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Glossary of Mathematical Terms

with a few examples and illustrations

By Roger Bickley

Walk-In Training 2011

Second Version

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LATSPHHOND

The mnemonic LATSPHHOND helps you to remember the names and sides of

the first 10 regular polygons:

1 L Line

2 A Angle

3 T Triangle

4 S Square

5 P Pentagon

6 H Hexagon

7 H Heptagon

8 O Octagon

9 N Nonagon

10 D Decagon

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2-D shapes Shapes in two dimensions i.e. which can be drawn on paper. Examples include triangles, squares, pentagons etc

3- D shapes Objects in 3 dimensions i.e. which have height, length and breadth, for example cubes, cuboids, prisms

12-hour clock Most domestic clocks have a clock face which shows the numbers 1 to 12

around the edge. Times are said (or written) using the term a.m. (ante-meridian i.e. before 12 oclock mid-day) or p.m. (post meridian i.e. after 12 oclock mid-day) to indicate whether the time is morning or afternoon e.g. 7:30 am is half-past seven in the morning and 9:20 pm is twenty past nine in the evening

24-hour clock This clock (usually) has two sets of numbers around the dial 1 to 12 around the outside and 13 to 24 around the inside giving two numbers for every hour e.g. 1 oclock is also 13 oclock, 6 oclock is also 18 oclock and so on except we dont say oclock but simply hours and the time is written differently so 6 oclock in the morning is 0600 hours on the 24 hour clock and 6 oclock in the evening is 1800 hours. The simple rule is that if the time is before 12 noon the hours stay as they are e.g. 0830 and if they are after noon we add twelve to the number e.g. 8:30 in the evening is 2030 hours

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Accuracy The level of detail required in an answer. For example there might have been 23102 people at a football match but many people would say there were around 23000.

The accuracy demanded of an answer will usually be determined in the question or problem e.g. write you answer to 2 DECIMAL PLACES or give your answer to one SIGNIFICANT FIGURE)

Acute angle An angle between 00 and 890

Adjacent Next to in angle terms this means angles next to each other, as in the illustration:

Algebra The branch of mathematics which uses letters to represent unknown quantities for example we might say three apples and four bananas cost 30 pence, and this could be written as:

3a + 4b = 30

Algebraic Fraction These are similar to normal fractions in that they have a NUMERATOR (top) and DENOMINATOR (Bottom) the difference being that both top and bottom are algebraic expressions:

4a 12d2 15g4 12a2 16d 60g7

Usually, as part of the problem-solving process, you will be required to

simplify such expression

Less than 900

a

b

Angles a and b are adjacent

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Alternate angles Angles which are the same size and are created by drawing a straight line across a pair of parallel lines

c

c

The angles c are equal

am Stands for ante-meridian and means times between midnight 00.00 and Noon 12.00 (see also p.m.). Times are written (for example) 2.35am which means

twenty-five minutes to three in the morning

Angle bisector A line which cuts an angle in half two equal angles:

Angle in a semicircle If you draw a diameter and then create a triangle from this diameter the angle opposite the diameter will ALWAYS be a right angle (900)

Angles in a Triangle The sum of the angles in ANY triangle is always 1800

Diameter Red squares

are 900

+ + = 1800

How you can cut up a triangle and arrange the angles in straight line to prove that the total is 1800

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Angles at a Point The sum of the angles at a point is ALWAYS 3600

Approximation When we are solving (or trying to solve) a maths problems we need to know the degree of accuracy needed. For example, if a nurse is drawing up drugs for a patient, accuracy is clearly vital.

However, if you only want an idea of the size of an answer you might get away with rounding the figures to give you a feeling for how large the answer

should be Example

3.1 x 4.9

If we havent got a calculator this involves writing the numbers down on paper and doing a, slightly messy, calculation always with the possibility of putting the decimal point in the wrong place! If however, we round the 3.1 down to 3 and round the 4.9 up to 4 we have a

much easier calculation and the answer to THIS calculation is 3 x 5 = 15 NOTE: THIS IS NOT THE ANSER TO THE PROBLEM

This does give us an idea of SIZE of the answer

The actual answer is 15.19 which is close enough to our approximation (15) to give us confidence that the answer is correct

Arc Length As its name suggests arc length if the length of an arc of a circle!

a

Length of arc = Angle a x 2r 360

NOTE: We are working out a fraction of the length of the circumference of the circle (2r)

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Area A measure of the space within a shape the area is sometimes given by a formula (e.g. the area of a square is l x l where l is the length of the side) and sometimes you need to work out the area by working out the areas of

small parts of the shapes and adding the separate areas together.

Area of a rectangle Area of rectangle = w x b where w is the length of one side of the rectangle and b is the length of the other side

Area of a parallelogram

h

w

Area of parallelogram = w x h

Area of a triangle

Area of triangle = 1/2 x b x h

Arc A part of the circumference of a circle

Arrowhead A quadrilateral (four-sided shape) in the shape of an arrow:

w

b

b

h

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Arithmetic Mean See below

Average (see also Mean) A quick measure of the general size of a set of data. Calculated by adding up all the separate piece of data and then dividing by the number of pieces of data.

For example the average of 2, 3, 7, 9, 14 is 2 + 3+ 7 + 9+ 14 divided by 5 = 7

Axes of symmetry A line drawn on a shape or graph so that the part of the shape on one side of

the line is a mirror image of the shape on the other side of the line

Bar chart A type of picture used to illustrate data. It is created by first

Bearing The direction of one point from another e.g. we might say that B is 900 in relation to A (which in fact would mean that B was due East of A)

A Vertical Bar Chart A horizontal bar chart

Axis of symmetry

B

A

900

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Bias When you do a statistical calculation you need to have all the possible outcomes to be equally likely (e.g. the faces of a die appearing upwards). If the events are not equally likely (e.g. you only ask women what football team

they support!) then there will be bias (less women than men are likely to support a particular football team)

BIDMAS A made up word standing for the order in which arithmetic operations are to be carried out. The letters stand for:

B Brackets (these are to be worked out first) I Indices D Division M Multiplication A Addition S Subtraction

Billion 1,000,000,000 (one thousand million)

Binary system The system of arithmetic which uses only the numbers 0 and 1 to represent any number (as against the 0,1,2,3,4,5,6,7,8,9 we use in then decimal system)

Bisect Cut into equal halves as in bisect and angle cut into two equal pieces

Box Plot A Box Plot (also known as a box and whisker plot) illustrates the distribution of a set of data including Lowest Value, Highest Value, UPPER QUARTILE, LOWER QUARTILE and MEDIAN

Example

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Highest Value

Lowest Value

Lower

Quartile

Upper

Quartile

Median

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Brackets Used to separate parts of an expression (e.g. in formulae). NOTE: that BIDMAS tells us to carry out the calculations within the brackets, before carrying out other actions.

5 + 3(7 2) = 5 + 3 x 5 = 5 + 15 = 20

Calculator An electronic machine used for carrying out calculations. Simple calculators can do normal addition, subtraction, multiplication and division, together with percentages and possible square roots. More complex calculators (often called

Scientific calculators, can do more complex mathematical tasks (like drawing graphs on the screen) and often can be programmed

Calendar A chart which shows the days of the week and months of the year in particular the 31 days in January, March, May, July, August, October, December; the 30 days in April, June. September and November and the

peculiarity of February which has 28 in it each year, except a Leap Year when it has 29 days

Cancelling fractions The process of making fractions smaller and more manageable by dividing the

top number (the numerator) by the same number as the denominator e.g. the fraction 8/12 can be simplified by cancelling down divide top and bottom by 4 and we have 2/3, the simplified fraction

Example: 6/12 can be simplified by dividing top and bottom by 6 to give 1/2 the fractions 6/12 and 1/2 are called Equivalent Fractions

Capacity, units of Units used to measure volume e.g. cubic centimetres (c.c), cubic feet etc

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Carry When doing out calculations such as addition, subtraction, multiplication and division when there is a remainder you move the reminder (carry it) in order to continue the calculation:

5 9

7

4 1 3

Cartesian Coordinates These are the points which you plot on a graph. The name is in memory of the mathematician Descartes

Remember: The point are in pairs in the order (x,y)

Celsius A unit of temperature measure 00c is freezing point of water 1000c is the boiling point of water

Census The process of collecting information on everybody in the UK. This process is carried out

every 10 years when there is a 1 at the end of the year! (2001, 2011, 2021 ....). This information is used by government to help them with policies

such as Health, Education etc

Centigrade A unit of measurement of temperature (also called Celsius). The melting point of ice is 00c and the temperature at which water becomes steam is

1000c

Centilitre A measurement within the metric system equal to 1/100 of a litre

6 7 4634 6 6

And there will

be another remainder of

4 to carry

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Centimetre A measurement of length within the metric system = 1/100 of a metre

Centre of a Circle This is the point inside a circle which is the same distance from the sides of the circle (another ways to look at it is that a circle is the LOCUS of a point which stays the same distance from a given point (the Centre of the Circle)

Centre of Enlargement See Enlargement, Centre of

Centre of rotation The point around which an object is to be rotated. In many/most cases the centre of rotation will be the centre of the object but it doesnt need to be so:

Certainty The mathematical probability of 1 (if the probability of event happening is 0 then the event is impossible - b) e.g. it is certain that the sun will rise tomorrow morning (probability 1 - a) and it is impossible that there will be a 30 February (probability 0)

Chance The likelihood of something happening e.g. the chance of getting a head when tossing a coin is

A

The triangle has been rotated 900

clockwise about the Centre of Ro-

tation A

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Changing the Subject (of a formula) Formulae are usually written with the aim of finding a particular unknown: e.g. y = 5x + 3

if we substitute x = (say) 5 in this equation we get y =5 x 5 + 3 = 28

so y = 28 But, if we want to find x, given y we need to rearrange the equation to put x

on its own:

x = (y 3) 5 This rearranging of the equation is called Changing the Subject (in the first equation, y is the subject and in the second x is the subject

Chord A line drawn across the inside of a circle,

touching the circle twice and which doesnt go through the centre.

