gcse maths - 2tier-foundation for edexcel a (collins, 2006)

This chapter will show you …● how to use basic number skills without a calculator Visual overview What you should already know ● Times tables up to 10 × 10 ● Addition and subtraction of

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This book provides indicators of the equivalent grade level of maths questions throughout The publisherswish to make clear that these grade indicators have been provided by Collins Education, and are not theresponsibility of Edexcel Ltd Whilst every effort has been made to assure their accuracy, they should beregarded as indicators, and are not binding or definitive.

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William Collins’ dream of knowledge for all began with the publication of his first book in

1819 A self-educated mill worker, he not only enriched millions of lives, but also founded

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C O N T E N T S

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Welcome to Collins GCSE Maths, the easiest way to learn and succeed inMathematics This textbook uses a stimulating approach that really appeals tostudents Here are some of the key features of the textbook, to explain why.

Each chapter of the textbook begins with an Overview TheOverview lists the Sections you will encounter in the chapter,the key ideas you will learn, and shows how these ideas relate

to, and build upon, each other The Overview also highlightswhat you should already know, and if you’re not sure, there is

a short Quick Check activity to test yourself and recap

Maths can be useful to us every day ofour lives, so look out for these Really Useful Maths!pages These double pagespreads use big, bright illustrations todepict real-life situations, and present ashort series of real-world problems foryou to practice your latest mathematicalskills on

EachSectionbegins first by explainingwhat mathematical ideas you areaiming to learn, and then lists the keywords you will meet and use The ideasare clearly explained, and this isfollowed by several examples showinghow they can be applied to realproblems Then it’s your turn to workthrough the exercises and improve yourskills Notice the different colouredpanels along the outside of the exercisepages These show the equivalent examgrade of the questions you are working

on, so you can always tell how wellyou are doing

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Working through these sections in the right way should mean you achieve your very best

in GCSE Maths Remember though, if you get stuck, answers to all the questions are atthe back of the book (except the exam question answers which your teacher has)

We do hope you enjoy using Collins GCSE Maths, and wish you every good luck inyour studies!

Brian Speed, Keith Gordon, Kevin Evans

Review the Grade Yourselfpages at thevery end of the chapter This will showwhat exam grade you are currentlyworking at Doublecheck What you should now know to confirm that youhave the knowledge you need toprogress

Every chapter in this textbook containslots of Exam Questions.These provideideal preparation for your examinations

Each exam question section alsoconcludes with a fully worked example

Compare this with your own work, andpay special attention to the examiner’scomments, which will ensure youunderstand how to score maximummarks

Throughout the textbook you will find

Puzzlesand Activities– highlighted in

the green panels – designed to

challenge your thinking and improve

your understanding

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You may use your calculator for this question

You should not use your calculator for this question

Indicates a Using and Applying Mathematics question

Indicates a Proof question

I C O N S

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This chapter will show you …

● how to use basic number skills without a calculator

Visual overview

What you should already know

● Times tables up to 10 × 10

● Addition and subtraction of numbers less than 20

● Simple multiplication and division

● How to multiply numbers by 10 and 100

4 Place value

and ordering numbers

5 Rounding

6 Adding and

subtracting numbers with

up to four digits

7 Multiplying and

dividing by single-digit numbers

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Adding with grids

1.1

Key words

addcolumn grid row

In this section you will learn how to:

● add and subtract single-digit numbers in a grid

● use row and column totals to find missing numbers

in a grid

Adding with grids

You need a set of cards marked 0 to 9

Shuffle the cards and lay them out in a 3 by 3 grid

You will have one card left over

Copy your grid onto a piece of paper Then add up each row and each column and write down their totals.

Finally, find the grand total and write it in the box at thebottom right

Look out for things that help For example:

• in the first column, 3 + 7 make 10 and 10 + 8 = 18

• in the last column, 9 + 4 = 9 + 1 + 3 = 10 + 3 = 13Reshuffle the cards, lay them out again and copy the new grid Copy the new gridagain on a fresh sheet of paper, leaving out some of the numbers

Pass this last grid to a friend to work out the missing numbers You can make it quite hard because you are using only the numbers from 0 to 9 Remember:

once a number has been used, it cannot be used again in that grid.

