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AQA GCSE - Maths > Key methods and terminology > Flashcards

Key methods and terminology Flashcards

1
Q

What are the factors of a number?

A

numbers that divide into it

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2
Q

What is the definition of BIDMAS?

A

B - brackets
I - indices
D - division
M - multiplication
A - addition
S - subtrwction

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3
Q

What does LCM stand for and what is included in this?

A
  • lowest common multiple
  • the lowest number that can be divisible by both numbers (to give an integer result)
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4
Q
  • What does HCF stand for?
  • what is included in this?
A
  • highest common factor
  • the biggest number that can divide into the numbers within the question
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5
Q

what are the first 10 prime numbers?

A

2,3,5,7,11,13,17,19,23,29

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6
Q

what are the first 10 square numbers?

A

1,4,9,16,25,36,49,64,81,100

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7
Q

What is the area of a triangle?

A

Area of triangle = 1/2 X base X height

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8
Q

what is the area of a square?

A

area of square = Base X height

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9
Q

what is the area of a circle?

A

πr2

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10
Q

what is the area of a trapezium?

A

area of trapezium = (a + b) X h / 2

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11
Q

what is the formula for compound interest?

A

A = P(1 + r/n)^nt

where:
A = final amount
P = initial principle balance
r = interest rate
n = number of times interest applied per time period
t = number of time periods

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12
Q

what are rational numbers compared to irrational numbers?

A

rational numbers can be written as: integers, fractions or terminating/recurring decimals
irrational numbers: they cannot be written as fractions; they are never ending, non repeating decimals.

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13
Q

how to multiply or divide fractions?

A

multiply: multiply the top and bottom separately to create a result fraction
divide: flip the second fraction and multiply with the first (Keep Flip Change)

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14
Q

How to add or subtract fractions?

A

find a common denominator and then subtract or add the numerators. The denominator (when they are equal) is not effected

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15
Q

How to write reoccurring decimals as fractions?

A

1) name the decimal a letter (for this example ‘x’)
2) multiply r by 10 to move the decimal place once
3) repeat this until you have the repetitive part one the left of the decimal place
4) subtract the equal decimals (and versions of r - like 100r - 10r)
5) divide to leave r, and cancel if possible

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16
Q

When estimating, what should you round the numbers to?

A

1 or 2 significant figures

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17
Q

What are bounds?

A

When a measurement is rounded to a given unit, the actual measurement can be anything up to half a unit bigger or smaller. These are the upper and lower bounds

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18
Q

What happens when a value is truncated?

A

When a measurement is truncated to a given unit, the actual measurement can be up to a whole unit bigger but no smaller.

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19
Q

What are the 3 rules of standard form?

A

1- the front number (before the decimal point) must always be between 1 and 10 (cannot be 10)
2 - the power of 10, n, is how far the decimal place moves
3 - n is positive for big numbers, n is negative for small numbers (it depends on which way the decimal place moves)

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20
Q

How to multiply and divide with standard form?

A

1) rearrange to put the front numbers and powers of 10 together in the equation
2) multiply or divide the front numbers and use the power rules to multiply or divide the powers of 10
3) make sure the answer is in standard form

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21
Q

how to add and subtract with standard form?

A

1) make sure the powers of 10 are the same - rewrite if not
2) add or subtract the front numbers
3) convert the answer to standard form if necessary

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22
Q

what are the seven simple rules of powers?

A
  1. when multiplying, add the powers
  2. when dividing, subtract the powers
  3. when raising one power to another, multiply them
  4. anything to the power of 1 will be itself
  5. anything to the power of 0 will be 1
  6. 1 to any power is 1
  7. with fractions, apply the power to both the top and bottom

with rules 1 and 2, they only work for powers of the same number

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23
Q

what happens with a negative power?

A

Negative powers - turn it upside down. This will make the power positive, and so you can then have the flipped numbers to the power of the (new) positive number

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24
Q

what happens with fractional powers?

