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Words have power

Persuasive Writing

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More and more questions in school entrance exams involve persuasive writing. This is the attempt to change people’s minds about an issue, change how they feel about it or persuade them to do something. It can take many forms:

  • Essays
  • Speeches
  • Letters
  • Newspaper articles

The format of each of these is obviously different, and you can read up on how to write essays, letters and newspaper articles elsewhere on this site. However, they all rely on similar techniques, which will be covered in this article:

  • Rhetoric (or poetic devices or literary techniques)
  • Anecdote
  • Facts
  • Statistics
  • Emotion
  • Call to action

Rhetoric

Strictly speaking, rhetoric is the name given to skills and techniques involved in public speaking. It was taught by the Greeks thousands of years ago, so some of the terms derive from Greek, but it still survives today. It largely overlaps with poetic devices and literary techniques, such as rhetorical questions, similes and metaphors.

Rhetorical techniques can be used in any form of persuasive writing, and you can see a long list and explanation of the main poetic devices in my article on the subject. However, these are the main ones to consider when it comes to persuasive writing:

  • Rhetorical questions (and hypophora)
  • Tricolon (or rule of three)
  • Anaphora
  • Epistrophe
  • Hyperbole
  • Alliteration
  • Simile
  • Metaphor
  • Imagery
  • Sentence structure

Anecdote

An anecdote is a story or account of an event used to illustrate your point. It’s a good idea to make it funny if you can to get readers on your side. For example, if you were arguing that mobile phones should be banned at school, you might tell the tale of diving into a swimming pool with your iPhone still in your pocket!

Facts

If you’re trying to persuade someone who disagrees with you, it’s no good to say what you think without any evidence to back it up. This is known as mere assertion. Facts are much more convincing and help prove your point.

For example, you might be asked to write an essay on what people should eat for breakfast. Yes, you could say what you like to have, but that’s just your own personal preference. It’s far better to be able to talk about the nutritional benefits, such as the fibre in muesli or the vitamin C in orange juice.

Statistics

If you read the newspaper, you’ll often see articles using statistics to support the writers’ arguments. Statistics are just numbers that show what’s happening in the world, but they’re useful because they offer a precise measurement of the quantity of something or its change over time.

You don’t need to be good at Maths to use statistics, but it helps if you’re familiar with fractions, decimals and percentages. These are just different ways of showing the share of something or the rate of change. They often come up in discussions of economics (the study of markets and business).

For example, you might read that the economy had grown by 2.1% in a given year or that the amount of Government debt was 100% of Gross Domestic Product (a way of measuring the size of the economy).

If you’re doing an entrance exam, you might not know that many statistics, so you won’t be able to include them in your persuasive writing. However, you can always revise the topic beforehand so that you’re fully prepared!

Emotion

People are often persuaded by emotion rather than logical argument. If you can make them laugh or cry, they’re more likely to take your side.

Telling jokes is obviously a good way to make someone laugh—although they have to be funny! Telling a sad story about someone suffering is a good way to make them cry—or at least sympathise with the victim and therefore become easier to persuade.

Call to Action (or CTA)

One of the best ways to win converts to your cause and create a positive result is to include a call to action. This is something that your reader can actually do to help, such as write a letter to the local MP or turn up for a protest march.

It’s a common technique in advertising, which is designed usually to get you to buy the product. Most adverts have a catchphrase, and it’s often a call to action that invites the customer to visit a website or buy a particular brand of product.

For example, the ride-sharing app Lyft told people to ‘Ride and save’, and a Mexican restaurant chain called Chipotle told their customers to ‘Get my guac’ (or guacamole).

Verdict

If you want to persuade someone to think, feel or do something a certain way, you can’t just rely on the message itself. You have to back it up with other techniques, such as rhetoric, anecdotes, facts, statistics, emotion and a call to action.

Whether your writing an essay, a letter, a speech or a newspaper article, all these skills can help you persuade your readers—and help get you a better mark!

