1.6.1 Introduction As to the propriety and justness of representing sums of money, and time, by parts of space, tho’ very readily agreed to by most men, yet a few seem to apprehend that there may possibly be some deception in it, of which they are not aware … (William Playfair (1786) The Commercial and Political Atlas, London) The political economist William Playfair, who developed many of the graphical r
1.5.3 Mathematical graphs: What could they mean? The values of the x- and y-coordinates in a graph sometimes relate to measurements of physical quantities: for example, in graphs of height against distance, or temperature against time. Physical quantities always have units associated with them, and these must be shown on the axes’ labels of the graph. In mathematics, however, values of x- and y-coordinates that have been calculated using a formula may simply be numbers: they may not have units atta
1.4.8: Graphical conversions: summing up This section started by looking at conversion graphs which were straight lines passing through the origin of the graph. The intercept in those cases was zero, and only one number – the gradient – was needed to describe the relationship between the quantities plotted on the horizontal and vertical axes. In the more general case, the graph is still a straight line with a constant gradient, but the line no longer goes through the origin. An extra number – the intercept – is used to pin t
4.7 Graphical conversions: What is the relationship between the Fahrenheit and the Celsius scales? To determine this, you first need to determine the gradient of the straight line inFigure 11. The new vertical scale goes from 0 to 180 as the Celsius scale on the horizontal axis goes from 0 to 100. So the gradient is and the relationship between the scales on th
1.4.6 Graphical conversions: So what is the relationship between the two scales? You know that if the line passes through the origin of a straight-line graph, then the gradient of the graph links the values on the horizontal and vertical axes. The general relationship is: on vertical axis = gradient x value on horizontal axis Now suppose the scale on the ve
1.4.5 Graphical conversions: How would you go about drawing a graph to convert from one scale to the First you need some data about corresponding temperatures on each scale. In the case of Celsius and Fahrenheit, there are two fixed points of reference: the freezing and boiling points of water. On the Celsius scale, the freezing point is defined to be 0°C; on the Fahrenheit scale, the freezing point is 32°F. So if you plot degrees Celsius on the horizontal axis and degrees Fahrenheit on the vertical axis of a graph, the freezing point of water is represented by a point with the coordinates
1.4.4 Graphical conversions: How is the constant of proportionality represented on a graph? One of the main features of a straight-line graph is that the line has a constant slope. The gradient of the slope is numerically equal to the constant of proportionality. For a 1 : 25 000 map, the constant of proportionality between ground distances in kilometres and map distances in centimetres is 0.25 km per cm. So the gradient of the corresponding graph is 0.25. A similar relationship holds for a 1 : 50 000 map. In this case, 1 cm on the map corresponds to 0.5 km on the ground, so t
4.3 Graphical conversions: How do you use the graph? Look at Figure 9. Start with the map distance on the horizontal scale, move vertically up until you reach the line, then move horizontally until you reach the vertical axis. The number at that point will give you the corresponding ground distance in kilometres. 1.4.2 Graphical conversions: drawing a straight-line graph This means, for instance, that if you double one value, the effect is to double the other, and if you third one value, the upshot is that the other is divided by three as well. And the fact that the relationship is directly proportional has an important consequence for the graph-it will necessarily be a straight line. Only two points are needed to draw a straight-line graph. Choosing one of the points is straightforward, it is the origin of the graph. Why? Because zero distance on the m 1.3.2 Time-series graphs: an example 1.3.1 Introduction The time-series plot is the most frequently used form of graphic design. With one dimension marching along to the regular rhythm of seconds, minute, hours, days, weeks, months, years, centuries, or millennia, the natural ordering of the time scale gives this design a strength and efficiency of interpretation found in no other graphic arrangement. (Tufte, E. (1983) The Visual Display of Quantitative Information, Gra 1.2 Every picture tells a story The main aim of this section is to give you practice in reading, interpreting and drawing a variety of graphs created for many different purposes.You will need graph paper for this section. Graphs occur in all sorts of different contexts and applications. Graphical representations can be used to show profiles of height plotted against distance for sections of the Peak District walk, for example. This section looks at three sorts of graphs: time-series graphs, conversion graphs and mathe 1.1.1 About the Unit This Unit teaches on drawing and interpreting graphs. It has eight sections, each dealing between them with 4 different types of graph. You will need to draw some graphs, so make sure you have a flat surface to work on. You will also need to have centimetre graph paper and your calculator to hand as you study the material. Section 3 is based on a video band, ‘Single-track minders’. This is split into four separate viewing sessions, each followed by an activity. You should expect to 1.1 A shared understanding All representations (including graphical ones) rely on shared understandings of symbols and styles to convey meaning. Like maps, graphical representations stress some features and ignore others. As you work through this Unit, bear in mind that graphs are selective representations of information. When you come across different graphs ask yourself what is being stressed and what is being ignored. In the newspapers, you are likely to find graphs used to present all sorts of information: ho Learning outcomes After studying this Unit you should be able to: Explain in English and by using examples, the conventions and language used in graph drawing to someone not studying the course Use the following terms accurately, and be able to explain them to someone else: ‘time-series graph’, ‘conversion graph’, ‘directly proportional relationship’, ‘“straight-line” relationship’, ‘gradient’, ‘intercept’, ‘x-coordinate’, ‘y-coordinate’, ‘coor Introduction Graphs are a common way of presenting information. However, like any other type of representation, graphs rely on shared understandings of symbols and styles to convey meaning. Also, graphs are normally drawn specifically with the intention of presenting information in a particularly favourable or unfavourable light, to convince you of an argument or to influence your decisions. This unit is from our archive and is an adapted extract from Open mathematics (MU120) which is no longer 3 Subtraction rules – order matters It’s important to remember that subtraction has different rules from addition. For example, when you add up numbers, it doesn’t matter what order you add them up in. So 6 + 4 is exactly the same as 4 + 6. The result is 10 in both cases. But in subtraction, order matters. So 6 – 4 is different from 4 – 6. With the first, you start with 6, subtract 4, and are left with 2. But with the second you start with 4 and if you subtract 6, which is a bigger number, you a Learning outcomes By the end of this unit you should be able to: subtract one number from another; subtract using decimals; practise your subtraction skills. 7 Dividing when you have to carry If the number you are dividing by does not go exactly (with no remainder) into the digit you are dividing into, you need to do something called carrying. Say you want to divide 952 by 7. The process is basically the same as in the previous section. First write it down on paper. Then, to do the calculation, you take each digit from the number being divided in turn, starting with the one on the far left, and see how many times the dividing number, 7 in this case, goes into it. The calcul 6 Dividing on paper If the numbers you want to divide are too large for you to do the calculation in your head, you can use a calculator. Alternatively, you can do the calculation on paper. In the example below, click on each step in turn to see how to divide 126 by 6.
