How to write better Directed Investigations in
maths
A guide-book.
S. Parker & D. Goodwin CCSS 2002.
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The aim of this booklet is to help you write better mathematics investigations and
projects.
WHAT IS A DIRECTED INVESTIGATION?
A Directed Investigation is a report describing the mathematical processes that you
have used to solve problems.
A Directed investigation will generally follow the format of;
(1) Introduction
(2) Mathematical Working
(3) Conclusion
(1)The ‘Introduction’ has information on what the report is about and what problems
need to be solved. You need to be clear about the main aims of your investigation.
These should be simply stated in your own words.
(2)The middle section will have all the mathematical working required to solve the
problem. This section may also contain tables and graphs to visually show results.
(3)The ‘Conclusion’ will contain discussions about the results that have been obtained
in the middle section.
The reader should be able to read your introduction and then be able to go straight to
your conclusion section and be able to know exactly what the Directed Investigation
has been about, and what results have been obtained. In the conclusion you are
summarising the main results from the middle section. Someone who may not be able
to follow your mathematical working in section 2 should however be able to read your
report and understand what it is that you have discovered.
In summary:
The Introduction will describe what the Investigation is about, and what you will be
solving. It will also briefly outline how you intend to go about doing it.
The middle section is all of the mathematical working.
The Conclusion summarises your report giving information about the problem that
you had to solve, the mathematical processes used to solve the problem, and
discussion on how you solved the problem.
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A Directed Investigation has the following layout:
The Introduction.
This must be placed at the start.
Where we state the aims or goals, and briefly
discuss how we will reach our goals.
We outline the problem as well.
The main body or middle part where we do most of the actual
mathematics. This is where you present data or number facts
based on what you have collected or found out and where you
think through the problem and answer the main questions. You
might solve equations or work on diagrams or graphs
The Conclusion section.
This must be placed at the end after the middle section.
Here we try to summarise our main findings from the
middle part and we see if we can apply what we found to
other cases. This is called generalising our results.
GATHERING INFORMATION TO WRITE AN INTRODUCTION
AND CONCLUSION:
Answer any of the following questions that are relevant to your Investigation. Make
sure all questions are answered in full sentence form. Use your responses to these
questions to help you gather information to construct an Introduction and Conclusion
for your Directed Investigation.
Introduction:
What is the aim of the Investigation?
How will you try to achieve the aim?
What mathematical processes will you use?
What forms of technology will you use in your work?
How will you be presenting your work? for example tables, graphs etc.
How will you gather information?
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How will you show comparisons in your work?
Answer the following questions after you have completed the mathematical working in the middle
section.
Conclusion:
Explain the process you used to investigate the aim of the investigation.
What results have you established?
Explain the mathematical processes that you have used.
Can you draw any conclusions from your results?
How did you solve the problem?
What problems if any, did you encounter in your working?
What conclusions could you draw from the information presented in tables and or
graphs?
What relationships have you established?
What comparisons have you established?
What have you learnt and discovered from this investigation?
Use your responses to the above questions as a starting point only to help
you construct an Introduction and Conclusion.
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INTRODUCTION ACTIVITIES
NUMBER SEQUENCES
73 – 21 – 2 is a number sequence with three terms which have been constructed
from the following rule;
Start with a two-digit number. The next number is the product of the digits of the
previous number. Stop when you get to a one-digit number.
Use the following terms to fill in the missing words for the introduction for the above
investigation.
TWO - DIGIT
PRODUCT
NUMBER SEQUENCES
DIRECTED INVESTIGATION
SEQUENCE
ONE - DIGIT
Introduction:
In this …………………………………………..I will be required to find
……………………………..consisting of three terms. The first number I will start
with will be a ……………………….number. The next number will be the
…………………of the digits of the number I started with. I will continue the
…………………by finding the product of the previous number until I finish with a
………………….number.
GOLDBACH’S CONJECTURE
Firstly find out the meaning of the word conjecture.
The mathematician Goldbach suspected that every even number could be written as
the sum of two prime numbers.
e.g. 4=2+2
6=3+3
8= 5+3
10 = 3+7
12 = 5+7
See if you can find two prime numbers that add up to 14, 16, 18, 20, 22, 24.
Does this mean that you have proved Goldbach’s conjecture?
Complete the following sentences to construct an introduction for the above
Investigation.
Introduction:
For this Directed Investigation, I will be investigating………………………………
Goldbach was a ……………………………… who suspected that
………………………………………………………………………………………..
I will need to find…………………………………………………..
I will use my number skills and show all calculations to establish whether I have
proved …………………………………………….
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After you have finished your main mathematical working out it’s time to start writing
your report. Here is some help to get you started.
