Complements #10 : Infinity

Although infinity doesn’t come up much at GCSE, it’s alluded to from time to time. When talking about experimental vs theoretical probability for example, or where the line ends for a function without limits. recurring decimals, the asymptote of the tan lines, or even the nth term of a sequence. So it seems apt that we look a little deeper into the rather difficult concepts behind infinity.

What is infinity?

Well, that depends. There’s a few kinds for starters. Let’s go with ‘traditional’ infinity, known sometimes as ‘potential infinity’. This is the infinity most of us know and probably don’t particularly love. Real numbers keep going, and going, and going, and going. They never end, they are endless, they are infinite. There is no ‘upper bound’. It is therefore boundless.

The symbol for (positive) infinity is the Lemniscate symbol, which is essentially a mobius strip, itself a never ending one-sided shape. ∞

3giphy

Negative infinity is just -∞ and  an infinitely small number, an infinitesimal number would be something like 1 / ∞

If we stop there, infinity is quite a manageable thing. However, if we delve a little deeper, it’s a hideously bizarre anomaly that is jam packed with counter intuitive mind blowing ridiculousness. So let’s do that!

Let’s compare odd and even numbers… to infinity!

There are infinitely many even numbers, but they can be put in ascending order. It’s fair to assume that each even number is ‘every other number’, as we’ve neglected to think about the odd numbers. So therefore, there are n even numbers for 2n numbers, where 2n must be infinity. Still with me? But we know there are in fact infinite even numbers, then there are 2 x infinite real numbers, but that can’t be right, because there’s no such thing as 2 x infinity. So… umm… now what?

Well, I guess we’ll have to accept what Galileo wrote about infinity:

“The attributes ‘equal’, ‘greater’ and ‘less’ are not applicable to infinite, but only to finite quantities’.

So maybe infinity is finite after all? Only a few nutters think like that!

But then we get ‘actual infinity’ which is kind of infinity, but with a defined limit. A box to put it all in if you will. Mathematicians like boxes to put things in.

Consider 1/2, 1/4, 1/8, 1/16, 1/32 … and so on. This is an infinite sequence, but you know it’s never going to get past 1. You could bound it using notation like this

{1/2, 1/4, 1/8, 1/16, …}

So now it’s an example of ‘actual infinity’, which causes umpteen paradoxes. So is there an ‘actual infinity’? Who the hell knows. Some say yes, some say no. Maths says yes so that you can manipulate things easier. Brain says no.

Another example of a sort-of-bounded infinity is the Sierpinski Triangle. Each triangle gets subdivided into smaller triangles, then they get subdivided, then they get… but it all stays inside the original triangle.

giphy

I’ll do a post all about fractals on another day, but they’re a great exploration into the infinite.

We’ve also stumbled across different sizes of infinity. Our fraction sequence is infinitely long, but in terms of SIZE, it’s nowhere near as big as an infinite sequence of real integers. It never gets past ‘1’ for starters.

So now we get to Cantor, who in 1891 started us all using the term ‘cardinalities’ to describe the different sizes of infinity.

He ended up constructing a whole series of sizes of infinity. The smallest of which were the natural numbers. He showed that there were more real numbers than natural numbers (infinitely more?!), even though there are infinite real and infinite natural numbers. What the hell is going on? Maybe the diagram below will help. If there’s infinite natural numbers, and infinite whole numbers, and infinite integers, and infinite rational numbers, and infinite real numbers, you can see that each ‘infinite’ is of a different size, as there’s simply more numbers to use!

So if there’s more of one than the other, are they in some way countable then? Countable infinites?! Well it turns out there’s plenty of those. Dr James Grime prefers the term ‘listable’ to countable. I’m inclined to agree.

Let’s see…

Listing (“counting”) Natural Numbers:

1, 2, 3, 4, 5, 6,… simple. I’m clearly not going to miss any of these out if I keep going…forever.

Listing Integers:

0, 1, -1, 2, -2, 3, -3 … simple… but arguably twice as big as the infinite natural numbers…?! So now we have a bigger countable infinity than the natural number infinity. But aren’t we not comparing sizes? Maybe we are now. I’m having trouble keeping up with myself here.

Listing Rational Numbers:

Hmm… trickier.

You have to be clever here, but it’s doable. If you just add 1 to the denominator of a fraction each time, you’ll never get to a whole number. Doh! There must be some kind of clever way of doing it… right? Right!

