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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).
Use straightforward examples to illustrate
how to solve simultaneous equations byelimination and by substitution.Class activity: Approximate the solution
to simultaneous linear equations by
graphical means.
Use straightforward examples to illustrate
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.
Use straightforward examples to illustrate
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
http://nrich.maths.org/public/leg.php
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).
Use straightforward examples to illustrate
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
6.3.1 Symmetry Properties
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
6.4.2 Symmetry Properties
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
http://nrich.maths.org/public/leg.php
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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.
Use straightforward examples to illustrate
how to solve problems using the sine andcosine rules.
Class activity: Solve two dimensionaltrigonometric problems in context.
Various problems at
http://nrich.maths.org/public/leg.php
Try the investigation at
http://nrich.maths.org/public/leg.php
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.
Use straightforward examples to illustrate
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.
Use straightforward examples to illustrate
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.
Use straightforward examples to illustrate
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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