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Dyffryn School Ysgol Y Dyffryn Mathematics Faculty Formulae and Facts Booklet Higher Tier Number Facts Sum Difference Product Share Prime number Square number Cube number Factor Common factor Highest Common Factor Multiple Common Multiple Lowest Common Multiple This means add. This means take away. This means multiply. This means divide. A number that has only two factors and can only be divided by itself and one. (1, 9, 21, 27 are not prime numbers.) The answer to when a number is multiplied by itself. The answer to when number is multiplied by itself 3 times. A number that can divide into another number e.g. the factors of 6 are 1, 2, 3 and 6 A number that can divide into two or more other numbers The largest common factor. A number that another number can divide into e.g. Multiples of 10 are 10, 20, 30 … A number that two or more numbers can divide into The smallest common multiple Averages Mean – Total divided by the amount of numbers. Median – Middle number of an arranged set. Mode – The most common number. Range – The difference between the largest and smallest numbers. Area Square = (length)2 Rectangle = Length × Width Triangle = Base × Height 2 Parallelogram = Base × Height Trapezium = (a + b) × height 2 Compound/composite shapes – Split these up into known shapes and add up the areas. Surface area of a cone = rl Surface area of a sphere = 4r2 Circle Diameter is the distance across the centre of a circle Radius = ½ diameter Area of a circle = × radius2 Circumference of a circle = × diameter For a semi-circle find the area as normal then half the answer Area of sector = × r2 360 Length of an arc = 360 × D Volume Volume of any prism = cross sectional area (Face area) × length Cube = (length)3 Cuboid = length × width × height Cylinder = r2 h Cone = 1/3 r2 h Sphere = 4/3 r3 Square based pyramid = 1/3 × area of the base × height Density Density = Mass Volume M D Average Speed Average speed = Distance Time V D S T Right angled triangles Pythagoras’ Theorem c2 = a2 + b2 or a2 + b2 = c2 (The hypotenuse is the longest side and is opposite the right angle) Trigonometry SOH CAH TOA Non-right angled Triangles Finding an angle Cosine rule Cosine rule to find an angle Sine rule Sine rule to find an angle Finding a side Cosine rule Must have all three sides. Cos A = b2 + c2 – a2 2bc Must know a side and an angle as well as the side opposite the angle you are trying to find. Sine A = Sine B = Sine C a b c Need to know two sides and the angle between the 2 sides. You are trying to find the third side. Cosine rule a2 = b2 + c2 – 2bc cosA Sine rule Need to know a side and an angle as well as the angle opposite the side you are trying to find. (look for pairs) Sine rule a = b = c Sine A Sine B Sine C Area of a triangle ½ absinC Foreign Exchange To convert from the original money to new money multiply To convert back divide Scale drawing To go from scale to real life multiply To go from real life to scale divide Bearings Bearings are always measured from the North and are measured in a clockwise direction. If a bearing is bigger than 180o, measure anticlockwise and subtract from 360. Estimated Mean Estimated mean = (Total frequency x midpoint) Total frequency Grouped frequency diagram Frequency polygon or total f × x Total f Bar chart with a proper scale Plot across to the midpoint and up to frequency. Join the points plotted in order with a ruler. Units (You have to know these) 1cm = 10mm 100cm = 1m 1000m = 1km 1g = 1000mg 1kg = 1000g 1 litre = 1000cm3 1 litre = 1000 millilitres 1 litre = 100 centilitres 1 mile 1.6 km 1m3 = 1000 litres 1ml = 1cm3 1 gallon = 4.5 litres 1 inch 2.5 cm Converting squared or cubed units cm2 mm2 × 102 or 100 2 m cm2 ×1002 or 10000 3 m cm3 × 1003 or 1000000 2 2 mm cm 102 or 100 cm2 m2 1002 or 10000 3 3 cm m 1003 or 1000000 Metric Units Weight Grams (g) Kilograms (kg) Tonnes Length Millimetres (mm) Centimetres (cm) Metres (m) Kilometres (km) Volume Millilitres (ml) Centilitres (cl) Litres (l) Grams are used for the weight of small items, for example the weight of a pound coin or the weight of an exercise book. Kilograms