Many students struggle with writing equations for linear functions, even with only 2 parameters to fill in (slope and the y-intercept are parameters for the equation). This approach make a connection between the table and graph with the equation. The relevant, real life context helps students.
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?”
Me: “20 what?”
Student: “20 cost”
Me: “What are you counting when you talk about cost?”
Me: “So the price is going up 20 what?”
Me: Show me this on the yellow” (student knows from before to write +$20)
Me: “What else is changing?”
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.