 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
Author(s): The Open University

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

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
Author(s): The Open University

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
Author(s): The Open University

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.

Author(s): The Open University

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
Author(s): The Open University

Representing ‘sums of money, and time, by parts of space’, 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, 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
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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 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
Author(s): The Open University

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
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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
Author(s): The Open University

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
Author(s): The Open University

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
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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
Author(s): The Open University 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
Author(s): The Open University