1. Exploring Square Roots

2. Think It Out
Every square is a rectangle.
Every rectangle is a square.
True
False
What are some properties we know about rectangles?
Opposite sides are parallel and congruent (equal)
4 right angles
The diagonals bisect each other
The diagonals are congruent
The diagonals bisect has opposite angles are equal

3. Think It Out >> > > >>
What are some properties we know about squares?
All four sides are congruent
4 right angles
The diagonals bisect each other at right angles
The diagonals are congruent
Opposite sides are parallel
>>
>
>
>>

4. Investigate…..
Using a piece of grid paper, make as many different rectangles as you can with each area:
4 square units
6 square units
8 square units
9 square units
10 square units
12 square units
16 square units
For how many areas above were you able to make a square?
4, 9, and 16 square units
How is the side length of these squares related to its area?
4 square units: side length = 2 units
9 square units: side length = 3 units
16 square units: side length = 4 units
The side length of a square multiplied by itself equals the area.

5. Perfect Square Or Square Number
Area of rectangle or square = length • width
Area = l · w
= 3 · 3
= 32
= 9 3 3
A number that is a square of an integer is called
Perfect Square Or Square Number

6. Perfect Square List the perfect squares for the numbers 1-12 1 2 3 4 56 7 8 9 10 11 12 1 4 9 16 25 36 49 64 81
100
121
144

7. Common Misconceptions
52 does not equal 5 x 2 = 10
It is 5 x 5 = 25
Likewise, if you see 53, it is not 5 x 3 = 15
it is 5 x 5 x 5 = 125
So what does 106 look like?
10 x 10 x 10 x 10 x 10 x 10
= 1,000,000

8. Classwork
Page 8 #4,5,11,12,16

9. Recall: a square # is a number that can be written as the product of a number and itself.
Ex. 9 is a square number (perfect square),
since 9 = 3 x 3 = 32

10. 25 81 Square Root = 5 = 9 Radical Sign
Is the inverse of the square number (x2 = x • x)
What is the square root of 16?
4 x 4 25
= 5
Radical Sign 81
= 9

11. Investigate…. complete the factors for the chart1 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  1 1   3  2
 11
 13
 17
 19
 23  5  9  7  4
 25
 10
 14
 15  8  6
 21
 22
 26
 16
 12
 18
 20
Which numbers have only two factors? What is special about these numbers?
 - 2,3,5,7,11,13,17,19, 23
Prime Numbers, have 2 factors
Note: 1 is not a prime # as only has 1 number
Which #s have an odd number of factors? What is special about these numbers?
 - 1,4,9,16,25
- Square numbers
What seems to be true about the factors of a square number (perfect square)?
- When a # has odd number of factors, it is a square number (perfect square)

12. 36 ÷ 6 = 6 Determining the SQUARE ROOT dividend divisor quotient
36 ÷ 6 = 6
dividend divisor quotient
Determining the
SQUARE ROOT
Example 1: Find the factors of 16.
16 ÷ 1 = 16
16 ÷ 2 = 8
16 ÷ 4 = ÷ 8 = 2
16 ÷ 16 = 1
The factors of 16 are: 1, 2, 4, 8, 16.
When placed in ascending order, the middle number 4, is the square root of 16, (√16 = 4).
When a number has an odd number of factors, it is a square number. When a number has an even number of factors, it is not a square number.

13. 136 has _______ factors, so it is an ______ number.
Ex. 2 Is a square number?
The factors of 136: 1,2,4,8,17,34,68,136.
136 has _______ factors, so it is an ______ number.
A square number has _____ number of factors;
therefore, 136 is _____ a square #.
Even
Odd
Not

14. Ex. 3 Find the square root of 121 using the side length of a square with area equal to the given number.
Area of a Square is 11 x 11 = 121
so = 11 11
121 11

15. Classwork
Page #6,7,10,13-15,17,19

16. Math fact:
The sum of any number of consecutive odd whole numbers, beginning with 1, is a perfect square
e.g. 1+3=4,
1+3+5=9,
=16

17. Square Root Recap….
The square root ( √ ) of a number is the number that when multiplied by itself results in the given number.
Example 1: Find √144.
  √144 = since 12 × 12 = 144.
We have also expressed the square root of a number as the side length of a square with area equal to the given number.
Example 2: Using a diagram, show that √25 is 5.
The side length of a square with area 25 units2 5 5

18. Investigate: Work with a partner
Investigate: Work with a partner. Use the number line below to place each square root on the number line to show its approximate value:
1 , 2 , 4 , 5 , 11 , 18 , 24 , 25

19. Square root of a non-perfect square
To estimate a square root:
Find the two consecutive perfect squares that the given number is between.
Find the square roots of these two perfect squares.
The square root of the given number will lie between these results.
The decimal place is then estimated by how close it is to either number.

20. Example 1: What is the √96?
Between which two consecutive perfect numbers is 96?
~9.8

21. Ex 2 What is the square root 57?
Between which two consecutive perfect numbers is 57?
57 is between the perfect squares _____ and _______.
Square root of ____ is ______ and _____ is _______ 
So, √57 is between ________________.
57 is a _______________ from 49 to 64,
so ~ ____________. 49 64
49 < 57 < 64 49 7 64 8
√49 < √57 < √64
7 & 8
7 < √57 < 8
little over halfway

22. Example 5: Estimate √20 to one decimal place.
16 < 20 < 25
√16 < √20 < √25
4 < √20 < 5
A good estimate of √20 is 4.4.

23. Classwork
p #1-5,7,10,11,13,16,21,22 (calculator can be used for #13 and 16 – symbol is beside the question).