This is a typical question from a Dulwich College 11+ Maths paper, and it asks you to draw a reflection of the triangle in the mirror line shown on the chart.
Dulwich papers tend to be a bit tricky, and this is not the easiest version of this kind of reflective symmetry question.
For a start, the mirror line is drawn at 45 degrees rather than being horizontal or vertical, and it doesn’t help that the diagram is a bit ‘squashed’, which means the mirror line is actually at around 40 degrees rather than 45!
So how should you do it?
The first thing to do is to imagine that you were looking at yourself in the mirror from, say, 30cm away.
Your reflection will appear ‘in’ the mirror, but it won’t be on the surface of the mirror, will it?
It’ll actually seem to be 30cm ‘behind’ the mirror – which is exactly the same distance as you are in front of it.
That’s important, and you’ll have to use that fact when you do the question.
The basic steps are these:
Plot the ‘vertices’ (or corners) of the reflected shape one by one by drawing a small cross in pencil.
Join them up using a ruler and pencil.
In order to plot each corner, you need to imagine that the corner is your face and that the mirror line is the mirror.
To see your reflection, you have to be standing right in front of the mirror – looking at an angle of 90 degrees to the mirror – so to ‘see’ the reflection of a corner, you have to do the same, looking at an angle of 90 degrees to the mirror line.
The distance from your face to the mirror is the same as the distance to the spot ‘behind’ the mirror where you see your reflection.
In the same way, the distance from the corner to the mirror line is the same as the distance to the spot ‘behind’ the mirror line where the reflected point should go.
If you use the diagram at the top of this article to help you, you should be able to see that the top of the triangle is one-and-a-half diagonal squares away from the mirror line.
That means you need to go another one-and-a-half diagonal squares the other side of the mirror line (continuing in the same direction) in order to plot the reflected point.
Now repeat this for the other corners of the triangle, which are four-and-a-half and three diagonal squares away from the mirror line.
Once you’ve done that, you can join up all three points using a ruler and pencil to make the reflected triangle.
Once you get the hang of it, you may not even need to plot all the corners: if it’s a simple shape like a square or a rectangle, then you might be able to draw it from scratch.
Just make sure you label the shape if the question asks you to.
One of the things that children taking Common Entrance exams at either 11+ or 13+ find most difficult to explain is humour. Here’s a quick guide to various different types with explanations, examples and a short quiz at the end.
Slapstick Comedy or Farce
This is a type of physical comedy that relies on the fact that we find it funny when other people hurt themselves. It’s called ‘Schadenfreude’ in German, and it really shouldn’t be funny…but it is!
Example: A man slips on a banana skin and falls over.
Deadpan or Dry Humour
This is any joke that’s told with a very matter-of-fact tone.
Example: “It can hardly be a coincidence that no language on earth has ever produced the expression ‘As pretty as an airport’.” The Long Dark Tea-time of the Soul, by Douglas Adams
Self-deprecation
This means putting oneself down in a self-mocking way.
Example: “If a book about failures doesn’t sell, is it a success?” Jerry Seinfeld
Toilet and Bodily Humour
What we do in the toilet or in the bedroom has given rise to a LOT of jokes over the years…
Example: “It’s just a penis, right? Probably no worse for you than smoking.” When You Are Engulfed in Flames, by David Sedaris
Puns, Wit and Wordplay
These are jokes based on double meanings or a play on words.
Example: “If not actually disgruntled, he was far from being gruntled.” The Code of the Woosters, by P.G. Wodehouse
Epigrams
An epigram is just a saying, and some sayings can be very funny – whether deliberately or not!
Example: “Always go to other people’s funerals, otherwise they won’t come to yours.” Yogi Berra
Dark Humour
Dark humour is usually about death or the gloomier aspects of life.
Example: I come from Des Moines. Somebody had to.” The Lost Continent: Travels in Small-Town America, by Bill Bryson
Sarcasm and Irony
Sarcasm is saying exactly the opposite of what you mean, but irony is much richer and more popular because the meaning for the reader can be anything from the literal truth of the statement to its exact opposite. It’s up to you…
Example: “It is a truth universally acknowledged that a single man in possession of a good fortune must be in want of a wife.” Pride and Prejudice, by Jane Austen
Innuendo
Finding a rude double meaning in a word or phrase is called innuendo.
Example: “Headline?” he asked. “‘Swing Set Needs Home,'” I said. “‘Desperately Lonely Swing Set Needs Loving Home,'” he said. “‘Lonely, Vaguely Pedophilic Swing Set Seeks the Butts of Children,'” I said.” The Fault in Our Stars, by John Green
Tongue-in-cheek
This expression just means the writer or speaker is being insincere in an ironic and/or mocking way.
Example: “In the beginning, the Universe was created. This has made a lot of people very angry and been widely regarded as a bad move.” The Hitchhiker’s Guide to the Galaxy, by Douglas Adams
Exaggeration and Hyperbole
Exaggeration can lead to a powerful punchline in a joke because it relies on shocking the reader with something unexpected.
Example: “In our family, there was no clear line between religion and fly fishing.” A River Runs Through It, by Norman Maclean
Parody and Mockery
Pretending to write in a certain style or copying the format of a particular writer or type of text can be done humorously – although the implied criticism may be affectionate.
Example: “It is a truth universally acknowledged that a zombie in possession of brains must be in want of more brains.” Pride and Prejudice and Zombies, by Seth Grahame-Smith and Jane Austen
Satire
This is making fun of something usually in religion, politics or current affairs.
Example: “They say the world is flat and supported on the back of four elephants who themselves stand on the back of a giant turtle.” The Fifth Elephant, by Terry Pratchett
The Surreal
‘Surreal’ just means absurd, nightmarish or like a fantasy.
Example: “As Gregor Samsa awoke one morning from uneasy dreams he found himself transformed in his bed into a gigantic insect.” The Metamorphosis, by Franz Kafka
Character Humour
Like a lot of sit-coms this form of humour relies on the personality of the characters. Things are funny because they are so typical of a certain type of person – often a stereotype.
Example: “As a boy, I wanted to be a train.” Machine Man, by Max Barry
Observational
A lot of stand-up comedy is based on observational humour, which means simply picking up on the typical habits of people in the world around us. We laugh because we recognise the behaviour and often the reason for it.
Example: “It’s a funny thing about mothers and fathers. Even when their own child is the most disgusting little blister you could ever imagine, they still think that he or she is wonderful.” Matilda, by Roald Dahl
Insults
The shock value of an insult lends itself to humour.
Example: Two whales walk into a bar. The first whale says to the other, “WOOOOOO. WEEEEEEEEOOOOO. WEEEEEEEEEEEEOOOOOOOOO.” The second whale says, “Shut up Steve, you’re drunk.”
Awkward Situations
If a situation is particularly cringeworthy or awkward, then it will often generate nervous laughter.
Example: “I don’t know how other men feel about their wives walking out on them, but I helped mine pack.” Breaking Up, by Bill Manville
Blue or Off-colour Jokes
Using rude words or swear words has the shock value that can generate humour.
Example: “If this typewriter can’t do it, then f*** it, it can’t be done.” Still Life With Woodpecker, by Tom Robbins
Sample Questions
How would you explain the humour in these lines?
“An unhappy alternative is before you, Elizabeth. From this day, you must be a stranger to one of your parents. your mother will never see you again if you do not marry Mr Collins, and I will never see you again if you do.” Pride & Prejudice, by Jane Austen
“There’s a door,” he whispered. “Where does it go?” “It stays where it is, I think,” said Rincewind. Eric, by Terry Pratchett
“It’s not because I want to make out with her.” “Hold on.” He grabbed a pencil and scrawled excitedly at the paper as if he’d just made a mathematical breakthrough and then looked back up at me. “I just did some calculations, and I’ve been able to determine that you’re full of s**t.” Looking for Alaska, by John Green
“I came from a real tough neighborhood. Once a guy pulled a knife on me. I knew he wasn’t a professional: the knife had butter on it.” Rodney Dangerfield
“A word to the wise ain’t necessary. It’s the stupid ones who need advice.” Bill Cosby
“To win back my youth, Gerald, there is nothing I wouldn’t do – except take exercise, get up early or be a useful member of the community.” A Woman of No Importance, by Oscar Wilde
“Some men are born mediocre, some men achieve mediocrity, and some men have mediocrity thrust upon them. With Major Major, it had been all three. Even among men lacking all distinction, he inevitably stood out as a man lacking more distinction than all the rest, and people were always impressed by how unimpressive he was.” Catch-22, by Joseph Heller
“Build a man a fire, and he’ll be warm for a day. Set a man on fire, and he’ll be warm for the rest of his life.” Jingo, by Terry Pratchett
“There are moments, Jeeves, when one asks oneself, ‘Do trousers matter?'” “The mood will pass, sir.” The Code of the Woosters, by PG Wodehouse
“There was a boy called Eustace Clarence Scrubb, and he almost deserved it.” The Voyage of the Dawn Treader, by CS Lewis
“I write this sitting in the kitchen sink.” I Capture the Castle, by Dodie Smith
“You can lead a horticulture, but you can’t make her think.” Dorothy Parker
“For a moment, nothing happened. Then, after a second or so, nothing continued to happen.” The Hitchhiker’s Guide to the Galaxy, by Douglas Adams
“For the better part of my childhood, my professional aspirations were simple – I wanted to be an intergalactic princess.” Seven Up, by Janet Evanovich
“It wasn’t until I had become engaged to Miss Piano that I began avoiding her.” Into Your Tent I’ll Creep, by Peter De Vries
“To lose one parent, Mr. Worthing, may be regarded as a misfortune; to lose both looks like carelessness.” The Importance of Being Earnest, by Oscar Wilde
You can use short multiplication if you’re multiplying one number by another that’s in your times tables (up to 12). However, if you want to multiply by a higher number, you need to use long multiplication.
Write down the numbers one on top of the other with the smaller number on the bottom and a times sign on the left (just as you would normally), then draw three lines underneath to hold three rows of numbers.
Multiply the top number by the last digit of the bottom number as you would normally.
Write a zero at the end of the next answer line (to show that you’re multiplying by tens now rather than units).
Multiply the top number by the next digit of the bottom number, starting to the left of the zero you’ve just added.
Add the two answer lines together to get the final answer.
Notes:
Some people write the tens they’ve carried right at the top of the sum, but that can get very confusing with three lines of answers!
Don’t forget to add the zero to the second line of your answer. If it helps, you can try writing it down as soon as you set out the sum (and before you’ve even worked anything out).
At 11+ level, long multiplication will generally be a three-digit number multiplied by a two-digit number, but the method will work for any two numbers, so don’t worry. If you have to multiply two three-digit numbers, say, you’ll just have to add another line to your answer.
Sample questions:
Have a go at these questions. Make sure you show your working – just as you’d have to do in an exam.
The ‘W’ words are useful if you’re trying to understand or summarise a story, but who, whom, who’s and whose tend to cause problems. Here’s a quick guide to what they all mean and how they can be used.
Who v Whom
Who and whom are both relative pronouns, which mean they relate to the person you’ve just been talking about. Note that they don’t relate to animals or things, just people. The difference is just one letter, but it signals that one of them stands for the subject (in the nominative case if you’ve ever done Latin) while the other stands for the object (in the accusative).
The subject of a sentence is the noun or pronoun that controls the verb, in other words the person or thing that’s ‘doing the doing’.
The object of a sentence is the noun or pronoun that is suffering the action the verb, in other words the person or thing that’s having something done to it.
For example, in the following sentence, ‘the girl’ is the subject, and ‘the boy’ is the object:
The girl tapped the boy on the shoulder.
We could also use pronouns, in which case ‘she’ is the subject, and ‘him’ is the object.
She tapped him on the shoulder.
Note that we use ‘him’ rather than ‘he’ in this case. That tells us that the boy is the object and not the subject. It’s the same with ‘who’ and ‘whom’. In fact, it’s the same letter – the letter ‘m’ – that tells us that ‘him’ and ‘whom’ are both the objects of the sentence, and that might be a good way to remember the difference.
For example, in the following sentence, ‘the girl’ is still the subject, so we use ‘who’:
They saw the girl who had tapped the boy on the shoulder.
In the next sentence, the boy is still the object, so we use ‘whom’:
They saw the boy whom the girl had tapped on the shoulder.
Note that neither who nor whom needs a comma before it in these cases. That’s because we are defining which people we’re talking about. It’s a bit like ‘which’ and ‘that’: ‘which’ describes things and needs a comma, but ‘that’ defines things and doesn’t. If we already know who people are and simply want to describe them, then we do use a comma.
They saw Patricia Smith, who had tapped the boy on the shoulder.
They saw Paul Jones, whom the girl had tapped on the shoulder.
In these cases, we know who the children are – Patricia and Paul – so all we’re doing is describing something that has happened. There is only one Patricia Smith and one Paul Jones, so we don’t need to define them. That means we need to use a comma in both cases.
I hope that all makes sense. Here are a few practice questions. Just decide in each case whether you should use ‘who’ or ‘whom’.
They talked to Jim, who/whom lived in Stoke.
He played football with the boy who/whom had red hair.
She was friends with the girl who/whom played volleyball.
Who/whom do you think will win the egg and spoon race?
Who/whom did they put in prison?
Who’s v Whose
The words ‘who’s’ and ‘whose’ are homophones, which is another way of saying they sound the same but mean completely different things. ‘Who’s’ is short for ‘who is’ or ‘who has’ while ‘whose’ is a possessive pronoun that means ‘of whom the’ or ‘of which the’. For example, take these two sentences:
Who’s going to the cinema tonight?
He was a big man whose hands were larger than dinner plates.
The first means ‘Who is going to the cinema tonight?’ whereas the second means ‘He was a big man of whom the hands were larger than dinner plates’. The only reason we don’t say those things is that they’re a bit of a mouthful, so it’s easier to use ‘who’s’ or ‘whose’.
I hope that’s clear now. Here are a few practice questions. Just decide in each case whether you should use ‘who’s’ or ‘whose’.
Who’s/whose in charge of the tennis rackets?
Who’s/whose bag is this?
He speaks to the woman who’s/whose behind the counter.
Homophones are words that sound the same even though they’re spelt differently and mean different things. Getting them right can be tricky, but it’s worth it in the end.
The reason why homophones are important is not just to do with the general need to spell correctly. Many people think getting them wrong is a ‘worse’ mistake than simply mis-spelling a word because it means that you don’t really know what you’re doing. Anyone can make a spelling mistake, but using completely the wrong word somehow seems a lot worse. That may not sound fair, but that’s just how a lot of people think, so it’s worth learning the common homophones so you don’t get caught out.
The subjunctive in French is generally used in the present tense after expressions such as ‘il faut que’ and some verbs that also take the word ‘que’ after them. These are generally the ones that express feelings or doubts (eg vouloir and craindre), especially when two parts of a sentence have different subjects, eg ‘I want her to be happy’ becomes ‘Je veux qu’elle soit contente’. Verbs ending in -er or -re have one set of endings, but -ir verbs have another (shown here in red):
Verbs Ending in -er, eg Donner (to Give)
Je donne(I may give) Tu donnes(You may give – informal) Il/elle donne(He/she may give) Nous donnions(We may give) Vous donniez(You may give – formal and/or plural) Ils/elles donnent(They may give – masculine or masculine and feminine/feminine only)
Verbs Ending in -re, eg Vendre (to Sell)
Je vende(I may sell) Tu vendes(You may sell – informal) Il/elle vende(He/she may sell) Nous vendions(We may sell) Vous vendiez(You may sell – formal and/or plural) Ils/elles vendent(They may sell – masculine or masculine and feminine/feminine only)
Verbs Ending in -ir, eg Finir (to Finish)
Je finisse(I may finish) Tu finisses(You may finish – informal) Il/elle finisse(He/she may finish) Nous finissions(We may finish) Vous finissiez(You may finish – formal and/or plural) Ils/elles finissent(They may finish – masculine or masculine and feminine/feminine only)
This article explains circle theorems, including tangents, sectors, angles and proofs (with thanks to Revision Maths).
Isosceles Triangle
Two Radii and a chord make an isosceles triangle.
Perpendicular Chord Bisection
The perpendicular from the centre of a circle to a chord will always bisect the chord (split it into two equal lengths).
Angles Subtended on the Same Arc
Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
Angle in a Semi-Circle
Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.
Proof
We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.
We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.
But all of these angles together must add up to 180°, since they are the angles of the original big triangle.
