1. Exploring Square Roots

2. PICK A 4 DIGIT # WITH DIFFERENT DIGITS
Find the greatest # that can be made with these digits.
Find the least # that can be made with these digits.
Subtract the least from the greatest.
Repeat steps 1,2 & 3 until you notice something interesting…….
6174

3. Think It Out What are some properties we know about rectangles?
Opposite sides are parallel and congruent
4 right angles
The diagonals bisect each other
The diagonals are congruent
What are some properties we know about squares?
All four sides are congruent
Opposite sides are parallel
The diagonals bisect each other at right angles
Every square is a rectangle.
True  
Every rectangle is a square.
False

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

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
52 is five multiplied by itself: 5 x 5 = 25
Likewise, if you see 53, it is not
5 x 3 = 15, rather, it is 5 x 5 x 5 = 125
What does 59 look like?
5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5

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

9. Symmetry
If something has SAME on both sides of the line it is said to have……

10. How many lines of symmetry for each shape?4 3 6 5 8

11. Regular Regular Polygons Equilateral Triangle Square Regular Pentagon
Regular polygons have lines of symmetry equal to the number of sides/angles that they possess.
Regular
Regular Hexagon
Regular Octagon

12. 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

13. 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

14. 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 27
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 is not a prime # as only has 1 number
Which numbers have an even number of factors, but more than 2 factors?
 - 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26
- Composite Numbers, have more than two factors
Which numbers have an odd number of factors?
 - 1,4,9,16,25
Which numbers in the chart are 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)

15. Determining the SQUARE ROOT 36 ÷ 6 = 6
36 ÷ 6 = 6
dividend divisor quotient
Example 1: Find the factors of 16.
16 ÷ 1 = 16
16 ÷ 2 = 8
16 ÷ 4 = 4
16 ÷ 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.

16. Ex. 2 The factors of 136: 1,2,4,8,17,34,68,136. Is 136 a square number?
136 has 8 factors,
so it is an even number.
A square number has odd number of factors; therefore, 136 is not a square #.
Also, there is no number times itself that gives 136.

17. Ex. 3 Find the square root of 121. 11 x 11 = 121, so 121 = 11
Also, the side length of a square with area 121 is 11 units. 11 11

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

19. 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

20. 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

21. 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:
2 , 4 , 5 , 11 , 18 , 24 , 25

22. To estimate a square root:
Square root of a non-perfect square is not a whole number but a decimal number.
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.

23. What is the square root 57?
57 is between the perfect squares 49 and 64.
49 < 57 < 64
√49 < √57 < √64
Square root of 49 is 7 and 64 is 8
7 < √57 < 8
So, √57 is between 7 and 8.
57 is a little over halfway from 49 to 64, so ~

24. Example 4: Which whole number is √96 closest to?
81 < 96 < 100
√81 < √96 < √100
9 < √96 < 10
So ~9.8

25. 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.

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