Circle Strictly a circle is formed by the path (locus) of a point moving at the same

distance from another point. This is achieved, usually, by means of a compass where the pencil stays the same distance form the point, as the compass is rotated.

Circumference The circumference of a circle is the distance around it

The circumference of a circle is calculated as follows;

Circumference 2 x x r Where r is the radius

Centre

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Circumscribe Literally (draw around) So a circumscribed circle around a triangle look like:

Class interval The upper and lower limit of a band of data e.g. if you were measuring peoples heights you might have an interval of 10cm all people between

Coefficient When you have an equation of formula you usually have unknowns (letters)

and these letters have numbers in forn of them. These numbers are called to COEFFICIENT of the unknown

Example

y = 5x2 + 7x - 9

the 5 is the COEFFIENT of x2 and 7 is the COEFFICIENT of x. NOTE: the -9 is

not a coefficient but is called the CONSTANT TERM

Collecting terms In algebra the process of simplifying calculations by collecting terms of the same type e.g. 3a + 5a 2a = 6a

Column Graph A graph where the data is shown as a series of columns (or BARS) this chart is also called a BAR CHART

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Common Factor A number (FACTOR) which will divide into two or more other numbers Example

5 will divide into 15, 325 and 1005 and so is a COMMON FACTOR

Common Multiple A number into which all of a set of number will divide

Example

Take the numbers 3, 4, 5 a COMMON MULTIPLE is 60 because 3, 4, 5 will all divide into 60. NOTE: another way of look at COMMON MULTIPLES is as a multiplication table = 60 = 3 x 20; 60 = 15 x 4; 60 = 12 x 5

Compass An instrument used for measuring direction. Most compasses

identify North, South, East, West, South-East, South-West. North-East, North-West

Compasses A instrument used for drawing circles, or parts of a circle.

Complementary Angle Two angles are complementary if they add up to 900

a a

b

b

In both cases a and b are complementary

angles

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Concave When a shape is bowed inwards rather than outwards (as in CONVEX) - think of the idea of a cave and that it goes into the rock.

Completing the Square The process of re-writing a quadratic so that it is solvable by making one side into a square!

Suppose we have f(x) = 2x2 + 3x 5

Step 1 add the coefficient of x and square it to BOTH sides And we get 2x2 + 3x 5 + (3/2)2 5 And we can write this as 2(x2 + 3/2x) 5 NOTE the number in the bracket is ALWAYS the coefficient of x or b/2a

Compound Interest When you INVEST money you expect to get back not only the money you invested, but some extra money (for you lending the money) this extra money is called the INTEREST and is usually worked out per year. But, if you invest your money for more than one year you will not only get the interest for the first year but also interest on the interest for the second year an so on Example

Suppose you invested 1000 for 3 years at 10%

At the end of the first year you would have 1000 + 1000 x 10% = 1000 +

100 = 1100

At the end of the second year you would have 1100 + 1100 x 10% = 1100

+110 = 1210

At the end of the third year you would have 1210 + 1210 x 10% = 1210 +

121 = 1331

Your investment of 1000 has now grown to 1331 with compound interest

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Concentric Circles A collection of circles of different sizes which all have the same centre:

Cone A three dimensional object which has a round base which tapers upward to a point (or VERTEX)

Congruence Meaning the same as in every detail so two shapes are only congruent if each side on one shape has a similar sized side on the other and each angle on one shape has an equivalent sized angle on the other.

Consecutive One after another e.g. the next three consecutive numbers after 5,6,7,8 are 9, 10, 11

Conjecture An educated guess i.e. a prediction based on evidence. So, you might say a pattern which I have found in a series of numbers suggests that the next number will be, and the nth number will be...... It is then up to you to prove your conjecture!

Constant A constant is a number, usually a real number, whose value is fixed i.e. doesnt vary. For example the number 5 in the equation

3x2 + 9x 5

is a constant, whereas x is a variable

Three versions of the cone

. All these circles have the same centre

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Construction A drawing which is made using (usually) only ruler and compass. Constructions are often of, for example, triangles or hexagons etc

Continuous data Data which cannot be split into distinct groups or categories, except by choosing ranges e.g. the height of people goes on and on and there is no starting and finishing points so we tend to say those people between 53 and 56 then those between 57 and 510 and so on

Convert To change from one form to another e.g. change from one currency to another ( to $) or from one form of measure to another e.g. miles to

kilometres

Conversion graph A graph used to convert one quantity into another e.g. converting money from one currency to another, or metric to imperial measures.

Convex A shape which bows outward as opposed to inward (concave)

A example of using the idea

of convexity in a mirror

Usually, you will use only ruler, pencil, compasses and possibly a protractor

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Co-ordinates The system of locating where a point is using pairs of numbers (x,y) where the first

number is taken to be horizontal (along the x-axis) and the second number to be

vertical (along the y-axis)

Correlation The relationship between two or more sets of data.

Correlation, negative A situation where there is little or no relationship between two sets of data e.g. the colour of a pupils eyes and their performance in a mathematics exam

Correlation, positive A situation where there is a close relationship between two sets of data e.g. the amount of homework done by students and their performance in examinations

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Corresponding angles Angle which have the same relationship to one another, usually created using parallel lines.

b

b

The angles b are equal

Cosine Rule A rule (or formula) to work out an angle or a side when you are given certain information e.g. two of the sides and the included angle

Counting numbers The normal numbers we use to count objects 1,2,3,4,5,6 ..

Cross-section A slice through a solid object, showing what the object is like from the inside.

Cube A three-dimensional object with all faces being squares

A cross-section of a tap showing how it looks from the

inside

b2 = a2 + c2 + 2ac Cos B

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Cube Numbers The cube numbers are so-called because they are all the VOLUMES of cubes which have INTEGER (whole number) lengths of side

So a cube with side 1 will have a volume of 1 x 1 x 1 = 1 (the first cube number)

A cube with a side of 2 will have a volume of 2 x 2 x 2 = 8 (the second cube number)

A cube with a side of 3 will have a volume of 3 x 3 x 3 = 27 (the third cube number)

The first 6 cube numbers are 1, 8, 27, 64, 125, 218

Cubic Graph A CUBIC GRAPH is a graph of a CUBIC FUNCTION that is a function which has a highest power of 3

Examples:

Cuboid A three dimensional box with opposite sides equal and all angles equal to 900

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Cumulative Frequency A rolling sum where you add up each of the preceding frequencies to get the next one. The example below shows how it is done

Frequency: Cumulative

Frequency:

4 4

6 10 (4 + 6)

3 13 (4 + 6 + 3)

2 15 (4 + 6 + 3 + 2)

6 21 (4 + 6 + 3 + 2 + 6)

4 25 (4 + 6 + 3 + 2 + 6 + 4)

Cummulative Frequency Graph A picture of the CUMMULATIVE FREQUENCY table:

Currency The general name given to the means by which goods and services are bought and sold in the UK the currency is the and in the United States the

dollar.

Cyclic Quadrilateral A quadrilateral which fits perfectly inside a circle:

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Cylinder An example of a RIGHT PRISM the cross-section is a constant-sized circle:

Data Facts usually collected using DATA COLLECTION SHEETS e.g. colour of eyes, age, car driven etc. NOTE: Not to be confused with INFORMATON which is DATA is connected to make sense e.g. John has blue eyes, is 27 and drives a Ford car

Data, continuous Data (numbers) which flow together without there being a break e.g. heights of children, temperatures etc

Data, discrete Data which is separate from other data e.g. the number of people who like a particular flavour of crisps, people with particular colour eyes etc

Data collection sheet

Decagon A ten-sided POLYGON . If all the sides are the same size it is called REGULAR

Data collection sheet to collect information about peoples attitudes to recycling

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Decimal A counting system which uses the number 10 (dec) as its base, so for example, all numbers in this system are multiples of powers of 10:

1 10 102 103 104 105 and so on

NOTE 1: !01 = 10 and 100 = 1

NOTE 2: any number to the power 0 is always 1

Decimal fraction A decimal fraction is simply a decimal which can be converted into a fraction

For example 0.2 is 1/5 and

0.3333333333333 is 1/3

Decimal Place The position of a digit after the decimal point:

In the number 24.5734 the digit 3 is in the third decimal place

Decimal Point A mark (or dot) which separates the whole number part of a number from the fractional part the digits to the RIGHT of the decimal point represent parts of a whole

Decimal system A system of arithmetic which uses the ten digits 0,1,2,3,4,5,6,7,8,9 to express

and carry out calculations dec meaning TEN

Decrease Make smaller or lessen. The result of a decrease is always a quantity which is smaller than the one you started with.

