GCSE Maths (Higher) Week 7 Results from assessment

GCSE Maths (Higher) Week 7 Results from assessment
New – compound interest More probability Introduction to algebra

Assessment questions

STARTER – non-calculator maths
Can anyone give me a non-palindromic three digit number? Write it down, then reverse it and write this down. Find the difference, then reverse this and add these two numbers together.

While we’re talking about palindromes ……
How many palindromic people in your family? Was it a car or a cat I saw? (The word itself is from the Greek palindromus, which means to run back again.)

Compound Interest

Definition

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

£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 = £ This is the total amount including interest: £

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

Just Maths Compound Interest Questions

What did we do last week? Slides handout?

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 ) and a company limited by guarantee registered in England and Wales (number ). 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 ) and a company limited by guarantee registered in England and Wales (number ). 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?

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?

Finally – the national lottery
º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º

Euromillions 5 main numbers 2 lucky stars º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º

Exam paper Question 18 1, 2, 3, 4, try 7, try 9 (look it up) 15, try 16, 18, 23, try 24 (look up minus powers) (done 6, 8, 10, 11) For question 5 area of a circle = πr2

New topic - algebra

Think of a Number A bit of maths magic?
Kindly contributed to by by Fiona Campbell City of Bristol College. N1/E2.3 Add and subtract two-digit whole numbers N1/E2.4 Recall addition and subtraction facts to 10. N1/E3.3 Recall addition and subtraction facts to 20 (but makes a good warm activity for any level)

Think of a number between 1 and 10

Double the result Hint: to double you by what?

Plus 5

Halve the result [Hint to halve something you divide by what?] ….

Add eight

Subtract 9

Subtract the number you first thought of

Your answer is 1 Try starting again with another number. Does it work for all numbers? Use big numbers and a calculator. How does it work?

Think of a number – any number (Sssh! – don’t tell anyone)

Double the new total

Subtract 2

Halve the new total

Take away the number you first thought of …

What’s your answer?

Does it work for a very large number?
Does it work for a fraction? Does it work for a decimal? Does it work for a negative number?

Think of a number – any number PS – you may need a pen & paper for this one 

Multiply the number by 3

Add 45 to the result

Double the new total

Divide your answer by 6

Take away the number you first thought of …

What’s your answer?

Does it work for a very large number?
Does it work for a fraction? Does it work for a decimal? Does it work for a negative number?

Think of a number between 1 and 10
Multiply this number by two. Add 5. Multiply the answer by 50. If you have already had your birthday this year, add 1767, otherwise add 1766. Now subtract the four digit year that you were born. (The first digit is the number you first thought of – the next two are your age.)

A3 Algebraic Expressions.avi

Write down the expression that means “Multiply n by 3”

Write down the expression that means “Multiply n by 3, then add 4”

3n + 4

Write down an expression for “Add 4 to n”

Add 4 to n, then multiply your answer by 3

3(n + 4)

Add 2 to n, then divide your answer by 4

n + 2 4

Multiply n by n

Multiply n by n, then multiply your answer by 4

Multiply n by 6 then square your answer

Think of a number …. Double it. Add 6. Divide it by 2. Take away the number you started with. What is your answer?

Can you show how this always works with algebra?

Substitution This is where you substitute a value into an expression or formula for the unknown value

Example If y = 4 and t = 6, work out the value of 7y – 3t

Example If q = 5, r = 2 and z = -3, work out the value of qr + z2

n = 4 Can you put the 6 cards in order? n = 9 – are they in the same order? n = 100?

