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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!

This is a nice example of using some maths to solve a puzzle from the mindyourdecisions youtube channel (screencaptures from the video).

How to Avoid The Troll: A Puzzle

In these situations it’s best to look at the extreme case first so you get some idea of the problem.  If you are feeling particularly pessimistic you could assume that the troll is always going to be there.  Therefore you would head to the top of the barrier each time.  This situation is represented below:

The Pessimistic Solution:

Another basic strategy would be the optimistic strategy.  Basically head in a straight line hoping that the troll is not there.  If it’s not, then the journey is only 2km.  If it is then you have to make a lengthy detour.  This situation is shown below:

The Optimistic Solution:

The expected value was worked out here by doing 0.5 x (2) + 0.5 x (2 + root 2) = 2.71.

The question is now, is there a better strategy than either of these?  An obvious possibility is heading for the point halfway along where the barrier might be.  This would make a triangle of base 1 and height 1/2.  This has a hypotenuse of root (5/4).  In the best case scenario we would then have a total distance of 2 x root (5/4).  In the worst case scenario we would have a total distance of root(5/4) + 1/2 + root 2.  We find the expected value by multiply both by 0.5 and adding.  This gives 2.63 (2 dp).  But can we do any better?  Yes – by using some algebra and then optimising to find a minimum.

The Optimisation Solution:

To minimise this function, we need to differentiate and find when the gradient is equal to zero, or draw a graph and look for the minimum.  Now, hopefully you can remember how to differentiate polynomials, so here I’ve used Wolfram Alpha to solve it for us.  Wolfram Alpha is incredibly powerful -and also very easy to use.  Here is what I entered:

and here is the output:

So, when we head for a point exactly 1/(2 root 2) up the potential barrier, we minimise the distance travelled to around 2.62 miles.

So, there we go, we have saved 0.21 miles from our most pessimistic model, and 0.01 miles from our best guess model of heading for the midpoint.  Not a huge difference – but nevertheless we’ll save ourselves a few seconds!

This is a good example of how an exploration could progress – once you get to the end you could then look at changing the question slightly, perhaps the troll is only 1/3 of the distance across?  Maybe the troll appears only 1/3 of the time?  Could you even generalise the results for when the troll is y distance away or appears z percent of the time?

Essential resources for IB students:

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.

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!

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.

Essential Resources for IB Teachers

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

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.

Statistics to win penalty shoot-outs

The last World Cup was a relatively rare one for England, with no heroic defeat on penalties, as normally seems to happen. England are in fact the worst country of any of the major footballing nations at taking penalties, having won only 1 out of 6 shoot-outs at the Euros and World Cup.   In fact of the 31 penalties taken in shoot-outs England have missed 10 – which is a miss rate of over 30%.  Germany by comparison have won 5 out of 7 – and have a miss rate of only 15%.

With the stakes in penalty shoot-outs so high there have been a number of studies to look at optimum strategies for players.

One study published in Psychological Science looked at all the penalties taken in penalty shoot-outs in the World Cup since 1982.  What they found was pretty incredible – goalkeepers have a subconscious bias for diving to the right when their team is behind.

As is clear from the graphic, this is not a small bias towards the right, but a very strong one.  When their team is behind the goalkeeper apparently favours his (likely) strong side 71% of the time.  The strikers’ shot meanwhile continues to be placed either left or right with roughly the same likelihood as in the other situations.  So, this built in bias makes the goalkeeper much less likely to help his team recover from a losing position in a shoot-out.

Shoot high

Analysis by Prozone looking at the data from the World Cups and European Championships between 1998 and 2010 compiled the following graphics:

The first graphic above shows the part of the goal that scoring penalties were aimed at.  With most strikers aiming bottom left and bottom right it’s no surprise to see that these were the most successful areas.

The second graphic which shows where penalties were saved shows a more complete picture – goalkeepers made nearly all their saves low down.  A striker who has the skill and control to lift the ball high makes it very unlikely that the goalkeeper will save his shot.

