My Mathematical Jewellery Collection

Today I'm going to a barbecue hosted by one of my colleagues. The theme is maths. We will be doing maths puzzles and games, and there is a prize for the most mathematical outfit. On reading this I exclaimed "What shall I wear?!" Not, as you might think, because I don't own any mathematical clothing. In fact it was because I own so much, that choosing was going to be difficult.

Two of my absolute favourite things in the world are maths and fashion. When you combine the two, you get mathematical fashion. And I LOVE it. I have a few mathsy clothing items, but I'm not going to show you those in this post. This post is dedicated to perhaps my favourite aspect of fashion: jewellery.

At my academy I'm known by both staff and students for having a massive collection of bold and bright rings and necklaces. A subset of this collection I will be showing to you today.

First up, my absolutely amazing, custom-made SOHCAHTOA necklace;

I bought this necklace from one of my favourite jewellery shops, Tatty Devine. As well as stocking quirky and unique perspex jewellery, they make name necklaces (like Carrie's from SATC). Seeing this feature gave me the genius idea of turning my favourite maths word into a necklace. I thought long and hard about what word (containing nine letters or fewer) to choose. I settled on SOHCAHTOA because it's so recognisable, not to mention useful. I've had several strangers see it and proudly exclain "I remember that! It means..." which is so nice to hear!

I think my favourite thing about this necklace is my students' reactions when I tell them I had it custom-made. I think to them it just confirms my status as geekiest teacher ever.

Onto my next necklace: The Infinity necklace:

I bought this necklace on Amazon after a lot of searching. I had decided I wanted a necklace in the shape of the infinity symbol, and then set about trying to find one, which is the complete opposite of how I normally shop, where I see something I never knew I wanted and then buy it immediately. This necklace is sterling silver and contains an actual real diamond, which sadly my poor eyesight inhibits me from seeing. Amazon assures me it is there, though.

Necklace number three: the Rubik's Cube:

Everyone loves my Rubik's cube necklace. I have been nearly strangled countless times as students grab it as  I walk away. Nobody can resist turning it a bit, which is why it is a jumbled mess. I am actually rubbish at solving them, although I keep telling myself I will learn the algorithms soon. My friend Emma appeared to learn overnight when she got one for her birthday, but knowing her she probably spent hours and hours learning it obsessively.

Oh, I almost forgot to tell you where this amazing necklace is from! I got it from Folksy, a brilliant website where random people sell all kinds of handmade and vintage stuff. I have bought so many things from there. The necklace was from a seller called Mary Quite Contrary.


And now, a bracelet:

I bought this bracelet quite recently from Folksy, from a seller called I heart my art. It was the only one of its kind, which makes me feel special. Check out her other items though. I like how the bracelet contains "e" because most of my students don't know about the exponential constant, so they'll ask questions about it. Any excuse to talk about e and logs! Although I bet most of them just assume it stands for Emma.

Next, my Origami necklace:

Boy was this hard to photograph! This little stunner really is origami: it was made from a single square of silver and folded into the iconic crane. I saw a very similar necklace on Anthropologie, one of the most beautiful shops in the world, but it cost way too much so I went looking elsewhere. I found this on Etsy, which is basically the American version of Folksy. I bought it from the hugely talented AllegroArts who handmakes all of these stunning origami pieces. This exact necklace can be found here.

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So that's my mathematical jewellery collection as it stands at the time of writing. I fully expect it to be twice the size this time next year!

I will leave you with one last photo: a little cutie I just had to buy from Ryman's as I was picking up my back-to-school stationery:

I just can't resist cute geekiness.

Emma x x x

Playing to Lose

(I was contemplating calling this post "Every Loser Wins", but I hate that song).

As all of you should know by now, I'm a winner. I win things. So when my friend Stacy challenged me to a game of Noughts and Crosses today, I was prepared to win. Until she said this:

"The winner is the person who loses. The aim is to NOT get three in a row".

As you can imagine, I was flummoxed. Winning is easy, losing is difficult. She told me to go first, which, now that I think about it, was a sneaky way of increasing my chance of winning - and hence losing. I obviously avoided the middle square. I figured the corner squares were also too good to use. So I opted for a side square. That decision was pretty easy. The rest was not so trivial. Stacy lost the game, and hence won. Apparently I am such a winner that even when I'm trying to lose, I win.

Please please please grab your partner/your child/your flatmate/the guy next to you on the bus and challenge them to lose a game of Noughts and Crosses. It's the only way you can really think about the strategy involved.

Have fun!

