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If you are a teacher then please also visit my new site: intermathematics.com for over 2500+ pages of content for teaching IB mathematics including worksheets, mock exams, investigations, enrichment tasks, technology guides, exploration support and investigations.

IB Maths Super Exploration Guide

Below you can download some comprehensive exploration guides that I’ve written to help students get excellent marks on their IB maths coursework.  These guides are suitable for both Analysis and also Applications students. Over the past several years I’ve written over 200 posts with exploration ideas and marked hundreds of IAs whilst working as an IB examiner.  It’s frustrating how many students throw away marks from not understanding the criteria correctly.  Hopefully this will make a big difference!

There are 3 guides:  (1) Exploration essentials (2) Statistical explorations (3) Modeling explorations.  

Content in the 3 guides includes:

  1. The new marking criteria,
  2. How to choose a topic,
  3. Examples of around 70 topics that could be investigated,
  4. Useful websites for use in the exploration,
  5. A student checklist for completing a good investigation,
  6. Common mistakes that students make and how to avoid them,
  7. Regression techniques (polynomial, trigonometric, exponential etc) 
  8. Non calculator methods for modelling (to show understanding)
  9. Technology support – how to use Desmos, Tracker and Geogebra effectively
  10. Pearson’s product and correlation investigation methods
  11. Chi-squared and Spearman’s rank investigations 
  12. Advanced statistical techniques (t-tests, Poisson, Bernoulli trials, normal distribution and more)
  13. Non calculator methods for statistics (to show understanding)
  14. And a lot, lot more!

Make sure you give yourself the best chance of doing well on your coursework.

Exploration Essentials Guide (63 pages)

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Give yourself a comprehensive understanding of all the essential information required to get an excellent mark on your coursework.  Understand the marking criteria – and exactly what is required in each strand to get top scores, learn very common student mistakes and how to avoid them, get guidance as to how to choose a good topic to maximise your chances of getting good grades, get an introduction to some excellent graphical packages that can be used for free to super-charge your coursework.

Part 2: Statistics for Explorations Guide (55 pages)

You can download a preview pdf here

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An essential resource for anyone thinking of doing some statistical analysis.  There is a dedicated Pearson’s Product section which is frequently done poorly without demonstration of good knowledge and understanding.  There are a large number of statistical tests explained with examples that fit straight into coursework ideas.

Part 3: Modelling for Explorations Guide (50 pages)

You can download a preview pdf here

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There are many modelling techniques for fitting curves to data – but students often rely on technology without demonstrating clear understanding.  Learn the various methods for fitting curves complete with how to lay this out in a coursework exploration to show a good knowledge of what mathematics you are using. 

Exploration Guide: Exploration Essentials Guide ($5.50)

You can receive the 63 page Exploration Guide full of all the essentials needed for excellent coursework marks.   Written by an experienced IB coursework moderator.  (This will be emailed to you the same day and is not an automatic download).

 

Exploration Guide: Exploration Essentials Guide and Statistics for Explorations and Modelling for Explorations ($7.50)

You can receive the 63 page Exploration Guide full of all the essentials needed for excellent coursework marks and also the Statistics for Explorations Guide (55 pages) and Modelling for Explorations Guide (50 pages).   All written by an experienced IB coursework moderator.  (This will be emailed to you the same day and is not an automatic download).

 

Super Bundle:  Exploration Guides and HL Paper 3 Analysis Pack ($12)

You can receive a super bundle of all the 3 Exploration Guides and also the Paper 3 HL Analysis Pack which contains 17 Paper 3 questions and full worked solutions.  (This will be emailed to you the same day and is not an automatic download).

 

Super Bundle:  Exploration Guides and HL Paper 3 Applications Pack ($12)

You can receive a super bundle of all the 3 Exploration Guides and also the Paper 3 HL Applications Pack which contains 6 Paper 3 questions and full worked solutions.  (This will be emailed to you the same day and is not an automatic download).

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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

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 Teachers

1) Intermathematics.com

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If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

  1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
  2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
  3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

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I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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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 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

Free HL Paper 3 Questions

P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.

Available to download here

IB Maths Super Exploration Guide

A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework.

Available to download here.

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