This article explains circle theorems, including tangents, sectors, angles and proofs (with thanks to Revision Maths).
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.
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.
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.
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.
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
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
The obvious question for a lot of amateur photographers is ‘How do I make money from photography?’ The answer, unfortunately, is that I don’t know. All I can do is tell you what I’ve done and give you a few ideas. I’m still learning the business after just four years, but my approach has always been to knock on as many doors as possible, whether it’s microstock, exhibitions, competitions, lessons or even talks. Every source of revenue has its part to play, and it’s just a question of working out where to focus your efforts. I make just under half my money from microstock/stock agencies and half from exhibitions, but everybody’s different.
Nick Dale Photography
I loved photography when I was a teenager. I bought (or was given) books on Henri Cartier-Bresson, Alfred Eisenstaedt, Ansel Adams and other great photographers, and I even bought myself an old Chinon CE-4 film SLR. I remember buying two 36-exposure films for it – one colour, one black and white – and using up every single frame in a couple of hours just taking pictures around the house! I took my camera on holiday to Majorca and the United States, developed pictures in a dark room at school and even talked to my mum about becoming a professional photographer. However, my mother said I could always take it up later – so that was that for 30 years! Fortunately, I was given a second chance in January 2013 when a friend of a friend invited me to climb Mount Kenya and go on safari with her and a couple of other people. I’d always wanted to go to Africa, but I’d foolishly been saving it for my honeymoon! As that didn’t seem very likely, I jumped at the chance.
My first digital camera was a Sony DSC-HX200V bridge camera, which means it had a good zoom range (both optical and digital), but not a very large sensor. As a result, it was only around £300 and therefore cheap enough for me to buy without worrying too much. Fortunately or unfortunately, a week in Kenya with people using proper Nikon SLRs gave me camera envy, and I bought a Nikon D800 SLR with a 28-300mm lens as soon as I got home!
And that was how it all started. I took hundreds of pictures in Kenya of the people, the landscape and especially the wildlife. When I got back, I bought an Apple MacBook Pro to work on them, upgraded the editing program to Aperture and then sent them off to various microstock agencies to see if they would help me sell them. It was hard at first, but getting the new camera helped, and I had a cash pile from remortgaging my flat in Notting Hill after another property purchase fell through, so I was able to go on plenty of trips to take more and more pictures.
An important breakthrough came when I sold a couple of prints for £100 each at my local tennis club’s Christmas Fair in November 2014, and another photographer told me about a cheap exhibition space called the Norman Plastow Gallery in Wimbledon Village. I’d always thought it would be very expensive to mount an exhibition, but this place was only £70 for a week, so I booked it as soon as I could! The only problem was that I didn’t have any actual prints to sell, and here I was very fortunate. I’d recently joined the Putney branch of London Independent Photography (or LIP), and there I’d met a very friendly and helpful chap called James, who’d offered to do all my printing for me at very low cost. After buying a few cheap, black, wooden frames from Amazon, I was all set. I invited all my friends to the exhibition in May 2015 – especially a group of tennis players from my club – and I ended up selling seven prints. As I was just starting out, I’d priced the small, medium and large framed prints at £80, £100 and £120 and the unframed ones at only £30, but I still managed to make £550 in total. The gallery hire charge was £200, and there were a few taxis to pay for plus incidental expenses, but the show actually turned a profit – unless you count the thousands of pounds I spent on buying camera equipment and flights to Kenya, Botswana, Antarctica and the Galápagos!
And there’s the rub. It’s relatively easy to generate revenue from photography, but actually making a profit out of it is another matter entirely. As a result, I have nothing but respect for the photographers I meet who have managed to make a career out of it. I’ve been on trips led by Paul Goldstein and Andy Skillen amongst others, and, in a way, that’s where I’d like to end up. Since that first show in Wimbledon Village, I’ve sold nearly 5,000 downloads through microstock agencies, sold 36 prints at solo exhibitions and art fairs, taught five photography students and given two or three talks to various clubs and societies. Overall, I’ve made around £12,000 from my photography – but that wouldn’t even have paid for my trip to Antarctica!
The problem is that everyone has a camera these days – even if it’s just an iPhone – and it’s almost ‘too easy’ to take pictures now that cameras are digital. The world is also a smaller place these days, with the arrival of cheap flights and a general rise in income and wealth. It takes a special talent to make it as a photographer, and part of that talent is being able to make the most of it.
What do I need to do first?
