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.
You can also use the graph to go from ground distance to map distance. Find the particular ground distance on the vertical scale, move horizontally across to the line and then move vertically down until you reach the horizontal scale. You can then read off the corresponding map distance in centimetres. If you could be bothered to do it – and had unlimited time (like forever!) – you could have drawn the same graph by working out every possible ground distance for every possible map distance between 0 and 5 cm, and plotting each coordinate pair. But the results of all those countless, individual, specific calculations are automatically included in just one straight line drawn using the knowledge that the relationship is a directly proportional one. A mathematical formula generalises a relationship by containing markers – words or symbols – which you replace by numbers for specific calculations. Likewise, a graph generalises the results of individual calculations, indicating by means of its shape the corresponding values in a relationship. A graph, therefore, represents the general form of a directly proportional relationship as well as allowing you to handle specific examples.
Activity 4: A litre of water …
When the metric system of weights and measures was introduced in the UK, the government information campaign included the rhyme: ‘A litre of water’s a pint and three-quarters’, to help people remember the conversion factor. The conversion constant is not specific to water (it merely helped the rhyme) and can be used for any fluid.
With the campaign slogan in mind, draw a graph on graph paper to convert between pints and litres. Use your graph to find (a) how many litres correspond to 3 pints, and (b) how many pints correspond to 4 litres.
Using the fact that there are 8 pints to 1 gallon, find the conversion factor between gallons and litres, and draw up a table of gallon/litre equivalents. Do you think it would be more helpful if you used a conversion graph rather than a table? Write down any advantages or disadvantages you can think of.
Figure 10 shows a graph to convert between pints and litres. 1 litre is about 1.75 pints (actually, a litre is closer to 1.76 pints, but stick with the rhyme!), so to be able to convert 4 litres, the scale along the horizontal axis of the graph needs to extend to at least 4×1.75=7. The graph has been drawn slightly bigger than the minimum size with the horizontal scale representing 0 to 10 pints and the vertical scale representing 0 to 5 litres. The graph was constructed by drawing a straight line from the origin (0,0) to the point (8.75, 5), representing the equivalence between 5 litres and 8.75 pints. You may have chosen different point and/or scale. Make sure that your graph has a title, and that the axes are scaled and labelled clearly and correctly.
From the graph, 3 pints is equivalent to just over 1.7 litres, and 4 litres is equivalent to 7 pints.
8 pints is equivalent to about 4.6 litres. So the conversion relationship is 1 gallon=4.6 litres.
Here are some ideas about the pros and cons of tables and graphs. Tables are easy to use and hold a lot of information in a compact way. But they do not give any visual impression of the general mathematical form of the relationship in the way that graphs do. In principle, graphs drawn from formulas can be used to convert between any two equivalent values, provided that the values lie within the range of the graph. Tables give only selected pairs of values, intermediate values have to be estimated. You may have thought of some other points.
Recall that two quantities being directly proportional to each other – such as map and ground distances, or pints and litres – means the amount of one is found simply by multiplying the other by a fixed conversion number. In Activity 4, for example, the formula describing the relationship (according to the rhyme) is:
volume in pints = 1.75 x volume in litres.
The number 1.75 is the constant of proportionality. It relates a volume measured in pints to the same volume measured in litres, and strictly it has units itself. In this case, the units are ‘pints per litre’.