Degree A measure of rotation there being 360 degrees to go around completely!

Denary Another name for the system which uses the number 10 as the base of all calculations also called the DECIMAL system

Denominator The number on the bottom of a fraction - in the fraction 3/4 the 4 is the

denominator

Diagonal A line drawn from one vertex (point) of a shape to another vertex.

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Diameter Any line drawn from one side of a circle to the other, and going through the centre

Dice The plural of the word DIE which is a six-sided cube with the numbers (usually) 1 6 on each of its sides

Difference of two squares If you have any two numbers lets call them x and y and you square them and subtract on square form the other, you get

x2 y2

you can factorise this expression to give you (x = y)(x y)

and this is true for ALL NUMBERS

Example

1002 992 = (100 + 99)(100 99) = 199 x 1 = 199 (easier than working out 1002 and 992!

Difference Table Used to work out the general formula for a sequence of numbers

Suppose we have a sequence:

7 12 17 22 27 32 37 ..... .....

To help us find the formula we create a DIFFERENCE TABLE:

n 1 2 3 4 5 6

nth term

7 12 17 22 27 32

1st difference 5 5 5 5 5

5n 5 10 15 20 25 30

Having got a first difference of 5 we know that the formula will start with 5n and if we try working out the sequence using 5n we see that, in every terms

we are short by 2 so we add 2 to the 5n to get the formula The formula then is 5n + 2 and this will give us any term in the sequence

All lines are di-

ameters

Die Dice

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Digit Another name for number often used to describe a part of a number or expression for example look at the third digit in the number

Direct Proportion When one quantity changes in a fixed way (or constant) in relation to another we say the two quantities are in DIRECT PROPORTION

Example The CIRCUMFERENCE of a circle is given by the equation c = x d The constant is (meaning that when we want to find the circumference of a circle all we have to do is multiply the diameter by The relation between two quantities which are in direct proportion is usually

written as: c d and you say it c is proportional to d To write it as an equation we can solve we write:

C = kd (where k is the constant in this case

Directed numbers Number which have a sign associated with them the sign will be either positive + or negative -

Discount The amount of money which is deducted from the price of an article, usually in order to stimulate sales. The discount is often stated as a percentage e.g. 10% off the original price.

Discrete data Data which stands alone and doesnt blend in with others. For example the number of pupils who support a particular team (you cant have 2.5 pupils!)

Division The technique for finding how many times one number will fit into another. Division is a method of repeated subtraction you can simply subtract one number from the other and count how many times you can do it

Divisor Any number which will divide into a given number (NOTE: We will always have 1 and the number itself as divisors)

Example 2, and 3 are divisor of 6

IMPORTANT NOTE: The full list of divisors for 6 is -2, -3, 2, 3 (in addition to 1, -1, 6 and -6)

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Dodecagon A POLYGON which has 12 sides. If all the sides are the same length the polygon is called REGULAR

Dodecahedron A 12-sides three-dimensional solid comprised of 12 pentagons

Edge The place where two edges meet usually a line, often sharp

Elimination The process of getting rid of some parts of a problem to make the problem easier to solve

Example

Take the SIMULTANEOUS EQUATIONS: 5x + 3y = 13

7x 2y = 12 We solve these equations by ELIMINATING one of the variables (x and y). In

this case it is easy to eliminate the y by multiplying the fist equation by 2 and the second equation by 3, to get

EDGES

An Australian 50 cent piece which has 12 sides (although they arent straight!)

Dodecahedron Net of Dodecahedron

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10x + 6y = 26

21x 6y = 36 If we now add these two equations we get:

31x = 62 And so x = 2

We solved these equations by eliminating one of the unknowns

Ellipse Another name for an ELLIPSE (see entry)

Enlargement The size of an object which is larger than the original. The making of the object larger is usually done according to a rule.

Enlargement, centre of The point from which an enlargement is carried out

Equally Likely Events which are equally likely are also described as Evens or 50-50

Equation A way of stating the relationship between two or more quantities. All

equations have an = sign in them and this means that what is on one side of the = is of the same value as that which is on the other side of the = sign

For example v = rt + rt2

Equiangular Having all angles equal as in a square, rectangle and any REGULAR POLYGON (see entry)

Equidistant The same distance from .......................

.

Centre of en-

largement

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Equilateral triangle A triangle which has all sides and all angles equal (all the angles are equal to 600 there are 1800 is all triangles and this triangles has three angles equal)

Equivalence Equivalence means that two or more expressions, shapes etc can, for all practical purposes, be treated as being exactly the same. The most common

examples are equations where the left-hand side is to be treated exactly the same as the right-hand side; and congruent shapes which are the same in every way.

Equivalent Equations When trying the solve an equation ,each line in the solution must contain and equivalent equation to the one above:

Example:

5x + 3 = 2x 9 5x 2x = -3 -9 3x = -12

x = -4

Equivalent Fractions Fractions are EQUIVALENT if you can cncel them down so that they all become the same when they are cancelled to their lowest form:

Example

8/12 10/15 200/300 24/36 14/21

Are all EQUIVALENT FRACTIONS because they can all be cancelled down to 2/3

Eratosthenes (sieve of) This is, in fact, a method for finding the PRIME NUMBERS (the method is

actually called an ALGORITHM)

You start off with a list of the numbers within which you want to find the PRIME NUMBERS (in the example we have picked numbers between 2 and 120)

The first prime is 2 so we would put a circle around this number

All angles equal to 600

All sides equal

b b

b

In working out the solution to the equation we have subtracted 2x from BOTH sides and

subtracted 3 from BOTH sides in each step the equation is equivalent to the one above

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We then look at all the multiples of 2 (4, 6, 8, 10, 12 and so on) and put a

cross through these (they are NOT primes because they are divisible by 2) The next number, after the first one (2) is also a prime, so put a circle around

it (3). Now look at each multiple of 3 and put a cross against each of those not already having a cross they cant be prime because they can be divided by 3)

So, far we have two primes 2 and 3

Start at the beginning of the list and find the first number which doesnt have a cross against it (5) this also prime, so put a circle around it No put a cross against all the multiples of 5 which dont already have a cross against them

We now have three primes 2, 3, 5

Carry on like this and you will end up with the list of primes shown in the illustration.

Escher A Dutch artist who created designs using mathematical shapes and ideas:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,

37, 41, 43, 47, 53, 59, 61, 67, 71,

73, 79, 83, 89, 97, 101, 103, 107, 109,

113

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Estimation The technique for approximating the answer to a calculation. NOTE that does not mean guessing the answer but involves rounding up parts of the information to make the calculation simpler and give an idea of the general size of the answer expected of the full calculation.

Eulers Formula Let us name the parts of a POLYHEDRON V = Vertices, E = Edges, F = faces

Then Euler Formula says that, for EVERY, polyhedron:

V E + F = 2 Example

Take a TETRAHEDRON:

It has 4 faces (F = 4)

It has 4 vertices (V = 4) It has 6 edges (E = 6)

So 4 6 + 4 = 2

Evaluate Sometimes you are given an expression say 5x + 3y and also the values of x and y and you have to work out the value of (EVALUATE) the expression

Suppose we have 7x 2y and x = 4 and y = 1

Then 7x 2y = 7 x 4 2 x 1 = 28 2 = 26

Even Chance Two events have an equal possibility of happening (heads or tails in the toss

of a coin) also known as evens or 50/50

Even number A number which is divisible by 2 and leaves no remainder. The first five even numbers are 0, 2, 4, 6, 8 and any other number ending in

one of the these numbers is also even.

Event An occurrence of a particular activity. If you are studying the numbers which appear on a die when throwing it every throw of the die is an event

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Exhaustive Events The total of all the possible outcomes of an experiment i.e. every possible result is covered. For example, if you throw a normal 6-sided die you MUST get 1, 2, 3, 4, 5 or 6

Expanding Multiplying out an expression and, usually, collecting all the like terms together:

Example: Expand 3x(2x 3) + 5x(x + 7) We get 6x2 9x +5x2 + 35x = 11x2 + 26x

Experimental Probability In order to work out the probability (likelihood) of an event (or events)

happening you need to make the event happen a large number of times and then divide the number of successes by the number of events e.g. you would find the probability of throwing a head by tossing the coin a large number of times (perhaps thousands) and dividing the number of heads by the number of tosses of the coin

Exponent The number to which another number is to be raised (otherwise called the number of times it is to be multiplied by itself)

Example

53 3 is the EXPONENT and means that 5 is to be multiplied by itself 3 times = 5 x 5 x 5 = 125

Expression A collection of unknowns and, possibly, numbers which means something. For example to hire a car might mean putting a deposit down of 10 and renting

at 25 per day.

