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.

Student Facing

Let’s learn about irrational numbers.

Required Materials

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.

Learning Targets

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.

CCSS Standards

Building On

Addressing

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.