In our 30-60-90 triangle lesson, we first find out what students already know about 30-60-90 triangles. We deliberately let them practice look for and make use of structure. Some students compose the triangle into an equilateral triangle and note that one side is double the length of the other (we eventually attended to precision to note that the hypotenuse is double the length of the shorter leg).
Some students rotated the triangle 180˚ about the midpoint of the hypotenuse; others rotated it 180˚ about the midpoint of the longer leg.
Other students decomposed the triangle by drawing the altitude to the hypotenuse and noted that they formed two additional 30-60-90 triangles.
Our lesson (partially from the Geometry Nspired activity Special Right Triangles) provided the opportunity for students to look for and express regularity in repeated reasoning using the equilateral triangle. What changes and what stays the same as you grab and move point B?
Recording side lengths in a table also provided the opportunity for students to look for and express regularity in repeated reasoning. But after our 45-45-90 lesson the day before, I thought it would be okay to skip that step and move to #3.
As soon as we recorded some of the student responses to #3, I realized I had made a mistake. They weren’t all getting that the longer leg is the shorter leg times √3. I had tried to rush to the result instead of giving students the time to notice when calculations are repeated, to evaluate the reasonableness of intermediate results, and to look for general methods and shortcuts.
We took a step back. They all used the Pythagorean Theorem to determine the altitude for an equilateral triangle with a side length of 10, and then in the little time remaining we used our technology to confirm the result and help us generalize the relationship between the side lengths of a 30-60-90 triangle.
And so the journey continues … learning more each day that providing students deliberate learning episodes steeped in using the Math Practices is much more effective than having them haphazardly figure out the math.
Easing the Hurry Syndrome 
teachmathmom
February 7, 2015 at 11:59 am
This is the way I had been introducing special triangles for years. It is so helpful for the students to build their reasoning to look for patterns. I am a math instructional coach now, and helping the geometry rewrite their curriculum to be in line with the content standards and practice standards. This topic was a big discussion topic in that process. We couldn’t see anywhere in the content standards were special triangles (or pythagorean triples) were expected in geometry. However, a background knowledge of them is expected for unit circle trig in “4th year classes” and there is mention of a proof for generating pythagorean triples in algebra 2. Also, rationalizing and/or simplifying radicals isn’t discussed anywhere either. Solving equations with radicals is, as well as working with complex numbers, but I think that is most,y algebra 2 as well. I agree that this is a wonderful place to look at the MPs of structure and repeated reasoning. It’s a hard sell for the team, though, when they feel the time crunch of content. I still feel teachers leaving out the practices to give direct instruction to “cover content” which is what we try to avoid. How have you ballanced the content vs. time to discover? WAs there content you have let go to be able to have “deeper” lessons? I feel that out teachers think that they’ve added so much with transformations and constructions, that if something isn’t specified in CCSS, they need to leave it out.
Thoughts or suggestions?
jwilson828
February 9, 2015 at 5:51 am
It has been a struggle to know some of what to leave out from our traditional course. I’ve written a few posts on that topic: https://easingthehurrysyndrome.wordpress.com/2012/10/26/implied-content/ and https://easingthehurrysyndrome.wordpress.com/2013/03/13/pythagorean-relationships/ that might be helpful. For now, one of our main guiding questions has been whether students will still need to know it for the ACT, as most of our students take the ACT for college entrance instead of the SAT. We ask our juniors and seniors about specific items & whether we should still cover them in geometry or whether it’s time to let them go, and they’ve been really good to help us know. We can also see a difference over the past 3 years in what students know now compared to what they used to know, and so we are able to leave out things now that we didn’t feel good about leaving out before (or at least spend less time on them in high school). At least for now, our geometry students don’t take a high stakes test at the end of the course – only our Algebra 1 students. But even in Algebra 1, our focus is on learning math using the Math Practices – we’re not willing to negotiate that, even to get everything covered. The students just aren’t going to have had all of the content by test-time. We’re doing our best to use the Major-Supporting-Additional labels on standards and focus on Major.
howardat58
February 7, 2015 at 12:37 pm
Nice one ! You realize that you are going to have to write a book eventually.
jwilson828
February 9, 2015 at 5:38 am
I don’t know about that 😉 but thank you!
Peggy Welch
February 7, 2015 at 10:42 pm
I’m writing an activity about how triangulation and trilateration are used for measurement with GPS. The thinking/investigating procedures in this activity will definitely be included.Thanks!
jwilson828
February 9, 2015 at 5:38 am
I’m glad this is helpful, Peggy!