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
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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.
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Comprehend the term “rational number” (in written and spoken language) to mean a fraction or its opposite.
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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
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I know what an irrational number is and can give an example.
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I know what a rational number is and can give an example.
CCSS Standards
Glossary Entries
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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.
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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
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