The last graphic also shows the risk involved in shooting high.  This data shows where all the missed penalties (which were off-target) were being aimed.  Unsurprisingly strikers who were aiming down the middle of the goal managed to hit the target!  Interestingly strikers aiming for the right corner (as the goalkeeper stands) were far more likely to drag their shot off target than those aiming for the left side.  Perhaps this is to do with them being predominantly right footed and the angle of their shooting arc?

Win the toss and go first

The Prozone data also showed the importance of winning the coin toss – 75% of the teams who went first went on to win.  Equally, missing the first penalty is disastrous to a team’s chances – they went on to lose 81% of the time.  The statistics also show a huge psychological role as well.  Players who needed to score to keep their teams in the competition only scored a miserable 14% of the time.  It would be interesting to see how these statistics are replicated over a larger data set.

Don’t dive

A different study which looked at 286 penalties from both domestic leagues and international competitions found that goalkeepers are actually best advised to stay in the centre of the goal rather than diving to one side.  This had quite a significant affect on their ability to save the penalties – increasing the likelihood from around 13% to 33%.  So, why don’t more goalkeepers stay still?  Well, again this might come down to psychology – a diving save looks more dramatic and showcases the goalkeepers skill more than standing stationary in the centre.

Test yourself

You can test your penalty taking skills with this online game from the Open University – choose which players are best suited to the pressure, decide what advice they need and aim your shot in the best position.

If you liked this post you might also like:

Championship Wages Predict League Position? A look at how statistics can predict where teams finish in the league.

Premier League Wages Predict League Positions? A similar analysis of Premier League teams.

Game Theory and Tic Tac Toe

The game of Noughts and Crosses or Tic Tac Toe is well known throughout the world and variants are thought to have been played over 2000 years ago in Rome. It’s a very simple game – the first person to get 3 in a row wins. In fact it’s so simple that it has been “solved” – before any move has been played we already know it should result in a draw (as long as the participants play optimal moves).

The way to solve Noughts and Crosses is to use combinatorial Game Theory – which is a branch of mathematics that allows us to analyses all different outcomes of an event.

This is the start of the game tree for Noughts and Crosses. We can expand this game tree to cover every possible outcome for the game. Once this complete tree is drawn, any participant can work through this tree to see what is their optimal move at any one time from any position.

An upper bound for the number of positions and number of different games is given by:

3 9= 19,683.  This is the total number of possible game positions in a 3×3 grid – as every square will either be a O, X or blank.

9! = 362,880. This is the total number of ways that positions can be filled on the grid. (First you have 9 choices of squares, then there are 8 choices of squares etc).  This counts each X and O as distinct from other X and Os.

9 choose 5 = 126.  This is the number of different combinations of filling the grid with 5 Xs and 4 Os.

However the analysis of this game tree can be significantly simplified by realising that many different positions are simply reflections or rotations of each other. By looking only for distinct positions (positions that are isometric under refection and rotation) we can, for example, see that there are actually only three distinct starting moves – as shown in the diagram above.

James Grime from Numberphile takes us through how to answer a related question, “How many different ways are there to completely fill a Noughts and Crosses board with 5 Xs and 4 Os – not including rotations and reflections?”  The solution above is a little complicated (it makes uses of Group Theory) but it is an excellent introduction to some uses of higher level mathematics.

This somewhat horrendous looking graphic actually contains the solution to playing Noughts and Crosses. You can use it to always achieve the optimal outcome for X. It works as follows:
1) The big red X in the top left hand corner represents your best first move. So you make this move first.
2) Next, you see what your opponent does and choose the grid with the big black O in the position they have chosen.
3) This new grid will have a big red X – this is your next optimal move.
4) You then remain in your subsection of the larger grid – and repeat the process.

If you liked this you might also like:

Game Theory and Evolution – do nice guys always finish last?

Knight’s Tour – an exploration of this 1000 year old mathematical puzzle.

Essential resources for IB students

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.

The video above is a great example of “mathemagic” – magic through maths.  Arthur Benjamin’s show at TED (using a mixture of mathematical tricks and savant like numerical ability) shows how numerical calculations can still produce a sense of awe and wonder.