Example Find the numbers missing from this grid.

819

7 1 9

6 2 0

40913

5218

86721

1781742

4

92

8

721

178

42

8 5 4

37818

56213

04913

8171944

9 2 8

4 6 7

0 5 3

9 8 7 6 5 4 3 2 1 0

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Find the row and column totals for each of these grids.

Find the numbers missing from each of these grids Remember: the numbers missing from each gridmust be chosen from 0 to 9 without any repeats

73

14

6216

1691136

1

714

5815

34

16

6152435

94

14

3

25

5819

1891138

2819

513

4010

16131342

1

12

29

4315

1612

09

815

2

4 10

1536

918

76

34

16

817

17

1338

96

325

784

089

615

743

019

863

724

38

459

671

562

917

384

071

826

345

71

406

853

075

819

642

160

829

753

Clues The two numbers missing from the second column must add up to 1, so they

must be 0 and 1 The two numbers missing from the first column add to 11, so theycould be 7 and 4 or 6 and 5 Now, 6 or 5 won’t work with 0 or 1 to give 17 acrossthe top row That means it has to be:

You can use your cards to try out your ideas

74819

1203

9

17

1711

74819

1203

95317

17111139

EXERCISE 1A

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Times table check

1.2

In this section you will:

● recall and use your knowledge of times tables

Special table facts

You need a sheet of squared paper

Start by writing in the easy tables These are the 1×, 2 ×, 5 ×, 10 × and 9 × tables.Now draw up a 10 by 10 tables square

before you go any further (Time yourselfdoing this and see if you can get faster.)Once you have filled it in, shade in all the easy tables You should be left withsomething like the square on the right

Now cross out one of each pair that

have the same answer, such as

3 × 4 and 4 × 3 This leaves you with:

Now there are just 15 table facts Do learn them

The rest are easy tables, so you should know all of them But keep practising!

×

12345678910

48 56

42 4936

64

×

12345678910

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Write down the answer to each of the following without looking at the multiplication square.

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Suppose you have to work out the answer to 4 + 5 × 2 You may say the answer is 18, but the correctanswer is 14.

There is an order of operations which you must follow when working out calculations like this The × is

always done before the +.

In 4 + 5 × 2 this gives 4 + 10 = 14

Now suppose you have to work out the answer to (3 + 2) × (9 – 5) The correct answer is 20

You have probably realised that the parts in the brackets have to be done first, giving 5 × 4 = 20.

So, how do you work out a problem such as 9 ÷ 3 + 4 × 2?

To answer questions like this, you must follow the BODMASrule This tells you the sequence in which you

must do the operations.

And to work out 60 – 5 × 32+ (4 × 2):

First, work out the brackets: (4 × 2) = 8 giving 60 – 5 × 32

+ 8Then the order (power): 32= 9 giving 60 – 5 × 9 + 8

● work out the answers to a problem with a number

of different signs

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Dice with BODMAS

You need a sheet of squared paper and three dice

Draw a 5 by 5 grid and write the numbers from 1 to 25

in the spaces

The numbers can be in any order.

Now throw three dice Record the score on each one

Use these numbers to make up a number problem

You must use all three numbers, and you must not put them together to make anumber like 136 For example, with 1, 3 and 6 you could make:

1 + 3 + 6 = 10 3 × 6 + 1 = 19 (1 + 3) × 6 = 24

6 ÷ 3 + 1 = 3 6 + 3 – 1 = 8 6 ÷ (3 × 1) = 2and so on Remember to use BODMAS

You have to make only one problem with each set of numbers

When you have made a problem, cross the answer off on the grid and throw the diceagain Make up a problem with the next three numbers and cross that answer off thegrid Throw the dice again and so on

The first person to make a line of five numbers across, down or diagonally is thewinner

