# Meaning of Slope

Many students understand graphing a line as ploting points and then connecting the dots. Begining at the y intercept, the go "over one and up m" to graph a line with slope m. This studio is designed to introduce a new meaning of constant rate of change more compatible with calculus: that for any change in x, the change in y is m times as big as the change in x. This new meaning of slope improves students' ability to imagine taking a limit as the change in x decreases to zero.

Part 1: Over 1 and up m
Students investigate "over one and up m" in a continuous setting, to demonstrate that this meaning of slope does not define a line.
[Touchable Demo]

Part 2: Slope
Students investigate change in x in a continuous setting, showing that for every change in x, the change in y is m times as big as the change in x.
[Touchable Demo]

Part 3: Testing the new meaning of Slope
If the new meaning of slope successfully defines a line, then for any non-linear function, there should be a change in x, for which the change in y is not always m times as big. In this activity, students are tasked with finding values that "break" non-linear functions, showing the new meaning of slope is robust.
[Touchable Demo]

Part 4: Using the new meaning of Slope
Two files showing how the new meaning of slope can be used with a static point (h,k) and a moving point (x,y) to develop an equation for point slope form (either y=mx-hm+k or (y-k)=m(x-h))
[Touchable Demo]

[Touchable Demo]