Elementary Maths
Unit 5:
Understanding fractions by asking
questions that challenge thinking
Introduction ...................................................................................................... 1
Learning outcomes .......................................................................................... 1
1 What’s so difficult about fractions?................................................................ 2
Talking fractions: using the language .................................................................... 2
2 Developing an understanding of fractions ..................................................... 3
3 Asking questions ........................................................................................... 5
4 Effective questioners give children time to think ........................................... 8
5 Summary .................................................................................................... 11
6 Resources ................................................................................................... 12
Resource 1: Learning about fractions in the classroom ....................................... 12
Resource 2: NCF/NCFTE teaching requirements ................................................ 12
References .................................................................................................... 13
Acknowledgements ........................................................................................ 14
Transcript ....................................................................................................... 15
Except for third party materials and otherwise stated, this content is made
available under a Creative Commons Attribution-ShareAlike licence:
http://creativecommons.org/licenses/by-sa/3.0/.
Introduction
In this unit you will think about how to introduce fractions to your students.
Students can see fractions as a very difficult topic to understand. There are
many reasons for this, but making sure that your students have rich and
varied experiences of working with fractions will help them to develop their
understanding.
In this unit you will explore the fact that a fraction only has meaning when
looked at in relation to a whole, and consider how to help your students to get
to know about different ways to read the symbolic representations of fractions.
Through activities you will also think about the value of asking your students
interesting and challenging questions, and of getting your students to ask
questions themselves.
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.
Following an activity, you will be able to read a commentary by another
teacher who did the same activity with a group of students. This will allow you
to evaluate its effectiveness and make a comparison with your own
experience.
Learning outcomes
This unit addresses learning outcomes in different areas relating to the
teaching and learning of mathematics, as follows.
Mathematical learning outcomes
In this unit you will develop your teaching of the following mathematical
concepts:
 that a fraction only has meaning in relation to a whole
 that finding a fraction can mean sharing out or dividing into parts
Pedagogical learning outcomes
After studying this unit, you should be able to:
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ask effective questions that are interesting and challenging
use embodiment as a tool for teaching.
NCF/NCFTE teaching requirements
This unit links to the teaching requirements of the NCF (2005) and NCFTE
(2009) outlined in Resource 2.
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1 What’s so difficult about fractions?
Pause for thought
Think for a moment about fractions. They occur a great deal in normal life; for
example, people regularly share out food between them, each of them taking
a fraction of the whole amount.
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Do you think it is likely that your students will have some intuitive ideas
about fractions? What are these likely to be? Think about how you may be
able to build on these understandings in teaching about fractions.
Could the students have built up misunderstandings about fractions as
well? Can you think of examples of such misunderstandings?
One of the reasons fractions can seem so difficult is that there is a lot to
understand. For example, half of something can be smaller than a quarter of
something else. An example of this is ‘half of six is three’ and ‘a quarter of
twelve is six’. So learning about fractions by folding pieces of paper or by
dividing circles may mislead students, especially if the paper is always the
same size. Students must be taught to ask ‘A fraction of what?’
Developing an understanding of fractions is not so different from learning to
understand other mathematical concepts. For example, very young children
are offered many different experiences as they learn to generalise the concept
of ‘three’.
Despite being older when they learn about fractions, elementary students
similarly will need a great many rich and varied experiences if they are to
begin to develop a good understanding of fractions.
Many students will have had experiences that help them to develop some
understanding of fractions. In her research, Nunes (2006) found that primary
school students already have insights into fractions when solving division
problems:
They understand the relative nature of fractions: if one child gets half
of a big cake and the other gets half of a small one, they do not
receive the same amount. They also realise, for example, that you
can share something by cutting it in different ways: this makes it
‘different fractions but not different amounts’. Finally, they
understand the inverse relation between the denominator and the
quantity: the more people there are sharing something, the less each
one will get.
