PREPARING
TO TEACH ANGLE
Prepared for the course team by Peter Gates andPete Griffin
Centre
Mathematics
Education
Project MATHEMATICS UPDATE Course Team
Gaynor Arrowsmith, Project Oficer, 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 Griffin, Author, Open University
Nick James, Liaison Adviser, Open University
Barbara Jaworski, Author, Open University
John Mason, Author and Project Leader, Open University
Ruth Woolf, Project Secretary, Open University
Acknowledgments
Project MATHEMATICS UPDATE was funded by a grant from the Department of Education and
Science. We are 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 t h a t they were working on parts of this pack.
The Open University, Walton Hall, Milton Keynes MK7 6AA.
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 15920 6
-
..
1.2
CONTENTS
0
INTRODUCTION
AI MS
WAYS OF WORKING
With this pack; With your colleagues
1
PREPARING YOURSELF
Getting started; Consolidating; Posing questions;
Looking at angle from a different angle; Looking back
2
3
INTERLUDE: A FRAMEWORK FOR PREPARING TO TEACH
PREPARING YOUR LESSONS
Using your textbook or scheme; Using contexts;
Measurement (a standard technique); Looking' back
4
SOME RESOURCES
5
6
THE LAST WORD
BIBLIOGRAPHY
0 INTRODUCTION
Have you ever noticed that pupils who seem to know about
angle have difficulties in interpreting diagrams involving
angles?
Have you ever wondered why a lot of pupils make errors when
measuring angles with a protractor?
Angle is often seen a s one of the 'smaller' topics in the secondary mathematics syllabus,
compared to, say, algebra or transformation geometry. However, there are many complexities
in the concept of angle that have exercised mathematicians for a long time and it is possible t o
spot the results of these complexities in the behaviour of pupils in the classroom.
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
angle?
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 angle and to the
challenges of teaching i t at the secondary level. And, along with work on other PREPARING TO
TEACH packs in the MATHEMATICS UPDATE series, it should also help you t o 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 it provide a model for how to teach angle.
Rather, it sets out to help you develop a framework with which to inform your thinking about
angle 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, you use. The framework outlined in this pack is
appropriate t o 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 angle.
To provide activities which illustrate the essential elements
of teaching angle.
To help teachers in assisting their pupils to develop a better
understanding of the nature of an angle.
To develop creative ways of using textbooks or schemes.
To encourage and promote a way 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 at 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 do the
mathematics. Alternatively, you could ask yourself what i t 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 as the very act of doing leads t o
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 as 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 that pupils made which highlighted an area of
uncertainty (or seemed to provide insight for others)?
Such experiences, which can prove t o 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 that are used throughout the pack (language patterns, imagery, standard
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 as
headings for this purpose.
The benefits of keeping such 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 are 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;
whole 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.
1
1
...
Ideas l Activities
II
Contexts
I
Imagery
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 that you intend to use). These symbols serve as a reminder that the notebook is
available to you so that you don't let something you have noticed pass you by:
Remember though that your notebook is for things that you notice and that you think are
important. You may feel that 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 - i t 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 and
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 is only vague and fuzzy in your
mind, and can help you recall what you have learned, even when no one else is present t o
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 the times when things
don't go right (or to write them off as 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 that of swapping
anecdotes (although this is useful).
All classroom experiences, when shared in the spirit of a conjecturing atmosphere, can
provide the possibility of focussing, as 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
This section focusses on the problems that pupils seem to experience with angle, and the
challenges involved in preparing yourself to teach the topic. Will the children that you teach
have the same ideas about angle a s you, or will there be many different ideas, possibly
misconceptions, which exist before you even begin to meet the topic together? It is invariably
useful and illuminating to tap these ideas and to get a feel for the various senses of angle.
Here you are invited to try to get a t those senses of angle by engaging in activities. You are
also invited to think about how you could devise similar activities for your pupils.
GETTING STARTED
Mathematical ideas and concepts are often understood from afar; rarely is there time to get
involved in a topic and to see i t from the inside. To be involved in some doing straightaway is
a useful starting point in coming to grips with any topic.
[NI As mentioned in the introduction, you should use your notebook to record anything which a t the
time seems important or relevant to you.
) FEELING ANGLES Form a group with two others and decide who will be the 'turnert. Give your
turner the following instructions.
Stand with your arm extended and pointing towards another member of the group.
Rotate on the spot until you are pointing towards the other member.
Rotate back and feel the turn (close your eyes if it helps).
Now get the turner to give the other members of the group instructions like:
Both move directly towards me.
Both move directly away from me.
One of you move nearer, one of you move away.
One of you move five steps away, one of you move two steps away.
Then repeat the turning instructions. Each time encourage the turner to feel the size of the turn
if this is possible.
Discuss in your group what you are doing. Try to record in pictures what the group has just
done and write down what you have noticed.
Now investigate what happens if the turner moves (forwards, backwards, from side to side)
while the others stand still.
You may wish to do this activity more than once: the first time to experience the doing of it;
the next to try and notice what you are doing. Do members of your group see angle in different ways? Were
there any insights about angle which came to you as you worked through the activity? (In case you want to
do the activity again, it is included in Section 4 under.the titles FEELING ANGLES and FEELING ANGLES
Comments
AGAIN .)
WHAT IS ANGLE? Think for a moment about what you believe to be involved in the topic of
angle. What words and phrases come to mind? Share these thoughts with your group.
Imagine now a group of pupils prior to you introducing the topic. What sort of experiences
might they have already gone through which involve the use of angle? What ideas about angle
might they already have? ' Try to write down a list of words and expressions that your pupils
might already associate with the idea of angle.
Comments You may find it useful to use the technique of brainstorming here. This involves freely
generating words, phrases and images and then writing them down so they can be reflected upon in order to
help the group develop feelings for angle.
Pupils use phrases like 'That's a sharp angle' or 'Let's look at it from a different angle'. To what extent does
the cultural background of your pupils affect the sort of language patterns which appear? There will always
be differences in the ways that different people come to the same topic because experiences are so varied.
Although such variety could cause problems in a class, it is possible to turn this to advantage by beginning a
topic with a sharing of ideas and experiences. Doing this activity in the classroom provides a way of sharing
ideas. Going through this activity yourself before beginning to teach a topic is a powerful way of putting
yourself in your pupils' shoes.
The idea of angle is embedded in our language, and words and phrases connected with angle
are used all the time in and out of the classroom. These are the language patterns associated
with the topic. They can help in the understanding of angle and they can hinder. That is why
an awareness of them is a good starting point when preparing to teach the topic.
When preparing to teach any topic i t is important to have a feeling for where the pupils are and
what conceptions, language patterns and images they already have. The only way to achieve
this is to encourage pupils to express their ideas in response to a question or stimulus. The type
of question asked is important. Questions aimed a t eliciting a correct answer rarely give a
child the opportunity to illustrate her understanding: either she knows the answer or she
doesn't! To gain insight into a child's thought processes on a topic it is important to ask
questions which highlight the critical conceptual issues involved in various aspects of the
topic. Successful use of questions in this way depends on encouraging the right atmosphere in
the classroom, one where being wrong isn't cause for shame or embarrassment, but where
pupils are prepared to say things they think could be correct. Developing a conjecturing
atmosphere in the classroom can help here.
F IMAGINING
This activity invites you to create some mental images. I t is therefore necessary
for someone, your tutor perhaps, to read out loud the following instructions.
Close your eyes.
Imagine a small, straight line.
Make it move around a little and then bring it to rest in the middle of your field of
vision.
Bring in another line and move i t around until the ends of your two lines meet.
Think of where they meet a s a hinge and open and close the hinge. Get used to
opening and closing the hinge.
Arrange it so that you have a very small angle between the two lines.
* Now make the angle between the two lines very large.
Make an angle of go0, 180°, 60°, 270°, 120°, 150°, 350°, 400".
Now open your eyes.
Discuss what you just imagined with one colleague. Draw some pictures if i t helps. Write
down some of the words and phrases to do with angle that were used in your discussion.
Compare your results with the rest of your group.
Comments
This activity is reproduced in Section 4 under the title IMAGININGif you want to use it again.
MORE THAN 360" When a pupil was asked to draw a diagram showing a 450" angle she said
'450" is more than 360" and you can't have angles more than a whole turn'.
What would you say to this pupil? What ideas about angle does this statement highlight?
b
Occasions where this type of situation is being discussed are precious because they allow the
possibility of real insight into a child's sense of angle.
Comments
It is worth developing the skill of inventing short but challenging questions like 'Can you
draw an angle of 450°? or 'Draw a triangle with two 100" angles'. If you only ever consider
the standard textbook questions such as
2
Find the missing
Draw the
following angles:
W, 60". 1200
then you are unlikely to get a view of the child's real perception of angle. Try also to use
questions which are puzzling, paradoxical, unusual, or impossible; it is often only then that
real learning occurs.
