Measurement of Volume
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While most of our discussion about measurement is in terms of linear measurement, we need to have skill and understanding of other forms of measurement as well. One of the more common attributes that we often measure is capacity (or volume). In our cooking we often add ingredients in terms of cups or fractions of cups. We purchase gasoline, water, and paint in gallons. Fresh fruit from a farm market is packaged in pints, quarts, pecks, and bushels. Bottled water, milk, and many other drinks are packaged in liters, quarts, ounces, or gallons. On a more abstract and sophisticated level, we can compute the volume of air in our home in order to do an energy audit (to determine ways to save money on heating and cooling costs). There are multiple examples in daily life where we need to measure and calculate volume. But how do we measure volume? How do we assign a numeric value to this abstract attribute?
We can, of course, sometimes measure volume in a direct way by pouring water or some other substance into a marked measuring cup. This rather straightforward technique conceals the difficulty of measuring volume, however. How did the manufacturer of the measuring cup know exactly where to make the marks to designate the specific quantities? Like so many other areas of mathematics, the issue boils down to units. What are the units of measure for volume? Since there is a direct relationship between volume of a substance and the weight of that substance, we can understand that units of volume may have originated in reference to specific quantities of weight. Rowlett (2001) tells us that originally a gallon was defined as the volume of eight pounds of wheat, for example. A well-known measurement rhyme, "A pint's a pound the world around," refers to the weight of a pint of water. Volume and weight are integrally connected.
At its core, volume is a way to represent or talk about three-dimensional space. Although most of the things we want to measure are not in the shape of cubes, we can always convert space into chunks that are cubes. The cube, a three-dimensional object that has equal height, width, and length, is a useful construction for making the abstract concept of volume more tangible. When we think about measuring liquids or gases (like air), it is not too difficult to picture these substances filling a container of any shape. Then it is not too difficult to imagine manipulating the full container into a cubic shape. From there it is not too difficult to understand that we could measure the length, width, and height of that cube-shaped container in order to obtain a measurement of volume. If we measured the length, width, and height of the cubic container in terms of inches, then our resulting volume measurement would be a number of cubic (three-dimensional) inches. If our measurements of length, width, and height were in terms of feet, our resulting volume measurement would be in terms of cubic (three-dimensional) feet.
Elementary Mathematics: Pedagogical Content Knowledge, by James E. Schwartz, is designed to sharpen pre-service and in-service teachers' mathematics pedagogical content knowledge. The five "powerful ideas" (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the text thoroughly integrates discussion of the five NCTM process strands.
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