I have never had the opportunity to teach slope before. This year is a first for several topics, among which was graphing functions  we did this last week, by using function tables to generate points, and I mentioned positive and negative slopes in passing. In getting more specific about slope, I knew that I didn't want to just tell the students about slope (and about yintercept)  I wanted them to figure out how the equation of a line can help them understand aspects of the graph of the line. But, I didn't know quite what to do. So, here's what I decided to do: I created a simple worksheet with four equations and their graphs and I simply asked students to find relationships between the numbers (and symbols) in the equations and the graphs of the equations. I didn't give much more direction than that. I had them each think about this, study the equations and graphs, and write their observation on their papers, without discussing with anyone, for about 5 minutes.
Then, I had them choose a partner to discuss their observations with, and to search for more ideas, for about five minutes. As they discussed, I circulated, listened, and asked questions. For the most part, they had written down how the negative/positive sign in front of the x relates to the slope, and many had identified the "added or subtracted number" as the yintercept. Some had noticed that when the coefficient is higher, the slope of the line is steeper. Next, I repaired the students using popsicle sticks, to allow them to share more ideas. At this point, I wrote on the board: "# that is added or subtracted" and "# in front of the x," and asked them to try to figure out what these numbers could tell them about the line (if they hadn't figured it out already). There weren't many students who made the connection that the slope tells how far to move horizontally and vertically between points, but there were several student whose observation was that the "m" is "how far apart" the points on the line were (they identified the points as where the line crossed the intersection of grid lines  I didn't put points on the lines for them). After the second pairing, I asked student to write their observations on the board and then we went through and discussed whether they were correct or not. Then we looked at the same lines graphed on the Smartboard, and we went through what the "m" tells us  we started with the fractional slopes and moved to the whole number slopes. In all, the entire lesson took about 35 minutes. I was really happy with the students' perseverance (for the most part) in trying to find what I wanted them to find:) I enjoyed their "aha" moments!
One "mistake" I made in the equations was that both equations with negative slopes also had negative yintercepts...this led some students to incorrect conclusions, so I changed that for next year. The fixed worksheet is here, if you'd like to use it:)
Today's thinking day is my favorite kind of day:)
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