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- Conservation, Transitivity, and Unit Iteration
- Partitioning
- Partitioning on a Number Line

In Session 1, we established that in order to measure something, we have to (1) select an attribute of the thing to be measured; (2) choose an appropriate unit of measure; and (3) determine the number of units. In conjunction with these three steps, many educators have noted that there are three components of measuring that contribute to students’ ability to make meaningful and accurate measurements: conservation, transitivity, and unit iteration.** Note 2**

**Conservation
**Conservation is the principle that an object maintains the same size and shape even if it is repositioned or divided in certain ways. If you understand this principle, you realize that a pencil’s length remains constant when it is placed in different orientations. For example, two pencils that are the same length remain equal in length when one pencil is placed ahead of the other:

You also realize that two differently shaped figures have the same area if they have the same component pieces. For instance, a jigsaw puzzle covers the same amount of space whether the puzzle is completed or in separate pieces.

**Transitivity
**When you can’t compare two objects directly, you must compare them by means of a third object. To do this, you must intuitively understand the mathematical notion of transitivity (if A = B and B = C, then A = C; if A < B and B < C, then A < C; if A > B and B > C, then A > C).

For example, to compare the length of a bookshelf in one room with the length of a desk in another room, you might cut a string that is the same length as the bookshelf. You can then compare the piece of string with the desk. If the string is the same length as the desk, then you know that the desk is the same length as the bookshelf.

Developmentally, conservation precedes the understanding of transitivity, because you must be sure that a tool’s length (area, volume, etc.) will stay the same when moved in the process of measuring.

**Unit Iteration**

In order to determine the correct unit for measurement, you must understand the attribute you are measuring. For instance, when measuring distance, a linear measurement is appropriate. When measuring area, you need two-dimensional units, such as squares, to cover the surface. When measuring volume, you need a three-dimensional unit.

Another key point to grasp is that the chosen unit influences the number of units. For example, weighing a package in grams results in a larger number of units (2,000 g) than weighing it in kilograms (2 kg). This inverse relationship — a larger number of smaller units — is a conceptually difficult idea.

Unit iteration is the repetition of a single unit. If you are measuring the length of a desk with straws, it is easy enough to lay out straws across the desk and then count them. But if only one straw is available, then you must iterate (repeat) the unit (straw). You first have to visualize the total length in terms of the single unit and then reposition the unit repeatedly.

**Problem A1**

Counts of a number of objects are exact (e.g., you can have either three chairs or four chairs around the table, not between three and four chairs), yet measurements cannot be made exactly. Why is that so? What makes a count different from a measure? **Note 3**

**Problem A2**

The units on measurement instruments, such as rulers and thermometers, run together; they are not distinct as are, for example, the number of books on a shelf.

- Why might this aspect of measurement cause confusion?
- How is understanding a length of 7 in. or a temperature of 63 degrees Farenheit different from understanding that you have seven balloons or 63 pennies?

**Problem A3**

Where else in mathematics is the concept of transitivity used? Give an example other than measurement.

“Conservation, Transitivity, and Unit Iteration” adapted from Chapin, S. and Johnson, A. Math Matters: Understanding the Math You Teach, Grades K-6. pp. 178-180. © 2000 by Math Solutions Publications. Used with permission. All rights reserved.

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.

**Problem A5**

Why is the concept of partitioning so important in measurement?

“Partitioning” adapted from Lamon, Susan J. *Teaching Fractions and Ratios for Understanding.* pp. 113-121. © 1999 by Lawrence Erlbaum Publishers. Used with permission. All rights reserved.

How many partitions of a number line are possible?

To use a rational number to describe how far a point on the number line is from 0, you can begin by partitioning the unit interval into an arbitrary number of equal parts. Each of those parts can then be partitioned into an arbitrary number of equal parts, and those, in turn, can be partitioned again.

This process is actually a composition of operations. You can use arrow notation to keep a record of your partitioning actions, as well as the size of the subintervals being produced.

For example, what if you wanted to locate 17/48 on a number line from 0 to 1? You would start by drawing the number line on a piece of paper and repeatedly folding it, making sure to mark the locations of 0 and 1 before you start folding:

Here’s one set of partitioning actions to find 17/48:

Video SegmentIn this video segment, the participants place a fractional value on a number line using the method of partitioning. They explore the reciprocal relationship that exists between partitioning and the number of units in a measure.Is there more than one way to do the partitioning to arrive at a particular fraction?You can find this segment on the session video approximately 2 minutes and 40 seconds after the Annenberg Media logo. |

**Problem A6**

Try these partitioning tasks: Note 6

a. |
Locate 7/24: |

b. |
Locate 3/8: |

**Problem A7**

Find another way (or ways) to locate the fractions in Problem A6 (a) and (b).

Start with a new number line.

