matheminutes

Jaro Jumble

noreply@blogger.com (owenelton) — Mon, 06 May 2013 14:29:00 +0000

It's not often that I openly admit to failing to solve a seemingly straightforward problem, especially one that I've posed myself. Perhaps in the interest of humility and letting it be known that maths teachers don't actually have all the answers, it is something that I should do more often.

As I was having my breakfast before school on Saturday I saw a news article on the BBC about a new online game called Jaro. Quite why providing free advertising for a money making gambling venture is the remit of our national broadcaster I don't know, but the premise that the game provides players with a choice of how much of their money goes to charity and how much into the prize pot (with the company taking a tidy 5% "administration charge") seemed to be a gimmick too tempting for the newsdesk to ignore.

Jaro: "Not a traditional lottery or raffle draw" (you're far less likely to win anything)

Players pay $10 for a gamecard and are pitched head to head in various rounds of a knock-out battle. The last man standing will receive the "global prize pool" of "a share of $1billion". Tempting. There are 27 rounds which make the chances of winning anything just 1 in 134 million. Claims that the game is actually tactical are made but, effectively, it relies on skill only as much as playing rock-paper-scissors; the odds are no better than 50-50 of beating an opponent despite the advertising to the contrary.

There are, however, some interesting mathematical questions that arise from the game.

For your $10 you will receive a Jaro game ticket that looks like this:

In the first round, the numbers 1 to 9 are randomly assigned to the 3x3 grid. In later rounds you fill them in yourself (this is where tactics supposedly play a part). The grid on the left are your numbers and your opponent tries to guess where they are. The grid on the right is your attack; this is your guess of where your opponent's numbers lie.

In each round, two tickets go head to head. Defence and attack are unleashed and the following scoring system applies for each cell:

100 points for a "direct hit" meaning a matching number.
10 points if there is a difference of 1 between defence and attack.
1 point if there is a difference of 2 between defence and attack.

Here is an example of player A's scoring against player B:

Player A scores two direct hits (green) and two near misses by one (yellow) leaving him with a total of 220 points. The respective calculation for Player B shows that he loses the round with only 121 points. Check it.

There are various questions we might ask ourselves about this game. The most obvious, I suppose, regards the probability of scoring 900 points. This is straight forward. There are 9! (362880) ways of arranging the numbers 1 to 9 in the grid, so only 1 in 362880 attacks will score this maximum. I reckon, if the tournament proceeds as planned, that this should occur about 740 times. How do you think I calculated that?

A slightly harder question is "What is the probability of matching no numbers". The number of ways of creating no matches is known as the number of derangements and, in this case, can be written !9. A cunning iterative formula is used to calculate these. It is derived by imagining that I've placed the number 1 in, say, the position of number 5, and splitting the resulting situation into two cases: Either 5 is placed in the position of number 1 or it's placed elsewhere. A little thought gives us the formula:

The probability of scoring no matches is !9/9! (a nice palindromic expression) which is about 0.368. This means that over a third of the time you will score fewer than 100 points.

The hard question, the one that has me stumped for a non-'brute force' solution, is this: What is the probability of scoring zero points? This is more difficult because different numbers behave in different ways. For example, to score zero, Player B can't place a number 9 in the positions of Player A's 7, 8, or 9, leaving six possible positions. The number 5, however, is banned from 3, 4, 5, 6, or 7, and so can only take one of four possible positions. This assymetry seems to be disastrous with regards an attempt to solve the problem neatly.

One way of restating the problem is as follows: How many ways are there of placing nine counters in the blue squares of the following grid, so that there is exactly one in each row and one in each column?

Quite often when approaching this type of problem, it's a good idea to start with a smaller grid and see if any patterns emerge. The smallest grid on which a placement is possible is a 6x6 grid; there is just one way of doing it. On a 7x7 grid there are eight ways. After this, however, the numbers get very big very quickly.

I am simply too lazy to write out a systematic list of all possibilities. If anyone is sufficiently coding-literate to write a programme that does it, or finds the flash of inspiration required to solve this elegantly, then please do get in touch.

The Geometry of Motherships

noreply@blogger.com (owenelton) — Fri, 12 Apr 2013 14:41:00 +0000

As I sat in the cinema to enjoy Tom Cruise's new film, Oblivion, I was struck by the screenwriter's reference to a platonic solid. All big sci-films have some sort of mothership; basically a portable planet that can be conveniently steered between solar systems carrying aliens and their weapons. In this particular film, the mothership has the rather cursory name, Tet. You can see it at the top of this still from one of the film's trailers.

It is presumably called Tet since it is based on a tetrahedron, a regular convex polyhedron constructed using four congruent equilateral triangles. In the above image, you might just be able to make out that each of the four corners has been sliced off. The resulting eight-faced shape is actually called a truncated tetrahedron, but presumably the name TruncTet was a little clumsy and didn't appeal to the writers.

Tetrahedron truncation

I then wondered if the other platonic solids had ever been used for motherships. With a little time on Google, I came across this; Star Trek's Borg Cube:

Borg Cube: Platonic mothership number 2.

That is where the trail ran cold, however. I failed to find any movie motherships based on octahedra, dodecahedra or icosahedra. I suspect that these haven't been used because they wouldn't have the same visual impact of the cube or tetrahedron. As the number of faces increases, the more spherical these solids appear, so you might as well just have yourself a Death Star:

Interestingly, the Death Star differs from a sphere in two distinct ways. Firstly, it has a conspicuous indentation that houses its planet-destroying laser. This means that it is easy to see that the solid is non-convex. Imagine a couple of points on opposite sides of the laser dimple. If these were joined by a straight line segment, this line would travel outside the Death Star. In order for a shape to be convex, any pair of points within it must be joined by a straight line that is also within the shape.

The purple line lies outside the Death Star and therefore demonstrates its non-convexity.

On closer inspection we see that, just like a real planet, the Death Star is a far cry from a perfect sphere. We see a number of buildings and channels, convenient for giving the smaller spacecraft a sense of speed, but making the edifice less spherical.

The bumpy surface of the Death Star.

Not every mothership is so easily describable. The lean-mean humanity-destroying machine from Indpendence Day appears to be a composition of three main parts; two inverted pyramids dangling beneath a slice of egg.

It's corridors, however, are based on triangular prisms.

It seems that Hollywood designers either like smooth curves or non-obtuse angles in creating their largest spacecraft. Where might they go in future? Well my vote would be for the Szilassi Polyhedron. It has seven faces but, more interestingly, contains a hole in the middle; it is topologically equivalent to a ring doughnut but looks far more threatening. Discovered by Hungarian mathematician, Lajos Szilassi, in 1977, I think it would make a beautifully intimidating mothership in a forthcoming flick.

The Szilassi Polyehdron: A mothership in the making?

Thatcher's Mathematical Legacy

noreply@blogger.com (owenelton) — Mon, 08 Apr 2013 15:21:00 +0000

With today's news of the passing of Baroness Thatcher, attention is correctly focused on her achievements in office: The warm relationship that she shared with Reagan and Gorbachev, her steely rhetoric and determination, and the fact that she is still Britain's only female Prime Minister. Something else, thought, that it is possible to remark on hindsight is the extraordinary change in mathematics education over which her government presided.

May 1979: Thatcher is elected Prime Minister of Great Britain

Just over a month after her election, a cohort of students sat their Mathematics O-Levels. These exams were aimed at the top twenty percent of the country's maths students. It isn't particularly trendy to focus solely on high achievers but, I would argue, the progress of this group in mathematics is important. These are the students who stand a chance of studying the subject at degree level and, later, use their skills and expertise in this field for the benefit of the economy.

Here is the first page of a paper from 1979:

If you are familiar with current Higher Level GCSE papers, the first thing you might notice is the density of the text; there are no diagrams provided at this stage and answers are not written on the question paper. Each question also seems a rather strange hotch-potch of topics. As far as the mathematical material is concerned, however, this all seems consistent with what a sixteen year old would be expected to deal with today. If I gave this to one of my classes I'd expect there to be a number of mistakes due to a lack of willingness to draw diagrams, but most would be able to have a good go.

It is section B of the paper, worth 45 of the overall 100 marks, that really poses an interesting challenge. Students are invited to answer three of five questions. One of them is as follows:

This is unlike anything that might appear on a GCSE paper today. The candidate will get all fifteen marks only if they are able to calculate this area

and this volume.

These are questions that are now commonly asked a year later. The other questions in the paper are similarly demanding requiring the ability to interpret and act upon intimidating paragraphs of text.

