Estimating Irrational Numbers

If we know our perfect squares and where they fall on a number line, then how do we estimate our non-perfect squares? Can we

estimate where these numbers fall on a number line?

WARM UP

1. Why is a rational number but an irrational number? Use the correct mathematical terminology to justify your answer.

2. Convert to a decimal fraction.

3. Is rational or irrational?

4. (-3)²

5. 4x + 6 = 2x - 8

Perfect Squares

Can you name all of your perfect squares 1 through 20?

Let’s try them (out loud and together)

√1

√ 4

√9

√16

√25

√36

√ 49

√64

√81

√100

√121

√144

OKAY – Now…… Let’s try the harder ones(they are in order – next time they won’t be)

√169

√196

√225

√256

√289

√324

√361

√ 400

If all perfect squares are RATIONAL …then what are non-perfect squares?

These are all rational numbers and can be found easily on a number line

What about Non-perfect squares?

ALL Non-perfect squares

are irrational

Estimating

Generally speaking, students have a difficult time estimating…students do not have a fully developed number sense and really don’t get estimating…

The goal of estimating is to get a number that it is close to….we are not looking for an exact number here…the exact number is not important…

Estimating using a number line

In the notes section of your notebook draw a number line.

Plot the perfect squares from 1 through 12 on the number line

When estimating the goal is to find out which whole number the perfect square is close to…

That will be your estimate… Let’s see what it looks like…

Number Line Your number line should look like this… To find the square root of a number find where it fits on a number line Then determine the two numbers it fits between Determine which number it is closest to and look at the perfect square

for that number

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Perfect Squares

The estimate of would be 9

Example

75

Find the

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Perfect Squares

The square root of 75 is between the square root of 64 and 81Since 75 is closer to 81 than 64 the nearest whole number would be 9

75

In the notes section of your notebook estimate each square root to the nearest whole number

1.

2.

3.

Your Turn

39

106

140

Solutions6

12

10

Classwork

Let’s work together first…

Problem Solving – These are to challenge your thinking….

Answer the following problem SHOW WORK!

1. I am a number. I am not zero. If I am squared, I’m still the same number. What number am I?

1

Answer the following problem SHOW WORK!

2. If a square bedroom has an area of 144 square feet, what is the

length of one wall?

12 feet

Answer the following problem SHOW WORK!

3. An artist is making two stained-glass windows. One window

has a perimeter of 48 inches. The other window has an area

of 110 inches. Which window is bigger?

The window with a perimeter of 48 inches.

Answer the following problem SHOW WORK!

4. A square garden has an area of 225 square feet. How much

fencing will a gardener need to buy in order to place fencing

around the garden?

60 feet