Approaching Mathematical Problems
Systematically
Description of this Guide
In this guide we will examine a systematic approach to problem solving. It won’t
give you a foolproof method for solving every problem you meet, you will still
have to think for yourself, but it will give you a framework to help you solve
problems. If you get in the habit of applying the framework to problem solving,
you will find you have increasing success in tackling problems and finding
solutions.
Learning outcomes
1. Understand and analyse the problem
2. Define the solution
3. Choose strategies for solution
4. Check your results
5. Find your own errors
6. Get effective help
Contents
1.0
Introduction
1.1
Types of problems
1.2
Active vs. passive study
1.3
Before you start on the worksheet
1.4
The problem solving process
1.5
Thinking and doing
1.6
Model answers - Who is the solution for?
2.0
Understanding the problem using a problem analysis chart
2.1
What are you trying to achieve?
2.1.1 Cracking the code
2.2
What are you given?
2.2.1 What else do you know?
2.3
Processing Required
2.4
Methods for a Solution
3.0
Strategies for solution
3.1
Some profitable strategies
3.1.1 Breaking the problem into parts
3.1.2 Making a picture
3.1.3 Working backwards
3.1.4 Solving a simpler, related problem
3.1.5 Getting a feel for the extremes
4.0
Carrying out your strategy
4.1
Knowing whether your solution is working
4.2
Spotting errors as you work
4.2.1 Common errors
4.3
Getting useful help
4.4
Checking your work
4.5
Putting the numbers in
4.6
Annotating your solution
5.0
Key Points
_____________________________________________________________________
The material in this guide is copyright © 2004 the University of Southampton. Permission is given for it to be copied for use
within the University of Southampton. All other rights are reserved.
Approaching mathematical problems systematically
Skills
Approaching Mathematical Problems
Systematically
1.0 Introduction to Solving Maths-Based Problems
In most subjects which involve a lot of maths, you will be asked to solve
problems both as part of your course work and in exams. If you want to improve
your marks, you need to take a systematic approach to solving problems.
Think about what happens when you try to solve the problems you are given, as
examples and on worksheets. Do any of these apply to you?
Yes or No
Comments
1. I look at the problem and I just don’t
know how to tackle it.
You may need to do some more work
understanding the topic before you start
to solve the problems. Sections 1.1 to
1.5 will help you to study your topic
effectively. If you have grasped the topic
then you may need to do more work to
understand the problem itself. Section 2
will help you with this.
2. I know how to do the problems, but I
make errors in algebra and/or arithmetic
as I work, so I never get the right
answer
You can lose marks through poor
accuracy in these areas. You need to
work on this just as much as you work on
new topics. Section 4.4 will give you
some useful advice.
3. I know what I am trying to find but I
don’t know how to go about it.
You need to develop some tools to help
you find solution strategies. There are
some standard approaches that are often
worth trying when you can’t see a way
forward. Have a look at Section 3.
4. I don’t know how to tell if I have the
right answer or not.
Sections 4.5 and 4.6 may be particularly
useful to you.
5. I can do the problem when I get the
sheet but when I come to revise I have
forgotten what I did.
You may like to look at Sections 4.5 and
4.6
Spending hours staring at the same problem and not getting anywhere or doing
all the problems on a worksheet but getting them all wrong are not efficient ways
for you to learn.
Improving your techniques and strategies for studying maths-based topics and
solving problems will help you to gain better marks and to learn more effectively
and efficiently. It may help to understand the different types of problems and
identify those you prefer or have difficulty with.
2
1.1 Types of Problems
Type of Problem
1 Problems testing memorisation.
There are some things that it is useful to
remember. An example might be your
multiplication tables. Problems to test your
memory might consist of a list of
multiplications to do without using a
calculator. These kinds of problems help
you to remember facts that can be useful
when you are solving more complex
problems.
E.g.:
Find all the factors of 56 is easier to do if
you can remember what 7x8=
This
problem
type is
Difficult /
OK
What to do
The best way to memorise things is by
repetition. Try always to find factors,
add numbers, work out products etc.
in your head before you reach for a
calculator. You can use the calculator
to confirm your answer.
Getting the facts at your fingertips often
involves a lot of repetition, but it is worth
it.
2 Problems testing skills.
There are some basic skills that you need
in all kinds of maths problems such as
rearranging formulae or adding fractions.
These kinds of problems help you to
practise skills so that you can use them in
more complicated problems as and when
you need them. As for problems of type 1,
they can involve doing the same thing
over and over until it is almost second
nature.
3 Problems requiring application of
skills to familiar situations.
Books designed for GCSE may be a
help with the basic skills or have a
look at www.mathcentre.ac.uk
Sections 2 to 4 of this guide will help
you to tackle this kind of problem.
In these problems you know the skills, and
facts you need and you understand from
your studies what the particular situation
involves in terms of theory. You have to
bring the facts, skills and theoretical
understanding together in the right way if
you are to solve the problem.
4 Problems requiring application of
skills to unfamiliar situations.
Sections 2 to 4 of this guide will help
you to tackle this kind of problem.
You know the necessary facts. You have
the necessary skills, but you haven’t met
the situation before. You need to learn
about the situation before you can proceed
and then, once you have the new theory,
you need to develop a strategy to solve
the problem using what you know.
3
Approaching mathematical problems systematically
5 Problems requiring you to extend
the facts, skills or theory you know
before applying them to an unfamiliar
situation.
Skills
Sections 2 to 4 of this guide will help
you to tackle this kind of problem.
Here you may know some of what you
need, but your first task is to identify what
you don’t know. Then you need to acquire
new skills, facts or theory to help you.
Finally you need to apply it all to a new
situation.
In early courses, you solved problems of types 1, 2 and 3. Now you will be
expected to tackle more problems of types 3 and 4, and (eventually) of type 5.
Each problem of types 4 or 5 usually requires you to use a multi-step approach,
and may involve several different maths skills and techniques. If you have
difficulties with the skills needed for problems of types 1 and 2 you should
consider doing some refresher work to get some practice. You can expect to
develop your ability with problems of types 3, 4, and 5 during your degree
programme.
1.2 Active vs. Passive Study
To be successful in studying maths-based subjects, you need to
be actively involved in managing your learning process and your study time.
Do you….
Yes/No
Comments
Recognise what you do and don't
know, and know how to get help
with what you don't understand?
You need to define your own needs and to make
sure you know how to get the support you need.
See ‘Being an Independent Learning’ guide.
Attend all your lectures every day?
In maths based subjects, each section builds on
your previous knowledge. It is important to keep
up steady progress with your work.
Participate actively in the class?
Get ahead by reading printed notes or textbooks
before lectures; try to work through some of the
problems before they are covered in class. Try to
anticipate what the lecturer's next step will be in
each worked example.
Ask questions in the class?
There are usually other students wanting to know
the answers to the same questions you have. If
the printed notes aren't clear about a complete
method, write additional comments on them
during the lecture.
Go to tutorials and ask questions?
You will be actively helping yourself by sorting
out problems as they arise, not leaving them
until later in the course.
Study regularly sorting out
problems as they arise?
This makes it much easier to prepare for exams
at the end of the semester.
Attempt all the problems on the
worksheets?
Math-based subjects are learned by doing
problems. The problems help you learn the
formulas and techniques you need to know, as
well as improving your problem-solving prowess
.
In maths-based subjects, practice in using new
methods, theories and techniques is the only way
to really grasp them. The more the better.
Find extra examples in textbooks
from the library?
4
1.3 Before you start on the worksheet

