BHASVIC MATHS DEPARTMENT

AS Mathematics

Learning Pack

Teacher: Angela (AMD)

Name: _____________________________

Block:________Tutor Group:___________

Welcome to AS Maths with Statistics Your Teacher is Angela

Contact me using email a.dufffy@bhasvic.ac.uk

Your Lessons

You will have 2 x double lessons per week on a Wednesday and a Thursday. You will have 36 lessons in which to learn the AS course then 16 lessons for revision, consolidation and exam practice after February half term. It is very important that you do not miss any of your 36 lessons. In your Wednesday lesson you will hand in your completed weekly assignment and have an assignment test. The test will consist of roughly 4 questions taken directly from the weekly assignment you’ve just handed in. You are therefore expected to get 100% on all weekly assignment tests. BRING THIS PACK TO EVERY LESSON

Work to be completed each week outside lessons

This will seem like a lot of work to you compared with your experiences from school. You will

need to be organised to meet all your deadlines.

Watch the videos for each lesson and fill in the corresponding pages of the pack, see the

planner at the front of this pack. This is usually about an hour’s work.

Complete the weekly assignment before Wednesday’s lesson. This is usually about 2 to 3

hours’ work and you will probably need help to get it 100% completed.

Expectations

Attend all lessons. If you need to miss a lesson, tell me in advance or contact me as soon

as possible to collect the missed work.

COMPLETE your weekly assignment each week by the deadline. This means there are no

questions you haven’t done and no questions with answers which do not match the correct

answers, which are given to you on the assignment. It is your responsibility to get help if

you need it.

Pass the weekly assignment test. This means getting at least 2/3 on the drill test (basic

skills test) and at least 9/10 on the randomly chosen assignment questions.

Try your best and believe in yourself.

Help and Resources

All videos and resources can be found on www.angelasalgebra.weebly.com or you can scan the QR codes in the pack. Learning Buddy in class Facebook: Join BHASVIC Maths Drop in with the doubles students (help in room 7 or the hall every day) A2 Mentor (maybe)

Core 1 GCSE Topics (in a bit more depth)

You should be already comfortable working with indices, surds, simultaneous equations and

quadratic equations, but we often find that students need a lot of practice to be confident with

these basic algebraic techniques.

You need to be able to sketch graphs of quadratics, cubics and reciprocal graphs, and sketch

transformations of any graph given to you.

Coordinate Geometry

At GCSE you are taught that y = mx + c but we don’t use that form at A Level. You will learn more

about how to construct the equation of a line and use the formulae for midpoint, gradient, distance

and triangle area to solve problems similar to those you’ve seen at GCSE.

Calculus

This is one of the few ‘new’ topics in Core 1 compared to GCSE. Calculus is the study of

instantaneous change. You will learn the basics such as gradients of curves and tangent lines

(differentiating) and un-differentiating (integrating).

Series & Sequences Notation

You might already understand the notation we use for recurrence relations as you’ve probably

seen it at GCSE. Sigma notation will be new to you and you’ll learn what 2r -1( )r=3

24

å means.

Arithmetic Series

You’ve almost certainly seen Arithmetic Series at GCSE (a sequence where you get from one

term to the next by adding on a fixed number). At AS you learn the equation for the nth term the

sum of the first n terms and solve problems involving Arithmetic Series. Usually this just turns into

an exercise in simultaneous equations.

C1 Intersection of lines and curves - Simultaneous Equations

Objective: Be able to use the substitution method to find the co-ordinates of the point(s) where two straight lines or a line and curve intersect

Write down the easier equation

Rearrange into y = or x =

Sub that the harder equation

Solve to find y (or x)

Use the easy equation to find x (or y)

Level 1 204

1032

yx

yx Little sketch of what you’re finding:

Level 2 204

1032

2

yx

yx Little sketch of what you’re finding:

Write down the easier equation

Rearrange into y = or x =

Sub that the harder equation

Solve to find y (or x)

Use the easy equation to find x (or y)

Level 3 204

1032

22

yx

yx Little sketch of what you’re finding:

C1 Graphs 1 - Sketching Graphs of Cubic Functions Objectives:

To be able to sketch graph transformations of

To be able to sketch graphs of factorised cubic functions

Graphs of Cubic Functions Activity: Sketch the following graphs:

a)

b)

Graphs of Factorised Cubic Functions Activity: Sketch the following graphs Sketching Checklist

Crosses y axis when x = 0, y =?

