Learning Math: Measurement
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.
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.
Answer the questions based on what you saw in the video:
- What concepts and skills did the teachers mention as being important for students to understand?
- What types of activities might be used to help students make sense of these concepts and skills?
- Are there related concepts or skills that will affect whether or not students can understand and use these ideas?
- 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?
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.
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?
- 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.
- 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.
- 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.
- 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.
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.
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.
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.