Igcse Maths 0580 Extended 2 Years

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Introduction

For many years, schools have been using Cambridge GCE curriculum as a preparation for students to further their studies. Now, as an approach to better

the education system in Brunei, a new curriculum called Cambridge IGCSE is introduced. Cambridge IGCSE is an international curriculum and is widely

recognised by higher education institutions and employers throughout the world. It enables students to gain skills in creative thinking, enquiry and problem

solving, and gives them excellent preparation for the next stage in their education.

Cambridge IGCSE uses a tiered approach so as to offer a diversity of routes for students of different abilities. Students will follow either a Core or an

Extended curriculum, depending on their examination performance. However, they can change level during the course according to their progress. Grading is on

an eight-point scale (A*-G) and grades A to E are equivalent to O level grades A to E. In some countries, IGCSE qualifications will satisfy the entry requirements

for university. In others, they are widely used as a preparation for A level and AS. Core curriculum students are eligible for grades C to G. Extended curriculum

students are eligible for grades A* to E.

Cambridge IGCSE offer a variety of Mathematics syllabus (syllabus with or without coursework) and Cambridge IGCSE Mathematics 0850 (without

coursework) has been chosen to be offered in schools in Brunei. Hence, students are assessed by written papers only.

This scheme of work is prepared for students who will follow the extended curriculum only. There are two sets of schemes of work. One set is to be

completed in 2 years and the other set in 3 years. Students who follow the 2 years scheme of work will sit for their exam in the year 2011. This scheme of work is

for those students taking 2 years course. The content is the same with the 3 years course but the time frame is different. This students have covered most of the

IGCSE syllabus in their lower secondary. The topics which are new to them are: Compound Interest, Functions, Locus, Vectors and Probability. In Statistics, they

have not studied Scatter diagrams and the meaning of positive, negative and zero correlation. Enlargement, Shear and Stretch are also included in the syllabus.This Scheme of Work focuses on enhancing their previous knowledge as well as introducing new topics. The suggested activities for teachers and students will

make their teaching and learning more related to real life situation. The suggested websites enable the teachers to get extra exposure besides the textbooks and

reference books.


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IGCSE MATHEMATICS 0580 (EXTENDED 2 YEARS)

SCHEME OF WORK FOR YEAR 9 (2010)

SUGGESTED NO. OFWEEKS

TOPICS/SUB-TOPICS OBJECTIVES SUGGESTED ACTIVITIES RESOURCES

3 1. NUMBERS

1.1 Number Facts

Identify and use natural numbers, integers(positive, negative and zero), prime

numbers, square numbers, common factors

and common multiples.

Identify and use rational and irrational

numbers, real numbers.

Revise positive and negative numbersusing a number line.

Define the terms factor and multiple and

use simple examples to find commonfactors and common multiples of two or

more numbers. Find highest common

factors and lowest common multiples.

Class activity: Identify a number from a

description of its properties, for example,

which number less than 50 has 3 and 5 as

factors and is a multiple of 9? Students

make up their own descriptions and test

one another.

Define the terms real, rational and

irrational numbers. Show that any

recurring decimal can be written as a

fraction. Show that any root which cannotbe simplified to an integer or a fraction is

an irrational number.

Investigation about prime numbers athttp://www.atm.org.uk/links/keystage

links.html

Information about rational and

irrational numbers at

http://nrich.maths.org/public/leg.php

1.2 Squares, Cubes and Roots Calculate squares, square roots and cubes

and cube roots of numbers. Use simple examples to illustrate squares,

square roots and cubes and cube roots of

numbers.Class activity: 121 is a palindromic

square number (when the digits arereversed it is the same number). Write

down all the palindromic square numbers

less than 1000.

1.3 Vulgar and Decimal

Fractions and Percentages Use the language and notation of simple

vulgar and decimal fractions and

percentages in appropriate contexts.

Recognise equivalence and convert between

Revise long multiplication, short and long

division, and the order of operations

(including the use of brackets). Use

examples which illustrate the rules for

multiplying and dividing by negative

Writing decimals as fractions at

http://www.ex.ac.uk/cimt/resource/de

cimals.htm


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these forms. numbers.

Class activity: Use four 4s and the four

rules for calculations to obtain all the

whole numbers from 1 to 20.

1.4 Directed Numbers Use directed numbers in practical situations. Use a number line to aid addition and

subtraction of positive and negative

numbers. Illustrate by using practical

examples, e.g. temperature change and

flood levels.

Weather statistics for over 16000

cities at

http://www.weatherbase.com/

1.5 Ordering Order quantities by magnitude and

demonstrate familiarity with the symbols =,

, >,


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Obtain appropriate upper and lower bounds

to solutions of simple problems (e.g. the

calculation of the perimeter or the area of arectangle) given data to a specifiedaccuracy.

Show how this information can be written

using inequality signs e.g.

2.95cm l< 3.05cm.

Class activity: Investigate upper andlower bounds for quantities calculated

from given formulae by specifying theaccuracy of the input data.

Extend the work on accuracy to include

calculating upper and lower bounds for

various perimeters and areas, givenlengths to a specified accuracy.

1.9 Standard Form Use the standard form A x 10 n where n is a

positive or negative integer, and 1A < 10. Use a range of examples to show how to

write numbers in standard form and vice-versa. Interpret how a calculator displays

standard form.Class activity: Use the four rules of

calculation with numbers in standard

form.