Therefore x + y + x + y = 180, in other words 2(x + y) = 180. and so x + y = 90. But x + y is the size of the angle we wanted to find.
Tangents
A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).
A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.
Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.
Angle at the Centre
The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.
Proof
You might have to be able to prove this fact:
OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a Similarly, ∠OXB = b
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a Similarly, ∠BOX = 180 – 2b Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX = 360 – (180 – 2a) – (180 – 2b) = 2a + 2b = 2(a + b) = 2 ∠AXB
Alternate Segment Theorem
This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.
Proof
You may have to be able to prove the alternate segment theorem:
We use facts about related angles
A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90. The angle in a semi-circle is 90, so ∠BCA = 90. The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180 Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90 But OAC + x = 90, so ∠OAC + x = ∠OAC + y Hence x = y
Cyclic Quadrilaterals
A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.
Area of Sector and Arc Length
If the radius of the circle is r, Area of sector = πr2 × A/360 Arc length = 2πr × A/360
In other words, area of sector = area of circle × A/360 arc length = circumference of circle × A/360
Common entrance exams have a time limit. If they didn’t, they’d be a lot easier! If you want to save time and improve your story, one thing you can do is to prepare three ‘off-the-shelf’ characters that you can choose from.
You can work on them beforehand, improving them and memorising them as you go. By the time the exam comes around, it’ll be easy to dash off 8-10 lines about one of your favourite characters without having to spend any time inventing or perfecting them.
Here’s what you need to do.
The first thing to say is that you need your characters to be a little out of the ordinary. Most pupils writing stories tend to write about themselves. In other words, 10-year-old boys living in London tend to write stories about 10-year-old boys living in London!
Now, that’s all very well, and the story might still get a good mark, but what you want to try and do is stand out from the crowd. Why not write a story about an 18-year-old intern at a shark research institute in the Maldives?!
To decide which one you’d rather write about, you just have to ask yourself which one you’d rather read about. One thing you can do to make sure your characters are special is to give them all what I call a ‘speciality’ or USP (Unique Selling Proposition).
It might be a superpower such as X-ray vision or mind-reading, or it might be a special skill such as diving or surfing, or it might be a fascinating back-story such as being descended from the Russian royal family or William Shakespeare – whatever it is, it’s a great way to make your characters – and therefore your stories – just that little bit more interesting.
Secondly, you should also make sure all your characters are different. Try to cover all the bases so that you have one you can use for just about any story. That means having heroes that are male and female, old and young with different looks, personalities and nationalities.
For instance, Clara might be the 18-year-old intern at a shark research institute in the Maldives, Pedro might be the 35-year-old Mexican spy during the Texas Revolution of 1835-6 and Kurt might be the 60-year-old Swiss inventor who lives in a laboratory buried deep under the Matterhorn! Who knows? It’s entirely up to you.
Thirdly, creating an off-the-shelf character is a great way to force yourself to use ‘wow words’ and literary techniques such as metaphors and similes. You may have learned what a simile is, but it’s very easy to forget to use them in your stories, so why not describe one of your heroes as having ‘eyes as dark as a murderer’s soul’?
If you use the same characters with similar descriptions over and over again, it’ll become second nature to ‘show off’ your knowledge, and you can do the same with your vocabulary. Again, why say that someone is ‘big’ when you can say he is ‘athletic’, ‘brawny’ or ‘muscular’?
Fourthly, try to stick to what you know. If you’ve never even ridden on a horse, it’s going to be quite tough to write a story about a jockey!
Alternatively, if you’ve regularly been to a particular place on holiday or met someone you found especially interesting, then use what you know to create your characters and their backgrounds. It’s always easier to describe places if you’ve actually been there, and it’s easier to describe people if you know someone similar.
So what goes into creating off-the-shelf characters? The answer is that you have to try and paint a complete picture. It has to cover every major aspect of their lives – even if you can’t remember all the details when you come to write the story. I’d start by using the following categories:
Name
Age
Job or education
Looks
Home
Friends and family
Personality
USP (or speciality)
Names are sometimes hard to decide on, so you might want to leave this one to last, but you just need to make sure it’s appropriate to the sort of character you’re creating. It wouldn’t be very convincing to have a Japanese scientist called Emily!
Age is fairly easy to decide. Just make sure your three characters are different – and not too close to your own age!
Job or education goes a long way to pigeon-holing someone. You can tell a lot from what someone does for a living or what they are doing in school or at university. You can include as much or as little detail as you like, but the minimum is probably the name and location of the school or college and what your characters’ favourite subjects are. You never know when it might come in handy!
Looks includes hair, eye colour, build, skin colour and favourite clothes. The more you describe your heroes’ looks, the easier it’ll be for the reader to imagine them.
Home can again be as detailed as you like, but the more specific the better. It’s easier to imagine the captain of a nuclear submarine patrolling under the North Pole than someone simply ‘living in London’…
Friends and family are important to most people, and it’s no different for the heroes of your stories. We don’t need to know the names of all their aunts, uncles, cousins and grandparents, but we at least need to know who they live with and who their best friends are.
Personality covers many things, but it should show what your characters are ‘like’ and what their interests are. Again, you don’t have to go into enormous depth, but it’s good to introduce the reader to qualities that might be needed later on in the story, such as athleticism or an ability to sail a boat.
USP (or speciality) covers anything that makes a character worth reading about. One of the reasons Superman is so popular is his superpowers: his ability to fly, his X-ray vision and the fact that he’s invulnerable. His greatest weakness is also important: Kryptonite. It’s the same for your characters. What can they do that most people can’t? What qualities can they show off in your stories? What will make them people we admire, respect and even love?
If you wanted to make Superman one of your off-the-shelf characters, this is what your notes might look like:
Name: Superman (or Clark Kent, Kal-El, The Man of Steel, The Last Son of Krypton, The Man of Tomorrow)
Age: Early 20s (when he first appears)
Job or education: News reporter at The Daily Planet in Metropolis
Looks: Tall, with a muscular physique, dark-haired, blue eyes
Home: Krypton, then the Kents’ farm in Smallville, Kansas, then Metropolis (or a fictionalised New York), where he lives in a rented apartment
Friends and family: Jor-El and Lara (biological parents)/Jonathan and Martha Kent (adoptive parents), Lois Lane (colleague, best friend, girlfriend), Jimmy Olsen (colleague), Perry White (boss as editor of The Daily Planet)
USP (or speciality): Superpowers, including invulnerability, super strength, X-ray vision, super hearing, longevity, freezing breath, ability to fly (but vulnerable to Kryptonite!)
Once you’ve created the notes for your three characters, you can write a paragraph of 8-10 lines about each of them. This is your chance to create something that you can easily slot into any of your stories, so use the past tense and stick to what the characters are like, not what they’re doing. That will be different in each story, so you don’t want to tie yourself down.
Here’s an example using Superman again:
Clark Kent led a double life. He wasn’t happy about it, but he needed his secret identity so that no-one would find out who he really was. He might have been a mild-mannered reporter for The Daily Planet with a crush on his partner, Lois Lane, but he was also a crime-fighting superhero: he was Kal-El, Superman and The Man of Steel all rolled into one!
His secret was that he’d actually been born on Krypton and sent to Earth as a baby to protect him from the destruction of his home planet. He’d been found by a childless couple living on a farm in Smallville, Kansas, and Jonathan and Martha Kent had adopted him as their own.
They didn’t know where he’d come from, but they’d provided him with a loving home as they watched him grow into a blue-eyed, dark-haired, athletic young man with a passion for ‘truth, justice and the American way’.
And they soon realised he was special when they saw him lifting a tractor with one hand…! He was faster than a speeding bullet, more powerful than a locomotive and able to leap tall buildings in a single bound! “Look! Up in the sky!” “It’s a bird!” “It’s a plane!” “It’s Superman!”
Try using your characters for stories you’re asked to write by your English teacher (or tutor, if you have one). The more often you use them, the better they’ll get as you change things you don’t like about them, bring in new ideas and polish the wording.
Next Steps
Try to create three off-the-shelf characters. Make them different ages, male and female and from different parts of the world. Start with the notes and then create a paragraph of 8-10 lines for each one in the past tense, ready to drop into any story…
Long division is on the syllabus for both 11+ and 13+ exams, so it’s important to know when and how to do it.
The basic idea is that it’s tricky to do short division when the number you’re dividing by (the ‘divisor’) is outside your times tables, ie more than 12. Using long division makes it easier by including a way of calculating the remainder using a proper subtraction sum.
It also makes it neater because you don’t have to try and squeeze two-digit remainders in between the digits underneath the answer line (the ‘dividend’).
So how does it work? Well, the only difference involves the remainder. In normal short division, you work it out in your head and put it above and to the left of the next digit in the dividend.
In long division, you work out the multiple of the divisor, write it down under the dividend and subtract one from the other to get the remainder. You then pull down the next digit of the dividend and put it on the end of the remainder, repeating as necessary.
To take the example at the top of the page, what is 522 divided by 18?
How many 18s in 5?
It doesn’t go
How many 18s in 52?
Two (write 2 on the answer line, and write 36 under the dividend with a line beneath it)
What’s 52 – 36?
16 (write it on the next line)
Pull down the next digit from the dividend (write it after the 16)
How many 18s in 162?
Nine (write it on the answer line, giving 29 as the answer, or ‘quotient’)
That’s the basic method, but here are a couple of tips to help you out.
The first is that you can make life easier for yourself by guessing round numbers. Working with numbers outside your times tables is tricky, so you can use ‘trial and error’ to come up with the right multiple of the divisor by trying ‘easy’ ones like 5 or 10. If it’s too big or too small, you can simply try again with a smaller or bigger number.
The second is that you can often divide the divisor by two to force it back into your times tables. Why divide by 18 when you can simply divide by nine and halve the result? You just have to be careful that you only deal in even multiples, eg 52 ÷ 18 is tricky, but the nearest even multiple of 9 is 4 (as 5 is an odd number and 6 x 9 = 54, which is too much), so the answer must be 4.
Writing a letter is not as easy as it might seem – especially if you have to do it during a Common Entrance exam! In this post, I’d like to explain the typical format of formal and casual letters and the decisions on wording that you’ll have to make.
First of all, here’s a quick list of the main parts of a letter that the examiner will be looking at:
Sender’s address
Date
Greeting
Text
Sign-off
Signature
Sender’s Address
It’s important to put the address of the sender (not the recipient!) at the top right of the letter (see above). The postman obviously doesn’t look inside the letter, so the address of the recipient needs to go on the envelope instead!
The only exception is if it’s a business letter intended to be posted in a window envelope. In that case, it needs to have the recipient’s address positioned above the sender’s address at just the right height so that it shows through the window when an A4 sheet is folded in three.
The address should really be aligned right, so you must remember to leave enough space for yourself when you start writing each line. Otherwise, it’ll look a bit of a mess…
Date
The date should be placed two or three lines below the sender’s address (again aligned right) in the traditional long format rather than just in numbers, eg 7th October 2018 rather than 7/10/18 (or 10/7/18 if you’re American!).
Greeting
Which greeting you use depends on the recipient. If you know the name of the person you’re writing to, then you should use ‘Dear’ rather than ‘To’, eg ‘Dear Mr and Mrs Dursley’. ‘To’ is fine for Christmas cards, but not for letters. You should also put a comma afterwards.
If you’re writing to a company or an organisation and you don’t know the name of the person, you have two options: you can either start the letter off with ‘Dear sir/madam’ or write ‘To whom it may concern’. This works better when it’s a reference for a job or a formal letter that may be circulated among several people.
Text
The text can obviously be whatever you like, but make sure you start it underneath the comma after the greeting. You should also use paragraphs if the letter is more than a few lines.
Sign-off
The sign-off is just the phrase you put at the end of the letter before your signature. If the letter is to a friend or relative, there aren’t really any rules. You can say anything from ‘Love’ to ‘Best regards’ or ‘Yours ever’.
Note that they all start with a capital letter and should be followed by a comma (or maybe a full-stop). If the letter is to someone else, the sign-off depends on the greeting: if you’ve used someone’s name in the greeting, you should use ‘Yours sincerely’, but it’s ‘Yours faithfully’ if you haven’t.
Signature
The signature is very important in letter-writing as it’s a simple way of ‘proving’ who you are, so you should develop one that you’re happy with. It should include your first name or your initial(s) plus your surname, eg Nick Dale or N Dale or NW Dale.
Your signature should be special, so it doesn’t need to be ‘neat’ or ‘clear’ like the rest of the letter. In fact, the prettier and the more stylish, the better!
And there you have it. This is only one way of writing a letter, and there are other ways of formatting the information, but these rules will at least give you the best chance of getting full marks in your Common Entrance exam!
Here’s a Maths trick a friend of mine saw on QI. Who knows? It might make addition and subtraction just a little bit more fun!
Find a book and write down the ninth word on p108, eg ‘becoming’.
Ask someone to write down a number with three different digits in it, eg 321.
Ask him to reverse it and take the smaller number away from the larger, eg 321 – 123 = 198.
Ask him to write down the answer and, again, reverse it, eg 891.
Ask him to add the two new numbers together, eg 198 + 891 = 1089.
Ask him to find the ninth word on p108 of the book, eg ‘becoming’.
Reveal the same word you wrote down earlier!
The trick is that the answer is always going to be 1089, whatever the number you first think of, so it should work every time – unless there’s a problem with the Maths!
Simultaneous equations help you work out two variables at once.
Why do we have simultaneous equations? Well, there are two ways of looking at it.
The first is that it solves a problem that seems insoluble: how do you work out two variables at once? For example, if x + y = 10, what are x and y? That’s an impossible question because x and y could literally be anything. If x was 2, then y would be 8, but if x was 100, then y would be -90, but if x was 0.5, then y would be 9.5 and so on.Simultaneous equations help us solve that problem by providing more data. Yes, we still can’t solve each equation individually, but having both of them allows us to solve for one variable and then the other.
The second way of looking at simultaneous equations is to imagine that they describe two lines that meet. The x and y values are obviously different as you move along both lines, but they are identical at the point where they meet, and that is the answer to the question.
The next question is obviously ‘How do we solve simultaneous equations?’ The answer is simple in theory: you just have to add both equations together to eliminate one of the variables, at which point you can work out the second one and then put it back into one of the original equations to work out the first variable.
However, it gets more and more complicated as the numbers get less and less ‘convenient’, so let’s take three examples to illustrate the three different techniques you need to know.
Simple Addition and Subtraction
The first step in solving simultaneous equations is to try and eliminate one of the variables by adding or subtracting them, but you can only do that if the number of the variable is the same in both. In theory, you could choose the first or the second term, but I find the one in the middle is the easiest, eg
4x + 2y = 10
16x – 2y = 10
Here, the number of the variables in the middle of the equations is the same, so adding them together will make them disappear:
20x = 20
It’s then simple to divide both sides by 20 to work out x:
x = 1
Once you have one variable, you can simply plug it back into one of the original equations to work out the other one, eg
4x + 2y = 10
4 x 1 + 2y = 10
4 + 2y = 10
2y = 6
y = 3
Answer: x = 1, y = 3
Multiplying One Equation
If the number of variables in the middle is not the same, but one is a factor of the other, try multiplying one equation by whatever number is needed to make the number of the variables match, eg
4x + 2y = 10
7x + y = 10
Multiplying the second equation by 2 means the number of the y’s is the same:
4x + 2y = 10
14x + 2y = 20
The rest of the procedure is exactly the same, only this time we have to subtract rather than add the equations to begin with:
10x = 10
x = 1
The next part is exactly the same as the first example as we simply plug in x to find y:
4x + 2y = 10
4 x 1 + 2y = 10
4 + 2y = 10
2y = 6
y = 3
Answer: x = 1, y = 3
Multiplying Both Equations
If the number of variables in the middle is not the same, but neither is a factor of the other, find the lowest common multiple and multiply the two equations by whatever numbers are needed to reach it, eg
4x + 2y = 10
x + 3y = 10
The lowest common multiple of 2 and 3 is 6, which means we need to multiply the first equation by 3:
12x + 6y = 30
…and the second by 2:
2x + 6y = 20
As the number of variables in the middle is now the same, we can carry on as before by subtracting one from the other in order to find x:
10x = 10
x = 1
Again, the final part of the technique is exactly the same as we plug x into the first of the original equations:
4x + 2y = 10
4 x 1 + 2y = 10
4 + 2y = 10
2y = 6
y = 3
Answer: x = 1, y = 3
Practice Questions
Job done! Now, here are a few practice questions to help you learn the rules. Find x and y in the following pairs of simultaneous equations:
I’ve talked to a few people who wanted to become private tutors, so I thought I’d write down a few tips for anyone who’s interested.
