PREPARING
TO TEACH RATIO
Prepared for the course team by Pete Griffin
Centre
for
Mathematics
Education
Project MATHEMATICS UPDATE Course Team
Gaynor Arrowsmith, Project Officer, Open University
Lynne Burrell, Academic Editor, Open University
Leone Burton, External Assessor, Thames Polytechnic
Joy Davis, Liaison Adviser, Open University
Peter Gates, Author, Open University
Pete Grifin, Author, Open University
Nick James, Liaison Adviser, Open University
Barbara Jaworski, Author, Open University
John Mason, Author and Project Leader, Open University
,
Acknowledgments
Project MATHEMATICS UPDATE was funded by a grant from the Department of Education and
Science. We a r e most grateful for comments from Ruth Eagle, Arthur Hanley, Michelle
Selinger, Eileen Billington, Gillian Hatch, Eric Love, Gill Close, and many others who may
not have realised a t the time that they were working on parts of this pack.
The Open University, Walton Hall, Milton Keynes MK7 6 M
First published 1988.
Copyright O 1988 The Open University.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted, in any form or by any means, without written permission from the publisher.
Printed in Great Britain by The Open University.
Further information on this and other Open University courses may be obtained from the Learning
Materials Services Office, Centre for Continuing Education, The Open University, PO Box 188, Milton
Keynes MK7 6DH.
ISBN 0 335 15921 4
1.2
CONTENTS
0
INTRODUCTION
AI MS
WAYS OF WORKING
1
With this pack; With your colleagues
PREPARING YOURSELF
Focus on language patterns; Focus on standard
misconceptions; Focus on contexts; Focus on imagery;
Methods and techniques; Divine proportion (a historical note);
Looking back
2
3
INTERLUDE: A FRAMEWORK FOR PREPARING TO TEACH
PREPARING YOUR LESSONS
Using your textbook or scheme; Time to reflect; Augmenting;
Using ratio in context; Ratio across the curriculum;
Looking back
4
SOME RESOURCES
5
6
THE LAST WORD
BIBLIOGRAPHY
0 INTRODUCTION
Why is it that pupils can often solve ratio problems set in one
context but have difficulty when the context is changed?
Have you ever noticed that the idea of ratio runs through
many topics in the mathematics curriculum and is found in
many different guises in textbooks?
Problems involving ratio have exercised the minds of mathematicians throughout the ages,
and ratio has always been regarded a s a core area in mathematics. Indeed, not so long ago,
most of the arithmetic that pupils experienced in school involved ratio, and many adults would
recognise the type of problems that follow a s the ones t h a t typified for them what school
mathematics was all about.
Tanya earns £2.60 per hour. How much will she earn for 40
hours' work?
If i t takes a man 12 days to reap a field, how long will it take 6
men?
If 17 yds cost £6 12s G d , what will 1 1 0 ~ ~cost?
ds
Problems of this sort are, by and large, not seen in mathematics classrooms today. However,
ratio continues to have a high priority in school mathematics; few areas of mathematics which
are now recognised by teachers a s standard topics on the syllabus have attracted as much
attention and research.
Have you ever felt that textbooks, schemes or workcards can
be limiting when preparing lessons?
What else can teachers use to prepare a series of lessons on
ratio?
0
Do you often feel that you want to develop your own classroom
activities but don't know where to start?
Do you sometimes feel the need for resources aimed
specifically a t preparing yourself better to teach a topic?
This pack should help you to come to an appreciation of the notion of ratio and to the challenges
of teaching it a t the secondary level. And, along with work on other PREPARING TO TEACH
packs in the MATHEMATICS UPDATE series, it should also help you to feel confident about what
is needed in preparing to teach any topic in secondary mathematics.
The pack does not provide ready-made classroom activities (although many of the activities
can be adapted for use in the classroom); nor does i t provide a model for how to teach ratio.
Rather, it sets out to help you to develop a framework with which to inform your thinking about
ratio and to prepare your lessons on this topic.
How you approach your teaching will obviously be affected by your own preferences and by the
particular textbook or scheme, if any, that you use. The framework outlined in this pack is
appropriate to any scheme, whether based on whole-class textbooks, individualised workcards
or booklets.
AIMS
To provide a framework within which to plan and prepare to
teach the topic of ratio.
To provide activities which illustrate the essential elements
of teaching ratio.
To help teachers in assisting their pupils to develop a better
understanding of the nature of ratio.
To develop creative ways of using textbooks or schemes.
To encourage and promote ways of working with colleagues
which can also be used in the classroom with pupils.
WAYS OF WORKING
WITH THIS .PACK
,
This pack contains two main sections: Section 1 PREPARING YOURSELF and Section 3
PREPARING YOUR LESSONS. The INTERLUDE (Section 2) should be read after you have worked
through Section 1 , and Section 4 (SOME RESOURCES) is intended to be dipped into whenever
appropriate. Section 5 THE LAST WORD is intended to be exactly that!
The two main sections are divided into subsections which invite you to enter into many
activities, either on your own, with colleagues or with pupils. Each of these activities should be
valuable, but you may find that there is insufficient time to try them all, in which case you
should aim to tackle a t least one activity from each subsection. There are a t least two ways of
becoming involved in an activity. You could work on the activity, that is, actually d o the
mathematics. Alternatively, you could ask yourself what it is about the activity which
provides insight into the idea of ratio.
You may find that doing the mathematics does not stretch you very much (depending on your
past experience) and that, in order to save time, you can imagine doing an activity before
moving on. However, it is worth actually getting involved a s the very act of doing leads to
insights which are important. On the other hand, there is always a danger when working
through a pack of this sort to become engulfed in the doing (of activities) and to miss the
opportunities for reflecting (on the activities). Such reflecting provides the sorts of vivid
experiences which are valuable when preparing to teach ratio (or indeed any topic).
Concentrating on the doing is all too easy whilst in the classroom a s well. Teaching is hectic
and often lessons go by without reflection upon questions such as the following.
What activities resulted in rich learning situations and what was it about them that
promoted such situations?
What were the questions that seemed to prompt fruitful discussion?
Were there any comments t h a t pupils made which highlighted an area of
uncertainty (or seemed to provide insight for others)?
Such experiences, which can prove to be extremely valuable, are easily lost. It is also difficult
to re-capture significant moments in order to use them in the process of preparing future
lessons.
For this reason, some activities are specifically labelled reflecting. They are intended to help
you pause, step back and assess the meaning of what you are doing. Of course, there may be
many other such instances and you should try to build these pauses into your way of working
wherever possible.
The pack also contains in the classroom activities which are intended to provide you with the
opportunity to try out some ideas or activities in your own classroom. You may wish to engage
all of the pupils in a particular group, only a handful of them or even some from one group and
some from another. Bear in mind that the purpose of all these activities is not to judge them as
good or bad (although such judgements may arise) but to focus on important aspects of
teaching.
The use of a notebook, in a fairly disciplined and systematic way, can help you focus on
significant moments in your own studies with this pack. The notebook can be used for two
purposes.
It can be used for accounts of moments and insights which may arise:
as a result of discussions or activities within your group;
a s a result of working on some part of the pack on your own a t home;
during your coffee break after a particular lesson or even during a lesson.
The six headings t h a t are used throughout the pack (language patterns, imagery, s t a n d a r d
misconceptions, root questions, contexts and standard techniques) and which are referred to
in detail in the INTERLUDE and on the BOOKMARK: Preparing to teach a topic could be used a s
headings for this purpose.
The benefits of k e e p i ~ gsuch a record are a t least twofold. Firstly, the existence of the notebook
and the will to use i t means that moments and insights a r e captured. Secondly, once written
down they become available for consideration and reflection either on your own or within your
group.
The other way of using a notebook is for more detailed ideas and activities which you feel you
would like to use in the classroom when you next have to teach ratio. These could include:
starting points for discussion during a lesson;
whule class, group or individual activities you can use in the classroom;
areas for investigation within the topic.
You may like to use the beginning of the notebook for your account of moments and insights
and the end for ideas and activities.
r
. .... .
I
Ideas / Activities
Standard techniques
Contexts
Root questions
I
I
1
I
Standard misconceptions
lmcgery
Language patterns
-
-
-
-
[NI You will find a notebook symbol [NI in places where you may have something useful to record
(sometimes this could be a note under one of the six headings and sometimes i t could be an idea
or activity t h a t you intend to use). These symbols serve a s a reminder that the notebook is
available to you so that you don't let something you have noticed pass you by.
Remember though t h a t your notebook is for things t h a t you notice and t h a t you think are
important. You may feel t h a t you do not need to record something every time you see a
notebook symbol (although i t might be worth asking yourself why); alternatively you may
want to make use of your notebook when there is no notebook symbol - it is important not to
restrict yourself to making notes only when the symbol occurs.
WITH YOUR COLLEAGUES
The nature of some activities requires collaboration with others, jointly exploring issues and
experimenting with ideas, both those of a mathematical nature and those concerned with
teaching. When working in a group with your tutor, you can use your colleagues to help you
work out the.meaning of activities, and you can assist them to do the same, provided you work
in a conjecturing atmosphere: a supportive atmosphere which involves listening to a n d
accepting what others say as a conjecture which is intended to be modified. (The TUTOR PACK
provides a fuller discussion of what is meant by a conjecturing atmosphere.) Also, don't
underestimate the value of talking to yourself - out loud if possible - when working alone.
Such 'expressing' can help you clarify for yourself what i s only vague and fuzzy in your
mind, and can help you recall what you have learned, even when no one else is present to
listen and encourage.
Finally, as with all packs in the MATHEMATICS UPDATE series, part of your time and attention
will be needed to explore the relevance of your work on this pack to your own classroom
practice, with subsequent reflection upon what happens when you try to use or adapt activities
and new ways of working with your pupils. There are a number of opportunities throughout the
pack to share and discuss classroom experiences with your colleagues. It is tempting to use
such discussions to convey only the successful moments and to ignore t h e times when things
don't go right (or to write them off a s unsuccessful lessons), resulting in a series of 'these
worked and these didn't' sessions. However, the purpose of these opportunities for discussion
is to provide a much broader exchange of views and comments than merely t h a t of swapping
anecdotes (although this i s useful).
All classroom experiences, when shared in the spirit of a conjecturing atmosphere, can
provide the possibility of focussing, a s a group, on the important issues in the teaching and
learning of mathematics. Being prepared to talk about your teaching and to listen to others in
a supportive way can help to air these issues.
Detailed tutor notes on how to run a series of meetings based on any of the MATHEMATICS UPDATE packs
are provided in the TUTOR PACK. Brief notes are also included in PM751 EXPRESSING GENERALITY.
1 PREPARING YOURSELF
Have you ever looked through examination questions and observed t h a t a number of them
seem to be testing the same idea? Here are some questions taken from the Southern Examining
Group specimen papers (1986).
The pie chart given below represents the number of drinks sold a t a school fite. A
total of 720 drinks were sold. How many drinks of tea were sold?
A frozen chicken weighing 8 Ib cost £3.76.
chicken.
Calculate the price per lb for frozen
Mrs. Robb insures her jewellery for £1600. The premium is 25p per £100 of cover.
What is her total premium?
A class of 30 pupils was asked 'How did you come to school?'. A list of results is
shown in the table below. What percentage of the class came by bus?
How did you come to school?
Bus
Car
Cycle
Walk
Number of pupils
15
3
8
4
Two men win f 50 000 for f l stake. Bill put up 60p of the stake and Ben put up the other
40p. They agreed to split their £50 000 winnings in the same ratio a s the amount they
paid. How much will Bill get?
SAME AND DIFFERENT Discuss these examination questions with some colleagues (ybu may
actually like to do them). What is the same and what i s different about them?
b
All of these questions seem to require an act of comparison: seeing a link between two
quantities and applying that link in order to determine a missing quantity. However, the type of comparison
that is necessary does vary from question to question.
Comments
Ratio is one method of comparing one thing with another and it h a s been said that a s many as
two-thirds of the questions which appear on examination papers at secondary school level are
based on this idea. For example, all of the following topics involve ratio in one way or another:
changing between different units of measurement; time and speed; exchange rates; recipes;
graphs of straight lines through the origin. Thus, many mathematical topics are essentially
contexts for dealing with ratio questions. Yet pupils have great difficulty with ratio. There
have been numerous studies which highlight t h e many a n d various errors a n d
misunderstandings associated with this topic.
This section deals with trying to get a sense of ratio, focussing on the difficulties for pupils and
the problems involved in preparing to teach t h e topic. There a r e a number of activities
intended to help you clarify your own lhoughts about ratio. However, most of these activities
can also provide the basis for pupils' work, a n d it is hoped t h a t you will use them in the
classroom in addition to working through them yourself.
