October 2006
Engagement in GCSE Mathematics
1
Introduction
In July 2005 an AST from Purbeck School attended an Ofsted
conference on Engagement in Mathematics at GCSE. Following
her feedback to the Inspector for Mathematics a working group
was established in January 2006. The working group consisted of
mathematics ASTs and teachers nominated by their head of
department from a selection of schools in Dorset. There was no
finance to support this initiative and I am very grateful to the
schools involved for releasing their teachers for meetings during
the school day.
It was agreed that in order to keep it manageable each teacher
initially identified one group in year 10 or 11 to work with. The
initial aim was to ensure that teaching in Key Stage 4 would reflect
what was working in Key Stage 3. Initially the group focused on
ensuring that the starter activity was always engaging and different
from the existing start to Key Stage 4 mathematics lessons. The
group met on a total of three occasions but there was the
opportunity to share and communicate between sessions through
email.
All teachers felt that there had been a positive response from both
the pupils and their colleagues. In fact there are contributions from
several teachers in Gillingham School. The ideas quickly spread to
be whole lesson ideas rather than just starter activities.
The booklet and CDROM have case studies with an activity outline
and the resources used. For ease of use the booklet has been
divided into 6 sections. Section 1 provides guidance on
questioning which will aid successful use of the activities in the
booklet. At the end of each subject section there are examples of
additional resources which the group have used but not written up
case studies for. The CDROM contains a master copy of the
booklet and copies of all the resources referred to in the booklet.
A copy of the booklet and resources will also be placed on the
county website and the SWGFL portal.
The working group hope that you will find this resource valuable
and would be grateful if you could forward your own ideas to be
added to the website resource bank.
Engagement in GCSE Mathematics
2
Thank you to all the staff involved and to the management of
Purbeck School for allowing Marie to attend the original course
and for hosting the working group. Thank you also to Gillingham,
Lytchett Minster and Shaftesbury schools for supporting the project
by releasing staff to attend the working group meetings. Thank you
also to Budmouth Technology College and Woodroffe for releasing
their ASTs to work on the project.
Please note : The activity sheets and case studies were completed under the
three tier system and may need adjustment to suit the new two tier
teaching for GCSE.
Working Group Members
Budmouth Technology College
Gillingham
Lytchett Minster
Purbeck
Queen Elizabeth
Shaftesbury
Woodroffe
Amy Harris *
Lucy Neil
Sally Burt
Marie Simmonds **
Jane Livesey * (now
Secondary Stratgey
Mathematics Consultant)
Clare Davenport
Jennie Golding *
* Dorset LA ASTs
** School based AST who attended the original Ofsted course.
Angela Easton
Inspector for Mathematics
Children’s Services
Dorset County Council
Engagement in GCSE Mathematics
3
Index
Page
Section 1
Questioning
5 -17
Section 2
Number
18 - 39
Section 3
Algebra
40 - 71
Section 4
Shape and space
72 - 78
Section 5
Handling Data
79 - 82
Section 6
Probability
83 - 101
Bibliography
Engagement in GCSE Mathematics
102
4
SECTION 1 – QUESTIONING
Effective Questioning and Intervention
Discussion paper by Angela Easton
‘Teachers may ask around 100 questions per hour’ but as teachers we must
try to ensure that our questioning is effective and enables pupils to make
gains in their learning.
Effective questioning and intervention can take place with individuals, groups
and whole classes. Good teacher listening skills are essential for effective
questioning; listening to what pupils actually say and responding by
encouraging and offering supportive statements or prompts. Pupils must be
encouraged to question as well as to answer.
Pupil discussion is an area where teachers must take care not to intervene
too often, too soon or too strongly as it prevents real discussion and the ability
of pupils to think and express themselves.
There are many types of questions and I believe we have come a long way
since educators were claiming that there were just two types of question
‘open’ and ‘closed’ (although the National Strategies for too long have
continued to use this simplistic idea). I found Don Stewart’s categories of fact
questions which test knowledge and thought questions which create
knowledge, particularly useful to analyse both my own questioning and that of
my colleagues when a head of department. He further divides these into 6
areas which he claims move from lower to higher order questions. See
appendix 1 in this section.
Teachers should plan their higher order questions which support and enhance
the learning taking place in the classroom. The focus of my planning was
always questions which would enable me to assess learning whether it was
for the beginning, end or during the main part of the lesson. Planning key
questions I believe is vital to ensure that the learning you aimed for during the
lesson has taken place.
I personally have always tried to be careful in the use of the word ‘why’
because although it is useful to explore pupils understanding it can also be
very threatening. To aid my use of this I would often have the pupils working
in pairs or groups so that a spokesperson was supported by at least one other
pupil.
Some teachers are naturally able to support and guide pupils’ learning
through the good use of questioning and others require support and guidance
in order to achieve this. The observation of good classroom practitioners in
mathematics always reveals a good relationship between the teacher and the
pupils and the effective use of a variety of types of questions as well as good
supporting strategies for pupils.
Engagement in GCSE Mathematics
5
In order for the activities in this booklet and contained on the CD-ROM to
have any impact in the classroom the teachers using them will need to be
able to support pupils working collaboratively, intervene appropriately and use
questions very effectively.
Many of these activities require pupils to discuss working in pairs or groups. D
James Dillon defines discussion as ‘ a group address to a question in
common’ The results of his research and that of David Wood suggests that
frequent questioning by the teacher inhibits rather than encourages
discussion i.e. the more teachers question the less pupils say. They went on
to specifically recommend against questioning by the teacher during a
discussion because they depress pupil thought and talk. Wood’s research
shows that pupils respond at greater length and with greater initiative (twice
as great) to statements rather than questions. They found that the less a
teacher interrogates pupils the more likely they are to listen to, make
contributions about and ask questions of pupils say.
In the appendix attached is some additional guidance to develop teachers
questioning skills. In addition in the booklet and on the CD-ROM you will find
guidance used by Gillingham school.
I hope you will find the questioning and discussion resources useful. Please
contact me if you would like to discuss or explore this further.
Two particularly useful books for developing questioning and discussion skills
published by ATM are: Questions and Prompts for Mathematical Thinking
Watson and Mason
ISBN 1 898611 05X
£9:00 (Member £6:75)
Thinkers
ISBN 1 898611 26 2
Bills, Bills, Watson and Mason
£10:00 (Members £7:50)
References to specific questions from the ‘Thinkers’ book has been added as
an appendix to this section of the booklet and in the question section of
resources on the CD-ROM
Angela Easton
September 2006
Engagement in GCSE Mathematics
6
Appendix 1
Lower Order Questions
Recall: of specifics; what seen/read. Data recall; facts,
procedures, classifications. Knowledge of theories. Examples
include ‘what’ ‘when’ ‘how’ ‘name’ ‘distinguish’
Comprehension: understanding of information as evidenced
by; comparing, simple descriptions, providing further examples
of principles, extrapolation etc Examples ‘compare…’ ‘Give an
example of …’ ‘predict …’ ‘rearrange …’ ‘hypothesise … ‘
Application: apply rules/knowledge and techniques to solve
simple ‘problems’ with single right/wrong answers. Interpret and
apply information. Examples ‘consider …’ ‘ tell us …’ ‘show your
working to …’ ‘ construct’ ‘ solve’ ‘demonstrate’ ‘show next’ ‘
check out’
Synthesis: put together parts or elements to make a complex
whole. Global predictions; build general theories; make
interesting juxtapositions of ideas and images; complex
reasoning; make broader classifications and categories.
Examples ‘write about …’ ‘create …’ ‘develop…’ ‘Suggest
reasons’ ‘derive’
Evaluation: judge the quality of the ideas; rationally based
opinions; discriminatory – pros and cons considered and
weighed. Examples ‘why’ ‘defend’ ‘is this the best…’ is this the
only…’
Higher Order Questions
Don Stewart Questioning January 1995 (adapted from Bloom’s taxonomy
1956)
Engagement in GCSE Mathematics
7
Appendix 2
Some General Questions
What is this about do you think?
Anything else you know about this?
Can you give me an example?
How did you make a start at this?
Can you explain to me what you think at this moment?
Can you suggest an alternative point of view?
How convinced are you by this argument?
Can you say that in a slightly different way? Be more precise?
What do you mean by …?
Do you think this is a reasonable argument?
What’s your view? What supports it?
What is this saying, do you think, in a nutshell?
What can you tell me about this?
That’s an interesting theory/decision – how did you come to it?
What similarities/differences do you think there are between these?
Can you think of a counter argument?
What is it you are not sure about?
What do you think will happen next?
How could we be clearer about this?
What else do you need to know?