During your private study time, read through the printed notes and your
own notes from the lecture.

Work through the worked examples in the notes. Don’t just read them –
try to do the examples yourself using the notes as a guide. Can you see how
and why each step is taken?

If you have problems following the examples, are they to do with the new
material or missing background knowledge (e.g. basic algebra or arithmetic
skills)? Have a look at the table in Section 1.1 to see if you can decide where
the difficulty arises and what type of problem it is.

Can you find a textbook or a friend to help you with your difficulties?

Make a list of the questions you need to ask at tutorials. Try to be
specific about what you don't understand. Mark the step where you got stuck.
Attend the tutorial and ask the questions.

Once you feel you have understood the topic you are ready to start on the
worksheet.
1.4 The Problem Solving Process
What are you being asked to
find?
What technique will you use to
try to solve the problem?
Do you understand all the
words in the problem?
Carry out the solution using
your chosen technique
What information and/or
relationships are you given in
the problem?
Will you need other facts or
relationships that are not
explicitly given in the
problem?
(expert knowledge)
Are you sure you know what
you are looking for? Did you
assume anything that was not
given? Were your assumptions
correct? Can you think of
another technique to try?
Is it working? Are you making
progress towards a solution?
Are you being accurate?
Can you check your answer?
No
Success?
Thinking
Doing
Yes
Hooray!
The problem solving process can be divided into 2 parts: thinking and doing. Within
each part you can systematically carry out a number of different tasks for each
problem you try to solve. We will look at the thinking tasks in more depth in sections
2.0 and 3.0. We will look at the doing tasks in more detail in section 4.0.
5
Approaching mathematical problems systematically
Skills
1.5 Thinking and Doing
In problem solving:
1. Understand the problem, work out what information you have been
given in the problem statement and what other information you will need.
2. Devise a strategy for solving the problem.
This is the thinking part of the process.
3. Carry out the strategy for solving the problem
4. Write down the steps for the solution
5. Check your answer
This is the doing part of the process.
1.6 Model answers - Who is the solution for?
A model answer can’t teach you to develop the skill of problem solving. You can
only learn this by practice through solving lots of problems. In fact, if you look at a
model answer too soon you will avoid doing the thinking for yourself. This doesn’t
help you to learn how to tackle these stages of the process.
If model answers or worked examples are available then they can be useful for
the third and fourth parts of the problem solving process; the doing parts. You
can always use them to check whether any final numerical answer you have
obtained is correct. If they use the same methods and techniques that you have
chosen, you can use them to check your method as well, but remember, there is
usually more than one way to do a problem. If you have used another method
then the model answer won’t help you know if it is valid or not.
You are given worksheets or sets of problems from textbooks so you can develop
your problem solving skills and apply the new techniques you are learning during
your course. If you are asked to submit the answers for marking, then the aim is
probably to give you feedback on your progress. So, the solution IS FOR YOU.
You may find that you can do a problem when the topic is new, but when you go
back a few weeks later, you can’t make sense of it.
You are the person who has to prepare for the assessment from the notes, so
take time to write down full solutions with explanations of what you were doing.
It may take a little more time initially, but it can save you hours of effort in the
long run.
You can also help yourself by making extra notes next to your solution.

Was it hard or easy? Why?

Did you have to look anything up or get help? With which bit? What
did you find out?

Can you identify any particular tricky bits? Did they relate to new
material or gaps in your basic skills (e.g. arithmetic or algebraic
manipulation)?

How is it different to the previous example? Or to the next
example?
6
Remember!
The aim is for you to learn, so write notes that will help you to recall what you
did.
You can find more ideas about annotating solutions in Section 4.6.
2.0
Understanding the problem using a problem analysis chart
The first step is to make sure you really understand your problem. This is the first
part of the thinking stage. It may help to use a Problem Analysis Chart – at
least to begin with.
Required result(s)
Write here what you are trying to
achieve (section 2.1)
Given Data
Write here a list of data you are given
(section 2.2)
Processing required
Write here any formulae you will need
(section 2.3)
Alternative solutions
Write here the different methods you
think might work to solve the problem
(section 2.4)
2.1 What are you trying to achieve?
If you are looking for something, you will never find it if you don’t know what it is!
The first step in any problem is to define what you are trying to find. In some
problems you will find a sentence that makes this clear. In other problems it is
harder to work out what is needed, and in some very complicated problems, working
out what will count as a valid solution is a whole problem in itself. You need to:

First, make sure you know what all the words in the problem mean.
If you find one you don’t understand then look it up in a dictionary.
Is it an everyday word or a specialist or technical word?