You must write down each problem and its answer so that they can be checked

Just put a line through each number on the grid, as you use it Do not cross it out sothat it cannot be read, otherwise your problem and its answer cannot be checked

This might be a typical game

First set (1, 3, 6) 6 × 3 × 1 = 18Second set (2, 4, 4) 4 × 4 – 2 = 14Third set (3, 5, 1) (3 – 1) × 5 = 10Fourth set (3, 3, 4) (3 + 3) × 4 = 24Fifth set (1, 2, 6) 6 × 2 – 1 = 11Sixth set (5, 4, 6) (6 + 4) ÷ 5 = 2Seventh set (4, 4, 2) 2 – (4 ÷ 4) = 1

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Work out each of these.

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Three dice are thrown They give scores of 3, 1 and 4.

A class makes the following questions with the numbers

Work them out

The ordinary counting system uses place value, which means that the value of a digit depends upon its

place in the number

In the number 5348the 5 stands for 5 thousands or 5000the 3 stands for 3 hundreds or 300the 4 stands for 4 tens or 40the 8 stands for 8 units or 8And in the number 4 073 520the 4 stands for 4 millions or 4 000 000the 73 stands for 73 thousands or 73 000the 5 stands for 5 hundreds or 500the 2 stands for 2 tens or 20You write and say this number as:

four million, seventy-three thousand, five hundred and twentyNote the use of narrow spaces between groups of three digits, starting from the right All whole and mixednumbers with five or more digits are spaced in this way

Place value and ordering numbers

1.4

Key words

digitplace value

● identify the value of any digit in a number

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Write the value of each underlined digit.

Copy each of these sentences, writing the numbers in words

a The last Olympic games in Greece had only 43 events and 200 competitors

b The last Olympic games in Britain had 136 events and 4099 competitors

c The last Olympic games in the USA had 271 events and 10 744 competitors

Write each of the following numbers in words

a 5 600 000 b 4 075 200 c 3 007 950 d 2 000 782

Write each of the following numbers in numerals or digits

a Eight million, two hundred thousand and fifty-eight

b Nine million, four hundred and six thousand, one hundred and seven

c One million, five hundred and two

d Two million, seventy-six thousand and forty

Write these numbers in order, putting the smallest first.

The correct order is:

1730 3071 3701 7031 7103 7130

EXERCISE 1D

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Copy each sentence and fill in the missing word, smaller or larger.

b Which of your numbers is the smallest?

c Which of your numbers is the largest?

Using each of the digits 0, 4 and 8 only once in each number, write as many different three-digitnumbers as you can (Do not start any number with 0.) Write your numbers down in order, smallestfirst

Write down in order of size, smallest first, all the two-digit numbers that can be made using 3, 5 and

8 (Each digit can be repeated.)

You use rounded information all the time Look atthese examples All of these statements userounded information Each actual figure is either

above or below the approximation shown here.

But if the rounding is done correctly, you can findout what the maximum and the minimum figuresreally are For example, if you know that thenumber of matches in the packet is rounded to thenearest 10,

• the smallest figure to be rounded up to 30 is

25, and

• the largest figure to be rounded down to 30 is

34 (because 35 would be rounded up to 40)

So there could actually be from 25 to 34 matches

● round a number

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What about the number of runners in the marathon? If you know that the number 23 000 is rounded tothe nearest 1000,

• The smallest figure to be rounded up to 23 000 is 22 500

• The largest figure to be rounded down to 23 000 is 23 499

So there could actually be from 22 500 to 23 499 people in the marathon

Round each of these numbers to the nearest 10

On the shelf of a sweetshop there are three jars like the ones below

Look at each of the numbers below and write down which jar it could be describing

(For example, 76 sweets could be in jar 1.)

m Which of these numbers of sweets could not be in jar 1: 74, 84, 81, 76?

n Which of these numbers of sweets could not be in jar 2: 124, 126, 120, 115?

o Which of these numbers of sweets could not be in jar 3: 194, 184, 191, 189?

80sweets (to the nearest 10)

EXERCISE 1E

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Round each of these numbers to the nearest 1000.