Here’s a way:

Listing *all* the Real Numbers? Well that includes Irrational Numbers. That causes a bit of a problem. How can I count the irrational numbers? They’re not fractions so… do I just randomly pick out a bunch of digits…to infinity…? Clearly these guys are a bit of a headache to be considered ‘countably infinite’.

Turns out we can’t count them… so they’re the biggest… infinity.

Are we finished? I didn’t mention any paradoxes yet! Here are some of the more famous ones:

Zeno’s Infinity Paradoxes: Dichotomy (quoted from wikipedia)

“Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.”

So what is the starting distance? Can Homer even begin his journey??

 Gabriel’s Horn:

1280px-GabrielHorn

Take the equation y = 1/x

From x=1 to x infinite, we rotate y around the x-axis. This creates a horn shape with a wide opening of radius 1 at x=1.

The horn has a finite volume and infinite surface area!

gabriel2loop

So if you fill the horn with a finite amount of paint, to fill the finite volume, it must cover the sides, but they’re infinite, so how can it? So maybe the finite paint has covered an infinite surface area, and you have paint left over, to make up the volume of the shape. So umm… how does that work??

Image result for confused

Hilbert’s Hotel

A Grand Hotel has an infinite number of rooms and infinite guests. So the rooms to guests ratio is 1 to 1. What happens if someone new comes along? Well the Hotel is full right? Nope. Everyone moves along one room, so room 1 goes to room 2 etc.

What if infinite new guests turn up?? BUT THE HOTEL IS FULL. Noop. Everyone could just go into the room that is double the room number they are already in, to make space for infinite guests.

So do we conclude that  ∞ + 1 = ∞

or 2∞ = ∞ ??

 

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Complements #9 LCM and HCF

With the exception perhaps of the quadratic formula, it seems to me that the method for finding the highest common factor and the lowest common multiple is the best (worst) example of really long methodology with zero understanding behind it. At least for students… and maybe for teachers too! So let’s take a closer look…

Lowest Common Multiple

The term itself means the first integer that your numbers will ‘go into’. In other words, the first integer to appear on what would be the times tables of each of your starting numbers. One might say this number is in fact… a common multiple (duh duh duuuuuh!)

You can of course just multiply the starting numbers together, and you will find *a* multiple of both of them, however it isn’t necessarily *the lowest* multiple.

Take 9 and 12 for example, 9 x 12 = 108. 108 is therefore a multiple of both 9 and 12, however the lowest multiple is 36 (9 x 4, or 12 x 3). You can rearrange both of those multiplication sums to make it clearer : 9 x 4 = 3 x 3 x 4, 12 x 3 = 3 x 4 x 3 = 3 x 3 x 4.

So the most laborious way to find the lowest common multiple would be to write out the times tables of both (or more than two if you’re feeling particularly ambitious) numbers until you find a matching answer. Bingo! But… that can be a long, long process, and mathematicians can be cleverer than that. We’ll come to a better method in a little while.

Highest Common Factor

Factors are numbers that you can multiply together to get your starting number. For example, if I start with 10, 5 x 2 = 10, so 5 and 2 are factors. Hence 10 appears in the times tables of both 5 and 2.

A Common Factor is a factor that is… errr… common to two (or more) starting numbers. For example, 2 is a common factor of both 4 and 10. In fact 2 is the Highest Common Factor (HCF) of both 4 and 10, because there is no greater integer that is a factor of both of these numbers.

We can write “2 is a factor of 10” as 2|10 … but we don’t in GCSE maths. But we do in the real world. “2 is a factor of 10” is also exactly the same as saying “2 is a divisor of 10”. In fact, Euclid called it a ‘common measure’. Bloody Euclid. Always different. Legend.

Students often get confused between the meaning of Highest Common Factor and Lowest Common Multiple, and often switch the terminology and think of the Highest Common Multiple and Lowest Common Factor. This is pretty daft in fairness, as there are infinite common multiples of any two (or more) numbers, and so Highest Common Multiple has no answer. Furthermore the Lowest Common Factor of any number is technically 1, which seems fairly pointless to work out if it’s *always* 1 (don’t get fussy on me with negatives and decimals).

Now that I’ve mentioned negatives, it’s a bit dangerous to ask “how many factors of 6 are there?” as instinct suggests these : 1 x 6, 2 x 3, thus 1, 2, 3 and 6 are factors.

However… -1 x -6, -2 x -3 also make 6, so they’re factors too. Or are they? It depends on who you talk to. Generally we only accept positive integers as factors, but some hardcore mathematicians argue negatives should be included.

