IB Mathematical Studies
An InThinking workshop for teachers new to the
course
Workshop leader - Jim Noble
Table of Contents
Course Overview ......................................................................................................................................... 4
Aims and Objectives .............................................................................................................................................. 4
FRIDAY 21 October 2011 ................................................................................................................................................. 4
SATURDAY 22 October 2011 .......................................................................................................................................... 4
SUNDAY 23 October 2011 ................................................................................................................................................ 5
Session 1 – Introductions ......................................................................................................................... 6
The IB Learner Profile.......................................................................................................................................... 7
What Documents? .................................................................................................................................................. 8
Session 2 – Popular Mathematics ....................................................................................................... 10
The Mathematics of War – Sean Gourley ..................................................................................................... 11
Mamogram Math – Diagnostic tests .............................................................................................................. 13
Chances Are ............................................................................................................................................................ 16
The False Positive Paradox .............................................................................................................................. 21
Session 3 – Internal Assessment......................................................................................................... 23
Example Projects.................................................................................................................................................. 24
Example Project 1 ................................................................................................................................................ 24
Example Project 2 ................................................................................................................................................ 25
Example Project 3 ................................................................................................................................................ 26
Marking Projects .................................................................................................................................................. 27
Session 4 – Internal Assessment 2 ..................................................................................................... 34
The Project Planner ............................................................................................................................................ 35
Choosing a Theme ............................................................................................................................................................ 37
Information ......................................................................................................................................................................... 39
Specific Potential............................................................................................................................................................... 41
Planning and Scheduling ............................................................................................................................................... 42
Reflection ............................................................................................................................................................................. 44
Checklist ............................................................................................................................................................................... 45
Session 5 – Exams .................................................................................................................................... 48
Paper 1 - Questions and Marking ................................................................................................................... 49
Paper 2 – Questions and Marking .................................................................................................................. 58
Common Errors .................................................................................................................................................... 64
Session 6/7 - Task Design ..................................................................................................................... 65
Visual Sequences .................................................................................................................................................. 66
Visual Representations................................................................................................................................................... 66
Common difference .......................................................................................................................................................... 67
A New Sequence ................................................................................................................................................................ 67
Finding terms ..................................................................................................................................................................... 68
Defining Variables ............................................................................................................................................................ 68
General Expressions for Sequences .......................................................................................................................... 69
What makes a good task? .................................................................................................................................. 70
Designing Activities............................................................................................................................................. 71
Stimulus 1 - Mathematics of War ............................................................................................................................... 72
Stimulus 2 - Sexy Maths – When it pays to play the odds ................................................................................ 73
Stimulus 3 - 100 metre runners and their split times ....................................................................................... 75
Stimulus 4 - Coordinate Geometry and the gradients of perpendicular lines ......................................... 77
Stimulus 5 - Exchange Rates ........................................................................................................................................ 77
Stimulus 6 - Modeling Temperature Fluctuations .............................................................................................. 77
Session 8 - Resources ............................................................................................................................. 78
Session 9 – Graphical Display Calculator......................................................................................... 79
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Calculator Skills - What should we know? .................................................................................................. 80
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Course Overview
Aims and Objectives
This workshop is designed for teachers who are new to the IB Maths Studies course and aims to
help delegates familiarise themselves with the make up and philosophy of the course. We will look
at the administrative and creative elements involved with delivering this course and delegates
should go away with lots of good ideas for their classrooms.
Maths Studies will start a new syllabus in September 2012. For many of us this involves teaching
both simultaneously for a year. A pre-course questionnaire will help find out about delegates
priorities, but the workshop will consider both syllabi as a running theme throughout.
Provisional Agenda
FRIDAY 21 October 2011
Session 1, 9:00-10:30, Introductions
To each other, to the course, to the IB learner profile and philosophy. This session should help
delegates to have a sound overview the requirements of the course.
Session 2, 11:00-12:30, Popular Maths - Maths in the media
Making use of popular maths and context. Bringing Maths Alive and making it topical.
Session 3, 13:30-15:00, Internal Assessment 1
Introduction to the Internal Assessment element of the course, marking and moderating.
Session 4, 15:15-16:45, Internal Assessment 2
Following on from the previous session, we will look at getting projects started, helping
students with ideas for projects and management strategies.
SATURDAY 22 October 2011
Session 5, 9:00-10:30, Exams
What are we working towards? Familiarisation with the paper 1 and paper 2 formats for the
external assessment at the end of the course.
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Session 6, 11:00-12:30, Task design
Teaching the course! The most important thing we do. In these sessions we will be looking at
good ideas for teaching the syllabus. Delegates will design tasks in the second session for us
all to take away.
Session 7, 13:30-15:00, Task design
As above - Teaching the course! The most important thing we do. In these sessions we will be
looking at
good ideas for teaching the syllabus. Delegates will design tasks in the second session for us
all to take away.
Session 8, 15:15-16:45, Resources
Books, websites, software and web 2.0. We will have a look at the resources we have at our
disposal to support the teaching of the course and how they can be used.
SUNDAY 23 October 2011
Session 9, 8:30-10:00, The Graphical Display Calculator
Getting familiar with this important tool for Maths Studies. What can be done? How is it done?
Session 10, 10:30-12:00, Roundup
This session is reserved for addressing issues we have not been able to during the workshop,
answering remaining questions and a final splurge on good ideas!
It is strongly recommended that delegates bring a wireless enabled laptop (and check the details
with their technicians) to take a full part in the course. Having reserved a place on the workshop,
delegates should receive a questionnaire to gather information about their backgrounds and
needs for the course. The course outline may vary from the above in response to information
gathered from the questionnaires.
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Session 1 – Introductions
Aim
The aim of this session is to introduce ourselves to each other and to the Mathematical Studies
course.
Session Outline
‘Human Progressions’ - Short mathematical activity to help delegates meet each other and get the
ball rolling.
‘Introductions’ – Taking a moment to introduce ourselves to each other at the start of the
weekend and share our thoughts about teaching Maths Studies ahead of the weekends activities!
‘About Maths Studies’ – Time designed to familiarise ourselves with the Maths Studies course and
its philosophy
Included in this guide
The IB learner Profile
A guide to IB Maths Studies Documentation
All other resources related to this session can be found on the course wikispace.
Additional resources required
Delegates will need both the subject guide and the TSM for projects to refer to for this session
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The IB Learner Profile
The aim of all IB programmes is to develop internationally minded people who, recognizing their
common humanity and shared guardianship of the planet, help to create a better and more
peaceful world.
IB learners strive to be:
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Inquirers
Knowledgeable
Thinkers
Communicators
Principled
Open-minded
Caring
Risk-takers
Balanced Reflective
 They develop their natural curiosity. They acquire the skills necessary to conduct inquiry and
research and show independence in learning. They actively enjoy learning and this love of learning
will be sustained throughout their lives.
 They explore concepts, ideas and issues that have local and global significance. In so doing, they
acquire in-depth knowledge and develop understanding across a broad and balanced range of
disciplines.
 They exercise initiative in applying thinking skills critically and creatively to recognize and approach
complex problems, and make reasoned, ethical decisions.
 They understand and express ideas and information confidently and creatively in more than one
language and in a variety of modes of communication. They work effectively and willingly in
collaboration with others.
 They act with integrity and honesty, with a strong sense of fairness, justice and respect for the dignity
of the individual, groups and communities. They take responsibility for their own actions and the
consequences that accompany them.
 They understand and appreciate their own cultures and personal histories, and are open to the
perspectives, values and traditions of other individuals and communities. They are accustomed to
seeking and evaluating a range of points of view, and are willing to grow from the experience.
 They show empathy, compassion and respect towards the needs and feelings of others. They have a
personal commitment to service, and act to make a positive difference to the lives of others and to
the environment.
 They approach unfamiliar situations and uncertainty with courage and forethought, and have the
independence of spirit to explore new roles, ideas and strategies. They are brave and articulate in
defending their beliefs.
 They understand the importance of intellectual, physical and emotional balance to achieve personal
well-being for themselves and others.
 They give thoughtful consideration to their own learning and experience. They are able to assess and
understand their strengths and limitations in order to support their learning and personal
development.
© International Baccalaureate Organization 2006
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What Documents?
The following is intended as a brief and useful guide to exactly what documents Maths Studies
teachers should have read and/or have to hand, what they are useful for and at what point they
are needed. These are documents published by the IB and are all available through the online
curriculum centre. All IB teachers should be registered with the online curriculum centre and have
the codes and passwords they need to access it.
The Subject Guide
This is the main port of call for all information about the course. Teachers should be very familiar
with the contents of this guide and have it always close to hand.
The Information Booklet
This is the formula booklet that students are given in their exams and contains a lot of very useful
formulae and general information about the different topics. As such, students are not expected
to memorise this information and expected more to know what to do with it. This is an invaluable
tool for students and teachers. Students are never expected to be without this or their calculators
and I always insist that they have both with them at all times. Total familiarity with this document
can make a big difference to performance overall!
TSM - Projects
This is long (188 pages) and useful document to be familiar with when preparing and running the
internal assessment part of the course. Mostly it is useful because it is published by the IB
themselves so it is not a question of opinion! It includes the following sections;
 Benefits of project work
 Related skills and strategies
 Choosing a topic
 Integration into the course of study
 Use of Technology
 Supervision of students
 Examples of students work
 Examples of projects
 Assessment criteria
 Assessment of projects
 FAQs
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TSM – GDC
This document is invaluable, particularly to those less familiar with the Graphical display
calculators. The document gives examples of the main features in use for the Maths Studies
course along with examples taken from the bank of specimen questions. It shows how to complete
these examples on either the Texas Instruments or Casio range of calculators.
Grade Descriptors
This is a short document that gives an insight into the meaning of the IB grades 1 to 7. It helps to
have and understand a broader set of descriptors and could be particularly useful if you regularly
use IB grades to grade work through out the course. It is also good to share with students.
Specimen Papers and Mark schemes
These are as the title suggests and are, as such a very useful resource for understanding the sorts
of questions that students will be required to answer and the mark schemes that judge how well
they have done so. These are freely available through the OCC.
Its worth noting here that a better source of such resources is the IB Questionbank software
published and provided by IB. This is something teachers should not really be without but does not
come free!
Guide to Procedures
This is a general guide across subjects to the details of procedures for running the course and for
the assessment. It is generally of more concern to IB coordinators, but it is helpful to know that it
is there and to know a little about some of these important details.
Procedures - Group 5
This is very pertinent to Maths teachers and only holds information about the different procedure
for group 5 subjects. Significantly it has information about and copies of the paperwork required
for the Internal Assessment procedures. Definitely good to have to hand.
Subject Reports
After each examination session, the IB publish a subject report that gives invaluable feedback on
the performance in exams and the work submitted for Internal Assessment. This is based on the
whole cohort who took part in the session and therefore gives teacher s a great opportunity to
compare their own students and practice internationally. There are always some striking
observations and advice that can be used to inform and improve the practice of students and
teachers.
Curriculum Reviews
Publications relating to work being done on the regular curriculum review are published on the
OCC as well.
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Session 2 – Popular Mathematics
Aim
The aim of this session is to share with delegates some sources for ‘Popular Mathematics’ and to
look at how the course can be enriched by looking at ideas in context. The group should discuss
the possible advantages of using resources of this type in attracting the attention and motivation
of students. There should also be a focus on exactly how these ideas can be applied in the
classroom.
Session Outline
‘The Mathematics of War’ – A short video from ‘Sean Gourley’ shows how mathematicians have
begun to analyse the mathematics of war. How can this inspire students to learn and use
mathematics?
‘False Positives’ – Inspired by some articles in the news, we look at this topical application of
conditional probability.
‘The most important video’ – A short look at this series of videos talking about our inability to
understand the exponential function!
‘A look at sources’ – Where can we find sources on popular mathematics that help to keep
teachers and students up to date and inspired.
Included in this guide
Some items related to the Mathematics of war
A copy of articles from John Allen Paulos and Steven Strogatz on ‘False Positives’
A worksheet for students on False Positives
All other resources related to this session can be found on the course wikispace
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The Mathematics of War – Sean Gourley
As you are watching this video, you may want to note down some thoughts for discussion using
the following questions as prompts
What do we think of this study?
Does this have a place in the course? Wider reading?
How could you integrate a study like this?
Data from mathematics of War
Spreadsheet from http://www.iraqbodycount.org/database/ contains data for over 22000
separate attacks, the location, target, weapons used, reported casualties and the source of the
information.
Tables from http://www.icasualties.org/Iraq/index.aspx
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Mamogram Math – Diagnostic tests
The Way We Live Now
Mammogram Math
By JOHN ALLEN PAULOS
Published: December 10, 2009
In his inaugural address, Barack Obama promised to restore science to its “rightful place.” This has
partly occurred, as evidenced by this month’s release of 13 new human embryonic stem-cell lines.