Crosses x axis when y = 0, roots at x = ?

Is there a repeated root?

Positive or negative ?

c)

d)

C1 Graphs 1 - Sketching Reciprocal Functions Objectives:

To be able to sketch graph transformations of

To understand the effect of an asymptote on the shape of the graph

Activity: Sketch the following graphs:

a)

b)

---------------------------------------------------------------------------------------------------------------------------------------------

Sketching Transformation of

(to be completed in class)

Sketching Checklist

Crosses y axis when x = 0, y =?

Crosses x axis when y = 0, x = ?

Identify the equations of the asymptote

Positive or negative

?

c)

d)

C1 Graph 2 - Transformations Objectives:

To be able to transform a graph using shifts, stretches and reflections

Understand the difference between an inside and outside transformation

Graph Transformations: Shifts

Activity Sketch on this graph a) f (x) + 2 b) f(x + 2)

Shifts Notation The plot (2.-3) moves to?

Which co-ordinate always changes

Outside f(x) + 2

f(x) – 2

Inside f(x+2)

f(x-2)

Graph Transformations: Stretches

Activity Sketch on this graph a) 2f(x) b) f(2x)

Stretches Notation The plot (2.-3) moves to?

Which co-ordinate always changes

Outside 2f(x)

Inside f(2x)

C1 Graph Transformations (continued) Graphs Transformations: Reflections

Activity Sketch on this graph a) - f (x) b) f(-x)

Shifts Notation The plot (2.-3) moves to?

Which co-ordinate always changes

Outside -f(x)

Inside f(-x)

Summary of Graph Transformations

Type Outside Description Inside Description

Shifts f(x) + a

f(x+a)

f(x) – a

f(x-a)

Stretches af(x)

f(ax)

Reflections -f(x)

f(-x)

C1 Coordinate Geometry 1 - Equation of a Line

https://www.youtube.com/watch?v=hij03saNV2U An equation of a line with a point (a, b) and a gradient m can be written as: Example: Find the equation of the straight line which passes through the points A( 4,5) and B (2,3) Activity: Pause the video. Complete the questions. Then play again to check. Find the equation of the line that passes through each of these pairs of points.

1. A(-2, 0) and B(1, 6)

2. R(-3, -5) and S(2, 5)

3. U(4, 2) and V(-2, 4)

4. D(-1, -3) and E(3, -1)

C1 Coordinate Geometry 2 – Gradients, , Distance, Triangle Area, Midpoint

https://www.youtube.com/watch?v=sdjosvOBQek

Gradients Gradients of parallel lines are the

Gradients of perpendicular lines. Example: Line 1 and Line 2 are perpendicular to each other. If line 1 has a gradient of 2, then line 2 has a gradient of (TIP! FLIP IT AND CHANGE THE SIGN!)

Length of a line

The distance between two points on a line can be calculated by using Pythagoras. Example: Find the distance between A(0, 4) and B( 2,0) Triangle Area

Area of a triangle is ½ bh

Always use the perpendicular height

Draw a sketch to help visualise it. Example: Find the area of the triangle ABC where A(0,4), B(2,0) and C (0,3) Midpoint

To find the midpoint of two points on a line add the x co-ordinates together and divide by two. Do the same with the y co-ordinates.

Example: Find the co-ordinates of the midpoints of the line AB, where A(0, 4) and B( 2,0)

GCSE Revision:Completing The Square solving Quadratic Functions Objective:

Be able to solve a quadratic function by completing the square Be able to complete the square when the co-efficient of x² is not 1.