Talking fractions: using the language
Encouraging the students to talk about fractions and use the vocabulary will
help them understand some of the difficult vocabulary associated with
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fractions. The questions you use should show the students how useful the
correct vocabulary is, so that everyone knows what is being referred to.
Once you have modelled some ways of talking about fractions and drawn
attention to how words are used, the more the students use the words
themselves, the more they will build their understanding of fractions. Asking
the students to make up questions to ask one another is a good way to get
them talking. Another way is to ask the students to explain the reasoning they
used to arrive at their answers.
Pause for thought
Think about what the students need to know about fractions, and make some
notes. For example:
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how to find out a fraction of a quantity
what fraction one quantity is of another
how to add together fractions.
For each of the ideas associated with fractions, think about the vocabulary
associated with those ideas and the way that that vocabulary is used to
express ideas, and make some notes. For example ‘half of ten’ means ‘divide
10 by 2’, but it can also mean ‘multiply 10 by 1/2’. The students might also see
10
/2, which has the same outcome and is thus equivalent in meaning but which
may be expressed as ‘10 divided by 2’ or ‘10 shared between 2 people’.
Think about some specific students in your class. What activities might help
them to understand the different ways that fractions can be expressed and the
different meanings given to those interpretations?
2 Developing an understanding of
fractions
The first activity focuses on students embodying the concepts of fractions –
that is, you will ask them to use their bodies to represent mathematical ideas.
If the students move themselves to make fractions of a whole, they will begin
to develop their concept of what a fraction is and how they can work with
fractions.
Before attempting to use the activities in this unit with your students, it would
be a good idea to complete all, or at least part, of the activities yourself. It
would be even better if you could try them out with a colleague as that will
help you when you reflect on the experience. Trying them for yourself will
mean you get insights into a learner’s experiences, which can, in turn,
influence your teaching and your experiences as a teacher.
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When you are ready, use the activities with your students. Once again, reflect
and make notes on how well the activity went and what learning happened.
This will help you to develop a more learner-focused teaching environment.
Activity 1: Embodying fractions
First create a space, and ask eight children to come to the front of the class or
somewhere where the rest of the class can see them.
1. Ask them to arrange themselves into a rectangle.
2. Ask someone else to divide the group into half.
3. Reform the rectangle, then ask another student to divide the group in
half in a different way.
4. Ask the students what is the same and what is different about the new
half of the group.
5. Now ask another student to divide the eight children into quarters
(fourths). Again ask whether there is a different way to do this division,
and what is the same and what is different about the new way of
dividing into quarters.
6. Now change the number of children and go through the process above
again. It may be that dividing into quarters is difficult but depending on
the chosen number, continue asking for 1/2, 1/4, 1/3 and so on, until a
fraction that cannot be done is reached. Ask the children why you
cannot find that fraction of these children. Dividing one child up into bits
is not allowed!
7. Ask the children to work in groups of 12. You could appoint a leader in
each group to note down ideas if the class does not split evenly into
groups of 12. Ask them to work out all the fractions they can divide 12
children into.
Case Study 1: Teacher Ravi reflects on using Activity 1
Teacher Ravi decided to use the ideas of embodiment to explore the ideas on
fractions that he had just started to introduce to his class. Here’s how he went
about it.
First, I invited eight students to come to the front of the class and to form
themselves into a rectangular shape where the rest of the class could see
them. I then asked student Anoushka to come and divide these eight students
in half, which was easy to do.
I then asked the class if the group of eight students could be divided in half in
another way. This proved to be a little challenging, as the students were used
to mathematics questions having just one answer, so they wondered at first if
Anoushka was wrong. They needed clarification about what ‘different’ meant
here. Of course, whichever way they divided the children in half, there were
always four children in each half. Since this was the answer I was looking for,
I gave them time to talk about these ideas.
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Next, I asked student Nita to come to the front and divide the group into
quarters. This time the children were able to suggest different ways to achieve
this, and they were happy there would always be two children in each part.