It is tempting to think that a statement like the one in MORE THAN 360" reveals a basic
misconception for the pupil and that i t can be resolved by making some sort of correcting or
clarifying remark. I t is worth resisting this temptation as such an interjection could very well
bring the conversation to a halt. Situations in which such things are said are extremely
valuable as they provide the opportunity for rich mathematical discussion.
SAME AND DIFFERENT
Consider these pupils' remarks:
540" is really just the same a s 180" because you just go all the way round to end up
where you started and then go another 180";
60" is really the same as 300".
Comment on these remarks. In each case, draw both angles for yourself and try to write down
some things that are the same about them and some that are different. What assumptions do the
statements make about angle? In what ways could such statements provide the basis for useful
discussion in the classroom?
b
Did you notice any difference in the way you pictured the angles in the first statement
compared to the second? Discuss this with your colleagues.
Comments
MORE THAN 360" and SAME AND DIFFERENT both invite the forming of a mental image of an
angle. By forming your own image of an angle and discussing it with others you are building
an understanding which is based on.your experience rather than someone else's. Also,
through discussing your own images and those of other members of the group it is possible to
gain further insigh ts.
REFLECTING Have any important insights about angle emerged from these activities? What,
[NI if anything, has the use of imagery in these activities provided that a pencil and paper exercise
would not?
b
CONSOLIDATING
When writing down your thoughts during earlier activities you may well have observed two
ways of thinking about angle. Angles can be thought of a s objects like the corner of a desk or.
one torn-off corner of a triangle. For example, you might point to the corner of a room and use
the word angle to describe some property of it. This idea is often used to describe an object - as
in angular - and so gives a measure of pointedness. Alternatively, angle can be pictured as a
measure of turn where a sense of movement is a n important part of its meaning.
This first way of thinking of angle involves a static sense of angle and the second a dynamic
sense. We invite you now to think about how your images of angle might help you to address
the question of static and dynamic and the link between the two.
) STATIC/DYNAMIC Go back to the list of words and phrases that you prepared for WHAT IS
ANGLE? and classify each one as either static or dynamic or both.
How much importance do you attach to pupils having a well-developed sense of both the static
and the dynamic properties of angle? .
Compare your views with those of your colleagues.
F
REFLECTING STATIC/DYNAMIC explores in more depth the language patterns associated with
[NI angle. Are there any words or phrases which give particularly vivid insights into the way you
and your colleagues see angle?
.
.
When thinking about preparing yourself to teach angle what benefit have you gained from
seeking out and discussing these words and phrases?
F
Comments Don't forget to use your notebook for recording these insights.
) IN THE CLASSROOM Try IMAGINING, MORE THAN 360° and SAME AND DIFFERENT with some
of your pupils. Afterwards try to reflect on what insight this gave you about pupils' notions of
angle and what insight the activities might have given them about angle.
b
Com m.ent s
You may need to modify the language used in classroom activities.
Pupils' experiences of angle are likely to be a lot less sophisticated than those of teachers and
difficulties will arise if they are made to shift attention from a dynamic feel of angle to a static
representation too quickly or with too little time for assimilation or reflection. Indeed, some
pupils may have already made a premature shift and so are stuck in a static sense of angle
without a n appreciation of the need for a dynamic sense when looking a t a diagram such a s the
following.
Some research has suggested that a lot of pupils' errors may be attributed to pupils being
presented with a generalization or an algorithm which doesn't relate directly to their own
experience. They become confused when things are talked about in a way with which they are
not familiar. For example, describing angle a s a measure of pointedness is fine when an
angle is acute, but if pupils are asked to consider an obtuse or reflex angle such representation
may lead to errors.
It is important that pupils be given the time to come to their own understanding of a topic. ORen
i t is falsely assumed that pupils of secondary age have already reached a level of abstract
understanding. If they are taken to a new level too fast they may fail to master some concepts
and misconceptions can arise.
F
THINKING BACK If you have taught angle before, try to recall some of the responses which
seem to occur over and over again and which suggest that there is some common gap in pupils'
understanding of angle.
Try to remember in as much detail a s you can, the specific tasks set or questions asked.
What exactly were your pupils' responses? I s there any evidence which suggests that these
responses might stem from the same misconception?
Are there any common experiences in your group?
F
It is unlikely that common misconceptions exist only in your own classroom. What you will
probably have found in discussing your findings with colleagues is that the mistakes pupils make are more
widespread than you might initially believe. If some standard misconception is identified within your group
it is worth pursuing and analysing its root. Use your notebook to record any thoughts, ideas, anecdotes, etc.
related to these misconceptions.
[NI Comments
What separates standard misconceptions from errors or mistakes and makes them more worthy of study is
the systematic and consistent way in which misconceptions produce responses of a certain type. In fact, a
misconception,just like a conception, is a process of thought (a useful thing to study), whereas an error or
mistake is merely the end product of that process and, as such, demonstrates little about the pupil's
understanding.
F
MISCONCEPTIONS The following diagrams illustrate some examples of typical statements
made by pupils of secondary age. Try first to identify the reasoning in each statement and
write down what this tells you about pupils' notions of angle.
A
B
Angle A is the smaller
A
B
Angle A is the larger
.
.
..
1. ( 4
. . ,
A
,
B
C
Angle A Is the only righf :angle
.
,.
.
,,
'L
A
,
,
Angle A measures 130"
The first two statements are likely to be attributable to the pupils not having a feel for, or
sense of, the way angles are recorded. They may have been introduced too quickly to static representation
and may have assumed that the size of an angle has something to do with the length of the arms or the size of
the arc. The third statement suggests that some pupils see right-angles only in a particular orientation; the
fourth seems to be confusing static and dynamic representations so that an angle is just any measurement
shown by the protractor scale, and the final statement suggests a confusion between angle and the amount
of space perceived in the triangles.
Comments
Between 1978 a n d 1982 the Assessment, of Performance Unit analysed responses from pupils on
various topics (APU 1978-82) a n d their results provide some insight into t h e r a n g e of
conceptions of a n angle.
) IN THE CLASSROOM U s e t h e APU questions, listed on pages 15-16, in your classroom to
explore misconceptions on angle. (Note t h a t t h e asterisked responses a r e t h e correct
responses.) You may like to investigate contrasts or comparisons between pupils of different
ages.
P
REFLECTING T h e l a s t three activities have been concerned with the standard misconceptions
[NI associated with angle, that is conceptions t h a t can lead to errors. Are there other standard
misconceptions of which you a r e aware b u t which have not been mentioned so far?
How do these misconceptions relate to t h e seeing of angles as static or dynamic a n d the ability
to switch from one to the other as appropriate?
Comments
There is clearly something about the notion of angle which causes pupils to have difficultyin
coming to an understanding of the underlying ideas. Being aware of these potential difficulties can help you
to respond in an appropriate manner when they appear in the classroom.
v
Item A2
Age
11
Put a r l c k by the c o r r e c t p n r v e r .
put a tick by t h e c o r r e c t answer.
~ n q l cA 1 s b l g g e r than Angle B
-
Anqle B i s b l q q c r man Angle A
------
-91s
YOU
A and Anqlc B a r e t h e same s i r e
can't t e l l
Other
-
-----------
Anqle A I s b i g g e r than Anqle B
33%
Anqlc
4X
4
I s b l g q e r than Anqle A
Anqlc A and Angle B a r e t h e same . I r e
52%
4%
You c a n ' t t e l l
6%
Other
-----------
-----------
Omit
1%
Omit
B
17%
' 61%
3%
2%
16%
1%
Comparing Angles when Line Length and Radius of Arc are both varied
Item AS, below, has been used with both 11 and 15 year olds.
Item A5
Put a ring round the l a r g e s t angle.
"--&p
2
-
4
Item A5 Responses
Age 11
.
Item B6
Age 11
"Measure the s i z e of this angle."
(Actual size 53O)
Responses:
* 53O
0
f 1
127O f l0
No response
Other
Question not put
47%
2%
7%
38%
6%
Age 15
Item B7
"Measure the s i z e of t h i s angle."
(Actual size 106~)
W
Responses:
No response
Other
Question not put
8%
detailed coding scheme was used for item B7. A total of 16% used
the wrong scale on the protractor; half these gave the supplementary
acute angle as their answer, the rest also misread the scale.
A
Source:APU (l978-82)
But it is not sufficient simply to know what misconceptions pupils have; teachers also need to
consider both what they result from, and how they can be circumvented. I t seems likely that a
lot of the misconceptions to do with angle may stem from a failure to understand the very
nature of an angle itself (and this seems to be supported in the results of some of the APU
work). I t is often assumed that, by secondary age, pupils will have gone beyond the need for
apparatus. However, in many cases it may be that pupils are still not aware of the basic notions
behind a topic. For example, in the case of angle i t seems common for pupils of all ages to
confuse 'angle' with 'arc' and 'vertex'.
