# A Level Mathematics - How accurate is a mathematical answer?

*Hints and tips - five minute read*

**Steven Walker, OCR Maths Subject Advisor**

Students often think that maths questions have a single correct answer. In the real world however, calculations are based on formulae generated from mathematical models, with assumptions made about which variables are significant and using measurements that are quoted only to a certain degree of accuracy.

In this blog I will give an overview of rounded and exact answers. I will also take a quick look at the different approach taken in maths and science for the approximate value for the acceleration due to gravity on Earth.

### Common mistakes

The most common mistake when rounding is to confuse decimal places with significant figures.

- Decimal places (dp) - count right from the decimal point.
- Significant figures (sf) - count from the first non-zero digit regardless.

Therefore 54.837 is 54.84 (2 dp), but 55 (2 sf). Likewise, 0.0548 is 0.05 (2 dp), but 0.055 (2 sf).

Another issue is with either over- or under-specifying a numerical result.

- Over-specification (e.g. by quoting most, or all, of a calculator display) suggests a level of accuracy that is probably not realistic.
- Under-specification may make it difficult to draw conclusions from the calculation.

Inaccuracies can also be caused through rounding iteratively, i.e. rather than directly rounding 0.0548 to 0.05 (1 sf), students have been seen to round 0.0548 to 0.055 (2 sf) and then round that to 0.06 (1 sf). This is why it is standard practice to over-specify intermediate results during a calculation and then only at the end round the answer to the required accuracy.

For example, when calculating A = π (5.14)^{2}, 5.14^{2} should not be rounded to 26.4. Instead the more accurate 26.4196 should be used (or at least 26.42 (4 sf)) to result in a more accurate A = 83.0 (3 sf).

### Exact answers

Mathematicians will often give an answer as a fraction, in surd form, as a multiple of π or e, in logarithm form, or just as an algebraic expression. In the two OCR suites of A Level Maths and Further Maths (Maths A/Further Maths A H230/H235/H240/H245 and Maths B (MEI)/Further Maths B (MEI) H630/H635/H640/H645), questions will often explicitly request answers to be given in ‘exact’ form. This indicates that a calculator generated decimal approximation will not be enough to gain full credit.

For more details of the question command words, see the specification for your qualification or this handy student poster.

### Numerical values in context

Whilst mathematicians may be happy to quote an exact result, scientists, accountants, engineers, etc. will need to quote a numerical value to an appropriate level of accuracy. Students often feel that all answers must be given to 3 significant figures, but in fact the context of the problem will define the level of accuracy needed.

It is common to quote calculated numerical values to the number of digits that are known, plus an extra one that is uncertain. For example, angles can be measured on a protractor to the nearest degree so quoting to one decimal place follows that rule.

Metric measurements are best given in significant figures so that there is no loss of accuracy if the units are changed. For example, 3.26 m is equal to 3.26 × 10^{2} cm and to 3.26 × 10^{-3} km.

Money will generally be given to 2 dp when dealing with everyday, going to the shops, pounds and pence amounts. 2 sf are often used however when the amounts get large, for example talking about transfer deals in the football premier league.

### Acceleration due to gravity

Students often ask why maths uses 9.8 ms^{-2} for the acceleration due to gravity on Earth, whereas science uses 9.81 ms^{-2}.

It is worth remembering that whilst g is a constant at a specific place, it is not constant everywhere on Earth (for more information, see this interactive map on the Physics Classroom website). Science is using a theoretical value calculated from the universal gravitation law, whereas maths is using a convenient approximate average (9.8 = 7^{2}/5) that is taken to be an ‘exact’ value for the purpose of the mathematical model used.

### Support across the curriculum

GCSE (9–1) Mathematics includes the topic of exact calculations and the topic of approximations and estimations. However, A Level students sometimes have difficulties with these topics (even those studying Maths).

GCSE (9-1) Mathematics |
Initial learning - 3.03 Exact calculations Initial learning - 4.01 Approximation and estimation Foundation tier - 3.03 Exact calculations Foundation tier - 4.01 Approximation and estimation Higher tier - 3.03 Exact calculations Higher tier - 4.01 Approximation and estimation |

Mathematics A - H230/H240 Mathematics B (MEI) - H630/H640 |
Student guide to bridging the gap between GCSE and AS/A Level |

Physics A - H156/H556 Physics B (Advancing Physics) - H157/H557 |
Physics mathematical skills handbook |

Biology A - H020/H420 Biology B (Advanced Biology) - H022/H422 |
Biology mathematical skills handbook |

Chemistry A - H032/H432 Chemistry B (Salters) - H033/H433 |
Chemistry mathematical skills handbook |

## Stay connected

What mathematical concept do you find is taught in different ways across the curriculum?

Let us know via the comments below, or you can email us at maths@ocr.org.uk, call us on 01223 553998 or Tweet us @OCR_Maths. You can also sign up for email updates to receive information about resources and support.

### About the author

#### Steven Walker, OCR Maths Subject Advisor

Steven joined OCR in 2014 during the major qualification reform period and now primarily focuses on supporting the Level 3 maths qualifications. Steven originally studied engineering before completing a PGCE in secondary mathematics. He began his teaching career with VSO in Malawi and has taught maths in both the UK and overseas.