FOCUS ON LANGUAGE PAllERNS
As h a s already been said, t h e essence of ratio can be thought of as m a k i n g a comparison
between two quantities. Pupils obviously have experience of comparing, b u t w h a t words do they
use? In other words, what a r e t h e accepted language patterns of comparison?
F
LANGUAGE Look back at t h e questions i n S A M E A N D DIFFERENT. In each case, explain to a
colleague w h a t s o r t of comparison you m a d e (write i t down if i t helps). Concentrate on t h e
language you used. Make a note of a n y words or phrases which help you to verbalise your
comparisons.
Comments Some of the questions seem to require a comparison of the two quantities within one situation
after which the same comparison is applied to the other situation, whereas some invite comparison between
one quantity in one situation and the corresponding quantity in the other situation. For example, comparison
within the situation in the first question leads to:
1st situation
2nd situation
120"
? drinks
X >360'
X
3
720 drinks.
Comparison between situations in the second question leads to:
1st situation
8 lb cost £3.76
2nd situation
1lb costs M.
Research h a s suggested for some time t h a t within a n d between comparisons embody different
ways of seeing ratio. An awareness of t h e language p a t t e r n s used to describe comparisons
may therefore help in preparing to teach t h e topic.
) NEWSPAPER Pick u p a newspaper or magazine a n d look for t h e language patterns of
comparison. T r y questioning your own experience: w h a t words a n d phrases associated with
comparison a r e p a r t of your lan p a g e ?
Devise a similar activity for your pupils t h a t highlights their language of comparison. Do this
in your classroom as soon a s possible.
b
It is likely that you came up with phrases like the following: 'voting was almost 5 to l';'three
for the price of two'; 'two free mugs with every litre of oil'; '8 out of every 10 owners said their cats preferred
it'.
Comments
Children are fascinated by comparisons and their everyday language reflects and reinforces this. For
example, they say: 'your cake is bigger than mine'; 'your apple is redder than mine'; 'your car is faster than
ours'; 'my Dad is bigger than yours'.
T h e words t h a t a r e used when talking about comparing one t h i n g with another inevitably
shape a n d a r e shaped by ways of t h i n k h g about comparison. Teachers m a y therefore have
ways of talking about comparison which assume some ways of comparing b u t not others.
F
TWO FOR ME, THREE FOR YOU Sonia i s sharing o u t some sweets with Marcia. S h e says,
'Two for you, three for me, two for you, three for me . . .'. S h e carries on like this until all the
sweets have gone. A third person looks on and, when the sharing is complete, says 'So Marcia
h a s got of t h e sweetsr. What does this tell you about, ratio? How could you use this a s s basis
for a n activity in t h e classi,oom'?
b
The confusion here lies in the difference between the ratio 2:3 which is comparing part with
2
part and the fraction g which is comparing part with whole. In terms of language patterns, it is evidence of
the difference between '2 for every 3' and '2 out of every 3'. This is a good example of how an awareness of
language patterns can help in coming to grips with ratio.
Comments
Ratio a n d fractions are often seen as different topics on the mathematics syllabus but the link
between them i s very strong. For example, when a quantity i s divided into two parts in the
ratio 2:3 or 3:5 or a:b, i t seems natural to ask what fraction each part is of the whole. So the
language patterns of ratio overlap with t h e language patterns of fractions. Similar links with
other topics on the syllabus mean t h a t the language patterns of ratio are particularly rich.
F
FINDING RATIO What other a r e a s of t h e mathematics syllabus involve t h e ideas of ratio?
What a r e the benefits of asking this question in the classroom?
Possibilities include: percentages, enlargements, map scales, speed, acceleration,
trigonometry, scale drawing, proportion, similarity.
Comments
Teachers often complain t h a t pupils fail to see the connections between situations involving
similar ideas, yet t h e way in which t h e syllabus i s partitioned often encourages t h i s
compartmentalised view of mathematics. Pupils need to be actively encouraged to see the
subject in a more holistic way.
For example, a ratio i s by definition without dimension, b u t an important idea which h a s
strong structural similarities with ratio, and therefore one which may give some insight into
the nature of ratio, is t h a t of rate. Strictly speaking, a ratio arises from t h e comparison (by
division) of two like quantities, while a r a t e results from comparing (again by division) two
unlike quantities such a s in 20 miles per gallon or 30 pence per pound.
Quite often a rate will encapsulate a quality or concept which is fundamental. Such rates then
become what might be called 'named ratios'; for example, speed i s miles per hour and pressure
is pounds per square inch. Yet another class of ratio arises from comparison of one unit with
another; for example, 2.54 centimetres per inch or 2.2 pounds per kilogram.
) NAMED RATIOS What other named ratios can you think of t h a t are in common use?
F
Some you might have written down are: fuel consumption (miles per gallon); crop yield
(volume or weight of crop per unit area); 0.22 gallons per litre; acceleration as change in speed per unit time.
Comments
RATE/RATIO When looking a t the above rates and named ratios, how easy is i t to get a sense
of multiplicative comparison? What is the same and what is different about comparing two
unlike quantities (as in rates) and comparing two like quantities (as in ratio)?
Comments By looking at these different forms of comparison, you may gain an insight into the essence
of ratio. For instance, you may feel that the way in which two quantities change yet go hand-in-hand with
each other is more evident in a rate. Having noticed this, you may look at ratio in a different light and with
more understanding.
Notice t h e prevalent use of the word 'per' in the examples quoted so far. The word 'rate' is
itself a t the heart of many language patterns which carry certain meanings. Rate i s often
associated with time but clearly this is not the essence of rate in many cases. It may therefore
be worthwhile examining what defines the link between the two quantities t h a t are being
compared in a named ratio. If time, speed or rate of change is not the quality t h a t runs through
all the comparisons, what might the link be?
SEEING RATIO
Consider the named ratio 'population density'; t h a t is, the number of people (or
animals) per unit area. Write down some contexts or situations in which appropriate units of
measurement might be:
animals per square metre;
animals per square centimetre;
animals per square millimetre.
What images do you use to help you do this activity? Do you have a picture of the animals or the
actual area?
P
Comments Many people, when doing this exercise, find themselves conjuring up pictures in their head.
These pictures or images vary quite considerably but seem to involve imagining say a square metre and then
asking themselves what sort of animal would fit several times (or even several hundred times) in that space.
Alternatively, people report considering different sizes of animals, and then trying to fit them into a square
metre if it seems reasonable.
The root idea here seems to be one of density: the number of animals which can be fitted into a
certain space. This is the quality t h a t remains invariant (or does not vary). How applicable is
this idea of density to other ratios and rates?
Look back at the list of rates that you made in NAMED RATIOS (you may feel t h a t you
can add some more now). How useful i s the idea of density in describing t h e link between the
two numbers? Do you have to broaden your definition of density in order to use it in this way?
DENSITY
Compare your findings with those of your colleagues.
Comments The idea of density or 'packedness' could be a way of describing comparison by division. For
example, when you divide 100 by 4 you are discovering that 4 fits into 100 twenty-five times and this could
be thought of as a measure of density. The density of 'twenty-five per unit' is the invariant feature for all
other pairs of numbers with the same ratio; for example, 50 and 2,500 and 20,12 and 0.48, etc.
REFLECTING What a r e the benefits of exploring all the language patterns associated with ratio
[NI a s a beginning to preparing to teach
the topic? What about encouraging pupils to explore their
existing language patterns? Does this help you to see ratio in a different way? Use your
notebook to record any useful ideas t h a t have arisen here.
FOCUS ON STANDARD MISCONCEPTIONS
b
INCREASING Consider t h e two numbers 4 a n d 8. Find a relationship which exists between
them. Increase each number by one (and then by 2,3,4, etc.) and each time investigate how the
relationship changes. Now multiply each number by 2 a n d then by 3, then 4, etc. a n d
investigate how t h e relationship changes. Now find a different relationship which exists
between the numbers and repeat the process.
Comments Children's early number sense is usually focussed on addition and difference; only later are
multiplication and division introduced. So addition becomes established as the primary arithmetic operation
and multiplication as secondary. Now this may seem logical because, with whole numbers, multiplication is
seen as synonymous with repeated addition, but should this really be so? Certainly, when considering
2 1.
multiplication of fractions, this comparison is not possible; for example, g x g s not repeated addition.
By developing multiplication solely a s repeated addition there i s therefore a danger of
stunting pupils' awareness, and pupils' undeveloped sense of comparison by ratio may well be
a result of this. For example, consider the following situation:
If these two rectangles
are the same shape. what
Is the height of the
larger one?
I I
To solve this by saying
3 + 3 + 3 + 3 = 1 2 and 9 + 9 + 9 + 9 = 3 6 ,
so the missing height is 36 cm,
is the result of a perfectly good strategy. However, if the addition strategy is seen a s the only
one available, misconceptions such as those in the following situation could arise:
If these two rectangles
are the same shape, what
is the height of the
larger one?
Using the same strategy as above,
3 + 3 + 3 + 3 + 1 = 1 3 and 9 + 9 + 9 + 9 + 1 = 3 7 ,
so the missing height is 37 cm!
Thus, one of the most common errors in ratio problems involves the use of the addition
strategy, which is based on comparing two quantities by difference.
Here is another example:
Work out how long the missing line should be If the dla~ram
on the right
Is to be the same shape but bigger than the one on the left.
Answer : 6 cm.
Source: Hart (1984)
STRATEGIES Answer t h e preceding question yourself and then spend a few moments with a
colleague discussing how you obtained your answer. Can you come u p with more t h a n one
strategy? What do you think was the strategy t h a t produced the answer above?
In order to obtain the correct solution, you may have focussed on the fact that 3 cm is 1;
1
times 2 cm and that therefore 7 cm must be l ~ t i m ethe
s height. Alternatively, since 7 cm is 2 ;times 3 cm the
height must be 2 fg times 2 cm. Or you might have used another strategy .
comments
The error seems to have come either from seeing the 7 cm side as 4 cm more than the 3 cm side or from
seeing 3 cm as 1 cm more than 2 cm.
There i s also evidence to suggest that, in certain situations, a strategy of successive doubling
(or possibly halving) may be used to compare two quantities. Suppose the last question was
changed to this:
Work out how long the upright must be to give a triangle the same shape
a s the small one, but bigger,on the base 7 cm long.
Answer : 5 cm.
Source: Hart (1984)
The diagram h a s been changed to one involving a triangle so t h a t pupils can check whether the
enlarged version i s the same shape or not by looking at the slope of the extra line, t h u s drawing
on their sense of similarity and proportion. A strategy which gives a n answer of 5 cm results
in a triangle roughly the same shape. Here the 3 cm h a s been doubled and then 1 added to give
7 cm and this perceived connection of 'double i t and add 1' h a s been applied to the 2 cm line to
give 5 cm. This new strategy, whilst not correct, i s not completely additive; i t h a s a n element
of multiplicativity in it.
The essence of ratio i s multiplicative and so i t is necessary to evoke in pupils a shift towards
multiplicative comparison. The last two examples demonstrate how pupils may be helped to
make a shift towards such a strategy. However, to see the connection between 3 cm and 7 cm a s
1
' 2 times
~
a s big' requires not only a n acceptance of finding a multiplicative comparison but
also a willingness to accept multiplying by a fraction.
D
IN THE C L A S S R O O M Use the two examples above with some of your pupils. What questions
would you ask? Would you modify the examples in any way? What i s the benefit of giving
such a n activity to pupils?
Comments There are inevitably many, varied strategies that pupils employ, the exact type depending on
the context and the numbers involved in the question. However, if pupils can be encouraged to reflect upon
the range of strategies that can be employed to give correct answers, the likelihood of getting trapped in one
type of strategy all the time is reduced.
Being aware of standard misconceptions (such a s acceptance only of a n addition strategy) is
very valuable in t h a t i t makes i t possible to offer valuable comments a n d t o respond to
classroom situations with stimulating questions. Misconceptions or, possibly more correctly,
conceptions t h a t may lead to errors, can often provide the opportunity for rich learning
experiences. They provide the motivation for devising activities which might bring pupils up
against surprises or situations t h a t seem to contradict their intuitions. Such questions and
activities (known as probes) can h e l p i n revealing incorrect s t r a t e g i e s a n d c a n offer t h e
opportunity for pupils to analyse t h e i r own senses of ratio.
F
PROBES T h e following extract relates t o two pupils engaged in t h e ~ r o b l e r nof writing a LOGO
program t o d r a w a rectangle.
They had succeeded in writing a program that would draw a rectangle of size 50
by 60. They then began to tackle the problem of modifying it so that it would
draw enlargements or reductions of this original rectangle. T o d o this they
decided to replace the 50 and the 60 by variables (they used words to stand for the .
numbers).
