1. MATHS
Week 12
Algebra

2. Starter!
How much does 1 pineapple weigh?
What about 1 apple?

3. What did we do last week?

4. Before we begin……

5. Have you done your directed study?

6. What are we going to do this week?
Algebra Terminology
Expand with two brackets
Factorise Quadratics
Laws of indices
Solving, writing & rearranging

7. Algebraic Terminology

9. Can you identify which is which?
You need to pick four different colours/highlighters. Can you colour code the grid to find which of the boxes contain expressions, formulae, identities and equations?

10. Answers

13. Expand (y + 10)
Expand h(h – 5)

14. Expanding two brackets using the FOIL method

15. You may be asked to expand and simplify two brackets that look like this:
(y + 6)(y + 3)

16. F O I L
irst
utside
nside
ast

17. Question – to do together
(x + 3)(x + 6)
F O I L

18. Question – to do together
(4x – 3)(3x – 4)
F O I L

19. Question – to do together
(y – 7)2
F O I L

20. Your Turn! Remove the brackets and simplify: (x + 4)(x + 1)

21. ANSWERS
x2 + 5x + 4
x2 – 5x + 6
x2 + 10x + 25
What do you notice about all these answers?
x² + bx + c
Do you know what these are called?

23. Factorising Factorise 5p – 25 Factorise mn + mt Factorise 9xy – 3y
Factorise w2 + 3wz

24. Factorising Quadratics

25. Factorising Quadratics
You can factorise quadratic expressions of the form x² + bx + c
Find two numbers whose products is + c and whose sum is +b
Use these two numbers, p and q, to write down the factorised form
(x + p) (x + q)

26. Example – do together

27. Example – do together

28. Example – do together

29. Example – do together

30. Your turn x² + 3x – 18 b) x² - 6x + 5 c) x² + 9x + 20 d) x² + 3x -28
e) x² + 2x – 24
f) x² + 8x + 7

31. Answers a) ( x - 3) (x + 6) b) ( x – 5) ( x – 1) c) ( x + 4) (x + 5)
d) ( x - 4) (x + 7)
e) ( x - 4) (x + 6)
f) ( x + 1) (x + 7)

32. Multiply out and simplify
(w – 3)(w + 3)
(x + 1)(x – 1)
(y + 4)(y – 4)
(z + 5)(z – 5)
What do you notice?

33. Factorise
x² - 36
b) x² - 49
c) y² - 144

34. Index Notation and Rules of Indices

35. 2 x 2 x 2 x 2 x 2 25 is read 2 to the power of 5 can be written 25
5 is the power or index
2 x 2 x 2 x 2 x 2
25 is read 2 to the power of 5
2 is the base
can be written 25

36. Example
Calculate 34
34 =
= 81
3 x 3 x 3 x 3

37. Write 7 x 7 x 7 x 7 x 7 x 7 in index form.
Example
Write 7 x 7 x 7 x 7 x 7 x 7 in index form. 76
7 x 7 x 7 x 7 x 7 x 7 =

38. Example
Simplify 23 x 24
23 x 24
= (2 x 2 x 2) x (2 x 2 x 2 x 2)
= 2 x 2 x 2 x 2 x 2 x 2 x 2
= 27
Provided the base numbers are the same you can add the powers when multiplying.

39. ax x ay = ax + y 43 x 45 = 4(3 + 5) = 48 Example Simplify 43 x 45
In general
ax x ay = ax + y

40. Example
Simplify 85 ÷ 83
85 ÷ 83 =
8 x 8 x 8
Provided the base numbers are the same you can subtract the powers when dividing.
8 x 8 x 8 x 8 x 8
= 82

41. ax ÷ ay = ax - y 57 ÷ 54 = 5(7 - 4) = 53 Simplify 57 ÷ 54 Example
In general
ax ÷ ay = ax - y
Example
Simplify 57 ÷ 54

42. 34 ÷ 34 =
3(4 - 4)
= 30
But 34 divided by itself is 1
a0 = 1
Example
30 = 1
In general anything to the power of zero is 1

43. Indices: Tick or Trash

45. Solving Equations

47. 2x = 18 What is x?

