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These 3 guides are all written by an experienced IB teacher with an MSc. in Mathematics, 10 years experience teaching IB Standard and Higher Level and who has worked as an IB examiner (IA moderation). They are suitable for all IB students.

Resource Number 1

The Exploration Guide talks you through:

  1. An introduction to the essentials about the investigation,
  2. The new marking criteria,
  3. How to choose a topic,
  4. Examples of around 70 topics that could be investigated,
  5. Useful websites for use in the exploration,
  6. A student checklist for completing a good investigation,
  7. Common mistakes that students make and how to avoid them,
  8. General stats projects advice,
  9. A selection of some interesting exploration topics explored in more depth,
  10. Teacher advice for marking,
  11. Templates for draft submissions,
  12. Advice on how to use Geogebra, Desmos and Tracker in your exploration,
  13. Some examples of beautiful maths using Geogebra and Desmos.

[If you don’t have a PayPal account you can just click on the relevant credit card icon].

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. [This is not an automatic download but will be emailed the same day].

$7.50

Resource Number 2

The Modelling for Explorations Guide is a 50 page pdf which talks you through various techniques useful for modelling explorations. The focus is on being able to use both calculator and non-calculator techniques to show good knowledge and understanding. Topics included are:

  1. Linear regression
  2. Quadratic regression
  3. Cubic regression
  4. Exponential regression
  5. Linearisation using log scales
  6. Trigonometric regression
  7. Normal distribution regression
  8. Extended technology guide to using Desmos for modelling (including measuring errors, plotting parametric curves, plotting on polar graphs, presenting your graphs perfectly).
  9. Extended technology guide to using Tracker for modelling (including how to achieve the correct scale and how to generate graphs of different functions such as velocity, acceleration, momentum and more).

[If you don’t have a PayPal account you can just click on the relevant credit card icon].

IB Maths Modelling for Explorations Guide

A 50 page pdf guide full of advice to help gain top marks for modelling explorations. [This is not an automatic download but will be emailed the same day].

$6.00

Resource Number 3

The Statistics for Explorations Guide is a 55 page pdf which talks you through various techniques useful for statistics explorations. The focus is on being able to use both calculator and non-calculator techniques to show good knowledge and understanding. Topics included are:

  1. Pearson’s Product investigations: Height and arm span
  2. Binomial investigations: ESP powers
  3. Poisson investigations: Customers in a shop
  4. 2 sample t tests: Reaction times
  5. Paired t tests: Reaction times
  6. Chi Squared: Efficiency of vaccines
  7. Bernoulli trials: Polling confidence intervals
  8. Spearman’s rank: Taste preference of cola
  9. Sampling techniques and experiment design.
  10. Extended technology guide to using Desmos for statistics (including plotting histograms, box plots, normal distribution curves, binomial curves, scatter graphs and more).

[If you don’t have a PayPal account you can just click on the relevant credit card icon].

IB Maths Statistics for Explorations Guide

A 55 page pdf packed full of examples for how to gain top marks for statistical investigations. [This is not an automatic download but will be emailed the same day].

$6.00

Compilation bundles

You can purchase a compilation of both the Exploration Guide and Modelling Guide and both the Exploration Guide and the Statistics Guide for a discount.

Exploration Guide AND Modelling for Explorations Guide

Both the Exploration Guide and the Modelling Guide bundled together.

$12.00

Exploration Guide AND Statistics for Explorations Guide

Both the Exploration Guide and the Statistics Guide bundled together.

$12.00

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Log Graphs to Plot Planetary Patterns

This post is inspired by the excellent Professor Stewart’s latest book, Calculating the Cosmos. In it he looks at some of the mathematics behind our astronomical knowledge.

Astronomical investigations

In the late 1760s and early 1770s, 2 astronomers Titius and Bode both noticed something quite strange – there seemed to be a relationship in the distances between the planets. There was no obvious reason as to why there would be – but nevertheless it appeared to be true. Here are the orbital distances from the Sun of the 6 planets known about in the 1760s:

Mercury: 0.39 AU
Venus: 0.72 AU
Earth: 1.00 AU
Mars: 1.52 AU
Jupiter: 5.20 AU
Saturn: 9.54 AU

In astronomy, 1 astronomical unit (AU) is defined as the mean distance from the center of the Earth to the centre of the Sun (149.6 million kilometers).

