Reflection: Rigor Properties of Parallelograms and Special Parallelograms - Section 1: Warm-Up: Logical Road Map

 

There are times when I wonder whether discussing proofs (or big problems, for that matter) are really worth taking up a significant amount of class time. Today's discussion was a reminder of how valuable this time is for my students. Even during the discussion, I could sense growth (maturation?) in their willingness to:

  1. Persist in the face of challenge 
  2. Appreciate different approaches 

Similar to the way in which I facilitated the discussion around Three polygons meet at Point B, today I sequenced student presenters in a particular order. Again, I used the subjective criteria, "I think this strategy will support the greatest number of students’ work as they try to deepen their understanding."

In today's lesson, the high leverage student strategy involved using:

  • the base angles of the isosceles triangle
  • corresponding angles
  • alternate interior angles

Since I have been promoting the flexible use of properties of special quadrilaterals in this unit, I also chose to promote the idea of using properties of isosceles trapezoids.  This approach was relatively novel (and only arose in one of my classes). Nonetheless, I thought it deserved public validation.

When the use of properties of isosceles trapezoids did not emerge in the other classes, I decided to treat this idea as an original scholarly contribution. I directed my students to focus on a diagram of the task, which I had projected using a document camera. Then, I silently numbered and labeled the “work” asking students to discuss reasons why the work made sense (or not) in small groups. (I was essentially asking students to re-imagine the parallelogram and isosceles triangle as forming an isosceles trapezoid). After approaching the idea in this way, it proved to be quite popular. It became a catalyst for building students’ interest in the reasoning of others and for motivating students to supply an argument explaining the contribution made by their own work. Our discussions of diagrams became much more engaging!

  Constructing Viable Arguments
  Rigor: Constructing Viable Arguments
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Properties of Parallelograms and Special Parallelograms

Unit 8: Discovering and Proving Polygon Properties
Lesson 6 of 9

Objective: Students will be able to apply angle relationships and properties of isosceles triangles and trapezoids in a proof.

Big Idea: In the Logical Road Map, students will make sense of multiple pathways to writing a proof.

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Subject(s):
Math, Geometry, polygons (Determining Measurements), properties of polygons, reasoning and proof, parallelogram
  90 minutes
prop of parallel li
 
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