6 algebra words Simplify Expand Factorise Substitute Solve (not yet)
Rearrange a formula/change the subject (not yet)

Simplifying means to make simpler – in other words, collect like terms together.
6a + 2a + a = 3t + 5u + 9t – 2u =

Your turn… Simplify the following expressions. a + a + b
3a – 2a + 4b + b 8a + 5a – 6a 12b + 5a + 3a – 8b 4a + 6b + 7c + 9a – 2b – c 9 x a 13 x b g x h 9 x j x j x j 6 x p x q x p x 2 2a + b 9a a + 5b 13b 7a gh 8a + 4b 9j3 13a + 4b + 6c 12p2q

John and Sarah are simplifying this expression
John and Sarah are simplifying this expression. 3a + 8b + 7a – 6b John says the answer is 2a + 2b and Sarah says the answer is 10a + 14b. What is the correct answer?

Expanding Brackets Expanding brackets is often called “multiplying out” brackets because we use multiplication to get rid of the brackets.

Expanding Brackets 5(a + 3)
To expand a bracket, multiply the term outside the bracket by everything inside the bracket x 5(a + 3) x 5a + 15

Expanding Brackets - together
7(n + 6) 8(p – 3)

Your turn… Expand the following brackets: 4(x + 5) 3(y + 9) 5(a – 2)
a(b + 3) n(n + 5) p(8 – p) 4x + 20 3y + 27 5a - 10 11r – 44 ab + 3a n2 + 5n 8p – p2

Expanding brackets and collecting like terms
Multiply out and simplify 3(5y – 3) – 2(4y + 5) 2(3x + 3) + 2(x - 5)

Exam paper Question no. 1

Plenary It’s Bingo Time!

Choose yourself 9 of the expressions below to go in your bingo grid.
6n2 + 2n 15n + 2 9n + 7 10n2 – 35n n2 + 6n 2n + 12 6n + 12 5n – 4 6n + 10  n2 + 6 6n2 + 10n 6n – 8n2 24 – 3n 5n – 2 15n + 6 7n – 21 8n + 12 15n + 30 2n2 + 5n 36n + 20 8n – 4

Instructions Expand the brackets and simplify the expressions.
If you have that answer in your grid, cross it off!

5(3n + 6)

n(n + 6)

6(n + 2)

7(n – 3)

3(5n + 2)

7n – – 2n

2n(3n + 5)

4n + 12 – 2n

4(2n + 3)

5(n – 3) + 11

6n + 2(n – 2)

2n(3n + 5)

5 + 2(n – 3)

2n(3 – 4n)

7n2 – n2 + 2n

2(3 + 5n) + 4(1 – n)

7(n + 3) + n – 5

5n(2n – 7)

8(n – 3) + 30

8 – 3(2 – 5n)

7(2n + 1) – 5n

n(2n + 1) + n(5 – 1)

n + 7(5n + 3) – 1

6(4 – 2n) + 9n

n(n + 4) + 2(3 – 2n)

Factorising expressions - together
Factorise 5p – 25 Factorise mn + mt Factorise 9xy – 3y Factorise w2 + 3wz Me

Factorising expressions – on your own
Factorise these expressions by removing a common factor. 4a + 6 9x – 6 7g2 – 2g 4b2 + 6b 12x2 + 3x 6ab – 9a You

A very common exam question
2(x + 16) + 4(x – 5) simplifies to a(x + b) Work out the values of a and b. (3 marks)

Homework Topic test 1st exam paper

Website of the week Help is at hand

5 9 x2 + 4 6 -2 x3 - 2 13 37 x2 + 7 ALGEBRA REVIEW X a 5m n 2p 4 20m b
3) Multiplying expressions Complete the table below 5) Function machines Find the output for each function machine Find the inputs for the function machine 1) Collecting like terms Simplify these expressions (a) 3m + 5n + 2n + 12m (b) 4p p - 2 (c) 8a + b + 5b - a (d) 4y + 3w + y – 7w (e) g – 4h + 8g + 6h (*f) ab + 2a + 5ab – 5b X a 5m n 2p 4 20m b 3h 6hp 3p 5 9 x2 + 4 6 -2 x3 - 2 13 37 x2 + 7 4) Expand brackets Expand 3(2y + 1) Expand 5(3m - 4) (*c) Expand 4(2w + 3) + 2(3w + 9) 2) Substitution a = 3, b = 5, c = -2, d = 10 (a) 4b (b) d (c) 3c (d) 5a + 2 (e) 3b (f) 3c + 1 (g) 2c (h) ab (i) 5² (*j) 4b – 6 2 Objective Mastered I can collect like terms    I can substitute positive values into expressions I can substitute negative values into expressions I can multiply an expression by an integer I can multiply two algebraic expressions I can expand brackets I can use a function machine to find an output when given an input I can use a function machine to find an input when given an output

