Exploring Square Roots

PICK A 4 DIGIT # WITH DIFFERENT DIGITS

1. Find the greatest # that can be made with these digits.

2. Find the least # that can be made with these digits.

3. Subtract the least from the greatest.

4. Repeat steps 1,2 & 3 until you notice something interesting…….

6174

Think It OutWhat 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• The diagonals are congruent• 4 right angles

Every square is a rectangle. True  Every rectangle is a square. False

Investigate…..• Using a piece of grid paper, make as many different rectangles as you can

with each area:

4 square units6 square units8 square units9 square units10 square units12 square units16 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 units9 square units: side length = 3 units16 square units: side length = 4 units

The side length of a square multiplied by itself equals the area.

Perfect Square

Area = l · w

= 3 · 3

= 32

= 93

3

A number that is a square of an integer is called

Area of rectangle or square:Area = length • width A = l • w

Perfect SquareList the perfect squares for the numbers 1-12

1 2 3456789101112

149162536496481100121144

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

Classwork • Page 8 #4,5,11,12,16

Symmetry• If something has SAME on both sides of

the line it is said to have……

How many lines of symmetry for each shape?

4 3 6

5 8

Regular

Regular Polygons

Equilateral Triangle Square Regular Pentagon

Regular Hexagon Regular Octagon

Regular polygons have lines of symmetry equal to the number of sides/angles that they possess.

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

Square RootIs the inverse of the square number (x2 = x • x)

Radical Sign

16 4√25 5

√81 9

What is the square root of 16?

4 x 4

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

1 1 1 1 1 1 1 1                                      

  2 3 2 5 2 7 2                                      

      4   3   4                                      

          6   8                                      

                                                     

                                                     

                                                     

                                                     

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 - 1 is not a prime # as only has 1 numberWhich 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 factorsWhich numbers have an odd number of factors? - 1,4,9,16,25Which numbers in the chart are square numbers?  - 1,4,9,16,25What 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)

Determining the SQUARE ROOT 36 ÷ 6 = 6 dividend divisor quotient

Example 1: Find the factors of 16. 

16 ÷ 1 = 16 16 ÷ 2 = 8 16 ÷ 4 = 416 ÷ 8 = 216 ÷ 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.

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.

Ex. 3 Find the square root of 121.

11 x 11 = 121, so = 11

Also, the side length of a square with area 121 is 11 units.

11

11

Classwork

• Page 15-16 #6,7,10,13-15,17,19

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, 1+3+5+7=16

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 = 12 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

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

Square root of a non-perfect square is not a whole number but a decimal number.

To estimate a square root:

1. Find the two consecutive perfect squares that the given number is between.

2. Find the square roots of these two perfect squares.

3. The square root of the given number will lie between these results.

4. The decimal place is then estimated by how close it is to either number.

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 ~ 7.5-7.6.

Example 4: Which whole number is √96 closest to?

81 < 96 < 100

√81 < √96 < √100

9 < √96 < 10

So ~9.8

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.

Classwork

• p.25-26 #1-5,7,10,11,13,16,21,22 (calculator can be used for #13 and 16 – symbol is beside the question).