Grade 9 Mathematics
Unit #1: Number Sense
Sub-Unit #3: Square Roots
Lesson
1
Topic
Whole Number
Perfect Square Roots
2
Square Roots of
Whole Numbers
Quiz #5 – Whole
Number Square Roots
Decimal Perfect

Square Roots

3
4
5
6
Square Roots of
Decimals
Fraction Perfect
Square Roots
Square Roots of
Fractions
Quiz #6 – Positive
Rational Square Roots
“I Can”
 Identify perfect squares
 Use technology to find square roots
 Use prime factorization to find perfect square roots over
100
 Estimate the square root of whole numbers that are nonperfect square roots by the use of benchmarks
Demonstrate your understanding of above topics
Identify properties that make a decimal a perfect square
numbers
Determine the square root of decimals that are perfect
square numbers
 Estimate the square roots of decimals that are nonperfect square numbers by the use of benchmarks
 Identify properties that make a fraction a perfect square
numbers
 Determine the square root of fractions that are perfect
square numbers
 Estimate the square roots of fractions that are nonperfect square numbers by the use of benchmarks
Demonstrate your understanding of the above topics
Lesson #1: Whole Number Perfect Square Roots
What do square roots represent?
Consider a square made up of 4 smaller squares:
This square has each side length made up of two sides of the smaller
squares. This is a representation of the square number 4 (how many
small squares it takes to make the big square) and its square root of 2
(how many sides of the smaller square are on each side of the big
square).
Consider a square made up of 9 smaller squares:
This represents the square number _________ with a square
root of ________.
Think about it: The square of any rational number (positive or
negative) will result in a positive number.
Ex)
( )
(
)
(
)
This is leads to why you cannot take the square root of a negative number. This will result in
an imaginary number which you will not learn about for a little while longer.
To determine if a number is a perfect square it must have a square root that is a rational
number. Remember that rational numbers are any number that can be written as a fraction.
{
}.
You are expected to know perfect squares with square roots up to 15:
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
Methods for determining a square root:
1. Guess and Check: You can determine the square root of a number by squaring whole
numbers until you find a square equal to the given value:
Ex: √
Think: 169 is greater than 100, therefore the square root will be greater than 10 (10 2 = 100).
Try 15:
Try 13:
2. Prime Factorization: Remember that a prime number has only two factors: itself and one.
Determine all the prime numbers between 1 and
100:
 Circle 1 and 2
 Cross off all multiples of 2
 Circle next uncrossed number and cross off
all multiples of that number
 Continue until all numbers are crossed off
To factor a number to primes use a factor tree;
continue until you have a multiplication expression
with only prime numbers.
Use the following rules for division to help you get started:
Dividing by 2
1.
2.
3.
All even numbers are divisible by 2. E.g., all
numbers ending in 0,2,4,6 or 8.
If so, the number is too!
For example: 358912 ends in 12 which is divisible
by 4, thus so is 358912.
Dividing by 5
Dividing by 3
1.
1.
2.
3.
Add up all the digits in the number.
Find out what the sum is. If the sum is divisible
by 3, so is the number
For example: 12123 (1+2+1+2+3=9) 9 is divisible
by 3, therefore 12123 is too!
Numbers ending in a 5 or a 0 are always divisible
by 5.
Dividing by 6
1.
If the Number is divisible by 2 and 3 it is divisible
by 6 also.
Dividing by 4
Dividing by 7 (2 Tests)
1.
Are the last two digits in your number divisible
by 4?

Take the last digit in a number.