Next, identify the required result. You might like to underline it or
write it in your own words in the top left of a problem analysis
chart. Different identification methods may better suit different
problems.
You must keep your goal in mind as you work through the problem.
Example : Identifying what you are trying to achieve
Two pickup trucks collected bags of rubbish. One collected 56 bags of
rubbish and the other collected 47 bags of rubbish. How many bags of
rubbish were collected altogether?
7
Approaching mathematical problems systematically
Skills
Required result(s)
Given Data
Total number of rubbish bags
Processing required
Alternative solutions
What about the next two problems? Simply identify what you are trying to
achieve:
1.
If Tom has three times as many apples as Susan and Susan has a quarter
as many as Joe, who has four, how many does Mary have if Mary has two
more than Tom?
Required result(s)
Given Data
Processing required
Alternative solutions
2.
Solve:
2x + 3y = 1
x – 4y = 17
Required result(s)
Given Data
Processing required
Alternative solutions
If you can’t get started on a problem, the most likely reason is that you
haven’t properly identified what you are looking for.
8
2.1.1 CRACKING THE CODE
In maths and science there are lot of specialist words. There are also words that
look like every day words, but which have a special meaning in the context of
maths.
Some words that can be used to identify the thing you are trying to find are…
Common Meaning
Find…
Work out an algebraic or numerical
expression
Evaluate…
Find a numerical value from a formula
Solve…
Find numerical values for any unknown
variables
Find a value for…
As for evaluate
Prove…
By logical reasoning show that
something is true
Derive…
Show how a particular formula or rule
is obtained from other rules or laws
Show that…
As for prove
Simplify…
Use algebraic manipulation to make an
expression less complicated
Express in terms of…
Put an expression in terms of a
specified parameter or symbol
Express in the form…
Put an expression in a particular form
Expand…
Make an expression more complicated
by e.g. using a series or multiplying
out brackets
Rearrange…
Change to give an expression or
equation a different form
Transpose…
Change the subject of an equation
An example from your own
subject
9
Approaching mathematical problems systematically
Skills
2.2 What are you given ?
The next step in solving a problem is to work out what information you have been
given in the problem statement.
You can make a list or write it in the top right-hand box of a problem analysis
chart.
You need to be very clear about what you do and don’t know. It is very easy to
make assumptions without really knowing you have made them. It may help to get
in the habit of writing down what you are given in your own words.
Let’s have another look at our problems from section 2.1.
Example : identifying what you have been given
Two pickup trucks collected bags of rubbish. One collected 56 bags of
rubbish and the other collected 47 bags of rubbish. How many bags of
rubbish were collected altogether?
Required result(s)
Given Data
Total number of rubbish bags
Truck 1 = 56 bags
Truck 2 = 47 bags
Processing required
Alternative solutions
What about the next two problems? Identify what you have been given.
1.
If Tom has three times as many apples as Susan and Susan has a quarter
as many as Joe, who has four, how many does Mary have if Mary has two
more than Tom?
Required result(s)
Given Data
The number of apples that Mary has.
Processing required
Alternative solutions
10
2.
Solve:
2x + 3y = 1
x – 4y = 17
Required result(s)
Given Data
A numerical value for x and a
numerical value for y
Processing required
Alternative solutions
2.2.1 WHAT ELSE DO YOU KNOW ?
The information you are given is often not the only information you need to solve
the problem. Frequently you are assumed to know something else as well.
The extra knowledge may be common sense or well known information or it may
be specialist knowledge you have obtained from studying your course. This is
your own expert knowledge.
Given
information
Needed for
problem
Expert
knowledge
Your
problem
Everything
you know
Needed for
problem
Does the question give you everything you need or not?
11
Approaching mathematical problems systematically
Skills
Example : Identifying the knowledge needed
Problem
Expert knowledge
needed
A round wheel with a radius of 37 cm rolls at a constant speed of 3
revolutions per second. How far does the axle of the wheel move in 8
seconds?
The writer of this problem is
assuming you know what is
meant by round, wheel, rolls,
radius, revolutions per second
and axle.
In the example above, round and wheel are everyday words while radius,
‘revolutions per second’ and axle are more technical. You need to know what all
these mean before you can solve the problem. This is your expert knowledge
that you bring to the problem.
What assumptions is the problem setter making about what you know in these
next two problems?
Problem
Expert knowledge
needed
1. Mary looked out of her farmhouse window and saw a group of
pigeons and donkeys passing by. She counted all the legs of the
pigeons and donkeys and found that the total number of legs added
up to 66. How many of each kind of animal (pigeons and donkeys)
passed by her window if the total number of animals is 24?
2. A car travels at 60 mph. How far does it travel in 30 minutes?
Once you have decided on your expert knowledge you can put this and any other
relevant formulae or facts you think you will need into the ‘processing required’
part of your Problem Analysis Chart (section 2.0).
2.3
Processing required
Back to the problems in 2.1.
Example : Processing required
Two pickup trucks collected bags of rubbish. One collected 56 bags of
rubbish and the other collected 47 bags of rubbish. How many bags of
rubbish were collected altogether?
12
Required result(s)
Given Data
Total number of rubbish bags
Truck 1 = 56 bags
Truck 2 = 47 bags
Processing required
Alternative solutions
No. for T1 + No. for T2
What about the next two problems? Identify the processes required to solve
them.
1.
If Tom has three times as many apples as Susan and Susan has a quarter
as many as Joe, who has four, how many does Mary have if Mary has two
more than Tom?
Required result(s)
The number of apples that Mary has.
Processing required
Given Data
No.
No.
No.
No.
apples
apples
apples
apples
that Tom has compared to Susan.
Susan has compared to Joe.
Joe has.
Mary has compared to Tom
Alternative solutions
2.
Solve:
2x + 3y = 1
x – 4y = 17
Required result(s)
A numerical value for x and a numerical value
for y
Processing required
Given Data
2 relationships between x and y
Alternative solutions
13
Approaching mathematical problems systematically
Skills
2.4 Methods for a Solution
Finally you need to put the solution methods you might try into the last box of the
chart. These could be standard methods for particular types of problem or they
might be one of the methods outlined in the next section.
Example: identifying other solutions
Two pickup trucks collected bags of rubbish. One collected 56 bags of
rubbish and the other collected 47 bags of rubbish. How many bags of
rubbish were collected altogether?
Required result(s)
Given Data
Total number of rubbish bags
Truck 1 = 56 bags
Truck 2 = 47 bags
Processing required
Alternative solutions
No. for T1 + No. for T2
Addition
Some suggestions are given for the two problems below – can you think of any
other possibilities?
1.
If Tom has three times as many apples as Susan and Susan has a quarter
as many as Joe, who has four, how many does Mary have if Mary has two
more than Tom?
Required result(s)
The number of apples that Mary has.
Processing required
T = 3S
4S = J
J =4
M=T+2
Given Data
No apples that Tom has compared to Susan.
No. apples Susan has compared to Joe.
No. apples Joe has.
No. apples Mary has compared to Tom
Alternative solutions
Substitution
Elimination
Working backwards
14
2.
Solve:
2x + 3y = 1
x – 4y = 17
Required result(s)
A numerical value for x and a numerical value
for y
Processing required
rearranging
substituting for x or for y
Given Data
2 relationships between x and y
Alternative solutions
Substitution
Elimination
Cramer’s Rule
You need to consider every area of the Problem Analysis Chart before you can hope
to solve your problem. For some problems, some of the sections will have very little
information while others will be much fuller. This doesn’t matter, it just depends on
the type of problem you are dealing with.
So from this section you know you have to:




familiarise yourself with the problem
make sure you know what you are looking for
identify what you are given
identify what else you might need to know
When you have done that you are ready to try some strategies to solve your
problems. Section 3 discusses a number of different methods.
3.0
Strategies for solution
Once you have a firm grasp of the problem it’s time to start solving it. You may
be able to think of one way to try to solve it, or you may be able to think of
several ways. It depends on how experienced you are with the subject matter
and also on the problem itself.
You need to be very systematic now. The process you need is shown below:
15
Approaching mathematical problems systematically
Skills
DEVISE
Select
your
solution
strategy
CARRY
OUT
Start
using it to
solve the
problem
REFLECT
Is it
going
OK?
CARRY
OUT
Finish it off
REFLECT
How did it
go? Was it
a good
method?
YES
YES
No
STOP
No
Go back
and try
another
method
Remember, you have only failed to solve the problem when you have run out of
strategies to try!
In the rest of Section 3 we will look at some general strategies that are often
worth trying.
In Section 4 we will look at carrying out your strategy and reflecting on its
success.
3.1
Some profitable strategies
Pólya was a mathematician, born in Budapest in 1887, who noticed that his
students often had trouble solving problems even though they knew a lot of
mathematics. To help them he tried to write a recipe for problem solving 1. In the
end he found that, while there were no hard and fast rules, there were a number
of strategies that all worked well at least some of the time. He couldn’t find a rule
to tell you which strategy to use in any particular case, but he said that all the
methods listed below (and some others) were worth considering. You just pick
the one that looks most promising for your particular problem.
The example problems are all quite simple, so that you don’t need specialist
knowledge in any particular subject to solve them, but the methods adapt well to
more complicated problems once you are used to using them.
3.1.1 BREAKING THE PROBLEM INTO PARTS
Many problems are easier to solve if you break them down into smaller parts, solve
each part separately and then build a final answer from the solutions to the smaller
parts.
The picture below shows a complicated machine with many parts. The problem is
how does this work ? If the input is a rotation of the red cog wheel, you might ask,
“What is the final output?”. Taking the machine as a whole this is not easy to
answer, but if you break the functions down into smaller parts and remember that
the output of one section is the input to the next, then perhaps you can make some
progress.
1
G Pólya, How to solve it, 2nd edition, Penguin Books, 1990, ISBN: 0140124993.
16
Red
Cog
Wheel
Mousetrap®
Game by
Milton Bradley
For instance, taking just the first section:
Player turns crank (A) which rotates gears (B) causing lever (C) to move and push
stop sign against shoe (D).
So for the first section the input is the turning of the crank and the output is the
movement of the shoe.
For the second section:
Shoe tips bucket holding metal ball (E) Ball rolls down rickety stairs (F) and into
drainpipe (G) which leads it to hit helping hand rod (H).
17
Approaching mathematical problems systematically
Skills
The shoe is the input and the ball hitting
the helping hand rod is the output.
For the third section:
Metal ball hitting helping hand rod (H) causes bowling ball (I) to fall from top of
helping hand rod through thing-a-ma-jig (J) and bathtub (K), to land on diving
board (L).
The metal ball hitting the helping hand rod is the input and the bowling ball landing
on the diving board is the output.
Finally:
Weight of bowling ball catapults diver (M) through the air and right into wash tub
(N), causing cage (O) to fall from top of post (P) and trap unsuspecting mouse.
The input is the falling bowling ball and the output is the trapped mouse.
Now we can work back stage by stage until we find that….the initial input is
rotating the handle and the final output is the trapped mouse
18
The same idea of breaking problems down into manageable steps works for a
variety of different, more complex problems.
Example
You wish to build a dam across a river. What design will you use for your dam?
The solution to this problem is made up from the solutions to a whole set of
related problems. First, consider some quite broad questions like those shown
in the table below. Can you think of some more that might apply?
Broad initial questions
What is the budget?
What is the dam for? It might be to control flooding, to generate hydro-electric power or it
may have more than one purpose.
What is the timescale for construction?
You might have thought of the environmental impact, the geology and the
topology of the area, communications with the build site, availability of materials,
annual rainfall etc.
These then lead to more specific questions like those below. You can’t answer
these until you have answered at least some of the first set of questions Can you
think of more of these questions?
More specific questions
Where will you put the dam?
What shape should it be?
What dimensions should it have?
19
Approaching mathematical problems systematically
Skills
You may have added questions about materials to use, number of workers
required, equipment needed etc.
Breaking the problem down into parts makes it more manageable. Often you can
solve the easier parts first and worry about the harder questions later. This helps
you to build your confidence about the problem.
3.1.2 MAKING A PICTURE
For almost all practical problems, and many more abstract ones, a drawing,
diagram or graph is useful. It helps you to summarise the problem and your
