CCSS-M G-CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
I wrote last year about spending a class period with my students trying to explain how we could show SSS is true using rigid motions. I was impressed that my students were using the practice make sense of problems and persevere in solving them, however, all in all, the majority of students didn’t successfully come up with a series of rigid motions to show that given three pairs of congruent sides, one triangle was congruent to another.
This lesson was inspired by an Illustrative Mathematics task.
I studied in preparation for this lesson again this year, trying to determine how I could get more students to meet the standard. I am using Usiskin’s Mathematics for High School Teachers – An Advanced Perspective, which I am glad to have from my graduate school days. I only wish I had been able to take the geometry seminar…I studied the first half of the text in a seminar on algebra.
Usiskin starts by proving two segments are congruent if there is a set of rigid motions to map one onto the other. Then proves that two angles are congruent if there is a set of rigid motions to map one onto the other. And then proves that SAS and SSS work.
Reading through his work gives me more confidence in how the standards have been deliberately thought out and sequenced.
CCSS-M 8.G.A.1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
We start with lines and segments and angles. And move up to triangles and other figures. It is perfect. Except that my current students didn’t have CCSS-M in grade 8. Eventually we can start with triangles, but I decided that maybe we should start with segments this year instead of jumping straight to SSS.
Well, we actually did start with a conversation about triangles. We have been learning which criteria are sufficient for proving triangles congruent by looking for regularity in repeated reasoning. But I wanted to remind students that we started talking about congruence this year in terms of rigid motions. So I showed a few triangles & asked how we could prove that they were congruent.
It didn’t take students long to recognize that we could show ∆APD ≅ ∆APE by reflecting one triangle about line AP.
What about these triangles?
It is easy for students to answer that we can show there is a rotation to map one triangle onto the other. But I wanted them to attend to precision. I wanted them to reason abstractly and quantitatively. What rotation will map one triangle onto the other?
A rotation of ∆APE about A by an angle measure of BAC.
Or actually by an angle measure of CAB. We learned something about our dynamic geometry software about angle direction and rotation that we didn’t previously know.
Now to segments. How can we show that segment AB is congruent to segment CD?
Almost everyone did a translation first. Some translated segment AB by vector BC to get segment CA’.
Others translated segment AB by vector AC to get segment CB’.
Everyone had a better understanding of why we can say both that segment AB is congruent to segment CD and that segment AB is congruent to segment DC.
Now that we have translated one of the segments so that one endpoint of one is now mapped to one endpoint of the other, how can we prove that the two segments are congruent?
At this point in the lesson, TI-Nspire Navigator’s Class Capture feature is invaluable. We have a teacher on maternity leave, and I am currently dealing with 60 students. There is no way I could keep up with what each student is doing. But with Class Capture, I was able to keep taking pictures of what was on each students’ screen to pay attention to interesting approaches. It helps me select which student work to discuss with the whole class. I even used the “Add to Stack” feature to have a record of some of the interesting work in case the student had moved to something else before we had time to discuss it as a class.
We went next to WA, who had shown that the two segments were congruent. (I could tell because the originally blue-colored segment had a black segment on top of it.) WA shared what she did with us. She rotated segment CA’ about point C by a measure of 41 degrees. How did you know to rotate the segment 41 degrees? I didn’t. I tried a different angle measure first, and that didn’t work, so I edited the angle measure until it worked. Ahh. That is perfect. I couldn’t have chosen a more perfect example for us to learn. So WA has done a great job of reasoning quantitatively. But I want us to move into reasoning abstractly. And here is why. The 41 degrees works now. But it won’t always work.
Another reason for TI-Nspire Navigator. I made WA the Live Presenter. And I asked her to move her original segment. This was the perfect opportunity to begin to understand the difference between the quantitative and the abstract. As soon as WA moved her original segment, the rotated segment no longer mapped to segment CD. 41 degrees was perfect for our original setup. But it doesn’t always work. What angle of rotation can we use to make our mapping always work?
Did everyone use a translation and a rotation to show that the two segments are congruent? BE couldn’t wait to tell us how he used rigid motions differently to show that the two segments were congruent. He used reflections. But I made him wait. I gave everyone a few minutes to work alone to see if they could create a sequence of reflections that would prove the two segments congruent. Then I gave students a few minutes to talk about their solution with a partner. Then BE shared his solution with the whole class.
Next we moved to SAS. Students had a paper version that they could use to plan their rigid motions.
They were much better prepared to tackle SAS after our conversation about the segments.
A few examples of student work:
And the great thing about our dynamic geometry software is that students are able to check their own work. If they can change the original triangle & still have a mapping of the pre-image onto the image, then they have been successful.
This lesson gives me hope. My students did so much better this year than last. And just think what could happen with students who have been through 8th grade CCSS-M. And so the journey continues …