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

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.

1.7 Every picture tells a story: summing up

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

1.6.2 Beware of first impressions

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):

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

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

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

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

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

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