GCSE Maths - Higher Lesson 4 Fractions, decimals and percentages - and finishing off from last week!

What do these inequalities say What do these inequalities say? What integer values satisfy the inequality?

Homework was finishing the topic test and the systematic listing one (answers were already on the back) Also finishing histogram sheet

Word of the day - integer What is an integer? An integer is a positive or negative whole number, including zero.

+, -, x, ÷ positive and negative integers Mathswatch Clip 99 (demonstrate Cdrom)

Negative number cards

Negative Number BINGO

Put one number in each box Put one number in each box. Pick any numbers from the list but no repeats. 15 -10 3 -21 -13 17 -9 11 18 -12 -15 -2 -4 22 6 25 7 20 -5 24 16 14 8 -3

8 -12

10 - -12

- 2 + 8

20 - -5

- 1 +8

-2 x -10

-2 -3

20 - -4

- 4 x -4

15 + -1

9 – 1

7 -10

16 + -1

-40 ÷ 4

-9 ÷ -3

7 x -3

-26 ÷ 2

-17 x -1

3 x -3

12 + -1

15 - -3

6 x -2

- 20 + 5

10 - 12

Practice Mr Corbett’s real life negative number questions – for homework?

What is the answer to this question? (give out a few basic calculators) 2 + 3 x 4 Now put it in your calculator – what is the answer?

Homework - bidmas activity

New vocabulary Factor Multiple Prime number Square Cube Root Power

http://www.visnos.com/demos/sieve-of-eratosthenes Use this to talk about factors, multiples as well as primes

Prime Numbers A prime number has exactly two factors – one and itself One is not a prime number because it has only one factor – itself Two is the first prime number Two is the only even prime number

Square numbers √ Square Square root Square Square root Area = 3 x 3 = 3² 3 9 =9 Square root √ Area = 4 x 4 Square = 4² 4 16 Square root = 16

Factors multiples puzzle

Answers

From these numbers: 1 3 8 12 16 20 13 15 2 17 24 6 Write down 1 3 8 12 16 20 13 15 2 17 24 6 Write down 4 prime numbers all the factors of 6 any multiples of 4

cube numbers cube cube Cube root Cube root Volume = 3 x 3 x 3 Volume = = 3³ = 4³ cube = 27 cube = 64 4 64 3 27 Cube root Cube root

Can you answer these questions? What is the square of 5? What is the square root of 36? Is there another one? What is the cube of 2? What is the cube root of 125? What are the powers of 10?

From these numbers: 121 49 20 19 800 1000 5 Find two square numbers 121 49 20 19 800 1000 5 Find two square numbers a cube number the square root of 400 the cube root of 125

For higher, you may need to estimate the values of powers and roots.

Estimate the square root of 43 Between which two integers does the square root of 150 lie? Between which two integers does the cube root of 100 lie? Estimate the value of 6.34 Estimate the value of Estimate the value of 2x = 40

Estimate the square root of 43 Between which two integers does the square root of 150 lie? Between which two integers does the cube root of 100 lie? Estimate the value of 6.34 Estimate the value of Estimate the value of 2x = 40

Finally - Factors, Multiples and Primes 8 quick questions Write down your answers on your whiteboard

1 Which of the following numbers divides into 49, without remainder? Divisors a: 3 b: 4 c: 7 d: 9

2 Including one and itself, how many factors has the number 6? Factors a: 2 b: 3 c: 4 d: 6

3 Which of the following is a multiple of 6? Multiples a: 1 b: 3 c: 9 d: 12

4 Common Factors a: 2 b: 4 c: 8 d: 36 Which of the following is a common factor of 12 and 18? Common Factors a: 2 b: 4 c: 8 d: 36

5 Prime Numbers a: 2 b: 15 c: 17 d: 29 Which of the following is not a prime number? Prime Numbers a: 2 b: 15 c: 17 d: 29

6 Prime Numbers a: 51 b: 53 c: 55 d: 57 What is the next prime number after 47? Prime Numbers a: 51 b: 53 c: 55 d: 57

7 Prime Factor Decomposition a: 2 x 15 b: 3 x 10 c: 5 x 6 d: 2 x 3 x 5 Which of the following shows the prime factors of 30? Prime Factor Decomposition a: 2 x 15 b: 3 x 10 c: 5 x 6 d: 2 x 3 x 5

