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# Fundamentals of Measurement

## Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.

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### In This Session

Part A: Measuring Accurately
Part B: The Role of Ratio
Part C: Precision and Accuracy
Homework

In Session 1, you began to explore what it means to measure. In this session, you will investigate the difference between a count and a measure and examine some of the important ideas essential to meaningful measurement. You will learn about the many uses of ratio in measurement and how scale models can help us understand relative sizes. Finally, you will investigate the constant of proportionality in isosceles right triangles and learn about precision and accuracy in measurement.

For the list of materials that are required and/or optional in this session, see Note 1.

### Learning Objectives

In this session, you will do the following:

• Understand the three components of measuring: conservation, transitivity, and unit iteration
• Consider the effects of partitioning a unit into smaller subunits
• Understand the use of proportional reasoning and ratio in measurement situations
• Use accuracy and precision to determine how much error is involved in any given measurement

### Key Terms

Previously Introduced

Measurement: Measurement is the process of quantifying properties of an object by comparing them with a standard unit.

Precision: The precision of a measuring device tells us how finely a particular measurement was made.

New in This Session

Accuracy: The accuracy of a measure (an approximate number) refers to the ratio of the size of the maximum possible error to the size of the number. This ratio is called the relative error. We express the accuracy as a percent, by converting the relative error to a decimal and subtracting it from 1 (and writing the resulting decimal as a percent). The smaller the relative error, the more accurate the measure.

Compensatory Principle: The compensatory principle states that the smaller the unit used to measure the distance, the more of those units that will be needed. For example measuring a distance in centimeters will result in a larger number of that unit than measuring a distance in kilometers.

Conservation: Conservation is the principle that an object maintains the same size and shape if it is repositioned, or divided in certain ways.

Partitioning: Partitioning is the division of something into parts. In measurement, partitioning is done with units: A meter is divided into centimeters, a gallon is divided into quarts, and so on. The level of partitioning used in a measurement affects the precision of that measurement. For example, a measurement taken with a meter stick divided into centimeters is more precise than a measurement taken with an unmarked meter stick.

Ratio: A ratio is a comparison between two quantities. A measurement is a type of ratio — it is a comparison with a unit. When we state that an object is eight inches long, we mean in comparison to the unit of one inch.

Scale: The scale used on a map or model is an example of a measurement ratio. A map with a scale of 1:250 indicates that one unit on the map is equal to 250 units in the actual distances represented by the map.

Scale Factor: A scale factor is a constant used to enlarge or reduce a figure. For example, if the sides of a triangle are enlarged to twice the length of the original triangle, we say the scale factor is 2.

Transitivity: Transitivity is a mathematical property stating that if A and B satisfy a relation and B and C satisfy the same relation, then A and C also satisfy the relation. Common examples include equality comparisons (if A = B and B = C, then A = C), inequality comparisons (if A < B and B < C, then A < C), and parallelism (if Line A is parallel to Line B, and Line B is parallel to Line C, then Line A is parallel to Line C). Transitivity allows objects to be compared indirectly, and allows measurements to be consistent in comparisons.

Unit Iteration: Unit iteration is the repetition of a single unit for a measurement. For example, someone wishing to measure the length of a field with only a meter stick would need to use unit iteration.

### Notes

Note 1

Materials Needed:

• Centimeter ruler