Example 1:

Complete the square in order to solve

Example 2 :

Complete the square in order to solve

C1 Quadratics 1 - The Discriminant Objectives:

Be able to use the discriminant to determine the type of roots of a quadratic function

Be able to solve discriminant problems

Quadratic Function Value of the Discriminant Corresponding graph of Quadratic function

Number and Type of Roots

C1 Quadratics 2 - Sketching Quadratics Objectives:

Be able to sketch the graphs of quadratic functions.

Be able to identify the co-ordinates of turning points and where the graph crosses both the y and x axes.

Sketching Checklist:

Crosses y axis, when x = 0

Crosses x axis, when y = 0

Turning point? Use completed square form to find the co-ordinates

C1 Quadratics 3 - Quadratic Inequalities Objective: Be able to solve quadratic inequalities

Checklist:

When y = 0, roots? x = , x =

When x = 0 y =

Sketch the quadratic graph

Identify and shade the region required

<0 or >0

Final solution using correct inequality notation

Checklist:

When y = 0, roots? x = , x =

When x = 0 y =

Sketch the quadratic graph

Identify and shade the region required

<0 or >0

Final solution using correct inequality notation

Class Activity: Describe the outlined region with an inequality, where x represents the range of numbers.

C1 Introduction to Differentiation Objectives:

To understand how to differentiate a function

To understand that the derivative is the gradient function of the curve and gives the gradient of the tangent to the curve at a particular point

To be able to use the gradient function to solve problems involving tangents and normals.

The gradient function gives the

gradient of a tangent to the curve

at a specific point on the curve

When the gradient function at a

specific point is ZERO. This means

that the tangent is a horizontal line

and at that point the curve has a

𝑑𝑦

𝑑𝑥

y =

C1 Differentiation 2 – Tangents/Normals Objectives:

To be able to calculate the gradient and equation of a tangent to a curve at a particular point and the gradient and equation of the associated normal

A NORMAL is _____________________ to its tangent at a particular point on a curve. On the graph draw the tangent to the curve at the point (x1, y1). Now draw on the normal to the curve at the point (x1,y1)

If the gradient function for this curve, gave the gradient of the tangent as 3. What would the gradient of the normal be?

What is the difference between the gradient on a straight line and a gradient measured on a curve?

C1 Arithmetic Progressions Objectives:

To be able to understand the notation of arithmetic progressions

To be able to solve problems involving n’th term

To be able to solve problems using the Sum to n terms formula rog

Finding the n’th term of an Arithmetic Progression In general : a + (a + d) + (a + 2d) + ........... a + ( n-1)d

Example: 2 + 5 + 8 + 11 + .......n’th term

a = , d =

C1 Recurrence Relations Objectives:

To understand the notation of recurrence relations

To be able to solve problems using recurrence relations

C1 Proof of Sum to n terms formula To be able to recall the proof of the sum to n terms ormula

Watch the video and write down the proof here In general :

a + (a + d) + ... + a + (n-2)d + a + ( n-1)d

C1 Sigma Notation Objectives:

To understand sigma notation

To be able to solve problems using sigma notation

Watch the video clip and complete the following examples:

6

3

12k

k

142

10

27k

r

10

4

3k

ku

4,5 11 uuu nn

4

2

2r

r

C1 Integration (Indefinate) Objectives:

To be able to differentiate a function

To be able to use the correct notation for integration

Integration is the reverse of differentiation. If we know the gradient function we can get back to the equation of the original curve by undoing the differentiation

If 14 xdx

dy then the original curve is

Example 1: To find the particular solution you need a point on the curve in order to solve for the constant term, c

So if 14 xdx

dy, find the equation of the original curve using the point (-1, 2)

General Formula Examples:

y = Don’t you forget

about me!

Don’t you forget

Don’t you forget

about me!