You may need to throw the die a lot more times, in order to get very close to the theoretical

probability

33

We can write the cost for any rental as 10 + 25d where d stands for the number of days rental If we put another unknown and an equal sign in we get an equation e.g. c = 10 + 25d

Exterior angles

`

Extrapolation Extending an idea or (in maths) a sequence, given the information for the first part of the sequence

Example

2 9 16 23 30 ... ... ...

The rule here is add 7 so we can EXTRAPOLATE the series by adding 7 each time, to get the series:

2 9 16 23 30 37 41 48

Faces The sides of a three-dimensional object e.g. a cuboid has six faces.

Factor(s) The numbers which will divide into another number e.g. the factors of 12 are 1, 2, 3, 4, 6, 12 because all these numbers will divide into 12 without leaving a remainder. NOTE: All numbers have the factors 1 and themselves

34

Factorising The process of breaking down a number or expression into parts which, when multiplied together, will give you the original number/expression.

Example 1 The number 24 can be written as 2 x 2 x 2 x 3 and the numbers 2 and 3 are the FACTORS and the number 24 has been FACTORISED Example 2

6x2 x 2 = (2x + 1)(3x 2)

(2x + 1) and (3x 2) are the factors

Fair/unfair (as in dice) A description usually applied to an object (or objects) which will be used in probability calculations. An object is fair if all possible outcomes (i.e. results) are equally likely e.g. in a fair die the probability of getting a 1, 2, ,3 ,4 ,5 or 6 are all equally likely

Fibonacci numbers Named after a 14 century mathematician the numbers come from adding each pair of numbers to get the next one. E.g. if the first two numbers are 0

and 1 the third number is 0 = 1 = 1, the second number is 1 + 1 = 2, the fourth number is 1 + 2 = 3 the next 2 + 3 = 5, the next 3 + 5 = 8 and so on. The first twelve Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,

89

Finite Decimals A FINITE DECIMAL is one which finishes there are no more digits to go after the last one written down:

Example:

0.3 0.5 0.125 0.875

Foot A unit of length measurement in the Imperial system. 3 feet make a yard and

there are 12 inches in a foot.

The number of dots on opposite sides of a die (the singular of dice) add up to 7

35

Formula A way of expressing a relationship between quantities using letters and numbers. e.g. we might say that the area of a rectangle is A = L x B where A is the Area, L is the length and B is the Breadth

Four Operations The four operations on numbers are Addition, Subtraction, Multiplication and

Division

Fractal A fractal is a geometric shape which is created by repeated, well defined, operations on a starting shape, with the same rule being applied to successive

results. For example, the Sierpinski Triangle starts as a filled in triangle and then triangles are subtracted from it as in the successive illustrations below:

Fraction A way of describing parts of a whole. A fractions has two parts the top part (called the Numerator) and the bottom part (called the Denominator). The bottom part tells how many parts the whole has been split into, and the top

part tell how many part we are to take.

Frequency The number of times something happens. In mathematics it might be the number of times the Ace of spades is drawn when a pack of cards is cut, of the number of times a 6 appears when a dice is thrown.

Frequency Distribution A graph showing the numbers of times each EVENT happens

The circle on the left has been cut into three pieces

and 2 of them coloured in giving us 2/3 coloured in

The circle on the right has been cut into 12 pieces with 8 slides coloured in giving us 8/12 coloured in And these two fractions are

clearly the same

36

Frequency polygon A shape drawn on a bar chart where the point of the shapes are the mid points of the bars.

Frequency Table A FREQUENCY TABLES records how often (frequently) a particular piece of data occurs

Class Interval Tally Frequency

0 9 111 4 10 19 1111 1111 1 11 20 29 1111 11 7 30 39 1111 5 40 49 1111 111 8 50 - 59 1111 1 6

60 - 69 11 2

Sum = 43

Function A formula or procedure which carries out a set series of operations e.g. ou might say take a number, double it and add 5

Function Machine A (mathematical) machine which takes an INPUT and operates on it according to a set of rules (adding, subtracting, multiplying etc) to give a result called the OUTPUT

Example:

In this FUNCTION MACHINE we have taken (INPUT) the number 5 and

subtracted 2 from it and then multiplied this answer by 3 to get the OUTPUT 9

NOTE: it is from this idea of a FUNCTION MACHINE that we get the idea of a FUNCTION although we write the function down in a much neater way. In the above example we would write f:x 3(x 2)

The frequency polygon

The total amount of data collected

37

Gallon A unit of liquid measurement within the Imperial system. There are 8 pints in a gallon and a gallon is approximately equal to ?? litres

Generate Produce e.g. an equation such as y=2x+3 will generate a series of numbers as you substitute for values of x

Gradient The SLOPE of a line = i.e. the angle at which it rises

Example

Gramme A unit of weight measurement within the metric system. A gramme is

approximately equal to ?? Ounces.

Graph A picture of a set of data see Bar Chart, Scatter Diagram etc

Greater than > A sign used to indicate which of two items in the larger e.g. we would write 5 > 3 meaning 5 is greater than 3, or 2 > -4 meaning 2 is greater than 4, or even x 4 > 6 meaning that x-4 is greater than 6

Greater than or Equal to () A symbol () used to show that an unknown (usually a letter) is BIGGER or EQUAL to a particular number Example:

X 5

This means that x is equal to 5 or BIGGER

Grouping data Collecting data together within given ranges e.g. you might group age data in bands of, say, -=5, 1-6. 7 12 years etc

HCF (Highest Common Factor)

3

3

The GRADIENT is calculated as the distance a line goes UP divided by how far it goes

ALONG In this case it goes up 3 and

along 3, so the GRADIENT is: 3 = 1

3

38

The Highest Common Factor (HCF) of a series of numbers is the largest

number which will divide into all of them, without leaving a remainder Example the HCF of 18 and 24 is 6

Height The distance from a base to an apex or vertex (the apex or vertex may be a point on a line)

Hendecagon An eleven sided figure (also called an ENDECAGON)

Heptagon A seven-sided shape (or polygon)

Hexagon A six-sided shape (or polygon)

Hire Purchase A method of buying things, where you pay a certain amount each week/month etc in order to pay for the goods.

Example

A shop offers the opportunity to buy a three piece suite for 600 cash or 21 per month for 30 months. At 21 per month for 30 months you will pay more

A hendecagon A 1 dollar coin issued in America in the

1970s but it was not successful despite being in the shape of a

hendecagon

Base Base

Height Height

39

than the original price of 600 this extra money is the INTEREST charged to you for being able to pay over a period of time

Histogram A histogram is a graphical display of frequencies. All the bars are next to each other with no spaces. A histogram is different from a bar chart in that it is the AREA of the bars which illustrates the measure rather than the height.

Horizontal A line going from left to right on the level (i.e. without being at any angle)

Hyperbola The curve created by cutting a cone through its base with a plane. Alternatively, the curve created when drawing the graph of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Horizontal comes from the word HORIZON the line which

separates the land/sea from the sky

40

Hypotenuse The line, in a right-angled triangle, which is opposite the right-angle

Icosahedrons 3 d object made up of 20 equilateral triangles, which have 30 edges and 12 vertices

Image The new shape created after operating on an original (called the object) - e.g. we might reflect a triangle in a mirror line the original is called the object and the new triangle is called the image

Imperial measure The old style system of measuring distance, weight etc. The principle units are ounces, pounds, stones (weight); inches, feet, yards, miles (distance), pints, gallons (liquid measure)

Impossibility If the mathematical probability of an event happening is 0 (zero) this means that it is impossible for the event to happen.. If the probability of the event

happening is 1, then it is certain that the event will happen.

Improper Fraction A fraction where the top number (the NUMERATOR) is larger than the bottom number (The DENOMINATOR)

0 1

Impossible Certain

hypotenuse

right angle

23

7

Icosahedron Net of an Icosahedron

41

Inch An Imperial Measure of distance. There are 12 inches in a foot and 36 inches in a yard.

Income tax Money paid to the government from earnings. Everybody has a certain amount of money on which they dont have to pay tax. A percentage of the remainder of their income is deducted and paid to the government (although some people work out their own tax, or get an accountant to work it out for them).

Increase Make bigger/larger

Independent Event Something which happens on its own and is not influenced or dependent on another event for example tossing a die and getting a number.

Index Notation When writing powers of numbers (i.e. the number of times a number has to be multiplied by itself) we write:

2 x 2 x 2 x 2 x 2 = 25 This way of writing powers is called Index Notation and the small number (in

this case the 5) is called the INDEX. The number being raised is called the BASE.