1.10 Ratio, Proportions and Rate

1.10.1 Ratio

1.10.2 Direct and Inverse

Proportions

1.10.3 Rate

1.10.4 Money

1.10.5Maps and Scales

1.10.6 Speed, Distance and Time

Demonstrate an understanding of the

elementary ideas and notation of ratio.

Divide a quantity in a given ratio.

Increase and decrease a quantity by a given

ratio.

Demonstrate an understanding of the

elementary ideas and notation of direct andinverse proportion.

Define the term ratio and use examples to

illustrate how a quantity can be divided

into a number of unequal parts.

Write a ratio in an equivalent form e.g. 6:8

can be written as 3:4, leading to the form

1:n .

Use straightforward examples to illustrate

how a quantity can be increased or

decreased in a given ratio, e.g. enlarging a

photograph. The idea of similar shapes can

be introduced here.Class activity: Investigate the ratio of the

length of one side of an A5 sheet of paperto that of the corresponding side of an A4

sheet of paper.

Solve problems involving direct

proportion by either the ratio method or

the unitary method.

Exchange rates can be found at

http://cnnfn.cnn.com/markets/currenci

es/


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Express direct and inverse variation in

algebraic terms and use this form of

expression to find unknown quantities.

Demonstrate an understanding of common

measures of rate.

Calculate using money and convert from

one currency to another.

Use current units of mass, length, area,volume, and capacity in practical situations

and express quantities in terms of larger orsmaller units.

Use scales in practical situations.

Calculate average speed.

Draw a graph to determine whether two

quantities (y and x ory and x2, etc.) are in

proportion.

Solve problems involving direct or inverse

proportion using the notationyxy =

kx and y 1/x y = k/x, where k is a

constant.

Solve straightforward problems involving

exchange rates. Up-to-date informationfrom a daily newspaper is useful.

Solve straightforward problems using

compound measures e.g. problems

involving rate of flow.

Use practical examples to illustrate how toconvert between: millimetres, centimetres,

metres and kilometres; grams, kilograms

and tonnes; millilitres, centilitres and

li tres. Use standard form where

appropriate.

Introduce the formula relating speed,

distance and time. Solve simple numerical

problems (which should involve

converting between units e.g. find speed in

m/s given distance in kilometres and time

in hours).

1.11 Time Calculate times in terms of the 24-hour and

12-hour clock

Read clocks, dials and timetables

Revise units for measuring time and use

examples to convert between hours,

minutes and seconds.

Use television schedules and bus/traintimetables to aid calculation of lengths of

time in both 12-hour and 24-hour clock

formats.

Class activity: Create a timetable for a

bus/train running on a single track line

between two local towns.

Work with world time differences.Class activity: Research and annotate a

Case study: scheduling aircraft at

http://www.ex.ac.uk/cimt/resource/sc

hedair.pdf

Time zone information athttp://www.ex.ac.uk/cimt/resource/ti


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world map with times in various cities

assuming it is noon where you live.

mezone.htm

1.12 Percentages Calculate a given percentage of a quantity.

Express one quantity as a percentage of

another.

Calculate percentage increase or decrease.

Solve simple problems involving

percentages, interpreting a calculator

display in calculations with money.

1.13 Personal and HouseholdFinance

1.13.1 Simple and Compound

Interest

1.13.2 Discount

1.13.3 Profit and Loss

Use given data to solve problems onpersonal and household finance involving

earnings, simple interest, compound

interest, discount, profit and loss.

Extract data from tables and charts.

Carry out calculations involving reverse

percentages, e.g. finding the cost price

given the selling price and the percentageprofit.

Solve simple problems using practicalexamples where possible, taking

information from published tables or

advertisements. (It is worth introducing arange of simple words and concepts here

to describe different aspects of finance,e.g. tax, percentage profit, deposit, loan.)

Use the formula I = PRT to solve a variety

of problems involving simple interest.Class activity: Research the cost of

borrowing money from different banks (ormoney lenders).

Revise: Work covered on percentages in

Topic 1.12.

Use simple examples to show how to

calculate the original value of something

before a percentage increase or decreasetook place.

Information about interest rates can be found from most banks. They

usually have their own web site in the

formathttp://www.bank name.com/

1.13 Use of a Calculator Use an electronic calculator efficiently.

Apply appropriate checks of accuracy.

Use rounding to 1sf or 2sf to estimate the

answer to a calculation. Check answers

with a calculator.Class activity: Investigate the percentage

error produced by rounding in calculationsusing addition/subtraction and

multiplication/division. (Percentage error

will need to be discussed beforehand)

4 2. ALGEBRA

2.1 Indices Use and interpret positive, negative,

fractional and zero indices.

Class activity: Revise writing an integer

as a product of primes, writing answersusing index notation.


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Use simple examples to illustrate the rules

of indices. Introduce negative indices, e.g.

21 = 2(23) =3

2

2

2=2

1and

20 = 2(3-3) =3

3

2

2= 1

Introduce fractional indices by relatingthem to roots (of positive integers), e.g. x

2

1

x2

1

=x1 so thatx2

1

= x .

Use the rules of indices to show howvalues such as 16

4

3can be simplified.

Class activity: By writing an integer as

the product of primes investigate how

expressions involving square roots can be

simplified. For example, the expression

4520 + can be written as 55 .

(This is not on the syllabus but it will broaden candidates mathematical

knowledge by introducing surds)

Solve simple exponential equations, e.g.

5x = 25, 3(x + 1) = 27, 2 x = 8.

2.2 Algebraic Representation and

Manipulation2.2.1 Expansion and

Simplification

2.2.2 Factorisation

2.2.3 Substitution

2.2.4 Changing the Subject of aFormula

2.2.5 Algebraic Fractions

Use letters to express generalised numbers

and express basic arithmetic processes

algebraically.

