0
RATE
OF
[H
by A. Orton,
Centre for Studies in Science and Mathematics Education
University of Leeds
Concepts associated with rate of change are not easy for
pupils to grasp. Fundamentally, rate of change is a manifestation of proportionality, and we know that ratio and
proportion present many difficulties for learners (see, for
example, Hart, 1981)1. In the classroom, the idea of gradient may be pursued diligently, but it still depends on
proportion. The technique of drawing any right-angled
triangle and calculating y-distance/x-distance requires that
the pupil does understand that the same ratio results
whatever the size of the right-angled triangle. Rate of
change is perhaps a more general concept than gradient and
may not receive as much emphasis in our teaching. It is
certainly not well understood by pupils.
The results described in this article were obtained from a
detailed large-scale study of pupils' and students' understanding of elementary calculus and the mathematical ideas
which underlie calculus (Orton, 1980)2. The data was
obtained through clinical interviewing, the one-to-one
teacher-pupil situation which is capable of yielding excellent information about conceptual understanding. Half of
those interviewed were from schools, the others were older
and came from colleges. All of the school pupils were from
sixth forms, so there are obvious implications in relation to
younger learners when one considers the results. Those
interviewed came from a very wide variety of school
backgrounds.
Difficulties concerning speed
One of the early tasks was as follows:
(i) It is found that a car travels the 5 miles across and
through a town in 15 minutes. At what rate per
hour is the car travelling?
(ii) Can you calculate exactly how far the car will have
Mathematics
in
travelled in the first 3 minutes, and if so, what is the
distance? If you do not think it is possible to
calculate this distance explain why not.
There were a number of interesting features to responses.
The majority of pupils were able to convert the information
given to produce 20 miles per hour, in answer to the request
for the rate per hour, but subsequently some pupils appeared to assume that this was a constant speed over the 15
minutes. Over one third of those interviewed gave an exact
answer to the question concerning the distance travelled in
the first 3 minutes. Consideration of what pupils said
suggests that there was real confusion between average
speed and constant speed. Furthermore, a number of pupils
who correctly stated that there was no way of knowing
exactly how far the car travelled in 3 minutes then gave as
their reason that it would take some time to accelerate from
rest up to a certain speed. The assumption that the car
started from rest was curious; no suggestion of this was
made in the question. Also, whether the car had to accelerate from rest or not, variation in speed through the
town due to traffic conditions was a valid point to make but
was completely ignored by this group of pupils. Intended
real-life situations can apparently give rise to peculiar
interpretations.
One would assume that pupils bring some knowledge and
conceptual structure to the ideas associated with speed
when it is studied in the context of rate of change in
mathematics lessons. Pupils seem to have some understanding of comparative speed, for example. They know that 70
miles per hour on a motorway will get them home much
more quickly than will 30 miles per hour on other roads.
They can make estimates of the speed of an overtaking
vehicle from the speed of the one being overtaken. On the
whole, they are interested in and therefore reasonably
School,
November
1984
23
motivated by studies of speed. Velocity, as rate of change of
displacement with time, must form an important aspect of
studies of rate of change. It seems, though, we must be
careful that we do not assume too much in terms of pupils'
abilities to sort out important ideas like variable speed,
constant speed, average speed and speed at an instant. We
should certainly devote time to discussion of these concepts.
Using difference tables
An unusual feature of some of the tasks was the inclusion
of simple difference tables, as in the task below.
whether we should do more work with pupils on differences. The idea and value of differencing can begin with
quite young children, simply in connection with number
patterns, and this is by no means a new idea (see, for
example, Sawyer, 1943)3. As a foundation for subsequent
approaches to certain algebraic ideas and to elementary
calculus studies of differences in number patterns would be
very valuable. In the context of rate of change, difference
tables might make a contribution in trying to generate an
overall understanding. Differences would certainly be
easier to use in rate of change if it was already a familiar idea
to the pupils.
The diagram represents a square growing at a constant rate
of increase to the length of its sides. For every second the
Linear graphs and differences
length of the side increases by 1 unit.
