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Extended Mathematics for Cambridge IGCSE

Student text

4. The scale of a map is 1 : 1000. What is the area, in cm2, on the mapof a lake of area 5000m2?

5. The scale of a map is 1 cm to 5 km. A farm is represented by arectangle measuring 1�5 cm by 4 cm. What is the actual area of the farm?

6. On a map of scale 1 cm to 250m the area of a car park is 3 cm2.What is the actual area of the car park in hectares?(1 hectare¼ 10 000m2)

7. The area of the playing surface at the Olympic Stadium in Beijing is35of a hectare. What area will it occupy on a plan drawn to a scale

of 1 : 500?

8. On a map of scale 1 : 20 000 the area of a forest is 50 cm2. Onanother map the area of the forest is 8 cm2. Find the scale of thesecond map.

1.6 Percentages

Percentages are simply a convenient way of expressing fractions ordecimals. ‘50% of $60’ means 50

100of $60, or more simply 1

2of $60.

Percentages are used very frequently in everyday life and aremisunderstood by a large number of people. What are the implicationsif ‘inflation falls from 10% to 8%’? Does this mean prices will fall?

Example

(a) Change 80% to a fraction.(b) Change 3

8to a percentage.

(c) Change 8% to a decimal.

(a) 80% ¼ 80

100¼ 4

5

(b)3

8¼ 3

8� 100

1

� �% ¼ 37 1

2%

(c) 8% ¼ 8

100¼ 0�08

Exercise 21

1. Change to fractions:

(a) 60% (b) 24% (c) 35% (d) 2%

2. Change to percentages:

(a) 14

(b) 110

(c) 78

(d) 13

(e) 0�72 (f ) 0�31

Percentages 23

3. Change to decimals:

(a) 36% (b) 28% (c) 7%

(d) 13�4% (e) 35

(f ) 78

4. Arrange in order of size (smallest first):

(a) 12; 45%; 0�6 (b) 0�38; 6

16; 4%

(c) 0�111; 11%; 19

(d) 32%; 0�3; 13

5. The following are marks obtained in various tests. Convert them topercentages.

(a) 17 out of 20 (b) 31 out of 40 (c) 19 out of 80

(d) 112 out of 200 (e) 2 12out of 25 (f ) 71

2out of 20

Example 1

A car costing $400 is reduced in price by 10%. Find the new price.

10% of $2400 ¼ 10

100� 2400

1

¼ $240

New price of car ¼ $(2400� 240)

¼ $2160

Example 2

After a price increase of 10% a television set costs $286.What was the price before the increase?

The price before the increase is 100%.

; 110% of old price ¼ $286

; 1% of old price ¼ $286

110

; 100% of old price ¼ $286

110� 100

1

Old price of TV ¼ $260

Exercise 22

1. Calculate:(a) 30% of $50 (b) 45% of 2000 kg(c) 4% of $70 (d) 2�5% of 5000 people

24 Number

Fully updated and refreshed to match the most recent Cambridge IGCSE syllabus

Questions are designed for the international classroom, with globally relevant context

Focused approach helps students absorb important theories without distraction

Lots and lots of graduated practice questions reinforce each concept, making sure that all variations are covered and understood

Examples are used throughout to clearly demonstrate exactly how problems are solved, ensuring comprehension

Free Student CD-ROM

Extended Mathematics for Cambridge IGCSE Teacher’s Resource Kit

Teacher’s Kit – in print and digital36 Mensuration

Exercise commentaryQuestions 1 to 4 and questions 6 and 7 in exercise 1 are basic examples whereas questions 5, 8 and 9 are more involved. More able students will naturally need less routine practice. Questions 10 to 14 test the students’ ability to work backwards while questions 15 to 20 test problem-solving. These could be used as extension questions.

Exercise 2 looks at triangles and parallelograms. Questions 13 to 17 are routine practice (diagrams given) while many of the other questions require an accurate sketch and the use of trigonometry. Depending on the ability of the students and their trigonometric knowledge, these could be used sparingly. The exercise could be supplemented to provide further (simple) examples for less able students. From question 24 onwards, students are problem-solving and working backwards to find missing dimensions. These questions could be used as extension questions for more able students or simply avoided.

CHAPTER 3 MENSURATION

Lessons 1 and 2 – Area

Textbook pages 92–7

Expected prior knowledge At this level, students should be familiar with the formulae for the area of simple geometrical figures. These lessons provide an excellent opportunity to revise these formulae and practice both problem-solving questions and ones where they have to work backwards.

Objectives31: Carry out calculations involving the area of a rectangle and triangle, the area of a parallelogram and trapezium (and kite).

StarterAsk students to write down the names of as many quadrilaterals as they can in 60 seconds. Pool answers as a group.

Lesson commentary● A fast paced question and answer session will test students’

recall of the formulae for the area of a rectangle and a triangle. Similarly, they should also be able to work out the areas of simple parallelograms and trapeziums. Although trapeziums may need further revision since the formula is less familiar. Encourage students to work together and check that students are happy when the orientation of the shape is ‘non-standard’.

● Introduce ‘reverse’ examples where missing dimensions need to be found and invite students to discuss the methods they might use. Rather than modelling a standard approach, encourage students to develop their own methods, the key objective being to get the correct answer.

● The second lesson will provide students with an opportunity to further consolidate this work and more able students will benefit from working on challenging questions.

PlenaryAsk students to create a compound shape and mark sufficient lengths to enable another student to work out the area of it. Students could then swap shapes and solve each other’s problems before discussing the answers at the end. Check students have correctly solved their problems and discuss any issues (not enough information, etc.)

The supplementary worksheet for Chapter 3 may be useful to support lessons 1, 4 and 5.

The extension worksheet for Chapter 3 may be useful to support lesson 3.

Mensuration 37

Lessons 3 and 4 – The circle

Textbook pages 97–101

Expected prior knowledge Students should have met the formulae for the area and circumference of a circle before and therefore these lessons provide an ideal opportunity to revise these and work on more difficult questions.

Objectives31: Carry out calculations involving the circumference and area of a circle.

StarterAsk students to write down as many parts of the circle (radius, circumference, sector, diameter, arc, etc.) that they can in one minute. They can then compare answers and you can have a short discussion as to what they have written down (making sure students can define them).

PlenaryA fast paced quiz which not only tests going forwards with the formulae but also includes examples which need to be worked backwards. Include a semicircle or quadrant. Students could be given 30 seconds to work out the answers on their calculators for each one.

Exercise commentaryQuestions 1 and 2 in exercise 3 are routine practice which may need to be supplemented, depending on the ability of the group. From question 3 onwards, students are working with semicircles, quadrants and compound shapes. Ensure that they are setting out their working clearly in order to communicate their solutions properly.

Questions 1 to 4 in exercise 4 are examples of working backwards while questions 5 and 6 require students to go from area to circumference or vice versa. From question 7 onwards, the questions test the students’ ability to solve problems involving the circumference and area of circles and could be used more extensively for able groups and for extension work.

Lesson commentary● Students’ familiarity with the formulae for the area and circumference of a circle could be given the diameter

or radius of a circle and then be asked to quickly work out the circumference or area using the standard formulae and write down the answers. Avoid using radius for circumference and diameter for area at this stage.

● Develop the use of the formulae for cases where diameter is given for area and radius for circumference to ensure that students are happy to work with either. This is often a common area for mistakes since students just without thinking ‘plug in’ the number given. Work also with a series of examples which go backwards. You could do this as a thought experiment. Ask questions like ‘what would happen if you were given the

circumference and asked to find the diameter?’ Try to avoid any more formulae here (d = C for example)

and get students to understand the process.

● At this point, further practice could be given to the students during a consolidation phase of the lesson.

● Introduce more ‘what if ...’ situations. These could be given out on prepared worksheets and could include area of semicircle, perimeter of quadrant, etc. and students could be given two or three of these (in pairs or groups if appropriate) and asked to justify their methods.

● The second lesson provides a good opportunity for further consolidation of this work.

Prior knowledge introductions flag up the knowledge required for each lesson, helping you prepare your students

Lesson commentaries provide useful guidance and structure for lessons that you can easily infuse with your own ideas

Plenary exercises accompany all lessons and add interactivity that will hold students’ interest

Lesson plans clearly indicate when to integrate each resource

into your lessons

Exercise commentaries

help you quickly separate the basics from the challenges, helping you stretch and support all of your ability levels

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A Revision Guide that will help all your students succeed in examsEndorsed by University of Cambridge International Examinations

David Rayner and Paul Williams take a clear, graduated approach that focuses on explanations, to ensure students completely grasp important concepts. Suitable for all ability levels, it guides students through the complete Cambridge IGCSE syllabus and provides a bank of questions to help students put their learning into practice.

Cambridge IGCSE Mathematics

Revision Guide

ExampleThe graph below shows the value of the Nigerian naira against the Kenyan shilling. Use the graph to fi nd (a) how many Kenyan shillings you would get for 50 Nigerian naira(b) how many Nigerian naira you would get for 60 Kenyan shillings.

30

10

20

60

50

40

10 40 60 80 1000 20 30 50 70 90

70

Ken

yan

Shill

ing

Nigerian naira

60 shilling

50 naira

3035

105

2025

15

40

1 4 6 80 2 3 5 7

y

x

Length of hiring (hours)

Am

ount

($)

3035

105

2025

15

40

1 4Length of hiring (hours)

Am

ount

($)

6 80 2 3 5 7

y

x

3 hours

$28

ExampleThe graph shows the amount that a shop charges for hiring a bike for up to 8 hours in a day.There is an initial charge and then an hourly charge.(a) What is the initial charge?(b) What is the hourly charge?(c) How much would it cost to hire a bike for 3 hours?(d) How many hours‘ hire would cost $28?

(a) The initial charge is the cost when the time is zero. This is $10.

(b) From the graph you can see that 1 hour costs $13.So the hourly charge is $3.

(c) Reading off from graph gives $19.(d) Reading off from graph gives 6 hours.

(a) Draw a vertical line up from 50 naira to the graph and then a horizontal line across to the vertical axis: 50 naira is about 33 shillings.

(b) Draw a horizontal line across from 60 shillings to the graph and then a vertical line down to the horizontal axis: 60 shillings is about 92 naira.

38

12 Graphs in practical situations

Rayn_Ch12_Graps.indd 38 13/10/09 15:36:32

Graphs in practical situations 12

39

Distance-time graphs

The gradient on a distance-time graph represents speed.

Exam question CIE 0580 June '08 Paper 1 Q22

The diagram shows the graph of Rachel’s journey on a motorway.Starting at A, she drove 24 kilometres to B at a constant speed.Between B and C she had to drive slowly through road works.At C she drove a further distance to D at her original speed.

(a) For how many minutes did she drive through the road works?(b) At what speed did she drive through the road works?

Give your answer in: (i) kilometres/minute(ii) kilometres/hour

(c) What is the total distance from A to D?

(a) She came to road works after 10 minutes and left them after 22 minutes. So she spent 12 minutes in the road works.

(b) (i) In 12 minutes her distance increased from 24 km to 34 km. So the road works were 10 km long.

Speed = 10 ___ 12

= 0.833 km min−1 (to 3 sf)

(ii) 0.833 … km min−1 = 0.833 … × 60 km h−1 = 50 km h−1 (c) Between A and B she travelled 24 km in 10 min. Between C and D she travelled 12 km in 5 min (since she was travelling at

the same speed as from A to B). Distance to C is 34 km so total distance from A to D is 46 km.

Dis

tanc

e (k

ilom

etre

s)

D

C 34

B 24

A 00 10

Time (minutes)

Not toscale

22 27

Speed-time graphs

Dis

tanc

e (m

)

15 20105x

10

0

20

30

40

50

60y

Time (s)

speed zero

speed constant

speed constant

speed increasing

speeddecreasing

What happens to the speed…Between 0 s and 2 s the boy is speeding up.Between 2 s and 5 s the boy is travelling at a constant speed.Between 5 s and 7 s the boy is slowing down.Between 7 s and 10 s the boy is stationary.Between 10 s and 12 s the boy speeds up.From 12 s to 15 s onwards the boy is travelling at a constant speed.From 15 s to 17 s the boy is slowing down.

Gradient on a speed-time graph represents acceleration.Area under a speed-time graph represents distance travelled.

This graph shows the speed of a toy car. From the graph you can tell how the car’s speed changes.

This graph shows the journey of a boy cycling. From the graph you can tell how the boy’s speed changes.

Exte

nded

Exte

nded

Rayn_Ch12_Graps.indd 39 13/10/09 15:36:33

Context is internationally focused, relevant to the international classroom

Exam questions prepare students for the real thing, ensuring preparedness and

building confidence

Information is presented in

bite-sized chunks, helping students

to focus