Which is the easiest/simplest view? Why do you think so?
Engagement in GCSE Mathematics
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What might we look at next?
What do you think this word means?
Let’s see, can we break this down a bit?
Where might you have gone wrong on this?
How do you work these out in general?
Engagement in GCSE Mathematics
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Appendix 3
Questioning Gambits (1) Things teachers say
I don’t like you shouting out, as it prevents others from joining in.
Yes, that seems right doesn’t it? Has anyone else anything to add?
OK, say it in your own words then.
Who hasn’t answered/spoken yet?
Shhh! (S)he can do it.
Did we do/what did we say about … I can’t remember.
You may confer with your partner.
Would you mind telling me, from your perspective, what you’ve done?
Where did we get to?
This seems important doesn’t it? If that is so what follows?
If you haven’t got a clear idea, half of one will do nicely.
I know it’s not easy to say what you are thinking but please try – I’m
interested.
What questions do you have about this work …?
Don’t just give me an answer try to give your reason as well.
Can anyone pick up on what … said?
What general issues have we/you been considering?
Let’s have a look at an idea you are not too sure about.
What weren’t you too sure about this lesson?
What do you need to remember/be aware of from this lesson?
What do you think we should look at next?
Engagement in GCSE Mathematics
10
Appendix 4
Questioning Gambits (2) interesting things teachers do
Gets pupils/groups to explain their views (after rehearsal) to the class. The
class not the teacher ask the questions.
Asks a regular stock of questions so that the pupils become familiar with them
and ask themselves as the teacher fades out.
Asks a question and then does not immediately simplify it with a string of even
simpler questions (some pupils are wise to this and wait)
Sits down when they have asked a question and waits for the answer.
Asks pupils to say ‘squeak’ or scratch their chin if they do not understand
something/were not listening at a vital point.
Tilt their head to one side/raises their eyebrows when looking at someone to
invite them to contribute.
Collects ideas on the board (generally without pupils’ names attached) to
record the development of ideas and to show interest.
Collects a range of answers from the class without indicating yes or no/right or
wrong.
Says when a question is hard – and doesn’t say when it is easy.
Writes tasks and questions on the board as well as saying them.
Asks the class to think of an example of a hard and an easy question on the
topic they have just covered.
Uses ‘inviting’ gestures (nods, smiles etc) to encourage talk – i.e. routine talk
maintenance work in any conversation
Uses tag questions at the end of statements (possibly with rising intonation) to
put them up for confirmation or to be modified e.g. ‘This is one reason for
using pie charts isn’t it?’
Uses rising intonation to turn statements into questions – requesting
confirmation.
Uses ‘attention’ beginnings e.g. ‘this is interesting…’ or ‘I’m interested in/by…’
teacher appears wondering and curious themselves.
Keeps an interesting tone to questions – doesn’t let them fizzle out at the end
of the sentence.
Engagement in GCSE Mathematics
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Moves around the classroom to change perspective when asking questionsmakes sure that they have looked at each pupil during the lesson and that no
pupil has been overlooked.
Greets pupils at the door as they enter, remembering interesting things e.g.
plays in school team etc setting the atmosphere
Stops any pupil who calls out, speaks across another or who whispers the
answer to another.
Thanks the pupils when it has been a ‘good lesson’.
Acknowledges effort and attempts by pupils to join in.
Engagement in GCSE Mathematics
12
Appendix 5
Listening to what pupils say – encouraging and supporting
‘Open/probing’ questions or phrases invite a lengthy response and encourage
the airing of sometimes tentative views. How the teacher responds will clearly
affect willingness to offer ideas.
Being encouraging, try to:
ƒ Suspend your own point of view (not guess what is in my head)
ƒ Listen to what they are saying
ƒ Try to see what they are seeing
ƒ Focus on content rather than delivery (listen hard and long without
interruption)
ƒ Look at them and make encouraging signs: nods or mms whether they
appear to be right or wrong.
ƒ Ask them to repeat what they have said or clarify a point
ƒ Give them time and be patient.
Four ‘C’s of active listening:
ƒ Confirm that you have received what is said – repeat things back to
check that you have got it right.
ƒ Check any assumptions that you have made: ‘It seems to me that …’
‘am I right in thinking that…’ ‘I suppose you are thinking that…’
ƒ Clarify any points that you are uncertain about: ‘When you said ... can
you say a little more about that?’ ‘What did you mean by…?’ ‘I didn’t
quite get what you said about …, can you say it again /in a different
way?
ƒ Consider that they have some reason for what they are saying and try
to find out what it is.
Paraphrasing
ƒ Helps them to listen carefully
ƒ Affirms that you are listening and interested in what they say.
ƒ Provides opportunity for correction (with care), editing and clarification.
ƒ Should focus on what they have said and be brief.
ƒ Can involve other class members.
Other thoughts
ƒ Move around the room so pupils can ask questions more privately.
ƒ Squat/sit down when talking to pupils individually or in small groups.
ƒ Asking ‘Are you sure?’ or ‘Why’ is often interpreted as ‘I have got it
wrong’
ƒ Asking ‘Is everyone OK on that?’ is a non-question!
ƒ Encourage pupils to build on/criticise what went before.
ƒ Talk to all pupils as intelligent beings.
ƒ Ask questions that you are interested in.
Engagement in GCSE Mathematics
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Appendix 6
Discussion: alternatives to questioning
Alternatives to questioning
ƒ Make statements – say what you think in relation to what the pupil has
just said.
ƒ Pupil questions – invite or allow a pupil to ask a question in relation to
what the speaker has just said.
ƒ Signals – indicate that you have heard what the pupil has said without
yourself taking the floor. Possibly pass the ‘floor’ around to someone
else, by brief phrase, gesture or word.
ƒ Silences say nothing at all but maintain a deliberate silence for at least
three seconds until the original speaker resumes or someone else joins
in.
What sort of statements might I make?
ƒ A thought that occurs - either complementary or a counter view
ƒ A reflective statement: try to re-phrase or just repeat what a pupil has
said giving the pupil the chance to confirm or expand on what they
have just said or another pupil the chance to intercede.
ƒ Simply describe your state of mind: ‘I’m confused by what you said
about…’ or ‘I disagree with what you are saying about…’ or ‘That
makes sense.’
ƒ Statement of interest, ‘I’m interested in what you said about ..’ or ‘I’m
interested in your definition of …’ or ‘It would help if I had an example
…’ or ‘I’d like to hear more on your views about …’.
ƒ Speaker referral: relate what one pupil has said to what another has
said, ‘That’s similar to what … was saying’ or ‘that seems to disagree
with what … was saying before.’
ƒ Self report, ‘I have trouble keeping track of the …’ or ‘I can’t quite
imagine what that means.’
Engagement in GCSE Mathematics
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APPENDIX 7
Questions to Extend Understanding and Reasoning
Page references ATM Thinkers Booklet where you will find some
very good question examples and stems. An invaluable book for all
departments.
Number
Page number
7
9
10
12
13
14
16
17
18
19
20
21
23
24
25
26
27
Algebra
Page numbers
7
9
10
12
14
16
18
19
20
22
23
24
25
Engagement in GCSE Mathematics
Question numbers
15, 22, 24,
1, 2, 3, 5, 6,
8, 9, 13,
4,
14, 15, 16, 17
1, 2, 4, 7, 10, 11, 13
2, 5, 6, 7, 8, 9,
2,3,4,5,6,7,8, 13, 15,
1 – 11, 13, 14, 18
1, 2, 3, 7, 8, 9, 10,
1, 2, 3, 4, 5, 6, 7, 8, 10, 17, 18, 19, 20, 22
1, 2, 3, 4, 5, 12
1, 2, 4, 5, 6, 7,
1, 5, 8,
1, 2, 3, 6, 9
1, 2, 3,
1, 2, 5, 6, 12
Question numbers
25, 26, 27, 28
8, 9, 10,
12, 14, 15, 16, 19, 20,
9
6, 14, 15, 18
10, 11, 16
15, 17,
6, 17,
13, 14, 15, 16,
6,8, 10, 11,
8, 9
12,13,
7,
15
27
7, 8, 9,
Shape and Space
Page numbers
Question numbers
7
16, 17, 18, 23, 29
9
4, 7, 11
10
10, 11, 17, 22
12
7, 8,
13
18, 19
14
3, 9, 16, 17, 19
16
3, 4, 14, 15, 17, 18
17
1, 9 10, 11, 14, 15,
18
16
19
4, 5, 12, 13, 14, 15,
20
9, 11,12, 21,
22
7, 9,
23
10. 11
24
3, 4, 7, 9, 10,
25
4, 5, 9, 10, 11
27
3, 13, 14
Handling Data – Statistics
Page numbers
Question numbers
7
19
10
21
12
5, 6,
19
11, 16,
24
11,
27
4,
Handling Data – Probability
Page numbers
Question numbers
7
20, 21
10
18,
14
12
16
12, 13,
17
12,
18
12
24
6,
Engagement in GCSE Mathematics
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How do you know these are good question
stems?
