This subsection looks at two ways of finding an â€˜averageâ€™. The first produces the mean, which is what was originally meant by â€˜averageâ€™, and what most people think of when they talk about an average. The second gives the median, which might more accurately be described as a â€˜typicalâ€™ or middle value. They will be illustrated using the following batch of heights.

The heights in metres (measured to the nearest centimetre) of a group of seven people are as follows
Author(s): The Open University

## Example 18 Making a lawn

Suppose you have some friends who are planning to put a new lawn in their garden. The lawn is to be 12 m by 14 m and they have a choice of either laying turf or sowing grass seed. You have been asked to help them decide between the two.

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

## Example 18: Making a lawn

Suppose you have some friends who are planning to put a new lawn in their garden. The lawn is to be 12 m by 14 m and they have a choice of either laying turf or sow
Author(s): The Open University

1 If tomatoes cost 75 pence per kg, how much would 1.45 kg cost in pounds (Â£)?

The formula is

cost of tomatoes = (price per kilogram) Ã— (number of k
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A formula is a rule or a generalisation. Word formulas â€“ formulas that use English words rather than mathematical symbols â€“ 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.

Author(s): The Open University

1 Read the following expression out aloud or write it out in full in words:

• (a) 3 Ã— 4 + 3 Ã— 5 = 3 (4 + 5).

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1 How would you add the following words to the list:

Â decimal, fraction, positive, negative.

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

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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 â€˜sumâ€™ might mean any calculation, but it has a precise mathematical meaning as exemplified by â€˜The sum of 456 and 789 is 1245â€™. Similarly, in English the word â€˜productâ€™ can have a variety of meanings, but
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1 Two labels have been omitted in the mathematics below. Where should they go to make sense of the argument?

• Since

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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),

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1 In the following two pieces of mathematical writing, remove or replace any inappropriate equals signs, and add link words and punctuation to help somebody else understand the mathematics.

• (a)

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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 â€˜Soâ€™, â€˜This impliesâ€™ or â€˜It follows thatâ€™ or â€˜Henceâ€™.

## Example 3

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

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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
Author(s): The Open University

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; j
Author(s): The Open University

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

Grateful acknowledgement is made to the following sources for permission to reproduce material within this product.

## Author(s): The Open UniversityLicense informationRelated contentExcept for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Timetables and distance-time graphs are different representations of scheduled train movements. They are both models which can be used to predict when trains will run, to analyse and compare different schedules when problems occur, and to design new operating schedules to meet new demands. Both models provide information which allows the company to operate safely and flexibly. The information is used by different groups of people:

• Author(s): The Open University

The main aim of this section is to show an application of distance-time graphs in the operation of a railway service.

You will need graph paper for this section.

This section uses the video â€˜Single track mindersâ€™ to illustrate how distance-time graphs are drawn and interpreted by the timetable planners of a small railway company, and shows the role of this graphical technique in planning a flexible service. Graphical representations of journeys have been used for over a centur
Author(s): The Open University

By drawing a distance-time graph, Alice has predicted that she and Bob will pass on the stretch of road between Newcastle and Nottingham. Using the OUâ€™s computer system, she sends an email message to Bob suggesting that they meet at a roadside restaurant about 275 km north of Milton Keynes (for Bob this will be 510 âˆ’ 275 = 235km south of Edinburgh). Bob acknowledges her email and the meeting is set up.

Alice guesses they will probably stop for about 30 minutes. But what effect will
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You should now be able to interpret distance-time graphs, and be able to use them to find information about the average speed, the distance travelled and the time taken for different sections of a journey. Given any two of these quantities you should be able to identify and use the appropriate formula to find the third.

An important feature of a straight-line graph is its gradient. The gradient, or slope, of a graph expresses a relationship between a change measured along the horizontal
Author(s): The Open University