 

 

 

 

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SOHCAHTOA

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SOHCAHTOA (pronounced ‘soccer-toe-uh’) is a useful ‘mnemonic’ to remember the definitions of sines, cosines and tangents. Amazingly, I was never taught this at school, so I just had to look up all the funny numbers in a big book of tables without understanding what they meant. As a result, I was always a bit confused by trigonometry until I started teaching Maths and came across SOHCAHTOA quite by accident!

The reason it’s called SOHCAHTOA is because the letters of all three equations make up that word – if you ignore the equals signs…

First of all, let’s define our terms:

  • S stands for sine (or sin)
  • O stands for the opposite side of a right-angled triangle
  • H stands for the hypotenuse of a right-angled triangle
  • C stands for cosine (or cos)
  • A stands for the adjacent side of a right-angled triangle
  • T stands for tangent (or tan)
  • O stands for the opposite side of a right-angled triangle (again)
  • A stands for the adjacent side of a right-angled triangle (again)

Sines, cosines and tangents are just the numbers you get when you divide one particular side of a right-angled triangle by another. For a given angle, they never change – however big the triangle is.

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

All these ratios were discovered by Indian and Arabic mathematicians some time before the 9th Century, but you can still use them today to help you work out the length of a side in a right-angled triangle or one of the angles.

Each of these formulas can be rearranged to make two other formulas. (If it helps, you can put the three values in a number triangle with the one in the middle at the top). Let’s take the sine formula first:

Sine = Opposite ÷ Hypotenuse means:

  • Hypotenuse = Opposite ÷ Sine
  • Opposite = Hypotenuse x Sine

As long as you know the angle and the length of the opposite side or the hypotenuse, you can work out the length of the other one.

  • Unknown: hypotenuse
    Known: opposite and angle
    • If one of the angles of a right-angled triangle is 45° and the opposite side is 5cm, the formula for the length of the hypotenuse must be opposite ÷ sin(45°). The sine of 45° is 0.707 (to three decimal places), so hypotenuse = 5 ÷ 0.707 = 7cm (to the nearest cm).
  • Unknown: opposite
    Known: hypotenuse and angle
    • If one of the angles of a right-angled triangle is 45° and the hypotenuse is 5cm, the formula for the length of the opposite side must be hypotenuse x sin(45°). The sine of 45° is 0.707 (to three decimal places), so opposite = 5 x 0.707 = 4cm (to the nearest cm).

Equally, as long as you know the the hypotenuse and opposite side lengths, you can work out the angle by using the ‘arcsine’ or ‘inverse sine’ function on your calculator, which works out the matching angle for a given sine and is written as sin-1, eg sin(45°) = 0.707, which means sin-1(0.707) = 45°.

  • Unknown: angle
  • Known: opposite and hypotenuse
    • If the opposite side of a right-angled triangle is 4cm and the hypotenuse is 5cm, the formula for the angle must be sin-1(4÷5), or the inverse sine of 0.8. The sine of 53° (to the nearest degree) is 0.8, so the angle must be 53°.

We can do the same kind of thing with the cosine formula, except this time we’re dealing with the adjacent rather than the opposite side.

Cosine = Adjacent ÷ Hypotenuse means:

  • Hypotenuse = Adjacent ÷ Cosine
  • Adjacent = Hypotenuse x Cosine

As long as you know the angle and the length of the adjacent side or the hypotenuse, you can work out the length of the other one.

  • Unknown: hypotenuse
    Known: adjacent and angle
    • If one of the angles of a right-angled triangle is 45° and the adjacent side is 5cm, the formula for the length of the hypotenuse must be adjacent ÷ cos(45°). The cosine of 45° is 0.707 (to three decimal places), so hypotenuse = 5 ÷ 0.707 = 7cm (to the nearest cm).
  • Unknown: adjacent
    Known: hypotenuse and angle
    • If one of the angles of a right-angled triangle is 45° and the hypotenuse is 5cm, the formula for the length of the adjacent side must be hypotenuse x cos(45°). The sine of 45° is 0.707 (to three decimal places), so adjacent = 5 x 0.707 = 4cm (to the nearest cm).

