Fraction and Decimal Concepts
Fractions
Up to this point, students have been taught about a fraction of a whole and a fraction of a group. The fraction when applied to a whole pizza or a rectangular area means to divide the object or area into 4 equal parts and consider 3 of these equal parts. The fraction when applied to a group of 16 soccer players means to divide the number of players into 4 equal groups (4 each) and consider 3 of the 4 equal groups (12 players). In fact, the idea of a fraction of a group is really the same as that of a fraction of a whole if the 16 soccer players are considered as one whole soccer team.
Another way to help students with fraction concepts is to relate fractions to a number line, in particular, fractions of an inch on a ruler. A unit fraction is defined as a fraction with numerator 1 (for example, , , , ). In a unit fraction, , one whole unit is divided into n equal parts. One of these smaller parts is the amount represented by the unit fraction. On the number line, the unit fraction represents the length of a segment when a unit interval on the number line is divided into n equal segments. The point located to the right of 0 on the number line at a distance from 0 will be .
The fraction can represent the quotient of m and n, or m ÷ n. If the fraction is defined in terms of the unit fraction , the fraction means m unit fractions . In terms of distance along the number line, the fraction means the length of m abutting segments each of length . The point is located to the right of 0 a distance m × from 0. The numerator of the fraction tells how many segments. The denominator tells the size of each segment.
A straightforward way to show that fractions represent a solution to a division problem is by using equivalent fractions. What is 35 ÷ 7? It is 5 because 35 equals 7 × 5. What is 5 ÷ 7? This is more difficult because 5 is not a multiple of 7. However, 5 = = 5 × × 7 = , and equals 35 unit fractions of . Just as 35 divided by 7 is 5, 35 unit fractions of divided by 7 is 5 unit fractions of . So 5 ÷ 7 = .
Equivalent Fractions
From the definition of fraction, students should understand that for each counting number m, m = and = 1. So there are many different names for the same point on the number line.
Equivalent fractions are fractions that represent the same point on the number line. As shown above for the number 1, there are many fractions that name the same amount. The number can represent the quantity of pizza given by 3 parts of a pizza cut into 4 equal parts. If each of the 4 parts is in turn cut into 3 equal parts, the pizza has been cut into 4 × 3 = 12 equal parts. The same quantity of pizza can now be written as 3 × × 3 = . This principle is summed up in the cancellation law for fractions that states m × × r = .
The cancellation law for fractions does not require any knowledge about multiplication of fractions. Once multiplication of fractions is understood, the cancellation law for fractions can be expressed as m × × r = × = × 1 = .
Simplest Form
A fraction is in simplest form if the numerators and denominators are as small as possible. A more formal way of stating this is to say that the numerator and denominator have no common factors other than 1. The usual way of simplifying fractions is to find the GCF of the numerator and denominator and then divide both the numerator and denominator by that GCF. Other ways include canceling all factors common to the numerator and denominator, dividing the numerator and denominator by the same number as many times as possible, or taking the prime factorizations of the numerator and denominator and canceling common factors.
Example:
Comparing and Rounding Numbers
An important application of place value is its use in comparing numbers. To compare decimals, students begin by comparing digits in each place value, starting from the left, until different digits appear. The number with the greater digit in that place is the greater number. For example, when comparing 0.238 and 0.2319, students note that the digits in the tenths and hundredths places are the same, but the digits in the thousandths place are different. The fact that 8 > 1 implies that 0.238 > 0.2319. Students should also note that 8 > 1 can be written as 1 < 8, which implies that 0.2319 < 0.238.
Place-value concepts are also applied when rounding whole numbers and decimals. When rounding a decimal to the nearest tenth, students look at the digit in the place to its right, the hundredths place. If the digit is 5 or more, the number is rounded to the next higher tenth; otherwise the digit in the tenths place is not changed. When rounding whole numbers, all places to the right of the rounded digit are changed to zeros. For example, 238,574 rounded to the nearest thousand is 239,000; to the nearest hundred is 238,600; and to the nearest ten is 238,570. When rounding to a decimal place, all digits to the right of the rounded place are dropped. For example, 6.0835 rounded to the nearest tenth is 6.1, to the nearest hundredth is 6.08, and to the nearest thousandth is 6.084.
Addition and Subtraction of Decimals
As students have learned in earlier grades, the use of a base-ten positional number system for writing numbers allows for the development of powerful algorithms for arithmetic operations. The algorithms that are used for whole numbers can also be used for the addition and subtraction of decimals. Digits are aligned according to place value, which means that the decimal points should be aligned. Then computation is completed from right to left.
Example: Add 33.78 + 29.83
Add the hundredths.
11 hundredths =
1 tenth + 1 hundredth
Add the tenths.
16 tenths =
1 one + 6 tenths
Add the ones.
13 ones =
1 ten + 3 ones
Add the tens.
When subtracting, appending one or more zeros after the decimal point may make computation easier.
Example: Subtract 17.32 from 31.5
Append a zero.
Regroup.
Subtract the hundredths.
Subtract the tenths.
Regroup.
Subtract the ones.
Subtract the tens.
Comparison of Fractions
Which is greater, or ? When comparing fractions with like denominators, it is enough to compare the numerators. Since 4 unit fractions of is greater than 3 unit fractions of , is greater than . Which is greater, or ? When comparing fractions with unlike denominators, it is usually necessary to find a common denominator. In the example, any common multiple of 12 and 8 will do. For simplicity, choose 12 × 8 = 96.
Now, = 7 × × 8 = and = 5 × × 12 = . Since 60 > 56, > .
Should students want to use only the least common denominator, suggest they try it both ways and decide for themselves which method is quicker or easier.
Repeating and Terminating Decimals
All fractions represent division and thus can be written as decimals. However, not all division ends, or terminates. A terminating decimal occurs when the division has a remainder of 0. An example is = 1 ÷ 2 = 0.5. Fractions like cannot be represented by a terminating decimal. The decimal equivalent for is 0.333…. This means the threes continue on and on indefinitely since there is always a remainder of 1. A complete explanation of why = 0.333… requires an understanding of infinite sums and is beyond most children in elementary school. However, students do understand patterns and can see the patterns that occur in repeating decimals. A repeating decimal occurs when a pattern of numbers repeats indefinitely after the decimal point, and the numbers are not all zeros. In writing repeating decimals, a bar is placed over the pattern of numbers that repeats. Thus,
Decimals
The fact that our decimal notation for whole numbers can be extended to represent rational numbers has many important consequences. Key to such an extension is the use of a decimal point to the right of the digit in the ones place. The place-value rule that each digit has a place value equal to 10 times that of the digit to the right extends to digits to the right of the decimal point. Each digit has a place value that is equal to one-tenth of the place value of the digit to its left. This can be represented as follows.
Ones | . | Tenths | Hundredths | Thousandths | Ten Thousandths |
---|---|---|---|---|---|
1 10^{0} = 1 |
× 1 or 10^{-1} = or 0.1 |
× or 10^{-2} = or 0.01 |
10 × 100 or 10^{-3} = or 0.001 |
10 × 10 or 10^{-4} = or 0.0001 |
The decimal 6.0542 is read six and five hundred forty-two ten thousandths. Note that the decimal point is read as “and.” When writing numbers that are less than zero, a 0 is usually placed to the left of the decimal point, such as in 0.572. However, the zero is not read; 0.572 is read as five hundred seventy-two thousandths.
Every decimal can be written as a fraction (3.75 = 3 ).
Teaching Model 4.3: Compare and Order Fractions and Decimals