The rather more routine question below, although basically a familiar graphical one, also incorporated differences.
Water isflowing into a tank at a constant rate, such that
Y'
for each unit increase in the time the depth of water
increases by 2 units. The table and graph illustrate this
6
5
situation.
4
Time (x) 0 1 2 3 4 5
3
2
Depth (y) 0 2 4 6 8 10
1
O
1st difference (depth) 2 2 2 2 2
1 23456
Fig. 1
y
10
We can tabulate increases in length (x) and
9
area (A) in this way:
8
Time
(seconds) 0 1 2 3 4 5
7
L.
Length (x) 0 1 2 3 4 5
6
5
Area (A) 0 1 4 9 16 25
cl
1st difference 1 3 5 7 9 Differences
2nd difference 2 2 2 2 in area
4
3
2
(i) Is the area changing at a constant rate?
1
(ii) Is the 1st difference in area changing at a constant
rate?
0
1
(iii) Is the 2nd difference in area changing at a constant
rate?
indicated that what is meant by "constant rate" was not well
understood in this context. There appeared to be confusion
between the existence of a number pattern and the presence
of a constant rate, thus 1, 3, 5, 7, 9, ... implied constant rate.
For whatever reason, more than half of the school pupils
stated "yes" to (i). Many more pupils failed to answer this
question than a corresponding question based on information supported by a graph. In part (iii) many pupils who
were able to state that the rate was zero went on to draw the
conclusion that it was not constant.
It is difficult to believe that the errors made by students in
answering these questions were entirely due to their presentation in difference table form. However, it is possible
that the emphasis on differences may have contributed to
the confusion. The curriculum question which arises is
Mathematics
3
4
5
x
Time
Fig. 2
When discussing this question, the origin of the differences was clearly explained to the pupils. It must be
admitted, however, that this may have been the first time
that many pupils had seen such information presented in
this way. Certainly, the basic underlying context of a
growing square should have been very familiar. Results
24
2
in
(i) How is the constant rate of flow reflected in the
differences?
(ii) How is this constant rate reflected in the graph?
(iii) What is the equation connecting x and y?
(iv) For a small increase in the time, h, the depth
increases by a corresponding small amount, k. What
is the relationship between h and k?
(v) What is the rate of increase in the depth when
x = 212 When x = T?
The meaning and implication of the first sentence did not
appear to have been grasped by the pupils. Even after good
responses to parts (i) and (ii), many pupils were unable to
cope with part (v). At x=21, a significant minority of
pupils responded with the y-value, 5, and not with the rate.
Even more pupils subsequently responded witl 2 T. Res-
ponses to a similar linear graph defined by an equation were
rather worse, one-third of the school pupils attempting to
give the y-value instead of the rate. It is possible that such
responses were from pupils who were working out the only
thing they could think of at the time. It is possible they were
School,
November
1984
side-tracked by being asked about equations in parts (iii)
and (iv), though if that is what happened here, then it must
happen frequently in solving mathematical problems. The
y
50
lesson was certainly clear, namely that sixth form Advanced
level mathematics students were not at all happy with rate
of change in the context of linear graphs. I should record, in
passing, that parts (iii) and (iv) produced some interesting
misconceptions too, but being algebraic in nature discussion of them in this article is not appropriate.
Graphical studies are obviously important in the teaching
of rate of change, but will not necessarily solve conceptual
difficulties. Problems pupils have in coping with graphs
have already been documented (see, for example, Kerslake
in Hart, 1981). Straight line graphs and their gradients are
always included in the curriculum for many pupils. It may
be very important that "real-life" situations are used to
provide data, for two reasons. Firstly, algebraic graphs are
very difficult for pupils in comparison with graphs built up
40
30
20
10
from tables of numbers. Secondly, "real-life" situations
ought to provide a basis for meaningful discussion with
pupils, and discussion appears to be vital in order to sort out
ideas in pupils' minds. In addition, we ought to seriously
consider using the terminology "rate of change" and
"rate", as much as possible and wherever appropriate, if we
hope that a broad understanding of the concepts will be
achieved.