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
What would happen (to the median)
if (we added x to the data)
What are the key points …
What made you decide to do it
that way?
What do you think they have done
wrong?
What mistakes do you think people
would make?
What (other) information do you
need to answer the question?
What have we been working on that
might help us with this problem?
What assumptions are we making?
What techniques would help?
Why do you think some people
might have a different answer?
Why is that wrong?
Why isn’t .. (half of 8 x 6, 4 x 3)
Why are these two (graphs) similar
/ different?
Why do all (of these equations)
give the same answer?
Why is this useful (to round
numbers)?
Why do you think that?
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
How do we know you are right
How can we be sure that?
How would you explain why (1.5 is
bigger than 1.14)
How would you explain to someone
(how to construct a triangle)
How do you know we have every
possibility (Probability)
How do we know that this is a
sensible/reasonable answer?
How do we know this is the best
solution?
How could you do this more
efficiently?
Write down values that satisfy
(x+y=10)
Give me an example of another
(equation with these values)
Is this always, sometimes or never
true?
In what ways can we ….
Write three questions that would
test what your neighbour has
learnt this lesson ..
Convince me that you’re right ….
Would this method always work?
Why?
Show me ways of representing (y =
5)
Find a number pattern and explain
it
How do you know we have every possibility?
© Gillingham School Mathematics Department
Engagement in GCSE Mathematics
17
SECTION 2 - NUMBER
Title: True/False
Area of Mathematics:
Entry level:
Any
Foundation*
But can be
adapted,
including to AL
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
A number of cards are produced – can be by students, but easier
to target if by teacher. 20 often works well. Each card reflects a
common misconception, eg ¼ = 0.4. An exemplar set is given.
Students work individually, in pairs or in groups to decide whether
the statement as given is True or False. During feedback they
have to justify their choice, with credit given for justification as well
as ‘correct’ placement. Can be done as a written exercise but rich
source of discussion.
Student production of cards slow but involves measuring as well
as lots of talking. Differentiates well: students want to make cards
as hard as they confidently can. Cards can be written ‘for year 8’ or
for other groups in the class.
‘How do you know?’ or ‘Convince me!’ used lots in this.
Name: Jennie Golding
School: Woodroffe
Date:
June 2006
*Please delete as appropriate
Please indicate if you have an attachment : - True or False cards
Engagement in GCSE Mathematics
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True or False Cards
0.7 = 7%
1/5 = 5%
6 is a factor of 12
2.8<2.08
If y-3=6 then y=9
500m = 5 km
The sum of 2 numbers is never
the same as their product
24 is a multiple of 6
9 is a prime number
¾ = 0.34
A square with perimeter 12 cm
has area 9 cm2
The angles in a quadrilateral
add up to 360°
30% of £15 is £4.50
47mm = 4.7cm
If 5z=20 then z=15
The mean of 3,3,4 and 6 is 3
Engagement in GCSE Mathematics
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Title: Review of Standard Form
Area of Mathematics: Standard Form
Entry level:
Intermediate*
Examination Board: AQA
Modular*
Outline of resource and how it was used including key
questions:
Brain storm starter, the class were asked to identify answers to two
questions, `what do you know about standard form’ and `what
questions could be asked’.
Class were given 3 minutes to think about it then ideas were
collected onto the board. The class identified information from
how to write numbers in standard form to questions on dividing
numbers in standard form.
This starter requires very little input from the teacher, and
promotes the class in being active learners and helps to reinforce
each technique needed.
We discussed ideas about how many marks might be given for
different questions and alternative ways that questions can be
written even though they are asking the same thing.
The class then worked on some exam questions and as a plenary
we worked through difficult standard form questions which involved
percentages etc.
Name: Marie Simmonds
School: Purbeck School
Date:
19/05/06
*Please delete as appropriate
Please indicate if you have an attachment : -
Engagement in GCSE Mathematics
Standard Form PowerPoint MS
Standard Form Word Worksheet MS
20
4.8x104
0.049
4.9x10-2
488.8
4.888x102
0.498
4.98x10-1
48700000
4.87x107
4.8
4.8x100
0.0000499
4.99x10-5
4.89x103
Engagement in GCSE Mathematics
4890
48000
21
Power point resource on CD see copy below
What do you know about
Standard Form?
What types of questions
might you be asked?
A resource sheet of examination questions at the appropriate level and for the
relevant board will also be needed.
Engagement in GCSE Mathematics
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Title: Understanding and applying the rules of indices
Area of Mathematics: Number and Algebra
Entry level:
Intermediate*
Examination Board: EdExcel
Linear*
Outline of resource and how it was used including key
questions:
Using the hexagonal jigsaw approach (as detailed in the Standards
Unit) I gave my students a blank with 8 equilateral triangles on it.
Having been taught the rules of indices in the previous lesson, I
asked them to design matching questions and answers which
when adjoined would create a jigsaw. The students worked in
groups and had been asked to design some appropriate questions
for homework. I encouraged the students to use the same variable
throughout their jigsaw to make the jigsaw harder to solve. Some
students stuck with questions and simplified answers such as y5 x
y3 = y8 whereas others were more inventive and posed two
questions which both simplified to give the same expression.
Some students used numbers as the base and evaluated some of
the answers to give a numerical jigsaw. The two spare triangles
were used for “red herrings”.
When groups were confident with their rough copy, I assigned a
new copy on coloured card ensuring I had a different colour for
each group in the class.
When cut, these jigsaws then provide a starter for a future lesson
and with the group’s name on the back, also bring about some
good discussion.
Name: Amy Harris
School: Budmouth Technlogy College
Date: June 2006
*Please delete as appropriate
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Title: Powers Review
Area of Mathematics: Number
Entry level:
Intermediate*
Examination Board: AQA
Modular*
Outline of resource and how it was used including key
questions:
Extended plenary task done in pairs.
Exam questions used in a quiz style plenary to assess the class’s
ability to answer exam questions on powers at the end of teaching
the topic.
Class were given a sheet to complete with the correct number of
answer spaces available to them, merit stickers used as prizes.
The questions were then given as homework/revision task a week
later to reinforce the techniques needed.
Name: Marie Simmonds
School: Purbeck
Date:
24/01/06
*Please delete as appropriate
Please indicate if you have an attachment : -
Engagement in GCSE Mathematics
Powers Review word MS
Powers Homework word MS
Powers Review Answer sheet MS
24
1.
Work out the value of
(a)
53
..................................................
(1)
(b)
104
..................................................
(1)
2.
Use the calculation
58.5 × 27 = 1579.5
to write down the answer to
(a)
585 × 27
...................................
(1)
(b)
1579.5 ÷ 27
...................................
(1)
(c)
585 × 0.027
...................................
(1)
3.
36 expressed as a product of its prime factors is 22 × 32
(a)
Express 45 as a product of its prime factors.
.......................................................
.......................................................
.......................................................
(3)
(b)
What is the Highest Common Factor (HCF) of 36 and 45?
.......................................................
.......................................................
(1)
(c)
What is the Least Common Multiple (LCM) of 36 and 45?
.......................................................
Engagement in GCSE Mathematics
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.......................................................
(1)
4.
(a)
407 u 2.91
0.611
Estimate the value of
.......................................................
................................................................................................
............
(3)
(b)
Write down the value of
64
1
2
.......................................................
(1)
5.
Work out the value of 53 ҟ 43.
..............................................................
..............................................................
(2)
6.
(a)
Work out
12 – (3 + 7)
.......................................................
(1)
(b)
Put brackets in each of these calculations to make them
correct.
(i)
18 – 4 – 2 = 16
(ii)
3
(iii)
20 ÷ 5 – 3 = 10
+ 4 × 5 = 35
(3)
7.
(a)
Work out the cube of 6.
.....................…………………
………………………………………………………………
(1)
(b)
Work out 0.22
Engagement in GCSE Mathematics
26
.....................……………………
(1)
(c)
A list of numbers is given below.
15
16
19
27
34
42
45
From this list, write down
(i) a cube number,
..........………………………
(1)
(ii) a prime number.
..........………………………
(1)
8.
Use the calculation
487 × 3.53 = 1719.11
to find the value of
(a)
487 × 0.0353
......................……………………
(1)
(b)
48700 × 0.00353
......................……………………
(1)
9.
