Download Internal Assessment_Exploration Guideline

IB Math Standard Level
THE MATHEMATICAL EXPLORATION – INTERNAL ASSESSMENT (IA)
STAGE 1: THE PROPOSAL DUE _________________________________

The Mathematical Exploration is report written about your exploration of a mathematical topic in which
you are genuinely interested. This is an individual assignment that must be completed with no
additional help. Your teacher will help guide you and must be consulted on a regular basis.

To begin your exploration, you must submit a proposal to your math teacher. Here are some helpful
(hopefully) hints:
 Take a subject of interest and investigate the math involved. Pay particular attention to Criterion
E (see below) and ask yourself if the level of math involved is appropriate.
 Think of something in real life and how math applies/determines it.
 What mathematics can be used to predict the way something works?
 What mathematics will predict the motion of something?
 See the list at the end of this document for ideas.
 Your topic must be approved by your teacher and must be unique (ie. no one else has your topic)

Your proposal must be written in essay form (no point form) and must clearly indicate:
 the topic you are investigating.
 why this topic is interesting to you.
 the mathematics that may be involved. Include enough detail such that the reader can easily
determine the level of mathematics. You do not need to investigate all of the math (this is for
September) just give the reader an idea of the math applicable to your exploration.
 what you are hoping to accomplish in the paper. What is your goal? What is the purpose of the
paper?

Mechanical Considerations:
 Length – 500 to 1000 words (Typed)
 1.5 to double spacing
 12 point font
Note: The exploration must be completed in consultation with your teacher. Although it is fine if you want to do
some research over the summer, please note that your teacher must be involved so it would be preferable if you
began your writing and most of your investigation in September.
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STAGE 2: THE PAPER – DUE _____________________________

The Mathematical Exploration is report written about your exploration of a mathematical topic in which
you are genuinely interested.

This is an individual assignment that must be completed with no additional help. Your teacher will help
guide you and must be consulted on a regular basis.

It will contribute 20% of your final IB mark and thus must be taken seriously.

The IA will earn a numerical score out of a total of 20 possible marks. See below for the 5 criteria that will
be assessed.

Must be completed by ________________________2014 (the exact date will be given to you by your teacher).

IA differs from the IB written exams (Paper 1 & 2 – 80%) in a number of ways:
 There are no strict time constraints.
 You choose your own mathematical topic (this must be unique to other students’) in which
you have a GENUINE personal interest. Your topic must be approved by your teacher.
 You are WRITING about mathematics, not just doing mathematical procedures.
 Regular discussion with your teacher is REQUIRED.

The Mathematical Exploration will require a significant amount of time and energy to complete successfully.
It should not be approached as an extended homework assignment. You will be required to do a
considerable amount of research, analysis, reading, consultation (with your teacher only), thinking,
writing, editing, mathematical work, problem solving and proofreading.

