Tag Archives: lesson

Long division

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?

  1. How many 18s in 5?
  2. It doesn’t go
  3. How many 18s in 52?
  4. Two (write 2 on the answer line, and write 36 under the dividend with a line beneath it)
  5. What’s 52 – 36?
  6. 16 (write it on the next line)
  7. Pull down the next digit from the dividend (write it after the 16)
  8. How many 18s in 162?
  9. 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.

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How to write a letter

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. 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!

Simultaneous equations

Simultaneous equations help you work out two variables at once.

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Why do we have simultaneous equations? Well, there are two ways of looking at it.

  1. 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.
  2. 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:

  1. 2x + 4y = 16
    4x – 4y = 8
  2. 3x + 2y = 12
    5x + 2y = 16
  3. 12x – 4y = 28
    3x – 2y = 5
  4. 2x – y = 12
    3x – 2y = 17
  5. 4x + 3y = 24
    5x – 2y = 7
  6. 4x + 3y = 31
    5x + 4y = 40
  7. x + 4y = 23
    5x – 2y = 5
  8. 4x + 3y = 37
    2x – 3y = -13
  9. 2x + 4y = 16
    3x – 5y = -9
  10. 2x + 4y = 20
    3x + 3y = 21

Basics of photography

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When you buy (or borrow), your first digital SLR, everything looks different, and it can be a bit worrying. What are all these buttons and dials for? Why is it so heavy? Where do I start? How do I change the shutter speed? All these are very good questions, and this is the place to find the answers!

Before we start, I should mention that I’m a Nikon user, and I have one D800 and one D810 camera body. The other major camera manufacturer is Canon, and they use slightly different terms for each function, but I’ll try and include both to make life easier.

Our first job is to cover the basics of photography: exposure and focus. Without understanding those two things, nothing else will make sense!

Exposure

Your first job as a photographer is to make sure that your images are well exposed, in other words, not too dark or too bright. Photographers talk about the ‘exposure triangle’, but that’s just a complicated way of saying that how dark or light a photograph is depends on three things: the shutter speed, the aperture and the ISO.

The level of exposure is measured in ‘stops’ or Exposure Values (EV), but what is a ‘stop’? Well, if you increase your exposure by a stop, the light is doubled (and vice versa). For example, if you lengthen your shutter speed from 1/200 of a second to 1/100 of a second, your shot will be twice as bright. They try to use round numbers, though, so the gap from 1/60 to 1/125 is obviously not quite right! The maths gets a bit more complicated when the gap is only 1/3 of a stop, but the idea is the same.

The built-in exposure meter in your camera will work out what the best exposure should be, but it has to make assumptions about the world that may not be true. To judge the ‘best’ exposure, the camera needs a starting point, and that is that the world is, by and large, 18% grey. If it assumes that to be true, then it can set the exposure accordingly. However, anyone who’s ever taken pictures of polar bears on the ice knows that that’s not always true! In order to make sure the camera is not fooled by very bright or very dark conditions, you need to use exposure compensation. If the scene is especially bright, you can dial in up to one or two stops of positive compensation. If it’s especially dark, you can do the opposite. It might take a few test shots to get it exactly right, but that’s better than coming home with lots of shots of grey bears!

Shutter speed (or Time Value if you have a Canon)

In the old days, cameras used film, and the shutter speed controlled how long it was exposed to the light in order to take the shot. These days, cameras are digital and have electronic sensors at the back, but the principle is still the same. The longer the shutter speed, the more light reaches the sensor and hence the brighter the image. The shorter the shutter speed, the less light reaches the sensor and hence the darker the image.

The shutter speed is measured in seconds and can be anything from 1/8000 of a second to 30 seconds or more. The amount of camera shake increases with the focal length, so the rule of thumb for general photography is to make sure your shutter speed is no less than the inverse of the length of your lens, eg if you’re using a 400mm lens, you should be using at least 1/400 of a second. Lens technology such as Nikon’s ‘Vibration Reduction’ or Canon’s ‘Image Stabilisation’ means that you might be able to get away with a couple of stops slower – ie 1/100 of a second – but that’s about it.

The reason why shutter speed is an important setting is that it controls how much (if any) motion blur there is in the image, and that is an artistic decision. Some people like shots of kingfishers catching a fish that look like they’re frozen in time, with every single water droplet sharp as a tack. Other people prefer shots of waterfalls shown with creamy torrents of water cascading over them. There isn’t a ‘right’ or ‘wrong’ answer. Just try both and see what you think.

Aperture

The aperture is simply the size of the hole in the lens through which light passes on its way to the sensor, and the principle is similar to that of the shutter speed. The bigger the aperture, the more light reaches the sensor and therefore the brighter the image. The smaller the aperture, the less light reaches the sensor and therefore the darker the image. The only thing difficult about it is the numbers, which often have a decimal point in them like f/5.6 or f/7.1. The reason the aperture is not always a nice round number is because it is what you get when you divide the focal length of the lens by the diameter of the hole. Neither of those numbers is necessarily going to be a nice round number, so the result of dividing one by the other certainly won’t be!

