You can use short multiplication if you’re multiplying one number by another that’s in your times tables (up to 12). However, if you want to multiply by a higher number, you need to use long multiplication.
Write down the numbers one on top of the other with the smaller number on the bottom and a times sign on the left (just as you would normally), then draw three lines underneath to hold three rows of numbers.
Multiply the top number by the last digit of the bottom number as you would normally.
Write a zero at the end of the next answer line (to show that you’re multiplying by tens now rather than units).
Multiply the top number by the next digit of the bottom number, starting to the left of the zero you’ve just added.
Add the two answer lines together to get the final answer.
Notes:
Some people write the tens they’ve carried right at the top of the sum, but that can get very confusing with three lines of answers!
Don’t forget to add the zero to the second line of your answer. If it helps, you can try writing it down as soon as you set out the sum (and before you’ve even worked anything out).
At 11+ level, long multiplication will generally be a three-digit number multiplied by a two-digit number, but the method will work for any two numbers, so don’t worry. If you have to multiply two three-digit numbers, say, you’ll just have to add another line to your answer.
Sample questions:
Have a go at these questions. Make sure you show your working – just as you’d have to do in an exam.
Homophones are words that sound the same even though they’re spelt differently and mean different things. Getting them right can be tricky, but it’s worth it in the end.
The reason why homophones are important is not just to do with the general need to spell correctly. Many people think getting them wrong is a ‘worse’ mistake than simply mis-spelling a word because it means that you don’t really know what you’re doing. Anyone can make a spelling mistake, but using completely the wrong word somehow seems a lot worse. That may not sound fair, but that’s just how a lot of people think, so it’s worth learning the common homophones so you don’t get caught out.
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! If you’re still not convinced, there are a number of reading lists on my Useful Links page.
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.
Nothing makes the heart of a reluctant mathematician sink like an algebra question.
Algebra is supposed to make life easier. By learning a formula or an equation, you can solve any similar type of problem whatever the numbers involved. However, an awful lot of students find it difficult, because letters just don’t seem to ‘mean’ as much as numbers. Here, we’ll try to make life a bit easier…
Gathering Terms
X’s and y’s look a bit meaningless, but that’s the point. They can stand for anything. The simplest form of question you’ll have to answer is one that involves gathering your terms. That just means counting how many variables or unknowns you have (like x and y). I like to think of them as pieces of fruit, so an expression like…
2x + 3y – x + y
…just means ‘take away one apple from two apples and add one banana to three more bananas’. That leaves you with one apple and four bananas, or x + 4y.
If it helps, you can arrange the expression with the first kind of variables (in alphabetical order) on the left and the second kind on the right like this:
2x – x + 3y + y
x + 4y
Just make sure you bring the operators with the variables that come after them so that you keep exactly the same operators, eg two plus signs and a minus sign in this case.
Here are a few practice questions:
3x + 4y – 2x + y
2m + 3n – m + 3n
p + 2q + 3p – 3q
2a – 4b + a + 4b
x + y – 2x + 2y
Multiplying out Brackets
This is one of the commonest types of question. All you need to do is write down the same expression without the brackets. To take a simple example:
2(x + 3)
In this case, all you need to do is multiply everything inside the brackets by the number outside, which is 2, but what do we do about the ‘+’ sign? We could just multiply 2 by x, write down ‘+’ and then multiply 2 by 3:
2x + 6
However, that gets us into trouble if we have to subtract one expression in brackets from another (see below for explanation) – so it’s better to think of the ‘+’ sign as belonging to the 3. In other words, you multiply 2 by x and then 2 by +3. Once you’ve done that, you just convert the ‘+’ sign back to an operator. It gives exactly the same result, but it will work ALL the time rather than just with simple sums!
Here are a few practice questions:
2(a + 5)
3(y + 2)
6(3 + b)
3(a – 3)
4(3 – p)
Solving for x
Another common type of question involves finding out what x stands for (or y or z or any other letter). The easiest way to look at this kind of equation is using fruit again. In the old days, scales in a grocery shop sometimes had a bowl on one side and a place to put weights on the other.
To weigh fruit, you just needed to make sure that the weights and the fruit balanced and then add up all the weights. The point is that every equation always has to balance – the very word ‘equation’ comes from ‘equal’ – so you have to make sure that anything you do to one side you also have to do to the other. Just remember the magic words: BOTH SIDES!