How Did I Start Out?
I started as a private tutor quite by accident. It was 2009, and I was finding it hard to get work as a freelance management consultant when I happened to read an article in the Telegraph called 10 Ways to Beat the Recession.
The author mentioned a few ways of earning some extra cash, including becoming an extra on film sets – which I was already doing – and working as a private tutor. I’d never done any proper teaching before, although I was a golf coach, and I’d coached skiing a few times in the Alps, but I thought I’d sign up with a couple of agencies and see what happened.
Within a week, I had two clients, and I’ve never looked back since!
What Qualifications do I Need?
The first and most important thing to say is that you don’t need any teaching qualifications! Yes, that’s right. You don’t need a PGCE, and you don’t need to have done any training as a teacher. As a private tutor, you are just that – private – so you don’t have to jump through all the Government hoops that a teacher in a state school would have to do.
Obviously, potential clients want the best person to teach their child, so you need to show some sort of academic record, but that can be as little as a degree in English – which is what I had when I started. Admittedly, I went to Oxford, which probably counts for a lot with Russian billionaires (!), but you don’t need to have an Oxbridge degree to become a tutor. Far from it.
However, what you probably will need is a criminal records check. This is just a piece of paper that certifies you haven’t been convicted of a criminal offence and was often known as a ‘CRB check’, although it’s now officially called an Enhanced Certificate from the Disclosure and Barring Service, or ‘DBS check’.
You can’t apply for an ‘enhanced certificate’ yourself, but your tuition agency can help you. In fact, they may require you to have one and even to renew it every year or two. It costs around £18 and can take up to three months to arrive, so it’s worth applying as early as possible.
Some agencies may charge up to £80 to make the application on your behalf, so be careful! You can find further information here.
What Subjects Can I Teach?
You can teach whatever you like! Agencies will just ask you which subjects you offer and at what level, so you have complete freedom to choose. I focus on English and Maths, which are the most popular subjects, but that’s mostly led by demand from clients. They are the main subjects at 11+ level, so that’s what most people are looking for help with.
What Age Children Can I Teach?
Again, the choice is yours. I’ve taught students from as young as five to as old as 75, but the peak demand is at 11+ level, when the children are around 10 years old. I make it a rule that I’ll only teach a subject to a level that I’ve reached myself, such as GCSE or A-level, but clients sometimes take you by surprise.
When I turned up to teach what I thought was going to be English to two boys, the nanny suddenly asked me to do Latin instead. When I said I hadn’t done any Latin since I was 15, she just said, “Oh, you’ll be fine…!”
What Preparation do I Need to do?
Research
One of the big attractions of tutoring for me is that the work is very enjoyable. I like teaching, and I like spending time with children, so it’s the perfect combination! The reason I stopped work as a management consultant was the constant stress, the persistent worry that I wasn’t up to the job, but teaching 10-year-olds never makes me feel like that.
Whether it’s English or Maths, I’m confident in my ability to teach and never worry about being asked an impossible question. However, that doesn’t mean you can walk into your first lesson without doing any preparation at all.
In my case, I wanted to teach English, so I needed to find out what kind of questions cropped up in 11+ and 13+ entrance exams and come up with a good method of answering them. Once I’d done that, I was ready.
Maths was a bit easier, but I still looked through a few papers to make sure there was no risk of being blind-sided by something I’d forgotten how to do or had never studied. Whatever the subject you’re offering, I suggest you do the same.
Past Papers
The other thing I needed to do was to find past papers to give to my pupils. That was a bit tricky in the early days until a kind parent gave me a collection of photocopied exams. After that, I carried a couple around with me to take to lessons, but it wasn’t a great solution, so I decided to create a website – this one.
Over time, I collected dozens of past papers and wrote various articles on how to do different kinds of question in Maths, English and French. Now, I don’t have to carry around anything with me or spend time dictating notes. I can simply ask my pupils to look it up online.
Setting up a website is pretty easy using WordPress or something similar, but you should feel free to use the resources on my past papers tab if you don’t want to go to the trouble yourself, and all my articles are available for free if you need them.
The main ones I use for English are about doing comprehensions and writing stories, but there are plenty more. The website proved unexpectedly popular, and I had over 28,000 visitors last year! The other advantage is that it generated enough business for me not to need agencies any more.
That means I can charge what I like, I don’t have to pay any commission, and I can have a direct relationship with all my clients without anybody acting as an intermediary – and often just getting in the way!
Business Cards
I know it sounds a bit old-fashioned, but having business cards is very useful. If you’re just starting out, nobody knows your name, so paying a few quid to market your services is one of the best investments you can make.
You never know when people will tell you they’re looking for a tutor, and it’s the easiest thing in the world to give them a business card. Even if you don’t have a website, it will at least tell them how to reach you, and you should get a lot more clients out of it.
How Can I Find Work?
Tuition agencies are the best place to start, but there are different kinds. Some are online and simply require you to fill out a form for them to check and vet, but others ask you to go through an interview, either over the phone or in person.
Either way, you need to put together a tailored CV that shows off your academic achievements and highlights any teaching experience you’ve had. This may not be very much at the beginning, but you simply need to show enough potential to get you through the door.
Once you’ve shown enough aptitude and commitment to get accepted by a few agencies, you’ll rapidly build up your experience on the job.
Here is a list of the tuition agencies I’ve been in touch with, together with contact details where available. I’m based in London, so there is obviously a geographical bias there, but some of the agencies such as Fleet Tutors offer national coverage, and you can always search online for others in your local area.
That’s obviously a long list, but, to give you an idea, I earned the most from Adrian Beckett (teacher training), Bespoke Tuition, Bonas MacFarlane, Harrison Allen, Keystone Tutors, Mentor & Sons, Personal Tutors and Shawcross Bligh.
Once you’ve been accepted by and started working for a few agencies, you’ll soon see the differences. Some offer higher rates, some the option to set your own rates, some provide a lot of work, some offer the best prospects of jobs abroad. It all depends what you’re looking for.
Where Will the Lessons Take Place?
When I first started tutoring, I had to cycle to all my clients. I put a limit of half an hour on my travel time, but it still took a lot of time and effort to get to my pupils. Fortunately, I’m now able to teach at my home, either in person or online using Skype and an electronic whiteboard, which means my effective hourly rate has gone up enormously.
Travel is still a little bit of a problem for most tutors, though, and I certainly couldn’t have reached my pupils without having a bicycle. I didn’t have a car, and public transport wasn’t really an option in most cases.
You just have to decide how far you’re prepared to go: the further it is, the more business you’ll get, but the longer it’ll take to get there and therefore the lower your effective hourly rate.
The other possibility, of course, is teaching abroad. I’ve been lucky enough to go on teaching assignments in Belarus, Greece, Hong Kong, Kenya, Russia, Switzerland and Turkey, and it’s a great way to see the world.
The clients can sometimes be a little bit difficult, and the children can sometimes behave like spoiled brats (!), but staying with a great client in a sunny getaway overseas can be a wonderful experience.
The only reason I don’t apply for more foreign postings is that I don’t want to let down my existing clients – going away for three weeks just before the 11+ exams in January would NOT go down well!
When Will the Lessons Take Place?
If you’re teaching children, lessons will usually be in the after-school slot between 1600 and 2000 or at weekends. That does limit the amount of hours you can teach, but it’s up to you how much you want to work.
I used to work seven days a week, but I eventually gave myself a day off and then another, so I now work Sundays to Thursdays with Friday and Saturday off.
During the holidays, you lose a lot of regular clients when they disappear to the Maldives or somewhere for six weeks (!), but others might ask for an intensive sequence of lessons to take advantage of the extra time available, and there’s obviously a greater chance of a foreign assignment.
All that means that the work is very seasonal, so you should expect your earnings to go up and down a bit and plan your finances accordingly.
What Should I do During the Lesson?
I generally teach from past papers, so I ask pupils to do a past paper for their homework and then mark it during the following lesson.
‘Marking’ means marking the questions, obviously, but it also means ‘filling in the gaps’ in the pupil’s knowledge. If he or she is obviously struggling with something, it’s worth spending a few minutes explaining the topic and asking a few practice questions.
I’ve written a few articles on common problem areas in English and Maths, such as commas and negative numbers, so I often go through one of those and ask the pupil’s parents to print it out and put it in a binder. After a few weeks, that collection of notes gradually turns into a ready-made revision guide for the exams.
If the parents want you to work on specific topics, that’s also possible. For example, one mother wanted to help her son with ratios, so she printed out dozens of past papers and circled the ratio questions for him to do. He soon got the knack!
I approach English in a slightly different way to begin with. There are two types of question in the 11+, comprehensions and creative writing, so I generally spend the first lesson teaching pupils how to do one of those. I go through my article on the subject online and then ask them to answer a practice question by following the procedure I’ve outlined.
They usually finish it off for their homework. After a few weeks of stories or comprehensions, I’ll switch to the other topic and do the same with that. I also ask pupils to write down any new words or words they get wrong in a vocabulary book because building vocabulary is very important for any type of English exam (and also for Verbal Reasoning).
I ask them to fold the pages over in the middle so that they can put the words on the left and the meanings on the right (if necessary). Every few weeks, I can then give them a spelling test. If they can spell the words correctly and tell me what they mean, they can tick them off in their vocab book.
Once they’ve ticked off a whole page of words, they can tick that off, too! I usually try to reinforce the learning of words by asking pupils to tell me a story using as many words as possible from their spelling test.
It can be a familiar fairy story or something they make up, but it just helps to move the words from the ‘passive’ memory to the ‘active memory’, meaning that they actually know how to use them themselves rather than just understand them when they see them on the page.
What Homework Should I Set?
Most children who have private lessons have pretty busy schedules, so I don’t want to overburden them. I generally set one exercise that takes around 30-45 minutes. That might be a Maths paper or an English comprehension or story, but it obviously depends on the subject and the level.
Just make sure that the student writes down what needs to be done – a lot of them forget! You should also make a note in your diary yourself, just so that you can check at the start of the next lesson if the work has been done.
What Feedback Should I Give the Parents?
I generally have a quick chat with the mother or father (or nanny) after the lesson to report on what we did during the lesson, what problems the child had and what homework I’ve set. This is also a good time to make any changes to the schedule, for instance if the family goes on holiday.
If that’s not possible, I’ll email the client with a ‘lesson report’. Some agencies such as Bonas MacFarlane make this a part of their timesheet system.
How Much Will I Get Paid?
When I first started, I had absolutely no idea how much I was worth, and I ended up charging only £10 an hour, which is not much more than I pay my cleaner! Fortunately, a horrified friend pointed out that it should be ‘at least’ £35 an hour, and I upped my rates immediately.
I now charge £60 an hour for private lessons, whether online or in person. Unfortunately, some agencies such as Fleet Tutors don’t allow you to set your own rates, so that’s one thing to bear in mind when deciding which agencies to work with.
However, they did provide me with quite a bit of work when I first started, so it’s swings and roundabouts. The pay scale often varies depending on the age of the student and the level taught, so you’ll probably earn more for teaching older students at GCSE level or above if the agency sets the prices.
If you have any private clients, you can obviously set whatever rate you like, depending on where you live, the age of your pupils, whether lessons are online or in person and so on. Personally, I only have one rate (although I used to charge an extra £5 for teaching two pupils at the same time), and I raise it by £5 every year to allow for inflation and extra demand.
Tutoring is more and more popular than ever these days, and I read somewhere that over half of pupils in London have private lessons over the course of their school careers, so don’t sell yourself short! You should be able to make around £25,000 a year, which is not bad going for a couple of hours’ work a day!
Foreign jobs are a little different, and there is a ‘standard’ rate of around £800 a week including expenses. That means your flights and accommodation are all covered, and you can even earn a bit more on the side if you decide to rent out your home on Airbnb while you’re away!
When it comes to day-to-day expenses such as taxis and food and drink, it’s important to negotiate that with the agency before accepting the job. It’s no good complaining about having to live in the client’s house and buy your own lunches when you’re in Moscow or Bratislava! It can be a dream job, but just make sure you look at it from every angle:
What subjects will I be teaching?
How many hours will I have to teach?
How many days off will I get per week?
Where will the lessons take place?
How do I get to and from my accommodation?
How long is the assignment? (I refuse anything more than three months.)
Where will I be staying? (NEVER at the client’s house!)
How old are the children?
Will I have any other responsibilities (eg ferrying the children to and from school)?
Do I need a visa?
What is the weekly rate?
What expenses are included (eg flights, accommodation, taxis, food, drink)?
How Do I Get Paid?
Most agencies ask for a timesheet and pay their tutors monthly via BACS payments directly into their bank accounts. That’s a bit annoying from a cash flow point of view, but there’s not much you can do about it – other than using a different agency.
When it comes to private clients, I generally ask for cash after the lesson, but it’s even more convenient if they can pay via standing order – as long as you can trust them! I once let a client rack up over £600 in fees because he tended to pay in big lump sums every few weeks, but then his business folded, and I had to use a Government website to try and chase him up.
Fortunately, his wife saw the email and paid my bill, but it took months to sort out. Normally, though, the worst that happens is that a client just doesn’t have the right change and promises to pay the following week, so you just need to keep track of who owes what.
As everyone knows, “Those who can, do; those who can’t, teach” – but that doesn’t stop me trying to do both!
Whatever kind of photographer you are and whatever kind of pictures you take, you always need to pay attention to composition. As an introductory guide (or a reminder), here are a few principles of composition to help you take better pictures. Just make sure you break all of them once in a while!
Rule of Thirds
The most common rule in photography is the rule of thirds. The aim of the game here is avoid taking pictures that are too symmetrical. For some reason, the human eye doesn’t like that, so it’s usually best to place the subject off-centre.
The rule of thirds is just one way to do that. Others include the golden ratio or the Fibonacci curve, and you can find them in Lightroom if you really want to, but the rule of thirds is the best and the simplest.
The idea is that you imagine that the viewfinder is divided up into thirds – both horizontally and vertically – and place the subject at the intersection of two of those invisible lines in order to give it more impact.
The lines also help you to place the horizon when you’re taking a landscape shot. If the horizon is in the middle of the frame, it looks a bit static. Instead, try to establish whether most of the interest is in the land or the sky.
If you want people to focus on the land, place the horizon on the lower imaginary line; if you want people to focus on the clouds in the sky, place it on the upper one. Just make sure that it’s straight!
‘The Decisive Moment’
Henri Cartier-Bresson was a French photographer considered a master of candid photography. He pioneered the genre of street photography. The Decisive Moment was the title of a book he wrote, and his idea was that timing is the secret of a good photograph.
This is obviously more important in certain types of photography (such as wildlife) than others (such as landscape), but it is still a useful guide to taking any kind of action shot.
Framing
Every photograph obviously has a frame, but have you ever tried using a ‘frame-within-a-frame’? Photographic frames come in all shapes and sizes, and so do the ones you find in real life. It might be the branches of a tree or a doorway or a window – the point is that it adds depth to a picture and focuses the viewer’s attention.
Negative Space
I don’t know why people call it ‘negative space’ rather than just ‘space’ (!), but the idea is that a picture with a single subject can look more balanced if there is empty space on the other side of the frame.
This is particularly useful for portraits if you want to stop them looking like ‘passport photos’! It’s also a good idea to allow space for a moving subject to move into. It just looks weird if a person appears to be ‘walking out of the frame’, so try to position the subject around a third of the way across in order to draw the eye into the picture rather than out of it.
Leading Lines
Leading lines are supposed to ‘lead’ the eye of the viewer into the frame – and ideally towards the main subject. They don’t have to be straight, but they tend to work best when they are. The obvious examples are railway tracks or a long, straight road stretching into the distance.
S-curves can do the same job as leading lines, but they also add dynamism and visual interest to a photograph, particularly if it’s a landscape. Again, it might be a road or a railway or even a winding river. All that matters is that the line is roughly in the shape of an S, meandering left and right into the distance.
Symmetry
The rule of thirds and others are meant to stop pictures looking too symmetrical, but sometimes symmetry suits the subject matter. If you have a reflection in the water or a human face, for example, you can’t really avoid it, so it’s sometimes best to make the most of it.
That might mean positioning the line where the water meets the line exactly in the centre of the frame or choosing a square aspect ratio for the picture to enhance the symmetry of a face.