Probably the best resource for “mathemagic” is the TES Word ebook from Stephen Froggatt.  This contains over 25 different maths magic tricks with full explanations about how to use them in a classroom setting.  As he says,

“Mathematics can be presented as a dry collection of rules and exercises (surely not!) or as a window through which can be seen explanations to many of the world’s mysteries. A magic trick provides the interest, and its explanation the demonstration of the power of mathematics to provide answers. Suddenly all that previous work on simplifying algebraic expressions comes into action when explaining why the Number You  Thought Of had to be seven.”

Magic Square Magic

As an example of one of this tricks, he describes how to ask a student his house number – upon answering, “46”, he immediately draws the following grid:

It’s then left to the students to find the connection between this grid and 46.  Each row adds to 46, as does each column, and both diagonals, and the 4 corners, and the 2×2 corner squares!  The impressive nature of this trick is the speed it can be calculated – and how it can be done with any given numbers.   The template needed is:

You simply need to substitute the the given value for N.  If you chose to reveal the secret it would be interesting to see if students could work out how to create their own grids with different template numbers.

Lightening Fast Multiplication

Another example from the book include how to multiply by 11 with lightening speed:

Write a large number on the board (eg. 3143221609) and race to see who can multiply this by 11 first.  The answer is 34575437699 and can be done in seconds.  Simply start with the end digit (in this case 9) and write that down, then working from right to left the next digit is 0+9 = 9.  The next digit is 6+0 = 6, the next digit is 1+6 = 7 etc.  Each time you just add the consecutive terms of the original number.  You finish by writing down the first term (in this case 3).

There are loads of other tricks in the free ebook to utilise.  The, “think of a number tricks,” are great for algebra topics, the magic cards make use of binary arithmetic and there is mobius magic for shape and space discussions.

Jan Honnens (also on TES here) has formalised some of this content into an investigation format with some great leading questions for students to follow.

Another example of a very powerful maths trick – which is very easy to do is given here:

And if all that isn’t enough, there’s a fantastic 96 page ebook pdf also free – available for download from here– which contains a large number of card and number tricks which make use of numerical and algebraic rules.

Essential resources for IB students:

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.

The Gorilla in the Room and Other Great Maths Investigations

These topics are a great way to add interest to statistics and probability lessons at KS3 and KS4 level, and also a good example of investigations that IB students can conduct. They also have a nice link to ToK – how can we believe what we see or what we hear?  To what extent should we trust our senses?  And it shows the power of statistics and empirical testing in trying to understand what is externally real and what is our own version of reality.

For each one, have the students make a hypothesis (if possible without giving the endings away!), then collect some data as to how the students react.  Then look at how the data could be collected in a larger scale experiment (or how the experiment could be modified).

The first one at the top of the page is the “Fa, Ba” test.  This is a really curious experiment that shows that what we “hear” is actually often influenced by what we see.

The second one is the amazing colour changing card trick by Richard Wiseman.  This is also a great way of showing how we often fail to see what is really in front of us:

The third video is even more impressive – though it doesn’t work on all students.  You have to set this one up so that all students are really intently concentrating on the screen – perhaps a prize for the student who gets the answer correct?  Also no talking!  Students have to count basketball passes:

The last one is a good test of whether students are “right brain” or “left brain” dominant.  They have to stare at a rotating woman – some students will see this going clockwise, others anticlockwise.  Some will be able to switch between the 2 views.  If they can’t (I initially could only see this going in an anti clockwise direction) near the end of the video it shows the woman rotating in a clockwise direction to help.  Then rewinding the video to the start – and as if by magic she had changed direction.

If you liked this post you might also like:

Even Pigeons Can Do Maths A discussion about the ability of both chimps and pigeons to count

Finger Ratio Predicts Maths Ability? A post which discusses the correlation between the two.

Cracking ISBN and Credit Card Codes

ISBN codes are used on all books published worldwide. It’s a very powerful and useful code, because it has been designed so that if you enter the wrong ISBN code the computer will immediately know – so that you don’t end up with the wrong book. There is lots of information stored in this number. The first numbers tell you which country published it, the next the identity of the publisher, then the book reference.