Emma x x x

Why We Actually Won The Olympics

Before this year, I'd never watched so much as one minute of an Olympic event. Not even China's opening ceremony last time (I have a phobia of fireworks). I am an incredibly competitive person, and the idea of watching a massive competition into which I'm not even entered, really holds no interest to me. Who cares what happens? The bottom line is, I'm not coming home with a shiny medal around my neck.

But then I went to spend a week in the house of a family who were interested in the Olympics, and (this is important) actually had television (we stopped buying our TV licence several years ago as our favourite shows only air in Japan anyway). I didn't have much else to do that week, no longer being a fifteen minute walk from a city centre (and hence, shops) so I found myself watching the Olympics A LOT. And I found myself enjoying it. I finally got it: sure, I wasn't going to win anything, but my country would, and in a way, me and my country are one and the same. I suddenly understood what Nick Hornby was on about in Fever Pitch (OK, I never actually read it, because, I enjoyed About A Boy and all, but seriously, a book entirely about football?) and I felt that I was Team GB. It was a great feeling.

And then we didn't win.

Those of you who know me well know that I don't particularly enjoy not winning. Please note this only applies to competitions I deem worthy: me always failing to win at Laser Quest, Mario Kart, and anything that involves Geography does not bother me in the slightest. Because they're stupid. But the Olympics was something I wanted to win, something I thought I was really good at (by me, I mean Team GB of course). I wasn't going to let stupid countries like China and the USA beat me. It's not fair: they already have crazy cheap designer clothes and fifty flavours of Pop Tarts, why do they have to have this too?!?

Naturally I did what I always do when I am told I have not won. I set about moving the goalposts, scouring the rule book, and  examining the data to try and find a way that means I actually won. But whichever way you count the medals, we were not higher than third place.

Rank by Gold	Country	Gold	Silver	Bronze	Total
1	United States of America	46	29	29	104
2	People's Republic of China	38	27	23	88
3	Great Britain	29	17	19	65

There was only one thing I could do: "To the math cave!!"

I'm sure that anyone who is even slightly mathematically minded automatically saw a problem with the scoring system. I don't mean the fact that it's all done on the number of Golds and not on a 5-3-1 system, because we still wouldn't win then. I'm sure that when most people looked at the scoreboard, their first thought was "Well of course those two countries are winning, they're absolutely massive". China and the USA are well known as having huge populations, along with India, Brazil and Indonesia (interesting point: this is probably the one geographical fact that I actually know without having to look it up. Maths teachers from my Academy will know why). Is it really fair that we are being compared to countries with 5 or even 21 times more people than us? (I had to look that up). If we randomly divided the USA into five pieces, and selected the best athletes from each piece, would they still be the best? And if we cut China into 21 pieces? I don't think so.

So, here comes the maths: I have created a table of the competing countries, and their number of golds per 10 million people. This is like doing population density (urgh, there is way too much geography in this post) but instead it's gold density. Except out of the number of people, not the area of land. Hmm. OK.

 

Country	Golds	Population (10 millions to 3sf)	G/10mill (2 dp)
USA	46	31.5	1.46
China	38	135	0.28
Great Britain	29	6.23	4.65

As you can see, we're actually 3 times better than the USA, and loads better than China. So, we actually should have w... Wait a minute. What?!?! Looking further down the list I can see some pretty disturbing numbers. I'd better give you the whole table. This is it ordered by golds:

 

Country	Golds	Population (10 millions to 3sf)	G/10mill (2 dp)
USA	46	31.5	1.46
China	38	135	0.28
Great Britain	29	6.23	4.65
Russia	24	14.3	1.68
North Korea	13	2.46	5.28
France	11	6.54	1.68
Germany	11	8.19	1.34
Hungary	8	0.996	8.03
Italy	8	6.08	1.32
Kazakhstan	7	1.68	4.17
Australia	7	2.27	3.08
Japan	7	12.8	0.55
New Zealand	6	0.443	13.54
Netherlands	6	1.68	3.57
Ukraine	6	4.56	1.32
Cuba	5	1.12	4.46
Jamaica	4	0.271	14.76
Czech Republic	4	1.05	3.81
South Korea	4	5	0.80
Iran	4	7.51	0.53
Croatia	3	0.429	6.99
Spain	3	4.62	0.65
South Africa	3	5.06	0.59
Ethiopia	3	8.43	0.36
Brazil	3	19.2	0.16
Lithuania	2	0.319	6.27
Norway	2	0.503	3.98
Denmark	2	0.558	3.58
Switzerland	2	0.795	2.52
Azerbaijan	2	0.924	2.16
Belarus	2	0.946	2.11
Romania	2	1.9	1.05
Poland	2	3.85	0.52
Kenya	2	4.27	0.47
Turkey	2	7.47	0.27
Grenada	1	0.0105	95.24
Bahamas	1	0.035	28.57
Trinidad and Tobego	1	0.132	7.58
Slovenia	1	0.206	4.85
Latvia	1	0.207	4.83
Georgia	1	0.45	2.22
Ireland	1	0.459	2.18
Serbia	1	0.712	1.40
Dominican Republic	1	0.945	1.06
Sweden	1	0.951	1.05
Tunisia	1	1.07	0.93
Venezuela	1	2.72	0.37
Uzbekistan	1	2.91	0.34
Uganda	1	3.29	0.30
Canada	1	3.49	0.29
Algeria	1	3.71	0.27
Argentina	1	4.01	0.25
Colombia	1	4.67	0.21
Mexico	1	11.2	0.09

