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Module 3 Super Learning Day Revision Notes November 2012

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In any exam Always Read the question at least twice Show ALL your working out Check your units eg. cm, cm² etc. Read the question again to make sure you have actually answered the question asked Check that your answer is sensible

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Revision Notes Shape and Measurement

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Interior and exterior angles of polygons In a REGULAR polygon Exterior angles add up to 360° 360 ÷ number of sides = exterior angle 180 – exterior angle = interior angle

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Regular polygons have n lines of symmetry Rotational symmetry of order n - where n is the number of sides

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Tessellations A tessellation is a tiling pattern with no gaps

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Learn names of shapes Triangles:- Equilateral, isosceles, right angled,and scalene Quadrilaterals:- Square, rectangle, parallelogram, rhombus, trapezium, kite and arrowhead.

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Perimeter The perimeter is the distance round the edge of the shape

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Area formulas Area of a Triangle = base x height ÷ 2 Area of a Rectangle = length x width Area of a parallelogram = length x height Area of a Trapezium = (a + b) ÷ 2 x height, ( where a and b are the lengths of the parallel sides) Area of a circle = πr²

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Volume Formulas Volume of cuboid = length x width x height Volume of a prism = Area of cross-section x length Volume of cylinder = area of circle x height

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Surface Area Work out the area of every face separately then add them together NOTE To find the surface area of a cylinder you need to add together the area of the 2 circles AND the rectangle that wraps round the cylinder. The length of the rectangle is equal to the circumference of the circle and the width is the height of the cylinder

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Views Plan Side elevation Front Elevation Plan view Side view Front view

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Conversions 1cm² = 100mm² 1cm = 10mm 1m³ = 1 000 000 cm³ 1m =100cm 100cm

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Congruent Means Alike in every respect

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Similar Means Same shape, Different size ( one is an enlargement of the other)

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Metric /Imperial conversions 1Kg = 2¼ lbs 1m = 1 yard (+10%) 1 litre = 1¾ pints 1 inch = 2.5 cm 1gallon = 4.5 litres 1 foot = 30 cm 1 metric tonne = 1 imperial ton 1 mile = 1.6 Km or 5miles = 8 Km

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Calculators and time Beware When using a calculator to work out questions with time make sure you enter the minutes correctly e.g. 30 minutes = 0.5 of an hour 15 minutes = 0.25 of an hour

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Density = Mass Volume e.g. g/cm³

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Speed = Distance Time e.g. Km/hour m/sec

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Geometry and Graphs

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Angle Facts An Acute angle is less than 90° A Right angle is equal to 90° An Obtuse angle is more than 90°, but less than 360° A straight angle is equal to 180° A Reflex angle is more than 180°

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Bearings Always Measure from the NORTH line Turn clockwise Use 3 figures (eg. 30° = 030°)

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Drawing Bearings To measure a bearing of B from A the North line is drawn at A. This is because the question says ‘from A’ N A B

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Angle Rules Angles in a triangle add up to 180° Angles on a straight line add up to 180° Angles in a quadrilateral add up to 360° Angles round a point add up to 360° The two base angles of an isosceles triangle are equal

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Parallel lines Look for ‘Z’ angles (Alternate angles) Look for ‘F’ angles (corresponding angles) Alternate angles are equal Corresponding angles are equal

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Transformations (ask for tracing paper to help you with these) Reflections Always reflect at right angles to the mirror line Diagonal mirror lines are sometimes called y = x or y = -x

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Rotations Always check for (or state) 1. The centre of rotation 2. The amount of turn 3. The direction (either clockwise or anti- clockwise)

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Enlargements Check the scale factor and centre of enlargement (if there is one) Draw construction lines from the centre of enlargement to help you draw the new shape Remember a scale factor of ½ will make the shape smaller

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Vector translations A vector translation slides the shape to a new position +x+x -y-y +y+y -x-x xyxy The bottom number y moves the shape up (or down if it is negative) The top number x moves the shape right (or left if it is negative)

