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KS3 Mathematics
A1 Algebraic
expressions
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Contents
A1 Algebraic expressions
A1.1 Writing expressions
A1.2 Collecting like terms
A1.3 Multiplying terms
A1.4 Dividing terms
A1.5 Factorizing expressions
A1.6 Substitution
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Using symbols for unknowns
Look at this problem:
+ 9 = 17
The symbol
stands for an unknown number.
We can work out the value of
.
=8
because
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8 + 9 = 17
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Using symbols for unknowns
Look at this problem:
–
The symbols
In this example,
For example,
and
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and
and
12 – 7 = 5
=5
stand for unknown numbers.
can have many values.
or
3.2 – –1.8 = 5
are called variables because their value can vary.
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Using letter symbols for unknowns
In algebra, we use letter symbols to stand for numbers.
These letters are called unknowns or variables.
Sometimes we can work out the value of the letters and
sometimes we can’t.
For example,
We can write an unknown number with 3 added on to it as
n+3
This is an example of an algebraic expression.
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Writing an expression
Suppose Jon has a packet of biscuits and he
doesn’t know how many biscuits it contains.
He can call the number of biscuits in the full
packet, b.
If he opens the packet and eats 4 biscuits, he can write an
expression for the number of biscuits remaining in the
packet as:
b–4
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Writing an equation
Jon counts the number of biscuits in the packet
after he has eaten 4 of them. There are 22.
He can write this as an equation:
b – 4 = 22
We can work out the value of the letter b.
b = 26
That means that there were 26 biscuits in the full packet.
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Writing expressions
When we write expressions in algebra we don’t usually use
the multiplication symbol ×.
For example,
5 × n or n × 5 is written as 5n.
The number must be written before the letter.
When we multiply a letter symbol by 1, we don’t have to
write the 1.
For example,
1 × n or n × 1 is written as n.
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Writing expressions
When we write expressions in algebra we don’t usually use
the division symbol ÷. Instead we use a dividing line as in
fraction notation.
For example,
n
n ÷ 3 is written as
3
When we multiply a letter symbol by itself, we use index
notation.
n squared
For example,
n × n is written as n2.
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Writing expressions
Here are some examples of algebraic expressions:
n+7
a number n plus 7
5–n
5 minus a number n
2n
2 lots of the number n or 2 × n
6
n
6 divided by a number n
4n + 5
4 lots of a number n plus 5
n3
a number n multiplied by itself twice or
n×n×n
3 × (n + 4)
or 3(n + 4)
a number n plus 4 and then times 3.
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Writing expressions
Miss Green is holding n number
of cubes in her hand:
Write an expression for the number of cubes in her hand if:
She takes 3 cubes away.
n–3
She doubles the number of
cubes she is holding.
2×n
2n
or
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Equivalent expression match
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Identities
When two expressions are equivalent we can link them with
the  sign.
x + x + x is
identically
For example,
equal to 3x
x + x + x  3x
This is called an identity.
In an identity, the expressions on each side of the equation
are equal for all values of the unknown.
The expressions are said to be identically equal.
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Contents
A1 Algebraic expressions
A1.1 Writing expressions
A1.2 Collecting like terms
A1.3 Multiplying terms
A1.4 Dividing terms
A1.5 Factorizing expressions
A1.6 Substitution
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Like terms
An algebraic expression is made up of terms and operators
such as +, –, ×, ÷ and ( ).
A term is made up of numbers and letter symbols but not
operators.
For example,
3a + 4b – a + 5 is an expression.
3a, 4b, a and 5 are terms in the expression.
3a and a are called like terms because they both contain a
number and the letter symbol a.
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Collecting together like terms
Remember, in algebra letters stand for numbers, so we
can use the same rules as we use for arithmetic.
In arithmetic,
5+5+5+5=4×5
In algebra,
a + a + a + a = 4a
The a’s are like terms.
We collect together like terms to simplify the expression.
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Collecting together like terms
Remember, in algebra letters stand for numbers, so we
can use the same rules as we use for arithmetic.
In arithmetic,
(7 × 4) + (3 × 4) = 10 × 4
In algebra,
7 × b + 3 × b = 10 × b
or
7b + 3b = 10b
7b, 3b and 10b are like terms.
They all contain a number and the letter b.
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Collecting together like terms
Remember, in algebra letters stand for numbers, so we
can use the same rules as we use for arithmetic.
