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 the constant of proportionality is 0.5 km per cm and the gradient of the corresponding graph is 0.5. The steeper gradient says, in effect, that you get more kilometres for your centimetres on this map.
In general, therefore, 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 relationship is:
value on vertical axis = gradient x value on horizontal axis
Different constants of proportionality give straight-line graphs with different gradients. The steeper the gradient, the greater the value on the vertical axis for a given value on the horizontal axis. Changing the scale on the vertical axis has an effect on the visual perception of steepness. Choice of scale can have a profound effect on the visual impact of a graph. But numerically, the gradient of the graph is unchanged by a simple change of axis scale.
Activity 5: Converting to metric
If you were to draw graphs to convert from (a) pounds to kilograms (1 pound = 454 grams), or (b) miles to kilometres (1 km = 0.621 miles), what would be the gradient in each case?
Make a quick sketch of the conversion graph in each case.
Did you find making the sketches helped you to answer the question? Or did you find they made the task more difficult? Make a few brief notes to record your response.
One pound is equivalent to 0.454 kg. So the conversion graph will go through the points (0,0) and (1,0.454), as in Figure 11a. The gradient is
A distance of 1 km is equivalent to 0.621 miles. So the conversion graph will pass through the points (0,0) and (0.621,1), as in Figure 11b. The gradient is
For many measures like those in Activity 5, a conversion graph will be a straight line starting at the point (0,0), because zero on one scale of measurement will also be zero on another. But this is not true for all measurement scales. Temperature, for example, can be measured using different scales which do not share the same zero point, because a temperature of zero degrees can be defined in different ways. Zero degrees Celsius, for instance, does not mean ‘no heat’.
For everyday use, most people tend to think in terms of either the Fahrenheit or the Celsius (or centigrade) scales. Which do you use? When you hear temperatures given in a weather forecast in degrees Celsius (written as °C), do these mean much to you, or do you try to get a feel for how hot or cold it is going to be by converting to degrees Fahrenheit (written as °F)? Older cookery books and ovens often quote temperatures in degrees Fahrenheit, whereas modern ones use degrees Celsius. How would you convert between the two temperature scales? What mental picture of the two scales would you use?
Celsius and Fahrenheit
The Celsius, or centigrade, temperature scale is named after Anders Celsius, a Swedish scientist who first devised a form of this scale in 1742. The word ‘centigrade’ means ‘one hundred steps’, and refers to the fact of expressly choosing 100 equal divisions between the boiling and freezing points of water.
Daniel Fahrenheit (1686–1736) was a German scientist who developed the first thermometers using the expansion of alcohol and mercury to indicate temperature.
The Celsius scale is defined in terms of the freezing and boiling points of water (at a particular standard air pressure). When water starts to freeze and form ice, its temperature is defined to be 0°C; when it boils and forms steam, its temperature is defined to be 100°C. Thermometers are calibrated using these fixed points – they are made so that they read 0 C when they are immersed in freezing water and 100°C when they are immersed in boiling water.
The Fahrenheit scale was originally set up by taking 0°F as the freezing temperature of a mixture of ice water and salt and 96°F (later adjusted to 98.6°F) as the ‘normal’ temperature of the human body. On this scale, pure water freezes at 32°F and boils at 212°F.
In science, temperatures are often quoted using the absolute or Kelvin scale on which zero (the so-called absolute zero, the lowest temperature theoretically possible) corresponds to about -273°C, or about -460°F. Here, zero on the Kelvin scale does mean ‘no heat’.