And this is it ordered by gold density:

 

Country	Golds	Population (10 millions to 3sf)	G/10mill (2 dp)
Grenada	1	0.0105	95.24
Bahamas	1	0.035	28.57
Jamaica	4	0.271	14.76
New Zealand	6	0.443	13.54
Hungary	8	0.996	8.03
Trinidad and Tobego	1	0.132	7.58
Croatia	3	0.429	6.99
Lithuania	2	0.319	6.27
North Korea	13	2.46	5.28
Slovenia	1	0.206	4.85
Latvia	1	0.207	4.83
Great Britain	29	6.23	4.65
Cuba	5	1.12	4.46
Kazakhstan	7	1.68	4.17
Norway	2	0.503	3.98
Czech Republic	4	1.05	3.81
Denmark	2	0.558	3.58
Netherlands	6	1.68	3.57
Australia	7	2.27	3.08
Switzerland	2	0.795	2.52
Georgia	1	0.45	2.22
Ireland	1	0.459	2.18
Azerbaijan	2	0.924	2.16
Belarus	2	0.946	2.11
France	11	6.54	1.68
Russia	24	14.3	1.68
USA	46	31.5	1.46
Serbia	1	0.712	1.40
Germany	11	8.19	1.34
Italy	8	6.08	1.32
Ukraine	6	4.56	1.32
Dominican Republic	1	0.945	1.06
Romania	2	1.9	1.05
Sweden	1	0.951	1.05
Tunisia	1	1.07	0.93
South Korea	4	5	0.80
Spain	3	4.62	0.65
South Africa	3	5.06	0.59
Japan	7	12.8	0.55
Iran	4	7.51	0.53
Poland	2	3.85	0.52
Kenya	2	4.27	0.47
Venezuela	1	2.72	0.37
Ethiopia	3	8.43	0.36
Uzbekistan	1	2.91	0.34
Uganda	1	3.29	0.30
Canada	1	3.49	0.29
China	38	135	0.28
Algeria	1	3.71	0.27
Turkey	2	7.47	0.27
Argentina	1	4.01	0.25
Colombia	1	4.67	0.21
Brazil	3	19.2	0.16
Mexico	1	11.2	0.09

So Grenada (a country I'm not even sure I've heard of) has stormed the league table with a whopping 95, making our 4.7 look pretty pathetic. We don't even make the top ten! Random places like Lithuania and Slovenia (who?) have done well, and places that I'm sure only exist for people to take holidays in have managed to come second and third!

I was really hoping that maths would prove we won the Olympics. I suppose this is why I should work out all the data BEFORE writing a blog post. It's not my fault: how was I supposed to know that our little Island actually has a lot of people on it? I felt so sure we were one of the smallest countries. I mean, we only have two types of Skittles over here! Big countries have at least five (including the sour ones, mmmm)! I am outraged.

And before anyone mentions it: the failure of this post has absolutely nothing to do with my lack of geographical knowledge.

The Point of This Post

This post has been a cluster-fudge of poor formatting and lazy researching, so you probably want the payoff now. Well here it is: whatever country you're in, think about doing the following activity at the start of the new term. Announce to the class: "I don't know what you've heard on TV, but Grenada actually won the Olympics". They'll be like, what? Show some pictures of the beautiful country of Grenada, and pictures of, uh...  *googles* nutmeg and mace and the uh... Grenada dove. Then, with or without giving them any extra information or instructions, but giving them access to the internet, get them to find out why Grenada actually won. I'm sure that intelligent pupils will be able to work it out, and you can give them lots of hints if need be. Extension: get them to prove that another country actually won the Olympics (e.g. by looking at the number of golds per GDP or something).

If you're worried the kids will google and find this blog and get the answer from here, fear not. As if any teenager is going to read through this long, dull piece! I'm surprised you've got this far, to be frank.

