Tag Archives: 11+

Verbal Reasoning

Verbal Reasoning (VR) tests were invented to test pupils’ logic and language skills – although they do sometimes includes questions about numbers. In order to do well in a VR test, the most important thing is to be systematic, to have a plan for what to do if the question is hard. Fortunately, there are plenty of past papers available online (including on this website!), so the types of question are well known. Here is a guide to the different kinds of problems and the best ways to approach them. I’m sorry that there are so many, but it’s best to be ready for anything…!

Insert a letter

One common type of question asks you to say which letter will start and finish two pairs of words, eg PRES( )TAND and WIND( )TAIN. Sometimes the answer is obvious (‘S’ in this case), but, if it’s not, the best thing to do is to look at all four words one after the other to see which letter might fit and then try that letter in the other words. If that doesn’t work, you might just have to go through every letter of the alphabet, but there are only 26, so it shouldn’t take too long! Bear in mind that there are different ways of pronouncing letters and different places to put the emphasis, so try writing down the likely options as well as saying them in your head.

Find the odd words

In this kind of question, you’re given five words, and you have to spot the two that don’t fit with the others, eg Lorry, Helicopter, Taxi, Bus, Plane. The best way is to try and find the three words that go together – whatever is left must be the odd ones out. Don’t just try to find a pair of words that go together. If you do, you might get the answer wrong if there’s another word that goes with them. You might also get it wrong because the ‘odd ones out’ don’t have anything in common. In this case, ‘Helicopter’ and ‘Plane’ ARE related, but they don’t have to be.

Alphabet Codes/Code Words

Here, you’ll be asked either to put a word into code or to decode a word. To do that, you’ll be given a word and the coded version, and it’s up to you to work out how the code works, eg STRAW might become UVTCY. Normally, you just have move one or two spaces forwards or backwards in the alphabet (in this case, it’s +2), but look out for other combinations. They might involve changing direction or a change to the number of spaces or a combination of both, eg -1, +2, -3, +4. The good news is that you’ll usually have an alphabet printed next to the question, so you can put your pencil on a letter and ‘walk’ forwards or backwards to get the coded version, but you can also write down the code underneath the word and write down how to get each letter with a positive or negative number – just make sure you don’t get confused between coding and decoding!

Synonyms (Similar Meaning)

Synonyms are words that have similar meanings, such as cold and chilly. In synonym questions, you’re given two groups of three words, and you have to find two synonyms, one from each group, eg (FILTER MATCH BREAK) (DENY DRAIN CONTEST). The first thing to do is to have a quick look at all the words to see if the answer’s obvious (MATCH and CONTEST, in this case). If it is, write it down. If it’s not, you have to be systematic: start with the first word in the first group and compare it with the first, second and third words in the other group. If that doesn’t work, repeat for the second and third words of the first group. Just be careful to think about ALL the possible meanings of a word, eg ‘minute’ can mean 60 seconds, but it can also mean very small! If you still can’t do the question (because you don’t know one or more of the words), try to work by process of elimination. That means narrowing down the options by getting rid of any pairs of words that definitely don’t mean the same. Once you’ve done that, feel free to guess which one of the leftover pairs is the answer. Guessing is fine in Verbal Reasoning: the only thing worse than a wrong answer is no answer at all!

Hidden Words

These questions ask you to find ‘hidden’ four-letter words between two other words in a sentence, using the last few letters from one word and the first few from the next, eg ‘The bird sat on the roof’. Again, scan the sentence quickly to see if the answer’s obvious. If it is, write it down. If it’s not, check every possibility by starting with the last three letters of the first word and the first letter of the second word, moving forward one letter at a time and then checking the next pair of words. You might want to put your fingers on each pair of words with a four-letter gap in the middle so that you can see all the options as they appear just by moving your fingers along the line. In this example, the possible words are theb, hebi, ebir, irds, rdsa, dsat, sato, aton, tont, onth, nthe, ther, hero and eroo, so the answer is obviously ‘hero’, but note that ‘tont’ is spread over three words (sat, on and the), and some words are not long enough to have the usual number of possibilities.

Find the Missing Word

These questions ask you to find a missing set of three letters that make up a word, eg There is an INITE number of stars in the sky. First of all, look at the word in capitals and try to work out what it’s meant to be in the context of the rest of the sentence. If it’s not obvious, try working out where the letters might be missing – is it after the first letter or the second or the third etc? Sometimes you might not know the word (‘INFINITE’ and therefore ‘FIN’ in this case), but, again, it’s worth a guess – just make sure your made up word sounds reasonable!

Algebra (Calculating with Letters)

This is one type of question that’s easier if you’re good at Maths! Algebra uses letters to stand for numbers and is a way of creating useful general formulas for solving problems. In Verbal Reasoning tests, you’ll generally have to add, subtract, multiply and/or divide letters, eg A = 1, B = 2, C = 3, so what is A – B + C? The first step is to convert the letters to numbers, and then you can simply work out the answer as you would in Maths. Just make sure you’re aware of BIDMAS/BODMAS. This is an acronym that helps you remember the order of operations: Brackets first, then Indices/Order (in other words, powers such as x squared), then Division and Multiplication and lastly Addition and Subtraction. Note that addition doesn’t actually come before subtraction – they belong together, so those sums should be done in the order they appear in the question, eg in this case, A – B must be done first (1 – 2 = -1) and then C added on (-1 + 3 = 2).

Antonyms (Opposite Meaning)

Antonyms are words that have opposite meanings, such as hard and soft. In antonym questions, you’re given two groups of three words, and you have to find two antonyms, one from each group, eg (GROW WATER WILD) (SLICE FREE TAME). The first thing to do is to have a quick look at all the words to see if the answer’s obvious (WILD and TAME, in this case). If it is, write it down. If it’s not, you have to be systematic: start with the first word in the first group and compare it with the first, second and third words in the other group. If that doesn’t work, repeat for the second and third words of the first group. Just be careful to think about ALL the possible meanings of a word, eg ‘minute’ can mean 60 seconds, but it can also mean very small! If you still can’t do the question (because you don’t know one or more of the words), try to work by process of elimination. That means narrowing down the options by getting rid of any pairs of words that definitely don’t mean the opposite to each other. Once you’ve done that, feel free to guess which one of the leftover pairs is the answer. Guessing is fine in Verbal Reasoning: the only thing worse than a wrong answer is no answer at all!

Complete the Calculation

This is another number question, and it again means you need to know BIDMAS/BODMAS. You’ll be given an equation (or number sentence), and you just have to fill in the missing number to make sure it balances, eg 24 – 10 + 6 = 8 + 7 + ( ). First of all, work out what the complete side of the equation equals, and then add, subtract, divide or multiply by the numbers in the other side to work out the answer (in this case, 24 – 10 + 6 = 20, and 20 – 8 – 7 = 5, so 5 is the answer). Don’t forget you’re working backwards to the answer, so you have to use the opposite operators!

Rearrange to make two new words

In these questions, you’re given two words, and you have to take a letter from the first word and put it in any position in the second word to leave two new words, eg STOOP and FLAT. Again, check first to see if the answer’s obvious, but then work through systematically, picking letters from the first word one by one and trying to fit it into each position in the second word. (In this case, the answer is STOP and FLOAT.) Remember that both the new words must make sense!

Number Relationship

This is another Maths question in which you’ll be given three sets of numbers in brackets with the middle one in square brackets. The middle number in the final set is missing, though, so you have to calculate it using the two on either side, based on what happens in the first two sets, eg (3 [15] 5) (2 [8] 4) (7 [ ] 3). The calculation will only involve the four basic operations (addition, subtraction, multiplication and division), but it gets much harder when the numbers appear more than once! In this example, all you need to do is multiply the outside numbers to get the answer (3 x 5 = 15 and 2 x 4 = 8, so 7 x 3 = 21), but you might get more complicated questions like this one: (16 [40] 8) (11 [27] 5) (4 [ ] 11). Here, you have to add the first number to itself and then add the other one (16 + 16 + 8 = 40 and 11 + 11 + 5 = 27, so 4 + 4 + 11 = 19). These kinds of questions can be very difficult, so try not to spend too long on them. If it takes more than a minute or so to answer a question, it’s time to move on. You can always come back later if you have time at the end of the test.

Alphabet Series/Sequence

These questions are a variation on number sequences in Maths – except using letters – and you answer them in the same way. You’re presented with several pairs of letters, and you have to fill in the blanks by working out what the patterns are, eg AB BD CF ??. The best way to do this is to focus on the first and second letters of each pair separately as there will always be a pattern that links the first letters of each pair and a pattern that links the second letters of each pair, but there usually won’t be a pattern that links one letter to the next. There’ll be a printed alphabet next to the question, so just do the same as you would for a number sequence question in Maths, drawing loops between the letters and labelling the ‘jump’ forwards or backwards in the alphabet, eg +1 or -2. Once you know what the pattern is, you can use it to work out the missing letters.

Analogies (Complete the Sentence)

In this type of question, you’re given a sentence that includes three possibilities for two of the words. You have to use logic and common sense to work out what the two other words should be, eg Teacher is to (bus, school, kitchen) as doctor is to (office, train, hospital). This is known as an analogy: you have to work out the relationship of the first word to one of the words in the first set of brackets in order to find the same relationship in the second half of the sentence. Again, the best way to do it is to have a quick scan to see if the answer’s obvious. If it is, write it down. If it’s not, go through the possibilities one by one, making sure to put the relationship into words. In this example, a teacher ‘works in a’ school, and a doctor ‘works in a’ hospital, so ‘school’ and ‘hospital’ are the answer.

Word codes

These are complicated! You are given four words and three codes, and you have to find the code for a particular word or the word for a particular code, eg TRIP PORT PAST TEST and 2741 1462 1851. Unfortunately, there’s no set way of doing these kinds of questions, so you just have to use a bit of logic and common sense. It’s useful to remember that each letter is always represented by the same number, so you can look for patterns in the letters that match patterns in the numbers, eg a double T in one of the words might be matched by a double 3 in one of the codes, so that means T = 3, and you can also find out the numbers for all the other letters in that word. In this example, TEST starts and finishes with the same letter, and 1851 starts and finishes with the same number, so TEST = 1851, which means T = 1, E = 8 and S = 5. You can then fill in those numbers for each of the remaining words, so TRIP = 1???, PORT = ???1 and PAST = ??51. Next, you should be able to see that the letter R is the second letter in TRIP and the third in PORT, and that’s matched by the number 4, which is the second number in 1462 and the third in 2741. That means R = 4, which means TRIP = 14??, PORT = ??41 and PAST = ??51. The only code starting with 14 is 1462, so TRIP = 1462, and the only code ending with 41 is 2741, so PORT = 2741 and the only code ending with 51 is 2351, so PAST = 2351. If PAST = 2351, that also tells us that A must equal 3, so you now know what each letter stands for, and you can answer any possible question they might throw at you. Phew!

Complete Word Pairs

These questions are similar to word codes but, fortunately, much easier! You are given three pairs of words in brackets, and you have to work out the missing word at the end by what has gone before, eg (SHOUT, SHOT) (SOLDER, SOLE) (FLUTED, ). The best way to go about it is to write down the position of the letters in the second word of the first two sets of brackets as they appear in the first. In other words, the letters from SHOT appear in positions 1, 2, 3 and 5 in the first word, and the letters from SOLE also appear in positions 1, 2, 3 and 5 in the first word, so the missing word must consist of the same letters from FLUTED, which means it must be FLUE. Now, you may not know that a flue is a kind of chimney, but don’t let that put you off. Just make sure you’ve got the right letters, and the answer must be right – even if you’ve never heard of it!

Another variation on this type of question contains a string of letters that appears in both words of each pair, just with a different letter or letters to start, eg (BLOAT, COAT) (CLING, DING) (SHOUT, ). The easy bit is to find the repeated set of letters (in this case OAT) and to see that the second letter is dropped each time, but you still need to work out why the first letter changes (from B to C and then C to D). That shouldn’t be too hard to work out, though, if you just go through the alphabet to find how many positions forwards or backwards you have to go (in this case, it’s +1, so the answer is TOUT).

Number Series/Sequences

These questions provide you with a series of numbers and ask you to fill in the blanks, which might be anywhere in the sequence, eg 1, 3, 5, 7, ?, ?. As with alphabet series, the best way to find the answer is to draw a loop between each pair of numbers and write down the change in value. In this case, it’s simple (+2 each time), so the answer is 9 and 11, but look out for more complicated sequences. It’s worth knowing the most common sequences, just so you can recognise them at once and don’t have to work them out. Here are a few of the commonest ones:

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)

Things get trickier when the sequence is actually a mixture of two separate sequences, eg 1, 3, 2, 5, 3, ?, ?. Here, the integers (1, 2, 3 etc) are mixed in with odd numbers starting with 3 (3, 5 etc), so you can’t simply find the difference between one number and the next – you have to look at every other number. In this example, the first missing number is the next integer after 1, 2 and 3, which is 4, and the second one is the next odd number after 3 and 5, which is 7.

