Ask, Don’t Tell (Part 1): Special Segments in Triangles

• # Ask, Don’t Tell (Part 1): Special Segments in Triangles

By Jennifer Wilson, Posted May 25, 2015 –

My 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 task. Where 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 work?

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?

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.