48. Solving equations
2x – 1 = 7

49. Solving equations
3x + 2 = 23

50. Solving equations
t + t + t = 4.5

51. Solving equations
x + 5 = 7 3

52. Solving equations
2(x – 4) = 12

53. Solving equations
3x + 7 = x + 12

54. Solving inequalities – same method
2x - 4 < 5x - 16

55. Solving inequalities
3(3x + 1) ≥ 21

56. Algebra to solve problems

57. Algebra - writing expressions
Ten quick questions

58. 1 A: x + 2 b: 2x c: x - 2 d: 2 I have x biscuits on a plate.
I eat 2. How many do I have left?
A: x + 2
b: 2x
c: x - 2
d: 2

59. 2
Write an expression for the sum of two numbers v and w.
a: v + w
b: vw
c: v - w
d: v ÷ w

60. 3
There are ‘t’ people in a queue. 7 people join. How many are in the queue now?
a: 7
b: t + 7
c: 7 - t
d: t - 7

61. 4 a: x + y b: 30x + 30y c: 5x + 6y d: 6x + 5y
A milkshake costs x pence and a fizzy drink costs y pence. I buy 5 milkshakes and 6 fizzy drinks. Write an expression for the total cost.
a: x + y
b: 30x + 30y
c: 5x + 6y
d: 6x + 5y

62. 5
I am p years old. 3 years ago my brother was twice my age. How old was my brother?
a: p - 3
b: 2(p – 3)
c: 2p
d: 6

63. 6
There are n sweets in a packet. I buy 4 packets. How many sweets do I have?
a: n + 4
b: 4n
c: n4
d: 4n + 4

64. 7
Cakes cost x pence and doughnuts cost y pence. I buy 5 cakes and 8 doughnuts. How much do I pay?
a: 5x – 8y
b: 8x + 5y
c: 40xy
d: 5x + 8y

65. 8
My sister is y years old. In 4 years time I will be twice my sister’s age. How old will I be?
a: y + 4
b: 2(y + 4)
c: 2y + 4
d: 2y

66. 9
There are d marbles in a bag. I give 3 to my friend. How many marbles do I have now?
a: 3
b: d + 3
c: 3d
d: d - 3

67. 10 a: v + w b: v - w c: w - v d: -vw
Write an expression for the difference between the numbers v and w.
a: v + w
b: v - w
c: w - v
d: -vw

68. Answers

69. 1 A: x + 2 b: 2x c: x - 2 d: 2 I have x biscuits on a plate.
I eat 2. How many do I have left?
A: x + 2
b: 2x
c: x - 2
d: 2

70. 2
Write an expression for the sum of two numbers v and w.
a: v + w
b: vw
c: v - w
d: v ÷ w

71. 3
There are ‘t’ people in a queue. 7 people join. How many are in the queue now?
a: 7
b: t + 7
c: 7 - t
d: t - 7

72. 4 a: x + y b: 30x + 30y c: 5x + 6y d: 6x + 5y
A milkshake costs x pence and a fizzy drink costs y pence. I buy 5 milkshakes and 6 fizzy drinks. Write an expression for the total cost.
a: x + y
b: 30x + 30y
c: 5x + 6y
d: 6x + 5y

73. 5
I am p years old. 3 years ago my brother was twice my age. How old was my brother?
a: p - 3
b: 2(p – 3)
c: 2p
d: 6

74. 6
There are n sweets in a packet. I buy 4 packets. How many sweets do I have?
a: n + 4
b: 4n
c: n4
d: 4n + 4

75. 7
Cakes cost x pence and doughnuts cost y pence. I buy 5 cakes and 8 doughnuts. How much do I pay?
a: 5x – 8y
b: 8x + 5y
c: 40xy
d: 5x + 8y

76. 8
My sister is y years old. In 4 years time I will be twice my sister’s age. How old will I be?
a: y + 4
b: 2(y + 4)
c: 2y + 4
d: 2y

77. 9
There are d marbles in a bag. I give 3 to my friend. How many marbles do I have now?
a: 3
b: d + 3
c: 3d
d: d - 3

78. 10 a: v + w b: v - w c: w - v d: -vw
Write an expression for the difference between the numbers v and w.
a: v + w
b: v - w
c: w - v
d: -vw

79. Writing our own equations

82. Mary and John have £50 altogether. Mary has £12 more than John
Mary and John have £50 altogether. Mary has £12 more than John. Write an equation for the money they have altogether and using the equation, work out how much money they have each.

83. Rearranging the Subject of a Formula

84. What does that mean?
The subject of the formula is the letter that is by itself.
To change the subject of a formula, we have to rearrange the other terms to get a certain letter by itself.

85. y = x + 5
If x = 6 what would y be? y = y = 11

86. y = x + 5 You have just rearranged formula!
If y = 15 what would x be? 15 = x + 5 We know that must be 10. But what is the calculation you have done? 15 – 5 = 10 This is the same as y – 5 = x
You have just rearranged formula!

