Learning Math: Measurement
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.
Answer the following questions based on your experiences in the course and what you saw in the video:
- What concepts and skills did the teachers mention were important for students to understand?
- Based on your own experiences, what content do students find difficult when studying area?
- What types of activities might be used to help students make sense of these concepts and skills?
- 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?
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.
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?
- 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
- Answers will vary. Students may have difficulty understanding measurement relationships, measurement formulas, indirect measurement, and the ideas of accuracy and precision.
- 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.
- 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.
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.
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.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.