Equally, as long as you know the the hypotenuse and adjacent side lengths, you can work out the angle by using the ‘arccosine’ or ‘inverse cosine’ function on your calculator, which works out the matching angle for a given cosine and is written as cos-1, eg cos(45°) = 0.707, which means cos-1(0.707) = 45°.

  • Unknown: angle
  • Known: adjacent and hypotenuse
    • If the adjacent side of a right-angled triangle is 4cm and the hypotenuse is 5cm, the formula for the angle must be cos-1(4÷5), or the inverse cosine of 0.8. The sine of 37° (to the nearest degree) is 0.8, so the angle must be 37°.

Finally, we can do the same kind of thing with the tangent formula, except this time we’re dealing with the opposite and adjacent sides.

Tangent = Opposite ÷ Adjacent means:

  • Adjacent = Opposite ÷ Tangent
  • Opposite = Adjacent x Tangent

As long as you know the angle and the length of the opposite or adjacent side, you can work out the length of the other one.

  • Unknown: adjacent
    Known: opposite and angle
    • If one of the angles of a right-angled triangle is 45° and the opposite side is 5cm, the formula for the length of the adjacent side must be opposite ÷ tan(45°). The tangent of 45° is 1, so adjacent = 5 ÷ 1 = 5cm.
  • Unknown: opposite
    Known: adjacent and angle
    • If one of the angles of a right-angled triangle is 45° and the adjacent side is 5cm, the formula for the length of the opposite side must be adjacent x tan(45°). The tangent of 45° is 1, so opposite = 5 x 1 = 5cm.

Equally, as long as you know the the opposite and adjacent side lengths, you can work out the angle by using the ‘arctangent’ or ‘inverse tangent’ function on your calculator, which works out the matching angle for a given tangent and is written as tan-1, eg tan(45°) = 0.707, which means tan-1(0.707) = 45°.

  • Unknown: angle
  • Known: adjacent and hypotenuse
    • If the adjacent side of a right-angled triangle is 5cm and the hypotenuse is 5cm, the formula for the angle must be tan-1(5÷5), or the inverse tangent of 1. The tangent of 45° is 1, so the angle must be 45°.

Sample Questions

  1. If one of the angles of a right-angled triangle is 20° and the opposite side is 15cm, how long is the hypotenuse?
  2. If one of the angles of a right-angled triangle is 35° and the hypotenuse is 7cm, how long is the opposite side?
  3. If the opposite side of a right-angled triangle is 3cm and the hypotenuse is 8cm, what is the angle?
  4. If one of the angles of a right-angled triangle is 75° and the adjacent side is 12cm, how long is the hypotenuse?
  5. If one of the angles of a right-angled triangle is 45° and the hypotenuse is 5cm, how long is the adjacent side?
  6. If the adjacent side of a right-angled triangle is 15cm and the hypotenuse is 2cm, what is the angle?
  7. If one of the angles of a right-angled triangle is 15° and the opposite side is 20cm, how long is the adjacent side?
  8. If one of the angles of a right-angled triangle is 45° and the adjacent side is 5cm, how long is the opposite side?
  9. If the adjacent side of a right-angled triangle is 15cm and the hypotenuse is 12cm, what is the angle?
  10. If the adjacent side of a right-angled triangle is 4cm and the hypotenuse is 18cm, what is the angle?

     

     

     

     

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Circle Theorems

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This article explains circle theorems, including tangents, sectors, angles and proofs (with thanks to Revision Maths).

Isosceles Triangle

Two Radii and a chord make an isosceles triangle.

Perpendicular Chord Bisection

The perpendicular from the centre of a circle to a chord will always bisect the chord (split it into two equal lengths).

Angles Subtended on the Same Arc

Angles subtended on the same arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angle in a Semi-Circle

angle in a semi-circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.

Proof

We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

Divide the triangle in two

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

Two isosceles triangles

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.

Tangents

A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).

