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What is the average distance between 2 points in a rectangle?

Say we have a rectangle, and choose any 2 random points within it.  We then could calculate the distance between the 2 points.  If we do this a large number of times, what would the average distance between the 2 points be?

Monte Carlo method

The Monte Carlo method is perfect for this – we can run the following code on Python:

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This code will find the average distance between 2 points in a 10 by 10 square.  It does this by generating 2 random coordinates, finding the distance between them and then repeating this process 999,999 times.  It then works out the average value.  If we do this it returns:

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This means that on average, the distance between 2 random points in a 10 by 10 square is about 5.21.

Generalising to rectangles

I can now see what happens when I fix one side of the rectangle and vary the other side.  The code below fixes one side of the rectangle at 1 unit, and then varies the other side in integer increments.  For each rectangle it then calculates the average distance.

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This then returns the first few values as:

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This shows that for a 1 by 1 square the average distance between two points is around 0.52 and for a 1 by 10 rectangle the average distance is around 3.36.

Plotting some Desmos graphs

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Because I have included the comma in the Python code I can now copy and paste this straight into Desmos.  The dotted green points show how the average distance of a 1 by x rectangle changes as x increases.  I then ran the same code to work out the average distance of a 10 by x rectangle (red), 20 by x rectangle (black), 30 by x rectangle (purple) and 100 by x rectangle (yellow).

We can see if we continue these points further that they all appear to approach the line y = 1/3 x (dotted green).  This is a little surprising – i.e when x gets large, then for any n by x rectangle (with n fixed), an increase in x by one will tend towards an increase in the average distance by 1/3.

Heavy duty maths!

There is actually an equation that fits these curves – and will give the average distance, a(X) between any 2 points in a rectangle with sides a and b (a≥b).  Here it is:

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I added this equation into Desmos, by changing the a to x, and then adding a slider for b.  So, when I set b=1 this generated the case when the side of a rectangle is fixed as 1 and the other side is increased:

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Plotting these equations on Desmos then gives the following:

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Pleasingly we can see that the points created by our Monte Carlo method fit pretty much exactly on the lines generated by these equations.  By looking at these lines at a larger scale we can see that they do all indeed appear to be approaching the line y = 1/3 x.

General equation for a square

We can now consider the special case when a=b (i.e a square).  This gives:

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Which we can simplify to give:

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We can see therefore that a square of side 1 (a=1) will have an average distance of 0.52 (2dp) and a square of side 10 (a=10) will have an average distance of 5.21  – which both agree with our earlier results.

Witness Numbers: Finding Primes

The Numberphile video above is an excellent introduction to primality tests – where we conduct a test to determine if a number is prime or not.  Finding and understanding about prime numbers is an integral part of number theory.  I’m going to go through some examples when we take the number 2 as our witness number.  We have a couple of tests that we conduct with 2 – and for all numbers less than 2047 if a number passes either test then we can guarantee that it is a prime number.

Miller-Rabin test using 2 as a witness number:

We choose an odd number, n >2. First we need to write it in the form:

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Then we have to conduct a maximum of 2 different tests:

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If either of the above are true then we have a prime number.

Testing whether n = 23 is prime.

First we need to write 23 in the following form:

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Next we need to check if the following is true:

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Remember that mod 23 simply means we look at the remainder when we divide by 23.  We can do this using Wolfram Alpha – but in this case let’s see how we could do this without a calculator:

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Therefore this passes the test – and we can say that it is prime.

Testing whether 1997 is prime

For 1997 we have:

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So we need to first test if the following is true:

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However using Wolfram Alpha we get:

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So this fails the first part of the test.

Trying the second part of the test, we need:

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We have already tested the case when r=0 (this gives the earlier result), so just need to look at what happens when r=1.  Again we use Wolfram Alpha to get:

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This passes the 2nd part of the test and so confirms that 1997 is prime.

What happens with 2047?

2047 is not prime as we can write it as 2 x 3 x 11 x 31.  However it is the first number for which the witness 2 gives a false positive (i.e we get a positive result even though it is not prime).  We write 2047 as:

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But we do indeed get:

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So we can call 2047 a pseudoprime – it passes this prime number test but is not actually prime.

Larger primes

For numbers larger than 2047 you can combine witnesses – for example if you use both 2 and 3 as your witness numbers (and regard a positive result as at least one of them returning a positive result) then this will find all primes for n < 1,373,653.

More interestingly for extremely large numbers you can use this test to provide a probability estimate for the likelihood that a number is prime.  Lots to explore here!

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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!

Have you got a Super Brain?

