What do geometry students need to know so that they can successfully master the following standard?
CCSS-M G-GPE.B.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
We recently started a unit on coordinate geometry. On the bell work, we put a question or two about calculating slope. Then we played with our version of the Math Nspired document Slope as Rate.
What do you notice?
What is true about the slope of a line that is increasing or decreasing, horizontal or vertical?
Then a Quick Poll. Can you relate slope to what we have been learning in geometry?
Would it help to reason quantitatively instead of abstractly?
Not much. 56% of the students now have it, compared to 50% when there were no given side lengths.
In Transformative Assessment in Action, James Popham says that one of the key “choice-points” in using formative assessment is to determine what performance level of students will cause you to make an “instructional adjustment” (pages 56, 74, 93). He suggests that making a decision ahead of time about the cut-off is more helpful than making that decision on the fly. I didn’t really think about this ahead of time – I’m learning to implement that practice – but because I consider these “challenge” questions that connect slope a bit differently to what we have been learning, I decided not to stop too long on these questions.
What happens when we put two slope triangles on the same line?
They are similar.
How do you know?
One student had her triangles arranged with one inside the other so that it was obvious that the corresponding horizontal and vertical sides were parallel, creating corresponding congruent angles when using the line as our transversal.
Another student arranged her triangles so that a dilation about the shared vertex of the two triangles was more obvious.
CCSS-M 8.EE.B.5 6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.
So let’s think about two parallel lines. What do we know about parallel lines?
They don’t intersect.
Their slopes are the same.
How do you know their slopes are the same?
Our teacher told us.
So guess what we get to do in this class?
Let’s start with two parallel lines in the coordinate plane. I’m going to give you a few minutes to think by yourself about how you might prove that the two lines have equal slope. I’ll watch why you think.
Many students used the math practice look for and make use of structure, adding auxiliary lines in their diagram.
Which diagram would you choose to share first? Quite a few students had created parallelograms out of their lines. But we didn’t have much time, so we didn’t talk about the parallelogram as a whole class.
We started with the quantitative reasoning. LB had put points on his lines and calculated the slope of each to show that they were equal. How can we extend that to work for all parallel lines?
I asked HJ to draw her picture for us.
What do we know?
Students asked for some labels so that it was easier to talk about the diagram.
It didn’t take long to prove that the triangles were similar.
How can we show that the slopes are equal?
CCSS-M G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Since we have shown the triangles are similar, then all corresponding pairs of sides are proportional. Now we have actually proven that the slopes of parallel lines are equal.
So what about the slopes of perpendicular lines? They are negative reciprocals. They are opposite reciprocals.
Okay. In some textbooks you would see that the product of the slopes of perpendicular lines is –1. Let me give you a minute to think that through.
Now we need to prove it. What do you see?
Let’s extend the lines. What do you see now?
It could be helpful to have one of the lines intersect the origin. Then all that is needed is a rigid motion to prove that it always works.
What do you see now?
You would think that I know by now to pay attention to my students. I was expecting to hear that they saw the right triangle that is highlighted in pink. After all, that is what I saw when I proved the slope criteria for perpendicular lines myself in preparation for the lesson (using the geometric mean relationships that occur because of the altitude drawn to the hypotenuse).
So I was surprised to hear a student say that she saw the blue triangle, which of course is a great example of making use of structure.
So that is where our lesson stopped. We had already spent two days on our one-day lesson on Lines. I had to spend at least a few minutes of the remaining class time on Triangles in the Coordinate Plane.
I offered Problem Solving Points to anyone who wanted to finish the proof outside of class.
One student’s proof is below.
The good news is that we made it farther in the lesson this year than last. We talked through a proof of the slope criteria for parallel lines last year but students didn’t have the opportunity to dig as deeply into as they did this year. At this rate, maybe next year we will get to do both.
And so the journey continues … being reminded each day to listen to what my students actually say instead of what I expect them to say.