screen-shot-2020-11-02-at-7.54.23-pm-1

You can download all 17 of the Paper 3 questions for free here: [PDF].

The full typed mark scheme is available to download at the bottom of the page.

Seventeen IB Higher Level Paper 3 Practice Questions

With the new syllabus just started for IB Mathematics we currently don’t have many practice papers to properly prepare for the Paper 3 Higher Level exam.  As a result I’ve put together 17 full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages of questions and 600 marks worth of content.  This has been specifically written for the Analysis and Approaches syllabus – though some parts would also be suitable for Applications.

 Below I have split the questions into individual pdfs, with more detail about each one. For each investigation question I have combined several areas of the syllabus in order to create some level of discovery – and in many cases I have introduced some new mathematics (as will be the case on the real Paper 3).

Topics explored:

Paper 1: Rotating curves: [Individual question download here.  Mark-scheme download here.]

Students explore the use of parametric and Cartesian equations to rotate a curve around the origin.  You can see a tutorial video on this above.  The mathematics used here is trigonometry (identities and triangles), functions and transformations. 

Paper 12: Circumscribed and inscribed polygons [Individual question  download here].

Students explore different methods for achieving an upper and lower bound for pi using circumscribed and inscribed polygons.  You can see a video solution to this investigation above.  The mathematics used here is trigonometry and calculus (differentiation and L’Hopital’s rule). 

Paper 2: Who killed Mr. Potato? [Individual question download here.]

Screen Shot 2020-11-09 at 11.49.25 AM

Students explore Newton’s Law of Cooling to predict when a potato was removed from an oven.  The mathematics used here is logs laws, linear regression and solving differential equations. 

Paper 3:  Graphically understanding complex roots [Individual question download here.]

Screen Shot 2020-11-09 at 11.49.37 AM

Students explore graphical methods for finding complex roots of quadratics and cubics.  The mathematics used here is complex numbers (finding roots), the sum and product of roots, factor and remainder theorems, equations of tangents. 

Paper 4: Avoiding a magical barrier [Individual question download here.]

Screen Shot 2020-11-09 at 11.49.50 AM

Students explore a scenario that requires them to solve increasingly difficult optimization problems to find the best way of avoiding a barrier.  The mathematics used here is creating equations, optimization and probability. 

Paper 5 : Circle packing density  [Individual question download here.]

Screen Shot 2020-11-09 at 11.50.08 AM

 Students explore different methods of filling a space with circles to find different circle packing densities.  The mathematics used here is trigonometry and using equations of tangents to find intersection points.

Paper 6:  A sliding ladder investigation [Individual question download here.]

Screen Shot 2020-11-09 at 11.50.17 AM

Students find the general equation of the midpoint of a slipping ladder and calculate the length of the astroid formed.  The mathematics used here is trigonometry and differentiation (including implicit differentiation).  Students are introduced to the ideas of parametric equations. 

Paper 7: Exploring the Si(x) function [Individual question download here.]

Screen Shot 2020-11-09 at 11.50.29 AM

 Students explore methods for approximating non-integrable functions and conclude by approximating pi squared.  The mathematics used here is Maclaurin series, integration, summation notation, sketching graphs.

Paper 8: Volume optimization of a cuboid [Individual question download here.]

Screen Shot 2020-11-09 at 11.50.41 AM

 Students start with a simple volume optimization problem but extend this to a general case of an m by n rectangular paper folded to make an open box.  The mathematics used here is optimization, graph sketching, extended binomial series, limits to infinity. 

Paper 9: Exploring Riemann sums [Individual question download here.]

Screen Shot 2020-11-09 at 11.50.53 AM

Students explore the use of Riemann sums to find upper and lower bounds of functions – finding both an approximation for pi and also for ln(1.1).  The mathematics used here is integration, logs, differentiation and functions

Paper 10 : Optimisation of area [Individual question download here.]

Screen Shot 2020-11-09 at 11.51.05 AM

Students start with a simple optimisation problem for a farmer’s field then generalise to regular shapes.  The mathematics used here is trigonometry and calculus (differentiation and L’Hopital’s rule)

Paper 11: Quadruple Proof [Individual question download here.]

Screen Shot 2020-11-09 at 11.51.13 AM

Students explore 4 different ways of proving the same geometrical relationship.  The mathematics used here is trigonometry (identities) and complex numbers. 

Paper 13:  Using the binomial expansion for bounds of accuracy [Individual question download here.]

Screen Shot 2020-11-09 at 11.51.40 AM

Students explore methods of achieving lower and upper bounds for  and non-calculator methods for calculating logs.  The mathematics used here is the extended binomial expansion for fractional and negative powers and integration. 