Having replaced the 60 by 'Jack', what should be done about the 502 Subtraction
was the obvious first try, so they replaced the 50 by 'Jack - 10' and then ran the
program a number of times putting in different values for 'Jack'. They were
disappointed and confused when they obtained a number of peculiar looking
figures which were obviously not scalings of the original rectangle.
Imagine t h a t you a r e i n t h e classroom while t h i s investigation i s t a k i n g place. You a r e aware
of t h e pupils' confusion a n d you a r e right n e x t to t h e m when t h i s look of confusion shows on
their faces. What, as a teacher, can you offer? Write down some questions you might ask.
Comments On the surface, this is evidence of the classic addition strategy. The move from 60 to 50 is to
subtract 10; in other words, when the 60 and the 50 are compared, it is the difference that is perceived first.
Instead of treating this as an error to be corrected, it might be useful to use it as a starting point for a
discussion, possibly along the following lines.
'How many 50s in 60?
Possible response: 'One and 10 left over'.
'So how many 50s do you need to make 60: less than one, exactly one or more than one?'
Possible response: 'More than one'.
'Can you tell me exactly how many 50s you will need?'
Possible response: 'Um, I'm not sure, it's more than one but it's not two because two lots of 50 would
be too much'.
'Would you like to guess a number and try it out? (A calculator is offered.)
Possible response: . . .
CALCULATOR E n t e r a number into your calculator. Multiply t h i s by a n o t h e r number. Keep
on multiplying b y t h i s s a m e n u m b e r over a n d over a g a i n a n d notice how quickly your
original n u m b e r increases. (Most calculators h a v e a 'constant' facility which enables you to
operate repeatedly on a number using just one key pressed over a n d over again.) W h a t about
m a k i n g your n u m b e r smaller by multiplying by something different? How quickly do your
n u m b e r s increase or decrease a n d can you m a k e them do so less quickly?
When doing this activity with pupils w h a t questions would you ask?
There is a tendency for some children to see enlarging as doubling and reducing as halving,
so that to 'double up' can be seen as synonymous with multiplying or even adding. This exercise could
therefore provide the basis for exploring multiplicative ways of increasing and decreasing which do not rely
on doubling and halving.
Comments
F
IDENTIFYING MISCONCEPTIONS Look at t h e tables bn pages 16-17. They show responses of 11a n d 15-year-olds to some t e s t i t e m s on ratio i n a survey carried o u t by t h e Assessment of
Performance U n i t (APU) between 1978 a n d 1982. T r y to identify for yourself t h e reasoning
behind each response (the asterisked responses a r e t h e correct responses). W h a t probes might
you u s e i n such situations?
Item P 5
Item P4
On t h e photograph J o h n ' s h e i g h t i s 5 c m and h l s
m o t h e r ' s i s 8 cm.
On tire p h o t o g r a p h ' i r e v o r ' s n e l g h t i s 5 cc, and h i s
m o t h e r ' s i s 8 an.
' i r e v o r ' s r e a l h e i g h t i s 100 cm.
H i s m o t h e r ' s r e a l h e i g h t i s 160 cm.
What i s J o h n ' s r e a l h e i g h t ?
Ratio
Involved
p4
-5= -
Responses
01:
=
8
160
X
p5
cm
Wnat i s h i s m o t h e r ' s r e a l h e i g h t ?
Possible Method
Age 1 1
---------- 0
Age 15
X
160
8
5
-
------
-5= - 100
8
42%
74%
~dditiveerror
9%
3%
Additive relation xi0
9%
5%
Other
31%
14%
Omit
9%
4%
100
Correct
157
130
160*
Correct
35%
71%
103
Additive error
10%
3%
130
Additive relation x10
6%
4%
300
Additive relation xlOO
20%
4%
31%
14%
8%
4%
X
C' r
Other
Omit
At age 1 1 a number of items which involve the relationships
j2 -- n Xa n d g3 = - X have been devized using different contexts.
24
Age 1 1
I
I
Context: p o i n t s
Ratio
Involved
I t e m P6
1tem p7
I n a game Jane g o t 3 p o i n t s
f o r every 8 p o i n t s t h a t
Judy g o t , and Judy g o t 2 4
p o i n t s . How many d i d Jane
get?
I
I
l
I
I
I
-3= - X
8
24
*
9
19 ( A d d i t i v e e r r o r )
Other
-----
Omit
57%
* 9
48%
1%
19 ( A d d i t i v e e r r o r )
15%
36%
1
Other
31%
1
Omit
6%
6%
1tem p9
I n a game Jane g o t 2 p o i n t s
f o r e v e r y 3 p o i n t s t h a t Judy
g o t , and Judy g o t 12 p o i n t s .
How many d i d J a n e g e t ?
I
I
I
I
I
3
I n t h e time t h a t Zena t a k e s t o
sharpen 3 p e n c i l s , Rachel can
sharpen 8.
When Rachel has sharpened 24
p e n c i l s , how many w i l l Zena have
done?
I
I
Item P8
2 --
Context: sharpening p e n c i l s
I n t h e time t h a t Zena t a k e s t o
sharpen 2 p e n c i l s , Rachel can
sharpen 3.
When Rachel h a s sharpened 12
p e n c i l s how many w i l l Zena have
done?
X
-
12
11 ( A d d i t i v e e r r o r )
Other
Omit
I
11 ( ~ d d i t i v ee r r o r )
38 %
I
Other
24%
I
Omit
3X
I
33%
7%
5%
Source: APU (1978-1982)
You may wish to use some of these test items with your pupils so that you can see how
standard misconceptions occur in your own classroom.
Comments
Items P4 and P5 seem to evoke a strategy which is not entirely additive in that it has an element of
multiplicativity in it. This may be because of the size of some of the numbers (in the 100s). In items P6 - P9
the context in which the question was set had a marked difference on both the number of correct responses
and the number of additive errors.
T h e desire to adopt a n addition strategy i s quite strong i n some pupils unless t h e size of t h e
numbers or t h e context i s arranged in order to prompt a shift towards some acceptance of
multiplication as a plausible alternative. Even then, t h e move to multiplying by a scale factor
can be strongly resisted, especially if t h e required factor i s not a whole number.
I t seems therefore as if there a r e at least three shifts in attention which pupils need to make in
order to appreciate the idea of comparison by a ratio for all types of problems:
the addition strategy must be seen to be appropriate in only a certain number of cases,
where the numbers to be compared are such that one is a whole number multiple of the
other (as in the example on page 13);
the idea of comparison using multiplication (not just successive doubling) must be
seen to be appropriate;
the idea of multiplying by a non-integer must be seen to be sometimes necessary.
F REFLECTING Look back over some or all of the activities in this subsection and reflect on how
[NI they may help your pupils to make the shifts in attention mentioned above.
What do you feel a r e the benefits of being aware of standard misconceptions before teaching a
certain topic? Do your colleagues in the UPDATE group agree with you? What about your
colleagues at school?
P
IN THE CLASSROOM One of the challenges in teaching ratio is to produce activities which
highlight oddities and apparent contradictions. I t is very useful to develop the skill of using
short, puzzling, impossible, paradoxical activities in order to help pupils see the conceptions
they already have about the topic. Here are some questions you might ask. Use them in your
classroom a s soon a s possible. What is their value? How effective a r e they in focussing on the
various conceptions of ratio?
Draw a rectangle. Increase each side by adding on the same amount and draw the
resulting shape. Keep increasing the size of the rectangle in tliis way and each time
draw the result. What do you notice about the shape?
Consider the two numbers 5 and 10. One is twice the other. Perform the same
operation on both of them so t h a t this 'twiceness' is conserved. Investigate this for
other examples where 'thriceness' or 'halfness' or anything else you choose is
con served.
I am three times a s old a s my son. What happens to this relation 'three times a s old'
a s the years go by? When my son's age doubles, what happens to mine?
Make a list of situations in which doubling one thing doubles another.
D
FOCUS ON CONTEXTS
As mentioned at the beginning of this section, ratio appears in many different contexts and
situations. This is often a source of great confusion to pupils because it seems as if there are so
many things to get to grips with all a t once. I t is only when some common features are seen
which link them all together that the ideas begin to seem manageable. You have already seen
how the understanding of ratio, fractions and rates are linked; in this subsection we will look
in a little more detail a t how ratio appears in a number of different situations and contexts.
REFLECTING Pause here for a moment and consider again what i t was t h a t you saw a s the
common link in SAME AND DIFFERENT on p 9. How easy was i t for you to see the questions a s
examples of the same topic? Are you really convinced that they are? What about your
colleagues?
F
P
CONTEXTS Consider the two numbers 3 a n d 9. Write down several ways of comparing them
(9 is three times 3 is but one way). Make a list of t h e different contexts in which you might be
called upon to compare them; for example, a s lengths of lines (3 cm a n d 9 cm), or two
temperatures (3 'C and 9 "C).How does the context affect how you perceive the comparison?
P
TEA-BAGS Do the following activities with a colleague with a view to trying them out with
your pupils. What questions would you ask? What sense of ratio is being brought out?
To make tea for four people I use three tea-bags. Write down some more recipes for
making tea of the same strength but for different numbers of people.
To buy three chocolate bars from the corner shop I need 60p. Write down some more
shopping lists for chocolate bars.
b
Comments By asking questions such as 'How many tea-bags for six people?', 'How many chocolate bars
can I buy for El?, 'What if I used two tea-bags?',you can talk about the different strategies that people use in
such situations. By collecting a number of different recipes and lists you can explore what is the same and
what is different about the two quantities. The quantities are changing but there is a sense in which they are
going hand-in-hand with each other.
P
RATIOIDIFFERENCE As discussed in the previous subsection, the root of many children's
problems with ratio lies in t h e confusion between comparing things with a d i f f e r e n c e
(subtractionladdition) a n d comparing things with a ratio (divisionlmultiplication). Make
two lists: one of quantities which you would naturally compare with a difference and one of
quantities where a ratio would be more appropriate for comparison. What i s i t about the two
lists t h a t distinguishes them?
You probably found that the actual numbers you were writing down were irrelevant and
that it was what the numbers represented that was important, for it is the context that provides evidence for
the nature of comparison. You may like to discuss with colleagues some other contexts in which the idea of
ratio naturally arises.
Comments
P
THREE TO TWO Consider a particular ratio, say 3 to 2 (3:2), and write down a number of
different contexts in which i t could be found.
Some examples are: '3 for me, 2 for you', 'the odds are 3 to 2 on', 'find a point P somewhere
along the line AB so that ARPB is 3:2','to make porridge, use 3 cupfuls of oats for every 2 cupfuls of water',
'half as much again'.
Comments
Sometimes, however, i t is the raw state of the numbers rather than what the numbers represent
t h a t signals one particular comparison a s opposed to another, and this can lead to errors. For
example, in the context
'John is 25, Sheila i s 32',
the relationship
'is 7 years older than'
is evident but
'John is 16, Sheila is 32'
may evoke the relationship
'is twice as old as'
not because the quantities represent ages but because they are 16 and 32. And the example
'2 tea-bags will make tea for 3 people'
may evoke the relationship
'number of people is 1 more than the number of tea-bags',
because the raw state of the numbers suggests this relationship much more strongly than
1
'1 cups per tea-bag'.
In both these cases there is a need to see through the numbers to the context within which they
are set, as i t is in the context that the essence of ratio is to be found.
F
IT TAKES A MAN What sense of comparison is evoked by each of the following examples?
How does the context affect this sense of comparison? What might be the result in each case for
pupils who are not aware of the context?
To make a pot of coffee for 8 people I use 6 tablespoons of coffee and 8 cupfuls of water.
How many tablespoons do I use for 4 people?
It takes 3 minutes to boil 1 egg. How many minutes does it take to boil 2 eggs?
It takes 2 men 1 day to dig a ditch. How many days would it take 4 men to dig a ditch
the same size?
Henry is 8 years old and Henrietta is 6 years old. How old was Henrietta when
Henry was 4?
Henry V111 had 6 wives. How many wives did Henry IV have?
F
Carraher, Carraher and Schliemann (1984)provide another example of how the method of
comparison may be determined purely on the raw state of the numbers, without reference to
what they represent. A child was required to find the size of a building given the size of its
shadow. To allow him to do this he was also given the size of the shadow cast by a pole of a
given size.
The child argued 'the shadow is the square of the size of the pole. For the same
reason the building is the square root of the size of its shadow'.
Pupils will go to great lengths to avoid certain multiplicative comparisons. For instance,
when asked to draw a larger version of a particular rectangle, so that the sides are connected
in the same way, they could produce many different drawings.
