Secondary Mathematics
Unit 13:
Word problems in mathematics:
identifying key terms and making
connections
Teacher Education
through School-based
Support in India
www.TESS-India.edu.in
http://creativecommons.org/licenses/
The TESS-India project (Teacher Education through School-based Support) aims to improve
the classroom practices of elementary and secondary teachers in India through studentcentred and activity-based approaches. This has been realised through 105 teacher
development units (TDUs) available online and downloaded in printed form.
Teachers are encouraged to read the whole TDU and try out the activities in their classroom
in order to maximise their learning and enhance their practice. The TDUs are written in a
supportive manner, with a narrative that helps to establish the context and principles that
underpin the activities. The activities are written for the teacher rather than the student,
acting as a companion to textbooks.
TESS-India TDUs were co-written by Indian authors and UK subject leads to address Indian
curriculum and pedagogic targets and contexts. Originally written in English, the TDUs have
then been localised to ensure that they have relevance and resonance in each participating
Indian state’s context.
TESS-India is led by The Open University and funded by UKAID from the Department for
International Development.
Version 1.0
Except for third party materials and otherwise stated, this content is made available under a
Creative Commons Attribution-ShareAlike licence: http://creativecommons.org/licenses/bysa/3.0/
Contents
Introduction
1
Learning outcomes
2
1
Reading word problems
3
2
From words to algebra and vice versa
7
3
Modelling a context mathematically
4
Making sense of the mathematics in the context of word
problems
12
5
Summary
15
6
Resources
16
Resource 1: NCF/NCFTE teaching requirements
16
10
References
17
Acknowledgements
18
Introduction
Introduction
Students can find understanding how to work with word problems hard to
do. They can become distracted by the narrative, which can seem to be
based in real life, but at the same time they seem a bit unbelievable. For
example, if 82 people are waiting for a lift that can only take nine people at
a time in a real office block, then at least one person, and probably several
more, will choose to walk up rather than wait.
However, in a mathematics word problem, such aspects of real life are
usually not considered relevant. Hence sometimes students need to be helped
to consider what is part of the context that is relevant to the mathematics
and what is not – in other words, to pay attention to what is important and
to disregard what is there as a distraction. Perhaps students need a set of
tools to make sense of word problems, to notice where the mathematics is
and to be aware that word problems are a case of modelling mathematical
ideas.
Word problems are usually an example of mathematical modelling. It may be
important to remind the students that this is how much mathematics is used
in careers beyond school: to model what happens (or may happen) in the
world so that complex situations and often awkward numbers can be
manipulated more simply and solutions to problems found. Students can
become aware of this by ‘decoding’ word problems, and also by making up
word problems themselves.
In this unit, you are invited to first undertake the activities for yourself, and
reflect on the experience; then try them out in your classroom, and reflect on
that experience. Trying 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. This will help you to develop a more learnerfocused 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.
Many of the word problems used in this unit have been adapted from word
problems in the NCERT textbooks for Class IX and X.
1
TDU 13
Word problems in mathematics: identifying key terms and making connections
Learning outcomes
After studying this unit, you should be able to:
.
help your students read, write and solve word problems
.
teach your students how to make sense of the mathematics in the context
of the word problem
.
show your students how to use word problems as mathematical
modelling tools.
The learning in this unit links to the NCFTE (2005, 2009) teaching
requirements specified in Resource 1.
2
1 Reading word problems
1 Reading word problems
‘Reading’ word problems involves ‘decoding’ the information in order to
understand what the mathematical model is and therefore what ideas are
needed to provide a solution. The very first step in this process involves the
reader scanning the problem to identify key words. and decide which
information seems relevant or irrelevant.
Although word problems may not contain irrelevant information, training the
mind to spot the relevance (or degree of relevance) of information in a word
problem is important as an aspect that relates directly to the mathematics
used beyond school. Helping students learn to do this becomes easier and
more efficient if some irrelevant (but related) information is contained in the
problems that they tackle.
Pause for thought
Thinking about your classroom, how are word problems perceived by
your students? Do they like them? Do they struggle with them? Why do
you think this is?
Thinking back about your experiences as a mathematics learner, how
did you perceive word problems? What helped you to understand how
to approach them?
The next task focuses on helping you train your mind to spot relevant
information and then to work at chunking the word problems into smaller
pieces of information.
Activity 1: Key words and chunking problems
Read each word problem and make a list of the key words or phrases
that can help you solve the problem. Then set out the problems in
simpler chunks of information.
3
TDU 13
Word problems in mathematics: identifying key terms and making connections
1 A square garden has a walking track that is eight feet wide along
its sides. If one side of the garden is ten metres long, find the
distance travelled by Hamid if he walks around the garden twice.
