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.