1.
Double and subtract the last digit in your
number from the rest of the digits.
Repeat the process for larger numbers.
Example: 357 (Double the 7 to get 14. Subtract
14 from 35 to get 21 which is divisible by 7 and
we can now say that 357 is divisible by 7.
2.
This one's not as easy, if the last 3 digits are
divisible by 8, so is the entire number.
Example: 6008 - The last 3 digits are divisible by
8, therefore, so is 6008.
Dividing by 9
NEXT TEST
Take the number and multiply each digit
beginning on the right hand side (ones) by 1, 3,
2, 6, 4, 5. Repeat this sequence as necessary
Add the products.
If the sum is divisible by 7 - so is your number.
Example: Is 2016 divisible by 7?
6(1) + 1(3) + 0(2) + 2(6) = 21
21 is divisible by 7 and we can now say that 2016
is also divisible by 7.
1.
2.
3.
Almost the same rule and dividing by 3. Add up
all the digits in the number.
Find out what the sum is. If the sum is divisible
by 9, so is the number.
For example: 43785 (4+3+7+8+5=27) 27 is
divisible by 9, therefore 43785 is too!
Dividing by 10
1.
If the number ends in a 0, it is divisible by 10.
Dividing by 8
Ex: Use prime factorization to find √
 Create the Factor Tree:
.
2916
 Write the number as a product of primes. **Write all primes that are the same
consecutively**
 Put the numbers into pairs
 The Square Root will be one of each pair.
Ex) Use prime factorization to find the following square roots:
a) 324
b) 1296
Ex) Use the Order of Operations to complete any indicated operations, and prime factorization
to calculate any square roots. **Beware of Invisible Brackets”
√
a)
√
)
b) √
√
Assignment: Use prime factorization; use your Assignments Section
1. √
5. √
9. √
2. √
3. √
6. √
10. √
7. √
11. √
8. √
12. √
4. √
Use the Order of Operations to complete any indicated operations, and prime factorization to
calculate any square roots. Look for any patterns that may help you.
13. √
√
14. √
15. √
√
16. (√
17. (√
)
)
18. (√
19.
20.
)
21.
√
22. √
√
√
√
√
√
√
Lesson #2: Square Roots of Whole Numbers
To estimate non-perfect square roots you need to know the two perfect squares that are
nearest to the non-perfect square root.
Method 1 – Use a number line and Benchmarks
Ex) Approximate the value of √ .
 Draw a number line with squares of numbers written above:
 Place the square root you are estimating between the numbers whose squares it is
between
 Use this placement to estimate the square root to one decimal place
Method 2 – Guess and Check
Ex) Approximate the value of √
.
Method 3 – Works well for numbers greater than 100
Ex) Estimate the value of √
 Group each digit into pairs, always starting at the decimal and working to larger place
values
√
 Estimate the square root of the highest non-zero pair. For every other pair place one
zero
√
Ex) Estimate each of the following square roots.
a) √
b) √
c) √
d) √
e) √
e) √
Assignment
 Estimate the following. If you are using benchmarks or guess and check estimate to
the nearest tenth.
1. √
2. √
8. √
3. √
4. √
10. √
11. √
5. √
12. √
6. √
13. √
14. √
15. √
16. √
9. √
7. √
17. √
18. √
19. √
 Estimate each square root to the nearest tenth and evaluate the following using all
order of operations rules.
20. √
21. √
22. √
23. √
√
√
24. (√ )(√ )
25. √
26. √
√
√
27.
√
√
√
Lesson #3: Decimal Perfect Square Roots
Multiply the following:
The following patterns exist for any decimal that will be a perfect square:
 A perfect square decimal will always have an even number of decimal points
 With the decimal removed the number would have been a perfect square
For example look at:
√
We know that
Since √
√
has an even number of decimal points and 144 is a perfect square number we an
determine that √
is a perfect square decimal.
Think:
 What is √
? (The number that would exist when the decimal is removed)
 How many decimal points existed? Divide that by 2 (This is why the number of
decimal places must be even!)
So, since √
√
This is the square root of the original number with half as many decimal places.
Ex) Determine the square roots below:
 What would the square root of the number be if it weren’t a decimal
 Divide the decimal places in half
 Write the square root of the number if it weren’t a decimal; place the decimal in the
appropriate place.
a) √
b) √
c) √
d) √
e) √
f) √
g) (√
)(
i) (√
)(√ )
)
h) (√
j) (√
)(√
)(√
)
)
Assignment - Evaluate the square root of each number:
1. √
7. √
13. √
2. √
3. √
8. √
14. √
15. √
4. √
9. √
10. √
5. √
11. √
6. √
12. √
Page 11-12 #4, 5abcd, 7fghij, 9ab, 10, 12, 14, 16
16. √
Lesson #4: Square Roots of Decimals
To determine the square root of a non-perfect square that is a decimal combine both methods
we used for determining the square root of a perfect square that is a decimal and the square
root of a non-perfect square.
Method 1 – Use of Benchmarks – Works well with decimals with few non-zero place values
Ex 1) Approximate the square root of √
 Look at the number without the decimal
 Use benchmarks (like before with whole numbers) to estimate the value of the square
root of the number without any decimal places
 Move the decimal place accordingly.
Method 2 – Highest digit estimation – Works well for most decimals with zero place holders.
Ex 2) Estimate the value of √
 Group each digit into pairs, always starting at the decimal and working to larger place
values
 Estimate the square root of the first non-zero pair after the decimal place. Place trailing
zeros after below all the following pairs
 Check your answer; if it is close consider it to be correct.
Ex 3) Estimate the following square roots using the most appropriate method.
a) √
b) √
c) √
d) √
e) √
f) √
Assignment: Page 18-19 #4, 7, 8a, 9, 10, 11aceg, 13a, 15, 16
Lesson #5: Fraction Perfect Square Roots
Remember the Exponent Law as applied to the Quotient of Powers:
( )
We can use this law to determine the square root of a fraction using the same strategy since a
square root is technically a power (don’t worry about why yet!)
So:
√
√
√
Ex 1) Evaluate √
 If the fraction is mixed turn it into improper (as you do with every operation including
fractions!)
 Apply the square root to both numerator and denominator
 Evaluate the square roots individually
 If the fraction was given to you in mixed form, turn it back into mixed form; if it was in
improper form you may leave it improper
Ex 2) Evaluate √
 Notice that neither of these numbers are perfect squares. If this happens you look to
reduce the fraction and see if there are perfect squares after a reduction.
 Apply the square root to both numerator and denominator
 Evaluate the square roots individually
Ex 3) Determine each square root below:
a) √
b) √
c) √
d) √
Assignment: Page 11 #3, 5efgh, 7abcde, 8efghjl, 9efgh, 13
Lesson #6: Square Roots of Fractions
You will be asked to find the square root of fractions when neither numerator nor denominator
can simplify to a perfect square.
Recall using benchmarks to estimate the square root of a non-perfect square:
√
You can use this same concept and expand it to fractions.
Ex) Estimate the value of √
 Break the square root up and apply the square root sign to both numerator and
denominator
 Estimate separately the square root of the numerator and denominator:
√
√
 Use these estimations in place of the square root in your fraction
 If you have decimals in your estimation, multiply by a power of 10 to make an
equivalent fraction with no decimals (the same process used in dividing decimals) and
reduce as necessary
**If you have a calculator you may check your estimations (don’t depend on the calculator; you
will not have one for the quiz!). If you are within a tenth you may consider your estimation
valid.
You may also choose to turn the fraction into a decimal and use your method of estimating
non-perfect decimal square roots.
Ex 2) Estimate the following
a) √
Assignment: Page 18 #6, 11bdfh, 12
Square Roots Review: Page 21 #1-9
b) √