Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.
In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.
Here’s a typical exchange – working through problems 11, 12:
- Me: “Look at the table, what’s changing?”
- Student: “the cost”
- Me: “How much is it changing?”
- Student: “20”
- Me: “20 what?”
- Student: “20 cost”
- Me: “What are you counting when you talk about cost?”
- Student: “money…dollars”
- Me: “So the price is going up 20 what?”
- Student: “Dollars”
- Me: Show me this on the yellow” (student knows from before  to write +$20)
- Me: “What else is changing?”
- Student: “People”
- Me: “By how much”
- Student: “1 people…person”
- Me: “write that on the green”
- Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
- Me: “What do you do next?” (they typically know to move over and up)
- Me: “Use green to highlight the over” (they highlight)
- Me: “How much did you go over?”
- Student: “1…1 person”
- Me: “Now what?” (Student goes up.)
- Me: “Highlight that in yellow.” (They highlight.)
- Me: “How much did it go up?”
- Student: “2…20…20 dollars”
- Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
- Me: “So what is the rate of change?”
- Student: “$20 and 1 person”
- Me: “Look at the problem at the top. What is the 20?”
- Student: “$20 per person.”
I point out that you can find this rate or slope in the equation, the table and in the graph.