A

A fractional power (a/b) is the ‘b’ root of the number provided

for example:
the power 1/2 means square root
the power 1/4 means fourth root

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25
Q

What are two-stage fractional powers?

A

With fractional powers where the numerator is not 1, split the fraction into a root and a power and do them in that order: root first, then power
for example:
64^(5/6) = sixth root of 64 to the power of 5 = 2^5 = 32

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26
Q

what is D.O.T.S?

A
  • Difference Of Two Squares
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27
Q

what are the 6 rules for manipulating surds?

A
  1. root a X root b = root a X b
  2. root a / root b = root a / b
  3. root a + root b = DO NOTHING! IT ISN’T root a + b
  4. (a + root b)^2 = (a + root b) X (a + root b)
  5. (a + root b) (a - root b) = a^2 - b
  6. rationalise the denominator
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28
Q

how to rationalise the denominator with surds?

A

It is when you get rid of the root on the bottom of the fraction:
a / root b = a / root b X rootb / rootb
= a X rootb / b

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29
Q

what are the six steps of solving an equation?

A

1 - get rid of any fractions
2 - multiply out any brackets
3 - collect all the letter terms on one side, and the number terms on the other
4 - combine all the like terms
5 - divide both sides by the number attached to the letter (you may have to factorise to do this as you may need to divide by the brackets)
6 - if you had the letter to the power of a number, root by that number to end up with a positive/negative result

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30
Q

how to factorise a quadratic?

A
  • rearrange into the format ax^2 + bx + c = 0
  • find 2 numbers that multiply to ac and add/subtract to b
  • rewrite the equation where bx is replaced by the 2 numbers that you found in step 2 (don’t forget x)
  • place these 4 values into a grid and factorise
  • factorise to find what each bracket is
  • then solve the equation by setting each bracket equal to 0 to see the value of x
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31
Q

How to complete the square?

A
  • rearrange the quadratic into the format ax^2 + bx + c = 0
  • write out the initial bracket: (x + b/2)^2
  • multiply out the brackets and compare to the original to find out what you need to add or subtract to complete the square
  • add or subtract the adjusting number to make it match the original

if a does not equal 1, you need to take out the factor of a first, and then multiply it back in later

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32
Q

how to find the nth term of a linear sequence?

A
  • find the common difference (this tells you what to multiply n by)
  • work out what to add or subtract to get to the original sequence using n (for example: 3n = 3,6,9)
  • write as one equation
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33
Q

how to find the nth term of a quadratic sequence?

A
  • find the difference between each term
  • find the second difference between the first differences
  • divide this value by 2 to find what to give the coefficient of n^2
  • subtract the n^2 term from the sequence to find the linear sequence
  • find the rule for the nth term of the linear sequence and add this to the n^2 term
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34
Q

What is the difference between an arithmetic and geometric sequence?

A

arithmetic - where you add or subtract the same amount each time
geometric - where you multiply or divide by the same amount each time

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35
Q

What is the fibonacci sequence?

A

where you add together the two previous terms to find the next term

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36
Q

What happens when you multiply or divide by a negative number with an inequality sign?

A

YOU MUST FLIP THE INEQUALITY SIGN

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37
Q

How to draw inequalities in a number line?

A

Use an open circle for greater than or less than, and a coloured in circle for greater than/equal to and less than/equal to

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38
Q

what is the general rule for quadratic inequalities?

A

if x^2 > a^2 then x > a OR x > -a
if x^2 < a^2 then -a < x < a

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39
Q

What is the method for showing inequalities on a graph?

A
  • convert each inequality to an equation (put an equals in place of inequality sign)
  • draw the graph for each equation (if it is > or < then draw a dotted line, but if it is ≥ or ≤ draw a solid line)
  • work out which side of the line you want (substitute a point into the inequality to see if it is on the correct side of the line)
  • shade the region this gives you
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40
Q

how to solve simultaneous equations by having the same coefficient?