Which of these sentences could be true and which must be false?

a There are 789 people living in Elsecar b There are 1278 people living in Hoyland

c There are 550 people living in Jump d There are 843 people living in Elsecar

e There are 1205 people living in Hoyland f There are 650 people living in Jump

These were the numbers of spectators in the crowds at nine Premier Division games on a weekend

in May 2005

a Which match had the largest crowd?

b Which had the smallest crowd?

c Round all the numbers to the nearest 1000

d Round all the numbers to the nearest 100

Give these cooking times to the nearest 5 minutes

Aston Villa v Man City 39 645Blackburn v Fulham 18 991Chelsea v Charlton 42 065

C Palace v Southampton 26 066Everton v Newcastle 40 438Man.Utd v West Brom 67 827Middlesbrough v Tottenham 34 766Norwich v Birmingham 25 477Portsmouth v Bolton 20 188

Welcome to Elsecar

Population 800

(to the nearest 100)

Welcome to Hoyland

Population 1200

Welcome to Jump

Population 600

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There are three things to remember when you are adding two whole numbers

• The answer will always be larger than the bigger number

• Always add the units column first.

• When the total of the digits in a column is more than 9, you have to carry a digit into the next column

on the left, as shown in Example 2 It is important to write down the carried digit, otherwise you may

forget to include it in the addition.

Subtraction

These are four things to remember when you are subtracting two whole numbers

• The bigger number must always be written down first

• The answer will always be smaller than the bigger number

• Always subtract the units column first.

• When you have to take a bigger digit from a smaller digit in a column, you must first remove 10 fromthe next column on the left and put it with the smaller digit, as shown in Example 3

Adding and subtracting numbers with up to four digits

1.6

Key words

additioncolumndigitsubtract

● add and subtract numbers with more than onedigit

EXAMPLE 2

+ 1 1 7 334691

1 6 7+ 2 5

1 9 21

EXAMPLE 3

Subtract: a 874 – 215 b 300 – 163

39010– 1 6 3

1 3 7

86714– 2 1 5

6 5 9

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Copy and work out each of these additions.

9 3

□ 7+ 3 □

8 4

5 3+ 2 □

□ 9

5375– 3547

8034– 3947

8432– 4665

8043– 3626

6254– 3362

580– 364

650– 317

638– 354

602– 358

673– 187

732– 447

572– 158

954– 472

908– 345

637– 187

56293+ 197

175+ 276

438147+ 233

4676+ 3584

483+ 832

287+ 335

317416+ 235

4872+ 1509

95+ 56

365+ 348

EXERCISE 1F

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There are two things to remember when you are multiplying two whole numbers.

• The bigger number must always be written down first

• The answer will always be larger than the bigger number

8 0 7 6– □ □ □ □

6 1 8 7

□ 4 □– 5 5 8

2 □ 5

□ □ □– 2 4 7

3 0 9

4 6 2– □ □ □

1 8 5

5 4 □– □ □ 6

3 2 5

□ 1 4– 2 5 □

3 □ 7

6 7 □– □ □ 3

1 3 5

8 5– □ □

2 7

□ 7– 3 □

5 4

7 4– 2 □

□ 1

3 5 7 8+ □ □ □ □

8 0 7 6

□ 4 □+ 3 3 7

7 □ 5

□ □ □+ 3 4 8

8 0 7

4 6 9+ □ □ □

7 3 5

5 4 □+ □ □ 6

8 2 2

□ 1 8+ 2 5 □

● multiply and divide by a single-digit number

EXAMPLE 4

Multiply 231 by 4 2 1 3

8521

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Note that the first multiplication, 3 × 4, gives 12 So, you need to carry a digit into the next column on theleft, as in the case of addition.

Division

There are two things to remember when you are dividing one whole number by another whole number:

• The answer will always be smaller than the bigger number

• Division starts at the left-hand side.