2 Kavita was given some money on her 16th birthday by her uncle.
She used the money to buy two pairs of jeans for Rs. 950 each
from a store that was offering a 20 per cent discount. After the
purchase, she still had Rs. 150 left. How much money did her
uncle give her?
3 Rita buys 3 kg of mangos and 12 bananas for Rs. 280 on
Tuesday. Three days later, from the same shop, Rahul buys 2 kg
of mangos and 18 bananas for Rs. 300. Write the equations that
can help find the cost of 1 kg of mangos and a dozen bananas.
Based on what you did, answer these questions:
1 For each key word or phrase you think is useful, provide your
reason why. Also, provide a reason for not using any information
that you disregarded.
2 What helped you decide which numerical data was useful and
which was not? Did you miss any useful information?
3 Was there some information that you found difficult to classify as
‘useful’ or ‘not useful’? How did you go about accepting or
rejecting this information? Why was this information difficult to
classify? What additional information about these could have
helped you decide better?
4 What did you find easy or difficult about chunking a word problem
into smaller pieces of information? Why was chunking some
information easy while the other difficult?
5 Did you visualise any of the above problems before identifying key
words or chunking? How was visualising helpful?
Case Study 1: Hamsa reflects on using Activity 1
My class have found using word problems very difficult. They like to
answer questions in mathematics quickly and get lots done, and word
problems always slow them down. Since being able to answer word
problems is so important in examinations, I decided to spend some time
helping my students learn to deal with them. First I asked them what
skills they needed when solving word problems. They could not answer
at first, so I asked them to discuss the question in pairs. Their answers
were that you have to:
4
.
read the problem carefully
.
recognise what is the context and what is important for solving the
mathematics
.
note down important words, numbers and information, perhaps on a
diagram
.
ignore irrelevant words and numbers
1 Reading word problems
.
think what mathematics must be used to solve it, then use it
.
check and decide whether the answer makes sense
.
give the answer in a way that relates it to the problem asked – for
example using appropriate units of measurement.
I then asked the class to look at the problems in Activity 1. They wanted
just to answer the problems, but I wanted them to think about the
process and use the problem solving skills we had come up with. So I
asked them to look at the second question together. When solving word
problems, it is important to rephrase the ‘critical’ information in your own
language – mathematically, if possible. For example, let us look at the
following word problem from the previous task.
Kavita was given some money on her 16th birthday by her uncle.
She used the money to buy two pairs of jeans for Rs. 950 each
from a store that was offering a 20 per cent discount. After the
purchase, she still had Rs. 150 left. How much money did her uncle
give her?
First I asked them to chunk the information. We came up with the
following chunks of important information:
.
Kavita was given some money.
.
She buys two pairs of jeans.
.
Each pair of jeans cost Rs. 950.
.
She still had Rs. 150.
.
How much money did her uncle give her?
Then we looked at the mathematics involved at each step. I asked the
class to come up with the meaning of each word/phrase and the
mathematical meaning it is trying to convey. I had to help them with the
first one, but after that they got the idea:
.
‘Kavita was given some money.’ This is the ‘unknown’ – say, x.
.
‘She buys two pairs of jeans.’ Whatever she spent has to be
subtracted from x.
.
‘Each pair of jeans cost Rs. 950.’ So she spent 2 × 950 = Rs. 1900.
.
‘She still had Rs. 150.’ This should be x – 1900.
.
‘How much money did her uncle give her?’ This is x.
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TDU 13
Word problems in mathematics: identifying key terms and making connections
Once we had done this, the students tackled the other two word
problems in the same way, chunking them and trying to write down the
mathematical expressions for what they were doing. They realised fairly
soon that to answer the first question they needed to know where
Hassan was walking on the path, and that the measurements were in
feet and metres, so some conversion would be needed. The third
question also resulted in some heated debate over whether it mattered
that the two visits to the shop were three days apart, so we definitely
tackled the question of irrelevant information and how difficult
constructing such problems could be.
I then sent the students off to have some fun devising word problems of
their own that contained some good mathematics and some irrelevant
information that was not obvious.
Pause for thought
Thinking of your classroom, what do you think are the barriers to
working through word problems with your students?
Thinking back to when you were learning mathematics in the classroom,
what do you think were (or could have been) the barriers to solving
word problems for you? What helped you to succeed in solving such
problems?
6
2 From words to algebra and vice versa
2 From words to algebra and vice
versa
One of the difficult ideas in word problems is about translating the words
into algebra and algebra into words. There are two parts to the next activity.