Introduction Sentence Starters
An Introduction is a paragraph at the beginning of the DI consisting of approximately
5 to 8 sentences. The Introduction discusses the aim of the DI, the mathematical
processes that will be used, and how you will present your information.
Below is a set of possible sentence starters.
In this Directed Investigation I will ……………
I will present my findings in the form of tables and graphs to show
……...………..……………………………
I will be using …………………to show……………………….
This Directed Investigation will involve …………………………………
I will use mathematical processes to …………………………………….
I will present my information on a computer using…………..
I will be investigating …………………………and I will be gathering information
from……………………………………..
I will present the data I have collected in ……………….
This investigation involves …………………………
I will show in this investigation how mathematical processes such
as…………………………….. can be used to solve the problem of ………………….
I will use the graphics calculator to ………………………………
From the mathematical processes used I will summarise my findings in
………………………….
I will be investigating ……………………and I will gathering information
from…………………..
I will present the data I have collected in ………………..
In this Investigation I used Statistics to…………………..
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An eight point checklist for your introduction section.
(tick each one after you’ve done it).
1
2.
3.
4.
5.
6.
7.
8.
Have you stated clearly the aim of the investigation?
Have you discussed how you will investigate the problem?
Have you described the mathematical processes you will use?
Have you described how you will present your information? For
example using tables graphs etc.
Have you discussed the processes will you use to draw
comparisons?
Have you discussed the possible relationships or conclusions
that you hope to establish from your results?
Have you noted all the specific questions that need answering?
Have you noted the marks scheme for each question or part? The
parts with more marks will need more work from you.
The Middle section.
This will be a quite different for each DI. Some DI’s are more directed than others
and have lots of small parts for you to answer in a step-by-step way. If you get stuck
make sure you talk to your teacher to get some helpful hints well before the due date
deadline.
What follows is an actual directed investigation so you can get an idea of what is
required.
An example DI showing the “middle section”.
“The Chicken Problem.
A year 10 maths directed investigation.”
The problem to be solved. Kim has 40 metres of fencing and wants to use it to form a
rectangular cage for some chickens. She wants to get the biggest area possible from
her piece of fence. The fence will be joined to a brick wall as shown in the diagram
below: Your task is to find the dimensions of the best possible shape for the cage, and
the maximum area that results.
existing wall
xm
xm
ym
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The Chicken Problem ( an example report)
Introduction
The aim of this investigation was to find out the length and breadth which
produced the biggest area for a chicken cage in the shape of a rectangle. To do
this I used the formula Area = length x breadth or A = L x B and then I made up
an EXCEL spreadsheet that used this. I used the spreadsheet to make a table
of values which increased the width by 1m each time, and also a graph based upon
this table of values.
Middle section:
1.
Firstly if we let x represent the breadth or width and we let y represent the
length then we can see that the total distance around the fence or its perimeter is
Perimeter = x + x + y
= 2x + y
given by simply adding the three sides
thus 40 = 2x + y
From this equation we get y = 40 - 2x
2.
We know that the area = L x B so this becomes
.
A = y x = (40 - 2x) . x
2
= 40x -2x
3.
Next I made up a spreadsheet which used this formula. The spreadsheet
table is shown on the next page and then there are formulas shown as well.
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A
B
The Chicken Problem
2 An
example
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a DI
area produced
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K
A
The Chicken
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width = x
0
=A5+1
=A6+1
=A7+1
=A8+1
=A9+1
=A10+1
=A11+1
=A12+1
=A13+1
=A14+1
=A15+1
=A16+1
=A17+1
=A18+1
=A19+1
=A20+1
=A21+1
=A22+1
=A23+1
=A24+1
B
Problem
example
C
of
D
E
a DI
area produced
=40*A5-2*A5^2
=40*A6-2*A6^2
=40*A7-2*A7^2
=40*A8-2*A8^2
=40*A9-2*A9^2
=40*A10-2*A10^2
=40*A11-2*A11^2
=40*A12-2*A12^2
=40*A13-2*A13^2
=40*A14-2*A14^2
=40*A15-2*A15^2
=40*A16-2*A16^2
=40*A17-2*A17^2
=40*A18-2*A18^2
=40*A19-2*A19^2
=40*A20-2*A20^2
=40*A21-2*A21^2
=40*A22-2*A22^2
=40*A23-2*A23^2
=40*A24-2*A24^2
=40*A25-2*A25^2
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5.
The formulas in my spreadsheet work like this. In cell A6 the breadth is
increased each time by 1 cm. In cell B5 the formula is =40*A5 – 2*A5^2 which
calculates the area according to the equation A = 40x – 2x2.
From the table we can see that the biggest area is 200 m2 and occurs when
6..
x = 10 metres.