Here is how it works:

Look at the 10 digit ISBN number. The first digit is 1 so do 1×1. The second digit is 9 so do 2×9. The third digit is 3 so do 3×3. We do this all the way until 10×3. We then add all the totals together. If we have a proper ISBN number then we can divide this final number by 11. If we have made a mistake we can’t. This is a very important branch of coding called error detection and error correction. We can use it to still interpret codes even if there have been errors made.
If we do this for the barcode above we should get 286. 286/11 = 26 so we have a genuine barcode.

Check whether the following are ISBNs

1) 0-13165332-6
2) 0-1392-4191-4
3) 07-028761-4

Challenge (harder!) :The following ISBN code has a number missing, what is it?
1) 0-13-1?9139-9

Answers in white text at the bottom, highlight to reveal!

Credit cards use a different algorithm – but one based on the same principle – that if someone enters a digit incorrectly the computer can immediately know that this credit card does not exist.  This is obviously very important to prevent bank errors.  The method is a little more complicated than for the ISBN code and is given below from computing site Hacktrix:

You can download a worksheet for this method here. Try and use this algorithm to validate which of the following 3 numbers are genuine credit cards:

1) 5184 8204 5526 6425

2) 5184 8204 5526 6427

3) 5184 8204 5526 6424

Answers in white text at the bottom, highlight to reveal!

ISBN:
1) Yes
2) Yes
3) No
1) 3 – using x as the missing number we end up with 5x + 7 = 0 mod 11. So 5x = 4 mod 11. When x = 3 this is solved.
Credit Card: The second one is genuine

If you liked this post you may also like:

NASA, Aliens and Binary Codes from the Stars – a discussion about how pictures can be transmitted across millions of miles using binary strings.

Cracking Codes Lesson – an example of 2 double period lessons on code breaking.

Essential resources for IB students:

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.

A maths song sung by current flavour of the month One Direction – follow the lyrics to arrive at the total.

Some of the best maths songs are by Learning Upgrade, such as, videos on circle formulae, fractions, exponents, the quadratic formula and the one below, “Mean, Median and Mode”:

Some other good maths songs:

Westerville South High School in Ohio have made some great rap-based maths songs such as the trigonmentry song, Getting Triggy With It

The Calculus Rhapsody is a fantastic  take on Queen’s famous song – and good for IB SL and HL students.

And James Blunt’s Triangle is good for a KS3 shape and space introduction.

One of my favourite resources is the Jeopardy quizzes. For those not familiar with the game (I think it’s American), it’s a gameshow, where you get to choose questions of different levels of difficulty, from a range of categories. I downloaded the template from TES – it’s a ready-made powerpoint which you can click on to take you to relevant questions, and then another click returns you to the home screen. The class can be split into teams, each team given a whiteboard and (say) 2 minutes to answer a question. Teams with the correct answer get those points for their teams. The challenge round adds a bit more excitement – and you can add any general questions or puzzles – such as dingbats or memory challenges (memorise pi to 10 places etc).

I’ve uploaded 20 of the quizzes onto TES here.  There’s 20 different ones –  KS3 Algebra, shape and space, fractions and general revision for different levels, GCSE and IGCSE topic specific quizzes on algebra, geometry, trigonometry, number, probability, matrices, functions, algebra 2 and linear graphs.  I have also done a small number of IB ones – which I’ll link to when I have a few more to share.

This is another interesting maths sequence puzzle:

When x = 1, y = 1, when x= 2, y = -1, when x = 3, y = 1,

a) if when x = 4, y = -1, what formula gives the nth term?

b) if when x = 4, y = 3, what formula gives the nth term?

Answer below in white text (highlight to see)

a) This is a nice puzzle when studying periodic graphs. Hopefully it should be clear that this is a periodic function – and so can be modelled with either sine or cosine graphs.

One possibility would be cos((n-1)pi)

b) This fits well when studying the absolute function – and transformations of graphs. Plotting the first 3 points, we can see they fit a transformed absolute value function – stretched by a factor of 2, and translated by (2,-1). So the function 2abs(x-2) -1 fits the points given.

This is a great resource from Mr Collins – Maths Pictionary. What I like about this is that it can be incorporated into a large number of classroom activities – from Jeopardy games, to starters to topic revision.  It can also be easily adapted to everything from KS3 to IB – and can be a great way of revising key vocabulary.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

### New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

### Free HL Paper 3 Questions

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