Compound Words (Form New Word

Here, you’re given two groups of three words, and you have to make a word by adding one from the first group to one from the second, eg (sleek pain seek) (search green killer). Again, it’s important to be systematic, so you have to start with the first word in the first group and try to match it with each word in the second group. If that doesn’t work, repeat as necessary for the next two words in the first group. In this case, ‘pain’ goes with ‘killer’ to make ‘painkiller’.

Create a Word (from the Letters of Two Others)

These questions give you two groups of three words with the middle one in brackets in the first group and missing in the second, eg arise (rage) gears paste ( ) moans. What you have to do is work out what the missing word is by finding where the letters in the word in brackets in the first group come from. They are all taken from the words outside the brackets, so it’s just a case of working out which letter in the words outside the brackets matches each letter in the word inside the brackets. Your best bet is to write down the second group of words underneath the first and go through each letter one by one. Just look out for letters that either appear twice in one of the words or letters that appear in both words outside the brackets. Those will obviously give you two different possible letters for the answer word, so you should probably write both of them one above the other until you’ve worked everything out and then simply choose the one that makes a proper word. In this example, the R from ‘rage’ might come from ‘arise’ or ‘gears’, so the first letter of the answer word is going to be either the second letter of ‘paste’ (A) or the fourth letter of ‘moans’ (N). The same is true of the A and E in ‘rage’. Once you work it all out, the letters are a or n, p or a, m and e or o, and the only sensible word is ‘name’.

Similar Meaning

These questions are slightly different from the synonym questions in that you have to choose a word out of five that has some similarity to or relationship with two pairs of words in brackets, eg (alter, amend) (coins, money) repair, trial, revue, change, passage. The two pairs of words in brackets usually have different meanings, so you have to look for a word with a double meaning. Again, have a quick look at all the words to see if the answer’s obvious. If it is, write it down. If it’s not, go through the five words one by one, comparing them to the words in brackets. It’s important to be open to the possibility of different meanings, so try to think laterally. In this example, for instance, the answer is ‘change’ as it can work as a verb meaning ‘alter’ or ‘amend’ but also as a noun meaning ‘coins’ or ‘money’.

Letter Relationships

For these questions, you’re given a sentence that describes the relationship between two pairs of letters – a little bit like the sentence analogies earlier. The final pair of letters is missing, so you have to work out what they are by finding the relationship between the first two pairs, eg CG is to ED as BW is to ( ). You should see an alphabet line to help you. The first relationship to look at is between the first letter of the first two pairs. In this case, you get from C to E by moving forward two places in the alphabet. That means you need to move two places on from B to get the first letter of the missing pair, which is D. Repeat this for the second letters, and you’ll find the other half of the answer. In this case, you get from G to D by going back three places, so you have to go back the same three places from W to get T. The overall answer is therefore DT.

Comprehension

The exact format of comprehension questions differs, but you’ll usually be given a lot of information about different people, and you’ll have to find the missing data. The subject could be people’s heights or ages, or it could be a schedule of events. For example, three children – Susan, George and Ryan – all left school at 1515 and walked home. Susan arrived home first. George arrived home five minutes later at 1530. It took Ryan 10 minutes longer than Susan to walk home. What time did Ryan get home?. The way to approach any of these questions is to build a complete picture of the situation by starting with something you know and then working from there – a bit like building a jigsaw. Start with the absolute data (about heights, ages or times) and then move on to the relative data (comparing other people’s heights, ages or times). One thing that often helps is to draw a timeline or simply write down the names of the children in order (of height, age etc). In this example, a timeline is probably your best option, starting at 1515 when the children left school and including George getting home at 1530. You can then add in Susan’s arrival time of 1525 (as she arrived five minutes before George) and finally Ryan’s arrival time of 1535 (as he arrived 10 minutes after Susan.

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Non-verbal Reasoning

Non-verbal reasoning tests are commonly found in Common Entrance exams at 11+ and 13+ level, and they’re designed to test pupils’ logical reasoning skills using series of shapes or patterns. It’s been said that they were intended to be ‘tutor-proof’, but, of course, every kind of test can be made easier through proper preparation and coaching.

Bond produces a lot of useful books of past papers, and there is also a Bond guide on How To Do Non-verbal Reasoning available from Amazon for £8.98. This article is partly a summary of that book, but it’s useful to know how Bond thinks pupils should be doing the questions as they’re the ones producing most of them!

The first thing to do is to describe the kind of questions that are involved. Here is the list taken from the back of one of the Bond papers:

  • Finding the most similar shape
  • Finding a shape within another shape
  • Finding the shape to complete the pair
  • Finding the shape to continue the series
  • Finding the code to match the shape
  • Finding the shape to complete the square
  • Finding the shape that is a reflection of a given shape
  • Finding the shape made when two shapes are combined
  • Finding the cube that cannot be made from a given net

Bond divides the questions into four different types:

  • Identifying shapes
  • Missing shapes
  • Rotating shapes
  • Coded shapes and logic

Each of these types is divided into various subtypes.

Identifying shapes

Types of question

  • Recognise shapes that are similar and different
  • Identify shapes and patterns
  • Pair up shapes

Sample questions

  • “Which is the odd one out?”
  • “Find the figure in each row that is most unlike the other figures.”
  • “Which pattern on the right belongs with the two on the left?”
  • “Which pattern on the right belongs in the group on the left?”
  • “Which shape is most similar to the shapes in the group on the left?”

Missing shapes

Types of question

  • Find shapes that complete a sequence
  • Find a given part within a shape
  • Find a missing shape from a pattern

Sample questions

  • “Which one comes next?”
  • “Which pattern completes the sequence?”
  • “Choose the shape or pattern the completes the square given.”
  • “In which larger shape or pattern is the small shape hidden?”
  • “Find the shape or pattern which completes or continues the given series.”

Rotating shapes

Types of question

  • Recognise mirror images
  • Link nets to cubes

Sample questions

  • “Work out which option would look like the figure on the left it it was reflected over the line.”
  • “Work out which of the six cubes can be made from the net.”

Coded shapes and logic

Types of question

  • Code and decode shapes
  • Apply shape logic

Sample questions

  • “Each of the patterns on the left has a two-letter code. Select the correct code for the shape on the right following the same rules.”
  • “Select the code that matches the shape given at the end of each line.”
  • “Which one comes next? A is to B as C is to ?”
    “Which pattern on the right completes the second pair in the same way as the first pair? A is to B as C is to ?”

Hints and tips

The Bond book goes into great detail about how to answer each individual type of question, but here we’ll only look at a few key things to look for:

  • Function
  • Location
  • SPANSS
  • Story
  • Symmetry

When looking for similarities between shapes, one thing to think about is the ‘function‘ of the objects shown. In other words, what are they for? If all but one of the drawings show kitchen equipment, then the bedside lamp must be the odd one out.

Another way of looking at it is to think about is the ‘location‘ of the objects shown. Where would you usually find them? If there is a rolling pin together with a lot of tools you’d find in the garage, then the tools ‘belong’ together in the same set.

Another useful way of working through a question is to use ‘SPANSS‘, which stands for Shape, Position, Angle, Number, Shading and Size (NOT ‘sides’, as some people have written online!). This is a list of all the possible things that can change in a diagram. Non-verbal reasoning questions demand that you’re very disciplined, logical and systematic when working through all the possibilities, so it’s useful to have a mnemonic such as SPANSS to help you tick off all the options.

If none of those works, another thing you can look for is a ‘story‘? For example, do the pictures show the steps you take to get ready for school in the morning, such as getting up, brushing your teeth, getting dressed and having breakfast?

Finally, look out for ‘symmetry‘. Could the images be reflections of each other, or could they show rotational symmetry – in other words, has one pattern simply been turned upside-down or turned 90 degrees?

I hope this brief outline has been useful. Beyond that, practice makes perfect, and a few lessons with a private tutor wouldn’t go amiss either…!

Creating off-the-shelf characters

Common entrance exams have a time limit. If they didn’t, they’d be a lot easier! If you want to save time and improve your story, one thing you can do is to prepare three ‘off-the-shelf’ characters that you can choose from. You can work on them beforehand, improving them and memorising them as you go. By the time the exam comes around, it’ll be easy to dash off 8-10 lines about one of your favourite characters without having to spend any time inventing or perfecting them.

Here’s what you need to do.

The first thing to say is that you need your characters to be a little out of the ordinary. Most pupils writing stories tend to write about themselves. In other words, 10-year-old boys living in London tend to write stories about 10-year-old boys living in London! Now, that’s all very well, and the story might still get a good mark, but what you want to try and do is stand out from the crowd. Why not write a story about an 18-year-old intern at a shark research institute in the Maldives?! To decide which one you’d rather write about, you just have to ask yourself which one you’d rather read about. One thing you can do to make sure your characters are special is to give them all what I call a ‘speciality’ or USP (Unique Selling Proposition). It might be a superpower such as X-ray vision or mind-reading, or it might be a special skill such as diving or surfing, or it might be a fascinating back-story such as being descended from the Russian royal family or William Shakespeare – whatever it is, it’s a great way to make your characters – and therefore your stories – just that little bit more interesting.

Secondly, ou should also make sure all your characters are different. Try to cover all the bases so that you have one you can use for just about any story. That means having heroes that are male and female, old and young with different looks, personalities and nationalities. For instance, Clara might be the 18-year-old intern at a shark research institute in the Maldives, Pedro might be the 35-year-old Mexican spy during the Texas Revolution of 1835-6 and Kurt might be the 60-year-old Swiss inventor who lives in a laboratory buried deep under the Matterhorn! Who knows? It’s entirely up to you.

Thirdly, creating an off-the-shelf character is a great way to force yourself to use ‘wow words’ and literary techniques such as metaphors and similes. You may have learned what a simile is, but it’s very easy to forget to use them in your stories, so why not describe one of your heroes as having ‘eyes as dark as a murderer’s soul’? If you use the same characters with similar descriptions over and over again, it’ll become second nature to ‘show off’ your knowledge, and you can do the same with your vocabulary. Again, why say that someone is ‘big’ when you can say he is ‘athletic’, ‘brawny’ or ‘muscular’?

Fourthly, try to stick to what you know. If you’ve never even ridden on a horse, it’s going to be quite tough to write a story about a jockey! Alternatively, if you’ve regularly been to a particular place on holiday or met someone you found especially interesting, then use what you know to create your characters and their backgrounds. It’s always easier to describe places if you’ve actually been there, and it’s easier to describe people if you know someone similar.

So what goes into creating off-the-shelf characters? The answer is that you have to try and paint a complete picture. It has to cover every major aspect of their lives – even if you can’t remember all the details when you come to write the story. I’d start by using the following categories:

  • Name
  • Age
  • Job or education
  • Looks
  • Home
  • Friends and family
  • Personality
  • USP (or speciality)

Names are sometimes hard to decide on, so you might want to leave this one to last, but you just need to make sure it’s appropriate to the sort of character you’re creating. It wouldn’t be very convincing to have a Japanese scientist called Emily!

Age is fairly easy to decide. Just make sure your three characters are different – and not too close to your own age!

Job or education goes a long way to pigeon-holing someone. You can tell a lot from what someone does for a living or what they are doing in school or at university. You can include as much or as little detail as you like, but the minimum is probably the name and location of the school or college and what your characters’ favourite subjects are. You never know when it might come in handy!

Looks includes hair, eye colour, build, skin colour and favourite clothes. The more you describe your heroes’ looks, the easier it’ll be for the reader to imagine them.

Home can again be as detailed as you like, but the more specific the better. It’s easier to imagine the captain of a nuclear submarine patrolling under the North Pole than someone simply ‘living in London’…

Friends and family are important to most people, and it’s no different for the heroes of your stories. We don’t need to know the names of all their aunts, uncles, cousins and grandparents, but we at least need to know who they live with and who their best friends are.

Personality covers many things, but it should show what your characters are ‘like’ and what their interests are. Again, you don’t have to go into enormous depth, but it’s good to introduce the reader to qualities that might be needed later on in the story, such as athleticism or an ability to sail a boat.

USP (or speciality) covers anything that makes a character worth reading about. One of the reasons Superman is so popular is his super powers: his ability to fly, his X-ray vision and the fact that he’s invulnerable. His greatest weakness is also important: Kryptonite. It’s the same for your characters. What can they do that most people can’t? What qualities can they show off in your stories? What will make them people we admire, respect and even love?

Once you’ve created the notes for your three characters, you can write a paragraph of 8-10 lines about each of them. This is your chance to create something that you can easily slot into any of your stories, so use the past tense and stick to what the characters are like, not what they’re doing. That will be different in each story, so you don’t want to tie yourself down.

Try using your characters for stories you’re asked to write by your English teacher (or tutor, if you have one). The more often you use them, the better they’ll get as you change things you don’t like about them, bring in new ideas and polish the wording.

Children’s reading list

Books, books, books…

I’m often asked by parents what books they should try to get their children to read, but I don’t think I’ve been much help so far, so this is my attempt to do better!

Tastes differ, obviously, so perhaps the best thing I can do is to list all the books that I loved when I was a boy. I can’t remember exactly how old I was when I read them, so you’ll have to use your common sense, but they did at least provide me with happy memories.

