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From Square Numbers 
to Square Roots 
Grade 8 Mathematics 
Mr. J. Lingley
Square # Review 
These numbers below are not square numbers. 
Which two consecutive square numbers is each 
number between? 
12 
40 
75 
200 
How do you know?
Square # Review 
The floor of a large square room has an area of 
144 m2. What is the length of a side of the 
room? How much baseboard is needed to go 
around the room?
Square # Review 
The floor of a large square room has an area of 
144 m2. What is the length of a side of the 
room? How much baseboard is needed to go 
around the room?
Is there a shorter way to find the side length?
144
What the 
heck is that? 
144
Square Roots 
The square root ( ) of a number finds the 
factor that when multiplied by itself will give 
you the square number. In other words it goes 
from area to side length. Back to the root. 
144 = 12 122 = 144 
A square root and a square are opposite operations.
Even donald knows something about square roots.
Exploring Square Roots 
Fun With Squares and Square roots! 
Name: Date: ! ! ! ! Class: 
In class we have been observing that any 
whole number multiplied by itself will give 
us a square number. Now it’s time to look 
at what the factors of those square 
numbers tell us. 
Factor: A number that divides exactly into 
another number. For example, 1,2,3 and 6 
are factors of 6. 
What are all of 
the factors of 10? 
Please help! 
Investigate! 
Working with a partner, complete the table below. Indicate all of the factors for a given 
whole number along the bottom of the table. Remember, that if a number multiplied by 
itself gives you the target whole number, only copy down that factor once. For 
instance: 9 = 3 x 3, however the factors for 9 are: 1, 3, 9 - not 1, 3, 3, 9.
us a square number. Now it’s time to look 
at what the factors of those square 
numbers tell us. 
Factor: A number that divides exactly into 
another number. For example, 1,2,3 and 6 
are factors of 6. 
Please help! 
Exploring Square Roots 
Investigate! 
Working with a partner, complete the table below. Indicate all of the factors for a given 
whole number along the bottom of the table. Remember, that if a number multiplied by 
itself gives you the target whole number, only copy down that factor once. For 
instance: 9 = 3 x 3, however the factors for 9 are: 1, 3, 9 - not 1, 3, 3, 9. 
6 8 
4 3 4 
2 3 2 5 2 7 2 
1 1 1 1 1 1 1 1 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 
start here! 
Questions: 
Calculators are permitted.
instance: 9 = 3 x 3, however the factors for 9 are: 1, 3, 9 - not 1, 3, 3, 9. 
6 8 
4 3 4 
2 3 2 5 2 7 2 
1 1 1 1 1 1 1 1 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 
start here! 
Questions: 
1. Which numbers have only two factors? What do you notice about these numbers? 
2. Which numbers have an even number of factors, but more than 2 factors? 
3. Which numbers have an odd number of factors?
Odd # of Factors Even # of Factors
Odd # of Factors Even # of Factors 
square number 
When a number has an odd 
number of factors, it is a square 
number. 
36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 9 Factors 
The square number can be 
always found in the middle.
Fill in this table... 
Square Root Square Number 
4 
64 
144 
7 
13 
100
Your Turn 
1. The factors of 136 are listed in ascending order. 
136 = 1, 2, 4, 8, 17, 34, 68, 136 
Is 136 a square number? 
2. Find: 
42 62 82 72 92 12 
25 64 81 
162

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From Square Numbers to Square Roots (Lesson 2)

  • 1. From Square Numbers to Square Roots Grade 8 Mathematics Mr. J. Lingley
  • 2. Square # Review These numbers below are not square numbers. Which two consecutive square numbers is each number between? 12 40 75 200 How do you know?
  • 3. Square # Review The floor of a large square room has an area of 144 m2. What is the length of a side of the room? How much baseboard is needed to go around the room?
  • 4. Square # Review The floor of a large square room has an area of 144 m2. What is the length of a side of the room? How much baseboard is needed to go around the room?
  • 5. Is there a shorter way to find the side length?
  • 6. 144
  • 7. What the heck is that? 144
  • 8. Square Roots The square root ( ) of a number finds the factor that when multiplied by itself will give you the square number. In other words it goes from area to side length. Back to the root. 144 = 12 122 = 144 A square root and a square are opposite operations.
  • 9. Even donald knows something about square roots.
  • 10. Exploring Square Roots Fun With Squares and Square roots! Name: Date: ! ! ! ! Class: In class we have been observing that any whole number multiplied by itself will give us a square number. Now it’s time to look at what the factors of those square numbers tell us. Factor: A number that divides exactly into another number. For example, 1,2,3 and 6 are factors of 6. What are all of the factors of 10? Please help! Investigate! Working with a partner, complete the table below. Indicate all of the factors for a given whole number along the bottom of the table. Remember, that if a number multiplied by itself gives you the target whole number, only copy down that factor once. For instance: 9 = 3 x 3, however the factors for 9 are: 1, 3, 9 - not 1, 3, 3, 9.
  • 11. us a square number. Now it’s time to look at what the factors of those square numbers tell us. Factor: A number that divides exactly into another number. For example, 1,2,3 and 6 are factors of 6. Please help! Exploring Square Roots Investigate! Working with a partner, complete the table below. Indicate all of the factors for a given whole number along the bottom of the table. Remember, that if a number multiplied by itself gives you the target whole number, only copy down that factor once. For instance: 9 = 3 x 3, however the factors for 9 are: 1, 3, 9 - not 1, 3, 3, 9. 6 8 4 3 4 2 3 2 5 2 7 2 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 start here! Questions: Calculators are permitted.
  • 12. instance: 9 = 3 x 3, however the factors for 9 are: 1, 3, 9 - not 1, 3, 3, 9. 6 8 4 3 4 2 3 2 5 2 7 2 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 start here! Questions: 1. Which numbers have only two factors? What do you notice about these numbers? 2. Which numbers have an even number of factors, but more than 2 factors? 3. Which numbers have an odd number of factors?
  • 13. Odd # of Factors Even # of Factors
  • 14. Odd # of Factors Even # of Factors square number When a number has an odd number of factors, it is a square number. 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 9 Factors The square number can be always found in the middle.
  • 15. Fill in this table... Square Root Square Number 4 64 144 7 13 100
  • 16. Your Turn 1. The factors of 136 are listed in ascending order. 136 = 1, 2, 4, 8, 17, 34, 68, 136 Is 136 a square number? 2. Find: 42 62 82 72 92 12 25 64 81 162