61 cents per ounce is a rate of change. Graph the line modeled by this (y intercept is 0) and it becomes slope of the line. In referring to algebra we often hear, “when will I ever need this?” My response is “all the time!” Our job as teachers is to make this connection for students.
Students can hit a road block at the steps that appear to be very simple. For example, in the problem below the students are prompted to find the highest point on the graph. Many think the graph refers to the entire coordinate plane and they pick 5 as the high point. It is the highest point on the y-axis but not the graph. I introduce the problem by highlighting the actual graph in pink and explain that this highlighted line is what is meant by the graph.
The use of color also helps students distinguish between the x and y axes and what the variables x and y represent in the context of the problem (# minutes and # kilometers in this problem) – see photo above. This problem also involves plugging in a # for x (blue) IN the function (red). In the photo below you see how I use color to help emphasize this.
This is an example of color coding (highlighting) to help make a calculus problem accessible. You don’t have to know calculus to see that the yellow sections (left and right of the 0) are going up while the green section is going down. Color coding breaks a whole into parts that are easier to see and understand – works in preschool all through calculus!
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.
I had a 7th grade student with PDD-NOS who was tested at a kindergarten math level. He struggled with graphing. I taught him to use Excel on the computer to generate graphs and gave him a standing direction to use it whenever we had a graph to complete. The top photo below shows a warm-up problem (“do now”) his class was given. A classmate’s artifact is shown at the right.
My student walked in, saw the assignment was a graph problem and jumped on the computer. Below is the graph he created on his own.