# 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]