Math

IB Maths HL workshop page 62

IB Mathematics Higher Level

Internal Assessment – The Exploration

~ Student Guide ~

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1. What is Internal Assessment in IB Mathematics Higher Level ?

Internal Assessment (IA) in Maths HL consists of a single internally assessed component (i.e. marked

by the teacher) called a mathematical exploration (or just the “Exploration”). The Exploration

contributes 20% to your overall IB score for the course.

2. What is the Exploration ?

Your Exploration is a written report (6-12 pages) involving a mathematical topic that interests you.

You will choose a topic in consultation with your teacher after conducting your own research.

3. How is the Exploration assessed ?

Your Exploration will earn a score out of 20 marks based on the following five criteria. Further details

for each criterion and guidance for addressing them is provided later in this guide.

Some important points to consider:

♦ In your Exploration you need to write about mathematics and not just do mathematics.

♦ Any idea, method, content, etc that is not your own must be cited at the point in the Exploration where it is used. Just listing your sources in a bibliography is not enough and may lead to the IB deciding that malpractice has occurred.

♦ The Exploration is an opportunity for you to learn more about a mathematical topic in which you are genuinely interested. You will be rewarded (personal engagement) for explaining your interest in the topic, and for demonstrating curiosity, creativity & independent thinking.

♦ Your audience is your fellow students – that is, you need to write your Exploration so that your classmates in Maths HL can read and understand it. Thus, it is not necessary to explain in great detail basic mathematics that will be familiar to a student in Math HL.

♦ You will be rewarded (reflection) for expressing what you think about the mathematics you are exploring. You should endeavour to pose your own questions and try to answer them using suitably sufficient level of mathematical ideas and procedures.

♦ You will be required to submit a complete draft of your Exploration – containing an introduction, conclusion and all planned content to sufficiently address all five criteria. You will receive feedback on the draft and then be given an opportunity to revise it to submit a final version.

♦ All of the work you do on your Exploration must be your own. When finished with your final version you will be required to sign a ‘declaration’ that states, “I confirm that this work is my own and is the final version. I have acknowledged each use of the words or ideas of another person, whether written, oral or visual.”

Criterion A max 4 marks Communication

Criterion B max 3 marks Mathematical Presentation

Criterion C max 4 marks Personal Engagement

Criterion D max 3 marks Reflection

Criterion E max 6 marks Use of Mathematics

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Maths HL Exploration Timeline

Stage Start date End date Deadline

1. Introduction / Preparation

2. Topic Choice

3. Writing Draft

4. Teacher Feedback (written)

5. Final Version

Notes:

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Step 1 Introduction / Preparation

Read the following two articles. The articles are not examples of IA Explorations but appeared in a professional journal for American math teachers and describe in detail how teachers might engage students in the exploration of a particular mathematical problem. Both articles illustrate good writing about mathematics at a level appropriate for HL Maths. We will discuss strengths and weaknesses of the articles in relation to expectations the Exploration you will write for your IA.

● article 1: Thinking out of the Box … Problem http://education.ti.com/images/online_courses/t3/calculus2/mod28/568-74_nov.pdf

● article 2: Rugby and Mathematics: A Surprising Link among Geometry, the Conics, and Calculus http://wesclark.com/rrr/rugby_and_math.pdf

Step 2 Choosing a topic for your Exploration

Listed on this page and the next page are 200 possible Exploration topics. Browse through the list and do some very quick research (perhaps 5 min spent looking at a Wikipedia page) on any topic that catches your interest. A quick look at some information about one of the topics may reveal some other topic (not on the list) which interests you. You will be given two weeks to organize a ‘short list’ of topics (3 to 5) that you will share with your teacher. You will need to consult with your teacher about any potential topic – regarding three important questions: (1) does the topic involve math at a suitable level for an HL Exploration? ; (2) is the topic narrow enough so that it can be treated sufficiently in a 6-12 page report? ; and (3) does the topic lend itself to demonstrating personal engagement (criterion C)? That is, can you envision some way that you could apply something of your own – your own viewpoint, your own examples, your own models (conceptual or physical), your own questions & ideas, etc.  Your topic must be approved by your teacher by the given deadline 

 200 Exploration ideas/topics  Algebra & Number Theory

Modular arithmetic Euler’s identity: 1 0 i

e    Goldbach’s conjecture

Chinese remainder theorem Probabilistic number theory Fermat’s last theorem Applications of complex numbers Natural logarithms + complex numbers Continued fractions Diophantine equations Twin primes problem Hypercomplex numbers General solution of a cubic equation Diophantine application: Cole numbers Applications of logarithms Odd perfect numbers Polar equations Euclidean algorithm for GCF Patterns in Pascal’s triangle Palindrome numbers Finding prime numbers Factorable integers of the form ak + b Random numbers Algebraic congruences Pythagorean triples Inequalities & Fibonacci numbers Mersenne primes Combinatorics – art of counting Magic squares & cubes Boolean algebra Loci and complex numbers Roots of unity Matrices & Cramer’s rule Divisibility tests Complex numbers & transformations Egyptian fractions Graphical representation of roots of complex numbers