The recent brouhaha over the guidelines put forth by the government task force on breast-cancer
screening, however, illustrates how tricky it can be to deliver on this promise. One big reason is
that people may not like or even understand what scientists say, especially when what they say is
complex, counterintuitive or ambiguous.
Data Source: USA Today/Gallup Poll, conducted among 284 women, Nov. 20-22.
Human Empire
As we now know, the panel of scientists advised that routine screening for asymptomatic women
in their 40s was not warranted and that mammograms for women 50 or over should be given
biennially rather than annually. The response was furious. Fortunately, both the panel’s concerns
and the public’s reaction to its recommendations may be better understood by delving into the
murky area between mathematics and psychology.
Much of our discomfort with the panel’s findings stems from a basic intuition: since earlier and
more frequent screening increases the likelihood of detecting a possibly fatal cancer, it is always
desirable. But is this really so? Consider the technique mathematicians call a reductio ad
absurdum, taking a statement to an extreme in order to refute it. Applying it to the contention
that more screening is always better leads us to note that if screening catches the breast cancers
of some asymptomatic women in their 40s, then it would also catch those of some asymptomatic
women in their 30s. But why stop there? Why not monthly mammograms beginning at age 15?
The answer, of course, is that they would cause more harm than good. Alas, it’s not easy to weigh
the dangers of breast cancer against the cumulative effects of radiation from dozens of
mammograms, the invasiveness of biopsies (some of them minor operations) and the aggressive
and debilitating treatment of slow-growing tumors that would never prove fatal.
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The exact weight the panel gave to these considerations is unclear, but one factor that was clearly
relevant was the problem of frequent false positives when testing for a relatively rare condition. A
little vignette with made-up numbers may shed some light. Assume there is a screening test for a
certain cancer that is 95 percent accurate; that is, if someone has the cancer, the test will be
positive 95 percent of the time. Let’s also assume that if someone doesn’t have the cancer, the
test will be positive just 1 percent of the time. Assume further that 0.5 percent — one out of 200
people — actually have this type of cancer. Now imagine that you’ve taken the test and that your
doctor somberly intones that you’ve tested positive. Does this mean you’re likely to have the
cancer? Surprisingly, the answer is no.
To see why, let’s suppose 100,000 screenings for this cancer are conducted. Of these, how many
are positive? On average, 500 of these 100,000 people (0.5 percent of 100,000) will have cancer,
and so, since 95 percent of these 500 people will test positive, we will have, on average, 475
positive tests (.95 x 500). Of the 99,500 people without cancer, 1 percent will test positive for a
total of 995 false-positive tests (.01 x 99,500 = 995). Thus of the total of 1,470 positive tests (995 +
475 = 1,470), most of them (995) will be false positives, and so the probability of having this
cancer given that you tested positive for it is only 475/1,470, or about 32 percent! This is to be
contrasted with the probability that you will test positive given that you have the cancer, which by
assumption is 95 percent.
The arithmetic may be trivial, but the answer is decidedly counterintuitive and hence easy to
reject or ignore. Most people don’t naturally think probabilistically, nor do they respond
appropriately to very large or very small numbers. For many, the only probability values they know
are “50-50” and “one in a million.” Whatever the probabilities associated with a medical test, the
fact remains that there will commonly be a high percentage of false positives when screening for
rare conditions. Moreover, these false positives will receive further treatments, a good percentage
of which will have harmful consequences. This is especially likely with repeated testing over
decades.
Another concern is measurement. Since we calculate the length of survival from the time of
diagnosis, ever more sensitive screening starts the clock ticking sooner. As a result, survival times
can appear to be longer even if the earlier diagnosis has no real effect on survival.
Cognitive biases also make it difficult to see the competing desiderata the panel was charged with
balancing. One such bias is the availability heuristic, the tendency to estimate the frequency of a
phenomenon by how easily it comes to mind. People can much more readily picture a friend dying
of cancer than they can call up images of anonymous people suffering from the consequences of
testing. Another bias is the anchoring effect, the tendency to be overly influenced by any initially
proposed number. People quickly become anchored to such a number, whether it makes sense or
not (“we use only 10 percent of our brains”), and they’re reluctant to abandon it. If accustomed to
an annual mammography, they’re likely for that reason alone to resist biennial (or even
semiannual) ones.
Whatever the role of these biases, the bottom line is that the new recommendations are
evidence-based. This doesn’t mean other right-thinking people would necessarily come to the
same judgments. To oppose the recommendations, however, requires facts and argument, not
invective.
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John Allen Paulos, professor of mathematics at Temple University, is the author most recently of
“Irreligion.”
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Chances Are
By STEVEN STROGATZ
Steven Strogatz on math, from basic to baffling.
Have you ever had that anxiety dream where you suddenly realize you have to take the final exam
in some course you’ve never attended? For professors, it works the other way around — you
dream you’re giving a lecture for a class you know nothing about.
That’s what it’s like for me whenever I teach probability theory. It was
never part of my own education, so having to lecture about it now is scary
and fun, in an amusement park, thrill-house sort of way.
Perhaps the most pulse-quickening topic of all is “conditional probability”
— the probability that some event A happens, given (or “conditional”
upon) the occurrence of some other event B. It’s a slippery concept, easily
conflated with the probability of B given A. They’re not the same, but you
have to concentrate to see why. For example, consider the following word problem.
Before going on vacation for a week, you ask your spacey friend to water your ailing
plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it
has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30
percent. (a) What’s the chance that your plant will survive the week? (b) If it’s dead when you
return, what’s the chance that your friend forgot to water it? (c) If your friend forgot to water it,
what’s the chance it’ll be dead when you return?
Although they sound alike, (b) and (c) are not the same. In fact, the problem tells us that the
answer to (c) is 90 percent. But how do you combine all the probabilities to get the answer to
(b)? Or (a)?
Naturally, the first few semesters I taught this topic, I stuck to the book, inching along, playing it
safe. But gradually I began to notice something. A few of my students would avoid using “Bayes’s
theorem,” the labyrinthine formula I was teaching them. Instead they would solve the problems
by a much easier method.
What these resourceful students kept discovering, year after year, was a better way to think about
conditional probability. Their way comports with human intuition instead of confounding it. The
trick is to think in terms of “natural frequencies” — simple counts of events — rather than the
more abstract notions of percentages, odds, or probabilities. As soon as you make this mental
shift, the fog lifts.
This is the central lesson of “Calculated Risks,” a fascinating book by Gerd Gigerenzer, a cognitive
psychologist at the Max Planck Institute for Human Development in Berlin. In a series of studies
about medical and legal issues ranging from AIDS counseling to the interpretation of DNA
fingerprinting, Gigerenzer explores how people miscalculate risk and uncertainty. But rather than
scold or bemoan human frailty, he tells us how to do better — how to avoid “clouded thinking” by
recasting conditional probability problems in terms of natural frequencies, much as my students
did.
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In one study, Gigerenzer and his colleagues asked doctors in Germany and the United States to
estimate the probability that a woman with a positive mammogram actually has breast cancer,
even though she’s in a low-risk group: 40 to 50 years old, with no symptoms or family history of
breast cancer. To make the question specific, the doctors were told to assume the following
statistics — couched in terms of percentages and probabilities — about the prevalence of breast
cancer among women in this cohort, and also about the mammogram’s sensitivity and rate of
false positives:
The probability that one of these women has breast cancer is 0.8 percent. If a woman has breast
cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does
not have breast cancer, the probability is 7 percent that she will still have a positive
mammogram. Imagine a woman who has a positive mammogram. What is the probability that
she actually has breast cancer?
Gigerenzer describes the reaction of the first doctor he tested, a department chief at a university
teaching hospital with more than 30 years of professional experience:
“[He] was visibly nervous while trying to figure out what he would tell the woman. After mulling
the numbers over, he finally estimated the woman’s probability of having breast cancer, given that
she has a positive mammogram, to be 90 percent. Nervously, he added, ‘Oh, what nonsense. I
can’t do this. You should test my daughter; she is studying medicine.’ He knew that his estimate
was wrong, but he did not know how to reason better. Despite the fact that he had spent 10
minutes wringing his mind for an answer, he could not figure out how to draw a sound inference
from the probabilities.”
When Gigerenzer asked 24 other German doctors the same question, their estimates whipsawed
from 1 percent to 90 percent. Eight of them thought the chances were 10 percent or less, 8 more
said 90 percent, and the remaining 8 guessed somewhere between 50 and 80 percent. Imagine
how upsetting it would be as a patient to hear such divergent opinions.
As for the American doctors, 95 out of 100 estimated the woman’s probability of having breast
cancer to be somewhere around 75 percent.
The right answer is 9 percent.
How can it be so low? Gigerenzer’s point is that the analysis becomes almost transparent if we
translate the original information from percentages and probabilities into natural frequencies:
Eight out of every 1,000 women have breast cancer. Of these 8 women with breast cancer, 7 will
have a positive mammogram. Of the remaining 992 women who don’t have breast cancer, some
70 will still have a positive mammogram. Imagine a sample of women who have positive
mammograms in screening. How many of these women actually have breast cancer?
Since a total of 7 + 70 = 77 women have positive mammograms, and only 7 of them truly have
breast cancer, the probability of having breast cancer given a positive mammogram is 7 out of 77,
which is 1 in 11, or about 9 percent.
Notice two simplifications in the calculation above. First, we rounded off decimals to whole
numbers. That happened in a few places, like when we said, “Of these 8 women with breast
cancer, 7 will have a positive mammogram.” Really we should have said 90 percent of 8 women,
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or 7.2 women, will have a positive mammogram. So we sacrificed a little precision for a lot of
clarity.
Second, we assumed that everything happens exactly as frequently as its probability suggests. For
instance, since the probability of breast cancer is 0.8 percent, exactly 8 women out of 1,000 in our
hypothetical sample were assumed to have it. In reality, this wouldn’t necessarily be true. Things
don’t have to follow their probabilities; a coin flipped 1,000 times doesn’t always come up heads
500 times. But pretending that it does gives the right answer in problems like this.
Admittedly the logic is a little shaky — that’s why the textbooks look down their noses at this
approach, compared to the more rigorous but hard-to-use Bayes’s theorem — but the gains in
clarity are justification enough. When Gigerenzer tested another set of 24 doctors, this time using
natural frequencies, nearly all of them got the correct answer, or close to it.
Although reformulating the data in terms of natural frequencies is a huge help, conditional
probability problems can still be perplexing for other reasons. It’s easy to ask the wrong question,
or to calculate a probability that’s correct but misleading.
Both the prosecution and the defense were guilty of this in the O.J. Simpson trial of 1994-95. Each
of them asked the jury to consider the wrong conditional probability.
The prosecution spent the first 10 days of the trial introducing evidence that O.J. had a history of
violence toward his ex-wife, Nicole. He had allegedly battered her, thrown her against walls and
groped her in public, telling onlookers, “This belongs to me.” But what did any of this have to do
with a murder trial? The prosecution’s argument was that a pattern of spousal abuse reflected a
motive to kill. As one of the prosecutors put it, “A slap is a prelude to homicide.”
Alan Dershowitz countered for the defense, arguing that even if the allegations of domestic
violence were true, they were irrelevant and should therefore be inadmissible. He later wrote,
“We knew we could prove, if we had to, that an infinitesimal percentage — certainly fewer than 1
of 2,500 — of men who slap or beat their domestic partners go on to murder them.”
In effect, both sides were asking the jury to consider the probability that a man murdered his exwife, given that he previously battered her. But as the statistician I. J. Good pointed out, that’s not
the right number to look at.
The real question is: What’s the probability that a man murdered his ex-wife, given that he
previously battered her and she was murdered by someone? That conditional probability turns
out to be very far from 1 in 2,500.
To see why, imagine a sample of 100,000 battered women. Granting Dershowitz’s number of 1 in
2,500, we expect about 40 of these women to be murdered by their abusers in a given year (since
100,000 divided by 2,500 equals 40). We can estimate that an additional 5 of these battered
women, on average, will be killed by someone else, because the murder rate for all women in the
United States at the time of the trial was about 1 in 20,000 per year. So out of the 40 + 5 = 45
murder victims altogether, 40 of them were killed by their batterer. In other words, the batterer
was the murderer about 90 percent of the time.
Don’t confuse this number with the probability that O.J. did it. That probability would depend on
a lot of other evidence, pro and con, such as the defense’s claim that the police framed him, or the
prosecution’s claim that the killer and O.J. shared the same style of shoes, gloves and DNA.
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The probability that any of this changed your mind about the verdict? Zero.
NOTES:
1. For a good textbook treatment of conditional probability and Bayes’s theorem, see:
S.M. Ross, “Introduction to Probability and Statistics for Engineers and Scientists,” 4th
edition (Academic Press, 2009).
2. The answer to part (a) of the “ailing plant” problem is 59 percent. The answer to part (b) is
27/41, or approximately 65.85 percent. To derive these results, imagine 100 ailing plants
and figure out (on average) how many of them get watered or not, and then how many of
those go on to die or not, based on the information given. This question appears, though
with slightly different numbers and wording, as problem 29 on p. 84 of Ross’s text.
3. The study of how doctors interpret mammogram results is described in:
G. Gigerenzer, “Calculated Risks” (Simon and Schuster, 2002), chapter 4. For more on the
O.J. Simpson case and a discussion of wife battering in a larger context, see chapter 8.
4. For many entertaining anecdotes and insights about conditional probability and its realworld applications, as well as how it’s misperceived, see:
J.A. Paulos, “Innumeracy” (Vintage, 1990);
L. Mlodinow, “The Drunkard’s Walk” (Vintage, 2009).
5. The quotes pertaining to the O.J. Simpson trial, and Alan Dershowitz’s estimate of the rate
at which battered women are murdered by their partners, appeared in:
A. Dershowitz, “Reasonable Doubts” (Touchstone, 1997), pp. 101-104.
6. Probability theory was first correctly applied to the Simpson trial by the late I.J. Good, in:
I.J. Good, “When batterer turns murderer,” Nature, Vol. 375 (1995), p. 541.
I.J. Good, “When batterer becomes murderer,” Nature, Vol. 381 (1996), p. 481.
Good phrased his analysis in terms of odds ratios and Bayes’s theorem, rather than the
more intuitive “natural frequency” approach presented here and in Gigerenzer’s book.
Good had an interesting career. In addition to his many contributions to probability theory
and Bayesian statistics, he helped break the Nazi Enigma code during World War II, and
introduced the futuristic concept now known as the “technological singularity.”
7. Here is how Dershowitz seems to have calculated that fewer than 1 in 2,500 batterers go on
to murder their partners, per year. On page 104 of his book “Reasonable Doubts,” he cites
an estimate that in 1992, somewhere between 2.5 and 4 million women in the United
States were battered by their husbands, boyfriends, and ex-boyfriends. In that same year,
according to the FBI Uniform Crime Reports, 913 women were murdered by their husbands,
and 519 were killed by their boyfriends or ex-boyfriends. Dividing the total of 1,432
homicides by 2.5 million beatings yields 1 murder per 1,746 beatings, whereas using the
higher estimate of 4 million beatings per year yields 1 murder per 2,793
beatings. Dershowitz apparently chose 2,500 as a round number in between these
extremes.
What’s unclear is what proportion of the murdered women had been previously beaten by
these men. It seems that Dershowitz was assuming that nearly all the victims were beaten,
presumably to make the point that even when the rate is overestimated in this way, it’s still
“infinitesimal.”
8. Good’s estimated murder rate of 1 per 20,000 women per year includes battered women,
so it was not strictly correct to assume (as he did, and as we did above) that 5 women out
of 100,000 would be killed by someone other than the batterer. But correcting for this
doesn’t alter the conclusion significantly, as the following calculation shows.
According to the FBI Uniform Crime Reports, 4,936 women were murdered in 1992. Of
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these murder victims, 1,432 (about 29 percent) were killed by their husbands or
boyfriends. The remaining 3,504 were killed by somebody else. Therefore, considering that
the total population of women in the United States at that time was about 125 million, the
rate at which women were murdered by someone other than their partners was 3,504
divided by 125,000,000, or 1 murder per 35,673 women, per year.
Let’s assume that this rate of murder by non-partners was the same for all women,
battered or not. Then in our hypothetical sample of 100,000 battered women, we’d expect
about 100,000 divided by 35,673, or 2.8 women to be killed by someone other than their
partner. Although 2.8 is smaller than the 5 that Good and we assumed above, it doesn’t
matter much because both are so small compared to 40, the estimated number of cases in
which the batterer is the murderer. With this modification, our new estimate of the
probability that the batterer is the murderer would be 40 divided by (40 + 2.8), or about 93
percent.
A related quibble is that the FBI statistics and population data given above imply that the
murder rate for women in 1992 was closer to 1 in 25,000, not 1 in 20,000 as Good
assumed. If he had used that rate in his calculation, an estimated 4 women per 100,000,
not 5, would have been murdered by someone other than the partner. But this still wouldn’t
affect the basic message — now the batterer would be the murderer 40 times out of 40 + 4
= 44, or 91 percent of the time.
9. A few years after the verdict was handed down in the Simpson case, Alan Dershowitz and
the mathematician John Allen Paulos engaged in a heated exchange of letters to the editor
of the New York Times. The issue was whether evidence of a history of spousal abuse
should be regarded as relevant to a murder trial, in light of probabilistic arguments similar
to those discussed in this post. Dershowitz’s letter to the editor and Paulos’s response
make for lively reading.
Thanks to Paul Ginsparg, Michael Lewis, Eri Noguchi and Carole Schiffman for their comments and suggestions.
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The False Positive Paradox