This is how the division was done:

• First, divide 3 into 4 to get 1 and remainder 1 Note where to put the 1 and the remainder 1

• Then, divide 3 into 11 to get 3 and remainder 2 Note where to put the 3 and the remainder 2

• Finally, divide 3 into 27 to get 9 with no remainder, giving the answer 139

Copy and work out each of the following multiplications

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Calculate each of the following divisions.

By doing a suitable multiplication, answer each of these questions

a How many days are there in 17 weeks?

b How many hours are there in 4 days?

c Eggs are packed in boxes of 6 How many eggs are there in 24 boxes?

d Joe bought 5 boxes of matches Each box contained 42 matches How many matches did Joe buyaltogether?

e A box of Tulip Sweets holds 35 sweets How many sweets are there in 6 boxes?

By doing a suitable division, answer each of these questions

a How many weeks are there in 91 days?

b How long will it take me to save £111, if I save £3 a week?

c A rope, 215 metres long, is cut into 5 equal pieces How long is each piece?

d Granny has a bottle of 144 tablets How many days will they last if she takes 4 each day?

e I share a box of 360 sweets between 8 children How many sweets will each child get?

CHAPTER 1:BASIC NUMBER

Edex_Found Math_01.qxd 13/03/06 14:38 Page 18

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Now circle another number that is not crossed out andcross out all the other numbers in the row and columncontaining this number Repeat until you have five numbers circled Add these numbers together What do you get? Now do it again but start with a differentnumber.

Magic squares

9 16

14 11 6 3

1 8

Now try to completethis magic squareusing every numberfrom 1 to 16

6 7 2

1 5 9

8 3 4

This is a magic square

Add the numbers in anyrow, column or diagonal

The answer is always 15.

N I N E

T WO+ T WO

F O U R

O N E+ O N E

T W O

Hints Letter sets

Think about numbers

Valued letters a

e other answers to each sum

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Fiona has four cards Each card has a number written

c Write the number that should be on the fifth card

Edexcel, Question 3, Paper 2 Foundation, June 2004

a Write the number seventeen thousand, two hundredand fifty-two in figures

b Write the number 5367 correct to the nearest hundred.

c Write down the value of the 4 in the number 274 863

The number of people in a London Tube Station onemorning was 29 765

a Write the number 29 765 in words

b In the number 29 765, write down the value of

i the figure 7

ii the figure 9

c Write 29 765 to the nearest 100

a i Write down the number fifty-four thousand andseventy-three in figures

ii Write down fifty-four thousand and seventy-three

to the nearest hundred

b i Write down 21 809 in words

ii Write down 21 809 to the nearest 1000

Look at the numbers in the cloud

a Write down the number from the cloud which is

i twenty eight million

ii two thousand eight hundred

b What number should go in the boxes to make the

calculation correct?

ii 2 800 000 ÷ = 280 000Murray and Harry both worked out 2 + 4 × 7

Murray calculated this to be 42

Harry worked this out to be 30

Explain why they both got different answers

The table below shows information about theattendance at two football grounds

a The total home attendance for Manchester United

in the 2004–05 season was 1 287 212 Write thenumber 1 287 212 in words

b The total home attendance for Chelsea in the

2004–05 season was 795 397 Write the number

795 397 to the nearest hundred

54 327 people watched a concert

a Write 54 327 to the nearest thousand

b Write down the value of the 5 in the number 54 327.

Work out the following Be careful as they are amixture of addition, subtraction, multiplication anddivision problems Decide what the calculation is anduse a column method to work it out

a How much change do I get from a £20 note if Ispend £13.45?

b I buy three pairs of socks at £2.46 each How

much do I pay altogether?

c Trays of pansies contain 12 plants each How manyplants will I get in 8 trays?

d There are 192 pupils in year 7 They are in 6 forms.

How many pupils are in each form?

e A burger costs £1.65, fries cost 98p and a drink is68p How much will a burger, fries and a drink costaltogether?

f A school term consists of 42 days If a normal school

week is 5 days, i how many full weeks will there be

in the term? ii How many odd days will there be?

g A machine produces 120 bolts every minute

i How many bolts will be produced by themachine in 9 minutes?

ii The bolts are packed in bags of 8 How manybags will it take to pack 120 bolts?

