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Grade 9 Math
Unit 5 – Square Roots & Surface Area
5.1 – Square Roots of Perfect Squares
To determine the area of a square or rectangle, we multiply the length by the width. In the case of a
square though, the length and width will always be the same. When we multiply a number by itself, we
say that we are taking the square of the number.
Ex. #1 Square all numbers from 1 to 15.
Answer:
12= 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102= 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
Ex. #2 Find the areas of the following squares.
a.
Area = 8 x 8 = 64 cm2
8 cm
b.
5 5 25 2
Area   
m
6 6 36
5 m
6
Finding the square root is the opposite of finding the square. Instead of taking a number multiplied by
itself to find the square, we have the square and need to find what number multiplied by itself will
equal that number.
Ex. #3 Complete the following table.
Area of a Square
Side Length (Square Root)
49
64
121
144
16/100
25/81
4/9
36/64
7
8
11
12
4/10
5/9
2/3
6/8 = 3/4
Ex. #4 Solve the following square roots.
a.
36 = 6
b.
4
= 2/3
9
Perfect Squares
A number that is the square of a whole number is called a perfect square.
Example: 16 is a perfect square because 16 = 42.
Fractions can also be perfect squares if both the numerator and the denominator are whole numbers.
Example: 25/100 is a perfect square because
25
5 1
 
100 10 2
When a fraction that is a perfect square is written as a decimal, then the decimal is also a perfect
square. The square root will be a terminating (ex. 3.5) or repeating (ex. 0.3333…) decimal. Numbers
that are terminating or repeating are called rational numbers.
Example: Solve the following square roots. State whether each answer is a rational number and
explain why.
100 10
a.
  5 Rational because it is a whole number, therefore is terminating.
4
2
b.
25
 Not rational because the decimal is not repeating or terminating. Also,
29
the denominator is not a perfect square.
c.
1 1
  0.3 Rational because it is a repeating decimal.
9 3
d.
25
5
  0.5 Rational because it is a terminating decimal.
100 10
e.
1.69 
f.
169 13
  1.3 Rational because it is a terminating decimal.
100 10
1
 0.5773502.... Not rational because it is not repeating or terminating. Also,
3
the denominator is not a perfect square.
Learning Activity 5.1
1. Check the numbers that are rational numbers. Explain why they are or are not rational numbers.
o 6
Yes - whole
o 5.5
Yes - terminating
o 4.123456…..No – non-terminating, non-repeating
o ¾
Yes – fraction, which also equals 0.75 which is a terminating decimal
o √25
Yes – equals 5 which is a whole number, and it is a perfect square
9
o
Yes – it is a perfect square and equals a fraction (3/4), which also equals 0.75
16
which is a terminating decimal.
2. What number has a square root of:
a. 3/8 = (3/8)2 = 9/64
b. 1.8 = (1.8)2 = 3.24
3. Is each fraction a perfect square? Explain your reasoning.
a. 8/18 = 4/9 Yes, because √4/9 = 2/3
b. 16/5
No. It cannot be reduced. Although the numerator has a square root that is a
whole number, the denominator does not. Therefore, 16/5 is not a perfect
square.
c. 2/9
No. It cannot be reduced. Although the denominator has a square root that
is a whole number, the numerator does not. Therefore, 2/9 is not a perfect
square.
4. Is each decimal a perfect square? Explain your reasoning.
a.
=
Yes!
625 25
25 5
Or, reduce the fraction first: 100  4  4  2 Yes!
627
b. 0.627 = 1000 It cannot be reduced, and neither the numerator nor the denominator
can be written as a product of equal factors. Therefore 0.627 is not a
perfect square.
Assignment
Do #1-16 p. 11
Grade 9 Math
Unit 5 – Square Roots & Surface Area
Student Copy
5.1 –
Square Roots of Perfect Squares
To determine the area of a square or rectangle, we multiply the ___________ by the ____________. In
the case of a square though, the length and width will always be the ____________. When we multiply
a number by itself, we say that we are taking the ____________ of the number.
Ex. #1 Square all numbers from 1 to 15.
Ex. #2 Find the areas of the following squares.
a.
8 cm
b.
5 m
6
Finding the __________ _________ is the _______________ of finding the ____________. Instead of
taking a number multiplied by itself to find the ____________, we have the square and need to find
what number ___________________ by _______________ will equal that number.
Ex. #3 Complete the following table.
Area of a Square
49
64
121
144
16/100
25/81
4/9
36/64
Side Length (Square Root)
Ex. #4 Solve the following square roots.
a.
b.
36
4
9
Perfect Squares
A number that is the square of a whole number is called a ______________ ______________.
Example: 16 is a perfect square because 16 = _____.
Fractions can also be perfect squares if both the _________________ and the ______________ are
whole numbers.
Example: 25/100 is a perfect square because
25

100
When a __________________ that is a perfect square is written as a _______________, then the
decimal is also a _________________ ____________. The square root will be a
_______________________ (ex. 3.5) or ____________________ (ex. 0.3333…) decimal. . Numbers
that are terminating or repeating are called _____________________ numbers.
Example: Solve the following square roots. State whether each answer is a rational number and
explain why.
100
a.

4
b.
25

29
c.
1

9
d.
25

100
e.
1.69 
f.
1

3
Learning Activity 5.1
1. Check the numbers that are rational numbers. Explain why they are or are not rational numbers.
o 6
o 5.5
o 4.123456…..
o ¾
o √25
9
o
16
2. Calculate the number whose square root is:
a. 3/8
b. 1.8
3. Is each fraction a perfect square? Explain your reasoning.
a. 8/18
b. 16/5
c. 2/9
4. Is each decimal a perfect square? Explain your reasoning.
a. 6.25
a. 0.627
Assignment
Do #1-16 p. 11