1. Algebra Expressions
Year 9

2. Note 1: Expressions
We often use x to represent some number in an equation. We refer to the letter x as a variable.
x + 5 means ‘a number with 5 added on’
x – 7 means ‘a number with 7 subtracted from it’
e.g.

3. Note 1: Expressions A few Rules:
A number should be written before a letter.
y x 2 = 2y
e.g.
Terms should be written in alphabetical order.
xyz and b x 4a = 12ab
e.g.
We don’t use the x or ÷ signs in algebra instead we write it like this:
5 x y = 5y x ÷ 9 =
e.g.

4. Activity: Expressions
Match each algebraic expression with the phrase
Five times a number
x – 8
3 + x
Three plus a number
A number multiplied by seven
Half of a number 7x
3x + 1
A number plus six
A number divided by nine 5x
A number minus eight
Three times a number plus one
x + 6

5. Activity: Expressions
Write an equivalent algebraic expression for each phrase
12x
Twelve times a number
1 + x
One plus a number 3x
A number multiplied by three
A quarter of a number
x + 100
A number plus one hundred
A number divided by nineteen
x – 4
A number minus four
IWB Ex 11.01 Pg
8x – 1
Eight times a number minus one

6. Note 2: Substitution
We replace the variable (letter) with a number and calculate the answer.
Algebra follows the same rules as BEDMAS!
e.g.
If a = 2, b = -3, c = 5 then calculate:
a + 5 3b
a + b + c
= 2 + 5
= 3 x -3 =
= 7
= -9
= 2 – 3 + 5
= 4

7. BEDMAS Note 2: Substitution Remember: e.g.
Do multiplication and division before addition and subtraction
Anything in brackets is worked out first
A number in front of the bracket means multiply
The fraction line means divide
e.g.
If d = 3, e = 7, f = -2 then calculate:
4(d + e)
5def – 2e =
= 4(3 + 7)
= 5 x 3 x 7 x -2 – 2 x 7 =
= 4 x 10
= -210 – 14
= 40
= -224
= 4

8. BEDMAS Note 2: Substitution Remember: e.g.
Do multiplication and division before addition and subtraction
Anything in brackets is worked out first
A number in front of the bracket means multiply
The fraction line means divide
e.g.
If d = 3, e = 7, f = -2 then calculate:
2(d - e)2
5f – 2d =
= 2(3 - 7)2
= 5 x -2 – 2 x 3
= 2 x (-4)2
= -10 – 6 =
= 2 x 16
= -16
= 32
= 2

9. Note 2: Substitution e.g. 5x Number of tyres = IWB
the number of cars
IWB
Ex Pg
Ex Pg 282
Puzzle Pg 283
b Number of tyres = 5 x 60
c 5 x 40 = 200
= 300

10. Starter - = 5 x 3 = 5 x -4 = 3 x -4 = 15 = -20 = -12 = 2 x -4 = 12 x 3
= -8
= 36
= -8
= 6 x 2
= 3 x 2
= 3 x 2 x -4
= 12
= 6
= -24

11. Note 3: Formulas
A formula is a mathematical rule that explains how to calculate some quantity.
e.g.
John baby sits for his neighbours. He charges a set fee of $10 plus $5 for every hour (h), that he baby sits. A formula to calculate this charge is given by:
Charge = $10 + $5h
Use the formula to calculate the amount John charges if he baby sits for:
5 hours
3 hours
= x 5
= x 3
= $ 35
= $ 25

12. Note 3: Formulas
e.g.
If John receives $65 how long did he baby sit for?
Charge = $10 + $5h
$65 = $10 + $5 x h
65 = h
= h
55 = 5h 5 5
11 = h
IWB
Ex Pg
Ex Pg
John baby sat for 11 hours

13. Starter
Multiply the base length by the height and divide by 2
= 20 cm2

14. Note 4: Multiplying Algebraic Expressions
Rules:
Multiply the numbers in the expression (these are written first)
Write letters in alphabetical order

16. Note 5: Adding & Subtracting Algebraic Expressions
Write in simplest form:
y + y + y + y
= 4y
a + a + a + a + a
= 5a
a + a + a + a + a + b
= 5a + b
a + a + a + a + a + b + b + b
= 5a + 3b
x + y + x + y + x + y + x
= 4x + 3y
x + y + y + y + y - x
= 4y

