# Classroom Case Studies, 6-8 Part A: The Concept of Area (25 minutes)

## Session 10: 6-8, Part A

To begin exploring what the teaching of measurement might look like in the classroom, participants in the Measurement course first re-examined the big ideas around one topic: area. They considered how students make sense of these ideas and discussed ways to present these concepts to middle school learners.

 Video SegmentIn this video segment, six teachers discuss some of the important concepts involving area encountered by students in grades 6-8. You can find this segment on the session video approximately 3 minutes and 41 seconds after the Annenberg Media logo.

### Problem A1

Answer the following questions based on your experiences in the course and what you saw in the video:

1. What concepts and skills did the teachers mention were important for students to understand?
2. Based on your own experiences, what content do students find difficult when studying area?
3. What types of activities might be used to help students make sense of these concepts and skills?
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 the topic 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

In the video segment, Mr. Cellucci and the other teachers discuss the importance of activities in which students actually measure objects, such as going outside to measure shadows and calculate heights. Is it important for older students to engage in measurement activities, or are they able to make sense of the material at an abstract level? In general, what role do manipulative materials play in helping students understand measurement concepts?

### Solutions

Problem A1

1. The teachers discussed the following ideas that would be important to explore with their students:
• The dynamic relationship between the perimeter and area of a shape
• The difference between estimating and physical measuring, which can further be tied to the ideas of accuracy and precision
• Experiencing area as a physical process of covering a two-dimensional surface in square units
2. Answers will vary. Students may have difficulty understanding measurement relationships, measurement formulas, indirect measurement, and the ideas of accuracy and precision.
3. To help students make sense of these concepts and skills, you can help them explore the relationship between area and perimeter of various shapes. They can also estimate the area of irregular shapes. They can see the effect that using different-sized square units has on a measurement and how using smaller units can help make a better approximation.
4. To deepen and extend their understanding of area, students can explore the effects of a change in dimension, surface area, and volume on the other attributes of a three-dimensional object.

Problem A2

Answers will vary. To deepen students’ understanding of area, you can have them examine the effect that changing dimensions will have on the surface area of a rectangular prism. Give students 24 unit cubes and challenge them to make a rectangular solid that has the least possible surface area and one that has greatest possible surface area.

Problem A3

Even older students can benefit from having the experience of physically measuring. Just as number play helps students develop number sense, measuring helps them develop measurement or unit sense. Measurement activities also help students see measurement shortcuts and develop measurement formulas. For example, filling a rectangular box with layers of unit cubes can help students see that a layer is equal to the area of the base, and also that volume can be determined by multiplying the area of the base by the height. The reason for conducting an activity like going outside to measure shadows is that it allows students to see how to apply indirect measurement concepts to the solution of problems. The skills involved relate measurement ideas to proportionality, which is another important topic for middle school students. Hands-on experiences like these help students form a strong conceptual foundation upon which to develop more abstract and complex forms of thinking and analysis.