(a)
Express 144 as the product of its prime factors.
Write your answer in index form.
.......................................................
.......................................................
.......................................................
.......................................................
.......................................................
.......................................................
(3)
(b)
Find the Highest Common Factor (HCF) of 60 and 144.
.......................................................
Engagement in GCSE Mathematics
27
.......................................................
.......................................................
(2)
10.
Write down the value of
3
27
.......................................................
(1)
Engagement in GCSE Mathematics
28
Title: Percentage/Fraction Fit
Area of Mathematics: Number
Entry level:
Foundation*
Or can be
adapted
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
Students work individually, in pairs or threes.
Preface either version with 2 or 3 examples explained by students.
Simpler version: Pairs of students have one component card, one
master card with spaces. They complete each sentence with one
card from each component set. Pairs have to convince each other,
since either may be called upon to explain their solution.
Harder version: Students create their own 6 (or 12, depending on
stickability and fluency) statements about percentages or fractions,
of the form ‘10% of 50 is 5’ on a master sheet (attached). The
components are then cut up, the words discarded and another
group fits as many components as they can into a blank grid. The
object of the game is to maximise the number of components
fitted. The game can of course be replayed, by this or another
group. It’s worth pointing out to the creators that they make the
questions re-fit harder by using similar numbers in several
statements. This version tends to differentiate itself, especially if
students are designing for someone else to play (and find hard!)
In both cases, statements can be checked using calculators; if
calculators are allowed in creation, statements can be made quite
tricky. Demanding 5 statements about fractions, 5 statements
about %s adds to difficulty.
Name: Jennie Golding
School: Woodroffe
Engagement in GCSE Mathematics
Date:
June 2006
29
FRACTION/PERCENTAGE FIT
of
Of
Of
Of
Of
Of
is
is
is
is
is
is
150
500
1
60
200
of
of
of
of
of
of
150
1000
50
6000
is
is
is
is
is
is
40
10
Use one card from each set to complete the sentences:
Set A
50%
Ҁ
1/5
100%
1/3
25%
200%
75%
½
1/10
¼
10%
Set B
36
100
600
80
10
500
200
1000
150
300
50
2000
There is also a master fraction/percentage fit table and cards included on the
CDROM.
Engagement in GCSE Mathematics
30
Percentage increase and decrease using decimal
multipliers
Area of Mathematics: Number
Entry level:
Intermediate*
Examination Board: EdExcel
Linear*
Outline of resource and how it was used including key
questions:
Starter or plenary activities
You can either use pupil whiteboards and use these boards purely
as the target boards containing answers to your questions OR you
can use them to play splat.
I play the first to five successful splats and have two pairs
competing at the same time. I use fly squatters as the splat sticks
and start by posing the questions myself but then ask the rest of
the class to provide the questions.
The first board is to reinforce the number bonds to 100 and hence
aid understanding that a 20% decrease can be calculated as 80%
of the original.
The second board is a selection of decimal multipliers which
provide both decreases and increases e.g. “Can you find me a
multiplier which would give an increase of 2%?” The answers 1.2
and 1.02 are deliberately placed to enhance discussion after the
activity.
Key question “Why do some of the decimal multipliers have 3dp?”
“What percentage changes would they give?” Which multiplier can
be used to calculate VAT?”
Title:
Name: Amy Harris
School: Budmouth Technlogy College
Date:
*Please delete as appropriate
Attached are the splat boards used either as OHT’s or as projected slides using a data
projector
Engagement in GCSE Mathematics
31
32
70
40
50
85
65
27
12
96
80
45
35
Engagement in GCSE Mathematics
32
1.045
1.03
1.055
1.075
1.068
1.02
1.2
0.75
0.985
0.98
1.025
1.05
1.07
1.04
1.175
0.9
0.8
0.85
1.095
1.01
Engagement in GCSE Mathematics
33
Title: Module 3 Quiz
Area of Mathematics: Number
Entry level:
Intermediate*
Examination Board: AQA
Modular*
Outline of resource and how it was used including key
questions:
This Quiz was used to utilise group work to reinforce some of the
techniques needed to answer module 3 questions. The class
answered them in groups of 2, 3 or 4 and there were prizes for the
best group in each round and overall.
The groups passed on their answers after each round to the next
group who marked it and passed it back. The excel document
ensures that the results are easily collated and totalled up.
The response from students was extremely positive and students
were motivated and engaged throughout.
I used this early on in the revision programme for module 3.
Name: Marie Simmonds
School: Purbeck
Date:
12/05/06
*Please delete as appropriate
Please indicate if you have an attachment : -
Module 3 Quiz PowerPoint MS
Module 3 Quiz results Excel MS
File available on the CDROM.
Engagement in GCSE Mathematics
34
NUMBER ADDITIONAL RESOURCES NO CASE STUDIES
PowerPoint Resources
Estimations to Check Arithmetic
If 24 x 78 = 1872
then 240 x 0.78 = 187.2
WHY?
What is the least number of steps needed to get from the first product to
the second?
If 24 x 78 = 1872
Then 18720 ÷ 7.8 = 2400
WHY? HOW?
If 24 x 78 = 1872
then 18720 ÷ 7.8 = 2400
WHY?
HOW?
Can you tell which ones are wrong?
AIM: TO USE ESTIMATIONS TO CHECK ARITHMETIC!
Which of these answers must be wrong? Why? Discuss in pairs.
1) 2.3 x 4.2 = 7.65
2) 5.7 x 5.6 = 24.7
3) 16.7 + 11.2 = 31.9
4) 24.1 ÷ 3.8 = 10
5) 30.1 ÷ 14.2 = 2.1
6) 2.4 x 3.1 = 7.8
7) 345.4 – 140 = 189.6
Fractions
What fraction, when you turn it into a decimal, starts like this:
0.0204081632 …?
What fraction, when you turn in into a decimal, starts like this?
0.0103092781 …?
Engagement in GCSE Mathematics
35
Truth
Which of these statements are true?
‘One fifth is the average of one fourth and one sixth.’
‘One third exceeds a quarter by one third of a quarter.’
‘One third of one fifth is greater than one fifth of one third.’
Pills
A pill for a certain illness must not be taken more than once in any period of
an hour, or more than 6 in any period of 12 hours.
What is the largest number of pills which could be safely taken in 18 hours?
Basic Challenge
Find a number where, if you take the last digit and put it at the front, the new
number is 50% bigger than the number you started with. For example you
could try 456, but this makes 645 which does not work because 50% bigger
would make the new number 687.
17
Split 17 and multiply its parts to make the largest and smallest answers
possible.
e.g.
17 = 4 + 5 + 8
17 = 8.5 + 8.5
4 x 5 x 8 = 160
8.5 x 8.5 = 72.25
Sequences
Find the next numbers in each of these sequences:
18
9
26
13
38
19
56
28
1
4
2
7
5
16
14
43
0
1
3
5
9
11
15
17
Engagement in GCSE Mathematics
14
41
21
…
…
…
…
…
…
36
NON-CALCULATOR ARITHMETIC TO MAKE
YOU THINK!
Quick quiz for the start or end of a lesson.
Questions can easily be varied for the topic of
ability. Could ask for the first to get a row,
column or diagonal.
Find 5/6 of 12
Add
together
3.6 and
17.98
Subtract
23.7 from
68.2
Evaluate 52
30x50
23x89
How much
change from
£20 when
you spend
£12.69
Find ¥49
Half of
10,480
I catch a train Double 341
at 10:30 and my
journey lasts
2hrs 5mins.
What time to I
arrive at my
destination?
-54 - -67
Evaluate
0.67x10
Engagement in GCSE Mathematics
1.2x4
Round 19.8
to the
nearest
whole.
3/6 + 5/6
37
Using Powers and Knowledge
Given that
13² = 169
133² = 17689
1333² = 1776889
and
16² = 256
166² = 27556
1666² = 2775556
Without using a calculator or a computer can you write down the answers to
13333² =
16666²=
Extension
Can you find another number between 10 and 20 which, if you keep repeating
the second digit, and squaring, has the same number pattern?
Engagement in GCSE Mathematics
38
ROUNDING TO ONE DECIMAL PLACE –
1 d.p.
Match these up, one has been done for you!
There is one question and answer which
don’t match up, which ones are they?
1.234
0.9
56.09
123.5
23.67
123.4
0.888
1.2
123.45
340.5
8.88
23.6
4.167
9.1
340.85
0.1
123.35
56.1
0.08
8.9
9.11
4.2
23.64
23.1
23.06
23.7
Engagement in GCSE Mathematics
39
SECTION 2 – ALGEBRA
Title: Snakes and Ladders
Area of Mathematics: Number or algebra or….