Paper 14: Radioactive Decay [Individual question download here.]

Screen Shot 2020-11-09 at 11.51.49 AM

Students explore discrete decay models, using probability density functions to investigate the decay of Carbon-14 and then explore the use of Euler’s method to approximate more complex decay chains.  The mathematics used here is integration, probability density functions and Euler’s method of approximation

 Paper 15: Probability generating functions [Individual question download here.]

Screen Shot 2020-12-02 at 12.06.22 PM

Students explore the use of probability generating functions to find probabilities, expected values and variance for the binomial distribution and Poisson distribution for predicting the eruption of a volcano.

Paper 16: Finding the Steiner inellipse using complex numbers [Individual question download here]

Screen Shot 2020-12-02 at 12.06.14 PM

Students use a beautiful relationship between complex numbers and an ellipse tangent to the midpoints of a triangle.  This relationship allows you to find the equation of an ellipse from coordinate points of a triangle.

Paper 17: Elliptical curves [Individual question download here]

Screen Shot 2020-12-02 at 12.06.06 PM

Students explore a method for adding points on an elliptical curve.  This has links with elliptical curve cryptography.

Mark-scheme download

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

IB HL Paper 3 Practice Questions and markscheme.

100 pages of preparatory questions with answers for the IB HL Analysis P3 exam. Please note this is not an automatic download and will be sent the same day.

$9.00

Compilation Bundles

IB HL Paper 3 Practice Questions and markscheme AND Exploration Guide

All the Paper 3 questions and mark scheme AND the 63 page Exploration Guide. The Exploration Guide includes: Investigation essentials, Marking criteria guidance, 70 hand picked interesting topics, Useful websites for use in the exploration, A student checklist for top marks, Avoiding common student mistakes, A selection of detailed exploration ideas, Advice on using Geogebra, Desmos and Tracker. And more! Please note this is not an automatic download and will be sent the same day.

$14.00

Screen Shot 2020-11-02 at 8.14.40 PM

Super Bundle! Paper 3 Practice Question and markscheme AND Exploration Guide AND Modelling Guide AND Statistics Guide

All the Paper 3 resources and also 3 separate guides to help teachers/students with the exploration. The Exploration Guide (63 pages) talks through all the essentials needed for excellent explorations, the Modelling Guide (50 pages) talks through both calculator and non-calculator methods for numerous regression techniques and the Statistics Guide (55 pages) talks through different statistical techniques that can be used in explorations. Also comprehensive sections on using Desmos to represent graphs and data effectively. Please note this is not an automatic download and will be sent the same day.

$20.00

3D Printing with Desmos: Stewie Griffin

Using Desmos or Geogebra to design a picture or pattern is quite a nice exploration topic – but here’s an idea to make your investigation stand out from the crowd – how about converting your image to a 3D printed design?

Step 1

Create an image on Desmos or Geogebra.  Remove the axes and grid pattern.  This image is a pre-drawn image already on Desmos available here.

Step 2

Take a screen capture image of your picture (jpeg, gif, png).  We need to convert this to a SVG file.  You can convert these for free at sites like picsvg.

Step 3

Lastly we need to use a 3D editing site .  You can join up with a site like Tinkercad for free.

Step 4

Making our 3D model.  We import our SVG file and we get the image above.  We can then resize this to whatever dimensions we wish – and also add 3D depth.

Lastly I would then save this file and send it to a 3D printer.  You can see the finished file below:

So, if we printed this we’d get something like this:

Screen Shot 2020-10-23 at 1.50.37 PM

3D printing the Eiffel Tower

Screen Shot 2020-10-22 at 7.16.45 PM

Let’s use another Desmos art work. The Eiffel Tower above was a finalist in their annual art competition drawn by Jerry Yang from the USA.

Screen Shot 2020-10-22 at 7.16.49 PM

This is then converted to the SVG file above.

Screen Shot 2020-10-22 at 7.16.54 PM

And this is the result on Tinkercad when I add some depth and change the colour scheme.  Let’s see what that would look like printed:

Screen Shot 2020-10-23 at 1.47.17 PM

Pretty good- we’ve created a cheap tourist souvenir in about 5 minutes!

Mathematical art

I thought I’d have a go at making my own mathematical art.  I started with using some polar coordinates to create this nice pattern:

Which then creates the following 3D shape:

Screen Shot 2020-10-23 at 1.52.26 PM

This topic has a lot of scope for exploration and links with art, design technology and engineering.  Thanks to our ever resourceful ICT wizz at school Jon for assistance, and also thanks for this excellent method which was posted by Ryan on Thingiverse. You can also explore huge numbers of ready made 3D templates on the site.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.