A medical test is accurate 99% of the time
The probability of a person having that condition is 1 in 10000
‘What is the probability that a person has the disease, given that they have tested positively?’
To answer the question, break the exercise down into parts,

What does 99% accuracy mean?

If 1,000,000 people are tested, how many are likely to have the condition?

How many of these people will test positively for the condition, and how many will test
negative?

How many will not have the condition?

How many of these people will test negative for the condition and how many positive?
Summarise your results in this table, then put your results in a Venn Diagram with one set for
People who have the condition and one for those that test positive (remember that the diagram
includes for regions so you should be able to put all your information on the diagram)
Have the condition
Do not have the
condition
Totals
Test Positive
Test Negative
totals
Venn diagram
Conclusions – answer the question

How many people tested positive for the condition?
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
How many of those people actually had the condition?

What is the probability that a person has the disease, given that they have tested
positively?
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Session 3 – Internal Assessment
Aim
The aim of this session is for delegates to become more familiar with the Internal Assessment
element of the course and some of the more pertinent details associated with it.
Session Outline
‘What is a project’ – Defining the task by looking at the details of what is expected from teachers
and students. We will use example projects to illustrate.
‘The Marking Criteria’ – Looking at the marking criteria to help define the task and trying it out
some sample projects
‘Questions’ – This is an area in which there are often many questions so time is allowed here for
this.
Included in this guide
2 example projects to be looked at
An elaboration of the marking criteria for internal assessment
All other resources related to this session can be found on the course wikispace
Additional resources required
Delegates will need both the subject guide and the TSM for projects to refer to for this session
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Example Projects
The following projects are intended to provide examples of the sorts of projects that can be
done and of how the markschemes are applied.
Below you will find the marks that were awarded to these projects with explanations. The
projects themselves will be available through the course wikispace and some may be printed
as an appendix to this guide.
Example Project 1
Name
Project 1
Project
Strand
Name
Mark
1
Introduction
2
2
Information /
measurement
2
3
Mathematical
analysis
2
4
Interpretation
of Results
2
5
Validity
1
6
Structure and
communication
2
7
Commitment
2
Total
Score
13
IB Maths Studies – Barcelona 2011
Punches
Justification
Title and description of task and plan are
included in the first 2 pages
The data is structured for use, but is limited in
quantity with only 34 people being tested. As
such it is judged insufficient in quantity.
Candidate attempts both simple and
sophisticated processes. Technically both chi
tests are incorrect with 25% of exp frequencies
less than 5. As such student gets 2 for
attempting sophisticated process.
For example, candidate correctly interprets the
scattergraph on page 6, despite chi² tests being
incorrectly interpreted.
On page 13 student discusses some limitations
with the data.
Project follows logical order and uses correct
notation consistently. Linking commentary and
the general flow of the project are not
substantial enough to make the project 'read
well'
Students struggles with the subject and worked
hard at a first attempt of a project related to
music, but wisely chose to abandon it and then
worked hard to meet commitments with this
one.
24
Example Project 2
Name
Project 2
Project
Crime Rates
Strand
1
Name
Introduction
Mark
2
2
Information /
measurement
3
3
Mathematical
analysis
4
4
Interpretation
of Results
2
5
Validity
1
6
Structure and
communication
3
7
Commitment
2
Justification
Clear title, description of task and plan
Candidate had found extensive and relevant data
sufficient in both quality and quantity. Data is
organised and ready for use. Candidate has
grouped data in to regions to help.
Simple processes averages, box and whisker
diagrams. Chi2 and regression lines done and
correct and relevant. The student has discussed
the validity of the chi2 test when big numbers are
used. Also, the candidate might consider that
these are crime frequencies and not rates. This
does not make the test irrelevant though and the
candidate is penalised in D for not considering this
in the interpretation. Processes are relevant and
correct.
Candidate correctly interprets most results in the
project. As mentioned above, the candidate does
not achieve 'comprehensive discussion' I would
have expected comment on the fact that data was
crime frequency and not rate for exampe.
Candidate considers validity of linear model
leading to zero or negative predictions. Candidate
also discusses the impact of big numbers on chi2
independence tests. On reflection this may even
have been enough to award 2 here.
Project follows logical structure, uses correct
notation and reads well. It is a good piece of work.
Note, the candidate does use * for multiply on a
couple of occasions. This does not detract from
the feel and I site the first example project in the
TSM from IBO where candidate does the same
and is still awarded 3. I am aware it is not always
appropriate to include whole database in the
body of the project, but in this case it is part of a
section on data and is not so long as to detract
from the flow of the project.
Candidate embraced the whole idea of project
work, met obligations and produced a good bit of
work.
Total
Score
17
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Example Project 3
Name
Project 3
Project
Governments
Strand
Name
Mark
1
Introduction
2
2
Information /
measurement
3
3
Mathematical
analysis
2
4
Interpretation
of Results
2
5
Validity
1
6
Structure and
communication
3
7
Commitment
2
Justification
Clear title. Introduction on page 3 and 4 gives clear
description of task and plan.
Candidate has collected data for 50 world leaders
that is suitable for the investigation. Candidate has
included a number of numerical and categorical
fields and has generated fields to measure 'change'
in GDP and life expectancy. The database is sufficient
in quality and quantity and structured appropriately
for use.
Although the candidate has made use of correlation
coefficient, there is not attempt to calculate this
manually and so I have judged that this does not
count as an attempt to use a sophisticate process.
Candidate is awarded 2 for using simple processes
that are 'mostly' correct.
Candidate has, for example, correctly interpreted the
correlation coefficient for the scatter graphs. There
are other examples of correct interpretations, but
not enough to be considered comprehensive
discussion.
On page 5 the candidate discusses problems with
data and limitations.
Project follows a logical order and correct notation
(although not much is used. The project reads well
despite its limited mathematical content (limiting
student to a mark of 2 for 'C')
Student worked well with an area of interest despite
struggling with the mathematics in general.
Total
Score
15
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Marking Projects
This section talks through the process of interpreting the criteria, encouraging the right kind of
content from students and awarding the marks. It is intended as practical elaboration on
information available from the IB based on teacher experiences.
A - Introduction
This Criterion is awarded 0, 1 or 2 marks
There is little excuse for students not getting a 2 in this criterion. The descriptors as they are
written really do not demand much of the student. It depends entirely on clarity both with the
description of the task and the plan for carrying it out. If students have a good idea and followed a
good plan then this is really just a case of spelling it out clearly and if this is not evident in the first
draft the teachers should point it out.
Often I find that a really good introduction goes well beyond these descriptors and will give some
brief background on the motivation for the project and its implications. I would urge students to
be as thorough as they can be with the introduction, so that they earn the marks in this criterion,
but more importantly so that they lay the foundation for a good project and increase potential
marks elsewhere. A good introduction can also contribute to the marks for Structure and
Communication.
This criterion does not usually present problems
B - Information/measurement
This criterion is awarded 0, 1, 2 or 3 marks
The most common issue here is likely to be choosing between 2 and 3 marks. The bottom mark of
zero marks is reserved for those who have made no attempt, whilst 1 is awarded where a
'fundamental flaw' exists. It is rare that a fundamental flaw is not unearthed before projects are
submitted by teachers. In most cases this should be spotted during the planning phase of the
project and teachers can offer guidance to help students plan again to put it right. Only in the case
of advice being repeatedly ignored is a student likely to carry this through.
So how to choose between 2 and 3? In practise a 3 is quite easily achieved with 50 ordered pairs
that are relevant to the exercise. To give the experience more value and to make it easier to
justify, I consider the following to help. The descriptors are identical but the elaboration below
each is intended to help differentiate. Both descriptors refer to the words quality and quantity but
there are no clear divisions between the two. The descriptor for 3 mentions depth and breadth
and I think there is an implied statement that the information and its means of collection is
generally sounder and more relevant. It may be helpful to read the section on 'Collecting
information' to help differentiate. The following are examples of the sort of things that would
persuade me to give 3 instead of 2 marks. (Assuming of course that the descriptor for is met)
Students spent time carefully planning and testing the data collection process, refining it before
putting it to full use
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