17. Note 5: Adding & Subtracting Algebraic Expressions
Write in simplest form:
y + 3y
= 4y
3x + 5x
= 8x
5x – 2x + 3x
= 6x
5x – 4x
= x
10x + x + 19x
= 30x
IWB
Ex Pg 285
Ex Pg 286
12p + 3r – 2p + 3r
= 10p + 6r

18. Note 6: Like and Unlike Terms
Like terms are terms that contain the exact same variables (letters) or combinations of letters.
e.g.
Like Terms
2x, 5x, 25x, -81x, 13x, 0.5x…
Remember: If we had written these terms properly (in alphabetical order), it would be more obvious that they are like terms.
xy, 2xy, -4xy, ½xy, -100xy,…
2abc, 4bac, 6 cab, 9abc, …..

19. Note 6: Like and Unlike Terms
Like terms are terms that contain the exact same variables (letters) or combinations of letters.
e.g.
Unlike Terms
2, 2x, 3y
3a, 7ab, 8b
2p, 4r, 10s

20. Note 6: Like and Unlike Terms
Expressions can have a mixture of LIKE and UNLIKE terms.
e.g.
3x + 5y
2a – 7b + 5a + 8b
= 7a + b
Like terms can be grouped together
and simplified
Unlike terms cannot be simplified
e.g.
5a + 2b – 3a + 6c + 4b
2a + 6b + 6c

21. Note 6: Like and Unlike Terms
2x + 10y
8x + 8y
4x + 1
6p + 9q
x + 1
IWB
Ex Pg 289
Ex Pg 292-3
Workbook
Ex E page 117
2x + 2 x
2x – 7

22. Starter Find the perimeter of this shape in terms of x
The perimeter is the sum of all side lengths
4x + 1
5x – 2
P = x + (2x + 1) + (2x + 3) + (4x + 1) + (5x – 2)
= 14x + 3
If the perimeter is 31 cm. What is the value of x and which side is the longest?
P = 14x + 3
28 = 14x
x = 2
31 = 14x + 3

23. Note 7: Powers (exponents)
Recall: When a variable (letter) is multiplied by itself many times, we use powers
e.g. Write the following in index form: p3
p x p x p = ________________
q x q x q x q x q = __________
s x s x s x t x t x t x t = _______
p x p x q x q x s x s = ________
s x t x t x t x s x s = __________ q5
s3t4
p2q2s2
s3t3

24. Note 7: Powers (exponents)
Substituting:
Evaluate the following when a =2, b = 7 and c = -3
5a2
= 5 x 22 a2
= 22
= 5 x 4
= 4
= 20
(b + c)2
= (7 - 3)2
= (4)2
2a2c2
= 2 x 22 x (-3)2
= 16
= 2 x 4 x 9
(5a)2
= (5 x 2)2
= 72
= 102
= 100

25. Note 7: Powers (exponents)
NOTE: b = b1
Multiplying:
Simplify: b3 x b4
= (b x b x b) x (b x b x b x b)
= b7
When multiplying power expressions with the same base, we add the powers.
an x am = am+n
e.g. e2 x e6
g8 x g
5m3 x 4m3
= e2+6
= g8+1
= 20m3+3
= e8
= g9
= 20m6

26. Note 7: Powers (exponents)
Dividing:
IWB
Ex Pg 346
Ex Pg 349
PUZZLE Pg 350
Simplify: =
= c3
When dividing power expressions with the same base, we subtract the powers.
am = am-n an
e.g.
g7 ÷ g
= g7-1
= f 6-3
= 5q7-3 =
= g6
= f 3
= 5q4

27. Starter
How would you calculate 7 x 83 in your head?
7 x x 3
= 581
What we have done in our head can also look like this: x x
7 (80 + 3) =
7 x 80
x 3
= 581

28. Note 8: Expanding Brackets
To expand (remove) brackets:
Multiply the outside term by everything inside the brackets
Simplify where possible
e.g. Expand:
a.) 4(x + y)
b.) −2(x – y)
c.) 5(x – y + 2z)
The Distributive Law
= 4x + 4y
= -2x + 2y
= 5x - 5y + 10z

29. Try These! e.g. Expand: = 8x + 8y = -8x – 8y = 4x - 4y = 5x – 15y
a.) 8(x + y)
b.) 4(x – y)
c.) 2(x – y)
d.) 3(-x + y)
e.) 9(x + y + z)
f.) -8(x + y)
g.) 5(x – 3y)
h.) -(x – 2y)
i.) -7(-x + 7y)
j.) -4(3x - y + 5z)
= 8x + 8y
= -8x – 8y
= 4x - 4y
= 5x – 15y
= 2x - 2y
= -x + 2y
= -3x + 3y
= 7x – 49y
= 9x + 9y + 9z
= -12x + 4y – 20z

