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GRAPHS AS BRIDGES BETWEEN MATHEMATICAL DESCRIPTION AND
EXPERIMENTAL DATA
Laurence Rogers, School of Education, University of Leicester, England
1. Why do physicists use graphs?
When physicists accumulate numerical data from practical experiments they need to store the data
in some form and this is traditionally done in a table of results. When it comes to analysing results,
although a table has a certain amount of use for comparing items of data and deriving further
information, the graph is a far more informative tool for this purpose. The visual impact of graphs
and their relational properties makes them extremely valuable for analysing and appreciating the
properties of data. It is implicit that a typical graph stores a large quantity of numbers, the data
pairs associated with each plotted point, but more importantly the graph contains valuable
information about the relationship between the variables represented by the data. Describing the
relationship between variables is an important but sophisticated skill for students to acquire and an
understanding of graphs, how they work and how they can be used, can be a key factor in helping
them develop this skill.
2. Information from graphs
The most obvious feature of a graph is its shape. When experimental data is presented as a graph,
the shape of the graph immediately conveys information in a qualitative manner without concerning
the observer with unnecessary, numerical detail. The shape of a graph can give pupils a quick
overview of what may be going on in an experiment; pupils can 'see' gradual or sudden changes,
continuity or discontinuity, the difference between a complex sequence of events and a progressive
trend. A range of features give valuable information about the variables: a gentle or steep gradient
indicates a slow or rapid change, the curvature of a line indicates a varying rate of change, peaks
and troughs indicate maxima and minima, and so on. Let us consider some common types of graph
shape and consider what scientific information pupils might be expected to infer from their
observation of the graph.
The first is a graph whose dominant feature is an upward trend. The upward trend indicates that as
one variable increases the other variable also increases. The classic example of this is a graph of
current against voltage for an electrical resistor. If the data is gathered using a data logger and
plotted using software, it is very easy to vary the plotting format to emphasise the relationship
between changes in the variables. A useful variant is to plot both voltage and current against time.
Inspection of the graph in Figure 1b leads to the simple observation that, as one variable increases,
so the other also increases. Thus the graph shape provides basic information about how the
variables are related.
Figure 1a. Graph of current against voltage
Figure 1b. Graph of current and voltage against
time
The second graph shows a general downward trend. This shows a relationship between two
variables which is the inverse of the first case, namely, as one variable increases the other variable
decreases. The example of pressure plotted against the volume of a fixed mass of gas illustrates this
type of relationship. Again, the use of software to plot the data provides a useful alternative format
for plotting the graph. Figure 2b makes it abundantly clear that the two variables vary in the
opposite sense.
Figure 2a. Graph of pressure against volume
Figure 1b. Graph of pressure and volume against
order of recording
3. Understanding the properties of graphs
The graphs in Figures 1a and 2a both illustrate progressive trends, without irregularities or
discontinuities. It is reasonable to expect that such smooth trends can be described by fairly simple
mathematical formulae. Clearly the straight line graph in Figure 1a can be described by the formula
y = mx + c and the constants m and c are readily evaluated by taking measurements from the graph.
Software provides useful tools for conducting these measurements speedily, using a curve fitting
facility or by adjusting a trial function to match the graph under observation. However, even more
usefully, software allows pupils to explore the properties of the graph to teach them the significance
of the linearity for precisely describing the relationship between the variables. The main tools for
this type of exploration are the cursors which provide automatic reading of data points on the graph
and which can calculate changes of both sets of variable simultaneously. From such explorations a
variety of statements may be made about the properties of a straight line graph:
1. Changes in the variables occur at a constant rate.
2. For a given increment in one variable, the other variable always increases or decreases in
equal steps. (For the case of a variable plotted against time, the size of the step for a given time
interval is always the same.)
3. This is independent of the magnitude of either variable.
4. The ratio between the increases or decreases in either variable is constant.
5. When this ratio is not unity, one variable changes more rapidly than the other.
6. The gradient is the same at all places on the graph. i.e. it is constant.
7. When the gradient is negative, an increase in one variable is accompanied by a decrease in the
other.