Ronald Welch

My favourite series of books when I was a child was the one written by Ronald Welch about the Carey family. He wrote about the men in the family over the course of around 500 years, from 1500 up to the First World War. Each novel focused on one character in one particular period – rather like Blackadder, and there was a clear formula: whatever the period, he would have to fight a duel, he would do something heroic and he would win the fair lady! The duels started with a dagger and a sword and then moved on to rapiers and then finally pistols as the years rolled on. I loved the military aspect to the books – as most boys would – and I read just about every single one I could get my hands on. Unfortunately, they’re almost impossible to find in print nowadays, but it’s always worth a look…

CS Forester

CS Forester wrote the ‘Hornblower’ novels. I was interested in both sailing and military history when I was young, and this sequence of novels about a naval officer called Horatio Hornblower in the Revolutionary and Napoleonic Wars from 1792-1815 was a perfect blend of the two.

Alexander Kent (Douglas Reeman)

Alexander Kent was the pen name of Douglas Reeman, who wrote a series of novels about Richard Bolitho. I first came across him after finishing all the CS Forester novels, and he provided a similar mix of nautical and military history during the same period. They weren’t quite as good as the Hornblower novels, but I still enjoyed them.

Enid Blyton

I didn’t read absolutely all the Enid Blyton books when I was a boy, but the one that I do remember is The Boy Next Door. Among other things, I loved the name of the character (‘Kit’), I loved the bits about climbing trees and I also loved the word ‘grin’, which I never understood but thought was somehow magical!

Roald Dahl

Again, I don’t remember reading all the Roald Dahl novels, but James and the Giant Peach left a big impression. The characters were so interesting, and the idea of escaping from home on an enormous rolling piece of fruit was very exciting to me in those days…!

Sir Arthur Conan Doyle

I read The Complete Adventures of Sherlock Holmes when I was a boy, and it’s probably still the longest book I’ve ever read. I remember vividly that the edition I read was 1,227 pages long! I listened to the whole thing again recently in a very good audiobook edition read by Stephen Fry, and it was just as good second time around. I loved the mystery of the stories, and I still read a lot of crime fiction even now. I’ve always had a very analytical mind, so Holmes’s brilliant deductions were always enjoyable to read about.

Charlie Higson

The Young Bond novels weren’t around when I was young, but I read the first few as an adult, and I enjoyed them. James Bond is a classic fictional creation that appeals to boys in particular, and I think I would’ve lapped it up as a teenager. The first one is called Silverfin. Once you’ve read it, you’ll be hooked!

Jane Austen

Jane Austen introduced me to irony with the immortal opening line from Pride and Prejudice, but the first of her novels that I read was actually Emma. I had to read it at school as part of my preparation for the Oxford entrance exam, and I didn’t like it at first. However, that was just because I didn’t understand what was going on. Once my English teacher Mr Finn had explained that the character of Emma is always wrong about everything, I found it very funny and enjoyable. They say that ‘analysing’ a book can sometimes ruin it, but in this case it was quite the opposite.

Ernest Hemingway

“If Henry James is the poodle of American literature, Ernest Hemingway is the bulldog. What do you think?” I was once asked that question in an interview at the University of East Anglia, and I had no idea how to reply! As it happens, Hemingway was one of my favourite authors. My interviewer called his style ‘macho’, but that wasn’t the appeal for me. I simply liked the stories and the settings. I particularly loved the bull-fighting scenes in The Sun Also Rises, and there was just a glamour to the characters and the period that I really enjoyed. If you don’t know where to start, try The Old Man and the Sea. It’s very simple and very short, but very, very moving.

Describing feelings

In many 11+ and 13+ exams, you have to talk about feelings. Yes, I know that’s hard for most boys that age, but I thought it might help if I wrote down a list of adjectives that describe our emotions. Here we go…

A bloke called Bob (actually Robert Plutchik) thought that people only ever felt eight different emotions:

His list is shown in this ‘wheel of emotions’. The basic eight feelings are:

  • Ecstasy
  • Admiration
  • Terror
  • Amazement
  • Grief
  • Loathing
  • Rage
  • Vigilance

If we had a think about all the adjectives that are associated with these categories (and sub-categories), we might come up with a list like this one:

  • Ecstasy
    crazy
    delirious
    ecstatic
    elated
    enthusiastic
    euphoric
    fervent
    glad
    happy
    joyful
    joyous
    mad
    overjoyed
    rapturous
    rhapsodic
    serene
    thrilled
    upbeat
  • Admiration
    accepting
    admiring
    adoring
    appreciative
    loving
    respectful
    trustful
    trusting
  • Terror
    afraid
    aghast
    alarmed
    apprehensive
    awed
    frightened
    frozen
    scared
    submissive
    terrified
  • Amazement
    amazed
    astonished
    astounded
    awe-struck
    bewildered
    dazed
    distracted
    dumbfounded
    flabbergasted
    impressed
    perplexed
    shocked
    staggered
    startled
    stunned
    surprised
    unprepared
  • Grief
    bitter
    grief-stricken
    grieving
    heart-broken
    melancholy
    mournful
    pensive
    pessimistic
    sad
    somber
    sorrowful
    sorry
    unhappy
    wistful
  • Loathing
    appalled
    bored
    disapproving
    disgusted
    outraged
    queasy
    tired
    weary
  • Rage
    angry
    aggressive
    annoyed
    contemptuous
    enraged
    exasperated
    furious
    heated
    impassioned
    indignant
    irate
    irritable
    irritated
    offended
    resentful
    sullen
    uptight
  • Vigilance
    anxious
    aware
    cautious
    circumspect
    expectant
    interested
    keen
    observant
    optimistic
    vigilant
    wary

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

Descriptive writing

Grand Central

Exams at 11+ and 13+ level always let you tell a story in the writing section, but they sometimes provide a picture and simply ask you to describe it or to ‘write about it in any way you like’. Writing a description is obviously different from writing a story, so it’s worthwhile pointing out the differences and the similarities.

When you write a story, the best way of doing it is probably to follow the five-step process that I outline in Story Mountains:

  1. Choose the title
  2. Brainstorm for ideas
  3. Plan your work
  4. Write it
  5. Check it

You can use a similar basic method for doing a description – except the planning stage obviously doesn’t involve creating a story mountain! – but what are the differences? Steps 1, 4 and 5 are pretty much the same, but you might want to have a look at these tips for the brainstorm and planning.

Brainstorm

When you’re brainstorming for a story, you have to think about characters, genre, period, setting and plot, and you also have to make sure there’s a ‘problem’ to solve so that your idea fits into a story mountain. However, descriptions don’t necessarily have all of those things in them, so you have to think about it in a different way.

The simplest form of description would simply involve describing what’s in a picture (or imagining what’s there if you’re just given the title). That might result in some very imaginative creative writing and open up the possibility of using some great vocabulary and all the poetic devices you can think of, including similes and metaphors. However, the very best descriptions have to have some kind of ‘hook’ to grab the reader’s attention, and that usually means a central character, situation or even a mini ‘plot’. You obviously need to describe exactly what’s in the picture, but what if you want to say more? What if the picture doesn’t have the things in it that you want. That’s a bit tricky, but you can always ask questions or just ask the narrator to imagine things.

Hal Morey’s picture of New York’s Grand Central Station is a good example. The shot has lots of elements to it, including the architecture, the people and the beams of light from the windows, so you could easily spend your whole time going over the picture in great depth, picking out each detail and thinking up the best words and metaphors to describe it. Vocabulary is important here, so you might make a list of the words that you planned to use. One good way of organising this is to think about the five senses: sight, hearing, taste, smell and touch. It’s sometimes difficult to get beyond the visual when you’re analysing a picture, so this method just forces you to think of the other aspects of a scene. Try to make separate lists for each of the senses. The visual vocabulary might include the following:

  • columns (not just walls)
  • vaulting (not just roof)
  • passengers (not just people)
  • spotlights (not just light)
  • cathedral (not just station)
  • nave (not just hall)
  • balustrade (not just railing)

It’s not enough to use words like ‘roof’ when more imaginative synonyms exist such as ‘vaulting’, so try to think of the very best words to use. After all, the examiner can’t tell that you know a word unless you use it! And the same goes for metaphors. Where did the word ‘cathedral’ come from? Well, the shape of the hall and all the windows are similar to what you’d find in a cathedral or a large church, so why not use that in a kind of ‘extended metaphor’? An extended metaphor takes a comparison and uses it more than once, so the main part of the hall might be the ‘nave’, the two large arched windows might be the ‘west windows’ and the people might be the ‘congregation’. Even better would be to link the metaphor to the purpose of the other building by saying something like this: “The congregation bustled to find their seats in the pews as they made their daily act of worship to the god of commerce.” Suddenly, you’ve gone from a bland description of what you can see for yourself to a new and imaginative way of looking at the world.

The other category of words on your checklist should be feelings. Why simply describe what people look like and not examine how they feel? As an example, use the context of Grand Central Station to imagine what’s going through the passengers’ minds? Are they bored, are they reluctant to go to work, or are they happy to be bunking off for a day at the beach?!

So what else can you do to take your writing to the next level? The answer is to introduce a main character or some kind of situation or miniature plot. You’re hardly ever ‘banned’ from using a plot in this kind of question, so there’s no problem with introducing one, but let’s stick with the idea of doing a description rather than a story. The Grand Central picture is again a good example. What would be the character or the situation or the plot here? Well, the obvious idea is to pick is a commuter who’s late for his train. You could introduce the description by focusing on one individual in the picture and explaining why he’s in a hurry. You could then have a kind of countdown clock as you described all the people and objects he sees as he rushes to make his train:

     Lionel Carey was in a hurry. He only had five minutes to make his train to get to the most important meeting of his life! He struggled along with his precious, old, leather briefcase, catching his fedora as it was blown off his head and cursing the long overcoat he had chosen to wear as it made him sweat uncomfortably and almost tripped him over. Now, what platform did he need?
     Four minutes to go…

And so on. This gives the passage a clearer focus and a sense of tension, excitement and mystery. Will Lionel catch his train? Where is he going? What is the meeting about? It just adds another layer to the description – and ideally leads to higher marks!

Alternatively, you can talk about things that are not in the picture by doing one of the following:

  • ask questions, eg if the picture was of the Colosseum in Rome, you could ask questions like Was this where the Roman gladiators stood before they made their way to their deaths in the arena?
  • create a section in which the narrator imagines objects or events, eg It was here that the emperor would stand before giving the thumbs-up or thumbs-down sign that would signal the fate of the gladiators.

Planning

It’s fairly obvious how to plan for a story because it has to have a plot, but how do you plan for a description? Do you just describe what’s in the picture, starting perhaps on the left and working  your way across? Or do you separate your work into five different paragraphs on each of the senses, with perhaps an extra one for the feelings of the travellers? Or can you introduce a timeline, charting the progress of an imaginary character – such as Lionel Carey, hurrying to catch his train? Each one might work, but you’ll probably get the best marks for something that engages the reader, and the best way of doing that is to have a central character and a carefully selected situation to place him in:

Lionel Carey in a hurry – needs to catch train for meeting, looks for platform

4 mins

Describe Lionel – importance of what’s in briefcase, mysterious ‘she’ he’s meeting
Describe station – architecture, light, people

3 mins

Describe people he sees
Bumps into coffee cart
Argues with staff
Will he ever see ‘her’?

2 mins

Describe trains – steam, smoke, whistle
Wrong platform – needs to run to Platform 16

1 min

Describe running, bumping into people, curses

Time’s up!
Too late – but wait! Train is delayed. He can give daughter Xmas present after all!

Whatever the picture or title, try giving this method a try. If you brainstorm and plan correctly, focusing on all five senses and people’s feelings and using a central character to add excitement and mystery, I’m sure you’ll do a good job.

What is a full sentence?

SVO

Teachers often tell pupils to use a ‘full sentence’ in their answers, but what is a full sentence?

Parts of a sentence

First of all, it’s important to know what all the words in the picture mean. (Note that the parts of a sentence are not always individual words, though they can be. For example, ‘she’ is the subject, but ‘a hot dog’ is the direct object even though it is three words.)

Subject

The subject of the sentence performs the action of the verb, in other words, “who’s doing the doing”. The girl in the picture is obviously the one doing the eating, not the hot dog!

Verb

The verb is often called the ‘doing word’, although some verbs like ‘being’ and ‘having’ don’t really involve much ‘doing’! Again, what’s being done is obviously shown by the word ‘ate’. There are two kinds of verb:

  • Transitive verbs need a direct object, like the word ‘ate’ in the picture
  • Intransitive verbs don’t need an object, like the word ‘swim’ in ‘They swim’.

Object

There are two kinds of object:

  • Direct objects are directly affected because they ‘suffer the action of the verb’. In other words, they have something done to them, like the hot dog in the picture.
  • Indirect objects are only indirectly affected, for example if they benefit from the verb like the teacher receiving an apple in the picture below.

Types of sentence

Now we know what the parts of a sentence are, we can talk about all the possible kinds of full sentence.