A
  • make the equations have one coefficient of the same letter equal (you may need to multiply or divide on equation)
  • subtract the equations (if both are positive or negative) or add the equations (if one is negative and one is positive) to get one letter and coefficient on its own)
  • then solve the equation normally to find what one of the values are
  • substitute this back into one of the equations to find the other unknown value
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41
Q

How to solve simultaneous equations by substitution when it involves a quadratic?

A
  • rearrange the quadratic equation so that it tells you what one of the unknowns are: y = 3x^2 - 3
  • substitute this into the other equation to get a new equation
  • rearrange this to get a quadratic equation and solve
  • substitute the first value back into one of the equations (pick the easiest)
  • then try with the second value in the same equation
  • you should have two pairs of answers, and write these clearly, using and to link the pairs
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42
Q

what four thins can be useful when answering a proof question?

A
  • any even number can be written as 2n
  • any odd number can be written as 2n + 1
  • consecutive numbers can be written as: n, n + 1, n + 2…
  • the sum, difference and product of integers is always an integer
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43
Q

How to disprove a statement?

A

Use an example that the statement does not work for (disproof by counter example)

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44
Q

What does a function do?

A

Takes an input, processes it and outputs a value

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45
Q

How to evaluate a function?

A

Put the numbers into the function and find the result

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46
Q

What should you do if you have to combine functions?
fg(x)

A
  • it is a composite function
  • composite functions are written as fg(x) which means “do g first, then do f using the result of g”
  • to find a composite function, rewritten fg(x) as f(g(x) then replace g(x) with the expression it represents:
    Given f(x) = 3 x + 2 and g(x) = x + 5, find fg(x)
    = f(g(x)) = f(x+5)
    = 3(x+5) + 2
    = 3x + 15 + 2
    = 3x + 17
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47
Q

How to solve an inverse function?

A
  • an inverse function reverse f(x)
  • write out the equation x =f(y)
  • rearrange to make y the subject
  • finally, replace y with f^-1(x)
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48
Q

What is the formula for a straight line equation and what do the letters represent?

A

y = mx + c
m = gradient
c = y-intercept

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49
Q

What is the formula for gradient of a line?

A

change in y / change in x

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50
Q

How to find the midpoint of a straight line?

A

Add the x-coordinates and divide by 2
Add the y-coordinates and divide by 2

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51
Q

How to use ratios to find coordinates?

A
  • find the difference between the coordinates of two points (A and B)
  • Now look at the ratio
  • the ratio tells you the third point (C) is a fraction of the way from A to B
  • find the fraction of each distance (difference between points)
  • now add these to the coordinates of the first point (A) to find C
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52
Q

what is the same about parallel lines?

A

Their gradients - in the equations y=mx+c, the m will be the same

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53
Q

What is the key fact about perpendicular lines and their gradients?

A

Their gradients are the negative reciprocal of each other

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54
Q

What is the equation for a circle with the centre (0,0)?

A

The equation for a circle with centre (0,0) and radius (r) is:
x^2 + y^2 = r^2

If you know the radius, substitute it in for r

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55
Q

What do reciprocal graphs look like?

A

= 1/x graphs
y = A/x or a y = A
These are all the same basic shape where the two halves of the graphs do not touch and do. it exist for x = 0. They are all symmetrical about the lines y = x and y = -x

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56
Q

what do k^x graphs look like?

A

y = k^x or y = k^-x
- They are exponential graphs, they are always about the x axis and go through the point (0,1)
- If k > 1, and the power is positive, the graph curves upwards
- if k is between 0 and 1, or the power is negative, the graph is flipped horizontally
- the graphs will start to increase very little and then increase rapidly

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57
Q

what do x^3 graphs look like?

A

y = ax^3 + bx^2 + cx + d
(b,c,d can be zero)
- cubic graphs have a wiggle in the middle (this can be flat or more pronounced)
- in x^3 it goes up from the bottom left
- in -x^3 it goes down from the top left

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58
Q

What are sin and cos graphs?

A

y = SIN x
y = COS x

sin graphs - waves (bounce between +1 and -1)
cos graphs - buckets (between +1 and -1)
sin graphs have a peak and a trough
cos graphs start at the top, dip and come back up again

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59
Q

How to solve equations using graphs?