The aperture is measured in f-stops, which typically start at f/2.8, f/4 or f/5.6 and continue up to f/22 and beyond. A ‘fast’ lens is one that has a wide maximum aperture such as f/1.4. Photographers like fast lenses as they allow pictures to be taken in low light and offer great flexibility.

The reason why the aperture is such an important setting is that it controls the depth of field, which is the amount of the subject that is acceptably sharp. The human eye is drawn to things it can see clearly, so making sure the subject is sharp and the background is an ideal way to focus the viewer’s attention on an animal, say, but a landscape photographer might want his image to be sharp all the way from the boat in the foreground to the mountains on the horizon. Again, there is no right answer; the important thing is to experiment and find what works for you.

ISO (or ASA if you’re still using a film camera!)

The ISO used to measure the sensitivity of the film being used, a ‘fast’ film with a high ISO being more sensitive than a ‘slow’ film with a low ISO. Now that most cameras are digital, we get the same effect, just with an electronic sensor instead of film. You might think that extra sensitivity is a good thing – and it is – but it comes at a cost. The higher the ISO, the ‘grainier’ or ‘noisier’ the image, in other words, the less smooth it is.

ISO is measured in ISO (funnily enough!), which just stands for International Standards Organisation. The lowest value is usually ISO 100, and the highest might be 12,800 or more, although the image quality at that value wouldn’t be acceptable to most professional photographers.

Focus

Your second job as a photographer is to make sure that the subject of your images is in focus. In the old days of film cameras, there was obviously no such thing as ‘autofocus’, and focusing had to be done by manually turning a ring on the lens, but today’s digital cameras have very good systems for making sure the images are sharp. In using the autofocus system, your job is first of all to choose the correct settings and secondly to make sure the camera is focusing on the right part of the frame.

There are lots of different focus settings, but the basic choice is between single area, shown as AF-S (or one-shot AF for a Canon), and continuous, shown as AF-C (or AI Servo for a Canon). Single area looks to focus on the area of the image under the little red square in the viewfinder (which you can move around the frame manually); continuous does the same but follows the actual subject if it moves. The best version of this on Nikon cameras is called ‘3D’. The other setting you can change is which button actually does the job of focusing. The shutter button does that on most cameras, but the disadvantage of doing it that way is that the camera stops focusing when you take a picture, which is bad news if you’re tracking a cheetah running at 60mph! The alternative is to use ‘back-button focusing’, which means separating the jobs of focusing and taking pictures. The shutter button still takes the picture, but the focusing is done by pushing a button on the back of the camera. (You have to set this up yourself, but I use the AF-ON button, which I can press with my right thumb.)

Camera guide (based on the Nikon D800)

This guide won’t go through every single setting on a DSLR, but it will show how all the main buttons work, not by saying what each one does but by answering the obvious questions. I hope that’s the easier way to learn!

(All the numbers used are taken from the diagram at the top of this article.)

How do I switch it on?

That’s simple. Just turn the power switch on the top right-hand side (1) to ‘ON’ (and back to ‘OFF’ when you’ve finished). If you turn it to the light bulb symbol, that just lights up the LCD display on top of the camera.

How do I set it to Manual?

There are lots of exposure modes on a camera, such as aperture-priority, shutter-priority and program, but using anything other than manual is a bit like buying a Ferrari with an automatic gearbox – you just don’t get as much control (or satisfaction). To select manual, press the ‘Mode’ button (50) and turn the main command dial on the back right of the camera (31). This allows you to set the shutter speed, aperture and ISO yourself, although I usually set the ISO to ‘ISO-AUTO’ by pushing the ‘ISO’ button on the top left of the camera (56) and at the same time turning the sub-command dial (21).

How do I make sure I’m shooting in RAW?

Press the ‘QUAL’ button (47) and turn the main command dial until the word ‘RAW’ appears on its own. The word ‘RAW’ doesn’t actually stand for anything, but everyone writes it that way to show that it’s a file format that contains the ‘raw’ data from the sensor. The alternative is JPEG (which stands for Joint Photographic Experts Group), but that’s a compressed file format and therefore should not be used. Note that RAW files don’t end in ‘.RAW’. It’s just a generic term, so each manufacturer has its own RAW extension, such as Nikon’s .NEF.

How do I set the white balance?

