Secondary Maths
Unit 4:
Finding factors: thinking and connecting
mathematics, and using a textbook
creatively
Introduction................................................................................................................... 1
Learning outcomes ..................................................................................................... 2
1 Some issues with learning from textbooks .......................................................... 3
2 Making connections to understand mathematics ............................................... 3
Preparation for Activity 1 ........................................................................................ 4
3 Working on finding a purpose for learning mathematics ................................... 7
Preparation for Activity 2 ........................................................................................ 7
4 Practising techniques and noticing differences between LCM and HCF ........ 9
Preparation for Activity 3 ........................................................................................ 9
5 Learning from the work of fictitious students ..................................................... 10
Preparation for Activity 4 ...................................................................................... 10
6 Adapting questions from textbooks ..................................................................... 12
Preparation for Activity 5 ...................................................................................... 12
7 Summary ................................................................................................................. 14
8 Resources ............................................................................................................... 15
Resource 1: NCF/NCFTE teaching requirements ........................................... 15
References ................................................................................................................. 16
Acknowledgements ................................................................................................... 16
Except for third party materials and otherwise stated, the content of this unit is
made available under a Creative Commons Attribution licence:
http://creativecommons.org/licenses/by/3.0/.
Introduction
Finding factors and multiples is an essential part of mathematics. The use of
these concepts starts from an early age, when young students are working on
multiplication and sharing. In school mathematics it is built on over the years
and these concepts are used in very high level mathematics as well.
In this unit, you will think about the concepts of factors and multiples, and use
the ideas of highest common factor (HCF) and lowest common multiple
(LCM). Through the activities you also think about how to extend your
students’ ability to think through mathematical ideas and make connections.
The textbook is often a mathematics teacher’s most valuable resource, but it
can constrain teaching. In this unit you will think about using the textbook
more creatively.
In this unit, you are invited to first undertake the activities for yourself and
reflect on the experience as a student; then try them out in your classroom,
and reflect on that experience as a teacher. Trying for yourself will mean you
get insights into a student’s experiences, which can in turn influence your
teaching and your experiences as a teacher. This will help you to develop a
more student-focused teaching environment.
The activities in this unit require you to work on mathematical problems either
alone or with your class. First you will read about a mathematical approach or
method; then you will apply it in the activity, solving a given problem.
Afterwards you will be able to read a commentary by another teacher who did
the same activity with a group of students so you can evaluate its
effectiveness and compare with your own experience.
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Learning outcomes
After studying this unit, you should be able to:
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specify what finding factors means in the case of numbers and compare
the methods with finding factors in expressions
explain the meaning of a highest common factor and a lowest common
multiple applied to numbers and expressions
turn textbook questions into richer and more interesting problems
focus on the process of doing mathematics instead of focusing on finding
answers
make connections between mathematical concepts and properties.
This unit links to the teaching requirements of the NCF (2005) and NCFTE
(2009) outlined in Resource 1.
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1 Some issues with learning from
textbooks
When students are using textbook questions, making connections is often lost
because the focus is on completing questions on one element. Making
connections is a very important part of understanding mathematics and is also
often the joyous part of mathematics.
Another issue with using textbooks is that the purpose of the learning is not
always made clear to the student and they are rarely helped to see how that
learning fits into mathematics as a whole. The student’s aim might be to do
what they are told to do, rather than learning mathematics as a connected
system. Further, the student or teacher might get so caught up in completing
the problems correctly that they lose any perspective of the learning that is
supposed to take place.
Pause for thought
Think about a recent mathematics lesson in your classroom. What learning
took place when students did exercises from the textbook? What mathematics
were they learning? How far were they thinking mathematically? How far were
they making connections between mathematical concepts and ideas? Why do
you think this is?
The activities in this unit start from taking problems and examples as they can
be found in any textbook. Asking questions like the following will move the
students from mechanically finding answers, to really thinking about what they
are doing:
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How did you find that answer?
What is the same and what is different in your answers to those
questions?
What is the same or what is different in your process of thinking?