Margaret Thatcher left office in 1990 just over two years after the first GCSE exams had been taken. Even though it was divided over two tiers the most able mathematics students did not face the same degree of challenge as they had a decade or so earlier. A 2008 report by think tank Reform mentions that the decline had set in near the beginning of Thatcher's reign: "In 1980 ... there is the clear beginning of a decline in difficulty. Fewer questions asked candidates to provide proof of concepts. Pupils were more led through the questions than in previous years." After this the report cites a sharp decline from 1990 as GCSEs came to the fore: "The ability to solve complex, multi-step arithmetical or algebraic problems was virtually absent from the GCSE papers".

The change to the landscape of mathematical education in the eighties led to what Reform rather dramatically calls "a lost generation of mathematicians". They conclude that "over the last thirty years the UK has lost sight of the value of mathematics. The losers have been pupils, teachers, employees, employers and education Ministers."

The changes to maths teaching that her government implemented will never be at the forefront of peoples' minds when Thatcher's legacy is mentioned. It is, however, certainly worth some sober reflection.

Eggquations

noreply@blogger.com (owenelton) — Sat, 30 Mar 2013 16:09:00 +0000

As you tuck into your chocolate eggs tomorrow will you, at any stage, wonder how you might model their shape? Well one man who has is Nobuo Yamamoto of Japan's TDDC Lab. On his webpage he has written seven different articles about equations of eggs. They make fascinating reading; I will share with you his first suggestion which results in the following shape (seen here from four different angles):

I was pretty impressed with this. They are convincingly eggy in my opinion.

Yamamoto's construction depends on three variables and is most easily expressed in parametric form. We start off in two dimensions. The following diagram demonstrates how a general point on the egg's largest cross section is determined.

The red dashed line is our egg cross section and the coordinates (x,y) are determined, as theta varies, by parametric equations based on (positive) constants l, a, and b. The crucial lengths that determine the coordinates for any particular value of theta are the horizontal red and diagonal blue lines above. The clever thing about these is that they also vary with theta. The red line begins with length l but that is reduced to -l as theta changes from zero to pi. Similarly, the blue line varies between a + b and a - b. This reduction in length allows the shape to be more pointy at one end to the other.

We can easily read expressions for the coordinates (x,y) from the picture above giving us parametric equations:

The transition from this two dimensional egg to the three dimensional one is also fairly easy to carry out. We must simply rotate the shape about the x-axis. We leave the x coordinate expression be, and split the y coordinate expression in two using a second parameter. This leaves us with equations

All that remains to be done is to plug these into some graphing software and have a play. You can use this Autograph Activity to save you some time. Have a very Happy Easter.

Ninja Flapjacks

noreply@blogger.com (owenelton) — Mon, 25 Mar 2013 16:36:00 +0000

Today is, undoubtedly, a slow news day. Possibly the slowest since October 2009, when the media went goofy over goggles, reporting that Adlington Primary School staff insisted that students wear safety specs in order to play conkers.

Students at Adlington Primary School play conkers...safely.

As of this morning, the press have gone barmy over baked goods, reporting with unrestrained relish that Castle View School in Essex have banned triangular flapjacks. The reason being that a year 7 pupil was injured after one such oaty projectile hit him square in the face.

My sympathies, of course, go out to this poor student and my thoughts are with his family. I would, however, like to point out that it is unlikely to have been the triangular property of the flapjack that was responsible for the injury.

The Independent mentions that a critic of the ban said, "a square flapjack has more sharp edges than a triangle shaped one." While this initially sounds like a reasonable argument, surely it means a pentagonal flapjack would be worse still, superceded by the hexagonal, the heptagonal, the octagonal and ultimately - pursuing this line of logic to its limit - the circular.

Surely what really matters is the 'number of corners' to 'perimeter of cross-section' ratio. We'll call this the Flapjack Potency Ratio, or FPR for short. It seems sensible that the higher this is, the more likely you are to be skewered. Suppose we begin with a square of unit length. Its FPR is a simple 1:1, as it would be for any regular polygon of unit side length. Was the offending triangle not equilateral? The BBC correspondent, chomping away at my license fee by broadcasting live from the scene, held up an isosceles right triangle. The lethal missiles had obviously been shaped by cutting squares across their diagonals. This, however, leads to a smaller FPR and so, surely, a relatively safer snack.

The square has the more dangerous Flapjack Potency Ratio

I would go further and suggest that the shape of the cross-section doesn't make a great deal of difference. If we're modelling the flapjacks as prisms then, surely, the way to make them safer is to make them thicker. Why do I say this? Well, I can only assume that the flapjack was thrown so that it was rotating about an axis through its centre of mass and perpendicular to its cross-section. We might find the centre of mass of a triangular flapjack by finding the intersection of the three angle bisectors of its triangular face.

A flapjack thrown in this manner uses a similarly devastating technique to that used by masked warriors in various subtitled martial arts films when they fling ninja stars or, more properly, hira-shuriken. These weapons do their damage by having sharpened leading edges which slice through their targets. Many of them also exhibit rather attractive rotational symmetries.

Shuriken exhibiting rotaional symmetry of orders 3, 4, 6, and 8.

You'll notice that the above weapons have their masses concentrated towards their centres. This, presumably, is in order that their rotation rate is higher for a certain fixed input of rotational kineric energy.

So, if I were to offer my advice to anyone wishing to make a safety flapjack, I would tell them to ignore the shape of the cross-section, to make them sufficiently thick so that they are not slicy, and to cut a hole in the middle so that relatively more energy is required to give them a deadly rotation. The banned triangles, as far as I am concerned, probably cause no more damage than the original squares in the midst of student on student warfare.

Mathematical Things to Do on a Rainy Day #011

noreply@blogger.com (owenelton) — Thu, 21 Feb 2013 11:35:00 +0000

Write a Mathematical Limerick

It strikes me that the one thing that the history of mathematics severely lacks is a series of limericks. These could summarise key events in a pithy manner and finally turn some of the worthy protagonists into household names, like that man from Nantucket.

So here's my request. Read the following guide and example, and leave your limerick in the comment space below.

Step 1
Choose a mathematician eg Evariste Galois.

Step 2
Identify their contribution to mathematics and/or what they're most famous for eg inventing Galois Theory and dying in a duel.

Step 3
Summarise the material from step 1 and step 2 in limerick form. eg:

The tale of young Galois' sad plight

Culminates in his very last night.

He worked on his Theory

'Til dawn when, quite weary,

He exited life in a fight.

Step 4

Post your composition in the comments box below.

Have fun.

Mapping Misconceptions

noreply@blogger.com (owenelton) — Wed, 20 Feb 2013 16:18:00 +0000

Every few weeks or so, during a lesson, a well-meaning and engaged student will raise his hand and suggest that I could make some algebraic progress in solving an equation by square-rooting both sides. I glance at the left hand side, which consists of a sum of squares, and I already know what's coming next: It is suggested that I can just make things a whole lot simpler by crossing out the squareds.

"No, no, no, no, no", I cry in anguish, frustrated that this misconception is holding us up again. "We can't do that", I'll say, "it's against the rules; you know that three squared plus four squared is five squared, so would you be happy square-rooting both sides, crossing out the squareds, and getting seven on the left?"

Erm, not cool!

There is no reply, just a well-meaning and engaged student who's now a little bit indignant because I appear to have invented a rule that has made the next twenty-five minutes a little bit harder. He is also confused that Mr Elton appears to have made his case using a single specific example, whereas he's always extolling the virtues of proving things in general.

So maybe I'm doing things wrong. Suppose that, instead of providing a counter-example, I showed the student the numbers for which his algebraic mishap would work.

Take this classic example:

Despite the ludicrous notion that the sum of two positive numbers might be smaller than each of them, this sort of mistake frequently crops up. Presumably it's because the denominators might be algebraic so the students aren't really thinking about them as numbers, thereby finding it difficult to distinguish between a permissible process and a fractional faux pas. Alternatively, perhaps, their minds are grappling instead with more pressing concerns. Like Calculus. Or the Harlem Shake.

So for which numbers, if any, is simply adding the denominators allowed? To find out, we must find pairs (x, y) that satisfy the equation:

Adding the fractions (properly) and rearranging gives:

This isn't looking enormously useful. The death knell for this particular example strikes when we convert to polar coordinates and get

The expression in the brackets is always positive, so we are left with the only possible pair (x,y) being (0,0). Since we are unable to plug these into our original expression (at least without beginning a conversation about simple poles), we are stuck with the rather sorry conclusion that this is the ultimate piece of algebraic bunkum. Not only is it not true for all x and y; it isn't actually true for any!

In this respect it is worse than

which, at least, is true whenever x or y is zero.