Entry level:
Foundation*
Can be adapted
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
Object of harder version is to make a set of cards for use in a
Snakes and Ladders game (and then swap, or play the game, to at
least check the cards work. Very often I shall then use them with a
Junior class). The answers to questions should therefore be whole
numbers, largely within the range 1 to 6. The activity differentiates
itself by outcome. The group would normally suggest starting
points: it is important to demonstrate how equations can be
formed, for example. Students can work individually, in pairs or
threes (more than that often leads to redundancy).
Materials: Each individual or group needs card, ruler and scissors
to design up to 40 cards on an A4 sheet (careful measuring
needed!); a set of Snakes and Ladders boards, and counters, is
needed for using the cards.
A demonstration set of cards is attached.
Easier version: students play pre-prepared game: cards can be
colour-coded according to difficulty or content. Players must agree
on solution before a move is made.
Name: Jennie Golding
School: Woodroffe
Date:
June 2006
*Please delete as appropriate
Please indicate if you have an attachment : -
Engagement in GCSE Mathematics
Snakes and ladders cards
Snakes and ladders board
40
Snakes and Ladders Cards
4+ ¨ = 6
1,2,3,_,5,6
2,4,6,¨,10 x+ 3 = 4
3×
=6
1 × 2 = 10
18 ÷ 2 = ¨ 4,y,10,13. 4 + x = 7
¨-3=5
30÷ †= 3 3+†+2=9
11 - † = 8 21 ÷ 3 =¨ 3 + y = 4
p+p+p=6 19,14,9,_,. p + 3 = 8
V-2=8
_,6,9,12,1
5
.×3=
15
0× 3 =
18
17 – x =
10
†-3=6
4,_,12,16.
5,x,15,20.. 4 × t = 12 60÷†=10
.
7–z=4
† × 3 = 3 0÷ 3 = 3 ¨ - 1 = 3
40,30,20
10 ÷ 1 = 5 ¨,14,21,28 p×p=25
,_
1,3,5,7,y,
12+†+†=24 X + x = 6 ¨+ ¨ = 8
11
Engagement in GCSE Mathematics
41
Using reasoning and collective memory to revise
equations and their graphs
Area of Mathematics: Algebra - graphs
Entry level:
Intermediate
Higher
Examination Board:
Modular Linear
Outline of resource and how it was used including key
questions:
Title:
Lesson Objectives
x
Maths – to consolidate knowledge of algebraic graphs and associated equations,
particularly the ability to use y=mx+c
x
Thinking skills – to develop skills in information processing, reasoning, enquiry and
evaluation.
Pupils work in groups of 4 – groups may be pre-selected to ensure a mix of abilities.
1. Each group is provided with a set of equations.
2. Each group selects one person to come to the front and look for a graph fro 10
seconds (change time if it is too long)
3. This person goes back to their group to discuss what they saw and to see if they can
match the graph to one of their equations.
4. The group then send another person to look at the graph- hopefully with a clearer
idea of what to look for (gradient, intercept).
5. Continue in this way until the group have refined what it is they need to look at and
most have decided which equation matches the graph!
After each matching activity, pupils will need to be told which the correct equation is and given
a minute or two to discuss how their strategy might change as a result of the answer.
There are nine sets of graphs and equations to use, starting simply and then building up to
more complex equations and including some quadratics, cubics etc.
This topic could be amended for work on transformations of graphs such as f(x) to –f(x)+a,
f(x+a), f(ax), a(f(x).
IF USING THE ATTACHED GRAPHS, MAKE SURE YOU CUT OUT THE EQUATION
WRITTEN UNDERNEATH THE GRAPH!!! (I couldn’t work out how to do it so I simply cut that
part off once I’d printed it!)
Name: Sally Burt
School: Lytchett Minster
Date: June 06
*Please delete as appropriate
Please indicate if you have an attachment : - PowerPoint
Engagement in GCSE Mathematics
Resource Sheet
Other
42
Brief Lesson Plan
Explain the task – On your desk you will have some equations
and on a desk at the front there will be a diagram of a graph
that matches one of your equations. Your task is to find the
correct equation for the graph. One member from each group
will be given 10 seconds to look at a graph, this person will
return to his/her group and you will be given a minute or two to
discuss any findings and plan the next visit which will be by a
different member of the group (another 10 seconds).
By working together you should be able to find the correct
equation.
Altogether there are nine graphs to be matched: we will do
them one at a time.
Towards the end of the lesson you will be asked to describe
the strategies you used to solve each task. There is a sheet of
paper on each desk for you to make notes on as you develop
strategies.
Plenary -Ask students to evaluate strategies used and their
successes and failures to summarise what information was
needed to match each graph to it’s correct equation.
(Intercept on y axis, gradient, positive or negative gradient,
shape of graph).
Engagement in GCSE Mathematics
43
Task One
Graph y = -x
Equations y = -x, y = x, y = -x – 1
Task Two
Graph y = -x
2
Equations y = -x,
2
y = x,
2
Task Three
Graph y = 6 - x
Equations y = 6 - x,
y = 2x
y = x + 6,
Task Four
Graph y = 3x + 4
Equations y = 3x + 4, y = 3x - 4,
= 4 - 3x
y=6
y = 4x + 3,
y
Task Five
Graph y- 5x = 1
Equations y - 5x = 1, y = 1 - 5x, y + 1 = 5x
Task Six
Graphs y = x2 + 4 and y = 3x2
Equations y = x2 + 4, y = 3x2, y = ½ x2,
Task Seven
Graphs y = 1,
x
Engagement in GCSE Mathematics
y = x2
and y=x3 - 3
44
y = 1,
x
y = x,
3
Engagement in GCSE Mathematics
y = x3,
y=x3 - 3
45
Engagement in GCSE Mathematics
46
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47
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48
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49
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50
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51
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52
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53
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54
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55
Title: Equations of Straight Lines
Area of Mathematics: Algebra - graphs
Entry level:
Foundation
Examination Board:
Linear
Outline of resource and how it was used including key
questions:
Pupils sort into 4 groups of their choice (should end up with some
sorting by coefficients of x some by constants etc
This can be followed by investigational work on finding the
significance of these values if this topic is new.
Can also be used for revision purposes, once pupils have sorted
cards into groups, give them copies of the graphs. This should
enable them to come to the correct conclusions.
Name: Sally Burt
School: Lytchett Minster
Date: June 06
*Please delete as appropriate
Please indicate if you have an attachment : - resource sheet of equations
Engagement in GCSE Mathematics
56
y = 2x – 1
y = 3x-4
y = 3x -1
y=½x+2
y = 2x + 2
y=½x–1
y = 2x – 4
y = ½ x -4
y=x–1
y = 2x + 3
y=x+2
y = 3x + 3
y=x–4
y = 3x + 2
Engagement in GCSE Mathematics
y=½x+3
y=x+3
57
A puzzle with Numbers
Title:
Area of Mathematics: Algebra
Foundation
Entry level:
Linear
Examination Board: Edexcel
Outline of resource and how it was used including key
questions:
This was used as a starter. It engaged students who did not really enjoy
algebra. Students practiced mental maths skills and found that the
answer was always 3. They then tried to use algebra to prove this.
Name: Clare Davenport
Date: June 2006
School: Shaftesbury School and Sports College
*Please delete as appropriate
Please indicate if you have an attachment : - PowerPoint
Resource Sheet
Other
A puzzle with numbers
•
•
•
•
•
•
•
•
Think of a number
Double it
Add 7
Multiply it by 5
Subtract 5
Divide by 10
Subtract the first number
What number do you have?
A puzzle with algebra
n
2n
2n + 7
5( 2n + 7)
10n + 35 – 5
10n + 30
10
• n+ 3 – n
•
•
•
•
•
•
Engagement in GCSE Mathematics
= 10n + 35
= 10n + 30
= n+3
=n
58
Title: Algebra Match Up Cards
Area of Mathematics: Algebra
Foundation
Intermediate
Entry level:
Linear
Examination Board: Edexcel
Outline of resource and how it was used including key
questions:
This was used as a starter. Students work in pairs to match up these
activities.
Print the resource sheet onto coloured card, laminate and cut up.
