By Jennifer Wilson, posted June 22, 2015 –

I used to tell my students the relationships between the legs and hypotenuses of special right triangles. Now I provide them the opportunity to figure out those relationships themselves.

We started our lesson practicing the Common Core’s Standard for Mathematical Practice 7: Look for and make use of structure. (See my post on SMP 7.) What do you know about a 45-45-90^o triangle? What can you figure out about a 45-45-90^o triangle?

• The triangle is right.

• The triangle is isosceles.

• The triangle is half of a square when I draw its diagonal.

• If I rotate it 90^o about the midpoint of a leg, the two triangles create a larger 45-45-90^o triangle.

• I can decompose one of them into two smaller 45-45-90^o triangles by drawing the altitude to the hypotenuse.

We moved into an activity about special right triangles that gave students the opportunity to practice SMP 8: Look for and express regularity in repeated reasoning. (See my post on SMP 8.) Using the Pythagorean theorem to answer the following questions, students recorded their calculations in a table.

• What stays the same?
• What changes?
• What if the side length is 10?

• What if the side length is x?

Instead of my telling them, my students told me to multiply the leg by the square root of 2 to get the hypotenuse. Instead of my telling them, they told me to divide the hypotenuse by the square root of 2 to get the leg.

In a journal reflection, one student wrote about using SMP 8 in this lesson:

The 30-60-90^o triangles unfolded similarly, with students noting that an altitude decomposes an equilateral triangle into two congruent 30-60-90^o triangles.

Students continued making connections between similar triangles and trig ratios.

At the end of this school year, I received a letter from one of my students in which she wrote: “Before being in your class, all of my teachers would just give me information on a big silver platter. But when I enrolled into your class, you taught me how to learn things by myself and not expect things to be just given to me. Although it kinda hurt my brain a lot I guess it made my synapses fire. I found out that when you memorize something, you’ll eventually forget it, but if you try to learn it and understand it, the info just sticks to you.”

Daniel Coyle has a similar argument in The Talent Code. When asked to remember pairs of words given visually, most people remember more words from the pairs of words that have a missing letter. Coyle suggests that you have to practice deeper to remember the words with a missing letter. Try taking one of the quizzes that proves his point.

“Ask, Don’t Tell” learning opportunities allow the mathematics that we study to unfold through questions, conjectures, and exploration. “Ask, Don’t Tell” learning opportunities begin to activate students as owners of their learning.

What #AskDontTell opportunities do and can you provide?

JENNIFER WILSON, http://www.easingthehurrysyndrome.wordpress.com, a National Board Certified Teacher, teaches and learns mathematics at Northwest Rankin High School and is a curriculum specialist at the Rankin County School District in Brandon, Mississippi. She is an instructor for TI’s Teachers Teaching with Technology (T³) program. She enjoys learning alongside the Illustrative Mathematics community, and she is a recipient of the Presidential Award for Excellence in Mathematics and Science Teaching.