  1. Verb only
    Strictly speaking, all you really need to make a full sentence is a verb. For example, ‘Sit!’ is a full sentence, even though it only has one word in it. That only works when you’re telling a dog – or a person! – what to do. Most of the time, you need a subject as well.
  2. Subject-Verb
    ‘He swims’ is a full sentence because it has a subject and a verb, but this only works because the verb is intransitive, which means it doesn’t need an object.
  3. Subject-Verb-Object
    The picture above shows the main parts of a simple sentence, which are the subject (S), the verb (V) and the object (O). The initial letters give us a typical pattern for a sentence, which is SVO. In this case, the object is a direct object, which means it’s directly affected but it can also be an indirect object, which may benefit indirectly. Here, the hot dog is the one that has to suffer being eaten – not the girl! – but it’s slightly different in the next picture. _
  4. Subject-Verb-Indirect Object-Direct Object (or Subject-Verb-Direct Object-Indirect Object)
    Here, the apple is being ‘given’, so the apple is the direct object, but the teacher also benefits indirectly, so she is the indirect object.

Common mistakes

  1. Punctuation
    Every sentence should end with either a full-stop, a question mark or an exclamation mark, but one common mistake is to put a comma in between sentences instead, eg He loved pizza, he always chose pepperoni. This is called the ‘comma splice’.
  2. Fragments
    A sentence that doesn’t have a subject, verb or object when it needs one is called a ‘sentence fragment’, eg Gave his teacher an apple. It’s obvious that it doesn’t make sense without the word ‘He’, but it’s easily done.
  3. Starting with conjunctions
    Teachers often tell their pupils not to start a sentence with ‘because’. When asked a question like ‘Why is Jack sad?‘, it’s easy to write ‘Because his dog died‘. That’s all right when you’re speaking in class – when people don’t care as much about their grammar – but not when you’re doing your homework. It’s not always wrong to do it, though. If you use ‘because’ to link two sentences together, that’s fine, eg Because it was so sunny, I had to wear sun cream.

 

You must be crazy!

2+2=5

Why? What’s wrong with that…?

Teachers and tutors ask pupils to check their work, but how can you do that in Maths without doing the whole sum all over again? Well, you can’t! So how are you supposed to check your work?

What you have to understand first of all is that checking everything is right is very different from checking nothing is obviously wrong. To check everything is right means doing the whole paper twice, but you obviously don’t have time to do that. Checking nothing is obviously wrong is much easier because it just means doing a ‘quick and dirty’ calculation in your head. It doesn’t guarantee that the answer is right, but it’s a good compromise. I call it ‘sanity checking’, which means making sure your answers are not crazy! Unfortunately, there isn’t one method that works for every question – it depends on what type of question it is – but here are a few examples:

Algebra

If you have to ‘solve for 𝑥’ and it’s a difficult question, try putting your answer back into the original equation and seeing if one side equals the other, eg if you think 𝑥 = 5, then that works for 2𝑥 + 6 = 16, but not for 3𝑥 + 2 = 5. That would be crazy!

Multiplication

Every multiplication sum starts with multiplying the last digit of each number together, so try doing that when you’ve got your answer and checking if the last digit of the result is equal to the last digit of the answer, eg 176 x 467 is going to end in a 2 because 6 x 7 = 42, which also ends with a 2. Your answer couldn’t end in any other number. That would be crazy!

Rounding

If you have any kind of sum that involves adding, subtracting, multiplication or division, an easy way to check it is to round the numbers to one or two significant figures (eg to the nearest hundred) and work out the answer in your head. If it’s close enough, then your answer is not obviously wrong. If it’s nowhere near, then you’ll have to do it again, eg 1.7 x 3.4 is close to 2 x 3, so the answer might be 5.78, but it wouldn’t be 57.8. That would be crazy!

Units

Most answers in Maths tests need some kind of unit, such as kg, m, cm or ml. Sometimes, the units are provided, but sometimes they’re not. If they’re not, you just need to make sure that you use the right ones, eg if the scale of a map is 1:100,000, the distance represented by 9.8cm is 9.8km, not 9.8m. That would be crazy!

 

Negative numbers

negative-numbers

Working with negative numbers can be confusing, but a few simple rules can help you add, subtract, multiply and divide successfully:

Adding and subtracting

If the operator and the sign are the same, then you must add the number, eg 4 – -3 = 4 + 3 = 7 or 4 + +3 = 7.

If the operator and the sign are different, then you must subtract the number, eg 4 + -3 = 4 – 3 = 1 or 4 – +3 = 1.

Here are a few practice questions:

  1. 14 – -3
  2. -4 + -7
  3. 21 – +8
  4. -8 + -7
  5. 6 – -4

Multiplication and division

If the signs of the two numbers are the same, then you must make the answer positive, eg -4 x -3 = 4 x 3 = 12 or +4 x +3 = 4 x 3 = 12.

If the signs of the two numbers are different, then you must make the answer negative, eg -4 x +3 = -12 or +4 x -3 = -12.

Here are a few practice questions:

  1. 1 × -3
  2. -8 ÷ -4
  3. 2 × -7
  4. -14 ÷ -7
  5. 6 × -4

Useful terms in Maths

maths

Maths is complicated, but a good first step on the road to understanding it is to get to know the most useful terms. There are lists in the front of the Bond books, but here’s my own contribution. I hope it helps!

Algebra: expressions using letters to represent unknown values, eg 2(x + 3) = 16.

Angles: there are three types of angle, depending on the number of degrees.

  • acute angles are between 0 and 90 degrees.
  • obtuse angles are between 90 and 180 degrees.
  • reflex angles are between 180 and 360 degrees.

Arc: part of the circumference of a circle.

Averages: there are three types of average, and they are all useful in different ways.

  • The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
  • The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
  • The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.

Chord: a straight line drawn between two points on the circumference of a circle.

Circumference: the distance all the way round the edge of a circle.

Congruent: triangles are congruent if they are the same shape and size, eg two right-angled triangles with sides of 3cm, 4cm and 5cm would be ‘congruent’, even if one is the mirror image of the other. You can prove that two triangles are congruent by using any of the following methods: SAS (Side-Angle-Side), SSS (Side-Side-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side) and RHS or HL (Right-angle-Hypotenuse-Side or Hypotenuse-Leg). If all three measurements of the angles and/or sides are equal, the triangles are congruent. You can only create a congruent copy of a triangle by translation, reflection or rotation. (Note: congruence is the same as similarity, except that the triangles have to be the same size.)

Cuboid: a solid with a rectangle for each of the six sides, eg a shoe box.

Denominator: the number on the bottom of a fraction, eg 2 is the denominator of ½.

Diameter: the length of a line drawn across a circle passing through the centre.

Equation: any line of numbers and operators with an equals sign in the middle, showing that the two sides balance, eg 4x + 12 = 34.

Factor: a number that goes into another number evenly, eg 6 is a factor or 12.

Fibonacci series: a sequence of numbers created by adding the previous two numbers together to get the next one, eg 1, 1, 2, 3, 5, 8, 13…

Formula: a way of calculating the answer to a common problem using letters or words, eg the formula for distance is speed x time (or D = S x T).

Highest common factor (or HCF): the highest number that goes into two other numbers evenly, eg the HCF of 12 and 18 is 6.

Improper fraction: a fraction that is greater than one (in other words, the numerator is greater than the denominator), eg 9/5.

Lowest common multiple (or LCM) / Lowest common denominator (or LCD): the lowest number that is divisible by two other numbers, eg the LCM of 6 and 8 is 24.

Multiple: a number that can be divided evenly by another number, eg 12 is a multiple of 6.

Numerator: the number on the top of a fraction, eg 3 is the numerator of ¾.

Order of operations: the sequence of doing basic mathematical sums when you have a mixture of, say, addition and multiplication. BIDMAS (or BODMAS) is a good way of remembering it, as it stands for:

  • Brackets
  • Indices/Order (in other words, squares, cubes and so on)
  • Division and Multiplication
  • Addition and Subtraction

Note that division doesn’t come ‘before’ multiplication, and addition doesn’t come ‘before’ subtraction – these operations have to be done in the order in which you find them. For division and multiplication, it doesn’t really matter as it doesn’t change the answer, eg 5 x 8 ÷ 2 = 20, whether you do 5 x 8 first or 8 ÷ 2. However, it does matter with addition and subtraction, eg 4 – 3 + 2 = 3 if you do the operations in order, which is correct, but you’d get the wrong answer of -1 if you did 3 + 2 first.

Operator: the sign telling you which mathematical operation to do. The most common ones are +, -, x and ÷.

Parallel: two lines are parallel if they will never meet, eg the rails on a railway line.

Perimeter: the distance all the way round the outside of a shape.

Perpendicular: at 90 degrees to each other.

Pi (or π): a constant used to work out the circumference and area of circles, often shown as 22/7 or 3.14 although it’s actually an ‘irrational’ number, which means it goes on for ever.

Prime factors: the lowest prime numbers that can be multiplied together to make a given number, eg the prime factors of 12 are 2² x 3.

Prime numbers: a number that can only be divided by itself and one, eg 2, 3, 5, 7, 11, 13…

Product: the result of multiplying two numbers together, eg 35 is the product of 5 and 7.

Quadrilateral: a four-sided shape such as the following:

  • Kite: a quadrilateral with two pairs of equal sides next to each other (or ‘adjacent’ to each other).
  • Parallelogram: a quadrilateral with opposite sides parallel to each other.
  • Rectangle: a quadrilateral with two opposite pairs of equal sides and four right angles.
  • Rhombus: a quadrilateral with equal sides.
  • Square: a quadrilateral with equal sides and four right angles.
  • Trapezium: a quadrilateral with one pair of parallel sides. (Note: an isosceles trapezium is symmetrical.)

Radius: the distance from the centre of a circle to the circumference.

Range: the highest minus the lowest value in a list, eg the range of the numbers 1-10 is 9.

Regular: a shape is regular if all its sides and angles are equal, eg a 50p piece is a regular (-ish!) heptagon.

Right angle: an angle of 90 degrees.

Sector: a ‘slice’ of a circle in between two radii.

Segment: a part of a circle separated from the rest by a chord.

Shapes: the name of each shape depends on the number of sides. Here are the first 12.

  • Quadrilaterals have four sides.
  • Pentagons have five sides.
  • Hexagons have six sides.
  • Heptagons have seven sides.
  • Octagons have eight sides.
  • Nonagons have nine sides.
  • Decagons have 10 sides.
  • Hendecagons have 11 sides.
  • Dodecagons have 12 sides.

Similar: triangles are similar if they are the same shape, but not necessarily the same size, eg a right-angled triangle with sides of 3cm, 4cm and 5cm is ‘similar’ to a right-angled triangle with sides of 6cm, 8cm and 10cm. (Note: similarity is the same as congruence, except that the triangles don’t have to be the same size.)

Square number: the result of multiplying any number by itself, eg 49 is a square number, as 7 x 7 = 49.

Square root: the number that has to be multiplied by itself to make another number, eg 6 is the square root of 36, as 6 x 6 = 36.

Sum: the result of adding two numbers together, eg 17 is the sum of 8 and 9.

Tangent: either a straight line that touches the circumference of a circle OR the length of the opposite side of a triangle divided by the length of the adjacent side

Triangles: there are four main types, each with different properties.

  • equilateral triangles have all three sides the same length and all three angles the same.
  • isosceles triangles have two sides the same length and two angles the same.
  • scalene triangles have three sides of different lengths with three different angles.
  • right-angled triangles have one 90-degree angle.

Variable: an unknown in algebra, eg x or y.

Algebra

Nothing makes the heart of a reluctant mathematician sink like an algebra question.algebra

Algebra is supposed to make life easier. By learning a formula or an equation, you can solve any similar type of problem whatever the numbers involved. However, an awful lot of students find it difficult, because letters just don’t seem to ‘mean’ as much as numbers. Here, we’ll try to make life a bit easier…

Gathering terms

X’s and y’s look a bit meaningless, but that’s the point. They can stand for anything. The simplest form of question you’ll have to answer is one that involves gathering your terms. That just means counting how many variables or unknowns you have (like x and y). I like to think of them as pieces of fruit, so an expression like…

2x + 3y – x + y

…just means ‘take away one apple from two apples and add one banana to three more bananas’. That leaves you with one apple and four bananas, or…

x + 4y

Here are a few practice questions:

  1. 3x + 4y – 2x + y
  2. 2m + 3m – 2n + 3n
  3. p + 2q + 3p – 3q
  4. 2a – 4b + a + 4b
  5. 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:

  1. 2(a + 5)
  2. 3(y + 2)
  3. 6(3 + b)
  4. 3(a – 3)
  5. 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:

  1. Multiplying out any brackets
  2. Adding or subtracting
  3. 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

Simple!

Here are a few practice questions:

  1. b + 5 = 9
  2. 3y = 9
  3. 6(4 + c) = 36
  4. 3(a – 2) = 24
  5. 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:

  1. (a + 1)(b + 2)
  2. (a – 1)(a + 2)
  3. (b + 1)(a – 2)
  4. (p – 1)(q + 2)
  5. (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:

  1. 30 – 3(p – 1) = 0
  2. 20 – 3(a – 3) = 5
  3. 12 – 4(x – 2) = 4
  4. 24 – 6(x – 3) = 6
  5. 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).