I then asked another group of students to come to the front, this time with six
students. This time I asked them to divide themselves into half in two ways. I
asked ‘Do you always get the same answer?’ ‘Yes sir!’ they said. Then I
asked ‘What other fraction can you divide yourselves into?’ They tried to
divide themselves into quarters but they could not, but what they did find was
that they could divide themselves into three parts and discussed what this
fraction was called.
I then put the class into groups of 12 and asked them what fractions they
could embody in their groups. One group came up with twelfths, but most
worked happily on halves, quarters, thirds and sixths.
Pause for thought
When you do such an exercise with your class, reflect after the lesson on
what went well and what went less well. Consider the questions that led to the
students being interested and being able to make progress, and those where
you needed to clarify things.
Such reflection helps with finding a ‘script’ that helps you to engage the
students and makes mathematics interesting and enjoyable for them. If they
do not understand or cannot do something, they are less likely to become
involved. Use this reflective exercise every time you undertake the activities,
noting, as Teacher Ravi did, some quite small things that made a difference.
Good questions to trigger this reflection are:
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How well 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?
3 Asking questions
Teachers ask a lot of questions in their work – some research suggests that
teachers ask up to 400 questions every day when they are teaching! It stands
to reason that the better the questions that teachers ask, the better their
teaching will be.
Much research has been undertaken about good questions, for example by
Wragg and Brown (2001) and Hattie (2008).The research concludes that
effective questions:
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are anchored in the learning of the lesson
build on the students’ previous knowledge
involve, interest and motivate the students
are sequenced to encourage higher-order thinking (but not too soon!)
enable the children to build their own knowledge
reveal misconceptions and misdirections
prompt and challenge thinking and reasoning
Pause for thought
Reflect on the questions you asked in the last lesson you taught.
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Did they challenge the students to think?
Could a small change have uncovered more about the students’ current
learning?
Could your questions have encouraged the children to build their own
learning more?
Activity 2: Asking effective questions about fractions
Part 1: Preparing to ask effective questions
If you can do this part of the activity with another teacher, you may find that it
is easier to do.
Think about the next lesson where you will teach on fractions. What is it you
want the students to know? Write some notes about that now.
What previous knowledge do you think they will need in order to understand
the ideas you want them to learn? Write a question which will enable you to
know whether or not they have that prior knowledge.
Think about some of the ways that fractions are used in the real world. Write a
question that might interest or engage the students because it is based on
something they know about and use.
Now write an easy question for the particular topic you have to teach and then
write a hard question. Write a sequence of questions that will challenge your
students – but not too much!
Think about all the ways that misconceptions can happen in fractions. Write
two or three questions that will help you check whether or not your students
have these misconceptions.
Now write a question that will encourage your students to reason their way to
a solution.
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Part 2: Using your effective questions in the classroom
Now you have written these questions, use them with a class.
Did you think that the class learned more because they used these
questions?
Don’t forget to use real objects to allow your students to work with ideas on
fractions and to approach challenging questions through a process of
reasoning.
Case Study 2: Teacher Shah questions the students to
check their understanding of fractions
Teacher Shah knows that students have a lot to understand with fractions and
that the language that is used sometimes leads to misunderstandings. So he
prepared some questions in advance.
When thinking about Part 1 of Activity 1, I decided I would use my normal
introduction to fractions by demonstrating fractions on the board as usual, but
to be very precise and repetitive in the questions and instructions I was going
to use. I wrote them down on a piece of paper, and put them on my desk so I
would not forget them.
These are the questions and prompts I prepared:
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Show me how you divide this circle into halves/quarters/eights.
o How do you know this is correct?
o Please describe your method clearly.
o Would anyone do it another way?
Show me on this circle one half/one quarter/one eighth.
o How do you know this is correct?
o Please describe your method clearly.
o Would anyone do it another way?
Show me on this circle one third/one sixth/one twelfth.
o How do you know this is correct?
o Please describe your method clearly.
o Would anyone do it another way?
Show me on this circle one third/one fifth/one seventh.
o How do you know this is correct?
o Please describe your method clearly.
o Would anyone do it another way?
