 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 2, Part A:
Measuring Accurately (45 minutes)

In This Part: 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.

 a. Why might this aspect of measurement cause confusion? b. 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.   Session 2: Index | Notes | Solutions | Video