F
RECTANGLE What sorts of comparisons could be made between the two sides of the rectangle
below? Draw some larger versions which retain these comparisons.
Comments Depending on what relationship is perceived there are a number of rectangles that could be
drawn; for example,
IN THE CLASSROOM Try RECTANGLE with some pupils. What questions would you ask?
How could such an exercise help foster an appreciation of ratio?
Com ments Questions you might ask include:
When comparing each rectangle with the original one, what is the same and what is different
each time?
Which one of the rectangles is most like the original one? Why?
Such questions may help to focus on the ratio 2:4 and how the two numbere can change while the ratio
doesn't.
Pupils are likely to d r a w on their intuitive sense of proportion or similarity i n order to answer
questions like those above, so i t might prove fruitful t o explore the meanings of these t e r m s a n d
to consider how t h e y relate to ratio.
SIMILAR/PROPORTION W h a t everyday language patterns involve t h e words 'similar' a n d
'proportion'? How do they differ from the more mathematical ways in which t h e s e words a r e
used?
P
Comments The everyday use of the word 'similar' is much less precise than the mathematical one. For
example, 'two pupils are of a similar age or are similar in height' conveys a meaning of 'almost the same'.
Alternatively, we might say that one pupil's solution to a problem is similar to another's and this conveys the
idea that there is some attribute in common without being specific about what the attribute is. However, for
two shapes to be similar in the mathematical sense one must be an enlargement of the other (using this term
also in its mathematical sense); that is, the ratio of the sides is the attribute which the two shapes have in
common.
With proportion also, the everyday meaning is far from clear. We say that 'a shape is not in proportion'
suggesting that, in some way, i t doesn't look right compared to some norm or standard, or we might ask
'what are the proportions of a particular shape?' when what we really want to know are its dimensions. In
rather more precise mathematical language, proportion is used to compare two shapes which are similar, as
in 'are these two rectangles in proportion?'. It is also used to compare the strengths of two mixtures, as in
'they are mixed in the same proportion'. So, as with the word 'similar', mathematical use of 'proportion'
refers to the invariance of a ratio in two situations.
COMPARISON W i t h a colleague, take a textbook o r a n examination p a p e r a n d look for
exercises, activities o r questions t h a t involve, i n some form o r another, t h e idea of comparison
with a ratio.
F
F REFLECTING W h a t insights have you gained while considering t h e contexts in which ratio
[NI arises? Are there a n y ideas or activities which would be worthwhile recording?
Comments Consideration of contexts a s a starting point from which to explore ratio is explored further
in Section 3, where you are invited to devise your own classroom activities.
FOCUS ON IMAGERY
We are all individuals. I t i s perfectly n a t u r a l in a class of 20 or 30 for t h e r e to b e m a n y
different w a y s of seeing a n d t h i n k i n g a b o u t ratio: t h e way i n which two q u a n t i t i e s can
increase hand-in-hand; a n intuitive feel for when two s h a p e s are in proportion; a n image of a
straight line through t h e origin; sorting of t h e form '2 for you, 3 for me', etc. As a teacher, i t i s
important to be aware of these so t h a t problem-solving strategies which arise from such images
of ratio m a y be analysed a n d so that you can give appropriate a n d helpful responses a n d probes
in your classroom.
) I M A G E OF RATIO When you perceive a ratio between two quantities, say 1 0 sweets and 15
sweets, do you have a picture of how they are related? What is your sense or image of ratio?
Does i t differ from those of your colleagues? Imagine a problem which involves finding the
ratio associated with 1 0 and 15. How do you solve such a problem? How much do you use your
image of ratio in your solution?
B
C o m m e n t s You may find it difficult to conjure up an image or sense of ratio at first, but it is worth
persevering at this stage if you can. If you do find it impossible, try to explain why to a colleague. Closing
your eyes and imagining the sweets in front of you (possibly in a row or in two piles) might help. Or you
may find that lines (l0cm and 15 cm long) or some other objects are more helpful.
Do you feel that there is a gap between the mathematics that you use when thinking about a ratio
problem and your image of ratio? I t is not uncommon for pupils to experience this gap. For
instance, a pupil may know that by dividing both 1 0 and 1 5 by 5 the two quantities are in the
ratio 2:3. Nevertheless the pupil may not have the image of l 0 and 1 5 as the result of '2 for you, 3
for me' sorting, in which case the division is likely to be seen a s meaningless manipulation of
numbers. If the mathematics of ratio in the classroom is seen a s distinct from images of ratio
in everyday situations, it is more likely t h a t mathematics will be seen a s separate from
experience and not related to the real world.
WHAT PUPILS SAY
Consider the following statements made by pupils.
Tom got 1 0 out of 1 5 for his test and Anne got 1 5 out of 20 for hers. So there wasn't
anything to choose between them because they both got 5 wrong.
I got a 10% reduction on the price of my coat and I made quite a saving. A 15%
reduction would have given me another E5 off.
The two fractions
2
3
5 and a are the same.
If you increase something by 10%and then decrease the answer by 10% you must end
up with the original number.
What do these statements tell you about pupils' senses of ratio? What would you say in
response? Compare your views with those of your colleagues.
C o m m e n t s You may have remarked that all these statements lack a sense of ratio: a feeling for how two
numbers are connected by a ratio, or how two numbers can change 'hand-in-hand'with each other.
Is there a way in which this sense or image of ratio can be explored in the classroom? Read the
following passage which is an account of a lesson in which the teacher, David Hewitt, a
member of the Secondary Mathematics Curriculum Group (19871,brought his pupils up against
the idea of ratio.
I begin by asking someone to help. Howard offers. I ask him what table he is
really good at and he replies the 5 times table.
DH: Let us practise: 3.
Howard: I don't understand. What a m I meant to do?
DH: You said you were good at the 5 times table. So let's practise it. I'm
saying 3.
Howard: 15 (said with some uncertainty).
DH: 8.
Howard: 40 (more confident).
DH: 5.
Howard: 25.
This continues for some time. I ask the whole class what is happening going
from me to Howard. They reply 'times 5'.
DH: Now let us try the other way round. You start (speaking to Howard).
Howard: 7.
DH: Ow! Now you have made it difficult for me. Say a number that
makes it easy.
Howard looks confused. Others in the class help by telling him he has to say
something i n the 5 times table.
Howard: 20.
DH: 4. And another.
Howard: 45.
DH: 9. Is this right? (to the whole class).
This continues for a while. I then ask what is happening going back from
Howard to me. They reply 'dividing by 5'.
Another pupil, John, is then brought in. He practises the 2 times table with Hewitt in the same
way. The following exchange then occurs:
John: 20.
DH: 10.
John: 14.
DH: 7. (Now I turn and look expectantly at Howard).
Howard: (Pause) . . . What? (He has a slightly confused look).
DH: I said 7.
Howard: Oh, 35.
(I look expectantly at John).
John: 18.
DH: 9. (Looking at Howard).
Howard: 45.
(Looking again at John).
John: 4.
DH: 2.
Howard: 10.
John: 6. (I write '6' on the board).
DH: 3.
Howard: 15. (I complete on the board '6 :157.
This continues with the notation going on the board as they say their numbers,
then.. .
John: 12.
DH: (to the whole class) Don't say anything but do you know what I will
say?
DH: What is Howard going to say?
Class: 30. ('12 :30' is written on the board).
The pupils are involved fully in the lesson and there seems to be a sense of everyone focussing
on the process of changing from one number to another, focussing on the link between the
numbers.
IN YOUR TUTOR GROUP Try this sort of exchange within your UPDATE group. Discuss with
your colleagues the feelings you experienced while engaged in the activity.
Comments Because very little, if anything, is written down during such an activity, it is possible to
develop a sense of how two numbers go hand-in-handand this contributes to an awareness of ratio. Such
experience is very valuable for pupils as it enables them to begin with their own feelings and interpretations
of ratio rather than with someone else's, which may not mean anything to them.
CUTnNG A LINE Imagine a straight line in front of you. Now divide this line into two lines
by a cut one-third of the way along. Compare the two lines that you have now obtained. What is
the ratio between them?
Try this for other cuts.
Can you now create an image of a ratio of your choice (say 2:3)by using this device of cutting a
line?
Compare your findings with those of your co!leagues.
Comments Think back to what you wrote down in IMAGE OF RATIO. Compare what you now feel with
what you were feeling then.
A dominant image of ratio for most pupils is that of similarity. As Freudenthal (1983)
remarks,
Without hesitation, children accept th.at objects on the blackboard are drawn ten
times as large as on the worksheet . . . they accept number lines where the same
interval means one unit, or ten, or one hundred, side by side. Children would,
however, immediately protest at structural modifications that violate the
similarity of the image.
Such images, with which a pupil's sense of ratio often begins, provide opportunities for
developing a n overall sense of ratio in a range of contexts.
REFLECTING This subsection explored aspects of imagery associated with ratio. What, for
you, is the importance of this? How does it help when preparing to teach this topic?
METHODS A N D TECHNIQUES
The range of methods and techniques associated with ratio results from certain root questions
which have arisen throughout the past, questions such a s the following.
Two ascetics lived a t the top of a cliff of height 100, whose base was distant 200 from a
neighbouring village. One descended the cliff and walked to the village. The other,
being a wizard, flew up a height and then flew in a straight line to the village. The
distance traversed by each was the same. Find the unknown height. (Brahmagupta
AD 630)
A bamboo 18 cubits high was broken by the wind. Its top touched the ground 6 cubits
from the root. Tell the lengths of the segments of the bamboo. (Brahmagupta AD 630)
A merchant pays duty on certain goods a t three different places. At the first he gives
l
1
1
gof
his goods, a t the second 7
(of the remainder) and a t the third g
(of the remainder).
The total duty is 24. What was the original amount of goods? (Anonymous 3rd-12th
century AD)
If the value of 2 oxen be that of 7 sheep, how many oxen will 49 sheep be worth?
(Anonymous)
REFLECTING What methods would you use to solve these question? How do you think such
problems were solved when they were first posed?
v
Comments Historically, algorithms for solution were question specific, but today they are more general.
One of the most common methods and one which was described, a t one time, in most textbooks
is the unitary method. For example, consider the following problems.
In 15 minutes I travelled 18 km. How far could I go in 25 minutes if I travelled a t the
same speed?
500 pencils cost £20. How much would 700 cost?
3 chocolate bars cost 63 pence. How much would I pay for 5?
A 300 g portion of chips contains 900 calories. How many calories would a 200 g
portion contain?
A solution to the first of these problems might proceed as follows:
I n 15 min I travel 18 km.
18
So in 1min I travel or 1.2 km.
Therefore in 25 min I will travel 25 X 1.2 = 30 km.
This method of dividing so that one of the numbers is 1 (or unity) and then multiplying by the
required amount is called the unitary method.
) WHAT METHOD? Use the unitary method to solve the other three problems listed above.
Would you have used this method if you had the choice? Which questions would you have
tackled differently? Why?
b
Sometimes such a method seems natural and sometimes it does not. Finding the cost of one
chocolate bar after being given the price of three seems perfectly sensible, but finding the cost of one pencil
having been told the cost of 500 does not. A far more workable method here would be to find the cost of 100
pencils and then to multiply by 7 to find the cost of 700. In practice, this modified version of the unitary
method is far more flexible. Did you use similar modifications of the unitary method throughout or did you
prefer totally different methods for different problems?
Comments
REFLECTING Have you gained anything by looking a t the unitary method? Look back a t some
of the questions that you did earlier in the pack and consider whether the unitary method would
have been a better method of solution than the one you actually used.
What aspects of ratio (language patterns, images, misconceptions or contexts) might be
emphasised or developed a s a result of introducing this method?
What are the benefits, if any, of introducing pupils to standard techniques of this sort?
Can you identify any other methods that you have used frequently throughout the pack? What
aspects of ratio are highlighted by consideration of these techniques?
Some research, which has offered a good deal of insight into the strategies and techniques
involved in the solving of ratio problems, was' undertaken by Noelting (1980) and his
associates. The test that he gave to a number of pupils involved the comparison of the relative
'oranginess' of two drinks made by combining a certain number of glasses of orange juice
with a certain number of glasses of water.
For example,
I
= orange
Whlch Is orangler In each case: A or B'?
Noelting found a number of different strategies employed by the pupils he tested. Some
strategies were successful in special cases but did not cope with all examples. For example,
some pupils based their decisions on how much orange was in each mixture and ignored the
amount of water. Some had strategies to cope with situations where the amounts of orange (or
water) were equal or when the amount of orange (or water) in one was a whole number
multiple of the amount of orange (or water) in the other. Of the strategies that were successful,
Noelting identified the two distinct ways of comparing that were mentioned earlier in this
pack:
comparison within: comparison between the amount of orange and the amount of
water in, say, mixture A, with the resulting ratio applied to mixture B;
comparison between: comparison between the amounts of the same variable (orange
or water) in each mixture.
