Ask, Don’t Tell (Part 3): Special Right Triangles
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 45-45-90o
triangle? What can you figure out about a 45-45-90o 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 90o about the midpoint of a leg, the two triangles
create a larger 45-45-90o triangle.
• I can decompose one of them into two smaller 45-45-90o 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 30-60-90o triangles unfolded similarly,
with students noting that an altitude decomposes an equilateral triangle into
two congruent 30-60-90o triangles.
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
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
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