1. ALGEBRA 1

2. Words and Symbols ‘Sum’ means add ‘Difference’ means subtract
‘Product’ means multiply
Putting words into symbols
Examples:
the sum of p and q means
p + q
a number 5 times larger than b means 5b
a number that exceeds r by w means
r + w
twice the sum of k and 4 means
2(k + 4)
y minus 4 means
y - 4
y less than 4 means
4 - y

3. The Language of Algebra
Word
Meaning
Example
Variable
A letter or symbol used to represent a number or unknown value
A = π r 2
has A and r as variables
Algebraic
Expression
A statement using numerals, variables and operation signs
3a + 2b - c
Equation
An algebraic statement containing an “ = “ sign
2x + 5 = 8
This is all precious
Inequation
An algebraic statement containing an inequality sign, e.g <, ≤, >, ≥
3x - 8 < 2

4. Terms
The items in an algebraic expression separated by + and - signs
4x, 2y2
3xy, 7
Like Terms
Two terms that have EXACTLY the same variables (unknowns)
4x and -7x
3x2 and x 2
NOT x and x2
Constant Term
A term that is only a number
-7, 54
Coefficient
The number (including the sign) in front of the variable in a term
4 is the coefficient of 4x2, is the
coefficient of
-7y

5. Simplifying Expressions
Multiplying algebraic terms.
Follow my rules and algebra is easy
Multiply the coefficients together.
Simplify the unknowns by writing the letters in alphabetical order
The terms do not have to be ‘like’ to be multiplied
examples: f x 4 =
4a x 2b = 4f 8 ab
-2a x 3b x 4c =
-24
abc
2 x a + b x 3 =
2a + 3b

6. Rules
Curvy
We usually don’t write a times sign e.g 5y not 5 x y, 5(2a + 6)
The unknown x is best written as x rather than x
Numbers are written in front of unknowns e.g. 5y not y5
Letters are written in alphabetical order
Instead of using a division sign ÷, we write the term as a fraction eg 6a ÷ y becomes

7. A number owns the sign in front of it
Like Terms
‘Like terms’ are terms which have the same letter or letters (and the same powers) in them. ie when you say them - they sound the same.
A number owns the sign in front of it
We can only add and subtract ‘like terms’
Examples:
5x + 7x = 12x
5a + 3b - 2a - 6b = 3a
- 3b
10abc - 3cab =
7abc (or 7bca
or 7cba etc)
-4x x x + 7x2 = 1
3x2
+ 2x
+ 3
Note: x means 1x

8. Like terms should sound the same
Collecting Like Terms
Like terms should sound the same
Adding like terms is like
adding hamburgers.
e.g
gives
You’ve started with hamburgers, added some more and you end up with a lot of hamburgers
2x + 3x = 5x
You started with x, added more x and end up with a lot of x, NOT x2

9. Substitution into formulae
Input, x 2
Putting any number, x into the machine, it calculates 5x - 7.
i.e. it multiplies x by 5 then subtracts 7 2 3 3
e.g. when x = 2 3
e.g Calculate 5a - 7 when a = 6 6
5 x =
30 - 7= 23
e.g. Calculate y2 - y + 7 when y = 4
Writing 4 where y occurs in the equation gives = 19

10. Substitution If x = 4 and y = -2 and z = 3 - eg 1: x + y + z = 4 + - 2=
eg 4: 2 x2 =
2 x 42
= 5
= 2 x 16
eg 2: xy (z - x) =
4 x -2
( 3 - 4)
= 32 x -1
= 4 x -2 x -1
eg 5: (2x)2 =
(2 x 4)2
= 8
= 82
= 64
eg 3: y2 =
( -2)2
= 4

11. Patterns Take the common difference to the rule.
Unit Number (n)
Number of cubes (c) 1 2 3 4 n
c = 30
100 49 3 +2 5 +2 7 +2 9 2 n
+ 1 61
201 24
Take the common difference
to the rule.
Multiply by the letter.
Look for what you need to add
or subtract to make it correct

12. Patterns - example 2 Unit Number (n) Matchsticks (m) 1 2 3 4 n m= 4041 6
+ 5 11
+ 5 16
+ 5 21
5 n
+ 1
201 8

13. Patterns - example 3 Unit Number (n) Number of dots (d) 1 2 6 3 10 414 n d= 18
102
+ 4
+ 4
+ 4
4 n
- 2 70 26

14. Indices in Algebra 24 24 means 2 x 2 x 2 x 2 = 16 a x a x a = a 3
index, power or exponent 24
base
a x a x a =
a 3
2 x a x a x a x b x b =
2 a 3
b 2
m x m - 5 x n x n =
m 2
- 5 n 2
When multiplying terms with the same base we add the powers
eg 1: y 3 x y 4 =
y 7 28
eg 2: x =
eg 3: 3 m 2 x 2 m 3 = 6
m 5

15. When dividing terms with the same base we subtract the powers.
eg 1: p 8 ÷ p 2 =
p 6 5
eg 2: 5
x 4 1 2 m
eg 3: 3

16. Expanding Brackets
Each term inside the bracket is multiplied by the term outside the bracket.
Remember:
means the terms are multiplied
example 1:
4 ( x + 2) = + 4x 8
example 2:
x ( x - 4) = -
x 2 4x
example 3:
3y ( y 2 + y - 3) =
3y 3
+ 3y 2
- 9y
example 4:
-2 (m - 4) =
-2m
+ 8
NB: -2 x -4 = +8
example 5:
x (x - 5) + 2 (x + 3)
= x 2
- 5x
+ 2x
+ 6
= x 2 - 3x + 6

17. Factorising This means writing an expression with brackets eg 1:
2x + 2y = 2
( ) x + y
eg 2:
3x + 12 = 3 ( x + 4 )
eg 3:
6x - 15 = 3 ( 2x -
5 )
eg 4:
4x2 + 8x = 4 x ( x
+ 2 )
x x x
NB: Always take out the highest common factor.
eg 5:
10d 2 - 5d =
5d ( 2d - 1)
eg 6:
12a 3b 4c a 2b 3c 3 + 8a 4b 4c 5
aaabbbbcc
aabbbccc
aaaabbbbccccc
= 4a 2b 3c 2 (
3a b
- 5 c
+ 2a 2 bc 3)