These strategies can be used in conjunction with different methods of solution a s illustrated in
the following examples.
Example l
B
A
Which Is orangier: A or B?
Between strategy: For one glass of juice, there are three glasses of water. For two glasses of
juice, there should be six glasses of water. As there are only five, the one on the right has a
stronger taste. Answer: B is orangier.
Within strategy: There is twice as much orange in B as there is in A but there is not twice as
much water. Answer: B is orangier.
Example 2
B
A
Which is orangier: A or B?
3
1
1
Between strategy: The amount of orange in B is F or 1~ times more than that in A. But 15 times
a s much water would make 4f glasses of water in B. There are only 4 glasses of water in B.
Answer: B is orangier.
3
4
15
Within strategy: On the right, there is 7 of juice for 7of water; that is, g of juice. On the left
14
there is only g of juice. Answer: B is orangier.
F
ALTERNATIVE STRATEGIES? Here is an account of another strategy for solving this type of
problem:
B
A
Which is orangier: A or fV
Strategy:
Begin by comparing the two ratios 5:4 and 3:2.
Subtracting corresponding numbers in these ratios gives 2:2 (5 - 3 and 4 - 2).
Now compare 3:2 and 2:2.
Subtracting a s before gives the ratio 1:O.
Now compare 2:2 and 1:O. Since the second ratio represents an orangier mixture
than the first ratio, so the second ratio in the first step represents an orangier mixture
than the first.
Answer: B is orangier.
Does this strategy always work? Why? Try i t out on a few examples. Invent some more
'orange juice' problems and explore further the range of possible strategies.
F
F
WITHIN OR BETWEEN Go back to the problems a t the beginning of this subsection on methods
and techniques. Which problems did you solve using a within comparison and which with a
between comparison?
REFLECTING What aspects of ratio arise from an awareness and consideration of the
strategies considered in this section?
P
DIVINE PROPORTION (A HISTORICAL NOTE)
The study of ratio and proportion has held a prominent position throughout the history of
mathematics. This subsection looks a t the emergence of the study of a particular ratio which
was consider'ed by some a s having an aesthetic quality and a relationship with harmony and
order in nature.
The ancient Greeks originally posed the problem: 'How can a line be divided into two portions
so that the ratio of the larger part to the smaller (called the mean ratio) is the same a s the ratio
of the whole line to the larger part (called the extreme ratio)?' This ratio is known as the
golden ratio or the divine ratio and the method of sectioning a line in this way is often referred
to as the golden section. As Bell (1945) reports,
It is said that some of the measurements of Greek vases, also the proportions of
temples, exemplify the golden section; and one prominent psychologist even
claimed to have proved that the pleasure experienced on viewing a masterpiece
alleged to be constructed according to the golden section is a necessary
consequence of the solid geometry of the rods and cones in the eye.
WHAT RATIO? Where would you put the point C on the following diagram so that C divides
the line AB in this special way? Try to get a feel for what this ratio might be.
Twenty-three centuries ago Euclid wrote proposition 11 of Book 2 of The Elements which
proved that a given line segment can be divided into two parts in this way and which also found a value for
the golden ratio. He later used this ratio to devise a ruler and compass construction of an angle of 72" and,
hence, of a regular pentagon.
Comments
GOLDEN RATIO? To obtain a value for the golden ratio yourself you may like to think of the
line split up as shown below.
C
A
A
?
(smaller part)
B
C
?
(larger part)
?
B
(whole)
First ascribe a letter to the unknown ratio (the Greek letter p (pronounced mew) is often used
for this). Next choose any length you like for CB and use the definition of the golden section to
find the lengths ofAC and AB. What does this tell you about the value of p?
Experiment with different lengths for CB.
If you choose a length of 1for CB then AC will be p and AB will be p2. Using the fact that the
smaller part plus the larger part equals the whole, gives the equation
Com ment s
1 +p=p2.
Different choices for the length of CB give rise to equations of a similar form. For example,
can each be reduced to
1+p=p2
by division.
Such equations are known a s quadratic equations (because of the power of two). There are a
number of methods of solving such equations; here, the solution, which is the golden ratio, has
(45 - 1)
the exact value of 7
This value cannot, however, be written a s a single decimal number,
although a calculator can be used to obtain an approximation. The reason for there not being
an exact decimal answer for p is that the decimal representation of 45 is non-terminating and
non-recurring (see PM752D APPROACHING INFINITY in the MATHEMATICS UPDATE series for
an exploration of non-terminating decimals).
The golden ratio was a source of great pleasure to mathematicians. For example, Bell (1945)
reports Kepler (1577-1 630) as saying
Geometry has two great treasures: one is the Theorem of Pythagoras: the other
the division of a line into extreme and mean ratio. The first we may compare to
a measure of gold; the second we may name a precious jewel.
One representation of the golden ratio is that of the golden rectangle in which the ratio of the
length to the width is the golden ratio. The golden rectangle was considered by some to be more
aesthetically pleasing than any other rectangle. I t certainly has a very pleasing property, in
that when the largest square is removed from it the rectangle that remains is also a golden
rectangle.
F
CONVINCING Convince yourself and then a colleague that this is always true.
F
The same argument can, of course, be applied to the remaining golden rectangle a s well and
so gives rise to a process of sectioning off squares to leave smaller and smaller golden
rectangles.
From this a rather pleasing curve can drawn by constructing a quarter circle inside each of
the successive squares as follows:
This curve is known, in mathematics, as a logarithmic spiral. An observation, reinforcing
for some the divine nature of this ratio, was that such a spiral frequently occurs in the
arrangements of seeds on the heads of some flowers, in the shells of snails and in certain cuts
of marble.
Let us now continue our exploration of this attractive and intriguing ratio by following a line
of investigation which leads to a very well-known series of numbers discovered by Leonardo
of Pisa.
Imagine a sequence of numbers generated by the rule:
the ratio of each number in the sequence to the number immediately before it is
always the golden ratio (p).
A GOLDEN SEQUENCE
Suppose the first number in the sequence is 1. Write down the next few terms.
b
Comments
The sequence obtained by following this rule is:
-
1, P, 9,
p3,p4,PS,* .
Such a sequence which is produced by continually multiplying by the same number is known as a geometric
progression.
b
FOR THE BOLD From the definition of the golden ratio, 1 + p = p2, so the third term in the above
sequence could be written as 1 + p, giving:
l,P,l+P,V3,P4
, a * a
Explore different ways of writing the other terms in the sequence using the equation 1 + p = p2.
Are there any patterns that emerge?
Comments By using this relationship, i t is possible to eliminate all powers of p from the sequence so that
a new and surprising connection between the terms emerges.
'
The resulting series is one form of the well-known Fibonacci series, discovered by Leonardo
of Pisa in 1202 in connection with the breeding of rabbits! His nickname was Fibonacci which
means 'son of good nature' and the name has stuck to the series. The series obtained by using
thefactthatl + p = p 2 i s
'9
l,p.,p+1,2p+1,3p+2,!5p+3,8p+5,13p+8
,...
and the usual form of the Fibonacci series is obtained by writing down only the coefficients of p
to give
Both forms of the sequence demonstrate that any term can be obtained by adding together the
two previous terms in the series.
Although the Fibonacci series is not of any great importance in pure mathematics, the fact that
it occurs both in nature and in art has made it the study of many mathematicians in history. If
you would like to explore further the occurrence of both the golden ratio and the Fibonacci
numbers, further references are included a t the end of this pack.
REFLECTING How does a historical perspective such as that described above contribute to the
process of preparing to teach ratio?
LOOKING BACK
This section dealt with some ways of coming to grips with ratio and of helping your pupils to do
SO.
P
REFLECTING Has your sense of ratio been modified in any way by working through this
section? If so, how?
Has working on this section, on your own and with your group, contributed to the process of
preparing yourself to teach ratio?
Have any awarenesses arisen for you while working through this section that relate to
preparing yourself t o teach any topic?
Compare your answers to these questions, and indeed to any others that have arisen for you, in
your group before moving on.
b
Comments You may have arrived a t some structure or framework for thinking of and teaching ratio as a
result of working through this section. What follows in the interlude is our framework, and its purpose is to
indicate the usefulness of such a structure in informing thinking about ratio. A framework like this can be
very helpful: what is important is that you use a framework in which you have faith and which relates to
your experiences. For this reason you may wish to change or modify some or all of what follows.
2 INTERLUDE: A FRAMEWORK FOR
PREPARING TO TEACH
The whole process of preparing to teach ratio is based on a progression from
developing your own expertise on the topic, building up a sort of internal resource which
you can call upon whenever you have to teach ratio (preparing yourself)
using external resources (textbooks or schemes) as a prompt or inspiration for devising
activities on ratio in your classroom (preparing your lessons).
We suggest that a framework with which to inform your thinking about ratio will be helpful in
preparing yourself, preparing your lessons and in progressing from one to the other. We call
this framework the PREPARING TO TEACH framework. It consists of the six headings
mentioned in the Introduction and introduced in Section 1, but these headings are simply
labels for a web of interconnecting and interacting aspects of any topic.
Language
patterns
Different
contexts
Standard
misconceptions
imagery or
sense of
Root
and methods
The framework is only of use when the headings have significance and meaning connected
with your experience. The interlude offers some general remarks about each heading,
indicating its role both in thinking about ratio, and in turning thinking into action in the
classroom.
LANGUAGE PAlTERNS
The language used to talk about ratio (the words and phrases used in everyday speech a s well
as the technical vocabulary found in textbooks) has an effect on pupils' understanding and
general sense of the topic. These words and phrases may be called the language patterns
(associated with ratio) and they can be particularly useful in helping to appreciate various
ways of seeing ratio.
Certain phrases like '1 for you, 2 for me' or 'half a s much again' have the idea of ratio
embedded in them and the discussion of such phrases is a way of building on pupils' existing
experience and extending it to achieve a richer understanding of ratio.
IMAGERY
There is a danger that pupils see the mathematics they encounter in maths lessons a s someone
else's, which they are required to take on board without the feeling that i t h a s anything to do
with them. The use of imagery is a way of enabling pupils to see mathematics a s something in
which they can have a hand in creating. Under imagery we therefore include all aspects of
thinking or inner mental activity which go to make up a sense of a topic. There may be vivid
pictures or just an awareness, there may be physical awareness derived from muscular
responses when using equipment (including the body), there may be strong aural associations,
and so on.
The act of trying to conjure up an image of something is closely linked to struggling to
understand the concept behind i t and to make it meaningful. Some pupils have great difficulty
with some ideas whereas others catch on very quickly. Problems are not necessarily caused
by a lack of ability but result from the fact that an idea does not relate to any part of a pupil's
experience.
Imagery is something which is individual: one person's mental image of, or feeling for, a
particular idea can quite easily be different from someone else's, and yet each can be valued
and shared. The introduction of activities where pupils can develop their own images of what
ratio is about is a way of allowing everyone to start from where they are and to move together to
an understanding of ratio.
STANDARD MISCONCEPTIONS
You may already know from observing the children in your school that there are some basic
errors and standard misconceptions t h a t certain children reveal in certain areas of
mathematics. These errors and misconceptions are independent of the teachers who have
taught them and they are often to be found a t all levels, from first year to sixth form! It is also
evident from research that these misconceptions are much more widespread than just in your
school. In classrooms up and down the country and, indeed, all over the world, similar errors
and misconceptions occur.
So it seems sensible to assume that the standard misconceptions associated with a particular
topic point to an underlying difficulty inherent in it. Therefore to study these standard
misconceptions can provide invaluable help each time you are preparing to teach a new topic.
I t is important to make the distinction, however, between standard misconceptions and what
might be called alternative conceptions. To take an example involving ratio:
To cook a meal for 12 people I need:
12 chops
6 tomatoes
18 potatoes
1mgpeas
How many potatoes will I need for 8 people?
The solution expected is, presumably, 12 potatoes (the ratio of people to potatoes is 2:3)and a
standard misconception is to see the number of potatoes as 6 more than the number of people
and therefore to give an answer of 14 potatoes. An alternative conception might be to see the '6
1
1
more' as pof the 1 2 people and therefore to add on pof the 8 people to obtain the answer 12. This
does not represent a misconception, just a different way of looking a t the connection between 12
and 18.
There may be many alternative conceptions of a particular mathematical idea and in fact
discussion of them between pupils and teacher can be a very valuable exercise.
There may be many alternative conceptions of a particular mathematical idea and in fact
discussion of them between pupils and teacher can be a very valuable exercise.