Calculus/Analysis & Functions Mean Value theorem Torricelli’s trumpet (Gabriel’s horn) Integrating to infinity Applications of power series Newton’s law of cooling Hyperbolic functions Fundamental theorem of calculus Brachistochrone (min.time) problem The harmonic series Second order differential equations l’Hopital’s rule and evaluating limits Torus – solid of revolution

Probability & Probability Distributions Normal distribution and natural phenomena The Monty Hall problem Monte Carlo simulations Random walks Insurance and calculating risks Poisson distribution and queues Determination of  by probability Lotteries Bayes’ theorem The birthday paradox

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Geometry Non-Euclidean geometries Cavalieri’s principle Packing 2D and 3D shapes Ptolemy’s theorem Hexaflexagons Heron’s formula Geodesic domes Proofs of Pythagorean theorem Tesseract – a 4D cube Minimal surfaces & soap bubbles Map projections Penrose tiles Tiling the plane – tessellations Morley’s theorem Cycloid curve Symmetries of spider webs Fractal tilings Euler line of a triangle Fermat point - polygons & polyhedral Pick’s theorem & lattices Conic sections Properties of a regular pentagon Nine-point circle Regular polyhedral Geometry of the catenary curve Euler’s formula for polyhedral Stacking cannon balls Eratosthenes’ - earth’s circumference Ceva’s theorem for triangles Area of an ellipse Constructing a cone from a circle Conic sections as loci of points Consecutive integral triangles Mandelbrot set and fractal shapes Curves of constant width Sierpinksi triangle Squaring the circle Polyominoes Reuleaux triangle Architecture and trigonometry Spherical geometry

Statistics & Modelling Logistic function & constrained growth Modelling growth of tumours Traffic flow Modelling epidemics/spread of a virus Correlation coefficients Hypothesis testing Modelling the shape of a bird’s egg Central limit theorem Modelling radioactive decay Modelling growth of computer power Least squares regression Regression to the mean Modelling change in record performances for a sport

Numerical Analysis Methods for solving differential eqns Linear programming Fixed point iteration Methods of approximating  Applications of iteration Newton’s method Estimating size of large crowds Generating the number e Descartes’ rule of signs

Logic & Sets Codes and ciphers Set theory and different ‘size’ infinities Mathematical induction (strong) Proof by contradiction Proving a number is irrational

Topology & Networks Knots Steiner problem Chinese postman problem Travelling Salesman Problem Königsberg bridge problem Handshake problem Möbius strip Klein bottle

Games & Game Theory The prisoner’s dilemma Sudoku Gambler’s fallacy Card games Knight’s tour in chess

Physical, Biological & Social Sciences Radiocarbon dating Gravity, orbits & escape velocity Biostatistics Mathematical methods in economics Genetics Crystallography Computing centres of mass Elliptical orbits Predicting an eclipse Logarithmic scales-decibel, Richter, etc Change in BMI for a person over time Fibonacci sequence and spirals in nature Concepts of equilibrium in economics

Miscellaneous Paper folding Designing bridges Mathematical card tricks Methods of approximating  Barcodes Applications of parabolas Curry’s paradox – ‘missing’ square Voting systems Terminal velocity Music – notes, pitches, scales, etc Towers of Hanoi puzzle Photography Flatland by Edwin Abbott (book) Art of M.C. Escher Harmonic mean Sundials Navigational systems A Beautiful Mind (film) The abacus Construction of calendars Slide rules Different number systems Mathematics of juggling Airline routes Global positioning system (GPS)

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Step 3 Write a complete draft

Before you start writing your Exploration be sure to carefully read through the details for all five of

the assessment criteria that is at the very end of this guide. Along with a brief description and

achievement level descriptors, there is also helpful guidance notes for each criterion.

A draft is not an abbreviated or incomplete version of your Exploration. It must be complete –

including an introduction, a conclusion and a bibliography – with sufficient content to address your

stated objective(s) and be in the range of 6 to 12 pages (spacing 1½, font Times New Roman). Your

Exploration needs to be logically organized; use appropriate mathematical terminology and notation;

include explanatory diagrams, graphs, tables, etc; contain citations to indicate where a source is

used; and focuses on the relevant mathematics. It is important to include your own thoughts,

questions, reflections & ideas when possible. Write in the first person, e.g. “I decided that the best

method is _____ because I realized that …”

Although the Exploration is an individual assignment and all the work must be your own, you are

strongly encouraged to regularly consult with your teacher. Your teacher can provide verbal

guidance and feedback while you are writing your draft.

