GCSE Maths (Higher) Week 6 Revision –ratio, histograms New – compound interest New - probability

A ratio question

+ - x ÷ mixed numbers and top heavy fractions Ratio questions Finish topic test Look at mathscast videos

ANSWERS I = 720 DVDs J = £15 K = 270g L = e.g. 2 : 10 A = widescreen B = 10 litres C = 4 litres D = 90° E = £1500 F = 2 : 1 G = 40 kilograms H = 8 : 2 : 1 I = 720 DVDs J = £15 K = 270g L = e.g. 2 : 10 M = 160 medium envelopes N = 4 : 8 (doesn’t simplify) O = 240 girls P = 90 : 1 Q = 15 ‘under twos’ R = £6 and £15 S = 450 students T = 14 more sweets U = 9 : 11

Topic test

(A percentage is a fraction out of 100)

Compound Interest

Definition

£2000 earning Compound Interest at 5% per year for 3 years Original Amount = 100% Compound Interest = 5% 100% + 5% = 105% = 1.05 3 £2000 x 105% 1.05 = £2315.25 This is the total amount including interest: £2315.25

Mathswatch clip 137 This is based on the method on the previous slide

Compound Interest Questions £10,000 earning Compound Interest at 1% per year for 3 years £8,650 earning Compound Interest at 2% per year for 5 years £5,000 earning Compound Interest at 0.5% per year for 4 years £10,000 earning Compound Interest at 1.5% per year for 6 years £8,000 earning Compound Interest at 3% per year for 7 years

The formula to calculate compound interest is: A = P x (1 + i)n

Starter Washing line activity Event cards, including baby being born cards Can we assign numerical values to the probability of all these events? How can you decide the probability of an event like “I will be struck by lightning this afternoon”? How can you calculate exactly the probability of getting a tail with one toss of a coin?

Throwing a dice What is the probability of throwing a 6 on a dice? Show your answer on the probability scale. What is the probability of throwing an even number on a dice? What is the probability of having a total score of 1 when the dice is thrown twice?

By the end of this next part of the lesson you will be able to: Use probability vocabulary, write probabilities as fractions, decimals or percentages and use a probability scale Understand that the sum of the probabilities of all possible mutually exclusive outcomes is 1 Understand the difference between theoretical and experimental probability Calculate probability using a tree diagram Identify and explain mutually exclusive and independent events Calculate relative frequency

Throw a dice What is the chance you’ll throw a 1? What is the chance you’ll throw 2 ‘1s’? Take 2 dice – what is the chance you’ll throw double 1? Why is it the same? Independent

AND/OR In probability calculations: AND = x (independent events) OR = + (mutually exclusive events)

AND = Multiplication Independent events The ‘AND’ rule Outcomes that are independent of each other. This means that one has no effect on the other (eg rolling two dice). The ‘AND’ rule P(A and B) = P(A) x P(B)

Tails never fail Stand up – throw a coin – if you get heads, sit down – if you get tails – stay standing – tails never fail! What is the probability of getting < > tails in a row? How can we work it out? One more go? What was the chance of getting another tail – independent event – nothing to do with what has gone before

What is the chance of getting a 4 with one dice?

What is the chance of getting a 6 on this spinner? 2 4 6 6 2

What is the chance of getting an odd number on this spinner? 1 2 3 4 5

What are the chances of getting a 6 on this spinner? 5 4 3 3 6

What is the chance of picking a blue ball out of the bag?

What is the chance of picking a purple counter out of the bag?

True or false? 15 questions – true or false? (Up to you how you do this, whether you print them out as cards and get the students in pairs/threes to put them in two columns, or do is as a class with T/F cards.)

A When you roll a fair six- sided dice, it is harder to roll a six than a four. False – a fair dice, so all numbers have equal chance

B Scoring a total of three with two dice is twice as likely as scoring a total of two. True – look at the combinations: next slide

True – look at the combinations: A score of two can only be obtained in one way – a 1 on each dice. A score of three can be obtained in two ways – 1 and 2 or 2 and 1, so the three is twice as likely.