28 28 000 280

2 800 000 2800

280 000 28 000 000

5 1

9 4

Team Total home attendance

in 2004–05 season

Manchester United 1 287 212

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The 2004 population of Plaistow is given as 7800 tothe nearest thousand.

a What is the lowest number that the populationcould be?

b What is the largest number that the population

could be?

a There are 7 days in a week

i How many days are there in 15 weeks?

ii How many weeks are there in 161 days?

b Bulbs are sold in packs of 6.

i How many bulbs are there in 12 packs?

ii How many packs make up 186 bulbs?

A teacher asked her pupils to work out the followingcalculation without a calculator

2 × 32+ 6

a Alice got an answer of 42 Billy got an answer of

30 Chas got an answer of 24 Explain why Chaswas correct

b Put brackets into these calculations to make them true

i 2 × 32

+ 6 = 42

ii 2 × 32 + 6 = 30The following are two pupils’ attempts at working out

3 + 52– 2Adam 3 + 52– 2 = 3 + 10 – 2 = 13 – 2 = 11Bekki 3 + 52– 2 = 82– 2 = 64 – 2 = 62

a Each pupil has made one mistake Explain whatthis is for each of them

b Work out the correct answer to 3 + 52– 2

WORKED EXAM QUESTION

Solution

a i 8763

ii 3678 iii 38 ×× 2 = 76

b i 3650

ii 3700

Start with the smallest number as the thousands digits, use the next smallest as the hundreds digit and so on Note the answer is the reverse of the answer to part (i).

Here are four number cards, showing the number 6387.

a Using all four cards, write down:

i the largest possible number

ii the smallest possible number iii the missing numbers from this problem.

3 6

3648 rounds down to 3600 Do not be tempted to round the answer to part (i) up to 3700.

There are three numbers left, 3, 7, 6 The 3 must go into the first box and then you can work out that 2 × 38 is 76.

A halfway value such as 48 rounds up to 50.

Start with the largest number as the thousands digits, use the next largest as the hundreds digit and so on.

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Quad bikes:

adults £21, children (6–15) £12.50

Paragliding:

adults £99

Windsurfing:

half day £59,full day £79

The table shows which activities they all chose Copy it and complete the "Totals" row Use it to work out the total cost of the activities.

Horse riding Water-jet Conventional Kayaking Coast jumping Windsurfing

boats boats

Alice Davies 11⁄ 2 hour beach ride ✓ ✓ 1 ⁄ 2 day ✗ 1 ⁄ 2 day

Totals

Horse riding:

11⁄2hour valley ride £28,

11⁄2hour beach ride £32

Water-jet boat:

adults £20, children (under 14) £10

Mr and Mrs Davies, their daughter,Alice (aged 15), and their son, Joe(aged 13), decide to take anactivity holiday The family want tostay in a cottage in Wales

The activities on offer are shownbelow

Mr Davies works out the total cost

of their holiday, which includes thecost of the activities, the rental forthe holiday cottage (in the highseason) and the cost of the petrolthey will use to travel to thecottage, for trips while they arethere, and to get home again

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Coast jumping:

Holiday cottage:

low season: £300mid season: £400high season: £550

Kayaking:

half day £29, full day £49

Diving:

adults £38

Conventional boat:

Diving Quad bikes Paragliding

Total:

Basic number

It is 250 miles from their home totheir holiday cottage They drive anextra 100 miles while they are onholiday Their car travels, onaverage, 50

miles to thegallon Petrolcosts £1 perlitre 1 gallon = 4.5 litres

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GRADE YOURSELF

Able to add columns and rows in gridsKnow the times tables up to 10 × 10Can use BODMAS to find the correct order of operationsCan identify the value of digits in different places

Able to round to the nearest 10 and 100Can add and subtract numbers with up to four digits Can multiply numbers by a single-digit numberAble to answer problems involving multiplication or division by

a single-digit number

What you should know now

● How to use BODMAS

● How to put numbers in order

● How to round to the nearest ten, hundred, thousand

● How to solve simple problems, using the four operations of arithmetic:addition, subtraction, multiplication and division

