Lesson 3Rational and Irrational Numbers
Learning Goal
Let’s learn about irrational numbers.
Learning Targets
I know what a rational number is and can give an example.
I know what an irrational number is and can give an example.
Lesson Terms
- irrational number
- rational number
- square root
Warm Up: Algebra Talk: Positive Solutions
Problem 1
Find a positive solution to each equation:
Activity 1: Three Squares
Problem 1
Draw 3 squares of different sizes with vertices aligned to the vertices of the grid.
For each square:
Label the area.
Label the side length.
Write an equation that shows the relationship between the side length and the area.
Activity 2: Looking for a Solution
Problem 1
Are any of these numbers a solution to the equation
Activity 3: Looking for
Problem 1
A rational number is a fraction or its opposite (or any number equivalent to a fraction or its opposite).
Find some more rational numbers that are close to
. Can you find a rational number that is exactly
?
Are you ready for more?
Problem 1
If you have an older calculator evaluate the expression
Explain why you might be suspicious of the calculator’s result.
Find an explanation for why
could not possibly equal . How does this show that could not equal 2? Repeat these questions for
an equation that even many modern calculators and computers will get wrong.
Lesson Summary
In an earlier activity, we learned that square root notation is used to write the length of a side of a square given its area. For example, a square whose area is 2 square units has a side length of
![A square with a blue square within turned so that its vertices are the midpoints of the outer square's sides. Side of the inner square are square root of 2.](../../../../../../embeds/16a55be2--8.8.B.3.revised.image.01.png)
A square whose area is 25 square units has a side length of
Here are some examples of rational numbers:
An irrational number is a number that is not rational. That is, it is a number that is not a fraction or its opposite.
![A number line with 10 evenly spaced tick marks. The first tick mark is labeled 0 and the sixth tick mark is labeled 1. An arrow points to the eighth tick mark and is labeled seven-fifths. A second arrow points to a point slightly to the right of the eighth tick mark and is labeled the square root of 2.](../../../../../../embeds/50df25f4--8.8.B2.Image.disc2.png)
In your future studies, you may have opportunities to understand or write a proof that