Point of View
I’m a wildlife photographer, and the most important rule of wildlife photography is to get down to eye-level with the animals. It makes a huge difference to the composition and elevates a quick snap to an intimate portrait.
Taking pictures at eye level sometimes means getting wet or muddy – especially if you’re taking pictures of insects on the ground! – but it’s the best way to go. The same applies to portraits, which usually look best taken at eye-level or above.
If you get down any lower than that, you take the risk of ending up with a close-up of the model’s nostrils!
Motion Blur
A photograph is just a static image, so it’s sometimes difficult to convey a sense of motion. One way to do that is to use a slower shutter speed in order to create motion blur. Different subjects require different shutter speeds, depending on how fast they are moving, so you might need to experiment a little bit to find that sweet spot between too much sharpness and too little.
You could start with 1/4 of a second for a pedestrian walking along the street, but a Formula One car would disappear if you didn’t cut that down to 1/250 or slower. If you want to go the whole hog, you might try the ‘slow pan’.
Panning just means moving the camera from side-to-side to keep a moving subject in the same part of the frame. The ‘slow’ bit relates to the shutter speed. What you get with a ‘slow pan’ should be a recognisable subject with relatively sharp eyes but blurred limbs (or wings) and a blurred background.
I warn you that this is a tricky business – I once took 1,500 slow pan pictures of guillemots in the Arctic and only kept four of them! – but it’s worth it when it works…
Depth of Field
Another crucial element in wildlife and other kinds of photography is depth of field. To make sure the focus is on the subject and separate it from the background, you can use a larger aperture (such as f/4 or f/2.8).
That will blur anything that’s not in the same plane as the subject while keeping the focal point sharp. The eyes are always the most important part of a portrait – whether it’s of an animal or a person – and we will always see something as being ‘in focus’ as long as they look sharp.
Depth of field is just as important in landscapes, but what we generally want now is sharpness all the way through the image, so it’s better to start with a smaller aperture such as f/11 or f/16.
Odd Numbers
One of the funny things about the way people see the world is that we seem to like odd-numbered groups of objects more than even-numbered ones. It doesn’t really matter why, I guess, but it’s an important point to remember when planning something like a still-life shoot. Just make sure you have three or five tomatoes rather than two or four!
Fill the Frame
Everyone has a camera these days because everyone has a mobile phone, but one of the problems with using your mobile to take pictures is that it’s hard to ‘fill the frame’. It’s all very well taking a selfie when you’re only a few inches from the lens, but trying to zoom in on a distant object or animal is difficult when you only have a few megapixels to play with.
It’s important to remember here the difference between ‘optical zoom’ and ‘digital zoom’. The optical version is what you get naturally with a DSLR lens when you zoom in by changing the focal length; the digital version is when a phone or a bridge camera fools you into thinking you’re zooming in by focusing on a smaller and smaller portion of the sensor.
It’s great when you look through the viewfinder or look at the back of the camera, but the image quality is a lot poorer. Anyway, the point is that what you really want to do is to make the subject dominate the image by making it as large as possible.
If you’re taking a picture of a cheetah, you don’t want it to be a dot in the corner of the frame! You can always crop the image later using Lightroom or another editing program, but that means losing pixels, so the quality will suffer.
It’s always better to get it right in camera if you can. You just need to be careful not to chop off body parts in the wrong place when you’re taking a portrait. Generally, it’s fine to crop in on someone’s face so that the top of the model’s head is not shown, but it’s not a good idea to crop people’s bodies at the joints.
It just looks odd if the edge of the frame coincides with the ankles, knees, waist, elbows, wrists or neck.
Aspect Ratio
For some reason, taking a picture in landscape format just seems more ‘natural’ than turning the camera 90 degrees for a portrait, but it’s important to choose the ‘right’ aspect ratio for the image.
A photographer once advised me to make sure at least a third of my pictures were in portrait format, but the point is to look at the subject and decide what’s best. If there are a lot of horizontal lines, then landscape is fair enough, but if there are more vertical lines – such as tree trunks in a forest – then you should probably choose portrait instead.
If you really want to emphasise the length (or height) of a subject, why not try a panorama instead? Different cameras have different set-ups, but the average aspect ratio of a DSLR is 3:2, which doesn’t suit all subjects. I’ve set up a 3:1 template in Lightroom to use for images in which nothing much is happening at the bottom and top of the frame.
Foreground Interest
When we see a beautiful view, most people’s instant reaction is to take a picture, but what we end up with a lot of the time is an image without any focus. Placing an object in the foreground can lead the eye into the frame and give the image balance. A picture taken on the beach, for instance, might be improved by getting down low in front of a weird rock or piece of driftwood.
Balance
Speaking of balance, it can be a good idea to have the main subject on one side of the frame and a smaller subject on the other. Again, it’s just a matter of what looks most satisfying to the human eye.
Juxtaposition
Old and new, blue and orange, large and small – all these are contrasts that a photograph can pick up on and emphasise. This kind of juxtaposition can be made the point of an image. Think of an elephant beside a mouse – it’s not a picture of an elephant or a picture of a mouse, it’s a picture of the contrast between the two.
Patterns, Textures and Colours
Sometimes, you don’t need a traditional ‘subject’ to make an image visually interesting. There are plenty of patterns in Nature or in the man-made environment; the trick is to find them amongst all the surrounding clutter.
Whether it’s the bark of a tree or paint peeling on a wall, you can sometimes get a very effective abstract image out of it. Black and white images tend to emphasise patterns and shapes, as there is no colour to distract the eye, but colours can form patterns as well – it just depends on the subject and your personal preference.
Simplicity
It’s hard to produce a visually striking image if there is no focal point, or if there are too many competing centres of attention. By creating a simple image – in terms of colour and/or composition – you can remove the distractions and focus on what’s important.
Background
To increase the focus on the subject of an image, it’s a good idea to remove any distractions in the background. It’s obviously not a good idea to take a picture of someone with a telegraph pole sticking out of his head (!), but it’s easy to pay too much attention to the subject and not enough to the background unless you consciously check the viewfinder.
One useful way to reduce the chances of an embarrassing blunder is to reduce the depth of field by increasing the size of the aperture. The traditional way of taking portraits of animals or people, for instance, is to use a ‘fast’ lens, which means one that has a very wide maximum aperture, and shoot wide open.
That reduces the depth of field, thus blurring the background and adding to the impact of the main subject. If you have lights in the background, you can even get a nice effect called ‘bokeh’, which works well for something like a bauble with Christmas tree lights in the background.
Humour
Whatever you’re photographing, there are always odd moments of humour to be found. People and animals are usually the best sources, but it doesn’t really matter what the subject is. If there’s a visual joke to be made, why not have a go?
I laughed when I saw these penguins together on South Georgia. It looked as if the female was confused by the rock. Was it an egg she was supposed to hatch, or was it just a rock? She spent about five minutes looking at it and examining it before the male came up and said something like, “Come on, darling. It’s just a rock…”
Breaking the Rules
Having said all that, it’s important to break the rules once in a while. Rules tend to set expectations, so breaking them can make an image seem fresh and original. Why should the horizon be straight?
Why should we see the whole face rather than just half of it? Why should the sky start two-thirds of the way up the frame? If you can’t answer these questions, then why not take a risk? It’s a bit like being a painter: you have to be able to follow the rules before you can break them!
If you’d like to know more or want to book a photography lesson with me, then please get in touch.
This would’ve been a great shot. It could’ve been a great shot. It should’ve been a great shot. But it wasn’t. Why? Motion blur. If you look closely, you can see that the whole body is slightly out of focus, and that was simply because I didn’t think to change my shutter speed.
I was parked in a jeep in Botswana when a herd of impala came chasing across the road. They were galloping fast, but there were five or six of them, so I did have time to focus on each of them, one by one, as they crossed the road in turn.
Unfortunately, I was using my default camera settings that were designed to capture animals that were standing still. I was using an 80-400mm lens, so I had my camera on 1/320 and f/8 with auto ISO. That would normally have worked, but not for a jumping impala! What I really needed was a shutter speed of at least 1/1000 of a second. I just didn’t think…
In order to avoid moments like that, here are my answers to a few obvious questions:
What Equipment Do I Need?
Good question. It’s obviously too late to do anything once you’re on safari, so it pays to get your equipment sorted out beforehand. People often ask me what camera I use, and it reminds me of a story I heard about Ernest Hemingway. He went to a photography exhibition in New York and was so impressed he asked to meet the photographer.
Hemingway: These pictures are great. What camera do you use?
Photographer: Well, I use a Leica with a 50mm lens for most of my shots. I’m actually a big fan of your work, too, Mr Hemingway. I’ve read all your books. Can I just ask: what typewriter do you use…?
The point is obviously that a good camera doesn’t necessarily make a good picture, and it’s mildly insulting to photographers if you ask about their equipment without complimenting them on their talent!
However, all other things being equal, a good camera can make life a lot easier for wildlife photography. I’d suggest getting a full-frame DSLR with a zoom lens with a maximum focal length of at least 300mm, preferably 400mm or more.
The problem with a bridge or DX camera is that you won’t get the quality you’re after, as they don’t have large enough sensors. I started off with a bridge camera and thought the zoom was great – until I saw the Nikon DSLR one of the other guys had!
I had a severe case of ‘camera envy’, so I emailed a friend of mine who was a professional photographer to ask what he would get. He recommended either Nikon or Canon, but Canon made photocopiers, so that was out of the question!
Instead, I bought myself a Nikon D800 – complete with 36.3 megapixels! – and it’s served me well ever since. I now also have a D810, which is an upgraded version of the D800. Having two cameras means I don’t have to worry about changing lenses.
Instead, I carry them both cameras on a SpiderPro holster that looks a bit like an old Western cowboy’s gun belt. I can take them out and put them back with just one hand, and I can lock them in place if I’m going on a boat ride or clambering over rocks and don’t want to take any chances.
As for lenses, I mainly use an 80-400mm on the D800 and rent an 800mm prime on the D810. They’re both made by Nikon, and for a very good reason. I tried a Sigma 50-500mm and then a Tamron 150-600mm lens, but the images just weren’t sharp enough.
I now manually check the autofocus of all my lenses using Reikan Focal automatic lens calibration software. All you do is print out a ‘target’ and set up your camera on a tripod to take pictures of it from a certain distance away.
Once you load the software, it guides you through the set-up and takes a number of exposures automatically, just asking you to change the manual focus adjustment anywhere from -20 to +20. When the routine is finished, it gives you a PDF report showing the optimal adjustment value – and that’s what persuaded me to use only Nikon lenses.
I’d been on a trip to Svalbard and wasn’t happy with my shots of the polar bears, which were all just a little bit soft. One of the other guys on the trip told me he did a manual focus check, and that’s when I started doing it, too.
It was only when I bought my new 80-400mm lens that I realised the huge difference in sharpness: the Sigma and Tamron were down at around 1400 on the numeric scale, and the Nikon was way up at 2200! In short, check your lenses.
They’re mass-produced items, so there’s always bound to be some slight variation in focus, and you’d rather fix it yourself than have to use it as an excuse when you don’t get the sharpness you want.
I also make sure I always pack a polarising filter together with a lens cleaning kit (with sensor swabs and cleaning fluid), a beanbag (for resting the lens on the windowsill of a jeep) and my laptop (so that I can download and work on my pictures in the evening).
If I’m going to be near a waterfall, like Iguazu or Victoria Falls, I’ll also take my tripod and a ‘Big Stopper’ neutral density filter to give me the chance of taking creamy pictures of the water with a long shutter speed.
What Else Can I Do Before I Leave?
Getting the right equipment (and changing the time zone on your camera!) is one thing, but you can help yourself out by booking the right holiday in the right location at the right time. Check when the ‘long rains’ are if you’re going to Africa.
Check when the peak season is for wildlife viewing. Check if it’s possible to visit when there’s a full moon or – even better – a harvest moon. You can ask all these questions (and more) to make sure you get the very most out of your trip.
One useful sight for African expeditions is Safari Bookings, which allows you to search for packages by location, duration and price. I also like to travel light. I hate the whole airport experience, so I avoid having to check any bags in by having a roll-aboard camera bag and packing all my clothing into a jacket that has a pocket in the lining that goes all the way round.
It looks a bit funny when you walk through customs – and some people just couldn’t do it – but it saves me an awful lot of time and bother.
What Should I Wear?
When it comes to clothing, I tend to cover up to avoid the sun and the insects. That means I wear green cargo pants (with lots of pockets!), a brown, long-sleeved shirt, a floppy hat and trainers. If I’m going on a walking safari, I’ll put on my hiking boots, and I might bring a jacket for those cool early morning starts. There’s a reason why I don’t wear bright colours. They don’t exactly frighten the animals, but you’ll get some funny looks if you turn up in hot pants and a Day-Glo pink T-shirt…!
What Should I Take With me on the Game Drives?
If you’re a keen photographer, you won’t want to miss anything while you’re out taking pictures from the 4×4, but that doesn’t mean you need to take the entire contents of your camera bag! I would simply take your camera(s) and your longest lens(es) plus a lens cloth, a couple of spare batteries and a bottle of water.
A beanbag might come in handy on certain vehicles, but that’s about it. You can apply sunscreen and/or insect repellent before you leave. When it comes to clothing, I tend to cover up to avoid the sun and the insects.
That means I wear cargo pants (with lots of pockets!), a long-sleeved shirt, a floppy hat and trainers. Oh, and don’t even think about wearing a day-glo orange or pink T-shirt…!
What Camera Settings Should I Use?
There’s an old photographer’s joke:
Fan to photographer: I love your pictures. What settings did you use?
Photographer to fan: f/8 and be there!
The point is that ‘being there’ is more important than any camera settings, but that doesn’t mean they don’t matter at all – as shown by my shot of the leaping impala.
Exposure
The ‘Exposure Triangle’ consists of the aperture, shutter speed and ISO value, and these are the only three ways you can change the brightness of the image: either having a bigger hole, keeping it open for longer or increasing the sensitivity of the sensor.
A lot of beginners stick to automatic as they don’t trust themselves to use manual settings, but they lose a lot of control by doing that. The camera doesn’t know how fast the animal is travelling or how much of it you want to be in focus, so how can it possibly decide the best combination of shutter speed and aperture?
Why not experiment a little and decide for yourself the kind of image you’re going to take? Now, you still have to make sure you get the correct exposure somehow, and I’m not suggesting you use the exposure meter and manually change the settings for each shot!
What I do is start off with a good set of general-purpose settings and set the ISO to automatic. That way, I get exactly the shutter speed and aperture I want, but the camera makes sure it’s correctly exposed.
The general rule is that you need a shutter speed the inverse of your focal length, so, If I’m using my 80-400mm lens at the top end of the zoom range, that means around 1/400th of a second. (Bear in mind, though, that you have to take into account the speed of the animal as well as how steady you can hold the camera!)
I generally like to take ‘portraits’ of the animals, so I want to throw the background out of focus to emphasise the eyes. That means a wide aperture such as f/5.6, but I’ve started using f/8 because my lens tests tell me that both my lenses perform at their sharpest at f/8, and I want the maximum sharpness I can get.
The problem comes, obviously, when there’s not enough light to use your default settings, or the animals are moving too fast. That’s when you need to take charge and make a difficult decision: which is the most important, the shutter speed, the aperture or the ISO?
If it’s a fast-moving animal, the shutter speed obviously takes priority. If the light level is dropping, then you probably want to compromise and change both aperture and shutter speed by 1/3 of a stop (or more). Most stock agencies don’t want pictures taken at high ISO values (640+), so that’s something to bear in mind if you’re trying to sell your work.
Autofocus
Manual focus has its place in macro photography and in low light conditions, but wildlife photography generally demands that we use one of the two methods of autofocus: single point (AF-S on the Nikon) or continuous (AF-C).
I generally keep my D800 with the wide-angle lens on single point, as I’ll be using it to take landscape shots, but I keep my D810 with the long zoom lens on AF-C 3D, as I’ll be using it to take pictures of animals.
In fact, sharpness is so important for wildlife shots that I use what’s called ‘back-button focusing’, which means setting up the camera so that I can focus by pressing the AF-ON button on the back with my right thumb.
The AF-C 3D setting continuously focuses on one particular point on the animal that you select when you first press the AF button, and it magically follows that point even if the animal is moving. It’s not perfect, but what it does mean is that you don’t have to worry about losing focus when you half-press the shutter and then take a picture.
By separating the focusing from releasing the shutter, you get the best chance of getting that all-important sharpness in the animal’s eye.
