My son had a doctor's appointment, so my morning classes didn't meet today. Maybe I came into my afternoon class with more energy, I don't know, but it felt really great. I gave a short mastery test on solving equations. Five questions; last one was a story problem. I think I'll break it up into two grades - one for solving equations, and one for the story problem. That way lots of them can know they've passed the equations portion. (Because I don't think many will have the story problem right.) After the test, I started our new unit on graphing, and it seemed to go really well. (Three of the students with difficult behavior had left after the test; that might have something to do with it.)
I started out with the function game. We had played it before, so this wasn't new. I chose a harder relationship than the ones I'd done before, and not many people were seeing it. Here's how it goes: I have one student up front as the scribe. I ask students for a number, and then I say "Casey said 4; I say 11." The scribe writes it down in a two-column table, and I call on someone else. Each time we play I've told them: "The most important rule of the game is - don't say the rule! If you think you know what's going on, say the number I'm going to say." After a few numbers are up, some of the students will call out the number I'm supposed to say. This time only a few people were getting it.
I decided to use this function game as my entree into graphing. I had meant for it to be one example among many, but when I saw that people weren't seeing the rule in my head, I thought maybe the graphing would help some people see it. And that would help them see the power of graphing! So I told them I wanted them to plot all the numbers we had on the board so far. Their first point would have an x-coordinate of 4, and a y-coordinate of 11. I walked around as they were doing that, and helped a few people with the typical difficulties (0 to 1 is often about twice as big as 1 to 2 and all the rest, or they put tick marks between the blue lines instead of on them so it's hard to be accurate, ...). After that I got to mention axes, quadrants, and all that. We got to see the linear relationship, and we got to talk about the rule (multiply by 3 and take away one) and what it would look like as an equation, y (or output) = 3 * x (or input) - 1. This was so much more fun than doing section 3.1 in the book, where all they do is plot points, identify coordinates, and plot data for (number of years since 1970, number of Walmart stores)!
We did another function game and its graph, and then we did a worksheet from Maria Andersen's Algebra Activities workbook (from the free teasers pack pdf).
Students kept asking me how to determine the equation of a line from points. (They didn't ask it that clearly.) And I kept asking them to wait until we had built up a bit more background. I answered that question just before class ended, by doing the function game a third time, and stopping after we had just two number pairs.
I think this is the best intro to graphing I have ever done. I hope it goes as well in my morning classes!