Spelling rules

The problem with the English is that we’ve invaded (and been invaded by) so many countries that our language has ended up with a mish-mash of spelling rules.

Keep calm and check your spelling

 

 

 

 

 

 

English is among the easiest languages to learn but among the most difficult to master. One of the problems is spelling. We have so many loan words from so many different languages that we’ve been left with a huge number of spelling rules – and all of them have exceptions! Contrast that with Spanish, for example, where what you see is generally what you get. The problem for students of English, then, is that it’s very difficult to find shortcuts to improve your spelling, and an awful lot of words just have to be learned off-by-heart. Considering that there are over a million words in English, that’s a big ask!

There are lists of spelling rules out there (including a good one at www.dyslexia.org), but I thought I’d put down what I think are the most useful ones.

  1. I before E except after C when the sound is /ee/.
    This is the most famous rule of English spelling, but there are still exceptions! Hence, we write achieve with -ie- in the middle but also ceiling, with -ei- in the middle, as the /ee/ sound comes after the letter c. The most common exceptions are weird and seize.
  2. If you want to know whether to double the consonant, ask yourself if the word is like dinner or diner.
    One of the most common problems in spelling is knowing when to double a consonant. A simple rule that helps with a lot of words is to ask yourself whether the word is more like dinner or diner. Diner has a long vowel sound before a consonant and then another vowel (ie vowel-consonant-vowel, or VCV). Words with this long vowel sound only need one consonant before the second vowel, eg  shinerfiver and whiner. However, dinner has a short first vowel and needs two consonants to ‘protect’ it (ie vowel-consonant-consonant-vowel, or VCCV). If the word is like dinner, you need to double the consonant, eg winnerbitter or glimmer. Just bear in mind that this rule doesn’t work with words that start with a prefix (or a group of letters added to the front of a word), so it’s disappoint and not dissapoint.
  3. If the word has more than one syllable and has the stress on the first syllable, don’t double any final consonant.
    This rule sounds a bit complicated, but it’s still very useful – particularly if it helps you spot your teacher making a mistake! We generally double the final consonant when we add a suffix starting with a vowel, such as -ing, but this rule means we shouldn’t do that as long as a) the word has more than one syllable and b) the stress is on the first syllable, eg focusing and targeted, but progressing and regretting. The main exceptions to this are words ending in -l and -y, hence barrelling and disobeying.
  4. When adding a suffix starting with a consonant, you don’t need to change the root word unless it ends in -y. This is among the easiest and most useful rules. There are loads of words ending in suffixes like -less-ment or -ness, but spelling them should be easy as long as you know how to spell the root word, eg shoe becomes shoelesscontain becomes containment and green becomes greenness. However, words ending in -y need the y changing to an i, so happy becomes happiness.
  5. When adding a suffix starting with a vowel to a word ending in a silent -e, the must be dropped unless it softens a or a g.
    An at the end of a word is often called a ‘Magic E’, as it lengthens the vowel before the final consonant, eg fat becomes fate. However, that job is done by the vowel at the start of the suffix when it is added to the word, so it needs to be dropped, eg race becomes racing and code becomes coded. The main exceptions come when the word ends with a soft or g, which need to be followed by an -e, an -i or a -y to sound like /j/ and /s/ rather than /g/ and /k/. If the suffix doesn’t begin with an e- or an i-, we still need the to make sure the word sounds right, eg managing  is fine without the -e, as the in -ing keeps the soft, but manageable needs to keep the -e to avoid a hard /g/ sound that wouldn’t sound right.
  6. The only word ending in -full is full!
    There are lots of words ending in what sounds like -full, but the only one that has two ls at the end is full. All the other words – and there are thankfully no exceptions! – end in -ful, eg skilfulbeautiful and wonderful.

 

Parts of speech

When is a verb not a verb? When it’s a part of speech.

Parts of speech

English exams often ask questions about the ‘parts of speech’. This is just a fancy term for all the different kinds of words, but they’re worth knowing just in case. Just watch out for words such as ‘jump’, which can be more than one part of speech!

Noun: a word for a person, place or thing

  • abstract noun: a word to describe an idea, eg peace
  • common (or concrete) noun: a word for a thing or object, eg table
  • proper noun: the name of a person, place etc, eg Nick, London
  • collective noun: the name of a group of animals, eg herd or flock

Tip: Make up a phrase or a sentence with ‘the’ in front of the word. If it makes sense, it’s probably a noun, eg He looked at the ______.

Pronoun: a word that stands in for a noun

  • personal pronoun: a word that shows a person or thing, eg he, she, them
  • possessive pronoun: a word that shows the owner of an object, eg his, their
  • relative pronoun: a word that ‘relates’ to the subject just mentioned, eg who, that, which

Tip: Make up a phrase or a sentence with a verb after the word (but without ‘the’ or ‘a’ in front of it). If it makes sense, it’s probably a pronoun, eg ______ looked at the wall.

Verb: a doing word, eg jumped, was, pays

Tip: Make up a phrase or a sentence putting the word after a pronoun such as ‘he’. If it makes sense, it’s probably a verb, eg He ______ it or He ______ in the garden.

Adjective: a word that describes a noun or pronoun, eg green or young

Tip: Make up a phrase or a sentence putting the word between ‘the’ and a noun. If it makes sense, it’s probably an adjective, eg The ______ book lay on the table.

Article: a word that introduces a noun

  • definite article: the
  • indefinite article: a or an

Tip: Make up a phrase or a sentence with the word in front of a noun. If it makes sense, it’s probably an article, eg ______ book lay on the table.

Adverb: a word that describes an adjective, adverb or verb, usually ending in -ly, eg really or quickly

Tip: Make up a phrase or a sentence with the word after a verb. If it makes sense, it’s probably an adverb, eg He ran ______ around the garden.

Preposition: a word that shows the position in time or space, eg in, at or after

Tip: Make up a phrase or a sentence about placing something somewhere, putting the word before the location. If it makes sense, it’s probably a preposition, eg She put the book ______ the table.

Conjunction: a word that connects two sentences together (sometimes called a connective), eg and, but or because

Tip: Make up a phrase or a sentence with two clauses joined by the word. If it makes sense, it’s probably a conjunction, eg He looked at the problem ______ decided to do something about it.

Interjection: an outburst or word people say when they’re playing for time, eg hey, well, now or so

Tip: Make up a phrase or a sentence that someone might say, putting the word at the start, followed by a comma. If it makes sense, it’s probably an interjection, eg ______, can we go to the mall?

You can test yourself by reading any passage in English and going through it word by word, asking yourself what parts of speech they all are. Why not start with this article? See how fast you can go. If you’re not sure, ask yourself the questions in each of the tips shown above, eg if you think it’s a noun, can you put it into a sentence with ‘the’ in front of it?

Capital!

I always sign Christmas and birthday cards with a capital ‘N’. It makes me feel like Napoleon…

Letter N

The three main things to check after writing anything are spelling, punctuation and capital letters, so when do you use capitals?

  1. For the important words in titles, either of individuals or pieces of writing, eg Chief Inspector of Schools or This Blog Post is Great!
  2. For proper nouns, eg Peter, Tuesday and Chelsea.
  3. For abbreviations, eg BBC (although acronyms you can actually pronounce only need one at the start, eg Nato, not NATO).
  4. For the first word in a sentence, eg This is a sentence.
  5. For the first word in direct speech, eg he said, “Hello.”
  6. The word “I”.

Speech marks

Speech marks, inverted commas, quotation marks, quote marks, quotes, 66 and 99 – does any other punctuation mark have so many names or cause so much confusion…?!

speech marks

Writing a story means striking a balance between what I call The Three Ds: Drama, Description and Dialogue. I’ve read quite a few stories from my pupils in which nobody talks to anyone – which is a bit odd! – but you need to know the rules of punctuation before you start.

  • Start a new paragraph whenever the speaker changes or someone stops talking.
  • Put speech marks before and after the actual words spoken, eg Hello, he said, NOT “Hello, he said.”
  • Start the first spoken word with a capital letter, eg she said, “This needs a capital letter,” NOT she said, “this needs a capital letter.”
  • Put either a comma, question mark, exclamation mark or colon between the speech and the ‘he said/she said’, eg “Don’t forget the comma,” he said, NOT “Don’t forget the comma” he said.
  • Put punctuation that belongs to the speech inside the speech marks, eg “The exclamation mark belongs inside!“, NOT “The exclamation mark belongs inside”! (The only exception comes with inverted commas, which look the same but are used with quotations rather than speech.)
  • Put a full-stop after the ‘he said/she said’ if it comes in the middle of the speech and the first part is a full sentence; otherwise, just put a comma, eg “This is a full sentence,” she said. “This is, too.” BUT “This is not a full sentence,” she said, “and nor is this.”
  • Don’t start the ‘he said/she said’ with a capital letter, even if it comes after a question mark or exclamation mark, eg “Don’t use a capital letter!” he shouted, NOT “Don’t use a capital letter!” He shouted.
  • If a speech lasts more than one paragraph, put speech marks before each paragraph and after the last one but NOT after the ones before.
  • Finally, don’t put ‘he said/she said’ after every single line of dialogue in a long conversation if it’s obvious who is speaking.

Sample questions

Format and put the correct punctuation and capital letters into the following lines of speech:

  1. I say john what time is it she asked
  2. hello she said my name is tara
  3. what are you talking about he cried I never said that
  4. hello he said whats your name Sarah she said Im Alan Nice to meet you you too
  5. I hate chocolate she said I only really eat chocolate ice-cream

Ratios

Hundreds of years ago, it was traditional to put dragons on maps in places that were unknown, dangerous or poorly mapped. Ratios are one of those places…

Here be ratios...!

Here be ratios…!

A ratio is just a model of the real world, shown in the lowest terms, but answering ratio questions can be just as scary as meeting dragons if you don’t know what you’re doing. The key to understanding ratios is to work out the scale factor. This is just like the scale on a map. If a map is drawn to a scale of 1:100,000, for instance, you know that 1cm on the map is the same as 100,000cm (or 1km) in the real world. To convert distances on the map into distances in the real world, you just need to multiply by the scale factor, which is 100,000 in this case. (You can also go the other way – from the real world to the map – by dividing by the scale factor instead.)

To work out the scale factor in a Maths question, you need to know the matching quantities in the real world and in the model (or ratio). Once you know those two numbers, you can simply divide the one in the real world by the one in the ratio to get the scale factor. For example:

If Tom and Katie have 32 marbles between them in the ratio 3:1, how many marbles does Tom have?

To answer this question, here are the steps to take:

  1. Work out the scale factor. The total number of marbles in the real world is 32, and the total in the ratio can be found by adding the amounts for both Tom and Katie, which means 3 + 1 = 4. Dividing the real world total by the ratio total gives 32 ÷ 4 = 8, so the scale factor is 8.
  2. Multiply the number you want in the ratio by the scale factor. If Tom’s share of the marbles in the ratio is 3, then he has 3 x 8 = 24 marbles.

The matching numbers in the real world and the ratio are sometimes the totals and sometimes the individual shares, but it doesn’t matter what they are. All you need to do is find the same quantity in both places and divide the real world version by the ratio version to get the scale factor. Once you’ve done that, you can multiply any of the ratio numbers to get to the real world number (or divide any real world number to get to the ratio number). Different questions might put the problem in different ways, but the principle is the same.

One complication might be having two ratios that overlap. In that case you need to turn them into just one ratio that includes all three quantities and answer the question as you normally would. For example:

If there are 30 black sheep, and the ratio of black to brown sheep is 3:2, and the ratio of brown to white sheep is 5:4, how many white sheep are there?

This is a bit more complicated, but the basic steps are the same once you’ve found out the ratio for all three kinds of sheep. To do this, we need to link the two ratios together somehow, but all the numbers are different, so how do we do it? The answer is the same as for adding fractions with different denominators (or for solving the harder types of simultaneous equations, for that matter): we just need to multiply them together. If we were adding fifths and halves, we would multiply the denominators together to convert them both into tenths. Here, the type of sheep that is in both ratios is the brown one, so we simply have to make sure the numbers of brown sheep in each ratio (2 and 5) are the same by multiplying them together (to give 10). Once we’ve done that, we can combine the two ratios into one and answer the question. Here goes:

  • Ratio of black sheep to brown sheep = 3:2

Multiply by 5

  • Ratio of black sheep to brown sheep = 15:10
  • Ratio of brown to white sheep = 5:4

Multiply by 2

  • Ratio of brown to white sheep = 10:8
  • Therefore, ratio of black sheep to brown sheep to white sheep = 15:10:8

Now that we have just one ratio, we can answer the question by following exactly the same steps as before:

  1. Work out the scale factor. The total number of black sheep in the real world is 30, and the total in the ratio is 15. Dividing the real world total by the ratio total gives 30 ÷ 15 = 2, so the scale factor is 2.
  2. Multiply the number you want in the ratio by the scale factor. If the number of white sheep in the ratio is 8, then there are 8 x 2 = 16 white sheep.

