I asked what else they could measure. The steps and risers were suggested, and they were off again. I had to troubleshoot measuring at this point (no, there are no fifths on any customary ruler I've ever seen), and we tied this back into 7th grade ratios. I asked students how we can compare two numbers and they remembered ratios. We set up the ratios for slope and reduced each one until we could compare them. Realizing how many different denominators we had, someone suggested dividing and comparing the decimals. We did so and came up with our final ranking.
In all of that discussion, the gem that emerged was a student's comment, "Couldn't we graph these?" Bingo! That was my in. The following day, we evaluated some functions and plotted the points. We connected the lines. I asked, "Can you tell which one is most steep? Least steep?" They chose their answers. I wondered aloud, "Wouldn't it be nice if we could compare these numerically?" They agreed. "Let's draw some steps on this line." The kids instinctively knew how to draw the slope triangles (something that's not always been super easy for my students in the past) and we were off and running in no time. We did a few examples of finding slope on graphs, they did a few themselves, and they were telling me how easy it was.
I honestly forget how I made this next segue, so forgive the gap. I showed students a graph where the lattice points weren't marked. When I asked about finding its slope, they were a bit lost. I asked them to tell me everything they knew about the graph. They told me it was "a function, a diagonal line, in the first and fourth quadrants, and negative." Out of ideas, they stopped. I drew a line that met their criteria but wasn't the same as the given line. "Does my line work?" They told me it didn't and I prodded them to give more criteria. A student told me an ordered pair that fell on the line. I drew a line intersecting through that point. Another student suggested a second ordered pair and I redrew my line on top of the given line.
Then we considered our criteria. We decided that it was important that we indicate the graph was a line and that it crossed through the two points we'd listed. The other criteria were redundant. I challenged them to find a way to calculate the slope of the line without counting. "How could you find out how long this part of the step is?" They decided to subtract the coordinates. "What should we do then?" Divide and simplify.
"What if I gave you points that didn't fit so nicely on the graph paper? You wouldn't want to count all these squares, I'm sure." I tossed an ordered pair on the board, something like (40, 125) and (100, 65). They told me to subtract the y's, subtract the x's, then write them as a fraction and reduce. Enter Mr. Sweeney's song. The best part of this song is that Mr. Sweeney's doppelganger is my school's math specialist and the kids can't help but exclaim, "That's Mr. A!" as soon as the video starts.
Anyway, it was an exciting morning in my classroom and I thought I'd share.