are used for the weight of heavy items, for example, the weight of a sofa or the weight of a person. Tonnes are used for the weight of very heavy items, for example The weight of an aeroplane. Millimetres are used for very small lengths, for example the width of a finger nail. Centimetres are used for small lengths, for example the length of an exercise book. Metres are used for distances such as the length of a room, the height of a person or the height of a door. Kilometres are used for long lengths such as the distance between Swansea and Cardiff. Millilitres are used for small volumes, for example the volume water in a teaspoon. Centilitres can be used to measure liquid in a glass. Litres are used for larger volumes, for example the volume of petrol in a tank. Angle Facts Angles on a straight line add up to 180o Angles round a point add up to 360o Angles in a triangle add up to 180o Angles in a quadrilateral add up to 360o Vertically opposite angles are equal Corresponding angles (F) formed in parallel lines are the same Alternate angles (Z) formed in parallel lines are the same Interior angles (C) formed in parallel lines add to 180 Circle Theorems Angle at centre twice one at circumference Angles subtended on the same arc are equal Opposite angles in a cyclic quadrilateral add to 180 Angle in semicircle is 90 The radius meets a tangent at 90 The angle between a chord and tangent creates an equal angle in the alternate segment Two tangents from one point outside the circle are equal in length A line drawn from the centre at 90 to a chord bisects the chord. Polygons The angle facts here are for REGULAR POLYGONS Polygon No of sides Sum of angles Interior angle Triangle 3 180o 60o o Quadrilateral 4 360 90o o Pentagon 5 540 108o o Hexagon 6 720 120o Exterior angle 120o 90o 72o 60o The sum of the interior angles can be found by using (n – 2) × 180. To then find each angle in a regular polygon divide what the sum of the interior angles is by the number of sides. The sum of the exterior angles in any polygon is 360o Shape Facts Triangle Types Equilateral Isosceles Right angle Scalene All sides are the same length, all angles are 60o Two sides the same length, two angles the same Triangle with a right angle in it All sides have different lengths Quadrilateral Types Square Rectangle Rhombus All sides the same length, all angles are 90o Opposite sides are the same length, all angles are 90o All sides are the same length, opposite sides are parallel, opposite angles are the same Parallelogram Opposite sides are parallel and equal in length, opposite angles are the same Trapezium One set of parallel sides Kite Two sets of equal sides, two horizontal angles are the same 3D Shapes Cube Cuboid Triangular prism Square based pyramid Tetrahedron All faces are congruent, this means the same (square) Either all faces are rectangles or 4 are rectangles and 2 are squares 2 faces are triangles and 3 are rectangles 5 vertices (corners), 4 faces are triangles and 1 is a square All faces are equilateral triangles. Dimensions 1 dimension = Length 2 dimensions = Area 3 dimensions = Volume Factorising 12a + 3 use one bracket, common factors 3x2 + 2x use one bracket, common factors 4a2b – 16b use one bracket, common factors x2 + 7x + 10 use two brackets, quadratic 9x2 – 16 use two brackets, difference of two squares 2 3x – 5x – 9 use two brackets, harder quadratics Quadratic formula x = -b + b2 – 4ac -- 2a Compound interest Use 1st year, 2nd year, 3rd year etc. Remember to add on the interest found each year to get a new total/balance for each year. The compound interest is the end total – what you started with. [or A = P(1 + R/100)n] Depreciation This is the same as compound interest but instead of adding the interest each year you subtract it. [ or A = P(1 - R/100)n] Percentage Change % change = change/original × 100 Histograms Frequency Density = frequency class width Frequency = frequency density × class width or Standard Form This is (a number between 1 and 10) × 10power The power is positive if the given number is > 1. The power is negative if the given number is > 0 but < 1. find the area of each bar