There must be a more fruitful blunder out there. Here's another that I've encountered recently:

Well why on earth shouldn't the circular functions be linear? (Don't answer that).

Let's set about solving the implied equation; the process is slightly more fiddly.

So, we have a family of diagonal parallel lines along which the equation is satisfied. That isn't all, however. In line four of the above method we set cosx and cosy not equal to one, so we need to see if these yield solutions. Whenever cosx is one, we know that sinx is zero. Checking the second line of the method shows that the equation will always be satisfied in these cases. This means that we must include a square grid in our picture of possible pairs (x,y).

A graph of the implicit map sinx + siny = sin(x+y)

So at last we've found a common algebraic slip that is actually valid for a (reasonably) interesting set of values. Next time you encounter such an identity failure, why not attempt to comfort the student with a picture of the values for which his assertion is correct.

Why Gove is Wrong about Long Division.

noreply@blogger.com (owenelton) — Wed, 06 Feb 2013 20:11:00 +0000

Michael Gove, the UK's Education Secretary, delivered a speech yesterday that paved the way for his curriculum of "core knowledge" that should be taught in primary and secondary schools. Some of his ideas seem fairly sensible; "scientific principles" will be taught in science; "grammar and punctuation" will be taught in English; "locational knowledge" will be encouraged in Geography.

For maths he would like the following:

"early memorisation of tables"
Fine. I'd love my students to arrive in Year 9 with sufficient numerical familiarity to ensure that they don't have to stretch themselves in order to pronounce the product of six and seven. If they are distracted by rudimentary arithmetic then the more engaging concepts that I am trying to explain will take a back seat in their minds.

"calculations with fractions"
Good. Fractions are challenging. Are they ratios? Are they points on the numberline? A failure to properly understand quotients can be a catastrophic hindrance with regards the algebra required at GCSE and beyond.

"written methods of long division"
I'm sorry, what? This has to be some sort of joke. Surely no sensible education policy advisor can suggest that long division is a core requirement in maths. It is an absolutely laughable inclusion!

Gove: Thumbs Up for Long Division

Let me argue my point, while trying terribly hard to keep a straight face.

Here is an example of long division:

As you can see, I have successfully divided 305753 by 31. The answer is 9863. Gove would like all students to be able to carry out this process by hand, which admittedly would have been very useful in, say, the 19th Century. These days, however, this archaic algorithm is completely and utterly obsolete. We have calculators built into our smartphones that will do the division for us. You might as well require all students to learn how to use a washboard and a mangle to do their laundry; at least in this case they might learn something useful about social history.

Long Division: An Analogy

So what could possibly be the arguments for including long division in the curriculum? Why might policy makers be under the misapprehension that it is a vital tool with which 21st Century students must equip themselves?

Perhaps they think students should be able to perform calculations manually. Well, I completely agree, but I don't think this is an argument for long division. I would heartily endorse the idea that we might ask students to aspire to write down answers to elementary mental arithmetic problems with little hesitation, but the whole point of long division is to assist more complicated calculation. Do we really need to insist that students carry out calculator-worthy calculations by hand? Wouldn't it be better to encourage a more estimation-based approach?

Here is a hypothetical situation:

If you're in a party of nine at a restaurant and your bill comes to £214.74 it is very useful to be able to say, "Well, splitting it nine ways comes to a little under £25 so let's all pay that and try to remember to be more generous with the tip next time we come here". This is a handy, quick, elegant, mathematically sound solution that requires minimum fuss and won't prove a sore point for the rest of the evening. Let's teach students to be able to do this.

Suppose, instead, that a keenly trained graduate from the school of Gove started hastily scribbling some long division on the back of a napkin. "Aha", he says, "a little long division shows me that nine divides the bill's total precisely. Everyone owes £23.86." After a little while longer, he'll calculate that £26.25 is the amount owed (to the nearest penny) with the inclusion of a 10% tip and everyone will begin to scrabble around in their pockets for 20p and 5p pieces. Sadly, the graduate probably didn't notice the rather elegant fact that the exact partitioning of the bill was inevitable since the sum of the digits of £214.74 is a multiple of nine. He's not a mathematician, though, he simply has the ability to carry out a set of instructions less reliably and more slowly than his phone.

When was the last time you impressed your friends with some long division over dinner?

Maybe, then, Gove wants students to be able to follow a set of numerical instructions. Maybe that's the point of long division. After all, it's a useful skill to have if you'd like to bake a cake, for example. Do the instructions really have to be in the form of long division, though? Why don't we teach students a different algorithm? Perhaps Euclid's Algorithm for calculating the greatest common divisor of two numbers would suffice. It has some distinct advantages over long division, after all: It has a historical context that will tick all sorts of cross-curricular boxes; the instructions really couldn't be simpler and they allow students to practise their mental arithmetic en route; it is not immediately obvious how to achieve the same result on a calculator; and, most beautifully of all, the algorithm forms the basis of a proof of the Fundamental Theorem of Arithmetic. How many students will leave school in ten years' time knowing how to divide longly but with absolute no idea that each positive integer has a unique prime factorisation? It would be a misrepresentation of mathematics on a national scale.

The first 2500 integers coloured according to their degree of prime factorisation

Maybe I've sold long division short; perhaps it can be extended and used to enrich students' mathematical experience just like Euclid's Algorithm. For example, we can use it to divide polynomials by polynomials (functions made by adding together different powers of x). I wouldn't recommend it, however, since it's frightfully messy and the same job can be achieved in a single step by systematically comparing coefficients. No, long division stops with long division. It is a subject that sews itself up; there is no means of extension besides increasing the size of the dividend and divisor and I certainly wouldn't be able to bring myself to command students do this as I have a mechanical device on my classroom desk that will perform the calculations with a few turns of a crank; just like a mangle.

Division machine complete with mangle action.

Perhaps I've overlooked some underlying elegance in the method of long division, or some deeply disguised application of the algorithm that is fundamental in the rich history of mankind's mathematical adventure. If so, I'd be delighted to hear about it. I would also eat my hat.

When Maths Becomes Unbelievable

noreply@blogger.com (owenelton) — Sat, 26 Jan 2013 11:53:00 +0000

Over the last five weeks of school, I've been guiding a Theory of Knowledge class through mathematics, one of six stipulated "Areas of Knowledge". For the uninitiated, Theory of Knowledge is one of the three central pillars of the International Baccalaureate Diploma's sacred hexagon, which all Learners of noble profile must conquer before drinking from the sweet chalice of success and receiving a positive integer no larger than 45.

If you are a teacher and you ever get the chance to approach your subject via a ToK agenda, I can heartily recommend jumping at the chance. I said "no" the first time. A year later, however, I had warmed to the idea and the experience of being released from the shackles of a syllabus and an exam to work towards (albeit for just two lessons a week) is utterly refreshing. One aim of mine has been to undo the students' pre-conceptions of mathematics; one that they share with the majority of the UK's population and that has been cultivated by the requirement to teach maths in a way that can be quickly and monotonously assessed on a large scale by those who often view the subject in the same way because that's how they were taught.

If maths really was all about completing the square, integrating by parts, adding fractions, and all the other pseudo-algebraic processes that we teach because they are easy to examine, then we might as well sit back with a glass of sherry and project a series of anodyne Khan Academy videos onto the SMART board while our robotic students gruellingly learn how to tick boxes. Because ticking boxes is what maths is all about.

But it isn't.

Maths is about the incredible journey into your imagination that happens when you make a series of assumptions and deduce their necessary consequences. Sometimes the results are completely incredible. Take the following example; I posted this on my ToK Class blog and asked students to comment. I hadn't mentioned the content previously in class. There was no warm up. They did not consult each other first. I simply asked them to consider the following scenario and questions:

Turning a Football into a Planet

I have a solid sphere the size of a football. Suppose I told you that I could split this sphere into a finite number of pieces and, just by rearranging these (a bit of rotation here, a bit of translation there) and without any stretching, I could create a solid sphere the size of the Earth.

You'd think I was talking nonsense, right?

Would you believe me if I said that the ability to do this had been mathematically proven in between the first and second world wars?

What lengths would I have to go to convince you that this is true? Would an internet site be sufficient? How about an academic paper? Who checks all this maths anyway?

The responses are entirely understandable:

ED: To be absolutely honest, I don't think you could ever make believe that this proposal was in fact true.

CS: I don't think that it's possible.

LS: This proposal is highly unlikely and seems absurd.

EY: I think this is unlikely to be true.

Some students would only be convinced if I could demonstrate the transformation in front of them:

ES: I think the only way to be convinced was if I had an actual demonstration but math papers wouldn't suffice.

