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# Fractions and Decimals

## Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.

In This Session:
Part A:
Fractions to Decimals
Part B:
Decimals to Fractions
Part C:
Ordering Fractions
Homework

In this session, you will explore the relationships between fractions and decimals and learn how to convert fractions to decimals and decimals to fractions. You will also learn to predict which fractions will have terminating decimal representations and which will have repeating decimal representations.

If you think about fractions and their decimal representations together, there are many patterns you can observe (which are easy to miss if you only think about them separately).

LEARNING OBJECTIVES
In this session, you will do the following:
• Understand why every rational number is represented by either a terminating decimal or a repeating decimal
• Learn to predict which rational numbers will have terminating decimal representations
• Learn to predict the period — the number of digits in the repeating part of a decimal — for rational numbers that have repeating decimal representations
• Understand how to convert repeating decimals to fractions
• Understand how to order fractions without converting them to decimals or finding a common denominator

### Key Terms

Previously Introduced:

prime number

A counting number is a prime number if it has exactly two factors: 1 and the number itself. For example, 17 is prime, 16 is not prime, and 1 itself is not prime, since it has only one factor.

rational numbers

Rational numbers are numbers that can be expressed as a quotient of two integers; when expressed in a decimal form they will either terminate (1/2 = 0.5) or repeat (1/3 = 0.333…)

New in This Session:

period

The period of a repeating decimal is the total number of digits in the group of digits that repeats. For example, 0.123123123. . . has a period of three digits (the repeating part is “123”), while 0.06151515. . . has a period of two digits (“15” — the “06” does not repeat and is not part of the period).

repeating decimal

A repeating decimal is a decimal that does not terminate but keeps repeating the same pattern. For example, 0.123123123. . . is a repeating decimal; the “123” will repeat endlessly. Any repeating decimal is equal to a rational number. For example, 0.123123. . . is equal to 123/999, or 41/333.

terminating decimal

A terminating decimal is a decimal that comes to a finite end, rather than repeating. For example, 0.5 and 0.381 are terminating decimals, while 0.123123123. . . is not. Any terminating decimal is equal to a rational number. For example, 0.381 is equal to 381/1,000.