How We Let Students Wrestle with Definitions

My colleague extraordinare over at The Mathy Murk, asked our Geometry team how we wanted to introduce the notion of midpoint. She was inspired by Dan Meyer's 3 Acts. I wasn't sure what the parameters were and was a little stuck by own notion of midpoint having to be something collinear. And thus the "notion of where is the best midpoint" was born. Please, oh, please comment on how you would make this lesson BETTER!

Students walk in and are handed a blank half sheet of paper:

Teacher: Draw two houses. (Check some of these beauties out)

 After 90 seconds: Teacher again: Put a dot on the midpoint between the houses.

Teacher: Come tape yours up on the whiteboard.
Teacher: Discuss with your table, which midpoint is the best.

 The odd thing was, that 99% of the students floated the midpoint.

Would you call this midpoint a floated midpoint or a foundational midpoint?

Only a very small handful of students put the midpoint on the ground lined up with foundation of the houses.
(One of the team came in at break, "Ack! no student (out of 30) put the point on the ground! But at least someone suggested it, thank goodness.")

Next we made a list of criteria and had a debate about the assertion:

The best midpoint is inline with ground and foundation.

Lots of pro and cons.

Phew. We talked about congruent segments, definition of between, what does middle mean, and finally, definition of midpoint. It was all a worthy discussion, AND, how could have it been better? Oh, and the best house of them all:

The Smurf house!

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  1. I think the lesson structure is good, but I wonder if you created the houses for students so that it is more challenging for students to decide where the midpoint is if you would end up with more variation in their choices?

    1. Hi David,

      Thanks for stopping by! Should we use the Smurf house in that case? : )

  2. This is great, Amy! Another extension might be to ask students, "based on where you put the midpoint...what is the midpoint of?" I know that's not quite what I mean, but when they're floating the midpoint, I'm curious about what two points that is the midpoint of? Great work!!!

  3. I think the students thought midpoint here meant of the entire structure, which means they answered a more complicated question. I think this is the centroid of a 2d geometric object, like a pair of houses.


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