AT: Just as the majority of my classmates or any ordinary people, in order to believe such extraordinary/fantastical theories, I need to first see the practical proof, the real evidence of a football-size solid sphere converting into a solid sphere the size of our planet.

LS: I would have to see it in person because it is very easy to doctor photos and videos while papers could also be falsified or have incorrect proof.

OS: I would never be able to believe it unless I saw it with my own eyes.

CS: I would not be convinced by an internet site, because nowadays you can manipulate pictures or videos so much, that you can easily prove the most impossible thing in the world....In order to be convinced I would ... need somebody to show it to me in front of me.

ED: I wouldn't be convinced by an internet site or mathematical equation.

So, plenty of students wouldn't believe an internet site. Presumably this also means that they would dismiss the Wikipedia page that discusses this particular result, known as the Banach-Tarski paradox.

What's that? Students won't necessarily believe everything they read on Wikipedia? Well, that's refreshing.

Wikipedia's (pretty darn misleading) illustration of the Banach-Tarski paradox.

The comments occasionally sounded rather like arguments surrounding religion, and one student hit this nail on the head:

HS: I think what this boils down to is the threshold required for a belief to take root.

The slightly confusing thing is that the Banach-Tarski result is dependent on the Axiom of Choice and I didn't have any objections when I described this assumption to the class: Of course I can pick something out of each of an array of boxes if each box contains at least one object. Even if there might be infinitely many boxes? Yeah, why not?

When on earth did maths become about believing things? No student has raised their hand in one of my regular maths classes and claimed "Sir, I don't believe you". That's probably because the thought that my exposition teeters precariously on some well-hidden assumption that cannot be proved truthful simply doesn't occur to them. Maybe because they're too busy ticking boxes. After all, if they put ticks in each of an infinite quantity of boxes and then assume that they can pick a tick from each box, then they can turn a football into a planet; and which teenagers don't wish they had a superpower?

The Mechanics of Bunny Suicide

noreply@blogger.com (owenelton) — Tue, 15 Jan 2013 20:32:00 +0000

Every time I have introduced a Mechanics course to a new group of sixth-formers, I have wrapped up the first lesson without really being satisfied that I have garnered the enthusiasm I had hoped for. Students may have drawn some pictures and placed forces where they think there are forces; but these have all been fairly basic, static, rather insipid scenarios and it has been all too easy for fuzzy thinking to go unchallenged or, worse, to be reaffirmed.

I think, however, that I might have found an answer. Next time, I'll use some of Andy Riley's remarkable Bunny Suicides. A number of the cartoons are absolutely perfect; by merely positioning black and white objects on a page, Mr Riley causes us subconsciously to unpick the physics of the next few moments.

Take the following example.

I think we can all imagine what's going to happen here. Look again, though. The image is richer than you think. Let's list the key A-level Mechanics syllabus ideas with the "cash register" sound effect: CHING.

The rockets are currently held in static equilibrium (CHING) by the reaction forces (CHING) acting on them as a result of their contact with the stakes - See Newton's Third Law (CHING). Once the fuses have lit the explosive material in the fireworks, they will be propelled by a thrust force (CHING) that, according to Newton's 2nd Law (CHING) causes them to accelerate (CHING) at a rate proportional to their mass (CHING) in the direction of the thrust. Assuming that the ropes are the same length, and possibly that they're light and inextensible (CHING) (CHING), there will become a time when they are no longer slack and contain some internal tension (CHING). At this point bunny's mission comes to fruition. Why doesn't the rope slip off the rockets? Well, that will be due to the friction (CHING) within the knot and the reaction force (CHING - we've had this one before but that was way back in the first sentence) of the inserted screws (clever bunny!) acting on the rope which, otherwise, according to Newton's First Law (CHING) would much prefer to stay on the ground.

It isn't exaggerating to suggest that this single cartoon can help introduce the majority of key notions in an A-level Mechanics course in one fell swoop.

Other concepts are covered elsewhere.

Connected particles...

...Projectiles...

...Moments.

This thorough textbook is not simply useful as an introduction, either. Later on in the course, we might use the Principle of the Conservation of Energy to discuss why Mr Riley may have taken some liberties with the Laws of Physics in this particular example:

Principia Mathematica it isn't; but it certainly beats drawing the forces acting on a book-underneath-another-book-resting-on-a-table as an ice-breaker!

A Galaxy Far Far Away.

noreply@blogger.com (owenelton) — Sun, 13 Jan 2013 18:21:00 +0000

A couple of days ago, I read that astronomers had discovered the largest spiral galaxy known to humankind. It is called NGC 6872 and lies a mind-boggling 212 million light-years away. The gargantuan object contains many infant stars, born as a result of a "recent" collision with another galaxy (IC 4970) about 130 million years ago. This figure doesn't seem correct, however; if it was, we would only receive information about the collision in 82 million years' time, so I presume the journalist means 212+130 million years ago - these intergalactic distances can be a right thorn in simultaneity's back!

A composite image of infra-red and ultra-violet emissions of the Whirlpool Galaxy (M51)

The galaxy pictured is a beautiful geometric object consisting of two main spiral arms, sweeping out a clockwise path from the central bulge. I thought it would be fun to attempt to recreate a spiral galaxy in Autograph. I'll assume that the galaxy is flat and the bulge in the middle is approximates a disc (rather than a bar like our own Milky Way). The arms are often found to approximate logarithmic spirals, which have general polar formula:

Here, a and b represent constants, r is the distance from the origin, and theta is the angle anti-clockwise from the positive x-axis. Setting a=b=0.1 and plotting the curve gives the following:

There are two problems with this. The first is that there is only one galactic arm. The other is that it sweeps out in an anti-clockwise direction. In order to address both of these, we need an equation in which we can off-set the relationship between r and theta. We might achieve this by converting the equation into parametric form.

A quick sketch of a right-angled triangle, with a hypotenuse running from (0,0) to an arbitrary point (x,y) will show you that:

Substituting for r from the earlier formula for a logarithmic spiral gives:

Now it's easy to make the arm journey clockwise, rather than anti-clockwise. We just change the sign of theta in the trigonometric functions:

We can even add a second spiral arm by inputting a second formula that perturbs the angle by pi:

All that remains is to colour the spiral more convincingly and work out how to make it rotate about its centre. The result of this is rather hypnotic as this Autograph Activity demonstrates.

2013: New Year Permutations

noreply@blogger.com (owenelton) — Mon, 31 Dec 2012 14:28:00 +0000

It's New Year's Eve here in the UK. In some parts of the world it's already 2013. The year 2012 was numerically interesting as it marked the end of the Mayan calendar. Some people predicted an apocalypse; matheminutes didn't. Permission to say, "I told you so".

Auckland, New Zealand sees in 2013

Can 2013 prove to be as rich a stimulant for mathematical discourse? On the face of it, not really. The number doesn't seem especially interesting. It isn't prime, being the product of 3, 11, and 61; we have four more years to wait for another prime year. It isn't square; we have twelve more years to wait for another of those. When we write it in binary, we get 11111011101 which is nearly palindromic but we'll have to wait another two years until it actually is.

This is all rather negative, though. Enough of what isn't true. What do we notice about 2013?

Well, it contains one each of the first four digits: 0, 1, 2, and 3. This is quite interesting. When was the last time this happened? Students of the famous Winchester College should be able to tell you, since it was a significant year for them. The last time a year contained each of these four digits was 693 years ago in 1320; the (possible) birth year of William of Wykeham who founded their school as well as New College, Oxford.

Clockwise from top left: William of Wykeham; Winchester College; New College, Oxford.

Before 1320, only eighteen years had passed since the previous occurence of these digits in 1302. In fact, we can systematically list all eighteen of the years that will contain each of these digits once:
1023, 1032, 1203, 1230, 1302, 1320, 2013, 2031, 2103, 2130, 2301, 2310, 3012, 3021, 3102, 3120, 3201, 3210.

I suppose that this alone isn't very interesting, but what if we were to calculate the differences between consecutive occurences? We get the sequence 9, 171, 27, 72, 18, 693, 18, 72, 27, 171, 9, 702, 9, 81, 18, 81, 9.

Do you notice any similarities between the numbers in this list? That's right. They are all multiples of 9. Is this just coincidence? Is this a special feature of the year 2013? We can find out with just a little thought.

First of all, it's useful to notice that we can get between any two years on the list by a sequence of swapping a pair of adjacent digits. For example, in order to get from 1320 to 2013, we can perform the following swaps:

Let's see what effect swapping adjacent digits has on the difference between the years. Suppose we have two numbers xy and yx. This is not "x times y" and "y times x" in the conventional sense, rather x and y are single-digits from the list 0,1,2,3 and the numbers might be pronounced "exty-wye and wyety-ex". Let's assume, without loss of generality, that x is bigger than y. Then:

So the difference is a multiple of 9.

