How to Teach Rational and Irrational Numbers in 8th Grade: Free Notes

8th grade math

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If you are planning how to teach rational and irrational numbers in 8th grade, here are the notes I use with my own students, and they are free: guided notes, an investigation on estimating irrational numbers, two practice pages and a full answer key.

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How to teach rational and irrational numbers: start with the decimals

A rational number can be written as a fraction a/b of two integers, with b not zero. Its decimal either stops (0.5) or repeats (0.333...). An irrational number cannot be written as a fraction: its decimal goes on forever without repeating, like pi or the square root of 2.

So the decimal is the test, and that is where the notes start. Students turn fractions into decimals, then repeating decimals back into fractions, then meet square roots and cube roots. Once they see that the square root of a perfect square is a whole number and the square root of any other whole number is irrational, they estimate those irrational square roots without a calculator.

The first page of the Real Numbers guided notes: rational numbers, terminating and repeating decimals

The lesson at a glance

Grade: 8th grade
Standard: Common Core 8.NS.A.1: know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number
Standard: Common Core 8.NS.A.2: use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions
Standard: Common Core 8.EE.A.2: use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational
You need: the notes, investigation and practice pages, one set per student, the answer key
Students will be able to: tell rational from irrational numbers, change repeating decimals to fractions, solve x2 = p and x3 = p, and estimate irrational square roots

Get the free notes and answer key

Before you teach

Print the student pages for each student: six pages of notes, three pages of investigation and two practice pages. Keep the answer key for yourself. The notes have a table of perfect squares and perfect cubes that students use all the way through, so have them keep it handy.

Rational and irrational numbers guided notes: fractions and decimals

The notes open with the key words: rational numbers, terminating decimals and repeating decimals, with examples. Then students write fractions as decimals (the fraction bar means divided by) and decimals as fractions, with worked examples and a Your Turn after each.

The Think About It page shows how to turn a repeating decimal into a fraction with an equation. For 0.777..., let x = 0.777..., multiply both sides by 10 to get 10x = 7.777..., subtract x from both sides to get 9x = 7, so x = 7/9. One repeating digit means multiply by 10; two repeating digits means multiply by 100.

Guided notes: square roots and cube roots

Next come square root, principal square root, perfect square, cube root and perfect cube, with the table of perfect squares and cubes. Students solve equations like x2 = 36 (two answers, 6 and -6) and x3 = 125 (one answer, 5), including ones with fractions.

Let's investigate: estimating irrational numbers without a calculator

The three-page investigation walks students through a way to estimate a square root that is not a whole number. Here it is with the square root of 20:

20 is not a perfect square, so the square root of 20 is irrational.
Find the two perfect squares it sits between: 16 < 20 < 25.
Take their square roots: the square root of 20 is between 4 and 5.
Find the fraction: (20 - 16) / (25 - 16) = 4/9.
Put the whole number and the fraction together and change it to a decimal: 4 and 4/9 is about 4.44.

The Let's Investigate page on estimating irrational numbers

Practice: 20 problems and a challenge

Two practice pages pull it together: fractions to decimals, decimals to fractions, square root and cube root equations, estimating irrational numbers to two decimal places without a calculator, and word problems. The investigation ends with a reflection question (is there another way to estimate?), and the practice ends with a challenge problem.

The answer key

Every page has its own answer key page, the notes and the investigation included, so you can project it while you teach or use it to check the practice.

Here are the free Rational and Irrational Numbers notes

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These notes are from Unit 1, Real Numbers, of the 8th grade math in my membership

Unit 1: Real Numbers
Unit 2: Exponents and Scientific Notation
Unit 3: Proportional Relationships
Unit 4: Nonproportional Relationships
Unit 5: Writing Linear Equations
Unit 6: Functions
Unit 7: Solving Linear Equations
Unit 8: Solving Systems of Linear Equations
Unit 9: Transformations and Congruence
Unit 10: Transformations and Similarity
Unit 11: Angle Relationships in Parallel Lines and Triangles
Unit 12: The Pythagorean Theorem
Unit 13: Volume
Unit 14: Scatter Plots
Unit 15: Two-Way Tables

See everything in the membership

Questions teachers ask about rational and irrational numbers

What is a rational and irrational number?

A rational number can be written as a fraction of two integers, like 3/4, -5 or 0.333... (which is 1/3). An irrational number cannot: its decimal never ends and never repeats, like pi or the square root of 2.

What is the difference between rational and irrational numbers?

Look at the decimal. If it stops or repeats, the number is rational. If it goes on forever with no repeating block, it is irrational. Square roots of perfect squares, like the square root of 49, are rational; square roots of other whole numbers are irrational.