 Submit a paper and electronic version of your draft to your teacher by the given deadline 

Step 4 Teacher feedback

Your teacher will provide written feedback on the draft of your Exploration. Be sure to ask questions

about any comments / feedback that you do not completely understand.

Step 5 Submit final version of your Exploration

From the time you receive written feedback on your draft you will have 6 weeks (3 school weeks & 3

weeks of the winter holiday) to revise your draft and complete the final version of your Exploration.

Before submitting your final version complete the student checklist on the next page →

 Submit a paper & electronic version of your final Exploration to your teacher by the deadline 

See “The Exploration – Top Tips” at the very end of this Student Guide.

It lists five important points that will help you with your Exploration.

It is absolutely critical that you are completely familiar with all five of the assessment

criteria. All of the details for the assessment criteria appear on pages 8-10 in this

Guide. Carefully read the Descriptors and Further Guidance for all of the five

criteria. Ask your teacher if you have questions or need further clarification.

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Mathematics HL Exploration  Student Checklist Student: _________________________________________ date: __________________

1. Is your report written entirely by yourself – and trying to avoid simply replicating

work and ideas from sources you found during your research?  Yes  No

2. Have you strived to apply your personal interest; develop your own ideas; and use

critical thinking skills during your exploration and demonstrate these in your report?  Yes  No

3. Have you referred to the five assessment criteria while writing your report?  Yes  No 4. Does your report focus on good mathematical communication – and read like

an article for a mathematical journal?  Yes  No

5. Does your report have a clearly identified introduction and conclusion?  Yes  No 6. Have you documented all of your source material in a detailed bibliography

in line with the IB academic honesty policy?  Yes  No

7. Not including the bibliography, is your report 6 to 12 pages?  Yes  No 8. Are graphs, tables and diagrams sufficiently described and labelled?  Yes  No 9. To the best of your knowledge, have you used and demonstrated mathematics

that is at the same level, or above, of that studied in IB Mathematics HL?  Yes  No

10. Have you attempted to discuss mathematical ideas, and use mathematics, with a

sufficient level of knowledge, understanding, sophistication and rigour?  Yes  No

11. Are formulae, graphs, tables and diagrams in the main body of text?  Yes  No

(preferably no full-page graphs; and no separate appendices)

12. Have you used technology – such as a GDC, spreadsheet, mathematics software,

drawing & word-processing software – to enhance mathematical communication?  Yes  No

13. Have you used appropriate mathematical language (notation, symbols,

terminology) and defined key terms?  Yes  No

14. Is the mathematics in your report performed precisely and accurately?  Yes  No

15. Has calculator/computer notation and terminology not been used?  Yes  No

( 2

y x , not ^ 2y x ;  , not  for approx. values;  , not pi; x , not abs(x); etc)

16. At suitable places in your report – especially in the conclusion – have you included

reflective and explanatory comments about the mathematical topic being explored?  Yes  No

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Criteria for HL Exploration (IA)

Criterion A: Communication (4 marks) This criterion assesses the organization and coherence of the exploration. A well-organised exploration has

an introduction, a rationale (a brief explanation of why the topic was chosen), describes the aim of the

exploration and has a conclusion. A coherent exploration is logically developed and easy to follow.

 Further Guidance 

▪ A complete exploration will have all steps clearly explained, and will meet its aim.

▪ Organization refers to the overall structure or framework, including the introduction, body, conclusion etc.

▪ A coherent exploration displays a logical development and is not difficult to follow (‘reads well’).

▪ A concise exploration remains focused on the overall aim and avoids irrelevant material.

▪ Key ideas and concepts need to be clearly explained.

▪ Graphs, tables and diagrams should be embedded in the text where most appropriate and not be

put in an appendix at the end of the document.

▪ The use of technology is not required but strongly encouraged where appropriate.

▪ It is absolutely critical that the use of a source is cited (footnoted) at the location where it is used.

▪ Your bibliography must list all sources (books, websites, etc) you consulted when writing your Exploration.

Criterion B: Mathematical Presentation (3 marks) This criterion assesses to what extent you are able to clearly and effectively use multiple forms of

mathematical representation such as formulae, diagrams, tables, graphs and models.

 Further Guidance 

▪ You are expected to use mathematical language (notation, symbols & terminology) when communicating

mathematical ideas, reasoning and findings.

▪ You should use appropriate technology such as graphic display calculators; and software such as equation

editors, spreadsheets, dynamic geometry, computer algebra, drawing and word-processing software along

with other mathematical software to enhance the presentation of mathematics in your Exploration.

▪ The meaning of key terms should clear and any variables or parameters should be explicitly defined.

▪ All graphs, tables & diagrams should be clearly labelled – and include captions where appropriate.

▪ Do not use calculator or computer notation unless it is software generated and cannot be changed.

