Clip 13/24: Quadrilateral Introduction - Part A
In the opening of the second day, the teacher shares students’ work sheets to illustrate how the students were thinking while investigating the quadrilateral. After sharing work, she has students write to a prompt about how well they keep track of their thinking. The students then share with the class their individual reflections.
The class is still learning how to prove a conjecture. The teacher had just given a test and many of the students were less than successful with proving theorems. The teacher wants to have all students successful with proving conjectures.
Each group begins to settle on the quadrilateral they will formally prove. The students use their prior knowledge of parallel lines and congruent triangles to approach the proofs of the quadrilaterals.
In creating a proof, the students first need to create a conjecture from their investigations and findings. Some of the groups struggle with taking a conjecture and determining what is given and what needs to be proved.
COMMENTARY BY CATHY HUMPHREYS: All students need opportunities to learn how to improve their work. In mathematics classes assignments are most often problem sets that require mainly (or only) the practicing of a procedure. The way to improve this kind of work is solely to learn how to execute the procedure correctly. When more is asked of students, however, as in this lesson, most students are in a realm in which they have had little experience. So I have used often used this teaching strategy – of photocopying work that demonstrates particular qualities - to create common expectations of high quality work. This gives students a way to assess their own work and improve it.
COMMENTARY BY COACH DAVID FOSTER: The introduction of the second day helped frame the task that students were to work on in this second session. Cathy shared examples of how students “kept track” of their own thinking. This helped students in their meta-cognitive efforts at understanding how they’re thinking and how to document their understandings. The quick write and sharing further solidified the nature of how they think and record findings. This is a powerful aspect of the instruction, that help us as observers understand how the students work and are capable of such independent and deep mathematical work.
Cathy reviewed how to prove a conjecture. She modeled how to prove that diagonals of different lengths, which bisect each other but are not perpendicular, form a parallelogram. She had the students use the definitions of parallelogram to help them focus on what was necessary for the proof. She had the students mark a figure drawn from the given to help reason through the proof. Students were successful in creating reasoning for the proof.
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