Ask, Don’t Tell (Part 1): Special Segments in Triangles
By Jennifer Wilson, Posted May 25, 2015 –
daughter, Kate, decided to make hot chocolate. She found a 1/3 measuring cup
and asked, “Where is the 2/3 measuring cup?” Without thinking, I almost said, “You
can just use that measuring cup twice,” but I caught myself. Instead I asked,
“Could you use the 1/3 measuring cup to get 2/3?” She thought for just a few
seconds and said, “Use this one twice!” I had almost short-changed Kate’s
opportunity to make her thinking visible by telling her what to do. Changing
“you can” into “could you” made all the difference.
How many times have I missed similar
opportunities with my students?
We want our geometry students to know and
use special segments in triangles along with their points of concurrency, and
we used to tell them how. More recently, however, we let the math content
unfold while students practice using appropriate tools strategically, constructing
a viable argument, and critiquing the reasoning of others.
We started with Placing a Fire
Hydrant from Illustrative Mathematics.
Three buildings are situated so that they could be the vertices of a triangle.
Where would you place a fire hydrant to serve all three buildings?
Students started on paper, using rulers,
folding, and compasses; then they moved to technology. I used what I have
learned from Smith and Stein’s 5
Practices for Orchestrating Productive Mathematics Discussions (NCTM 2011) to
monitor, select, and sequence the student work for our whole-class discussion.
My students came into this lesson not knowing
the vocabulary associated with special segments in triangles, so I purposefully
included some incorrect solutions for placing the fire hydrant equidistant from
the buildings to bring out that new vocabulary. Throughout
the lesson, students learned about medians, centroids, midsegments,
perpendicular bisectors, circumcenters, and circumscribed circles—because of
the work that they themselves had done.
Caroline had constructed the
perpendicular bisectors of each side of the triangle. She had measured from
their intersection, the circumcenter, to each building to show that they were
equidistant. When she presented her solution to the class, she used our interactive
geometry software to change the location of the buildings, showing that the
fire hydrant was still the same distance from each building. Gabe asked, “Why
would we put the fire hydrant there?” Caroline stopped, and we all took a good
look at the setup.
Caroline moved the buildings again to
exaggerate how ridiculous it would be to place a fire hydrant that far away. The
technology made the students realize that the circumcenter is not always the
most efficient place for the fire hydrant, even if it is equidistant from the
three buildings. We began to explore when it makes sense to put the fire
hydrant equidistant from the buildings and when it does not.
During the next class, we used
the Locating a Warehouse
would you place a warehouse that needs to be equidistant from three roads that
form a triangle?
Students worked on paper for a few
minutes before they set their initial placement of the warehouse. Their conjectures
gave me so much to think about.
Is there a point outside the triangle equidistant from the three
roads? One student defended her point: “I
drew a circle with that point as center that touched all three roads.”
How do you know that the roads are the same distance from the
center? “They are all radii of the
circle. They are perpendicular to the road from the center.”
Could a point inside the triangle of roads be correct? What is
significant about the point that will be the same distance from all three sides
of a triangle? Most students thought about perpendicular bisectors. Is the
circumcenter equidistant from the vertices and the roads? Another student
insisted that the point needed to lie on an angle bisector. Would that always
The mathematics unfolded
through questions, conjectures, and exploration. “Ask, Don’t Tell” opportunities activate students as owners
of their learning.
What #AskDontTell opportunities do and can you provide?
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