Key Stage 4 maths · Number

GCSE maths standard form, indices and limits of accuracy

In Years 10 and 11 the number strand builds on everything pupils learned in Key Stage 3. They list possibilities systematically, calculate with roots and integer indices, leave answers exact using fractions and π, work with very large and very small numbers in standard form, solve ratio problems that involve fractions, and describe the range of values a rounded or truncated number could have. This is content for all pupils, whichever GCSE tier they take.

Standard form, indices and accuracy, explained

Systematic listing means writing out all the possibilities in an organised order so that none is missed or repeated. If a pupil wants every two-digit number that can be made from the digits 3, 5 and 7 without repeating a digit, they fix the tens digit and vary the ones: 35, 37, 53, 57, 73, 75. That gives six. The same habit is used later for probability sample spaces and for counting arrangements, and it is the best defence against the common error of stopping when the list “feels” finished.

Pupils calculate with roots (square and cube roots) and with positive, zero and negative whole-number indices. The index laws are a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n) and (a^m)^n = a^(mn), with a^0 = 1 and a^(−n) = 1 ÷ a^n, so 2^(−3) = ⅛. They also calculate exactly with fractions and multiples of π. A circle with radius 6 cm has area 36π cm² and circumference 12π cm, and these are the exact answers; 113.1 cm² and 37.7 cm are only approximations. Likewise ⅔ + ⅕ is exactly 13/15, not 0.867.

Standard form is a way to write very large or very small numbers as A × 10^n, where A is at least 1 but less than 10 and n is an integer (a positive or negative whole number, or zero). So 4,500,000 = 4.5 × 10^6, and 0.00032 = 3.2 × 10^(−4). A positive power shows a large number, a negative power a small one. Pupils multiply and divide numbers in standard form by handling the A parts and the powers of ten separately, add and subtract by writing both numbers with the same power of ten, and also learn to enter and read standard form on a scientific calculator.

Ratio problems now often involve fractions. A ratio of 3 : 5 means the parts are ⅜ and ⅝ of the total, so sharing £240 in that ratio gives £90 and £150. If boys make up ³⁄₇ of a class, the ratio of boys to girls is 3 : 4. Pupils switch between ratios and fractions, and use the unitary approach of finding the value of one part and then scaling it up.

Limits of accuracy describe the range of values a rounded or truncated number could have come from. A length of 7.4 cm rounded to 1 decimal place could be anything from 7.35 cm up to, but not including, 7.45 cm, written 7.35 ≤ l < 7.45. If it was truncated instead, the range is 7.4 ≤ l < 7.5. Pupils need to interpret these ranges, for example deciding whether a measurement is reliable enough to support a conclusion.

Words your child will hear

Systematic list: An organised list in a clear order so that every possibility appears once.
Index (plural indices): The small raised number showing how many lots of a number are multiplied together, so 5^3 = 5 × 5 × 5.
Standard form: A number written as A × 10^n, with 1 ≤ A < 10 and n an integer.
Exact answer: An answer left in terms of π, a fraction or a root, with no rounding.
Truncate: Cut a number off after a given number of digits without rounding. 3.789 truncated to 1 d.p. is 3.7.
Error interval: The range of values a rounded or truncated number could have started as.
Reciprocal: What you multiply a number by to get 1. The reciprocal of 4 is ¼, and a negative index gives a reciprocal.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Apply systematic listing strategies
Calculate with roots, and with integer indices
Calculate exactly with fractions and multiples of π
Calculate with numbers in standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer
Identify and work with fractions in ratio problems
Apply and interpret limits of accuracy when rounding or truncating

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

A worked example

Work out (3.2 × 10^5) × (4 × 10^(−2)). Give your answer in standard form.

Multiply the decimal parts together.
```
3.2 × 4 = 12.8
```
Multiply the powers of ten by adding the indices.
```
10^5 × 10^(−2) = 10^(5 + (−2)) = 10^3
```
Put the two results together: 12.8 × 10^3. This is not yet in standard form because 12.8 is bigger than 10.
Write 12.8 as 1.28 × 10, then combine the powers of ten.
```
12.8 × 10^3
= 1.28 × 10 × 10^3
= 1.28 × 10^4
```
Check with ordinary numbers: 3.2 × 10^5 = 320,000 and 4 × 10^(−2) = 0.04. Then 320,000 × 0.04 = 12,800, which is 1.28 × 10^4.

(3.2 × 10^5) × (4 × 10^(−2)) = 1.28 × 10^4

Common mistakes and how to help

Leaving the first number outside the range 1 to 10

After multiplying, pupils often get an answer such as 12.8 × 10^3 or 0.5 × 10^4 and leave it, because the calculation itself felt finished.

What helps: Remind your child that standard form has a rule about A: at least 1 and less than 10. Ask “is the first number between 1 and 10?” as the last step, and adjust the power of ten to compensate.

Thinking a negative index makes a negative number

The minus sign in 10^(−2) or 2^(−3) looks like it affects the sign of the answer, giving −100 or −8.

What helps: A negative index means a reciprocal: 2^(−3) = 1 ÷ 2³ = ⅛, and 10^(−2) = 0.01. So 3 × 10^(−2) is 0.03, a small positive number.

Adding the powers of ten when adding numbers in standard form

The rule “add the indices” works when multiplying, so it is applied to every operation. 3 × 10^4 + 2 × 10^3 then gets written as 5 × 10^7.

What helps: To add or subtract, write both numbers with the same power of ten first: 3 × 10^4 + 0.2 × 10^4 = 3.2 × 10^4. Checking against 30,000 + 2,000 = 32,000 shows why.

Getting the ends of an error interval wrong

Pupils halve the rounding unit correctly but write the upper limit with ≤, or add and subtract the whole unit instead of half.

What helps: For a length of 7.4 cm to 1 d.p., the unit is 0.1, half is 0.05, and the limits are 7.35 and 7.45. The upper limit uses < because 7.45 itself would round up to 7.5.

How to help at home

Real numbers in standard form. The Moon is about 3.84 × 10^5 km away and the Sun about 1.5 × 10^8 km. Ask your child to write each as an ordinary number and say which is further and by roughly how many times. The speed of light, about 3 × 10^8 metres per second, is another good one.
Practise on the calculator. Ask your child to show you how they enter standard form on their scientific calculator and how to read the display. Then check a calculation such as 3 × 10^8 × 2 × 10^3 = 6 × 10^11 both on the calculator and by hand, since misreading the display is an easy way to lose marks.
Make a list the organised way. How many different outfits from 3 tops and 4 pairs of trousers? How many different ways can three friends sit in a row? Ask for the list written in an order, then talk about how they know none is missing.
Limits in the real world. A phone is advertised as 150 mm long to the nearest millimetre. What is the shortest and longest it could be? (149.5 mm up to, but not including, 150.5 mm.) Do the same with a baking weight or a journey time.

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