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This chapter will show you …

● how to add, subtract, multiply and order simple fractions

● how to cancel fractions

● how to convert a top-heavy fraction to a mixed number (and vice versa)

● how to calculate a fraction of a quantity

● how to calculate a reciprocal

● how to recognise a terminating and a recurring decimal fraction

Visual overview

What you should already know

● Times tables up to 10 × 10 ● What a fraction is

Reminder

A fraction is a part of a whole The top number is called

the numerator The bottom number is called the

denominator So, for example, –43means you divide awhole thing into four portions and take three of them

It really does help if you know the times tables up to

10 × 10 They will be tested in the non-calculator paper,

so you need to be confident about tables and numbers

13half of 16 14 half of 8 15 half of 20 16 a third of 9

17a third of 15 18 a quarter of 12 19 a fifth of 10 20 a fifth of 20

Equivalence Cancelling Fraction of a quantity Quantity as a fraction Fraction

Adding Subtracting Multiplication

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What fraction is shaded in each of these diagrams?

Draw diagrams as in question 1 to show these fractions.

27

45

110

19

89

16

58

15

23

34

● recognise what fraction of a shape has beenshaded

● shade a given simple fraction of a shape

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Fractions that have the same denominator (bottom number) can easily be added or subtracted

511

711

38

58

16

46

310

910

59

89

17

47

25

35

29

79

27

57

16

56

13

23

510

810

48

78

15

45

14

34

511

211

19

49

36

26

15

35

17

47

310

210

28

58

15

35

46

16

59

29

37

27

13

510

310

15

25

38

18

24

14

58

28

78

710

410

310

simple fractions

2.2

Key words

denominatornumerator

● add and subtract two fractions with the samedenominator

Just add or subtract the

numerators (top numbers).

The bottom number stays thesame

EXERCISE 2B

Trang 34

a Draw a diagram to show

b Show on your diagram that =

c Use the above information to write down the answers to these

a Draw a diagram to show

b Show on your diagram that =

c Use the above information to write down the answers to these

10

12

310

12

110

12

510

12

34

12

14

12

24

Recognise equivalent fractions

2.3

Key word

equivalent

● recognise equivalent fractions

Making eighths

You need lots of squared paper and a pair of scissors

Draw three rectangles, each 4 cm by 2 cm, on squared paper

Each small square is called an eighth or –18

Cut one of the rectangles into halves, another into quarters and the third into eighths

You can see that the strip equal to one half takes up 4 squares, so:

=

These are called equivalent fractions.

48

12

1

4 14 14 14 1

1 2 2

1 8 1 8

Trang 35

1 Use the strips to write down the following fractions as eighths.

You need lots of squared paper and a pair of scissors

Draw four rectangles, each 6 cm by 4 cm,

on squared paper

Each small square is called a twenty-fourth or 24––1.Cut one of the rectangles into quarters, another into sixths, another into thirds and the remaining one into eighths

You can see that the strip equal to a quarter takes up 6 squares, so:

= This is another example of equivalent fractions

This idea is used to add fractions together

For example:

+ can be changed into:

24

424

624

16

14

624

14

1 4 1 4 1 4 1 4

12

34

38

14

18

12

14

12

38

18

34

38

14

34

14

1 8 1 8 1 8 1 8

Trang 36

Use the strips to write down each of these fractions as twenty-fourths.

15

34

14

310

110

35

15

14

310

710

25

35

12

110

45

34

15

14

58

13

16

58

34

16

38

12

56

18

13

58

18

23

18

16

14

18

13

12

78

58

38

34

56

23

18

13

16

EXERCISE 2C

Trang 37

Equivalent fractions are two or more fractions that represent the same part of a whole.