In arithmetic,
2 + (6 × 2) – (3 × 2) = 4 × 2
In algebra,
x + 6x – 3x = 4x
x, 6x, 3x and 4x are like terms.
They all contain a number and the letter x.
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Collecting together like terms
When we add or subtract like terms in an expression we
say we are simplifying an expression by collecting
together like terms.
An expression can contain different like terms.
For example,
3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b
= 7a + 8b
This expression cannot be simplified any further.
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Collecting together like terms
Simplify these expressions by collecting together like terms.
1) a + a + a + a + a = 5a
2) 5b – 4b = b
3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6
= 2c + 2d + 9
4) 4n + n2 – 3n = 4n – 3n + n2 = n + n2
5) 4r + 6s – t
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Cannot be simplified
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Algebraic perimeters
Remember, to find the perimeter of a shape we add
together the length of each of its sides.
Write an algebraic expression for the perimeter of the
following shapes:
2a
Perimeter = 2a + 3b + 2a + 3b
3b
= 4a + 6b
5x
4y
x
5x
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Perimeter = 4y + 5x + x + 5x
= 4y + 11x
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Algebraic pyramids
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Algebraic magic square
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Contents
A1 Algebraic expressions
A1.1 Writing expressions
A1.2 Collecting like terms
A1.3 Multiplying terms
A1.4 Dividing terms
A1.5 Factorising expressions
A1.6 Substitution
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Multiplying terms together
In algebra we usually leave out the multiplication sign ×.
Any numbers must be written at the front and all letters should
be written in alphabetical order.
For example,
4 × a = 4a
1×b=b
We don’t need to write a 1 in front of the letter.
b × 5 = 5b
We don’t write b5.
3 × d × c = 3cd
We write letters in alphabetical order.
6 × e × e = 6e2
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Using index notation
Simplify:
x + x + x + x + x = 5x
Simplify:
x × x × x × x × x = x5
x to the power of 5
This is called index notation.
Similarly,
x × x = x2
x × x × x = x3
x × x × x × x = x4
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Using index notation
We can use index notation to simplify expressions.
For example,
3p × 2p = 3 × p × 2 × p = 6p2
q2 × q3 = q × q × q × q × q = q5
3r × r2 = 3 × r × r × r = 3r3
2t × 2t = (2t)2
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or
4t2
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Grid method for multiplying numbers
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Brackets
Look at this algebraic expression:
4(a + b)
What do do think it means?
Remember, in algebra we do not write the multiplication sign, ×.
This expression actually means:
4 × (a + b)
or
(a + b) + (a + b) + (a + b) + (a + b)
=a+b+a+b+a+b+a+b
= 4a + 4b
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Using the grid method to expand brackets
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Expanding brackets then simplifying
Sometimes we need to multiply out brackets and then simplify.
For example,
3x + 2(5 – x)
We need to multiply the bracket by 2 and collect together
like terms.
3x + 10 – 2x
= 3x – 2x + 10
= x + 10
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Expanding brackets then simplifying
Simplify
4 – (5n – 3)
We need to multiply the bracket by –1 and collect together
like terms.
4 – 5n + 3
= 4 + 3 – 5n
= 7 – 5n
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Expanding brackets then simplifying
Simplify
2(3n – 4) + 3(3n + 5)
We need to multiply out both brackets and collect together
like terms.
6n – 8 + 9n + 15
= 6n + 9n – 8 + 15
= 15n + 7
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Expanding brackets then simplifying
Simplify
5(3a + 2b) – 2(2a + 5b)
We need to multiply out both brackets and collect together
like terms.
15a + 10b – 4a –10b
= 15a – 4a + 10b – 10b
= 11a
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Algebraic multiplication square
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Pelmanism: Equivalent expressions
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Algebraic areas
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Contents
A1 Algebraic expressions
A1.1 Writing expressions
A1.2 Collecting like terms
A1.3 Multiplying terms
A1.4 Dividing terms
A1.5 Factorising expressions
A1.6 Substitution
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Dividing terms
Remember, in algebra we do not usually use the division
sign, ÷.
Instead we write the number or term we are dividing by
underneath like a fraction.
For example,
(a + b) ÷ c
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is written as
a+b
c
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Dividing terms
Like a fraction, we can often simplify expressions by
cancelling.
