**TOK Resorces **

1. Plus Maths – Probably the best maths website on the internet – absolutely packed with articles that explore ToK issues, real life mathematics and careers in maths.

2. Maths Illuminated 13 interactive units which explore everything from infinity to non-Euclidean geometry.

3. Mrs Davis’ ToK – A ToK site put together by Mrs Davis with a number of pdfs, powerpoints and videos

4. Tok Talk A great ToK site with a large number of ToK resources including some maths ToK articles.

5. Larry Ferlazzo’s ToK site Hundreds of links and ideas for ToK maths articles in the news.

**IB Mathematics**

1.3D Function Grapher A useful 3D grapher if you don’t have Autograph.

2. 3D Function Online Archimy

3.A collection of JAVA apps for teaching calculus, vectors and complex numbers. JAVA apps for use with IB maths.

4.GeoGebra Tutorial GeoGebra isn’t especially intuitive – this is how to use it.

5.A large collection of GeoGebra Applets Some great apps on everything from the unit circle, differentiation from first principles, volume of revolutions etc.

6.GeoGebraTube with even more Geogebra applets. See above!

7.Mandelbrot and Julia set generator. A great extension for lessons on complex numbers.

8.Learning Maths Forum A good forum to use in addition to the OCC site.

9.IB STATS Bulletin Very useful statistical information on grade distributions.

10.Nick’s Mathematical Puzzles A fantastic collection of difficult maths puzzles – HL or very able SL.

11.Oxford Education Math HL Support material A great addition to the new Oxford texbooks – full worked answers and calculator resources.

12.Oxford Education Math SL Support material

13.Oxford Education Math Studies support material.

14.Risps A large collection of extension problems for SL and HL students.

15.Online curriculum centre.The first stop resource for all IB documents and a great forum for support.

16.Graphing Calculator Tutorials How to use the TI-84.

17.IB Maths past papers Free past papers from Freeexampapers.com.

18.Investopedia.com – stock market simulator. A great resource to link with financial maths and maths careers.

19.Big number scientific calculator For when the TI-84 just isn’t powerful enough!

20. Youtube Numberphile channel About 100 short Youtube videos explaining everything from infinity to prime encryption to Graham’s number. Also indexed here.

21. Educational Freeware. Some great freeware and website links.

22. Maths Fail. A great blog with everything from maths jokes to puzzles to games and news.

23. Using the Internet to Spice up your Maths Lessons. This is an incredible resource which maps all possible areas of maths resources online.

24. Maths Tube . An excellent resource – which has grouped together a huge number of top quality maths videos.

25. Over 30,000 maths contest problems. This is a blog post which groups together all the world’s maths challenge materials – both questions and solutions.

**General Mathematics sites**

1.Emaths A comprehensive collection of resources – in particular a lot of KS3 resources, tarsia and SATS papers.

2. NCTM Illuminations A large number of searchable enrichment activities, plus a large number of apps (under activities) to help with graphing.

3.Maths is fun A large number of maths puzzles – algebra, number and logic.

4.Mymaths (Paid) login required – but it contains a huge number of resources for KS3 and IGCSE.

5.General Nrich A constantly updated and searchable site with lots of graded maths problems. 6.CIMT Lots of resources for real life maths, code breaking and puzzles.

7. TES Maths resources Probably the biggest teacher resource site on the net. Search for it and someone’ll have done a lesson handout already.

8.MrBarton A fantastic resource with powerpoints, handouts, tarsias excel sheets etc for all KS3 and IGCSE topics.

9.Mathematics Illuminated Units Entire online units which explore everything from infinity, topology, game theory and extra dimensions.

10. Plus Maths A massive site with a large number of maths articles exploring careers, real life maths and current mathematical projects. Also lots of puzzles to try.

11.World Maths Day A chance for students to compete in lightning fast arithmetic calculations against contestants from across the world. (March each year)

12.Great Maths Teaching Ideas A mixture of blog posts, videos and maths teaching resources.

13. Maths Careers. An excellent resource to show the huge range of jobs that mathematicians can enter.

14. Wisweb A great collection of apps – especially useful for 3D visualisation.

15. Maths Fail. A great blog with everything from maths jokes to puzzles to games and news.

16. Using the Internet to Spice up your Maths Lessons. This is an incredible resource which maps all possible areas of maths resources online.

17. Over 30,000 maths contest problems. This is a blog post which groups together all the world’s maths challenge materials – both questions and solutions.

18. Crypto Corner is the newest and best code making and code breaking website online – it’s got a huge amount of code information and also allows you to generate your own codes.

