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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 students:

1) Revision Village

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

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

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

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

2) Exploration Guides and Paper 3 Resources

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

 

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 1600+ 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.  To see the style of content you can download a pdf of the eight Analysis investigation here.

Please note you do not need a PayPal account to pay – simply click on the relevant card to pay by credit card.

HL Analysis students

You can buy a student Paper 3 pack – which includes 8 full investigation questions (around 240 marks) and full worked solutions through the PayPal link below.  There is another bundle offer at the bottom of the page.

Student Paper 3 Investigation pack for HL Analysis 

Questions include:

  1. Rotating curves
  2. Who killed Mr Potato?
  3. Graphically understanding complex roots
  4. Avoiding a magical barrier
  5. Circle packing density
  6. A sliding ladder investigation
  7. Exploring the Si(x) function
  8. Volume optimization of a cuboid 

Buy the Student P3 Analysis pack [with MS]

This includes 8 investigation questions and full worked solutions (57 pages of content). You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$9.00

Student P3 AA pack [with MS] AND Exploration Guide

This includes all the investigation material and also a 60 page Exploration Guide. You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$14.00

HL Applications students

You can buy a student Paper 3 pack – which includes 6 full investigation questions (around 180 marks) and full worked solutions through the PayPal link below.  

Student Paper 3 Investigation pack for HL Applications

Questions include:

  1. Investigating BMI
  2. Who killed Mr Potato?
  3. Life’s a Beach
  4. Hare vs. Lynx
  5. Rolling Dice
  6. Avoiding a Magical Barrier

Buy the Student P3 Applications pack [with MS]

This includes 6 investigation questions and full worked solutions (41 pages of content). You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$9.00

Student P3 AI pack [with MS] AND Exploration Guide

This includes all the investigation material and also a 60 page Exploration Guide. You can pay below. If you don’t have a PayPal account please click the relevant credit card. Please note this is not an automatic download – I will email it to you the same day.

$14.00

Examples of Analysis questions

Paper 1: Rotating curves: 

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.

Click to access rotate-curve.pdf

Paper 2: Who killed Mr. Potato?

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.

Click to access Who-killed-mr-potato.pdf

Paper 3:  Graphically understanding complex roots 

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.

Click to access graphically-understanding-complex-roots.pdf

Paper 4: Avoiding a magical barrier 

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.

Click to access Avoiding-a-magical-barrier.pdf

Super bundle

You can also buy the following content as a super bundle:

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

All the Paper 3 Analysis 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

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

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

Teacher resources

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If you are a teacher then please also visit my new site:  intermathematics.com.  This new site has been designed specifically for teachers of mathematics at international schools.  Some of the content includes:

  1. Over 120 original pdf worksheets designed to cover the Analysis and Interpretations SL and HL syllabus.  Each worksheet is designed to last between 40 minutes and 1 hour and combines skills practice and past paper style questions.  All the pdf worksheets have fully typed up answers for ease of marking.
  2. Twenty original Paper 3 investigations.  Each investigation is designed to last between 90 minutes and 2 hours.  Fully worked solutions are also provided.
  3. An Exploration Guide, Modelling for Explorations Guide and Statistics for Explorations Guide to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. 35 flip chart quizzes specifically covering the IGCSE 0580 syllabus – each with around 20 pages and worked solutions to exam questions
  5. Full worked solution notes for both the IGCSE 0580 syllabus and the IGCSE 0606 Additional syllabus.
  6. Enrichment and engaging classroom activities, such as maths club activities, treasure hunts, quizzes, investigations and more.
  7. Mock exam papers: I have made a Paper 1 and Paper 2 SL (both 80 marks) and Paper 1 and Paper 2 HL full mock exam paper (both 110 marks) for the new syllabus.

You can see an example of some of the content included below.  Explore the site!  Feel free to contact me with any questions.

1. Introduction to the site with example content (pdf here).

Click to access introduction-pack-2.pdf

2. 8 original Paper 3 investigations (pdf here).

Click to access paper-3-exploration-questions-eight-questions.pdf

Example Paper 3 markscheme for Rotating Curves:

Click to access rotate-curve-ms.pdf

Example worksheet 1: Box and whiskers

Click to access 6box-and-whisker-and-cumulative-frequency.pdf

Example worksheet 2: Area between curves

Click to access 3area-2-curves-and-ftc.pdf

Example mark scheme for integrating factor differential equations:

Click to access 14integrating-factor-ms.pdf

Example mark scheme for sketching with a GDC

Click to access 6-sketching-graphs-hw-ms.pdf

Example: Modelling Guide preview

Click to access modelling-guide-preview.pdf

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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 1600 pdf pages of resources to support IB teachers.

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 Exploration Guides

Three comprehensive pdf guides to help you get excellent marks on your maths exploration coursework.  Over 150 pages of advice from an IB examiner.

Available to download here.

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