For any topic, pupils have a collection of experiences which need to be connected together, and
to do this they need the time to explore their ideas by doing activities which bring them up
against the basic conceptual ideas behind the topic, and by talking about these ideas to others.
Developing language patterns is a part of reaching an understanding about concepts. Keeping
pupils in a situation long enough for them to develop language patterns, and encouraging them
to develop their own ways of talking about angle can therefore be a first step towards helping
them to make connections about angles for themselves.
) ANGLE IN OTHER SUBJECTS Where does the idea of angle arise in other subjects? Can you use
such observations to develop a task which gives your pupils an opportunity to talk about angle?
F
If your main teaching subject is not maths, you may like to begin by thinking about this
activity from the point of view of your main subject.
Comments
Remember that one of a maths teacher's main tasks in teaching the topic of angle is to help
pupils recognise the significance of a static representation of a n angle, the sort of
diagrammatic representation they will have to become adept a t manipulating. Instead of
presenting pupils with a ready-made picture of an angle they need to be encouraged to find
their own ways of writing about and recording an angle.
Using personal images is a way of helping pupils begin to come to terms with formal
mathematical recording and notation. Having them develop their own recording strategy is a
way of bringing them up against the difficulties of representing what is a dynamic idea
(turning) in a static representation. If they are given the time to come to this realization for
themselves they are more likely to have a sound basis upon which to build.
POSING QUESTIONS
One of the activities which is fundamental to mathematics is that of asking questions. New
mathematical ideas arise as a result of people asking questions and then struggling to answer
them. The mathematical topics that are familiar in school have not always been present but
have emerged through history a s a result of this process.
Quite a bit about the nature of angle and the number of different ways in which i t is seen is
conveyed by the fact that the question of when the concept was first introduced is far from
settled and the original definitions many and varied. For example, you may well have
pondered on the reason for the use of the term 'arm' when talking about angle and, in fact, 'das
Bein' (the leg) is used in the same context in Germany. This prompted Kline (1982)to express
the view that the concept of angle emerged a s a result of observing angles formed a t the elbows
and knees.
As you might expect, the fields of astronomy and geometry gave rise to many questions a t the
heart of the idea of angle, though originally it was distances or lengths of time that were being
investigated - not angles - because there was no vocabulary for angular measure a t that time.
The early Greeks had a notion of angle which included the idea of angles in solids and on
curved surfaces a s well a s those in a plane formed by straight lines (with which people are
more familiar). Furthermore, there was considerable debate among Greek philosophers as to
whether angle was a quality or a quantity and some early definitions of angle highlight this.
the rate of divergence of two lines at their point of intersection (Plutarch);
the' deflection or breaking of a single line (Aristotle);
contracting of a surface or a solid at one point under a broken line or surface
(Apollonius);
the inclination to one another of two lines in a plane which meet one another and
which do not lie in a straight line (Euclid).
Aristotle's deflection and Apollonius's contracting seem to indicate a quality, while
Plutarch's rate of divergence admits the possibility of something which could be measured.
And although Euclid's definition suggests a quality, i t is clear from his work that he
definitely regarded angle as something quantifiable.
The first commonly used unit of angular measure was the right-angle and this was mainly
used in building and marking out plots. Even before the time of the Greeks, the concept of
perpendicularity and the use of plumbline devices were important and so the right-angle
became the standard unit. In early astronomical records the right-angle and fractions of a
right-angle were used. With complete revolutions of planets taking rather a long time the
desire, in astronomy, for a smaller unit soon emerged but it is not entirely clear where the idea
of 360 divisions of a complete turn originated.
Somewhere between 4000 BC and 3000 BC the Babylonians began to study astronomy and its
relation to the calendar and the seasons and produced a mass of data from observations, but no
record of Babylonian calculations using anything akin to degrees has been found.
Jones (1953) suggests that 360 may have arisen from a rough estimate for the number of days in
a year when considered as 12 thirty-day months. This view is supported by the observation that
early Chinese geometers divided the circle into 365 parts as they took this to be the number of
days in a year. Hypsicles (about 180 BC) was the first Greek astronomer to divide the circle of
the zodiac into 360 parts by following the Chaldean practice of dividing each of the 12 signs into
30 parts. This particular usage may well have made popular this form of division and hence
the systematic use of the degree.
Ptolemy (about AD 150) constructed detailed astronomical tables and used subdivisions of the
degree. Almost certainly, because of the Babylonian sexagesimal number system, he used 60
as the divisor. According to the Latin translation he subdivided his degrees into 60 'partes
minutae primae' and each of these into 60 'partes minutae secundae'. I t is from this that the
sub-units 'minutes' and 'seconds' are derived so that one degree equals 60 minutes and one
minute equals 60 seconds.
Although the desire for a unit of angular measure stems from the consideration of the rotation
of planets in assumed circular orbits (a dynamic notion), i t is interesting to note that all
astronomical records are based on linear measurements such as arc length or the length of a
chord of a circle (static notions). And, in fact, i t is still unclear whether the degree was
originally a 360th of the circumference of a circle or a 360th of a full turn.
What emerges from all this is the feeling of people trying to make sense of the things around
them and, through asking questions, seeking solutions and refining definitions, coming to a
clearer understanding. Such questions might be named root questions, as they lie a t the heart
of the topic being studied.
REFLECTING What has the above discussion contributed to your sense of angle? Is there any
way in which an awareness of the root questions behind angle could be of benefit in the
classroom?
If pupils are to come to a clearer understanding of angle then asking questions is important
and, in order to ask useful and illuminating questions about angle, i t is necessary to
introduce situations which prompt the use of angle and discussion about angle.
I t is also important for pupils to be aware of the standard techniques and methods that are used
in angle (and indeed in any topic) and to become proficient in their use. Without the
knowledge and confident use of these techniques it is impossible to have a full and rich
understanding of angle. Working on questions stimulated by everyday situations in which
angle arises can provide a basis for analysing such techniques.
P
SWING Imagine a child's swing moving to and fro. Try to devise some questions about this
situation that involve the idea of angle. Try to make the questions engaging to ensure that they
encourage activity. Don't worry too much initially about whether the questions are good or
bad; remember that you are working in a conjecturing atmosphere and that the very act of
devising questions is intended to spark off discussions within your group about the usefulness
of such an activity in your classroom.
b
Possible questions include:
What is the maximum angle you can swing through?
If the seat sweeps through 40°, what angle does the child's feet sweep through?
You may feel that such questions seem a little vague and difficult to get started on. You may also feel that
you weren't given enough information in the activity even to get started yourself. That is deliberate. By
exploring situations yourself you become much more involved in them. The questions become yours and
you are likely to have some investment and motivation in coming up with solutions.
COm ments
WORKING ON QUESTIONS Take one of the questions that you developed in SWING and work
on i t for about half an hour. It may not be necessary to go out and find a swing to do this, though
i t may be fun! You could, of course, use a model or a diagram.
b
REFLECTING This subsection considered how you might generate and then work on questions
[NI which arise while looking a t angle in various contexts. Such contexts may stem from other
subjects in the curriculum, from some historical.anecdote, or from everyday situations.
What sort of questions did you find yourself asking?
Did you get stuck? If so, what did you do about it?
Are there any differences between the sorts of questions that you considered here and the
questions on angle that you might find in a textbook?
LOOKING
ANG LES FROM
DIFFERENT ANGLE
This section has looked a t the topic of angle in a number of different ways but so far it is likely
that you have been fairly familiar with the topic itself, if perhaps a little rusty. Finally,
however, we want to try to present you with a situation in which you confront angle from a
different direction (or from a different angle!). In working through what follows, keep track
of the sorts of things you are doing. You may be going through processes that are similar to
ones that some pupils go through when they first experience angle.
A NEW ANGLE
Imagine a short line fixed a t one end. Rotate i t about the fixed end so that the
free end travels through an arc which is the same length as the length of the line.
Stop here for a moment and draw what you have just imagined.
This is now going to be used a s the basic unit of turn (instead of the degree). I t is, at least, a s
sensible a s using the rotation of the earth around the sun!
This measure of t u r n is called 1 radian because of t h e use of radials or radii in its
construction. Here are some questions which seem natural to ask a t this stage. Attempt them
a s best you can.
What happens if the length of the line is twice a s long?
What statements can be made about 10 radians?
How many radians in a complete turn?
How many radians in a right-angle?
Comments It is possible that you have just been introduced to a new way of measuring an angle. You
have probably (depending on your previous experience) been put into a situation where you were
experiencing something completely new. This may not be so dissimilar from the sorts of experiences offered
to pupils every day. You probably found it fairly easy to make some sort of response to the first two
questions but you may have found the last two quite difficult. However, you will have time to consider these
questions again as we will return to them later. Meanwhile, don't move on until you feel you have really
worked on the questions as much as you can.
