Quadratics and rate-of-change functions


Shannon and the students are working on a unit of quadratics and rate of change functions. On the first day of the unit, the class looked at a constant rate of change function, and constructed the linear function that has that rate of change. Then the students were asked to repeat the exercise with a rate of change function that was a step function of step size 1 that approximated a line. Their result was a piecewise linear approximation to a quadratic curve. Shannon used graphing calculator to demonstrate how smaller and smaller intervals would produce a smoother looking curve.

Figure 1: A rate of change step function produces a piecewise linear 'original' function.

A Classroom moment

This is a clip from the second day of the unit. Prior to this clip, the students have been given a graph of the function x2 on grid paper. The students have been asked to fill in a table by going through the graph in intervals of step size one, and calculating the average rate of change of the graph for each interval. Average rate of change was not a new idea to the students. It had been discussed in prior units. Each row of the table consists of the end points of the interval and the rate of change for that interval.

Start Point End Point Avg. Rate of Change
(-4, 16) (-3, 9) -7
(-3, 9) (-2, 4) -5
(-2, 4) (-1, 1) -3
(-1, 1) (0,0) -1
(0,0) (1,1) 1
(1,1) (2,4) 3
(2,4) (3,9) 5
(3,9) (4,16) 7

In this clip, Shannon asks the students to graph the results on their table. Specifically, she asks them to "graph the rate of change on the y axis... with their corresponding x values. But see how this rate of change has two corresponding x values? Take the left one."

Looking at student work

As Shannon moves from student to student, she notices that some students have sketched step functions and others plotted a series of points. Few students did as she anticipated -- plot the points and then connect them. The students' graphs are actually a welcome, even if unanticipated, development. Those students who drew step functions seemed to have thought about the function's rate of change at values within the interval and used the best information they had -- that it didn't change (hence a constant function over each interval). Those students who plotted points seemed to see no reason to go beyond the information given -- meaning that no information was given about the rates within the intervals, so they couldn't sketch anything.

Discussing students' work

Shannon called the class together to discuss their reasoning for the graphs they sketched, and to discuss how to construct a more accurate graph for the original function.

A Follow-up Interview

Shannon had not anticipated that students would graph step functions in their answers. Rather, she had expected that the students would connect their plotted points to form a line. As the video above shows, several students did not connect the points: either leaving them as points, or graphing a step function.

Two days later, in this follow up interview, Sharon and Carlos ask one of the students about his choices to graph a step function. Caution: the audio is much louder for this clip