White Balance
You can always change it in Lightroom later (or another image-processing software package), but I generally still try to update my white balance setting as the light changes. It saves time later, and it follows the general principle of trying to get everything right in camera. Messing around in Lightroom should always be a last resort.
Quality (RAW)
Shoot in RAW. There. Is. No. Alternative.
Other Settings
One of the confusing and frustrating thing about the DSLR is the number of settings there are and the fact that you can’t ‘reset’ everything in one go. It would be wonderful if there were one button that would do everything, but there isn’t.
There are mechanical as well as electronic settings, so it’s impossible to assign one button to change both. As it is, it’s worth having a mental checklist to go through before you go out on the game drive and even while you’re out there. The main settings to monitor are the following:
Mode (Manual, unless you’ve never picked up a camera before…)
Shutter speed (1/focal length, although Vibration Reduction means you might get away with up to four stops ‘slower’)
Aperture (f/5.6 or f/8, depending on where your lens’s sweet spot is)
ISO mode (I generally use ‘auto’)
Exposure compensation (0 – unless you’re photographing a very bright or dark scene)
Autofocus (AF-C 3D for wildlife)
White balance (Daylight – if it’s your typical African sunny day!)
Active D-lighting (Auto or off unless you’re taking a picture into the sun and want detail in the shot – it’s a kind of in-camera HDR to squeeze the histogram for images that would be too contrasty otherwise)
Lens lock (off, obviously – you don’t want to miss a shot because you can’t zoom in!)
What Should I Do While We’re Driving Around?
It’s all very well chatting to the guy next to you and having a laugh, but you’re there to take pictures, so you should follow these guidelines if you don’t want to be disappointed:
Always keep an eye out. I try to sit in the front seat so that I get a better view and can let the driver and the rest of the group know if I see something. If it’s not particularly interesting or too far away to get a good shot, I’ll just point or say, “Impala,” but I’m always ready to pat the driver on the shoulder or tell him to stop if there’s the prospect of a good sighting. One of the best sightings I had in Botswana came from the cook’s assistant sitting in the back of the jeep. As we were driving along, he suddenly said something in Setswana to our driver, who stopped and then backed up to see what was going on. After another incomprehensible conversation, I was shown a spotted eagle owl sitting on a branch not 10 yards away!
Don’t be shy. The guide will often be the one to spot an animal or a bird, and he or she will usually stop without having to be asked. However, if you spot something and want to take a picture, it’s important to stand up for yourself.Just tap the driver on the shoulder or ask him to stop. You always remember the shots you missed more than the shots you made, so be brave!
Be prepared. A lot of game drives involve looking at nothing in particular for hours on end, but that doesn’t mean you shouldn’t be ready at all times. You never know when someone will spot a white rhino or a leopard, so you need to make sure you have your camera(s) to hand with the right shutter speed, aperture and other settings dialled in.I tend to use 1/000 of a second at f/8 with auto ISO, but it depends on the light level. In the early mornings, you often have to make some awkward compromises. Just remember, though, that it’s better to get a sharp shot at a high ISO than a blurred one at ISO 100 and 1/60!
Keep the noise down. An animal or bird might seem quite far away, but they spook quite easily, so do make sure you don’t speak too loudly – or shout out something in your excitement! The other guests will thank you for it…
Keep still. You’re usually in a jeep with three or four other people, all wanting to take the best photographs they can, so you have to be sympathetic with your movements.If someone’s trying to take a picture, try to move as carefully and slowly as you can – or just wait for them to finish. You don’t want to rock the vehicle or jog an elbow and ruin the perfect shot!
Be polite. Tempers sometimes fray in the excitement of a game drive, when everyone wants to get the best possible view of the animals, but it’s worth keeping cool and being aware of those around you.
If you take too long over a shot or you accidentally get in the way of someone else, just apologise and move on. People go on safari to enjoy themselves and have a good time, not harbour festering grudges over the guy who thought it was all about him…!
Take care of your kit. I always cover my lenses with dust- and waterproof covers when I’m shooting. It might not seem necessary in some countries and in some climates, but you never know when you might have a sudden shower or get a cloud of dust in your face from the jeep in front.
I also take a lens cloth and/or a dust blower with me on game drives, and it’s worth checking your lens every now and then to make sure it’s not getting dusty. It’s hard to tell sometimes when you have a lens hood on, but it’s very easy for lenses to get dirty during the course of a long game drive.
I found Botswana particularly dusty, and there was a lot of dust in the air in Tadoba that gradually stuck to my camera and turned my lens cloth red whenever I used it!
What Makes a Good Photograph?
Dust, air and spume. That’s the Holy Trinity of wildlife photography, according to Paul Goldstein, who is a wildlife photographer and also a great speaker and raconteur.
I’ve been on two trips he’s led to Svalbard to see the polar bear and Tadoba in India to see the tiger, and I’ve seen several of his presentations.
The idea is that ‘dust’ is thrown up by the movement of the animals and gives you a sense of dynamism and energy, ‘air’ means that an animal is in the air and about to land – so we have a sense of anticipation and expectation – and ‘spume’ is the spray that is thrown up by movement in water.
That’s just Paul’s view, and there are obviously other aspects to the question. One thing that he also points out is the difference between a ‘record shot’ and a ‘photograph’. To him, a ‘record shot’ is just a snapshot, a picture that records exactly what’s in front of you, but a ‘photograph’ is something that obeys the rules of composition and has been consciously constructed by the photographer to provoke an emotional reaction.
There aren’t that many rules of composition in wildlife photography, but it’s worth bearing them in mind when you’re out shooting. Here are a few of the common ones:
Fill the frame. Robert Capa once said: “If your pictures aren’t good enough, you aren’t close enough.” People don’t want to have to search the image for the animal, so zoom in or ask your driver to get closer so that you can make it the centre of attention!
Use leading lines. Where available, they can lead the eye of the viewer into the image, for instance in a picture of an impala on the horizon crossing a road leading into the distance.
Use the Rule of Thirds. Human eyes don’t like things that are too symmetrical – unless you can manage a perfect reflection – so try to put the focal point of your shot off-centre. That adds dynamism and a different kind of balance.
Focus on the eyes. People don’t care if 99% of an animal is out of focus as long as the eyes are sharp.
Wait for ‘the decisive moment’. A guide in the States once compared my shots to those of another guy on the trip. He said that Stefan’s were always technically perfect, very sharp and with gorgeous, saturated colours, but mine were all about the moment.
I take that as a compliment. It means you have to wait for the right moment to take the shot. Don’t just keep clicking away like a Japanese tourist by Big Ben. Compose your shot and then wait for the animal to do something to make it more memorable. It could be a sneeze, a yawn – anything! – but it will mark your picture out as special.
Here are a couple of examples:
If a lion is walking across the road in front of your jeep, don’t take the shot until it steps forward with the leg that’s furthest away from you. That means it will have to turn its body and show more of its chest in the shot, which makes a better shot.
Try to capture pictures of birds in flight. Portraits are all very well, but an action shot is usually better. Given how quickly birds take off, the best way to capture them with their wings spread is just before they land. Find a bird on a branch and take a few ‘portrait’ shots, but don’t give up when it flies away.
A lot of birds have a ‘favourite’ branch, so it’s worth focusing on it and waiting for the bird to come back. If it does, take a series of shots in continuous mode, starting when the bird is just about to land.
That’s the best way to capture the prize, which is a picture of the bird with its wings spread, showing off all its plumage. Just make sure you have reasonable depth of field (at least f/8) and a high enough shutter speed (at least 1/1000).
Tell a story. The tagline to this website is ‘Every picture tells a story’, and that’s a goal we should all aspire to when taking pictures. What are we trying to say? What mood are we trying to create? What’s the emotion behind the shot?
It’s not always easy, but picking exactly the right composition can create humour, joy, sorrow, horror and any number of other powerful reactions – which is just what we want.
Break the rules – selectively! Obeying the rules will give you a nice, balanced image, but Paul for one hates ‘nice’, and I can see his point. Sometimes, the best way of creating a strongly emotional image is to break a rule or two. You have to do it sparingly – and consciously – but it sometimes gives you that much more of a chance of creating a genuinely arresting image.
One of his favourite techniques is the ‘slow pan’, which means following a moving animal or bird with a slow shutter speed and taking a number of shots as it goes past.
The idea is to create a sense of movement by blurring the background and the legs or wings of the animal or bird while keeping the body and especially the eyes sharp. It’s a technique that’s very difficult to master.
You have to do a lot of experimentation, and it helps to have a tripod! I once went on a boat trip in Svalbard and took 1,504 pictures of guillemots using the slow pan – but I only kept four of them! It sounds like a lot of effort, but it’s worth it in the end.
I live in an Art Deco mansion block in Putney, and every year the porters put a Christmas tree in the entrance hall. Last year, I took some pictures of some of the baubles, inspired by an email from one of the photographic magazines about how to capture bokeh lighting. This year, the tree and the baubles were different, so I decided to have another go.
The Location
Ormonde Court, Upper Richmond Road, London SW15 6TW, United Kingdom, around 2100 on 12 December 2014.
Manfrotto 190XProB tripod with 496RC2 universal joint head
Hähnel HRN 280 remote release.
I’ve just managed to remortgage my flat in Notting Hill, so I’ve been investing in a few photographic supplies. Ever since a German called Stefan took a magnificent shot of Old Faithful at night using flash, I’ve wanted a proper flashgun. Well, now I have one. I bought the Nikon SB-910 Speedlight a couple of weeks ago, and it arrived just in time for this shoot. I didn’t know whether I’d need it or not, but I was prepared to experiment.
The Settings
Manual ISO 100
f/5.6
1 second
105mm
Tungsten white balance
Single-point auto-focus
The Technique
In the last of these posts, I mentioned how I’d got used to taking a tripod with me in almost all circumstances, and last night was no exception. Last year, I was generally pleased with my shots of the baubles, but the ISO was far too high.
I was using my tripod, funnily enough, but to hold the bauble rather than my camera! This year, I decided I would definitely mount the camera on the tripod, but that left me with nothing to hold the baubles.
I thought about using a light stand from my flash kit, but I needed something horizontal rather than vertical so that I could hang the decorations from it. I then had the idea of using my golf clubs. I could stand the bag in the lobby and balance one of the clubs on top, held in place by the other clubs.
As it turned out, I’d forgotten that the bag would be at an angle of 45 degrees, so my original plan didn’t work, but I simply pulled my 4-iron half-way out and hung the first bauble from that. It was a silver reindeer, but the green wire loop wasn’t very long, and I wouldn’t have been able to get the shots I wanted without the golf club getting in the frame.
I needed a piece of string. I thought about going back to my flat, but leaving my golf clubs and my camera unattended in the entrance hall didn’t seem like a sensible idea! Fortunately, I was wearing trainers, so I just used one of the laces.
It took a few gos to get each bauble to point in the right direction and remain still – particularly as there was a stream of curious residents opening the front door on their way home from work! – but I managed in the end. Phew!
I took lots of pictures of the silver reindeer, a red bauble with a spiral pattern on it and the red star shown above, and I played around with the flash settings to try to make the background a bit darker.
Sadly my new flash was so powerful that I couldn’t manage that – even with -3.0EV of exposure compensation! There might’ve been a better way, but it was the first time I’ve ever used a flashgun, so I’m still a newbie.
The main problem I had in taking the shots was actually getting enough depth-of-field. The reindeer was fine, but the round baubles and even the star were proving a nightmare. If I focused on the front of the bauble, the metal cap and wire loop were out of focus, but, if I focused on those, the rest of the bauble was out of focus.
I’m an absolute stickler for sharpness in my images, so I wasn’t sure what to do. In the end, I stopped down a little bit and hoped that f/5.6 would be a small enough aperture to keep everything acceptably sharp. I tried ‘chimping’ (or checking the shots on the LCD screen) a few times, but it was tricky to tell.
My problem was a kind of Catch-22: the three variables controlling depth-of-field are normally the focal length, the aperture and the relative distances of the camera to the subject and the subject to the background.
I couldn’t change to a wide-angle lens, as I needed to limit the background to just the Christmas tree; I couldn’t change to a much smaller aperture without making the bokeh circles of the blurred Christmas lights in the background too small; and I couldn’t change the relative positions of the camera, bauble and tree without changing the composition completely.
Hmm…
As you can see from the shot above, the two arms on the right of the red star didn’t turn out completely sharp, but it was ‘good enough for Government work’. Shutterstock obviously didn’t accept it – they’re very hot on sharpness! – but I did win an award on Pixoto for the sixth best image uploaded to the Christmas category!
The Post-processing
I made three changes to this shot:
I had the camera on ‘Tungsten’ white balance, as I’d just read somewhere that I should use the amber filter on the flashgun when shooting indoors in order to avoid a clash of different light sources. However, it turned out that the shot looked a lot warmer with the ‘Flash’ white balance, and that was just the look I was after at Christmastime.
A lot of my images end up being quite dark, and I’m not sure whether it’s just because I’m lucky to spend a lot of time in very sunny places or whether there’s a problem with my camera! In this case, I actually had to push the exposure up by +2EV in Aperture to make it look like all the others. I have a feeling that’s because I changed from f/2.8 to f/5.6 to get more depth-of-field but forgot to lengthen the shutter speed to compensate. Silly me…
I was desperately trying to frame the shot perfectly so I wouldn’t have to crop, but the balance of the bauble with the ‘negative space’ on the right wasn’t quite right, so I cropped in slightly to position the star a third of the way into the frame.
I’m a photographer (among other things), and this is the first of a series of posts about my favourite photographs. I’ll tell you how I took them and break down the shot into the idea, the location, the equipment, the settings, the technique and any post-processing.
The Idea
When I took this shot, I was at a Battle of Hastings re-enactment at Battle Abbey in Sussex. I was there to take pictures of the battle scenes between enthusiasts dressed up as Normans and Saxons, and I had no idea there was going to be a falconry display until I bought my ticket and was given a flyer with the plan for the day.
The golden eagle is my favourite bird (isn’t it everyone’s?!), so I was very excited to be able to see one in action. The falconers from Raphael Historical Falconry put on a couple of displays with a variety of birds, including a gyrfalcon and a Harris hawk, but the golden eagle was the highlight.
Afterwards, I wandered over to their tent, and I was able to get within just a few feet of all the birds. The falconer was happy to chat with the spectators with a bird on his arm (so to speak!), and later he fed and watered the birds outside. That gave me the chance to set up my tripod and get a few good close-ups, and this was the best of the lot.
The Location
Battle Abbey, High Street, Hastings and Battle, East Sussex TN33 0AD, United Kingdom, around 1500 on 11 October 2014.
The Equipment
Nikon D800 DSLR camera
Sigma 50-500mm F4.5-6.3 APO DG OS HSM lens
Manfrotto 190XProB tripod with 496RC2 universal joint head
Hähnel HRN 280 remote release.
I was a bit worried about using my ‘Bigma’ to take this picture, as I hadn’t been very impressed with it on my trip to Spitsbergen to see the polar bears. Admittedly, the bears were usually a few hundred yards away, and no zoom lens is at its best when it’s at its longest focal length, but I was disappointed that my shots were so soft.
As a result, I did a manual focus check and discovered that the calculated auto-focus fine tune setting was a whopping -12! Armed with this new improvement to the sharpest tool in my box, I was ready for anything…
PS They call it the ‘Bigma’ as it’s made by Sigma, and it’s enormous!
The Settings
Auto ISO 110
f/9
1/250
500mm
Daylight white balance
Single-point auto-focus
I had the camera on Manual with ISO on Auto, which I thought was appropriate for a day when lots of things would be happening, and I’d be taking candid shots without much opportunity to sit down and check my settings. However, I should probably have set the ISO to its optimum value of 100 for this shot, as I had plenty of time.
The Technique
I’m generally a travel and wildlife photographer, but I normally don’t use a tripod as it gets in the way and doesn’t work too well in a Land-Rover moving at 40mph! However, I learnt a new perspective from a professional photographer called Mark Carwardine.
He happened to be on a cruise to Spitsbergen that I went on a few months ago, and he was always carrying around his tripod with the legs fully extended – even on the Zodiac inflatables that we used to land on the islands.
I thought to myself, If he can do it, so can I! After that, I’ve tried to use a tripod wherever possible. I love really sharp wildlife shots, and a 36.3-megapixel DSLR and a tripod make a winning combination.
Another important thing about wildlife shots is to get down to the level of the animal or bird you’re shooting. You can see from this shot that I’m right at eye-level with the eagle, and that gives the sense of power and intimacy I was looking for.