Name: Clare Davenport
Date: June 2006
School: Shaftesbury School and Sports College
*Please delete as appropriate
Please indicate if you have an attachment : - PowerPoint
Engagement in GCSE Mathematics
Resource Sheet
Other
59
Match
Ł
4(3b² - 2a)
Match
Ł
3a(2 + 4b)
Match
Ł
a(12a + 4b)
Match
Ł
-7(q² - s)
Match
Ł
2(2a + 3b)
Match
Ł
-12s² + 9t
Match
Ł
30m – 5mn
Match
Ł
-32f – 48g
Match
Ł
-7q² + 7s
Match
Ł
9m(-m+nb)
Match
Ł
3p(2p-4r)
Match
Ł
-20c + 24d
Match
Ł
-(6v – 7u)
Match
Ł
-4(5c – 6d)
Match
Ł
-6m² - 4m
Match
Ł
12b² - 8a
Match
Ł
5(6m –mn)
Match
Ł
8(-4f- 6g)
Match
Ł
12a² + 4ab
Match
Ł
4x(x + 2y)
Match
Ł
-9m² + 9bmn
Match
Ł
-10t – 5u
Match
Ł
4s(2s – 4c)
Match
Ł
12x² - 18x
Match
Ł
-2(3m² + 2m)
Match
Ł
4a + 6b
Match
Ł
5(6a + 3b)
Match
Ł
6a + 12ab
Match
Ł
30a + 15b
Match
Ł
8s² -16cs
Match
Ł
4x² + 8xy
Match
Ł
6p² - 12pr
Match
Ł
-6v + 7u
Match
Ł
-3(4s² - 3t)
Match
Ł
6x(2x – 3)
Match
Ł
-5(2t + u)
Engagement in GCSE Mathematics
60
Title: Excel Match up Game
Area of Mathematics: Algebra
Lower
Intermediate
Higher
Entry level:
Linear
Examination Board: Edexcel
Outline of resource and how it was used including key
questions:
This is used as a game at the end of a lesson. The questions can
obviously be changed according to ability. We have a house system
with colours so when a student correctly chooses a pair, it can be filled
that colour.
Name: Clare Davenport
Date: June 2006
School: Shaftesbury School and Sports College
Engagement in GCSE Mathematics
61
Title: Quadratic Maze
Area of Mathematics: Algebra
Intermediate
Higher
Entry level:
Linear
Examination Board: Edexcel
Outline of resource and how it was used including key
questions:
Thi sheets is an excellent activity to reinforce understanding. Use as a
starter or during a lesson. Students work from IN to OUT by shading
boxes which have multiples of (x-2)
Name: Clare Davenport
Date: June 2006
School: Shaftesbury School and Sports College
Please indicate if you have an attachment : - PowerPoint
Engagement in GCSE Mathematics
Resource Sheet
Other
62
Find your way through this quadratic maze.
Shading multiples of (x - 2)
You can only move vertically or horizontally.
Factorising the quadratics will help!
IN
x² - x – 2
x² + x - 6
x² + 2x - 3
x² + x - 2
x² - 9
2x² - 5x – 3
x² - 4x + 4
x² - 4
2x² - x - 6
x² - x - 6
x² - x – 2
5x² - 21x + 4
x² + 3x + 2
3x² - 5x - 2
3x² - x - 2
2x² + 3x + 1
x² + 5x - 14
x² - 5x + 6
x² - 3x + 2
3x² - 7x - 6
x² - 2x – 15
5x² - 12x + 4
x² + 6x + 9
x² + 7x + 12
2x² - 7x - 4
x² + 4x + 4
x² - 8x + 12
x² + 5x + 6
x² - x - 12
x² - 6x + 8
x² - x – 2
x² - 7x + 10
2x² + 9x + 4
3x² - 10x - 8
x² - x - 2
x² - 10x + 16
2x² - 7x + 6
x² - 2x - 8
5x² + 13x + 6
3x² - 2x - 8
x² x – 2
x² + 8x + 15
3x² - 4x - 4
x² - x - 2
x² + 4x - 12
x² + 3x – 10
2x² - 3x - 2
2x² + 3x - 14
x² - 3x - 10
5x² - 16x + 12
OUT
Engagement in GCSE Mathematics
63
Title:
Equation Sort
Area of Mathematics:
Algebra
Foundation
Intermediate
Entry level:
Linear
Examination Board: Edexcel
Outline of resource and how it was used including key
questions:
Print these equations onto card. PGet students to sort the equations in
any grouping, ask them to justify their grouping. You could ask
students to group these groups in groups. You could then get students
to work out a point which lines on each line. Students could plot these
graphs onto a grid.
Name: Clare Davenport
Date: June 2006
School: Shaftesbury School and Sports College
y = 3x
y=x+2
y=x+1
y=x
y=4
y = 10
y = 2x + 2
y = 4x
y = -2
y=7
y = 2x - 4
y = 2x
y=2–x
y=x+4
y=7-x
y = 2x - 4
y=½x+4
y=x–4
y = 2x + 4
y = 10 - x
y = 4x + 2
y=½x+2
y=0
y = 3x + 2
Engagement in GCSE Mathematics
64
Title
Swatting Flies!!
Area of Mathematics:
Algebra
Foundation
Intermediate
Entry level:
Linear
Examination Board: Edexcel
Outline of resource and how it was used including key
questions:
This is a very light hearted way of revising any type of algebraic graph.
It would help if you had Autograph, but you could do this on the
board or on paper. Sketch a set of axes and mark several flies. Give the
students an appropriate number of lives in which to swat them. This
means so that a graph crosses through them! On the picture below you
can see how the line y = 2 has squished a fly. You could set conditions
about the type of graph they can use.
Name: Clare Davenport
Date: June 2006
School: Shaftesbury School and Sports College
Engagement in GCSE Mathematics
65
ADDITIONAL ALGEBRA RESOURCES WITHOUT A CASE
STUDY
PowerPoint Resources
P and Q
If p and q are two numbers between 0 and 1, then p+q - pq will also be a
number between 0 and 1. Why?
Guess my number
I think of a positive number, add one, multiply my answer by itself, take away
twice the number I first thought of, and my answer is 26. What number did I
think of?
Which two are they?
Two of these expressions always have the same value, whatever number you
choose n to be. Which two are they?
n²- 1
7n + 5
Engagement in GCSE Mathematics
(n+1)(n–2)
2n² + n
n² - n - 2
66
Brackets Grid Starter
Use as a quick starter to revise multiplying expressions with
Intermediate Year 10.
X
2a – b + c
-3
a2
a+b
2c -4b
-4b
b–c
a 2 – b2
Engagement in GCSE Mathematics
67
Dominoes Like Terms
7a+2a+a
2r+2w 5t -6s –s +4t
-s
2n
b+6b-2b
10a
t-t + t + t + s 9t – 7s
+ s-s
2x+4y+y+8
x
5b
7h+h+5h+h
-h
2t+s
7g-3g-7+3 10x+5 2w+8-10-w
y
13h
10t+6s+5t2s-t
w-2
6k-10k
16j-3j-5j
Engagement in GCSE Mathematics
4g-4
7k+m+n+km+n
14t+4s 5r+4t+6d+r3d
-4k 7y+8-4y-9
8k+2n
6r+4t+3
d
68
n+n+n-nn+n
8j
r+r+s+w+ws-s
3y -1
S.Burt
Engagement in GCSE Mathematics
69
Dominoes multiplying terms
5j x 4l
6a x 7c
k3
h4m5
axaxa
42ac 2w x 2w x 2w
20jl
2b x 3c
a3
c2 x c3
8w3
2f x f
6bc
2d2 x 3d2
c5
8h x 5h
2f2
(7p)2
6d4
2m x 3n x
4p
40h2
rxrxtxt
49p2
2d x 3d x 2e
x 2e
r2t2
y x y x y x 24mn
y
p
Engagement in GCSE Mathematics
70
k2 x k
y4
h2 x h2 x m3 x
m2
24d2
e2
S.Burt
Engagement in GCSE Mathematics
71
SECTION 3 – SHAPE AND SPACE
Title: Angle – True or False
Area of Mathematics: Angle rules
Entry level:
Foundation*
Intermediate*
Higher*
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
An activity with diagrams and statements. Students, in pairs, need
to decide/work which are true and which are false.
I used this with Intermediate Year 10 to revise angle work
completed during Key Stage 3. There might be too many cards but
it would be easy to reduce the number or simply take out the one
involving parallel lines for the less able students.
I set a homework based on GCSE questions to complete the
revision before we moved on to circle theorems.
Angle\Angle true or false.xls
PS I think there are a couple of spelling errors on the work sheets.
Name: Sally Burt
School: Lytchett Minster School
Date:
8/6/06
*Please delete as appropriate
Resource Sheet ‘Shape and angle
true or false’ Excel resource – paper copy in school master copy only.