Construct simple and complicated

expressions and equations.

Expand products of algebraic expressions.

Revise simple algebraic notation, e.g. ab

andx2.

Class activity: Revise transforming

simple formulae.

Use straightforward examples (with both

positive and negative numbers) to

illustrate expanding brackets. Extend thistechnique to multiplying two brackets

together - use a 2x2 grid to help

understanding.

Class activity: Use algebra to show that

the solution to the following problem is

Information and worksheets on many

aspects of algebra at

http://www.algebrahelp.com/workshe

ets.htm


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Manipulate directed numbers; use bracketsand extract common factors.

Factorise where possible expressions of the

form ax + bx + kay + kby, a2x2 b2y2,

a2 + 2ab + b2, ax2 + bx + c.

Substitute numbers for words and letters in

formulae.

Transform simple and complicated

formulae.

Manipulate algebraic fractions, e.g.

3

2

2

1,

10

9

4

3

,3

5

4

3,

2

)5(3

3

2,

2

4

3

+

xx

aa

abaxxxx

Factorise and simplify expressions, e.g.

65

2

2

2

+

xx

xx

always 2. Think of a number, add 7,

multiply by 3, subtract 15, multiply by ,

take away the number you first thought

of. Investigate similar problems.

Use straightforward examples (with both positive and negative numbers) to

illustrate factorising simple expressions.

Extend this technique to factorising

quadratic expressions, including spotting

expressions which are the difference oftwo squares.

Substitute numbers into a formula

(including formulae that contain brackets).

Class activity: Investigate the difference

between simple algebraic expressions

which are often confused. For example,find the difference between 2x, 2 + x and

x2 for different values ofx.

Transform simple/complex formulae,

e.g. rearrange y = ax + b to make x the

subject; x2 + y2 = r2, s = ut + at2,

expressions involving square roots, etc.

Use examples to illustrate how to simplify

algebraic fractions - build on the workwith fractions in Topic 1. Transform

formulae involving algebraic fractions,

e.g.vuf

111+=

Factorising quadratic expressions athttp://www.bbc.co.uk/schools/gcsebit

esize/maths/algebraih/index.shtml


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2.3 Solutions of Equations and

Inequalities

2.3.1 Linear Equations

2.3.2 Simultaneous Equations

2.3.3 Quadratic Equations

2.3.4 Linear Inequalities

Solve simple linear equations in one

unknown.

Solve simultaneous linear equations in two

unknowns.

Solve quadratic equations by factorisation

and either by use of the formula or bycompleting the square.

Solve simple linear inequalities.

Use straightforward examples to show

how to solve simple linear equations, e.g.3x + 2 = -1.

Revise how to solve linear equations

(including expressions with brackets).


how to solve simultaneous equations byelimination and by substitution.Class activity: Approximate the solution

to simultaneous linear equations by

graphical means.


how to solve quadratic equations byfactorisation, by using the quadratic

formula and by completing the square(real solutions only).

Construct equations from information

given and then solve them to find the

unknown quantity. This could involve the

solution of linear, simultaneous orquadratic equations.


how to solve simple linear inequalities.

Start by showing that multiplying or

dividing an expression by a negative

number reverses the inequality sign.

Try the Pyramid investigation at


Information about inequalities and

graphs at

http://www.projectgcse.co.uk/maths/i

nequalities.htm

3. GRAPHS I

3 3.1 Straight Line Graphs Calculate the gradient of a straight line fromthe coordinates of two points on it.

Calculate the length of a straight line.

Calculate the coordinates of the midpoint ofa straight line segment from the coordinates

of its end points.

Interpret and obtain the equation of a

Using examples which illustrate bothpositive and negative gradients, show how

to calculate the gradient of a straight line

given only the coordinates of two points

on it.

Class activity: Revise drawing a graph of


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straight line graph in the form y=mx+c.

Determine the equation of a straight line

parallel to a given line.

y=mx+c from a table of values.

Starting with a straight line graph show

how its equation (y=mx+c) can beobtained.

3.2 Linear Programming Represent inequalities graphically and use

this representation in the solution of simple

linear programming problems (the

conventions of using broken lines for strict

inequalities and shading unwanted regionswill be expected).


how to solve linear programming

problems by graphical means. Construct

inequalities from constraints given and

show that a number of possible solutionsto a problem exist, indicated by the

unshaded region on a graph.

Information about inequalities and

graphs at

http://www.projectgcse.co.uk/maths/i

nequalities.htm

2 4. FUNCTIONS

4.1 Evaluation of Functions

4.2 Inverse Functions

4.3 Composite Functions

Use function notation, e.g. f(x) = 3x - 5,

f: x 3x - 5 to describe simple functions,and the notation f-1(x) to describe their

inverses.

Form composite functions as defined by

gf(x) = g(f(x)).

Define f(x) to be a rule applied to values

ofx. Evaluate simple functions for specificvalues, describing the functions using f(x)

notation and mapping notation.

Introduce the inverse function as an

operation which undoes the effect of a

function. Evaluate simple inversefunctions for specific values, describing

the functions using f-1(x) notation andmapping notation.

Using linear and/or quadratic functions,

f(x) and g(x), form composite functions,

gf(x), and evaluate them for specific

values ofx.

5. GRAPHS II

3 5.1 Graphs of Functions Construct tables of values for functions of

the form ax + b, x2 + ax + b, a/x (x 0)

where a and b are integral constants; drawand interpret such graphs.