C In a lottery, the six numbers 3, 12, 26, 37, 44, 45 are more likely to come up than the six numbers 1,2,3,4,5,6. False – each number has the same chance

The probability of two heads is therefore When two coins are tossed there are three possible outcomes: two heads, one head, or no heads. The probability of two heads is therefore False – look at the combinations: next slide

False because there are four outcomes: HH, HT, TH, TT So the probability of HH is one out of four, or

There are three outcomes in a football match: win, lose or draw. The probability of winning, if the teams are equally matched, is therefore False, and I have tried to explain why...(next slide)

Suppose team A score 1 goal, and team B scores other possible numbers: A, B 1, 0 1, 1 1, 2 1, 3 1, 4 etc Only one of these results is a draw, the rest are all win or lose, so the draw is least likely. Check some football scores on a Saturday, it is FALSE

F In a ‘true or false’ quiz with ten questions, you are certain to get five right if you just guess.  False – you would expect five right, but because of chance it won’t happen every time.

G After tossing a coin and getting a head five times in a row, the next toss is more likely to be a tail than a head. False – only gamblers believe their luck will change. The probability will be 1/2 each time, whatever happened before.

H In a group of ten people the probability of two people being born on the same day of the week is 1. True. There are only seven days in a week, so some pair must have been born on the same day.

I My friend has four sons. If she has another baby, it is more likely to be a boy than a girl. False, same as with the coin, unless you look at scientific results, which might suggest various things.

J The probability of getting exactly three heads when I toss a coin six times is ½. False – and there are a large number of possible outcomes. The actual probabilities are shown on the next page.

My drawer contains 4 blue socks, 7 red socks, and 3 yellow socks My drawer contains 4 blue socks, 7 red socks, and 3 yellow socks. What is the probability that I pull out a blue sock first? If I pull out 2 socks, what is the probability I’ll choose a blue sock and a red sock? What is the probability that I won’t choose red?

Mutually Exclusive Events OR = Addition Mutually Exclusive Events Events are said to be mutually exclusive if they cannot happen at the same time. The ‘OR’ rule P(A or B) = P(A) + P(B)

My drawer contains 4 blue socks, 7 red socks, and 3 yellow socks My drawer contains 4 blue socks, 7 red socks, and 3 yellow socks. What is the probability that I pull out a blue sock first? What is the probability I’ll choose a blue sock or a red sock?

Independent, dependent or mutually exclusive? Independent: the outcome of 1 situation does not affect the outcome of a following situation - think of drawing a card from a deck, returning it, then picking another card. Dependent: the outcome of 1 situation affects the outcome of a following situation - think of drawing a card from a deck, not returning it, then picking another card. Mutually exclusive: situation 1 and situation 2 cannot happen at the same time. For example, flipping a coin. It can land on heads, or it can land on tail, but it can't land on both.

Independent or dependent events? (calculate independent ones) A head on a 10p coin and a head on a £1 coin Choosing a girl for my football team, then choosing another girl Rolling a six on one dice and rolling a five on another Picking an ace from a pack, keeping it and picking another ace Choosing a red ball from a bag, replacing it, then a black ball Choosing a yellow sweet from a bag, eating it and choosing a red sweet Choosing a king from a pack of cards, replacing it and choosing another king

Mutually exclusive - an example What is the likelihood of picking a red card and a Queen from a pack of cards?

A MUCH MORE DIFFICULT QUESTION: I throw a coin and a dice. What is the probability of scoring a head or a six?

I throw a coin and a dice. What is the probability of scoring a head or a six?

The probability of scoring a head or a six is

The probability of scoring a head = The probability of scoring a six = The probability of scoring a head or a six =

Because one case got counted twice – the ‘head and six’ These two events are NOT mutually exclusive – they can happen at the same time. So we need to subtract the probability of both happening.

S7 sheet 1 spinners

Back to targets Can you now identify and explain mutually exclusive (OR +) and independent (AND x) events?

A triangular spinner has sections coloured white (W), green (G) and blue (B). The spinner is spun 20 times and the colour it lands on each time is recorded. W W B G G W B G G W G B G B G W G B G B Complete the relative frequency table. Colour White (W) Green (G) Blue (B) Relative Frequency

Suppose I roll a dice – what is the probability that I score: A six? What do we mean by ‘expected outcomes’? Suppose I roll a dice – what is the probability that I score: A six? An odd number? A number greater than 4? If I rolled the dice 60 times, how many times would you expect me to score: 2 of 4 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

The probability that a new car that is produced in the UK is green is 0.05 If 1.5 million cars are produced in the UK each year, how many of these would you expect to be green? 3 of 4 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

One for you to do: A bag contains only red counters and blue counters. There are 90 red counters in the bag. The probability of choosing a red counter from the bag is 0.3 How many blue counters are in the bag?

One for you to do: The probability that Colin is late for work, on any given day = 0.2 What is the probability he is late two days in a row?

Lottery How many numbers to choose from? Black Red (WINNERS!)