The basic fraction, –34in Example 1, is in its lowest terms This means that there is no number that is a

factor of both the numerator and the denominator

cancelling

2.4

Key words

denominatorlowest termsnumerator

● create equivalent fractions

● cancel fractions, where possible

EXAMPLE 1

Complete these statements

a Multiplying the numerator by 4 gives 12 This means 12––16is an equivalent fraction to –34

b To change the denominator from 5 to 15, you multiply by 3 Do the same thing to the

numerator, which gives 2 × 3 = 6 So, 2

–5= ––156

□15

25

□16

× 4

34

EXAMPLE 2

Cancel these fractions to their lowest terms

a Here is one reason why you need to know the times tables What is the biggest number

that has both 15 and 35 in its times table? You should know that this is the five timestable So, divide both top and bottom numbers by 5

= =

You can say that you have ‘cancelled by fives’

b The biggest number that has both 24 and 54 in its times table is 6 So, divide both the

numerator and denominator by 6

= =

Here, you have ‘cancelled by sixes’

49

24 ÷ 6

54 ÷ 6

2454

37

15 ÷ 5

35 ÷ 5

1535

245415

35

Trang 38

Copy and complete each of these statements.

12

□

□25

□20

18

□

□20

□16

6

□

□15

□12

6

□

□10

□8

□20

×□

710

□40

×□

58

□12

×□

34

□18

×□

23

□20

×□

35

□9

×□

13

□20

×□

× 2

310

□28

× 4

37

□18

× 3

56

□15

× 3

45

□40

× 5

38

□12

× 3

14

□20

× 4

25

This shows that = , = and =

In order, the fractions are:

, , 56

34

23

912

34

812

23

1012

56

9

12

68

34

8

12

69

46

23

10

12

56

34

2356

Trang 39

Copy and complete each of these statements.

A fraction such as 9–5is called top-heavy because the numerator (top number) is bigger than the

denominator (bottom number) You may also see a top-heavy fraction called an improper fraction.

A fraction that is not top-heavy, such as –45, is sometimes called a proper fraction The numerator of a

proper fraction is smaller than its denominator

310

25

13

56

710

45

34

910

14

13

16

712

34

23

12

25

710

58

12

34

23

56

12

4212

2114

3648

1827

69

1812

50200

4260

721

1025

820

1421

2535

1521

1215

416

1020

2835

1416

510

39

68

1218

515

46

Top-heavy fractions and mixed numbers

2.5

Key word

mixednumberproperfractiontop-heavy

● change top-heavy fractions into mixed numbers

● change a mixed number into a top-heavy fraction

Make all denominators thesame, e.g –12, –65, –23, is –36, –65, 4–6

Trang 40

Converting top-heavy fractions

You need a calculator with a fraction key, which will look like this ƒ

Your calculator probably shows fractions like this orThis means –23or two-thirds

Key the top-heavy fraction –95into your calculator 9 ƒ 5

The display will look like this

Now press the equals key = The display will change to:

This is the mixed number 14–5.(It is called a mixed number because it is a mixture of a whole number and a properfraction.)

Write down the result: = 1Key the top-heavy fraction –84into your calculator 8 ƒ 4

Now press the equals key = The display will change to:

This represents the whole number 2 Whole numbers are special fractions with adenominator of 1 So, 2 is the fraction –21

Write down the result: =

• Now key at least ten top-heavy fractions and convert them to mixed numbers.Keep the numbers sensible For example, don’t use 37 or 17

• Write down your results

• Look at your results Can you see a way of converting a top-heavy fraction to amixed number without using a calculator?

• Test your idea Then use your calculator to check it

Converting mixed numbers

Key the mixed number 2–34into your calculator 2 ƒ 3 ƒ 4

Now press the shift (or I) key and then press the fraction key ƒ

The display will change to: 1 1 4

4 3 2

21

84

2

4 8

45

95

5 4 1

5 9

3 2 3

2

Tiêu đề	GCSE Maths - 2Tier-Foundation for Edexcel A
Tác giả	Brian Speed, Keith Gordon, Kevin Evans
Trường học	Collins Education
Chuyên ngành	Maths
Thể loại	Textbook
Năm xuất bản	2006
Thành phố	London

Định dạng
Số trang	606
Dung lượng	14,86 MB