30. Lets do some more e.g. Expand: = 8x + 32 = -8x – 64 = 4x – 8y
a.) 8(x + 4)
b.) 4(x – 2y)
c.) 2(x – 10)
d.) 3(2x + 9)
e.) -(x + 2y - z)
f.) -8(x + 8)
g.) 5(5x – 3)
h.) -(x – 11)
i.) -7(x + 12)
j.) -2(x - y + 14)
= 8x + 32
= -8x – 64
= 4x – 8y
= 25x – 15
= 2x - 20
= -x + 11
= 6x + 27
= -7x – 84
= -x – 2y + z
= -2x + 2y – 28

31. Starter e.g. Expand: = 2x + 2y = -8x – 16 = 4x – 4y = -15x + 30
a.) 2(x + y)
b.) 4(x – y)
c.) 2(x – 3y)
d.) 3(x + 30)
e.) -(2x - 4y + z)
f.) -8(x + 2)
g.) -5(3x – 6)
h.) -(x – 23)
i.) -3(-2x + 3)
j.) -5(x - 2y + 1)
= 2x + 2y
= -8x – 16
= 4x – 4y
= -15x + 30
= 2x – 6y
= -x + 23
= 3x + 90
= 6x – 9
= -2x + 4y – z
= -5x + 10y – 5

32. Note 8: Expanding Brackets
The terms on the inside can also be multiplied by a variable on the outside.
e.g. Expand:
a.) a(x + y)
b.) 2a(a + b)
c.) 5x2(x2 – x + 2)
= ax + ay
= 2a2 + 2ab
= 5x4 – 5x3 + 10x2

33. Your turn! e.g. Expand: = ax + 4a = x9 + x8y = bx – 5by = 15x2y2 – 5xy
a.) a(x + 4)
b.) b(x – 5y)
c.) x(x – 15)
d.) y(2x + 2)
e.) x(x2 + 2y – 8)
f.) x5(x4+ x3y)
g.) 5xy(3xy – 1)
h.) -x(x – 11)
= ax + 4a
= x9 + x8y
= bx – 5by
= 15x2y2 – 5xy
= x2 – 15x
= -x2 + 11x
= 2xy + 2y
= x3 – 2xy – 8x

35. Note 8: Expanding Brackets and Collecting Like Terms
Expand the brackets first, then simplify!
e.g. Expand & Simplify
a.) 4(2x + y) + 3(x + 5y)
= 8x + 4y + 3x + 15y
* Collect like terms
= 11x + 19y
b.) 4(5x - y) – 3(x – 10)
= 20x - 4y – 3x + 30
* Collect like terms
= 17x – 4y + 30

36. Your turn! = 5x + 5y + 2x + 2y = 7x + 7y = 2x + 2y + 8x + 4y
IWB - odd only
Ex 15.02, Pg 389
Ex Pg 390
Ex Pg 391
Ex 15.06, Pg 393
Ex Pg 397
= 2x + 2y + 8x + 4y
= 10x + 6y
= 6x + 3y + 6x + 12y
= 12x + 15y
= 12x + 18y – 10x – 4y
= 2x + 14y
= 2x x + 24
= 6x + 30

37. Factorising 3x + 6 3 (x + 2) Expanding Factorising
Factorising is the reverse procedure of expanding.
Expanding
3x + 6
3 (x + 2)
Factorising

38. Note 9: Factorising (put in brackets)
Factorising is the reverse process of expanding.
We want to put brackets back into the algebraic expression
find the highest common factor and write it in front of the brackets
e.g. Factorise
3x + 3y
4x – 4y
7x + 7y + 7z
= 3( )
x+y
= 4( )
x – y
= 7( )
x + y + z

39. Try These e.g. Factorise: = 6( ) a+b = 7( ) x+1 = 7( ) x+2 = 3( )
e.g. Factorise:
a.) 6a + 6b
b.) 3p – 3q
c.) 4x + 4y
f.) 7x +7
g.) 7x + 14
h.) 24x + 36
= 6( )
a+b
= 7( )
x+1
= 7( )
x+2
= 3( )
p – q
= 4( )
x+y
= 12( )
2x+3
d.) 6x + 12
= 6( )
x+2
You can check that your answer is correct by expanding
= 24( )
x+y
e.) 24x + 24y