We can apply this result to any two consecutive years in the above list; since there are only two digits changing at a time, then the difference between the years is just a multiple of nine multiplied by some power of ten.

Now, to find the difference between 1320 and 2013, we just need to perform the subtraction 2013 - 1320. We can cunningly rewrite this as so:

Each bracketed subtraction is between two years that differ only by two swapped adjacent digits. We know, therefore, that each bracket contains a multiple of nine. When you add up multiples of nine, then you get another multiple of nine.

So, no, it's not just some coincidence. The differences between the years containing 0,1,2, and 3 are all multiples of nine for a very good reason.

Is this a special feature of 2013? Well, no again, actually. When we used x and y above, there was no need to restrict them to just 0, 1, 2, and 3. They don't even need to have different values. This means that this result generalises to any two integers that contain the same digits but in a different order (or permutation as mathematicians usually say).

For example 746204 - 440672 = 305532, which is 33948 lots of nine. Try it for yourself. It will work every time. We've proved it!

The Seven Grimm Numbers

noreply@blogger.com (owenelton) — Thu, 20 Dec 2012 14:02:00 +0000

Today, Google is celebrating the 200th anniversary of the first publication of a volume of folk tales collected by the Brothers Grimm. They are doing so with a delightful scrolling doodle illustrating the tale of Little Red Riding Hood (or Rotkäppchen in the more efficient compound style of the German language).

Famous last words: "Why Grandma, what big teeth you have!"

I am not alone in having been absolutely entranced by these stories when I was younger; the magical world where you could be conned by a canine, complimented by a mirror, live in a gingerbread house, or (something which particularly encouraged me) be awesome even if you're only the size of a thumb was, literally, wonderful.

But what has it got to do with maths?

Well, quite a few of the titles of Grimms' Fairy Tales contain a number: The Twelve Brothers, The Three Little Men in the Wood, The Two Travellers, et cetera. In fact, thirty-one of them do; between them they contain thirty-three numbers. You might imagine that the distribution of these numbers is fairly even among low positive integers but this is far from the case as the following frequency chart shows.

Gosh, there's a whole lot of "three".

If we were to give a name to the numbers that do occur in the titles, we might call them Grimm Numbers.

The Grimm Numbers are 1,2,3,4,6,7, and 12.

What mathematical properties do these numbers have? Well, apart from 7, it is simply a list of the factors of 12. Seven seems quite an important addition though since this is how many Grimm Numbers there are. Why do the factors of twelve and the number seven make good fairy tale titles? It seems particularly curious that poor old number five doesn't get its own tale.

A search in the brilliant Online Encyclopaedia of Integer Sequences gives us some other interesting properties of the Grimm Numbers:

The Grimm Numbers are the first seven integers, n, such that 2ⁿ+3 is prime: Substituting the Grimm Numbers into this expression give the primes 5, 7, 11, 19. 67, 131, 4099.

They are also the first seven integers, n, such that 9ⁿ contains no zeroes: Nine to the power of each of the Grimm Numbers respectively is 9, 81, 729, 6561, 531441, 4782969, 282429536481. Not a zero in sight, while every other digit is represented.

A little more obscure is a fact involving modular arithmetic. The Grimm Numbers are the first seven integers, n, such that -3 is a square number modulo n. Modulo 12 is easy to understand as it is just like doing arithmetic using the hours on an analogue clock. -3 is the same as 9 (because 9 o'clock is 3 hours before 12 o'clock) and 9 is a square number. We also get a square number on clocks whose little hand rotates once every 1, 2, 3, 4, 6, and 7 hours, although I wouldn't recommend these for telling the time; particularly the 7-hour clock!

So, hooray for the Brothers Grimm. They brought us hours of childhood magic and now, a little curious mathematical icing on the cake.

Stars in the East

noreply@blogger.com (owenelton) — Mon, 17 Dec 2012 16:12:00 +0000

I can think of two famous individuals whose birth traditionally gave rise to unusual astronomical events. One is the late North Korean leader, Kim Jong-Il, whose nativity on Mount Baekdu, according to the Democratic People's Republic of Korea, was accompanied by shooting stars. More astronomically dubious, however, is one of the most enduring aspects of the traditional Christmas story.

"Three Kings from Persian lands afar

To Jordan follow the pointing star:

And this the quest of the traveller's three,

Where the new-born King of the Jews may be."

Every six year-old who has been in a nativity play will be able to tell you that a brilliant star shone over the stable where Jesus was born, leading three wise men from the East to Bethlehem bearing gifts of gold, frankincense, and myrrh.

Church of the Nativity, Bethlehem

These days, Bethlehem lies in the controversially occupied West Bank and, for the record, a star would be jolly useful for finding Manger Square. The Palestinian Territories are a rather hot place in which to try and geographically accustom yourself, and there's always the risk of ending up where you shouldn't:

No, I don't think the three wise men came this way.

A rock marks the supposed location of Jesus' birth and visitors who'd like to touch the relic face an unholy scramble to jostle their way into position.

The traditional birthplace of Jesus.

The star that helped orientate the men from the Orient, presumably remained over Bethlehem for at least the twelve days between Chirstmas and Epiphany. If the carol "Little Donkey" is to be considered trustworthy, the star also led Joseph and the heavily pregnant Mary to the stable; an extraordinary feat of cosmic endurance given that, over a few hours, stars do this:

Source: nathanwillsphotography.org

Or rather, the rotation of the earth about its axis, makes the stars appear to travel in circular paths. Except, that is, for the stars directly above the north and south celestial poles. Granted, a bright new star at a pole would have remained in the same point in the sky for a long time but it would also have led the three wise men North, contradicting this carol:

"O, Star of wonder, star of night,

Star with royal beauty bright,

Westward leading, still proceeding

Guide us to thy Perfect Light"

It is quite possible, however, for a body to orbit the earth and remain above a fixed point on the surface. This type of orbit is called Geostationary. If we consider Newton's model of gravitation we can see how such an orbit can be achieved.

Any object orbiting the earth purely under the influence of gravity, must do so in a near-circular plane with the earth at its centre. We model the earth as a point mass of nearly 6 septillion kg. At first, this seems rather a rash assumption, but remember, Newton's Laws of gravity were perfectly sufficient for the purposes of landing men on the moon. Let's imagine, something of mass mkg orbiting the earth. It feels a force proportional to the mass of the two bodies and the inverse square of the distance between them. Using Newton's 2nd Law of motion in the direction directly from the orbiting body to the Earth, we can write down

where M is the mass of the Earth, G is Newton's gravitational constant, r is the distance between the Earth and the orbiting body, and a is the acceleration of the orbiting body.

Now, assuming that this orbit is nearly a circle of constant radius, and that the body is travelling at a constant speed, we may use the fact that

where the lower-case omega repesents the angular velocity of the orbiting body.

Substituting this into the initial equation, and solving for r gives:

Now, G and M pre-determined. We also wish the satellite to be fixed above a point on the Earth's surface so its angular velocity must equate to one revolution every day, that means a value for omega of about 0.00007 radians per second. The above equation, therefore, tells us precisely how far away the centre of the orbiting body must be: 35786km above the stable. If this is a star, it would be very bright indeed; this length is about a twentieth the radius of our Sun. The birth of Jesus would have been as toasty as the apocalypse that we're all looking foward to later this month.

There's another problem with the "star above the stable" scenario. Bodies in Geostationary orbit move in circles whose centre coincides with the centre of the Earth. In order to be fixed above a point on the Earth's surface, this terrestrial point must also rotate in such a circle. There are infinitely many points that fit this description, but they all lie on the equator. Bethlehem has a latitude of 31.7 degrees North. It isn't even close!

Bethlehem: Distinctly not on the equator.

To be fair to the chap who wrote Matthew's Gospel (the one from which all these scientifically suspect carol lyrics have been inferred), this star is not alleged to have been stationary over the stable for nearly the length of time that "We Three Kings" would have us believe. Crucially, the wise men say, "we have seen his star in the East", which, having travelled West, means that they were doing quite the opposite of following it. Only after they had visited Herod, the star obligingly "came to rest over the place where the child was".

A distant 21st Century Bethlehem as seen from Herodion

Anyway, it really doesn't matter that I find Kim Jong-Il's natal shooting star story far more feasible than the possibility of a star in Geostationary orbit over Bethlehem. The one atop my Christmas tree still looks wonderfully festive.

