Private: Learning Math: Measurement
Classroom Case Studies, 6-8 Part B: Reasoning About Measurement (40 minutes)
Session 10: 6-8, Part B
In This Part
- Exploring Standards
- Analyzing a Case Study
The National Council of Teachers of Mathematics (NCTM, 2000) has identified measurement as a strand in its Principles and Standards for School Mathematics. In grades pre-K-12, instructional programs should enable all students to do the following:
- Understand measurable attributes of objects and the units, systems, and processes of measurement
- Apply appropriate techniques, tools, and formulas to determine measurements
In grades 6-8 classrooms, students are expected to do the following:
- Understand both metric and customary systems of measurement
- Understand relationships among units and convert from one unit to another within the same system
- Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume
- Use common benchmarks to select appropriate methods for estimating measurements
- Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision
- Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles, and develop strategies to find the area of more complex shapes
- Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders
- Solve problems involving scale factors, using ratio and proportion
- Solve simple problems involving rates and derived measurements for such attributes as velocity and density
The NCTM (2000) Measurement Standards suggest that “frequent experiences in measuring surface area and volume can also help students develop sound understandings of the relationships among attributes and of the units appropriate for measuring them. For example, some students may hold the misconception that if the volume of a three-dimensional shape is known, then its surface area can be determined. This misunderstanding appears to come from an incorrect over-generalization of the very special relationship that exists for a cube: If the volume of a cube is known, then its surface area can be uniquely determined. For example, if the volume of a cube is 64 cubic units, then its surface area is 96 square units. But this relationship is not true for rectangular prisms or for other three-dimensional objects in general. … Students can reap an additional benefit by considering how the shapes of rectangular prisms with fixed volume are related to their surface area. By observing patterns in the tables they construct for different fixed volumes, students can note that prisms of a given volume that are cubelike (i.e., whose linear dimensions are nearly equal) tend to have less surface area than those that are less cubelike” (NCTM, 2000, pp. 242-243).
Principles and Standards for School Mathematics © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or redistributed electronically or in other formats without written permission from NCTM. standards.nctm.org
Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.
Analyzing a Case Study
To continue the exploration of what measurement topics look like in a classroom at your grade level, you will watch a video segment of a teacher who took the Measurement course and then adapted the mathematics to his own teaching situation. We will begin by looking at some of the content addressed in the videotaped lesson. Note 3
In the videotaped lesson, Mr. Cellucci challenges students to construct a rectangular prism with a volume of 72 cm3. Students must find a prism with this volume that has the smallest and largest surface area. Generate a list of dimensions for rectangular solids that have a volume of 72 cm3, and then calculate the surface area for each of the different solids. What do you notice about the relationship between the dimensions of a prism and its resulting surface area?
Mr. Cellucci chose the volume of the rectangular prism to be 72 cm3 as compared to 71 or 73 cm3. What might be the purpose of using 72 cm3 as the volume?
As you watch this video segment, think about how both the lesson and the teacher are assisting students in making sense of the relationships between surface area and volume described in the NCTM Standards.
You can find this segment on the session video approximately 17 minutes and 26 seconds after the Annenberg Media logo.
- Students use different methods to discover the dimensions that result in a solid with a volume of 72 cm3. What problem-solving strategies are students using to find the shape with the smallest surface area?
- Having found the shape with the smallest surface area, the students draw the net of the solid on a white board and build the solid from paper. What are the instructional benefits of examining this solid as a two-dimensional net and a three-dimensional solid?
Mr. Cellucci also asks his students to find the rectangular solid with the largest surface area. How does this task support the NCTM Measurement Standards?
This lesson focuses on the fact that if volume remains constant (in this case, 72 cm3), the surface area of shapes constructed with that volume can vary. Do you think the mathematical purpose of the lesson is clear? What other factors make a lesson successful?
Often we want students to generalize what they have learned. How did Mr. Cellucci use a summary discussion to move students toward generalizing? What generalizations did his students mention?
The purpose of the video segments is not to reflect on the teaching style of the teacher portrayed. Instead, look closely at the methods the teacher uses to bring out the ideas of measurement while engaging his students in activities.
With a Volume of 72 cm3
|1 by 1 by 72||290 cm3|
|1 by 2 by 36||220 cm3|
|1 by 8 by 9||178 cm3|
|2 by 2 by 18||152 cm3|
|2 by 4 by 9||124 cm3|
|3 by 4 by 6||108 cm3|
|4 by 4 by 4.5||104 cm3|
|4.16 by 4.16 by 4.16||103.8336 cm3|
Volume does not uniquely determine the size of a rectangular prism. In terms of size, as the dimensions of the rectangular prism become more similar, the surface area decreases. Said another way, the surface area of a rectangular prism is minimized as its shape approaches a cube.
Since 72 has many factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), Mr. Cellucci’s choosing 72 cm3 for the rectangular prism’s volume allowed for a range of rectangular prisms to be constructed.
- Some students noticed that rectangular prisms that looked more like cubes had smaller surface areas. Other students focused on the dimensions and saw that as the dimensions “got closer,” the surface area “got smaller.” One group hypothesized that a perfect cube would minimize surface area, and then set out to determine the necessary dimensions and to construct that cube.
- Drawing the two-dimensional net and three-dimensional solid helped students assign appropriate dimensions and visualize what the rectangular prism would look like before they constructed it. Drawing the net of the three-dimensional solid was also helpful in understanding and determining surface area. Students saw that the total area of the two-dimensional net was the surface area of the rectangular prism and that the net is, in effect, “wrapped around” the solid. By working with the nets, students also focused on the number of identical faces and started to realize they could find the area of certain faces and then multiply to get the total surface area.
One of the standards listed for grades 6-8 is “Develop strategies to determine the surface area and volume of selected prisms,…” which this problem asks students to do (repeatedly). They must determine surface areas, and they must, in a sense, determine volumes in reverse — using a volume and coming up with dimensions of a prism that yield it. In addition, this problem directly addresses the common misconception pointed out in the standards and cited in the previous part, “Exploring Standards.”
The mathematical purpose of the lesson is clear. Students understand that they can create a range of rectangular prisms with the same volume. Some of the factors that make the lesson successful include the following:
- The lesson is hands-on. Students get to measure, construct, and fill the prisms with rice.
- Students organized data in a table and looked for patterns. Some students will more readily notice number patterns than geometric patterns.
- Students moved from making two-dimensional nets to three-dimensional solids. Creating nets for the rectangular prisms helped students to visualize them first.
The lesson allowed students to enter into the task at a variety of points. Mr. Cellucci also underscored the practical applications by suggesting situations in which you would want to be “economical” and minimize packaging materials.
In the summary discussion, Mr. Cellucci helped students focus on the dimensions, as well as the surface area, of their rectangular prism by prompting them for the dimensions and recording them in a chart. This allowed students to focus on relationships among the dimensions and between dimensions and surface area. The resulting generalizations were the following:
- As the dimensions get closer or the “lengths more similar,” the surface area gets smaller.
- A cube will have the smallest surface area.
- You can find the dimensions of a cube by taking the cube root of the volume. (This was generalized from the students’ understanding of dimensions and area of a square and of taking a square root.)
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.