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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:
In grades 6-8 classrooms, students are expected to do the following:
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.
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
Problem B1
In the videotaped lesson, Mr. Cellucci challenges students to construct a rectangular prism with a volume of 72 cm^{3}. 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 cm^{3}, 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?
Problem B2
Mr. Cellucci chose the volume of the rectangular prism to be 72 cm^{3} as compared to 71 or 73 cm^{3}. What might be the purpose of using 72 cm^{3} as the volume?
Video Segment
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. |
Problem B3
Problem B4
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?
Problem B5
This lesson focuses on the fact that if volume remains constant (in this case, 72 cm^{3}), 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?
Problem B6
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?
Note 3
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.
Problem B1
Rectangular Prisms With a Volume of 72 cm^{3} |
||
Dimensions | Surface Area | |
1 by 1 by 72 | 290 cm^{3} | |
1 by 2 by 36 | 220 cm^{3} | |
1 by 8 by 9 | 178 cm^{3} | |
2 by 2 by 18 | 152 cm^{3} | |
2 by 4 by 9 | 124 cm^{3} | |
3 by 4 by 6 | 108 cm^{3} | |
4 by 4 by 4.5 | 104 cm^{3} | |
4.16 by 4.16 by 4.16 | 103.8336 cm^{3} |
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.
Problem B2
Since 72 has many factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), Mr. Cellucci’s choosing 72 cm^{3} for the rectangular prism’s volume allowed for a range of rectangular prisms to be constructed.
Problem B3
Problem B4
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.”
Problem B5
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 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.
Problem B6
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: