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Mathematics for Cambridge IGCSE David Rayner These internationally trusted Mathematics texts are now part of a full course complete with teacher’s resources
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Fully differentiated Student Books with brand new CD-ROMs
New Teacher’s Resource Kits for a truly complete Cambridge IGCSE Mathematics course
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Help all of your students achieve their full potential in exams. Teachers all over the world trust David Rayner’s straightforward and practice-based approach, which really helps students focus on the Mathematics and boost exam potential. Plus, new digital material means that students can now choose how they learn, with even more practice questions, revision support, and a whole suite of past exam questions. Accompanying teacher support and Revision Guide now available.
Endorsed by University of Cambridge International Examinations
Incorporating everything IGCSE Mathematics teachers say they need, these supportive kits will help you align your teaching with the most recent Cambridge IGCSE syllabus. Plus, they include customisable CD-ROMs so you can tailor your teaching resources to meet the needs of your students – whether that be ability, context or the freedom to easily add in your own questions!
Student text Fully updated and refreshed to match the most recent Cambridge IGCSE syllabus
Percentages
4. The scale of a map is 1 : 1000. What is the area, in cm2 , on the map of a lake of area 5000 m2 ? 5. The scale of a map is 1 cm to 5 km. A farm is represented by a rectangle measuring 1�5 cm by 4 cm. What is the actual area of the farm?
Questions are designed for the international classroom, with globally relevant context
6. On a map of scale 1 cm to 250 m the area of a car park is 3 cm2 . What is the actual area of the car park in hectares? (1 hectare ¼ 10 000 m2 ) 7. The area of the playing surface at the Olympic Stadium in Beijing is 3 of a hectare. What area will it occupy on a plan drawn to a scale 5 of 1 : 500?
23
24
3. Change to decimals: (a) 36% (d) 13�4%
8. On a map of scale 1 : 20 000 the area of a forest is 50 cm . On another map the area of the forest is 8 cm2 . Find the scale of the second map.
(b) 28% (e)
(c) 0�111; 11%;
1 9
36
(c) 7%
3 5
(f )
4. Arrange in order of size (smallest first): (b) 0�38; (a) 12 ; 45%; 0�6
6 ; 16
7 8
4%
(d) 32%; 0�3;
(d) 112 out of 200
(e) 2 12 out of 25
Prior knowledge introductions flag up the knowledge required for each lesson, helping you prepare your students
(f ) 7 12 out of 20
Example 1 A car costing $400 is reduced in price by 10%. Find the new price.
1.6 Percentages
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
10% of $2400 ¼
10 2400 � 100 1
New price of car ¼ $(2400 � 240) ¼ $2160
Example
Example 2
(a) Change 80% to a fraction. (b) Change 38 to a percentage. (c) Change 8% to a decimal.
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%.
80 4 ¼ 100 5 3 100 � % ¼ 37 12 % 8 1
(a) 80% ¼
;
110% of old price ¼ $286
3 (b) ¼ 8
;
1% of old price ¼ $
;
100% of old price ¼ $
(c) 8% ¼
8 ¼ 0�08 100
1. Change to fractions: (a) 60% (b) 24%
The extension worksheet for Chapter 3 may be useful to support lesson 3.
Textbook pages 92–7
Lesson commentaries provide useful guidance and structure for lessons that you can easily infuse with your own ideas
(c) 35%
(d) 2%
(b) 101 (e) 0�72
1. Calculate: (a) 30% of $50 (c) 4% of $70
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’.
●
(b) 45% of 2000 kg (d) 2�5% of 5000 people
Free Student CD-ROM
Extended Mathematics for Cambridge IGCSE
Plenary exercises accompany all lessons and add interactivity that will hold students’ interest
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. Objectives 31: Carry out calculations involving the circumference and area of a circle.
Objectives
●
(c) 78 (f ) 0�31
Lessons 3 and 4 – The circle
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.
Starter Ask students to write down the names of as many quadrilaterals as they can in 60 seconds. Pool answers as a group.
286 100 � 110 1
Exercise 22
2. Change to percentages: 1 4 1 3
Lessons 1 and 2 – Area
Old price of TV ¼ $260
Exercise 21
(a) (d)
286 110
Examples are used throughout to clearly demonstrate exactly how problems are solved, ensuring comprehension
Mensuration The supplementary worksheet for Chapter 3 may be useful to support lessons 1, 4 and 5.
31: Carry out calculations involving the area of a rectangle and triangle, the area of a parallelogram and trapezium (and kite).
¼ $240
Percentages are simply a convenient way of expressing fractions or 50 of $60, or more simply 12 of $60. decimals. ‘50% of $60’ means 100 Percentages are used very frequently in everyday life and are misunderstood by a large number of people. What are the implications if ‘inflation falls from 10% to 8%’? Does this mean prices will fall?
Teacher’s Kit – in print and digital
Mensuration
CHAPTER 3 MENSURATION
1 3
5. The following are marks obtained in various tests. Convert them to percentages. (a) 17 out of 20 (b) 31 out of 40 (c) 19 out of 80
2
Lesson plans clearly indicate when to integrate each resource into your lessons
Number
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. Plenary Ask 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.)
Exercise commentary Questions 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.
Starter Ask 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).
37
Exercise commentary Questions 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. ●
● ●
●
Exercise commentaries Develop the use of the formulae for cases where diameter is given for area and radius for circumference to help quickly ensure that students are happy to work with either. This is often a common area for mistakesyou since students just without thinking ‘plug in’ the number given. Work also with a series of examples which go backwards. separate the basics You could do this as a thought experiment. Ask questions like ‘what would happen if you were given the C circumference and asked to find the diameter?’ Try to avoid any more formulae here (d =the for example) from challenges, and get students to understand the process. helping At this point, further practice could be given to the students during a consolidation phase of theyou lesson. stretch Introduce more ‘what if ...’ situations. These could be given out on prepared worksheets and could include and support all of 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. your ability levels The second lesson provides a good opportunity for further consolidation of this work.
Plenary A 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.
Extended Mathematics for Cambridge IGCSE Teacher’s Resource Kit
These new student CD-ROMs provide everything your students need to achieve the best possible result in exams
Plus, the Teacher’s Resource Kit comes with a free CD-ROM for structured and effective lessons with quick and painless planning!
✓ Flexible approach – students can practise and revise how they learn best, using the book or the CD-ROM
✓ Repeated lesson plans from the Teacher’s book in digital form
✓ Lessons are extended and solidified with even more practice questions and conceptual support ✓ Dedicated ‘Revision Section’ ensures students have the skills they need to succeed in exams
✓ Differentiated, customisable worksheets, levelled for all abilities ✓ All the answers and past exam questions from the Student Book ✓ Clearly outlined links with the most recent Cambridge IGCSE syllabus, so you can be sure your teaching is completely up-to-date
✓ A whole suite of past Cambridge IGCSE exam papers ensures there are no surprises come exam time Extended Mathematics for Cambridge IGCSE: Student CD-ROM
Concerned about covering the complete syllabus? Use the Teacher’s CD-ROM to see exactly how the Student Book matches the syllabus, complete with page references.
A Revision Guide that will help all your students succeed in exams Endorsed 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. Exam questions prepare students for the real thing, ensuring preparedness and building confidence Graphs in practical situations
Distance-time graphs
Extended
12 Graphs in practical situations Example
The gradient on a distance-time graph represents speed.
The graph below shows the value of the Nigerian naira against the Kenyan shilling. Use the graph to find (a) how many Kenyan shillings you would get for 50 Nigerian naira (b) how many Nigerian naira you would get for 60 Kenyan shillings.
Exam question
70 60 shilling
Kenyan Shilling
60 50 40 30
0
50 naira
10
20
30
40 50 60 Nigerian naira
70
80
90
100
(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.
60
40 y 35 30 25 20 15 10 5 0
x 8
speed constant
30 20
speed constant speed decreasing speed increasing
$28
0
2 3 5 6 4 Length of hiring (hours)
0
5
10 Time (s)
15
7
x 8
Gradient on a speed-time graph represents acceleration. Area under a speed-time graph represents distance travelled. This graph shows the speed of a toy car. From the graph you can tell how the car’s speed changes.
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13/10/09 15:36:32
The next step Evaluate free for 30 days or place your order Tick to place a firm order Tick to evaluate free for 30 days
22 10 Time (minutes)
27
x 20
Speed-time graphs
3 hours
1
A 0
What happens to the speed… Between 0 s and 2 s the boy is speeding up. Between 2 s and 5 s the boy is travelling at a constant speed. Between 5 s and 7 s the boy is slowing down. Between 7 s and 10 s the boy is stationary. Between 10 s and 12 s the boy speeds up. From 12 s to 15 s onwards the boy is travelling at a constant speed. From 15 s to 17 s the boy is slowing down.
speed zero
40
10
Rayn_Ch12_Graps.indd 38
Title
7
Distance (m)
2 3 4 5 6 Length of hiring (hours)
B 24
y
50
1
C 34
This graph shows the journey of a boy cycling. From the graph you can tell how the boy’s speed changes.
Extended
40 y 35 30 25 20 15 10 5 0
Amount ($)
The 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 $19. (d) Reading off from graph gives 6 hours.
Amount ($)
Example
D Not to scale
(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. 10 = 0.833 km min−1 (to 3 sf) Speed = ___ 12 (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.
20 10
Cambridge IGCSE Mathematics Revision Guide
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?
Distance (kilometres)
Context is internationally focused, relevant to the international classroom
12
Rayn_Ch12_Graps.indd 39
Information is presented in bite-sized chunks, helping students 39 to focus 13/10/09 15:36:33
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