Complex Numbers as Matrices – Euler’s Identity

Euler’s Identity below is regarded as one of the most beautiful equations in mathematics as it combines five of the most important constants in mathematics:

I’m going to explore whether we can still see this relationship hold when we represent complex numbers as matrices.

Complex Numbers as Matrices

First I’m I’m going to define the following equivalences between the imaginary unit and the real unit and matrices:

The equivalence for 1 as the identity matrix should make sense insofar as in real numbers, 1 is the multiplicative identity.  This means that 1 multiplied by any real number gives that number.  In matrices, a matrix multiplied by the identity matrix also remains unchanged.  The equivalence for the imaginary unit is not as intuitive, but let’s just check that operations with complex numbers still work with this new representation.

In complex numbers we have the following fundamental definition:

Does this still work with our new matrix equivalences?

Yes, we can see that the square of the imaginary unit gives us the negative of the multiplicative identity as required.

More generally we can note that as an extension of our definitions above we have:

Complex number multiplication

Let’s now test whether complex multiplication still works with matrices.  I’ll choose to multiply the following 2 complex numbers:

Now let’s see what happens when we do the equivalent matrix multiplication:

We can see we get the same result.  We can obviously prove this equivalence more generally (and check that other properties still hold) but for the purposes of this post I want to check whether the equivalence to Euler’s Identity still holds with matrices.

Euler’s Identity with matrices

If we define the imaginary unit and the real unit as the matrices above then the question is whether Euler’s Identity still holds, i.e:


First I can note that:

Next I can note that the Maclaurin expansion for e^(x) is:

Putting these ideas together I get:

This means that:


Next I can use the matrix multiplication to give the following:

Next, I look for a pattern in each of the matrix entries and see that:

Now, to begin with here I simply checked these on Wolfram Alpha – (these sums are closely related to the Macluarin series for cosine and sine).

Therefore we have:

So, this means I can write:

And so this finally gives:

Which is the result I wanted!  Therefore we can see that Euler’s Identity still holds when we define complex numbers in terms of matrices.  Complex numbers are an incredibly rich area to explore – and some of the most interesting aspects of complex numbers is there ability to “bridge” between different areas of mathematics.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.

Sierpinski Triangle: A picture of infinity

This pattern of a Sierpinski triangle pictured above was generated by a simple iterative program.  I made it by modifying the code previously used to plot the Barnsley Fern. You can run the code I used on repl.it.  What we are seeing is the result of 30,000 iterations of a simple algorithm.  The algorithm is as follows:

Transformation 1:

xi+1 = 0.5xi

yi+1= 0.5yi

Transformation 2:

xi+1 = 0.5xi + 0.5

yi+1= 0.5yi+0.5

Transformation 3:

xi+1 = 0.5xi +1

yi+1= 0.5yi

So, I start with (0,0) and then use a random number generator to decide which transformation to use.  I can run a generator from 1-3 and assign 1 for transformation 1, 2 for transformation 2, and 3 for transformation 3.   Say I generate the number 2 – therefore I will apply transformation 2.

xi+1 = 0.5(0) + 0.5

yi+1= 0.5(0)+0.5

and my new coordinate is (0.5,0.5).  I mark this on my graph.

I then repeat this process – say this time I generate the number 3.  This tells me to do transformation 3.  So:

xi+1 = 0.5(0.5) +1

yi+1= 0.5(0.5)

and my new coordinate is (1.25, 0.25).  I mark this on my graph and carry on again.  The graph above was generated with 30,000 iterations.

Altering the algorithm

We can alter the algorithm so that we replace all the 0.5 coefficients of x and y with another number, a.

a = 0.3 has disconnected triangles:

When a = 0.7 we still have a triangle:

By a = 0.9 the triangle is starting to degenerate

By a = 0.99 we start to see the emergence of a line “tail”

By a = 0.999 we see the line dominate.

And when a = 1 we then get a straight line:

When a is greater than 1 the coordinates quickly become extremely large and so the scale required to plot points means the disconnected points are not visible.

If I alternatively alter transformations 2 and 3 so that I add b for transformation 2 and 2b for transformation 3 (rather than 0.5 and 1 respectively) then we can see we simply change the scale of the triangle.

When b = 10 we can see the triangle width is now 40 (we changed b from 0.5 to 10 and so made the triangle 20 times bigger in length):

Fractal mathematics

This triangle is an example of a self-similar pattern – i.e one which will look the same at different scales.  You could zoom into a detailed picture and see the same patterns repeating.  Amazingly fractal patterns don’t fit into our usual understanding of 1 dimensional, 2 dimensional, 3 dimensional space.  Fractals can instead be thought of as having fractional dimensions.