Show me on this circle three quarters/six eights.
o How do you know this is correct?
o Please describe your method clearly.
o Would anyone do it another way?
I drew the circle using chalk. I then invited students to come to the board, and
asked them the questions. Having the questions written down really helped
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me to focus and helped to avoid diversions from what I had intended to do. I
also noticed that as a result there was less ‘teacher talk’ and more student
talk and student work.
Now watch the video in Resource 1 about fractions. If you are unable to watch
the video, it shows a teacher questioning and prompting the students to make
sure that they understand fractions. You may also find it useful to read the
video’s transcript.
4 Effective questioners give children
time to think
Mary Budd Rowe (1986) researched the ‘wait time’ that teachers allowed after
asking a question. ‘Wait time’ is the length of quiet time that teachers allow
after asking a question before they expect a student to answer, or before they
rephrase the question or even answer the question themselves. Her team
analysed 300 tape recordings of teachers asking questions over six years.
They found the mean wait time was 0.9 seconds.
If you ask a question that requires the students to think, is this really enough
time for them to think? Or is it only enough time to instantly react?
The teachers in Budd Rowe’s research were trained so that they became able
to increase their wait time to between three and five seconds. The increased
wait time resulted in:
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an increased length of student response
an increased number of unsolicited, but appropriate, replies
a decreased failure to respond
an increased confidence of response
an increase in the incidence of students comparing their answers to
those from another student
the number of alternative explanations offered multiplying.
In other words, the students had time to think and that increased the level of
discussion that went on in the classroom, which in turn meant the teachers
learnt more about their students thinking and were able to act on any
misconceptions. Increasing wait time is not easy to do and can feel odd when
you start, but if your students are to think, they must be given sufficient time.
The next activity links together many of the ideas that have been discussed so
far. It suggests that you:
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ask the students to work with concrete objects to answer some
challenging questions
work together so that they can support one another
give them more time to think.
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Activity 3: Learning about fractions
This activity is an example of the kinds of rich activity that students need to
build their understanding of fractions. For this task you will need a quantity of
paper plates, or card cut into rectangles of the same size.
Arrange the students to work in groups of three or four, and give them a pile
of paper plates or cards.
1. Ask them first to show you half of a plate, then a quarter of a plate. It is
important not to tell them to take one plate here; let them think this out
for themselves.
2. Then ask each group to show you ‘half of six’ using the plates.
Make sure everyone is able to do this before carrying on.
3. Ask the students to suggest several fraction problems that they can
solve using the plates. Each time ask the class to show you the
solution using the plates. If they don’t suggest questions using just one
plate then prompt them.
The idea is to give everyone time to play a little with fractions and to think
about what fractions are.
4. Ask the students what is the same and what is different about the two
types of questions they have worked on so far. This will help the
students recognise some of the different ways that fractions are used in
mathematics.
Now move on to problems that mix the two ideas.
5. Ask the class to find 1/4 of 12, then ask them to find 1/4 of 13.
6. When the class have had time to solve this, ask one group to explain
the process they went through in order to solve the second problem.
7. Now ask the students to suggest how they could record the problem
and the answer using mathematical notation. Spend time on this, as it
is important that the students recognise the relationship between the
way the problems are written and what they are doing with the plates.
8. Now ask the class to solve other ‘hard fraction problems’ using the
plates, for example 1/5 of 12, or 1/4 of 10, which require two plates to be
shared out.
9. Again, ask for suggestions and discuss how these ideas might be
written down.
10. Ask each group to make up an easy problem and a hard problem with
fractions to give to another group to do. Ask each group to record their
answers.
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Case Study 3: Teacher Akash reflects on using Activity 3
Teacher Akash decided to instruct his class to work in groups to help them to
think about some challenging ideas in fractions. He put the class into groups
that he thought would work well together of between four and six students.
Note that where he used paper plates, squares of card or paper is an
alternative.
I gave each group 12 paper plates. The plates were to help to support the
students’ thinking that finding fractions is about sharing out equally.