My Final Conclusion

That last bout of googling I did to find out the main exports and national bird of Grenada revealed something interesting to me: Grenada is a Commonwealth country. Do you know what that means? Yep, it's owned by Queen Elizabeth. Who? Yep, the Queen of England. So technically, technically, we won the Olympics. I knew it!!!

Emma x x x

Why Is It Called a Quadratic?

Today I had a really weird experience. I taught a group of students different from any other that I've taught.

Adults.

It was a lot of fun, and something I'd love to do regularly. Adults are just so... reasonable. The best thing about the session is that I got some really interesting questions. Like this one: why is it called a quadratic equation when the highest power is 2, not 4? I'm ashamed to admit I couldn't answer off the top of my head. I had to have a long think about it (and then I googled it, natch).

The reasoning is this: quad in this case doesn't really mean four, it means square. The formula for the area of any quadrilateral will be a quadratic equation, and for every quadratic equation, there exists a quadrilateral whose area is described by it. Similarly, a polynomial of order three is called a cubic not a tresic or something, because it is the formula for the volume of a cube or cuboid.

The pattern breaks down here, because there was no Latin word for hypercube (a four dimensional cube) and they probably weren't too bothered about finding the hyper-volume of one either, so they just called a polynomial of order four a quartic (as in four) and five a quintic, six a sextic, etc.

I know this question has been plaguing you your entire life, so I'm glad to have cleared this up for you! Now go out an enjoy the sunshine whilst it lasts!

Emma x x x

True or False PART II

The other day I asked you all a true or false question:

I brought this up at school today and it created quite a bit of discussion.


True or False:

If x^a = x^b  then a = b. 

Let me know your answer in the comments below!

I'll do a full post on this once I have some answers.

Emma x x x 

 Thank you to everyone who replied. As well as asking on my blog, I asked my students, my colleagues, and some non-mathematical people, including my dad. There are a few interesting things to discuss regarding this question. 


The "logic" of the question


The question I posed contained no existential or universal quantifiers. However, some people reading the question assumed that there were implied quantifiers which completely changed the meaning of the question. For example:


If x^a = x^b  for all values of x then a = b. Is this statement true or false? I think it's true.


The other thing some people confused was the logic of the phrase "if... then...". I would normally use an implication symbol instead but a) I don't know how to type that (my html is an fml) and b) I thought that it's likely not everyone would know its meaning. What surprised me was that even phrasing it this way, there were some people who misunderstood. I received some responses along the lines of "it can't be false because it works for [example]". They didn't understand that for this statement to be false only one example has to fail. This sort of logic isn't taught until A level maths. 


The types of number allowed in the question


 It is obvious that if we restrict x, a and b to natural numbers excluding 0 and 1, the statement is true. A lot of people, seeing the letters a and b, assumed they had to be integers. Just for fun, I asked the question to different people using different letters. When I used y and z, they were less likely to assume they were integers than when I used n and m. 

When x = 0 or 1, a and b can be anything*, rendering my question trivial. So to make things more fun, I told some people that they weren't allowed to use zero or one. This means to find a counterexample you have to go into complex numbers.

 Complex numbers and e

The question had first occurred to me when I was teaching a Further Pure 2 lesson, aand we had to solve the equation e³ = e^z which prompted one of my students to say, surely the answer's just z = 3? I had already gone through the solution on the board and we had come to the answer which by the way is z = 3 + 2kπj (for integer k). My student didn't believe this could be true, and I even started doubting it myself. I couldn't see how z could be anything other than 3. e is just a number, after all. It's kind of amazing when you think about it. That is what is so awesome about complex numbers, they always manage to surprise you. The reason why it works is because you can write

e^3+2kπj  as   e³  x e^2kπj   and it can be shown that e^2kπj = 1.

When I told somebody that this was the type of counterexample I was looking for, they protested, saying that I couldn't use e because e is a function. I replied that e wsan't a function, it was a number, 2.7182818... a bit like pi. They still weren't happy with this, and changed their track. x stands for a variable, but e isn't a variable, it's a constant. I found this very hard to explain: x is a variable and e is one instance of that variable. That particular instance doesn't work, so the statement is false. The person still wasn't convinced.

So, a simple question, but it generated a lot of discussion. It highlighted to me how many people (including maths teachers) have misconceptions with the whole idea of proof and logic and  numbers. 

Wow, this was a pretty heavy post for the last day of the school year! 

Fellow teachers, enjoy your freedom!

Emma x x x

PS If you're wondering why I call it j instead of i, it's because at my school we follow the MEI A level programme, and their links with engineering means it makes sense to use the letter preferred by those in industry. I also followed this syllabus when I studied A level myself, so j is what I grew up with. It was so weird at uni where everyone called it i! 