assumptions about it in a systematic way. Diagrams, graphs and drawings for
problem solving don’t need to be a beautiful work of art, but they do need to be
informative to you, the user.
Example : visualising your problem
40 students when out for the night. 14 fell in the river, 13 got over
excited, 16 got lost on the way. Three of the over excited people fell in the
river. Five of them fell in the river and got lost. Eight got over excited and
also were lost. Two experienced all 3 mishaps. How many students
escaped with none of these mishaps?
Try using the diagram below to help you with this problem.
Fell in the river
Got lost on the way
Got over excited
Don’t forget the overlapping parts can be used to deal with people where more
than one mishap occurs to a person.
Did you find that 11 people escaped with no mishap?
20
Try the following:
Doug and Anne both work part-time at the corner shop. The shop is open
seven days a week? Doug works one day and then has 2 days off before
he works again. Anne works one day and then has 3 days off before she
works again. Doug & Anne both worked on Wednesday, 1st of August. On
which other days in August did they both work at the same time?
Diagram
Did you draw a calendar? You should have found that they worked together on
13th and 25th August.
You can draw any diagram that helps you to understand the problem you are
considering. It may be a sketch of equipment, a map, a graph or bar chart, a pie
chart, a calendar, a Venn diagram, a picture, a spider diagram, a mind map or
something else….
3.1.3 WORKING BACKWARDS
If you know what you are trying to find, it’s sometimes easier to start from the
answer and work backwards.
Example : working backwards
If Tom has three times as many apples as Susan and Susan has a quarter
as many as Joe, who has four, how many does Mary have if Mary has two
more than Tom?
Let T, S, J and M be the number of apples that Tom, Susan, Joe and Mary
each have.
Start with the answer you need.
21
Approaching mathematical problems systematically
M=?
Skills
(1a)
Now ask yourself – what do you know about M?
M=T+2
(1b)
So far so good – you could solve it if you knew what T was. What do you
know about T?
T = 3S
(2)
So
M = 3S + 2
(1c)
Even better! If you knew what S was you’d have your answer. What do
you know about S?
S = J/4
(3)
So
M = 3(J/4)+2
(1d)
Finally…
J=4
(4)
So
M = 3(4/4)+2
(1e)
M=5
Mary has 5 apples.
Working back from the answer gives a structure to the problem and the
relationship between its different statements which is not apparent from the
problem statement. Each of the equations labeled (1x) is the same equation with
a bit more information in it.
You can solve the next problem by setting up a set of simultaneous equations and
solving them. Alternatively you can try working backwards from the required
answer. Try both.
The next problem can also be solved forwards or backwards. The answer is at the
bottom of the page.
Anne, Bob and Cath play a certain game. The player who loses each round
must double the money of the other players. In Round 1 Anne loses and
gives Bob and Cath as much money again as each of them has already. In
Round 2 Bob loses and gives Anne and Cath as much money again as each
of them already has. Cath loses in Round 3 and gives Anne and Bob as
much money as they each have. They decide to stop at this point and
discover they each have £24. How much money did they each start the
game with?
To work forwards let Anne, Bob and Cath have £x, £y and £z respectively at the
start of the game. Now work forwards constructing an equation for each step of
the game. Finally solve your equations to find values for x, y, and z.
To work backwards, fill in the blank spaces. in the table below.
Solution
How much money did each have after round 3?
Anne
Bob
Cath
24
24
24
12
48
How much money did each of them have at the end
of round 2?
How much money did each of them have at the end
6
22
of round 1?
How much money did each of them have at the start
of the game?
12
Try this similar problem for yourself.
Mark breeds hamsters as a hobby. In March the number of hamsters
increases by 10%. At the beginning of April he buys 6 new hamsters and
at the end of April he sells one quarter of all the hamsters he has. May is a
good time for breeding and 35 new hamsters are born during the month.
At the end of May, Mark sells half of all the animals he has. So far in June,
9 hamsters have been born and Mark now has 70 Hamsters. How many
did he start with on March 1st?
For the game questions the starting amounts were Anne 39, Bob 21 and Cath 12.
Mark started with 100 Hamsters.
3.1.4 SOLVING A SIMPLER, RELATED PROBLEM
Sometimes we are faced with a problem that looks too complicated to solve
straight away. We can often make some progress by working out what makes it
hard by solving similar, but simpler problems until we can see a pattern
emerging.
Find the value of
2  4  6  8  10  12  14  16  18  20  22  24  26  28  30  32  4  36  38
3  6  9  12  15  18  21  24  27  30  33  36  39  42  45  48  51  54  57
Why do you think this problem looks hard? You probably think it looks hard
because there are a lot of numbers on the top and bottom of the fraction. It
would be easy to make a mistake when you added them all up.
The most obvious way to simplify it is to reduce the size of the problem.
What do you get if you calculate:
2
?
3
24
?
36
246
?
369
Have you spotted a pattern yet? Do you need to try some more to convince
yourself?
23
Approaching mathematical problems systematically
Skills
Did you guess the solution of the first problem to be
2
3
?
Try this technique on the next two problems
The factors of 360 add up to 1170. What is the sum of the reciprocals of
the factors?
You are trying to find a value for
1 1 1 1
1
1
1
    ......