So where do we draw the line? If we accept decimals as factors, then there are infinite factors for every integer. So… we don’t.

Algebraically, often we factorise which in essence is the process of creating a pair (or more!) of factors of an algebraic term:

x2 + 4x + 3 = (x + 3)(x + 1)

So (x + 3) and (x + 1) are both factors of x2 + 4x + 3, because they multiple together to equal it.

Ok let’s confuse things further with Prime Factors!

Prime Factors

Prime Factors are, confusingly, not really related much to factors. If you think back to Complements #4: Prime Numbers – you read that right? 😉 Then you’ll recall that every single integer is made up of prime numbers multiplied together. They’re the building blocks of other composite numbers.

So in essence, every number has a unique combination of prime numbers multiplied together as its mathematical DNA.

For example, 15 = 3 x 5. No other number has the DNA of 3 x 5. 16 = 2 x 2 x 2 x 2 etc.

So if any pair of numbers has a Common Factor… then that Common Factor’s DNA (prime factors) must appear *within* the DNA of both of our original numbers. I think it needs a diagram:

hcf

From the picture above, you can see that 9 is a common factor of 27 and 36, and that it is made up of the prime factors 3 and … well, another 3. Those 3’s also appear in the composition of the other two numbers. They are common prime factors.

Every common factor will always be made solely from prime factors that are common to both of your original numbers. From above, 3 is also a common factor, which again is made from a common prime factor within 27 and 36.

The Highest Common Factor is made from the maximum amount of common prime factors from your original numbers. In the case above, 9 is both a common factor AND the highest common factor.

These shared common factors are represented nicely in a Venn diagram – hence why we use them in schools to find the HCF:

cpf

Now this handy Venn serves a second purpose. We can find the Lowest Common Multiple from it too!

First, appreciate that the lowest common multiple of any two prime numbers, is those two numbers multiplied together (LCM of 3 and 5 is 15, LCM of 7 and 2 is 14 etc). So because we’ve already separated the common prime factors in our Venn Diagram, we’re left with the uncommon ones (the 3 on the left, and two 2’s on the right). They’re the problem. By multiplying them together we’ll get their lowest common multiple. However, we must also multiply by the *common* prime factors too as they are part of the DNA of our originals.  Our original numbers are essentially going to become factors of this new multiple, and so must be made up *only* of prime factors included in the multiple.

3 x 2 x 2 is just the common multiple of our uncommon prime factors. Confusing.

Whereas (3 x 2 x 2) x (3 x 3) will get us our lowest common multiple of 27 and 36.

So LCM = HCF (the product of the common prime factors) x the product of the uncommon prime factors.

Phew! There are other ways of course too…

We’ve stuck so long with the example of 27 and 36, so we’ll continue. Back to the Venn first of all…

cpf

So the HCF is 3 x 3 = 9, and the LCM is (3 x 3) x (3 x 2 x 2) = 108.

Now 108 is also 27 x 2 x 2, and it’s also 36 x 3. So if you wanted, rather than multiplying everything in the Venn together for LCM, you could just multiply one of your original numbers by the other original’s uncommon prime factors. You’re essentially just doing the same thing in a different way.

In fact, prime factorisation has other uses too. You can check (and find) whether a number has an integer square / cube root from its prime factors. Consider 441. If you break it down as a product of its prime factors, you get 3 x 3 x 7 x 7. There are two pairs with nothing left over. The square root of 3 x 3 is 3, and of 7 x 7 is 7.
The square root of (3 x 7) x (3 x 7) must also be 3 x 7, which is 21. So the square root of 441 is 21. Similarly if you don’t ONLY have pairs of prime factors (or, I suppose, any multiple of 2 of each prime factor), then you will not get an integer as the square root.

You can use exactly the same idea to find cube roots (you would need every prime factor to be in triplet). How clever!

Then there’s Euclid’s Algorithm… Told you he was a legend.

We don’t teach Euclid’s Algorithm, probably because its awesomeness would make heads explode, and nobody wants that. Nobody wants that.

It’s a real shame we don’t teach it, because it fits nicely with primary school division, where we get remainders and keep them and love them and cherish them.

Secondary comes along and we BANISH them, and BEAT THEM UNTIL THEY DIE. The remainders that is, not the children.

Here’s how Euclid’s Algorithm works:

Example: Find the HCF of  50604 and 10206.

