G-GPE 5, under “Use coordinates to prove simple geometric theorems algebraically”, says that students should “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).”
I read and re-read this learning objective several times before it occurred to me that we weren’t just exploring the slopes of parallel and perpendicular lines to make conclusions about their relationships – we were actually asked to prove the slope criteria for parallel and perpendicular lines. And then I had to think even longer to realize that I don’t remember ever proving that parallel lines have equal slopes. It just seems like something that I have always known.
I recently came across learning objective 8.EE.6 while working on a project.
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.
And so luckily it didn’t take me long to make the connection between the progression of how students should think about slope triangles from grade 8 through high school geometry.
We started with the Math Nspired Algebra Slope as Rate activity.
Points A and B are dynamic on this page. In the midst of talking about what we mean by slope, I happened to move the points so that the slope of the line containing A and B was 1.
One of the students noted to the whole class that the slope triangle was a 45°-45°-90° triangle. So we talked about the slope of the hypotenuse of a 45°-45°-90° triangle with horizontal and vertical legs.
It made me think to ask about the slope of the hypotenuse of a 30°-60°-90° triangle with horizontal and vertical legs. So we paused for a moment, and I set up a Quick Poll to ask what is the slope.
You can tell from the poll that I had the diagram drawn elsewhere. We had decided to put the 30° angle for the angle formed by the hypotenuse and horizontal leg. You can also tell from the poll, that no one correctly got the slope. One of the Standards for Mathematical Practice is to reason abstractly and quantitatively. They weren’t quite ready to reason abstractly.
So without giving them the answer, I added one piece of information. I honestly don’t remember now which piece of information it was, but I think it was a possible length of the hypotenuse.
Now that students had a quantity, about half were successful at giving the slope of the hypotenuse.
By this point, I was in a hurry. We had strayed off topic. I know that I should have spent more time on the slope problem to get everyone with us. Instead, I let a student explain how she got her answer and we spent just a little time looking at the responses of the Quick Poll in Navigator to correct some of the incorrect thinking. (I look forward to seeing what students do on their summative assessment when I change the angle formed by the hypotenuse and horizontal leg to 60°.)
And then we moved on. I was trying to get my students to the point of proving that the slopes of parallel lines are equal, and we hadn’t even made it to talking about the relationship between two slope triangles on the same line. So we looked at this page that I had added (with the help of Jeff McCalla). What relationship do two slope triangles on the same line have to each other? Students immediately recognized that the triangles would be similar. How do you know?
They first asked to move point C to coincide with point A. They had seen these triangles before – and so it didn’t take long to show that the triangles were similar by AA~. And then because I am really trying to change our first instinct for proving triangles are similar from AA~, SSS~, and SAS~ to thinking about whether one triangle is a dilation (or some series of rigid motions + a dilation) of the other, we talked about that as well. Then we generalized to what happens when A and C are not the same point.
By this time we were ready to move to parallel lines (or at least I was).
What Standards of Mathematical Practice did students have the opportunity to enter into with this problem?
- Look for and make use of structure (Would they think to add auxiliary lines – slope triangles for each of the parallel lines? Where would they place the slope triangles? Did it matter?)
- Reason abstractly and quantitatively (Were they ready for abstract? Or did we need to start with some numbers?)
But I failed. We raced through talking about what a proof would look like, but we didn’t really prove it. And the next day, another lesson was calling our name (or at least my name).
This is hard work. And even though I try every day to “ease the hurry syndrome”, I’m not there yet. I keep hoping for next year when I will have the opportunity to try this again with a new group of students. It will be easier then, right?
At least the journey continues ….