First, I set them the task of dividing the plates into quarters. I asked several of
the groups to talk about the process of dividing into quarters. Then I asked
them to divide their 12 plates into thirds. When they had done this, I once
again asked the students to explain how they did it. I made sure that everyone
was comfortable sharing out the objects that they were working with, in this
case the plates. The students enjoyed working and collaborating together in
groups and completing the task.
I then decided that the class were ready for a more challenging question. I
gave each group one more plate, so that they then had 13 plates, and again
asked them to divide the plates into quarters and then thirds. This time the
students discovered that they needed to subdivide the extra plate in order to
share out the plates equally into quarters and thirds.
This time I spent more time on the feedback session in order to make sure
that everyone understood the reason why one of the plates had to be
subdivided. I then asked the class to divide the plates into thirds, and this time
I offered them scissors as well. Several students gave some good reasons
why they needed to divide up the extra plate, but working in groups helped
them all to try out their ideas first before telling the whole class.
Now watch the video in Resource 1 about attempting this activity. If you are
unable to watch the video, it shows a teacher asking their students to
complete this task and also taking the time to ask the class for feedback on
how they divided the plates up. You may also find it useful to read the video’s
transcript.
Pause for thought
After trying out the ideas of Activity 3 in a lesson, reflect on what went well
and what went less well.
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How well 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 like Teacher Akash did? If so, what
was your reasoning for doing so?
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5 Summary
This unit has focused on teaching fractions, but you have also looked at how
to ask questions that require students to think and the importance of giving
students sufficient time to think.
In studying this unit you have thought about how to enable your students to
develop their ideas about fractions and about the necessity of providing rich
and varied activities if students are to learn, understand and use ideas about
fractions.
You have also seen how reflecting on learning, and how learning happens, is
important in becoming better at teaching.
Pause for thought
Identify three techniques or strategies you have learnt in this unit that you
might use in your classroom, and two ideas that you want to explore further.
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6 Resources
Resource 1: Learning about fractions in the
classroom
You should watch the video below in its entirety. If you want to watch only the
material relating to the section of the unit you are working on, refer to the
timings below:
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
Students showing their understanding of fractions: 0:00–2:18
Students using plates to develop their understanding of fractions: 2:19–
6:21
Video content is not available in this format. You will find the transcript at the
end of this unit.
Having watched the video, you should now return to the section of the unit
you were working on.
Resource 2: 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 learners 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 learn important mathematics and see mathematics is
more than formulas and mechanical procedures.
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References
Budd Rowe, M. (1986) ‘Wait time: slowing down may be a way of speeding
up!’, Journal of Teacher Education, vol. 43, pp. 44–50. Abstract available
from: http://jte.sagepub.com/cgi/content/abstract/37/1/43 (accessed 3
February 2014).
Hattie, J. (2008) Visible Learning: A Synthesis of Over 800 Meta-analyses
Relating to Achievement. New York: Routledge.
National Council of Educational Research and Training (2005) National
Curriculum Framework (NCF). New Delhi: NCERT.
National Council for Teacher Education (2009) National Curriculum
Framework for Teacher Education (online). New Delhi: NCTE. Available from:
http://www.ncte-india.org/ publicnotice/ NCFTE_2010.pdf (accessed 3
February 2014).
National Council of Educational Research and Training (2012) Mathematics
Textbook For Class IX. New Delhi: NCERT.
National Council of Educational Research and Training (2012) Mathematics
Textbook For Class X. New Delhi: NCERT.
Nunes, T. (2006) Fractions: Difficult but Crucial in Mathematics Learning,
Teaching and Learning Research Brief, Economic and Social Research
Council, UK. Available from:
http://www.tlrp.org/pub/documents/no13_nunes.pdf (accessed 3 February
2014).
Wragg, E. and Brown, G. (2001) Questioning in the Secondary School.
London: Routledge Falmer.
