Language, notation and formulas

Introduction

An integral part of learning mathematics involves communication.

Writing mathematics is a specific skill which needs to be developed and practised: there is a lot of difference between putting down a few symbols for your own use and writing a mathematical solution intended for someone else to read. In attempting mathematical questions, you may previously have written down very little, just enough, perhaps, to convince yourself that you could answer the questions. This may suffice now, but you may want to use your notes and solutions for revision so you will want them to be self-contained, able to stand on their own and easy to read. This will also be the case if you are writing mathematics for somebody else to read.

This OpenLearn course provides a sample of level 1 study in Mathematics.

Learning outcomes

After studying this course, you should be able to:

• lay out and, where appropriate, label simple mathematical arguments

• understand the precise mathematical meaning of certain common English words

• understand and use common mathematical symbols

• write clear, unambiguous mathematical solutions using appropriate notation

• identify and modify some sources of ambiguity or inappropriate use of notation in a mathematical solution.

1 Communicating mathematics

1.1 Writing mathematics

How do you write mathematics that can be read and easily understood by anyone else, such as a tutor or another student?

It takes practice, but the following guidelines may be helpful.

• Ensure that what you write consists of sentences. Too many people believe that mathematics is a language which is entirely made up of unfamiliar symbols. It is not. Many symbols act in part merely as abbreviations which, when read, can be translated back into spoken words.

• Ensure that there are sufficient words of explanation so that the reader is led, line by line, through your proposed solution or argument.

1.2 Talking and writing mathematics

One way of testing whether or not you are conforming to the first guideline, is to read your solutions through aloud. Speaking aloud involves you in translating every symbol on the page into its verbal equivalent. If you find yourself needing to say more than is written on the page, you may need to expand your written account. To give you practice at this and at assessing the quality of some written mathematics, work through the animation below. The actual mathematics used is not important; just concentrate on the style of its presentation.

Working through the following animation should take you about forty minutes. The relevant frames are shown with the appropriate extract and you will need to answer using pen and paper. After each extract you will need to click on the play button to hear the next audio.

2 Reading mathematics

2.1 Layout

As mentioned in the animation in Section 1.2 writing mathematics has a lot in common with writing English. When you write mathematics, you should write in the equivalent of sentences, with full stops at the end. As in English, each new statement should follow on logically from the previous one or it should contain an indication that a new idea is being introduced. However, laying out mathematics differs from laying out English: because mathematics is more condensed than written English, it is often best to start each statement on a new line so that the various steps involved are readily apparent.

In writing mathematics you can use a mixture of words and symbols – symbols in mathematics are just shorthand for words. If you do not know the correct symbol, use the word itself.

Above all, make sure that you write so that somebody else can understand – that somebody might be yourself at a later date! It often helps to read aloud the mathematics that you have written to see if it makes sense.

2.1.1 Try some yourself

2.2 Using the equals sign

As mentioned previously, one of the most misused mathematical symbols is the equals sign, =. It stands for the verb ‘equals’ or the phrase ‘is equal to’ or ‘which equals’, and so it should only come between two things that are equal.

2.2.1 Try some yourself

Activity 2

Here is a poor example of mathematical writing, although the final answer is correct. Rewrite it, correcting the layout and the mathematical punctuation.

Answer

There are many possible alternatives – here is one of them.

Since 5² = 25, 4² = 16 and 3² = 9,

hence

2.3 Link words

A lot of people use the equals sign wrongly in places where another word or phrase might actually make the meaning clearer. Sometimes a link word or phrase is useful at the beginning of a mathematical sentence: examples include ‘So’, ‘This implies’ or ‘It follows that’ or ‘Hence’.

Example 3

Add link words and punctuation to the calculation below to help somebody else to follow it.

Answer

Notice that mathematical calculations are often easier to read if all the equals signs are aligned underneath one another.