8 Prime factor decomposition a: 23 x 5 b: 22 x 10 c: 2 x 4 x 5 Which of the following shows the prime factors of 40 in index form? Prime factor decomposition a: 23 x 5 b: 22 x 10 c: 2 x 4 x 5 d: 4 x 10

Answers

1 Which of the following numbers divides into 49, without remainder? Divisors a: 3 b: 4 c: 7 d: 9

2 Including one and itself, how many factors has the number 6? Factors a: 2 b: 3 c: 4 d: 6

3 Which of the following is a multiple of 6? Multiples a: 1 b: 3 c: 9 d: 12

4 Common Factors a: 2 b: 4 c: 8 d: 36 Which of the following is a common factor of 12 and 18? Common Factors a: 2 b: 4 c: 8 d: 36

5 Prime Numbers a: 2 b: 15 c: 17 d: 29 Which of the following is not a prime number? Prime Numbers a: 2 b: 15 c: 17 d: 29

6 Prime Numbers a: 51 b: 53 c: 55 d: 57 What is the next prime number after 47? Prime Numbers a: 51 b: 53 c: 55 d: 57

7 Prime Factor Decomposition a: 2 x 15 b: 3 x 10 c: 5 x 6 d: 2 x 3 x 5 Which of the following shows the prime factors of 30? Prime Factor Decomposition a: 2 x 15 b: 3 x 10 c: 5 x 6 d: 2 x 3 x 5

8 Prime factor decomposition a: 23 x 5 b: 22 x 10 c: 2 x 4 x 5 Which of the following shows the prime factors of 40 in index form? Prime factor decomposition a: 23 x 5 b: 22 x 10 c: 2 x 4 x 5 d: 4 x 10

Add them up! 7 or 8? Brilliant 5 or 6? Jolly good show 3 or 4? Some good, but revision needed 1 or 2? Oh dear, have you been paying attention? 0 Whoops!

For after the break ……

What are we going to do now? Fractions Order fractions Find a fraction part of a quantity The four operations: + - x ÷ Percentages Find a percentage part of a quantity Write one value as a percentage of another value Percentage increase and decrease (simple/compound interest) Reverse percentages Decimals Order decimal numbers Convert between fractions, decimals and percentages

Fractions Finding a fraction part Equivalent fractions & simplifying Ordering fractions Adding and Subtracting fractions Multiplying and Dividing fractions

Finding a fraction part of a number You need to remember the following: Divide by the bottom THEN Times by the top

Example: What is 5/7 of 350 ? First, divide by the bottom: 350 ÷ 7 = 50 Then, times your answer by the top: 50 x 5 = 250 So 5/7 of 350 is 250.

Your Turn…

Can you guess which thing One of these things, it doesn’t belong. 3 8 𝑜𝑓 32 1 2 𝑜𝑓 24 Can you guess which thing One of these things, it doesn’t belong. Before I finish my soonnnggg. One of these things is not like the other thing is not like the other things 2 5 𝑜𝑓 20 6 7 𝑜𝑓 14

3 8 𝑜𝑓 32 1 2 𝑜𝑓 24 2 5 𝑜𝑓 20 6 7 𝑜𝑓 14

Simplify the fractions… Customers have paid the following amounts to begin to pay off their overdraft. Simplify the fractions below. £300 paid of £450 £50 paid of £600 £400 paid of £1000 £60 paid of £120 £30 paid of £40 £45 paid of £90 £110 paid of £450

Equivalent Fractions Equivalent means equal to, so equivalent fractions are fractions that are the same. ½ = 2/4 = 3/6 = 4/8 = 5/10 (These could all be simplified to ½ therefore they are equivalent)

Ordering fractions - together We use equivalent fractions to put fractions in order: We find the common denominator and use it to place the fractions in order, smallest to largest

Your turn Using the common denominator place the fractions in order, smallest to largest.

A fraction problem to solve together Anna, Bobby and Cheryl order a large 24 piece pizza. Anna eats 1/6 of the pizza. Bobby has 5 pieces and Cheryl eats a quarter. What fraction of the pizza is left? 3/8 is the answer

Another fraction problem – “The builder’s day” A builder works for of a day. If he sleeps 7 ½ hours per day what fraction of the day does he have left ? 13/48

Yet another – “The TV Programme” A TV programme lasting an hour has ⅖ of it dedicated to adverts. The rest of the programme is split equally between current affairs and sport. How long is spent on Sport ? 18 minutes

Adding and Subtracting Fractions To add or subtract fractions, you need to have a common denominator – both fractions need to have the same value on the bottom. 1 2 + 2 5 =

Or you can use the “kiss & smile” method to add and subtract fractions There’s a really clear handout for this method plus x and ÷ and top-heavy etc.

