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A1 Algebraic expressions
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A1.1 Writing expressions
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The birth of algebra
Al-Khwarizmi (780-850 C.E) was a
mathematician who lived in Baghdad.
He is known as the ‘father of algebra’
because he wrote the first book on solving
equations, ‘Hisab al-jabr w'al-muqabala’.
What does the word al-jabr remind you of?
“al-jabr" means "completion" and is the
process of removing negative terms
from an equation.
His book was meant to teach people maths for practical use,
and it was written entirely in words, without any symbols.
These were developed later, by other mathematicians.
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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 8 + 9 = 17
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Using symbols for unknowns
Look at this problem:
–
The symbols
In this problem,
Two examples are,
and
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and
=5
stand for unknown numbers.
and
can have many values.
12 – 7 = 5
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 ×.
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.
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.
n ÷ 3 is written as
n
3
When we multiply a letter symbol
by itself, we use index notation.
n squared
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
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She doubles the number
of cubes she is holding.
2 × n or 2n
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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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A1.2 Collecting like terms
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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.
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.
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 lengths of each of its sides.
Write algebraic expressions for the
perimeters 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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