Please indicate if you have an attachment: -
Engagement in GCSE Mathematics
72
Title: Transformation Sorting
Area of Mathematics: TRANSFORMATIONS
Entry level:
Foundation*
Intermediate*
Higher*
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
Used as a starter to revise transformations with an Intermediate
Year 10 group but could be used with any ability if only some of
the cards are used.
Class worked in groups and were asked to group the cards
according to the four transformations. They should have found that
some of the cards didn’t fit into a single group, some as they
involve skews, some as they are combined transformation.
This can be followed by a discussion on how to tell what
transformation has taken place and recap on how to define
transformations fully (E.g. Rotation is defined by giving a direction,
the number of degrees and the centre of rotation).
This can then be followed by formal work on combined
transformations for Higher Students
Transformations and vectors\transformation sorting.doc
Name: Sally Burt
School: Lytchett Minster School
Date:
8/6/06
*Please delete as appropriate
Please indicate if you have an attachment : -
Engagement in GCSE Mathematics
Resource Sheet
73
1 Shape
Image
Image
2 Shape
3 Shape
Image
Image
4 Shape
5 Shape
Image
6 Shape
7 Shape
Image
Image
8 Shape
Image
9 Shape
Image
10
Shape
Image
11 Shape
Image
12 Shape
Image
13 Shape
Image
Engagement in GCSE Mathematics
14 Shape
74
15 Shape
Image
Image
16
Shape
17 Shape
Image
18 Shape
Image
19 Shape
Image
20 Shape
Image
Image
22 Shape
21 Shape
23 Shape
Image
Image
Image
24 Shape
S.Burt
The images on the work sheet resource are all within the boxes difficulty has
been experienced importing it into Word.
Engagement in GCSE Mathematics
75
Title: Pythagoras’ theorem – a right angled triangle or not?
Area of Mathematics: Shape, space and measure
Entry level:
Intermediate*
Examination Board: EdExcel
Linear*
Outline of resource and how it was used including key
questions:
Right angled triangle or not?
Having taught Pythagoras’s theorem the previous lesson I used
the attached OHT to aid discussion of whether the 11 triangles
detailed were right angled triangles or not. A lively discussion
followed with much use of calculators with some students jotting
working out down. The Pythagorean triples were soon spotted and
the similarity between these triangles identified.
You could use this as either a starter or a plenary or even as a
homework.
Name: Amy Harris
School: Budmouth Technology College
Date:
*Please delete as appropriate
Please indicate if you have an attachment : - Resource Sheet
Engagement in GCSE Mathematics
76
A right angled triangle or not?
Check these out
h2
a 2 b2
A
b
h
3
4
5
9
12
15
4
5
7
6
8
10
1
2
3
5
7
8
5
8
9.4(1dp)
4
6
7.8(1dp)
4
8
8.9(1dp)
1
1
1.4(1dp)
30
40
50
Engagement in GCSE Mathematics
decision
77
ADDITIONAL SHAPE AND SPACE RESOURCES WITHOUT A
CASE STUDY
Power Point
Point T
The distances of a point T from the corners of an equilateral triangle are 3, 5
and 7 cms. What is the size of the equilateral triangle?
Engagement in GCSE Mathematics
78
SECTION 4 – HANDLING DATA - STATISTICS
Title: The Averages song
Area of Mathematics: Handling data – calculating averages
Entry level:
Foundation*
Intermediate*
Examination Board: OCR
Modular*
Outline of resource and how it was used including key
questions:
A simple song to help remember which average is which.
This should be sung to the Eastenders theme tune.
Commonest is called the mode
Can be 2 or 3 depends on the data
Median is in the middle
If you line ‘em up from smallest to the biggest
Add ‘em up, share ‘em out
Then you have found the mean of the data
Name: Lucy Neil
School: Gillingham
Date:
July 06
*Please delete as appropriate
Theme music for Eastenders is on the CDROM.
Engagement in GCSE Mathematics
79
Title:
Tongue twisters
Handling Data (scatter graphs, and
correlation)
Entry level:
Foundation*
Intermediate*
Higher*
Examination Board: OCR
Modular*
Outline of resource and how it was used including key
questions:
In pairs students should time how long it takes each other to say
the tongue twister below.
Area of Mathematics:
‘This is a can opener, a can opener can open any can can a
opener can. If a can opener cannot open a can it cannot be a can
opener can it?’
They should make several attempts and then take an average. Ask
the students who was the quickest? Why they think they were the
best? You could also ask the slowest what they think. Ask the rest
of the class what they think they might be able to affect how quick
they are at tongue twisters.
Ÿ Ability (later do a short IQ test)
Ÿ Size of mouth (measure width of mouth when grinning
widely)
Ÿ Height of mouth when open
Ÿ Volume of liquid in a mouthful (measure water in a
measuring jug from the science department)
Ÿ Quick wittedness (time how long it takes to answer 20 quick
mental arithmetic questions)
Ÿ Other ideas?
Record the results for each member in the class. Then draw
scatter graphs to check if there is any correlation.
Name: Graham Holdaway
School: Gillingham
Date:
July 06
*Please delete as appropriate
Engagement in GCSE Mathematics
80
Title: Correlation
Area of Mathematics: Handling Data
Entry level:
Foundation*
Intermediate*
Higher*
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
Use as an introduction/recap to correlation with any ability.
Give pupils sets of cards and ask them to match (in pairs) those
that they believe have a relationship.
Then put the ‘matched pairs’ into two groups – strong relationship,
weak relationship.
There are lots of ways the cards could be matched so this could
generate some discussion.
Students could also be asked to find the pair which is least likely to
have a relationship.
Name: Sally Burt
School: Lytchett Minster
Date:
June 06
*Please delete as appropriate
Please indicate if you have an attachment : - Resource sheet –Is there a
relationship?
Engagement in GCSE Mathematics
81
The size of
feet
The price
of a car
Handspan
The age of
a car
Height of a
person
The length
of a car
Mock
results for
Maths
Paper 2
Marks out
of 10 in a
Times
Tables Test
Marks
award by
Judge B in
a dancing
competition
Price of
House
Ability to
spell
Ability to
read
Ability to
sing
Ability to
play the
piano
Size of
mouth
How loud
you can
shout
Number of
bedrooms in
a house
Number of
hours
worked
Length of
leg
Colour of
front door
of a house
Wages
Weight of a
person
How fast
you can run
Number of
times you yawn
in a day
Maths set
Age of a
person
Length of
arm
Size of bed Length of
hair
Engagement in GCSE Mathematics
Intelligence
Hair colour
Mock results
for Maths
Paper 1
Marks out of
10 in a Mental
Maths
Marks award
by Judge A in
a dancing
competition
Age of House
Number of
hours of sleep
82
SECTION 5 – HANDLING DATA - PROBABILITY
Title: The Monty Hall Problem
Area of Mathematics: Handling data - probability
Entry level:
Intermediate*
Higher*
Examination Board: OCR
Modular*
Outline of resource and how it was used including key
questions:
Read the article from ‘The Curious Incident of the Dog’ (as much
or as little from P 78-79)
Before reading out an explanation, act out the game using some
prizes (chocolate or forfeit) and ask students what they should do
and what the chances are of winning a car/chocolate. They will
usually say P(win car/chocolate) = ½ .
Then explain that you should change because the P(win
car/chocolate) = 2/3
Ask students to work out why it is better to swap (here you might
want to go through the comments made on page 79-80.
Draw a probability tree to show how it works. See page 81 for an
example. Fill in the probability on each branch. As the plenary you
can carry out several trials of the problem with different members
of the group and check that the probability of winning when you
change does tend towards 2/3.
Alternatively you might just want to use this lesson as a game and
rather than referring to the book just set up the lesson as a game
show, record the results, do enough times so that if you swap
P(win) is 2/3. Then discuss with students are they just a lucky
class and draw a probability tree to show that the p(win if swap) is
2/3 rather than ½.
Name: Rachel Day
School: Gillingham
Date:
*Please delete as appropriate
The pages form the book are included as jig file on the CD-ROM
Engagement in GCSE Mathematics
83
Title: The Tree of Life
Area of Mathematics: Probability
Entry level:
Intermediate*
Higher*
Examination Board: OCR
Modular*
Outline of resource and how it was used including key
questions:
Draw a probability tree to show what you would do on the
weekend. This can be as detailed as you want, I used
P(go away for the weekend) =0.4 P(don’t go away) = 0.6
Friday evening
P(go to pub) = 0.7 P(don’t go to pub) = 0.3
Saturday
P(go shopping) = 0.65 P(don’t go shopping) = 0.35
Saturday evening
P(stay in) = 0.85 P(don’t stay in) = 0.15
Sunday
P(work) = 0.6 P(no work) = 0.4
As a whole class activity, draw a probability tree. Then ask
questions:
1. What is the probability of going out both nights?
2. What is the probability of going out at least one night?
3. What is the probability of going shopping on Saturday and
out in the evening?