) REFLECTING You have now done a n activity and possibly done quite a bit of mathematics on
[NI the way. Now reflect on some of the elements of A NEW ANGLE which may be relevant to your
teaching.
What were your feelings when you were asked to do the activity?
If you found the questions difficult or challenging, why was this?
Did you try to answer the questions through a static or a dynamic picture of a radian?
Were you aware of any new techniques or methods t h a t you were using?
What relevance do these questions have for your teaching of angle?
Compare your reactions with those of your colleagues.
It is possible that you found A NEW ANGLE difficult because it involves a way of measuring
turn with which you are not familiar. You might have felt some reluctance because you could not realise the
purpose of measuring in that particular way. You might also have felt that the introduction of some new
technique in addition to a new idea was difficult to cope with. This underlies the need for the learner to be
familiar with the measuring unit. Although the idea of turning is quite basic to you, you may not have had a
clear sense of what a radian looks like or feels like. A pupil may well have a feel for angle as a measure of
turn but the introduction of degrees as a unit of measure may seem so sudden and artificial that it causes
problems where there were no problems before.
Comments
Let us work a little more on the idea
of measuring with the radian a s a unit of turn.
(Remember t h a t if you are experiencing difficulty this may well provide you with valuable
insight into pupils' perspective of being introduced to a new unit of measurement.) The radian
is not just a measure we have come up with for the purposes of this pack. It i s a standard unit of
angle measure, in fact, the standard unit in higher mathematics. Indeed, if you have a
scientific calculator you may have wondered what the 'rad' means when you press the 'mode'
key (the method of doing this may be different, depending on your caIculator).
WHAT
IS A RADIAN? HOW BIG IS IT?
These questions could be answered by simply giving a definition (and a t this stage you may
like to say, in your own words, what such a definition might be). But this doesn't get u s much
further. What i s needed is a way of comparing radian measure with something which i s more
familiar. For example, consider the following line of thought:
How could I draw an angle of 1 radian? What do I need?
A circular ruler, a protractor graduated i n radians.
I don't have either of these. What do I have?
A protractor graduated i n degrees.
What do I need now?
A method of converting radians to degrees.
What do I know?
The definition of a radian.
) REFLECTING Go back to the sketch of one radian t h a t you made as a result of A NEW ANGLE
and redraw it so t h a t the radii are bigger and then redraw i t again with the radii smaller.
b
Comments
Notice how the idea of a radian shows that the lengths of 'arms' do not alter the measure of
turn.
The diagram above seems to be related to a part of a circle.
The circumference of a circle is,Zm, so how many radian-sized segments could you fit into a
whole circle? You may like to make (out of card) a number of radian-sized segments and a
circle on which to fit them. A piece of string might help.
) REFLECTING AGAIN Look back a t the questions listed in A NEW ANGLE. Are they easier
now? Try to come to a satisfactory conclusion about each one.
b
) TEXTBOOK There are many activities, which pupils are asked to do and which involve the
use of degrees, where t h e introduction of the idea of a degree h a s been quite swift and
seemingly matter of fact. The following page from a school textbook introduces the idea of a
degree. Imagine t h a t the page has been modified so that the measure of turn which has been
offered is the radian not the degree. What words and phrases would be different? How would
you feel if presented with this exercise to do as a n introduction to the radian?
An angle is an amount of turn.
Here are some things w h ~ c hhave turned.
For each picture write down:
(a) the fraction of a turn
(b) whether clockwise or anticlockwise
" means degrees
Fractions of a turn are too clumsy to describe some angles.
Over 4000 years ago the Babylonians realized this.
They divided a full turn into 360 equal parts.
We call each part a degree ("1.
So,
lfull turn = 360"
How many degrees in a:
(a) f turn (b) i t u r n (c) $turn
(d) turn
(e) i t u r n ?
A f turn is a common angle.
You will see it all around you.
We call it a right angle.
1 right angle = 90".
This is its special mark.
These lines are perpendicular
to each other.
1. Describeordraw5differentrightanglesinyourclassroom.
2. How many degrees in:
(c) 4 right angles
(a) 2 right angles
(d) $right angle?
(b) 3 right angles
Source: Graham, D. and Graham, C. (1981)
In the light of your responses to the questions above, re-write the page so that it provides a better
introduction to the degree.
REFLECTING What do you feel you have gained from looking a t angle in this different way?
What insights has it offered into ways of introducing your pupils to the degree as a unit of turn?
CALCULATING Make sure your calculator has some means of switching from degree
measure to radian measure (it may be accessed in lots of different ways) and begin by setting
it to degree measure. By entering an angle and then pressing the sine button, find the values
of the sine of 90" (usually written as sin SO0), sin 180°, sin 270" and sin 360".
Now change to radian measure and repeat this exercise with the same numbers as before (that
is, 90,180,270 and 360). What do you notice? Why does it happen?
Comments The lack of realization of this difference between sin x (in radians) and sin x (in degrees) and
other ways in which the use of the radian affects the results of calculations is a source of many errors in
sixth-form mathematics.
WORKING WITH RADIANS U s e y o u r calculator (set on degree m e a s u r e ) t o m a k e a
comparison between a n angle (X) a n d t h e sine of t h a t angle (sin X) for a small angle. For
example, you may like to s t a r t with 1" and compare 1" with sin l". Repeat this as you decrease
the angle to O.lO,O.OlO,O.OOlO,etc. What happens as the angle gets smaller a n d smaller?
Now change to radian measure a n d repeat t h e exercise. W h a t can you say about t h e connection
between x and sin x as x gets smaller a n d smaller this time?
b
SECTORING T h e circumference of a circle is equal to 2nr. Using this information can you
find a formula for t h e length of t h e a r c of a sector if the size of its angle i s known in degrees?
How does this formula change if the size of the angle is given in radians?
What about a formula for t h e area of a sector?
b
LOOKING BACK
This section h a s dealt with some ways of getting a sense of angle and of helping your pupils to
do so.
REFLECTING H a s your sense of angle been modified i n a n y way by working through this
section? If so, how?
Has working on this section, on your own a n d with your group, contributed to t h e process of
preparing yourself to teach angle?
Have a n y awarenesses arisen for you while working through t h i s section t h a t relate to
preparing yourself t o teach any topic?
Compare your answers to these questions, a n d indeed to a n y others t h a t have arisen i n your
group, before moving on.
Comments You may have arrived at some structure or framework for thinking of and teaching angle 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 angle. 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 angle 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 angle (preparing yourself)
using external resources (textbooks, schemes, etc.) a s a prompt or inspiration for
devising activities on angle in your classroom (preparing your lessons).
We suggest that a framework with which to inform your thinking about angle 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
Techniques
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 angle, and in turning thinking into action in the
classroom.
LANGUAGE PATTERNS
Have you ever heard people utter phrases like 'Be careful, that desk has particularly sharp
angles' or 'I can't score a goal from here, the angle is too thin'? Are they just throw-away
phrases or do they convey anything about the way people see angle?
How do pupils interpret phrases like 'the angle at A' or 'the angles on a straight line add up to
180°'?
The language used to talk about angle (the words and phrases used in everyday speech as well
as the technical vocabulary found in textbooks) has an effect on people's understanding and
general sense of the topic. These words and phrases may be called the language patterns
(associated with angle) and they can be particularly useful in helping to appreciate various
ways of seeing angle.
To be able to share ways of talking about angle and to look closely a t what we believe angle to be
about is a valuable starting point. This facilitates building on what is already known about
angle and encourages the deliberate and consistent use of particular language patterns in the
classroom in order to help pupils to express their understanding of concepts in such a way as t o
be understandable by others.
IMAGERY
Under imagery we 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. Angle is typical of many mathematical
topics in being essentially a dynamic idea, but which becomes static when represented on
paper, and the imagery heading is a reminder that behind words and diagrams there are
meant to be associations and a sense of the dynamic.
The act of trying to conjure up an image of something is closely linked'to struggling t o
understand the concept behind it 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. A simple activity like imagining a series of angles can help to create a valuable
shared experience for the group so that all subsequent angle activities are based on something
in which everyone has had personal involvement.
Notice also how, a t the beginning of Section 1, the language patterns and imagery associated
with angle began to raise questions about misconceptions that pupils might have.
STANDARD MISCONCEPTIONS
You may already know from observing the children in your school that there are some basic
errors and standard misconceptions that 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 invalhable help each time you are preparing to teach a new topic.
For example, in tests on 15-year-olds the APU (1981) found the following when asking about the
definition of an angle.
2.62
Finally pupils were asked to explain in words what an angle is. 4 per
cent gave responses judged to be acceptable and examples of these are
given below:
'The degree of turn between two intersecting lines.'
'Amount of rotation between two lines where they meet.'
2.63
Around 30 per cent of pupils defined an angle as the distance, area or
space between two lines and examples of such responses are:
'Distance measured between two lines.'
'The gap in between where two lines meet.'
'Area between two lines.'
