The students use the graphing calculator program to solve the problem of finding the equation of a line with a rate of change of -2 passing through the point  (-3,4). In this case x increases by 3 so y changes by “negative 2 times as much.”  Ms. Coombs then reminds students where the function is changing from.  In essence, the students know the -2x, they are working covariationally to find b.

Prior to this clip, students were given the point (5, 2) with the rate of change -7 and asked to find the corresponding linear function using the graphing calculator program.  Here, we see Ms. Coombs working with a pair of students to determine the initial value, b, by coordinating the change in x with a change in y. In this process students struggle with the meaning behind what has become a mantra of “y changes by ___ times as much.”