2.3.1 Try some yourself

2.4 Labelling lines

Sometimes you may want to refer to mathematical sentences or phrases further up your work. You can label such sentences and then refer back by label. Thus, Example 3 could be laid out as follows.

So, from (1) and (2),

2.4.1 Try some yourself

Activity 4

Two labels have been omitted in the mathematics below. Where should they go to make sense of the argument?

• Since

  

  it follows that

  

  Now

  

  Hence

  

  So, from (1) and (2),

  

Answer

Since

it follows that

Now

Hence

So, from (1) and (2),

2.5 Vocabulary

In mathematics, some words are used in a more precise way than in English. It is important that a mathematical argument is unambiguous; therefore words that can be used in several contexts in English usually take only one meaning in mathematics. For instance, in English the word ‘sum’ might mean any calculation, but it has a precise mathematical meaning as exemplified by ‘The sum of 456 and 789 is 1245’. Similarly, in English the word ‘product’ can have a variety of meanings, but in mathematics the product of two numbers is the result obtained by multiplying those numbers together.

Below is a list of mathematical words and phrases, together with their mathematical meanings and examples of their use. You will have come across some of them before. Read them carefully and try to see the connections with the meanings that the words have in everyday English. You may well wish to copy the table into your notes and add to it in your future studies.

2.5.1 Try some yourself

Activity 5

How would you add the following words to the list:

 decimal, fraction, positive, negative.

For each one, give the mathematical meaning and an example of its use.

Answer

There are many different correct answers to this question. However, make sure that the meanings you have given are equivalent to the ones below.

Word	Mathematical meaning	Example of use
Decimal	A number expressed in terms of tenths, hundredths, etc.	A quarter expressed as a decimal is 0.25.
Fraction	One whole number over another.	0.75 expressed as a fraction is .
Positive	Greater than zero.	2 is a positive number.
Negative	Less than zero.	−2 is a negative number.

2.6 Making sense of symbols

Mathematical symbols are a shorthand way of writing words or phrases that crop up often in mathematical writing.

2.6.1 Arithmetical symbols

You will have already met the symbols for the basic arithmetical operations, which are +, −, × and ÷, but you may not have met some of the alternative ways of writing × and ÷.

To recap, the main symbols for arithmetical operations are:

There are other alternatives to × and * for multiplication. Sometimes, provided there is no risk of ambiguity, no symbol is used, just as sometimes the words ‘multiplied by’ or ‘times’ can be omitted in speech: for example, two fours or five fives means two times four or five times five, respectively. So, instead of writing 3 × (4 + 5), you can just write 3(4 + 5) and save one symbol. Mathematicians like to be as concise as possible! Brackets avoid ambiguity in mathematical expressions.

An alternative to ÷ and / for division is a fractional notation: for example, means ‘add three and six, and then divide by four’. The slash symbol / is sometimes used for fractions: thus, three-quarters may be written as 3/4. Many calculators use this notation to display fractions.

2.6.2 Try some yourself

2.7 Powers and roots

There are several symbols for powers and roots: for instance, 2⁴ means ‘2 to the power 4’. An alternative to 2⁴ is 24, where the symbol means ‘to the power’. This symbol is often used by computers and calculators.

A related symbol is the square root symbol, as in meaning the positive square root of 25 (that is, 5). Symbols for other roots, are exemplified by meaning the cube root of 8 (that is, 2).

Raising to the power ⁻1 gives the symbol for a reciprocal: thus, 6⁻¹ means the reciprocal of 6, or . The symbols used by calculators for scientific notation often involve the letter E: for example, 3.1E2 means 3.1 × 10², or 310.

2.7.1 Try some yourself

2.8 Relationship symbols

There are also symbols that show the relationship between numbers or quantities. Two common ones are = and , but there are several other symbols of this type.

Symbols indicating relationships include:

• = means ‘equals’;

• ≠ means ‘does not equal’;

• means ‘is approximately equal to’;

• > means ‘is greater than’;

• ≤ means ‘is less than or equal to’;

• ≥ means ‘is greater than or equal to’.