When two or more numbers WITH THE SAME BASE are multiplied together we ADD the INDICES

Example:

37 x 35 = 35+7 = 312

When we divide two numbers WITH THE SAME BASE we subtract the indices

Example 58 divided by 53 = 58-3 = 55

Inequalities An inequality shows a relationship between two quantities the relationship might show that they are equal, or that one is larger than the other. The symbols used are > (greater than) < (less than) >= (greater than or equal to) 2 2x >= 4 (this means that x is greater than or

equal to 2)

42

Infinite Forever a term used to indicate that something doesnt end. For example there is no end to the natural numbers there is no largest number they are INFINITE

Infinite Decimal A decimal which goes on forever, and doesnt end e.g. if you divide 1/7 you get 0.14285714285714 ....... and this decimal goes on forever it is INFINITE

Input/Output Terms often used in the context of computers you enter (INPUT) data into the system (or formula), it is processed (a calculation is carried out) and you

get OUTPUT otherwise known as the answer

Inscribe Draw inside for example draw a circle inside a triangle:

Installment If you take out a finance agreement to buy something, you will pay it back at

regular intervals and each payment is called an INSTALMENT

Integer Any whole number -34, 4, 6, 12345 etc

Intercept Where a line crosses (intercepts) an axis in a graph either the X or Y axis

y-intercept

43

Interest The money you receive, in addition to the amount invested, when you invest money for example in a savings account

Example You invest 200 at 5% for 1 year

At the end of the year you would receive your 200 back PLUS 5% of 200 = 10 INTEREST making 210 in all

Interior angles The angles on the inside of a closed shape

Interpolation Inserting terms in a sequence which are missing using the rule for the sequence

Example

Take the sequence

2 4 6 ... ... ... 14 16 18 The rule is clearly add 2 so we can INTERPOLATE the missing numbers using the rule, to get:

2 4 6 8 10 12 14 16 18

Interquartile Range Quartiles divide a set of data into four with 25% of the data less than the LOWER QUARTILE and 75% of the data being less than the UPPER QUARTILE

The INTERQUARTILE RANGE is simply UPPER QUARTILE LOWER QUARTILE

a

e

d

c

b

44

Intersection, point of The point where two or more lines cross

Intersecting lines Two lines which cross. Opposite angles are always equal where two straight line cross

Inverse The opposite - e.g. if I add 3 to a number the inverse operation would be to subtract 3 (and I would get back to where I started from!)

Inverse Proportion In non-mathematical terms this might be said the more one thing happens, the less another does

Example Suppose we have a rectangle with sides of 4 and 6 then the area will be 4 x 6 = 26

If we want to draw another rectangle with the same area, but double the length of one of the sides, say, 12, then we need to reduce the length of the other side by a half (2) to keep the area the same

Interval Usually, the gap between values. For example when creating graph of a

function you would choose the gap (interval) between values against which to plot the graph:

Investment When money is loaned to a company (e.g. when you buy shares in the company) you hope to get back more money than you put in!. This way of making money is called an INVESTMENT

The intervals here are 2 the difference between two

each pair of values

Here

the inte

rval is

5 t

he

diffe

rence

betw

een e

ach

pair o

f

num

bers

45

Irrational numbers Irrational numbers are numbers which cannot be written as fractions, such as pi and 2. In decimal form these numbers go on forever and the same pattern of digits are not repeated.

Isometric A type of drawing which shows objects in three dimensions

Isometric paper Paper on which dots or lines are printed in diamond formation. The paper is particularly useful for drawing 3-dimensional objects

Isosceles triangle A triangle which has two sides equal and two angles equal

Key A guide to what symbols on a diagram or graph mean e.g. you might use stick-men on a scatter diagram, so you need to say (in the key) what each stick-man stands for

Kilo The prefix to a word which means 1000 e.g. kilogram 1000 grams; kilometre 1000 metres

Kilogram A measure of weight equivalent to 1000 grams

Kite A quadrilateral with two pairs of adjacent sides equal.

Base angles equal

Two sides equal

a a

46

Laws of Indices These are the rules which tell you how to multiply and divide numbers written as indices WHICH HAVE THE SAME BASE

Multiplication ADD THE INDICES

56 x 53 = 56+3 = 59

Division SUBTRACT THE INDICES

85 82 = 85-2 = 83

LCM (Lowest Common Multiple) The smallest number into which two or more number will divide, without leaving a remainder. For example, the LCM of 12 and 18 is 36

Leap Year A year which occurs every four years and which has 366 days I it, as against the 365 days in a normal year. Leap years all have year numbers where the last two digits are divisible by 4 e.g. 1896, 1900, 1904, 1956, 200 were all Leap Years.

Length A one-dimensional measure, usually along a line e.g. 6cm

Less than < A symbol used to show that of two quantities one is larger than the other. The larger quantity is always on the right of the symbol

5 < 12 -2 < 1 1 < 4 -5 < -3

Less than or Equal to () A symbol used to show that of two quantities one is larger or equal to the other. The smaller quantity is always on the left of the symbol

3x + 1 7 means EITHER 3x + 1 < 7 or 3x + 1 = 7

LHS (Left Hand Side) Usually refers to the left-hand side of an equation and often said in a sentence like whatever you do to the left-hand side of an equation, you must do exactly the same to the right-hand side

47

Like Terms Parts of an equation or expression which are essentially the same e.g. in the expression 5a2 + 3ab 2a2 + 6c2 + 7ab the 5a2 and 2a2 are like terms and the 3ab and 7ab are also like terms

Likely/unlikely An unmathematical way of describing the probability of an event happening always between 0 and 1 in probability terms

Line of best fit On a scatter graph, a straight (or sometimes curved) line which

goes through the dots and creates the best path through the dots.

Line of symmetry A line (often dotted) which divides a shape so that one side of the line is a mirror image of the other side.

Line graph, straight The graph of equations of the

form y=mx+c is a straight line.

Linear equation An equation of the form y=mx+c where m is the coefficient of x and c is the

constant. Examples include y=3x-6; y = 7-5x. The reason it is called a linear equation is because the graph of the equation is a straight line.

1 2-1-2

1

2

3

-1

-2

Here we have the graph of the linear equation y = 2x + 3

48

Linear Graph A graph which is a straight line and is drawn form a LINEA EQUATION (se above)

Litre The main metric measure of liquid volume with all other measures being derived form the basic word millilitre, centilitre etc

Loci The locus (plural loci) is a path which a point or line travels when you make a

rule. For example the locus of a point which stays the same distance form a fixed point is a circle

Long division The process of dividing one number by another number, without using a calculator.

Lower Quartile (Q1) The Lower Quartile of a set of data is

(n + 1) 4th value Example

What is the Lower Quartile of the data:

21, 15, 3, 7, 9, 10, 4, 2, 6, 1, 15 First put them in order:

1, 2, 3, 4, 6, 7,9, 10, 15, 15, 21

The Lower Quartile is given by: (11 + 1) 4 = 3rd value, which is 3

.

The path travelled b a point which stays a constant distance from a fixed point

.

.

. .

.

.

49

Lowest terms We use the expression lowest terms often when we are working with fractions and want the numbers as small as possible, by cancelling down (finding numbers which will divide into top and bottom. The lowest terms is

reached when there is no number which will divide into top and bottom.

Magic Square A magic square is one in which all the numbers in any line (vertical, horizontal, diagonal add up to the same number

Map/Map Scales A scale picture of a real place (or places) - with actual distances on the ground being shown as smaller scaled images on paper. Most maps show the

scale of the paper picture to the real objects - with, for example. 1:50000 means that 1cm on the paper picture corresponds to 50000 cm on the ground.

Mass

8 3 4

1 5 9

6 7 2

In this magic square each row, column and diagonal

adds up to 15

Maps come in a wide variety of scales the smaller the number on the right the

greater the small detail shown on the map for example a map with a scale of

1:50000 will have 50000 inches represented by just one inch on the map

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Mean The technical term for Average calculated by adding up all the numbers and then dividing by the number of numbers.

Take the numbers: 3 12 5 7 9 11 2

Adding them together gives us 49

There are 7 numbers, so the mean is 49/7 = 7

Median A measure of the range of a set of data. It is calculated by putting all the numbers in order (it doesnt matter whether the order is ascending or descending) and then finding the middle number this is the median. If the number of numbers is even, find the two middle numbers add them together and divide by 2 to get the median

Example: Data is 27 12 5 17 35 then, in order, this is: 5 12 17 27 35 and the Median is the middle number 17

Metre The basic metric measure of distance (and where the word metric comes

from!). All other distances come form this basic root millimetre, kilometre etc

Metric system The system of measurement which uses grammes, metres etc.

Mile A measurement of distance within the Imperial system. There are 1,760 yards in a mile and a mile is approximately equal to 1.6 kilometres

Milli A measure of size in the METRIC SYSTEM e.g. milligram, millimetre, millilitre milli means one thousandths so there are for example 1000 milligrams in a gram

Milligram A units of measurement within the metric system. There are 1000 milligrams

in a gramme.