This may have been in the design of a questionnaire where the student has focused on
getting the questions just right to suit the aims of the project. A good questionnaire will
probably go through a few drafts and a couple of tests.
It may be in the design of an experiment to make sure the really interesting results are
recorded. They may have done a brief test and then refined the experiment or
measurements
Even where secondary data was used, a student may have experimented with some data
to look for preliminary findings before refining their choice of filed headings
Students compiled secondary data thoughtfully from multiple sources so that it would be richer
rather than just using a single source


Its one thing to find an enormous and interesting database, but another entirely to extract
exactly the information you need from it to suit the aims of the project and identify what is
missing and go looking for it elsewhere.
An appropriately hand picked data set can demonstrate a greater awareness
Students used collected information to generate, by calculation, new information that they
could use for their projects


A simple example is when a student collects weight and height for a group of people and
uses it to generate body mass index.
Another example I like is when students have attempted to use the results of a
questionnaire to generate a numerical index for that response. One such example is where
multiple choice responses to a questionnaire where given numerical values that were
combined to give a number on a scale from -1 to 1 which represented extremes of the
political spectrum. The aim of the project was to look at peoples political persuasions and
the reasons for them
C - Mathematical processes
This Criterion is awarded 0, 1, 2, 3, 4 or 5 marks
This is the criterion with the most marks at stake, it hinges most on understanding and use of
Mathematics, and not surprisingly it is, as a result, the one with the most to consider! The award
of marks is focussed around two key factors;


The distinction between simple and sophisticated processes
The correctness and relevance of those processes used
Simple & sophisticated processes
It is easier to start by describing sophisticated processes and then assuming that those that aren't
sophisticated are simple!
The descriptors suggest the following as examples of sophisticated processes;


volumes of pyramids and cones
analysis of trigonometric and exponential functions
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


optimization
statistical tests
compound probability
This list is not described as exhaustive and is open to interpretation as it stands. Clearly it is easier
if students have used one of the above and if they are warned that trying to do so is important for
marks in this strand. They are all elements of the syllabus and the project is a great opportunity to
practise them in real applications. Some project ideas, will be less obviously suitable for some of
these techniques and this is where it can get a little grey.
Correctness & Relevance
Ironically, I find students almost more likely to make elementary mistakes with 'simple processes'
with things like bars touching or not or mean averages calculated too quickly by hand. Students
will put categorical data on a bar chart and discuss it as though it is sequential. The same student
might then completely nail an independence test! Such things make these descriptors harder to
interpret because each one assumes the previous!
Students should try to avoid graphing or averaging the first set of data they see because they
know how to and stick to plans they made about what they are going to do and why. Simple
processes should be used sparingly to avoid irrelevance.
Correctness is obviously easier to establish than relevance and it is appropriate for teachers to
point out where there are mistakes without correcting them.
Relevance is where the most interesting decisions are! Hopefully this is covered during the
planning phase of the project. If students have been able to identify the specific potential in a idea
then they will have planned to use different techniques for different reasons related to the aims of
their project and so relevance is likely to be covered. Students must demonstrate an
understanding of why the chosen process was appropriate to the task and related to the aim.
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Once judgments are made about the above then the award of marks here becomes a little easier;
1. student attempts simple processes - correctness not required
2. simple processes are largely correct OR a sophisticated technique has been attempted with
some errors
3. a sophisticate process is carried out AND all the processes are largely correct
4. as for 3 but with added requirement that processes are relevant
5. a number of relevant sophisticated techniques are carried out
There are some remaining subjective elements, for example if


A student has correctly and relevantly applied a sophisticated technique and likewise some
simple techniques, but has also included a number of irrelevant techniques that contain
some mistakes. This could be addressed in the draft stages by advising students to remove
the irrelevant material.
How many errors are too many? At this point I would suggest it depends on whether the
errors appear careless or conceptual.
A potential pitfall that remains in this scheme is for a student who relevant applies a sophisticated
technique but makes a mistake. Accuracy is rated higher than relevance in this case because this
student will go from a 4 to a 2 with this mistake! A possible error that can slip through might be
that the student has not considered the limits of validity for a chi 2 independence test and included
expected frequencies of less than 5 for example.
D - Interpretation of results
This Criterion is awarded 0, 1, 2, or 3 marks
As with the other criteria, a mark of 0 or 1 is really reserved for the very poorest of efforts with
students only being required to show a recognition for the need to interpret results and
attempting to do so for a score of 1. Really there is little point in the whole exercise if students are
not interested in trying to interpret results and it should be emphasised from early on that this is
the point of the exercise.
So what remains is again the decision between 2 and 3. The descriptor for 2 is easily met and
justified with students only needing one interpretation and/or conclusion that is consistent with
their findings, even if the process was incorrectly used or performed. As such the leap between a 2
and a 3 here is huge, with the descriptors detailing a 'comprehensive discussion of interpretations.
This is likely to result in 2 being the most common grade, but it is important to consider that which
might persuade the award of a 3. They key word, that makes it hard to award a 3 is
'comprehensive'. The implication is that all aspects of the project have been correctly and
insightfully interpreted. In order to award a 3 here I would look for the following elements;

The project needs not to be too simple or one dimensional as this would prohibit the
award of a 3. This implies that students will have broken their investigation up into smaller
parts that allow for more detailed interpretation.
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



Students have gone beyond simply stating the results, eg, 'the average boys score was
higher than the average girls score therefore we conclude that the boys were more
proficient'
The presence of 'inferences' that show that a student has begun to look for possible
reasons why the results may have turned out the way they did. These are 'possibilities' and
effectively point to new areas for investigation that could help to get under the skin of the
results.
Students have not used too many general statements about results that are too sweeping
and ignore some of the subtle differences.
Students link the interpretations from different parts of their investigation together in
discussion.
E - Validity
This Criterion is awarded 0, 1 or 2 marks
This is probably one of the most difficult areas to communicate to students. The notion of whether
or not it was valid to use a particular piece of Mathematics for particular investigation is really
quite a sophisticated one to understand. As a result I suspect that the award of 2 marks here is
rare and that 1 will be the most common by far.
Students can be awarded 1 here for recognising the limits of validity in the information they have
process and the subsequent results of their investigations. Students might point out that they
needed more information or information from wider sources and so on and that as such their
conclusions could be debated.
For students to be awarded 2 marks here then they would be expected to comment on the
Mathematical processes themselves. 'Was it valid to use an independence test with this data?' and
associated questions about expected frequencies for example. 'At what level of correlation is it
valid to use a line of best fit on a scatter graph?' These and similar questions need 'significant
discussion'.
F - Structure and communication
This Criterion is awarded 0, 1, 2, or 3 marks
This criterion stands in some ways alone because it is not necessarily related to the Mathematics
of the project and draws on an entirely different set of skills. As such is it is possible to score well
here having not so in areas and vice versa. From the outset it is important that students
understand that they are required to 'tell the story' of their investigation and this is where they
should benefit if they have been able to.
A student is really only awarded 0 marks here if they did not complete the work or hand anything
of any significance in.
Notation & Terminology
A key term for the next two marks is 'appropriate notation and terminology' and it is really worth
spending some time with students on this so that they understand what it means and the
importance of expressing themselves correctly. Of course, it could be debated at length and so the
goal becomes being as tidy as possible so that there is little reason to argue. Here are some points
to focus on;
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





It helps if they have written formulae correctly and not necessarily, as it was easiest to do
on a computer. Software packages will often use inappropriate short cuts.
Expressing multiplication should not be done with the ' * ' symbol. Numbers in standard
form should not expressed with 'e'
It helps if the correct words are used for the processes and not muddled up. Don’t say 'Null
when you mean 'Alternate'
Charts and graphs are given correct titles. Bar graph, frequency polygon or Histogram, line
graph or scatter graph
Axes are labeled and scale are accurate
Keys and legends are appropriately used and presented
Logical Development
This is about whether or not the report reads progressively. Does one section lead neatly to the
next? In one sense it is really a test of whether or not students projects made sense and comes
back to the initial planning. In another it is simply a judgment about whether or not the report
tells a story in logical order! Here are some indicators for logical development;






beginning, middle and end - these are key features that should stand out
a table of contents can really help to show the structure of the project
project split into smaller sections that are clearly labeled
each section makes reference to the previous and the next
conclusions refer back to aims and hypotheses
There is no rambling or irrelevant chatter or investigation at different points in the project.
Communication
Although in the title of the criterion, it is not explicitly mentioned in the descriptors, it is worth
noting that a report may have logical development but it is hard to see because of the way in
which it is communicated. The following points about communication skills can help to distinguish
between 2s and 3s;