When students are given the opportunity to think about the process of
learning and making connections, they will actually learn. However, students
might be uncomfortable with these kind of questions, as they might not have
been required to think in this way before. Therefore, a support system will
need to be put in place; for example, let the students work in groups or pairs.
2 Making connections to understand
mathematics
By Class IX, teachers assume that their students will have a solid knowledge
of what factors and multiples are, and be able to apply them in different
settings. Students should be able to expand this knowledge further. The
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National Curriculum Framework 2005 requires students to study the HCF and
LCM of numbers, and to be able to apply this knowledge when working with
expressions. These topics and concepts are thus studied at different times,
and in different years. Because study is completed over time, students can fail
to see connections between the different aspects that they study, meaning
that their knowledge becomes fragmented. Students often have to rely on
memorisation at each stage rather than understanding of the underlying
principles. This leads to a lack appreciation of the power of factors and
multiples.
Preparation for Activity 1
The next activity aims to address this fragmentation by focusing on the
mathematical thinking processes involved in finding factors of numbers and
expressions. The students are asked to make connections between what a
factor is, factors of numbers and factors in expressions. The activity requires
students to exchange their ideas with other students – they could be working
in pairs or small groups.
Activity 1: Finding factors of numbers and expressions
List the various factors of the following numbers and expressions.
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60
3xy
15
12x2y3
3x4 – 27x4
2x2 – 8x + 8
1. What factors did you find? Do you consider these factors to be all of a
same ‘type’? Discuss with your neighbour or within your group.
2. Why do you think some numbers only have two factors?
3. How did you find these? Describe your own method of finding the factors
to the others in your group. Did you all use the same methods? Did all the
methods work well with all questions?
4. If you had coloured pencils or pens, could you come up with another
method using colour-coding?
5. Write your own definition or description of what a factor is, with examples
of where it can be found.
Case Study 1: Teacher Tanseem reflects on using Activity 1
To offer mutual support and to give more opportunities for coming up with a
rich collection of ideas, I put students in groups of four. I told them to move if
necessary so that discussion could take place easily. The students came up
with the factors very easily in the case of pure numbers but there were a lot of
arguments about the ones involving expressions. This happened in most of
the groups, so I thought it might be helpful to discuss these questions as a
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whole class. First I asked them to prepare a short presentation for the whole
class; this is what they came up with:
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They quite accurately identified the factors of 60 as 1, 2, 3, 4, 5, 6,
10, 12, 15, 20, 30 and 60 – although some groups missed out a few
of them, they quickly wrote down the ones they had missed. This led
to a discussion about the need to be systematic, and what
approaches could help them to be systematic.
In the second question, some groups identified only three factors:
3xy, 3 and x and y.
Many groups were quite good at noticing prime factors and
explaining that these were special because they could only be
divided by themselves and by 1.
I really wanted the students to notice the similarity between factors of
numbers and expressions. As a first step, instead of asking, ‘In what way are
prime factors different from non-prime factors?’ – to which I expected they
would give the explanation they had already given – I asked, ‘In what way are
non-prime factors different from prime factors?’
They were a bit baffled by that, so I added, ‘OK, look at the factors of 60 as an
example. Now start your sentence with: “Non-prime factors are different
because …”.’ There were several volunteers to have a go at answering this,
but it sounded neither clear nor concise – which I did not think was conductive
to good discussions or learning. I decided to give them a ‘speaking frame’, like
they sometimes give writing frames in language learning. So I told them, ‘Try
saying it first to your neighbour and then I will ask you to say it to the whole
class.’
They practised verbalising their thinking in that way with each other for a few
minutes. I also noticed that some were writing what they had said down in
their exercise books, probably so as not to forget what to say when asked.
The speaking frame really seemed to have helped because when we came to
sharing their sentences, their responses were clear and concise and they
used mathematical language in a precise way. They had managed to both
notice and express that non-prime factors were numbers that could be
decomposed further into products of other numbers until only prime factors
were used. I was seriously impressed! But they still had not made the
connection between these kind of factors and the factors of expressions.