A

Plot both graphs and see where the graphs cross each other

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60
Q

What is the equation for a translation of a graph on the y-axis?

A

y = f(x) + a

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61
Q

What is the equation for a translation of a graph on the x-axis?

A

y = f(x - a)

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62
Q

What is the equation for reflections of a graph?

A

y = -f(x)
y = f(-x)

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63
Q

Describe the points of a distance-time graph?

A
  • at any point, gradient = speed
  • the steeper the graph, the faster it is going
  • flat sections are where it has stopped
  • if the gradient is negative, it is coming back
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64
Q

Describe the points of a velocity-time graph?

A
  • at any point, gradient = acceleration
  • negative slope is deceleration (slowing down)
  • flat sections are steady velocity
  • area under graph = distance travelled
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65
Q

How to find the average gradient?

A
  • Draw a straight line between the two points (these are given in the question)
  • find the gradient of the straight line
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66
Q

How to estimate rate at a given point?

A
  • Draw a tangent that touches the curve at the given point
  • the gradient of the tangent is the same as the rate at the chosen point
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67
Q

How to deal with changing ratios?

A

1) write the ratios as equations
2) turn the ratios into fractions
3) solve the equations simultaneously
example:
The ratio of male to female pupils on a trip is 5:3, four male teachers and nine female tea chefs are going. the ratio of males to females is 4:3. How many female pupils are going?
let m be the number of male pupils and f be the number of female pupils
m:f = 5:3
(m+4);(f+9) = 4:3

m/f = 5/3 and m+4/f+9 = 4/3
3 m = 5f and 3m + 12 = 4f + 36
f = 24

24 female pupils are going on the trip

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68
Q

What is the equation for direct proportion using y and x?

A

y is directly proportional to x
y = kx

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69
Q

What is the equation for inverse proportion using y and x?

A

y is inversely proportional to x
y = k/x

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70
Q

What are the golden rules for direct and inverse proportion?

A

direct proportion: divide for one, then times for all
inverse proportion: times for one, then divide for all

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71
Q

When working with percentages and it says of, what does it mean?

A

of means times (multiply)

72
Q

How to find the multiplier for an increase and decrease?

A

increase: convert the percentage to a decimal and add to 1
decrease: convert the percentage to a decimal and subtract from 1

73
Q

How to express x as a percentage of y?

A

divide x by y, and multiply 100

74
Q

How to find percentage change?

A

percentage change = (new - old) / old X 100

75
Q

How to find the original value of something?

A

1) write the amount as a percentage of the original
2) divide to find 1% of the original value
3) multiply by 100 to give the original value

76
Q

What are the common metric conversions?

A

1cm =0.394 in 10mm
1m = 100cm
1km = 1000m
1kg = 1000g
1 tonne = 1000kg
1 litre = 1000ml
1 litre = 1000cm^3
1cm^3 =1 ml

77
Q

What are the common imperial conversions?

A

1 yard = 3 feet
1 gallon = 8 pints
1 stones = 14 pounds
1 pound = 16 ounces
1 foot = 12 inches

78
Q

what are the common metric-imperial conversions?

A

1kg = 2.2 pounds
1 foot = 30cm
1 gallon = 4.5 litres
1 mile =1.6km

79
Q

what does 1 m^2 equal?

A

100cm X 100cm = 10000cm^2

80
Q

how to calculate speed?

A

speed = distance / time

81
Q

how to calculate density?

A

density = mass / volume

82
Q

how to calculate pressure?

A

pressure = force / area

83
Q

what are the 5 rules for angles?

A
  • angles in a triangle add up to 180 degrees
  • angles on a straight line add up to 180 degrees
  • angles in a quadrilateral add up to 360 degrees
  • angles round a point add up to 360 degrees
  • isosceles triangles have 2 sides the same and 2 angles the same
84
Q

what is the rule regarding vertically opposite angles?

A

vertically opposite angles are equal

85
Q

What is the rule regarding alternate angles?

