Experience First

This experience can begin in groups.  The students will start by reasoning about the relationship between time and blocks for each person, hopefully noticing that Justin and Sarah are traveling at the same speed, Rishi isn’t moving, and Jayla is moving away from Coldstone.  They will discover parallel lines by examining the slopes and y-intercepts of Justin and Sarah.  While monitoring the students, have them be specific about the language in number 4.  They learned equations of lines yesterday, so they should be using proper language like “slope” and “y-intercept,” but also be able to put it in context.

In number 5, we highlight a common misconception about perpendicular lines—the slopes are not just opposites of each other, but opposite reciprocals.  Make sure the students notice that the sign of the slope indicates the direction they are traveling relative to Coldstone.

In numbers 6 and 7, we want the students to make the connection between Rishi not moving and having a slope of 0 (no rise).

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Formalize Later

In order to connect the informal definition of slope (rise over run) to horizontal and vertical lines, we describe horizontal lines as having “no rise” and vertical lines as having “no run.”  Be sure to show mathematically how no rise is 0 divided by a number, which is always 0, and no run is a number divided by 0, which is undefined.

The definitions of parallel and perpendicular lines should also pertain to the definition of slope.  Parallel lines never intersect, have the same slope, but different y-intercepts.  It’s important to mention different y-intercepts so the students don’t think two lines that are the exact same are considered parallel.
For perpendicular lines, you can show that opposite reciprocals are fractions that multiply to equal negative one during the debrief, or just give several examples of two slopes that are opposite reciprocals of each other so they can see that the sign changes and the fraction “flips.”