87. y = x + 5 became y – 5 = x What do you notice?
The +5 has moved side and
the + has turned to –

88. y = 5a
This means y = 5 times ‘a’ If a = 8 what would y be? y = 5 x 8 y = 40

89. y = 5a You have just rearranged formula!
If y = 20 what is a? 20 = 5 x a We know that must be 4. But what is the calculation you have done? 20 ÷ 5 = 4 This is the same as y ÷ 5 = a
You have just rearranged formula!

90. y = 5a became y ÷ 5 = a What do you notice? The 5 has moved side and
the multiply has turned to divide

91. When changing the subject of a formula the term that needs to be the subject has to be by itself. Everything else needs to be moved to the opposite side and do the inverse operation. + a would become – a x y would become ÷ y

92. y = a + b - c
At the moment y is the subject but I want to make a the subject. I will leave it where it is Everything else needs to be moved to the other side and reversed I need to move c to the other side and + it instead of - y + c = a + b I need to move b to the opposite side and – it instead of + y + c – b = a

93. This means we want to rearrange the formula so it says
Rearrange the formula to make a the subject
b = 5a + 21
b – 21 = 5a
b – 21 = a
This means we want to rearrange the formula so it says
a =
÷5 ÷5 5
Our answer should say ... a = b – 21 5

94. This means we want to rearrange the formula so it says
Rearrange the formula to make t the subject
h = t
h – 13 = 7t
h – 13 = t
This means we want to rearrange the formula so it says
t =
÷7 ÷7 7
Our answer should say ... t = h – 13 7

95. Hint: You need to expand the brackets first
Exercise One
Rearrange each formula to make s the subject
u = 11s + 3
w = 8s + p
q = 3s + 4t
7s + m + t = l
a = 3(s + 4)
4 = g(s – 7)
22 + 5s = g
Hint: You need to expand the brackets first

96. Rearrange each formula to make s the subject
u = 11s + 3  s = u - 3
w = 8s + p  s = w – p
q = 3s + 4t  s = q – 4t
7s + m + t = l  s = l – m – t
a = 3(s + 4)  s = a - 12
4 = g(s – 7)  s = 4 + 7g
22 + 5s = g  s = g - 22 11 8 3 7 3 g 5

97. This means we want to rearrange the formula so it says
Rearrange the formula to make v the subject
e = 3v + t
5e = 3v + t
5e – t = 3v
5e – t = v
This means we want to rearrange the formula so it says
v = 5
x x5
- t t
÷ ÷3 3
Our answer should say ... v = 5e – t 3

98. Topic Test
13 minutes

99. Answers

103. 12 Quick multiple choice questions
And finally
12 Quick multiple choice questions

104. Recognise Words Used In Algebra
Identity
Equation
Expression
Formula
Recognise Words Used In Algebra

105. 6x
A : Identity
B : Equation
C : Expression
D : Formula

106. 6x
A : Identity
B : Equation
C : Expression
D : Formula

107. 4x = 18
A : Identity
B : Equation
C : Expression
D : Formula

108. 4x = 18
A : Identity
B : Equation
C : Expression
D : Formula

109. 6x + 12 = 3(2x + 4) A : Identity B : Equation C : Expression
D : Formula

110. 6x + 12 = 3(2x + 4) A : Identity B : Equation C : Expression
D : Formula

111. v = u + at
A : Identity
B : Equation
C : Expression
D : Formula

112. v = u + at
A : Identity
B : Equation
C : Expression
D : Formula

113. a² + 2a = a ( a + 2 ) A : Identity B : Equation C : Expression
D : Formula

114. a² + 2a = a ( a + 2 ) A : Identity B : Equation C : Expression
D : Formula

115. A = π r²
A : Identity
B : Equation
C : Expression
D : Formula

116. A = π r²
A : Identity
B : Equation
C : Expression
D : Formula

117. 9b - l0
A : Identity
B : Equation
C : Expression
D : Formula

118. 9b - l0
A : Identity
B : Equation
C : Expression
D : Formula

119. 2x² + 3x – 7 = 0
A : Identity
B : Equation
C : Expression
D : Formula

120. 2x² + 3x – 7 = 0
A : Identity
B : Equation
C : Expression
D : Formula

121. An expression is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

122. An expression is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

123. A formula is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

124. A formula is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

125. An identity is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

126. An identity is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

127. An equation is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

128. An equation is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C
D : 8x + 2 = 2(4x + 1)

129. Directed Study

130. Revise You have your next assessment next week so make sure you revise
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