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

Answer

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.

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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 â€˜equalsâ€™;

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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 previously, one of the most misused mathematical symbols is the equals sign, =. It stands for the verb â€˜equalsâ€™ or the phrase â€˜is equal toâ€™ or â€˜which equalsâ€™, and so it should only come between two things that are equal.

Example 2

Which of the equals signs should not be
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1 Read the following expression out aloud or write it out in full in words:

• (a) 21 + 34 = 55

Answer

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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
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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
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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.

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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:

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Now watch the video.

Video, Click to watch part three