1 2 3 4
120 180 360
Why is the problem hard? How can you change the problem to make it easier?
What about trying the same thing with the factors of 12 or of 15 and looking for a
pattern?
Choosing a related problem that gives you insight into the original problem is
something you can only learn with practice. However, whenever you can use this
method it is likely to save you a substantial amount of time and also gives you a
great feeling of satisfaction.
For the sum of the reciprocals of the factors the answer is 1170/360.
3.1.5 GETTING A FEEL FOR THE EXTREMES
Often, when you can’t see the answer to a problem straight away it helps to try to
estimate the biggest and/or the smallest values the answer could have.
Example: getting a feel for the answer
In a drawer there are 8 blue socks, 6 green socks and 12 black socks. What
is the smallest number of socks that must be taken from the draw without
looking at them to be certain of having 2 black socks?
In this case you could consider the worst case scenario. You might not pick a
black sock before you had picked all the blue and green socks. That would take a
minimum of 14 picks. To then get 2 black socks you would need a further 2 picks.
In this case then, to be certain to get 2 black socks you must pick out a
minimum of 16 socks.
Of course you might be lucky and get two black socks straight away, but it isn’t
absolutely assured unless you go for all 16 selections.
Jo took 5 maths tests this semester. Each test was marked out of 100. Jo’s
average was 90 over the 5 tests. What was the lowest possible score that
Jo could have earned on any one test?
To solve this, first work out the Jo’s total score and then think about the
maximum score for each test.
24
You should be able to reason that the lowest possible test score Jo could have got
was 50.
A car is driving along a road at a constant speed of 55 mph. The driver
notices a second car exactly half a mile behind. The second car passes the
first, exactly 1 minute later. How fast was the second car travelling
assuming its speed was constant?
In the problem, the first car has a speed of 55 mph, but what would happen if it
was travelling extremely slowly? Can you generalise from this special case?
You should be able to show that the second car must have been travelling at 85
mph.
4.0 Carrying out your strategy
At this stage of your solution you should have worked out what you are trying to
find, what you have been told, what else might be useful and you should have
one or more ideas about how you could proceed. You are ready to try out a
solution.
If you are lucky, the first method you try will work out. If not, you need to know
when to stop trying one way and to try another. Is it the wrong method or have
you made a mistake along the way? You need to develop the skills to spot
mistakes in your method. In the end, if none of your methods work, you need to
know how to get really useful help that means you might do better with the next
problem.
Finally you need to record what you did in a way that helps you when you come
to look at the work again, perhaps to prepare for an assessment.
4.1
Knowing whether your solution is working
You may start out with one or more solution methods in mind. You pick the most
likely one and start your solution. How do you know if it is working? At what point
should you abandon it and try another way?
Unfortunately there is no hard and fast rule about this. Like the ability to think of
a variety of methods, knowing if your method is going to work comes with
experience. However there are some questions that you may ask yourself that
can help:
Useful questions to check your progress
Comments
Are you confident that each step is correct as you take
it?
See section 4.2 for advice about spotting
errors as you work.
Can you imagine what the next step will be? And the
next? And the one after that?
You should be looking forwards after
each step. If you are heading in the right
direction, things should be becoming
clearer. You should be more aware after
each step how you are going to proceed
to your goal. The looking forward is like
playing a strategy game. The best chess
players know all the options for many
moves ahead; a novice knows some
moves for the next step. You can develop
this skill with practice.
25
Approaching mathematical problems systematically
Have you used all the information you wereSkills
given?
In text book problems you are usually
given no extraneous information, so if
you have a use for it all, it’s probably a
good sign. In real life problems this may
not be true, so use this question with
care.
4.2 Spotting errors as you work
If you make errors in algebra or arithmetic, you will loose marks even if your
overall solution follows a sound method. You may only be penalised a small
amount for each error, but the lost marks can add up to a substantial penalty.
Good accuracy saves these marks and makes it easier for you to check that your
solution works.
If you think you know how to solve a problem it’s easy to rush ahead at top
speed writing down the answer. Taking a more measured pace will help you to
maintain your accuracy. It might seem slower, but if you only have to do the
problem once to get it right it will save you time in the long run. Think tortoise
not hare!
A frequent cause of errors is miscopying from one line to another. Reading each
line aloud and saying what you have done can help because you spot changes in
numbers more readily. Of course, in an exam you wouldn’t be able to do this, but
doing it during private study can train you not to make this kind of error very
often.
Accurate working is a habit. Checking for errors as you work trains you
to spot errors more readily.
Can you spot the errors in the calculations below? Try talking to yourself aloud
about what is happening at each stage. Does it help you to find the mistakes?
Can you spot the errors?
3x+1 = 25
2x = 24
x = 12
Say to yourself…
Answer
Three x plus one equals
25.
Taking the 1 to the
other side and
subtracting it from 25
leaves 24 on the right
so then two x equals
24 and dividing by 2 to
leave x on its own tells
me that x equals 12.
(you should have found 1
error)
24x – 3y -2(x+y) + 5 = 10x + 5y - 4
24x – 3y -2x +2y +5 = 10x + 5y – 4
22x-y = 10x + 5y + 1
12x = 6y + 1
(you should have found 2
errors)
26
3(8x+6) = 10
27x+18 = 10
27x = - 8
x = -8/27
(you should have found 1
error)
The more you practise with your own work, the better you’ll get at this.
4.2.1 COMMON ERRORS
It is impossible to list every error you might make and why, but if you can avoid
the ones in the table you will be doing well. They are all based on common
misconceptions or errors that are found in many students’ work.
Wrong
RIGHT!
Comments
3(2x + 1)
= 6x + 1
= 6x + 3
The 3 outside the
brackets multiplies
every term inside the
brackets
2x + 1 – (x + 2)
=x+3
=x-1
The minus sign
operates on every
term inside the
brackets
  x  3dx