You must keep a log book recording the work you put into your exploration. It must be shown to your
teacher during all student-teacher consultations (these are mandatory). The log book must include:
 any books, website, magazines, that you consult
 the date, time (number of hours) and description of all activities related to your exploration
 the date, time and description of all discussions with your teacher
note: a template will be provided for you; you may use a different format but it must be on paper
when you bring it to your meeting with your teacher
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IA Assessment Criteria – It is very important that you familiarize yourself with these assessment
criteria and refer it them when you are writing your report.
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Suggestions in components necessary for a successful exploration:
Criterion A: Communication
 You must include:
 An introduction in which you should discuss the context of the exploration
 A rationale which should include an explanation of why you chose this topic.
 A description of the aim of the exploration which should be clearly identifiable.
 A conclusion.
 Must “read well” which means it must be logically developed, easy to follow and concise (avoid
irrelevancies).
 Graphs, tables and diagrams should be clearly labelled.
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 6 to 12 pages using a 11 -12 point easily read font (ie. Times New Roman)
 References must be cited. Your exploration should contain appropriate footnotes and a bibliography.
Criterion B: Mathematical Presentation
 Use appropriate mathematical language (notation, symbols and terminology)
 Use multiple forms of mathematical representation such as formulae, diagrams, tables, charts, and graphs.
 Choose and use appropriate ICT tools such as graphic display calculators, mathematical software,
spreadsheets, databases, drawing and word-processing software.
 Define key terms and explicitly define variables.
 Express your results to an appropriate degree of accuracy.
 Include scales and labels on graphs; include concise, descriptive headings on tables.
 Do not use calculator notation (ie. use 2x and not 2^x)
Criterion C: Personal Engagement
 You should choose a topic that you are genuinely interested in as it will be easier to display personal
engagement.
 Suggestions for demonstrating personal engagement (this must be in your paper):
 Thinking and working independently; Thinking creatively
 Addressing your personal interests
 Presenting mathematical ideas in your own way, using simple language to describe complex ideas
 Asking questions, making conjectures and investigating mathematical ideas
 Looking for and creating mathematical models for real-world situations
 Considering historical and global perspectives
 Exploring unfamiliar mathematics
Criterion D: Reflection
 Although reflection may be seen in the conclusion to the exploration, it may also be found throughout the
exploration. This assesses how well you review, analyse and evaluate your exploration.
 You can show reflection by:
 Discussing the implications of your results
 Considering the significance of your findings and results
 Stating possible limitations and/or extensions to your results
 Making links to different fields and/or areas of mathematics.
Criterion E: Use of Mathematics
 The mathematics you explore should be either part of the syllabus, or at a similar level (or beyond).
 If the level of mathematics is inadequate, your maximum achievement level will be two marks.
 Your mathematics must clearly demonstrate that you fully understand the mathematics.
Selecting a Topic- your topic should be mathematics that is of genuine interest to you
The following are a list of ideas/topics. Some of these could be a title or focus of a Mathematical Exploration, while
others will require you to investigate further to narrow your focus.
Modular arithmetic
Applications of complex numbers
General solution of a cubic equation
Patterns of Pascal’s Triangle
Pythagorean triples
Loci and complex numbers
Egyptian fractions
Algebra and Number Theory
Goldbach’s conjecture
Diophantine equations
Applications of logarithms
Finding prime numbers
Mersenne primes
Matrices and Cramer’s rule
Complex numbers and transformations
Probabilistic number theory
Continued fractions
Polar equations
Random numbers
Magic squares and cubes
Divisibility tests
Euler’s identity: ei  1  0
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Chinese remainder theorem
Twin primes problem
Odd perfect numbers
Factorable sets of integers of the form
ak+b
Combinatorics – art of counting
Fermat’s last theorem
Hypercomplex numbers
Euclidean algorithm for GCF
Algebraic congruences
Natural logarithms of complex number
Diophantine application: Cole numbers
Palindrome numbers
Inequalities related to Fibonacci numbers
Boolean algebra
Roots of unity
Recurrence expressions for phi (golden
ratio)
Fermat’s little theorem
Graphical representation of roots of
complex numbers
Prime number sieves
Geometry
Cavalieri’s principle
Hexaflexagons
Proofs of Pythagorean theorem
Map projections
Morley’s theorem
Fractal tilings
Pick’s theorem and lattices
Packing 2D and 3D shapes
Heron’s formula
Minimal surfaces and soap bubbles
Tiling the plane – tessellations
Cycloid curve
Euler line of a triangle
Properties of a rectangular pentagon
Conic sections
Regular polyhedral
Stacking cannon balls
Nine-point circle
Euler’s formula for polyhedral
Ceva’s theorem for triangles
Conic sections as loci of points
Mandelbrot set and fractal shapes
Squaring the circle
Architecture and trigonometry
Geometric structure of the universe
Consecutive integral triangles
Curves of constant width
Polyominoes
Spherical geometry
Rigid and non-rigid geometric structures
Calculus/analysis and functions
Torricelli’s trumpet (Gabriel’s horn)
Newton’s law of cooling
Differential equations
Integrating to infinity
Fundamental theorem of calculus
L’Hopital’s rule and evaluating limits
Non-Euclidean geometries
Ptolemy’s theorem
Geodesic domes
Tesseract – a 4D cube
Penrose tiles
Symmetries of spider webs
Fermat point for polygons and
polyhedral
Tangrams
Geometry of catenary curve
Eratosthenes – measuring earth’s
circumference
Constructing a cone from a circle
Area of an ellipse
Sierpinksi triangle
Reuleaux triangle
Gyroid – a minimal surface
Mean value theorem
Applications of power series
Brachistochrone (minimum time)
problem
Hyperbolic functions
Projectile motion
Traffic flow
Modelling epidemics/spread of virus
Central limit theorem
Modelling radioactive decay
Regression to the mean
The Monty Hall problem
Insurance and calculating risks
Lotteries
Normal distribution and natural
phenomena
The harmonic series
Why e is base of natural logarithm
function
Statistics and modelling
Logistic function and constrained growth
Modelling the shape of a bird’s egg
Modelling change in record
performances for a sport
Least squares regression
Modelling growth of computer power