Now those two numbers are big and hideous. Using our Factor Tree / Venn thing could take a while.

Euclid’s Algorithm goes like this:

50604 / 10206 = 4(10206) r. 9780

Euclid proved that the HCF of, say 50604 and 10206 is in fact the same as the HCF of 10206 and 9780 (obviously using the example above). The proof can be found here if you’re nerdy.

So the same rule still applies, so we do it all again:

10206 / 9780 = 1(9780) r. 426

or 10206 = 1(9780) + 426

9780 = 22(426) + 408

426 = 1(408) + 18

408 = 22(18) + 12

18  = 1(12) + 6

12= 2(6) + 0

So HCF of 50604 and 10206 is the same as HCF of 6 and 12, which is 6.

Don’t believe me? Here’s the GCSE method:

hcf

How clever! Such fun.

Complements #8 Zero

hqdefault

This post was inspired by a training session I attended on Saturday, where I watched in awe as a video was played of a lady teaching an entire lesson to Year 2 students (aged 6) about zero. And only about zero. Very impressive.

So where to begin… the beginning of zero I suppose. Zero began a lot later than all the other numbers. An afterthought of convenience that changed our numerical system forever. Some 5000 years ago, traders in Mesopotamia began using double dashes like this // to mean “nothing goes here” when writing out large numbers.

Consider this for a moment, before zero, the typical structure of numerical symbolism was to use new symbols or symbol repetition as you ascended the number line. Roman numerals are a great example. X is ten, L is 50 etc. The downside to doing things this way around, is that you either have infinite symbols, or ridiculously long combinations of symbols for relatively small numbers.

For example: LXXXVIII is 88, and MCMLXXXVII for 1987. So it was a bit of a stroke of genius when someone started using a ‘gap’ symbol. It allowed you to write 101 as ‘one gap one’, or ‘1 // 1’. Now you can use a mere 9 symbols, and just show everyone that sometimes there’s nothing in your ‘hundreds’ column, or ‘tens’ column using your new // symbol. Handy. Eventually this evolved into our special zero. Although not before various disappearances and banishings, and it wasn’t considered a NUMBER until much later. But we’ll save those for another day.

Zero evolved further to become the most important gate keeper since Zuul the gatekeeper of Gozer.

300px-Zuul_vinz_2

Zero is the gatekeeper between the positives… and the negatives (“Booooo, hissssss”).

Zero annoys a lot of mathematicians because it doesn’t play by all of the rules. It needs special care and gets a timeout pass for certain operations.

First and foremost, we don’t divide by zero. No no no no no. Dividing by zero is like the end of the Matrix Trilogy. No-one knows what just happened, it doesn’t make sense, you’re a bit scared and you wish you stayed at home. You can’t even divide zero by zero. Your calculator will likely display NaN (Not a Number …  or perhaps Now Annihilation Nears) How annoying.

And look at this crazy mess:

0 x 6 = 0, 0 x 7 = 0, but 6 doesn’t equal 7 ! But we get the same answer. Zero is really messing with things. No other numbers get that kind of nutty result.

The craziness is easier to see if we switch to algebra

ab = d, ac = d, but b  c.

Bloody mind bending zero!

Here are some more properties that may ease your mind, or make you scream uncontrollably.

Zero is the only number that is neither positive or negative

Zero is even

Zero is neither prime nor composite.

Anything multiplied by zero is zero… or is it…?

Anything divided by zero is undefined… and ends the universe.

x ± 0 = x

There are infinite zeroes hidden in front of every number. (e.g. 3 = 03 = 003 = 00000003)

Zero factorial, written as 0! is 1. If you’re unfamiliar with ‘factorial’, 4! = 4 x 3 x 2 x 1 = 24. But 0! = 1… but anything multiplied by 0 = 0…

WHAT?!?!?!

Anything to the power of zero is also 1.

e.g. 2500 = 1

WHAAAAT?!?!

baby-crying

(I need an adult!!)

Zero is melting my brain. Let’s look a little closer at those two counter intuitive examples.

Why is 0! = 1 ?

If you’re impatient, the key reasoning is highlighted pretty well below:

rew

And as for 2500 = 1, well that’s linked to the laws of indices.

Anything to the power of zero equals one

I’ll leave you with this final mind bender…

-30 = -1 but (-3)0  = 1

I’ll let you ponder that one on your own 🙂

Complements #7 The Pythagorean Theorem

The Pythagorean Theorem is possibly the first (and in my opinion, only) piece of beautiful maths in GCSE. Not only is it a majestic piece of mathematical wonder, it carries with it great history, real-life application, visual beauty and it opens the door to a wide range of geometry.