My first instinct was simply to tell them – to just pass on my knowledge to
them. But they were so full of enthusiasm and engagement, from being asked
to come up with their ideas, that I could not just tell them. I wanted them to
discover the mathematics themselves, and experience the joy of discovering,
seeing the beauty of mathematical structures and their connectivity. But what
question could I pose so that they could become aware of this connection
between factors of numbers and factors of expressions? So many questions
and approaches came up in my mind, but they were too complicated, or
simply disguised ‘telling them what it is’ questions! What if I just told them
what I really wanted them to do? So I said:
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‘I really want you to discover the connections and similarities and differences
between finding factors of numbers, and finding factors of expressions. For
example, between the factors of 60 and the factors of 3xy. And I don’t just
want to tell you, I want you to think about it and discover it for yourself. So
think for a moment about all these discussions we have had during this
lesson, about the difference between prime and non-prime factors, and look at
the factors you have found for 60 and 3xy. What is the same, what is
different? Have you got all the factors for all? Have a go at it in your groups.’
And they did find the missing factors for 3xy. They found the factors for 3x4 –
27x2: from re-writing it as 3x2(x2 – 9) and then going onto identifying the
factors as 3, x, x2, x2 – 9, (x2 – 9), x2(x2 – 9) and 3x2(x2 – 9).
Only then did we move on to think about methods used. We had to do that in
the next lesson because we had run out of time. This was actually not a bad
thing, because I was able to ask them to think back to the previous lesson,
and look in their exercise books what they had done and what their thinking
had been, asking them to re-enter that thinking. Coming up with descriptions
of methods went rather smoothly, probably because we had done so much
thinking about it already. We shared the different methods with the whole
class, and they corrected each other if the method was lacking or
overcomplicated. We actually ended up with a ‘Suggested method of Mrs A’s
Class’, and wrote that on a large piece of paper that was then displayed on
the wall.
Pause for thought
When you do an exercise like this with your class, reflect afterwards on what
went well and what went less well. Consider the questions that led to the
students being interested and being able to get on, and those where you
needed to clarify. Such reflection always helps with finding a ‘script’ that
encourages the students to find mathematics interesting and enjoyable. When
your students struggle too much to understand, they are less likely to engage.
Use this reflective exercise every time you implement an activity from this unit,
so that you can recognise, as Teacher Tanseem did, some quite small things
that made a difference.
Good questions to trigger your reflections are:
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How did the lesson go with your class?
What questions did you use to probe your students’ understanding?
Did you feel you had to intervene at any point?
What points did you feel you had to reinforce?
Was there anything you would do differently another time?
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3 Working on finding a purpose for
learning mathematics
In the previous activity the students were asked to think about different
methods they could use to factorise, and the limitations of such methods.
They were given the opportunity to think about these processes and learn
from that experience. It can be very useful to discuss the different ways of
thinking students are using for different reasons:
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It allows you to find out any misconceptions.
It can be liberating to find out that there are several ways of
approaching a problem and thus removing the stress of having to
remember a single ‘correct rule’.
Students can realise that they can construe mathematics
themselves, and that their way of thinking (or at least some of their
thinking) is valid.
Students can become more in charge of their own thinking and
learning.
The learning purposes of the activities can become clearer.
Another question that students often ask is: ‘Why do I have to learn this
mathematics, apart from being able to sit tests? When will I ever use it outside
the classroom?’ Such questions should perhaps not be dismissed out of hand,
or considered rude or impolite. They should be discussed and answered.
Imagine being asked to cook a meal but never experienced feeling hungry,
the joy and pleasure of eating, or the sense of creativity that can be
experienced from the process of cooking – for example, from experimenting
with tastes and consistencies. The task of cooking is a chore, because it is not
clear why you have to do it. Just being told why, without ever having
experienced the need for it yourself, might not be sufficient – therefore, the
student needs to experience the intrinsic pleasure of mathematics.
Preparation for Activity 2
The next activity aims to address this dilemma. It offers a mathematical
problem that requires the application of what they have learned previously,
but from a slightly different perspective so the students will have to think.