A
  • it is angles found in a Z shape
  • alternate angles are the same
86
Q

what is the rule regarding co-interior (allied) angles?

A
  • they are angles within a C or U shape
  • co interior (allied) angles add up to 180 degrees
87
Q

what is the rule regarding corresponding angles?

A
  • these are angles found in an F shape
  • corresponding angles are the same
88
Q

what is the sum of exterior angles for a polygon?

A

sum of exterior angles = 360 degrees

89
Q

what is the sum of interior angles for a polygon?

A

sum of interior angles = (n - 2) X 180

90
Q

what is the interior angle for a polygon?

A

interior angle = 180 degrees - exterior angle

91
Q

how to calculate the exterior angle for regular polygons only?

A

exterior angle = 360 / n

92
Q

circle theorems: describe the rule “A tangent and a radius meet at 90 degrees”

A

A tangent is a line that just touches a single point on the circumference of a circle. A tangent always makes an angle of exactly 90 degrees with the radius it meets at this point

93
Q

circle theorems: describe the rule “A tangent and a radius meet at 90 degrees”

A

A tangent is a line that just touches a single point on the circumference of a circle. A tangent always makes an angle of exactly 90 degrees with the radius it meets at this point

94
Q

circle theorems: describe the rule “Two radii form an isosceles triangle”

A

unlike other isosceles triangles they don’t have the marks on the sides to say they are the same, the fact that they are both radii is enough to make it an isosceles triangle within the circle

95
Q

circle theorems: Describe the rule “The perpendicular bisector of a chord passes through the centre of the circle”

A

A chord is any line drawn across a circle. No matter where you draw a chord, the line that cuts it exactly in half (at 90 degrees) will go through the centre of the circle

96
Q

circle theorems: Describe the rule “The angle at the centre of a circle is twice the angle at the circumference”

A

The angle subtended at the centre of a circle is exactly double the angle subtended at the circumference of the circle from the same two points (two ends of the same chord)

angle subtended at is the angle made at

97
Q

circle theorems: Describe the rule “The angle in a semicircle is 90 degrees”

A

A triangle drawn from the two ends of a diameter will always make an angle of 90 degrees where it hits the circumference of the circle, no matter where it hits.

98
Q

circle theorems: Describe the rule “Angles in the same segment are equal”

A

All triangles drawn from a chord will have the same angle where they touch the circumference. Also, the two angles on opposite sides of the chord add up to 180 degrees

99
Q

circle theorems: Describe the rule “Opposite angles in a cyclic quadrilateral adds up to 180 degrees”

A

A cyclical quadrilateral is a 4 sided shape with every corner touching the circle. Both pairs of opposite angles add up to 180 degrees.

100
Q

circle theorems: Describe the rule “Tangents from the same point are the same length”

A

Two tangents drawn from an outside point are always equal in length, creating two congruent right angled triangles

101
Q

circle theorems: Describe the rule “The alternate segment theorem”

A

The angle between a tangent and a chord is always equal to the angle in the opposite segment (the angle made at the circumference by two lines drawn from the ends of the chord)

102
Q

what are the four conditions to prove two triangles are congruent?

A

1) SSS - three sides are the same
2) AAS - two angles and a corresponding side match up
3) SAS - two sides and the angle between them match up
4) RHS - a right angle, the hypotenuse and one other side all match up

103
Q

What are the three conditions to show that two triangles are similar?

A

1) all the angles match up
2) all three sides are proportional
3) any two sides are proportional and the angle between them is the same

104
Q

what is the difference between similarity and congruence?

A

congruence - same shape, same size and same angles
similarity - same shape, different size

105
Q

how to describe a translation?

A
  • specify it is a translation
  • give the vector that it moves by
106
Q

how to describe a rotation?

A
  • specify it is a rotation
  • give the angle of rotation
  • the direction of rotation
  • the centre of rotation
107
Q

how to describe a reflection?

A
  • specify it is a reflection
  • give the equation of the mirror line
108
Q

how to describe an enlargement?