Clear labels and headings let the reader know where they are at any given point during the
report. New sections begin on new pages.
Items on the page are suitable spaced out, clear and not overlapping.
Interpretations are made near to the Mathematics, eg a commentary on a graph is written
on or near a graph so that the reader can be comparing the two.
Keys, titles, legends etc are appropriately used to help the reader link words to tables and
diagrams.
Fonts and sizes etc are consistent.
Use of footnotes
Presence of references and bibliography
No unnecessary rambling
If students can manage the 'Notation and Terminology' OR 'Logical development' to their projects
then they should comfortably get to a 1.
If students manage BOTH then they qualify for a 2.
If students manage BOTH and tick a lot of the communication boxes then they could get a 3.
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G - Commitment
This Criterion is awarded 0, 1 or 2 marks
This criterion is a triumph for 'trust in the teacher' and I for one am really pleased that it was
included! I don’t think there is a real need for elaboration for this criterion at this stage. I think its
very important to show students these descriptors from the outset and remind them that it really
is their control. Despite the obvious potential for abuse I urge teachers to use this
criterion properly and give students the grade they most deserve. Imagine being asked to justify
why you gave a grade here. Its only a couple of marks, but a real chance to reward students that
show responsibility!
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Session 4 – Internal Assessment 2
Aim
The aim of this session is to consider ways in which teachers can introduce and manage the
Internal Assessment. We will look at how students can come up with ideas, how teachers and
students can work together effectively and at some of the tools available to help.
Session Outline
‘Project Ideas’ – How to get students inspired to think of great ideas for their projects.
‘Management Strategies’ – Ideas for organising the whole IA process
‘Tools’ – Google forms and Wikispaces for IA
‘Data Sources’ – Sharing sources for data students could use in their projects
Included in this guide
The Project Planner – A series of activities that can help students and teachers manage the IA
process
All other resources related to this session can be found on the course wikispace
Additional resources required
Delegates may need the TSM for projects to refer to for this session
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The Project Planner
Planning Activities
The following are a list of activities and tasks that help students to plan and keep track of their
projects as well as to help teachers monitor progress. The tasks can be given as the teacher sees
fit. Either give them all at once and let students manage or give them a set times with set due
dates. Any combination of the above can be done to best suit the teacher and the class.
Choosing a theme
Without a doubt the most difficult part of the process for students is choosing an appropriate
theme or topic for the project. This task is aimed at getting students to think in a structured way
about this and help them to narrow down their choices. Students narrow down and submit 3
outline ideas from which an informed choice can be made. At this stage the outline should include
brief references to the following;





Theme
Questions/hypotheses/aims
Information
Potential for analysis
Possible outcomes
Information
Its crucial for students to be able to identify exactly what information they are going to collect and
how they are going to collect it. Statistically based or not, there is a need for collection of
information. That information may and is likely to be a number of related data sets, but could also
be a series of samples or calculations. Either way there needs to be a plan for 'what?' and 'how?'
and this plan needs to consider the nature, quality and quantity of this information.
Identifying specific potential
In the first activity students are asked to consider possible mathematical activity as part of their
drafts. Having narrowed down their project ideas, they are then asked to identify very specific
potential for mathematical analysis that involves the using the information collected to address
the questions and hypotheses in the theme.
Plan and Schedule
In many ways, if and when the above have been thoroughly completed the only difficult part left is
the making and scheduling of a sound plan. The only thing then left to do is execute the plan.
Often students will start the main body of work before they have this sound plan or really sound
answers to the previous two tasks. As such the task becomes more difficult and the result less
coherent. It is really worth taking the time on this stage, being as specific as possible about dates
and times and to be as realistic as possible in doing so! This task aims to get students to do just
that!
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Reflection
It is hard to be prescriptive about reflection and the main aim of this task is just to reinforce the
idea that reflection at all stages is relevant and important. For possible use 'mid project' this task
asks students to reflect carefully on how their project is going.
Check
Towards the end of the project period, it is important that students look with great care at the
nature and quality of their work to check that they have done what they set out to do and that
they have what they need to get their marks. This task is just a checklist for students to use to help
them decide answers to the above. Depending on the students this is a possible opportunity for
some peer assessment by asking students to do this for each other.
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Choosing a Theme
The aim of this task is to help choose, narrow down and decide upon a project idea that really
suits you and has the potential to help you achieve your best. Its a difficult task but can be made
easy with a little structure as suggested below. Ultimately you will be asked to submit 3 basic
project outlines with the format given below. The idea is to work up to this by completing the
tasks below! Each question comes with an example of how it could be answered to help you.
You should complete this task either by hand using pen and paper and clear headings to show
each section or digitally, by using the linked word document as a template.
Your Interests
Please write down as many of your personal interests as you can think of below. Try to make as
varied a list as possible. This can include anything related to one of your hobbies or favourite
things to do. It might be something you feel passionate about at least strongly enough to have
debated on it in the past. It might be something that has always fascinated or interested you,
whether or not you have ever looked deeply at it.
Your interests and Mathematics
Write down any connections there may be between any of your interests and any part of
Mathematics. Is there any inherent mathematics? Could mathematics be used to analyse or model
it in anyway?
Connections to hypotheses
Could any of the above connections give rise to making a hypothesis? Are there any questions you
could ask about this connection that Mathematics could be used to answer? Have you always
believed something about this connection to be true but never been able to prove it? Write down
any such thoughts, questions or hypotheses.
Themes


A 'theme' in this context refers to a topic from which a number of smaller related
mathematical investigations may arise.
A 'theme' may be a question, hypothesis or headline
Based on your responses to the previous tasks can you identify three possible 'Themes' for a
project?
IB Maths Studies – Barcelona 2011
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Outline Plan
For each of these themes the aim is for you to sketch out an outline plan for a project on the
theme that will help to judge its potential as an idea. One of these is likely to develop in to your
project. As a structure, please use the following questions as headings.
1. What is the 'theme'?
2. Describe the nature of potential investigation. What questions could be asked and
answered with mathematics?
3. What information would be collected and how? The more detail added here the better.
Some specific ideas a required to really help judge the potential of the idea
4. What potential is there for mathematical analysis? What maths would you use, with what
data for what purpose? Again, some specific references are needed here.
5. What might you expect to find out and conclude from this exercise? What would be the
point of doing it? What interesting findings might come out?
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Information
In order to answer any questions there are in your project theme or idea you will need to have
some information, measurement or data to help you do it. Having picked a theme or idea, you
now need to be very specific about what that information could be in this case. The following
questions should help you to do that! Done properly, much of what you write here can be used as
part of your written report.
Points on Information
Before you make decisions and answer questions about your information, you should consider the
following list of points about good information for projects;
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Must not be one dimensional (ie temperature change over the last 50 years, lots of
numbers but they are all temperatures)
Must be largely numerical - categorical fields are important as well but not enough on
their own
Must be sufficient in quantity – ie a survey might be best with a minimum of 200 responses
If you find data on the internet, it must be easily transferred into a spreadsheet for huge
time saving and flexibility benefits
If you plan a survey, you must spend time working soundly on you questionnaire, you cant
go back and add/change questions once you have done it
Where possible, surveys should involve some kind of measurement
Where possible it is good if you can use your information to generate new information (for
example, collect height and weight to generate body mass index)
Information collected experimentally needs to be collected under defined and consistent
conditions
What is the nature of your information?

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
data from a questionnaire
the results of some experiments
measurements
calculations
data from the internet
other...
Specifics
Specifically describe the information you will collect. This might include the questions you plan to
ask, the measurements you plan to take etc. The key word here is specific. Avoid general phrases
here. Say exactly what you intend to collect. It may be easier to list this.
How?
Outline here exactly the plan you would have for collecting this information. If you were planning
a questionnaire then you would need to describe how you would create it, administer it, collate
IB Maths Studies – Barcelona 2011
39
the results and so on. Again, this is only worth doing if you add specifics. If you are measuring then
you need to think about the practical aspects of when and how this can be done and what
equipment you would need. If you are experimenting then you will need to consider practical
aspects as well. If you are calculating then you need to think about the systematic methods
required to be efficient. Time spent answering this carefully is really worth it later!
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Specific Potential
This is a major milestone in your projects. Identifying the specific aspects of the mathematical
analysis that you will do is very important. If you cant do that at this stage then you may need to
reconsider the project idea. The main aim is to consider what mathematical processes can be used
in conjunction with what information to answer questions or investigate your theme or idea.
Again, the key word here is specific. You should make a list below of potential ideas. Each idea
should include...




a specific reference to a mathematical process
a specific reference to information
a specific reference to the question or theme
The following are some examples to demonstrate and refer to fictional ideas or themes;
I will calculate the mean average sprint time of 12 year old boys and compare it to that of 13, 14
and 15 year old boys to see if there is a steady increase.
I will plot the mean average sprint times of the different age groups of boys on a scatter graph to
look for correlation. I will then compare this with a similar scattergraph for girls
I will calculate the probability of being dealt 2 cards that total all the numbers from 2 to 21 to
determine which are the most and least likely hands to be dealt.
I will categorise my data so that I can perform a chi2 test for independence to determine of
gender is independent of political preference
It is better to have a longer list that you can narrow down to the most relevant tasks than a
shorter one with not enough on it.
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Planning and Scheduling
At this point, you should probably have;



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a theme or idea that interests you and has potential for mathematical investigation
a description of the related information that you are going to collect in order to investigate
along with how you plan to collect it
a list of very specific potential ideas for investigation
Now you just need to need to make a sound plan about how you are going to complete all of this
in the time frame available to you without leaving too much to the last minute and causing you
and your teacher too much stress. Few of us stick rigidly to our plans but having them in place
always makes progress more steady and likely. The key to successful planning is to be completely
aware of what is requires and when its required by and being realistic about what you can do in a
given amount of time.
Time Frame
Before you start, you should have answers to the following questions so that you can see the task
in terms of how much time is available to complete it.



When is your project due to be completed by?
How, when and at what points is your teacher planning to give you feedback on your work
that you can use to advance?
How much time in a week should you/could you be spending on your projects? Your
teacher may or may not be expecting you to manage this along with other homework, but
should be able to help you answer this question by building in the class time that may also
be available.
Ways of working
Individuals will manage their time differently but it is important to have some kind of strategy to
do this. The 'Do things as they come up' approach is generally unsuccessful at IB because of the
numerous requirements of the diploma program and the associated demands on time!
In general, it is recommended that you identify a regular slot or slots each week for working on
your project and then identify the tasks that you are going to try and complete in those slots.
Some of those slots will be at home and some in class and they will be of different time lengths.
What has to be done?
You have done most of this already in the build up to this point. The next stage requires you to
make a list of the tasks associated with the project these tasks should generally be of the following
kinds;
 those associated with collecting and organising information
 those related to the carrying out of mathematical analysis
 those about writing your report, describing what you did, how you did it, what happened
and how you interpreted the results
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The Plan
So here it is! Make a list of tasks associated with your project and put in them in the chronological
order they need to be done in and where possible try to identify when you plan to complete them!
Use a table or spreadsheet to do this and submit to this to your teacher who may choose to
display them in your classroom. You could put another copy on your fridge or above your desk at
home and so on. The point is that it has to be looked at regularly if it is to be of any use!
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Reflection
This is a brief task to be carried out mid project with aim of getting you to reflect on how your
project is progressing and consider some improvements you could make. It consists simply of a list
of questions and ideas for you to try from which ideas for improvement may emerge.
Progress Questions
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How well are you sticking to your plan?
What improvements could you make here?
Have you found out anything interesting from doing your project?
What and why was it interesting?
What are your biggest difficulties?
Have you been asking for help?
Will it be ready?
Talk about it!
Try explaining the aims and processes behind your project to a friend and/or family member in
conversation. Set aside a good 15 minutes and encourage them to ask you questions. Ultimately
you have to submit a written report that reads well and that demonstrates your interest and
enthusiasm for the project. Practising this in conversation can really help you to focus your
thoughts. You may need to make some notes straight after the conversation.
Ask one of your classmates to do the above for you. Listening to someone elses project story is a
great experience. Not only does it give them experience described above but it gives you the
opportunity to see what a listener/reader needs in order to fully understand an idea.
Both of these experiences may give rise to some new ideas for you as well as helping you to focus.
Newspaper!
Imagine writing a newspaper article about your project.
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What would the headline be?
What would the subheadings be?
Why would people read it?
What would be its main message?
What bits of your project would you definitely include?
Which bits would need to be better explained?
Which bits would you leave out?
This exercise can really help you to think about putting the theme across to a reader and to focus
on the structure you need to sue when writing the report.
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Checklist
This is intended as a guide that you can use to check the work you have done and if you have
included important elements that can help you get better marks on your projects. The following
are suggestions only and should be considered in the second half of the project period.
Table of contents
This may be the last thing you do because it needs to be an accurate reflection of what is in the
project. It helps to score on structure and communication because it helps the reader see you
project as a whole and also because it makes you think about your structure. The contents could
include the headings and sub-headings you have used in the project in a sensible order and these
headings should be descriptive enough to be clear but not so long as to appear as paragraphs. As a
rule of thumb, reading the table of contents ought to give the reader a quick summary of your
project.
Introduction
You really should get the highest mark in this strand! The best introductions/plans are often
written retrospectively after you know what you have done but based firmly on the plan you made
at the outset. Don’t be afraid to be wordy in this part but make sure your introduction reads well
and generally introduces the story you are about to tell.
Things to you could include to help you achieve this;