Millilitre There are 1000 millilitres in a litre

51

Millimetre A unit of measurement within the metric system. There are 1000 millimetres in a metre.

Millilitre 1000th of a litre I.e. 1000 millilitres = 1 litre

Minimum Value The smallest value which an equation, expression or graph can achieve. In graphical terms it is usually where the curve is at its lowest.

Minute A measure of time there are 60 minutes in an HOUR

Mirror line An imaginary line drawn through a shape which divide the shape so that the image on one side of the line is a mirror image of the shape on the other side of the line.

Mixed number A number which has a whole part and a fraction part

For example: 5 is mixed 5 is the whole part and the fraction

Mode A measure of the shape of a set of data. The Mode of a set of data is the most frequently occurring number

The mirror line

Here we can see that the minimum value is -10

52

Take the set of data:

2 5 7 12 3 2 6 8 2 17

The MODE of this set of data is 2

Mebius Strip Take a strip of paper and twist one end through 1800 and attach this end to

the other end of the strip and you have a Moebius Band. The strange thing about the Moebius band is that it only has one side if you put a pencil on any part of the band and draw along the strip you will eventually come back

to where you started without having taken you pencil off the paper!

Mode One of the AVERAGES used to describe a set of data

The MODE is simply the most frequently occurring piece of data Example

3 4 7 3 9 1 4 6 3 9 0

In the above set of data the MEAN is 3 because it is the most frequent number

Multiplication A shorter way than repeated addition. E.g. we could add 5 + 5 + 5 + 5 + 5 + 5 + 5 but since there are 7 5s we say 7 x 5 = 35 I.e. we multiply 5 by 7

Multiples The numbers which are multiples of a given number e.g. 4, 8, 12, 16, 20, 24 and so on are all multiples of 4. You will alos know these series of numbers as your multiplication tables!

Mutually Exclusive Events Two or more events are mutually exclusive if the both CANNOT occur e.g. if you toss a coin you cannot get BOTH a Head and a tail. However, if you threw a die you could get a number which was even AND larger that 3 e.g. 4, so these two events (evenness and larger than 3) are NOT mutually

exclusive.

53

Natural Numbers The NATURAL NUMBERS are the COUNTING NUMBERS 1, 2, 3, 4, ..... Zero is not usually included as a Natural Number although there is no complete agreement on whether it should be included or not!

Nth term Suppose we have a sequence of numbers as follows:

1 3 5 7 9 11 13 . It is straightforward to see that the next one will be 15, the next 17, the next

19 and so on But we need to be able to have a general expression for any term in the

sequence, and this expression is called the nth term So in the example above we have an equation which says that

Nth term is 2n-1

And when we substitute for n=1, n=2, n=3 etc we get the series

Negative number All numbers less than zero (0). If you add the positive digit part of a negative number to the negative number you always get 0 e.g. if we take 45 and add the digit part (45) we get 45+45 = 0

Nets The flattened two-dimensional shapes which when folded make 3 dimensional solids

Nonagon A nine-sided polygon

The net

of a cuboid

A regular nonagon has internal angles

which total 7 x 1800 = 12600

Which gives us the internal angle as 1260/9

= 1400

54

Normal The NORMAL to a curve at a particular point is a line at right-angles to the tangent to the curve at that point

Normal Distribution In Statistics this is a measure of how a set of results are spread around a

central point i.e. the MEAN. In a simple version of the distribution the results show a symmetrical distribution around the mean and the shape of the graph is a bell (so, the curve is often called a Bell Curve). The way in which the results are distributed around the peak (the top of the bell curve) is called the Standard Deviation.

Number line A line used to help work with positive and negative numbers. The line has the

positive numbers to the right, the negative numbers to the left and the number zero (0) in the middle

Number sequence A series of numbers which is linked by a rule e.g. 3, 7, 11, 15 .. Has the rule add 4

Numerator The number on top of a fraction e.g. in the fraction 4/7 the number 4 is the NUMERATOR

Object The shape of thing which are going to operate on e.g. we might have the drawing of a square and we are going to move it (translate it) to another

position the original square is called the object, and the square in its new position is called the image.

Normal

55

Oblong Another (and non-mathematical) name for a rectangle

Object/image If, you enlarge an shape (say a triangle) then this is called the object. The result of the enlargement is called the image.

Obtuse angle A angle which is greater than 900 and less than 1800

Octagon An eight sided polygon. An octagon with all sides and

angles equal is called a REGULAR OCTAGON

Octahedron A 3D solid with 8 sides made up of 8 equilateral triangles, four of which meet at each vertex

Odd number Any whole number which cannot be divided by 2 without leaving a remainder

e.g. 1,3,5,7,9,11,13,15,17...

d is an ob-

tuse angle d

object image

Octahedron Net of Octahedron

56

Ogive The graph you get when you join the points in a CUMULATIVE FREQUENCY CURVE

Opposite angles Angles which are opposite one another in a shape or diagram in cases where the opposite angles are as a result of two line crossing, the angles are equal.

Order 1 The order of an equation or expression is the largest power present in the

equation/expression For example

The expression 3x6 5x3 + 2x2 8x + 9 is of power 6 because this is largest power of x

Order 2 To order a series of numbers means to put them in a sequence where each one is larger (or smaller) than the one before called Ascending and Descending

For example the following numbers are in Ascending order:

3.1, 3.15, 4.01, 4.1, 5.23, 5.27

Order of rotational symmetry The number of times a shape will fit onto itself when it is rotated through 3600

a a The angles a are equal

57

Ordered Pairs Simple two numbers, written together e.g. (2,3). They are called ORDERED PAIRS because the order matters so (2,3) (3,2)

Origin The point where the X and Y axes cross (in, for example a graph). The point is usually (0,0)

Ounce An IMPERIAL MEASURE of weight. There are 16 ounces (oz) in a POUND (lb)

Outcome (usually) The result of carrying out a series of experiments often associated with trying to work out the probability of an event happening.

Oval Another name for an ELLIPSE

Palindromic Number A number which reads the same forwards and backwards e.g. 1234321 NOTE: If you take that last digit of each of the first 9 square numbers and

put them together they form a palindromic number

1 4 9 16 25 36 49 64 81

X

Y

. (0,0)

Ordered pairs are plotted with the

first number being along the X AXIS and the second along the Y AXIS

58

Parabola The curve created when the graph of a quadratic equation is plotted, or the shape created when you slice a cone through its slope and down to the base:

Parallel lines Line which do not meet (or which are the same distance apart forever!) and are usually shown as parallel with one or more ticks on them:

Parallelogram A quadrilateral which has opposite sides equal and parallel (NOTE: the opposite angles are also equal)

Parallelogram, area of (See Area of parallelogram)

Pascals triangle The sequence of numbers (often drawn as a triangle) which show the

coefficients of the variable in the expansion on (1+x)n

59

Pentagon A five-sided figure (or polygon).

Percentage The number of 1/100ths (as expressed as a fraction) - e.g. 25% means 25/100 = 1/4

Percentage change The amount (measured as a percentage) by which an amount (e.g. the price

of an article) goes up or down e.g. if an item cost 100 and was re-priced as 120 there would have been a 20% change (upwards!)

Percentage profit The amount of profit (measured as a percentage) made by someone selling

goods or services e.g. is a television is bought for 200 and sold for 250 they there has been a 50/200 x 100 = 25% profit

Perimeter The distance around the outside of an object e.g. around a square it is the sum of the lengths of the sides as is the perimeter of a rectangle

Perpendicular Or vertical as it is often called. A line which is at 900 to the horizontal

6

5 5

5.5 5.5

Perimeter = 5 + 5 + 5.5 +

5.5 + 6 = 27

60

Perpendicular Bisector A line which cuts exactly in half another line an at right angles to the line

Pictogram A chart which uses small pictures to show how data.

Pie chart A pictorial representation of data in the form of a circle with slices cut out of it to show the proportions.

Pint An IMPERIAL MEASURE of liquid. 8 pints equal 1 gallon

Place Value Every digit in both whole numbers and decimals has its own individual value called its place value. The value of a digit is a power of 10

Example the value of the integer 7 in the number 6783 is 700 and the value of the integer 7 in the number 34.076 is 7/100

Plan The view looking down on a three-dimensional object

Plane (mathematical) A plane can be thought of as a 2-dimensional sheet (say of paper). Often such a plane will be used to cut a 3-dimensional object, as in the case of the

cone:

BBC

CH4

BBC2

ITV

Perpendicular bisector

61

Plane of Symmetry A slice through a 3-dimensional object so that the two halves are identical

Platonic Solids The name comes from the mathematician and philosopher PLATO. There are five of them:

pm Meaning post-meridian pm indicates times between 12 noon and midnight

Points of the compass The main four points are North, South, East, West and these are, clockwise

from North South 1800; East 900; West 2700

Polyhedron The general term given to geometric 3 dimensional shapes (examples of polyhedra include DODECAHEDRON, TETRAHEDRON .....

