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**IB Maths Exploration Guides**

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

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

**Guides**

**Exploration Essentials Guide **(63 pages)

Give yourself a comprehensive understanding of all the essential information required to get an excellent mark on your coursework.

Understand the marking criteria – and exactly what is required in each strand to get top scores, learn very common student mistakes, get guidance as to how to choose a good topic to maximise your chances of getting good grades, and get an introduction to some excellent graphical packages

Download here.

**Statistics for Explorations Guide **(55 pages)

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

Download here

**Modelling for Explorations Guide **(50 pages)

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

Download here

If you are a teacher and find this useful then please also visit my new site: intermathematics.com for over 3000+ pages of content for teaching IB mathematics including worksheets, mock exams, investigations, enrichment tasks, technology guides, exploration support and investigations. Thanks!

Andrew

This is inspired from the great site, Practical Cryptography which is a really good resource for code making and code breaking. One of their articles is about how we can use the Chi Squared test to crack a Caesar Shift code. Indeed, if you use an online program to crack a Caesar shift, they are probably using this technique.

This is the formula that you will be using for Chi Squared. It looks more complicated than it is. Say we have the following message (also from Practical Cryptography):

*AOLJHLZHYJPWOLYPZVULVMAOLLHYSPLZARUVDUHUKZPTWSLZAJPWOLY ZPAPZHAFWLVMZBIZAPABAPVUJPWOLYPUDOPJOLHJOSLAALYPUAOLWSH PUALEAPZZOPMALKHJLYAHPUUBTILYVMWSHJLZKVDUAOLHSWOHILA*

We first work out the frequency of each letter which we do using the Counton site.

We next need to work out the expected values for each letter. To do this we first need the expected percentages for the English language:

Then we can count the number of letters in the code we want to crack (162 – again we can use an online tool)

Now, to find the expected number of As in the code we simply do 162 x 0.082 = 13.284.

The actual number of As in the code is 18.

Therefore we can do (13.284-18)^{2}/18 following the formula at the top of the page.

We then do exactly the same for the Bs in the code. The expected number is 162 x 0.015 = 2.43. The actual number is 3.

Therefore we can do (3-2.43)^{2 }/2.43

We do this same method for all the letters A-Z and then add all those numbers together. This is our Chi Squared statistic. The lower the value, the closer the 2 distributions are. If the expected values and the observed values are the same then there will be a chi squared of zero.

If you add all the values together you get a Chi Squared value of ≈1634 – which is quite large! This is what we would expect – because we already know that the code we have received has letter frequencies quite different to normal English sentences. Now, what a Caesar Shift decoder can do is shift the received code through all the permutations and then for each one find out the Chi Squared value. The permutation with the lowest Chi Squared will be the solution.

For example, if we shift every letter in our received code back by one – using the Counton tool (so A goes to Z etc) we get:

*ZNKIGKYGXIOVNKXOYUTKULZNKKGXROKYZQTUCTGTJYOSVRKYZIOVNKX YOZOYGZEVKULYAHYZOZAZOUTIOVNKXOTCNOINKGINRKZZKXOTZNKVRG OTZKDZOYYNOLZKJGIKXZGOTTASHKXULVRGIKYJUCTZNKGRVNGHKZ*

We can then do the same Chi Squared calculations as before. This will give a Chi Squared of ≈3440 – which is an even worse fit than the last calculation. If we carried this on so that A goes to T we would get:

THECAESARCIPHERISONEOFTHEEARLIESTKNOWNANDSIMPLESTCIPHER SITISATYPEOFSUBSTITUTIONCIPHERINWHICHEACHLETTERINTHEPLA INTEXTISSHIFTEDACERTAINNUMBEROFPLACESDOWNTHEALPHABET

and a Chi Squared on this would show that this has a Chi Squared of ≈33 – ie it is a very good fit. (You will get closer to zero on very long code texts which follow standard English usage). Now, obviously we could see that this is the correct decryption without even working out the Chi Squared value – but this method allows a computer to do it, without needing the ability to understand English. Additionally a codebreaker who spoke no English would still be able to decipher this code, on mathematics alone.

The Practical Cryptography site have a tool for quickly working out Chi Squared values from texts – so you can experiment with your own codes. Note that this is a slightly different use of Chi-Squared as here we are not comparing with a critical value, but instead comparing all Chi Squared to find the lowest value.

If you liked this post you might also like:

Code Breakers Wanted by the NSA – A look at some other code breaking techniques.

RSA Public Key Encryption – The Code that Secures the internet – How understanding RSA code is essential for all people involved in internet security.