The first part gives your students the opportunity to practise matching words
and algebraic expressions and vice versa in an enjoyable way; the second
part asks the students to make their own word problems from some algebraic
equations.
Activity 2: Words and algebra
Part A
Make flash cards in two different colours (green and orange have been
used in Figure 1). Leave one side of the flash card plain so that the
students can write on that side. On the green cards, write an arithmetic
statement in English (or the language of instruction followed in your
school). On the orange cards, write the corresponding statement using
mathematical symbols and operations. Make the cards relevant to the
kinds of work that your students are engaged with. This could be
trigonometry, circles or any other aspect of mathematics – see the
examples in Figure 1.
Figure 1 Words and algebra flashcards.
If you have 30 students in your class, you will need 15 pairs of cards.
Randomly distribute the cards to the students. Each student has to find
the student who has the card that completes his or her pair.
Part B
For each equation write as many context-based word problems as you
can. For example, for the equation y = 3x you could write ‘Kavita’s feet
are three times as long as her baby brother’s’.
Here are some examples of equations:
.
y = 3x
.
x + y = 150
.
3x – y = 22
.
2x + 3y = 88
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TDU 13
Word problems in mathematics: identifying key terms and making connections
.
A = 16p
.
32 = x(y + 2)
For which equations was writing word problems more difficult than
others? Why do you think these were difficult?
For each equation, which one of your word problems was most realistic?
Why? Can you try to make the other word problems more realistic?
Case Study 2: Hamsa reflects on using Activity 2
My students had enjoyed using Activity 1 and their confidence was
growing, but they still had difficulties writing down the algebra for ideas
such as the ‘she still had Rs. 150’ in the problem that we did together.
Therefore, I decided to use the cards from Activity 2 just as they were
so that I could see what would happen.
There were 44 in my class, so I made 22 pairs so that everyone had
one. There was a lot of noise while they moved about trying to find their
pair, but it was over quickly. Once they had all paired up, I got them to
sit down and work together on two context-based word problems that
would result in the algebra on their cards, which they wrote on the back
of their cards. Each pair then joined another pair and gave one another
their context-based word problems to write the mathematical phrase in
English and then to write the algebra. When the students disagreed
about anything, we discussed the ideas as a class and then I asked for
some ‘really good’ word problems to share with the class.
Everyone seemed to get a lot out of using this idea. They had people to
ask if they were not sure and they all heard and worked with lots of
examples connecting algebra with words and with contexts.
For the Part B, I asked the students to continue to work in the groups of
four that they had formed earlier and to write at least four problems for
each equation. When each group had come up with something for every
equation, I used the questions in a class discussion. I had written them
on the board earlier and had told the class that I would use them later,
but they were still surprised that I didn’t want them to read out the word
problems they had written. Asking ‘Which was most difficult and why?’
meant that the students had to think about how they thought about their
mathematics – that is called ‘metacognition’, I think. Asking them to take
this overview also meant I became more aware of what they found
difficult and therefore where more practice would be needed; for this
class it was the one with brackets. The question about whether they
were ‘realistic or not?’ was also useful, I felt. They had to think about
what they could model with this level of mathematics and saw why word
problems can be unrealistic at times.
8
2 From words to algebra and vice versa
Pause for thought
When you do such an exercise 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 make connections between topics.
Such reflection always helps with finding ways to help you to engage
the students and help them to find mathematics interesting and
enjoyable. If they do not understand and cannot do something, they are
less likely to become involved. Use this reflective exercise every time
you undertake the activities, noting as Teacher Hamsa did some quite
small things that made a difference.
Good questions to trigger this reflection are:
.
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?
9
TDU 13
Word problems in mathematics: identifying key terms and making connections
3 Modelling a context
mathematically
Identifying the mathematical ideas that will be needed to solve word
problems, or modelling the context mathematically, can be difficult for
students. The word problems they encounter in textbooks usually only need
the mathematics that has just been studied – so thinking about what
mathematics is needed to model the situation is seldom required, except in
examinations. The next exercise focuses on thinking about what mathematics
is needed and how to express the problem in a way that the mathematics can
be used to solve the problem.
Activity 3: Identifying the mathematical model for a
word problem
Here are three word problems:
1 Yamini and Fatima, two Class IX students, together contributed
Rs. 100 towards the Prime Minister’s Relief Fund to help the
earthquake victims. Write a linear equation that matches this text.
2 Mary wants to decorate her Christmas tree. She wants to place
the tree on a wooden box covered with coloured paper with
picture of Santa Claus on it. She must know the exact quantity of
paper to buy for this purpose. If the box is 80 cm long, 40 cm
wide and 20 cm high, how many 40 cm square sheets of paper
would she require?