Simple!

Here are a few practice questions:

  1. One hundred paintings have to be selected for an art exhibition. If the ratio of amateur paintings to professional paintings has to be 2:3, how many amateur paintings and professional paintings have to be selected?
  2. The ratio of brown rats to black rats is 1:2. If there are 24 black rats, how many brown rats are there?
  3. Peter has 20 blue pens. How many red pens must he buy if the ratio of blue to red pens has to be 2:3?
  4. There are 35 children in a class and 15 are boys. What is the ratio of boys to girls?
  5. There are 15 girls and 12 boys in a class. What is the ratio of girls to boys? Give your answer in its simplest form.
  6. A newspaper includes 12 pages of sport and 8 pages of TV. What is the ratio of sport to TV? Give your answer in its simplest form.
  7. Anna has 75p, and Fiona has £1.20. What is the ratio of Anna’s money to Fiona’s money in its simplest form?
  8. Sam does a scale drawing of his kitchen. He uses a scale of 1:100. He measures the length of the kitchen as 5.9m. How long is the kitchen on the scale drawing? Give your answer in mm.
  9. A recipe to make lasagne for 6 people uses 300 grams of minced beef. How much minced beef would be needed to serve 8 people?
  10. A recipe for flapjacks requires 240g of oats. This makes 18 flapjacks. What quantity of oats is needed to make 24 flapjacks?
  11. Amit is 12 years old. His brother, Arun, is 9. Their grandfather gives them £140, which is to be divided between them in the ratio of their ages. How much does each of them get?
  12. The angles in a triangle are in the ratio 1:2:9. Find the size of the largest angle.
  13. In a certain town, the ratio of left-handed people to right-handed people is 2:9. How many right-handed people would you expect to find in a group of 132 people?
  14. Twelve pencils cost 72p. Find the cost of 30 pencils.
  15. Jenny buys 15 felt-tip pens. It costs her £2.85. How much would 20 pens have cost?
  16. If three apples cost 45p, how much would five apples cost?
  17. Sam is 16 years old. His sister is 24 years old. What’s the ratio of Sam’s age to his sister’s age? Give your answer in its simplest form.
  18. A map scale is 1:20000. A distance on the map is measured to be 5.6cm. What’s the actual distance in real life? Give your answer in metres.
  19. A recipe for vegetable curry needs 300 grams of rice, and it feeds 4 people. How much rice would be needed for 7 people?
  20. £60 is to be divided between Brian and Kate in the ratio 2:3. How much does Kate get?

Working with fractions

Me again...

Me again…

People don’t like fractions. I don’t know why. They’re difficult to begin with, I know, but a few simple rules will help you add, subtract, multiply and divide.

Adding and subtracting

Adding and subtracting are usually the easiest sums, but not when it comes to fractions. If fractions have the same denominator (the number on the bottom), then you can simply add or subtract the second numerator from the first, eg 4/5 – 3/5 = 1/5. If not, it would be like adding apples and oranges. They’re just not the same, so you first have to convert them into ‘pieces of fruit’ – or a common unit. The easiest way of doing that is by multiplying the denominators together. That guarantees that the new denominator is a multiple of both the others. Once you’ve found the right denominator, you can multiply each numerator by the denominator from the other fraction (because whatever you do to the bottom of the fraction you have to do to the top), add or subtract them and then simplify and/or convert into a mixed number if necessary, eg 2/3 + 4/5 = (2 x 5 + 4 x 3) / (3 x 5) = (10 + 12) / 15 = 22/15 = 1 7/15.

  1. Multiply the denominators together and write the answer down as the new denominator
  2. Multiply the numerator of the first fraction by the denominator of the second and write the answer above the new denominator
  3. Multiply the numerator of the second fraction by the denominator of the first and write the answer above the new denominator (after a plus or minus sign)
  4. Add or subtract the numerators and write the answer over the new denominator
  5. Simplify and/or turn into a mixed number if necessary

Sample questions

  1. 1/5 + 2/3
  2. 2/7 + 3/5
  3. 4/5 – 2/3
  4. 7/8 – 3/4
  5. 5/8 – 2/3

Multiplication

This is the easiest thing to do with fractions. You simply have to multiply the numerators together, multiply the denominators together and then put one over the other, simplifying and/or converting into a mixed number if necessary, eg 2/3 x 4/5 = (2 x 4) / (3 x 5) = 8/15.

  1. Multiply the numerators together
  2. Multiply the denominators together
  3. Put the result of Step 1 over the result of Step 2 in a fraction
  4. Simplify and/or turn into a mixed number if necessary

Sample questions

  1. 1/5 x 2/3
  2. 2/7 x 3/5
  3. 4/5 x 2/3
  4. 7/8 x 3/4
  5. 5/8 x 2/3

Division

Dividing by a fraction must have seemed like a nightmare to early mathematicians, because nobody ever does it! That’s right. Nobody divides by a fraction, because it’s so much easier to multiply. That’s because dividing by a fraction is the same as multiplying by the same fraction once it’s turned upside down, eg 2/3 ÷ 4/5 = 2/3 x 5/4 = (2 x 5) / (3 x 4) = 10/12 = 5/6. You can even cut out the middle step and simply multiply each numerator by the denominator from the other fraction, eg 2/3 ÷ 4/5 = (2 x 5) / (3 x 4) = 10/12 = 5/6.

  1. Multiply the numerator of the first fraction by the denominator of the second
  2. Multiply the numerator of the second fraction by the denominator of the first
  3. Put the result of Step 1 over the result of Step 2 in a fraction
  4. Simplify and/or turn into a mixed number if necessary

Sample questions

  1. 1/5 ÷ 2/3
  2. 2/7 ÷ 3/5
  3. 4/5 ÷ 2/3
  4. 7/8 ÷ 3/4
  5. 5/8 ÷ 2/3

Simplifying fractions

By the way, to simplify a fraction, try dividing the denominator by the numerator first, eg 9/18 = 1/2. If that works, you don’t have to do anything else. If not, try dividing by the first few prime numbers, ie 2, 3, 5, 7 and 11. You don’t need to try the other numbers, because they’re all multiples of the primes, so they won’t work if the others don’t, eg 4 won’t work if 2 doesn’t work. Ideally, the quickest way would be to divide the numerator and denominator by the highest common factor (or HCF), but you don’t know what that is at the beginning, so it would take time to work it out. This way is a good compromise.

  1. If possible, divide the numerator and denominator by the numerator
  2. If the numerator doesn’t go exactly, start dividing by the smallest prime number that will go into both numbers, starting with 2, 3, 5, 7 and 11
  3. Repeat Step 2 until the only number that goes into the numerator and denominator is 1

Sample questions

  1. Simplify 14/28
  2. Simplify 8/24
  3. Simplify 4/12
  4. Simplify 27/36
  5. Simplify 30/50

Turning improper fractions into mixed numbers

To turn an improper fraction into a mixed number, simply divide the numerator by the denominator to find the whole number and then put the remainder over the original denominator and simplify if necessary, eg 9/6 = 1 3/6 = 1 1/2.

  1. Divide the numerator by the denominator
  2. Write down the answer to Step 1 as a whole number
  3. Put any remainder into a new fraction as the numerator, using the original denominator
  4. Simplify the fraction if necessary

Sample questions

  1. What is 22/7 as a mixed number?
  2. What is 16/5 as a mixed number?
  3. What is 8/3 as a mixed number?
  4. What is 18/8 as a mixed number?
  5. What is 13/6 as a mixed number?

Turning mixed numbers into improper fractions

To turn a mixed number into an improper fraction, multiply the whole number by the denominator of the fraction and add the existing numerator to get the new numerator while keeping the same denominator, eg 2 2/5 = (10 + 2)/5 = 12/5.

  1. Multiply the whole number by the denominator of the fraction
  2. Add the answer to the existing numerator to get the new numerator
  3. Write the answer over the original numerator
  4. Simplify if necessary

Sample questions

  1. What is 2 2/7 as an improper fraction?
  2. What is 3 2/3 as an improper fraction?
  3. What is 4 1/4 as an improper fraction?
  4. What is 5 1/5 as an improper fraction?
  5. What is 3 2/9 as an improper fraction?

There you go. Easy peasy lemon squeezy!

A picture is worth a thousand words

"Where does this grey piece go...?"

“Where does this grey piece go…?”

Have you ever tried to do a 10,000-piece jigsaw of grey seals on a grey rock in a grey sea under a grey sky? Tricky, isn’t it? The key to doing a jigsaw puzzle or writing a description is attention to detail. Everything matters: the mood, the setting, the period, the action, the characters – everything that our five senses can tell us and more. To try and produce the very best description you can when faced with any English exam that includes a composition question, it’s important to approach it in the right way.

Choose the right question (1 minute, assuming you have 30 minutes for the whole question)

  • Common Entrance exams at either 11+ or 13+ usually offer a choice between a story, an essay or a description, perhaps of a photograph or illustration. If you choose the description, make sure you know something about the subject or at least have a good enough stock of synonyms or specialist vocabulary to describe it properly. If the picture is of a horse in a tack room bit you don’t know anything about horse riding, leave well alone!
  • We always write more imaginatively and at greater length if we’re writing about something we like or enjoy, so try to find a question that gets an emotional reaction out of you. If you like beautiful things, then you might respond better to a photograph of a sunset in the Maldives than a montage of burnt-out cars in Beirut!

Brainstorm ideas (5 minutes)

  • This is where attention to detail is most important. Before you take the exam, make sure you have a mental checklist of all the aspects of a scene that you might need to describe. Once you start planning your description, take a sheet of paper and divide it up into sections (or use a mind map with different bubbles for sight, hearing, taste, touch and smell). Make notes based on all five senses. Even if you can’t think of anything to write about one of them, such as ‘taste’, use your imagination. If it’s a seascape, is there the tang of salt on the air? Can the sailors smell land?
  • Think creatively about how to approach the description. A little bit of plot or context goes a long way to creating an appropriate mood, so think about topping and tailing the piece with one or two sentences about the observer. What’s the point of view of the narrator? Is he or she a spy, a pilot, a soldier or a scientist? Is he or she deaf or blind? Would it be better to write in the first or third person?
  • Even if you don’t decide to tell part of a story, think carefully about what the atmosphere of the scene should be. Every situation will demand a different mood or tone. Do you want it to be peaceful, suspenseful or frantic? What’s about to happen? Is it an ambush, an escape plan or a drug deal?
  • The atmosphere should be reflected in the vocabulary you choose to describe the scene. Make a list of the words that have the right associations or connotations, including ‘wow words’ or ‘golden words’ that you think might impress the examiner. Think of as many synonyms as you can. Why ‘destroy’ when you can ‘annihilate’? Why should a tree be plain old ‘green’ when it could be ‘verdant’?
  • Writing descriptively is not the same as writing an essay or doing a comprehension. You don’t need to be brief and matter-of-fact all the time. Think of different poetic devices you can use to make the characters or objects jump off the page. Can you get over the atmosphere or the emotion better with a simile, a metaphor or onomatopoeia? Is the sun in the Sahara desert ‘rather warm’ or ‘as hot as the furnace in the forge of Hell’?

Write a plan (5 minutes)

  • Failing to plan means planning to fail. I’ve read hundreds of compositions written by pupils of all ages, and it doesn’t take long to realise when something hasn’t been planned. You don’t need a plot here necessarily, but you do need some sort of structure. Look at the ideas that you came up with during your brainstorm and decide how to group them together. Draw up a brief outline on a clean sheet of paper, listing the different paragraphs and including bullet points for each with the subject matter, key concepts or particular words that you want to use. What needs to go first? How are you going to finish? Are you going to take each sense in turn? Should you describe the different parts of the scene one by one, the lake followed by the mountains and then the village?

Write the description (whatever time you have left, less 5 minutes to check at the end)

  • Stick to the plan. It’s all very well having a plan, but getting your head down and writing the whole answer without looking at it is no better than not having one in the first place! If you do have a good idea that you want to include, by all means add it to the plan, but make sure you don’t get carried away and write too much about one topic, leaving too little time for all the others.
  • You don’t get any marks for answering a question that’s not even on the exam paper, so make sure you don’t get tempted to wander off the beaten path. Re-read the question now and again. Are there any special instructions? Are you doing exactly what you’ve been asked to do? Are you covering every part of the question?
  • Write as quickly as you can. I could never write as much as I wanted to, and one of the professors at my Oxford interview actually complained about it! I had to tell him I hoped that I was giving him ‘quality rather than quantity’, but I wish I’d been able to hand in six sides rather than four! You don’t have much time, so don’t spend a whole minute searching for the perfect word when another will do. You can always come back to it later when you check your answer.

Check your work (5 minutes)

  • Check for spelling, punctuation and capital letters (as you should for any piece of writing in English).
  • Check you haven’t made any other silly mistakes, either grammatical or stylistic. Make sure you’ve said what you want to say, and feel free to cross out the odd word and replace it with a better one if you can. Just make sure your handwriting is legible!