Adapting and exploring maths challenge problems is an excellent way of finding ideas for IB maths explorations and extended essays.  This problem is taken from the book: The first 25 years of the Superbrain challenges.  I’m going to see how many different ways I can solve it.

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The problem is to find all the integer solutions to the equation above.  Finding only integer solutions is a fundamental part of number theory – a branch of mathematics that only deals with integers.  

Method number 1: Brute force

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This is a problem that computers can make short work of.  Above I wrote a very simple Python program which checked all values of x and y between -99 and 99.  This returned the only solution pairs as:

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Clearly we have not proved these are the only solutions – but even by modifying the code to check more numbers, no more pairs were found.

Method number 2: Solving a linear equation

We can notice that the equation is linear in terms of y, and so rearrange to make y the subject.

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We can then use either polynomial long division or the method of partial fractions to rewrite this.  I’ll use partial fractions.  The general form for this fraction can be written as follows:

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Next I multiply by the denominator and the compare coefficients of terms.

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This therefore gives:

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I can now see that there will only be an integer solution for y when the denominator of the fraction is a factor of 6.  This then gives (ignoring non integer solutions):

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I can then substitute these back to find my y values, which give me the same 4 coordinate pairs as before:

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Method number 3: Solving a quadratic equation

I start by making a quadratic in x:

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I can then use the quadratic formula to find solutions:

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Which I can simplify to give:

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Next I can note that x will only be an integer solution if the expression inside the square root is a square number.  Therefore I have:

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Next I can solve a new quadratic as follows:

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As before I notice that the expression inside my square root must be a square number.  Now I can see that I need to find m and n such that I have 2 square numbers with a difference of 24.  I can look at the first 13 square numbers to see that from the 12th and 13th square numbers onwards there will also be a difference of more than 24.  Checking this list I can find that m = 1 and m = 5 will satisfy this equation.

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This then gives:

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which when I solve for integer solutions and then sub back into find x gives the same four solutions:

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Method number 4: Graphical understanding

Without rearranging I could imagine this as a 3D problem by plotting the 2 equations:

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This gives the following graph:

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We can see that the plane intersects the curve in infinite places.  I’ve marked A, B on the graph to illustrate 2 of the coordinate pairs which we have found.  This is a nice visualization but doesn’t help find our coordinates, so lets switch to 2D.

In 2D we can use our rearranged equation:

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This gives the following graph:

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Here I have marked on the solution pairs that we found.   The oblique asymptote (red) is y = 2x-1 because as x gets large the fraction gets very small and so the graph gets closer and closer to y = 2x -1. 

All points on this curve are solutions to the equation – but we can see that the only integer solution pairs will be when x is small.  When x is a large integer then the curve will be close to the asymptote and hence will return a number slightly bigger than an integer.

So, using this approach we would check all possible integer solutions when x is small, and again should be able to arrive at our coordinate pairs.

So, 4 different approaches that would be able to solve this problem.  Can you find any others?

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.

Paper 3 investigations

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Teacher resources:

If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pages of content for teaching IB mathematics including worksheets, mock exams, investigations, enrichment tasks, technology guides, exploration support and investigations.

Student resources:

I have made separate student packs for both Applications students and Analysis students.   You can download a 41 page pdf with 17 Paper 3 questions here.  I have also provided the full worked solutions below which can be purchased for a small cost.

Individual questions

Rotating curves (Analysis)

Download the question pdf here

Download the full worked solutions pdf 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.

Approximating pi with polygons (Analysis)

Download the question pdf here

Download the full worked solutions pdf here.

Students follow in the footsteps of the great Greek mathematicians to derive approximations for pi using a mixture of trigonometry and calculus.

Who killed Mr. Potato? (Analysis and Applications)

Download the question pdf here.

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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.

Graphically understanding complex roots (Analysis)

Download the question pdf here.

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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.

Avoiding a magical barrier (Analysis and Applications)

Download the question pdf here.

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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.

Life’s a Beach (Applications)

Download the question pdf here.

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Purchase options

It takes a lot of time to create these resources – so if you would like to support the site (and prepare well for the exams!) you can also purchase a full exam pack with full typed solutions below. 

Analysis Students

(1) Analysis HL Paper 3 pack ($7.50)

17 full Paper 3 investigations and full worked solutions with typed markscheme.  Please note this is not an automatic download and will be emailed to you within a day.