Activity 2: Why bother learning about HCF and LCM?
1. If I eat one third of a cake and you eat half of the same cake, how much of
the cake have we eaten and what part is left over?
2. How can you work out your answer? What is your method? Are there other
ways to work it out? Is drawing a picture helpful?
3. Now think about your method. Can you explain each step of your method
in terms of:
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a. why what you are doing is mathematically correct
b. why you are doing it that way.
4. In what way do you think this task is related to finding factors as you did in
Activity 1?
5. Look back at parts 3, 4 and 5 of your work on Activity 1. Do you want to
add anything to your answers for those parts?
6. Design three more tasks, preferably with different contexts, where you
would use HCF or LCM.
Case Study 2: Teacher Tanseem reflects on using Activity 2
The students worked in pairs and started the question confidently, thinking
they would have no problems with it. I first showed them parts 1 and 2 of the
activity. They worked out the answer and described their methods in terms of
an algorithm. We shared these ideas with the whole class. Then they came to
part 3 of the activity, which stopped them in their tracks. They kept repeating
that they had given the rule and they had learned – that this was the rule and
that I had told them so! I tell you, this did cause some soul-searching on my
part! But I insisted and kept asking how they knew they were allowed to do
each step and why they were doing each step. I asked them to imagine their
little sister keeping asking that question, ‘Why?’, and that she would not be
happy with ‘Because I tell you’ as an answer.
They struggled, until someone shouted out (actually not allowed in my
classroom) that this was the inverse of what we were doing earlier – that we
were looking for common multiples because otherwise we could not compare
like with like, or add the parts together. So we had a discussion about whether
we could or should have any common multiples, and the advantages of going
with a low common multiple. After which, I revealed part 4 of the activity –
perfect scaffolding for that part of the activity!
We shared the ideas for tasks, and discussed whether or not they would give
a purpose to learning about and using LCM and HCF. There was no absolute
agreement on this, but I felt happy with it because the students had started
thinking about the purpose of using LCM and HCF, and had made an attempt
at coming up with their own ideas for applications.
Pause for thought
When you try this activity in your class, ask yourself the following questions in
order to recognise how you might improve the teaching another time or
transfer what you learned to other contexts:
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How did the lesson go with your class?
What questions did you use to probe your students’ understanding?
Did you feel you had to intervene at any point?
What points did you feel you had to reinforce?
Did you modify the task in any way like Teacher Tanseem did? If so,
what was your reasoning for doing so?
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4 Practising techniques and noticing
differences between LCM and HCF
Preparation for Activity 3
Thus far students have worked on becoming aware of the processes involved
with working out common factors and common multiples, and with the
purpose of the tasks (learning) and the mathematics. The next activity is,
again, very similar to an activity that can be found in textbooks. What is
perhaps different is that it gives students a mixture of problems in terms of
having to find common multiples and factors in one activity, and mixing up
numbers and expressions. The other difference is the request to make notes
on the methods they have used. The aims of these modifications are to make
them aware of connections between topics, noticing the differences and
sameness, and for the mathematical thinking processes involved to become
explicit. Again, to help them overcome the feeling that they are out of their
comfort zone, it might help to let them work in pairs or small groups on this.
Activity 3: Practising techniques and noticing differences
between LCM and HCF
Try to find common factors and multiples of the following:
1.
2.
3.
4.
5.
48 and 72
x2 and 3xy
√18 and √32
(a – b)2 and (a – b)3
(a2 – b2) and (a3 – b3)
Add to your notes in your exercise book about what methods you use to work
these out.
Case Study 3: Teacher Faraz reflects on using Activity 3
The students did the first question with great confidence. The second one
provoked some discussion but the third one was left by most. For this third
question I gave a hint of getting factors within the root sign, and then some of
them got the answer almost at once. The fourth question resulted in a bit of
discussion, but they came up with an answer. However, for the last question,
some pairs came up with a2 – b2 as the common factor and a3 – b3 as the
multiple.