Press the ‘WB’ button 57 and turn the main command dial (31) to whatever is right for the lighting conditions. The icons aren’t very easy to see, but the options are:

  • Incandescent (ie light bulbs)
  • Fluorescent
  • Direct sunlight
  • Flash
  • Cloudy
  • Shade
  • Choose colour temp
  • Preset manual

The white balance tells the camera the colour of the light you’re working with. It’s a bit like working out what colour the curtains are at the cinema. The camera can’t tell the difference between something white that’s lit by red light and something red that’s lit by white light, so the white balance setting just makes sure it makes the right call. If you can’t quite see the icons or want to set up a custom white balance or preset, you can always go through the menu system. However, if you’re shooting in RAW, you can always change the white balance later on your computer, so don’t feel bad about sticking with ‘AUTO’!

How do I set the focus mode?

First of all, make sure your lens is not set to ‘M’, or manual focus, and that the focus mode selector (18) is set to ‘AF’, or auto focus. After that, press the AF-mode button (17) and at the same time turn the main command dial (31) to choose single area or – preferably – continuous. If you want the 3D option, you press the same button but at the same time turn the sub-command dial (21) until the LCD screen shows ‘3D’.

How do I set up back button focusing?

Press the ‘MENU’ button (46), scroll to the menu item with the pencil icon, select ‘a Autofocus’ and then set ‘a4 AF activation’ to ‘AF-ON only’. Half-pressing the shutter-release button won’t work any more, so don’t forget to focus by pressing (and holding) the AF-ON button (30) with your right thumb while you shoot.

How do I set the shutter speed?

Half-press the shutter-release button (3) if the shutter speed is not illuminated in the viewfinder or on the LCD screen and then turn the main command dial (31).

How do I set the aperture?

Half-press the shutter-release button (3) if the aperture is not illuminated in the viewfinder or on the LCD screen and then turn the sub-command dial (21).

How do I set the shutter-release button to continuous shooting?

Press the release button next to the ‘D800’ symbol and turn the release mode dial (48) to ‘CH’, or Continuous High. The D800 can shoot five frames a second.

How do I move the focus point in the viewfinder?

Turn the focus selector lock switch (34) to the dot symbol (rather than ‘L’ for lock) and use the multi selector to move the focus point anywhere within the central area of the viewfinder.

How do I check the depth-of-field?

Press the depth-of-field preview button (20).

How do I set the shutter-release button to continuous shooting?

Press the release button next to the ‘D800’ symbol and turn the release mode dial (48) to ‘CH’, or Continuous High. The D800 can shoot five frames a second.

How do I add exposure compensation?

Press the exposure compensation button (52) and at the same time turn the main command dial (31) to add or subtract as many stops of compensation as you need.

How do I bracket my shots?

Press the ‘BKT’ bracketing button (55) and at the same time use the main command dial (31) to choose the number of frames (3-9) and/or the sub-command dial (21) to choose the exposure interval (from 0.3 to 1 stop).

How do I shoot video?

You have to use the monitor rather than the viewfinder for this, so first of all turn the live view selector (36) to the film camera icon, press the live view button and then, when you’re ready, press the red movie-record button to start (and stop) video recording.

How do I look at my pictures?

Just press the playback button (22) and scroll through the images using the multi selector (32). To zoom in, either use the playback zoom in/zoom out buttons (43, 44) or set up the multi selector centre button to zoom immediately to 100%. This is very useful to check that images are acceptably sharp. To do that, press the ‘MENU’ button (46), select ‘f Controls’, then ‘f2 Multi selector centre button’ and set ‘Playback mode’ to ‘Zoom on/off’ with ‘Medium magnification’.
To play videos, just press the multi selector centre button (32).

How do I delete my pictures?

Just press the delete button (23). If you want to delete all the pictures on the memory card, the best way is to format it. Press the ‘MENU’ button, select ‘Format memory card’ and then select the appropriate card, either the small, thin Secure Digital (SD) card or the thicker, bigger Compact Flash (CF) card.

Rules of composition

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!

Moth on a daisy
‘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.

Bear about to catch salmon in mouth


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.

Juvenile rufescent tiger heron stretching its wings


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!

Close-up of lion lying in golden light


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…

Male azure hawker dragonfly flying through undergrowth


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.

Cobra held by snake charmer


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.

Close-up of golden eagle head with catchlight


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.

Pig peeping out from behind wall


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.

Malachite kingfisher on dead branch facing camera


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…”

King penguin stepping over rock with another


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.

Good luck…

Number sequences

2-4-6-8 ain't never too late...

2-4-6-8 ain’t never too late…

Number sequences appear in Nature all over the place, from sunflowers to conch shells. They can also crop up either in Maths or Verbal Reasoning, and both are essential parts of 11+ and other school examinations. The trick is to be able to recognise the most common sequences and, if you find a different one, to work out the pattern so that you can find the missing values (or ‘terms’).

Common sequences

Here are a few of the commonest number sequences. For each one, I’ve given the rule for working out the nth term, where n stands for its position in the sequence.