Other symbols that are sometimes used include:

• means ‘is approximately equal to’ (an older version);

• ≡ means ‘is exactly equivalent to’;

• means ‘implies’;

• means ‘therefore’;

• means ‘because’.

Another type of symbol that you come across are abbreviations for units of measurement: for example, m for metres, kg for kilograms, and h for hours, and in imperial units ft for feet

You will meet many symbols in your mathematical studies. It is important that you are sure what each one means in any given context, so that you can read it and use it appropriately.

2.8.1 Try some yourself

3 Formulas

3.1 Word formulas

A formula is a rule or a generalisation. Word formulas – formulas that use English words rather than mathematical symbols – are so much a part of life that people often use them without realising that they are doing so. Here are some examples.

• The cost of a purchase of oranges is the price per orange times the number of oranges.

• The total cost of petrol is the price of petrol per litre times the number of litres.

• The average speed is the distance travelled divided by the time taken.

• The number of tins of paint needed to cover a wall is the area of the wall divided by the area covered by one tin of paint.

These word formulas can be written more clearly using mathematical symbols. Notice how brackets can be used for clarity.

• Cost of oranges bought = (price per orange) × (number of oranges).

• Cost of petrol = (price of petrol per litre) × (number of litres).

• Average speed = (distance travelled)/(time taken).

• Number of tins of paint needed = (area of wall)/(area covered by one tin).

Sometimes letters are used as shorthand in formulas.

3.1.1 Try some yourself

3.2 Using formulas

Formulas are important because they describe general relationships, rather than specific numerical ones. For example, the tins of paint formula applies to every wall. To use such a formula you need to substitute specific values for the general terms, as the following examples show.

Example 8

If the price of petrol is 85 pence per litre and a driver buys 25.9 litres, what is the cost?

Answer

The formula is

The price per litre is 85 pence and the number of litres is 25.9. So

Rounded to the nearest penny, this is 2202 pence or £22.02.

Example 9

Suppose you are travelling on a UK motorway in a coach and you want to know how far the coach has travelled in the previous quarter of an hour. You glance over the shoulder of the driver and you see that the speed is a steady 70 miles per hour. Estimate how far you have travelled on the motorway in the last quarter of an hour, assuming a constant speed of 70 miles per hour.

Answer

The formula gives

3.2.1 Try some yourself

Activity 13

You are planning to paint three rooms with total wall areas of 56, 38 and 40 square metres, using paint that comes in tins which claim to cover 15 square metres per tin. How many tins will you need for each room? And how many in total?

Answer

For the first room, with a wall area of 56 square metres,

For the second room, with a wall area of 38 square metres,

For the third room, with a wall area of 40 square metres,

The total number of tins required will depend on whether you are planning to paint all the rooms in the same colour or not. If you are, the number of tins required will be

so you will need to buy 9 tins. If all the rooms are to be different colours, you will need to buy 4 tins for the first room, 3 for the second and 3 for the third – a total of 10 tins.

3.3 Units and formulas

As already noted, the units used in a formula must be consistent. For instance, in Example 9, the speed is in miles per hour, so the time travelled must be in hours and the distance travelled must be in miles. If one of the quantities is not in the correct units, then you need to change it before using the formula.

Example 10

Suppose you have been travelling in a car at a roughly constant speed of 50 miles per hour for 45 minutes. How far would you have travelled?

Answer

You need to change the time into hours in order for the units to be consistent. Thus, 45 minutes is hour. Then, from the formula,

3.3.1 Try some yourself

Activity 15

If you travel in a car at a constant speed of 60 km per hour, how far would you have travelled after 1.5 hours, after 2 hours 40 minutes and after three and a half hours?

Answer

The formula is

distance travelled = (speed) × (time).

So, after 1.5 hours,

distance travelled = (60 km per hour) × (1.5 hours) = 90 km.