The Hausdorff dimension is a measure of the “roughness” or “crinkley-ness” of a fractal.  It’s given by the formula:

D = log(N)/log(S)

For the Sierpinski triangle, doubling the size (i.e S = 2), creates 3 copies of itself (i.e N =3)

This gives:

D = log(3)/log(2)

Which gives a fractal dimension of about 1.59.  This means it has a higher dimension than a line, but a lower dimension than a 2 dimensional shape.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.


Sphere packing problem: Pyramid design

Sphere packing problems are a maths problems which have been considered over many centuries – they concern the optimal way of packing spheres so that the wasted space is minimised.  You can achieve an average packing density of around 74% when you stack many spheres together, but today I want to explore the packing density of 4 spheres (pictured above) enclosed in a pyramid.

Considering 2 dimensions

First I’m going to consider the 2D cross section of the base 3 spheres.  Each sphere will have a radius of 1.  I will choose A so that it is at the origin.  Using some basic Pythagoras this will give the following coordinates:

Finding the centre

Next I will stack my single sphere on top of these 3, with the centre of this sphere directly in the middle.  Therefore I need to find the coordinate of D.  I can use the fact that ABC is an equilateral triangle and so:

3D coordinates

Next I can convert my 2D coordinates into 3D coordinates.  I define the centre of the 3 base circles to have 0 height, therefore I can add z coordinates of 0.  E will be the coordinate point with the same x and y coordinates as D, but with a height, a, which I don’t yet know:

In order to find I do a quick sketch, seen below:

Here I can see that I can find the length AD using trig, and then the height DE (which is my a value) using Pythagoras:

Drawing spheres

The general equation for spheres with centre coordinate (a,b,c) and radius 1 is:

Therefore the equation of my spheres are:

Plotting these on Geogebra gives:

Drawing a pyramid

Next I want to try to draw a pyramid such that it encloses the spheres.  This is quite difficult to do algebraically – so I’ll use some technology and a bit of trial and error.

First I look at creating a base for my pyramid.  I’ll try and construct an equilateral triangle which is a tangent to the spheres:

This gives me an equilateral triangle with lengths 5.54. I can then find the coordinate points of F,G,H and plot them in 3D.  I’ll choose point E so that it remains in the middle of the shape, and also has a height of 5.54 from the base. This gives the following:

As we can see, this pyramid does not enclose the spheres fully.  So, let’s try again, this time making the base a little bit larger than the 3 spheres:

This gives me an equilateral triangle with lengths 6.6.  Taking the height of the pyramid to also be 6.6 gives the following shape:

This time we can see that it fully encloses the spheres.  So, let’s find the density of this packing.  We have:

Therefore this gives:

and we also have:

Therefore the density of our packaging is:

Given our diagram this looks about right – we are only filling less than half of the available volume with our spheres.

Comparison with real data

[Source: Minimizing the object dimensions in circle and sphere packing problems]

We can see that this task has been attempted before using computational power – the table above shows the average density for a variety of 2D and 3D shapes.  The pyramid here was found to have a density of 46% – so our result of 44% looks pretty close to what we should be able to achieve.  We could tweak our measurements to see if we could improve this density.

So, a nice mixture of geometry, graphical software, and trial and error gives us a nice result.  You could explore the densities for other 2D and 3D shapes and see how close you get to the results in the table.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.

Screen Shot 2020-06-05 at 10.14.13 PM

Martingale II and Currency Trading

We can use computer coding to explore game strategies and also to help understand the underlying probability distribution functions.   Let’s start with a simple game where we toss a coin 4 times, stake 1 counter each toss and always call heads.  This would give us a binomial distribution with 4 trials and the probability of success fixed as 1/2.

Tossing a coin 4 time [simple strategy]

Screen Shot 2020-06-05 at 9.27.47 PM

For example the only way of losing 4 counters is a 4 coin streak of T,T,T,T.  The probability of this happening is 1/16.  We can see from this distribution that the most likely outcome is 0 (i.e no profit and no loss).  If we work out the expected value, E(X) by multiplying profit/loss by frequencies and summing the result we get E(X) = 0.  Therefore this is a fair game (we expect to neither make a profit nor a loss).

Tossing a coin 4 time [Martingale strategy]

Screen Shot 2020-06-05 at 9.34.19 PM

This is a more complicated strategy which goes as follows:

1) You stake 1 counter on heads.
b) if you lose you stake 2 counters on heads
c) if you lose you stake 4 counters on heads
d) if you lose you stake 8 counters on heads.