Mathematical Things to Do on a Rainy Day #010

noreply@blogger.com (owenelton) — Wed, 12 Dec 2012 10:08:00 +0000

Create a fractal fractal

Here's something that occured to me the other day. You could use an online photomosaic maker such as Mosaically to create a fractal fractal.

Iteration 1:

Choose a picture of a fractal. I went for a Sierpinski triangle that I created in MSpaint for this post.

Next load up some suitably mathsy pictures. I decided to let the algorithm devour the illustrations that have so far appeared on this blog.

If you follow the instructions on the website, and wait a few hours for your picture to cook, you might end up with something like this:

As you can see, the image is made up entirely of tiny copies of the photos that were uploaded (and some colour-related cheating, but that's not the point). Zooming in a little reveals some of the detail:

Pleasingly, you can just about make up some Sierpinski Triangle detail used as tiles.

Iteration 2 (I haven't actually done this because I don't have the patience):

Create a large quantity of these fractal mosaics. Use them as tiles to make a fractal mosaic fractal mosaic.

Iteration 3:

Similarly, create a fractal mosaic fractal mosaic fractal mosaic.

... et cetera ad infinitum.

Dodec(k) The Halls

noreply@blogger.com (owenelton) — Sun, 09 Dec 2012 14:34:00 +0000

Suppose that you wanted to decorate your Christmas tree using mathematically motivated means. Suppose, also, that you had at your disposal a number of sporting students eager to get cutting and sticking in the last maths lesson of term. Well, the result might look like this:

There are four types of decoration photographed here. Each is pretty easy to make. You'll simply need the instructions linked to this post and some stuff along these lines:

Decoration 1: Twelve Days of Christmas Rhombic Dodecahedron

This requires a little bit of cutting and gluing to begin with. After this, however, it all neatly folds up and satisfyingly slots together with no further sticking required. The template is one I modified from the extraordinarily excellent website CutOutFoldUp. You can download a pdf of the Christmassified version by clinking here.

Students making the rhombic dodecahedron.

Decoration 2: Fractal Snowflake

OK, admittedly, this isn't actually fractal. It is, however, the fourth iteration of the Koch Snowflake so it's a few steps nearer to a polygon of infinite perimeter than an equilateral triangle. It's a little bit fiddly to cut out but you can save some time by taking advantage of the rotational symmetry. A template is available in pdf form by clicking here.

Students making the fractal snowflake.

Decoration 3: Möbius Strip Paper Chain

This is the easiest of the lot. All you'll need to do is to cut an A4 sheet of paper into 3cm strips parallel to its shorter side. Each strip is then turned into a link in the chain by giving it a twist and securing the ends together with sellotape. Don't spend too much time wondering whether to stick the sellotape on the inside or the outside though!

A single twist is required to make each link in the chain.

Decoration 4: Intersecting Tetrahedra Star

This is another design from CutOutFoldUp. Click here for instructions. The model creates the illusion of containing five intersecting regular tetrahedra and is deceptively simple to make by taping together 20 identical modules. In actual fact, the star pictured here only contains 19, since one was omitted so that the top of the tree could be inserted.

Students creating some tree-topping tetrahedra.

The only things that remain to be done are decorating the tree and having a very merry mathematical Christmas. Mince Pis all round!

Penrominoes

noreply@blogger.com (owenelton) — Sat, 17 Nov 2012 21:26:00 +0000

Every so often someone in some field comes up with an elegantly simple idea that changes the way everyone else looks at something. So it was with Roger Penrose's invention of the Penrose Tiling; a beautiful way to fill an infinite plane with two shapes such that the pattern never repeats itself. As I explored in this previous post, the implications of his idea reached as far as last year's Nobel Prize for Chemistry.

A Penrose Tiling - based locally on rotational symmetry of order 5.

I thought it would be an interesting idea to come up with a game based on this tesselation so I produced some Penrominoes and some rules:

A pile of Penrominoes

There are 54 tiles in total; 27 darts and 27 kites, each containing 3 coloured areas in one of 3 possible colours.

The aim of the game is to take turns placing Penrominoes adjacent to those already there, scoring points for the number of edges connected. Lengths and colours of edges must match.

A game of Penrominoes.

Full instructions and a set of Penrominoes to print out are available by clicking this link.

Have fun playing! I'd be really interested to know if you can think of any improvements that might be made; the game is fairly entertaining but far from perfect.

The Teacher's Dilemma

noreply@blogger.com (owenelton) — Fri, 02 Nov 2012 11:07:00 +0000

In 2011 we all knew where we stood with GCSE results. The BBC website reported the 23rd annual increase of A* to C grades. I sat swearing at the news as yet another political type told us that this showed that students were getting brighter and teachers were getting better. It doesn't, of course, any more than the existence of Hurricane Sandy proves that climate change is accelerating. Come on politicians, we're not idiots!

BBC graphic from 2011 (bizarrely missing data from that year): "Proof" that boys have got twice as clever in twenty years and girls haven't improved as dramatically but have increased the gender gap or something.

This year, however, something different happened. 2012 showed the "first fall in GCSE grades in exam's history". Funnily enough, no-one dared to come on the TV and tell us that this was because students had got less bright and teachers were getting worse. School's Minister Nick Gibb made almost precisely the same statement about "hard work" as he had the year before. Quite soon, however, it appeared that some exams had been affected more than others. The crucial C grade in English had had its bar raised by a significant margin between January and June due to pressure from the govenment's qualifications watchdog, Ofqual. This made it unfair if you had sat your exam in , say, June.

Michael Gove, the Education Secretary, distanced himself saying that the grade boundary decisions were "made by the exam boards entirely free from political pressure". Now you'll excuse me, Mr Gove, if I don't believe you here. This hiding behind a fog of bureaucracy has been a favourite trick of politicians as old as politics itself. Adolf Hitler, for example, famously never put pen to paper to sign any document pertaining to the Final Solution. I suspect that he may have applied some pressure "off the record" to grease the wheels, though.

This morning, Ofqual (a week after a group of Head Teachers launched legal action against them) have issued a report about what went wrong. This news channels delighted in stoking up the existing animosity by pointing out that the report cites teachers' marking of coursework as the root of all the problems. Apparently everyone marked too generously.

Glenys Stacey on the news saying that it isn't the teacher's fault that their marking was to blame.

As a maths teacher I find it utterly hilarious, although pretty worrying, that this should have come as a surprise. In theory, exam boards' moderation should neutralise this phenomenon but, as Ofqual's report admitted, this "could have been run more tightly". There is a very good mathematical reason why a lack of effective moderation will cause all hell to break loose; this is because the marking of coursework is a version of the most famous scenario in Game Theory, the Prisoner's Dilemma.

Let me explain by simplifying things a little. Suppose that just two schools (A and B) are competing for league table position and each school contains one teacher responsible for the coursework results. Each set of results can be marked either strictly or generously. Here are the four possible outcomes.

Let's assign some scores to each of these. If A and B are both strict then everyone gets the marks that they deserve and everyone's reasonably happy. Let's say that A and B both get 5 points.

If A is generous and B is strict then, in the absence of effective moderation, school A goes shooting up the league table, beating its rivals. Let's say that A gets 10 points and B gets -10. Similarly, the reverse is possible with B's generous marking winning it 10 points to strict A's -10.

Finally, we have the situation where both A and B mark generously. As we have learnt, this results in complete carnage; incompetent looking exam boards and fuming head teachers. Not many people are happy so let's say that A and B both get -5 points in this instance.

Now, let's think about the possible outcomes from the perspective of school A:

If A's marks are strict then their mean outcome is -2.5 whereas a mean of +2.5 awaits generous marking. They should mark generously.

The marking of coursework nationwide is effectively a multiplayer version of this game. In an ideal world, everyone would stick strictly to the mark scheme and the system would police itself. Given, though, that we tend to think about our own student's welfare before the entire nation's education, it is only rational that we teachers will act in the interest of our students and err on the generous side. Therefore, it is not a surprise to me that a huge quantity of teachers do so. It is a surprise, however, that it caught the exam boards napping; this is basic economics; you don't need any more than a critical mind and an ounce of common sense to consider it a distinct possibility.

Monobrow Man Marks Movember

noreply@blogger.com (owenelton) — Thu, 01 Nov 2012 09:16:00 +0000

Honestly, I'd love to spend this month growing a moustache for charity. It would look fabulous!

artist's impression

My priority, however, has to be to the mathematical wellbeing of my students; I'm concerned that, instead of raising their hands to ask sensible questions, they would be preoccupied with queries such as, "Sir, shouldn't you concentrate on growing some hair on *top* of your head?" et cetera.

