As much as possible, I want the students to know a few unifying ideas about triangles. That’s why I dumped medians, orthocenters, and a lot of the relatively obscure triangle facts—not because they aren’t useful, but because most of my students will never use them again and I want them to have plenty of time working with the geometry they’ll need forever. My top students get practice thinking through challenging problems but I don’t throw a lot of random facts into this mix.
This has a lot to do with my own preferences. I like to remember the big things and look up the little, and I don’t personally see much value in making students jump through hoops to remember obscure math facts. (History is a different matter.) I do want them to remember the important math facts (Pythagorean theorem, special right ratios, area formulas, and so on), so when I say, underscored five times, remember this, my students won’t treat it as one of a random flood. That doesn’t mean they’ll all remember it, alas. A distressing number of my students still look at me perplexedly when I ask them the formula for the area of a triangle. It is to weep. But that’s all the more reason to keep the memorizing to a minimum.
I taught special right triangles as a ratio, rather than a pattern. This worked very well for most students, and it also provided for continuity when we moved into similarity. Same process, but now the ratio isn’t fixed. (And this should make the move to trigonometry, which is also based on ratios, part of the same continuity).
I gave them a test after special rights and then a test after similarity. I tested much of the same material in both. (After break, I’m going to give them a test on other things from first semester, to see how their retention is.) Here are the tests and the correct percentages for each.
I just noticed that I forgot to fix the typo in question 14 before I made this image. The kids were given corrections before they started the test. (My tests often have minor typos, as I create them from scratch each time. My kids know this and are encouraged to check with me if they think they find something wrong. They are usually incorrect—that is, there was no typo—but I’m confident that no one is sitting silently perplexed for the wrong reasons.)
- Questions 1 and 2: see what I mean about area? I was actually pleased with those numbers. They are still forgetting to take half.
- Questions 3 and 4: very good understanding of perimeter and algebra (I don’t think my diagonals are accurate on that pentagon).
- Question 5: A straightforward test of Triangle Inequality, something I’ve emphasized. It’s frequently on the SAT and ACT and besides, it’s a useful limit to remember. And they did!
- Question 6 and 7: The students had to know that triangles had 180 degrees AND that a “not acute” triangle must have one angle of 90 degrees or higher. Another 20% knew that triangles had 180 degrees, but didn’t catch the second criteria.
- Questions 8, 9, 10–straightforward special right questions.
- Questions 11 and 12 are a case of unintended consequence. I thought 11 would be the easy one, as we’d just reviewed congruent triangles and I’d reminded them (again) of the SSA (see the bad word? Yeah. Not a congruence shortcut). Most chose the SAS answer—the correct answers were all top students. While question 12 didn’t have great results, more students hroughout the ability spectrum answered it correctly.
- Questions 13-15—more special right questions. (see note above about the glitch on 14).
The last five were probably too difficult; I wanted to see how my top students would do. I ended up dropping the last five and grading students on the first 20.
Average curved score: 68% (51% uncurved)
This was a much more solid performance, I thought. I’d increased the difficulty, and the average was just a little higher (70% curved, 54% uncurved).
In both tests, the students had a terrible time dealing with radicals and the Pythagorean Theorem, which is what led to a review. But they showed increased mastery of special rights, and their understanding of similar triangles was better than I’d been hoping for.
These percentages are the actual answers given. I went through the D and F tests and reviewed questions 15, 17, and 20. In all three cases, I’d put a spin on the question. So if a D or F student saw that the angle was 60 degrees but didn’t realize I was asking about the unmarked part, did the proportion correctly in question 17 but forgot to convert to feet, or solved for x and then forgot to plug it in, I gave them credit for that work.
Any geometry teacher knows that I’m going veeeeeery slowly. I have two major topics to cover before state tests—trig and circles—and two minor ones, volume/solids and regular polygons. At the same time, I’m definitely not giving my students an easy time. I’m working them hard and I think they’re doing challenging math. In fact, that’s the most consistent beef my students have about me—they think my tests are “weird”. But they also appreciate my curving, and they also, I think, accept my assurances that this is what actual standardized tests look like, so they may as well man up.