## Chapter 2: The Real Number System

**BIG IDEAS**

Daily tasks and activities make use of real numbers and their operations.

**CHAPTER OVERVIEW**

This chapter is about the numbers that are used in algebra, the operations between them, and the properties of these operations. It begins with a discussion of integers in lesson 1. The number line is introduced to clarify the concept of absolute value, and is used as a device to help explain addition and subtraction of integers. With the introduction of negative numbers, subtraction can now be expressed in terms of addition. It is a good idea to emphasize this concept and to give many exercises on this. The rules for multiplying and dividing signed numbers are discussed as well.

Lesson 2 is about rational and irrational numbers. Two equivalent definitions of rational numbers are discussed. First, rational numbers are real numbers that can be expressed as a quotient of two integers, where the divisor is not zero. Thus, a rational number can be written as a fraction. Not all fractions, however, are rational. For example, √2/3 is a fraction but it is not rational. Second, rational numbers are real numbers whose decimal form is either terminating or repeating. Real numbers that are not rational are called irrational numbers. Several examples of irrational numbers are given in the textbook.

Lesson 3 is about operations and their properties. It begins with a discussion on the use of grouping symbols in mathematics. When using variables, the product of x and y can be written as x · y, xy, (x)(y), x(y), or (x)y. However, extra care should be observed in using these notations when numbers are involved. For example, 3(–2) means the product of 3 and –2, so 3(–2) = –6. However, this cannot be written as (3) – 2 since (3) – 2 means 3 – 2, which is equal to 1. Notice that the parentheses in (3) – 2 are unnecessary.

Properties of operations are also discussed in the last lesson. In the textbook, the word opposite is used to refer to the additive inverse of a number, while the word reciprocal is used to refer to the multiplicative inverse of a nonzero real number.

The chapter ends with a review of operations on fractions.

Lesson 2 is about rational and irrational numbers. Two equivalent definitions of rational numbers are discussed. First, rational numbers are real numbers that can be expressed as a quotient of two integers, where the divisor is not zero. Thus, a rational number can be written as a fraction. Not all fractions, however, are rational. For example, √2/3 is a fraction but it is not rational. Second, rational numbers are real numbers whose decimal form is either terminating or repeating. Real numbers that are not rational are called irrational numbers. Several examples of irrational numbers are given in the textbook.

Lesson 3 is about operations and their properties. It begins with a discussion on the use of grouping symbols in mathematics. When using variables, the product of x and y can be written as x · y, xy, (x)(y), x(y), or (x)y. However, extra care should be observed in using these notations when numbers are involved. For example, 3(–2) means the product of 3 and –2, so 3(–2) = –6. However, this cannot be written as (3) – 2 since (3) – 2 means 3 – 2, which is equal to 1. Notice that the parentheses in (3) – 2 are unnecessary.

Properties of operations are also discussed in the last lesson. In the textbook, the word opposite is used to refer to the additive inverse of a number, while the word reciprocal is used to refer to the multiplicative inverse of a nonzero real number.

The chapter ends with a review of operations on fractions.

**DESIRED RESULTS**

**Content Standards**

The learner demonstrates understanding of key concepts on the real number system.

**Performance Standard/s**

The learner is able to apply various procedure and manipulations on the different subsets of the set of real numbers.

**CHAPTER OBJECTIVES**

At the end of the chapter, the students should be able to . . .

• identify numbers as natural numbers, whole numbers, integers, rational numbers, irrational numbers, or real numbers;

• determine the absolute value of real numbers;

• perform fundamental operations on integers;

• classify real numbers as rational or irrational;

• perform operations on fractions and reduce the answers to lowest terms;

• solve problems based on real-life situations involving real numbers; and

• determine the uses of real numbers in real-life situations.

• identify numbers as natural numbers, whole numbers, integers, rational numbers, irrational numbers, or real numbers;

• determine the absolute value of real numbers;

• perform fundamental operations on integers;

• classify real numbers as rational or irrational;

• perform operations on fractions and reduce the answers to lowest terms;

• solve problems based on real-life situations involving real numbers; and

• determine the uses of real numbers in real-life situations.

**PERFORMANCE TASKS**

Create a Word Problem Minibook that contains at least five word problems and the solutions for each.

**HOW TO INTRODUCE THE CHAPTER**

1. Assess the students’ prior knowledge about numbers as they learned in elementary. Ask what topics or lessons they still remember about numbers and have them briefly explain each one. Form a concept map by writing every topic or lesson that they have mentioned. Ask them to help you place the topic or lesson they have mentioned onto the concept map. Take note of gaps and misconceptions.

2. On a sheet of paper, ask the students to write their answer to the question “How useful are numbers?” Take note of the extent of their understanding on the use of real numbers.

3. Tell the students what they will learn in the chapter.

4. Discuss with the students the performance task for the chapter. Discuss with them the rubric for assessment of the performance task. (Please see Culmination/Transfer on pages 36 to 38 of this TLG.)

2. On a sheet of paper, ask the students to write their answer to the question “How useful are numbers?” Take note of the extent of their understanding on the use of real numbers.

3. Tell the students what they will learn in the chapter.

4. Discuss with the students the performance task for the chapter. Discuss with them the rubric for assessment of the performance task. (Please see Culmination/Transfer on pages 36 to 38 of this TLG.)

**Lesson 1: Integers**

Lesson 2: Rational and Irrational Numbers

Chapter Wrap-Up

Lesson 2: Rational and Irrational Numbers

Chapter Wrap-Up