How tragic then, that it ends up being so poorly received by students, who often sit puzzling over where the hippopotamus is, which side is the ‘b side’, and trying to find the barely used square-root button on their calculator.

Pythagoras-2

Pythagoras is fairly lucky to have his name attached to the theorem, as the maths behind it had been used for many, many years before him. However kind of like patenting, he was the first to *formally* prove it, despite it being in use all over the place, and being independently discovered by various mathematicians all over the world.

pythagorean_squares_example

Ironically, the proof almost unanimously used in classrooms across the land, the so-called ‘Bride’s Chair’ proof (see above) is in fact NOT a Pythagorean Proof at all. It’s Euclid’s proof (almost). The *only* Pythagorean proof is this one:

Pythagoras-proof-anim.svg

This is the exact proof that Pythagoras came up with.

However there are literally hundreds of alternative proofs of the theorem to choose from should you so desire. And you should, because some are wonderfully clever.

Students will understand the properties of similar triangles, so why not prove it using those:

proof

Or a proof by tessellation

In fact, if you think carefully about the construction of a2 + b2 = c2, you may come to realise that you don’t even need squares at all. You just need similar shapes, whose side length is the length of the side of the right angled triangle (in fact, you don’t even need that last bit!)

circleindex

Any shape at all really… as long as each iteration is similar.

pythagoras-scaling-5

OK so now for some other interesting things linked (sort of) to the Pythagorean Theorem…

1. The Vecten Configuration

If you join the remaining vertices of the squares (brides chair), the new triangles are all of equal area to the area of the original triangle. This is in fact true for any triangle whatsoever (which I think is cooler than the Pythagorean Theorem… this one isn’t held back by specific triangles)

350-i-samearea

So in the picture above, all the turquoise triangles have equal area. Sadly we don’t use this fact in GCSE, but we should.

Following on from this, we can deduce the Finsler-Hadwiger Theorem, which looks like this:

FinslerHadwingerTheorem_1000

Put two squares together (ABCD and AB’C’D’) as shown, with one vertex touching, and you can create two triangles of equal area (also shown, DAB’ and BAD’), and by joining the midpoints of DB” and BD’ with the centres of the original squares, you get another square. Maybe it’s clearer on Geogebra…

geogeb

You can see the equal areas for both triangles in the side menu.

Finally, if you extend the sides of the squares from the Vecten configuration, you get Grebe’s Theorem (it’s not even on Wolfram!) which states that if you draw all those lines, you get two similar triangles that are homothetic (think “centre of enlargement”)

grebe

I could go on… but I won’t. Instead I’ll just show off my cool Pythagoras poster I made for my office:

ByJ4G_DIAAA2ESC

Complements #6 Polygonal Numbers

Here we’ll look a bit closer at square and triangular numbers, but I’ll offer a bit of insight into polygonal numbers in general.

Polygonal numbers are numbers that can be represented as points on a plane arranged into a polygon (and they fall under a broader umbrella known as figurate numbers. The idea dates back to ancient Greece, where lots of mathsy things were done with pebbles and sand.

You’ll be familiar with triangle numbers:

DoingInvestigations_chp6_2

and square numbers :

Both have an nth term. Square numbers (also known as “perfect squares”) have the formula n2, and triangular numbers have the formula n(n+1)/2. Both can be derived using a simple sequence table, and it’d be worth doing so with your students. Obviously we’re talking quadratic sequences, not linear.

The term ‘squared’ is in fact slightly different to a square number. Squaring a number is the application of the formula n x n and is perhaps best thought of alongside cubed etc rather than polygonal numbers, whereas a “square number” is a number that can form a square using units. For clarity, consider a triangle number. I don’t say ‘4 triangled’ when I apply n(n+1)/2. Furthermore, (2.4)2 is not a ‘square number’.

Anyway, each polygonal number has it’s own strict formula that can be derived by school students if you so feel the need.

Now, there are some interesting relationships between square numbers, and between triangle numbers, and between square AND triangle numbers.

Let’s talk about Gauss.

