Key Stage 4 maths · Number

GCSE higher tier maths: surds, recurring decimals and bounds

The National Curriculum marks some number content as extra material for more highly attaining pupils, broadly the GCSE higher tier. Pupils count possibilities with the product rule, estimate and calculate with roots and fractional powers, work exactly with surds, convert recurring decimals to fractions and back, and calculate with upper and lower bounds. These topics reward careful, exact working.

Surds, recurring decimals and bounds, explained

The product rule for counting says that if one choice can be made in a ways and a second choice in b ways, then there are a × b ways to make both. Three tops, four pairs of trousers and two pairs of shoes give 3 × 4 × 2 = 24 outfits. A four-digit PIN with no repeated digit can be chosen in 10 × 9 × 8 × 7 = 5,040 ways, because each choice removes one digit from the next. This is quicker than writing a list, and pupils still use listing to check small cases.

Pupils estimate powers and roots of any positive number by bracketing it between numbers they know. Since 7² = 49 and 8² = 64, the square root of 50 is just over 7. Since 3³ = 27 and 4³ = 64, the cube root of 30 is a little over 3. They also calculate with fractional indices, where the denominator names the root and the numerator the power, so 27^(2/3) = (∛27)² = 9 and 16^(−3/4) = 1 ÷ (⁴√16)³ = 1 ÷ 8 = ⅛.

A surd is a root that cannot be written as a whole number or a fraction, such as √2, so it is kept in root form to stay exact. Pupils simplify them by pulling out square factors (√12 = √4 × √3 = 2√3 and √50 = 5√2), expand brackets such as (3 + √2)(3 − √2) = 9 − 2 = 7, and rationalise denominators so that no root remains underneath. 6 ÷ √3 becomes 6√3 ÷ 3 = 2√3 by multiplying top and bottom by √3. When the denominator is like 2 + √3, they multiply top and bottom by 2 − √3, so the denominator becomes 4 − 3 = 1.

Recurring decimals are decimals with a repeating pattern, such as ⅓ = 0.333… or 4/11 = 0.363636…. They are rational numbers, which means they can always be written as fractions, unlike surds. To convert one, pupils multiply to line up the repeat, subtract to cancel the endless part, and solve. For 0.777…, 10x − x = 7.777… − 0.777…, so 9x = 7 and x = 7/9. Going the other way is just division. A fraction in its simplest form terminates (stops) only if its denominator has no prime factors except 2 and 5.

Upper and lower bounds extend limits of accuracy to calculations. If each side of a rectangle is given as 8 cm and 5 cm to the nearest centimetre, the true lengths lie between 7.5 and 8.5, and 4.5 and 5.5. The largest the area could be is 8.5 × 5.5 = 46.75 cm², and the smallest is 7.5 × 4.5 = 33.75 cm². By convention the upper bound is written as 8.5, even though a value of exactly 8.5 would have rounded up. For subtraction and division, pupils need to mix bounds: the largest value of a − b uses the upper bound of a and the lower bound of b.

Words your child will hear

Product rule for counting: If there are a ways to make one choice and b ways to make another, there are a × b ways to make both.
Surd: A root that cannot be simplified to a whole number or fraction, such as √7. It is irrational.
Rationalise: Rewrite a fraction so that its denominator contains no surd, for example 1 ÷ √2 = √2 ÷ 2.
Recurring decimal: A decimal with an endlessly repeating digit or group of digits, such as 0.2727…
Upper bound: The top of the range a rounded number could have come from. For 8 cm to the nearest cm it is 8.5 cm.
Lower bound: The bottom of the range a rounded number could have come from. For 8 cm to the nearest cm it is 7.5 cm.
Irrational number: A number that cannot be written as a fraction of two whole numbers, such as π or √2.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Apply systematic listing strategies, including use of the product rule for counting
Estimate powers and roots of any given positive number
Calculate with fractional indices
Calculate exactly with surds; simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators
Change recurring decimals into their corresponding fractions and vice versa
Apply and interpret limits of accuracy, including upper and lower bounds

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

A worked example

The recurring decimal 0.272727… has the digits 2 and 7 repeating. Write it as a fraction in its simplest form.

Let x stand for the decimal.
```
x = 0.272727…
```
Two digits repeat, so multiply by 100 to move the decimal point two places and line the repeats up.
```
100x = 27.272727…
```
Subtract the first equation from the second. The endless decimal parts cancel exactly.
```
100x = 27.272727…
  x =  0.272727…
99x = 27
```
Divide both sides by 99 to get x.
```
x = 27/99
```
Simplify by dividing top and bottom by 9. Check by dividing 3 by 11 on a calculator: the display shows 0.272727….
```
27/99 = 3/11
```

0.272727… = 3/11

Common mistakes and how to help

Multiplying by 10 when two digits repeat

Pupils learn the method with a single repeating digit, such as 0.777…, where multiplying by 10 works, and apply it unchanged to patterns of two or more digits.

What helps: Match the multiplier to the length of the repeat: 10 for one digit, 100 for two, 1,000 for three. Check that the digits after the point line up exactly before subtracting.

Squaring a surd incorrectly, such as (2√3)² = 2 × 3 = 6

Pupils know that squaring a root gives the number inside it and forget that the 2 outside must be squared as well.

What helps: Write it out in full: (2√3)² = 2√3 × 2√3 = 2 × 2 × √3 × √3 = 4 × 3 = 12. Then check against a calculator value for 2√3 (about 3.46, which squares to about 12).

Using lower bounds for every number to find a lower bound

It seems logical that the smallest answer comes from the smallest inputs, and this does work for adding and multiplying positive numbers.

What helps: For a − b and a ÷ b the smallest answer needs a small a and a large b. Ask “what makes the answer as small as possible?” each time instead of following a fixed pattern.

Believing 0.333… is only close to one third, not equal to it

The decimal never seems to finish, so it feels like an approximation. The same doubt arises with 0.999… and 1.

What helps: Show the algebra: if x = 0.333…, then 10x − x = 3, so x = ⅓ exactly. Adding ⅓ + ⅓ + ⅓ = 1 on one side and 0.333… × 3 = 0.999… on the other shows why 0.999… = 1.

How to help at home

Count codes and combinations. How many three-letter codes use different letters from the alphabet? (26 × 25 × 24 = 15,600.) How many four-digit PINs allow repeats? (10,000.) Ask your child to explain why the answers differ.
Bracket a root without the calculator. Pick a number such as 70 and ask for its square root between two whole numbers (8 and 9). Then ask for a better guess, such as 8.4, and check it by squaring.
Explore recurring decimals. Ask your child to divide 1 by 7, 2 by 7 and 3 by 7 on a calculator and look for a pattern in the repeating digits. Ask them to predict 4 divided by 7 before checking, then try the same with ninths and elevenths.
Weigh up the errors. If three ingredients are each weighed to the nearest 10 g, the total could be out by up to 15 g in either direction. Talking this through with a kitchen scale makes bounds feel real.

Builds on

Key Stage 3: Place value, ordering and number properties
Key Stage 3: Powers, roots and standard form
Key Stage 3: Fractions, decimals and percentages
Key Stage 3: Rounding, estimation and error