Rational and Irrational Numbers

Lesson 3

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

Warm Up

• 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 _______.

Three Squares

Activity 1

• MLR8: Discussion Supports
• Anticipate, Monitor, Select, Sequence, Connect

There are 9 vertices in each 2-by-2 grid.

“Draw 3 squares of different sizes with vertices aligned to the vertices of the grid.”

This has vertices aligned to the grid, but is not a square.

This looks like a square, but doesn’t have vertices aligned to the grid.

Let’s share your ideas and reasoning!

Looking for a Solution

Activity 2

• MLR1: Stronger and Clearer Each Time
• Think Pair Share

Reminder:

1.5 is equivalent to 3/2

Begin working on your own. (2-3 min)

Then, we’ll share in teams and a whole-class dicussion.

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

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

rational numbers!

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Here are some examples of rational numbers:

Looking for √2

Activity 3

• MLR2: Collect and Display

rational number

a fraction or its opposite

Examples:

• 9/8
• ﹣9/8
• terminating decimals, like 0.7 = 7/10

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A rational number is a fraction or its opposite

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

Begin working on the task! (5 min)

1. Find some more rational numbers that are close to √2.
2. Can you find a rational number that is exactly √2?

What strategies did you use to find √2?

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

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

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

Are you ready for more?

While we have collected some evidence that supports the claim that √2 is irrational, we have not actually proved the claim.

If I told you that there are no purple zebras, and you spent your whole life traveling the world and never saw a purple zebra, does it mean I was right?

No, it’s possible you just failed to find a purple zebra.

While we have collected some evidence that supports the claim that √2 is irrational, we have not actually proved the claim.

So if we spent our whole lives testing different fractions and never quite got one whose square is 2, does that mean there are no such fractions?

No, maybe you just haven’t found it yet!

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.

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

Cool Down