ALGEBRA REVIEW 4) Formulae Substitution The formula t = v – u a
a = 3, b = 5, c = -2, d = 10 (a) 3b – d (b) ac + b (c) b(4a – 3) (d) a(2b + c) 4 4) Formulae The formula t = v – u a is used to calculate time (t). If v = 21, u = 6 and a = 3, calculate what t will be? If v = 14, u = 2 and t = 3, calculate what a is? Expand brackets Expand 3(4y – 5) Expand m(3m + 5) Expand 4(2w – 3) + 2(3w + 9) Expand 5(2y + 1) – 3(3y – 2) Expand (p + 3)(p + 5) Shape algebra Calculate the perimeter and area of this rectangle 3 4n - 1 Objective Mastered I can expand single brackets    I can expand separate brackets I can expand double brackets I can factorise expressions into single brackets I can substitute into expressions I can substitute into formulae I can solve algebra problems in shapes 2) Factorise Factorise b – 10 Factorise a² + 3a Factorise a – 15ab

Work with general iterative processes
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

Two-way tables and frequency trees.

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 ) and a company limited by guarantee registered in England and Wales (number ). 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 ) and a company limited by guarantee registered in England and Wales (number ). 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 ) and a company limited by guarantee registered in England and Wales (number ). Our registered address is AQA, Devas Street, Manchester M15 6EX.

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 ) and a company limited by guarantee registered in England and Wales (number ). 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 ) and a company limited by guarantee registered in England and Wales (number ). Our registered address is AQA, Devas Street, Manchester M15 6EX.

What did we do last time? 1.

Last bit of probability for now - tree diagrams

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

Probability revision quiz HIGHER
In teams of 3 work together to answer all the questions

Finding probabilities 1 (words)
5 points 10 points 15 points P(roll an odd number on a dice) is P(it will rain tomorrow) is P(baby born is a girl) is P(being younger tomorrow) is P(win lottery) is P(sun rising tomorrow) is

Finding probabilities 2 (fractions)
5 points (dice) 10 points (cards) 15 points P(red prime)= P(black)= P(4)= P(even)= P(red picture card)= P(less than 5) =

Finding probabilities 3 (OR rule)
5 points (dice) 10 points (cards) 15 points P(prime or square)= P(6 or 7)= P(4 or 5)= P(2, 3 or 4)= P(red 3 or black queen)= P(2 or 3) =

Finding probabilities 4 (2 events)
5 points (2 dice) 10 points 15 points P(2 numbers the same)= P(5 or 6)= P( 7 )= P(13)= P(less than 4) = P(even) = You have 1 minute to list all the outcomes of 2 dice before the questions come up.

Finding probabilities 5 (biased dice)
1 2 3 4 5 6 0.2 0.1 0.05 x 0.15 5 points 10 points 15 point P(not 4)= P(6)= P(2)= P(1 or 3)= P(not 2)= P(odd) =

Finding probabilities 6 (tree diagrams)
Copy down this information: The probability it rains on a Monday is 0.3 The probability it rains on Tuesday is 0.25 5 points 20 points 30 points P( it rains on both days) Draw a tree diagrams to show all outcomes P( it does not rain on Monday)= P(It does not rain on Tuesday)=

Bonus question (40 points)
Based on the following tree diagram what is the probability I pick 2 different colours?

Relative frequency in a graph

PLENARY Who wants to be a millionaire? millionaire_probability.ppt

GCSE Maths (Higher) Week 7 Results from assessment

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