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9th & 10th Grade Math - Properties of Quadrilaterals

Clip 3/24: Tuesday Introduction - Part C


Cathy Humphreys introduces the Properties of Quadrilaterals task. The lesson is introduced by explaining the students’ roles and responsibilities to carry out the investigation. The teacher models how to gather information for the tinkering stage of the investigative process. She also poses the math problem in the context of a kite making company. The kite making company uses sticks to make kites. The investigation involves how two sticks will be selected and positioned to determine the shape of a kite. She creates a purpose for exploring the key factors (length, intersection point, and angle position) that define the shape of a quadrilateral. She introduces manipulatives (different and like length strips with holes and a brad) and demonstrates how one quadrilateral might be determined by the arrangement of the diagonals. She demonstrates how the stick might be used to draw a rhombus.

Teacher Commentary

COMMENTARY BY CATHY HUMPHREYS: I have posed this problem as a “real-world” situation, which it clearly is not. The idea of the kite factory motivates a context for the investigation but it is a fanciful context, indeed. There are many truly real-world applications of mathematics (e.g., the mathematics of global warming). These applications are especially important for students to experience in order to understand the relevance and importance of mathematics in their lives.

I also want to emphasize the importance of the process of private individual thinking prior to the group addressing a task. This, as Ori said, ensures individual ideas and different ways of thinking that enrich the group’s approach to the problem.

As I closely watched video of the small-group discussions (an opportunity which a teacher rarely has), I noticed that although the students had individually made some conjectures, there was no emphasis made on recording these; and this also obstructed examination of exactly what the person had said. When I do this lesson again I think I will have one student at each group record for the group: 1) “What have you found out so far?” and 2) “What are you wondering about?” This way they will have a better starting point for their group work.

Finally, I would like to comment about the structure of the group work during this task. The idea of these roles comes from Elizabeth Cohen’s work on Complex Instruction. I have, however, not employed these roles in the true spirit and intent of Complex Instruction, in which the roles are assigned so that all students have the opportunity to perform each of the roles. Assigning the roles can help negate status issues that may arise from certain students always selecting a specific or preferred role.

COMMENTARY BY COACH DAVID FOSTER: Cathy states explicitly how students are responsible for investigating, working with others, making sense of the mathematics and sharing their findings. She describes the thinking the students will experience in the investigation, reminding them of the process they have both experienced in the past and how it is depicted in the process on their poster. She uses the notion of tinkering to help the students understand that they will have to work at making sense of the situation before they come to understanding it completely. Cathy poses the math problem, with a purpose that student relate to and can understand. She introduces the manipulative before having the students begin their investigation. She gives two sets of manipulatives to each group of four to foster collaborative exploration.