Buy a camera If you want to make money out of photography, your first job is to get yourself a decent camera, and that means a digital SLR (or DSLR). The easiest way to earn cash is through so-called microstock agencies – which means selling pictures online in exchange for royalty payments – and they usually require shots to be taken with a camera that has at least 12 megapixels, if not more. You can obviously try to sell holiday snaps from your ‘back catalogue’, but, as I found out to my cost, it ain’t easy. Once you’ve decided to buy a DSLR, the two main brands to choose from are Nikon and Canon. There isn’t much between them these days, and the only reason I chose Nikon is that I didn’t want a camera from a company that made photocopiers! They both make good lenses, but, unfortunately, they have different mounts, so one you go with one or the other you’re locked in. I have various lenses ranging from an 18-35mm wide angle zoom to a 105mm macro lens for close-up work to an 80-400mm mid-range zoom, but I also rent an 800mm lens from Lenses for Hire whenever I go on a major wildlife photography trip.
Buy a laptop If you don’t have one already, buying a decent laptop is great for photography. I take mine with me on all my trips, and it means that I can work on my images every evening after I get back from a shoot or a game drive. I should warn you, though, that the so-called RAW files from digital cameras are very large (in the case of my camera over 40MB each!), so I’d recommend getting as fast a processor as possible and as much memory and hard disk space as you can afford. You should also arrange a back-up system: the last thing you need is for your life’s work to disappear thanks to a software glitch! You could use an external hard drive, but I prefer backing up to the cloud just to be on the safe side. I use CrashPlan, which automatically detects any added, edited or deleted files and backs up the changes in real time, but there are other similar products out there.
Subscribe to Lightroom Adobe Lightroom Creative Cloud is the choice of professionals and serious amateurs for organising and editing their photographs. It only costs around £8 a month (including Photoshop), and it’s a very powerful tool, as well as being relatively easy to use once you’ve mastered the basics. Digital photographs never come out of the camera looking perfect, so it’s always a good idea to try and improve the contrast, highlight and shadow areas and anything else you need to. If you’re selling through agencies, you’ll also need to add titles, captions and keywords (plus any other fields you’re asked to fill in), and all that is possible with Lightroom. It’s a pain to do for each individual photograph, but you can ‘synchronise’ any changes you make across a number of pictures, and you only need to do it once. If you’ve never used it before, I suggest you to do what I did and watch Anthony Morganti’s series of free YouTube videos on Lightroom. He takes you through all the functionality, and it’s an easy way to learn.
Start taking pictures If you’re a wildlife photographer, this is just a euphemism for ‘spend thousands of pounds on trips to long-haul destinations’! However, you don’t have to travel far to take pictures. Whether you’re a landscape, portrait, Nature, fashion, wildlife, wedding or sports photographer, there’s always something photogenic not far from home, and you simply have to have the enthusiasm (and discipline) to be able to get out there and take more and better shots. Quality and quantity are both important. The quality of your images is ultimately what matters, but even a shot that’ll never win a competition might earn you money on a microstock site. I give my shots three stars if they’re good enough for Facebook, four if they’re good enough to be sold via agencies and five if they’re good enough to go on my website.
Start marketing your work As a photographer, you have to learn to talk the talk as well as walk the walk. If you want to be taken seriously, you have to cover the basics, which means building a website, printing out business cards and having an active presence on social media. You can’t expect to win a bid for a photo shoot if you’re still using an old Hotmail address! Personally, I have this website powered by SquareSpace plus a Facebook ‘fan page’, a YouTube page, a LinkedIn account and a Twitter feed, all of which are printed on the back of my business cards. I post articles on my blog about photography trips, exhibitions and useful techniques (which also appear on Facebook, LinkedIn and Twitter), and I tweet and retweet a ‘Shot of the week’ (which gets fed through to my Facebook account as well).
Yes, but how can I make money?