past few decades
Probability and probability distributions
Monte Carlo simulations
Poisson distribution and queues
Bayes’ theorem
Medical tests and probability
Torus – solid of revolution
Modelling growth of tumours
Correlation coefficients
Hypothesis testing
Modelling the carrying capacity of the
earth
Random walks
Determination of  by probability
Birthday paradox
Probability and expectation
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The prisoner’s dilemma
Poker and other card games
Zero sum games
Knots
Travelling salesman problem
Mobius strip
Codes and ciphers
Proof by contradiction
Linear programming
Applications of iteration
Generating the number e
Radiocarbon dating
Biostatistics
Computing centres of mass
Fibonacci sequence and spirals in nature
Concepts of equilibrium in economics
Column buckling – Euler theory
Paper folding
Mathematical card tricks
Applications of parabolas
Flatland by Edwin Abbott
Photography
Sundials
Construction of calendars
Mathematics of juggling
Origami
Design of product packaging
Games and game theory
Sudoku
Knight’s tour in chess
Topology and networks
Steiner problem
Konigsberg bridge problem
Klein bottle
Logic and sets
Set theory and different ‘size’ infinities
Zeno’s paradox and Achilles and the
tortoise
Numerical analysis
Fixed-point iteration
Newton’s method
Descartes’ rule of signs
Physical, biological and social sciences
Gravity, orbits and escape velocity
Genetics
Elliptical orbits
Predicting an eclipse
Mathematics of the ‘credit crunch’
Miscellaneous
Designing bridges
Curry’s paradox – ‘missing’ square
Music – notes, pitches, scales…
Terminal velocity
Art of M.C. Escher
Navigational systems
Slide rules
Global positioning system (GPS)
Napier’s bones
Mathematics of weaving
Gambler’s fallacy
Billiards and snooker
Chinese postman problem
Handshake problem
Mathematical induction
Four colour map theorem
Methods of approximating 
Estimating size of large crowds
Methods for solving differential
equations
Mathematical methods in economics
Crystallography
Logarithmic scales – decibel, Richter, etc.
Change in a person’s BMI over time
Branching patterns of plants
Mathematics of rotating gears
Barcodes
Voting systems
Towers of Hanoi puzzle
Harmonic mean
The abacus
Different number systems
Optical illusions
Celtic design/knotwork
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Planning – Mind Mapping
Stimuli
sport
algorithms
sine
e
space
volcanoes
games
codes
tiling
viruses
play
biology
physics
archaeology
cell phones
musical harmony
electricity
orbits
diet
symmetry
the internet
population
health
π
business
chemistry
computers
music
motion
water
food
Euler
architecture
communication
agriculture
dance
geography
economics
information technology in a global society
Choose a stimulus and create your own mind map here.
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Checklist
Item
Yes
Partially
Is the work entirely yours?
Have you chosen a topic that you are interested in and developed your own ideas? Is it evident in your
exploration?
Have you explained the reason why you have chosen your topic in your exploration?
Is the aim of your exploration included in your introduction?
Do you have an introduction and conclusion? Is your exploration organized?
Have you defined key terms/variables?
Have you used appropriate mathematical language (notation, symbols and terminology) consistently
throughout your exploration?
** Calculator/computer notation should not be used. **
Have you used more than one form of mathematical representation? Are all graphs, tables and
diagrams sufficiently described and labeled?
Are formulae, graphs, tables and diagrams in the main body of the text? No full-page graphs and no
separate appendices.
Have you used technology to enhance your exploration?
Have you explained what you are doing at all times? Explanatory comments should be seen throughout
your exploration?
Have you used mathematics that is commensurate with the Standard Level course (or beyond)?
Is the mathematics in your exploration correct?
Have you reflected on your finding at appropriate places in your exploration, particularly in your
conclusion?
Have you considered limitations and extensions in your reflection?
Have you considered the assessment criteria when writing your exploration? Have you self assessed
your exploration?
Is your exploration approximately 6 to 12 pages long?
Have you referenced your work in a bibliography?
Have you had someone else read your exploration to ensure that the communication is good? Does it
have flow and coherence? Is it easily understandable? Does it read well?
Have you completed your self-assessment?
Have you submitted a first draft to your teacher and used the feedback to improve your report?
If your teacher requires you to submit your report to turnitin.com, have you done so?
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No
Authenticity
Plagiarism
This includes copying quotes, information and ideas, directly or paraphrased, from books and websites.
Collusion
This includes working closely with another student such that the work between the two students is
similar.
Ensuring academic honesty
To prevent plagiarism, you need to cite your sources correctly and include any sources in your
bibliography. If you have questions on how to properly cite your sources, seek advice from your teacher
or from the school librarian.
To prevent collusion, you should discuss ideas with other students, but you should never giver another
student your work, either in print or electronically.
Turnitin
To help detect plagiarism, some schools use the website turnitin.com. Students at these schools need to
submit their first draft and final copy of their exploration through turnitin.com. In order to access the
class' turnitin.com page, you will need the following information:
Class ID:
Join password:
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Recommended Technology
Some examples of technology include:



any kind of calculators, the internet, data logging devices
word processing packages, spreadsheets, graphics packages
statistics packages or computer algebra packages.
Great software for working with graphs, diagrams, functions, spreadsheets, statistics, calculus and much,
much more.
www.geogebra.org
A modern, easy-to-learn, programming language that is great for writing simulations. There are loads of
tutorials available: just google “python tuts”.
www.python.org
Not sure how to use your TI-83/84. The Baltimore County Community College website has a set of
excellent video tutorials that are mainly statistics-oriented, but that are also useful for a general
familiarisation.
http://faculty.ccbcmd.edu/elmo/math141s/TIVideo/TIWebpage.htm
An online graph plotter with graphing capabilities similar to those of your graphical calculators.
www.fooplot.com
A really powerful search / CAS engine. (For example, type “find antiderivative of f(x) = 3x” into the search
bar.)
www.wolframalpha.com
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