If you win, the your next stake is always to go back to staking 1 counter.

For example for the sequence: H,H,T,T 

First you bet 1 counter on heads.  You win 1 counter
Next you bet 1 counter on heads.  You win 1 counter
Next you bet 1 counters on heads.  You lose 1 counter
Next you bet 2 counters on heads.  You lose 2 counters

[overall loss is 1 counter]

For example for the sequence: T,T,T,H 

First you bet 1 counter on heads.  You lose 1 counter
Next you bet 2 counter on heads.  You lose 2 counters
Next you bet 4 counters on heads.  You lose 4 counter
Next you bet 8 counters on heads.  You win 8 counters

[overall profit is 1 counter]

This leads to the following probabilities:

Once again we will have E(X) = 0, but a very different distribution to the simple 4 coin toss.  We can see we have an 11/16 chance of making a profit after 4 coins – but the small chance of catastrophic loss (15 counters) means that the overall expectation is still zero.

Iterated Martingale:

Here we can do a computer simulation.  This is the scenario this time:

Screen Shot 2020-06-05 at 10.01.23 PM

We start with 100 counters, we toss a coin for a maximum of 3 times. We then define a completed round as when we get to a shaded box.  We then repeat this process through 999 rounds, and model what happens. Here I used a Python program to simulate a player using this strategy.

Screen Shot 2020-06-05 at 10.06.15 PM

We can see that we have periods of linear growth followed by steep falls – which is a very familiar pattern across many investment types.  We can see that the initial starting 100 counters was built up to around 120 at the peak, but was closer to just 40 when we finished the simulation.

Let’s do another simulation to see what happens this time:

Screen Shot 2020-06-05 at 10.14.13 PM

Here we can see that the 2nd player was actually performing significantly worse after around 600 rounds, but this time ended up with a finishing total of around 130 counters.

Changing the multiplier

We can also see what happens when rather than doubling stakes on losses we follow some other multiple.  For example we might choose to multiply our stake by 5.  This leads to much greater volatility as we can see below:

Multiplier x5

Screen Shot 2020-06-06 at 6.18.05 AM

Here we have 2 very different outcomes for 2 players using the same model.  Player 1 (in blue) may believe they have found a sure-fire method of making huge profits, but player 2 (green) went bankrupt after around 600 rounds.

Multiplier x1.11

Screen Shot 2020-06-06 at 6.28.07 AM

Here we can see that if the multiplier is close to 1 we have much less volatility (as you would expect because your maximum losses per round are much smaller).

Screen Shot 2020-06-06 at 5.20.07 PM

We can run the simulation across 5000 rounds – and here we can see that we have big winning and losing streaks, but that over the long run the account value oscillates around the starting value of 100 counters.

Forex charts

We can see similar graphs when we look at forex (currency exchange) charts.  For example:

Screen Shot 2020-06-06 at 6.42.53 AM

In this graph (from here) we plot the exchange between US dollar and Thai Baht.  We can see the same sort of graph movements – with run of gains and losses leading to a similar jagged shape.  This is not surprising as forex trades can also be thought of in terms of 2 binary outcomes like tossing a coin, and indeed huge amounts of forex trading is done through computer programs, some of which do use the Martingale system as a basis.

The effect of commission on the model

Screen Shot 2020-06-06 at 7.16.35 PM

So, to finish off we can modify our system slightly so that we try to replicate forex trading.  We will follow the same model as before, but this time we have to pay a very small commission for every trade we make.  This now gives us:

E(X) = -0.000175. (0.0001 counters commission per trade)

E(X) = -0.00035. (0.0002 counters commission per trade)

Even though E(X) is very slightly negative, it means that in the long run we would expect to lose money.    With the 0.0002 counters commission we would expect to lose around 20 counters over 50,000 rounds.  The simulation graph above was run with 0.0002 counters commission –  and in this case it led to bankruptcy before 3000 rounds.

Computer code

Screen Shot 2020-06-06 at 6.53.33 AM

The Python code above can be used to generate data which can then be copied into Desmos.  The above code simulates 1 player playing 999 rounds, starting with 100 counters, with a multiplier of 5.   If you know a little bit about coding you can try and play with this yourselves!

Screen Shot 2020-09-12 at 1.36.32 PM

I’ve also just added a version of this code onto repl.  You can run this code – and also generate the graph direct (click on the graph png after running).  It creates some beautiful images like that shown above.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.

Screen Shot 2020-04-15 at 10.20.21 AM

Time dependent gravity and cosmology!

