A difference of two squares

1
A difference of two squares

• Mr Fakhfakh

2
Learning objective

• Understand how to calculate a difference of two
  squares by using knowledge of square numbers

3
Assessment objective

• By the end of this lesson you will be able to
• calculate the answer to questions such as 4012
  4002 without a calculator
• prove a difference of two squares geometrically
  and algebraically

4
Square numbers???
5
Square numbers???

• 1 36 121
• 4 49 144
• 9 64 169
• 16 81 196
• 25 100 225

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Work out

• 22 - 12
• 32 - 22
• 42 - 32
• 52 - 42

7
Work out

• 22 - 12 3
• 32 - 22 5
• 42 - 32 7
• 52 - 42 9

8
Work out

• 32 - 12
• 42 - 22
• 52 - 32
• 62 - 42

9
Work out

• 32 - 12 8
• 42 - 22 12
• 52 - 32 16
• 62 - 42 20

10
Work out

• 42 - 12
• 52 - 22
• 62 - 32
• 72 - 42

11
Work out

• 42 - 12 15
• 52 - 22 21
• 62 - 32 27
• 72 - 42 33

12
What do you notice?
42 - 12 15 52 - 22 21 62 - 32 27 72 - 42
33

• 22 - 12 3
• 32 - 22 5
• 42 - 32 7
• 52 - 42 9

32 - 12 8 42 - 22 12 52 - 32 16 62 - 42 20
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So

• Difference of two squares sum of two numbers x
  difference of two numbers

Let a first number Let b second number a2
b2 (a b) x (a b)
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Proof of a difference of two squares

• Geometric proof

15
a
b
a
b
Area of each of the squares is?
16
a
Area a2
Area b2
b
a
b
17
a
Area a2
b
Area b2
a
b
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Area ?
a
b
a
b
19
Area a2 b2
a
b
a
b
20
Area a2 b2
a
b
?
b
21
Area a2 b2
a
b
a-b
b
22
Area a2 b2
a
b
a-b
b
23
Area a2 b2
a
b
a-b
b
24
?
a
b
a-b
b
25
a-b
a
b
a-b
26
a-b
a
b
a-b
27
a-b
a
b
a-b
28
a-b
a
b
a-b
29
a-b
a
b
a-b
30
a-b
a
b
a-b
31
a-b
a
b
a-b
32
a-b
a
b
a-b
33
a-b
a
b
a-b
34
a-b
a
b
a-b
35
a-b
a
b
a-b
36
a-b
a
b
a-b
37
a-b
a
b
a-b
38
a-b
a
b
a-b
39
a-b
a
b
a-b
40
a-b
a
b
a-b
41
a-b
a
b
a-b
42
a-b
a
b
a-b
43
a-b
a
b
a-b
44
a-b
a
b
a-b
45
a-b
a
b
a-b
46
a-b
a
47
a-b
a
48
a-b
a
49
a-b
a
50
a-b
a
51
a-b
a
52
a-b
a
53
a-b
a
54
a-b
a
55
a-b
a
56
a-b
a
57
a-b
a
58
a-b
a
59
a-b
a
60
a-b
a
61
a-b
a
62
a
a-b
?
63
a
a-b
ab
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Area ?
a-b
ab
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Area (a b)(a - b)
a-b
ab
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Area (a b)(a - b)
a-b
ab
Therefore a2 b2 (a b)(a - b)
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Proof of a difference of two squares

• Algebraic proof

68
Algebraic proof

• a2 b2 (a b) x (a b)

69
Algebraic proof

• Recap
• Multiplying 23 by 25 using the grid method

70
23 x 25
71
23 x 25
72
23 x 25
73
23 x 25
74
23 x 25
75
23 x 25
76
23 x 25
77
23 x 25
78
23 x 25
79
23 x 25

• 400
• 100
• 60
• 15

80
23 x 25 575

• 400
• 100
• 60
• 15
• 575

81
(x 1) (x 3)
82
(x 1) (x 3)
83
(x 1) (x 3)
84
(x 1) (x 3)
85
(x 1) (x 3)
86
(x 1) (x 3)
87
(x 1) (x 3)
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(x 1) (x 3)
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(x 1) (x 3)
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(x 1) (x 3)
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(x 1) (x 3)
(x 1) (x 3) x2 x 3x 3 So (x 1) (x
3) x2 4x 3
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(a b) (a - b)
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(a b) (a - b)
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(a b) (a - b)
95
(a b) (a - b)
96
(a b) (a - b)
97
(a b) (a - b)
98
(a b) (a - b)
99
(a b) (a - b)
100
(a b) (a - b)
101
(a b) (a - b)
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(a b) (a - b)
(a b) (a - b) a2 ab ab b2
(a b) (a - b) a2 ab ab b2
(a b) (a - b) a2 b2
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We have our proof!
(a b) (a - b) a2 b2
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What is

• 5012 5002
• 5022 5002
• 5032 5002
• 1052 1022

105
Learning objective

• Understand how to calculate a difference of two
  squares by using knowledge of square numbers

106
Assessment objective

• By the end of this lesson you will be able to
• calculate the answer to questions such as 4012
  4002 without a calculator
• prove a difference of two squares geometrically
  and algebraically

107
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