Your motivation for taking on the project
Background information on the theme and this may include some references to existing
material that you have taken from books or the internet.
A clear outline of your plan. This could be bullet pointed or each part of the plan could be a
separate heading. Under each heading you would write specifically what data you would
take and specifically what you would do with it and then what you be looking for. You
could/should also include a prediction of what you might expect to find (it doesn’t matter if
it did not turn out that way). You could be drawing heavily on planning work you did in the
early stages of the project here.
You can use separate headings or bullet points to make this section stand out to the
reader. This also helps the award of marks for structure and communication
Information
This need not be enormous but it is a good idea that you include a section on your information. In
this section you could include the following,
 A summary of what information you have (the data itself need not be included here, it can
be at the back). You should just describe carefully what you have collected
 A statement about how the data was collected, where you found it etc.
 If you used a questionnaire for some or all of your data you should discuss how you
decided on the questions and what the important features of the questions were.
 You should talk about problems you encountered collecting data, either in finding it or
designing questions and collecting it.
IB Maths Studies – Barcelona 2011
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


If you have generated extra data from the data you have collected then you should point
that out. (This is true if you have added an extra column to your database and put a value
in it that you have calculated form the existing data)
Be sure to mention the limitations of this data, how you could/would improve it if you
were to do it again.
Questions you could have asked or data that would have complemented what you already
had. Be equally sure to use the word ‘validity’ at this point. Eg, how accurate is your data
likely to be.
Mathematical processes
It may be helpful at this stage to give a brief outline of the techniques you plan to use and how
they work. For example if you are using regression lines and correlation coefficients then you
could have a brief section on what these things are, how they are calculated and interpreted. (e.g.
if r is 1 then that is perfect positive correlation if it is 0 then you have absolutely no correlation).
For standard deviation, you might just show a full calculation once and the results thereafter.
These are particular to your project and so will take on different forms but there should be some
common themes. For example, in your table of contents, one might expect to see subheadings for
each separate investigation you did here. These sub-headings would be repeated in your project
and under each you could…





Describe briefly what you are going to do (this may in some ways be a repeat of what you
wrote in your introduction). What data, what process? Etc.
The process – if you have calculated a statistic, explain how or refer to the beginning of this
section as described above. The same for graphs and diagrams. Make sure diagrams and
results are clearly labelled
Make a statement about your finding (interpretation marks) and try to make a relevant
inference. ( for example a simple statement of mean average means less unless you
attempt to explain why it is with remarks like, that seems lower/higher than I might have
expected, perhaps this is because….. etc.)
Make a statement about the validity of your findings. For example was your sample big
enough, is the process rigorous enough, can you really make conclusions from this process.
The more coherent discussion about the last 2 points the better! Make sure that these subinvestigations are visibly separate and that the statements are clearly matched to the
relevant statistics and diagrams.
Interpretation/conclusions
In your introduction you should have stated a general aim to investigate a theme. In the
mathematical processes section you have performed a series of smaller investigations and made
summary statements about all of them. In a section called conclusions, you should attempt to
thread all of these Statements together and begin to answer the questions you have put in your
introduction. If you made hypotheses, you should remark on their correctness. You can also
comment again in this section about changes that could be made to your approach.
IB Maths Studies – Barcelona 2011
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Validity
This is a tricky concept and one that requires a good deal of thought. Its a good idea to keep a log
somewhere of occasions when validity comes up in discussion. References to validity generally
come under the following two headings;


the nature of the information used questions the validity of the conclusions reached
the process used or the way it was used or the place in which it was used was not valid
The latter is generally the more difficult to spot. An example might be to comment on having
considered validity based on the expected frequencies in an independence test and changed your
categories as a result. Another would be reading from regression line outside the range of the data
used or plotting a line of best fit when there is no correlation to speak of.
Structure and Communication
This is a must for maximum marks, consider the following bullet points when thinking about
structure and communication










Title, table of contents, page numbers
Use a consistent system of headings and sub headings
Start new sections on a new page with a clear heading/subheading
Make sure graphs are titled with axes labeled
Make sure graphs are interpreted on the same page – you could do so with textboxes and
arrows. I should not have to flick between pages to see if interpretations are consistent
with graphs
Include a bibliography – you must reference your sources – everybody will have something
to reference
Include footnotes (these are things that might disrupt the flow in the main body of the
text, but that should be included)
Consider the logical flow of the project – does the order make sense? Are you telling a
story?
Use correct mathematical symbols – no ‘*’ for multiply etc – calculator or computer
notation is not accepted.
Make sure working out is coherently shown eg not 3 x 4 = 12 + 5 = 17. When you mean 3 x
4 + 5 = 17. A good rule is not to use = twice on the same line.
IB Maths Studies – Barcelona 2011
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Session 5 – Exams
Aim
The aim of this session is to become familiar with the external assessment tools used by the IB at
the end of the course and how they are assessed.
Session Outline
‘Paper 1’ – Some exposure to Paper 1 style questions, how they are answered and how they are
marked
‘Paper 2’ – Some exposure to Paper 2 style questions, how they are answered and how they are
marked
‘Common Errors’ – Looking at typical mistakes to help students avoid them
Included in this guide
Paper 1 and 2 questions with associated markschemes
The Marking Codes – a brief guide to what is meant
Common Errors – a list of common exam errors made by students
All other resources related to this session can be found on the course wikispace
Additional resources required
Delegates will need the pack of Specimen Questions to refer to for this session
IB Maths Studies – Barcelona 2011
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Paper 1 - Questions and Marking
Markscheme Instructions
Paper 1
Abbreviations








M Marks awarded for Method
A Marks awarded for an Answer or for
Accuracy
C Marks awarded for Correct answers
R Marks awarded for clear reasoning
ft Marks that can be awarded as follow
through from previous results in the question
d Mark awarded at the examiner’s discretion
Method of Marking




All marking done with a red pen
The maximum mark is awarded for a correct answer on the answer line.
If the answer is wrong, marks should be awarded for the working according to the
markscheme.
Working crossed out should not be awarded any marks.
Using the Markscheme





As A marks are normally dependent on the preceding M mark being awarded, it is usually
not possible to award (M0)(A1)
Similarily (A1)(R0) cannot be awarded for an answer which is accidentally correct for the
wrong reasons given
Accept alternative methods
Unless the question specifies otherwise, accept equivalent forms.
As this is an international examination, all valid alternative forms of notation should be
accepted.
Accuracy of Answers






Unless otherwise stated in the question, all numerical answers must be given exactly or
correct to 3 significant figures.
A penalty known as an Accuracy penalty (AP) is applied if an answer is either
rounded incorrectly to 3 significant figures or
rounded correctly or incorrectly to some other level of accuracy.
This penalty is applied to the final answer of a question part only
The accuracy penalty is applied at most once per paper.
IB Maths Studies – Barcelona 2011
49
Other Penalties




Level of Accuracy in Finance Questions
The level of accuracy will be specified and a penalty known as a Financial Accuracy Penalty
(FP) is applied if the answer is not given to the required accuracy. This is applied as most
once per paper.
Units in Answers
A penalty known as a Unit Penalty (UP) is applied if an answer does not include the correct
units. This applies to both missing units and to incorrect units.
The Unit Penalty is applied at most once per paper.
Graphic Display Calculators

Candidates will often be obtaining solutions directly from their calculators. They must use
mathematical notation not calculator notation. No method marks can be awarded for
incorrect answers supported by calculator notation.
IB Maths Studies – Barcelona 2011
50
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IB Maths Studies – Barcelona 2011
52
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IB Maths Studies – Barcelona 2011
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IB Maths Studies – Barcelona 2011
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IB Maths Studies – Barcelona 2011
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Paper 2 – Questions and Marking
Markscheme Instructions
Paper 1
Abbreviations








M Marks awarded for Method
A Marks awarded for an Answer or for
Accuracy
C Marks awarded for Correct answers
R Marks awarded for clear reasoning
ft Marks that can be awarded as follow
through from previous results in the question
d Mark awarded at the examiner’s discretion
Method of Marking




All marking done with a red pen
The maximum mark is awarded for a correct answer on the answer line.
If the answer is wrong, marks should be awarded for the working according to the
markscheme.
Working crossed out should not be awarded any marks.
Using the Markscheme





As A marks are normally dependent on the preceding M mark being awarded, it is usually
not possible to award (M0)(A1)
Similarily (A1)(R0) cannot be awarded for an answer which is accidentally correct for the
wrong reasons given
Accept alternative methods
Unless the question specifies otherwise, accept equivalent forms.
As this is an international examination, all valid alternative forms of notation should be
accepted.
Accuracy of Answers






Unless otherwise stated in the question, all numerical answers must be given exactly or
correct to 3 significant figures.
A penalty known as an Accuracy penalty (AP) is applied if an answer is either
rounded incorrectly to 3 significant figures or
rounded correctly or incorrectly to some other level of accuracy.
This penalty is applied to the final answer of a question part only
The accuracy penalty is applied at most once per paper.
Other Penalties

Level of Accuracy in Finance Questions
IB Maths Studies – Barcelona 2011
58




The level of accuracy will be specified and a penalty known as a Financial Accuracy Penalty
(FP) is applied if the answer is not given to the required accuracy. This is applied as most
once per paper.
Units in Answers
A penalty known as a Unit Penalty (UP) is applied if an answer does not include the correct
units. This applies to both missing units and to incorrect units.
The Unit Penalty is applied at most once per paper.
Graphic Display Calculators

Candidates will often be obtaining solutions directly from their calculators. They must use
mathematical notation not calculator notation. No method marks can be awarded for
incorrect answers supported by calculator notation.
IB Maths Studies – Barcelona 2011
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IB Maths Studies – Barcelona 2011
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Common Errors
Here are errors that occur frequently on exam papers.





