**Number**

1. More with Daniel Tammet – An autistic Savant who “sees” numbers (synathesia).

2. Funny Youtube video “proving” 25 divided by 5 = 14.

3. Youtube video with a “proof” that 1 = 2

4. Excellent Maths Illuminated unit on primes

5. Youtube video of Fibonacci numbers in nature.

**Algebra**

1. Nrich discussion on the compounding chess squares legend.

2. Carl Sagan on the speed of light and time dilation.

**Geometry shapes and measures**

1. Marcus du Sautoy TED talk about universal symmetry in nature.

2. Excellent Maths Illuminated unit on symmetry

3. Maths Illuminated resources on the fourth dimension.

4. Carl Sagan video on flatland and tesseracts.

5. Youtube video on how to imagine the 10th dimension.

6. Michael Kaku youtube video on string theory and multi-dimensions.

7.3D shapes solid shapes and their nets

8.App for transformations

9.Nets of cubes

10.Nets of shapes

11.Shapes that tessellate

12.Symmetry face research

13.symmetry mona lisa

14.TANGRAMS

**Statistics and Probability**

1. Wikipedia on Lottery mathematics.

2. Game theory and expected value for Deal or No Deal.

3. Youtube double slit experiment and wave function collapse.

4. Heisenberg’s Uncertainty Principle, Schrodinger’s Cat and probability as the fabric of reality.

5. Kaku video discussing determinism, free will and probability.

6. Collection of resources demonstrating misleading data.

7. Youtube BBC video discussing determinism, probability and chaotic systems.

8. Collection of Plus Maths articles discussing a variety of chaotic systems in real life.

9. A short BBC Horizon clip which discusses the Computer Simulation hypothesis – which is made by considering conditional probability.

10.Reaction Time baseball

11.Reaction Time startle

12.Sheep Reaction time BBC

13.Data handling actury life expectancy

14.Horse race 2 dice.

15.Gapminder data comparison

16.Guardian datablog

17.OECD stats

18.UN stats

19.Census at school data

20.Dice Roller

21Data handling actury life expectancy

**Maths Games and Puzzles**

1.A large number of Maths challenge puzzles.

2.Math and Logic Games

3.Make 24

4.The Ultimate Puzzle Site – Puzzles, riddles, quizzes, and tests.

5.Create and Solve Substitution Ciphers online

6.Morse Code Translator

7.Matchstick Puzzles

8.Some more Matchstick Puzzles

9.Math and Logic Puzzles from Mathisfun

10.Official Maths challenge materials

11.More maths puzzles from Galileo

12. Math Enrichment topics from Mathwire

13.Open-Ended Math Problems

14.Puzzles & Brain Teasers

15.Puzzles from Freepuzzles

16.The Ultimate Puzzle Site – Puzzles, riddles, quizzes, and tests….

17.Countdown Maths app.

**Art**

1. Mandelbrot image Youtube video.

2. Short BBC documentary on fractals and Mandelbrot

3. Youtube video describing how to generate Julia sets.

4 Short BBC documentary looking at Escher’s maths through art.

**Music**

1. Donlald Duck in Math Magicland video re Pythagoras and music.

2. Youtube video of Bach’s Crab Cannon looped on a Mobius strip.

**History**

1. Youtube video on Newton’s beliefs and eccentricities.

**IGCSE topics**

1. A large number of Plus Maths articles on vectors in the real world.

2. Youtube video – honey bees use vectors dancing to describe the position of flowers.

**Youtube**

1.Crazy Vectors 747 Crosswind Landing

2.Donald in Mathmagic Land

3.GCSE – Distance-Time Speed-Time Graphs

4.How to Draw a Fibonacci Spiral

5.ESP card trick

6.Learning upgrade fractions song

7.learning upgrade quadratics song

8.Maths trick

9.O2 learn – Dr Maths presents Getting Triggy With It

10.Mandlebrot

11.TED-Ed Subjects

12. Math Learning Upgrade song – circles.

13. Learning Upgrade song – mean, median, mode

14.YouTube – maths trick

15. Cartoon video explaining the use of prime numbers in excryption.

**Non-maths games**

1.memory chimp

2.Optical Illusions1

3.Echalk optical illusions

Great Maths Blogs This is a work in progress which I will be adding to over the next few weeks. For now, see:

Number Loving is both a blog and resource bank run by 2 UK teachers, it hosts a large amount of lesson content and is a fantastic resource.

Matheminute is an amazing maths blog – absolutlely packed full of interesting lesson ideas for extra-curricular maths.

Teaching Mathematics is a really thought provoking and well designed blog with a lot of lesson content ideas and also reflection on teaching practice.

