Estimating Irrational Numbers 8NS2

E S T I M AT I N G
I R R AT I O N A L
NUMBERS
Presented by G. Laws
8th Grade Math
JCMS
STANDARDS
• 8.NS.2 – Use rational approximations of irrational numbers to complare
the size of irrational numbers, locate them approximately on a number
line and estimate the value of expressions involving:
• Square roots and cube roots to the tenths
• to the hundredths.
ESSENTIAL QUESTION
How do I use rational numbers to approximate the

value of irrational numbers, and locate them on a
number line?
TARGET STATEMENT
I CAN estimate the value of irrational numbers.
I CAN approximately locate irrational numbers on

a number line.
E X P R E S S I N G R AT I O N A L N U M B E R S A S
DECIMALS
1. A rational number is any number that can be written as a ratio
in the form , where a and b are integers and b is not zero.
Rational #
; b
Examples: 4 and 0.5 are rational numbers.

4 can be written as 0.5 can be written as .
T E R M I N AT I N G A N D R E P E AT I N G D E C I M A L S
2. A rational number can be written as a terminating decimal or a
repeating decimal.
3. Terminating decimals are rational numbers that stops before or

after the decimal point. Example. 1.5, .6, .001
4. Repeating decimals are rational numbers that repeats after the

decimal point. Example. .333…; ;
T E R M I N AT I N G D E C I M A L S A S F R A C T I O N S
5. Every terminating decimal can be written as a fraction.
6. You can use place values to find the fraction that is equivalent to any
terminating decimal.
Terminating Decimal Place Value Fraction
0.4 Four tenths
0.75 Seventy-five
hundredths
0.386 Three hundred eighty-six
thousandths
= 0.4 Four tenths
R E P E AT I N G D E C I M A L S A S A F R A C T I O N
7. Every repeating decimal can be written as a fraction.
Example# 1 : Write 0.
Step 1. Let x represent the repeating
Let x = 0. decimal
Step 2. Since the repeating decimal is

10 x = 0.
in the tenths place, multiply both sides
by 10.
10x – 1x = 3. – 0.
Step 3. Subtract 1x from the left side
and 0. from the right side.
R E P E AT I N G D E C I M A L S A S A F R A C T I O N
Example# 1 : Write 0.
Step 4. Solve for x by dividing 9 on
9x = 3 = both sides of the equation.
9 9 Note: For a repeating decimal to be
repeating, subtract 1 from the decimal
𝟏 place value to make the denominator either
𝟎 .𝟑=
𝟑 9, 99, 999…etc. Examples:
= = 25/33
𝟓𝟏𝟐
𝟎 .𝟓𝟏𝟐=
𝟗𝟗𝟗
I R R AT I O N A L N U M B E R S
8. Irrational numbers are not rational.

9. Irrational numbers are non-terminating (don’t stop) and non-
repeating (don’t repeat) decimals
10. They cannot be written as a fraction.
11. Non- perfect squares are square roots that are irrational.
• Example: , , ..etc
12. Non-perfect cubes are cube roots that are irrational.
• Example: , , …etc
E S T I M AT I N G I R R AT I O N A L N U M B E R S
11. You can use perfect squares to help estimate an irrational
numbers value like the .
Step 1: What two perfect squares do
√1 √2 √4 the falls between?
1 2
Step 2: What two consecutive
So, the is between 1 and 2 integers do the falls between?
Step 3: Is it closer to 1 or 2?
13. How do you estimate the value of the ?
√1 √2 √4 Step 4: 1st estimate the value by

1 squaring two or more terminating
1.4 2
decimals.
Estimate: 1.32,1.42 and 1.52
Step 5: Multiply each square number to
find a rational number close to
1.3 1.4 1.5
x 1.3 x 1.4 x 1.5 Step 6: The value of is
1.69 1.96 2.25
(to the nearest tenths)
14. You can use perfect cubes to help estimate an irrational

numbers value like the .
Step 1: What two perfect cubes do
√3 27 √3 60 √3 64 the falls between?
3 3.91 4 Step 2: What two consecutive

integers do the falls between?
So, the is between 3 and 4
Step 3: Is it closer to 3 or 4?
𝟑 .𝟕𝟏𝟑=3.71∙ 3.71 ∙3.71=51.06
Step 4: Estimate the value by cubing
𝟑 .𝟖𝟏𝟑=3.81∙ 3.81 ∙3.81=55.03
two or more terminating decimals.
𝟑 .𝟗𝟏𝟑=3.91∙ 3.91 ∙3.91=𝟓𝟗 . 𝟕𝟕 Estimate to the nearest hundredths
S U M M A RY
 Look over your notes; add questions you may have missed.
 Write down some important facts about the lesson.
 Is there something you don’t understand about the lesson?
 Can you answer the essential question?
 Can you do the target statement

YOUR TURN
1. Change the following decimals into fractions. Make sure you reduce them to
the lowest term.
a = b = c = d. =
2. What two integers does the square root or cube root fall between on a
number line.
is between ___ and ___ b is between ___ and ___
is between ___ and ___ is between ___ and ___

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E S T I M AT I N G

How do I use rational numbers to approximate the

I CAN approximately locate irrational numbers on

Examples: 4 and 0.5 are rational numbers.

3. Terminating decimals are rational numbers that stops before or

4. Repeating decimals are rational numbers that repeats after the

Step 2. Since the repeating decimal is

8. Irrational numbers are not rational.

√1 √2 √4 Step 4: 1st estimate the value by

14. You can use perfect cubes to help estimate an irrational

3 3.91 4 Step 2: What two consecutive

 Write down some important facts about the lesson.

 Is there something you don’t understand about the lesson?

 Can you answer the essential question?

 Can you do the target statement

is between _ and _ b is between _ and _

is between _ and _ is between _ and _

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