3 The Shanti Sweets Stall was placing an order for making
cardboard boxes for packing its sweets. Two sizes of boxes were
required: a larger box with dimensions of 25 cm × 20 cm × 5 cm,
and a smaller box with dimensions of 15 cm × 12 cm × 5 cm. For
all the overlaps, 5 per cent of the total surface area is required
extra. If the cost of the cardboard is Rs. 4 for 1000 cm2, find the
cost of cardboard required for supplying 250 boxes of each kind.
For each word problem given:
.
draw an illustration for the problem
.
identify the unknowns in the problems
.
identify what you know
.
find relationship between unknowns and knowns
.
represent the relationship mathematically.
Case Study 3: Daksha reflects on Activity 3
I showed the class the three problems from Activity 4 and asked them
not to solve them, but to work through the five steps in modelling the
10
3 Modelling a context mathematically
mathematics in the situation. Of course, several of the students set
about solving the problems: when they put up their hands to give the
answer, I asked them if they had a question and to stand up so that the
whole class could hear it and help answer it. That confused them a bit
until I reminded them about what they were supposed to be doing!
I remembered then that I had read about using a ‘no-hands up’ policy in
class [Black et al., 2003]. I thought about what I had read and noticed
that when Jagadev put up his hand to answer, he stopped working and
stopped thinking. The students who were talking quietly with a partner
were thinking mathematically. I decided there and then to make it a rule
in class that ‘hands-up’ was only to be used for asking a question. I
would expect everyone to continue to think about and discuss the work
with others until I told them to stop, not to compete to be the first to say
‘I’ve finished’. When I wanted an answer I asked a specific person,
because then everyone would have to continue thinking. I think when
we all remember we are a ‘hands-up only to ask’ class, there is a lot
more thinking going on and if Jagadev or others finish their work, they
are now more likely to find someone to discuss the work with further as
there is no need to compete to be the first to answer or finish. This has
made all the class more cooperative and confident as well.
All this focus on word problems means that my students are now happy
to talk about mathematical situations. They ask ‘Could we model that
like this?’ and no longer leave out the word problems at the end of the
exercise. They have been known to ask to work together on some ‘hard
word problems’ recently and I heard Sunita saying she wanted to be a
model – a mathematical model!
Pause for thought
Think about the pedagogical approach of ‘no-hands up’: did your pupils
put their hands up with an answer to the problem when you had asked
them to think about the process? Do you think a ‘no-hands up’ rule
except to ask a question would help encourage your students to work
more collaboratively and therefore do more thinking and learning?
Think about who answers questions normally in your classroom. Is it
just a few? Are you sure that everyone answers over a few lessons? If
some do not answer any questions, are you sure they understand the
work? If you are not sure about the answers to these questions, tick a
class list every time a student answers a question during one week. Are
the ticks evenly spread?
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TDU 13
Word problems in mathematics: identifying key terms and making connections
4 Making sense of the
mathematics in the context of word
problems
Word problems have been around for a very long time. Consider these two
problems:
.
It takes three men six hours to dig a ditch. How long does it take two
men to dig the same ditch? (Traditional)
.
Suppose a scribe says to thee, four overseers have drawn 100 great
quadruple hekat of grain, their gangs consisting, respectively, of 12,
eight, six and four men. How much does each overseer receive? (Problem
68, Rhind Mathematical Papyrus, c. 1700 BCE)
The style that these problems are written in makes them unlikely to be
mistaken for anything other than ‘word problems’, and you probably can
make sense of the mathematics without understanding the context. The
context is not in itself important in many word problems. This can even be
true when the context tries harder to be relevant
Suppose you read the following word problem: ‘Sally made 32 cakes. One
eighth of them were chocolate; how many were not?’ What responses or
reactions might be possible other than just trying to work it out? Someone
might reasonably ask: ‘Why might I want to know this?’ or ‘Is it plausible
in my experience?’, or even ‘Can I imagine myself asking such a question
outside a mathematics classroom?’
Word problems are often essentially mathematical problems dressed up in
everyday language, but they are there to help students understand that
mathematics can model the real world and they can act like mathematicians
when they do this. This is why it is important that students begin to realise
the power of mathematics in real-world problems is about its ability to
model complex situations and extract the essential elements from them.
Focusing on the process of making sense of a complex situation and
modelling it mathematically can also help students focus on the ‘making
sense’ aspect of word problems. Activity 4 also looks at how to help
students know how to find out what they need to know more independently.