Knowing the difference between the number sets N, Z, Q and R
Knowing the difference between significant figures and decimal places
The ‘exclusive or’ and mixing up inverse, converse and contrapositive
Factors, multiples, etc in set theory questions
Using the correct Compound Interest formula and using it properly
Knowing the difference between Simple Interest and Compound Interest
Using the Finance function correctly and knowing what to write down on the answer
paper.
‘Show that’ questions
Conditional Probability questions
Drawing Venn diagrams
3-Dimensional problems (eg identifying the angle between two planes)
When to use the GDC – ie for writing down mean, standard deviation, correlation
coefficient or regression line.
Forgetting to set the GDC in degrees
What information to include when sketching a graph from the GDC.
Finding the intersection of 2 curves using the GDC.
How to write down a range of values
Horizontal and vertical asymptotes and their equations.
Finding the derivative of functions such as 2/x or x/4
Using the formula Period = 360°/b
Mixing up amplitude and period
Using the equation of the axis of symmetry
Writing the equation of a line in the form
ax + by + d = 0. Particularly when a, b and d are integers.
Stem and leaf diagrams
The difference between range and interquartile range.
Outliers
How to use the polysmlt function on the GDC (TI users)
How and when to use the Solver on the GDC.
Writing down the correct unit in answers.
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Session 6/7 - Task Design
Aim
The aim of this session is to remind ourselves of the elements of a good activity. How do we match
the philosophy of the IB with the syllabus content and our own philosophy of teaching? This
session gives us an opportunity to create and collaborate with colleagues and perhaps look in
more detail at some of the elements of the syllabus.
Session Outline
‘Visual Sequences’ – An activity designed to promote mathematical thinking in the teaching of
arithmetic sequences.
‘What makes a good task?’ – Our collective thoughts are put together on this question
‘Sharing good ideas’ – As a group we will share some of best teaching ideas that might be used to
teach the Maths Studies syllabus
‘Task Design’ – In small groups we aim to produce some good ideas and activities that we can all
take away
Included in this guide
Visual Sequences – some related resources
What makes a good task? – a mind map
A design brief
Some Stimuli for task design
All other resources related to this session can be found on the course wikispace
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Visual Sequences
Visual Representations
Consider the following sequence
1, 5, 9, 13
In your groups, using multilink cubes. Create a visual representation of this sequence. If you have
enough cubes trying creating two different representations. Set them up on your table as though
it were an exhibition. When your teacher is sure that all exhibits are ready you will be invited to
go around the room to look at the other exhibits and try to explain them. You should ask each
other questions about the decisions made.
When the class comes back together, there should be a discussion of the different exhibits and an
attempt to note the merits and limits of each one. Make some notes/sketches/photographs in the
space below to prepare you for this discussion.
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Common difference
Use an example you have seen or a new one of your own
The ‘common difference’ from one term to another in this sequence is 4. Give an example of an
exhibit that shows a Pattern that grows by 4 cubes each time in a systematic way. Include a sketch
or a photograph
A New Sequence
Consider the following Sequence
1, 7, 13, 19
Construct an exhibit that shows each term as that number of cubes with the common difference
being added systematically from one term to the next. Show a sketch/photograph below.
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Finding terms
Consider the first sequence again
1, 5, 9, 13
How many cubes would it take to make the next term?
How many cubes would it take to make the 10th term? How did you calculate this?
How many cubes would it take to make the 100th term? How did you calculate this?
Defining Variables
Consider the first sequence again
1, 5, 9, 13
- For this question you need to consider that
- The first term (1) is called U1, the second (5) U2, the third (9) U3 and so on,
- The common difference (4) is represented by ‘d’
Explain why the following statement is true U3 = U2 + d
Express U4 in terms of U1 and d (only)
Express U10 in terms of U1 and d (only)
Express Un in terms of U1, n and d (only)
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General Expressions for Sequences
Consider the sequence defined by
Un = 3n + 1
By substituting n = 1, 2, 3 and 4 into the general expression, find the first four terms (U 1, U2, U3 and
U4) of the above sequence
What is the value of ‘d’ (common difference) for this sequence?
Given that you now know U1 and d, use your answer given in the last section to make your own
expression for Un in terms of U1, d and n.
How does your expression for Un compare with the one given above?
Following these activities and subsequent discussions you should attempt a number of related
problems from your preferred source.
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What makes a good task?
In groups delegates will work with a blank grid and collect their own ‘Ingredients of a good task’
mind map. There is no underlying implication that all of the elements must be present for an
activity to be good, but rather that for a given unit of work that it is desirable to include as many
as possible!
The following is the result of the same exercise with a previous group! Please feel free to add, edit
or annotate the diagram below.
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Designing Activities
In smaller groups we will work on designing activities given stimuli as starting points. The stimuli
are intended as just that and should not stop another burning idea taking precedence. A brief for
the design task is included below
The Brief
Each group should aim to produce the following by the end of the session
Title of Activity
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The stated aim of the activity referring to learning objectives
An outline description of the activity
The resources needed for the activity
Elements of differentiation
Expected Outcomes
If time allows then any resources created would be great!
My aim is to post these on the course wikispace so that we can all refer back to them and
hopefully use them!
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Stimulus 1 - Mathematics of War
Based on the idea presented in session 2 and some of the resources provided. Design an activity
or series of activities that could be based on this theme
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Stimulus 2 - Sexy Maths – When it pays to play the odds
Sexy maths: when it pays to play the odds
Mathematicians, and the laws of probability, can tell you whether to have a flutter, or keep
hold of your money?
Marcus du Sautoy
Let’s start by playing a game. I roll a dice and pay you in pounds the number that appears on it.
How much would you be prepared to pay to play? If you pay £1 you cannot lose, and if you pay £6
you cannot win but at what point do the odds tip from my advantage to yours?
It was the correspondence between two of the greats of mathematics, Fermat and Pascal, that led
to the discovery that you could apply mathematics to analyse games of chance. Previous
generations had not dreamt of such a connection. Mathematics is a subject of certainties and
truths. How can it apply to the analysis of chance and randomness? The mathematics of chance
crystallised with the publication by Swiss mathematician Jakob Bernoulli of Ars Conjectandi, or The
Art of Conjecture. It is here that you find the formula for the fair price that you should pay for any
game.
Suppose there are N possible outcomes (in our dice game, N=6). You win W (1) pounds if outcome
1 occurs (ie, £1 for a roll of 1). This happens with probability P (1) (in this case, 1/6). Similarly,
outcome 2 occurs with probability P (2) in which case you win W (2) pounds (in our game, £2 for a
roll of 2, again with a probability of 1/6). On average, a game earns you W (1) x P (1) +…+W (N) x P
(N) pounds each time you play, which in our dice game equals £1 x 1/6 + £2 x 1/6… + £6 x 1/6 =
£3.50. So if I offered you less than this to play, then you’re going to be the winner in the long run.
The formula seemed sound until Jakob Bernoulli’s cousin Nicolaus, in an almost oedipal act, came
up with the following game: I toss a coin. If it lands heads I pay you £2 and the game ends. If it
lands tails then I toss again. If the second toss is heads I pay you £4. If it is tails I toss again. Each
time I toss, the payout doubles. So if I toss six tails followed by a head I’ll pay you 2 x 2 x 2 x 2 x 2 x
2 x 2 = 2 to the power of 7 = £128. How much would you be prepared to pay to play Nicolaus’s
game? Four pounds? Twenty pounds? One hundred pounds?
Well, there’s a 50 per cent chance that you’ll win only £2. After all, the probability that it lands
heads on the first toss is 1/2. So P (1) = 1/2 and W (1) = 2. But you’re hoping for a long run of tails
followed by a head to get as big a prize as possible. The probability that you get a tail followed by a
head will be 1/2 x 1/2 = 1/4. But this time you win £4. So the second outcome has P (2) = 1/4 but
W (2) = 4. As you keep going the probabilities get smaller but the payout bigger. For example, six
tails followed by a head has a probability of (1/2) to the power of 7 = 1/128 but wins you 2 to the
power of 7 = £128.
If you stopped the game after seven tosses then you would lose only if there were seven tails in a
row. Using Jakob’s formula the average payout would be W (1) x P (1) +…+ W (7) x P (7) = (1/2 x 2)
+ (1/4 x 4) +…+ (1/128 x 128) = 1 + 1 +…+ 1 = £7. It is worth playing the game, therefore, if anyone
offers you less than £7 to play.
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But here is the sting. Nicolaus is prepared to play the game indefinitely until a head appears.
You’re a winner every time. So how much will you pay to play the game?
There are infinitely many options now. The formula says that the average payout will be 1 + 1 + 1
+… namely infinity pounds! If anyone offers to play this game with you, it’s worth playing
whatever the cost to play. In the long run the maths says that you will come out on top. But why is
it that most of us wouldn’t play the game for anything more than about £10?
It’s called the St Petersburg Paradox after Nicolaus’s cousin Daniel who, while working at the
Imperial Academy of Sciences in St Petersburg, came up with the first explanation of why no
rational person would pay any price to play the game. The answer is what any billionaire will tell
you. The first million you earn is worth so much more than the second million. You shouldn’t put
in the formula the exact amount you win but what that prize is worth to you. In this way the price
to play this game will vary according to how you value the outcomes. Daniel’s resolution goes far
beyond just the curiosity of a mathematical game: it is essentially the foundation of modern
economics.
Go to www.mathematik.com/ Petersburg/Petersburg.html for an online simulation of the game
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Stimulus 3 - 100 metre runners and their split times
Usain Bolt 100m 10 meter Splits and Speed Endurance
August 22, 2008 by Jimson Lee · 52 Comments
Usain Bolt won the 100 meters because of his speed endurance.
I’ve said this all along, unless you are running a 40 yard dash or 50 meter sprint, sprinting the 100,
200, or 400 meters is all about speed endurance… reach your top speed, and maintain it. The
winner of two athletes with the same top end speed will be the one who decelerates the least.
Most world class 100 meter men reach their top speed within 50-60 meters. Women reach their
top end speed a bit earlier, so more of their race is speed endurance.
I have collected 10 meter segment splits for the last 20 years. And yes, I am including Ben Johnson
and Tim Montgomery because they still ran those times, supplementation included. I am looking
for relative comparisons.
In the chart below, RT = reaction time and is included in the 0-10m segment.
Disclaimer: These are not IAAF official splits but splits extracted from high speed video analysis
Until Bolt came along, 0.83 was the fastest top end speed recorded. 0.83 seconds per 10 meters
translates to 12 meters per second (m/s) or almost 27 miles per hour (mph) or 43 kilometers per
hour (kph).
Ben Johnson’s time of 9.79 could be extrapolated at 9.72 if he didn’t slow down and celebrate,
assuming 0.85 seconds rate for the last 20 meters (0.2 + 0.5)
If you extrapolate Usain Bolt’s last 10 meter segment, without the chest thumping, it would be fair
to say he would have ran 0.84 or 0.85 seconds, making his 100m World Record 9.63 or 9.64.
Jimmie R. Markham of 400meteroval.com submitted this nice 3D graph:
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Also, a 9.64 doubled plus or minus +/- 0.2 seconds = 19.28 for 200 meters, which is the pretty
close to his 19.30 World Record.
It is a known fact that Bolt (or his coach) was concentrating his efforts in the 200 and 400 meters
over the past few years. He only took the 100 meters seriously this year, which is a scary thought.
Hence, 200/400 training involves 3 main components: speed, speed endurance, and special
endurance.
Usain Bolt is all the proof we need.
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Stimulus 4 - Coordinate Geometry and the gradients of
perpendicular lines
A square consists of 2 pairs of perpendicular parallel sides. As such, the gradients of two opposite
sides are equal and the gradients of 2 adjacent sides multiply together to give -1.
Used in the context of a Cartesian coordinates grid, how could this develop into en engaging
activity that helped students to discover this relationship?
Stimulus 5 - Exchange Rates
Amazon.uk or .com or . fr, different places, different currencies, different prices. I’ve often
wondered how the pricing structures work for organisations such as amazon, or apple etc. Could
this be used as a context for an activity on using exchange rates?
Stimulus 6 - Modeling Temperature Fluctuations
Paris and Helsinki. What does the average temperature cycle for these two cities look like (Try
searching on Wolfram Alpha – ‘Average temperature Paris Helsinki). Does this lend itself to
modeling with trig functions? How could this be turned into a classroom activity?
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Session 8 - Resources
Aim
The aim of this session is to distribute and share some of the best resources available for Maths
teaching and in particular the teaching of Maths Studies
Session Outline
This session will simply be a show and tell of good resources in the following categories;
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Texts
Web 2.0
Connections
Software
Apps
Websites
Included in this guide
All resources related to this session can be found on the course wikispace
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Session 9 – Graphical Display Calculator
Aim
The aim of this session is to become more familiar with the role and workings of the Graphical
Display calculator in teaching this course.
Session Outline
