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
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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
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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): The Open University

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
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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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In summary, this section has looked at time-series graphs, conversion graphs and mathematical graphs. Like all representations, graphs draw from a range of common conventions and styles to convey meaning. From a mathematical point of view, graphs give a visual impression of the relationship between two (or sometimes more) variables; but bear in mind that this impression is largely under the control of whoever draws the graph. When you are drawing graphs for yourself or others, you need to cho
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Representing â€˜sums of money, and time, by parts of spaceâ€™, as Playfair put it, may indeed seem obvious and readily agreed, but nevertheless graphics showing the rise and fall of profits, expenditure or interest rates over time often need to be approached carefully. As the inventor of the bar chart (or bar graph), Playfair might well have raised a quizzical eyebrow at the example in Author(s): The Open University

As to the propriety and justness of representing sums of money, and time, by parts of space, thoâ€™ 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 â€¦

(William Playfair (1786) The Commercial and Political Atlas, London)

The political economist William Playfair, who developed many of the graphical r
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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 axesâ€™ 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
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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 â€“ the gradient â€“ 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 â€“ the intercept â€“ is used to pin t
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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
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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
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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 0Â°C; on the Fahrenheit scale, the freezing point is 32Â°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
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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
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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.

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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
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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
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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
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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, â€˜Single-track mindersâ€™. This is split into four separate viewing sessions, each followed by an activity. You should expect to
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