ROOT QUESTIONS
If you have spent any time in a mathematics classroom you will be all too familiar with the
question 'Why are we doing this?'. A common response might be 'because it's on the
syllabus'. While this may be true to a certain extent, it certainly cannot be the whole story and
neither is it likely to satisfy the reasons behind the question which could be anything from real
curiosity to disenchantment. Furthermore, it is unlikely to motivate many pupils! Most
mathematical topics become identified as such because they represent a technique or approach
that has been developed for dealing with a range of questions o r problems which our
civilization has faced. At the root of mathematics is the asking of questions and new
mathematical ideas arise out of the struggle to answer new questions or old questions in a new
way. So to study these root questions which originally inspired the desire to develop and refine
the concept of ratio into what has now become a 'methematical topic' is an important step
towards developing a broad, well-balanced view of the topic and, indeed, of mathematics in
general.
CONTEXTS
When teaching a topic you presumably want your pupils to be able to recognise both the sorts of
situations where that topic arises and the types of questions which can be dealt with by it.
Furthermore, you want them to apply the techniques appropriately. If they have no idea of the
sorts of questions the topic is intended to answer then they are unlikely to be able to meet either
objective. If the model of mathematics pupils are presented with is that of a series of sets of
questions unrelated to anything outside of the mathematics classroom then i t should be no
surprise if they are reluctant to become involved in problem solving and investigating. Pupils
may get the feeling that mathematics actually takes place outside of them and that it represents
some mysterious language into which they need to be initiated. Everyday contexts in which
the idea of ratio arises are many and varied and provide not only opportunities for practising
important techniques but also a rich source of investigative work.
STANDARD TECHNIQUES AND METHODS
When looking a t certain topics in mathematics, the same sorts of techniques are being
encountered over and over again. The ideas of multiplying by a scale-factor, division of one
number by another and the use of fractions and decimals are all techniques that will be
required in many of the situations where ratio appears. The awareness of these techniques
and methods and making them part of one's own mathematical behaviour is a valuable step in
the process of getting to grips with the topic.
USING THE FRAMEWORK
By considering all of these aspects of ratio it is possible to build up a picture of the essential
elements of the topic. This not only encourages you to come to a full sense of ratio yourself but
also helps you to appreciate the levels of awareness that your pupils need to encounter if they are
to construct their sense of ratio successfully.
The PREPARING TO TEACH framework is meant to inform your thinking about ratio and not
merely to generate a mechanical sequence of questions. Any framework can be used
mechanically, but if used in this way the results will be mechanical. Each heading
contributes an aspect and the aspects overlap; they are not intended to imply mutually distinct
features of the topic. Used creatively, each aspect can be a useful reminder to look a t the topic
from several directions.
The framework provides a useful structure for analysing what was, or could be, gained from
any activity on ratio: from those to be found in your textbook or scheme to those that you might
devise yourself. For example, you may find i t instructive to ask the following questions.
Does the activity provide enough time for pupils to become fluent in discussing
aspects of ratio, that is, to develop their language patterns?
Do pupils have the opportunity to evoke an image of ratio?
Is there the possibility of the teacher providing searching questions or probes in order
to highlight and correct standard misconceptions?
Are there opportunities for practising standard methods and techniques?
Can the activity be linked to historical or other root questions in order to provide
added stimulation and motivation?
Are there opportAities for pupils to abstract the underlying features or essence of the
activity so that they can pose their own similar questions in other contexts?
In the next section you are invited to engage in activities intended to stimulate you to devise
your own activities for use in your classroom. Questions such a s those above should help with
this process.
3 PREPARING YOUR LESSONS
USING YOUR TEXTBOOK OR SCHEME
One point of reference when beginning to prepare activities on ratio will be the textbook or
scheme that you have available in your school. But no matter how detailed, comprehensive or
well-written this may be, it can only ever be a resource which helps teachers to use their skills
more effectively. The skill of using such printed material creatively is one well worth
cultivating.
This section therefore considers:
what a textbook or scheme can provide;
what mathematics pupils are doing when working through particular bits of this
material;
shortcomings in a textbook or scheme;
what can be done to augment a textbook or scheme.
:
Do you remember David Hewitt's lesson in Section 1 (see pp 22--23)? Go back and read through
it again if necessary. Such activities are not found in textbooks; indeed, no textbook exercise
could possibly provide this sort of experience. Yet i t seems to be a valuable exercise in
thinking about ratio. A textbook presents a way of seeing and talking about ratio, but it is
someone else's way of seeing and talking, and pupils need to begin with a concept of ratio
which is based on their own feelings and experiences.
As discussed in Section 1 , many topics taught in school have a t their root the idea of ratio. If
;ou were to find a reference to ratio in the index of a textbook, however, you may find that it
refers to a rather narrower set of contexts and techniques than those considered in this pack.
So, how can you use your textbook or scheme when preparing to teach ratio? What can it
provide?
TEXTBOOK What sorts of contexts are dealt with in your textbook or scheme under the
heading of ratio? What particular words and phrases associated with ratio are used in the
relevant chapter or section?
B
Textbooks and schemes obviously vary. Contexts that often occur include the following:
dividing a given sum of money between two or more people in a given ratio, as in 'Tejinder and Natalie
share E3 in the ratio 2:3, how much do they each receive?'; extracting a ratio from a situation and then
simplifyingit, as in 'there are 15 girls in a class of 25 pupils, what is the ratio of girls to boys?'.
Comments
The language of dividing and sharing is often prevalent in such situations. Some specific words and
phrases that we found in textbooks were: 'change to their simplest form'; '2 parts sand and 1 part cement';
'proportional division'; 'unfair sharing'; 'write the ratio 1to 3 as a fraction'; 'simplify to the smallest possible
numbers'; FQ is :of QR,and PQ is :of RS'; 'draw a larger rectangle the same shape as'; 'pedals will turn at
half the speed of the bicycle's wheels'.
Pupils will often use the words and phrases that they come across in their textbook or schemc to
help them construct their sense of the topic. The images of map scales, percentage increase,
rates of pay, enlargements, similarity, and many others (you may have noted others in
FINDING RATlO.in Section 1 , p 11) can play an important part in the development of a sense of
ratio, yet such topics may appear under other headings, not necessarily linked by pupils to
ratio.
) LOOKING ELSEWHERE You may like to try TEXTBOOK again, concentrating on a section or
chapter on one of the topics mentioned above or indeed on any other that you can think of that
deals with aspects of ratio.
'
In the context of the PREPARING T O TEACH framework, most textbooks offer the following:
technical and, perhaps, everyday language patterns associated with the topic;
some typical contexts in which the topic arises and the standard techniques that may
often go hand-in-hand with these contexts;
an indication of the standard misconceptions that may occur;
diagrams, worked examples and exercises.
Language patterns can give rise to certain images of ratio; the awareness of ratio in a number
of different contexts allows pupils to experience a wider range of language patterns; contexts
invite techniques. All offer the opportunity of gaining a richer sense of ratio. It is in this way
that the strands of the PREPARING T O TEACH framework are interwoven. By addressing one
strand, another may come into focus which can then be explored.
IDENTIFYING RATIO Look a t the following pages taken from a textbook. In what sense are both
these extracts about ratio? What elements of the PREPARING TO TEACH framework can be
identified in each extract?
1
denom~natar
he munoer o
equal size part51
3
h
3
Dave 1s an oilman
Heearns C l 0 for every hour heworkr
We say hts rale of pay 1s C 1 0 per hour
;EZ!r%er
of
parts w e are
thmkrng 01)
\
a fractlon
3 a\ One o l the rectangles has ) 01
i c area c o l a u r ~ d
Whrch 1s 11)
W r f e down the rate of pay tor these
b) Whlch rectangle tias more cf
m area coloured than the other t w o '
cl The area o f A is 32cm2
C
Copy and Complete
(1)
(I,)
(1181
4
&or32cni1=
01 32cm2 =
80132cm2=
U
a) One of the crcles has
4 u l 81s area coloured
W h ~ 8sh 81)
b) W I w h circle has more o f
81s area coloured than the
Olher l w o '
C)
LOPVand cornplele for circle C
n
Fraction "la e d coloured
g
.
e-
2 Mary worksa (we-day week
She earns f400each week
W h r l I S her rate 01 pay per day,
Whar
4
IS
ooooa
her rate o l pay per week7
Svlv~eearnsf 1 6 800 d ~ ~ n u a l l v
She gets the samearnoun! each month
What I S her rate o f pay per month'
5 Alan IS patd f 5 per hour o n weekdavsand C8 per hour on
r
Saturdays and Sunddvs
Last week he worked 8 hours per day Monday to Frldav
On Saturday he worked 6 hours
H o w much dld heearn,
Challenge-
5 Draw d c ~ r c l e w l l hradius 3 c m
:
Shade ~n 01
6
ilula worked 1; hours
She earned C6 75
What IS her rale 01 pay per hour'
~lsarea
Source: NMP (1987)
Cornments
The idea of ratio runs through so many areas of the secondary mathematics curriculum, yet
two topics such a s fractions and rates of pay are often seen a s completely separate by teachers as well a s by
pupils.
TIME TO REFLECT
In the Introduction, i t was observed that there is a danger of getting lost in a lot of doing, where
the activities are seen as an end in themselves and the purpose for doing them is missed. It is
often comforting to do a task without thinking, but this rarely contributes to effective learning.
It is likely t h a t certain components of the task will be automated and, obviously, teachers
would not wish pupils to reflect each time they divided one number by another! However,
encouraging pupils to reflect on the whole task and to make connections between tasks is an
essential part of a teacher's task. I t is therefore worth giving time in lessons to questions like:
What type of question is this? Can you find any other questions of this type?
What do you mean by 'this type of question'?
What is the same and what is different about the tasks that you have just completed?
What mathematics have you been doing?
What has this lesson been about?
By reflecting in this way, pupils can come to the idea that a ratio not only occurs in one slot on
the syllabus but runs through a great deal of mathematics. So mathematics can be seen a s a
unified subject and not one consisting of arbitrary compartments.
An often expressed view is 'I never really understood that until I came to teach it'. This is
often because coming to teach a topic is the first time that one reflects on what is known and
how it came to be known. We would probably all recognise the criticism that is levelled at the
so-called armchair philosopher who thinks all the time about the problems of the world but
never looks a t the practicalities. But i t is just a s important not to get trapped on a treadmill of
doing and fail to allow time to step back and think about what we are doing and why we are
doing it.
A textbook or scheme, when i t is viewed in a questioning manner, provides one way of
introducing this process of reflection to your
What can also arise from this for you is
the desire to augment activities. In this way the textbook or scheme becomes a catalyst for the
creation of more activities rather than something which dictates the range of activities
available.
) FRACTION/RATIO One of the extracts on the next page comprises questions about fractions;
the other concerns ratio. In what ways are the two sets of exercises the same and in what ways
are they different? What parts of the PREPARING TO TEACH framework come into focus while
working on this activity?
Comments 'There are many different ways of 'giving' a set of exercises to pupils. The most c.)mmon is
to discuss the questions first and then to ask pupils to work through them and to write down their eolutions.
However, it is also valuable to ask questions about the questions:
What are these questions about?
What do they have in common?
Which is the eaaieet, which is the hardest and why?
This activity of taking two exercises from different chapters in a textbook and then looking at similarities
and differences provides an opportunity for reflection. Furthermore, the responses need not be written
down; useful group and class discussion can arise from such a classroom activity.
More paint mixing
C10 You can see
f of this wlndow.
Ray makes a grey colour.
H e mixes 2 tins of black and 3 tins of whlte.
How many squares
are there
In the wlndow?
So the re10 of black to white
i s 2 to 3.
I
F1 One day Ray wants twlce as much grey palnt.
H e uses 4 tins of black.
How many tins of white does he use?
I F2
C12 You can s e e $ o f t h s tram
4
9
Next day he wants a lot of grey palnt.
He uses 10 tins of black.
How many tins of white does he use'
U0
l
F3 Next time he uses 12 tlns of whlte.
How many tins of black does he use?
How many carrlagesare rhere In rhP whole tram'
C13
YOU
can see301 thrs chimney
How tall lsthechmney'
- .
U U U
>Y
/
L
,
l
,,,
X">.
{.-$
F4 Ray wrltes out a table.
A T h e table shows how much black paint
and white paint he has to mlx.
