So You’re Saying we Should Teach Congruence and Similarity before Teaching Slope?
From the Field — NCTM 2015
This was a question was posed at the end of my session, “Learning Slope via Rate not Rote,” at this year’s NCTM. I wasn’t sure if I should jump for joy or shed a tear. It was obvious from the question that my presentation made sense to the person asking. “YES! Absolutely teach these geometry concepts before introducing the concept of slope to students” was my response. The person that asked was just making sure she heard the message correctly, and I appreciated her for that.
What made me want to shed a tear was the fact that the question needed to be asked at all. The CCSS-M for Grade 8 requires students to use similar triangles to explain why the slope between any two distinct points on a non-vertical line in the coordinate plane is m. Further, students are expected to derive the equations y = mx and mx + b. Traditionally, students memorized the slope formula and the equations y = mx and y = mx + b. This is the rote learning I am referring to in the title of my session. Many teachers, myself included, taught this way and were flummoxed by assessment results. What do you mean my students didn’t score well on slope problems? They are so easy! All students need to do is plug in some coordinates and do the math, or simply draw the slope triangle and count. Test results made it obvious that rote learning is not the best way to go.
Learning slope via rate requires students to know about ratios, equivalent ratios, constant of proportionality, and most importantly properties of similar triangles. Specifically that similar triangles have corresponding sides that are equal with respect to the values of their ratios. That is, if two pairs of corresponding sides of similar triangles are R’ Q’: RQ and P’ Q’ : PQ, then we would expect:
Further, if the given sides represent the similar right triangles drawn from a line on the coordinate plane, the value of the ratio is the slope of the line! Once this is understood completely, deriving the equation y = mx is simple, and with an understanding of basic rigid motions, y = mx + b is just a translation of y = mx, b, units along the y-axis.
To the woman that asked the question, ‘So you’re saying we should teach congruence and similarity before teaching slope?’ I say thank you. You articulated the objective of the session with a single question. As teachers we need to recognize the need for change, and make sense of how we are supposed to go about making the change. Teaching concepts of congruence and similarity before teaching slope will greatly increase students’ understanding of a topic that has mystified them for far too long.