Microstock Microstock agencies are online intermediaries that accept work from photographers and then market those images to potential clients such as creative directors of newspapers, magazines and other buyers. The advantage of using them is that it’s ‘making money while you sleep’, in other words, it’s a passive income that you can build over time as you add more and more shots to your portfolio. Some agencies sell a lot of images but with low royalty rates, some the reverse, but here is the list of the ones I’ve used (in descending order of sales): Getty Images/iStock Shutterstock Adobe/fotolia DepositPhotos 123RF Bigstock PIXTA SolidStockArt Dreamstime EyeEm Canstock photodune ClipDealer Panthermedia Pixoto featurePics Mostphotos Pond5 500px Redbubble Alamy Yay Micro Stockfresh Crestock Zoonar Lobster MediaI should mention that not all agencies will accept you, and not all your shots will be accepted by any agency that does, but you shouldn’t take it personally. I’ve had over £4,000 in microstock sales in the last four years, but my overall acceptance rate is only 41%! Even if your pictures are accepted, of course, that doesn’t mean they’ll sell. I’ve had 5,120 downloads from microstock sites, but only 1,521 individual shots have ever been sold out of a total of 4,389. The rest of them are just sitting there, waiting for a buyer. Every now and then, though, you take a picture that goes viral: I’ve sold my jumping penguin (see above) 705 times! The basic process is similar across all agencies. You add titles, captions and keywords to all your pictures and then export them as JPEG files to upload to each individual agency via their websites or an FTP service using a program like Filezilla. You then typically add the category, country or other data for each of them and submit them for approval. The agencies then approve the ones they like and reject the ones they don’t. After that, it’s just a question of watching the money rolling in! A useful way of doing that is by downloading an app called Microstockr. All you need to do is to set up your various agencies on the accounts page and then check the dashboard every now and then for any sales you’ve made. It’s very addictive! Sales should come quite soon after each batch is uploaded, but you may have to wait a while for payment. Most agencies have a ‘payment threshold’ of $50 or $100, which means your first payment (usually through PayPal) might take months to arrive. You’ll also need to keep adding more pictures. Buyers tend to sort images according to what’s most recent, so you definitely get diminishing returns from your shots, however good they are. The other thing to say is that, with dozens of agencies and hundreds or even thousands of images, it gets very confusing. As a result, I’ve created a spreadsheet to keep track of the whole thing. With filenames down the left and agency names across the top, I know if each file has been uploaded (‘u’), submitted (‘s’) or accepted (‘y’) and how many times it’s been sold. I keep a record of the dollar value of all the image downloads on a separate financial spreadsheet. I suggest you do the same.
Stock agencies In the good old days, it was much easier to make a living out of stock photography, mainly because the royalty rates were a lot higher. The difference between ‘stock’ and ‘microstock’ is simply the average price level. Stock agencies want to differentiate themselves from microstock agencies (and everything else out there on the web) in order to charge a higher price, so they generally ask for exclusive agreements over one to five years and set a higher standard for acceptance. I use Design Pics, and you can see that they sell my images for hundreds of dollars rather than just a few dollars for the microstock agencies. My general strategy is to offer Design Pics the first pick of my pictures before sending the leftovers to all the microstock agencies. (I’ve also submitted some flower images to flowerphotos and a few marine wildlife shots to SeaPics, but I haven’t seen any sales from them so far.) Due to the long sales and reporting cycle, I didn’t see my first sale from Design Pics until more than a year after I’d signed up, but sales are starting to trickle in now, so it just takes a bit of patience. If you’re looking for a list of stock agencies, I recommend buying a copy of 2017 Photographer’s Market, which is the equivalent of the Writers’ and Artists’ Yearbook. It has comprehensive coverage of the industry, including helpful articles and a wealth of phone numbers and email addresses for magazines, book publishers, greeting card companies, stock agencies, advertising firms, competitions and more. I suggest buying the Kindle electronic version, and then you can download everything on to your laptop. I did that and then simply emailed every stock agency on the list – Design Pics was the only one to say yes!
Competitions If you just want the ego boost of seeing yourself winning a competition, then I suggest you sign up with Pixoto and enter the contests with the lowest number of entrants. It’s a peer-to-peer site, and you can organise your own competitions, so there’s a very good chance of winning something! That’s exactly what I did, and I ended up with the Judge’s Award in four competitions. However, there isn’t much prestige to something like that, and it certainly doesn’t earn you any money. Alternatively, you can scour the 2017 Photographer’s Market for competitions, bearing in mind your chances of winning, the cost of entry, the potential prizes and the subject matter. The UK national press is a good place to start, too, and I recently won £250 in Wex Photographic vouchers in the weekly Sunday Times/Audley Travel Big Shot competition.