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Acknowledgements
The content of this teacher development unit was developed collaboratively
and incrementally by the following educators and academics from India and
The Open University (UK) who discussed various drafts, including the
feedback from Indian and UK critical readers: Martin Crisp, Els De Geest,
Anjali Gupte, Clare Lee and Atul Nischal.
Except for third party materials and otherwise stated, this content is made
available under a Creative Commons Attribution-ShareAlike licence:
http://creativecommons.org/licenses/by-sa/3.0/.
Video: thanks are extended to the heads and students in our partner schools
across India who worked with The Open University in this production.
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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Transcript
Narrator
In this video the teacher uses group work to
encourage students to actively explore how
collections of objects such as nuts or dishes can be
divided in to equal parts or fractions.
Teacher
I am sure everybody has heard the story of a
monkey and two cats.
Students
Yes ma’am.
Teacher
Yes, who knows the story? Okay Anushka, come.
Student Anushka There were two cats fighting for a bread and then a
monkey passed by, the cats told him to act as a
judge that both the cats get equal amount of food.
So, the monkey divided, two parts, he ate one. Okay,
this is larger, because one part was.
Teacher
The monkey divided the chapati into.
Students
Two parts.
Teacher
Two parts.
Student Anushka And then he saw that this part is not smaller then he
ate another part and this is how he went on eating
and eating and he ate the chapati.
Teacher
Now monkey said, oh now this part is bigger, this is
smaller, let me eat little from this also. He ate little
from this, now which part is bigger.
Students
One.
Teacher
Then he ate one part from this and then which part is
bigger, then he ate one part from this. So, he ate
whole chapati. What was wrong here?
Students
Monkey didn’t divide it as equal.
Teacher
He didn’t divide it as equal. The division was not
equal. So, today we are talking about fractions.
Students
Fractions.
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Teacher
We are talking about fractions. Fractions are equal
parts of a whole. What do you mean by half? Two
parts of a whole. What is one-fourth?
Students
Four parts of a whole.
Teacher
Four equal parts. Four.
Students
Equal parts.
Teacher
Fractions are equal parts. Okay, now
Narrator
The 35 students in the class are being divided into
five groups, so how many should there be in each
group?
Teacher
Five groups. What is one-fifth of thirty five? One fifth
of thirty five is
Students
Seven.
Teacher
This group has walnuts cover, okay. This group has
chestnuts. Now, this is your paper plate. Your group
has a chapati and this group again has paper plate.
Now the task is one-fourth of your collection.
Students
[murmur]
Narrator
While each group is working on their task, the
teacher moves around the groups to monitor their
progress and makes suggestions or ask questions if
necessary.
Students
[murmur] We have 56 chestnuts and one-fourth of
this is 14. So ma'am one-fourth of 56 is 14. We have
21 plates, we have divided 21 plates into…..
Teacher
Now look, they have 21 plates, it’s very difficult to do
one-fourth of 21, see how smartly they have done.
Show all the pieces. What is one-fourth of twenty?
What is one-fourth of 20?
Students
Five.
Teacher
So what they have done, they have taken five plates
Students
[murmur]
Teacher
One-fourth of the….. So clap for them.
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Narrator
After this group has solved their first task, the
teacher sets them another more challenging one.
Teacher
Can you do one-tenth of your collection?
Students
These are those 12 pairs of two plates then the last
remaining plate we will divided into 10 parts. Half like
this, one-fourth, yes. One, two, three, four…. One,
two, three four, eight parts, then eight parts, then one
two parts.
Teacher
Or you can divide one into half, total there will be 10
parts.
Students
Wait, wait, wait. Done, done, done. Yes ma'am.
Teacher
What was your first step?
Student
With the five plates we divided all the five plates into
half and this make 10 two and a half each. So, 25
one-tenth is two and a half.
Teacher
One-tenth of twenty five is two and a half or 2.5.
Narrator
Telling a story to the whole class is an excellent way
to energise the start of a lesson.
Students enjoy using everyday objects in the
classroom, it helps them to understand mathematical
concepts.
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