Great Maths Teaching Ideas does what its name suggests, providing a massive number of teaching ideas – helpfully sorted into relevant areas of maths.

Math and Multimedia is another brilliant blog – which contains a huge amount of ideas and resources for using ICT in maths lessons. There are comprehensive Geogebra tutorials as well as information on using wikispace, Wolfram, and creating your own blogs. All this and a large, indexed collection of maths posts.

GFSMaths contains some great lesson ideas and a lot of visual content from lessons

Mr Barton Maths has a comprehensive list of both maths blogs and websites

White Group Maths is a site by a Singaporean teacher which provides a good number of higher level maths (suitable for high achieving HL students) worked examples and some nice online quizzes.

IB Maths Stuff - This is a great resource with free lesson notes on a large number of IB SL topics. It’s really well designed and well worth a look.

**Tok Maths Resources**

Hopefully this page will be useful to:

1. Provide varied 10 minute starter material at the beginning of lessons to stimulate students’ interest in maths.

2. Provide a wealth of resources for students interested in writing about mathematics for their ToK essays.

3. Provide ideas and resources for teachers giving any ToK lectures on mathematics.

4. Provide ideas about future careers involving mathematics

**General ToK**

1. Tok blog posts – An ever expanding number of blog posts on everything from the nature of reality to the philosophy of mathematics.

2. A huge number of articles from Plus Maths exploring mathematics and the nature of reality. For example, read Plus Maths article, “Is God a Mathematician?”

3. Max Tegmark’s paper on physics, parallel universes and whether maths describes reality or is reality. Also a Youtube video explaining Tegmark’s views.

4. A detailed article on the question of mathematics and reality.

5. A ToK website with some good resources by Mrs Davis.

6. The Numperphile video channel. For example, “Do numbers exist?”

7. Larry Ferlazzo’s Maths ToK blog links – A fantastic resource with a large numbers of links to news reports relating to ToK maths.

8. Lancaster School ToK - A nice general ToK resource with some good ideas across the ToK curriculum.

**1.1 Basic Algebra**

**TOK:**Mathematics and the knower. To what extent should mathematical knowledge be consistent with our intuition?

**TOK:**Mathematics and the world. Some mathematical constants (pi, e, , Fibonacci numbers) appear consistently in nature. What does this tell us about mathematical knowledge?

**TOK:**Mathematics and the knower. How is mathematical intuition used as a basis for formal proof? (Gauss’ method for adding up integers from 1 to 100.)

**Aim 8:**Short-term loans at high interest rates. How can knowledge of mathematics result in individuals being exploited or protected from extortion?

1. Are the Constants of Nature Really Constant?

2. A mathematical look at Fibonacci Numbers in Nature

3. Youtube video of Fibonacci numbers in nature.

4. A discussion on the speed of light and why time is not constant.

5. Carl Sagan on the speed of light and time dilation.

6. Brain Greene TED talk about universal “fine tuning”.

7. Excellent Maths Illuminated unit on primes

8. Daniel Tammet – An autistic Savant who “sees” numbers (synathesia)..

9. Nrich article on Gauss

10. Nrich discussion on the compounding chess squares legend.

11. Plus Maths article on compound interest.

12. Plus Maths article on radioactive decay.

13. Cesaro Summation – does the series 1-1+1-1…= 1/2 ?

**1.2 Logs**

**TOK:**The nature of mathematics and science. Were logarithms an invention or discovery?

1. Open University discussion about the invention of logs by Napier.

2. Wikipedia article about the various log scales used in real life.

**1.3 Counting Principles**

**TOK:**The nature of mathematics. The unforeseen links between Pascal’s triangle, counting methods and the coefficients of polynomials. Is there an underlying truth that can be found linking these?

**Aim 8:**How many different tickets are possible in a lottery? What does this tell us about the ethics of selling lottery tickets to those who do not understand the implications of these large numbers?

1. Maths is Fun discussion on Pascal’s Triangle, the Chinese equivalent and links to binomial distribution.

2. Wikipedia on Lottery mathematics.

3. Game theory and expected value for Deal or No Deal.

4. Excellent Maths Illuminated unit on combinatorials

**1.5 Imaginary numbers**

**TOK:**Mathematics and the knower. Do the words imaginary and complex make the concepts more difficult than if they had different names?

**TOK:**The nature of mathematics. Has “i” been invented or was it discovered?

**TOK:**Mathematics and the world. Why does “i” appear in so many fundamental laws of physics?

1. A brief history of complex numbers.

2. Interactive applications demonstrating the use of i in modelling real world problems.

**1.6 Imaginary numbers**

**TOK:**The nature of mathematics. Was the complex plane already there before it was used to represent complex numbers geometrically?

