In this student centered lesson, students will discover with guidance from the teacher the formula for finding area of a parallelogram. Students will do this by recalling prior knowledge and by using cut-outs of both rectangles and parallelograms.
Lesson: Area of a Parallelogram (8.4)
New Jersey Core Curriculum Content Standards
4.2A Geometric Properties
1. Understand and apply properties of parallelograms
2. Use logic and reasoning to make and support conjectures about
geometric objects
4.2E Measuring Geometric Objects
1. Develop and apply strategies for finding area of a parallelogram
2. Develop and apply strategies for finding area by combining triangles
and rectangles
Focus: Problem-Solving and Reasoning and Proof, Lesson Based on Research
Article (Attached)
Learning Objectives
· Students will identify the relationship between rectangles and parallelograms, given examples of each shape.
· After the task has been completed, students will be able to calculate the area of a parallelogram in two different ways.
Lesson Strategy
The teacher will introduce the lesson by discussing the formula for the area of a rectangle with the class. The teacher will then give each student a rectangle and parallelogram (cut out of paper). These shapes will have the same area. Then, students will be encouraged to answer questions regarding similarities and differences between the figures, and the teacher will give students the formal definition of parallelogram. The teacher will then lead the students in cutting off a triangular portion of the parallelogram, and having the students rearrange the two portions of the parallelogram to look exactly like the rectangle. Again, discussion will take place in which the students, guided by the teacher, should discover the formula for area of parallelogram (which is the same formula for area of a rectangle). The students will then be asked to draw a parallelogram on dot paper. The teacher will instruct students to divide this parallelogram into two triangles and a rectangle, and add up each individual area. Finally, discussion will again take place in which both methods of calculating the area of a parallelogram are compared. Students will do a few problems on the worksheet Area of a Parallelogram (1), and complete the remaining problems for homework.
Visual learners will truly benefit from this lesson because they will picture how the portions of the parallelogram should be rearranged to form a new figure, and divide a figure into smaller shapes. Auditory learners will be adept in the classroom discussion, and following directions for folding and cutting the parallelogram. Kinesthetic learners will enjoy physically rearranging the shapes to create a rectangle.
Adaptations: For students who have difficulty following the directions about folding and cutting the parallelogram, the teacher might have to draw a line on which the students should cut. Some students could have trouble envisioning how a parallelogram can be altered to form a rectangle, so the teacher might have to display how to rearrange the pieces for them. When teaching the second way of determining area of a parallelogram, students might need extra guidance in order to discover that parallelograms can be divided into triangles and rectangles. If students complete the activity quickly, the teacher should review some exercises in the textbook about calculating area, or allow students to begin their homework worksheet early.
Materials
· Pencils
· Parallelogram and Rectangle Cut-Outs
· Scissors
· Dot paper
· Colored pencils
· Worksheet: Area of a Parallelogram (1)
Preparation
The teacher must cut out enough rectangles and parallelograms for each student to have one of each shape. The teacher must practice folding the paper in order to instruct the students to do the same. Also, the teacher must prepare the worksheet Area of a Parallelogram (1). In case the lesson finishes earlier than expected, the teacher should choose some appropriate textbook exercises to review with students prior to assigning homework.
Procedure
Before
· Review topic of area of a rectangle with students
· Ask for volunteers to help pass out materials to each student
- Rectangle and Parallelogram Cut-Outs
- Scissors
- Dot Paper
- Colored Pencils
- Worksheet: Area of a Parallelogram (1)
Activity
· Discuss with students the similarities between two figures
· Provide students with definition of parallelogram
· Lead students in folding and cutting parallelogram to form rectangle
- Allow time for students to discover how the pieces of the parallelogram can be rearranged to form a rectangle
- Ask students how the terms length and width relate to the terms base and height
· Discuss with students the formulas for area of each figure
· Instruct students to draw parallelogram on graph paper
· Instruct students to divide parallelogram into triangles and rectangles, and sum the individual areas
- Instruct students to use different colored pencils to shade in the areas of each triangle and rectangle to distinguish between the shapes
- Instruct students to perform each calculation separately and record their steps
Summary and Reinforcement
· Compare and contrast the two ways of calculating area of parallelograms with students
- Discuss with students when each method is more appropriate
- Ask students if it is always possible to use the second method in calculating the area of a parallelogram
· Assist students in completing the first few exercises on Area of a Parallelogram (1) worksheet
· Instruct students to complete this worksheet for homework
Assessment
The teacher will assess the students’ understanding of how area of a rectangle and area of a parallelogram are similar through class discussion. The teacher will observe students’ work and calculations following the second task, to determine if they found the correct area of the parallelogram. Assessment during discussion will again take place during a summary of the lesson, in which the class will compare and contrast these methods. The Area of a Parallelogram (1) homework worksheet will serve as another form of assessment during the next class.
Resources
Duke, Charlotte. “Tangrams and Area”. Mathematics Teaching in the Middle School. Vol. 3, May 1998.
Strutchens, M. E. “Assessing Geometric and Measurement Understanding using Manipulatives.” Mathematics Teaching in the Middle School. Vol. 6, March 2001.
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