In our universe we have a gravitational constant – i.e gravity is not dependent on time.  If gravity changed with respect to time then the gravitational force exerted by the Sun on Earth would lessen (or increase) over time with all other factors remaining the same.

Interestingly time-dependent gravity was first explored by Dirac and some physicists have tried to incorporate time dependent gravity into cosmological models.  As yet we have no proof that gravity is not constant, but let’s imagine a university where it is dependent on time.

Inversely time dependent gravity

The standard models for cosmology use G, where G is the gravitational constant.  This fixes the gravitational force as a constant.  However if gravity is inversely proportional to time we could have a relationship such as:

Screen Shot 2020-04-15 at 10.28.17 AM

Where a is a constant.  Let’s look at a very simple model, where we have a piecewise function as below:

Screen Shot 2020-04-15 at 10.28.35 AM

This would create the graph at the top of the page.  This is one (very simplistic) way of explaining the Big Bang.  In the first few moments after t = 0, gravity would be negative and thus repulsive [and close to infinitely strong], which could explain the initial incredible universal expansion before “regular” attractive gravity kicked in (after t = 1).  The Gravitational constant has only been measured to 4 significant figures:

G = 6.674 x 10-11m3kg-1s-2.

Therefore if there is a very small variation over time it is possible that we simply haven’t the accuracy to test this yet.

Universal acceleration with a time dependent gravitational force

Warning: This section is going to touch on some seriously complicated maths – not for the faint hearted!  We’re going to explore whether having a gravitational force which decreases over time still allows us to have an accelerating expansion of the universe.

We can start with the following equation:

Screen Shot 2020-04-15 at 1.44.09 PM

To work through an example:

Screen Shot 2020-04-15 at 1.46.19 PM

This would show that when t = 1 the universe had an expansion scale factor of 2.  Now, based on current data measured by astronomers we have evidence that the universe is both expanding and accelerating in its expansion.  If the universal scale factor is accelerating in expansion that requires that we have:

Screen Shot 2020-04-15 at 1.49.45 PM

Modelling our universe

We’re going to need 4 equations to model what happens when gravity is time dependent rather than just a constant.

Equation 1

Screen Shot 2020-04-15 at 1.50.45 PM

This equation models a relationship between pressure and density in our model universe.  We assume that our universe is homogenous (i.e the same) throughout.

Equation 2

Screen Shot 2020-04-15 at 1.50.54 PM

This is one of the Friedmann equations for governing the expansion of space.  We will take c =1 [i.e we will choose units such that we are in 1 light year etc]

Equation 3

Screen Shot 2020-04-15 at 1.51.00 PM

This is another one of the Friedmann equations for governing the expansion of space.  The original equation has P/(c squared) – but we we simplify again by taking c = 1.

Equation 4

Screen Shot 2020-04-15 at 1.51.06 PM

This is our time dependent version of gravity.

Finding alpha

We can separate variables to solve equation (3).

Screen Shot 2020-04-15 at 1.58.20 PM

Substitution

We can use this result, along with the equations (1) and (4) to substitute into equation (2).

Screen Shot 2020-04-15 at 2.00.23 PM

Our result

Now, remember that if the second differential of r is positive then the universal expansion rate is accelerating.  If Lamba is negative then we will have the second differential of r positive.  However, all our constants G_0, a, B, t, r are greater than 0.  Therefore in order for lamda to be negative we need:

Screen Shot 2020-04-15 at 2.05.57 PM

What this shows is that even in a universe where gravity is time dependent (and decreasing), we would still be able to have an accelerating universe like we see today.  the only factor that determines whether the universal expansion is accelerating is the value of gamma, not our gravity function.

This means that a time dependent gravity function can still gives us a result consistent with our experimental measurements of the universe.

A specific case

Solving the equation for the second differential of r is extremely difficult, so let’s look at a very simple case where we choose some constants to make life as easy as possible:

Screen Shot 2020-04-15 at 2.14.02 PM

Substituting these into our equation (2) gives us:

Screen Shot 2020-04-15 at 2.14.10 PM

We can then solve this to give:

Screen Shot 2020-04-15 at 2.14.19 PM

So, finally we have arrived at our final equation.  This would give us the universal expansion scale factor at time t, for a universe in which gravity follows the the equation G(t) = 1/t.

Screen Shot 2020-04-15 at 2.22.58 PM

For this universe we can then see that when t = 5 for example, we would have a universal expansion scale factor of 28.5.

So, there we go – very complicated maths, way beyond IB level, so don’t worry if you didn’t follow that.  And that’s just a simplified introduction to some of the maths in cosmology!  You can read more about time dependent gravity here (also not for the faint hearted!)