'The spacing between two lines which meet at a fixed point.'
These .were not considered acceptable since no mention was made of how the
space between the lines is measured. It can be seen from the written tests that
.
some 20 per cent of pupils judged the size of angles on irrelevant features such
as the size of the arc which labels the angle or the length of the bounding lines.
2.64
A further 10 per cent simply said a n angle was where two lines meet and
others said a n angle was a corner. Nearly 10 per cent defined a n angle
i n terms of degrees, for example:
'Number of degrees between two straight lines.'
'So many degrees, like a circle is 360.'
'An amount o f degrees.'
2.65
Almost one quarter did not respond to this question and many were
surprised at being asked to explain the concept o f a n angle. Indeed one
pupil commented
'People talk about angles but do not mention what they are. '
Source:M U (1981)
What is surprising here is not so much the failure of the vast majority of the pupils suweyed to
articulate correctly what an angle is, but just how similar most of these responses are to those
you could hear in most classrooms up and down the country any day of the year!
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 satisfjl the reasons behind the question which could be anything from real
curiosity to disenchantment. Furthermore, i t 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 or problems which our
civilization h a s 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 angle into what has now become a 'mathematical 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 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 angle 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. For example, when considering various contexts in which
angle occurs, techniques such as: using a protractor; subtracting a number from 90 or 180; etc.
may be needed time and time again. 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 angle i t 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 angle 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 angle successfully.
3 PREPARING YOUR LESSONS
USING YOUR TEXTBOOK OR SCHEME
One point of reference when beginning to prepare activities on angle 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, i t 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
'L'
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.
Textbooks and schemes provide lots of information, facts and definitions with regard to angle,
but this is often not the way that one wants to introduce angle. The feel for angle cannot be
gleaned from such printed material alone.
For example, i t is likely that young children will have been brought to the concept of angle
through a notion of turning. They have probably already used words such a s turn, twist, rotate,
spin, etc. to describe angle. However, when communicating an angle to someone else, a way
of denoting an angle is needed and this almost automatically becomes a static, diagrammatic
representation. Most mathematics textbooks a t secondary level assume pupils will have
reached this higher level of representing angle, even though they may include diagrams
which seem to indicate motion, diagrams like
Such diagrams can be confusing; for example, in diagram B the arrow shows the direction of
turn but many pupils will still give the angle as the smaller, acute angle. So the way angles
are usually pictured may follow a convention but may not fit in with pupils' ideas of what an
angle looks like. Conventions, often to be found in textbooks etc., have to be arrived a t after
pupils have had a chance to explore their own way of picturing angles.
F REFLECTING Think back to FEELING ANGLES in Section 1and, as vividly a s possible, imagine
[NI yourself doing the activity again. Try to identify moments that particularly stand out for you.
What were the experien&s and insights that you gained from the activity?
Were there any moments later on in the section that drew on this initial experience?
How could this activity be used in the classroom?
F
Comments
FEELING ANGLES was placed at the beginning of the pack partly in order to encourage you to
share your sense of angle with those of your colleagues and partly in order to provide some common
experience for all members of your group that we hoped would be valuable when various aspects of angle
were introduced. How does this introductory activity compare with that to be found in your textbook or
scheme?
The PREPARING TO TEACH framework, introduced in Sections 1 and 2, provides a useful
structure for analysing what was, or could be, gained from a n y activity on angle: from
activities such a s FEELING ANGLES or from those to be found in your textbook or scheme. 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 angle, that is, to develop their language patterns?
Do pupils have the opportunity to evoke an image of angle?
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 opportunities for pupils to abstract the underlying features or essence of the
activity so that they can pose their own similar questions in other contexts?
Throughout this section you are invited to engage in activities intended to stimulate you t o
devise your own activities for use in your classroom. Questions such as those above should
also help with this process.
First, how does the use of printed material affect the way in which pupils are introduced to
angle? To attempt to answer this question and to help you gain more insight into the role of
textbooks or schemes in the teaching of mathematics, you are invited to explore what exactly is
provided by such materials.
) TEXTBOOK Pick up a textbook and search through the index for references to angle. What do
you find? What did you expect to find?
Spend a.few moments reflecting on your findings and then write down what you think this
textbook will provide on the topic of angle.
Your response to this activity may very well depend upon what you wanted from the
textbook before you opened it. If you wanted a set of questions then that is probably what you will have
written down; if you wanted something other than this you may well have seen things differently. Most
textbooks contain more than just a bank of questions to be answered; they also contain written explanations,
diagrams and pictures, ideas for discussion and practical work, and possibly other things.
Comments
In the context of the PREPARING TO 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.
Pupils are likely to interpret the technical terms and conventions which invariably appear in
textbooks in lots of different ways. A textbook's use of particular words, symbols or diagrams
can unintentionally mislead pupils unless you are aware of them. For example, in the
following extract pupils are invited to match up equal angles.
Source: SMP 11-16 (1983)
This may well signal to some pupils that area is somehow important in the measurement of
angle. The author obviously did not intend this, but no textbook can guard completely against
such misinterpretations. If pupils encounter only printed material then i t i s extremely
difficult for teachers to guard against such misinterpretations which, in turn, could lead to the
sorts of misconceptions that you saw earlier.
Another example: younger pupils may consider t h a t naming a n angle as a right-angle
naturally (and in some ways logically) leads to the idea of a 'left-angle'.
This may not be caused by any misconception but rather a failure to appreciate the specific
language pattern. Textbooks are full of technical vocabulary and conventions and you need to
be aware of their possible effects.
LANGUAGE Go through the following three excerpts and try to identify the technical terms
which pupils will encounter when working on these pages and which are related in some way
to angle. In what ways are these terms used?
Until now, we have measured angles In r ~ g hangles
t
and fract~onsof
a right angle. This 1s not always convenient, just as it IS not convenient to measure all weights In k~logrammesor all lengths I n
k~lometres.T o measure smaller angles we need a smaller unit. A
complete turn is dwded into 360 degrees, wrltten 360". T h ~ sunlt
was chosen a long time ago by the Babylon~ans.
Exercise
A complete turn = 360"; a stra~ght angle = 180";
0'.
a right angle = 9
How many degrees are there between the hands o f a clock at
R
3 p m
9
b l p m
c 2 p m
d 4 p m
Where i n your d~agramwould you mark o R a scale In degrees so as
to be able to measure the angle between O A and OB?
c6pm?
Through how many degrees does the minute hand o f a clock turn
when i t moves from
c 61010
d 91012
b 2104
a 3104
g71010
h5toII?
r 8 t o I Z
/ I t 0 6
10 What fraction o f a complete turn ts
a 90'
b 45'
c 180"
d
4
Make a drawing o f F~gure8 In your jotter. Take a piece o f thread.
or hold your ruler on edge, and keep one end fixed at 0. Stretch i t
in the direct~onOA. Turn the thread or ruler about 0 Into the
position OB. What happens to the angle between OA and O B as
you turn?
Place the thread or ruler where you thmk the slze o f the angle
between O A and the thread or ruler would be.
c 130"
d 180"
45"
b 90"
I n F~gure8 we name the angle between O A and O B e~therAOB
or BOA. We always put the letter at the vertex In the middle. The
symbol L IS used to represent 'angle', and we wrlte L AOB. L BOA,
etc.
A protractor is an lnslrument for measuring angles In degrees.
Very often i t IS semicircular i n shape but t h ~ sIS not essent~alas long
as an accurate scale In degrees is marked on 11. I n fact you may have
a protractor on the back o f your ruler. I f you cannot see how to use
your protractor your teacher w ~ lgu~de
l
you There are two scales on
it, and 11IS Important that you use the correct one. So the abhty to
make a rough estlmate o f the slze of an angle IS valuable.
270°?
3 Drawing and measuring angles
Draw a h e OA about 6cm long and place your protractor approxlmately In the posrllon o f the semlc~rcleIn F~gure8.
Swmg the thread or ruler round to make angle AOB = Zoo, then
40", 80", W",150" and 180".
I n order t o measure angles i n degrees we need a scale for cornpanson,
just as we measure lengths by comparison wtth the markmgs on a
ruler or tape measure.
Use your ruler and protractor to draw the angles below. I n each case
start by drawmg a hormontal arm about 6 cm long.
L A B C = 45"
b
L D E F = 90"
c
L G H K = 120"
Repeat questlon 3, startlng wtth a h e wh~ch1s not honzontal.
Use your ruler and protractor to draw angles of.
20"
b 72"
c 166"
Letter and name the angles yourself,
Esrin~arethe size In degrees o f each angle In F~gure9, giv~ngyour
answer hke th~s:L D E F = . ".
Source: Scottish Mathematics Group (1983)
Angles. Bwrings.Tri~ngles,fbIygons
/
6A Angles
Measuringangles
A p r o t r a c t o r IS used t o measure angles
On your protractor find.