Don’t you forget

Core 2 Circles You need to know the equation for a circle and the fact that tangents to circles are perpendicular to the radius. Then you can combine this with your C1 coordinate geometry knowledge to solve problems involving circles and lines. Really fun. Calculus More differentiation, this time about finding maximum and minimum points on a graph. Also, a nice application of integration – finding areas under curves. You’ll learn the trapezium rule which is a formula for finding the approximate area under a curve and isn’t as much fun as integrating. TOOLS First you’ll learn how to work in radians. They’re much better than degrees (some people think centimetres are better than inches but not everyone. EVERYONE thinks radians are better than degrees). We made up the word TOOLS, but it’s a handy way to remember the 5 equations you need to solve arc of circle/triangle problems. Always check your calculator – degrees or radians?! Trigonometry You already know how to solve equations like sinq = 0.5 by pressing shift sin on your calculator, but at AS you learn how to find not just one but many answers, and also how to solve more complicated equations like sin2q +2cosq =1 Logarithms You know how to solve x2 = 3but do you know how to solve 2x = 3? No, you don’t. That’s why you need to learn about logarithms. There are four equations to learn which you’ll then need to practice using to solve more complicated equations. Factor & Remainder Theorem You know that 3 is a factor of 12, and that 7 divided by 3 has remainder 1. In AS you learn how to understand the same statements but for algebraic expressions instead of numbers. Binomial Expansion

Can you expand 3+2x( )2? What about 3+2x( )

6? The Binomial Expansion is a

formula which allows you to expand out multiple brackets really easily. Geometric Series Like Arithmetic Series but instead of adding on a fixed number to get from one term to the next, you multiply by a fixed number. You’ll learn the formulae for nth term and sum of n terms and also, if the fixed number is between -1 and 1, how to add up infinity many terms of the sequence.

C2 Differentiation: Stationary (Turning) Points and the Second Derivative ( 6 minutes) Objectives: Be able to calculate the first derivative in order to find the stationary points for a given function Be able to calculate the second derivative in order to test if the stationary points are of a maximum or minimum point on the curve.Understand that stationary and turning points are synonymous

You can check the gradient of the

tangent using only 𝑑𝑦

𝑑𝑥 before and after

the turning point to see how the

gradient is changing in order to

determine if a given point is a minimum

or a maximum.

Alternatively the second derivative

evaluates for a particular point if the

rate of change of the gradient is

positive or negative hence determining

the nature of the stationary point.

The second derivative:

The tester function. 𝑑2𝑦

𝑑𝑥2

This is found by differentiating

the first derivative 𝑑𝑦

𝑑𝑥

If 𝑑2𝑦

𝑑𝑥2 < 0 (NEGATIVE), the turning

point is a _________________

If 𝑑2𝑦

𝑑𝑥2 > 0 ( POSITIVE), the turning point

is a _____________________

If 𝑑2𝑦

𝑑𝑥2 = 0, then you would need to

check the change in the gradient before

and after the turning point in order to

see if if might be a POINT OF INFLEXION

C2 Differentiation: Optimisation Problems Objectives: Be able to write an algebraic equation to solve a problem Be able to change the subject of an algebraic equation Be able to differentiate a function and solve in order to find a minimum or maximum value Be able to use the second derivative to test a value in order to prove that it is either a maximum or minimum value.

1. Write down an equation for the surface area

of the cylindrical tin

2. Make “h” the subject of your equation.

3 a) write down the equation for the volume of a

cylinder

b) substitute in your expression for “h” from step 2.

C2 Differentiation: Optimisation Problems (continued) Objectives: Be able to write an algebraic equation to solve a problem Be able to change the subject of an algebraic equation Be able to differentiate a function in order to find a minimum or maximum value Be able to use the second derivative to test a value in order to prove that it is either a maximum or minimum value.

4. You are now ready to differentiate your expression for volume with respect to the

radius in order to find the value of the radius which will give the maximum volume.

THINK BIG BIRD!