If you follow all these steps, you may not have the greatest ever description in the history of English literature, but you’ll have given it your best shot! If it helps, challenges or inspires you, here’s one of my favourites. It was written by James Joyce and comes in the final paragraph of his short story The Dead:

“Yes, the news­pa­pers were right: snow was gen­eral all over Ire­land. It was falling softly upon the Bog of Allen and, fur­ther west­wards, softly falling into the dark muti­nous Shan­non waves. It was falling too upon every part of the lonely church­yard where Michael Furey lay buried. It lay thickly drifted on the crooked crosses and head­stones, on the spears of the lit­tle gate, on the bar­ren thorns. His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the liv­ing and the dead.”

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…

Commas

Commas

Comma or apostrophe?

If you had the chance to take a contract out on one punctuation mark, most people would probably choose the comma. Unfortunately, that’s not possible although modern journalists are doing their best to make it into an optional extra!

Punctuation should be there to help the writer and the reader, and the comma is no exception. If I know the rules for using commas, I expect one in certain situations and not in others. If there isn’t one when there should be, or there is one where there shouldn’t be, then I end up getting confused. I may even have to re-read the passage to make sure I understand it. There are certainly ‘grey areas’ when even experts don’t know whether a comma is required or merely optional, but those should be the exception rather than the rule. You might say that nobody has the right to decide what grammatical rules are ‘correct’ and that the plethora of rules I go by were taught to me back in the 1970s, but clarity comes first in my view, so here goes…

  • Lists are the obvious example of using a comma. In the old days, people used to use what’s called an ‘Oxford comma’ before the word ‘and’, but we don’t any more, eg ‘I went to the market and bought apples, pears and bananas’. There are some circumstances when using the Oxford comma makes the sense of the text clearer, but most people would agree that you don’t need it. The list may also be a list of adjectives before a noun, eg ‘It was a juicy, ripe, delicious peach’.
  • Conjunctions (or connectives) make two sentences into one ‘compound’ or ‘complex’ sentence with two separate clauses.
    • Coordinating conjunctions‘ are used to make a ‘compound’ sentence when the clauses are equally important, and the two ‘main clauses’ should always be separated by a comma, eg ‘The sun was warm, but it was cooler in the shade’. There is a useful way of remembering the coordinating conjunctions, which is to use ‘FANBOYS’. This consists of the first letter of ‘for’, ‘and’, ‘nor’, ‘but’, ‘or’, ‘yet’ and ‘so’.
    • Subordinating conjunctions‘ are used to make a ‘complex’ sentence when there is a main clause and a subordinate clause. (Subordinate just means less important.) If the sentence starts with a subordinating conjunction, the clauses need a comma between them, eg ‘Even though it was very hot, he wasn’t thirsty’. However, if the subordinate clause comes at the end, there is no need for a comma, eg ‘He wasn’t thirsty even though it was very hot’. There are lots of subordinating conjunctions, such as ‘after’, ‘although’ and ‘because’, but the easy way to remember it is to ask yourself if the conjunction is in FANBOYS. If it is, it’s a coordinating conjunction; if it’s not, it’s a subordinating conjunction.
  • Which (but not that) needs a comma before it when used as a relative pronoun, eg ‘The sky, which was tinged with orange, was getting darker before sunset’ or ‘He looked up at the sky, which was tinged with orange’. If you don’t know whether to use which or that, the word ‘which’ describes something, whereas the word ‘that’ defines it. The rule about commas also applies to ‘who’ when it comes to describing people, although you still use the same word whether you’re defining or describing someone. Relative pronouns such as ‘which’, ‘that’ and ‘who’ all create a relative clause, which is a type of subordinate clause, so the sentence will be a complex sentence.
  • Openers are a useful way of starting a sentence, usually in order to specify a particular time or place, eg ‘At half-past three, we go home to tea’ or ‘At the end of the road, there is a chip shop’. The subject of the sentence (ie the noun or pronoun that governs the verb in the main clause) should come first. If it doesn’t, you should put a comma after whatever comes in front of it.
  • Direct speech needs something to separate what’s actually said from the description of who said it, and this is normally a comma (although it can sometimes be a question mark or exclamation mark if it’s a question or a command), eg
    “Hello,” he said.
    …or…
    He said, “Hello.”
    The tricky bit comes when the description of the speaker comes in the middle of what’s being said. Here, the rule is that a comma should be used after the ‘he said’/’she said’ if the speaker hasn’t finished the sentence yet, eg
    “On Wednesday evening,” he said, “we’re planning to go to the cinema.”
    When the sentence is over, though, you need a full-stop afterwards, eg
    “I like chocolate biscuits,” she said. “They’re so delicious.”
  • Vocatives and interjections and are simply interruptions to a normal sentence – usually when someone is speaking – to incorporate a name or an exclamation, such as ‘well’ or ‘now’. They should therefore be separated with one or more commas – even if that leads to a long list of words followed by commas, eg
    “Well, now, Mum,” he said, “let me explain.”
  • Certain adverbs fall into the same boat, such as ‘however’, ‘nevertheless’, ‘furthermore’ and the humble ‘too’, and should be separated by commas, eg ‘She played on the swings and the roundabout, too.’
  • Extra information (or ’embedded clauses’ or ‘interrupters’) is sometimes added to a sentence to describe something or someone. If the sentence would still make sense without it, you should put commas before and after the phrase to separate it from the rest of the sentence, eg ‘He stood, cold and alone, before his fate.’
  • Eg and ie are useful shorthand to mean ‘for example’ (exempli gratia in Latin) and ‘that is’ (id est) and should be preceded by a comma, eg ‘He knew lots of poetic devices, eg metaphors and similes.’
  • Names and places sometimes need a comma to separate their different parts. If the day comes after the month and before the year, it should have one, eg ‘December 7, 1941′. If someone has a qualification or letters after his or her name, you should use a comma, eg ‘John Smith, PhD’. If a town is followed by a state or country, the state or country should be separated by commas, eg ‘He lived in Lisbon, Portugal, for five years.’
  • Numbers need commas to separate each power of a thousand. Start on the right at the decimal point and work left, simply adding a comma after every three digits, eg 123,456,789.0.
  • Repetition of a word or phrase also demands a comma, eg “Half a league, half a league, half a league onward…”

Sample questions

Put the correct punctuation in the following sentences:

  1. I like music shopping and dancing
  2. The food was good but he didn’t like the service
  3. The book arrived after she went to the shops
  4. He put on a jacket that was thick enough to keep out the cold
  5. She called her mother which is what she usually did on Sunday evenings
  6. At the end of the road he saw a fox
  7. These apples are expensive he said
  8. What are you doing she cried I need those biscuits for the charity bake sale
  9. When Im on my own she admitted I watch a lot of daytime TV
  10. Could you help me please David he asked
  11. Fortunately he was experienced enough to avoid capsizing the boat
  12. He stood nervous and bashful in front of the prettiest girl hed ever seen
  13. He loved 19th Century novels eg Emma
  14. She was persona non grata ie she wasnt welcome
  15. He lived in Paris France
  16. The Germans invaded Poland on September 1 1939
  17. He watched 1001 Dalmations
  18. All he could see was rain rain rain

Story mountains

Story mountains

Are we nearly there yet…?

Everyone needs a route map, whether it’s Hillary and Tenzing climbing Mount Everest or an English candidate writing a story. One of the ways of planning a story is to create a story mountain, with each stage of the tale labelled on the diagram. The drawing doesn’t have to be any more than a big triangle, but the five stages help to provide a good structure. The story mountain is only part of the process, however. Even before the exam, you could invent two or three interesting characters to use or practise telling a particular story – perhaps an old fairy tale in a modern setting. It’s always good to be prepared, and it’s too late by the time you sit down in the exam hall. If you’re taking an 11+ or 13+ English paper, you should have around half an hour left for the composition after doing the reading comprehension. The routine to follow includes choosing the right question, brainstorming ideas, creating the story mountain, writing the story and checking your work afterwards. Depending on the total length of the exam, you should plan to leave yourself a set amount of time for each stage (shown in brackets, assuming you have a total of 30 minutes).

  1. Choose the right question (less than 1 minute).
    Sometimes you won’t be given a choice, but you will always have different options in a proper 11+ English exam. One might be a description (often based on a drawing or photograph), and another might be an essay on a factual subject, but there will always be the chance to write a story, either based on a suggested title or in the form of a continuation of the passage from the reading comprehension. The important thing here is to try to find a topic you know a bit about and – in an ideal world – something you’d enjoy writing about. If you you’ve never ridden a horse, it would be pointless trying to write a story all about horse racing!
  2. Brainstorm ideas (5 minutes).
    Some pupils go straight into writing the story at this point. The story might occasionally be quite good, but the danger is that you don’t give yourself the chance to come up with the best possible ideas, and you certainly won’t make it easy for yourself to structure it when you don’t have a plan to help you. Whether in business or at school, the best way of coming up with ideas is to spend some time brainstorming. That means coming up with as many ideas as possible in a limited time. There’s no such thing as a bad idea, so try to think positively rather than crossing out anything you don’t like. It takes time to come up with well thought-through ideas for a story, so you might not think of more than a couple, but that at least gives you the chance to pick the better one. If you’re having trouble, think about the different elements you can change: the plot, the characters, the setting, the period and the genre. Those are the basics, and imagining a particularly good character or setting might just provide the clue you’re looking for, and you can always change what kind of story it is – a thriller will look a lot different from a romance or a comedy!
  3. Create a story mountain (5 minutes).
    Once you’ve decided on an idea, you can create your story mountain. You don’t actually have to draw a mountain or a triangle, but you do need to map out the five main stages of the story. You don’t need to write full sentences, just notes that are long enough to remind you of your ideas.

    Opening (or introduction)
    The best way to open a story is probably to start ‘in the middle’. Most fairy stories start with something like ‘Once upon a time, there lived a beautiful princess locked up in a tower…’ The trouble with this kind of ‘exposition’ is that it’s a bit boring! Nothing actually happens. Far better to think of the most exciting moment in your story and start from there. Why not begin with the handsome prince trying to climb up the tower but screaming “Aaaaaaagggghhh!!!” as he falls to his death?! Once you’ve written a paragraph or so grabbing the reader’s attention, you can then introduce the main characters, where they live, when the story is set and so on.

    Build-up (or rising action)
    The build-up should describe what the main character is trying to do. For instance, is he or she robbing a bank, escaping from prison or fighting off an alien invasion

    Problem (or climax)
    Every story needs drama, which is really just conflict. If you show what the hero’s trying to do in the Build-up, the Problem is just what gets in the way. It might be guilt at leaving a friend behind, say, or a prison warder spotting the escaping convicts or a searchlight lighting up the yard. Whatever it is, it’s a problem that needs to be solved.

    Solution (or falling action)
    The solution to the problem is what the hero tries to do to fix it. It may not work, but it’s usually the best option available.

    Outcome (or ending)
    Not many 10-year-old boys like happy endings, so the plan doesn’t always have to come off! If you want your hero to die in a hail of bullets like Butch and Sundance, that’s up to you. Another way to end a story is to use a ‘cliffhanger’. In the old days, that meant the hero was literally hanging on to the edge of a cliff, and the reader would obviously want to know if he held on or not. These days, it just means adding another mystery or problem that needs to be fixed. For example, the hero could escape from prison…only to find a police car chasing him!

  4. Write the story (15 minutes or more, depending on the length of the exam).
    Now for the important bit!

    Stick to the plan
    The most important thing to remember is to stick to the plan! It’s very tempting to get carried away when you’re writing and follow wherever your imagination leads you, but the downside is that your story probably won’t have a proper beginning, middle and end, and you might run out of time trying to get the plot back on track.

    Balance the Three Ds
    You should also strike a balance between the Three Ds: Drama, Description and Dialogue. Every story has a plot, so drama will always be there, but a lot of pupils focus so much on what’s happening that there is very little if any description or dialogue. Readers want to imagine what people look like and how they feel, so you have to give them something to go on. People also generally have a lot to say when they get emotional or find themselves in tough situations, so you won’t be able to capture that unless they talk to one another in your story.

    Show off your vocabulary
    This is also a chance to show off your vocabulary. Including a few ‘wow words’ (or ‘golden words’) such as ‘annihilate’ instead of ‘destroy’ will impress the examiner no end – as long as you know how to spell them!

    Use poetic devices
    What’s the difference between ‘in the evening’ and ‘on a night as dark as a murderer’s soul’? If you think one of these is a little bit more descriptive and atmospheric than the other, then why not use poetic devices such as similes, metaphors and personification in your own writing? Just make sure the comparison is appropriate. If you’re describing a picnic, things might be ‘as black as Bovril’ instead!