Construct tables of values and draw graphs

for functions of the form axn where a is a

rational constant and n = -2, -1, 0, 1, 2, 3

and simple sums of not more than three of

these and for functions of the form ax where

a is a positive integer.

Draw linesx = constant andy = constant.

Draw a straight line graph from a table ofvalues.

Use simple examples to show how to

calculate the gradient (positive, negative

or zero) of a straight line from a graph.

The gradient should be expressed as a

fraction or a decimal. Use these results to

consider the gradient of the line x =constant.

Graphing linear equations at

http://www.math.com/school/subject2

/lessons/S2U4L3GL.html


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Find the gradient of a straight line graph.

Solve linear and quadratic equations

approximately by graphical methods.

Estimate gradients of curves by drawing

tangents.

Solve associated equations approximately

by graphical methods.

Show how the solutions to a quadratic

equation may be approximated using a

graph. Extend this work to show how thesolution(s) to pairs of equations (e.g.y =x2

- 2x - 3 andy =x ) can be estimated usinga graph.

Class activity: Computer packages such

as Omnigraph or Derive are useful here.

Draw quadratic functions from a table ofvalues.

Draw functions of the form

xaax

x

a

x

a,,,

3

2where a is a constant,

from tables of values. Recognise common

types of function from their graphs, e.g. parabola, hyperbola, quadratic, cubic,

exponential.

Use straightforward examples to find the

gradient at a point on a curve. Extend thisto find the equation of the tangent at apoint on a curve.


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2 5.2 Graphs in Practical Situations

5.2.1 Conversion Graphs

5.2.2 Travel Graphs

Demonstrate familiarity with Cartesian

coordinates in two dimensions.

Interpret and use graphs in practical

situations including travel graphs andconversion graphs, draw graphs from givendata.

Apply the idea of rate of change to easy

kinematics involving distance-time andspeed-time graphs, acceleration and

deceleration.

Calculate distance travelled as area under alinear speed-time graph.

Revise coordinates in two dimensions.

Class activity: For candidates studying

the core syllabus, draw a picture byjoining dots on a square grid. Drawx and

y axes on the grid and note the coordinatesof each dot. Ask another student to draw

the picture from a list of coordinates only.

Solve straightforward problems using

compound measures e.g. problemsinvolving rate of flow.

Draw and use straight line graphs to

convert between different units e.g.

between metric and imperial units or

between different currencies.

Draw and use distance-time graphs to

calculate average speed (link tocalculating gradients). Interpret

information shown in travel graphs. Draw

travel graphs from given data.

Class activity: Draw a travel graph for thejourney to and from school. Answer a set

of questions about the journey, e.g. what isthe average speed on the journey to

school?

Introduce the formula relating speed,

distance and time. Solve simple numerical

problems (which should involve

converting between units e.g. find speed in

m/s given distance in kilometres and timein hours).

Revise how to calculate the area of a

rectangle and the area of a right angled

triangle.

Draw and use speed-time graphs to

calculate acceleration and deceleration.Use straightforward examples to show that

the area under a linear speed-time graph is

equivalent to the distance travelled.

Information on speed, distance and

time athttp://www.mathforum.org/dr.math/fa

q/faq.distance.html


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2 6. GEOMETRY

6.1 Fundamental Properties Use and interpret the geometrical terms:

point, line, parallel, bearing, right angle,

acute, obtuse and reflex angles,

perpendicular, similarity, congruence.

Use and interpret vocabulary of triangles,quadrilaterals, circles and polygons.

Classifying angles at

http://www.math.com/school/subject3

/lessons/S3U1L4GL.html

6.2 Polygons

6.2.1 Symmetry Properties

6.2.2 Angle Properties

Recognise rotational and line symmetry(including order of rotational symmetry) in

two dimensions and properties of triangles,

quadrilaterals and circles directly related to

their symmetries.

Calculate unknown angles using the

following geometrical properties:

(a) angles at a point,

(b) angles on a straight line and

intersecting straight lines,(c) angles formed within parallel lines,

(d) angle properties of triangles andquadrilaterals,

(e) angle properties of regular polygons.

Define the terms line of symmetry andorder of rotational symmetry for two

dimensional shapes. Revise the

symmetries of triangles (equilateral,

isosceles) and quadrilaterals (square,

rectangle, rhombus, parallelogram,

trapezium, kite).Class activity: Investigate tessellations.

Produce an Escher-type drawing.

Revise basic angle properties by drawing

simple diagrams which illustrate (a), (b)

and (c). Define acute, obtuse and reflex

angles; equilateral, isosceles and scalene

triangles.

Define the terms (irregular) polygon andregular polygon. Use examples that

include: triangles, quadrilaterals,

pentagons, hexagons and octagons.

By dividing an n-sided polygon into a

number of triangles show that the sum ofthe interior angles is (n 2)180 . Show

also that each exterior angle isn

360

.

Solve a variety of problems that use these

formulae.Class activity: Draw a table of

information for regular polygons. Use as

headings: number of sides, name, exterior

angle, sum of interior angles, interior

angle.

Pictures of tessellations produced byEscher at

http://library.thinkquest.org/16661/


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6.3 Circles


6.3.2 Angle Properties

Use the following symmetry properties of

circles:

(a) equal chords are equidistant from thecentre,

(b) the perpendicular bisector of a chordpasses through the centre,

(c) tangents from an external point are

equal in length.

Calculate unknown angles using thefollowing geometrical properties:(a) angle in a semi-circle,

(b) angle between tangent and radius of a

circle,

(c) angle at the centre of a circle is twice

the angle at the circumference,

(d) angles at the same segment are equal,

(e) angles in the opposite segments are

supplementary; cyclic quadrilaterals.