 

 

 

Animated GIF - Find & Share on GIPHY

The Tusi couple – A circle rolling inside a circle

Numberphile have done a nice video where they discuss some beautiful examples of trigonometry and circular motion and where they present the result shown above: a circle rolling within a circle, with the individual points on the small circle showing linear motion along the diameters of the large circle.  So let’s see what maths we need to create the image above.

 Projection of points

Animated GIF - Find & Share on GIPHY

We can start with the equation of a unit circle centred at the origin:

Screen Shot 2020-06-18 at 7.01.45 PM

and we can then define a point on this circle parametrically by the coordinate:

Screen Shot 2020-06-18 at 7.01.49 PM

Here t is the angle measured from the horizontal.

If we then want to see the projection of this point along the y-axis we can also plot:

Screen Shot 2020-06-18 at 7.01.53 PM

and to see the projection of this point along the x-axis we can also plot:

Screen Shot 2020-06-18 at 7.02.06 PM

By then varying t from 0 to 2 pi gives the animation above – where the black dot on the circle moves around the circle and there is a projection of its x and y coordinates on the axes.

Projection along angled lines

Screen Shot 2020-06-18 at 7.32.46 PM

I can then add a line through the origin at angle a to the horizontal:

Screen Shot 2020-06-18 at 7.35.25 PM

and this time I can project so that the line joining up the black point on the edge of the large circle intersects the dotted line in a right angle.

In order to find the parametric coordinate of this point projection I can use some right angled triangles as follows:

Screen Shot 2020-06-18 at 7.57.20 PM

The angle from the horizontal to my point A is t.  The angle from the horizontal to the slanted line is a.  The length of my radius BA is 1.  This gives me the length of BC.

Screen Shot 2020-06-18 at 7.53.30 PM

But I have the identity:

Screen Shot 2020-06-18 at 7.53.34 PM

Therefore this gives:

Screen Shot 2020-06-18 at 7.53.37 PM

And using some more basic trigonometry gives the following diagram:

Screen Shot 2020-06-18 at 8.00.53 PM

Therefore the parametric form of the projection of the point can be given as:

Screen Shot 2020-06-18 at 7.40.17 PM

Adding more lines

Screen Shot 2020-06-18 at 8.03.32 PM

I can add several more slanted lines through the origin.  You can see that each dot on the line is the right angle projection between the line and the point on the circle.  As we do this we can notice that the points on the lines look as though they form a circle.  By noticing that the new smaller circle is half the size of the larger circle, and that the centre of the smaller circle is half-way between the origin and the point on the large circle, we get:

Screen Shot 2020-06-18 at 8.07.58 PM

Screen Shot 2020-06-18 at 8.10.16 PM

We can the vary the position of the point on the large circle to then create our final image:

Animated GIF - Find & Share on GIPHY

We have a connection between both linear motion and circular motion and create the impression of a circle rolling inside another.

You can play around with this demos graph here.  All you need to do is either drag the black point around the circle, or press play for the t slider.

More ideas on projective geometry:

Screen Shot 2020-06-18 at 8.16.30 PM

Ferenc Beleznay has made this nice geogebra file here which shows a different way of drawing a connection between a moving point on a large circle and a circle half the size. Here we connect the red dot with the origin and draw the perpendicular from this line to  the other edge of the small circle.  The point of intersection of the two lines is always on the small circle.

Further exploration 

There is a lot more you can explore – start by looking into the Tusi Couple – which is what we have just drawn – and the more general case the hypocycloid.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.

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

The Martingale system

The Martingale system was first used in France in 1700s gambling halls and remains used today in some trading strategies.  I’ll look at some of the mathematical ideas behind this and why it has remained popular over several centuries despite having a long term expected return of zero.

The scenario

You go to a fair ground and play a simple heads-or-tails game.  The probability of heads is 1/2 and tails is also 1/2.  You place a stake of counters on heads.  If you guess correctly you win that number of counters.  If you lose, you double your stake of counters and then the coin is tossed again.  Every time you lose you double up your stake of counters and stop when you finally win.

Infinitely deep pockets model:


You can see that in the example above we always have a 0.5 chance of getting heads on the first go, which gives a profit of 1 counter.  But we also have a 0.5 chance of a profit of 1 counter as long as we keep doubling up our stake, and as long as we do indeed eventually throw heads.  In the example here you can see that the string of losing throws don’t matter [when we win is arbitrary, we could win on the 2nd, 3rd, 4th etc throw].  By doubling up, when you do finally win you wipe out your cumulative losses and end up with a 1 counter profit.