Finally, I’ve learnt from a couple of portrait shoots the value of the ‘catchlight’. This is the reflection of the light source that you see in the eye of your subject. It’s just as important with wildlife as with people, and I was lucky enough to get a break in the clouds that allowed the sun to provide the perfect catchlight. Lucky me!
The Post-processing
I changed from a PC to a Mac a few years ago, so I do all my post-processing in Aperture. I suppose I should upgrade to Lightroom or Adobe Camera Raw or Photoshop, but iPhoto was the default image-processing software on the Mac, and Aperture was the cheapest upgrade!
I only had two changes to make to this shot:
Even at 500mm, I still wasn’t quite close enough for the bird’s head to fill the frame, so I had to crop in later. I’ve found from experience that 6.3 megapixels is the minimum size that the major online photo libraries accept, so I never go below 6.4 MP (to avoid rounding errors), and that’s the new size of this file.
In the end, the automatic ISO setting was close enough to the optimum of 100, but the shot was slightly overexposed due to the dark colours of the eagle’s feathers and the grassy background, so I had to reduce the exposure by 0.5EV.
Teachers and tutors ask pupils to check their work, but how can you do that in Maths without doing the whole sum all over again? Well, you can’t! So how are you supposed to check your work?
What you have to understand first of all is that checking everything is right is very different from checking nothing is obviously wrong. To check everything is right means doing the whole paper twice, but you obviously don’t have time to do that.
Checking nothing is obviously wrong is much easier because it just means doing a ‘quick and dirty’ calculation in your head. It doesn’t guarantee that the answer is right, but it’s a good compromise.
I call it ‘sanity checking’, which means making sure your answers are not crazy! Unfortunately, there isn’t one method that works for every question – it depends on what type of question it is – but here are a few examples:
Algebra
If you have to ‘solve for 𝑥’ and it’s a difficult question, try putting your answer back into the original equation and seeing if one side equals the other, eg if you think 𝑥 = 5, then that works for 2𝑥 + 6 = 16, but not for 3𝑥 + 2 = 5. That would be crazy!
Multiplication
Every multiplication sum starts with multiplying the last digit of each number together, so try doing that when you’ve got your answer and checking if the last digit of the result is equal to the last digit of the answer, eg 176 x 467 is going to end in a 2 because 6 x 7 = 42, which also ends with a 2. Your answer couldn’t end in any other number. That would be crazy!
Rounding
If you have any kind of sum that involves adding, subtracting, multiplication or division, an easy way to check it is to round the numbers to one or two significant figures (eg to the nearest hundred) and work out the answer in your head. If it’s close enough, then your answer is not obviously wrong. If it’s nowhere near, then you’ll have to do it again, eg 1.7 x 3.4 is close to 2 x 3, so the answer might be 5.78, but it wouldn’t be 57.8. That would be crazy!
Units
Most answers in Maths tests need some kind of unit, such as kg, m, cm or ml. Sometimes, the units are provided, but sometimes they’re not. If they’re not, you just need to make sure that you use the right ones, eg if the scale of a map is 1:100,000, the distance represented by 9.8cm is 9.8km, not 9.8m. That would be crazy!
Maths is complicated, but a good first step on the road to understanding it is to get to know the most useful terms. There are lists in the front of the Bond books, but here’s my own contribution. I hope it helps!
Algebra: expressions using letters to represent unknown values, eg 2(x + 3) = 16.
Angles: there are three types of angle, depending on the number of degrees.
acute angles are between 0 and 90 degrees.
obtuse angles are between 90 and 180 degrees.
reflex angles are between 180 and 360 degrees.
Arc: part of the circumference of a circle.
Averages: there are three types of average, and they are all useful in different ways.
The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.
Chord: a straight line drawn between two points on the circumference of a circle.
Circumference: the distance all the way round the edge of a circle.
Congruent: triangles are congruent if they are the same shape and size, eg two right-angled triangles with sides of 3cm, 4cm and 5cm would be ‘congruent’, even if one is the mirror image of the other. You can prove that two triangles are congruent by using any of the following methods: SAS (Side-Angle-Side), SSS (Side-Side-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side) and RHS or HL (Right-angle-Hypotenuse-Side or Hypotenuse-Leg). If all three measurements of the angles and/or sides are equal, the triangles are congruent. You can only create a congruent copy of a triangle by translation, reflection or rotation. (Note: congruence is the same as similarity, except that the triangles have to be the same size.)
Cube: the result of multiplying any number by itself twice, eg 8 is the cube of 2, as 2 x 2 x 2 = 8.
Cube root: the number that has to be multiplied by itself twice to make another number, eg 2 is the cube root of 8, as 2 x 2 x 2 = 8.
Cuboid: a solid with a rectangle for each of the six sides, eg a shoe box.
Denominator: the number on the bottom of a fraction, eg 2 is the denominator of ½.
Diameter: the length of a line drawn across a circle passing through the centre.
Equation: any line of numbers and operators with an equals sign in the middle, showing that the two sides balance, eg 4x + 12 = 34.
Factor: a number that goes into another number evenly, eg 6 is a factor or 12.
Fibonacci series: a sequence of numbers created by adding the previous two numbers together to get the next one, eg 1, 1, 2, 3, 5, 8, 13…
Formula: a way of calculating the answer to a common problem using letters or words, eg the formula for distance is speed x time (or D = S x T).
Highest Common Factor (HCF): the highest number that goes into two other numbers evenly, eg the HCF of 12 and 18 is 6.
Improper fraction: a fraction that is greater than one (in other words, the numerator is greater than the denominator), eg 9/5.
Lowest Common Multiple (LCM) or Lowest/Least Common Denominator (or LCD): the lowest number that is divisible by two other numbers, eg the LCM of 6 and 8 is 24.
Multiple: a number that can be divided evenly by another number, eg 12 is a multiple of 6.
Numerator: the number on the top of a fraction, eg 3 is the numerator of ¾.
Order of operations: the sequence of doing basic mathematical sums when you have a mixture of, say, addition and multiplication. BIDMAS (or BODMAS) is a good way of remembering it, as it stands for:
Brackets
Indices/Order (in other words, squares, cubes and so on)
Division
Multiplication
Addition
Subtraction
Note that addition doesn’t come ‘before’ subtraction – these operations have to be done in the order in which they occur in the sum, and it makes a difference to the answer, eg 4 – 3 + 2 = 3 if you do the operations in order, which is correct, but you’d get the wrong answer of -1 if you did 3 + 2 first.
Operator: the sign telling you which mathematical operation to do. The most common ones are +, -, x and ÷.
Parallel: two lines are parallel if they will never meet, eg the rails on a railway line.
Perimeter: the distance all the way round the outside of a shape.
Perpendicular: at 90 degrees to each other.
Pi (or π): a constant used to work out the circumference and area of circles, often shown as 22/7 or 3.14 although it’s actually an ‘irrational’ number, which means it goes on for ever.
Prime factors: the lowest prime numbers that can be multiplied together to make a given number, eg the prime factors of 12 are 2² x 3.
Prime numbers: a number that can only be divided by itself and one, eg 2, 3, 5, 7, 11, 13…
Probability: the chance of something happening, calculated as the number of ways of getting what you want divided by the total number of possible outcomes, eg the chance of a coin toss being heads is ½ as there is one ‘heads’ side but two sides in total. To work out the probability of a sequence of events, you have to multiply the individual probabilities together, eg the chance of a coin toss being heads twice in a row is ½ x ½ = ¼
Product: the result of multiplying two numbers together, eg 35 is the product of 5 and 7.
Quadrilateral: a four-sided shape such as the following:
Kite: a quadrilateral with two pairs of equal sides next to each other (or ‘adjacent’ to each other).
Parallelogram: a quadrilateral with opposite sides parallel to each other.
Rectangle: a quadrilateral with two opposite pairs of equal sides and four right angles.
Rhombus: a quadrilateral with equal sides.
Square: a quadrilateral with equal sides and four right angles.
Trapezium (or Trapezoid): a quadrilateral with one pair of parallel sides. (Note: an isosceles trapezium is symmetrical.)
Radius: the distance from the centre of a circle to the circumference.
Range: the highest minus the lowest value in a list, eg the range of the numbers 1-10 is 9.
Regular: a shape is regular if all its sides and angles are equal, eg a 50p piece is a regular (-ish!) heptagon.
Right angle: an angle of 90 degrees.
Sector: a ‘slice’ of a circle in between two radii.
Segment: a part of a circle separated from the rest by a chord.
Shapes: the name of each shape depends on the number of sides. Here are the most common ones.
Triangles have three sides.
Quadrilaterals have four sides.
Pentagons have five sides.
Hexagons have six sides.
Heptagons have seven sides.
Octagons have eight sides.
Nonagons have nine sides.
Decagons have 10 sides.
Hendecagons have 11 sides.
Dodecagons have 12 sides.
Similar: triangles are similar if they are the same shape, but not necessarily the same size, eg a right-angled triangle with sides of 3cm, 4cm and 5cm is ‘similar’ to a right-angled triangle with sides of 6cm, 8cm and 10cm. (Note: similarity is the same as congruence, except that the triangles don’t have to be the same size.)
Square number: the result of multiplying any number by itself, eg 49 is a square number, as 7 x 7 = 49.
Square root: the number that has to be multiplied by itself to make another number, eg 6 is the square root of 36, as 6 x 6 = 36.
Sum: the result of adding two numbers together, eg 17 is the sum of 8 and 9.
Tangent: either a straight line that touches the circumference of a circle OR the length of the opposite side of a triangle divided by the length of the adjacent side
Transformations: there are three main kinds of transformation: reflection, rotation and translation.
With reflection, you need to state the formula of the mirror line, eg the shape has been reflected in the line y = 4.
With rotation, you need to state the number of degrees, the direction and the centre of rotation, eg the shape has been rotated 90 degrees clockwise around the point (4, 3).
With translation, you need to state the change in the x and y values, eg the shape has been translated four units up and three units to the right.
Triangles: there are four main types, each with different properties.
equilateral triangles have all three sides the same length and all three angles the same.
isosceles triangles have two sides the same length and two angles the same.
scalene triangles have three sides of different lengths with three different angles.
Nothing makes the heart of a reluctant mathematician sink like an algebra question.
Algebra is supposed to make life easier. By learning a formula or an equation, you can solve any similar type of problem whatever the numbers involved. However, an awful lot of students find it difficult, because letters just don’t seem to ‘mean’ as much as numbers. Here, we’ll try to make life a bit easier…
Gathering Terms
X’s and y’s look a bit meaningless, but that’s the point. They can stand for anything. The simplest form of question you’ll have to answer is one that involves gathering your terms. That just means counting how many variables or unknowns you have (like x and y). I like to think of them as pieces of fruit, so an expression like…
2x + 3y – x + y
…just means ‘take away one apple from two apples and add one banana to three more bananas’. That leaves you with one apple and four bananas, or x + 4y.
If it helps, you can arrange the expression with the first kind of variables (in alphabetical order) on the left and the second kind on the right like this:
2x – x + 3y + y
x + 4y
Just make sure you bring the operators with the variables that come after them so that you keep exactly the same operators, eg two plus signs and a minus sign in this case.
Here are a few practice questions:
3x + 4y – 2x + y
2m + 3n – m + 3n
p + 2q + 3p – 3q
2a – 4b + a + 4b
x + y – 2x + 2y
Multiplying out Brackets
This is one of the commonest types of question. All you need to do is write down the same expression without the brackets. To take a simple example:
2(x + 3)
In this case, all you need to do is multiply everything inside the brackets by the number outside, which is 2, but what do we do about the ‘+’ sign? We could just multiply 2 by x, write down ‘+’ and then multiply 2 by 3:
2x + 6
However, that gets us into trouble if we have to subtract one expression in brackets from another (see below for explanation) – so it’s better to think of the ‘+’ sign as belonging to the 3. In other words, you multiply 2 by x and then 2 by +3. Once you’ve done that, you just convert the ‘+’ sign back to an operator. It gives exactly the same result, but it will work ALL the time rather than just with simple sums!
Here are a few practice questions:
2(a + 5)
3(y + 2)
6(3 + b)
3(a – 3)
4(3 – p)
Solving for x
Another common type of question involves finding out what x stands for (or y or z or any other letter). The easiest way to look at this kind of equation is using fruit again. In the old days, scales in a grocery shop sometimes had a bowl on one side and a place to put weights on the other.
To weigh fruit, you just needed to make sure that the weights and the fruit balanced and then add up all the weights. The point is that every equation always has to balance – the very word ‘equation’ comes from ‘equal’ – so you have to make sure that anything you do to one side you also have to do to the other. Just remember the magic words: BOTH SIDES!
There are three main types of operation you need to do in the following order:
Multiplying out any brackets
Adding or subtracting from BOTH SIDES
Multiplying or dividing BOTH SIDES
Once you’ve multiplied out any brackets (see above), what you want to do is to simplify the equation by removing one expression at a time until you end up with something that says x = The Answer. It’s easier to start with adding and subtracting and then multiply or divide afterwards (followed by any square roots). To take the same example as before:
2(x + 3) = 8
Multiplying out the brackets gives us:
2x + 6 = 8
Subtracting 6 from BOTH SIDES gives us:
2x = 2
Dividing BOTH SIDES by 2 gives us the final answer:
x = 1
Simple!
Here are a few practice questions:
b + 5 = 9
3y = 9
6(4 + c) = 36
3(a – 2) = 24
4(3 – p) = -8
Multiplying Two Expressions in Brackets (‘FOIL’ Method)
When you have to multiply something in brackets by something else in brackets, you should use what’s called the ‘FOIL’ method. FOIL is an acronym that stands for:
First Outside Inside Last
This is simply a good way to remember the order in which to multiply the terms, so we start with the first terms in each bracket, then move on to the outside terms in the whole expression, then the terms in the middle and finally the last terms in each bracket.
Just make sure that you use the same trick we saw earlier, combining the operators with the numbers and letters before multiplying them together. For example:
(a + 1)(a + 2)
First we multiply the first terms in each bracket:
a x a
…then the outside terms:
a x +2
…then the inside terms:
+1 x a
…and finally the last terms in each bracket:
+1 x +2
Put it all together and simplify:
(a + 1)(a + 2)
= a² + 2a + a + 2
=a² + 3a + 2
Here are a few practice questions:
(a + 1)(b + 2)
(a – 1)(a + 2)
(b + 1)(a – 2)
(p – 1)(q + 2)
(y + 1)(y – 3)
Factorising Quadratics (‘Product and Sum’ Method)
This is just the opposite of multiplying two expressions in brackets. Normally, factorisation involves finding the Highest Common Factor (or HCF) and putting that outside a set of brackets containing the rest of the terms, but some expressions can’t be solved that way, eg a² + 3a + 2 (from the previous example).
There is no combination of numbers and/or letters that goes evenly into a², 3a and 2, so we have to factorise using two sets of brackets. To do this, we use the ‘product and sum’ method.
This simply means that we need to find a pair of numbers whose product equals the last number and whose sum equals the multiple of a. In this case, it’s 1 and 2 as +1 x +2 = +2 and +1 + +2 = +3.
The first term in each bracket is just going to be a as a x a = a². Hence, factorising a² + 3a + 2 gives (a + 1)(a + 2). You can check it by using the FOIL method (see above) to multiply out the brackets:
(a + 1)(a + 2)
= a² + 2a + a + 2
=a² + 3a + 2
Subtracting One Expression from Another*
Here’s the reason why we don’t just write down operators as we come across them. Here’s a simple expression we need to simplify:
20 – 4(x – 3) = 16
If we use the ‘wrong’ method, then we get the following answer:
20 – 4(x – 3) = 16
20 – 4x – 12 = 16
8 – 4x = 16
4x = -8
x = -2
Now, if we plug our answer for x back into the original equation, it doesn’t balance:
20 – 4(-2 – 3) = 16
20 – 4 x -5 = 16
20 – -20 = 16
40 = 16!!
That’s why we have to use the other method, treating the operator as a negative or positive sign to be added to the number before we multiply it by whatever’s outside the brackets:
20 – 4(x – 3) = 16
20 – 4x + 12 = 16
32 – 4x = 16
4x = 16
x = 4
That makes much more sense, as we can see:
20 – 4(4 – 3) = 16
20 – 4 x 1 = 16
20 – 4 = 16
16 = 16
Thank Goodness for that!