sin (x + y)
= sin x + sin y
(x + y)2
x y
log(x + y)
x2
 3x  c
2

x2
 3x  c
2
You can’t simplify
this without more
information
= x2 + y2
You can’t simplify
this without more
information
 x y
You can’t simplify
this without more
information
You can’t simplify
this without more
information
= log x + log y
The minus sign
operates on every
term inside the
integral
One of the first rules of
maths you learn is
3(2x + 1) = 6x + 3.
It seems natural to
assume it works in
cases other than
multiplication. Mostly it
doesn’t! Try it – let x
= 30o and y = 60o.
Does sin 90o = sin 30o
+ sin 60o?
As for the case above,
squaring doesn’t follow
the same rules as
multiplication. Try it –
let x = 2 and y = 3.
Find x + y then square
your answer and
compare with the
answer from squaring
x, squaring y and then
adding.
This is another similar
case. Test it out using
x=2 and y=3 again.
And again. This is
especially confusing
because
log (xy) = log x + log y
is true. Make sure you
have learnt your log
rules really well.
27
Approaching mathematical problems systematically
Skills
Wrong
RIGHT!
Comments
1
x y
1 1
 
x y
You can’t simplify
this without more
information
d (uv )
dx
=
 uvdx
  udx   vdx
log x
 log x
sin 3x
= 3sin x
And another of the
same type of
confusion. Once more –
try it with x=2 and
y=3.
Another case of the
same. If it was as
simple as the wrong
answer – we wouldn’t
need the chain rule.
This is like the one
above. We have lots of
rules for integrating
products depending on
the form of the
product, but it is rarely
as simple as the wrong
case shown here.
This is a different type
of confusion from
above – but it also
comes from working
mainly with numbers in
your early studies. For
multiplication 2x3 =
3x2. The order you do
things doesn’t matter.
However, mostly
order matters. You
can’t change the order
of operations just
because you feel like it.
Evaluate the
expression in both
cases with x = 2. Same
answer each time? No!
– so the two
expressions can’t mean
the same thing.
As above – you can’t
just change the order.
Convince yourself by
considering x = 30o
du dv

dx dx
v
du
dv
u
dx
dx
The right answer
here depends on u
and v. Maybe you
need to substitute or
to use the chain rule.
You’ll have think
further about the
form.
You can’t simplify
this without more
information
You can’t simplify
this without more
information
28
Wrong
(3 x  7)( 2 x  9)  (4 x  1)
(3 x  7)( x 2  1)

(2 x  9)  (4 x  1)
( x 2  1)
RIGHT!
Comments
You can’t simplify
this without more
information
This is a confusion
about cancelling. Think
about numerical
fractions. There we say
that 62
 13 . The top and
bottom have a
common factor. We can
do the same for
510
25
x
4x

6
3x  1 2 x  5
x  4x 
sin 2 x
 sin( x 2 )
sin 1 x
6(3x  1)( 2 x  5)

1
 cosec x
sin x
x(2 x  5)  4 x(3x  1)
 6(3x  1)( 2 x  5)
 (sin x) 2
 arcsin x

5(1 2)
25
 152  53
Notice that the 5
multiplies every term
on the top. In our
algebraic example, if
we are to cancel, the
same must be true.
There must be a factor
which multiplies every
term on the top and
the same factor which
multiplies every term
on the bottom.
The rule for equations
is: whatever you do to
the right-hand side you
must also do to the
left-hand side. If you
multiply the right by
(3x+1) you must also
multiply the left by
(3x+1). That means
every term on the left
and right gets
multiplied by (3x+1).
For the first term then,
the x is multiplied by
(3x+1) and divided by
(3x+1): a common
factor so you can
cancel.
Here the notation is
meant to distinguish
between “take the sine
of x and then square
the answer” and “take
x, square it and find
the sine of the
answer”. Only the first
is correct
This notation is used to
mean the inverse
operation of taking the
sine; the arcsine or
inverse sine. It does
not mean the
reciprocal of the sine
(known as the cosec)
even though in other
cases the power -1
does mean “one over”.
This is really confusing
and inconsistent – so
watch out.
29
Approaching mathematical problems systematically
4.3
Getting useful help
Skills
In maths based subjects, the new material builds on your previous studies, so
anything you don't understand now will make future material difficult to
understand.
Get help as soon as you need it. Don't wait until a test is near.
Use the resources you have available

Ask questions in class. That way you get help and stay actively
involved in the class.

Attend the tutorials. Lecturers like to help students who want to help
themselves.

Ask friends, members of your study group, or anyone else who can
help. The classmate who explains something to you learns just as
much as you do; he/she must think carefully about how to explain the
particular concept or solution in a clear way. So don't be reluctant to
ask a classmate.

All students need help at some point, so be sure to get the help you
need.
Don't be afraid to ask questions. Any question is better than no question at all (at
least your lecturer/tutor will know you are confused). But a good question will
allow your helper to quickly identify exactly what you don't understand.
Not very helpful comment:
Likely Outcome
"I don't understand this section."
The best you can expect in reply to such a
remark is a brief review of the section, which
may miss out the particular thing(s) which
you don't understand.
Good comment:
Likely Outcome
I don't understand why f(x + h) doesn't equal
f(x) + f(h)."
This is a very specific query that will get a
very specific response and hopefully clear up
your difficulty.
Okay question:
Likely Outcome
How do you do question 17?
Someone may tell you how to do it, but then
you will learn nothing about the problem
solving process. Alternatively they will give a
brief hint that may solve you difficulties or
may not.
Better question:
Likely Outcome
Can you show me how to get started on
question17?
Someone can give you a targeted hint and
then let you try to finish the problem on your
own.
Good question:
Likely Outcome
This is how I tried to do question 17. What
went wrong?
The focus of attention and explanation is on
your thought process and your
misconceptions.
As soon as you get help with a problem, try to work through another similar
problem by yourself. This reinforces your understanding.
You control the help you get. Helpers should be coaches. They should encourage
you, give you hints as you need them, and very occasionally show you how to do
problems. But they should not, nor be expected to, actually do the work you
need to do. They are there to help you work out how to learn maths for yourself.
30
You will get more from tutorial sessions if you follow these guidelines:

When you go to see your lecturer, your study group or to a tutorial, have a
specific list of questions prepared in advance. You should run the session
as much as possible.

Do not allow yourself to become dependent on a tutor. The tutor cannot
take the exams for you. You must take care to be the one in control of
tutoring sessions.

You must recognize that sometimes you do need some coaching to help
you through, and it is up to you to seek out that coaching.
4.4
Checking your work
When you have finished a problem you need to try to check if you have made any
errors.

Check your reasoning

Check the algebraic or arithmetic manipulation
Often people talk about “going over” their solution. When this means looking back
at each line to see if it appears right it is often a very ineffective way to check your
work. After all, if you made a mistake the first time, you are quite likely to make it
again the next time.
Identify an independent way to check you work – you could:

substitute your answer back into the first line of the problem, or

work backwards from the answer statement to the problem
statement, or

put it away for an hour or two and then go back and try it again – do
you get the same answer? Getting the same answer the second
time can increase your confidence in your answer. Getting different
answers tells you at least one of them is wrong so it is worth your
while to take some time to consider why.
Just as it isn’t easy to find a solution method in the first place, it isn’t always easy
to see how to check your answer independently, but with practice you can train
yourself to be better at devising an appropriate method. You should get in the habit
of thinking of a check for every problem you solve and carrying it out.
31
Approaching mathematical problems systematically
Skills
Problem
Answer
How can you
check it?
Was the answer
right?
(you don’t need to
solve the whole
problem again!)
Solve for x and y:
2x + 3y = 1
x – 4y = 17
x=5, y=-3
Solve for x:
x2 – x – 6 = 0
x = 3 or x = -2
Find:
 (x
2
 2 x  3)dx
A group of people
agree to pay equal
shares to rent a
cottage for the
weekend. Each pays
£15. If there had been
another four people in
the group then the cost
per head would only
have come to £10. How
many people were
there in the group?
4.5
x3
 x 2  3x  C
3
8 people
Putting the numbers in
In many problems you have a choice whether to work with symbols to begin with
and to put numbers in later or whether to work with numbers from the start.
Which of the two solutions below might be most useful when you came to revise
the topic?
32
A hinged trapdoor of mass 15 kg and length 1 m is to be opened by
applying a Force F at 90o to the door surface at the opposite end to the
hinge. Calculate the magnitude of the force F. Assume the acceleration
due to gravity is 10 ms-2
Using Symbols first:
Using numbers first:
Let the mass of the door be m, then the weight
of the door is mg where g is the acceleration
due to gravity and let l be the length of the
beam
If mass of door is 15 kg, weight of door can be
taken as 150 N
F
hinge
l
2
F
hinge
0.5m
l
2
mg
Taking moments about the hinge:
0.5m
150 N
Taking moments about the hinge:
l
mg   F  l  0
2
mg
F
2
150  0.5  F  0
75  F  0
F  75N
Then substituting for m and g:
F
15  10
 75 N
2
Now look at the problem below and compare it with the problem above. You should
be able to see the similarities even if you haven’t studied this topic before.
A uniform beam of length 2 m is attached to a wall at one end by a hinge. The
mass of the beam is 10 kg. The beam is supported at the other end by a rope
which is attached to the ceiling vertically above the end of the beam. Find the
tension in the rope. Assume the acceleration due to gravity is 10 ms-2
Using Symbols first:
Using numbers first:
Let the mass of the beam be m, then the
weight of the beam is mg. Let g be the
acceleration due to gravity and let l be the
length of the beam
If mass of beam is 10 kg, weight of beam can
be taken as 100 N
T
hinge
T
hinge
1m
l
2
mg
l
2
1m
100 N
Taking moments about the hinge:
Taking moments about the hinge:
l
mg   T  l  0
2
mg
T
2
100 T  2  0
100  2T  0
T  50N
Then substituting for m and g:
T
10  10
 50 N
2
33
Approaching mathematical problems systematically
Skills
Continuing to use the symbols until close to the end highlights the similarities
between the two problems and allows you to see patterns easily. The more
patterns you can spot, the less you need to memorise methods. It is much harder
to spot the similarities and patterns when you put the numbers in early in the
solution.
4.6
Annotating your solution
Solving problems is very satisfying in its own right, but you are studying for a
qualification, and eventually you will probably need to complete an assessment.
You can help yourself prepare for assessment if you take a little extra time to
think about your solutions as you find them and to make notes.
Often, you are so excited to find a good method, that it is tempting to scribble
down the minimum amount to check your solution will work and then to hurry on
to the next problem. When you revise your work, this is often very hard to
decipher.
Make sure you write down what you are doing at each stage. This doesn’t need to
mean a lot of writing. Just a quick note will do.
Which solution of this problem will be best to revise from?
Solve for x and y
2x + 3y = 1
x – 4y = 17
Solution 1
2x + 3y = 1
x – 4y = 17
x  17  4 y
2(17  4 y )  3 y  1
34+11y = 1
11y = -33
y = -3
x=5
Solution 2
2x + 3y = 1
x – 4y = 17
(1)
(2)
Rearranging (2) gives:
x  17  4 y (3)
Substituting (3) into (1) gives:
2(17  4 y )  3 y  1
then simplifying and rearranging gives:
34+11y = 1
11y = -33
y = -3
Substituting the value for y back into (3)
gives:
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x=5
So the solution is x = 5 and y = -3
It doesn’t take much extra time to make the notes and in the long run it saves
you time trying to puzzle out what you meant when you come back to the
problem in the future.
Once you have solved a problem, take a moment to reflect on the solution.



Was anything especially hard?
Did any particular step cause you a problem?
How was it different from the previous problem?
Taking a moment to note these things down next to your solution can really help
when it comes to revision.
The solution you write is there for you to learn from. Don’t be scared to write
notes on it in a way that helps you.
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Approaching mathematical problems systematically
Skills
Would you find the notes below useful for revision or not?
Solve for
x
x
4x

6
3x  1 2 x  5
Multiplying by
x
3x + 1:
Remember to multiply every term on
both sides of equation by the same
thing. Then first term can be
simplified by cancelling.
4 x(3 x  1)
 6(3 x  1)
2x  5
2x – 5:
x(2 x  5)  4 x(3x  1)  6(3x  1)( 2 x  5)
Multiplying by
Multiplying out brackets:
2 x 2  5x  12 x 2  4 x  6(6 x 2  15x  2 x  5)
Made a mistake first
time – remember: minus
sign operates on
everything inside
bracket.
Rearranging and collecting like terms:
46 x 2  69 x  30  0
Finding roots of equation using quadratic formula:
x
69  10281
92
Be careful – several steps
summarised between previous
line & here – safer to write
them all out in an exam!
And simplifying:
x = 1.85 or x = -0.352
In the end it’s just a quadratic equation, but it was harder than the others on the sheet because I had to
rearrange it to start with. It didn’t look like a quadratic and I found the fractions a bit tricky to sort out
– perhaps some revision on fractions would be a good idea before the exam!
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5.0
Key Points
This guide can’t give you a recipe for solving every problem you meet, but if you
follow the advice, it can help you to take a systematic approach to solving
problems. The key points can be summarised as a list of questions you can ask
yourself as you tackle any new problem.

What am I trying to find?

Do I understand all the words in the problem?

What information am I given?

What other things do I know that might be relevant?

What strategies am I going to try out?

Am I making progress towards the answer?

Have I used all the information?

Can I check my answer?

Do I need to get help?

Do I need to note any special points to help with my revision later?
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