C2 Co-ordinate Geometry: Circles ( 11 mins) Obectives: Recall and use the general form of the equation of the circle to solve circle problems Be able to complete the square to find the centre and radius of a circle from an equation of a circle Use circle theorems/facts to solve problems

General Equation of a circle

with a centre (a,b) and radius, r

How do you find the centre and radius of the circle when given the equation of a circle?

Circle Facts. Copy the diagrams!

Most important top tip:

C2 TOOLS: TOOLS Summary Page

Objectives: Understand radian measure Recall and Use formula for arc length and arc sector (IN RADIANS) Solve problems with triangles, sectors and segments using TOOLS kit COMPLETE THIS PAGE WITH ALL THE FORMULA YOU NEED TO LEARN.

T Triangle Area

O Sector Area

O Arc Length

L Cosine Rule

S Sine Rule

C2 TOOLS: Radians and Arc length(CONTINUED)

Radian Measure: If the radius of the sector is equal to the arc length, then the angle of the sector measures one radian.

Degrees Radians

180

360

90

60

45

30

The Arc Length and Area of a Sector Formula you will learn only work if the angle is in radians.

C2 TOOLS: Area of a Sector(CONTINUED)

C2 TOOLS: Area of a Segment

Area of Segment = Area of Sector – Area of Triangle Using previous formulae, in general this can be written as:

C2 The Sine and Cosine Rule: Sine rule

C2 The Sine and Cosine Rule: Cosine Rule

C2 Trig Identities and Equations: Trig Equations

Objectives: Be able to sketch the graphs of sine, cosine and tangent functions Be able to use the graphs to solve simple trig equations Be able to change the extend the range of trig function graphs

Examples: Show how to solve these trig equations.

2

3sin x 3600 x

2

3sin x 20 x

2

3cos x 3600 x

2 2tan x 3600 x

2

130sin x 3600 x

C2 Trig Identities and Equations: Trig Equations

LEARN: 1cossin 22 xx

ALSO LEARN: xx

xtan

cos

sin

03sin2sin 2 xx

0sinsin 2 xx

01sincos 2 xx

0cos4sin xx

C2 Exponentials and Logarithms: Exponentials and Logarithms ( 4 minutes)

How to change between log form and index form

Index form BP = A Log form logBA = P

2

3 = 8

32 = 9

log525 = 2

log101000 = 3

42 = 16

7-1 = 1/7

30 = 1

log5125 = 3

log100.1 = -1

log101000 = 3

C2 Exponentials and Logarithms: Rules of Logarithms ( 7 minutes)

Objectives: Learn and use the rules of logarithms to simplify and solve log equations

Addition rule

Subtraction rule

Power rule

Change of base rule

Example: In General:

Example: In General:

Example: In General:

Example: In General:

C2 Exponentials and Logarithms: Solving log equations ( 22 minutes)

How to solve log equations of the type 2x = 5 (index form) How to solve log equations of the type logx 5 = 7 (log form)

Solving easy index equations

Solve

1. 2x = 3 2. 3x = 10

3. 4x = 5 4. 6x = 7

Is it in the form BP= A ?

Solving harder index equations

56772 xx

03339 1 xxx

C2 Algebra and Functions: Factor/Remainder Theorem (11 minutes)

2

142 3

x

xx

1

423

x

xxx

The Remainder Theorem

The Factor Theorem

C2 Algebra And Functions: Factorising Cubics (5 minutes)

Factorising Cubics Factorise 51472 23 xxx using the fact that 12 x is a factor

C2 Integration: Definite integration (13 minutes)

Objectives: Be able to integrate a function between two limits using correct notation

6

4

2

3 243 dx

xx

2

1

2

32 dxx

1

3

23 53 dxxx

x

y

C

O

P

A

R

C2 Integration: Area Under A Curve (easy)

Find the area bound by the graph y = x2 + 2, the lines x = 1, x = 5 and the x axis