Polygon, irregular Any polygons which doesnt have all its sides and angles equal.

Examples of regular polygons

62

Polygon, regular A polygon )many-sided shape) which has all sides equal and all angle equal

Polygon The general name given to any closed shape e.g. square, rectangle, pentagon, hexagon etc are all examples of polygons

Polynomial Function Poly means many so a polynomial function means any function which is made up of variables and constants and the operations of addition, subtraction and multiplication and whole number powers

For example 5x2 3x + 9 is a polynomial, but 3x2 4/x + x2/3 isnt because the constant 4 is divided by the variable x and the power (or exponent) of x is 3/2 not a whole number

Population The complete set of people or things which we sample. For example the

population could be all the pupils in a class, the coloured balls in a bag, all the people who live in Liverpool

Positive number All numbers greater than 0. Every positive number has an opposite such that when you add the positive number to its opposite you get 0 (zero). This opposite is called the negative of the number. For example the opposite of 3.25 is 3.25 because when you add them together you get 0

Pound 1. A measure of weight in the IMPERIAL SYSTEM of measures

1 pound (lb) = 16 ounces (oz)

2. A unit of money ()

Power The number of times a number is to be multiplied by itself shown as a small superscript number e.g. 53 means multiply the number 5 by itself 3 time 5 x 5 x 5 = 125

An irregular pentagon

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Prime Factor A number which will divide into another number which is also a PRIME NUMBER

The first seven PRIME NUMBERS ARE 2, 3, 5, 7, 11, 13, 17

Prime numbers Numbers which cannot be divided by ant other numbers apart form themselves and 1 the prime numbers up to 100 are are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Prime Factor Decomposition The prime factors are 2, 3, 5, 7, 11, 13, 17 and so on.

Any number can be written as a series of prime factors multiplied by each

other (some factors may be repeated) Example: 24 = 2 x 2 x 2 x 3 and 100 = 2 x 2 x 5 x 5

Priority of operations The order in which mathematical operations are to be carried out often remembered by the acronym BODMAS Brackets, Over, Division, Multiplication, Addition, Subtraction

Prisms A three-dimensional object which has the same cross-section throughout its

length

Prism, volume of The volume of a prism = area of cross section x length

Probability The likelihood (or chance) of an event happening is called the probability of

the event happening.

For example a pack of cards has 52 cards

A 2 3 4 5 6 7 8 9 10 J Q K

And 4 suits = Hearts, Clubs, Diamonds, Spades

So, for example, there are 4 Aces, 4 twos, 4 threes etc

A triangular prism

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Therefore the probability of getting an ace is 4/52 = 1/13 (when cancelled

down)

Probability scale A measure of probability usually a line with 0 at one end (impossible) and 1 at the other end (certain) along which every event must lie

Probability of an event The likelihood of an event happening defined as

number of successes Possible number of outcomes

Proper/improper fraction A proper fraction is one where the numerator (the top number) is smaller than the denominator (the bottom number). so 5/12 is a proper fraction whereas 21/5 is an improper fraction

3/7 is a proper fraction

12/7 is an improper fraction

Proportion Two quantities are in proportion if they increase (or decrease) at a similar rate (or ratio as it is called). For example 3 pints of blue and 1 pint of white

paint will make enough paint to paint two walls of a square room. In order to paint all four walls we need to double the amount of blue and the amount of white paint 6pints of blue and 2 pints of white paint.

Protractor An instrument, often semi-circular for measuring and drawing angles

Pyramid A three dimensional object which has a flat base of three or four sides and an apex to which flat, triangular. Sides rise form the base. A p[pyramid with a

triangular base is also called a tetrahedron.

The entrance to the Louvre museum in Paris is a square-based pyramid

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Pythagoras Theorem The theorem which states that The square on n the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides

Pythagorean Triples In a right-angled triangle (according to PYTHAGORAS):

a2 = b2 + c2 Where a is the hypotenuse

When all three numbers are INTEGERS the three numbers are called

PYTHAGOREAN TRIPLES (because it you draw a triangle with the sides of these lengths it will be a right-angled one!)

Examples

a b c a b c a b c a b c

7 24 25 8 15 17 12 35 37 9 40 41

15 112 113 16 63 65 20 21 29 28 45 53

28 195 197 33 56 65 44 17 125 48 55 73

60 91 109 65 72 97 133 156 205 140 171 221

In all the cases of the triples above a2 + b2 = c2

Quadrant A quarter of a whole e.g. the section which has only the positive/positive co-ordinates, the quadrant which has the positive/negative co-ordinates etc

Quadratic Equation An equation where the highest power is 2

h b

a

As an equation Pythagoras Theorem states:

h2 = a2 + b2

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Examples

5x2 + 3x 2 = 0 8x2 + 2 = 0 7-5x2 = 0

Quadrilateral Any four-sided figure examples are square, rectangle, rhombus, parallelogram, trapezium, kite.

Qualitative A feature of a person or object which cannot be (easily) measured e.g. beautiful. elegant

Quantitative A feature of a person or object which can be measured height, weight etc

Quartile There are three quartiles which divide sorted data into four equal parts The first quartile (lower quartile) cuts off the bottom 25% of the data

The second quartile is the Median and cuts the data in two

The third quartile (upper quartile) cuts of the bottom 75% or highest 25% of the data

Questionnaire A form used to record information (often from questions asked!).

Quotient The result of dividing one umber by another e.g. if we divide 12 by 3 we get the answer 4 which is the quotient

Radius The distance between the centre of a circle and the circumference

Examples of quadrilaterals

Radius)

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Random An event which happen without any planning or outside influence often used to indicate a trial where there outcome is unknown (e.g. throwing a die)

Random Sample When a survey is being conducted it is, usually, impossible to survey the

entire population you are interested in so you take a representative and random sample. This means sampling things or people who have been chosen randomly, without any pattern to their choice.

Range The range of set of numbers is the difference between the largest and

smallest number

Ratio A means of sharing things e.g. sweets might be shared in the ration of 3:2 this means that there are 5 shares (3+2) and one person will get 3 shares and the other person 2 shares

Rational numbers A rational number is a number which can be written as a fraction where

numerator and denominator are integers (where the top and bottom of the fraction are whole numbers). For example 1/2, 4, 1.75 (=7/4).

Raw Data Data which has simply been collected (for instance ion questionnaires) and not yet been collated or analysed.

Rectangle, area of The area of a rectangle is length x breadth

Rectangle A quadrilateral which has opposite sides equal and all angles equal to 900

The external angles of a rectangle are each equal to 900 making their total 3600

Reciprocal The reciprocal of a number is that number turned upside down e.g. the reciprocal of 2/3 is 3/2 and the reciprocal of 7 is 1/7

Recurring Decimal A recurring decimal is one which has a number, or set of numbers, repeating after the decimal point e.g. 1/9 = 0.111111111and 1/7 = 0.142857142857142857

W

B

Area = W x B

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Rectangular Prism A prism with a cross section of a rectangle

Reflection The image of a shape as seen in a mirror (or an imaginary, drawn mirror

Reflex angle An angle which is greater than 1800 and less than 3600

Regular (as in polygon) All the sides and angles are the same

f

A square A regular octagon

A regular hexagon

An equilateral triangle

Volume of prism = (2a + 3) x a x

3a

Surface area of prism = 2 ((2a + 3) x a)) + 2(a x 3a) +

2((2a + 3) x 3a))

2a + 3

a

3a

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Remainder What is left over after you have carried out an operation. For example if you were adding 37 and 59 you would start with 7 + 9 and get 16. This is one TEN and 6 left over (REMAINDER) the 6 would go in the answer and the 1 would be carried

Resultant of Two Vectors The resultant of two vectors is diagonal of a parallelogram drawn using the two vectors as adjacent sides

Revolution The verb to revolve means to go around; so a REVOLUTION is a complete

circle. A revolution takes place around a central point the AXIS OF REVOLUTION rather like the axel on a car wheel

Rhombus A four-sided figure (quadrilateral) with all sides equal and opposite angles equal (NOTE: all the angles in a rhombus are NOT equal)

F1 and F2 are the

vectors and Fnet is the resultant

The blades of a wind turbine REVOLVE around the spindle in the centre

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RHS (Right Hand Side) Usually refers to the right-hand side of an equation and remembering that what you do to the RHS you must do the same to the LHS

Right Angle An angle of exactly 900

Right-angles triangles A triangle which has one angle of 900. The side opposite the right-angle is

called the hypotenuse

Right Prism A three-dimensional object whose cross-section is the same all the way through

Examples

The VOLUME of a right prisms is:

Area of the cross-section x length

Roman Numerals A way of writing numbers which is said to go back to the Romans (hence the

name!)

hypotenuse

Some examples of right prisms take a cross-section parallel to the base and the cross section will be the same all the way

through the shape

Unorthodox shapes such as this house are also right-prisms

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Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

I II III IV V VI VII VIII IX X XI XII XIII XIV XV

Root A number which, when multiplied by itself a given number of times will equal a given number

Example the square root of 16 is 4 because 4 x 4 = 16 The third root of 8 is 2 because 2 x 2 x 2 = 8

The square root of a number is written as (for example) v16 = 4

Rotation The turning of a shape, often by a given number of degrees e.g. a shape might be rotated by 900, or 1800 etc

Rotational symmetry The number of times a shape will fit onto itself when it is rotated

Rotation symmetry, order of The number of times a shape will fit onto itself as it is rotated.

