# ## Units of Measure

Measurement
Measurement is the process of determining a number that represents a particular attribute of an object. This process begins by defining a “unit object” that has a measure of 1. Once a unit of measure is defined, measurement is the process of determining how many copies of this unit fit, without overlap, along a side of the given object. A number that specifies a quantity in terms of a unit of measurement is called a denominate number.

Suppose a line segment S is designated as a segment of unit length, with a measure of 1. Then the length of another line segment T is determined by how many copies of S, or parts of S, fit alongside T without overlap. In the illustration below, T has a length of 2 units. The accuracy of a measurement depends upon the size of the unit of measure being used. The smaller the unit, the more precise the measurement. For instance, a measurement rounded to the nearest sixteenth of an inch is more precise than a measurement rounded to the nearest eighth of an inch.

Abbreviations are sometimes used in denominate numbers. These abbreviations do not have periods after them, except that “in.” is used as the symbol for inch to avoid confusion with the word in. These abbreviations are used for both singular and plural units of measure, so “ft” is the symbol for foot and feet, and “m” is the symbol for meter and meters. Students need to memorize the abbreviations of measures studied in this chapter. For area and volume, the abbreviations used involve an exponent of 2 or 3, such as 4 in.² (4 square inches) for area and 4 ft³ (4 cubic feet) for volume. The use of abbreviations such as “sq. in.” or “cu. ft” is discouraged.

Area is measured in unit squares. A unit square is a square where each side has a length of one unit. In this chapter, the area of rectangles is studied. A rectangle that is 2 units by 3 units has an area of 6 square units because it can be covered by 6 unit squares without any overlap. A rectangle that is m units by n units has an area of m × n square units. Students should realize that area measurements always contain the word square. Thus, square feet, square yards, and square meters all refer to area while feet, yards, and meters refer to length.

Customary System of Measurement
In the customary system of measurement used in the United States, the common units to measure length include inch, foot, yard, and mile. Students should know the following equivalencies and abbreviations. Remind students that abbreviations do not take the plural s.

Length
12 inches (in.) = 1 foot (ft)
3 feet (ft) = 1 yard (yd)
36 inches (in.) = 1 yard (yd)
5,280 feet (ft) = 1 mile (mi)
1,760 yards (yd) = 1 mile (mi)

When the length and height of a rectangle are known, the perimeter and the area of the rectangle can be found. The perimeter of a rectangle (the distance around it) can be found either by finding the sum of the lengths of the sides or by using the formula P = 2l + 2w. Since perimeter is a length or distance, the unit of measure is inches, feet, yards, or miles. The area of the rectangle is found by using the formula A = l × w. Perimeter
P = 2l + 2w
= 2(6) + 2(9)
= 30
The perimeter is 30 feet or 30 ft.
Area
A = l × w
= 6 × 9
= 54
The area is 54 square feet or 54 ft².

In the customary system of measurement, the common units of measure and abbreviations for weight and capacity are shown below. Students should know the following equivalencies and abbreviations.

Weight
16 ounces (oz) = 1 pound (lb)
2,000 pounds (lb) = 1 ton (T)

Capacity
8 fluid ounces (fl oz) = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)

Metric System of Measurement
The metric system of measurement is based on 10 and powers of 10. The prefixes used for length, capacity, and mass tell what part of the basic unit is being considered.

 kilo- hecto- deka- deci- centi- milli- 1,000 100 10 0.1 0.01 0.001 kilo- hecto- deka- deci- centi- milli- k h da d c m as in km for kilometers as in hm for hectometers as in dam for dekameters as in dm for decimeters as in cm for centimeters as in mm for millimeters

To change from one unit to another, multiply or divide by a power of 10. To change from a larger unit to a smaller unit, multiply by the appropriate power of 10. To change from a smaller unit to a larger unit, divide by the appropriate power of 10.

Examples:
Change centi- to kilo-.
Think: small to large, so divide
Count 5 moves left, so divide
by 105, or 100,000.
Change deka- to deci-.
Think: large to small, so multiply
Count 2 moves right, so multiply
by 102, or 100.

The common unit lengths include millimeters, centimeters, decimeters, meters, and kilometers. Students should know the following equivalencies and abbreviations. Remind students that abbreviations do not take the plural s.

Length
10 millimeters (mm) = 1 centimeter (cm)
1,000 millimeters (mm) = 1 meter (m)
10 centimeters (cm) = 1 decimeter (dm)
100 centimeters (cm) = 1 meter (m)
10 decimeters (dm) = 1 meter (m)
1,000 meters (m) = 1 kilometer (km)

In the metric system of measurement, the common units of measure and abbreviations for mass* and capacity are shown below. Students should know the following equivalencies and abbreviations.

Mass
1,000 milligrams (mg) = 1 gram (g)
1,000 grams (g) = 1 kilogram (kg)
1,000 kilograms (kg) = 1 metric ton (t)

Capacity
1,000 milliliters (mL) = 1 liter (L)
10 deciliters (dL) = 1 liter (L)

*Note that the metric system uses a measure of mass rather than weight. Often these terms are used interchangeably, but there is a difference. Mass measures the amount of matter in an object. Weight measures the gravitational pull on the object. In space an astronaut, who is “weightless” still has the same mass as on Earth.

Time
When adding or subtracting denominate numbers involving time, students must remember that there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. When finding elapsed time, it is also important for them to notice whether the times are a.m. or p.m.

Teaching Model 6.6: Add and Subtract Measurements