0
1
2
3
4
Another major point which we expect many pupils to
grasp in a graphical study of rate of change concerns the
distinction between straight lines and curves. For a curve an
average rate of change may be calculated as for linear
graphs, though it seems likely that the idea of average rate of
Fig. 3
problem might be deeper than we imagine, given responses
to the following question.
The diagram shows a circle and a fixed point P on the
circle. Lines PQ are drawn from P to points Q on the
circle and are extended in both directions. Such lines
across a circle are called secants, and some examples are
shown in the diagram. As Q gets closer and closer to P what
happens to the secant?
change is difficult for pupils to grasp. In addition, for
curves, there is also the idea of rate of change at a point on
the curve, with the likelihood that for every point on the
curve the rate of change will have a different value. One
difficulty for us, when teaching about rate of change at a
point and average rate of change, is that the distinction is
blurred by first studying straight lines. What we study first
very often fixes ideas in pupils' minds. If we try to draw a
distinction between average rate and rate at a point with
01
Q2
Q3
Q4
linear graphs we may lead pupils to believe there is no
distinction for graphs in general. The following task in the
study was one of the ones based on a curve.
A ball is rolled from rest down a hill. The distance
travelled from the top (y) and the time (x) are found to
be related in the way shown by the table of values below:
x
0
0
2
1
8
2
3
18
x
The tangent as the limit
Curves and differences
y
5
4
32
5
50
1st diff. (y) 2 6 10 14 18
P
Fig. 4
The idea of the rotating secant was meant to relate to the
approach to differentiation often used and so was considered to be an important task in giving further evidence
concerning level of understanding of the tangent as the limit
as Q approaches P. It seems very significant that over 40 per
cent of all those interviewed were unable to state that the
2nd diff. 4 4 4 4
secant eventually became a tangent, despite considerable
encouragement, through further questioning, to say more
about what happened, until they ran out of things to say
The values of (x, y) are graphed in Figure 3.
about the situation. There appeared to be considerable
(i) Is the rate of change of y constant?
(ii) Is itpossible to measure the rate at whichy is changing
at x= 2-? at x = X? If so, how?
Part (i) is similar to the earlier task concerning the
growing square, and there were many incorrect responses.
Hardly any pupils were able to answer part (ii), though it
seems extremely likely that they would have encountered all
of the ideas involved, namely, joining up points on a curve,
drawing a tangent at a point on a curve and calculating the
confusion in that the secant was ignored by many students;
they appeared to focus only on the chord PQ, despite the
fact that the diagram and explanation were intended to try
to ensure that this did not happen. Typical responses
included, "The line gets shorter", "It becomes a point",
"The area gets smaller", and, "It disappears".
This situation appears to be one where pupils need more
help than might have been expected. It might ease the
problem to refer to the secant RS, say, where R and S are
end-points of the secant, rather than refer to the secant by
gradient of a tangent. The results certainly suggested a very
using P and Q, the end-points of the chord. It is clearly an
limited grasp of rate of change in these circumstances. The
important, though small, point and I think it is helpful to be
Mathematics
in
School,
November
1984
25
aware of the potential confusion inherent in the task. It
level of understanding and in view of the importance of rate
seems to me that a number of textbooks run straight into the
trap here by using chords PQ and not even using the secant.
of change in mathematics, they suggest that we should
experiment with any teaching ideas which we think might
help. Some suggestions are included in this article. It might
be very important to try to incorporate discussion of
appropriate situations within the teaching programme and
A simple computer demonstration of the rotating secant
might be a very helpful teaching aid.
Average rate of change from graphs
the tasks of this article might provide very suitable vehicles
for discussion of rate of change.
The final task I wish to include was originally considered
to be routine since it only tested the use of the ratio change
in y/change in x.
The graph ofy for a certain equation, for x = 0 to x = 6, is
shown below
References
1. Hart, K. M. (Ed.) (1981), Children's Understanding of Mathematics:
11-16, John Murray.
