The goal of this activity is to sharpen students’ focus on slope. As students place points on an imaginary line, the activity asks them to estimate first, then calculate, and then notice proportionality.
Use student ideas here to define slope as a ratio of change in y-coordinates to change in x-coordinates. By the time students get to the end of the activity, they should have a number of ways of talking about this, but it’s unlikely they’ll write a fraction with Δy in the numerator and Δx in the denominator. They’ll be ready for you to introduce this idea.