No matter though, I have a friend that I introduced via an earlier post who has decided to grow some facial foliage instead. Yes, Monobrow Man is back:

Monobrow Man: More charitable than me!

He has taken upon himself to mark Movember. Being made of equations, it is only natural that his moustache contains a number of variables that enable its design to be altered. Here are just a few examples:

I think he looks very dashing indeed. You can see what designs you can come up with by trying out this Autograph Activity. What sort of equations do you think I've used to create the moustache? Can you come up with your own equations to create a wider range of upper-lip decoration?

If you enjoyed playing with Monobrow Man's topiary then why not head over to the Movember website and make a donation?

Multiplication Magic

noreply@blogger.com (owenelton) — Wed, 31 Oct 2012 16:45:00 +0000

I'm excited. A pocket calculator that I recently bought on ebay arrived today:

Granted, it doesn't look much like a calculator but, I promise you, this particular model was widely used in England during the middle of the 20th Century. It was invented by a grocer called Otis King and was manufactured until 1972.

You can get a better idea of how it works if I extend it to reveal three spirals, each 66 inches (168 cm) in length.

This device can be used to carry out nasty mulitplication and division very quickly and pretty accurately. To anyone of my parents' generation this will be old hat; it's an easy-to-use, rather theatrical version of a slide rule. To me, however, it's a bit like magic.

Here's how to calculate 206 x 182 with just a flick of this wand.

On the bottom cylinder, we move the white mark until it shows 206:

Next we normalize the top cylinder by adjusting it until it reaches a special point on the scale that reads ONE:

Now we move the black casing until this top notch points to 182:

Finally, we take a look at where the bottom notch has ended up:

This is just under 375. Now, with a little bit of thought, we might expect 206 x 182 to be shy of 40000 (since this is 200 squared). That means, I reckon that 206 x 182 = 37500 (to 3 significant figures).

The actual answer is 37492. Pretty neat, huh? It's almost as quick as typing it into an electronic calculator!

How on earth does it work, though? The secret (as with all slide rules) lies in the fact that the scales are logarithmic. Instead of labelling 1,2,3,...,10 with an equal distance between each, the numbers are marked according to where their logarithm lies. The following diagram shows a comparison with a linear scale. On the logarithmic scale, 3 is placed where 4.77 would lie on the linear scale since log (3) is about 0.477. Why isn't there a zero on the logarithmic scale?

The Otis King calculator works by comparing two logarithm scales. Let's demonstrate how this works by looking at 2x4. The first step is to carry out the normalization process and move the top scale so that it starts at the number on the bottom scale (2 in this case) that we are multiplying. The start point is marked here with a red vertical line.

Now we move this red line until it reads 4 on the upper scale. The answer can now be read straight off the bottom scale.

What we have physically done is added log(4) to log(2). This provides the correct answer since log(2) + log(4) = log(8). In fact, the system relies on the fact that for any positive a and b, log(a) + log(b)= log(ab).

This is an identity that students these days don't encounter until the sixth form. Previous generations used it when they were plenty younger but, I suppose, that's progress for you!

When NOT to Cancel Fractions

noreply@blogger.com (owenelton) — Sun, 28 Oct 2012 13:28:00 +0000

I have lost count of the number of times that I have seen a fractional answer such as 6/4 given at the end of a student's solution, reached for my red pen, ringed the offending ink, and written, "=3/2 simplify!!!" or some other expression of frustration. Furthemore, I'm sure that I'm often guilty of writing "Always cancel your fractions." at the top of the page.

If you have been on the receiving end of this apparently neurotic obsession with 'simplification' then this post might provide you with some ammunition for a response. Interested? I thought you might be.

Let's begin by thinking about how to draw a regular polygon; a heptagon for example. We mark seven points at equal intervals about a circle. Starting off with one of the points, we join it to an adjacent point using a straight line. Then we do the same to the point we've reached, and so on until we return to the beginning. Voila: a regular heptagon.

What would happen, however, if we tried something a little different; beginning with the seven equally spaced points but, instead of connecting adjacent points, missing one out?

We could then do the same again, from the point we've reached:

Continuing until we've reached the beginning, we produce the following shape:

It's still seven-sided. It's equilateral and equiangular since all sides are the same length and the interior angles at the vertices are identical. This means that it is regular, so should we call it a regular heptagon?

It does differ from a regular heptagon in two important ways: It is self-intersecting and non-convex. In order to distinguish it from its convex brother, it is known instead as a heptagram.

What would happen if, instead of joining every second point, we joined every third point?

The result is another, pointier, heptagram.

A problem arises here; if I'm talking to someone about a heptagram, how will they know to which I'm referring? I could say "the more pointy" or "the less pointy" one, but things will get particularly cumbersome if I want to talk about star polygons with a greater number of points.

Thankfully Ludwig Schläfli, a 19th century Swiss with a talent for mathematics and languages, devised a system for describing regular shapes in many dimensions. It just so happens that this notation can be usefully applied to the labelling of star polygons. The Schläfli Symbol for describing a regular polygon consists simply of a pair of curly brackets containing a rational number. The Schläfli Symbols for the three shapes we've seen so far are as follows:

The numerator describes the number of points that the polygon has, and the denominator tells you which which points are joined; in the pointiest heptagram, every 3rd point is joined whence the denominator 3.

You might think that we could write the symbol for a regular heptagon as {7} instead of {7/1} and you'd be right, that's fine.

Could we "cancel" {9/3} to {3} though? Absolutely not! The polygon {3} is an equilateral triangle but {9/3} is a nine-sided shape with every 3rd vertex joined. It is an enneagram and looks like this:

You might spot, however, that this shape is not really a polygon at all, but the superimposition of three equilateral triangles.

We can, therefore, say that 3{3} is equivalent to {9/3}, but that "cancelling" and claiming that {9/3}={3} is erroneous in this context. There is, of course, a relationship between the two. We can say that {3n/n}=n{3} since any bracketed fraction that would cancel to 3 (were cancelling permitted) represents a number of equilateral triangles piled on top of eachother. Can you prove this?

So, if your teacher ever tells you that you must always cancel your fractions then, if you're feeling brave, offer the following response: "I believe that your mantra, while well meant, is fundamentally fallible since you cannot possibly argue that the enneagram with Schläfli Symbol {9/3} is identical to an equilateral triangle."

Just don't tell them who put you up to it. Deal?

Campaign Against Numerical Excessification

noreply@blogger.com (owenelton) — Sun, 14 Oct 2012 14:31:00 +0000

Last night, as I leaned back to enjoy one of the X-factor's advert breaks (usually a pleasant hiatus during the barage of vocal mediocrity and flashing lights) one particular commercial caught my eye for all the wrong reasons. I narrowly avoided upending a takeaway chicken methi all over my colleague's carpet as I baulked at what I was seeing from L'Oreal.

The product in question was this: The new Volume Million Lashes Excess mascara.

L'Oreal Paris: Guilty of Number Abuse

Now, I'm not claiming to be an expert in eye makeup; I very rarely wear any. I'm sure that this is a perfectly good product and, no doubt, its well-resourced marketing campaign will make it seem sufficiently different from other near-identical mascaras to make it a commercial success. I do, however, object to its name. It smacks of flagrant number abuse.

The applicator within is called the millionizer brush. "Millionizer" isn't a real word but, if it did exist, we could extrapolate from words such as "vaporize" and "prioritize" to imagine that it would mean "to make the quantity of something into one million". Mascara doesn't actually add more eyelashes, though; it just makes them appear thicker; so we can dismiss this bizarre claim and assume that the "Million" figure refers to the pre-mascarred quantity of lashes.

One of these is a made up word.

A human face might actually have about 600 eyelashes in total. I didn't know this, I took the data provided on an Xtreme Lashes Studio webpage and used the midrange. This, incidentally, is the website from which I've learnt everything I know about eyelash extensions.

600 is 999400 short of 1000000, yet this product seems to suggest that the two figures are in the same ball park. Perhaps you think it doesn't matter. Perhaps they're just both big numbers and a company should be able to multiply by a few inconsequential powers of ten to reach a nicer sounding word. As far as I'm concerned, however, this casual excessification (I can invent words too) has damaging consequences. It betrays an accepted worldwide lack of precision regarding large numbers. It's as though once we've counted to a certain point, all the remaining integers fall into the same pot of "big numbers with funny names" that we are free to pick and choose from regardless of their correspondence with reality.

This is not a new phenomenon, however. Some animal names are equally misleading. Check out this chap:

It's a centipede.

Why is it called a centipede?

Because it has one hundred legs.

Does it really?

Count them. It hasn't. It doesn't come close.

