I have been surprised while leading professional development on more than one occasion this year when talking about a 45°-45°-90° triangle as half of a square and a 30°-60°-90° triangle as half of an equilateral triangle to learn that many teachers have not thought of these special right triangles in this manner before. I think that the Special Right Triangles Geometry Nspired activity is an excellent way to provide students the opportunity to look for regularity in repeated reasoning. After they explore the TNS document and record their observations, you can move students towards reason abstractly and quantitatively, generalizing and proving that multiplying the leg of a 45°-45°-90° triangle by the square root of 2 will result in the length of the hypotenuse, multiplying the shorter leg of a 30°-60°-90° triangle by the square root of 3 will result in the length of the longer leg, and multiplying the shorter leg of a 30°-60°-90° triangle by 2 will result in the length of the hypotenuse. My students still need practice with calculating area and perimeter of two-dimensional figures as they prepare for current standardized tests even though the area of two-dimensional figures is not explicitly listed as a high school geometry standard, so I used this opportunity to let them practice what they learned about special right triangles with problems involving area and perimeter.
The other teachers at my school enjoyed these lessons, based off of the Geometry Nspired activity, and our students had the opportunity to enter into the Standards of Mathematical Practice as they were making sense of the relationships among the lengths of the sides of special right triangles.
I was excited to read Kaci’s journal about look for regularity in repeated reasoning.
Kaci embodied this practice, noticing when calculations are repeated and looking for general methods and for shortcuts, but I was most impressed that Kaci also looked for and made use of structure, seeing complicated things being composed of several objects. It is significant that Kaci recognized that the square root of 2 will always be “in” or “part of” the hypotenuse, even though it might not always look like the square root of 2. Kaci’s reflection has challenged me to make sure that all of my students recognize what she has as our journey continues ….