Gauss-10DM-Vorderseite-300dpi

You’re possibly familiar with the age-old tale of Gauss being asked to sum the numbers 1-100, and he did it super quick with no working out (smart git). He did it by figuring out the sum of 1-100 is 1 + 100, 2+ 99, 3 + 98 etc rather than just mindlessly adding them sequentially. If you hadn’t noticed, that’s just 50 x 101 = 5050

Now, what’s the 100th triangle number?  100(100+1)/2 = 5050. Coincidence? No sir.

Let’s take the sum of all numbers up to 6. You can work that out in your head as 6+5+4+3+2+1 = 3 x 7 (Gauss) = 21.

6(6+1)/2 = 21. Let’s visualise what’s going on:

tr

So essentially, the sum of all consecutive integers ‘n’ *is* the triangle number ‘n’.

Fun huh? Similarly, the sum of consecutive odd numbers is a square number.

sq

So if I said ‘sum the first eight odd numbers together’ i’d just do 8 x 8.

Or if I said ‘sum the odd numbers from 1 – 50’ I’d just do (50 / 2)2
(there are 50/2 odd numbers between 1 and 50)

And with some jiggery pokery you could answer ‘sum the odd numbers between 120 and 210’ but I’ll let you ponder that one on your own.

So there are lots of fun patterns using square and triangle numbers. That’s just a couple of them.

Finally we can also prove visually that the sum of two consecutive triangle numbers is always a square number:

sqtr

So whenever you’re talking about triangle numbers, be sure to stack them in a right-angled triangle rather than a kind of pyramid. It makes for easier proof visualisations. I wonder if the new exam emphasis on proofs will allow for visual proofs? Hmm…

Random aside:  there is actually a different ‘type’ of square and triangle number which has a similar name but different properties. The ‘centered square‘ numbers, and ‘centered triangle‘ numbers look like these:

CenteredTriangularNumber_600 CenteredSquareNumber_600

Different patterns, different numbers involved, and different formulae. Pretty.

Now, before we finish, you may have been wondering about the term ‘polygonal numbers’. It alludes to OTHER shape-based numbers. And there are many other types. Infinite in theory. They get a bit boring and nameless later on, but here are a few:

Pentagonal Numbers:

500px-Polygonal_Number_5

Hexagonal Numbers:

500px-Polygonal_Number_6

Nonagonal Numbers:

NonagonalNumber_800

And they all have ‘centered’ versions too, which are prettier:

280px-Centered_nonagonal_number.svg

(centered nonagonal numbers)

Eventually things get dull and we call them ‘23-gonal‘ etc. 1166 is a 23-gonal number, but who cares.

Some numbers are extra special, and they are more than one shape! 36 is both square and triangular for example. I find that strangely exciting. As do many geeky maths people. There are a few mysteries attached to multi-polygonal numbers (I made that term up, at least I thought I did. Turns out it’s in use.) 9801 for example is both square and pentagonal (apparently). But is there a number that is triangular, square  AND pentagonal? These are the nerdy things mathematicians try to discover. (hint: no-one has found one yet).

I’ll leave you with this little puzzle:

Arrange the numbers from 1 – 16 in such a way that each pair sums to equal a square number. (Taken from this amazing book by Matt Parker)

index

(that’s a flippin’ cube, not a square).

Complements #5 : Fractions

Ugh. fractions. Is it me, or are these really a lot more difficult than people assume?? I feel like they’re taught way too early, and that a lot of the complications that arise from them are partly due to the necessary approaches to communicate the in’s and out’s of fractional operations at such an early age.

Aaaanyway let’s begin…

One of the most likely things to completely throw students (and teacher explanations) when it comes to fractions is this: they have multiple meanings in a variety of contexts.

Students are usually first introduced to them as part-to-whole comparisons. Metaphors such as slices of pizza are often used simply because fractions are a fairly abstract concept to get your head around, and when you’re six or seven years old, or any other age for that matter, abstract leads to metaphor.

Later on fractions become divisional sums. No longer is there two slices “out of” three  making up a whole pizza. Instead, using exactly the same notation we’re now taking the integer 2 and dividing it by 3 to get 0.6 recurring.

*Hint, show students that the divide symbol is a secret fraction. The dots are numerator and denominator. This isn’t “a thing”, the symbol was originally used for subtraction, but it’s a nice spooky coincidence.

Then finally just to really kill them off, we change our minds again and they become part of the ratio party:

rat

This post will be focused primarily on number operations using fractions.

Too often students are shown a simplified way to add / subtract fractions in schools without ever exploring why it works, where it comes from, or what is actually happening mathematically.