I thought these answers needed to be explored. I knew that just telling them
that they were wrong would not help them to understand how to give a correct
response in the future, so I asked the ones who gave these answers to come
up to the board and explain their reasoning. This resulted in a good
discussion: as I has expected, there were a few who had thought of the
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factors. Throughout this discussion, the students willingly multiplied out many
expressions, probably far more than they would have done if they were doing
a standard exercise. Ultimately there was a consensus and –judging by the
smiles in the room – everyone had enjoyed tackling some complex
mathematics
Pause for thought
You should be familiar now with reflecting on your implementation of unit
activities in the classroom. You may have developed a reflection log or diary
in which you make notes more regularly to keep a record of your professional
development and to generate your own list of ‘tips’ for better teaching
practice. Regarding this activity in your class:
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How did the lesson go?
What questions did you use to probe your students’ understanding?
What points did you feel you had to reinforce?
What points did you feel you had to reinforce?
Did you modify the task in any way like Teacher Faraz did? If so,
what was your reasoning for doing so?
5 Learning from the work of fictitious
students
By now, the students have had to think quite a lot about the mathematical
processes involved in finding common factors and multiples, in order to
analyse and describe their own methods.
Preparation for Activity 4
The next activity will show another way to encourage students to think
critically about what they are doing and what they are leaning. It asks them to
critically evaluate the working methods of some fictitious students and come
up with reasons why the students’ way of working is mathematically valid or
not. Using work from fictitious students often works well in exposing possible
misconceptions, because it avoids emotional reactions and feelings of
embarrassment. Since it is not work from the students themselves or any of
their classmates, they can treat it without concern for anyone who may be
embarrassed or upset about what they are saying. The task is further
enriched because one of the methods offered is not incorrect as such, but
could be (that is, it has limitations).
Activity 4: Using fictitious students’ work
Study the methods used by Meena, Deepak , Aditya and Aneesh to solve the
following:
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a. Meena did it thus:
b. Deepak thought this was what should be done:
c. Aditya was sure this was the method:
d. Aneesh solved it in this way:
Discuss each of these methods.
1. Are they mathematically correct, incorrect or partly correct?
2. Why?
3. How do you know?
Case Study 4: Teacher Faraz reflects on using Activity 4
The students were grouped in pairs so that there could be input and ideas
from both students. I decided not to go for bigger groups this time because I
thought the work needed quite detailed reading and discussion. Pair
groupings offer a closer working environment where the work could be more
thoroughly examined by each student.
I could hear a lot of arguing, with students questioning each other and asking
for reasons when they disagreed. After they had had sufficient time to get to
grips with each item, we discussed the methods of the fictitious students one
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by one at the board. For each one I asked a student to come to the board to
be the teacher and guide the discussions. The first attempt was slightly
chaotic, with some students shouting out, but after I told them there was to be
no shouting out at all and hands-up only, the discussion really improved and
seemed valuable. I stood at the side, sometimes offering another thought, but
with actually very little intervention on my part.
Pause for thought
Teacher Faraz obviously had some very excited students in the class who
reacted well to the way this lesson was taught. You may want to introduce
other dimensions into your reflection that comment on the level of
engagement by students or any challenges you had in managing behaviours
or noise levels.
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How did it go with your class?
What questions did you use to probe your students’ understanding?
Did you feel you had to intervene at any point?
What points did you feel you had to reinforce?
What would you do differently another time?
6 Adapting questions from textbooks
Preparation for Activity 5
The last activity in this unit requires students to tackle some harder questions
and to identify the different places in their mathematics course where they
have to use HCF and LCM, and hence get equivalent numbers before they
can add or subtract. They will also be asked to notice what is the same and
what is different in these questions, so they can become aware of which
methods and processes to use in which situation.
Activity 5: Adapting questions from textbooks
Look at the following problems:
a.
b.
c.
d.
1. Are there any similarities in the above questions? What ‘mathematical
differences’ can you see?
2. How would you solve these problems?
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3. Are there any one correct or incorrect ways of doing these?
4. Look at exercises in the NCERT textbook in different topics and see if you
can identify the exercises where you have encountered such questions.