A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.

angle with a tangent

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Tangents from an external point are equal in length

Angle at the Centre

Angle at the centre

The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

Proof

You might have to be able to prove this fact:

proof diagram 1

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b

proof diagram 2

Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a
Similarly, ∠BOX = 180 – 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX
= 360 – (180 – 2a) – (180 – 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB

Alternate Segment Theorem

Alternate segment theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.

Proof

You may have to be able to prove the alternate segment theorem:

proof of alternate segment theorem

We use facts about related angles

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.

Area of Sector and Arc Length

A sector

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360

 

 

 

 

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Children’s Reading List

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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.

 

 

 

 

If you’re looking for past papers with answers, especially in the run-up to 11+/13+ exams, GCSEs or A-levels, you can visit my Past Papers page and subscribe for just £37.99 a year.

Useful Terms in Maths

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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 around the edge of a circle.

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

Cube: the result of multiplying any number by itself twice, eg 8 is the cube of 2, as 2 x 2 x 2 = 8.

Cube root: the number that has to be multiplied by itself twice to make another number, eg 2 is the cube root of 8, as 2 x 2 x 2 = 8.

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

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

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

Dividend: the number being divided in a division sum, eg in 24 ÷ 4 = 6, the dividend is 24.

Divisor: the number to divide by in a division sum, eg in 18 ÷ 6 = 3, the divisor is 6.

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).

Frequency density: the frequency per unit for the data in each class of values, defined as frequency ➗ class width, eg if 10 people were between 180 cm and 190 cm tall (180 ≤ x < 190), the frequency density would be 10 ➗ (190 – 180) = 10 ➗ 10 = 1.

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

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

Irrational number: a number that can’t be written as a fraction and has an infinite number of decimal places without any repeating sequences, eg pi (or π).

Lowest Common Multiple (LCM) or Lowest/Least Common Denominator (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.

Natural number (or counting number): a positive whole number, eg 1. (Some people include zero, some don’t.)

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 or PEMDAS) is a good way of remembering it, as it stands for:

  • Brackets/Parentheses
  • Indices/Order/Exponents
  • (in other words, squares, cubes and so on)
  • Division
  • Multiplication
  • Addition
  • 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 they occur in the sum, and it can make a difference to the answer, eg 4 – 3 + 2 = 3 if you do the calculations from left to right, which is correct, but if you did 3 + 2 first, you’d get -1, which is the wrong answer.

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. On diagrams, parallel lines are shown with matching arrows—one for the first pair, two for the second etc.

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

Perpendicular: at 90 degrees to each other.

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

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

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

Probability: the chance of something happening, calculated as the number of ways of getting what you want divided by the total number of possible outcomes, eg the chance of a coin toss being heads is ½ as there is one ‘heads’ side but two sides in total. To work out the probability of a sequence of events, you have to multiply the individual probabilities together, eg the chance of a coin toss being heads twice in a row is ½ x ½ = ¼

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

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

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

Quotient: the answer to a division sum, eg in 12 ÷ 4 = 3, the quotient is 3.

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.

Rational number: a positive or negative number that can be written as a fraction, including zero and all whole numbers, eg 1 or ½.

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

Right angle: an angle of 90 degrees.

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

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

Shapes: the name of each shape depends on the number of sides. Here are the most common ones.

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

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

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

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

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

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

Transformations: there are three main kinds of transformation: reflection, rotation and translation.

  • With reflection, you need to state the formula of the mirror line, eg the shape has been reflected in the line y = 4.
  • With rotation, you need to state the number of degrees, the direction and the centre of rotation, eg the shape has been rotated 90 degrees clockwise around the point (4, 3).
  • With translation, you need to state the change in the x and y values, eg the shape has been translated four units up and three units to the right.

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

  • equilateral triangles have all three sides the same length and all three angles the same. On diagrams, lines of similar length are shown with matching tick marks going across them—one for the first pair, two for the second etc.
  • 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.

Vinculum: the line between the numerator and denominator in a fraction (also called the fraction bar).

 

 

 

 

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