Here are a few practice questions:
30 – 3(p – 1) = 0
20 – 3(a – 3) = 5
12 – 4(x – 2) = 4
24 – 6(x – 3) = 6
0 – 6(x – 2) = -12
Other Tips to Remember
If you have just one variable, leave out the number 1, eg 1x is just written as x.
When you have to multiply a number by a letter, leave out the times sign, eg 2 x p is written as 2p.
The squared symbol only relates to the number or letter immediately before it, eg 3m² means 3 x m x m, NOT (3 x m) x (3 x m).
Apostrophes. The difference between feeling you’re nuts and feeling your nuts.
The apostrophe is tricky. It means different things at different times. This article is meant to clear up any confusion and help you use apostrophes, which might mean you get straight As in your exams – or should that be A’s?!
The main reason for using apostrophes is to show a contraction, which is a word made up of two other words shunted together – the apostrophe just stands for the missing letter(s), eg didn’t = did not, could’ve = could have and won’t = will not.
The second most common usage is in showing the possessive, in other words showing that something belongs to someone (or something). This is where it gets tricky, because where you put the apostrophe depends on how many things you’re talking about. If the noun is plural and ends with -s, you just need to put an apostrophe on the end of the word. In all other cases, you should put ‘s, eg two horses’ hooves, BUT a horse’s hooves or the children’s books or St James’s Palace.
The other occasion when you might find an apostrophe is in the plural of individual letters or numbers. Somehow, it just looks better, eg he got three A’s at O-level back in the 1980’s.
If you think you’ve mastered the rules, try taking this quiz! Alternatively, here are a few sample questions. Just choose the correct option.
He stole James’s/James’/Jameses book.
She marked the childrens/children’s/childrens’ homework.
He didnt/didn’t/did’nt mind at all.
They wont/wo’nt/won’t be back in time.
The two girls/girl’s/girls’ bags were next to each other.
You need to get three As/As’/A’s to get into Oxford.
I love the clothes we used to wear in the 1970s/1970’s/1970s’.
The problem with the English is that we’ve invaded (and been invaded by) so many countries that our language has ended up with a mish-mash of spelling rules.
English is among the easiest languages to learn but among the most difficult to master. One of the problems is spelling. We have so many loan words from so many different languages that we’ve been left with a huge number of spelling rules – and all of them have exceptions!
Contrast that with Spanish, for example, where what you see is generally what you get. The problem for students of English, then, is that it’s very difficult to find shortcuts to improve your spelling, and an awful lot of words just have to be learned off-by-heart. Considering that there are over a million words in English, that’s a big ask!
There are lots of lists of spelling rules on the web, but I thought I’d put down what I think are the most useful ones.
I before E except after C when the sound is /ee/. This is the most famous rule of English spelling, but there are still exceptions! Hence, we write achieve with -ie- in the middle but also ceiling, with -ei- in the middle, as the /ee/ soundcomes after the letter c. The most common exceptions are weird and seize.
If you want to know whether to double the consonant, ask yourself if the word is like dinner or diner. One of the most common problems in spelling is knowing when to double a consonant. A simple rule that helps with a lot of words is to ask yourself whether the word is more like dinner or diner. Diner has a long vowel sound before a consonant and then another vowel (ie vowel-consonant-vowel, or VCV). Words with this long vowel sound only need one consonant before the second vowel, eg shiner, fiver and whiner. However, dinner has a short first vowel and needs two consonants to ‘protect’ it (ie vowel-consonant-consonant-vowel, or VCCV). If the word is like dinner, you need to double the consonant, eg winner, bitter or glimmer. Just bear in mind that this rule doesn’t work with words that start with a prefix (or a group of letters added to the front of a word), so it’s disappoint and not dissapoint.
If the word has more than one syllable and has the stress on the first syllable, don’t double any final consonant. This rule sounds a bit complicated, but it’s still very useful – particularly if it helps you spot your teacher making a mistake! We generally double the final consonant when we add a suffix starting with a vowel, such as -ing, but this rule means we shouldn’t do that as long as a) the word has more than one syllable and b) the stress is on the first syllable, eg focusing and targeted, but progressing and regretting. The main exceptions to this are words ending in -l and -y, hence barrelling and disobeying.
When adding a suffix starting with a consonant, you don’t need to change the root word unless it ends in -y. This is among the easiest and most useful rules. There are loads of words ending in suffixes like -less, -ment or -ness, but spelling them should be easy as long as you know how to spell the root word, eg shoe becomes shoeless, contain becomes containment and green becomes greenness. However, words ending in -y need the y changing to an i, so happy becomes happiness.
When adding a suffix starting with a vowel to a word ending in a silent -e, the e must be dropped unless it softens a c or a g. An e at the end of a word is often called a ‘Magic E’, as it lengthens the vowel before the final consonant, eg fat becomes fate. However, that job is done by the vowel at the start of the suffix when it is added to the word, so it needs to be dropped, eg race becomes racing and code becomes coded. The main exceptions come when the word ends with a soft c or g, which need to be followed by an -e, an -i or a -y to sound like /j/ and /s/ rather than /g/ and /k/. If the suffix doesn’t begin with an e- or an i-, we still need the –e to make sure the word sounds right, eg managing is fine without the -e, as the i in -ing keeps the g soft, but manageable needs to keep the -e to avoid a hard /g/ sound that wouldn’t sound right.
The only word ending in -full is full! There are lots of words ending in what sounds like -full, but the only one that has two ls at the end is full. All the other words – and there are thankfully no exceptions! – end in -ful, eg skilful, beautiful and wonderful.
When is a verb not a verb? When it’s a part of speech.
English exams often ask questions about the ‘parts of speech’. This is just a fancy term for all the different kinds of words, but they’re worth knowing just in case. Just watch out for words such as ‘jump’, which can be more than one part of speech!
If you struggle to remember what they all mean, think about the words themselves. Sometimes, there’s a clue in the way they sound, eg adverbs describe verbs, pronoun sounds like noun, preposition contains the word position and a conjunction is the ‘junction’ between two sentences.
Noun
A noun is a word for a person, place or thing
abstract noun: a word to describe an idea, eg peace
common (or concrete) noun: a word for a thing or object, eg table
proper noun: the name of a person, place etc, eg Nick, London
collective noun: the name of a group of animals, eg herd or flock
Tip: Make up a phrase or a sentence with ‘the’ in front of the word. If it makes sense, it’s probably a noun, eg He looked at the ______.
Adjective
An adjective is a word that describes a noun or pronoun, eg green or young
Tip: Make up a phrase or a sentence putting the word between ‘the’ and a noun. If it makes sense, it’s probably an adjective, eg The ______ book lay on the table.
Verb
A verb is a doing word, eg jumped, was, pays
Tip: Make up a phrase or a sentence putting the word after a pronoun such as ‘he’. If it makes sense, it’s probably a verb, eg He ______ it or He ______ in the garden.
Pronoun
A pronoun is a word that stands in for a noun
personal pronoun: a word that shows a person or thing, eg he, she, them
possessive pronoun: a word that shows the owner of an object, eg his, their
relative pronoun: a word that ‘relates’ to the subject just mentioned, eg who, that, which
Tip: Make up a phrase or a sentence with a verb after the word (but without ‘the’ or ‘a’ in front of it). If it makes sense, it’s probably a pronoun, eg ______ looked at the wall.
Article
An article is a word that introduces a noun, ie a, an or the.
Strictly speaking, an article is just one kind of ‘determiner’, a word that introduces a noun:
Tip: Make up a phrase or a sentence with the word in front of a noun. If it makes sense, it’s probably an article, eg ______ book lay on the table.
Adverb
An adverb is a word that describes an adjective, adverb or verb, usually ending in -ly, eg really or quickly
Tip: Make up a phrase or a sentence with the word after a verb. If it makes sense, it’s probably an adverb, eg He ran ______ around the garden.
Preposition
A preposition is a word that shows the position in time or space, eg in, at or after
Tip: Make up a phrase or a sentence about placing something somewhere, putting the word before the location. If it makes sense, it’s probably a preposition, eg She put the book ______ the table.
Conjunction
A conjunction is a word that connects two sentences together (sometimes called a connective), eg and, but or because.
‘Coordinating conjunctions‘ are used to make a ‘compound’ sentence when the clauses are equally important, and the two ‘main clauses’ should always be separated by a comma, eg ‘The sun was warm, but it was cooler in the shade’. There is a useful way of remembering the coordinating conjunctions, which is to use ‘FANBOYS’. This consists of the first letter of ‘for’, ‘and’, ‘nor’, ‘but’, ‘or’, ‘yet’ and ‘so’.
‘Subordinating conjunctions‘ are used to make a ‘complex’ sentence when there is a main clause and a subordinate clause. (Subordinate just means less important.) If the sentence starts with a subordinating conjunction, the clauses need a comma between them, eg ‘Even though it was very hot, he wasn’t thirsty’. However, if the subordinate clause comes at the end, there is no need for a comma, eg ‘He wasn’t thirsty even though it was very hot’. There are lots of subordinating conjunctions, such as ‘after’, ‘although’ and ‘because’, but the easy way to remember it is to ask yourself if the conjunction is in FANBOYS. If it is, it’s a coordinating conjunction; if it’s not, it’s a subordinating conjunction. Alternatively, subordinating conjunctions are sometimes known as ‘WABBITS’ because some of the commonest ones start with those letters (when, where, while, after, although, before, because, if, though and since).
Tip: Make up a phrase or a sentence with two clauses joined by the word. If it makes sense, it’s probably a conjunction, eg He looked at the problem ______ decided to do something about it.
Interjection
An interjection is either an outburst like hey or a word people say when they’re playing for time, eg well or now.
Tip: Make up a phrase or a sentence that someone might say, putting the word at the start, followed by a comma. If it makes sense, it’s probably an interjection, eg ______, can we go to the mall?
Quiz
You can test yourself by reading any passage in English and going through it word by word, asking yourself what parts of speech they all are.
Why not start with this article? See how fast you can go. If you’re not sure, ask yourself the questions in each of the tips shown above, eg if you think it’s a noun, can you put it into a sentence with ‘the’ in front of it?
Here’s a quick quiz…
What are the nine parts of speech? (9 marks)
What do they all mean? (9 marks)
What are the four different kinds of noun (4 marks)
What are the three different kinds of pronoun (3 marks)
What are the two kinds of article? (2 marks)
What are the two kinds of conjunction? (2 marks)
What are the two words that help you remember the different kinds of conjunction? (2 marks)
Speech marks, inverted commas, quotation marks, quote marks, quotes, 66 and 99 – does any other punctuation mark have so many names or cause so much confusion…?!
Writing a story means striking a balance between what I call The Three Ds: Drama, Description and Dialogue. I’ve read quite a few stories from my pupils in which nobody talks to anyone – which is a bit odd! – but you need to know the rules of punctuation before you start.
Start a new paragraph whenever the speaker changes or someone stops talking.
Put speech marks before and after the actual words spoken, eg “Hello,“ he said, NOT “Hello, he said.”
Start the first spoken word with a capital letter, eg she said, “This needs a capital letter,” NOT she said, “this needs a capital letter.”
Put either a comma, question mark, exclamation mark or colon between the speech and the ‘he said/she said’, eg “Don’t forget the comma,” he said, NOT “Don’t forget the comma” he said.
Put punctuation that belongs to the speech inside the speech marks, eg “The exclamation mark belongs inside!“, NOT “The exclamation mark belongs inside”! (The only exception comes with inverted commas, which look the same but are used with quotations rather than speech.)
Put a full-stop after the ‘he said/she said’ if it comes in the middle of the speech and the first part is a full sentence; otherwise, just put a comma, eg “This is a full sentence,” she said. “This is, too.” BUT “This is not a full sentence,” she said, “and nor is this.”
Don’t start the ‘he said/she said’ with a capital letter, even if it comes after a question mark or exclamation mark, eg “Don’t use a capital letter!” he shouted, NOT “Don’t use a capital letter!” He shouted.
If a speech lasts more than one paragraph, put speech marks before each paragraph and after the last one but NOT after the ones before.
Finally, don’t put ‘he said/she said’ after every single line of dialogue in a long conversation if it’s obvious who is speaking.
Sample Questions
Format and put the correct punctuation and capital letters into the following lines of speech:
I say john what time is it she asked
hello she said my name is tara
what are you talking about he cried I never said that
hello he said whats your name Sarah she said Im Alan Nice to meet you you too
I hate chocolate she said I only really eat chocolate ice-cream
“If I’d known I’d have to go back to school, I’d never have become a teacher!”
The QTS numeracy and literacy tests are not very popular, but trainee teachers still have to pass them before they can start teaching in the state sector, so I thought I’d try and help out. There is always more than one way of doing a Maths question, but I hope I’ll demonstrate a few useful short cuts and describe when and how they should be used.
The point of short cuts is that, even though you may have to do more sums, they’ll be easier sums that can be done faster and more accurately. The numeracy test consists of two sections – mental Maths and interpreting charts – and I’m going to focus on the first of these.
Fractions to Percentages – Type 1
There are a number of typical types of questions in the numeracy test, and a lot of them involve multiplication – so knowing your times tables is an absolute must! One of the most common kinds of question involves converting fractions to percentages.
These are just two ways of showing the same thing, but to answer these questions you’ll need to try different approaches. First of all, have a look to see if the denominator (or the number on the bottom of the fraction) is a factor or a multiple of 100.
If it is, you can simply multiply or divide the numerator (the number on the top) and the denominator by whatever it takes to leave 100 on the bottom. Any fraction over 100 is just a percentage in disguise, so you just need to put the percentage sign after the numerator, eg what is the percentage mark if:
a pupil scores 7 out of a possible 20? Answer: 20 x 5 = 100, so 7 x 5 = 35%.
a pupil scores 18 out of a possible 25?
a pupil scores 7 out of a possible 10?
a pupil scores 9 out of a possible 20?
a pupil scores 130 out of a possible 200?
Fractions to Percentages – Type 2
If the denominator is not a factor of 100, check if it’s a multiple of 10. If it is, you can convert the fraction into tenths and then multiply the top and bottom by 10 to get a fraction over 100, which, again, is just a percentage in disguise, eg what is the percentage mark if:
A pupil scores 24 marks out of a possible 40? Answer: 40 ÷ 4 = 10, so 24 ÷ 4 = 6 and 6 x 10 = 60%.
A pupil scores 12 marks out of a possible 30?
A pupil scores 32 marks out of a possible 80?
A pupil scores 49 marks out of a possible 70?
A pupil scores 24 marks out of a possible 60?
Fractions to Percentages – Type 3
If neither of the first two methods works, that means you have to simplify the fraction. Once you’ve done that, you should be able to convert any common fraction into a percentage in your head. The most commonly used fractions are halves, quarters, fifths and eighths, so it’s worth learning the decimal and percentage equivalents off-by-heart, ie
½ = 0.5 = 50%
¼ = 0.25 = 25%
¾ = 0.75 = 75%
⅕ = 0.2 = 20%
⅖ = 0.4 = 40%
⅗ = 0.6 = 60%
⅘ = 0.8 = 80%
⅛ = 0.125 = 12.5%
⅜ = 0.375 = 37.5%
⅝ = 0.625 = 62.5%
⅞ = 0.875 = 87.5%
To simplify the fractions, check first to see if the numerator goes into the denominator. If it does, you can simply divide both numbers by the numerator to get what’s called a ‘unit fraction’, in other words, a fraction with a one on top, eg 4/8 divided by four top and bottom is ½.
By definition, a unit fraction can’t be simplified, so then you just have to convert it into a percentage.
If the numerator doesn’t go exactly, divide it by the first prime number (two) and then try to divide the denominator by the result, eg 6 ÷ 2 = 3, so 6/9 divided by three top and bottom is 2/3.
If that doesn’t work, try dividing the numerator by the next prime number (three) and so on and so on…
This will guarantee that the fraction ends up in the lowest possible terms, at which point it should be in the list above, which means you can easily convert it into the correct percentage, eg what is the percentage mark if:
a pupil scores 7 out of a possible 28? Answer: 7 goes into 28 four times, so the fraction is 1/4, which is 25%.
a pupil scores 27 out of a possible 36? Answer: 27 doesn’t go into 36, but 27 ÷ 3 = 9, so 27/36 divided by 9 top and bottom makes 3/4, which is 75%.
a pupil scores 24 out of a possible 48?
a pupil scores 8 out of possible 32?
a pupil scores 9 out of a possible 24?
Multiplying Three Numbers Involving Money
There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it, eg 1 x 2 x 3 is the same as 3 x 2 x 1.