This defenceless critter has been the victim of numerical excessification. According to the relevant Wikipedia entry, so-called centipedes can have many more than a hundred legs but they all have an odd number of pairs. This means, of course, that they can never have exactly one hundred (unless some have been mislaid) since it is a multiple of four.

Our treatment of millipedes is equally misleading. No millipede currently known to science has more than 750 legs. I agree that 750 is an impressive quantity of limbs but there's no need for the excessification. Confusingly, some millipede species have significantly fewer legs than some species of centipede. I'd much prefer it if we just grouped all these creatures together and called them polypedes, although I suppose any classicists reading this would cry foul at the juxtaposition of Greek and Latin roots in a single term.

From Biology to Cosmetics, numerical exessification is becoming an increasing part of our lives. If the trend continues, then before long you will be faced with trillions of cases every day and will simply be immune to it. Therefore, I announce a move to stamp it out; to dredge this mire of cardinal* sin. I call on teachers everywhere to be aware of the CANE; the Campaign Against Numerical Excessification.

Next time a student claims he couldn't do his homework because he had "a million other pieces of work to do", invoke the CANE.

When a student is late for your lesson because he "had to come miles from Chemistry", remind him of the CANE.

When a student's excuse for walking across the grass is that he's "just seen thousands of others do it". Turn to the CANE.

They will thank you in the long run.

*Mathematical pun intended. Billions of points if you spotted it.

91st Carnival of Mathematics

noreply@blogger.com (owenelton) — Thu, 11 Oct 2012 08:44:00 +0000

Welcome to the 91st Carnival of Mathematics.

It's an absolute pleasure to be hosting it here at matheminutes. There have been some cracking submissions this month but first, in order to uphold the venerable tradition of CoM, here are some aspects of the number 91 itself.

91, at first glance, doesn't seem particularly noteworthy. It isn't quite prime, it's in No Man's Land approximately half way between two squares, and shares a similar relationship with the cubes. This casual veneer is, however, grossly misleading; if you have 91 things then you can manoeuvre and stack them in all manner of interesting patterns:

We are spoilt for choice! The purple picture shows that 91 must be the smallest integer that can be written as the sum of six distinct squares. The orange picture shows that it can also be written as the sum of cubes. In fact it's the smallest number that can be written in two different ways as the sum of cubes (but only if you allow negative numbers).

Furthermore, 91 is the smallest Fermat Pseudoprime of base 3. This means that 91 is the smallest integer to pass a particular primality test based on Fermat's Little Theorem. To demonstrate that it passes, we have to find 3^91-1and show that it is exactly one more than a multiple of 91.

Now, 3⁹⁰is 8727963568087712425891397479476727340041449

and 95911687561403433251553818455788212527928 ninety-ones is 8727963568087712425891397479476727340041448. Job done!

Why is this a test for primality? Well, all prime numbers, p, satisfy this test for all integer bases smaller than p. This isn't a fail-safe method for showing that a number is prime, however. There are some numbers, known as Carmichael Numbers, that aren't prime and satisfy all these Fermat primality tests - but more about those in the 561st Carnival of Mathematics, hitting your iPad 42 in December 2051.

Continuing a prime theme, Peter Rowlett brought the fantastic new series of podcasts known as Relatively Prime to my attention with this post. If you have any interest in mathematics (and I imagine you probably do if you have read this far) then these are essential listening.

Relatively Prime's four superlative podcasts. There are more on the way.

Peter has also written two related posts about the state of maths in the higher education sector. The first discusses the affiliation between lecturing and research, and the second provides an interesting insight into the attitudes of graduates towards the mathematics that they often find themselves involved in professionally.

Let's move on to some maths that's pleasing on the eye. Evelyn Lamb has written an absolutely adorable post on Fractal Kitties, in which she delightfully explains a general result relating Julia sets to two dimensional forms, feline and otherwise.

Complex Cat: A Julia set of a 301 degree purrrrlynomial.

Felix Mendelssohn may have had his Songs Without Words but Mr Honner has provided the mathematical equivalent with a neat graphical proof inspired by a "recent faulty New York State math exam question [that] suggested that knowing that two lines were perpendicular could be enough to conclude that containing planes were also perpendicular". His proof shows this to be fallacious.

How about this for another picture: Thony Christie has submitted this wonderfully rich title page belonging to Cavalieri's book, Trigonometria, of 1643.

There are graphs galore in this entertaining post from Kate Nowak as she investigates how "math was wrong" in its prediction for the date of registration for facebook's billionth member. In other graphing news, the inaugural Autograph Newsletter was released recently, offering free dynamic mathematics resources for use in the classroom. If you're a maths teacher (or even if you're not) then sign up here to get future fortnightly editions delivered directly to your inbox.

Dan Meyer has written this post that demonstrates the incredible educational potential of the interactive graphics of an iPad. Surely his ideas are not in the realms of fantasy; please send this post to every programmer you know so that someone might code the piece of software, offering autonomy for teachers, that he is after; cohort upon cohort of students would be incredibly grateful.

Teachers having ability to create bespoke interactive activities for their classes; wouldn't that be great!

Tony Mann reflects on the importance of the passing of time when learning new mathematics in this thought provoking post. Why does maths seem simpler when we return to it after a while? Does this and should this influence how we learn it and teach it? In my experience, logarithms are one of those things that students certainly have to come back to in order to grasp properly. Colin Beveridge, the Mathematical Ninja, has provided a cunning party trick to exploit this.

Finally for this month's carnival, here's something completely different: In this post, Egan Chernoff highlights a destination in the virtual world, Second Life. Wheeler Island is a virtual manifestation of the David Wheeler Institute for Research in Mathematics Education.

Wheeler Island in Second Life

That's it for the 91st Carnival of Mathematics. Thanks very much for reading. The 92nd will be hosted by Frederick at White Group Mathematics.

Mathematical Things to Do on a Rainy Day #009

noreply@blogger.com (owenelton) — Tue, 09 Oct 2012 19:02:00 +0000

Create a Spoof Children's Book

Step 1: Choose a Topic

Step 2: Compose Book - Start off Simple

Step 3: Work your way towards some ludicrously complicated examples.

Step 4: Find an internet-based company to print your book and post it to you.

Step 5: Enjoy your work

Step 6: Put your book up for sale in a vain attempt to recuperate some of the cost of buying your own copy.

You can buy Shapes here.

Monkey Maths

noreply@blogger.com (owenelton) — Sat, 29 Sep 2012 14:47:00 +0000

"Roll up! Roll up! Come and see the world's most mathematical monkey!"

The box in which this toy is packaged makes the following proud claim:

This monkey can calculate.

Set feet to point at two numbers, fingers will locate their product.

I entirely agree with the second sentence. Wherever you move the monkey's feet, his fingers undeniably display the result of the integers' multiplication. If a human and the monkey gave me different answers, I would trust the monkey.

I don't, however, agree with the first sentence. The monkey cannot calculate. Instead, it simply uses a clever method to look up the answer. All 78 possible products of the first twelve integers are given their own location in the triangular product space behind by the monkey's limbs. As the left leg is moved to the smaller of the two numbers, the monkey's fingers move from left to right. As the right leg is moved closer to or further from the left leg, his fingers move down and up respectively.

This is not the same process as is undertaken when a pocket calculator does multiplication. Can you imagine the memory that would be required to list every result of every calculation that it is possible to enter, and the time it would take to look this up?

The monkey doesn't calculate.

He could, however, diversify. Suppose that I replaced the numberline and the triangular product space with the following diagram:

According to an online colour mixing application, the row of colours along the bottom are (from left to right) the primary colours, the secondary colours (formed from two units of primary colour), and the seven different combinations of primary and secondary colours (each three units in volume). Placing the monkey on top would allow you to see a mixture of any pair of colours in this bottom row. His fingers would point to the resulting hue.

Is the monkey "doing art"?

No. Of course he isn't. Just as he isn't "doing maths" in his original function. In fact, the monkey is merely regurgitating the results of my online colour mixing; I'd be rather irked if he began taking the credit.

If recalling the answers to times-tables isn't "doing maths", then why on earth are children expected to recite them as part of the mathematics national curriculum? Well, because it's important that they begin to understand how numbers relate to one another and start to see some of the structures underlying the integers. In school, it is most effective to teach by example, which is why we don't tend to pile in with groups, rings, and integral domains in primary school.

Arithmetic comes first. Grasping this will give students a chance at mastering school algebra; a useful tool but, more importantly, yet another small example of structures that exist in a more general mathematical universe.

The poor monkey will never have a hope of seeing beyond his feeble square-dozen world. He breaks down outside this comfort zone; look at what happens when he attempts to multiply 5.5 and 9.5: He's 3.75 out!