Find the shaded area

2 33 1

2 4y x x

C2 Integration: Area Under A Curve (easy) (CONTINUED)

Find the shaded area

1 5y x x x

y

O x2

C

1 5

A

R

O

B

x

y

C2 Integration: Area Under A Curve (harder)

Find the area shaded between the curves 9y x and2 2 3y x x

y

x

C

L

R

O

R is the region bounded by 26y x x and 2y x

Find the area of R

x

y

O

A

B

C

C2 Integration: Area Under A Curve (even harder)

The curve3

22 6 10, 0y x x x passes through the point A(1, 6) and has a

minimum turning point at B.

y

x

A

B

NO

y x x x= – 8 + 203 2

R

C2 Integration: Area Under A Curve ( even harder)

(Continued)

Find the shaded region

Write down the formula for the trapezium rule:

C2 The Trapezium Rule

C2 The Trapezium Rule (continued)

Example Question

C2 Geometric Progressions: Geometric Sequences and the nth term

Examples of Geometric Sequences 2, 4, 8, 16 1000, 100, 10, 0.1 Each term in a geometric sequence has been multiplied by a common ratio, r to generate the next term. The first term is normally given the letter, a

Finding the n’th term of a geometric progression:

Finding the 20th

term in the GP, 2 + 4 + 8 + 16

Finding the number of terms in the GP, 2 + 4 + 8 + ... 256

C2 Geometric Progressions (Continued)

C2 Geometric Progressions: Sum of geometric progression ( 3 minutes)

Proof of the sum of a geometric progression

Find the tenth term of the series 10 + 5 + 2.5 + ...

Find the number of terms in the GP 10000 +1000 +...+ 1

C2 Binomial Expansion: Binomial Expansion

7

4

3

2

1

0

ba

ba

ba

ba

ba

ba

Finding the coefficients

Method 1:

Method 2:

Method 3:

432 x

71 x

4

22

x

42 px

nx21

Statistics 1 GCSE Data Handling Revision of finding the mean, median, quartiles and standard deviation from raw data and data in frequency tables. You will also need to be able to interpret the skewness of the data by using a variety of methods. You will need to be able to interpret histograms and calculate the dimensions of bars in a histogram. Most of this should be a revision of what you have studied at higher tier GCSE. In addition you will need to understand how coding of data affects the mean and the standard deviation. Probability You will need to be able to use possibility space diagrams, two way tables and tree diagrams; this should be revision of what you have already studied. In addition we will look at Venn Diagrams and their notation and how they can be used to solve probability problems. We will also look at relationships such as mutually exclusive and independent events and rules affecting conditional probability. Discrete Random Variables We will look at the probability distributions of random variables such as rolling a dice etc. We will see how we can work out the expected value and variance of a particular event occurring. Normal Distribution We will look at the normal distribution; this shows the particular distribution of certain continuous variables such as height and time. The normal distribution can be used to estimate probabilities of the likelihood of certain events occurring. Correlation and Regression Following on from your look at Scatter Diagrams at GCSE we move it on by trying to put a numerical value on the strength of correlation that the scatter diagram suggests, using the Product Moment Correlation Coefficient. We then will look at the use of Regression to try and give the line of best fit an equation in the form of y = mx + c.

S1 GCSE Data Handling

Median and Quartiles from a list

Example: Stem and Leaf Diagram

0 6 7 8

1 0 2 3 4 7 7 7 8 9

2 1 3 4 5 5 7

3 1 1 2 6 6 9

4 1 5 5 6 9

5

6

7 9

Address: Where to find the quartiles? 𝑛 𝑖𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠

Lower Quartile, 𝑄 𝑛

4

Median, 𝑄

Upper Quartile, 𝑄

Rounding Rules:

If the address not a whole number then ______________________

Address is a whole number then __________________ ______________________

Example: Copy down example of

the quartiles

𝑄 𝑄

Inter- Quartile Range (IQR)

Example:

𝐼𝑄𝑅 . 𝑥 𝑄

𝐼𝑄𝑅 . 𝑥 𝑄

Outliers: extreme values or anomalies

that unnecessarily skew the data

Example:

GCSE Data Handling Mean and Standard Deviation from a list

Number of Goals Scored by Palace

6

2

3

3

5

4

Number of Goals Scored by Brighton

0

2

1

2

1

1

Mean 𝒙 𝒙

𝒏 Example:

Standard Deviation 𝜎 Example: Variance =

Mean: 𝒙 𝒙

𝒏 Example:

Standard Deviation: 𝜎 Example: Variance =

Comparisons: Palace Brighton

variance is the mean of the x’s squared minus the mean squared

GCSE Data Handling : Skew

Type Positive No Skew Negative

Tail Tail in higher values (on x axis)

Symmetrical

Tail in lower values(on x axis)

Position of mean?

Box Plots and quartiles

Example: What’s the skew

GCSE Data Handling Box plots

𝐼𝑄𝑅 . 𝑥 𝑄

𝐼𝑄𝑅 . 𝑥 𝑄

Outliers: extreme values or

anomalies that unnecessarily skew the data

GCSE Data Handling

This video shows you how to find the mean and standard deviation of grouped and ungrouped frequency tables.

Mean and Standard Deviation from a frequency table

Ungrouped Frequency

Maths test mark (x) Number of people (f)

1 6

2 5

3 7

4 4

Grouped Frequency

Height Frequency x fx x² f x²

0-4 2

5-10 4

11-16 6

17-20 5

21-30 5

Find the mean and the standard deviation

Mean: 𝒙 𝒇𝒙

𝒇 Example:

Standard deviation: 𝜎 Example:

Find the mean and the standard deviation

Mean: 𝒙 𝒇𝒙

𝒇 Example:

Standard deviation: 𝜎 Example:

Data Handling Effects of Coding on Mean and Variance

X, is the number of goals scored in a netball match

1,2,3,4,5

Calculate the mean and variance of 1,2,3,4,5

Calculate the mean and variance of 11,12,13,14,15e

Calculate the mean and variance off 10,20,30,40,50

GCSE Data Handling

This video shows you how to work out the median and quartiles from a frequency table called INTERPOLATION

Median and Quartiles from a frequency table

Upper Quartile

n = t = gw = gf = gb = median =

n = t = gw = gf = gb = lower quartile =

n = t = gw = gf = gb = upper quartile =

GCSE Data Handling Median and Quartiles from a frequency table (CONTINUED)

Pause the video clip and have a go yourself. Then play the video and check if you are right.

GCSE Data Handling

This video shows you the key features of

Histograms.

Histograms

Data Handling This video shows you how to find the dimensions of a Histogram

Histograms: How to find the dimensions of bars

IMPORTANT!

!!

CHECK THE

BAR WIDTH!

S1 Probability - Tree diagrams

This video reminds you how to use a tree diagram

A bag contains 3 blue balls and 5 red balls. Two are selected at random without replacement.

Find the probability that

a) They are both blue

b) There is one of each colour

Pythagoras tree…look it up!

There are 5 black socks and 3 reds in a bag. I pick 2 socks without replacement. Find the

probability I get

a) Two of the same coloured sock

b) At least one red sock

S1 Probability - Sample space diagrams

This video introduces the idea of a sample space diagram

A dice house

Two dice are thrown and their scores added. Find the probability of getting

a) A total of 7

b) A total of 8 or more

c) A score of 5 or more given the first score was even

Two coins are thrown. What is the probability of getting two tails?

Spinner A has 2,3,4 and spinner B has 1, 2,3,4,5. They are spun and their scores are added.

Find the probability of

a) A total of 6

b) A total of 7 or more

c) A total of 4 or more given that spinner A was an even number

Venn Diagrams

Formulae to Learn:

Addition Rule

Mutually Exclusive:

Independent:

S1 Probability Introduction to Venn Diagrams

𝑃 𝐴 ∪ 𝐵′

∪

OR =

AND =

Example 1 A school has 97 students in year 12, 55 take Maths and 32 take Chemistry. 31 take neither. How many take both?

