After yesterday's lesson, students have seen paragraph proofs and two-column proofs. I offer students several proof formats so they can choose the format that best enables them to organize their reasoning as they work towards a logical conclusion. Since two-column proofs are new, I begin today's lesson by asking my students to re-visit the proof they practiced yesterday. Then, I will introduce a third style for presenting a proof. I model for students how flowchart proofs work, emphasizing the pairing of statements and reasons and how these are followed to a conclusion. I highlight the way the arrows connect necessary information to where they are needed in the proof and how the organization of the flowchart proof provides a visual representation of one’s thinking during a proof-writing exercise.
Then, I give students two different flowchart templates** that they can use to complete statements and/or reasons needed for the proof. Since I want this practice to offer students a chance to really reflect on the progression of ideas in their thinking, I ask students to go back and narrate some of the process for themselves in writing. For example, I want students to write a note to themselves about how one statement/reason bubble lead to another (and why that order matters), or why multiple statement/reason bubbles are needed to prove triangles are congruent. This is the kind of thinking that reinforces for students how they can check their own reasoning and proof writing.
** Flowchart templates are from Discovering Geometry, Key Curriculum Press.
At this point in the lesson, I want students to be able to practice using the different proof formats so they can find the best fit for their own work. For this practice, students will work individually at first, then with a partner. Time to work on their own gives students time to think freely, unimpeded by another student’s ideas, suggestions, or critique. I circulate the room during this time, monitoring students’ progress on each of these proofs and checking in with individuals who need more help. When it seems like most students have finished at least one proof, I ask them to then check in with their partner, trading papers to see what is similar or different in their proofs, to identify opportunities to improve their proof writing.
After both partners have finished their proofs, they call me over to check their work. If they have successfully written their proofs, I hand them the proof challenge, which asks them to prove the diagonals of a parallelogram bisect each other.
The Isosceles Triangle Investigation gives students an opportunity to formalize their inklings about the symmetry line of isosceles triangles. In this group investigation, students investigate a case of an isosceles triangles (acute, right, obtuse, equilateral), to see what else they can prove to be true about the angle bisector of the vertex angle. Using Geometric construction, students compare their individual discoveries and make conjectures about the angle bisector of the vertex angle of an isosceles triangle (MP1, MP2). As a final product, they glue their individual cases on a mini group poster, write out their group’s conjecture, then check in with me.
If the group has successfully drawn the correct conclusions from their investigation, I move them onto the final part of the lesson, which requires them to prove their conjecture. Like other lessons, I give students a note taker, which is where they will prove that for any isosceles triangle, the angle bisector of the vertex angle is also the median and the altitude.
Like all of our investigations, at the end of this lesson we formalize our discoveries through a whole-class discussion in which we take notes. We record the conjectures groups made during the isosceles triangle investigation and we take time to prove that the angle bisector of the vertex angle in isosceles triangles is also a median and an altitude.