  5. Check your work (4-5 minutes)
    If there’s one tip that beats all the rest, it’s ‘Check your work’. However old you are and whatever you’re doing, you should never finish a task before checking what you’ve done. However boring or annoying it is, you’ll always find at least one mistake and therefore at least one way in which you can make things better. In the case of 11+ or 13+ exams, the most important thing is to test candidates’ imagination and ability to write an interesting story, but spelling and grammar is still important. Schools have different marking policies. Some don’t explicitly mark you down (although a rash of mistakes won’t leave a very good impression!); some create a separate pot of 10 marks for spelling and grammar to add to the overall total; and some take marks off the total directly – even if you wrote a good story. Either way, it pays to make sure you’ve done your best to avoid silly mistakes. If you think you won’t have time to check, that’s entirely up to you. You’ll almost certainly gain more marks in the last five minutes by correcting your work than trying to answer one more question, so it makes sense to reserve that time for checking. If you do that, there are a few simple things to look out for. You may want to make a quick checklist and tick each item off one by one.

    Spelling
    This is the main problem that most Common Entrance candidates face, but there are ways in which you can improve your spelling. Firstly, you can look out for any obvious mistakes and correct them. It can help to go through each answer backwards a word at a time so that you don’t just see what you expect to see. Secondly, you can check if a word appears anywhere in the text or in the question. If it does, you can simply copy it across. Finally, you can choose another, simpler word. If you’re not quite sure how to spell something, it’s often better not to take the risk.

    Capital letters
    This should be easy, but candidates often forget about checking capitals in the rush to finish. Proper nouns, sentences and abbreviations should all start with capital letters. If you know you often miss out capital letters or put them where you don’t belong, you can at least check the beginning of every sentence to make sure it starts with a capital.

    Punctuation
    This simply means any marks on the page other than letters and numbers, eg full-stops, commas, quotation marks, apostrophes and question marks. Commas give almost everybody problems, but you can at least check there is a full-stop at the end of every sentence.

    Other grammar
    It’s always useful to read through your story to make sure everything makes sense. It’s very easy to get distracted first time round, but it’s usually possible to spot silly mistakes like missing letters or missing words on a second reading.

Sample questions

Try going through the whole five-step process to write a story based on the following choice of titles:

Left behind

A summer’s day

The ghost from the future

Saying sorry

The lie

The race

Lost boy

A fresh start

The voice in the darkness

The Noah’s Ark

Smoke

Silence

The hot afternoon

My father was furious

The swimming lesson

Caravanning

The choice

The garden

Sleeping

Twins

Junk food

The picnic by the lake

A gift

Great things come from small beginnings

Saying goodbye

The person in the queue

Through the window

The photograph

The long hot summer

The joke

The loner

The dare

The first day of term

Crossing the line

Weird habits

Mirror

Show and tell

Going underground

Echo

A visit from uncle

‘The room was so quiet that I noticed the clock ticking’

(Write a story that opens with this sentence.)

Every day, she sat alone by the upstairs window
(Write a story that opens with this sentence.)

Nobody’s perfect
(Write a story that uses this as its final line.)

Remember the iceberg!

Remember the iceberg!

What, that tiny thing…?

To pass Common Entrance, you have to be a scuba diver. Only a small part of any iceberg is visible above the waves, and only a small part of any answer to a question is visible in the text. To discover the rest, you have to ‘dive in’ deeper and deeper…

When I tutor Common Entrance candidates at either 11+ or 13+, I explain how to approach the two main types of question in the entrance exam: the reading comprehension and the composition.

Most 11+ papers last an hour or an hour and a quarter, and the marks are equally divided between the comprehension and the composition. That means half an hour or so for the comprehension. The 13+ exam is a little different and may involve two papers, one covering a prose comprehension and the other a poetry comprehension and a story. Whatever the format, it’s very important to read the instructions on the front cover. They will tell you exactly what you have to do and – crucially – how much time to spend on each section.

When it comes to doing a comprehension, I recommend a five-step process that involves reading the passage, reading the questions, reading the passage again, answering the questions and then checking your work. Reading the text may take five minutes or so, reading the questions a minute or two, re-reading the text another five minutes and checking your work another five minutes after that, which leaves only 15-20 minutes to answer the questions. If there are 25 marks available, that means around 30-45 seconds per mark. The number of marks available for each individual question will tell you how much time you have in total, eg two minutes for a four-mark question.

  1. Read the passage (5 mins).
    The text is usually taken from a short story, a novel or a poem. Whatever it is, the most important thing to do is to make sure you understand it and remember the main points. Rather than reading it as fast as you can – just to get it over and done with – you should go as slowly as you would if you were reading it out loud and make sure you understand everything. Re-read any bits where you get stuck and ask yourself the W questions as you go along: who, what, where, when, why and how? It may help to repeat  the story or the message to yourself – just to make sure everything makes sense.

  2. Read the questions (1 min).
    Once you’ve read the passage, it’s time to read the questions. Again, understanding and memorising them are more important than sheer speed. When you re-read the text, you’ll need to look out for answers to all the questions, and you won’t be able to do that if you can’t remember what they were! If it helps to jog your memory or draw your eye to the most important bits, you can underline key words and phrases in the questions or in the text itself, eg if you have to give the meaning of the word ‘annihilate’ in line 25.

  3. Read the passage again (5 mins).
    Some people suggest only reading the passage once, but the danger of doing that is that you’re not so familiar with it, which means you can’t answer so many questions off the top of your head and often have to hunt through the text for the answer. What that means is that you effectively end up reading most of the passage three or four times just looking for the bit you need! Reading the text twice is probably a good compromise between speed and memorability, and it also gives you the chance to underline or mark the answers to any of the questions that you happen to spot as you go through.

  4. Answer the questions (15-30 mins, depending on the length of the exam).
    This is obviously the main task, but there are a few things to remember:

    Read the question carefully
    . You don’t get any marks for answering the wrong question, so make sure you know exactly what you have to do. If that means reading the question twice, then that’s a price worth paying.

    Use the mark scheme as a guide. Most exam papers will let you know how many marks are awarded for each question, so it’s worthwhile bearing that in mind when you’re writing your answers. There’s no point spending ten minutes on a question that’s only worth one mark, and it would be daft to write only one sentence for a question worth ten marks. If it helps, you can always work out how much time you have for each question. For example, if you have 15 minutes to answer questions worth 30 marks, you’ll have 30 seconds for each mark, which means two minutes for a four-mark question.

    Follow any instructions to the letter
    . All these hints and tips are useful, but they are only general rules. Occasionally, examiners will let you off the hook and tell you that you don’t need to use full sentences, eg for the meanings of words. Just be sure to abide by what they say. If you’re told to answer a question ‘in your own words’, that means you can’t use any of the words in the text (except ‘filler’ words such as ‘the’ and ‘of’ or words that don’t have any obvious alternative, eg ‘lightship’ or the names of the characters). You need to show that you understand what’s written in the text, and you’ll actually be marked down for using quotations, even though that’s what’s usually needed.

    Refer to the text.
    I know the point of reading the text twice is to try and remember it, but you can’t possibly expect to remember the answers to all the questions and all the quotations to back them up! The answer is always in the text, so don’t be afraid to spend a few seconds going back over it to make sure you get the answer right and are able to support it with the right evidence.

    Use full sentences. Even if a question is as simple as ‘What is Jack’s dog’s name?’, the answer should be ‘His name is Rover’ rather than just ‘Rover’. The only time you don’t need to use a full sentence is when the question gives you special permission.

    Make sure any definitions you’re asked for fit exactly in the context. Words have different meanings, so you must check to see whether you have the right meaning and the right part of speech, eg ‘catch’ can be a verb meaning to fall ill or a noun meaning a fish! Nouns also vary in number, and verbs vary in tense and person, so it’s easy to lose marks by putting down ‘destroy’ rather than ‘destroys’, for example.

    Use PEE (Point, Evidence, Explanation) for longer answers. This is a good way of structuring your answers. The ‘point’ should be a short sentence answering the question as briefly as possible. The ‘evidence’ should be a quotation or another reference to the text. Finally, the ‘explanation’ should make it clear how the evidence backs up your argument.

    Answer ‘how’ questions be talking about the language. Comprehensions often start with a simple one-mark question such as ‘In what country is this passage set?’ This is a ‘what?’ question, a question about content, about facts. However, there is another kind of question, the ‘how?’ question, which is all about language. Suppose you’re asked, ‘How does the writer explain how Jack feels after losing his dog?’ What do you have to do? What you definitely shouldn’t do is say how Jack feels. The question is not ‘What are Jack’s feelings?’ You’re not being asked for facts but for an analysis of the techniques the author uses. That might include using poetic devices such as metaphor, simile or personification, or it might refer to sentence structure or the use of different parts of speech such as adjectives or verbs. The important thing to remember is that a ‘how?’ question is generally going to be about language, not content.

    Use the right tense. Most of the time, you should use the ‘eternal present’ to talk about the text, but the most important thing is to use the same tense as the question. Sometimes, passages are about historical events, so the past is more appropriate. For example, if the text comes from The Diary of Anne Frank, it wouldn’t make sense to talk about the Second World War as if it were still going on!

    Don’t repeat the whole question in your answer. 
    In primary school, teachers often tell their pupils to do this to make sure they’ve understood the question, but it takes too long when you’re older. I’ve seen children spend a whole minute carefully copying down most of the question before they’ve even thought about the answer! The best approach is probably to imagine what the whole answer would look like and then simply start writing from the word after ‘because’, eg Jack was crying when he made the long walk home from school on Friday because…’His dog had just died, and he missed him very much.’

    Answer all parts of the question. Examiners will sometimes try to catch you out by ‘hiding’ two questions in one. You should pay particular attention to these questions, eg ‘How do you think Jack feels about losing his dog, and how do you think you would feel if you lost your favourite pet?’ It would be easy to answer the first part of the question and then forget about the rest, so be careful!

    Don’t waste time with words you don’t need. You never have enough time in exams, so it’s pointless trying to pad out your answers by including waffle such as ‘it says in the text that…’ or ‘the author writes that in his opinion…’ Far better to spend the time thinking a bit more about the question and coming up with another quotation or point to make.

    Use quotations. Whether or not you’re using PEE and whether or not the question asks you to ‘make reference to the text’, you should generally try to back up your arguments with a short quotation or example. Just make sure you use quotation marks or inverted commas, copy the quotation out accurately and drop the key words into a sentence of your own, eg Jack feels ‘devastated’ by the loss of his dog. If the quotation is too long, you can miss words out by replacing them with an ellipsis (…). If it needs a slight change to make sense, you can put the change in square brackets. That’s quite useful if the question is in the present tense and the story is written in the past, eg Jim ‘love[s] strawberries’ rather than Jim ‘loved strawberries’.

    Remember the iceberg! As you can see from the picture, the vast majority of an iceberg remains hidden from view, and it’s the same with the answers to questions in a reading comprehension. Don’t be satisfied by what you can see on the surface. That won’t get you full marks. Like a scuba diver, you have to dive in deeper to find the rest…!

  5. Check your work (5 mins).
    If there’s one tip that beats all the rest, it’s ‘Check your work’. However old you are and whatever the subject, you should never finish a piece of work before checking what you’ve done. However boring or annoying it is, you’ll always find at least one mistake and therefore at least one way in which you can make things better. In the case of 11+ or 13+ comprehensions, the most important thing is to test candidates’ understanding of the passage, but spelling and grammar is still important. Schools have different marking policies. Some don’t explicitly mark you down (although a rash of mistakes won’t leave a very good impression!), some create a separate pot of 10 marks for spelling and grammar to add to the overall total and some take marks off each answer directly – even if you got the content ‘right’ . Either way, it pays to make sure you’ve done your best to avoid silly mistakes. If you think you won’t have time to check, make sure you manage your time. You’ll almost certainly gain more marks in the last five minutes by correcting your work than trying to answer one more question, so it makes sense to reserve that time for checking. If you do that, there are a few simple things to look out for. You may want to make a quick checklist and tick each item off one by one.

    Correct and complete answers. 
    This is the most important thing to check, and it takes the longest. Make sure that each answer is correct (by referring back to the text if necessary) and that each part of the question has been covered. Quite a few of my students have needlessly lost marks in practice tests by forgetting to look at all the pages, so you should always check you haven’t missed out any questions.

    Spelling. 
    This is the main problem that most Common Entrance candidates face, but there are ways in which you can improve your spelling. Firstly, you can look out for any obvious mistakes and correct them. It can help to go through each answer backwards a word at a time so that you don’t just see what you expect to see. Secondly, you can check if a word appears anywhere in the text or in the question. If it does, you can simply copy it down from there. Finally, you can choose another simpler word – if you’re not quite sure how to spell a word, it’s sometimes better not to take the risk.

    Capital letters. This should be easy, but candidates often forget about checking capitals in the rush to finish. Proper nouns, sentences and abbreviations should all start with capital letters. If you know you often miss out capital letters or put them where they don’t belong, you can at least check the beginning of each answer to make sure it starts with a capital.

    Punctuation. This simply means any marks on the page other than letters and numbers, eg full-stops, commas, quotation marks, apostrophes and question marks. Commas give almost everybody problems, but you can at least check there is a full-stop at the end of every sentence.

    Other grammar. It’s always useful to read through your answers to make sure everything makes sense. It’s very easy to get distracted first time round, but it’s usually possible to spot silly mistakes like missing words when you read everything again.