Copy the table and fill In the mlsslng numbers.
m
20
Source: SMP 11-1 6 (1983).Fractions 2 and Ratio 1
When an author writes a series of examples for a textbook or scheme, there is almost always a
thread t h a t runs through them. The phrase 'carefully graded examples', which is used in
some textbooks, suggests that the questions have been chosen for some specific purpose and set
in a particular order. However, i t is quite common for pupils t o tackle each question
separately, possibly even using a completely different strategy to answer each one, and thus to
have no sense of the overall structure of the material:
Mathematics is about looking for links and commonalities between specific happenings; it is
about seeking a generality from a number of particular instances (see PM750 LEARNING AND
DOING MATHEMATICS and PM751 EXPRESSING GENERALITY also in the MATHEMATICS
UPDATE series). So looking a t a collection of ratio examples and trying to find a common link
or theme is not only valuable a s a way of coming to a fuller understanding of ratio, but also
provides a model for sound mathematical activity.
RECIPES Look a t the following extract, which invites the reader to study a number of recipes
for coffee, and decide which ones are stronger and which ones are weaker than Basil's recipe.
Work through this activity then rank questions A to G in order of difficulty.
Do you think your pupils would rank the questions in this order?
Are there any questions that you answered using the same strategy? If so, group them together.
For each group explain a different strategy which you could have used. What would be the
benefits of asking your pupils to do this?
You may well find that such an activity can highlight pupils' misconceptions about ratio.
Alternatively, you may notice some more general things about using this kind of technique in your
classroom. Use your notebook to record your observations.
[NI Comments
l
Mth a b i e d
7
Here are some more reclpes for coffee
a) D e c ~ d ebetween you w h c h ones
make stronger coffee than Bas~l's
r.
4 cups of water
5 coffee bags
3 cups of water
3 coffee bags
"
I1
7 coffee bags
I
KI cups of water
20 cups of water
I
39 coffee bags
1 2 coffee bags
7 cups of water
6 coffee bags
b) W h ~ c hreclpes make weaker coffee than Bas~l's?
c) W r ~ t ea reclpe of your o w n
It should make stronger coffee than Bas~l's,
but weaker than reclpe C
You can make ~tfor as many cups of
coffee as you l ~ k e
OOu
00
0
w
I
e the s a m t r e n t h
use the same proportion of water and coffee
4 cups. 3 coffee bags
8 cups, 6 coffee bags
.-------/1I
source: NMP (1987)
AUGMENTING
Often, a textbook exercise suggests some other areas of investigation in addition to the ones
explicit in the exercise and this may prompt you to augment the activity in some way.
FRACTIONS Consider the extract overleaf on fractions.
collection of activities is intended to introduce?
What ideas do you think this
Do you think that by working through the material pupils will come up against these ideas?
What actual mathematics might pupils be engaged in? What degree of modification could you
introduce and why?
B5 Work ~heseour (a) f of 24
B6 Work these our
A (a)-/of16
(b)fof12
(b)
B8 Worktheseout. (a) f of l5
2 of24
(b)
J of l 5
B9 Work these out
(=)+of4
(d)fof8
(elf0140
(a)
of 9
(b)
4 of 30
B10 Copy and complere
(c)
4 of 3
(d)
of 24
(e)
4 o f 18
$ of 20 =
There are 15 plgs In another p~g-sty
'What 1s { o f IS?' asks Luke.
'Work It out like thrc,' says Amos 'Frrst you work our 4 of 15
Then you mult~plyby 3 '
B11 Workour g o f 15.
Source: SMP 11-16 (1983), Fractions 2
The questions concentrate on finding a fraction by thinking of it as a combination of two
operations: first multiplying by the numerator, then dividing by the denominator. The idea of multiplying
by a single fractional number less than one and obtaining a smaller result could well be missed.
Comments
Here is one way of augmenting such an exercise: 'Take a number, divide it by 4 and then multiply it by 3.
Write down your result and compare it with the original number. Which is smaller? How much smaller is
it? Investigate this for different starting numbers.'.
In order t h a t your textbook or scheme becomes a useful tool a n d one which you use rather t h a n
one whose use i s dictated to you, we suggest t h a t you ask yourself these three questions for a n y
activity:
What a r e the pupils supposed to do?
Do I w a n t more from the activity?
If so, how would I augment it?
For example, how might t h e following situation be investigated by pupils?
In a warehouse you a r e given a 20% discount on some goods b u t you have to pay VAT
at 15%. Which do you want calculated first: VAT or discount?
Asking the three questions listed above could result in these ideas:
What a r e the pupils supposed to do?
Notice that £1.00 increased by 15% gives £1.15, which, decreased by 20%, gives 92p, and
that £1.00 decreased by 20% gives 80p, which, increased by 15%, gives 92p. They are
then intended to observe that similar findings (that is, order does not matter) occur for
other starting values.
Do I want more from this activity?
The realisation that this general principle is true and an understanding of the scale
factors associated with percentage increases and decreases.
How could I augment it?
Pose the questions: 'What single percentage reduction is equivalent to an increase of
15% followed by a decrease of 20%?', or 'In general, how can you define the single
percentage increase (or decrease) which is equal to two successive percentage
increases (or decreases)?'.
A S K I N G QUESTIONS Take a ratio activity from your textbook or scheme a n d ask yourself the
three questions listed above. Discuss with colleagues any modifications you make to the
activity.
P
F REFLECTING How can the PREPARING TO TEACH framework help in the process of using your
[NI textbook or scheme to prepare or augment activities on ratio?
F
USING RATIO IN CONTEXT
Mathematics i s about asking questions. Often, new mathematical ideas a n d fresh thoughts
arise out of situations which invite t h e posing of questions. One way of stimulating
interesting ratio questions for the classroom i s to look a t some of the contexts in which the idea
of ratio occurs. For example, consider the following situation:
An A3 sheet of paper can be divided into two sheets of A4. A4 paper is the same shape
a s A3 b u t smaller. (Do this for yourself in order to convince yourself of the validity
of this statement. What exactly is meant by 'the same shape'?)
) THE S A M E SHAPE What questions occur to you a s a result of looking a t this context of ratio?
Write down some questions which may provide starting points for investigation. Don't worry
about whether the questions are 'good' or 'bad' at this stage and try not to work on them in any
way before you write them down. The object of this activity is merely to have in front of you as
many questions a s occur to you.
Questions that occurred to us were:
What size is A3?
How big is A5, A6, A7, . . .,A2, Al,AO?
What other paper sizes behave in this way?
What if the original piece of paper is divided into 3 equal pieces or 4 or 5 or . . .?
Com mento
Each of these questions points to other questions. Even 'What size is A3?' begs the question 'Why, what's
special about A3?. That is the essence of mathematical questions; they are the beginning of a line of enquiry,
not the signalling of an end point or a conclusion.
P
W O R K I N G ON QUESTIONS Now, with a colleague, work together on some of these questions
(or others t h a t you have produced). If, in the process of working on one question, another one
emerges, write t h a t down a s well.
P
REFLECTING How easy did you find i t to move from the original context to a question on
which you could work? What about your colleagues?
-P
GEARS Two cogs are adjacent to each other such that their teeth fit together. When one is
rotated, the other one rotates. What questions arise from this context? Write them down a s
they occur to you. What if a third cog is introduced?
P
Comments You may have found that engaging in the gears activity allowed you to see ratio in a
different way. A different context can often throw up new language patterns or new images of ratio or
indeed can highlight any of the six aspects of the PREPARING TO TEACH framework, and so can provide
variety and richness. For this reason, it is important for pupils to experience new topics in a number of
different contexts.
P
DIVIDING A LINE How do you divide a line of any length into, say, five equal parts, without
measuring its length and performing what might be a complicated division? The following
method, which involves the use of ratio, provides a way of doing this.
Draw a line, which is easy to divide into five equal portions (say, 5 cm), parallel to
the given line, and mark off five equal portions on this line.
Now mark a point a t some distance from the line that you drew and draw lines
passing through the point, the marked off portions and the given line, a s shown in
the diagram below.
The given line will now be divided into five equal portions.
Could this method be used to divide a line in a given ratio, say, 3:2 or even 5:3:7?
What aspects of ratio arise from this context?
P
BRAINSTORM Brainstorm with some colleagues and try to come up with some more contexts
of ratio which could result in questions on which you can work.
P
IN THE CLASSROOM Try some of these ideas in your classroom as soon a s possible. Look out
for opportunities where your pupils can be invited to make up their own questions.
REFLECTING In the lagt few activities we asked you to generate and then to work on questions
that may arise from various contexts and situations. Consider now the features of posing your
own questions as opposed to using textbook questions.
What is the same and what is different?
Did any particular techniques, or ways of talking about and seeing ratio, emerge from these
activities?
In general, what aspects of the PREPARING TO TEACH framework emerged while generating
and working on these questions?
RATIO ACROSS THE CURRICULUM
There is a long history of conflict between science and mathematics departments, centred on
the teaching of ratio and proportion. On the one hand, the scientists say that pupils can't d o
ratio so that many important scientific ideas are lost in the midst of calculational confusion.
On the other hand, mathematicians say that they have indeed taught ratio and that pupils find
the topic very difficult. The basic idea that the mathematics is needed before i t is possible to do
the science does not take into account the complementary role of both disciplines.
I t might be said that the only reason for computing ratios is to find a simple relationship
between two quantities that holds i n general. Scientists seek properties which do not vary as a
result of certain actions. For example,
chemists seek quantities associated with chemical reactions that are invariant
under changes in the relative amount of reactant present;
physicists seek properties of materials and objects that do not vary when the amount
of material is changed (e.g. density, elasticity);
environmental scientists seek properties of species invariant under changes in the
actual population or the area inhabited.
Far from the mathematics and the science being separate in these situations, i t is the science
that provides the context through which the mathematical idea becomes more readily grasped
and it is through the understanding of the mathematical idea that the basic scientific principle
can be understood.
What follows is the account of a lesson, given to 12- and 13-year-olds of average ability, on the
heating of copper carbonate and the subsequent mass loss, as observed by Jack Rainsden, a
Teacher Fellow a t the Open University during 1985-1986.
Whilst carrying out the experiment the children busied themselves with the
weighings and the recording, the heating and observation, until they came to
the number crunching stage. Up to this point what was expected was clear; the
instructions were executed faithfully a n d there was a very positive
commitment by the pupils., I a m sure they enjoyed 'doing' the chemistry but I
sensed that they had no strong feelings about what it was they were attempting to
reveal and why. Then came the calculation and I was struck by the difficulty
that some pupils experienced in trying to relate the numbers to the experimental
activity. Their expectations o f how to perform mere usually vague and
unconvincing. Even after they were given stroke by stroke calculator
instructions there were still errors and a good deal of puzzlement. Talking to
the pupils about the calculation resulted always i n questions: 'Do I subtract
these numbers?'; 'Do I put this i n the calculator first?'; 'How do I find out how
m u c h green powder?'. Further questions about the significance o f their
calculation simply produced blank expressions. The pupils were quite unable
t o see a n y relevance i n t h i s calculation to the heating experiment just
completed.
They were i n no doubt that heating the green substance chan.ged it! They were
sure, for the most part, that quite apart from the colour there had been some
change i n mass too. They were able to talk about the water that was driven off
and the collection o f carbon dioxide - so the chemical messages were being
received. However, the mathematical conclusion, which alone brings out the
prime aim of the lesson - the idea of constancy of chemical purity - had not
been grasped.
The mathematics (the sense of a ratio between the mass before and after heating) can help
pupils to see the relevance of the heating experiment, and the science (the way in which the
masses of copper carbonate and, indeed, any substance before and after heating are related)
both help students to appreciate the idea of ratio. I t seems difficult to separate out the two
disciplines here.
) OTHER SUBJECTS If you have experience of teaching a subject other than mathematics, try to
think of some instances where mathematics impinges on this subject, a s i t did in the
chemistry lesson described above (it may be ratio or i t may not). What diffxulties or problems
did your pupils experience? What difficulties or problems did you experience?
While thinking about such instances, ask yourself what this subject could have gained from
looking a t the mathematics and what mathematics could have gained by looking a t this
subject.
b
Comments It is worth being aware of the emergence of mathematical ideas, such as ratio, in other
subjects, as this can provide contexts from which you may be able to develop stimulating questions as you
experienced earlier in this subsection.
LOOKING BACK
This section looked at ways of using your textbook or scheme creatively in order to give you
and your pupils the opportunity both to reflect on the mathematics that is being done and to
appreciate the way in which the ideas of ratio extend into many different areas of the
secondary mathematics syllabus. It also considered some of the contexts in which ratio occurs
and used them a s a prompt to help you devise activities for your classroom.
While working through this material you may have found or devised quite a few activities
and/or questions that may prove useful in your classroom. How useful will these activities
actually be and how will you use them?
REFLECTING Choose some activities which you think will:
NI
generate the most discussion;
provide pupils with the opportunity to develop a sense of ratio or to develop their own
questions;
provide opportunities for you to ask useful questions and to discuss misconceptions;
offer the chance to practise and develop techniques and methods.