**TOK:**Mathematics and the knower. Why might it be said that e^ip + 1 = 0 is beautiful?

1. Plus Maths article on Euler’s Identity.

2. Introduction to Julia and Mandelbrot sets

3. Mandelbrot image Youtube video.

4. Youtube video describing how to generate Julia sets.

5. A video explaining how to derive Euler’s equation. (Requires integration and complex numbers).

6. Wikipedia article on Euler’s Identity

**1.4 and 1.7 Induction**

**TOK:**Nature of mathematics and science. What are the different meanings of induction in mathematics and science?

**TOK:**Knowledge claims in mathematics. Do proofs provide us with completely certain knowledge?

**TOK:**Knowledge communities. Who judges the validity of a proof?

**TOK:**Reason and mathematics. What is mathematical reasoning and what role does proof play in this form of reasoning? Are there examples of proof that are not mathematical?

1. Blog posts on Fermat’s Last Theorem and the Million Dollar Maths problems

1. Fermat’s Last Theorem.

2. The Clay Institute Millennium Problems

3. P = NP – the most accessible of the Millenium Problems

4. Plus Maths article on The Travelling Salesman problem (and links to P = NP)

5. Introduction to binary operations (1+1 =! 2)

6. Scientific induction and deduction.

7. Funny Youtube video “proving” 25 divided by 5 = 14.

8. Youtube video with a “proof” that 1 = 2

9. 3 incorrect mathematical proofs

10. Wikipedia article explaining the Goldbach conjecture – which is assumed but not proved.

11. Youtube video explaining Goldbach’s conjecture visually.

12. Discussion of how the series 1-1+1-1… can be “proved” to equal different answers.

**1.8 Solutions in 3D**

**TOK:**Mathematics, sense, perception and reason. If we can find solutions in higher dimensions, can we reason that these spaces exist beyond our sense perception?

1. Maths Illuminated resources on the fourth dimension.

2. Carl Sagan video on flatland and tesseracts.

3. Youtube video on how to imagine the 10th dimension.

4. Michael Kaku youtube video on string theory and multi-dimensions.

5. Blog Posts on fractals, the Koch Snowflake and Dragon Curves (fractional dimensions)

**Functions**

**2.1 Concepts of functions**

**TOK:**The nature of mathematics. Is mathematics simply the manipulation of symbols under a set of formal rules?

**2.2-2.5 Graphing functions**

**TOK:**Mathematics and knowledge claims. Does studying the graph of a function contain the same level of mathematical rigour as studying the function algebraically (analytically)?

1. Youtube video on the inverse square law.

2. A (very detailed) discussion about how testing for violations to the inverse square law are at the cutting edge of physics.

3. A discussion of the time dilation graph.

4. Carl Sagan on the speed of light and time dilation.

5. 60 seconds in Thought – Twin paradox (number 5)

**2.6 Quadratics**

**Aim 8:**The phrase “exponential growth” is used popularly to describe a number of phenomena. Is this a misleading use of a mathematical term?

1. Youtube video Joel Cohen: An Introduction to Demography (minutes 3-9)

2. TED talk: Folding a piece of paper to the Moon.

3. Introduction to tertration – the next hyper operation after exponentials.

**Trigonometry**

**3.1-3.4 Triangles**

**Int:**The origin of degrees in the mathematics of Mesopotamia and why we use minutes and seconds for time.

**TOK:**Mathematics and the knower. Why do we use radians? (The arbitrary nature of degree measure versus radians as real numbers and the implications of using these two measures on the shape of sinusoidal graphs.)

**TOK:**Mathematics and knowledge claims. If trigonometry is based on right triangles, how can we sensibly consider trigonometric ratios of angles greater than a right angle?

**Appl:**Triangulation used in the Global Positioning System (GPS).

**Int:**Why did Pythagoras link the study of music and mathematics?

1. Donlald Duck in Math Magicland video re Pythagoras and music.

2. Youtube video of Bach’s Crab Cannon looped on a Mobius strip.

3. Babylonian Mathematics.

4.An explanation of why we use radians.

**3.5-3.6 Periodic graphs**

**TOK:**Mathematics and the world. Music can be expressed using mathematics. Does this mean that music is mathematical, that mathematics is musical or that both are reflections of a common “truth”?

1. A number of Plus Maths articles on the link between maths and music.

2. Excellent Maths Illuminated unit on the mathematics of sound and waves

**3.7 Cosine rule**

**TOK:**Mathematics and knowledge claims. How can there be an infinite number of discrete solutions to an equation?

**TOK:**Nature of mathematics. If the angles of a triangle can add up to less than 180°, 180° or more than 180°, what does this tell us about the “fact” of the angle sum of a triangle and about the nature of mathematical knowledge?