2. Orton, A. (1980), A Cross-Sectional Study of the Understanding of
Elementary Calculus in Adolescents and Young Adults, Ph.D. thesis,
University of Leeds.
3. Sawyer, W. W. (1943), Mathematician's Delight, Penguin Books.
Y
8
6
C
J---
5
A
World Studies and Mathematics
4
For the academic year 1984-85, I have been seconded to the
3
D
World Studies Teacher Training Centre at York University for
the purpose of developing links between the areas of World
Studies and Mathematics, including the use of computers. In
particular I wish to develop materials for the teaching of
statistics using research data relating to social and economic
factors on a worldwide basis including military expenditure.
The project will include the production of materials from a
multicultural perspective and also an evaluation of textbooks
and computer software for sexist, ethnocentric and aggressive
bias. In addition I wish to include a bibliography of any
materials currently available of a related nature.
H
2
1
E
o
1
2
3'\
4
G
/Is
6
~
x
F
I would be pleased to hear from anyone who would be
interested in this project and particularly interested in details of
any noteworthy practice in this field.
Fig. 5
Brian Hindson,
What is the average rate of change of y with respect to x,
World Studies Centre, University of York
(i) From A to B?
(ii) From B to E?
(iii) From A to J?
Tutorial Software Ltd.
A surprising number of pupils found this question
difficult. Over one-third of those interviewed could not
obtain the correct answer to (i), and the success rate
declined through subsequent parts. In (ii) one-quarter of
the students omitted the negative sign; others calculated an
incorrect ratio by dividing the correct y-increment by the x-
coordinate of E. Nearly one-half of the students were
thrown into complete confusion by (iii). A larger number
gave no answer at all, but a smaller group consisted of those
who gave an answer involving 6 and 5, the coordinates of J
being (6, 5), though it is surprising how many different ways
6 and 5 can be combined to form an answer!
It seemed that this task involved a situation which was
very useful for discussion purposes. Talking about places
on a curve where the function is increasing, where it is
decreasing and where it is increasing or decreasing most
rapidly may be very important in the context of rate of
change. At the, same time we need to bring in numerical
measures of the gradient, where it is positive and where it is
negative, where it is numerically large or small. The ideas of
SENIOR SCHOOL
EDUCATIONAL PROGRAMS
Each of our program packs is designed to help you to teach
by running a simulation which uses the ful l sound and graphics capabilities
of the computer to capture the pupil's interest. The simulations are based
upon the real laws and success both demands and teaches a clear under-
standing of the processes.
Each program pack is now housed in a sturdy, hard backed binder, and the
menu driven programs require no previous computing experience. Each is
supplied on cassette for the most popular school computer, the BBC B, and
the detailed teaching notes include instructions for transferring to disc if
required. The following are available now:
MATHS 1:
TRY-ANGLES
Draughts style teaches angles
ratios, tan, sin, cos. 25 levels
MATHS 2:
COORDINATES
Battleship style teaches x and y in
four sectors, directed numbers
stationary points, turning points, maximum and minimum
points need to be introduced whilst the opportunity is there
and long before any formal algebra/calculus treatment of
the issues. In so doing, however, the results to the task
above give a clear indication of where pupils are likely to
make errors and where they might need particular help.
Whatever ideas we might have to improve pupils' under-
standing of rate of change through amended teaching
approaches we cannot be sure they will be effective. The
facts of this article consist of the responses of pupils to
particular questions. These responses indicate a very low
26
Mathematics
in
Now available for BBC B and ELECTRON. Each pack
contains main program, extra sell test program and Core
Facts book for only M11.95 or any two for E19.95.
HOW TO PLACE YOUR ORDER
Simply senda listof your requirements, with an official order
if necessary and a cheque if possible (full receipts are supplied) to:
TUTORIAL SOFTWARE LTD., FREEPOST, WIRRAL, MERSEYSIDE L81 1AB.
School,
Please state BBC B or ELECTRON
November
1984