PPS I now have a contents page! Check it out!

*OK, not completely true, depending on how you define zero to the power of zero.

True or False...?

I brought this up at school today and it created quite a bit of discussion.

True or False:

If x^a = x^b  then a = b. 

Let me know your answer in the comments below!

I'll do a full post on this once I have some answers.

Emma x x x 

 

The Punctuation of Maths

I do love a bit of cross-curricularism. Although maths will always be my first love, I must admit to being passionate about language too. One of my (many) skills is being able to identify the Latin/Greek/Whatever root of a word and explain the original meaning. Actually, that's just given me an idea for another post...

But anyway, this post isn't about words, it's about punctuation. I am still quite confused about when to use commas, semi-colons, colons, emdashes, endashes etc, although I do find them interesting. Language is very mathematical when you think about it.* But I'm not talking about English grammar here (which is probably a good thing, as I began this sentence with a conjunction), I'm talking about the punctuation of maths.

All of you reading this now, write down what a half is as a decimal. Have you all done it? Yes, even you. Good. Now, type what you just wrote into the search bar on your browser's toolbar (assuming you have one) without pressing enter. If you prefer, type it in a Notepad file. Done? OK.

Compare what you have written and typed. Are they the same? Look specifically at the decimal point. I'm guessing all of you used a full stop for your typed version. What about your handwritten version? Is your point at the bottom, resting on the line, or hovering in the middle? (Some of you may have used a comma instead. I'll get to you later).

In the US, it has always been the norm to use a full stop (or as they call it, a "period") as the decimal mark. So we can assume this is wrong. Kidding! (I have a few readers from the US who I must try not to insult). In Britain (and in British Empire nations) however, the mid dot, or "interpunct", was the standard symbol. However, the full stop was OK to use in typing and printing. The mid dot can be easier to read on lined paper because the point can't be hidden by the line.

There is a problem with the mid dot though. To me, such a mark indicates multiplication (although as far as I can tell this is never used in schools), and the top guys at the SI agree, because they rejected it as the standaard decimal mark.

In the end (well, in 2003), The 22nd General Conference on Weights and Measures declared that "the symbol for the decimal marker shall be either the point on the line or the comma on the line" (yes, comma users, I'll get to you), meaning that officially decimal points and full stops have the same appearance. Some older British people may angrily disagree with this decision. For once I'm not bothered.

OK, the people I was ignoring: some people use commas instead of dots for the decimal mark. Namely, people from non-British Europe, and also some random places like South Africa. This is because originally, a short, vertical mark dash was used as the decimal mark. This evolved into either the comma or the dot. France preferred the comma (presumably because it's more phallic) and the rest of Europe then took sides.

Now, some more writing for you to do. Write down, in figures, the number twenty seven million, five hundred and sixty four thousand, two hundred and fourteen. Done? Good.

Look between the seven and the five. What's there? A comma? A gap? Nothing? (For the Europeans: a dot?)

I was always taught at school that you should just leave a space between every three digits, and not put a comma. I had a few reasons given to me for doing this: because a comma could get confused with a decimal point (really?) or because Europeans might think it's a decimal (fair enough). Because I was taught this in school, I get really annoyed when I see people using commas. It's not what you're supposed to do!

I have researched this to check that I am correct, and I have found that the International Bureau of Weights and Measures states that "for numbers with many digits the digits may be divided into groups of three by a thin space, in order to facilitate reading. Neither dots nor commas are inserted in the spaces between groups of three". Officially this space is supposed to be a half space, although I don't know how to do that on MS Word.

I think most maths teachers in the UK would know better than to use commas to separate digits (they jolly well should!) but non-maths teachers may not know this rule or appreciate the importance of it and hence you might find commas in long numbers in Geography lessons. It should be part of a school's numeracy policy that every teacher conforms to this. Otherwise it would be as bad as a non-English teacher using an aberrant apostrophe (which I'm sure NEVER happens). I fully realise that by saying this I am practically begging people to criticise my grammar on this blog. Bring it on!!!

I will leave you with a nice little anagram puzzle:
Rearrange this expression to make another one: "I'm a dot in place".

Emma x x x


*One punctuation thing that annoys me because of its lack of mathematical sense is the " quotation mark. In a book, if someone is talking for a long time, there can be a paragraph break in the middle of their speech. You do not close the speech marks at the end of the paragraph, but you do have to put an open speech mark symbol at the start of the next paragraph. This annoys me because it's like having an open bracket that is never closed. In programming, such a thing is, to my knowledge, always disallowed.