**Int:**The use of triangulation to find the curvature of the Earth in order to settle a dispute between England and France over Newton’s gravity.

1. Wikipedia page explaining the history of Euclidean and non-Euclidean geometry.

2. Plus Maths article on Non Euclidean geometry, art, fractals and Indra’s Pearls.

3. Plus Maths article on why triangles don’t always add up to 180 degrees.

4. Youtube video on triangulation and parallax

5. Excellent Maths Illuminated unit on non-Euclidean geometry

**Vectors**

**Vectors 4.1**

**TOK:**Mathematics and knowledge claims. You can perform some proofs using different mathematical concepts. What does this tell us about mathematical knowledge?

**Aim 8**: Vectors are used to solve many problems in position location. This can be used to save a lost sailor or destroy a building with a laser-guided bomb.

1. A large number of Plus Maths articles on vectors in the real world.

**4.2 Scalar product**

**TOK:**The nature of mathematics. Why this definition of scalar product?

**4.3 Vector equations**

**TOK:**The nature of mathematics. Why might it be argued that vector representation of lines is superior to Cartesian? Appl: Physics SL/HL 6.3 (magnetic force and field).

Youtube video – honey bees use vectors dancing to describe the position of flowers.

**4.7 Vector planes in 3 dimensions**

**TOK:**Mathematics and the knower. Why are symbolic representations of three-dimensional objects easier to deal with than visual representations? What does this tell us about our knowledge of mathematics in other dimensions?

1. Maths Illuminated resources on the fourth dimension.

2. Carl Sagan video on flatland and tesseracts.

3. Youtube video on how to imagine the 10th dimension.

4. Michael Kaku youtube video on string theory and multi-dimensions.

5. Excellent Maths Illuminated unit on extra dimensions

6. Blog Posts on fractals, the Koch Snowflake and Dragon Curves (fractional dimensions)

**Statistics and Probability Core**

**5.1 Statistics and Probability**

**TOK:**The nature of mathematics. Why have mathematics and statistics sometimes been treated as separate subjects?

**TOK:**The nature of knowing. Is there a difference between information and data?

**Aim 8:**Does the use of statistics lead to an overemphasis on attributes that can easily be measured over those that cannot?

**Appl:**Misleading statistics in media reports.

1. Youtube double slit experiment and wave function collapse.

2. Heisenberg’s Uncertainty Principle, Schrodinger’s Cat and probability as the fabric of reality.

3. Kaku video discussing determinism, free will and probability.

4. Collection of resources demonstrating misleading data.

5. Youtube BBC video discussing determinism, probability and chaotic systems.

6. Collection of Plus Maths articles discussing a variety of chaotic systems in real life.

7. TED talk about statistics and misconceptions in real life.

8. Excellent Maths Illuminated unit on chaos.

9. 60 seconds in thought – Schodinger’s cat (number 6)

**5.2 Basic probability**

**Aim 8:**Why has it been argued that theories based on the calculable probabilities found in casinos are pernicious when applied to everyday life (eg economics)?

1. Article on casino probabilities.

2. Detailed Plus maths article on the importance of statistics in trials, and the “Prosecutor’s fallacy”

**5.4 Conditional probability**

**Appl:**Use of probability methods in medical studies to assess risk factors for certain diseases.

**TOK:**Mathematics and knowledge claims. Is independence as defined in probabilistic terms the same as that found in normal experience?

1. An article on the need for Bayesian testing for alternative medicine.

2. A short BBC Horizon clip which discusses the Computer Simulation hypothesis – which is made by considering conditional probability.

3. Nick Bostrom’s Computer Simulation hypotheis paper

4. Monty Hall Problem on conditional probability.

**5.5 Discrete and continuous**

**TOK:**Mathematics and the knower. To what extent can we trust samples of data?

1. Derren Brown tosses 10 heads in a row. Discuss how this is possible?

2. Derren Brown revealed solution (7 mins in). Discuss hypothesis testing versus data mining.

3. Ben Goldacre article on data mining in the pharmaceutical industry 4. Benford’s Law – how probability is used in accountancy to catch fraud.

5. More explanation about Benford’s Law.

6. Youtube video on Benford’s Law.

**5.6 Binomial Distribution**

**TOK:**Mathematics and the real world. Is the binomial distribution ever a useful model for an actual real-world situation?

1. Online ESP test – test ESP claims using binomial model.

2. A fake ESP trick.