Use the PREPARING TO TEACH framework in this way to inform your thinking about your
chosen activities and how you might use them in the classroom. Compare notes with your
colleagues and justify your choice.
4 SOME RESOURCES
The activities included in this section are not intended to be used in any particular order, nor
indeed to provide any sort of framework for the range of ideas involved in ratio. They are
intended to be used in two ways. You could try them for yourself and discuss with a colleague
(or colleagues) what ideas of ratio are involved; alternatively, you could try them with a group
of pupils, concentrating on appropriate ways of working in order to promote a conjecturing
atmosphere in the classroom.
F
MIXING IT UP To make a jug of orange squash I use two beakers of orange and five beakers of
water.
Orange
Water
What happens to the strength of the mixture if I add an extra beaker of orange?
What happens if I add an extra beaker of water?
What happens if I add one of each, two of each, etc.?
Source: Noelting (1980)
b
b
ONE DAY I'LL CATCH UP My Dad is four times older than me. When will he be three times as
old?
When will h e be twice as old?
When will I catch up?
F
QUICK PERCENTAGES If you want to increase something by 1 0 8 , it's easy, you just multiply i t
by a magic number!
Does this work? What is this 'magic' number? Does i t work for other percentages?
What about decreasing?
b
VAT In a warehouse you are given a 20% discount on some goods but you have
15%. Which do you want calculated first, VAT or discount?
F
to payVAT a t
THE METRIC FOOT A metric foot (if such a thing existed) would be t h e same length a s an
imperial foot b u t would be made u p of 1 0 smaller units. What would you call these smaller
units?
Design a conversion table which will change inches into these units and vice versa.
P
ESTIMATION Nadia estimates the length of two objects - a lollipop stick a n d a wooden pole.
Lollipop stick
Wooden pole
Actual length
Estimate
12 cm
10 cm
2m
1 m80cm
Which is the best estimate? Why?
Could this form the basis of a practical, measurement activity which could bring out t h e idea of
ratio?
CHOCOLATE In a room, there a r e three tables on which there a r e bars of chocolate a s
illustrated below.
Ten children a r e lined up outside the room. One a t a time they enter the room and sit a t any
table. They all like chocolate, so a t the end the chocolate a t each table is shared out between
them. Once t h e pupils have s a t down (you may like to imagine such a n arrangement or
actually do i t with your colleagues), how is the chocolate shared out?
If one pupil i s chosen and invited to sit somewhere else (you could imagine t h a t you a r e t h a t
pupil), where would he or she sit?
Is there a fair arrangement so t h a t all children will receive an equal amount of chocolate?
If not, why not and what could you change so t h a t a fair arrangement i s possible?
Source: ATM (1987)
P
TEXAS DOUGHNUTS Sign in Texas Homecare: 'Doughnuts 13p each or 5 for 50p'. How much
for four doughnuts?!
NO PROBLEM(?) Sean reckons t h a t percentage reduction is easy: 'All you do is take £10 off if
it's a 10% reduction, take £15 off if it's a 15%reduction, etc.'.
What do you say if you are trying to explain to Sean why this doesn't work?
The trouble is, it does work sometimes. When and why?
P
PENDULUM Make a pendulum which will swing back to its starting position in one second.
Now arrange i t so that it swings back in two seconds. What did you change to do this?
P
ROLLING Investigate how far a block of wood can be rolled on a cylindrical roller.
What happens when you change the size of the roller?
FRACTION CHANGE Is each of the the following statements true or false?
Fractions get bigger when you make the numerator bigger.
Fractions get smaller when you make the denominator bigger.
Fractions stay the same when you make both the numerator and the denominator
bigger.
b
FAMILY COMPARISONS Write down a s many statements a s you can which compare the four
members of the family drawn below.
GARDEN PAVING If I want to surround my garden with tiles laid according to the following
pattern, how many of each type of tile do I need?
5 THE LAST WORD
It is difficult to write a pack which has direct application to all classrooms in all schools and
for all ages of pupil. PREPARING TO TEACH RATIO set out to explore some of the issues related to
the teaching of ratio, although we have tried to do this in such a way as to raise general
principles involved in teaching any topic in mathematics.
Central to the pack is the PREPARING TO TEACH framework, with its six headings. This
framework was used to help inform thinking on important aspects of preparing to teach ratio.
It was also used to direct thinking about what underlies activities on ratio. This process of
reflection is an important one in both the teaching and the learning of mathematics.
SO WHAT IS RATIO? Try to construct the sense of ratio which you now have after working
with this pack by addressing the following questions.
What is a ratio? Is i t a pair of numbers? Is it a fraction? Is it a scale-factor?
What different contexts give rise to different perceptions of ratio?
b
PUlllNG THINGS TOGETHER What, for you, are the general principles involved in the twin
processes of preparing yourself and preparing your lessons?
How have the headings of the PREPARING TO TEACH framework helped to inform your
thinking while working on this pack?
How could you use this pack to explore the teaching of other mathematical topics?
b
AND FINALLY As your final activity on this pack discuss with colleagues and then prepare
your first two lessons on ratio.
BIBLIOGRAPHY
APU (Assessment of Performance Unit) (1978-1982) Mathematical Development. A Review
of Monitoring in Mathematics Part 1.
ATM (Association of Teachers of Mathematics) (1987) Working Investigationally. Ideas for
Znvestigational Approaches to Standard Topics. A Resource for GCSE, An ATM Activity
Book.
Bell, E. T. (1945, first published 1940) The Development of Mathematics, McGraw-Hill.
Carraher, T. N., Carraher D. W. and Schliemann A. D. (1984) 'Can Mathematics Teachers
Teach Proportions?, a paper presented a t the Fifth International Congress on Mathematical
Education, Adelaide, Australia
Freudenthal, H. (1983) Didactical Phenomenology of Mathematical Structures, D. Reidal
Publishing Company, p 191.
Hart, K. M. (1984) Ratio: Children's Strategies a n d Errors. A Report of the Strategies and
Errors in Secondary Mathematics Project, NFER-Nelson.
Heath, T. L. (1956, second edition) The Thirteen Books of Euclid's Elements, Vol. 1, Dover
Publications.
NMP (1987) Mathematics for Secondary Schools, Book 2, Blue Track, Longmans.
Noelting, G. (1980) 'The Development of Proportional Reasoning and the Ratio Concept' Parts
I and I1 in Educational Studies in Mathematics, Vol. 11,pp 217-253,331463.
Secondary Mathematics Curriculum Group (1987) 'The Place of Algebra in the Curriculum'
in Mathematics Teaching, Vol. 121, December 1987, pp 21-26.
SMP 11-16 (1983) Fractions 2, Level 1(b) Number, Cambridge University Press, pp 10-11, P 16.
SMP 11-16 (1983) Ratio 1, Level 2 (a) Number, Cambridge University Press, p 16.
Southern Examining Group (1986) GCSE Syllabus Mathematics (with Centre-based
Assessment) 1988 Examination.
FURTHER READING ON FlBONACCl NUMBERS
Adamson, B. (1978) 'Understanding Fibonacci Numbers' in Mathematics in Schools, Vol. 7,
No. 5.
Gardner, M. (1971, first published 1961) More Mathematical Puzzles and Diversions, Penguin
Books.
Gardner, M. (1982, first published 1968) Mathematical Circus, Penguin Books.
Ghyka, M. (1977) The Geometry of Art and Life, Dover Publications.
Northrop, E. P. (1978, first published 1944) Riddles in Mathematics, Penguin Books, pp 55-6.
Verob'er, N. N. (translated in edition Sneddon I. N.) (1961) Popular Lectures in Mathematics,
Vol. 2, Fibonacci Numbers, Pergamon Press.
W
Begin by reviewing in your mind, or by brainstorming with colleagues, any
associations and aspects of t h e topic. We recommend keeping a notebook for
each major topic so that you c a n remind yourself quickly of the underlying ideas
and of particularly effective activities.
If you find yourself running o u t of ideas, or with a sense of something missing
or incomplete, then refer to t h e framework to see if there is an aspect that you
have overlooked or underplayed. Try looking in a textbook for more ideas.
If you find yourself with too many ideas to cope with, try organising them under
the framework headings. N o t e that any particular idea may relate to several
headings.
The six components of t h e framework provide touchstones to seek out
deficiencies in whatever scheme you use. You can then generate or be on the
lookout for new activities which will support pupils to make real contact with
the essence of the topic.
PREPARING YOURSELF
The framework shown overleaf has proved useful for thinking about what is
involved in a topic for oneself, and for deciding what strengths and weaknesses
are present in whatever scheme of work is being used. It does not constitute an
algorithm for teaching and, i f used mechanically, will only produce mechanical
results. The framework assumes that you have established a mathematical
atmosphere in which to work with your pupils, and that you have organised the
classroom to this end.
Copyright O 1988 The Open Unive
SUP 17195 3
By being aware of the language pat
remember to take time over the intr
in mind that pupils come to lessons
and language patterns of the topic.
By thinking about underlying imag
give pupils direct contact with key
language patterns, and thus to try to
By indicating where a topic come
deals with, you can help pupils to g
and its applications.
By providing opportunities to enc
important techniques, you can help
Pupils who participate in formulati
are more likely to remember it later.
The six components which compri
pairs, corresponding to behaviour (l
(root questions and different conte
misconceptions). They overlap and
should not be used mechanically.
PREPARING YOUR LESSONS
PREPARING TO TEACH A TOPIC
STANDARD
MISCONCEPTIONS
Being aware of incomplete imagery, inaccurate language
patterns and pupils' attempts to make sense of what they
hear and see, makes it possible to devise activities to help
pupils avoid or overcome common errors and difficulties.
The range of applications
of the basic ideas; the
variety of settings which
are unified by the topic.
LANGUAGE
PAllERNS
The verbal side of behaviour, needed to master the topic.
Pupils will be using some of the technical words already
but perhaps without mathematical meaning.
Part of the behaviour
topic - what pupils are
can automate. Techn
pupils recognise when
and METHODS
TIO
ROOT
IMAGERY or
SENSE OF
The inner sense of t
made between conte
language patterns a
STUCK?
Good! RELAX and ENJOY it!
Now something can be learned
Sort out
What you KNOW
and
What you WANT
-
SPECIALISE
GENERALISE
Make a CONJECTURE
Find someone to whom to explain
why you are STUCK
WHAT TO DO
WHEN YOU ARE STUCK!
The following suggestions do not constitute an algorithm. They have been found helpful by others, but
they will only help you if they become meaningful. The way to learn about being stuck is to notice not
only what helped to get yougoing again, but also what contributed to you getting stuck in the first place.
Such 'learning from experience' is then available for use in future situations.
Recognise a n d a c k n o w l e d g e that you are STUCK
Record this in your working as STUCK! In so doing, you will have broken out of the familiar experience
of going round in circles, retreading unfruitful ground, and will have focussed your energy and
attention on devising a strategy to get unstuck.
) Write down t h e headings KNOW and WANT
Under KNOW, make a list of everything that you know that is relevant. Where helpful, replace
technical terms with your own words and include some examples.
Under WANT, write down your current question in your own words. You may need to go back t o the last
time you wrote down a CONJECTURE to see ifyou have lost sight of where you are in the question. Hence
the value of recording conjectures a s you work.
Your new task is to construct a bridge, an argument linking KNOW and WANT. Sometimes the act of
listing under these headings will be sufficient to free you from what it was that was blocking progress.
Sometimes you will need to narrow down your question to a sub-question that you feel you can tackle,
or you may find i t useful to articulate the prompts 'If only I can show/geffdo . .'. If you are unable to
progress, you may need to SPECIALISE further.
.
SPECIALISE
Replace generalities in KNOW and WANT with particular examples or cases with which you are
confident. Try to get a better picture of what is going on in the particular cases, with an eye to
generalising later.
SPECIALISING has two functions:
to enable you to detect an underlying pattern which can lead to a generalisation, perhapsin the form
of a CONJECTURE;
to simplify a question which is givingyou trouble to a form in which progress can be made, leading
to a fresh insight on your original problem.
Be SYSTEMATIC, and collect the data or examples together efficiently. Apattern is less likely to emerge
from random specialising, or from a jumble of facts and figures. Draw clear diagrams where
appropriate.
SPECIALISE DRASTICALLY by simplifying wherever possible in order to find a level a t which progress
can be made. Sometimes this involves temporarily relaxing some of the conditions in the question.
If you are still stuck, you m a y need to TAKE A BREAK
Simply freeing your attention from the problem - or explaining it to someone else - can lead to a falling
away of the block.
Copyright O 1988 The Open University
1.2
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