All 5 entries tagged Maths

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June 10, 2011

The best way to teach solving equations?

I think this is one of the things that maths teachers argue over the most.

There's the "move it to the other side" method, the "do it to both sides method" and (one I'd not heard of until recently) the "cover up method".

Method 1 is what I was taught and what I use myself. I knew that it was not a very good method as it does not explain why you're doing that and how it works. I defended this method forcefully, sure that my way of thinking must be the best way (I'm arrogant like that). But since teaching both methods at my placement school, and teaching different methods to my one-to-one tutees, I've come to realise that the second method is much less confusing, more intuitive, and more memorable.

Here is how I would write a model solution:

3x + 5 = 10

    - 5      -5

3x       = 5

3x/3    = 5/3 (written as fractions)

And then I'd cross out the two threes in the fraction on the left because they cancel each other out.

Writing the -5 underneath makes it really easy for pupils to see what they're doing and it's really hard to make a mistake.

Doing the dividing bit as fractions is good because it encourages pupils to leave things as fractions, it makes division easier (instead of actually dividing, just cancel down the fraction as far as you can in easy steps), and it means that the whole side will be divided and not just the last term.

That is the method I would recommend for higher-tier pupils.

The cover-up method is one I cam across recently and I think it's really good because it's the way a pupil would intuitively try to work out the answer.

In equations like 10 - x =7, most pupils can just "see" that the answer is 3. In equations like 3x + 4 = 10, it's harder to just seei it, but using the cover up method you can. Cover up the 3x, and say "what add 4 gives you 10?" They'll say 6. Then you uncover the 3x and say, "well then 3x must be 6. So what's one x?" And they'll say 2 (after painstakingly counting on their fingers, if they're anything like my pupils!)

I wouldn't teach higher tier pupils this because it's hard to apply it to more complicated equations (although it obviously still works). But for lower ability pupils it seems to be less confusing.

I'm glad I'm starting to realise what methods I'm going to be teaching because I want to be consistent, so that pupils don't get confused with lots of different methods.


May 04, 2011

How Lewis Carroll Would Vote Tomorrow

I've mentioned in a previous post that I love Lewis Carroll. He combines two of my favourite things: books and maths (Lewis Carroll is the pen name of mathematician Charles Dodgson. Please keep up!)

Dodgson became involved in college elections in the early 1870s at Oxford university where he was a professor. He became interested in the theory of voting, of the accuracy and fairness of different voting systems.

First Past The Post

Dodgson was not a fan of this voting system. He claimed "the extraordinary injustice of this Method may be very easily demonstrated". He then gives an example to show how stupid it is:

Suppose there are 11 electors and 4 candidates a, b, c and d. Each elector ranks the four candidates in order of preference. The 11 columns here show their choices:

a
a
a
b
b
b
b
c
c
c
d
c
c
c
a
a
a
a
a
a
a
a
d
d
d
c
c
c
c
d
d
d
c
b
b
b
d
d
d
d
b
b
b
b

It's easy to see that a is considered best by three of the electors and second best by the rest. But in actual fact, it is b who ends up winning, even though he/she was considered the worst by seven voters.

I don't think Dodgson looked at "Alternative Vote", alothough he did write about lots of other systems.

The Method of Elimination

In this method, each voter chooses their favourite, and then the one who gets the fewest votes is eliminated, and the process is repeated (a bit like Big Brother? The TV show, not the Orwellian thing). This method at first seems pretty flawless. However, consider the following situation:

b
b
b
c
c
c
d
d
d
a
a
a
a
a
a
a
a
a
a
a
b
c
d
c
d
b
b
b
c
c
b
d
d
c
d
c
d
d
d
b
b
c
c
b

Notice that a is everybody's first or second choice, and hence appears to be the best candidate. However, he/she will be eliminated first. c will be elected instead.

The Method of Marks

In this method, each voter is given a specified number of marks that they can divide between the candidates. Then the candidate who gets the most marks wins. Dodgson said that this method would be perfect as long as the voters divided their marks fairly: giving most to their favourite but some to the candidates that they wouldn't mind electing. But Dodgson commented that "since we are not sufficiently unselfish and would assign all our votes to our favourite candidate, the method is liable in practice to conicide with that of the simple majority [first past the post] which has already been shown to be unsound".


So how would Lewis Carroll vote tomorrow? I think he would vote no to AV. Because I think he'd think that AV is better than the current system, but still not good enough. If we vote in AV, we'll have no chance of getting Single Transferable Vote, because we'd have to spend a few decades getting used to AV before referendumming again, and I think he'd think STV was a far superior system.


Emma x x x

All quotes are from Robin Wilson's "Lewis Carroll in Numberland", a book I highly recommend.



May 01, 2011

Explaining the Fourth Dimension


Has a kid ever asked you what the fourth dimension is? What it looks like? Whether it could possibly exist?

No, no kid has ever asked me either. But I really wish one would one day.

Anyway, I was reading this book by one of my favourite authors, the hilarious Scottish crime writer Christopher Brookmyre. The book is called Pandaemonium and for goodness' sake don't recommend it to any of your pupils. The "c" word appears frequently, there's a lot special hugging* in it, there's really really gross violence and gore, and it's just generally very offensive. If it was released as an audiobook, it would be read by Frankie Boyle.

I loved every page.