A
  • specify it is an enlargement
  • give the scale factor
  • the centre of enlargement
109
Q

how to calculate scale factor for an enlargement?

A

scale factor = new length / old length

110
Q

give four key facts about scale factors:

A
  • if the scale factor is bigger than 1, the shape gets bigger
  • if the scale factor is smaller than 1, it gets smaller
  • if the scale factor is negative, the shape will be on the other side of the enlargement centre
  • the scale factor also tells you the relative distance of old points and new points from the centre of enlargement
111
Q

how to calculate the area of a sector?

A

area of sector = x /360 X area of full circle

112
Q

how to calculate the length of an arc?

A

length of arc = (x / 360) X circumference of full circle

113
Q

How to find the area of a segment?

A

1) find the area of the sector
2) find the area of the triangle, and subtract it from the sectors’ area

114
Q

How to calculate the surface area of a sphere?

A

Surface area of sphere = 4πr^2

115
Q

how to calculate the surface area of a cone?

A

surface area of cone = πrl + πr^2

where r is the radius, and l is the slanted height of the cone

116
Q

How to calculate the surface area of a cylinder?

A

area of cylinder = 2πrh + 2πr^2

where r is the radius, and h is the height of the cylinder

117
Q

How to calculate the volume of a sphere?

A

volume of sphere = 4/3 π r^3

118
Q

how to calculate the volume of a pyramid?

A

volume of pyramid = 1/3 X base area X vertical height

119
Q

How to calculate the volume of a cone?

A

volume of cone = 1/3 X πr^2 X vertical height

120
Q

How to calculate volume of frustum?

A

= volume of original cone - volume of removed cone

121
Q

What are the four types of Loci?

A
  • “a fixed distance from a given point” - this is a circle
  • “a fixed distance from a given line” - tic tac shape
  • “equidistant from two given lines”
  • “equidistant from two given points”
122
Q

How to draw bearings using key words?

A

1) FROM - put your pencil on the diagram at the point you are going from
2) NORTH LINE - at the point you are going from, draw a north line (it may already be there)
3) CLOCKWISE - draw in the angle clockwise from the nor5h line to the line joining the two points

123
Q

how to find the distance between two points?

A

Use pythagoras’ theorem:
- sketch the right angled triangle
- find the lengths of shorter sides by subtracting coordinates
- use Pythagoras’ to find the length of the hypotenuse (this will be the distance)

124
Q

What are the three trigonometry formulas?

SOH CAH TOA

A

sin = opposite / hypotenuse
cos = adjacent / hypotenuse
tan = opposite / adjacent

125
Q

what are the exact trig values for sin?

A

sin 0 = 0
sin 30 = 1/2
sin 60 = root3 / 2
sin 45 = 1 / root2
sin 90 = 1

126
Q

what are the exact trig values for cos?

A

cos 0 = 1
cos 30 = root3 / 2
cos 60 = 1 / 2
cos 45 = 1 / root2
cos 60 = 1/2
cos 90 = 0

127
Q

what are the exact trig values for tan?

A

tan 0 = 0
tan 30 = 1 / root3
tan 45 = 1
tan 60 = root3
tan 90 = undefined

128
Q

how to label the triangle for the sine and cosine rule?

A

label the angles a lowercase “A,B,C” and the opposite length a capital “a,b,c”

129
Q

what is the sine rule?

A

a / sinA = b / sinB = c / sinC

130
Q

what is the cosine rule?

A

a^2 = b^2 + c^2 - 2 bc cosA

131
Q

What is the area of a triangle using sin?

A

Area of triangle: 1/2 ab sinC
- you can use this when you know two sides and the angle between them

132
Q

what are the vector notations?

A

1)you can have column vectors: (2)
(5)
- column vectors are how far something moves on the x above how far on the y axis. It is not written with a line between, and the one thing is written in one set of big brackets
2) it can be represented by a letter, if handwritten it is underlined “a”, or if typed, it is in bold “a”
3) it can also be written in as the two points and an arrow to show the direction the vector is moving in

133
Q

What do vectors have?

