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Classroom Case Studies, 3-5 Part A: The Concept of Area (25 minutes)

Session 10: 3-5, Part A

To begin to explore what the teaching of measurement might look like in the classroom, participants in the Measurement course first revisited a problem on area presented during Session 6. The participants then considered how children make sense of these ideas and discussed ways to present area concepts to elementary school students.

 Video Segment In this video segment, four teachers discuss some of the important concepts involving area that are encountered by students in grades 3-5. When planning instructional sequences, teachers need to consider what mathematical skills and concepts students need to understand and what activities will help them develop that understanding. You can find this segment on the session video approximately 2 minutes and 20 seconds after the Annenberg Media logo.

Problem A1

Answer the questions based on what you saw in the video:

1. What concepts and skills did the teachers mention as being important for students to understand?
2. What types of activities might be used to help students make sense of these concepts and skills?
3. Are there related concepts or skills that will affect whether or not students can understand and use these ideas?
4. Thinking back to the big ideas of this course, what are some other ideas that students should encounter to help extend and deepen their understanding of area?

Problem A2

Choose one of the concepts that you listed for Problem A1 and describe an instructional activity that you might use to help students grasp that concept.

Problem A3

What role do manipulative materials play in making sense of these mathematical ideas? Do they support or hinder students’ mathematical understanding of conservation of area?

Solutions

Problem A1

1. Teachers talked about the importance of students understanding that area is the measure of the amount of space covered as well as what a square unit is. They talked about how, when asked to build a 3-by-5 rectangle, students will often create just a border and not cover the middle of the rectangle. The teachers would like for students to see the connection between area and multiplication as well as count the perimeter accurately.
2. Having hands-on experiences where students are building rectangles with manipulatives such as tiles helps them to see and understand area. They are physically covering the rectangle and counting square units to determine area. Drawing on grid paper also helps students to visualize the concept of area. Either way, students can see the rectangular arrays that are formed and connect area to multiplication. For example, in a 6-by-8 rectangle, students can see six rows of eight tiles or eight rows of six tiles, and the concept of multiplication is brought to the forefront for them.
3. As mentioned above, learning about area provides students with an opportunity to deepen their understanding of multiplication. Determining the area of a rectangle becomes a context in which students can “see” multiplication. Looking at it another way, having a firm understanding of multiplication will also help students in their study of area. When considering the interpretation of the meaning of multiplication as an array of length times width, it is clear that students’ knowledge of both area and multiplication are developing at the same time.
4. Students should also look at conservation of area, appropriate type and size of units of measurement, relationships between perimeter and area, strategies for determining areas of irregular shapes, and surface area.

Problem A2

Answers will vary. One way to allow students to explore area as a covering is to have them find the area of an irregular shape (for example, the outline of a pair of scissors) on grid paper. They can then determine the number of square units that it covers.

Problem A3

Using manipulative materials is essential for giving students the opportunity to see and feel area. Understanding that area is the measure of the amount of surface covered is much easier when students are actually covering rectangular surfaces with square unit tiles. The idea of conservation is also made concrete to students when they can actually hold the amount of area in their hands by using manipulative materials. If students are given 12 tiles to make various rectangles, they can convince themselves that all the rectangles will have the same area — even though some may look bigger or smaller — because they were all made with the same number of tiles. Manipulative materials are an essential tool for learning about area.