Rational and Irrational Numbers

Rational and

Irrational Numbers

Lesson 3

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Let’s learn


about irrational



numbers!

Today’s Goals

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

❏ I know what an irrational number is and can give an
example.

Algebra talk:

Positive solutions

Activate: Warm Up 3.1

●

MLR8: Discussion Supports

Could 8 be a solution? Why or why not?

x2 = 49

Find a positive solution to each equation:

I noticed that ________.

First, I _______ because
_______.

Looking for a

Solution

Comprehend/ Apply: Activity 3.3

●

MLR1: Stronger and Clearer Each Time

●

Think Pair Share

Which of these numbers are a solution to the equation x² = 2?

rational
number

a fraction or its
opposite

Remember we can always
write a fraction in the form
a/b, where a and b are whole numbers (and b is not zero).

They can be positive or negative!

a
b

Organize: Activity 3.3 Synthesis

Because fractions and ratios are closely related ideas, fractions and their opposites are called rational numbers!

Here are some examples of rational numbers:

Looking for √2

Comprehend/ Apply: Activity 3.4

●

MLR2: Collect and Display

rational
number

a fraction or
its opposite

Examples:
➔ 9/8

➔ ﹣9/8

➔ terminating decimals, like

0.7 = 7/10

Let's Practice

Multiple Choice

Which of the following is a rational number?

• 0

• $$\sqrt[]{5}$$

• $$\pi$$

• $$\sqrt[]{12}$$

Multiple Choice

Which phrase does not describe a rational number?

• integer number

• repeating decimal

• terminating decimal

• non-repeating, non-terminating decimal

Multiple Choice

Which term describes all four of these labeled numbers?

• irrational numbers

• rational numbers

• mixed numbers

• integers

Multiple Choice

Which number is rational?

• $$-\sqrt[]{5}$$

• $$\pi$$

• $$\sqrt[]{10}$$

• $$0.4$$

Multiple Choice

Which of the following numbers is rational?

• $$0.31311.....$$

• $$\sqrt[]{5}$$

• $$\sqrt[]{16}$$

• $$\sqrt[]{27}$$

A rational number is a fraction or its opposite
(or any number equivalent to a fraction or its opposite).

1. Find some more rational numbers
that are close to √2.

2. Can you find a rational number that is exactly √2?

Begin working on the task! (5 min)

What strategies
did you use to find √2?

No number exists for √2 because it
is an irrational
number!

Organize: Activity 3.4
Synthesis

An irrational number is a number that is not rational; it is not a fraction or its opposite.

The square root of a whole number is either a
whole number or irrational, so √10, √67, etc. are
all irrational.

Big Idea:

We haven’t learned enough to prove for sure that √2 is not equivalent to a fraction.

For now, we just have to trust that there are numbers on the number line that are not equivalent to a fraction and that √2 is one of them!

However, it is possible to get very close estimates with fractions.

More Practice

Multiple Choice

Which set contains an irrational number?

Multiple Choice

Which number is irrational?

• 0.65665665......

• $$0.\overline{78}$$

• 2.35

• $$\frac{22}{7}$$

Multiple Choice

Which number is irrational?

• $$\frac{3}{16}$$

• $$\frac{2}{9}$$

• $$\sqrt[]{4\times4}$$

• $$\sqrt[]{2\times9}$$

Multiple Choice

A computer programmer knew there was an error in his formula because the output was an irrational number. Which number could have been the output?

• 75.345345345345345

• 86.98765876576546543

• 76.000000000000008

• 00000.000000000000

Today’s Goals

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

❏ I know what an irrational number is and can give an
example.

Types of
Solutions

Apply/Summarize: Cool Down 3.5

Open Ended

In your own words, define a rational number.

Provide three examples of a rational number.

Open Ended

In your own words, define an irrational number.

Provide two examples of a rational number.