Using your protraclar,
draw these angles In your book
a)
25"
4 150'
b) 125"
c)
/l
250'
g) 80"
50"
d) lW0
h) 280"
Label each one 'acute'. 'obtux' or 'reflex'.
On most protractors there are two sets of numbers. so ~tIS ~mportont
2 Uslng a ruler and pcncll only, draw these angles as accurately as you can
quarlcr of a rcvolutlon
b) a thud of a revolutm
C) a stxth of a revolutlon
e) a
Measure accurately the angles you have drawn and wrlte down by how many
degrees your angle vanes from the correct one In each case
J
to estlmace first
d) two thlrds of a revolutlon
twelfth of a rcvolutmn
a) a
vertex
Th~sangle IS
approwmately 45'
Usmg a ruler and pcncd only, draw thcw angles as accurately a$ you can
a ) 20"
b)
c) 110")
145"
e) 230"
n 320"
Measure each angle accurately and wnte down by how many degrees your
angle 8s tncorrect
Label each one 'acute', 'obtusc' or 'reflex'
Make sure the centre of the orotractor IS on the vertex
and the base h e ;1 on one arm ofthe angle./
Reod offthe sue of the angle where the other arm cuts the scale (42')
Compare the measurement wlth your est~mate(42" t 45')
4 Draw for yourself four trregular pentagons (5 mdes) Carefully measure the
angla In each and find the total You should find that the lour totals arc
appro~matelythe same I f they arc not. check your measurements What do
you thtnk that the total should bc?
I John measured t h ~ sangle as 138". Mary made ~t 142"
Whatdvd they do wrong'
5
M)"
8
Repeat qucruan 4, th~stlme drawmg four seven-slded figures.
1 Est~moteand then measure these angles
6
Wrln dawn the smallest angle between the hands of a clock at
atlack
b) 8 o'clock
c) 1 10
d)3a
e)620
o) 3
7 Wnte down the angle turned through by the mlnute hand of a clock In
a) l0mm
d ) 55mm
b) 25mm
e) 65mm
c)
45mm
/l 90rmn
8 The spadometer an a car IS graduated from 0 to IM)km/h In 280"
a) What angle has the pointer turned through from rest to a speed of
1) 2Okm/h
11) 32 km/h
W ) 80km/h
m ) lWkm/h?
b) As the car accelerates from 24 km/h the potnter on the speedo mete^
turns through 70" What speed has then k e n reached7
3 Measure the angles of the tr~anglesand quadr~loteralsyou
constructed for Chapter 4on pages 14and I5
Source: Dallison and Rigby (1982)
Source: Wyuill (1985)
b
Comments Specific technical terms we identified were: protractor; angle; acute; obtuse; reflex; degree;
irregular pentagon; complete revolution; base line; estimate; approximately; vertex; arm; triangle;
quadrilateral; construct; arc; level; line segment; right-angle; straight-angle; horizontal; vertical;
perpendicular; swing. You may not have included some of these but included others.
The way that these terms are used demonstrates just how much pupils are often assumed to understand
alongside their learning of mathematical knowledge. Language patterns and techniques that we noticed
here include: draw these angles; measure these angles; smallest angle; angle turns through; arm of angle;
cuts the scale; draw a line segment lOcm long; triangle ABC; measure angle c; measure and record; fraction
of a right-angle; a complete turn = 360"; name these angles; angle AOB = 20"; L DEF = 90".
) REFLECTING Technical terms can evoke certain images and give rise to certain techniques
I:Nl so t h a t by studying one-aspect of the PREPARING TO TEACH framework others will naturally
arise. What aspects of imagery became apparent to you a s you were working on LANGUAGE?
Apart from those specifically mentioned, what other techniques are suggested by the language
patterns you observed?
Are there any other strands of the framework which emerge from looking a t these language
patterns?
IN THE CLASSROOM Textbooks clearly assume t h a t pupils can talk about angle in a large
number of ways. But only after focussing first on pupils' existing language patterns i s i t
sensible to look a t those in a textbook or scheme.
Take an appropriate part of your textbook or scheme and give the following task to some of your
pupils.
List all the words and phrases to do with angle.
You may not understand the meaning of some words or phrases; don't worry, just
make your list under two headings: understand and don't understand.
What sort of discussion would you want to take place after this activity? What, in your view, i s
the value of such a n exercise?
Pupils have their own ways of talking about angle; other ways are often used in textbooks,
workcards, etc. Sometimes there is a close link between the two and sometimes it seems to pupils that a
completely different language is being used in their textbook or scheme.
Comments
There are many ways of using the activities and exercises on angle to be found in your
textbook or scheme, because there is inevitably a'process of transferring what is in the textbook
into what you (as teacher) and your pupils do in the classroom. The level of involvement that
you choose to have in this process can vary from almost nothing, such a s 'turn to page 45,
exercise 2(a), and do the first 10 questions!', to a great deal (for example, where the activity in
the classroom is of your own devising but uses the textbook a s the prompt or inspiration), with
many levels in between. A way of increasing your level of involvement i s to look at the
textbook in a n enquiring fashion, to adopt an attitude where you are constantly questioning the
purpose of text and activities.
..
d
TURNING Look a t the following extract. What ideas do you think this collection of activities
is intended to introduce?
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?
6Turnkrg
C or anhclodrmse 3
C or anhclcclrmse 3
a) hhe see the nunshle tunung clockmse
b) Fmd the mouse.
hshesee h e tumsble tunung clcckmse
Thetumsblestertskehs
It makes one full m anhclaclrmse.
Q
7
7
-
It ends k e Uus
How many people have
gone mto the m?
3 ThetumsblesmhkeUus.
Four peopleare m n g to go m.
%me ofthem go throwh lhe hunstlle,
It ends up W e llus
How many people have gone through?
1
Thm hunshe has made a hall manhcloclrmse
/through
i
rThink
The~emthls~~llUlllllgcloclrarirs
6
Eech frameshows a new p h o n aRer one person hasgone through
~pFJ'T'J~~~;
I
A
C
D
E
F
G
~ p e m n ~ v e s h e t u m s b l e a ~ e r m c m)
~cc~(~
Copy and wrnplete lhe table
Source: NMP 1(1987)
[NI Comments
When you are provided with a textbook, particularly if i t i s attractively presented and well
explained, i t is tempting to focus completely upon a n activity and not ask yourself what is the purpose of the
activity or what the pupils will actually be doing. When you begin to make this shift i n attention the textbook
becomes a much more useful tool. What aspects of the PREPARING TO TEACH framework are relevant both
to the activities i n the extract and to any modifications that you might have made? Use your notebook both
to inform your thoughts and to record them.
.
.
-
,
,
) ANGLE CALCULATIONS Look a t the following extract on a;gle calculbtions. In whit'sense
are all these examples similar and in what sense are they different?*.Arethere some.that are
more difficult than others? Rank the examples in order of difficulty.
.
. .
F
-"
Donot we a pmnacror.
1 Write down the slze of each angle marked ?
'
.
A
2 Wnte down the s u e of each angle marked ?.
Remember
L
!.
E
L
a
a
)
/
3 Write down the slze of each angle marked 7.
1 Wnte down the s u e of each angle marked ?
2 Wnte down the stze of the angle marked
a)
b)
c)
0
Q
Q
3 Thmk of a manner's compass.
a) Wnte down the sue of e~ther
angle between Nand S.
b) Wnte down the sue
of the smaller angle belween
(I) Nand NE
(11) NE and SE.
(UI)SW and E
C
W
S
Source: NMP 1 (1987)
b
When you ask yourself these sorts of questions you begin to get a sense of the purpose behind
activities and can therefore use them in the way you want them to be used rather than have their use
dictated to you.
Comments
For any activity, we therefore suggest you ask yourself these three questions.
,
,
What are the pupils supposed to do?
Do I want more from the activity?
If so, how could I augment it?
For example, consider again the following three examples from the 'think it through' box in
ANGLE CONNECTIONS.
What are the pupils supposed to do?
Notice that 360"is a full turn and divide this equally between a number of sectors.
Do I want more from this activity?
Perhaps a deeper look at regular polygons; for example, what about the other angles in
the last shape (the interior angles)?
How could I augment it?
Investigate the connection between the size of the angle at the centre and the interior
angles.
Imagine that you have given the set of exercises in ANGLE CALCULATIONS to
your pupils. You do not necessarily want the questions to be done (although you may decide
that this is useful) but you do want your pupils to arrive a t a feeling for the reason behind them.
How would you structure such an activity? What instructions would you give and how would
you expect the lesson to proceed?
AUGMENTING
Which strands of the PREPARING TO TEACH framework are evident in the excerpt in ANGLE
CALCULATIONS and which could be used to inform your thinking with regard to augmenting
the text?
SHAPES Look a t the following extract which investigates shapes from isosceles triangles.
What mathematics would you expect pupils to use while doing this activity? (You may wish to
do it yourselfl. How would you introduce the activity? How would you organise the lesson?