Highest
point
The turkey problem
250
area of cage in square metres
200
150
100
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width (x) in metres
7.
From the EXCEL graph we can see that the highest point is 200m2 when
x = 10 metres
8.. The actual sides for the biggest area are breadth (x) = 10 metres, and length y
= 20 metres., 2 x x = 20 metres plus the y =20 gives 40 metres total which checks.
Conclusion .
In this investigation I have found out that the biggest area for 40 metres of fencing
would be 200 m2 and that this happens when the breadth is 10 metres and the
length is 20 metres. I have learnt how to use EXCEL to help me in a maximisation
problem. A general rule would seem to be that to get the maximum area we should
make the length twice the size of the breadth so that for say 80 metres of fence I
would make the length 40 metres and the breadth 20 metres.
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The Conclusion section.
To conclude something means that you are saying something important about what
you have found out. For example you may have found out or discovered a pattern in a
table of values, or you may have discovered an equation or rule that fits with most of
your data. You may have noticed how a graph changes its shape or position, and how
this is related to changes in its equation.
Some typical sentences that we use in a conclusion section are shown below:
(1)
In this investigation I have noticed that if we double the length of the sides of
the rectangle then the area is….
(2)
From table 2 I was able to see that if the value for a in y = ax2 is increased then
the shape of the parabola becomes much thinner. Notice how much thinner y = 4x2 is
compared to y = 2x2.
(3)
The formula that I found that best fitted the data was….
(4)
I have discovered that there is a rule linking the position of the parabola and
its equation.
(5)
Every time the original equation had a zero or x-intercept, we saw that this
was associated with a gap in the graph of its reciprocal.
(6)
From the main table of measurements I have noticed that the numbers in the
last column C2 are equal to the sum of the numbers in the two previous columns A2
and B2.
(7)
In summary then I have found out that for nearly all of the products it is
cheaper to buy them in bulk quantities. There were however two big exceptions.
These were…
(8)
I presented my findings in the form of…..
Usually the verbs used in a conclusion section are in the past tense because you are
looking back over all of your work.
Some good verbs to use are listed below
Present tense
notice
observe
associate
Past tense
noticed
observed
associated
measure
discover
measured
discovered
Meaning
you saw something
you saw something carefully
something was linked or
connected
how big something was
what you found out or
uncovered
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A conclusion section is also where we talk about how there might have been some
problems or even errors in our investigation. Often errors occur when we have not
been able to measure something as accurately as we would wish. So if one or two bits
of information don’t fit the main pattern you should discuss why this has happened.
Another important part of the conclusion is where you try and generalise to other
situations. For example a student has been investigating rectangular shaped fences for
sheep, and has concluded that for 1000 metres of fence, the best shape to make to get
the greatest area, is a square shape. The student could also generalise that this would
also be true for 2000 metres of fence, or for any other length of fence, not just a 1000
metre one. This is also called conjecturing.
Finally in a conclusion section you make statements about what you personally have
learnt, and whether the investigation itself could have been done better. For example
you might state, “I have learnt that by using a table of values it is easier to see the
main patterns. This DI could have been improved if we had been able collect prices
from more than one supermarket and over a longer time period than just two weeks.”
How long should your conclusion be? Well that depends on the task and what your
teacher expects. Have a close look at the marking scheme. Usually there is a mark per
point made in a conclusion section. So if it is worth 6 marks, try to make 6 points or
separate statements. A conclusion section most times would be about half a page long,
sometimes a bit more, sometimes a bit less.
A 10 point checklist for your conclusion section. (tick each one
after you’ve done it).
1.
2.
3.
4.
5.
6.
7.
Have you presented all of your main findings or discoveries
Have you referred in particular to the tables, graphs, drawings or other
data in the main body?
Have you discussed or explained any errors or exceptions
Have you been able to generalise to other similar situations?
Have you stated what you have learnt?
Have you discussed how the DI could be changed for the better?
Have you spell checked your work? The button
in word does this
Ask someone else if unsure about the spelling of any words.
8. For WBLA have you checked that you have written at least 250 words
9. Have you checked that your sentences start with capital (A,B, C etc)
letters, and finish with full stops.●
10. Have you asked someone else to read it in order to check that the
English reads fluently, and then actually used their improvements?
To write mathematics such as equations in a document in a word processor like
WORD is very time-consuming. You probably are better off just writing things by
hand or perhaps combining hand written equations and graphs by just leaving spaces
in your main WORD document. On the CCSS computer network we have got two
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programs called FX graph and FX draw which can both be useful for inserting graphs
and maths diagrams into a WORD document. You can also use equation editor as an
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inserted object in order to put complicated equations such as ∫ 3 x − 2 into your
x
document.
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