1. Super Learning Day Revision Notes November 2012
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. - where n is the number of sides
Regular polygons have
n lines of symmetry
Rotational symmetry of order n
- where n is the number of sides

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

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

8. 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. Work out the area of every face separately then add them together
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. Conversions Think 1cm² = 100mm² 1m³ = 1 000 000 cm³ 1cm = 10mm

14. Means Alike in every respect
Congruent
Means
Alike in every respect

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

16. 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
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. A vector translation slides the shape to a new position+y
The top number x moves the shape right (or left if it is negative) x y -x +x
The bottom number y moves 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 a CIRCLE •

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
Add
Square root
Square
Subtract
Square root
To find the hypotenuse
To find a short side

37. x and y Coordinates A graph has 4 different regions1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-10 x y
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 For example
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)

39. Straight Line Graphs 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-10 x y
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
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, m is 3 and c 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 by finding 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 , 4, 6, 8, 10,……
Odd numbers , 3, 5, 7, 9, 11,…..
Square numbers 1, 4, 9, 16, 25,….
Cube numbers , 8, 27, 64, 125,….
Powers of , 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 … ….
The RULE
‘add 3 to the previous term’ ‘Subtract 6 from the previous term’
2. ADD or SUBTRACT a CHANGING NUMBER
e.g …… ……
‘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 ……
÷ ÷ ÷2
‘Divide the previous term by 2’
ADD THE PREVIOUS TWO TERMS
e.g
‘Add the previous two terms’

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 …
‘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 x² y y²
Invisible + sign xy term x² term y term y² term number term

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

52. 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

53. 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)
= p² p p
= 6p² -10p -4

54. 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 !!!

55. 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 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 go into all three terms a common factor
15x4y

56. 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 x x – 4
Times it by Subtract 4
Example 2:- to find y, square x, divide it by 3 and then subtract 7
Start with x x² x² x² - 7
square it divide by subtract 7
Multiply x Divide x Square x (x²) Add or subtract a number
So y = 3x - 4
So y = x² - 7 3

57. 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

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

59. 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)

60. 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 =
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 ( 7 ) 18

61. Solving Equations To solve equations just follow these 3 rules
Always do the same thing to both sides of the equation
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
Keep going until you have a letter on its own

62. Examples of Solving Equations 1
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
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

63. 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)

64. 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

65. 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

66. 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’

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

68. Trial and Improvement
This is a good method for finding an approximate answer to equations that don’t have simple whole number answers
Method
Substitute two initial values into the equation that give one answer that is too small and one answer that is too large.
Choose your next value in between the previous two, and put it into the equation.
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.
Now take the exact middle value to decide which is the answer you want.

69. The solution lies between 3.5 and 3.6 but is nearer to 3.6
Trial and Improvement
Example
A solution to the equation x³ + 3x = 56 lies between 3 and 4. x²
x³+3x
Comment 3
= 36
Too small 4
= 76
Too large
3.5 =
3.6 =
3.55 =
The solution lies between 3.5 and 3.6 but is nearer to 3.6