There are three main types of operation you need to do in the following order:
Multiplying out any brackets
Adding or subtracting from BOTH SIDES
Multiplying or dividing BOTH SIDES
Once you’ve multiplied out any brackets (see above), what you want to do is to simplify the equation by removing one expression at a time until you end up with something that says x = The Answer. It’s easier to start with adding and subtracting and then multiply or divide afterwards (followed by any square roots). To take the same example as before:
2(x + 3) = 8
Multiplying out the brackets gives us:
2x + 6 = 8
Subtracting 6 from BOTH SIDES gives us:
2x = 2
Dividing BOTH SIDES by 2 gives us the final answer:
x = 1
Simple!
Here are a few practice questions:
b + 5 = 9
3y = 9
6(4 + c) = 36
3(a – 2) = 24
4(3 – p) = -8
Multiplying Two Expressions in Brackets (‘FOIL’ Method)
When you have to multiply something in brackets by something else in brackets, you should use what’s called the ‘FOIL’ method. FOIL is an acronym that stands for:
First Outside Inside Last
This is simply a good way to remember the order in which to multiply the terms, so we start with the first terms in each bracket, then move on to the outside terms in the whole expression, then the terms in the middle and finally the last terms in each bracket.
Just make sure that you use the same trick we saw earlier, combining the operators with the numbers and letters before multiplying them together. For example:
(a + 1)(a + 2)
First we multiply the first terms in each bracket:
a x a
…then the outside terms:
a x +2
…then the inside terms:
+1 x a
…and finally the last terms in each bracket:
+1 x +2
Put it all together and simplify:
(a + 1)(a + 2)
= a² + 2a + a + 2
=a² + 3a + 2
Here are a few practice questions:
(a + 1)(b + 2)
(a – 1)(a + 2)
(b + 1)(a – 2)
(p – 1)(q + 2)
(y + 1)(y – 3)
Factorising Quadratics (‘Product and Sum’ Method)
This is just the opposite of multiplying two expressions in brackets. Normally, factorisation involves finding the Highest Common Factor (or HCF) and putting that outside a set of brackets containing the rest of the terms, but some expressions can’t be solved that way, eg a² + 3a + 2 (from the previous example).
There is no combination of numbers and/or letters that goes evenly into a², 3a and 2, so we have to factorise using two sets of brackets. To do this, we use the ‘product and sum’ method.
This simply means that we need to find a pair of numbers whose product equals the last number and whose sum equals the multiple of a. In this case, it’s 1 and 2 as +1 x +2 = +2 and +1 + +2 = +3.
The first term in each bracket is just going to be a as a x a = a². Hence, factorising a² + 3a + 2 gives (a + 1)(a + 2). You can check it by using the FOIL method (see above) to multiply out the brackets:
(a + 1)(a + 2)
= a² + 2a + a + 2
=a² + 3a + 2
Subtracting One Expression from Another*
Here’s the reason why we don’t just write down operators as we come across them. Here’s a simple expression we need to simplify:
20 – 4(x – 3) = 16
If we use the ‘wrong’ method, then we get the following answer:
20 – 4(x – 3) = 16
20 – 4x – 12 = 16
8 – 4x = 16
4x = -8
x = -2
Now, if we plug our answer for x back into the original equation, it doesn’t balance:
20 – 4(-2 – 3) = 16
20 – 4 x -5 = 16
20 – -20 = 16
40 = 16!!
That’s why we have to use the other method, treating the operator as a negative or positive sign to be added to the number before we multiply it by whatever’s outside the brackets:
20 – 4(x – 3) = 16
20 – 4x + 12 = 16
32 – 4x = 16
4x = 16
x = 4
That makes much more sense, as we can see:
20 – 4(4 – 3) = 16
20 – 4 x 1 = 16
20 – 4 = 16
16 = 16
Thank Goodness for that!
Here are a few practice questions:
30 – 3(p – 1) = 0
20 – 3(a – 3) = 5
12 – 4(x – 2) = 4
24 – 6(x – 3) = 6
0 – 6(x – 2) = -12
Other Tips to Remember
If you have just one variable, leave out the number 1, eg 1x is just written as x.
When you have to multiply a number by a letter, leave out the times sign, eg 2 x p is written as 2p.
The squared symbol only relates to the number or letter immediately before it, eg 3m² means 3 x m x m, NOT (3 x m) x (3 x m).
