Dyffryn School Ysgol Y Dyffryn Mathematics Faculty

Transcription

1 Dyffryn School Ysgol Y Dyffryn Mathematics Faculty Formulae and Facts Booklet Higher Tier

2 Number Facts Sum This means add. Difference This means take away. Product This means multiply. Share This means divide. Prime number A number that has only two factors and can only be divided by itself and one. (1, 9, 21, 27 are not prime numbers.) Square number The answer to when a number is multiplied by itself. Cube number The answer to when number is multiplied by itself 3 times. Factor A number that can divide into another number e.g. the factors of 6 are 1, 2, 3 and 6 Common factor A number that can divide into two or more other numbers Highest Common Factor The largest common factor. Multiple A number that another number can divide into e.g. Multiples of 10 are 10, 20, 30 Common Multiple A number that two or more numbers can divide into Lowest Common Multiple 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 = 4 r 2 Circle Diameter is the distance across the centre of a circle Radius = ½ diameter Area of a circle = radius 2 Circumference of a circle = diameter For a semi-circle find the area as normal then half the answer Area of sector = Length of an arc = 360 r2 D 360 Volume Volume of any prism = cross sectional area (Face area) length Cube = (length) 3 Cuboid = length width height Cylinder = r 2 h Cone = 1 / 3 r 2 h Sphere = 4 / 3 r 3 Square based pyramid = 1 / 3 area of the base height Density Density = Mass Volume D M V Average Speed Average speed = Distance Time S D T

3 Right angled triangles Pythagoras Theorem c 2 = a 2 + b 2 or a 2 + b 2 = c 2 (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 Must have all three sides. Cosine rule to find an angle Cos A = b 2 + c 2 a 2 2bc Sine rule Must know a side and an angle as well as the side opposite the angle you are trying to find. Sine rule to find an angle Finding a side Cosine rule Cosine rule 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. a 2 = b 2 + c 2 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 180 o, measure anticlockwise and subtract from 360. Estimated Mean Estimated mean = (Total frequency x midpoint) Total frequency or total f x Total f Grouped frequency diagram Frequency polygon Bar chart with a proper scale Plot across to the midpoint and up to frequency. Join the points plotted in order with a ruler.

4 Units (You have to know these) 1cm = 10mm 100cm = 1m 1000m = 1km 1g = 1000mg 1kg = 1000g 1 litre = 1000cm 3 1 litre = 1000 millilitres 1 litre = 100 centilitres 1 mile 1.6 km 1m 3 = 1000 litres 1ml = 1cm 3 1 gallon = 4.5 litres 1 inch 2.5 cm Converting squared or cubed units cm 2 mm or 100 m 2 cm or m 3 cm or mm 2 cm or 100 cm 2 m or cm 3 m or Metric Units Weight Grams (g) Kilograms (kg) Tonnes Length Millimetres (mm) Centimetres (cm) Metres (m) Kilometres (km) 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. Volume Millilitres (ml) Centilitres (cl) Litres (l) 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.

5 Angle Facts Angles on a straight line add up to 180 o Angles round a point add up to 360 o Angles in a triangle add up to 180 o Angles in a quadrilateral add up to 360 o 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 Exterior angle Triangle o 60 o 120 o Quadrilateral o 90 o 90 o Pentagon o 108 o 72 o Hexagon o 120 o 60 o 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 360 o Shape Facts Triangle Types Equilateral Isosceles Right angle Scalene All sides are the same length, all angles are 60 o Two sides the same length, two angles the same Triangle with a right angle in it All sides have different lengths Quadrilateral Types Square All sides the same length, all angles are 90 o Rectangle Opposite sides are the same length, all angles are 90 o Rhombus 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.

6 Dimensions 1 dimension = Length 2 dimensions = Area 3 dimensions = Volume Factorising 12a + 3 3x 2 + 2x 4a 2 b 16b x 2 + 7x x x 2 5x 9 use one bracket, common factors use one bracket, common factors use one bracket, common factors use two brackets, quadratic use two brackets, difference of two squares use two brackets, harder quadratics Quadratic formula + x = -b -- b 2 4ac - 2a Compound interest Use 1 st year, 2 nd year, 3 rd 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 find the area of each bar Standard Form This is (a number between 1 and 10) 10 power The power is positive if the given number is > 1. The power is negative if the given number is > 0 but < 1.