Key Stage 4 maths · Working mathematically

GCSE higher tier maths: algebraic proof, surds and fractional indices

The National Curriculum marks some GCSE content as additional material for more highly attaining pupils, broadly the higher tier. For working mathematically this means extending number to fractional indices, calculating exactly with surds, simplifying algebraic fractions, moving between graphs and equations of exponential and trigonometric functions, and writing algebraic proofs. This page explains what each involves and how to support a teenager meeting it.

Surds, indices and proof (higher), explained

Fractional indices extend the rules for powers that pupils already know. A power of one half means a square root, so 25^(1/2) = 5, and a power of one third means a cube root, so 8^(1/3) = 2. More generally a^(m/n) means take the nth root, then raise to the power m. Hence 8^(2/3) = (∛8)² = 2² = 4. A negative index gives a reciprocal: 16^(−1/2) = 1 ÷ √16 = ¼. Doing the root first keeps the numbers small, which is why teachers usually recommend that order.

A surd is a root that cannot be written as a whole number or an exact fraction, such as √2 or √3. Leaving it as √2 keeps an answer exact, whereas 1.414… is only an approximation. Pupils learn to simplify (√12 = 2√3 because 12 = 4 × 3), to expand brackets such as (2 + √3)(2 − √3) = 4 − 3 = 1, and to rationalise denominators so that 6 ÷ √3 becomes 2√3. Surds also appear inside algebra, for example in solving equations with exact answers. Questions signal this with phrases such as “give your answer in surd form” or “find the exact value”.

Algebraic fractions are fractions with algebra in the numerator, the denominator or both. The central idea is that pupils can only cancel factors, not terms, so they must factorise first. For example, (x² − 9) ÷ (x² + x − 6) factorises to (x − 3)(x + 3) over (x + 3)(x − 2), and cancelling the common bracket (x + 3) leaves (x − 3) ÷ (x − 2). Adding and subtracting algebraic fractions uses a common denominator, just as with number fractions, and equations involving them are solved by multiplying through.

Pupils at this level also move freely between ways of representing exponential and trigonometric functions: a table of values, a graph, an equation and a real context. An exponential function such as y = 2^x passes through (0, 1), always stays above the x-axis and grows faster and faster. It models compound growth and, with a base between 0 and 1, decay. The graphs of y = sin x and y = cos x repeat every 360° and stay between −1 and 1, while y = tan x repeats every 180° and, between 0° and 360°, has breaks at 90° and 270°, where it is undefined. Pupils learn exact values such as sin 30° = ½ and tan 45° = 1, and use them to read information from a graph or identify the equation of a given curve.

Finally, algebra is used to prove. Pupils represent whole numbers in general form, such as 2n for an even number and 2n + 1 for an odd one, then expand, simplify and factorise to show a result must be true for every case. A good proof states what the letters mean, shows each step in a logical order, and finishes by saying why the final expression proves the claim, for instance “this is 8 times a whole number, so it is a multiple of 8”.

Words your child will hear

Fractional index: A power that is a fraction. The index ½ means square root and ⅓ means cube root.
Surd: An irrational root left in exact form, such as √5 or 3√2.
Rationalise: Rewrite a fraction so that the denominator has no surd, for example 1 ÷ √3 = √3 ÷ 3.
Algebraic fraction: A fraction whose numerator or denominator contains letters, such as (x + 1) ÷ (x − 2).
Exponential function: A function where the variable is the power, such as y = 3^x, giving rapid growth or decay.
Exact value: An answer given using symbols such as √2 or π, or as a fraction, with no rounding.
Identity: A statement true for every value of the letters, written with the symbol ≡.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Extend their understanding of the number system to include fractional indices
Select and use appropriate calculation strategies, including exact calculations involving surds
Extend their understanding of algebraic simplification and manipulation to include expressions involving surds and algebraic fractions
Move freely between different numerical, algebraic, graphical and diagrammatic representations of exponential and trigonometric functions
Use algebra to support and construct proofs

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Prove that the sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8.

Test a couple of cases first to see what to aim for.

3² + 5² = 9 + 25 = 34 = 32 + 2
7² + 9² = 49 + 81 = 130 = 128 + 2

Write the numbers in general form. Let the first odd number be 2n + 1, where n is a whole number. The next odd number is 2n + 3.

Expand each square.

(2n + 1)² = 4n² + 4n + 1
(2n + 3)² = 4n² + 12n + 9

Add the two results and collect like terms.
```
8n² + 16n + 10
```
Show the multiple of 8 by splitting 10 into 8 + 2, then factorise the first three terms.
```
8n² + 16n + 8 + 2
= 8(n² + 2n + 1) + 2
= 8(n + 1)² + 2
```
Conclude: 8(n + 1)² is 8 times a whole number, so it is a multiple of 8. Adding 2 gives a number that is 2 more than a multiple of 8, for every whole number n.

The sum of the squares is 8(n + 1)² + 2, which is always 2 more than a multiple of 8.

Common mistakes and how to help

Adding surds as if they were whole numbers, or splitting a root over a sum

The root sign looks as if it can be shared out over a sum, so √(9 + 16) gets written as √9 + √16 = 3 + 4 = 7. The correct value is √25 = 5.

What helps: Test with numbers where the answer is known. Only products split: √(a × b) = √a × √b. √2 + √3 does not equal √5, which a calculator quickly confirms.

Cancelling terms instead of factors in algebraic fractions

Cancelling is familiar from number fractions, so (x + 3) ÷ 3 is simplified to x + 1 by crossing out the 3s.

What helps: Substitute a number. With x = 3, (x + 3) ÷ 3 is 2 but x + 1 is 4, so the cancelling is wrong. Factorise the top and bottom so that whole brackets are common, and cancel only those.

Reading a fractional index as a division or a multiplication

Seeing 8^(1/3), pupils work out 8 ÷ 3 or 8 × ⅓ because the fraction is the only thing on view.

What helps: Remember that the bottom number of the index names the root and the top number is the power. Practise with roots pupils know: 27^(1/3) is 3, and 4^(3/2) = (√4)³ = 8.

Giving a decimal when an exact answer is required

Calculators show decimals by default, so a result like 3.46 for 2√3 seems perfectly good.

What helps: Check the wording. “Exact” or “in surd form” means leaving √ or π in the answer. A decimal approximation will usually not earn full marks, and rounded values also drift during multi-step working.

How to help at home

Test with numbers. Whenever your child simplifies an expression, suggest substituting x = 2 into the original and the simplified version. If the results differ, there is an error. It is a quick check that works at any level.
Explore powers on a calculator. Compare 9^0.5 with √9, and 8^(1/3) with the cube root key. Seeing the same numbers appear makes fractional indices feel less like a trick.
Spot exponential growth and decay. Look at examples such as savings with compound interest, a rumour that doubles in reach each day, or a car that loses the same percentage of its value every year. Ask your child to describe each with an equation and a rough sketch graph.
Ask them to teach you a proof. Ask your child to explain a proof they have done, and to say what each letter stands for. If they can tell you why each line follows, they understand it.

Builds on

Key Stage 3: Developing fluency
Key Stage 3: Reasoning mathematically
Key Stage 3: Solving problems