To put this into context, imagine you had never added two fractions together. I tell you that you simply turn the numbers upside down whilst patting your head. You look at me like I’m mad, but do it obligingly. When you get your (weird) answer, I jump for joy and tell you WELL DONE, you’re AMAZING!!! Then give you twenty more questions. Each time I see those lovely sixes becoming nines, and fives becoming undefined symbols I pat you on the head and weep with joy. The method is eventually embedded. The student has no idea what the hell is going on, but they get ‘happy happy joy joy’ response when they do it, so they keep doing it.

That’s essentially what’s going on for a lot of students.

So let’s look at what’s actually being done mathematically.

First, consider why we can’t work with two fractions with different denominators without altering them:

Here are a few representations of 1 as different fractions:

fb1

Otherwise known as fraction strips / bars. Easy to visualise, easy to see that two halves make 1, eight eighths make 1 etc. I think it’s important to use the word ‘one’ instead of ‘a whole’. We never say ‘a whole’ anywhere else in mathematics, so why bother here? Just adding to the confusion (pun intended).

What’s also nice about the above picture, (taken from this interactive resource) is that you can visualise the relationships between sizes of fractions with different denominators.

fs2

So we see that 1/4 + 1/4 is equal to 2/4, which nicely lines up exactly with 1/2, because, well, it’s equal to 1/2. But notice that the 1/3’s don’t line up quite so nicely. At this point I should mention that I think it’s *really* important to talk about equivalent fractions in some depth. Metaphors are great here. “would you be any more full of chocolate if I cut my bar into six equal pieces and gave you three, or if I cut it into two pieces and gave you one piece?” or “which would you prefer to receive? Half of all of my money, or just three sixths of it?” etc. I fear a lot of students don’t really see that equivalent fractions are ‘worth’ the same amount. </digression>

So if we take 1/2 + 1/3 it’s not quite so straight forward:

fs3

Nothing lines up nicely. The answer cannot be in terms of halves, nor can it be in terms of thirds.

fs4

Lucky for us though, the handy sixths line up nicely both with halves and thirds. A half is equal to 3/6, and a third is equal to 2/6, so now the sum is achievable without switching to decimals.

But the point is that before we converted to a different denominator (specifically, one whose denominator is a common multiple of both our originals), this sum was going nowhere.

If you object to visual representations (you MONSTER!), then you can show students the logic behind the need to convert denominators like this:

1/2 + 1/3 = 1÷2 + 1÷3

The rules that dictate the order of operations do not allow us to do the addition first, so we cannot ‘add the tops’ or any such like.

However:

3/6 + 2/6 = 3÷6 + 2÷6 = (3+2)÷6

Ooh, see now we’ve trumped those division signs with brackets. In your face, divisor.  

I die a little every time I hear a teacher say “because it just does”.

Subtraction is no different to addition, so let’s move onto multiplication.

“Just times the tops and times the bottoms”

Job done.

Aaah you know me better than that by now. Although even the trusty BBC Bitesize website tells you to ‘just’ do that. (Fix it in the new version @tessmaths !)

How often have you seen teachers jump straight in with this:

1/3 x 1/4

When it would be far more intuitive and understandable to a child if you started with these:
frf
frrr

and showed solutions in this way:

frr

These are taken from this webpage, which I like a lot. I rarely see teachers even have a conversation about what 1/3 x 1/4 means. Let’s put the record straight. It means a third of a quarter, or a quarter of a third  (multiplication is commutative don’t you know!). Without having that discussion, students are less likely to have any intuition about their answer ‘looking’ right.

This Khan Academy video does a pretty good job of showing all of what I just said:

Before we move onto division, let’s just look at one more case:

Ever wondered why this works?

simplify-before-multiply-example1

Well, that’s linked to commutative properties of multiplication again.

fcc

Last one… division.

“Just flip the second fraction then multiply”

Just get out.

Here’s the fact you need to know : any number multiplied by its reciprocal is equal to one.

Regardez:

1/3 x 3 = 1

1/3 x 3/1 = 1

7 x 1/7 = 1

7/1 x 1/7 = 1

etc etc

So… here we go.

recip

Thank you, and good night *tosses microphone into crowd, leaves*.

Complements #4 : Prime Numbers

silly_prime_joke

Prime numbers get a really rough deal in schools. They’re often brushed over or bunched together with triangle and square numbers for a single hour lesson if they’re lucky.

“A prime number is divisible by itself and one, such as? … Great, that’s that one sorted”

Sigh.