As 2 hours 40 minutes is hours, or hours, the formula gives

For three and a half hours (3.5 hours), the formula gives

distance travelled = (60 km per hour) × (3.5 hours) = 210 km.

Activity 16

3 Suppose you are selling a new lawn-edging strip and need to calculate the perimeters of lawns in order to quote prices to prospective customers.

The formula for the distance round the perimeter of a rectangular lawn is

perimeter = 2 × length + 2 × breadth.

Use this formula to find the perimeter of a lawn that is 10 m by 8 m.

Answer

The length of the lawn is 10 m and its breadth is 8 m. So, from the formula,

The perimeter of the lawn is 36 m.

3.4 Converting units by using a formula

It is sometimes necessary to convert from one unit of measurement to another. You can use a formula in order to do this.

Sometimes the conversion formula that is given is in the opposite direction from the one you want. However, it is possible to turn such formulas round. For example, since 1 km = 1000 m, it follows that

Example 12

Turn the formula 1 mile = 1.609 km round to obtain 1 km in miles. Give your answer to 3 significant figures.

Answer

The formula can be written as

Dividing by 1.609 gives

3.4.1 Try some yourself

Activity 18

1 Use the formula 1 mile = 1.609 km to find 12 miles in kilometres rounding your answer to 3 significant figures.

Answer

Activity 19

Turn the following formulas round to obtain centimetres in terms of inches, and grams in terms of pounds giving your answer to 4 decimal places:

1 inch = 2.54 cm;

1 Ib = 453.6 g (to 4 s.f.).

How many pounds are there in a kilogram?

Answer

From 2.54 cm = 1 inch, it follows that

From 453.6 g = 1 lb, it follows that

Since 1 kg is 1000 g,

4 The format of questions

4.1 Understanding the question

Before you can attempt a question, you must read and understand it. This may sound obvious but you will need to know, before you start, what is expected by way of an answer. In particular, you will need to know the meaning of the instructions contained in the question. This section contains a discussion of the precise meanings attached to words like ‘find’, ‘show’, ‘write down’ and ‘determine’ in mathematics questions. The different types of instruction are illustrated by posing several questions all based on the same initial problem.

4.1.1 ‘Write down’ question

The number [4] at the right-hand side means there are four marks awarded for a correct solution. The marks give an indication of the length of answer expected. For example, if a question is in two parts, the first of which is allocated two marks and the second eight marks, then you would expect the first part to have a shorter and more straightforward solution than the second. Here, four marks would be awarded for stating the answer correctly, as in the solution below, because all you were asked to do was to ‘write down’ the area.

4.2 A better solution

Here is an improved solution which shows working.

4.3 ‘Find’ questions

The word ‘find’ implies some reasoning is required. Although the following solution contains an error (so it does not get full marks), it is nevertheless worthy of credit. Note how marks are awarded for correct reasoning and justification.

4.4 ‘Show that ’ questions

In this type of question you are given the answer! All the marks are allocated for correct reasoning and justification.

Note that the question says ‘show that’. So no marks are allocated for the final cost. This is reflected in the way the marks have been awarded in the following solution to Example 15

4.5 Questions and answers

The following table summarises some of the types of instructions you will encounter.

You may like to copy this table into your own notes and add to it if you encounter other question wording in the future.

Maximise

5 Approaches to problem solving

You should not expect always to be able to read a problem and then just write down the answer. When you are faced with a written mathematical question or problem to solve, read it carefully. It is important that you get to grips with the question in two ways: first, that you absorb the information given; and second, that you find out what the question is really asking. Your solution will link the two. This method can be summarised by the following questions.

• What do I know?

• What do I want?

Try to decide not only what you know from the question itself, but also what you know from past experience which is at all relevant to the problem.

When you consider what you want, first decide whether you have to (a) find an answer or (b) show that something is true.

In order to get to grips with what a question is really about, it can often be helpful to begin by drawing a diagram.