Think about keeping the format of the activity the same but changing some of the conditions.
Can you create a different activity? Does this new activity have the possibility of encouraging
some conjecturing and testing?
Shapes from isosceles triangles
"
You need somecard and a protractor
Drawthese ~rorcelestrrangleson card and cut them out
a) Use tr~angleA
Follow these mstructlons
-.
Mark
a
polnt
Placethe trlangle
hke fhls
Draw around ct
Turn the trmgle
Repe
Draw around ~t
unt~lyou have
a 'perfect polygon
b) Fmd out whlch of the other
I S O S C ~trlangles
I~S
make perfect polygons
Think it through
Mark thesue of each angle of your tr~angle
b) Wrltedown what ~ s r p e c ~about
al
trlangles whlch make perfect polygons
Source: NMP 2 (1987)
Comments Pupils will probably turn the triangle through an angle a s they build their 'perfect' polygon.
They may also notice the angle a t the vertex of each triangle. They may or may not make a link between
these two angles and the sort of polygon (if any) that is formed.
Questions you might ask yourself (or indeed your pupils might ask) include:
Can the triangles be overlapped?
What if there's a gap between them?
How many times can you rotate the triangle before getting back to where you started?
What do you think will happen if these questions are asked?
An alternative approach to creating a new activity might involve inventing new methods of recording
information. In this case, diagrams could be given a form of notation based on the information which
formed their construction. For example, (60°, 30") could represent the pattern produced by a triangle with a
60' angle in it being rotated by 30° each time. What further mathematical work might this stimulate?
Another possibility: LOGO could be used to enhance the investigation if i t is available in your school.
) AUGMENTING
Choose a n exercise or activity involving angle from a textbook o r scheme of
your choice, preferably one t h a t you a r e using with pupils. Ask yourself:
What are the pupils supposed to do?
Do I w a n t more from t h e activity?
If so, how could I augment it?.
,
W h a t i d e a s a n d activities arise from these questions? How a r e these ideas different from the
ones t h a t you could h a v e j u s t lifted from t h e textbook? W h a t a r e t h e benefits of t h i s kind of
activity for you?
Comments You may feel, depending on your situation, that your textbook or scheme does not provide a
sufficient variety or balance of activities or that it is lacking in activities of a particular type and therefore this
is what motivates the way in which you make modifications. But whatever your feelings, i t is important to
have a view about what you want from your activities, and to identify what it is that constitutes a balanced
classroom atmosphere.
/
USING CONTEXTS
Sections 1 a n d 2 mentioned t h a t t h e process of observing t h e contexts i n which angle occurs i s
not only a way of informing thinking on t h e topic b u t also a valuable source of activities for t h e
classroom. In this subsection we examine what these contexts might be a n d how you may go
about constructing activities from them.
Sport, i n particular any sport which involves shooting at a goal, i s a rich source of activities on
angle. Consider t h e situation i n hockey where a player i s i n possession of t h e ball somewhere
in front of t h e goal. I t i s often said t h a t if t h e goalkeeper moves forward, away from t h e goal
line, t h e n h e or she h a s narrowed the angle a n d therefore made i t more difficult for t h e player
to score. W h a t does this mean? Where i s t h e angle t h a t h a s been narrowed?
) HOCKEY T r y to write down some questions which arise o u t of trying to get a sense of what is
m e a n t by narrowing the angle. As with SWING in Section 1,do not strive at this stage to refine
wording, simply try to g e t down as many questions as you can. (A diagram m a y help here.)
B
The art of devising questions like this is very much a matter of letting go and allowing
questions to emerge without mentally vetting them beforehand. The important thing is to have a number of
questions to work on, because it is during the process of working on them that useful modifications can be
made so that new, better questions emerge. The following questions occurred to us.
Where else could the goalkeeper stand so that the shot angle is the same?
How would the keeper move in order to block the shot completely, that is, naqow the angle to
nothing?
Where would the keeper move to halve the angle, divide the angle by 3, etc.?
Comments
F
WORKING ON QUESTIONS Now, preferably with a colleague or group of colleagues, choose
one of these questions (or one you have devised yourself) a n d work on i t together.
b
) REFLECTING What, for you, a r e t h e benefits of working on such a question together? Did any
[NI ideas a r i s e which you could t a k e into t h e classroom? Discuss with your colleagues a n y
difficulties you found i n moving from t h e context given in HOCKEY to t h e questions on which
you actually worked.
b
Comments The essence of mathematics lies in the asking of questions and the need to ask questions
arises from situations. It is for this reason that we feel that the searching for different contexts and exploring
them for interesting and fruitful questions is an extremely valuable exercise for pupils as well as for teachers.
SNOOKER Consider the game of snooker. When a ball hits the cushion i t rebounds so t h a t t h e
angle i t s path makes with the cushion i s t h e same before a n d after t h e instant of contact. (Once
again, you may feel t h a t you need to draw a diagram to understand w h a t i s going on here.)
W h a t questions arise out of this context? W h a t if there i s just one ball on t h e table? What if
there a r e a number of balls? W h a t if you a r e snookered (that is, you want to h i t one ball onto
another b u t a third ball i s in t h e way)?
Comments Remember, let the questions flow out if you can (without bothering too much at this stage
about whether they are good or bad) so that they become available to be worked on further. This is when
useful ideas tend to emerge.
) LOGO The programming language LOGO which, among other things, allows the creation of
pictures on the screen by typing simple commands, can be a rich environment for
experiencing angle. If you are familiar with LOGO it may be worth considering its role in the
context of preparing to teach angle. What aspects of LOGO make i t useful for the teaching of
angle?
How do the headings of the PREPARING TO TEACH kamework help in seeing how LOGO can
contribute to the teaching of angle?
Comments You may like to work with LOGO in a group before considering these questions. You may
find this valuable even if you are not familiar with the programming language.
) FIELD OF VISION Imagine looking a t an object in the distance, say a tree, and holding your
thumb up in front of you so as to obscure completely the tree from your field of vision. How is it
possible for a thumb to block out a tree? What questions arise from this context?
) MORE CONTEXTS Brainstorm with colleagues and try to come up with some more contexts
that may result in interesting questions on angle. Work on one or more of these questions.
MEASUREMENT (A STANDARD TECHNIQUE)
One technique that may arise from working on all these contexts, and probably the major
technique within the topic of angle, is that of measuring an angle. We have already
demonstrated in Section 1 that the introduction of a unit of measure for angle may be fraught
with difficulties. Is the a c t of measuring also a potential area for confusion and
misconception? This subsection investigates the following questions related to angle
measurement.
Do pupils see the need for a measure of angle?
If asked to devise their own instrument for measuring angle what would i t look
like?
What are the relative merits of semi-circular and circular protractors and rotatable
angle measurers?
DUPLICATING ANGLES A pupil rotates on the spot, as in FEELING ANGLES and so moves through
an angle. If another pupil is asked to duplicate this (that is, to rotate the same amount a s the
first pupil), what does this second pupil need to know?
Try this activity with some pupils. What sort of questions could arise from the situation? How
could the activity provide the basis for a discussion about the need for a measure of angle?
b
Comments The second pupil may want to know the length through which the first pupil's arm has
moved. Alternatively, she or he may use the right-angle as a way of comparing (that is, to see whether the
turn was more or less than one right-angle or two right-angles).
REFLECTING If pupils were asked to make their own instrument which would measure both of
the angles produced i n DUPLICATING ANGLES and so check t h a t they were both the same, what
sort of device do you think would be produced?
C o m m e n t s If DUPLICATINGANGLES resulted in discussions about the need to look at an amount of turn
rather than any linear measurement, it is likely that most measuring devices would involve some sort of
moveable part. It is therefore significant to note that many of the popular angle-measuringinstruments used
in schools do not have any moving part.
There are many different shapes and sizes of protractor but they can be divided into three main
types:
the semi-circular half- or 180" protractor;
the circular or 360" protractor;
the full, rotatable protractor.
The last of these three (which consists of a rotatable plastic disc mounted, concentrically, on a
larger plastic disc) i s t h e only one t h a t h a s a moveable part. Close (1982) found t h a t pupils
performed f a r better in measuring angles with full rotatable protractors t h a n with halfprotractors. Furthermore, she identified some of the most common errors that occurred when
the pupils t h a t she studied used a half-protractor. These were:
placing the 90" mark either vertically or horizontally;
placing t h e 90" mark either along one of the arcs or in between a s if to bisect the
angle;
using the wrong scale;
misinterpreting the sense or direction in which the scale is graduated;
for reflex angles, measuring the non-reflex part.
REFLECTING AGAIN How important i s the feeling of movement when measuring a n angle?
Think about what you do when you measure an angle with a half-protractor. Do you have a
sense of movement? You may like to measure an angle now just to reflect on exactly what you.
do. Discuss this with your colleagues.
