Does the Area of the Quadrilateral Change?
From APEC HRDWG Wiki
The lesson “Does the Area of the Quadrilateral Change?” was captured on video for the APEC Education Network (EDNET) project called Classroom Innovations through Lesson Study. The lesson is an example of using the Lesson Study process of professional development in the teaching of Mathematics.
Lesson Overview
This lesson was taught by Akihiko Takahashi during the Chicago Lesson Study Conference, May 13, 2006. In the previous lesson, students explored methods for determining the area of rectangular combinations and parallelograms. Here, the teacher uses polystrips™ to construct an 8×10 unit rectangle. Students are asked if the area remains the same or changes as the polystrips™ are shifted slightly to make a new parallelogram (with constant base but slightly less height). Some students report that the area remains unchanged, while others say that the area gets smaller. Working with partners, students use the methods learned previously to check these claims. As the lesson develops, students discover that the area of a parallelogram can be calculated simply as base times height, thus revealing the common equation for the area of a parallelogram: Area =base×height.
The goals of the lesson are:
- Deepen students’ understanding of the concept of area through problem solving.
- Develop the formula for finding the area of parallelogram.
- Help students become good problem solvers by encouraging students to:
- use their prior knowledge to examine a problem situation to develop their ability to use logical reasoning to make conjectures; and,
- examine and justify solutions presented by their peers in order to find a solution to the problem.
- Provide opportunities for students to recognize the importance of working with their peers in order to deepen their understanding of mathematics.
[Demonstration Practice  ] (see Resource Types)
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| Lesson Plan | Lesson Video in MPEG 4
(Video Clips and Highlights Available below. Right-click and select "Save Target As" to download the file.) |
Video Clips and Highlights from the Lesson
The video clips are selected from the full List of Episodes. The Full Lesson Video may be downloaded for further study.
The following video is an example of a research lesson used during the lesson study cycle. Wiki users may use this video to experience a part of the cycle and can hold a post-lesson discussion with their colleagues after watching this video to continue the lesson study process.
| Description of Video Clip | Video Links |
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Students are reminded of the methods that were used in the previous lesson to find the area of parallelograms. Then rectangle made of polystrips™ is introduced. After discussion, students agree that the initial shape is a rectangle. Then students are asked to determine the area of this rectangle. Students are shown that area is length times width: A=l×w. |
1. Preparing the Problem
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The teacher shifts the polystrips™ slightly. Students are asked if the area stays the same or changes. Students views differ and discussion follows. Some students are seen changing their positions. |
2. Posing and Understanding the Problem
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Working in table groups, students determine the area of the parallelogram. We focus on to students who disagree about the solution. |
3. Working on the Problem
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Three groups present to the class. Two new questions are exposed: If the area is no longer 80 units, but 70 units, where did the 10 go? What is changing, the length of the side or height? |
4. Discussion, Part I
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An unexpected development, students express a belief that the length of the side of the parallelogram is changing. |
5. Discussion, Part II
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Students distinguish between perimeter (which stays the same) and area (which changes). The teacher highlights the measurement of height, and students identify that area of a parallelogram depends on the height and length of the base. |
6. Summary
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A computer simulation illustrates dynamically the relationship of height and area. The lesson reaches the conclusion that students no longer need to count unit squares when all they really need to know is the base and height. |
7. Conclusion
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