A
  • vectors have direction (the way it moves) and magnitude (the size)
134
Q

how to multiply vectors by a scalar?

A

multiplying a vector by a positive number changes the vectors’ size but not its direction. It scales the vector. If the number is negative, then the direction gets switched.

135
Q

How to add or subtract vectors?

A
  • You can describe movements between points by adding and subtracting known vectors
  • “a + b” means “go along a then b”
  • to add or subtract column vectors, add/subtract the top to the top and the bottom to the bottom.
136
Q

how are vectors used along a straight line?

A
  • you can use vectors to show that points lie on a straight line
  • you need to show that the vectors along es h part of the line point in the same direction - they are scalar multiples of each other
  • this means you can write them as fractions of each other
137
Q

how can vectors involve ratios?

A

Ratios are used in vectors to tell you the lengths of different sections of a straight line. If you know the vector along part of that line, you can use this to find other vectors along the line.

138
Q

what are all probabilities between?

A
  • they are always between 0 and 1
  • a probability of zero means it will never happen, and one will mean it definitely will happen
139
Q

what is the formula to calculate the probability of something?

A

probability = number of ways for something to happen / total number of possible outcomes

140
Q

what will a probability always add to?

A
  • It will always add to one
  • P(event happens) + P(event doesn’t happen) = 1
141
Q

what is the probability rule?

A

The number of ways to carry out a combination of activities equals the number of ways to carry out activity multiplied together.

142
Q

how to calculate relative frequency?

A

relative frequency = frequency / number of times you tried the experiment

143
Q

how to find the expected/theoretical frequency?

A

expected frequency of a result = probability X number of trials

144
Q

what is the difference between independent and dependent events?

A
  • two events are independent if one event happening does not affect the probability of the other happening
  • if one affect does affect the probability of the other, the events are dependent
145
Q

what is the and rule?

A

If two events (A and B) are independent, then:
P(A and B) = P(A) X P(B)

It is the probability of both events happening

146
Q

What is the OR rule?

A

for two events (A and B):
P(A or B) = P(A) + P(B) - P(A and B)

it is the probability of at least one of the events happening

147
Q

What are mutually exclusive events?

A

It is two events that cannot happen at the same time
P(A or B) = P(A) + P(B)

148
Q

what are the four tree diagram facts?

A
  • on any set of branches which meet at a point, the probabilities add up to 1
  • multiply along the branches to get the end probabilities
  • the end probabilities should add to 1
  • to answer any question, add up the relevant end probabilities
149
Q

what are conditional probabilities?

A
  • the conditional probability of A given B is the probability of event A happening given that event B happens
  • a lot of questions that say “without replacement” is most likely to be a conditional probability question
  • if events A and B are independent then P(A given B) = P(A) and P(B given A) = P(B)
150
Q

what is the and rule for two events that are dependent?

A

P(A and B) = P(A) X P(B given A)

151
Q

Describe venen diagrams?

A
  • they use sets: these are collections of things “called elements”
  • sets can be written in different ways, but always be in a pair of curly brackets: {}
  • n(A) means the number of elements in set A
  • on a venn diagram, each set is represented by a circle containing the elements of the set or the number of elements in the set
  • this ξ means universal set
152
Q

what are the key notations for probability?

A
  • P(A U B) means everything in set A or B
  • P(A ∩ B) means everything in set A and B
  • P(A’) means everything not in set A
153
Q

What is a sample used for?

A

To find out about a population or amount

154
Q

What is a random sample?

A
  • this means every member of the population has an equal chance of being in it
155
Q

What does increasing the size of the sample increase?

A

you increase the reliability

156
Q

How do you do a random sample?

A

1) assign a number to every number of the population
2) create a list of random numbers - by using a computer, calculator or picking out of a bag
3) match the random numbers to members of the population

157
Q

How to spot bias within a sampling method?

A

consider:
- when, where, and how the sample is taken
- how many members are included

1) if certain groups are excluded, it is not random. This can lead to BIAS from things like age, gender, different interests

158
Q

What are the two types of data?

