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Module 3

Module 3. Super Learning Day Revision Notes November 2012. In any exam Always. Read the question at least twice Show ALL your working out Check your units eg. cm, cm ² etc. Read the question again to make sure you have actually answered the question asked

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Module 3

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  1. Module 3 Super Learning Day Revision Notes November 2012

  2. 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

  3. Revision Notes Shape and Measurement

  4. 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

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

  6. Tessellations A tessellation is a tiling pattern with no gaps

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

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

  9. 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

  10. 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

  11. 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

  12. Views Plan view Plan Side view Side elevation Front Elevation Front view

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

  14. Congruent Means Alike in every respect

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

  16. 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 Metric /Imperial conversions learn

  17. 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

  18. Density = Mass Volume e.g. g/cm³

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

  20. Geometry and Graphs

  21. 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°

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

  23. 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 B A

  24. 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

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

  26. 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

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

  28. 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

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

  30. 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

  31. Locus 1 The locus of points which are A FIXED DISTANCE from a GIVEN POINT Is simply aCIRCLE •

  32. 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

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

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

  35. Pythagoras

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

  37. y 10 9 8 7 6 5 4 3 2 1 x 0 -10 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -7 -5 -1 -1 -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’ x negative y positive x positive y positive •(-4,3) •(4,2) •(-6,-2) •(7,-5) x negative y negative x positive y negative

  38. 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 For example 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)

  39. y 10 9 8 7 6 5 4 3 2 1 x 0 -10 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -7 -5 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Straight Line Graphs 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 = xgoes up from left to right and the line y = -xgoes down from left toright x = -5 y= x y = 2 y= -x

  40. 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

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

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

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

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

  45. Numbers and Algebra

  46. 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,…..

  47. Number Patterns and Sequences There are five different types of number sequences • 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’

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

  49. 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

  50. 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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