This leads to something of a paradoxical situation, despite only having a 1/2 chance of guessing heads we end up with an expected value of 1 counter profit for every 1 counter that we initially stake in this system.

So what’s happening?  This will always work but it requires that you have access to infinitely deep pockets (to keep your infinite number of counters) and also the assumption that if you keep throwing long enough you will indeed finally get a head (i.e you don’t throw an infinite number of tails!)

Finite pockets model:

Real life intrudes on the infinite pockets model – because in reality there will be a limit to how many counters you have which means you will need to bail out after a given number of tosses.  Even if the probability of this string of tails is very small, the losses if it does occur will be catastrophic –  and so the expected value for this system is still 0.

Finite pockets model capped at 4 tosses:

In the example above we only have a 1/16 chance of losing – but when we do we lose 15 counters.  This gives an expected value of:

Finite pockets model capped at n tosses:

If we start with a 1 counter stake then we can represent the pattern we can see above for E(X) as follows:

Here we use the fact that the losses from n throws are the sum of the first (n-1) powers of 2. We can then notice that both of these are geometric series, and use the relevant formula to give:

Therefore the expected value for the finite pockets model is indeed always still 0.

So why does this system remain popular?

So, given that the real world version of this has an expected value of 0, why has it retained popularity over the past few centuries?  Well, the system will on average return constant linear growth – up until a catastrophic loss.  Let’s say you have 100,000 counters and stake 1 counter initially.  You can afford a total of 16 consecutive losses.  The probability of this is only:

but when you do lose, you’ll lose a total of:

So, the system creates a model that mimics linear growth, but really the small risk of catastrophic loss means that the system still has E(X) = 0.  In the short term you would expect to see the following very simple linear relationship for profit:

With 100,000 counters and a base trading stake of 1 counter, if you made 1000 initial 1 counter trades a day you would expect a return of 1000 counters a day (i.e 1% return on your total counters per day).  However the longer you continue this strategy the more likely you are to see a run of 16 tails – and see all your counters wiped out.

Computer model

I wrote a short Python code to give an idea as to what is happening. Here I started 9 people off with 1000 counters each.  They have a loss limit of 10 consecutive losses.  They made starting stakes of 1 counter each time, and then I recorded how long before they made a loss of 10 tosses in a row.

For anyone interested in the code here it is:

The program returned the following results.  The first number is the number of starting trades until they tossed 10 tails in a row.  The second number was their new account value (given that they had started with 1000 counters, every previous trade had increased their account by 1 counter and that they had then just lost 1023 counters).

1338, 1315
1159, 1136
243, 220
1676, 1653
432, 409
1023, 1000
976, 953
990, 967
60, 37

This was then plotted on Desmos. The red line is the trajectory their accounts were following before their loss.  The horizontal dotted line is at y = 1000 which represents the initial account value.  As you can see 6 people are now on or below their initial starting account value.  You can also see that all these new account values are themselves on a line parallel to the red line but translated vertically down.

From this very simple simulation, we can see that on average a person was left with 884 counters following hitting 10 tails.  i.e below initial starting account.  Running this again with 99 players gave an average of 869.

999 players

I ran this again with 999 players – counting what their account value would be after their first loss.  All players started with 1000 counters.  The results were:

31 players bankrupt: 3%

385 players left with less than half their account value (less than 500): 39%

600 players with less than their original account value (less than 1000): 60%

51 players at least tripled their account (more than 3000): 5%

The top player ended up with 6903 counters after their first loss.

The average account this time was above starting value (1044.68).  You can see clearly that the median is below 1000 – but that a small number of very lucky players at the top end skewed the mean above 1000.

Second iteration

I then ran the simulation again – with players continuing with their current stake.  This would have been slightly off because my model allowed players who were bankrupt from the first round to carry on [in effect being loaned 1 counter to start again].  Nevertheless it now gave:

264 players bankrupt: 26%

453 players left with less than half their account value (less than 500): 45%

573 players with less than their original account value (less than 1000): 57%

95 players at least tripled their account (more than 3000): 10%

The top player ended up with 9583 counters after their second loss.

We can see a dramatic rise in bankruptcies – now over a quarter of all players.  This would suggest the long term trend is towards a majority of players being bankrupted, though the lucky few at the top end may be able to escape this fate.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

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.

Screen Shot 2018-03-19 at 4.42.05 PM.png

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!

Screen Shot 2021-05-19 at 10.05.18 AM

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

Screen Shot 2021-05-19 at 6.32.13 PM

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.

Website Stats

  • 8,105,994 views

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.

Recent Posts

Follow IB Maths Resources from British International School Phuket on WordPress.com