Alternatives to this tree depending on the level of the group would
be to include conditional probabilities, and ask more detailed
questions.
E.g. What is the probability of going away and working on a
Sunday?
Then ask the students to draw their own life tree – to make it
simpler to start with, it should only be one evening and use
complementary events (i.e. P(watch TV), P(don’t watch TV)) rather
than a variety of options.
They should the write a list of questions for their partner to
calculate the probability of certain outcomes.
Name: Lucy Neil
School: Gillingham
Date:
July 06
*Please delete as appropriate
Engagement in GCSE Mathematics
84
Title: Biased or Not?
Area of Mathematics: Data collection
Entry level:
Foundation*
Intermediate*
Higher*
Examination Board: Any
Modular* Linear*
Outline of resource and how it was used including key
questions:
Haven’t used this yet as have written it for the new 2 Tier Scheme
for Sept. I hope to use it as an introduction to collecting unbiased
data so it will be used before any formal work takes place –
hopefully students will be able to decide when data is unbiased as
a result of the task!
Name: Sally Burt
School: Lytchett Minster School
Date:
8/6/06
*Please delete as appropriate
Please indicate if you have an attachment : -
Engagement in GCSE Mathematics
biased or not worksheet
85
Decide which of the following ways of collecting data is fair. If you think
the sample would be biased, write down what is wrong and how you could
improve it.
1. Mr Fit wants to find out how
7. An engineer, Miss Roads, is
many people in his area regularly
trying to find out how busy a
played sport. He decided to stand
certain stretch of road is. Each
outside the local Sports Centre to
day, he counts the number of cars
ask people.
passing a certain point between 2
p.m. and 3 p.m. so that he can write
a report.
2. An engineer, Mr Pelican is trying 8. Mr Softy, the Headteacher,
to find out how busy a pedestrian
wants to find out if pupils are happy
crossing is used. Each day, he
with the design of their school tie.
counts the number of people using
He asks ten males in Year 10.
it for twenty minutes 8.30 a.m.,
1.30 p.m. and 4.30 p.m. so that he
can write a report.
3. A local radio station is carrying
9. As part of a school project,
out a survey to find out how popular David wants to find out how older
people feel about modern music. He
the new cinema is. They send Mr
decides to ask his parents, uncles,
Sound to stand outside the local
supermarket and ask fifty adult
aunts and grandparents.
males, fifty adult females and fifty
teenagers.
4. Mrs Story, the school librarian,
10. A car manufacturer wants to
wants to find out about the reading find out if there have been any
habits of children in her school.
problems with the new car they
One lunchtime, as the children
have just launched. They send out a
come into the library, she asks
questionnaire to every third person
them how many books they have
who has purchased the car.
read in the last two weeks. She
uses this information to work out
the average number of books read
by the children in the school.
Engagement in GCSE Mathematics
86
5. Ms Beauty works for a famous
skin care firm who have just
launched a new product called ‘Get
rid of Wrinkles’. She is asked to
carry out a survey to see how
people feel about the product. She
asks ten women who she thinks are
about 25 years old.
6. Mr Green is doing a survey to
find out how much time people
spend gardening. He visits a local
gardening centre and asks people as
they are leaving.
Engagement in GCSE Mathematics
11. To find out how popular the
school dinners are, Mrs Mean asks
15 boys and 15 girls from each year
group.
12. Miss Wood works for a DIY
magazine. She wants to find out
how many people do their own
decorating. She goes to the high
street on Wednesday morning and
again on Saturday morning and asks
as many people to answer her
questions as she can in an two
hours.
87
ADDITIONAL HANDLING DATA RESOURCES PROBABILITY
WITHOUT CASE STUDIES
Random Sample PowerPoint
Imagine!
• Two students from this group are to be given a packet of
sweets.
• Another two are going to have to help me with some
work during their break time.
• I need to select these students at random.
• Write down as many ways as you can think of that would
really make my selection random
LEARNING OBJECTIVE:
To know what a random sample is
•
To know how to select a random sample.
To know how to select a ‘Systematic Sample.’
Engagement in GCSE Mathematics
88
Stratified Sample PowerPoint
LEARNING OBJECTIVE:
To know how to collect an unbiased sample
A committee of 10 people is required to represent a small
village with a population of 800 women and 200 men, What
would be a ‘fair’ sample?
How many men, how many women?
• What if 20 people were needed for the committee?
• What if you wanted a committee of 25 from a village
where there are 1600 men and 1000 women?
• What if you wanted to select 20 people to stand on the
School Council where there are 80 in Yr9,
• 105 in Yr 10 and 115 in Yr 11.
Engagement in GCSE Mathematics
89
PowerPoint
What could this table be about?
What could the table be about?
What facts could you write down from the information given?
(You should be able to find nine!)
Blue
Brown
Boys
12
10
Girls
14
15
• Now write some probabilities that you could find from
the table and then find them!
Engagement in GCSE Mathematics
90
Give students following sets of data with some possible
statistical measures.
Students have a set time to find out as many incorrect values
as possible - the idea being that they begin by using common
sense such as the mean in Set A can’t be 9 as it is only just
within the data (not because there isn’t a 9)
SET A
MEAN = 6
4, 1, 7, 10, 8, 8, 5, 6,
7, 4
MODE = 8
MEAN = 9
Engagement in GCSE Mathematics
91
SET B
MEAN = 3 ¼
1.2, 4.5, 7.8, 2.3,
2.4, 1.3
RANGE = 6.6
There is no
MODE.
MEDIAN = 2.35
Engagement in GCSE Mathematics
92
SET C
RANGE = 34
34.6, 28.2, 31.7, 34.8,
29.2, 30, 29.6, 34.6
MEDIAN = 30.85
MODE = 34.7
There is no MODE
Engagement in GCSE Mathematics
93
SET D
MEAN = 310
310, 330, 329, 319,
325, 348, 317, 325,
337, 340, 329, 311
RANGE = 325
RANGE = 38
MEDIAN = 329
Engagement in GCSE Mathematics
94
SET E
MEAN = 13.67
SHOE
SIZE
5
6
7
FREQUENCY
14
15
12
MEAN = 5.95
MEDIAN = 6
RANGE = 2
Engagement in GCSE Mathematics
95
SET F
RANGE = 0.9
2.3, 2.5, 2.9, 2.1 2.0,
2.6, 2.7, 2.3, 2.3, 2.5,
2.4, 2.3, 2.8, 2.7
MEAN = 2.85
MODE = 2
Engagement in GCSE Mathematics
96
SET G
MEAN = 52.8
NUMBER
OF
MATCHES
51
52
55
NUMBER
OF BOXES
2
7
4
MEAN = 4.3
RANGE = 4.3
RANGE = 4
Engagement in GCSE Mathematics
97
SET H
MODE = 38
32, 38, 36, 36, 32, 34,
35, 38
RANGE = 6
MEAN = 32
MEAN = 38
Engagement in GCSE Mathematics
98
SET I
MEDIAN =
19.5
HANDSPAN FREQUENCY
(nearest cm)
16cm – 17
16
cm
18 cm – 19
21
cm
20cm – 21
29
cm
RANGE = 13
MODE = 21
MEAN = 22
Engagement in GCSE Mathematics
99
RANGE = 5
Engagement in GCSE Mathematics
100
Mutually Exclusive Events
These cards refer to finding probabilities. They can be sorted into two groups. Can you
sort them? How did you sort them?
Has blue eyes or
fair hair.
Good at singing or Pick an Ace or a
good at dancing.
Heart from a
pack of cards.
Roll a prime
Rode a bike or
Pick an even
number on or
caught the bus to number or a
even number on
school last
multiple of 3.
dice.
Tuesday.
Girls with long
Has blue eyes or Roll a 2 or a
hair or girls with brown eyes.
number greater
blonde hair.
then 4 on a
dice.
Choose a toffee Win a football
Wears glasses
or a mint from a match or draw a or wears a
bag of sweets.
football match.
brace.
Engagement in GCSE Mathematics
I’ll have chips or
chops for tea.
Wore trainers to
school or wore
shoes to school
Pick an Ace or a
Jack from a pack
of cards.
Wore glasses or
contact lenses to
school yesterday.101
Bibliography
Improving learning in mathematics – mainly software
resources
Standards Unit materials
Questions and Prompts for Mathematical Thinking
ATM book Watson and Mason 1998
Thinkers
ATM book Bills, Bills, Watson and Mason
2004
Engagement in GCSE Mathematics
102