Even numbers: 2, 4, 6, 8 etc… Rule: 2n
Odd numbers: 1, 3, 5, 7 etc… Rule: 2n – 1
Powers of 2: 2, 4, 8, 16 etc… Rule: 2ⁿ
Prime numbers: 2, 3, 5, 7 etc… Rule: n/a (each number is only divisible by itself and one)
Square numbers: 1, 4, 9, 16 etc… Rule: n²
Triangular numbers: 1, 3, 6, 10 etc… Rule: sum of the numbers from 1 to n
Fibonacci sequence
: 1, 1, 2, 3 etc… Rule: n₋₂ + n₋₁ (ie each successive number is produced by adding the previous two numbers together, eg 1 + 1 = 2, 1 + 2 = 3)

Here are a few questions for you to try. What are the next two numbers in each of the following sequences?

  1. 14, 16, 18, 20…
  2. 9, 16, 25, 36…
  3. 3, 6, 12, 24…
  4. 7, 11, 13, 17…
  5. 5, 8, 13, 21…

Working out the pattern

The best way to approach an unfamiliar sequence is to calculate the gaps between the terms. Most sequences involve adding or subtracting a specific number, eg 4 in the case of 5, 9, 13, 17 etc. Sometimes, the difference will rise or fall, as in 1, 2, 4, 7 etc. If you draw a loop between each pair of numbers and write down the gaps (eg +1 or -2), the pattern should become obvious, enabling you to work out the missing terms.

  • If the missing terms are in the middle of the sequence, you can still work out the pattern by using whatever terms lie next to each other, eg 1, …, 5, 7, …, 11 etc. You can confirm it by checking that the gap between every other term is double that between the ones next to each other.
  • If the gaps between terms are not the same and don’t go up (or down) by one each time, it may be that you have to multiply or divide each term by a certain number to find the next one, eg 16, 8, 4, 2 etc.
  • If the gaps go up and down, there may be two sequences mixed together, which means you’ll have to look at every other term to spot the pattern, eg 1, 10, 2, 8 etc. Here, every odd term goes up by 1 and every even term falls by two.

Generating a formula

At more advanced levels, you may be asked to provide the formula for a number sequence.

Arithmetic sequences

If the gap between the terms is the same, the sequence is ‘arithmetic’. The formula for the nth term of an arithmetic sequence is xn ± k, where x is the gap, n is the position of the term in the sequence and k is a constant that is added or subtracted to make sure the sequence starts with the right number, eg the formula for 5, 8, 11, 14 etc is 3n + 2. The gap between each term is 3, which means you have to multiply n by 3 each time and add 2 to get the right term, eg for the first term, n = 1, so 3n would be 3, but it should be 5, so you have to add 2 to it. Working out the formula for a sequence is particularly useful at 13+ or GCSE level, when you might be given a drawing of the first few patterns in a sequence and asked to predict, say, the number of squares in the 50th pattern. You can also work out the position of the pattern in the sequence if you are given the number of elements. You do this by rearranging the formula, ie by adding or subtracting k to the number of elements and dividing by 𝒳. For example, if 3n +2 is the formula for the number of squares in a tiling pattern, and you have 50 squares in a particular pattern, the number of that pattern in the sequence = (50-2) ÷ 3 = 48 ÷ 3 = 16.

Quadratic sequences

If the gap between the terms changes by the same amount each time, the sequence is ‘quadratic’, which just means there is a square number involved. The formula for a quadratic sequence is 𝒳n² ± k, where 𝒳 is half the difference between the gaps (or ‘second difference’), n is the position of the term in the sequence and k is a constant that is added or subtracted to make sure the sequence starts with the right number, eg the formula for 3, 9, 19, 33 etc is 2n² + 1. The differences between the terms are 6, 10, 14, so the second difference is 4, which means you need to multiply the square of n by 4 ÷ 2 = 2 and add 1, eg for the first term, n = 1, so 2n² would be 2, but it should be 3, so you have to add 1 to it.

Geometric sequences

If each term is calculated by multiplying the previous term by the same number each time, the sequence is ‘geometric’. The formula for the nth term of a geometric sequence (or progression) is ar(n-1), where a is the first term, r is the multiplier (or ‘common ratio’) and n is the position of the term in the sequence, eg the formula for 2, 8, 32, 128 etc is 2 x 4(n-1). The first term is 2, and each term is a power of 4 multiplied by 2, eg the fourth term = 2 x 4(4-1) = 2 x 43 = 2 x 64 = 128.

Here are a few questions for you to try. What is the formula for the nth term in each of the following sequences?

  1. 14, 16, 18, 20…
  2. -1, 3, 7, 11…
  3. 4, 6, 10, 16…
  4. 9, 7, 5, 3…
  5. 2, 6, 18, 54…