40. Try These e.g. Factorise: = 2( ) 4a+3b = 7( ) x+7 = 9( ) x+7 = 3( )
e.g. Factorise:
a.) 8a + 6b
b.) 12p – 3q
c.) 4x + 8
f.) 7x + 49
g.) 9x + 63
h.) 45x + 81
= 2( )
4a+3b
= 7( )
x+7
= 9( )
x+7
= 3( )
4p – q
= 4( )
x+2
= 9( )
5x+9
d.) 6x + 30
= 6( )
x+5
You can check that your answer is correct by expanding
= 29( )
x+1
e.) 29x + 29

41. Starter Factorise: = 4(a+2b) = 3(p – 2q +r) = 4(x + 2y + 3z)
Factorise:
a.) 4a + 8b
b.) 3p – 6q + 3r
c.) 4x + 8y + 12z
= 4(a+2b)
= 3(p – 2q +r)
= 4(x + 2y + 3z)
d.) 6x + 21
= 3(2x +7)
IWB - odd only
Ex Pg 400
Ex Pg 401
Ex Pg 402
Ex Pg 403
Ex Pg 405
e.) 24x - 32
= 8(3x – 4)

42. Extension – More exponent rules
How do we simplify an exponential term raised to another exponent? × ×
e.g. (2y3)2
e.g. (3a4)3
= (2y3) × (2y3)
= (3a4) × (3a4) × (3a4)
= 4y6
= 27a12
Notice that there is a shortcut to get the same result
= 22y2×3
= 33a4×3
= 4y6
= 27a12

43. Extension – More exponent rules
(am)n = amn
1.) Index the number. 2.) Multiply each variable index by the index outside the brackets. 3.) If the bracket can be simplified, do this first.
e.g. Simplify × ×
(2x2)3
(-4h2g6)2
= 23 x23
= (-4)2h2×2g6×2
= (4x)2
= 8x6
= 16h4g12
= 16x2

44. Extension – Expanding 2 Brackets
QUADRATIC EXPANSION When we expand two brackets we use:
F – first (multiply the first variable or number from each bracket)
O – outside (multiply the outside variables together)
I – inside (multiply the two inside variables together)
L – last (multiply the last variable in each bracket together)
Simplify, leaving your answer with the highest power first to the lowest power (or number) last.
F O I L
e.g. (x + 4) (x – 2)
= x x x - 8
= x2 + 2x - 8

45. Extension – Expanding 2 Brackets
QUADRATIC EXPANSION
e.g. (x + 10) (x + 1)
e.g. (x + 3) (x – 5)
= x2 + 10x + x + 10
= x2 + 3x - 5x - 15
= x2 + 11x + 10
= x2 - 2x - 15
e.g. (x - 4) (x + 4)
e.g. (x - 3) (x – 8)
= x2 - 4x + 4x - 16
= x2 - 3x - 8x + 24
= x2 - 16
= x2 - 11x + 24
Notice the middle term cancels out
DIFFERENCE OF SQUARES

46. Extension – Expanding 2 Brackets
QUADRATIC EXPANSION
e.g. (x – 5) (x + 4)
e.g. (x + 7) (x – 9)
= x2 - 5x + 4x - 20
= x2 + 7x - 9x - 63
= x2 - x - 20
= x2 - 2x - 63
e.g. (x - 9) (x + 9)
e.g. (x - 2) (x – 6)
= x2 - 9x + 9x - 81
= x2 - 2x - 6x + 12
= x2 - 81
= x2 - 8x + 12
Notice the middle term cancels out
DIFFERENCE OF SQUARES

47. Extension – Factorising Quadratics
e.g. Factorise simple quadratics:
a.)
x2 + 6x + 8
Find 2 numbers which multiply to 8 and add to 6
(x+2) (x+4)
4 and 2
Put these number into brackets
The factors are (x+2) and (x+4)
b.)
x2 - 5x - 24
Find 2 numbers which multiply to -24 and add to -5
(x-8) (x+3)
-8 and 3
Put these number into brackets
The factors are (x-8) and (x+3)
c.)
x2 + 5x - 24
Find 2 numbers which multiply to -24 and add to 5
(x+8) (x-3)
8 and -3
Put these number into brackets