The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning and then multiply the remaining three numbers together, eg a number of pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get a certain number of correct spellings, and the average amount of sponsorship is shown for each.
How many pounds would the class expect to raise for charity if the basic sum is:
20 x 30 x 5p? Answer: 2 x 3 x 5 = 6 x 5 = £30.
40 x 500 x 7p?
30 x 400 x 6p?
50 x 40 x 8p?
60 x 20 x 9p?
Division by Single-digit Numbers
This is what I call the ‘wedding planner problem’. There are three ways of doing this type of question:
Method A: Use the ‘bus stop’ method to divide the total number of guests by the number of seats per table – remembering to add one if there is a remainder.
Method B: Go straight to the end of your times tables by multiplying the number of seats by 12, then calculating the remainder and dividing by the number of seats per table, again remembering to add one if there is another remainder.
Method C: Use trial and error by estimating the number of tables needed using a nice, round number such as 5, 10 or 20 and working out the remainder as before.
Dining tables seat 7 children. How many tables are needed to seat 100 children? Answer: Method A) 100 ÷ 7 = 14 r 2, so 14 + 1 = 15 tables are needed. Method B) 7 x 12 = 84, 100 – 84 = 16, 16 ÷ 7 = 2 remainder 2, 12 + 2 + 1 = 15 tables. Method C) 10 x 7 = 70, which is too small, 20 x 7 = 140, which is too big, 15 x 7 = 70 + 35 = 105, which is just right as there are only 5 seats to spare.
Dining tables seat 6 children. How many tables are needed to seat 92 children?
Dining tables seat 5 children. How many tables are need to seat 78 children?
Dining tables seat 9 children. How many tables are needed to seat 120 children?
Dining tables seat 6 children. How many tables are needed to seat 75 children?
Division by Two-digit Numbers
If the number of seats is outside your times tables, the best option is just to use trial and error, starting with 5, 10 or 20, eg
It is possible to seat 40 people in a row across the hall. How many rows are needed to seat 432 people? Answer: 40 x 10 = 400, 432 – 400 = 32, so one more row is needed, making a total of 10 + 1 = 11 rows.
It is possible to seat 32 people in a row across the hall. How many rows are needed to seat 340 people?
It is possible to seat 64 people in a row across the hall. How many rows are needed to 663 people?
It is possible to seat 28 people in a row across the hall. How many rows are needed to seat 438 people?
It is possible to seat 42 people in a row across the hall. How many rows are needed to seat 379 people?
Percentages to Fractions
This is a type of question that looks hard at first but becomes dead easy with the right short cut. All you need to do is to work out 10% first and then multiply by the number of tens in the percentage.
Another way of saying that is just to knock one zero off each number and multiply them together, eg a test has a certain number of questions, each worth one mark. For the stated pass mark, how many questions had to be answered correctly to pass the test?
?/30 = 40% Answer: 3 x 4 = 12 questions (ie 10% of 30 is 3 questions, but we need 40%, which is 4 x 10%, so we need four lots of three, which is the same as 3 x 4).
?/40 = 70%
?/50 = 90%
?/80 = 70%
?/300 = 60%
Ratio – Distance
There are two ways of converting between different units of distance from the metric and imperial systems:
Method A: Make the ratio into a fraction and multiply the distance you need to find out by that same fraction, ie multiply it by the numerator and divide it by the denominator. (Start with multiplication if doing the division first wouldn’t give you a whole number.)
Method B: Draw the numbers in a little 2 x 2 table, with the figures in the ratio in the top row and the distance you need to find out in the column with the appropriate units, then find out what you need to multiply by to get from the top row to the bottom row and multiply the distance you have to find out by that number to fill in the final box.
8km is about 5 miles. How many kilometres is 40 miles? Answer: Method A) 8:5 becomes 8/5, and 40 x 8/5 = 40 ÷ 5 x 8 = 8 x 8 = 64km. Method B) Miles km 5 8 x 8 40 8 x 8 = 64km
6km is about 4 miles. How many kilometres is 36 miles?
4km is about 3 miles. How many kilometres is 27 miles?
9km is about 7 miles. How many miles is 63 kilometres?
7km is about 4 miles. How many kilometres is 32 miles?
Ratio – Money
You can use the same methods when converting money, except that the exchange rate is now a decimal rather than a fraction. Just remember that the pound is stronger than any other major currency, so there will always be fewer of them. It’s easy to get things the wrong way round, so it’s worth spending a couple of seconds checking, eg
£1 = €1.70. How much is £100 in euros? Method A) 100 x 1.70 = €170. Method B) £ € 1.00 1.70 x 100 100 1.70 x 100 = €170
£1 = €1.60. How much is £200 in euros?
£1 = €1.50. How much is €150 in pounds?
£1 = €1.80. How much is €90 in pounds?
£2 = €3.20. How much is £400 in euros?
Time – Find the End time
The most useful trick to use here is rounding. If the length of a lesson is 45 minutes or more, then just round up to the full hour and take the extra minutes off at the end. This avoids having to add or subtract ‘through the hour’, which is more difficult.
If the lessons are less than 45 minutes long, just work out the total number of minutes, then convert into hours and minutes and add to the start time, eg
A class starts at 9:35. The class lasts 45 minutes. What time does the class finish? Answer: 9:35 + 1 hour – 15 minutes = 10:35 – 15 minutes = 10:20.
A class starts at 11:45. There are three consecutive classes each lasting 25 minutes and then half an hour for lunch. What time does lunch finish? Answer: 11:45 + 3 x 25 + 30 = 11:45 + 75 + 30 = 11:45 + 1 hour and 15 minutes + 30 minutes = 13:30.
Lessons start at 11:15. There are two classes each lasting 40 minutes and then lunch. What time does lunch start?
Lessons start at 2:00 in the afternoon. There are four 50-minute classes with a 15-minute break in the middle. What time does the day finish?
Lessons start at 9:40. There are two classes of 50 minutes each with a break of 15 minutes in between. What time do the classes finish?
Time – Find the Start Time
It’s even more important to use rounding when working backwards from the end of an event, as subtraction is that bit more difficult, eg
A school day finishes at 3:15. There are two classes of 50 minutes each after lunch with a break of 15 minutes in the middle. What time does lunch end? Answer: 3:15 – 2 hours + 2 x 10 minutes – 15 minutes = 1:15 + 20 minutes -15 minutes = 1:20.
A school day finishes at 4:30. There are two classes of 40 minutes each after lunch. What time does lunch finish? Answer: 4:30 – 2 x 40 = 4:30 – 80 minutes = 4:30 – 1 hour and 20 minutes = 3:10.
Lunch starts at 1:05. There are two classes before lunch of 55 minutes each. What time do the classes start?
Lunch starts at 1:15. There are three classes before lunch of 45 minutes each. What time do the classes start?
A school bus arrives at school at 8:45. It picks up 20 children, and it takes an average of four minutes to pick up each child. What time is the first child picked up?
Percentage to Decimal
A decimal is a fraction of one unit, but a percentage is a fraction of 100 units, so, to convert from a percentage to a decimal, you just need to divide by 100, eg
What is 20% as a decimal? Answer: 20 ÷ 100 = 0.2.
What is 30% as a decimal?
What is 17% as a decimal?
What is 6% as a decimal?
What is 48% as a decimal?
Multiplying Decimals
Decimal points can be confusing, so the best way to do these sums is to take out the decimal point and put it back at the end. You just need to remember to make sure there are the same number of decimal places in the answer as in both numbers in the question, eg
1.5 x 1.5 Answer: 15 x 15 = 10 x 15 + 5 x 15 = 150 + 75 = 225, but there are two decimal places in the numbers you’re multiplying together, so the answer must be 2.25.
3 x 4.5
4.7 x 8
7.5 x 7.5
2.5 x 6.5
Multiplying Decimals by a Power of 10
Because we have 10 fingers, we’ve ended up with a ‘decimal’ number system based on the number 10.
That makes it really easy to multiply by powers of 10, because all you have to do is to move the decimal point to the right by a suitable number of places, eg one place when multiplying by 10, two when multiplying by 100 etc. (You can also think of it as moving the digits in the opposite direction.)
This type of question is therefore one of the easiest, eg
4.5 x 10 Answer: 45.
3.8 x 100
7.6 x 1000
4.6 x 100
3.5 x 10
Percentage of Quantity
Finding a percentage is easy if it ends with a zero, as you can start by finding 10% (Method A). If you happen to know what the fraction is, you can also divide by the numerator of that fraction (Method B), so 20% is 1/5, so you just need to divide by five, eg
Find 20% of 360 Answer: Method A) 360/10 x 2 = 36 x 2 = 72. Method B) 360 ÷ 5 = 72 (or 360 x 2 ÷ 10 = 720 ÷ 10 = 72).
Find 20% of 45
Find 30% of 320
Find 60% of 60
Find 80% of 120
Multiplication
Just because this is the ‘mental Maths’ section of the test doesn’t mean that you can’t work things out on paper, and these simple multiplication sums can be done like that.
Alternatively, you can use ‘chunking’, which means multiplying the tens and units separately and adding the results together, and the short cut for multiplying by five is to multiply by 10 and then divide by two, eg
23 x 7 Answer: 20 x 7 + 3 x 7 = 140 + 21 = 161.
42 x 5 Answer: 42 x 10 ÷ 2 = 420 ÷ 2 = 210
34 x 6
56 x 8
34 x 8
Short Division
Again, working these sums out on paper is probably quicker (and more reliable), although the easiest way to divide by four is probably to halve the number twice, and the short cut for dividing by five is to multiply by two and then divide by 10.
Hundreds of years ago, it was traditional to put dragons on maps in places that were unknown, dangerous or poorly mapped. Ratios are one of those places…
Here be ratios…!
A ratio is just a model of the real world, shown in the lowest terms, but answering ratio questions can be just as scary as meeting dragons if you don’t know what you’re doing. The key to understanding ratios is to work out the scale factor.
This is just like the scale on a map. If a map is drawn to a scale of 1:100,000, for instance, you know that 1cm on the map is the same as 100,000cm (or 1km) in the real world. To convert distances on the map into distances in the real world, you just need to multiply by the scale factor, which is 100,000 in this case.
(You can also go the other way – from the real world to the map – by dividing by the scale factor instead.)
To work out the scale factor in a Maths question, you need to know the matching quantities in the real world and in the model (or ratio). Once you know those two numbers, you can simply divide the one in the real world by the one in the ratio to get the scale factor. For example:
If Tom and Katie have 32 marbles between them in the ratio 3:1, how many marbles does Tom have?
To answer this question, here are the steps to take:
Work out the scale factor. The total number of marbles in the real world is 32, and the total in the ratio can be found by adding the amounts for both Tom and Katie, which means 3 + 1 = 4. Dividing the real world total by the ratio total gives 32 ÷ 4 = 8, so the scale factor is 8.
Multiply the number you want in the ratio by the scale factor. If Tom’s share of the marbles in the ratio is 3, then he has 3 x 8 = 24 marbles.
The matching numbers in the real world and the ratio are sometimes the totals and sometimes the individual shares, but it doesn’t matter what they are. All you need to do is find the same quantity in both places and divide the real world version by the ratio version to get the scale factor.
Once you’ve done that, you can multiply any of the ratio numbers to get to the real world number (or divide any real world number to get to the ratio number). Different questions might put the problem in different ways, but the principle is the same.
One complication might be having two ratios that overlap. In that case you need to turn them into just one ratio that includes all three quantities and answer the question as you normally would. For example:
If there are 30 black sheep, and the ratio of black to brown sheep is 3:2, and the ratio of brown to white sheep is 5:4, how many white sheep are there?
This is a bit more complicated, but the basic steps are the same once you’ve found out the ratio for all three kinds of sheep. To do this, we need to link the two ratios together somehow, but all the numbers are different, so how do we do it?
The answer is the same as for adding fractions with different denominators (or for solving the harder types of simultaneous equations, for that matter): we just need to multiply them together. If we were adding fifths and halves, we would multiply the denominators together to convert them both into tenths.
Here, the type of sheep that is in both ratios is the brown one, so we simply have to make sure the numbers of brown sheep in each ratio (2 and 5) are the same by multiplying them together (to give 10). Once we’ve done that, we can combine the two ratios into one and answer the question. Here goes:
Ratio of black sheep to brown sheep = 3:2
Multiply by 5
Ratio of black sheep to brown sheep = 15:10
Ratio of brown to white sheep = 5:4
Multiply by 2
Ratio of brown to white sheep = 10:8
Therefore, ratio of black sheep to brown sheep to white sheep = 15:10:8
Now that we have just one ratio, we can answer the question by following exactly the same steps as before:
Work out the scale factor. The total number of black sheep in the real world is 30, and the total in the ratio is 15. Dividing the real world total by the ratio total gives 30 ÷ 15 = 2, so the scale factor is 2.
Multiply the number you want in the ratio by the scale factor. If the number of white sheep in the ratio is 8, then there are 8 x 2 = 16 white sheep.
Simple!
Quiz
Here are a few practice questions:
One hundred paintings have to be selected for an art exhibition. If the ratio of amateur paintings to professional paintings has to be 2:3, how many amateur paintings and professional paintings have to be selected?
The ratio of brown rats to black rats is 3:2. If there are 16 black rats, how many brown rats are there?
Peter has 20 blue pens. How many red pens must he buy if the ratio of blue to red pens has to be 2:3?
There are 35 children in a class and 15 are boys. What is the ratio of boys to girls?
There are 15 girls and 12 boys in a class. What is the ratio of girls to boys? Give your answer in its simplest form.
A newspaper includes 12 pages of sport and 8 pages of TV. What is the ratio of sport to TV? Give your answer in its simplest form.
Anna has 75p, and Fiona has £1.20. What is the ratio of Anna’s money to Fiona’s money in its simplest form?
Sam does a scale drawing of his kitchen. He uses a scale of 1:100. He measures the length of the kitchen as 5.9m. How long is the kitchen on the scale drawing? Give your answer in mm.
A recipe to make lasagne for 6 people uses 300 grams of minced beef. How much minced beef would be needed to serve 8 people?
A recipe for flapjacks requires 240g of oats. This makes 18 flapjacks. What quantity of oats is needed to make 24 flapjacks?
Amit is 12 years old. His brother, Arun, is 9. Their grandfather gives them £140, which is to be divided between them in the ratio of their ages. How much does each of them get?
The angles in a triangle are in the ratio 1:2:9. Find the size of the largest angle.
In a certain town, the ratio of left-handed people to right-handed people is 2:9. How many right-handed people would you expect to find in a group of 132 people?
Twelve pencils cost 72p. Find the cost of 30 pencils.
Jenny buys 15 felt-tip pens. It costs her £2.85. How much would 20 pens have cost?
If three apples cost 45p, how much would five apples cost?
Sam is 16 years old. His sister is 24 years old. What’s the ratio of Sam’s age to his sister’s age? Give your answer in its simplest form.
A map scale is 1:20000. A distance on the map is measured to be 5.6cm. What’s the actual distance in real life? Give your answer in metres.
A recipe for vegetable curry needs 300 grams of rice, and it feeds 4 people. How much rice would be needed for 7 people?
£60 is to be divided between Brian and Kate in the ratio 2:3. How much does Kate get?
Teaching Greek children is like watching France play rugby: you never know what you’re going to get…
Stoa of Attalos: the Athenian version of the local mall
I just spent two weeks in Greece preparing a Greek boy and his twin sisters for 10+ and 12+ entrance examinations at a school in England. Highlights included spending a long, sunny weekend at a holiday home in Lagonissi, spending another long, sunny weekend skiing near Delphi – I wonder if the oracle saw that one coming! – and seeing the Parthenon every day from my hotel balcony.
Political refugees take many forms, but, personally, I prefer shipping magnates fleeing with their adorable (if strong-willed) families from Communist governments in the Mediterranean…
When it comes to incest, folk dancing and teaching kids who burst into tears every few minutes, my advice is: “Just say no.”
And all I got was this lousy sherbet…
I was due to spend nine weeks in Turkey early in 2015, teaching a 12-year-old boy Maths and English, but all I ended up with was a two-week holiday in the Ankara Sheraton and a client who refused to pay! I only managed to do four lessons before it became clear that things weren’t going to work out, so all I could do was take pictures of the food in the restaurant and the view out of my hotel window.
Given the circumstances, all I can do now is publish a few of the pictures I took. And learn an important lesson: if everything about a job from the very beginning seems wrong, it’s probably better just to say no…
