## Rational Numbers

**What We Are Teaching and Why**

According to the Principles and Standards for School Mathematics (NCTM, 2000), students at sixth grade should start to deepen their understanding of various number systems. For some students that will include expanding the study of number systems to include all rational numbers. Thus far students have learned that there are different sets and subsets of numbers.

Classification | Examples | Definition |
---|---|---|

Counting Numbers | 1, 12, 332, 999, | Numbers used to count |

Whole Numbers | 0, 1, 2, 3… | Zero, and all counting numbers |

Integers | ^{–}23, ^{–}11, 0, 1, 44 |
Zero, positive whole numbers, and their opposites |

Rational Numbers | ^{–} , ^{–}0.45, 3, 5.14 |
Any number that can be written in the form , where a and b are integers and b is not equal to zero |

Students have actually studied positive rational numbers since kindergarten. The study of negative rational numbers provides a chance for middle school students to expand their understanding of our number system.

**Rational Numbers**

A rational number is a number that can be expressed in the form a ÷ b or
, where a and b are integers and b ≠ 0. They are called rational because they are the ratio, or quotient, of two integers. Rational numbers include integers as well as positive and negative fractions. Each rational number can be associated with a point on a number line.

All properties of whole numbers also hold for rational numbers. One additional property is the Additive Inverse Property, which states that the additive inverse of a number is the opposite of that number and that the sum of a number and its additive inverse is 0. In computations with rational numbers, the same rules are used as for integers.

**Teaching Model 22.4:** Order of Operations With Rational Numbers