Prime numbers deserve more attention. Below is a nice version (ish) of the Fundamental Theorem of Arithmetic, proven in 300BC by Euclid the Rock God of Maths (unofficial title). Not only that, but it’s also a kind of version of the Sieve of Eratosthenes. The theorem states that every whole number greater than 1 is either a prime or a product of primes. In other words each natural number is built from a unique set of prime numbers (You can of course, derive a number’s unique combination of prime factors by using a factor tree blah blah you knew that bit.). Whereas the Sieve of Eratosthenes is a (fairly laborious) method of finding primes, by finding all the multiples of a prime (known as composite numbers), then moving along to the next number that wasn’t marked by any prime before it. It was quite handy back in the day to find low prime numbers.

pr

Sieve_of_Eratosthenes_animation

There are other sieves too. The sieve of Atkin who later went on to create a crazy diet (err, wrong Atkin), the sieve of Sundaram and Euler’s sieve, which is basically the same as the Sieve of Eratosthenes. All of these sieves suffer the same stumbling setback – they take ages to find primes much larger than a few hundred. And primes become much more sporadic the further up the number line you dare to tread.

Anyhoo, the fact that prime numbers are essentially the basis of all other numbers makes them rather special. They are considered the ‘atoms’ of mathematics. If we knew them all we could make a kind of Periodic Table for maths. Sadly we don’t. But that’s also why primes are so fun.

Their mystery also makes them appealing. Is there a prime pattern?

If you fill out a 10 x 10 grid with prime numbers, you’d be forgiven for thinking there’s nothing spectacular to behold:

aoKCC

(meh)

but if you change that grid to 6 across …

pat

Oooh… patterny. Everything (bar 2 primes) is now in two distinct columns. That’s because primes can’t be multiples of 3, or even numbers, so that leaves numbers one above and one below multiples of 6. Of course not ALL numbers one above or one below 6 are primes otherwise we’d have this whole prime thing pretty nailed by now.

But there IS a pattern. And it’s so close… yet so far.

Throughout history people have cracked pieces of the puzzle. One of the most famous examples is by Marin Mersenne in the 17th Century. He claimed to have found a large quantity of prime numbers using the formula 049087cbf961a9d755e7fa8c067c49bb although a lot of them were later proven to not be prime numbers. However, any number that fits that formula and *isn’t* prime, still gets the fancy title of ‘Mersenne Number’.  The ones that *are* prime numbers get the fancier title of ‘Mersenne Prime‘. Despite the minor errors along the way, a Mersenne Number is far more likely to be prime than any other randomly selected number.

There are also further mysteries surrounding ‘Twin Primes‘. These little buggers are pairs of prime numbers that have a difference of 2. Such as 37 and 39, 3 and 5,
3,756,801,695,685 x 2666689 + 1 and 3,756,801,695,685 x 2666689 – 1…

Are there infinite twin primes? Nobody knows. In fact, the gap between primes excites mathematicians a lot too (they’re very excitable people). The Riemann Hypothesis generalises the distribution of prime numbers, and is possibly the most coveted conundrum in mathematics. Solve it, and have your name written in gold in the world of mathematics forever. When the British mathematician G. H. Hardy went on a dangerous sea voyage in the 1920’s, he sent a note claiming to have proven the Riemann Hypothesis, apparently so that God would not allow him to die.

Surely God would smite him for being a big fat liar?? Anyway…

And so finally we come to the Ulam Spiral. The Ulam spiral is a visual representation of integers, like this:

200px-Ulam_spiral_howto_all_numbers.svg

…which when you look only at the primes, reveals a distinct pattern. Explained nicely (again) by those clever folk over at numberphile in the video clip below (watch full screen):

Are you not fascinated by these bizarre creatures yet? I’m not talking about Dr James Grime, I’m talking about prime numbers.

Are prime numbers useful in society? Sure, Prime Factorisation is the basis of some common methods of cryptology. Essentially, use very large numbers that are the product of two prime numbers (known as a semiprime) as ‘locks’ and their prime factors as ‘keys’.

Something that often stumps people is why ‘1’ isn’t a prime number. Well, it used to be but it was considered so useless (prime factorisation can become infinitely long for example) and required so many special case exclusions, that mathematicians finally succumbed and changed their minds about it.

I’ll leave you with this: There is currently a $250,000 prize for the first person to discover the first billion digit prime. Good luck with that. It at least gives you an indication of the excitement that undiscovered primes cause in mathematicians.

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