5. What is same and what is different in these questions?
Pause for thought
How did it go with your class? What questions did you use to probe your
students’ understanding? Did you feel you had to intervene at any point?
What points did you feel you had to reinforce? Did you modify the task in any
way? If so, what was your reasoning for doing so?
Continuous and comprehensive evaluation (CCE)
Description
CCE logo.
When it comes to improving learning, the most important part of CCE is the
formative aspect. Finding out what students can and cannot do or understand,
and then changing the next learning activity in order to address their gaps in
knowledge, is formative assessment. It is formative assessment that has been
shown to really make a difference to students’ learning.
Formative assessment is threaded through the case studies in this unit
particularly strongly, because the tasks that the students are asked to do
reveal misconceptions and gaps in knowledge to the students themselves
while at the same time giving the students ways to learn what they need. For
example, in Activity 3, each student was encouraged to talk about their
responses and try out ideas until they understood what the correct response
was and why. In Activity 4, they were encouraged to explore work that
contained misconceptions and errors in working; this allowed them to think
about their own ideas and talk through with another student where they saw
any difficulties. If they could not work out the misconception with their partner,
they had the opportunity to listen to others later.
In each case the students were encouraged to evaluate their own thought
processes and check them against what others were offering – and therefore
know what they needed to learn. They were then immediately offered the
opportunity to extend their learning by listening to and joining in a thoughtful,
reflective discussion.
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7 Summary
This unit explored ideas on HCF and LCM in order to illustrate how one idea
is threaded through many parts of the mathematical curriculum. The
requirements for teaching from the NCF (2005) and NCFTE (2009) are often
seen as ambitious goals, but the ideas in this unit show that using textbook
questions creatively will allow students to perceive relationships and to see
structures. If the students use reasoning, argue the truth or falsity of
statements, and look for similarities and differences, they will begin to
understand and see the web of ideas in mathematics. Understanding that
fluency with one idea met in the early years of schooling is necessary in order
to work with much more complex ideas later helps students know why they
are being asked to engage with ideas that seem to have no immediate
application.
Identify three techniques or strategies you have learned in this unit that you
might use in your classroom and two ideas that you want to explore further.
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8 Resources
Resource 1: NCF/NCFTE teaching
requirements
This unit links to the following teaching requirements of the NCF (2005) and
NCFTE (2009), and will help you to meet those requirements:
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View students as active participants in their own learning and not as
mere recipients of knowledge; how to encourage their capacity to
construct knowledge; how to shift learning away from rote methods
Let students see mathematics as something to talk about, to
communicate through, to discuss among themselves, to work
together on
Let students use abstractions to perceive relationships, to see
structures, to reason out things, to argue the truth or falsity of
statements
Engage with the curriculum, syllabuses and textbooks critically by
examining them rather than taking them as ‘given’ and accepted
without question.
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References
National Council of Educational Research and Training (2005) National
Curriculum Framework (NCF). New Delhi: NCERT.
National Council of Educational Research and Training (2009) National
Curriculum Framework for Teacher Education (NCFTE). New Delhi: NCERT.
National Council of Educational Research and Training (2012a) Mathematics
Textbook for Class IX. New Delhi: NCERT.
National Council of Educational Research and Training (2012b) Mathematics
Textbook for Class X. New Delhi: NCERT.
Acknowledgements
This teacher development unit was first scoped in a workshop of teachers and
teacher educators in India. The process for the development of this unit
involved coaching on writing OERs (open educational resources) and the
initial authoring work was by Els De Geest, Anjali Gupte, Clare Lee and Atul
Nischal.
Except for third party materials and otherwise stated, the content of this unit is
made available under a Creative Commons Attribution licence:
http://creativecommons.org/licenses/by/3.0/.
The material acknowledged below is Proprietary, used under licence and not
subject to any Creative Commons licensing.
Grateful acknowledgement is made to the following:
CCE logo: www.cbse.nic.in
Every effort has been made to contact copyright owners. If any have been
inadvertently overlooked, the publishers will be pleased to make the
necessary arrangements at the first opportunity.
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