Rounding Tidying up a number to a pre-determined level of accuracy e.g. we might round 199 to 200, or 4.003 to 4

Rule A set of steps which have been laid down and which have to be followed in a

particular sequence e.g. take a number, double it, add 3 and divide by 2. Often a rule is given in short-hand by a formula

a

b

c d

e

This (regular) pentagpn has

order of rotational symmetry 5 you can turn it through 720 each time and it will occupy the same

space

This parallelogram has order of rotational symmetry 2 you can turn it through 1800 and it occupies its original position

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Scale A measure used to represent something which is larger by something which is smaller:

Scale drawing A smaller drawing of an actual (and much bigger) object e.g. the plans of your house!

Scale factor The amount by which a shape has been increased or decreased in size. If the number is greater than 1 this means that the shape has been made larger; if the number is less than 1 the shape is made smaller. If the number is equal

to 1 the shape remains the same.

Scalene triangle A triangle which has all sides all angles of different sizes.

Scatter diagrams A diagram showing two sets of different data e.g. the amount of homework done by pupils and their success in examinations, which allows a comparison between the two sets of data

Number of children

Nu

mb

er

of

we

ed

s

Graph showing the number of

weeds in the garden and the

number of children in the family

Map scales tell you what the relationship is between distance on the ground and distance on

the map 1:100000 means on cm on the map represents 100000 cm on the ground

Model cars may be made to, say,

a scale of 1:40 meaning that they are exactly the same as the real thing but 1/40 of the size!

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Scientific Notation Another name for STANDARD FORM

Second

The two types of measure are not connected

Sector

Segment A portion of a circle bounded by two points on the circumference

Semicircle

1. One sixtieth of a minute in time

i.e. there are 60 minutes in an

hour and 60 seconds in a minute 2. One sixtieth of a minute in circular

measure there are 3600 in a circle, sixty minutes in a degree and 60 seconds in a minute.

The yellow area is called a SECTOR. The part of the

CIRCUMFERENCE of the circle which is on the edge of the sector is called an ARC

A SEMICIRCLE is half a

circle bounded by a DIAMETER

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Sequence A series of numbers which are related to one another by a rule e.g. the numbers 2 4 8 16 32 are related by the rule multiply by 2

Similar Figures Two (or more) shapes which have EXACTLY THE SAME SHAPE but DIFFERENT SIZE. In maths terms they have the same size angles but different length sides

Simple Interest The amount of extra money you get when you invest money and when you dont get interest on the interest (which you do with COMPOUND INTEREST) The formula for working out Simple Interest is

P x T x R 100 where P in the amount, T is the Time in years and R is the rate of

interest (as a percentage)

Example 2000 invested at 3% simple interest for 5 years would earn:

2000 x 5 x 3 = 30000 = 300 in interest 100

so you would have 2000 + 300 = 2300

Simplest form Simplest form is the name given to a (usually) fraction which cannot be cancelled down any further

For example 8/12 can be cancelled down to 2/3 (by dividing top and bottom by 4) but it cant be cancelled any further, so is in simplest form

Simplifying Making an expression, fraction etc as simple as possible b collecting together terms, or cancelling down (in the case of fractions)

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e.g. 3x + 5y 2y + 7x y + 8x can be simplified to 18x + 2y and 18/24 can be simplified to 3/2 by dividing top and bottom by 6

Simultaneous Equations Two linear equations which have a common solution. For example the equations 5x + 2y = 16 and 3x y = 3 have the solution x = 2 and y = 3

Sine Take any angle in a RIGHT-ANGLED TRIANGLE; the sine of that angle is

the ratio of the OPPOSITE SIDE side to the HYPOTENUSE

Sine Rule A collection of expressions which allow you to work out the angles of a triangle, using the sines of the angles and the lengths of sides

Solution The value(s) which, when substituted in an equation, makes the equation true e.g. the solution to 2x + 3 = 9 is x = 3 because when we replace the x with the value 3 the equation is tru

Sphere A three-dimensional object which has a cross-section always of a circle

a = b = c sinA sinB SinC

hypotenuse

a

opposite

adjacent

Sin a = opposite hypotenuse

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Standard Form A way of writing very large or very small numbers using index notation (powers of 10) e.g. 56000000 can be written 5.6 x 107 and 0.0000000056 can be written 5.6 x 10-9 IMPORTANT NOTE the number used (in this case 5.6 must itself be between 1 and 10

Stem and Leaf Diagrams A way of grouping data into classes particularly useful since the original data can still be seen

Example: the original data is shown below 154, 143, 148, 139, 143, 147, 153, 162, 136, 147, 144, 143, 139, 142, 143,

156, 151, 164, 157, 149, 146 NOTE: the data is first of all organised into classes of equal width in this case 5 (136 139 etc). Then the diagram is simplified to look as follows

Subject, change of Making a particular unknown (x, y etc) the subject of an equation means

putting it (the unknown) on its own on the left-hand side of the equation, with everything else on the other side. Then the unknown will be the subject of the equation.

Example suppose we have y = 3x + 7. In this case y is the subject. Suppose

we want to change it so that x is the subject. We will get x = (y 7)/3 and here x is the subject.

Substitution Substitution is what you do when you replace a VARIABLE or UNKNOWN with a value ( a number)

Example

We have an expression 5x + 3y When we SUBSTITUTE x = 2 and y = 4 we get this expression equal to 22

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Surd A way of writing square roots, without actually working them out Examples

None of these numbers has an INTEGER (whole number) square root.

However, we can write them (or some of them) in terms of simpler roots:

This way of writing square roots is called write them in SURD form. This

format allows the solution of problems without trying to work out square roots which do not have an exact answer (e.g. the square root of 2)

Surface Area The collected areas of the outside faces/area of a shape:

Examples:

Symmetrical See Line of symmetry

Significant figure The digits to the right of any 0 (zero) in a number - e.g. 123 has 3 significant figures; 0.0034 has two significant figures

Solids A three-dimensional object. The most common ones are: Tetrahedron, Square-based pyramid; Cone; Cylinder; Cube; Cuboid, Hexagonal prism..

2 15 8 32 200

8 2 2 32 4 2 200 10 2

Surface area is 6 x the

area of a side (all the sides have the same area)

Surface area is:

2 x area of the sides + 2 x area of the tops + 2 x area of the front

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Solving (equations) Finding the value of variable(s) in an equation which make the equation true (solve it) - e.g. for the equation 3x + 2 = 8 if we put x = 2 in it we get x 2 + 2 = 8 and so the solution to the equation is x = 2

Square A quadrilateral having all sides equal and all angles equal (to 900)

Square numbers The sequence of numbers obtained by multiplying the whole numbers by themselves. The first 6 square numbers are 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9; 4 x 4 = 16; 5 x 5 = 25

NOTE: Square numbers get their name from the fact that they are the areas of squares which have whole number (integer) length sides

Square root Given any number the square root is the number which, when multiplied by itself will equal the given number e.g. given the number 64 the square root is 8 because 8 x 8 = 64

Statistics A branch of mathematics which is concerned with presenting and analysing

data with a view to understanding or presenting it better. E.g. a set of data can be described by using measures such as Mean, Mode, Median and data

can be illustrated using bar and pie charts

4cm

4cm

The area of this square ios 4 x 4 =

16cm2

If we know that the area of this square is 49cm2 then we can work out the length of the sides (which are all the same) by

working out the square root of the area = 7cm

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Straight line There are 1800 in a straight line. You can solve problems using this piece of information in the diagram below, if we know that one angle is 1450 we can work out the other to be 350 by subtracting the 1450 from 1800

Substitution The process of replacing letters in an equation or expression with numbers, often to find out an answer.

If we have an expression such as 5x + 3y and we substitute x = 3 and y = 4

we get: 5 x 3 + 3 x 4 = 27

Subscript A number of letter which appear to the right of another number of letter, but

smaller and a little lower. We write chemical formulae using subscripts e.g. H2SO4 the 2 and 4 are subscripts

Superscript A letter or number which appear to the right of another letter or number but

smaller and higher. Superscripts are used to denote powers e.g. 52 means 5 x 5

Symmetry, line of An artificial line which has a perfect mirror image of an object (or part of an object) on each side of the line

NOTE that the image on the right is an exact copy of the image on the left but back to front.

NOTE: also that each point on one image has a corresponding point on the other image exactly the same distance from the line of symmetry

a a b b

1450 350

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Tally marks Ticks you write down to help you when you are counting e.g. if you were to count the number of different coloured cars passing you,. You would have a list of the possible colours and put a tick against a colour ever