Draw simple diagrams to illustrate the

circle symmetry.

Use diagrams to introduce the angleproperties (a) and (b). Solve a variety of problems which involve the angle

properties.

Class activity: Investigate cyclic

quadrilaterals. For example, explain why

all rectangles are cyclic quadrilaterals.

What other quadrilateral is cyclic? Is itpossible to draw a parallelogram that is

cyclic? etc.

6.4 Solids

6.4.1 Nets


Use and interpret vocabulary of simple solid

figures including nets.

Recognise symmetry properties of the prism

(including cylinder) and the pyramid

(including cone);

Illustrate common solids, e.g. cube,

cuboid, tetrahedron, cylinder, cone,sphere, prism, pyramid, etc. Define the

terms vertex, edge and face.

Starting with simple examples draw the

nets of various solids. Show, for example,

that the net of a cube can be drawn in

different ways.

Class activity: Draw nets on card andmake various geometrical shapes.

Define the terms plane of symmetry and

order of rotational symmetry for three

dimensional shapes. Use diagrams to

illustrate the symmetries of cuboids(including a cube), prisms (including a

cylinder), pyramids (including a cone) andspheres.

Explore geometric solids and their

properties athttp://www.illuminations.nctm.org/im

ath/3-5/GeometricSolids/

6.5 Congruency Discuss the conditions for congruent

triangles. Point out that in namingtriangles which are congruent it is usual to

For information and activities about

congruent triangles and shapes, searchfor congruent at


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state letters in corresponding order, i.e.

ABC is congruent to EFG implies that

the angle at A is the same as the angle at

E.

http://www.learn.co.uk

6.6 Similarity

6.6.1 Areas of Similar Triangles

and Figures

6.6.2 Volumes and Surface

Areas of Similar Solids

Use the relationships between areas of

similar triangles, with corresponding results

for similar figures and extension to volumes

and surface areas of similar solids.

Introduce similar triangles / shapes. Use

the fact that corresponding sides are in the

same ratio to calculate the length of an

unknown side.

4 7. TRIGONOMETRY

7.1 Pythagoras Theorem

7.2 Trigonometric Ratios

Apply Pythagoras theorem and the sine,

cosine and tangent ratios for acute angles to

the calculation of a side or of an angle of a

right-angled triangle (angles will be quotedin, and answers required in, degrees and

decimals to one decimal place).

Use simple examples involving the sine,

cosine and tangent ratios to calculate the

length of an unknown side of a right-

angled triangle given an angle and thelength of one side.

Class activity: Use trigonometry to

calculate the height of a building or tree.

You will need to discuss how to measure

the angle of elevation practically.

Use simple examples involving inverse

ratios to calculate an unknown angle giventhe length of two sides of a right-angled

triangle.

Revise Pythagoras theorem using

straightforward examples.

Class activity: Solve problems in contextusing Pythagoras theorem and

trigonometric ratios (include work withany shape that may be partitioned into

right-angled triangles).

Class activity: Calculate the area of a

segment of a circle given the radius and

the sector angle.

Draw a sine curve and discuss its

properties. Use the curve to show, for

example, sin 150 = sin 30 . Repeat for

the cosine curve.

Revise Pythagoras theorem at

http://www.bbc.co.uk/schools/gcsebit

esize/maths/shapeih/index.shtml

Try the Degree Ceremony

investigation at



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7.3 Angle of Elevation and

Depression Solve trigonometrical problems in two

dimensions involving angles of elevation

and depression, extend sine and cosinefunctions to angles between 90o and 180o.

Define angles of elevation and depression.


how to solve problems using the sine andcosine rules.

Class activity: Solve two dimensionaltrigonometric problems in context.

Various problems at


Try the investigation at


7.4 Sine Rule

7.5 Cosine Rule

7.6 Area of a Triangle

Solve problems using the sine and cosine

rules for any triangle and the formula areaof triangle = absinC.

Rearrange the formula for the area of a

triangle (bh) to the form absinC.Illustrate its use with a few simple

examples.

7.7 Bearings Interpret and use three-figure bearings Discuss how bearings are measured and Maps of the world at


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measured clockwise from the north (i.e.

000o - 360o ).

written. Use simple examples to show how

to calculate bearings, e.g. calculate the

bearing ofB from A if you know the

bearing ofA fromB.Class activity: Use a map to determine

distance and direction between two places,etc.

http://www.theodora.com/maps

7.8 Three-Dimensional

Problems Solve simple trigonometrical problems in

three dimensions including angle between a

line and a plane.

Introduce problems in three dimensions by

finding the length of the diagonal of a

cuboid and determining the angle it makeswith the base. Extend by using more

complex figures, e.g. a pyramid.

28. CONSTRUCTION AND

LOCI

8.1 Construction of Simple

Figures Measure lines and angles.

Construct a triangle given the three sides

using ruler and compasses only.

Construct other simple geometrical figures

from given data using protractors and set

squares as necessary.

Construct angle bisectors and perpendicular

bisectors using straight edges andcompasses only.

Read and make scale drawings.

Class activity: Reinforce accurate

measurement of lines and angles through

various exercises. For example, each

student draws two lines that intersect.Measure the length of each line to the

nearest millimetre and one of the angles to

the nearest degree. Each student should

then measure another students drawing

and compare answers.

Show how to construct a triangle using a

ruler and compasses only, given thelengths of all three sides; bisect an angle

using a straight edge and compasses only;

construct a perpendicular bisector using a

straight edge and compasses only.

Class activity: Construct a range of

simple geometrical figures from givendata, e.g. construct a circle passing

through three given points.

Use a straightforward example to revise

the topic of scale drawing. Show how to

calculate the scale of a drawing given a

length on the drawing and the

corresponding real length. Point out thatmeasurements should not be included on a

scale drawing and that the scale of a

drawing is usually written in the form 1 :

Information and ideas for teachers on

geometric constructions at

http://www.forum.swarthmore.edu/lib

rary/topics/constructions/


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n Class activity: Draw various situations

to scale and interpret results. For example,

draw a plan of a room in your house to

scale and use it to determine the area ofcarpet needed to cover the floor, plan an

orienteering course, etc.

8.2 Loci and Intersection of Loci Use the following loci and the method of

intersecting loci for sets of points in two

dimensions:

(a) which are at a given distance from agiven point,

(b) which are at a given distance from a

given straight line,

(c) which are equidistant from two given

points,(d) which are equidistant from two given

intersecting straight lines.

Draw simple diagrams to illustrate (a), (b),

(c) and (d). Use the convention of a

broken line to represent a boundary that is

not included in the locus of points.Class activity: A rectangular card is

rolled along a flat surface. Trace out the

locus of one of the vertices of the

rectangle as it moves.


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IGCSE MATHEMATICS 0580 (EXTENDED 2 YEARS)

SCHEME OF WORK FOR YEAR 10 (2011)

SUGGESTED NO. OFWEEKS

TOPICS/SUB-TOPICS OBJECTIVES SUGGESTED ACTIVITIES RESOURCES

2 9. MATRICES

9.1 Elements, Columns, Rows and

Order of Matrix

9.2 Matrix Operations

9.3 Determinant and Inverse

Display information in the form of a matrix

of any order.

Calculate the sum and product (whereappropriate) of two matrices.

Calculate the product of a matrix and a

scalar quantity.

Use the algebra of 22 matrices includingthe zero and identity 22 matrices.

Calculate the determinant and inverse A-1 of

a non-singular matrix A.

Use simple examples to illustrate that

information can be stored in a matrix. For

example, the number of different types ofchocolate bar sold by a shop each day for a

week. Define the order/size of a matrix asthe number of rows x number of columns.

Class activity: Investigate networks -

recording information in a matrix. (This is

not on the syllabus but it will broaden

candidates mathematical knowledge of

matrices)

Explain how to identify matrices that youmay add/subtract or multiply together. Use

straightforward examples to illustrate how

to add/subtract and multiply matrices

together.

Define the identity matrix and the zeromatrix. Use simple examples to illustrate

multiplying a matrix by a scalar quantity.


how to calculate the determinant and the

inverse of a non-singular 2x2 matrix.

Class activity: Investigate how to use

matrices to help solve simultaneous

equations.


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4 10. TRANSFORMATION

10.1 Simple Transformations

10.1.1 Translation

10.1.2 Reflection

10.1.3 Rotation

10.1.4 Enlargement

10.1.5 Shear

10.1.6 Stretch

Construct given translations of simple

plane figures.

Reflect simple plane figures in horizontal

or vertical lines.

Rotate simple plane figures about the

origin, vertices or mid points of edges of

the figures, through multiples of 90.

Construct given enlargements of simpleplane figures.

Recognise and describe reflections,

rotations, translations and enlargements.

Draw an arrow shape on squared paper.

Use this to illustrate: reflection in a line

(mirror line), rotation about any point(centre of rotation) through multiples of 90o

(in both clockwise and anti-clockwisedirections) and translation by a vector.

Several different examples of each

translation should be drawn. Use the word

image appropriately.Class activity: Using a pre-drawn shape on

(x,y) coordinate axes to complete a numberof transformations using the equations of

lines to represent mirror lines and

coordinates to represent centres of rotation.

Work with (x,y) coordinate axes to show

how to find: the equation of a simple mirror

line given a shape and its (reflected) image,

the centre and angle of rotation given ashape and its (rotated) image, the vector of

a translation.

Draw a triangle on squared paper. Use this

to illustrate enlargement by a positive

integer scale factor about any point (centreof enlargement). Show how to find the

centre of enlargement given a shape and its(enlarged) image. Draw straightforward

enlargements using negative and/or

fractional () scale factors.

Use straightforward examples to illustrate a

shear and a stretch. Using a shape and its

image drawn on (x,y) coordinate axes showhow to find the scale factor and theequation of the invariant line.

Class activity: Starting with a letter E

drawn on (x,y) coordinate axes, perform

combinations of the following

transformations: translation, rotation,

reflection, stretch, shear and enlargement.

Try the investigation at

http://nrich.maths.org/public/leg.ph

p

For further information abouttransformations search for 'rotation',

'enlargement', 'reflection' or

'translation' at

http://www.learn.co.uk


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10.2 Combined Transformations Use the following transformations of the

plane: reflection (M); rotation (R);

translation (T); enlargement (E); shear (H);

stretching (S) and their combinations.(If M(a) = b and R(b) = c the notation

RM(a) = c will be used; invariants underthese transformations may be assumed).

10.3 Matrix Transformations Identify and give precise descriptions of

transformations connecting given figures;

describe transformations using co-ordinatesand matrices (singular matrices are

excluded).

Use a unit square and the base vectors

0

1

and

1

0

to identify matrices which

represent the various transformations met

so far, e.g.

01

10represents a rotation

about (0,0) through anti-clockwise. Work

with a simple object drawn on (x,y)coordinate axes to illustrate how it is

transformed by a variety of given matrices.

Use one of these transformations to

illustrate the effect of an inverse matrix.

Work with a rectangle drawn on (x,y)coordinate axes to illustrate that the area

scale factor of a transformation isnumerically equal to the determinant of the

transformation matrix. For example use the

matrix

20

02.

3 11. STATISTICS

11.1 Data Representation

11.1.1 Pictogram

11.1.2 Bar Chart

11.1.3 Pie Chart

11.1.4 Simple Frequency

Distribution

11.1.5 Histogram

11.1.6 Scatter Diagram

Collect, classify and tabulate statistical

data.

Read, interpret and draw simple inferencesfrom tables and statistical diagrams.

Construct and use bar charts, pie charts,

pictograms, simple frequency distributions,

histograms with equal intervals and scatter

diagrams (including drawing a line of bestfit by eye), understand what is meant by

positive, negative and zero correlation.

Use simple examples to revise collecting

data and presenting it in a frequency (tally)

chart. For example, record the differentmakes of car in a car park, record thenumber of letters in each of the first 100

words in a book, etc. Use the data collected

to construct a pictogram, a bar chart and a

pie chart. Point out that the bars in a bar

chart can be drawn apart.

Use a simple example to show how discrete

data can be grouped into equal classes.

Download newspaper stories -

worldwide coverage at

http://www.newsparadise.com/

Try the Bat Wings problem at

http://nrich.maths.org/public/leg.ph

p


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Construct and read histograms with equal

and unequal intervals (areas proportional to

frequencies and vertical axis labelledfrequency density).

Draw a histogram to illustrate the data (i.e.

with a continuous scale along the horizontal

axis). Point out that this information could

also be displayed in a bar chart (i.e. withbars separated).

Class activity: Investigate the length ofwords used in two different newspapers and

present the findings using statistical

diagrams.

Record sets of continuous data, e.g. heights,weights etc., in grouped frequency tables.Use examples that illustrate equal and

unequal class widths. Draw the

corresponding histograms (label the vertical

axis of a histogram as frequency density

and point out that the area of each bar is

proportional to the frequency). Show howto calculate frequencies from a given

histogram and how to identify the modalclass.

11.2 Mean, Median and Mode Calculate the mean, median and mode for

individual and discrete data and distinguishbetween the purposes for which they are

used.

Calculate the range.

Calculate an estimate of the mean for

grouped and continuous data.

Identify the modal class from a grouped

frequency distribution.

Design and use a questionnaire collect

results and present them in diagrammaticform. From data collected show how to

work out the mean, the median and the

mode. Use simple examples to highlight

how these averages may be used. For

example in a discussion about average

wages the owner of a company with a few

highly paid managers and a large work

force may wish to quote the mean wagerather than the median. Point out how the

mode can be recognised from a frequency

diagram.

Use straightforward examples to show how

to calculate an estimate for the mean ofdata in a grouped frequency table.

Class activity: Survey a class of students -heights, weights, number in family, etc.

Use different methods of display to help

analyse the data and make statistical

inferences.

Compare the median and the mean

interactively athttp://www.standards.nctm.org/docu

ment/eexamples/chap6/6.6/index.ht

m


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11.3 Cumulative Frequency Construct and use cumulative frequency

diagrams.

Estimate and interpret the median, percentiles, quartiles and inter-quartile

range.

Explain cumulative frequency and use a

straightforward example to illustrate how a

cumulative frequency table is constructed.

Draw the corresponding cumulativefrequency curve. Point out that this can be

approximated by a cumulative frequencypolygon.

Use a cumulative frequency curve to help

explain percentiles. Introduce the names

given to the 25th, 50th and 75th percentilesand show how to estimate these from agraph. Show how to calculate the range of a

set of data and how to estimate the inter-

quartile range from a cumulative frequency

diagram.

2 12. PROBABILITY

12.1 Definition of Probability Calculate the probability of a single eventas either a fraction or a decimal (not a

ratio).

Understand and use the probability scale

from 0 to 1.

Understand that the probability of an event

occurring = 1 the probability of the eventnot occurring.

Understand probability in practice e.g.

relative frequency.

Discuss probabilities of 0 and 1, leading tothe outcome that a probability lies between

these two values.

Class activity: Calculate probabilities

based on experiment. For example,

investigate whether a coin is biased.

Use theoretical probability to predict the

likelihood of a single event. For example,find the probability of choosing the letter M

from the letters of the word

MATHEMATICS.

Various problems involvingprobability at

http://www.nrich.maths.org/public/l

eg.php


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12.2 Probability of Combined Events

12.2.1 Possibility Diagrams

12.2.2 Tree Diagrams

Calculate the probability of simple

combined events, using possibility

diagrams and tree diagrams whereappropriate (in possibility diagrams

outcomes will be represented by points on agrid and in tree diagrams outcomes will be

written at the end of branches and

probabilities by the side of the branches).

Use simple examples to illustrate how

possibility diagrams and tree diagrams can

help to organise data.

Use possibility diagrams and tree diagramsto help calculate probabilities of simple

combined events, paying particular

attention to how diagrams are labelled.

Solve straightforward problems involvingindependent and dependent events, e.g. picking counters from a bag with and

without replacement.

2 13. SETS

13.1 Set Language and Notation

13.2 Set Operations

13.3 Venn Diagrams

Use language, notation and Venn diagrams

to describe sets and represent relationshipsbetween sets as follows:

Definition of sets, e.g.