‘Calculators and Emulators’ – A brief look at the main two types of calculator available and
emulators that teachers can use to help students learn about their calculators
‘What do I need to know?’ – A skill by skill guide to what is needed that will give us an opportunity
to practise and ask questions
Included in this guide
‘What do I need to Know’ – a written guide to the skills are required by Maths Studies students.
All other resources related to this session can be found on the course wikispace
Additional resources required
Delegates will need the TSM on GDCs to refer to for this session
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Calculator Skills - What should we know?
As I teacher, what I most want to know, is what skills should my students be fluent with? Where
can the calculator save them time and limit error? and what are they expected to do with their
calculators in certain circumstances? The calculators lend themselves more to some topics than
others. Some of the functions are essential and an enormous help, whilst others can get in the way
of understanding. The following is a list by Topic of the skills that students should have to make
the best of their calculators. Where appropriate there is a little commentary to go with the skill to
help teachers think about how and when to use it. Please use the contents menu at the top right
to help you jump down to the topic you are looking at.
General
Mode/Set up
A few basic settings can be decided here and so familiarity is important. The most important is
perhaps the switch between Radians and Degrees. Radians is often the default setting and since it
is never required in the Studies course, students should get in to the habit of changing to Degrees.
Resetting the memory
Required for all exams, students should be able to reset the memory and then adjust the settings
accordingly afterwards. Get in to practice with this early is my advice.
Adding and Removing applications
Depending on the access to ICT at different schools, you may or may not make use of the myriad
applications that can be downloaded on to the GDC. Either way it is important to know how to
manage these and prepare the GDC for exams by only having 'allowed' applications.
Basic Calculations
Silly though it may sound, if students have been used to a normal scientific calculator then it can
take some time to get used to performing basic calculations on the GDC. The screen is of course
bigger and records more of what you did. The order in which keys need to be pressed differs on
scientific calculators. Many of the operations are under menus rather than printed on buttons.
Even something as simple as 'enter' in stead of '=' can cause confusion, so it is worth putting in
some practice.
Question 3 – Make the following calculations
Question
Your answer
Find the cube root of 6 to 3 sf
Calculate 6 to the power of 2.5
Find the 5th root of 20 to 3sf
If y = x2 what is y if x = -3.4?
Work out (6.4 + 3.7)/(-2.8 +
11.9)
If Cos A = 0.9, find A to 3sf
Change 0.345 into a fraction in
its simplest form
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Equation Solving
The TI has an equation solver where solutions can be found for equations equal to zero, and this
can be useful although its worth noting that it only gives you the solution that is closets to your
guess and so some understanding is required. The application POLYSMLT is allowed for further
solving. The Casio has a built in Equation solver that is a bit more versatile. Students should know
how to get the best of equation solving out of their calculators.
Number and Algebra
Rounding
Using 'Float' or 'Fix' from the calculator's memories will round answers to a given degree of
decimal places. this, though, for me is a good example of a skill I prefer students to do manually. it
is crucial that they understand it, given the importance of appropriate accuracy in exams. Answers
should be given to 3 significant figures unless otherwise stated or unless the answer is money and
one or two other exceptions. As a result I feel that students should be very fluent with this and all
rounding skills. The calculator is a distraction here in my view
Standard Index Form /Scientific Notation
The 'SCI' mode can convert numbers to standard form and obviously the reverse can be done by
simply entering the number in standard form (provided the number can be displayed on the
screen). Both of these can be useful for quick answers that limit the possibility for error. Most
important, though, is that students recognise what is meant by 2.34 E-3 when it appears on the
calculator. As such I find this another example of when sound understanding of scientific notation
should be the main aim.
Sequences
Using 'List - Opps' or 'Recursion Mode' can help generate the different terms of a sequence, but
my preference here is the consider the sequences as functions 2where the domain is limited to
integers. The former detracts from the understanding of the structure of sequences and the latter
reinforces these skills. Answers to questions looking for either the nth term or the value of n that
corresponds can be found from the tables that go with the functions.
Simultaneous Equations
Here is where the calculator comes in to its own! This is a very important and often underused
skill. Solving simultaneous equations algebraically is a high order skill with lots of potential for
error. If the equations are linear and in the form ax + by = c then students should use either
'POLYSMLT' which is a loadable, permissible application for TI or 'EQUATION' for Casio. If
equations are in the form y = .... then they can be plotted using the grapher and then the
coordinates of the intersections can be found.
Sets, Logic and Probability
Truth Tables
The List functions on the GDC can be used to look at truth values for certain logical propositions. T
and F are denoted by 1 and 0 and the possible values for these are entered in the first lists as you
would manually. Truth values can then be calculated one column at a time for combinations of
'and' 'or' 'xor' (exclusive disjunction) and 'not'. This can be very useful for long propositions where
the potential for error is large and where a mistake early in the piece changes everything. There
are two significant issues with this functionality though. Firstly, the 'implication' function is not
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available and so any problem involving implication cant be solved. As a result, it would be very
dangerous to rely on the calculator here. Secondly, these ideas are very approachable and can and
should be understood by Maths Studies students. From my experience it is implication that
traditionally causes the most difficulty and the calculator cant help here. I advocate showing
students this function, but I do so with a serious health warning and prefer my students to be able
to do these manually.
Functions
This is one of the areas where the GDC is absolutely crucial and knowing the subtleties of the
calculator equally so! A word of warning! Because the calculators often use detail approximation
techniques you sometimes get answers like 1.00000005 where the answer should be 1. The ability
to check here and make sense is important.
Graph Plotting and adjusting the viewing window
Students must know that they can enter a number of functions in to the graphing menu if they are
in the form y = f(x). (Worth spending some time explaining how p = g(x) can be translated). This
functions can be plotted individually or simultaneously and the screen of the calculator provides a
'window' looking on to this function. The issue then becomes making sure that the window shows
the parts of the function you want to look at. For example its possible that the window will not
show any part of the function at all! The detective work that is then required in changing the
minimum and maximum x and y values and the scale of the window really helps students get to
grips with the Cartesian plane. It is really worth spending time on this function. You can use
various 'Auto', 'zoom' and 'Standard' functions to help, but there is no substitute for being able to
adjust the boundaries of the window yourself and the understanding this gives. Many of the other
functions rely on being able to see the right part of the function in the window!
Tables of values
Another very useful skill that its worth taking the time to help students get to grips with. Students
are not expected to read values from the graphs on their screens. There need to know how to
'Calculate' (see following items) and to read from their tables. Essentially the Calculator holds a
enormous, interactive table of values for each function. The starting point and the steps of the
table can be changed. Students should know how to find y values that correspond to x values and
vice- versa. For example, a quadratic that is equal to 0 can be solved by looking for y values of 0
and the corresponding x values. If the solutions are integers or even terminating decimals then
this can be done using the tables. Likewise when looking for the given term of a geometric
sequence that exceeds a certain amount, the tables can be used.
Calculating Zeros
Students should know how to get the coordinates of the 'x - intercepts', 'roots' or 'zeros' of a given
function. This never seems particularly intuitive and you need to help the calculator localise your
search, as there are likely to be more than one intercept. This is a good alternative to using tables,
particularly when the 'x - values' are not integers and need rounding.
Calculating Intersections
As mentioned before in section 2 in Number and Algebra, the intersections of two functions
provide the simultaneous solutions and the GDC can do this as well. Like the finding of zeros, the
calculator just needs help localising the search.
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Local Maximums and Minimums
Again, with help localising the search, the GDC can find you these coordinates. This is useful for
finding, or checking the equation of the axis of symmetry for quadratics and for checking answers
produced using calculus.
Geometry and Trigonometry
Degree Mode
As mentioned above, students are required to clear the 'RAM' from their calculators before their
exams and this can set the calculator to 'Radian' mode by default. Students must be in the habit of
resetting this!! I try to reset RAM regularly through the course so that Students get used to this.
Statistics
Another area in which the GDC is essential. The statistical capabilities of the calculator go way
beyond those required for this course and there is, in my view, a line past which it is not always
helpful for students to go.
Editing Lists
Like the tables for functions, its very important to be fluent with these. In many ways they are
even more important because here is where the raw data is entered! Students should know how
to clear, edit and sum the lists and how to create new lists that are functions of the existing lists
and sums.
Grouped frequency Tables
The distinction between 1 and 2 variable data can be a huge sticking point and grouped frequency
can add further confusion but save time! Students need to be aware that a second list can be the
frequency for items in the first list and as such 2 lists are still modelling 1 variable data.
1 Variable data
Either as 1 list or 2 (where the second is frequency) students should know how to calculate the 1
variable statistics like Mean, Standard Deviation and the Quartiles. Students need to distinguish
the symbols for Standard Deviation so as they don’t choose the unrequired 'Unbiased Estimate'.
There are still occasional questions that ask students to demonstrate their understanding of how
these statistics are calculated manually. As such it is important to cover it and quite nice to use the
list functions to calculate and check the values given. This is good practice of using lists and
understanding Mean and Standard Deviation.
2 Variable Data - Linear Regression and Correlation Coefficient
Students are expected to analyse bivariate data that can be plotted on a scattergraph by entering
in 2 lists and performing a linear regression analysis. This is easy enough, but there is a small hitch
with the TI calculators. In order to see all the results, calculators must have 'Diagnostics On' which
can be done through the catalogue menu.
Chi2 Test for Independence
Students need to know how to set up a contingency table for observed data as a Matrix and then
how to ask the calculator to calculate the expected frequencies, degrees of freedom and the chi2
statistic. This is a very useful function, but students need to be able to do all of these calculations
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manually as well. Exam questions may ask students to 'show' how an expected frequency is
calculated or that the degrees of freedom are...... My preference is to calculate the chi 2 statistic
manually a number of times before showing students how to do it on their calculators.
Statistical Graph Plotting
This is the grey area for me. Students can plot frequency polygons, histograms, box and whisker
diagrams and scattergraphs on their calculators, but I am not sure of when it is really useful for
them to do so. It would certainly be useful when learning about the concepts, but the GDC is a
very poor second to other statistical software packages like 'Autograph' for example. It may
depend on the access you have to ICT and your own preferences but this does not come under
'Need to Know' in my view.
Introductory Differential Calculus
Measuring Gradients
Students should know how for any given function they can read the gradient at any given value of
x. This is mostly useful for checking manual calculations using the derivative of the type students
are more likely to see on exams. It could also be a useful way of investigating patterns with
gradients when introducing the topic.
Equations of Tangents
This is a very useful function that gives the equation of a tangent to a curve for any given value of
x and students should know it. I do teach it along side the manual calculation though as it is an
application of calculus that could be tested manually.
Functions Skills
I often find that the calculus unit 1 part calculus and 4 parts Algebraic manipulation and
understanding functions. As such the sills listed under the Functions heading are all essential for
this module as well!
Financial Mathematics
Simple currency conversions, buying and selling rates and understanding of the role of commission
are surprisingly confusing. There is no specific function on the calculator that does this. I wish
there was, but do feel that this is a fundamental life skill that students are likely to need. I always
tell them to imagine its their money! The 'Finance' application needed is the TVM solver
TVM Solver for Investments and Loans
Obviously the two are quite different and I like to take the time to point out how difficult it is to
make a calculation for what a monthly repayment should be for a given loan and period! This
application can be used to calculate the unknown variables for different investments and loans
and is very useful indeed. It does, however, require very careful description and demonstration
and, as a teacher, it is really worth trying it out a few times before you show a class. Students
should know weather to enter amounts as positive and negative, how to calculate the number of
time periods and most importantly how to be consistent defining the variables.
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