Exhibitions Putting on an exhibition may seem like a big deal if you’ve never done it before, but it doesn’t have to be expensive or time-consuming. The Norman Plastow Gallery where I started out is cheap, but it’s slightly off the beaten path, and you have to man the exhibition yourself, which is obviously impossible for most full-time employees. You realise pretty soon as a freelance photographer that the most expensive item on your tab is often the opportunity cost of NOT doing what you usually do when you take time off. As a tutor, for instance, I could easily have earned £1,000 during the two weeks of my first exhibition, but them’s the breaks… If you’re looking for a list of galleries, www.galleries.co.uk is a useful starting point. London is obviously the best place to look, but exhibition spaces there don’t come cheap. I recently looked for galleries to use for an exhibition, and the ones in central London regularly quoted me thousands of pounds for a week! Everything is negotiable, though, so don’t give up. I started out with 15 prints at my first solo show, but I also printed out a few postcards and greetings cards. You might not make as much money out of them, but at least you’ll get something from punters who can’t afford a print. There are some who say that cards are just a distraction, but it’s so difficult to tell. I’ve had exhibitions with and without cards on sale, and it doesn’t seem to make much of a difference. However, the main reason for an exhibition is to sell prints, so that should be the focus. One of the problems you’ll almost certainly have is knowing how to price your work. Choosing your favourite shots is easy enough – although getting a second opinion from a friend is a useful exercise – but how much should you charge? I started off at £80 for an A3 print and ended up three years later at £2,000 for a 53″ x 38″ print, so you’ll just have to suck it and see. Andy Skillen suggested a mark-up of two-and-a-half times your printing and framing costs to make sure your cashflow remained positive, but that’s just a rule of thumb.
Photo shoots Proper professional photographers make most of their money from photo shoots, but clients aren’t easy to find. If you’re a wedding photographer, I suppose you can put up flyers at various local venues such as churches and registry offices, but, for the rest of us, it’s just a question of plugging away, taking as many good shots as we can and putting them online so that as many potential clients can see them as possible. It would be a dream to be able to rely on commissions from wealthy clients who called us up whenever they wanted pictures of something. A photographer told me once about a group of directors who asked him for a picture of five hippos in a lake looking at the camera. He sent them all the hippo shots he had, but they weren’t happy. In the end, he told them if they didn’t want to compromise on the picture, then they’d have to send him on an all-expenses-paid trip to Zambia for a week. Which they did! He got the shot within a couple of days and then spent the rest of the trip taking pictures for himself! That sounds like a nice way to make a living, doesn’t it? However, until we’re well established enough with a good enough reputation to get those kinds of jobs, all we can do is keep on snapping and use the networks that we have. I’ve worked for a milliner, a local councillor, a businesswoman and others, but all my photo shoots have come from friends of friends or personal contacts. I’m not very good at networking – and it’s certainly not something I enjoy unless it happens naturally – but it’s very important in this business.
Lessons I work as a private tutor as well as a photographer, so I guess it was an obvious fit to offer photography lessons. It’s finding the students that’s the real problem, though. One of my tuition agencies provided me with a couple of clients, while the rest came from connections I made at exhibitions and talks. You never know when you might meet just the right person, so it’s important to keep a few cards in your wallet just in case.
Talks If you don’t mind public speaking, then giving a slideshow and talk on photography is an enjoyable way to earn some pocket money. Camera clubs and other groups won’t generally pay more than £100 (if anything at all!), but it’s also a useful chance to take along a few prints to sell and to hand out business cards. I got started after meeting a very nice woman on an Antarctic cruise, and I’ve now given talks at her branch of the WI, two camera clubs and a local library. If you want to be proactive about it, I’d simply Google camera clubs (or WI branches!) and email all of them to see what happens. As my mum used to say, you have to cast your bread upon the waters…even if it sometimes comes back a soggy mess!
Photography trips One final way of making money is to lead photography trips. A lot of photographers do it to supplement their income, and it’s a good way to reduce your travel budget. I recently put together a list of tour operators and emailed them all one afternoon to find out if it could work, and I soon received a call from the founder of Gane & Marshall, asking me to lead a trip to Tanzania! I offered my services for free in exchange for the chance to go on an all-expenses-paid photographic safari. Now all we have to do is find at least five people to come on the trip and make it economic. Fingers crossed!
I hope all that was useful. If you have any more questions, please drop me a line at email@example.com. It’s not easy becoming a professional photographer, but we can at least take pictures as a hobby while we wait for our big break.
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!
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.
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.
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)
Choose colour temp
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.
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.
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.
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.
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.
Shoot in RAW. There. Is. No. Alternative.
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.
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…
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
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.
There are three main types of operation you need to do in the following order:
Multiplying out any brackets
Adding or subtracting
Multiplying or dividing
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
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’.