A = {x:x is a natural number}

B = {(x,y):y = mx + c}

C = {x: a x b}D = {a, b, c, .....}

Notation:

number of elements in set A n(A)

.... is an element of ....

.... is not an element of .... Complement of the set A A'

The empty set Universal set

A is a subset of B A B

A is a proper subset of B A B

A is not a subset of B A B

A is not a proper subset of B A BUnion of A and B A B

Intersection of A and B A B

Revise: Properties of numbers covered in

Topic 1.

Give examples from work already covered

to illustrate the language and notation of

sets. Distinguish between a subset and a

proper subset.

Draw Venn diagrams and shade the regions

which represent the sets A B, A B, A'

B, A B', A' B, A B', A' B' and

A' B' . Show that (A B) ' is the same as

A' B' and that (A B) ' is the same as A'

B' .

Use Venn diagrams to solve problemsinvolving sets.

Information and references to

activities for teachers athttp://www.mathworld.wolfram.co

m/VennDiagram.html

3 14. VECTORS

14.1 Vector Representation

14.2 Addition and Subtraction of

Vectors

Describe a translation by using a vector

represented by

y

x, ora; add and subtract

vectors and multiply a vector by a scalar.

Use the concept of translation to explain a

vector. Use simple diagrams to illustrate

column vectors in two dimensions,

explaining the significance of positive and

negative numbers. Introduce the various

Interactive work on vector sums at

http://www.standards.nctm.org/docu

ment/eexamples/chap7/7.1/part2.ht

m


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14.3 Multiplication by a Scalar

14.4 Column Vectors

14.4.1 Magnitude

14.4.2 Parallel Vectors

Represent vectors by directed line

segments.

Use the sum and difference of two vectors

to express given vectors in terms of twocoplanar vectors.

Calculate the magnitude of a vector.(Vectors will be printed as AB or a and

their magnitudes denoted by modulus signs,

e.g. AB or a . In their answers to

questions candidates are expected to

indicate a in some definite way, e.g. by an

arrow or by underlining thus AB ora.

Use position vectors.

forms of vector notation.

Show how to add and subtract vectorsalgebraically and by making use of a vector

triangle. Show how to multiply a columnvector by a scalar and illustrate this with a

diagram.

Use simple diagrams to help show how to

calculate the magnitude of a vector(Pythagoras theorem may have to berevised).

Define a position vector and solve various

straightforward problems in vectorgeometry.

2 15. NUMBER SEQUENCE

Continue a given number sequence.

Recognise patterns in sequences and

relationships between different sequences,generalise to simple algebraic statements

(including expressions for the nth term)

relating to such sequences.

Define a sequence of numbers. Work with

simple sequences, e.g. find the next twonumbers in a sequence of even, odd,

square, triangle or Fibonacci numbers, etc.

Find the term-to-term rule for a sequence,

e.g. the sequence 3, 9, 15, 21, 27, .... has a

term-to-term rule of +6

Find the position-to-term rule for asequence, e.g. the nth term in the sequence

3, 9, 15, 21, 27, .... is 6n - 3 .

Class activity: Square tables are placed in

a row so that 6 people can sit around 2

tables, 8 people can sit around 3 tables, and

so on. How many people can sit around n

tables?

Various problems involving

sequences of numbers athttp://nrich.maths.org/public/leg.ph

p


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3 16. MENSURATION

16.1 Perimeter and Area

16.1.1 Common Figures

16.1.2 Composite Figures Carry out calculations involving the

perimeter and area of a rectangle and

triangle, the circumference and area of acircle, the area of a parallelogram and a

trapezium.

Revise, using straightforward examples,

how to calculate the circumference and area

of a circle, and the perimeter and area of arectangle and a triangle. Extend this to

calculating the area of a parallelogram anda trapezium.

Class activity: Using isometric dot paper

investigate the area of shapes that have a

perimeter of 5, 6, 7, . units.

Calculating areas of parallelograms

and trapeziums at

http://www.bbc.co.uk/schools/gcsebitesize/maths/shape/index.shtml

16.2 Arc Length and Area of Sector Solve problems involving the arc length

and sector area as fractions of the

circumference and area of a circle.


how to calculate arc length and the area of

a sector.

16.3 Volume and Surface Area16.3.1 Common Solids

16.3.2 Composite Solids Carry out calculations involving the

volume of a cuboid, prism and cylinder and

the surface area of a cuboid and a cylinder.

Solve problems involving the surface area

and volume of a sphere, pyramid and cone(given formulae for the sphere, pyramid

and cone).

Use nets to illustrate how to calculate the

surface area of a cuboid, a triangular prism,

a cylinder, a pyramid and a cone. Show

how to obtain the formula r(r+l) for the

surface area of a cone. Calculate the surface

area of a sphere using the formula 4r2.


how to calculate the volume of various prisms (cross-sectional area length).

Calculate the volume of a pyramid(including a cone) using the formula

3

1 area of base perpendicular height.

Calculate the volume of a sphere using the

formula3

4r3 .

Class activity: Find the surface area andvolume of various composite shapes.

Class activity: An A4 sheet of paper can

be rolled into a cylinder in two ways.

Which gives the biggest volume? If the

area of paper remains constant but the

length and width can vary investigate whatwidth and length gives the maximum

cylinder volume.

Calculating volumes and surface

areas at

http://www.bbc.co.uk/schools/gcseb

itesize/maths/shapeih/index.shtml

Try the dipstick investigation athttp://www.ex.ac.uk/cimt/resource/d

ipstick.htm


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Igcse Maths 0580 Extended 2 Years

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