A

Quantitative data - measures quantities using numbers. For example heights of people

Qualitative data - it is descriptive, it uses words rather than numbers - for example pet names

159
Q

What are the two types of quantitative data?

A

discrete data - it is discrete if the numbers can only take exact values, like the number of customers in a shop in one day
continuous data - if the number can take any value in a range, it is continuous data. Like heights and weights are continuous measurements

160
Q

What are the definitions of mean, median, mode and range?

A

mode - most common
median - middle value (when data is ordered in size)
mean - total of items / number of items
range - difference between the lowest and highest value

161
Q

how to find averages from grouped frequency tables?

A
  • add a 3rd column and enter the mid-interval. slur for each class
  • add a 4th column to show ‘frequency X mid-interval values’ for each class
  • You need to use this to find the modal class and the class containing the median, not exact values, The range can also be estimated too (using the class boundaries)
162
Q

How to find the estimated range of grouped frequency tables?

A

Find the difference between the highest and lowest class boundsriesv

163
Q

What are the:
upper quartile
lower quartile
median
interquartile range

A

upper quartile - 75% of the way into the data
lower quartile - 25% of the way into the data
median - 50% of the way into the data
interquartile range - difference between the upper and lower quartile (and contains the middle 50% of values)

164
Q

How to draw a box plot?

A
  • draw on the lower and upper quartile values and draw the box
  • draw the median line (this should be within the box)
  • mark on the minimum and maximum points and draw on horizontal lines to connect them to the box
165
Q

What does cumulative frequency mean?

A

It just means adding it up as you go along - the total frequency so far

166
Q

How to draw a cumulative frequency graph?

A
  • add a “cumulative frequency” column to the table - and fill it with the running total of the frequency column
  • plot points on the graph using the highest value of each class and the cumulative frequency
  • join the points with a smooth. urge or straight lines

make sure the cumulative frequency goes up the y-axis, and the x-axis is the collection name (like height)

167
Q

How to find the median, upper/lower quartiles and interquartile range on a cumulative frequency graph?

A

median - go halfway up the side, across to the curve, then down and read the bottom scale
lower/upper quartiles - go 1/4 and 3/4 up the side, across to the curve, then down and read off the bottom scale
interquartile range - the distance between the lower and upper quartiles

168
Q

How to calculate frequency density?

A

frequency density = frequency / class width

169
Q

How to find the frequency from a histogram?

A

frequency = frequency density X class width
frequency = area of bar

170
Q

What are the types of correlation in a scatter graph?

A

strong correlation - when the points make a fairly straight line
weak correlation - when the points don’t line up properly, but you can still draw a line of best fit through them

positive correlation - when the points move from the bottom left upwards
negative correlation - when the points move from the top left downwards

no correlation - when the points do not have any pattern (strong or weak, positive or negative)

171
Q

How to use a line of best fit to make a prediction?

A
  • draw a line of best fit (crossing through as many points as possible), and continue until the end of the graph
  • use the extended line to make an estimate for the value you have been asked about
172
Q

How can line graphs show time series?

A
  • with time series, a basic pattern often repeats itself - seasonality
  • the time taken for the pattern to repeat itself (measured from peak to peak, or trough to trough) is called a period
  • you can also look at the overall trend - look at the peaks and troughs
173
Q

What do pie charts show?

A

-proportion
- the total of everything = 360 degrees

174
Q

what are frequency polygons are used to show?

A
  • grouped data
  • the frequency of each class is plotted against the mid-interval value and the points are joined with straight lines
175
Q

what do stem and leaf diagrams help with?

A

Spreading the data

176
Q

How to compare data sets using box plots?

A
  • You can read the median and work out the IQR and range. Remember to say what these values mean in context of the data
  • A larger spread means that the values are less consistent, and larger variation
177
Q

How to compare data sets using histograms?

A
  • analyse the shape
  • compare the centres
  • examine the spread
  • consider overlapping sreas

AQA GCSE - Maths (1 decks)

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