2.6.2 Try some yourself 1 Read the following expression out aloud or write it out in full in words: (a) 3 × 4 + 3 × 5 = 3 (4 + 5). 2.5.1 Try some yourself 1 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. 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 2.4.1 Try some yourself 1 Two labels have been omitted in the mathematics below. Where should they go to make sense of the argument? Since 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), 2.3.1 Try some yourself 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) 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’. 2.2.1 Try some yourself 1 Here is a poor example of mathematical writing, although the final answer is correct. Rewrite it, correcting the layout and the mathematical punctuation. 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 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; j Acknowledgements 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. 1.9.6 On the right lines: summing up 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: 1.9.1 Introduction 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 ‘Single track minders’ 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 1.8.11 A mathematician’s journey: using the model for planning 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 OU’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 − 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 1.8.8 Reading distance-time graphs: summing up 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 1.8.7 Distance-time graphs: a mathematical story Distance-time graphs are a means of replacing a description given in words by a mathematical description of the same event. What follows is a narrative account: that is, a description in the form of story about a bicycle ride. Read the story and then think about how you would use this account to produce a mathematical model of the ride in the form of a distance-time graph. Sunday started a bit cloudy. The temperatu 1.8.6 The final graph The three separate lines are combined into one overall distance-time graph representing the entire journey, as shown in Figure 44. The times for the sections are added together, so that the scale on the horizontal axis shows the total time that has elapsed since leaving Paris. Similarly, the distances of the sections are combine 1.8.4 Distance, time and speed: an example The Eurostar train service that connects London and Paris via the tunnel under the English Channel (la Manche) covers a distance of about 380 km in three hours in 1996. Assuming a constant speed, what would the distance-time graph of this journey look like? Take the Gare du Nord (the Northern Station) in Paris as the start and measure time and distance from there. The vertical axis on Author(s): 1.8.3 Distance, speed and time: assumptions The formulas for speed, distance and time are all examples of mathematical models. Here, you should bear in mind that such models stress some aspects of travelling but ignore others. Building a mathematical model involves making some assumptions, and usually this involves disregarding those inconvenient aspects of real-world events which can not easily be fitted into a mathematical description. Take, for example, the model s = d/t used to calculate speed. Dividing a 1.8.1 Introduction The main aim of this section is to introduce the distance-time graph as a mathematical model of a journey. Like any mathematical model, a distance-time graph stresses some features of the situation it claims to represent and ignores others. Bear this in mind as you work through this section, and note for yourself which aspects of a journey are described graphically, and which do not feature in the model. You will need graph paper for this section.
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