To the tutored eye, these descriptions are clearly equivalent to or follow from each other. Their
significance here is that they are individually testable using software tools: when a cursor is moved
across the graph, changes in the variables may be read automatically and the rate of change
calculated; measurements may taken from any selected part of the graph; 'x' and 'y' cursors may be
locked together, easily showing the relative changes in two variables; the gradient at a cursor may
be read automatically. Through a variety of explorations, pupils may learn to associate a
characteristic set of properties with the linear graph so that when experimental data yields a straight
line, they will have a certain understanding of how the variables relate to each other, how changes
in one variable are associated with predictable changes in the other. Thus for ohmic resistors, the
straight line indicates behaviour which may be described by phrases like “equal increases in voltage
lead to equal increases in current”, or “the ratio between the voltage and current does not vary”.
The straight line graph is a common occurrence in the analysis of experimental data and its unique
properties are easily described informally and mathematically. However, it is only one shape of
many which can describe variables which increase simultaneously. The example of the ohmic
resistor points to another relationship which is certainly not linear, that between the power
dissipation and the current flowing. Figure 3 shows the curve which is characteristic of this
relationship.
Figure 3. Graph of power vs. current for an ohmic resistor
Although power increases when current increases, the ratio between the increases is clearly not
invariant as it was for the straight line graph. Software tools allow the properties of this shape of
curve to be explored:
Cursors are used to calculate successive increases in power for a given increase in current. For an
upward curve it can be expected that the increase in power for a given increase in current is larger
according to the value of the current. For this particular curve, the difference between successive
increases in power is always the same. This is a unique characteristic of the quadratic curve. The
use of a software curve fitting facility identifies the formula and indeed confirms that this curve is a
parabola described by y = ax2. Further use of cursors shows that the power increases according to
the square of the current.
Again, through a variety of explorations, pupils may learn to associate a characteristic set of
properties with this particular graph shape which can in principle be applied to any experimental
data yielding a parabola. For example the braking distance of a motor car can be related to the
square of its velocity, demonstrating that a required braking distance does not increase in simple
proportion to the velocity of a car.
The previous example of an inverse relationship between the pressure and volume of a gas can be
analysed with a similar range of software tools. Informally the downward slope of the graph
(Figure 2a) shows that increases in one variable are associated with decreases in the other. The use
of cursors can confirm that there is a definite pattern to this trend which allows predictions to be
made. For example when the volume is reduced by half its value, the pressure doubles in value.
pupils can gain a feeling for this relationship by looking out for this pattern. As for the previous
graph shape, a software fitted curve identifies the characteristic formula for inverse proportionality,
in this case expressing Boyle’s Law.
The inverse square law is another common relationship with a downward curve graph which at first
sight appears to be very similar to the previous example. Exploration with cursors soon reveals the
distinctive properties of this curve which has much more rapid changes to the gradient. A casual
view of a curve for exponential decay might also suggest a similar relationship, but curve fitting and
cursor reading techniques soon reveal its unique ‘constant ratio’ properties.
4. Graph shape as an indicator of a relationship
The discussion in this paper has argued that, when experimental data is represented by a graph, this
facilitates three levels of description of the data:
• informal qualitative description based on observations of graph shape e.g. “the temperature
falls quickly at first and then slowly rises.”
• informal quantitative description based on numerical exploration of the data using software
cursor tools e.g. “at double the speed the distance is four times greater”
• mathematical description using formulae evaluated by curve fitting and curve matching
techniques.
Pupils should be encouraged to develop all these levels of description and build up a notion of the
characteristic properties of each graph shape so that when graphs of experimental data are observed
these properties can be applied to the variables concerned.
References
Barton R., Computer-aided graphing: a comparative study’, Journal of Information Technology for Teacher Education,
6, (1), (1997), 59-72.
Newton L.R., Graph talk: some observations and reflections on students data-logging, School Science Review, 79,
(287), (1997), 49-54.
Rogers L.T., Probing the Hidden Secrets of Graphs, Hands-on Experiments in Physics Education - Proceedings of the
GIREP, Conference in Duisburg (1998).