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Loci A Locus (more than one are called Loci) is simply:- A path that shows all the points which fit a given rule There are only 4 to remember

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Locus 1 The locus of points which are A FIXED DISTANCE from a GIVEN POINT Is simply a CIRCLE

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Locus 2 The locus of points which are A FIXED DISTANCE from a GIVEN LINE This locus is an oval shape It has straight sides and ends which are perfect semicircles

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Locus 3 The locus of points which are A EQUIDISTANT from TWO GIVEN LINES This is the Angle Bisector (use compasses!)

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Locus 4 The locus of points which are A EQUIDISTANT from TWO GIVEN POINTS B A This locus is the perpendicular bisector of the line AB

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Pythagoras

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The square on the hypotenuse is equal to the sum of the squares on the other two sides Hypotenuse a b h² = a² + b² Square Add Square root Square Subtract Square root To find the hypotenuse To find a short side Remember

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0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 x and y Coordinates A graph has 4 different regions Always plot the x value first followed by the y value x goes across y goes up/down ‘in the house and up the stairs’ (4,2) (-4,3) (-6,-2) (7,-5) x positive y positive x negative y positive x negative y negative x positive y negative

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Midpoint of a line Midpoint is just the middle of the line! To find it just add the x coordinates together and divide by 2 Then add the y coordinates together and divide by 2 You have just found the midpoint If A is (2,1) and B is (6,3) Then the x coordinate of the mid point is (2 + 6) ÷ 2 = 4 And the y coordinate is (1 + 3) ÷ 2 = 2 So the mid point is (4,2) For example

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Straight Line Graphs 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 x = -5 y = 2 x= a is a vertical line through ‘a’ on the x axis y = b is a horizontal line through ‘b’ on the y axis Don’t forget that the y axis is also the line x = 0 and the x axis is also the line y = 0 The diagonal line y = x goes up from left to right and the line y = -x goes down from left to right y= x y= -x

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Straight Line Graph:– y = mx + c In the equation y = mx + c The m stands for the gradient and the c is where the line crosses the y axis

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Using y = mx + c to draw a line 1.Get the equation in the form y = mx + c 2.Identify ‘m’ and ‘c’ carefully (eg. In the equation y = 3x + 2, m is 3 and c is 2) 3.Put a dot on the y axis at the value of c 4.Then go along one unit and up or down by the value of m and make another dot 5.Repeat the last step 6.Join the three dots with a straight line

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Finding the equation of a straight line 1.Use the formula y = mx + c 2.Find the point where the graph crosses the y axis. This is the value of c 3.Find the gradient by finding how far up the graph goes for each unit across. This is the value of m. 4.Now just put these two values into the equation

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Quadratic Graphs 1.Fill in the table of values 2.Carefully plot the points 3.The points should form a smooth curve. If they don’t they are wrong! 4.Join the points with a smooth curve 5.The graph should be ‘u’ shaped

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Simultaneous equations with graphs 1.Do a table of values for each graph 2.Draw the two graphs 3.Find the x and y values where they cross 4.This is the solution to the equations

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Numbers and Algebra

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Special Number Sequences Even numbers 2, 4, 6, 8, 10,…… Odd numbers 1, 3, 5, 7, 9, 11,….. Square numbers 1, 4, 9, 16, 25,…. Cube numbers 1, 8, 27, 64, 125,…. Powers of 2 2, 4, 8, 32, 64,….. Triangle numbers 1, 3, 6, 10, 15, 21,…..