Achievement Level Descriptor

0 The Exploration does not reach the standard described by the descriptors below.

1 The Exploration has some coherence.

2 The Exploration has some coherence and shows some organization.

3 The Exploration is coherent and well organized.

4 The Exploration is coherent, well organized, concise and complete.

Achievement Level Descriptor

0 The Exploration does not reach the standard described by the descriptors below.

1 There is some appropriate mathematical presentation.

2 The mathematical presentation is mostly appropriate.

3 The mathematical presentation is appropriate throughout.

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Criterion C: Personal Engagement (4 marks) This criterion assesses the extent you engage with the exploration and make it your own. This includes

independent thinking, creativity, addressing personal interest and presenting math ideas in your own way.

 Further Guidance 

▪ It is important to choose a topic in which you are genuinely interested.

▪ If it is necessary to include mathematical work from a source such as a textbook in your Exploration then

you should endeavour to insert your own comments and description of the work as much as possible.

▪ Ways to show personal engagement include: investigating your own questions & conjectures; making up

your own examples; presenting ideas & results in your own words; creating your own models or functions.

Criterion D: Reflection (3 marks) This criterion assesses how well you review, analyze and evaluate your exploration. Although reflection may

be seen in the conclusion, it should also exist throughout the exploration. Reflection may be demonstrated

by considering limitations or extensions, and relating mathematical ideas to your own previous knowledge.

 Further Guidance 

▪ Simply describing results represents limited or superficial reflection. To achieve a score higher than 1 you

will need to provide deeper and more sophisticated consideration of methods and results.

▪ Ways of showing meaningful reflection include: linking results to the aim of your Exploration; commenting

on what you have learned; considering limitations; or comparing different mathematical approaches.

▪ Ways of showing critical reflection include: considering implications of results; discussing strengths and

weaknesses of methods; considering different perspectives; making links between different areas of math.

▪ Substantial evidence is likely to mean that reflection is present throughout the exploration.

Criterion E: Use of Mathematics (6 marks) This criterion assesses to what extent you use mathematics in your exploration. The mathematics explored

should either be part of the syllabus, or at a similar level, or beyond. It should not be completely based on

mathematics listed in the prior learning topics. If the level of mathematics is not commensurate with the

course, a maximum of two marks can be awarded for this criterion. A piece of mathematics can be regarded

as correct even if there are a few minor errors so long as they do not cause a disruption to the flow of

mathematics or lead to an incorrect or inaccurate result.

Achievement Level Descriptor

0 The Exploration does not reach the standard described by the descriptors below.

1 There is evidence of limited or superficial personal engagement.

2 There is evidence of some personal engagement.

3 There is evidence of significant personal engagement.

4 There is abundant evidence of outstanding personal engagement.

Achievement Level Descriptor

0 The Exploration does not reach the standard described by the descriptors below.

1 There is evidence of limited or superficial reflection.

2 There is evidence of meaningful reflection.

3 There is substantial evidence of critical reflection.

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 Further Guidance 

▪ It is critical that you clearly demonstrate that you understand the mathematical concepts and methods that

you write about in your Exploration.

▪ Sophistication in mathematics may include understanding & use of challenging math concepts, looking at a

problem from different perspectives and seeing underlying structures to link different areas of mathematics.

▪ Rigour involves clarity of logic and language when making mathematical arguments and calculations.

▪ Precise mathematics is error-free and uses an appropriate level of accuracy at all times.

1. Choose a topic in consultation with your teacher that: (i) you’re interested in, (ii) involves math at a level

suitable for Math HL, (iii) is narrow enough for 6-12 pages, (iv) has opportunities for personal engagement.

2. Your Exploration must have an aim or objective which involves doing some mathematics. It is important

to maintain a focus on the overall aim/objective and a focus on mathematical concepts and methods.

3. Although all the work on your Exploration must be your own, do not hesitate to ask your teacher for

advice and feedback at any stage. Your teacher will provide written feedback on your draft.

4. Be sure you fully understand the expectations of the five assessment criteria, and refer back to them while

you are planning and writing your Exploration.

5. The Exploration is an opportunity to complete a significant assessment item (20% of IB score) while not

under the pressure of timed exam conditions. Take advantage of the opportunity by following instructions,

meeting deadlines, engaging & reflecting in your own way, and enjoying some math you are interested in.

Achievement Level Descriptor

0 The Exploration does not reach the standard described by the descriptors below.

1 Some relevant mathematics is used. Limited understanding is demonstrated.

2 Some relevant mathematics is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated.

3 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Good knowledge and understanding are demonstrated.

4 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication expected. Good knowledge and understanding are demonstrated.

5 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication and rigour expected. Thorough knowledge and understanding are demonstrated.

6 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and reflects the sophistication and rigour expected. Thorough knowledge and understanding are demonstrated.

The Exploration – Top Tips