The compensatory principle states that the smaller the subunit you use to measure the distance, the more of those subunits you will need; the larger the subunit, the fewer you will need. When multiples of two different subunits cover the same distance, different fraction names result. There is only one rational number associated with a specific distance from 0, so these fractions are equivalent. **Note 7**

For example, when measuring the diameter of a pencil using two different subunits, we would have the following:

But 1/4 and 2/8 are equivalent fractions, so these are the same measurements.

**Problem A8**

State the compensatory principle in your own words. What type of relationship exists between the size of a measuring unit and the number of that unit needed to measure a property?

**Note 2**

Take a few minutes to read the information about conservation, transitivity, and unit iteration. Whereas adults conserve measures, we can sometimes become confused (as with the tangram activity in Session 1) by a visual image. Transitivity is used in algebra and geometry (for example, as justification for steps in a proof) as well as in measurement, when comparing the equality of a number of measures. Examining the concept of units leads us to consider the kind of units that are used when we count versus when we measure.

**Note 3**

If you are working in a group, discuss Problems A1-A3 together. When discussing Problem A2, consider the fact that young children first learn about numbers using discrete quantities. How does that differ from measurement, which is never exact (discrete), as we can infinitely divide continuous quantities?

**Note 4**

Rational numbers are what is known as a dense set: A dense set is such that for any two elements you choose, you can always find another element of the same type between the two.

To learn more about the concept of density, go to *Learning Math: Number and Operations,* Session 2.

**Note 5**

To learn more about rational numbers and the part-whole interpretation of fractions, go to *Learning Math: Number and Operations,* Session 8.

**Note 6**

If you are working in a group, work in pairs on both parts of Problem A6. First use the fraction given to find one unit; then consider how you can use partitioning and equivalence to locate the desired fraction.

**Note 7**

The compensatory principle is an important mathematical idea. The idea of an inverse relationship between the size of a unit and the number of units can be examined numerically (e.g., the area of a surface that is 1 m^{2} can also be expressed as 10,000 cm^{2}). An inverse relationship can also be shown graphically. A linear inverse relationship produces a straight line that is drawn diagonally from the upper left to the lower right in the first quadrant. Be sure to reflect on or discuss other inverse relationships when working on Problem A8.

**Problem A1**

Counts are exact; they are not on a scale, nor are they ratios. In a count, the unit is absolute. In contrast, measurements are not exact; the units are relative, and typically they don’t directly match what we’re measuring. For example, a person’s height, measured in centimeters, is very unlikely to be an exact number of centimeters, so we approximate. A measurement is continuous, not discrete; someone can be 180 cm tall, 181 cm tall, or any number in between.

**Problem A2**

- It is not possible to just “count” inches or centimeters, since the result of a measurement may not be an exact number in those units. Also, depending on the measuring device used, the unit of the measurement can change; for example, the same measurement could be expressed as 6 (in.), 0.5 (ft.) or 1/6 (of a yard).
- Again, it’s a question of relative vs. absolute: When we hear that the temperature is 63 degrees, this means that the temperature has been rounded off to the nearest whole number. When we count that we have 63 pennies, there is no rounding off; we have exactly 63 pennies.

**Problem A3**

Transitivity is used in many places — in parallelism, for example. If lines A and B are parallel, and lines B and C are parallel, then lines A and C are parallel.

**Problem A4**

Answers will vary. One possible answer is that in part-whole interpretation, the number of parts that the whole is divided into is predetermined, whereas in measurement, you can vary the number of equal parts according to whatever is most appropriate for your measurement situation.

**Problem A5**

Partitioning is important in measurement, because the measurements taken depend entirely on the partitioning. The example of timing a swim meet is relevant here, since the partitioning of time determines the measured times in the event (to the nearest second, 100th of a second, and so on). There are an infinite number of possible partitions of the number line, since we can always break any partition into a smaller one.

**Problem A6**

a. |
Since the number line between 0 and 1 is already partitioned into 12 equal parts, we will need to partition the 12ths into two equal parts so that each is 1/24. Then, since 1/3 = 8/24, count one partition to the left of 1/3. |

b. |
The number line between 0 and 1 is partitioned into 18 equal parts now (since 1/6 is three partitions over). To locate 3/8, partition each of the 18 parts into four equal parts so that each is 1/72 (so 3/8 = 27/72). Since 1/6 = 12/72, count over to 2/6 (i.e., 24/72), then three partitions beyond it. |

**Problem A7**

Answers will vary. In either case, it is also possible to start with a new number line and make partitions different from the ones you made before. For example, to locate 7/24, you could partition the number line into thirds and then partition those into eighths, which would also result in 24ths (as 3 and 8 are both factors of 24). Other combinations with other factors of 24 are also possible.

**Problem A8**

Write and reflect