Understanding Simple Algebra

Eleven-Plus Preparation Specialists

As part of the 11+, you’ll need to understand simple algebra. Depending on the school that you’re applying to, there may be a need for more complex work too – however, this is to be expected of more selective schools. This article focuses on the simple algebra that you’ll need for any 11+ examination.

Understanding Simple Formulae

You should be aware of, and able to use, simple formulae to solve common problems. An example that you should be aware of is:
1/2bh – this is ½ * base * height, which is used to find the area of a triangle.

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Using algebra to represent sequences

A linear sequence is a group of numbers, which are ordered according to a particular rule. A linear sequence is one that increases by the same amount each time, like 3,6,9,12. A non-linear sequence might increase or decrease by a different amount each time, like 1,2,3,5,8,13 (where we are adding the previous two numbers). Whilst you can expect both types of sequence in the 11+, you’d be expected to represent only linear numbers using algebra. To represent numbers in sequences, we use the term n, where ‘n’ represents the term. As an example, the rule ‘2n+2’ would give the following:
4,6,8,10, etc

Looking at this from another angle, the numbers
1,3,5,7,9 would be represented by the rule 2n-1

A simple way of working out these problems whilst learning how to do them is to use a table, in which you write the value of n in one column and the value of the part of the sequence that it represents in the other column.

You may need to find specific numbers in a sequence. As an example, you might be given
1,3,5,7,9, and then asked to find the 12th term in the sequence. You would therefore use 2n-1, and multiply 12 by 2, before subtracting 1; you therefore find that the 12th term is 23.

Finding solutions to equations with two unknowns

Imagine that you are told that x+y = 10. With no more information, what could x and y be?
Possible combinations of values are therefore 1,9 ; 2,8; 7,3; 6,4 ; 4,6 ; 3,7 ; 2,8 ; 1,9

Remember the numbers must be different as they are represented by different letters.

You must also be able to do slightly more complex iterations of this that involve e.g. 2x, or 2y.

Using letters to represent unknowns

If you are told that a number of dogs, all of whom have four legs, have a total of 32 legs, and are asked how many dogs there must be, you should be able to understand that this can be written

4x = 32
Therefore x = 8

As we have used x to represent the unknown – in this case, the number of dogs.

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Ways to practise

You should familiarise yourself with these questions through generating as many sequences of numbers as you can and practising representing them using algebra. Try to describe the sequence first, describe what the next number would be, then use algebra to represent the sequence. Remember, if you are struggling then identifying the sequence is the first part – and this will always get you some marks. Think about the relationship between different numbers in the sequence, and see how they vary. As an example, is there a clear doubling, or tripling, or a clear simple addition. The goal should be to recognise any number in the sequence using algebra. Thinking about more general algebra questions, try to represent everyday problems using algebra. How many days you need until you can buy something, the price of something – all these can be represented using algebra and doing so will help you become much more used to using it.

Complex Algebra

As mentioned above, some schools will expect a much higher level of algebra – even down to using simultaneous equations, for example. You should therefore ensure that you practise at this level if applying to a particularly selective school or looking to achieve a scholarship.
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