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Number Patterns and Sequences There are five different types of number sequences 1.ADD or SUBTRACT the SAME NUMBER e.g. 2 5 8 11 14 …. 30 24 18 12…. +3 +3 +3 +3 -6 -6 -6 The RULE ‘add 3 to the previous term’ ‘Subtract 6 from the previous term’ 2. ADD or SUBTRACT a CHANGING NUMBER e.g. 8 11 15 20…….. 53 43 34 26…… +3 +4 +5 -10 -9 -8 The RULE ‘ Add 1 extra each time to the previous term’. ‘Subtract 1 extra each time from the previous term’

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3.MULTIPLY by the SAME NUMBER EACH TIME e.g. 5 10 20 40…… x2 x2 x2 The RULE ‘Multiply the previous term by 2’ 4.DIVIDE by the SAME NUMBER EACH TIME e.g. 400 200 100 50…… ÷2 ÷2 ÷2 The RULE ‘Divide the previous term by 2’ 5.ADD THE PREVIOUS TWO TERMS e.g. 1 1 2 3 5 8 + + + + The RULE ‘Add the previous two terms’

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Finding The nth Term To find the nth term you can use the formula dn + (a – d) Where ‘d’ is the difference between the terms And ‘a’ is the first number in the sequence e.g. 3 7 11 15 19… ‘d’ is 4 ( because you add 4 to get the next term ) and ‘a’ is 3 ( that is the first number ) This means that (a – d) is (3 – 4) = -1 So the nth term, dn + (a – d) is 4n - 1

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Algebra Terms A term is a collection of numbers, letters and brackets, all multiplied /divided together Terms are separated by + and – signs Terms always have a + or a – sign attached to the front of them E.g. 4xy + 5x² - 2y + 6y² + 4 Invisible + sign xy term x² term y term y² term number term

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Simplifying (Collecting Like Terms) EXAMPLE Simplify 2x – 4 + 5x + 6 x terms number terms 7x +2 So = 7x + 2 1.Put bubbles round each term, making sure that each bubble has a + or – sign. 2.Then move the bubbles so that LIKE TERMS are together 3.Collect the like terms using the number line to help you 2x – 4 + 5x + 6+2x +5x -4 + 6 = 2x – 4 + 5x + 6

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Multiplying out Brackets The thing outside the bracket multiplies each separate term INSIDE the bracket When letters are multiplied together they are just written next to each other e.g. pq Remember R x R = R² Remember, a minus outside the brackets reverses all the signs when you multiply

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Expanding Double Brackets Remember to multiply everything in the second bracket by each term in the first bracket ( 2p – 4 ) ( 3p + 1 ) = (2p x 3p) + (2p x 1) + (-4 x 3p ) + ( -4 x 1) = 6p² + 2p - 12p -4 = 6p² -10p -4

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Squared Brackets Example (3p + 5)² Write this out as two brackets (3p + 5)(3p + 5) (3p + 5)(3p + 5) = 9p² +15p +15p +25 = 9p² +30p +25 The usual wrong answer is 9p² +25 !!!

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Factorising This is the exact opposite of multiplying out brackets Take out the biggest NUMBER that goes into all terms For each letter in turn take out the highest power that will go into EVERY term Open the brackets and fill with all the bits needed to reproduce each term E.g. +20 x²y³z -35x³yz² 5x²y (3x² + 4y²z - 7xz² ) Biggest number that Highest powers z wasn’t in all terms so that will divide into of x and y that will it can’t come out as 15, 20, and 35 go into all three terms a common factor 15x 4 y

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Writing Formulas These questions ask you to write an equation The only things you will have to do are:- Example 1:- to find y, you multiply x by 3 and then subtract 4 Start with x 3x 3x – 4 Times it by 3 Subtract 4 Example 2:- to find y, square x, divide it by 3 and then subtract 7 Start with x x² x² x² - 7 3 3 square it divide by 3 subtract 7 Multiply x Divide x Square x (x²) Add or subtract a number So y = 3x - 4 So y = x² - 7 3

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Formulas from words This time you change a sentence into a formula Example:- Froggatt’s deep-fry CHOCCO- BURGERS (chocolate covered beef burgers) cost 58 pence each. Write a formula for the total cost, T, of buying n CHOCCO-BURGERS at 58p each. In words the formula is Total cost = Number of CHOCCO-BURGERS x 58p Putting letters in, it becomes:- T = n x 58 It would be better to write this as T = 58n

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BODMAS B Brackets O Other (like squaring ) D Divide M Multiply A Add S Subtract Remember this is the order for doing the sums.

