 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 2, Part A:
Measuring Accurately

In This Part: Conservation, Transitivity, and Unit Iteration | Partitioning
Partitioning on a Number Line

 Let's look more closely at the idea of a unit and how one goes about partitioning that unit into subunits. How are rational numbers (fractions and decimals) interpreted in measurement situations? Imagine that you are timing a swim meet. If you timed a 100 m backstroke race to the nearest hour, you would not be able to distinguish one swimmer's time from another's. If you refined your timing by using minutes, you still might not be able to tell the swimmers apart. If the swimmers were all well trained, you might not be able to decide on a winner even if you measured in seconds. In high-stakes competitions among well-trained athletes (the Olympics, for example), it is necessary to measure in tenths and 100ths of seconds. Now suppose that you are working on a project that requires some precision. You need to determine the exact length of a strip of metal in inches. Holding the strip up to your ruler, with one end at 0, you see that the other end lies between 4 and 5 in.: Note that only the right end of the metal strip is shown here. What would you say its length is? You might think to yourself, "The length is between 4 9/16 and 4 10/16, so I'll call it 4 19/32." These situations illustrate the measurement interpretation of rational numbers. A unit of measure can always be divided into finer and finer subunits so that you can take as accurate a reading as you need. On a number line; on a graduated beaker; on a ruler, yardstick, or meterstick; on a measuring cup; on a dial; on a thermometer -- some subdivisions of the unit are marked. The marks on these common measuring tools allow readings that are accurate enough for most general purposes, but if the amount of the object you are measuring doesn't exactly meet one of the provided hash marks, it certainly doesn't mean that you can't measure it. Rational numbers provide us with a means to measure any amount of stuff. Note 4 If meters will not do, we can partition into decimeters; when decimeters will not do, we can partition into centimeters or millimeters -- and so on. When we talk about rational numbers as measures, the focus is on successively partitioning the unit. Certainly partitioning plays an important role in other models and interpretations of rational numbers, but there is a difference. In measurement, there is a dynamic aspect; instead of comparing the number of equal parts you have to a fixed number of equal parts in a unit, the number of equal parts in the unit can vary, and what you name your fractional amount depends on how many times you are willing to keep up the partitioning process. In the above example, you've seen how the units were first divided into 16 equal parts and then into 32 equal parts (the fractional amount was thus expressed in 16ths or 32nds, respectively). If necessary, you could further partition the unit into 64 or more equal parts, each time refining the precision of your measurement. Problem A4 In your own words, clarify the difference between the measurement interpretation of rational numbers and the part-whole interpretation of rational numbers. Note 5  Part-whole interpretation of rational numbers refers to dividing one or more units into equal-sized parts. You can think of it as pieces of a pie -- 3/4 would mean three equal-sized slices from a total of four.   Close Tip Part-whole interpretation of rational numbers refers to dividing one or more units into equal-sized parts. You can think of it as pieces of a pie -- 3/4 would mean three equal-sized slices from a total of four. Problem A5 Why is the concept of partitioning so important in measurement?   Session 2: Index | Notes | Solutions | Video