12
4 Making sense of the mathematics in the context of word problems
Activity 4: Making sense of the context and the
mathematics in a word problem
Read each problem and answer the questions that follow.
1 Savitri had to make a model of a cylindrical kaleidoscope for
her science project. She wanted to use chart paper to make the
curved surface of the kaleidoscope. What area of chart paper
would she require, if she wanted to make a kaleidoscope with a
length of 25 cm and a 3.5 cm radius?
2 Ramesh and Mahesh together can row a boat at a speed of 12
kph. If they want to row the boat at 15 kph, they need to apply
180 N more force. What is the total weight of the boat and its
two occupants in water?
3 Lalita was awarded a 5 per cent increment in her annual salary
because of her contribution to the water conservation project
for her company. If her base salary was Rs. 3.5 lakh per annum,
what is her revised monthly salary?
For each problem, consider the following:
.
Do you know the meaning of each of the words or phrases
highlighted in bold? Are there some terms or phrases that are new
for you? Do you feel that these will be relevant to solving the
problem?
.
How would you go about learning what these words or phrases
mean, or the mathematical ideas you would need to work with them?
.
Rephrase the words and phrases highlighted in bold to simplify the
word problem. If a particular term, word or phrase is not required,
you may omit it. Which terms did you find difficult to rephrase? Why?
Case Study 4: Daksha reflects on Activity 4
I am pleased that I used these three problems with my class. I have to
say it was hard work getting them to focus at the start – they just
seemed to find problems and to say that they were stuck. However, I
persevered. I told them to work in pairs, which I always think is helpful if
the work is unfamiliar and likely to require a lot of thinking. I reminded
them that they just had to note down anything they did not know the
meaning of and then think about how they could find out about these
ideas.
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Word problems in mathematics: identifying key terms and making connections
After everyone had done some thinking, we shared ideas about what we
could use to find out about the ideas. They first said, ‘Ask the teacher’,
but I banned that for this exercise and asked them to be more
imaginative. One said, ‘Use the internet’, another ‘Look it up in the
textbook’, so I suggested that they look up what they could in their
textbooks and that if they brought me a note of anything they couldn’t
find, I would be their internet search engine for today! I made sure to be
awkward and only give information on what was actually ‘entered in the
search bar’ in order to make them think about what they really needed
to know.
Once everyone felt they had the information they needed, they went
onto the rephrasing exercise. It seemed that this was easy now because
the class were working collaboratively and learning together by this
time.
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 like Teacher Daksha did? If so, what was your
reasoning for doing so?
14
5 Summary
5 Summary
This unit has concerned learning to work with word problems in their widest
sense. It has discussed the relationship between contexts and the
mathematics that can model that context. The suggestion is that focusing on
the process of moving from words to algebra and vice versa will enable
students to understand that the world can be modelled by mathematics. This
is important because when mathematics is used in real life, it is used to
allow complex problems to be simplified and worked with so that solutions
can be suggested with confidence.
Working with word problems can give meaning to mathematics that is based
in the student’s personal experience and therefore can make them active
participants in their own learning as they use mathematics to model their
own situations.
Identify three approaches have you learned from this unit that you might use
in your classroom and two ideas that you want to explore further.
15
TDU 13
Word problems in mathematics: identifying key terms and making connections
6 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:
16
.
View learners as active participants in their own learning and not as mere
recipients of knowledge; to encourage their capacity to construct
knowledge; to ensure that learning shifts away from rote methods.
.
View learning as a search for meaning out of personal experiences and
knowledge generation as a continuously evolving process of reflective
learning.
.
Support students to learn to enjoy mathematics rather than fear it.
.
Connect school knowledge with community knowledge and life outside
the school.
References
References
Black, P., Harrison, C., Lee, C., Marshall, B. and Wiliam, D. (2003) Assessment
for Learning: Putting it into Practice. Buckingham, UK: Open University Press.
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 at http://www.
ncte-india.org/publicnotice/NCFTE_2010.pdf (Accessed 16 January 2014).
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.
17
TDU 13
Word problems in mathematics: identifying key terms and making connections
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: 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/
The material acknowledged below is Proprietary and used under licence (not
subject to Creative Commons Licence). Grateful acknowledgement is made
to the following sources for permission to reproduce material in this unit:
Problem solving: © almagami/iStockphoto.com.
Jeans: © Chris Rubber Dragon/iStockphoto.com.
Activity 3: © Nikhil Gangavane | Dreamstime.com.
Activity 4: © Andrew Turner/Flickr made available under Creative
Commons, Attribution licence: http://creativecommons.org/licenses/by/
2.0/deed.en.
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.
18