For example,
3
n
n3 ÷ n2 = 2
n
2
6p
6p2 ÷ 3p =
3p
1
1
1
n×n×n
=
n×n
6×p×p
=
3×p
=n
= 2p
1
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2
1
1
1
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Algebraic areas
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Hexagon Puzzle
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Contents
A1 Algebraic expressions
A1.1 Writing expressions
A1.2 Collecting like terms
A1.3 Multiplying terms
A1.4 Dividing terms
A1.5 Factorizing expressions
A1.6 Substitution
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Factorizing expressions
Some expressions can be simplified by dividing each term by
a common factor and writing the expression using brackets.
For example, in the expression
5x + 10
the terms 5x and 10 have a common factor, 5.
We can write the 5 outside of a set of brackets and mentally
divide 5x + 10 by 5.
(5x + 10) ÷ 5 = x + 2
This is written inside the bracket.
5(x + 2)
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Factorizing expressions
Writing 5x + 10 as 5(x + 2) is called factorizing the
expression.
Factorize 6a + 8
Factorize 12 – 9n
The highest common
factor of 6a and 8 is 2.
The highest common
factor of 12 and 9n is 3.
(6a + 8) ÷ 2 = 3a + 4
(12 – 9n) ÷ 3 = 4 – 3n
6a + 8 = 2(3a + 4)
12 – 9n = 3(4 – 3n)
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Factorizing expressions
Writing 5x + 10 as 5(x + 2) is called factorizing the
expression.
Factorize 3x + x2
The highest common
factor of 3x and x2 is x.
(3x +
x 2)
÷x=3+x
Factorize 2p + 6p2 – 4p3
The highest common factor
of 2p, 6p2 and 4p3 is 2p.
(2p + 6p2 – 4p3) ÷ 2p
= 1 + 3p – 2p2
3x + x2 = x(3 + x)
2p + 6p2 – 4p3
= 2p(1 + 3p – 2p2)
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Algebraic multiplication square
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Pelmanism: Equivalent expressions
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Contents
A1 Algebraic expressions
A1.1 Writing expressions
A1.2 Collecting like terms
A1.3 Multiplying terms
A1.4 Dividing terms
A1.5 Factorising expressions
A1.6 Substitution
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Work it out!
4 + 3 × 0.6
43
–7
8
5
===–17
133
5.8
28
19
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Work it out!
7 × 0.4
22
–3
6
9
2
====–10.5
31.5
1.4
21
77
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Work it out!
0.2
12
–4
3
9
2
+6
===6.04
150
22
15
87
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Work it out!
2( –13
3.6
18
69
7 + 8)
===23.2
–10
154
30
52
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Substitution
What does
substitution
mean?
In algebra, when we replace letters in an expression or
equation with numbers we call it substitution.
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Substitution
How can 4 + 3 ×
be written as an algebraic expression?
Using n for the variable we can write this as 4 + 3n
We can evaluate the expression 4 + 3n by substituting
different values for n.
When n = 5
4 + 3n = 4 + 3 × 5
= 4 + 15
= 19
When n = 11
4 + 3n = 4 + 3 × 11
= 4 + 33
= 37
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Substitution
7×
can be written as
7n
2
2
7n
We can evaluate the expression
by substituting different
2
values for n.
When n = 4
7n
2
= 7×4÷2
= 28 ÷ 2
= 14
When n = 1.1
7n
2
= 7 × 1.1 ÷ 2
= 7.7 ÷ 2
= 3.85
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Substitution
2
+6
can be written as
n2 + 6
We can evaluate the expression n2 + 6 by substituting
different values for n.
When n = 4
n2 + 6 = 42 + 6
= 16 + 6
= 22
When n = 0.6
n2 + 7 = 0.62 + 6
= 0.36 + 6
= 6.36
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Substitution
2(
+ 8)
can be written as
2(n + 8)
We can evaluate the expression 2(n + 8) by substituting
different values for n.
When n = 6
2(n + 8) = 2 × (6 + 8)
= 2 × 14
= 28
When n = 13
2(n + 8) = 2 × (13 + 8)
= 2 × 21
= 41
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Substitution exercise
Here are five expressions.
1) a + b + c = 5 + 2 + –1 = 6
2) 3a + 2c = 3 × 5 + 2 × –1 = 15 + –2 = 13
3) a(b + c) = 5 × (2 + –1) = 5 × 1 = 5
4) abc = 5 × 2 × –1= 10 × –1 = –10
22 – –1
b2 – c
5)
=
=5÷5=1
a
5
Evaluate these expressions when a = 5, b = 2 and c = –1
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Noughts and crosses - substitution
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