Anyway, I'll try and get to the point. There's this bit in it where CB explains the concept of a fourth dimension really nicely. I loved it so much I highlighted it and wrote a little note (I LOVE my new Kindle!!). I'll try and paraphrase it here:

Imagine there are some ants crawling across the duvet on your bed. They are only aware of two dimensions: walking forwards and walking sideways. There is no up or down for them. So if you picked one of the ants up and suspended it in the air, all the other ants would think the ant had vanished. They would have no idea where it went. And then if you put the ant back down a few centimetres from where you picked it up, the ants would all think it had teleported.

Maybe there's a higher dimension out there that we can't comprehend. Maybe there's something out there that could pick one of us up and put us down somewhere else, and it would look like we've teleported. If we could access this fourth dimension, think of the possibilities: we could perform surgery without breaking the skin.

Back to the ants on the duvet: imagine picking up two opposite corners of the duvet and bringing them together. The ants are still only moving in two dimensions, but now they can walk off one edge of the duvet and return at the opposite end, walking in the opposite direction. This involves moving their world around in three dimensions, but keeping it as a two dimensional world, and without them noticing. Could something similar happen to us? Could we walk off one "end" of the universe and end up on the other side? Is dying walking off the end? Does God live in the fourth dimension?

Well, I found it interesting.


Emma x x x

*credit goes to Lizzie Bowen, drama-with-English trainee, for that very useful phrase.



February 25, 2011

Alice in Numberland

You may have noticed that the title of my blog is a nod towards Lewis Carroll's oeuvre, Alice's Adventures In Wonderland. This is pretty much my favourite book of all time. Why? Because it's FULL OF MATHS!

Lewis Carroll is the nom de plume of Charles Dodgson, who was a maths lecturer at Oxford. He was actually more famous for being a children's photographer, but his maths was pretty good too. He was really obsessed with Euclid's Elements and wrote textbooks to go alongside it. He loved logic too, which is what you see most of in his books.

"if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic".

Tweedledee says that. It's one of my favourite quotes from the book. The best logic puzzles appear in his other books, like The Hunting of the Snark.

Here's another favourite quote (it's just as Alice is falling down the rabbit hole):

"I'll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is - oh dear! I shall never get to twenty at this rate!"

Alice's wrong calculations can be explained by using different base systems:

in base 18, 4*5= 20 = 18^1 + 2 = 12

in base 21, 4*6= 24 = 21^1 + 3 = 13

if you carried on the pattern, you'd get:

base 24: 4*7 = 14

base 27: 4*8 = 15

...

base 39: 4 *12 = 19

base 42: 4*13= 42^1 + 10 = 1X (where X is the symbol for 10 in base 42)

So Alice is right, she'll never get to 20!

Interestingly, it breaks down at base 42. Lewis Carroll seems to have a slight obsession with the number 42, it appears everywhere in all of his books. It has been suggested that Douglas Adams used the number 42 in the Hitchhiker's Guide to the Galaxy because of this. Adams named his chapters of the radio show "fits" which is also what Carroll did, suggesting he's a fan.

My favourite maths thing that Dodgson did though, was his work on voting systems. This is really interesting, it's got some great maths in it, and if you tackled it in a lesson you would be hitting the moral, social and ethical aspects of maths.

So if anyone asks you to come up with ideas for a maths/English connected curriculum day, suggest studying some of Lewis Carroll's work. I love it when literature and maths combine, as they are my two favourite things. Another good mathsy book is Mark Haddon's The Curious Incident of the Dog in the Nighttime, which I love, but the maths in that is very separate from the story, and I don't think it would be good to use in class.

One of my favourite authors of all time is Louis Sachar, who wrote Holes, which is very commonly studied at secondary schools. He also wrote some great maths books, the Sideways Arithmetic from Wayside School series, so if you have a year 8 class studying Holes, it would be cool to do some puzzles from that to run alongside it.

Finally, another of my favourite authors of all time is Koji Suzuki, the Japanese author of Ring, which was made popular by the abomination that is the film The Ring. There are two sequals, Spiral and Loop, and all three books are outstanding on their own or as a series. They include some brilliant codebreaking linked to genetic codes, and some discussion of meta-mathematics which is just plain awesome. I'm not saying bring them into your lesson, but suggest them to sixth formers who like reading and maybe need showing the coolness of maths. And you should read them yourself because they're sooo good!


Emma x x x


PS if you want to find out more about the mathematical life of Lewis Carroll, I recommend Robin Wilson's Lewis Carroll in Numberland. I have a copy if you want to borrow it. I also have copies of all the books mentioned above, but some of them are at my parents' house in Leicester, so I can't get them until Easter.


February 22, 2011

Birthday Equations

This idea started as a stupid facebook status but is starting to sound like quite a useable rich task.

I noticed that my birthday this year (today: 22/02/11) forms an equation 22/2=11. I invited my facebook friends to work out when the next time was that their birthday would form an equation. Quite a few people replied, including non maths teachers.

Then Lydia posed an interesting question: are there people out there whose birthdays will never form an equation?

Initially I thought no, because I thought addition would always work. But if you were born 01/01/03 then you'd never have a birthday equation.

So far in all my 22 years, I've only had two birthday equations. 22/02/00 was one because 22 mod 2 = 0.

When are my next ones?:

22/02/20 22-2=20

22/02/24 22+2=24

22/02/44 22*2=44

I'm having trouble thinking of other operations. 22 choose 2 is 231 which is too big. 22^2 is 484 which is too big.


I really know how to enjoy my birthday don't I?


Emma x x x

PS this rich task is copyright Team Hopper. When Team Hopper publishes their first book, this will be one of the activities included. You may not reproduce this task without the permission of Team Hopper.


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