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Substitution To substitute numbers into an expression or a formula all you need to do is replace the letters with their values and work out either the solution of the equation or the value of the expression (don’t forget to use BODMAS)

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Examples of substitution Example of substituting in a formula If P = 3Q + 7 what is P when Q is 8? P = 3Q + 7 P = 3 x 8 + 7 P = 24 + 7 P = 33 Example of substituting in an expression If a = 3, b = 5and c = 7 what is the value of a² - b + 2c a² - b + 2c (3)² - 5 +2( 7 ) 9 - 5 + 14 18

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Solving Equations To solve equations just follow these 3 rules 1.Always do the same thing to both sides of the equation 2.To get rid of something just do the opposite. The opposite of + is – and the opposite of – is + The opposite of x is ÷ and the opposite of ÷ is x 3.Keep going until you have a letter on its own

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Examples of Solving Equations 1 Solve 5x = 15 5x means 5 x x so do the opposite and divide both sides by 5 x = 3 Solve p/3 = 2 p/3 means p ÷ 3 so do the opposite and multiply both sides by 3 p = 6 Solve 4y – 3 = 17 The opposite of –3 is +3 so add 3 to each side 4y = 20 The opposite of x 4 is ÷ 4 so divide both sides by 4 y = 5

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Examples of Solving Equations 2 Solve 2( x + 3 ) = 11 The opposite of x 2 is ÷ 2 so divide both sides by 2 x + 3 = 5.5 The opposite of + 3 is –3 so subtract 3 from both sides x = 2.5 Solve 3x + 5 = 5x + 1 There are x’s on both sides so subtract 3x from both sides 5 = 2x + 1 The opposite of +1 is –1 so subtract 1 from each side 4 = 2x The opposite of x 2 is ÷ 2 so divide both sides by 2 2 = x (or x= 2)

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REARRANGING FORMULAE You do exactly the same for this as for solving equations. When you are asked to make a letter the subject of the formula then that letter needs to be by itself on one side of the equation

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REARRANGING FORMULAE Example: Rearrange the formula 2(b – 3) = a to make b the subject of the formula We want to get rid of the times 2 outside the bracket and the opposite of times 2 is divide by 2 So b – 3 = a 2 The opposite of –3 is +3 so b = a + 3 2

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Inequalities There are 4 inequalities and you need to learn the symbols. > means ‘greater than’ < means ‘less than’ ≥ means ‘greater than or equal to’ ≤ means less than or equal to’

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Inequalities There are 3 types of inequality questions 1. Solving inequalities ( these are just like solving equations) 2. Drawing an inequality – use a number line and remember to fill in the blob if there is ≤ or ≥ 3. Writing values that satisfy an inequality. Just use common sense (draw a number line to help you if you need to)

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Trial and Improvement This is a good method for finding an approximate answer to equations that don’t have simple whole number answers Method 1.Substitute two initial values into the equation that give one answer that is too small and one answer that is too large. 2.Choose your next value in between the previous two, and put it into the equation. 3.After only 3 or 4 steps you should have 2 numbers which are to the right degree of accuracy but differ by the last digit. 4.Now take the exact middle value to decide which is the answer you want.

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Trial and Improvement x²x² x³+3x Comment 327 + 9 = 36Too small 464 +12 = 76Too large 3.542.875 +10.5 = 53.375Too small 3.646.656 +10.8 = 57.456Too large 3.5544.738875 + 10.65 = 55.388875Too small Example A solution to the equation x³ + 3x = 56 lies between 3 and 4. The solution lies between 3.5 and 3.6 but is nearer to 3.6

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