# Lesson 3

Rational and Irrational Numbers

### Lesson Narrative

In previous lessons, students learned that square root notation is used to write the side length of a square given the area of the square. For example, a square whose area is 17 square units has a side length of $$\sqrt{17}$$ units.

In this lesson, students build on their work with square roots to learn about a new mathematical idea, irrational numbers. Students recall the definition of rational numbers (MP6) and use this definition to search for a rational number $$x$$ such that $$x^2 = 2$$. Students should not be left with the impression that looking for and failing to find a rational number whose square is 2 is a proof that $$\sqrt{2}$$ is irrational; this exercise is simply meant to reinforce what it means to be irrational and to provide some plausibility for the claim. Students are not expected to prove that $$\sqrt{2}$$ is irrational in grade 8, and so ultimately must just accept it as a fact for now.

In the next lesson, students will learn strategies for finding the approximate location of an irrational number on a number line.

### Learning Goals

Teacher Facing

• Comprehend the term “irrational number” (in spoken language) to mean a number that is not rational and that $\sqrt{2}$ is an example of an irrational number.
• Comprehend the term “rational number” (in written and spoken language) to mean a fraction or its opposite.
• Determine whether a given rational number is a solution to the equation $x^2=2$ and explain (orally) the reasoning.

### Required Preparation

It would be useful throughout this unit to have a list of perfect squares for easy reference. Consider hanging up a poster that shows the 20 perfect squares from 1 to 400. It is particularly handy in this lesson.

### Student Facing

• I know what an irrational number is and can give an example.
• I know what a rational number is and can give an example.

Building On

Building Towards

### Glossary Entries

• irrational number

An irrational number is a number that is not a fraction or the opposite of a fraction.

Pi ($$\pi$$) and $$\sqrt2$$ are examples of irrational numbers.

• rational number

A rational number is a fraction or the opposite of a fraction.

Some examples of rational numbers are: $$\frac74,0,\frac63,0.2,\text-\frac13,\text-5,\sqrt9$$

• square root

The square root of a positive number $$n$$ is the positive number whose square is $$n$$. It is also the the side length of a square whose area is $$n$$. We write the square root of $$n$$ as $$\sqrt{n}$$.

For example, the square root of 16, written as $$\sqrt{16}$$, is 4 because $$4^2$$ is 16.

$$\sqrt{16}$$ is also the side length of a square that has an area of 16.