C o m m e n t s We have stressed throughout this pack the need to be aware of the' two ways of representing
angle, the static and the dynamic, and to be able to switch from one to the other. If the dynamic aspect of
angle is not always evident while measuring then the mere act of measuring may reinforce the static and
make it more difficult to make this switch.
ANGLE MEASURER After trying DUPLICATING ANGLES with your pupils, ask them to make
their own angle measurer with which they can check whether the two angles a r e of a similar
size.
b
[NI C o m m e n t s
You may like to devise other activities which involve the use of such angle measurers. Check
how many of the five common errors identified by Close still occur when they are used. Relate your findings
to the headings in the PREPARING TO TEACH framework. Don't forget to use your notebook for any
moments, insights, ideas or activities that arise from this.
REFLECTING What, for you, are the important messages t h a t arise from this subsection on
measuring? How would you structure some activities which attempt to result in the successful
use of angle-measuring devices? How do these activities compare with those devised by your
colleagues?
b
LOOKING BACK
This section has dealt with ways of developing activities for the classroom through the use of a
textbook or scheme and through considering the contexts in which the idea of angle arises.
REFLECTING
What does your textbook or scheme provide?
What role has the PREPARING TO TEACH framework played in augmenting this and in posing
questions?
P
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 angle. 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 angle 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
FEELING ANGLES Form a group with two others and decide who will be the 'turner'. Give your
turner the following instructions.
Stand with your arm extended and pointing towards another member of the group.
Rotate on the spot until you are pointing towards the other member.
Rotate back and feel the turn (close your eyes if i t helps).
Now get the turner to give the other members of the group instructions like:
Both move directly towards me.
Both move directly away from me.
One of you move nearer, one of you move away.
One of you move five steps away, one of you move two steps away.
Then repeat the turning instructions.
Discuss in your group what you are doing. Try to record in pictures what the group has just
done and write down what you have noticed.
How pupils are working together is as important as what they are doing. When discussing,
recording and writing, there are opportunities for specialising, generalising and conjecturing. You can play
a vital role in encouraging these processes.
Comments
FEELING ANGLES AGAIN Form a group of three as in FEELING ANGLES but investigate now
what happens if the turner moves'while the others stand still.
What happens if the turner moves forwards, backwards, from side to side?
Again discuss in your group what you have done; record it in pictures and words.
b
F
IMAGINING
Get someone to read out loud the following instructions.
Close your eyes.
Imagine a small, straight line.
Make i t move around a little and then bring i t to rest in the middle of your field of
vision.
Bring in another line and move i t around until the ends of your two lines meet.
Think of where they meet a s a hinge and open and close the hinge. Get used to
opening and closing the hinge.
Arrange i t so that you have a very small angle between the two lines.
Now make the angle between the two lines very large.
Make an angle of go0, 180°,60°,270°,120°,150°,350°,400".
Now open your eyes.
Discuss what you just imagined with one colleague. Draw some pictures if i t helps. Write
down some of the words and phrases to do with angle that were used in your discussion.
Which angles were difficult to imagine and which were easy? Why?
F
IMAGINING TRIANGLES
What is the largest possible angle in a triangle?
What is the smallest?
Can you draw a triangle where one angle is the sum of the other two?
Can you draw a triangle where all the angles are the same?
Try t o make up some similar questions of your own about triangles and answer them. Are
there any general rules that arise?
F
ANGLES IN PICTURES What angles can you see in the following picture?
Measure a s many angles as you can. What do you notice? Are there any conjectures or
statements that you could make?
Can you find any ways of testing your conjectures or convincing someone else that your
conjectures are true?
) RIGHT-ANGLES What is the largest number of right-angles that a polygon can have?
) FINDING A PATH What is your path if you walk so that your direction is always a t 90" (or 45",
or 60°,or . . .) to a fixed point? (Source: ATM )
This activity is best investigated in groups of three or four. Do try it for yourself. It provides
the opportunity for experiencing both the static and the dynamic sense of angle.
Comments
F
SCORING A GOAL
What is the shooting angle?
. .
Shooter
Where else could the footballer stand to have the same,shooting angle?
b
) ANGLE OF VISION
Select a pupil who must stand looking forward. She instructs two others to
stand a t the extreme edge of her vision. What is the angle of vision? Does everyone have the
same angle of vision? Get the rest of the class to line up behind each other so that they illustrate
the angle of vision.
b
ESTIMATING ANGLES A child stands up and holds out one arm, pointing a t an object. He then
turns to point to another. Estimate the angle.
The child is now given a metre rule to point with and goes through the same routine, pointing to
the same objects. Estimate the angle now.
I t may be useful to draw the situation. Get pairs or groups to write an account/argument to
convince someone of what they think happened.
What does this activity illustrate?
b
HIDING FROM THE TEACHER Get pupils to sit in seats in rows in front of you. Ask them to
rearrange themselves so t h a t each is hidden behind someone else (can't be seen by the
teacher). Encourage them to imagine this before doing it. Then get them to draw the situation
and to measure some of the angles.
F
F
BLIND SPOTS
Imagine sitting in a car. What is your angle of vision? Do it.
Do all cardcoaches have the same angle of vision?
What angle is obscured by the car body a t the edges of the windscreen?
Expand this activity into one on blind spots.
F
WALKING ANGLES As a child I used to like walking past a telegraph poldamp post in such a
way that someone on the other side of the street (or a car) was hidden.
Expand this into an activity which encourages the exploration of angle and allows the
opportunity for conjecturing, generalising and convincing.
W
5 THE LAST WORD
It is difficult to write a pack which has direct application t o all classrooms in all schools and
for all ages of pupil. PREPARING TO TEACH ANGLE set out to explore some of the issues related to
the teaching of angle, 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 angle.
It was also used t o direct thinking about what underlies activities on angle. This process of
reflection is an important one in both the teaching and the learning of mathematics.
F
PUllING 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?
F
F
AND FINALLY As your final activity on this pack, discuss with your colleagues and then
prepare your first two lesscns on angle.
6 BIBLIOGRAPHY
APU (Assessment of Performance Unit) (1978-82) A review of monitoring in mathematics,
part l , pp 259,262,26%269.
APU (Assessment of Performance Unit) (1981) Secondary survey report no 2, HMSO, pp 16-17.
ATM (Association of Teachers of Mathematics) Investigation card no 39 'Angling' in A way
with maths, the children's workshop manual.
Close, G. (1982) Children's understanding of angle a t the primarylsecondary level,
Polytechnic of the South Bank.
Dallison, K. J. and Rigby, J. P. (1982) 0 & B maths, Book 1, Oliver and Boyd, p 52.
Graham, D. and Graham, C. (1981) Maths for you, Book 1,Hutchinson, p 15.
Jones, P. S. (1953) 'Angular measure
Teacher, October 1953, pp 419-426.
- enough
of its history to improve its teaching', Maths
Kline, M. (1982, first published 1953) Mathematics in western culture, Penguin Books.
NMP 1 (1987) Mathematics for secondary schools, Book 1, Longman, pp 5051,120-121.
NMP 2 (1987) Mathematics for secondary schools, Book 2, Longman, p 39.
Scottish Mathematics Group (1983, first published 1971) Modern mathematics for schools, Book
l , Blackie Chambers, pp 90-91.
SMP 11-16 (1983) Angle l, l(b) Space, Cambridge University Press, p 5.
Wyvill, R. (1985, first published 1983) Nuffield maths 6 pupils' book, Longman, p 46.
PREPARING YOURSELF
Begin by reviewing in your mind, or by brainstorming with colleagues, any
associations and aspects of the topic. We recommend keeping a notebook for
each major topic so that you can remind yourself quickly of the underlying ideas
and of particularly effective activities.
If you find yourself running out of ideas, or with a sense of something missing
or incomplete, then refer to the 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. Note that any particular idea may relate to several
headings.
The six components of the 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.
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, if 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 Universi
SUP 17184 0
By being aware of the language patte
remember to take time over the intro
in mind that pupils come to lessons
and language patterns of the topic.
By thinking about underlying images
give pupils direct contact with k e y id
language patterns, and thus to try to ci
By indicating where a topic c o m e s f
deals with, you can help pupils t o get
and its applications.
By providing opportunities to encou
important techniques, you can help pu
Pupils who participate in formulating
are more likely to remember it later.
The six components which comprise
pairs, corresponding to behaviour (la
(root questions and different contexts
misconceptions). They overlap a n d in
should not be used mechanically.
PREPARING YOUR LESSONS
PREPARING TO TEACH A TOPIC
STANDARD
MISCONCEPTIONS
DIFFERENT
CONTEXTS
Being aware of incomplete imagery, inaccurate languagz
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
PATTERNS
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 ar
can automate. Techn
pupils recognise whe
TECHNIQUES
and METHODS
ROOT
QUEST'IO
IMAGERY or
SENSE OF
The inner sense of t
made between conte
language patterns a