Maximise

If you are stuck, it may help to review your work. Try to write out your thoughts as clearly as possible so that you can easily see each step and the ideas they are based on.

You may need to leave the problem for a while and then come back to it, in order to see it more clearly. In this case, it is vital that you have written down what you have done so far. It may also be a good idea to talk to someone about it. The act of describing a problem to someone can often be enough to help you see what you should do to arrive at a solution. If there is nobody to talk to, try ‘talking to yourself’.

Always consider whether the answer you have come up with ‘makes sense’; that is, step back a bit and see whether it fits in with the other information that you have.

6 Writing out solutions

6.1 How a question is marked

When you tackle a mathematical question, it is often helpful to think about how it would be marked.

To give yourself some practice at this, in this section you will:

• read and criticise sample solutions;

• then read what the tutor wrote about them; and

• see if any of the tutor's comments are applicable to your own solutions.

A billions problem

First write out a full solution to the following problem.

6.1.2 Look at another solution

Now look at the sample solution below: consider both mathematical accuracy and style.

Sample Solution A to Example 16

Maximise

Click on ‘Reveal answer’ to see the tutor's marking.

Answer

Tutor's comments on Sample Solution A

Maximise

6.2 Another ‘billions’ solution

Pretend that you are the marker of another solution to the same problem. How would you mark Solution B?

Sample Solution B to Example 16

• (a) 10⁹

• (b) 10⁹ ÷ 10² ÷ 10³ = 10⁴

• (c) 10⁹ ÷ 60 ÷ 60 ÷ 365 = 761

• (d) = 4.980 × 10⁶ ÷ 10⁹ = 4.980 × 10⁻³

Click on ‘Reveal answer’ to see the tutor's marking.

Answer

Tutor's comments on Sample Solution B

Maximise

6.3 Solving the riddle of St Ives

Write out your own solution to the following problem.

Now comment on the sample solution which follow as if you were the tutor. Finally, use the tutor's comments to assess your own solution and comments.

Sample Solution A to Example 17

Click on ‘Reveal answer’ to see the tutor's marking.

Answer

Tutor's comments on Sample Solution A

Maximise

6.4 Mark another St Ives' question

Now pretend your are the tutor marking a rather briefer response to the St Ives' riddle.

Sample Solution B to Example 17

Maximise

Click on ‘Reveal answer’ to se the tutor's marking.

Answer

Tutor's comments on Sample Solution B

Maximise

Notice that the tutor suggests a clearer way of displaying the results. Solution A's method is also clearer. You will notice that the student lost marks for not really ‘showing’ that there were 7⁴ kits, even though he or she probably understood this; the marks are for giving good, convincing reasoning.

6.5 Solutions to ‘making a lawn’

Learning from the marking of the previous questions, write out a good solution to the following problem.

Now look at the sample solutions and mark them as if you were the tutor. Finally, use the tutor's comments to assess your own solution and comments.

Sample Solution A to Example 18

Maximise

Click on ‘Reveal answer’ to see the tutor's marking.

Answer

Tutor's comments on Sample Solution A

Maximise

6.5.1 Another ‘making a lawn’ solution

Sample Solution B to Example 18

Click on ‘Reveal answer’ to see the tutor's marking.

Answer

Tutor's comments on Sample Solution B

Maximise

7 OpenMark quiz

Hopefully, thinking about how your solution to a mathematical problem might be marked, will help you to produce better solutions for yourself, as well as for somebody else. Now try the quiz and see if there are any areas you need to work on.

Conclusion

This free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.

Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:

Course image: Tom Brown in Flickr made available under Creative Commons Attribution 2.0 Licence.

The content acknowledged below is Proprietary and is used under licence.

The following material is proprietary (and not subject to Creative Commons Licence) and used under licence (see terms and conditions):

Section 4.5: Sesame;

Section 5: Syndication International, Mirror Group Newspapers.

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