Anyways, the point of this post is to talk about this activity that I did earlier in the year with Geometry classes. Rather than giving my students the side to side ratios for special right triangles (45-45-90 / 20-60-90), I wanted them to discover the ratios on their own using prior knowledge (Pythagorean Theorem). Now, as this was early on in my student teaching at the placement, they were VERY resistant to not being told what to do or just being told the formula. However, I am a huge fan of discovery/inquiry based teaching methods and this was a great version of this. Basically, I gave my students these packets (one triangle on each day) where they developed a conjecture about the side ratios for each triangle. For the most part, it is now firmly placed in their minds. Anyways, I have a few screen shots of the worksheets just for your reference.
Basically, the 45-45-90 triangles were slightly easier, thus why we started with these first, but the the students struggle with radicals. Overall, they can see that the sides will be the same (because of the angles) and then they find the ratio. I would have added something about how to go backwards more firmly as this was a struggle.
Starting with an equilateral triangle, you can draw a bisector to create a 30-60-90 triangle. The short and hypotenuse are very easy to see for the students, but again, the radical throws students for a loop. Again, I would also emphasize how to go backwards after deriving this.
But again, I truly love discovery and the students are super proud of themselves after they notice the property. So good luck using discovery/inquiry in your classroom but it really does make a world of difference, especially with common core coming into play :)