Our wifi has been on the fritz for several weeks (it seems like the issue may be resolved for the time being, but it was pretty hairy for a while there) which was a pain in the you-know-what. But like all storm clouds, it had its silver lining.

From my first year at my school, seven years ago, whenever I taught triangle congruence I did it using an interactive Geometer’s Sketchpad file that had students manipulate virtual triangle configurations to try to make non-congruent triangles. This year I knew I needed a back-up plan in case the wifi was down and they couldn’t download the document onto their laptops. So it was back to good ol’ rulers and protractors!

Talking the idea over with my office-mate, I decided that I didn’t care whether the wifi was out or not, I was going to do the paper and protractor version anyways.

Kinesthetic learning! Using appropriate tools with precision! Decorating triangles!

The decorating was very important, btw.

I had read this blog post, which for some reason I didn’t clip to Evernote (why?!?), that talked about giving students the following scenario:

Let’s say I ask Suzy to create a triangle with sides measuring 3 inches, 4 inches and 5 inches for homework. Now, I know Suzy is really amazing with a ruler and scissors, so she is going to come back tomorrow with a really great triangle whose sides are 3, 4 and 5 inches long. Let’s say I also ask Stewart to do the same task. Stewart’s skills with ruler and scissors are equally amazing, so his triangle will also be great.

I want to know, will Suzy’s and Stewart’s triangles be exactly the same or could they be different?

The class discussed this, as well as the second scenario, which is the same except that now the triangle being made is 45 degrees, 45 degrees and 90 degrees. Then the teacher threw out another scenario (don’t remember what it was) and kept doing this until a student suggested “Let’s try it!” Then, out came the rulers, protractors and scissors and away they went.

I didn’t wait for students to suggest the cutting–I gave them time to discuss the first and second scenarios and then I had them construct the two triangles, decorate them and cut them out. In my first class I had them tape their triangles to pieces of paper all at the end of class (it was kind of a last-minute thought) but for the second class I had them do this for each triangle as they went along. I taped up papers labeled “1”, “2”, etc. around the room and wrote out the triangle’s criteria next to the paper on the whiteboard. This worked really well; students went up and taped their triangle to the paper as they finished and then they moved on to the next triangle. I was able to take down papers that everyone had finished and put up the new ones as we went along, which helped keep everyone moving forward without having stress to rush through because they hadtofinishinthreeminutesohmygahhhh!

There were multiple (!) exclamations of “this is fun” from (*unexpected!*) students as they were cutting and measuring and taping and decorating. When class was over we had multiple pages worth of triangles for many of the criteria and a full page worth of all the rest.

That afternoon/next morning I selected triangles to put on the “final sheet”–really I just took off triangles so that there was one page per set of criteria–and photocopied each page. This is where the decorations came in handy. That was my selection criteria–not accuracy, but aesthetics! [also what will show up on a photocopy best]

Day two was our day to come up with some conclusions: Was SAS enough information to determine whether two triangles would be congruent? What did SAS mean, really?

I had students **Trust, But Verify** one another’s work. Which means I passed out the photocopies of the triangle sheets and check to see whether each triangle on the page satisfied the given criteria. This means students needed to measure the side lengths and angle measures as well as figure out *if they were in the correct order*. So, not only were students getting additional practice with protractors–something some of them really needed–but they were forced to confront the idea that AAS meant something specific about where the 30 degree angle went and realize that which angle the 5 inch side was touching made a difference.

I didn’t anticipate that this activity would address this misconception, but it did. Big time! Students were figuring out on their own that if the 5″ side was touching the 40 degree angle rather than the 30 degree angle this was not the same triangle as the one that was doing the reverse.

By the end of the class we had a set of criteria for determining triangle congruence.

And I could positively kiss the kid who, when we were talking about SSA, asked:

Isn’t this like that problem we did in our last unit?

Yes, little darling.

Yes, it absolutely is.