3 doors – behind which is the car? http://www.youtube.com/watch?v=mhlc7peGlGg http://www.stayorswitch.com/ http://www.nytimes.com/2008/04/08/science/08monty.html?_r=0#

There is a 1/3 chance of the car being behind door number 1 and a 2/3 chance that the car isn’t behind door number 1. After Monty Hall opens door number 2 to reveal a goat, there’s still a 1/3 chance that the car is behind door number 1 and a 2/3 chance that the car isn’t behind door number 1. A 2/3 chance that the car isn’t behind door number 1 is a 2/3 chance that the car is behind door number 3.

How could we display the information below in an easier format How could we display the information below in an easier format? A school chess club has 70 members of which 40 are boys. Students play in competitions on a regular basis. Last month, 13 girls and 11 boys played in competitions. 2 of 4 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

A two way table might be helpful…. Played in competition last month Did not play in competition last month Boys 11 29 40 Girls 13 17 30 24 46 70 What other methods might we use to display the information? 3 of 4 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

11 40 Boys 29 70 13 Girls 30 17 Use a frequency tree Played in competition 11 40 Boys Didn’t play in competition 29 70 Played in competition 13 Girls 30 17 Didn’t play in competition 4 of 4 Copyright © 2015 AQA and its licensors. All rights reserved. AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

x of x Version 3.0 Copyright © AQA and its licensors. All rights reserved.

x of x Version 3.0 Copyright © AQA and its licensors. All rights reserved.

In Year 6 at a local primary school there are 120 students In Year 6 at a local primary school there are 120 students. The ratio of boys to girls is 9:6. The girls were twice as likely to own a mobile phone as they were to not own a mobile phone. The ratio of boys who own a mobile phone to those who don’t own a mobile phone is 5:3 2 of 3 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

45 72 Boys 27 120 32 Girls 48 16 Use a frequency tree Owns a mobile phone 45 72 Boys Doesn’t own a mobile phone 27 120 Owns a mobile phone 32 Girls 48 Doesn’t own a mobile phone 16 3 of 3 Copyright © 2015 AQA and its licensors. All rights reserved. AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

TREE DIAGRAMS First coin Second coin H H T H T T

H T

H T

Imagine choosing a ball from this bag and then replacing it Imagine choosing a ball from this bag and then replacing it. If you did this three times, what's the probability that you would pick at least one green ball? What’s the best method to use to answer this question? What if you didn't replace the ball each time?

Replacing it - if you did this three times, what's the probability that you would pick at least one green ball?

Not replacing it - if you did this three times, what's the probability that you would pick at least one green ball?

Two cards are drawn from a pack with replacement 13 52 39 A spade Not a spade 1st card 2nd card

Two cards are drawn from a pack without replacement 13 52 39 A spade Not a spade 1st card 2nd card

Exam question 11

Looking back at targets Do you understand that the sum of the probabilities of all possible mutually exclusive outcomes is 1? Do you understand the difference between theoretical and experimental probability? Can you calculate probability using a tree diagram?

Relative frequency in a graph

PLENARY Who wants to be a millionaire? millionaire_probability.ppt

Reviewing targets Can you now: Show the probability of something happening on a probability scale? Calculate relative frequency? Do you understand that the sum of the probabilities of all possible mutually exclusive outcomes (outcomes that don’t happen at the same time – like raining and not raining) is 1? Draw a tree diagram?

Website of the week Help is at hand https://www.sites.google.com/site/mathscasts/mathscast-videos

Work with general iterative processes https://allaboutmaths.aqa.org.uk/attachments/5626.pdf Will need to make this into a slide sequence

Percentage increase and decrease – how would you do that?

My monthly gas bill of £64 is going up by 15% - what is the new price My monthly gas bill of £64 is going up by 15% - what is the new price? The value of my car which I bought new for £3,600 has gone down by 35% - what is the new value?

Percentage Profit/Loss (another form of percentage increase/decrease) If a value has increased or decreased by an amount and the question asks what this would be as a percentage, this is how you would work it out: actual increase/decrease x 100 original amount

Example

Your turn

Reverse Percentages

There is a 20% sale on in Topshop. The bag I want is now £60. What was the original cost of my bag? % 60

In a sale, everything is reduced by 30% In a sale, everything is reduced by 30%. If an armchair costs £175 in the sale, how much did it cost before the sale? % 175

A mouse increases its body weight by 15% A mouse increases its body weight by 15%. If it now weighs 368g, what was the mouse’s original weight? % 368