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I have produced 3 pdf guides and also an in-depth online course with Udemy to assist students in getting the best possible marks in their coursework.  They are suitable for all IB students.  You can buy as single guides or in bundles (scroll to the end of the page to see these).

Teacher resources 

If you are a teacher then please also visit my new site: intermathematics.com.  This new site has all the resources below as well as 100 pdf worksheets with full worked solutions  covering the whole of the SL and HL Analysis syllabus, 20 paper 3 investigations with full worked solutions, 35 flipchart quizzes for IGCSE students and full worked solution notes for IGCSE Extended and Additional Maths. 

Student resources

1. Exploration Guide:

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

Click to access exploration-guide-preview.pdf

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. And lots more!

Buy the Exploration Guide

You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$9.00

2.  Modelling for Explorations Guide:

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.

Click to access modelling-guide-preview.pdf

The Modelling for Explorations Guide talks you through:

  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 
  9. Extended technology guide to using Tracker for modelling 

Buy the Modelling for Explorations Guide

You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$8.00

3. The Statistics for Explorations Guide

This 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. 

Click to access statistics-guide-preview.pdf

The Statistics for Exploration Guide includes:

  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).

Buy the Statistics for Explorations Guide

You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$8.00

4. Udemy course: Getting a 7 in IB coursework

I’ve teamed up with Udemy – the world’s leading provider of online courses to create a comprehensive online guide to the exploration.  It includes 9 tutorial videos totaling 2 hours 30 minutes of essential information designed to ensure you get the best possible grade.

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The video tutorials cover

1) The tools required to pick an excellent topic,

2) Looking at how to gain a deep understanding of the criteria points,

3) Non calculator technique to demonstrate thorough understanding,

4) Exploring top tips for making beautiful graphs and modeling using Desmos,

5) Comparing “good” versus “bad” examples of coursework.

6) Achieving a Level 7 – what you need to do to hit the top criteria levels.

You can sign up for this course for 40% off the standard price by using the coupon: NEWTON

5. Bundle resources

Exploration Guide + Modelling Guide

This will be emailed to you and is not an automatic download. If you don’t have a PayPal account please click the relevant credit card.

$15.00

Exploration Guide + Stats Guide

This will be emailed to you and is not an automatic download. If you don’t have a PayPal account please click the relevant credit card.

$15.00

Exploration Guide + Stats Guide + Modelling Guide

This will be emailed to you and is not an automatic download. If you don’t have a PayPal account please click the relevant credit card.

$20.00

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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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About

All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

New website for International teachers

I’ve just launched a brand new maths site for international schools – over 1000 pdf pages of resources to support IB and IGCSE maths lessons.

Explore here!

1-1 coursework tuition

My colleague Dr Taylan Celtik and his team offer well renowned and professional 1-1 coursework tuition.

Find out more here

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths exploration coursework.

Available to download here.

Getting a 7 in IB Maths Exploration Coursework

I have just made a Udemy online tutorial course for the exploration.  This includes nine tutorial videos of essential information designed to ensure you get the best possible grade.

Use the code NEWTON for a 40% discount.

IB HL Paper 3 Practice Questions (120 page pdf)

Eight P3 investigation questions and fully typed mark scheme (around 240 marks)

Available to download here

Modelling Guide for Explorations

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.

Available to download here.

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.

Available to download here.

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.

Available to download here.

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