 

(2) Analysis HL Paper 3 pack + Super Exploration Guide ($12)

17 full Paper 3 investigations and full worked solutions with typed markscheme and also 3 pdf guides to support students in achieving excellent marks in their explorations.  The guides take you through exploration essentials, technology support, common mistakes, statistical methods, regression techniques and lots more.  Written by an experienced IB Mathematics coursework moderator. Please note this is not an automatic download and will be emailed to you within a day.

 

Applications Students

(1) Applications HL Paper 3 pack ($6.50)

6 full Paper 3 investigations and and full worked solutions with typed markscheme.  Please note this is not an automatic download and will be emailed to you within a day.

 

(2) Applications HL Paper 3 pack + Super Exploration Guide ($12)

6 full Paper 3 investigations and full worked solutions with typed markscheme and also 3 pdf guides to support students in achieving excellent marks in their explorations.  The guides take you through exploration essentials, technology support, common mistakes, statistical methods, regression techniques and lots more.  Written by an experienced IB Mathematics coursework moderator.  Please note this is not an automatic download and will be emailed to you within a day.

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!

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 Teachers

1) Intermathematics.com

Screen Shot 2021-08-21 at 1.07.49 PM

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

Screen Shot 2021-12-01 at 1.19.14 PM

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.

Teacher resources

I have collected together below a lot of (hopefully!) useful resources to support IB teachers teaching IB Maths Analysis SL and HL and also Applications SL and HL.

This is a small portion of the content on  my new site: intermathematics.com which has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2500 pages of pdf content for the entire SL and HL syllabus for both Analysis and Applications.  

Exploration coursework

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1. 63 page guide to the exploration process.  [Download].

Paper 3 investigations. 

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1. Exploring curvature.  [Questions].  [Markscheme].

2. Rotating curves.  [Questions].  [Markscheme].

3. Approximating pi with polygons.  [Questions].  [Markscheme].

4. Life’s a Beach with Markov chains.  [ Questions]

4. Eight Paper 3 questions.  [Questions].

Sequences

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1. Mixed sequences past paper style questions. [Questions].  [Markscheme].

2. A sequences puzzle.  [Questions].  [Markscheme].

3. A mini sequences investigation. [Questions].

Logs

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1. Laws of logs treasure hunt. [Questions].  [Markscheme].

2. Radioactive decay investigation. [Question].  [Markscheme].

3. Benford’s Law investigation. [Question].  [Markscheme].

Binomial expansion

1. Fractional and negative indices worksheet.  [Questions].  [Markscheme].

Complex numbers 

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1. Converting between complex numbers.  [Questions].  [Markscheme].

2. Julia Sets investigation.  [Questions].  [Markscheme].

Functions

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1. Quadratic functions bridge building.  [Questions].  [Markscheme].

2.  Mixed past paper style questions.  [Questions].  [Markscheme].

3. Viete’s Theorem investigation.  [Questions].  [Markscheme].

4. Where to build a hospital with Voronoi diagrams.  [Questions].  [Markscheme].

Trigonometry and geometry

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1. Hours of sunlight investigation [Questions].  [Markscheme].

2. Treasure hunt trig graphs [Questions].  [Markscheme].

3. Trig past paper style questions GDC.  [Questions].  [Markscheme].

4. Encryption with matrices investigation. [Questions].  [Markscheme].

Probability and statistics

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1. Normal distribution classroom investigation.  [Questions].  [Markscheme].

Calculus

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1. Optimisation worksheet. [Questions].  [Markscheme].

2. Circle rolling down a cosine curve investigation.  [Questions].  [Markscheme].

Enrichment

1. School code breaking resource pack.  [Download]

TOK

1. TOK links to support IB Maths TOK teachers when delivering this subject. [Link]

Other great resources (other teachers)

Analysis:

1) The Analysis and Approaches formula book for SL and HL. (Reworked and in a beautiful one page format).

2) Full Analysis and Approaches lecture notes from Christos Nikolaidis.  (Extremely comprehensive and well written).

3) A beautiful mind-map for the entire Analysis and Approaches SL course produced by the IB.

4) A fantastic 55 page pdf for Paper 3 Investigations from IB teacher Daniel Hwang – these are pretty challenging – great for Level 6/7 HL students!

5) Full Analysis course notes from Mac Maths.  Beautifully handwritten notes.

Applications:

1) The Applications and Interpretations formula book for SL and HL. (Reworked and in a beautiful one page format).

2) Full Applications and Interpretations lecture notes from Christos Nikolaidis.  (Extremely comprehensive and well written).

3) A beautiful mind-map for the entire Applications and Interpretations SL course produced by the IB.

4) Full Applications course notes from Mac Maths.  Beautifully handwritten notes.

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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 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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