Your turn

Multiplying Fractions Multiplying fractions is nice and easy – you just multiply across! 3 7 x 4 9 =

Your turn

Dividing Fractions To divide fractions, you first need to flip the second fraction and multiply them. 5 8 ÷ 2 3 =

Your turn

+, -, x and ÷ top heavy fractions and mixed numbers

+

-

x

÷

Homework for next 2

Percentages Finding a percentage part without a calculator Finding a percentage part with a calculator Writing one number as a percentage of the other

Without a calculator Remember how to find the following amounts and you can then work out any percentage: 50% = divide by 2 25% = divide by 2 again 75%? 10% = divide by 10 20%? 5%? 15%? 1% = divide by 100

Without a calculator 20% of 400 = 15% of 60 = 90% of 180 =

How do you find a percentage with a calculator? What is 34% of 812?

How do you find a percentage with a calculator? What is 27% of 540?

Make some up

Writing one number as a percentage of the other First write it as a fraction and then multiply it by 100 to turn it into a percentage. Freddie scored 25 out of 40 on his Maths test. What percentage did he get right?

Example: A DVD contains a film and some extra clips. The film lasts 120 mins. The clips after the main film last an extra 30 mins. What % of the DVD is taken up with: (a) the film (b) the clips?

Your turn

Percentage Increase and Decrease

BRITISH GAS INCREASE PRICES BY 15% Mr and Mrs Jones, BILL 3 months = £150 BRITISH GAS INCREASE PRICES BY 15%

You could do something like “percentages codebreaker” here to practise increase or decrease, or the following slide or something else.

Percentage Profit/Loss 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

Going for Gold A fractions, decimals & percentages problem to solve together The times, in seconds for a 400m hurdle race are shown below. 23.56 23.561 23.605 23.585 23.405 23.6 23.495 24.761 23.706 24.716 24.005 23.995 23.805 23.5 23.509 24.003 24 23.995 23.67 23.91 • The top 15% get a gold award • The bottom two fifths do not get an award. • Twice as many get bronze awards as silver awards

Which times get a gold award? 23.405, 23.495, 23.5

Which times get a silver award? 23.509, 23.56, 23.561

Which times get a bronze award? 23.585, 23.6, 23.605, 23.67, 23.706, 23.805

So how do we convert between fractions, decimals and percentages? Fractions to decimals? Top of fraction divided by bottom Decimals to percentages? x 100 Fractions to percentages Both of the above, one after the other Decimals to fractions 1 decimal place – over 10 2 decimal places – over 100 etc Percentages to decimals ÷ 100 Percentages to fractions a fraction of 100 and cancel if you can MAKE YOUR OWN NOTES ON THE HOMEWORK SHEET

Match the Fractions and the Percentages 35% 1/50 120% 7/20 24% 4/5 2% 6/5 80% 6/25

Please can you complete the equivalencies table showing the fraction, decimal and percentage for each amount.

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

Reverse percentage practice – “scaffolded” sheet or exam questions

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

Exam practice here – or the questions on the next 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

Help is at hand Website of the week www.mathsgenie.co.uk/gcse.html

Work with general iterative processes https://allaboutmaths.aqa.org.uk/attachments/5626.pdf Start with this next week because it’s kind of revision

Homework Mr Corbett’s real life negative number questions The next 2 slides – negative numbers Bidmas matching Equivalences table 4 operations for fractions with mixed numbers Topic test – basic fractions Look at mathsgenie website

The number in the centre equals = -5 + -3 The number that goes in the top right box is 5 less than -5 The number in the bottom left is -3 x -3 The number in the top left is the number in the centre x -1 add -13 The number in the middle of the bottom row is 6 less than the number above it There is a number 6 in one of the middle row boxes There is a number equal to -4 x -5 next to the box containing -5 When you add up the numbers in the right hand column you get -27 The number below -5 is -6-10

Copy & Complete; subtract the right hand square from the left to get the square below... 5 3 -1 6 4 2 4 -9 -33