S1 Probability Introduction to Venn Diagrams (Continued)

Find the probability distribution for S the score on a die.

A die is thrown until either a six appears or I’ve thrown it three times. Find the probability

distribution for T where T is the number of throws.

S1 Discrete random variables DRVs - Probability distributions

This video explains what a DRV is and shows you how to find the probability

distribution

S1 Discrete random variables DRVs – Probabilities total 1

This video shows you the important fact that the total probability is 1.

Not much to write here!

Just remember 1)( xXP The sum of the probabilities = 1!!!!!!!!!

S1 Discrete random variables DRVs - F(X)

This video shows you the cumulative distribution function F(X)

Here is a probability distribution

X 1 2 3 4 5

P(X =

x)

1/10 4/10 2/10 1/10 2/10

F(3) =

F(2.7) =

Find the cumulative distribution function

X 1 2 3 4 5

F(X)

Can you go back the other way?

Now imagine we were given a cumulative distribution table

X 1 2 3 4 5

F(X) 1/10 2/10 4/10 8/10 10/10

and asked to find the probability distribution table

X 1 2 3 4 5

P(X=x)

F(X) =

This just means it’s the

sum of the probability and

all the probabilities below

it

S1 Discrete random variables DRVs - E(X)

This video shows you how to find the expected value or mean of a DRV.

The probability distribution is:

X 1 2 3 4

P(X =

x)

10/60 20/60 15/60 5/60

Find E(X)

X 1 2 3 4

P(X =

x)

1/6 2/6 2/6 1/6

Find E(X)

S1 Discrete random variables DRVs - Var(X)

This video shows you how to calculate the variance of a DRV.

X 1 2 3 4

P(X =

x)

10/60 20/60 15/60 5/60

Find Var (X)

X 1 2 3 4

P(X =

x)

1/6 2/6 2/6 1/6

Find Var (X)

Did you remember

to minus the mean

squared?

Variance squariance

S1 Normal distribution – An introduction

This video gives an introduction to the Normal Distribution

Continuous Data such as height of an adult, time taken to run a race etc tend to have a NORMAL DISTRIBUTION. If data is known to have a NORMAL DISTRIBUTION, there are certain characteristics we can assume:

It is a SYMMETRICAL distribution and is centered on the mean of the data.

99.8% of the data lies within 3 standard deviations of the mean

The mean and standard deviation can be used to standardise the distribution in order to estimate probabilities of events using z values

A normal distribution with a mean of 0 and a standard deviation of 1 has probabilities already calculated for it. These are the z values.

How to standardise? The formula to standardise is z =

Example:

S1 Normal distribution – Standard Normal Distribution

This video shows you how to use the z values for the standard normal

distribution

S1 Correlation & Regression - Product moment correlation coefficient (pmcc)

This video looks at scatter graphs and how we calculate and interpret the

pmcc.

Find the PMCC and comment on it.

English

marks

Maths

marks

105 5

100 2

108 8

100 1

102 2

104 4

These are in the formula book

Find the PMCC and comment on it.

Maths

marks

English

marks

5 7

6 5

7 3

8 2

S1 Correlation & Regression – The regression line

This video shows you how to find an equation for the regression line (line of

best fit)

Mins

spent

revising

Maths

marks

105 5

100 2

108 8

100 1

102 2

104 4

Find the equation of the least squares regression line y on x

Interpret a

Interpret b

The equation of the least squares regression

line y on x is

y = a + bx where and

Use your line to find the Maths mark if you revise for

101 hours

110 hours

Hours

spent

revising

English

marks

5 7

6 5

7 3

8 2

Find the equation of the least squares regression line y on x

Interpret a

Interpret b

Use your line to find the English mark if you revise for

6.5 hours

10 hours