Now, at first glance there does not appear to be any obvious relationship here – it’s definitely not linear, but how about geometric? Well dividing the term above by the term below we get r values of:

1.8, 1.4, 1.5, 3.4, 1.8

4 of the numbers are broadly similar – and then we have an outlier of 3.4. So either there was no real pattern – or there was an undetected planet somewhere between Mars and Jupiter? And was there another planet beyond Saturn?

Planet X

Mercury: 0.39 AU
Venus: 0.72 AU
Earth: 1.00 AU
Mars: 1.52 AU
Planet X: x AU
Jupiter: 5.20 AU
Saturn: 9.54 AU
Planet Y: y AU

For a geometric sequence we would therefore want x/1.52 = 5.20/x. This gives x = 2.8 AU – so a missing planet should be 2.8 AU away from the Sun. This would give us r values of 1.8, 1.4, 1.5, 1.8, 1.9, 1.8. Let’s take r = 1.8, which would give Planet Y a distance of 17 AU.

So we predict a planet around 2.8 AU from the Sun, and another one around 17 AU from the Sun. In 1781, Uranus was discovered – 19.2 AU from the Sun, and in 1801 Ceres was discovered at 2.8 AU. Ceres is what is now classified as a dwarf planet – the largest object in the asteroid belt between Jupiter and Mars.

Log Plots

Using graphs is a good way to graphically see relationships. Given that we have a geometrical relationship in the form d = ab^n with a and b as constants, we can use the laws of logs to rearrange to give log d = log a + n log b.

Therefore we can plot log d on the y axis, and n on the x axis. If there is a geometrical relationship we will see us a linear relationship on the graph, with log a being the y intercept and the gradient being log b.

(n=1) Mercury: d = 0.39 AU. log d = -0.41
(n=2) Venus: d = 0.72 AU. log d = -0.14
(n=3) Earth: d = 1.00 AU. log d = 0
(n=4) Mars: d = 1.52 AU. log d = 0.18
(n=5) Ceres (dwarf): d = 2.8 AU. log d = 0.45
(n=6) Jupiter: d = 5.20 AU. log d = 0.72
(n=7) Saturn: d = 9.54 AU. log d = 0.98
(n=8) Uranus: d = 19.2 AU. log d = 1.28

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We can use Desmos’ regression tool to find a very strong linear correlation – with y intercept as -0.68 and gradient as 0.24.  Given that log a is the y intercept, this gives:

log a  = -0.68

a = 0.21

and given that log b is the gradient this gives:

log b = 0.24

b = 1.74

So our final formula for the relationship for the spacing of the n ordered planets is:

d = ab^n

distance = 0.21 x (1.74)^n.

Testing the formula

So, using this formula we can predict what the next planetary distance would be. When n = 9 we would expect a distance of 30.7 AU.  Indeed we find that Neptune is 30.1 AU – success! How about Pluto?  Given that Pluto has a very eccentric (elliptical) orbit we might not expect this to be as accurate.  When n = 10 we get a prediction of 53.4 AU.  The average AU for Pluto is 39.5 – so our formula does not work well for Pluto.   But looking a little more closely, we notice that Pluto’s distance from the Sun varies wildly – from 29.7 AU to 49.3 AU, so perhaps it is not surprising that this doesn’t follow our formula well.

Other log relationships

Interestingly other distances in the solar system show this same relationship.  Plotting the ordered number of the planets against the log of their orbital period produces a linear graph, as does plotting the ordered moons of Uranus against their log distance from the planet.  Why these relationships exist is still debated.  Perhaps they are a coincidence, perhaps they are a consequence of resonance in orbital periods.   Do some research and see what you find!

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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IB Maths Resources by Andrew Chambers

All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner). Please contact here for information on webinar training or for business ideas.

IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

Modelling Guide


IB Exploration Modelling Guide 

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

Available to download here.

Statistics Guide

IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis and Applications students!

Available to download here.

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