24610552 maths igcse scheme of work 0580 2010year10

SCHEME OF WORK – IGCSE MATHEMATICS (0580) YEAR 9 2010

Suggested no. of weeks

Topics /

Sub – topicsAssessment Objectives Suggested Activities / Approaches Supplementary

Resources

10 Weeks1. NUMBERS

1.1. Number Facts

1.2. Square, square roots and cubes

1.3 Equivalence and Conversion

1.4 Ordering

1.5 The Four Rules

1.6 Approximation and Estimation

1.7 Limits of Accuracy

1.8 Standard Form

1.9 Ratio, Proportion and Rate

1.9.1 Ratio

1.9.2 Direct and Inverse Proportion

1.9.3 Rate

1.9.4 Money

1.9.5 Scales

1.9.6 Speed, Distance and Time

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.

Calculate squares, square roots and cubes and cube roots of numbers.

Use directed numbers in practical situations.

Use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts; recognise equivalence and convert between these forms.

Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, ≥, ≤ .

Use the standard form A x 10n where n is a positive or negative integer, and 1≤ A < 10.

Use the four rules for calculations with whole numbers, decimal fractions and vulgar (and mixed) fractions, including correct ordering of operations and use of brackets.

Make estimates of numbers, quantities and lengths, give approximations to specified numbers of significant figures and decimal places and round off answers to reasonable accuracy in the context of a given problem.

Give appropriate upper and lower bounds for data given to a specified accuracy (e.g. measured lengths).

Revise positive and negative numbers using a number line.

Define the terms factor and multiple and use simple examples to find common factors 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 term prime number (1 is not prime). Write any integer as a product of primes.

Class activity: Investigate Goldbach’s conjecture.

Define the terms real, rational and irrational numbers. Show that any recurring decimal can be written as a fraction. Show that any root which cannot be simplified to an integer or a fraction is an irrational number.

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 are reversed it is the same number). Write down all the palindromic square numbers less than 1000.

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.

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 numbers.

Class activity: Use four 4’s and the four rules for calculations to obtain all the whole numbers from 1 to 20.

Use a number line to describe simple inequalities and ranges of values e.g. x ≥ 3, -2 ≤ x < 5, etc.

Class activity: Given a list of quantities (e.g. a list of fractions and decimals), order them by magnitude making use of inequality signs.

Use a range of examples to show how to write numbers in standard form and vice-versa.

Class activity: Use the four rules of calculation with numbers in standard form.

IGCSE Mathematics (2nd edition) by Ric Pimentel and Terry WallPg 1 – 23 Pg 24 – 40 Pg 41 – 50 Pg 51 – 68 Pg 69 – 79

IGCSE Mathematics by Karen MorrisonPg 1 – 47

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Topics /


Resources

1.10 Time

1.11 Percentages

1.12 Personal and Household finance

1.12.1 Simple Interest and Compound Interest

1.12.2 Discount

1.12.3 Profit and Loss

1.13 Use of an Electronic Calculator

Demonstrate an understanding of the elementary ideas and notion of ratio direct and inverse proportion; divide a quantity in a given ratio

Express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities; increase and decrease a quantity by a given ratio.

Demonstrate an understanding of common measures of rate; use scales in practical situations, calculate average speed.

Carry out calculations involving reverse percentages, e.g. finding the cost price given the selling price and the percentage profit.

Use an electronic calculator efficiently; apply appropriate checks of accuracy.

Use current units of mass, length, area, volume, and capacity in practical situations and express quantities in terms of larger or smaller units.

Calculate times in terms of the 24-hour and 12-hour clock; read clocks, dials and timetables.

Calculate using money and convert from one currency to another..

Use given data to solve problems on personal and household finance involving earnings, simple interest and compound interest, discount, profit and loss; extract data from tables and charts.

Revise equivalent fractions. Use this idea to aid addition and subtraction of fractions. Revise multiplication and division of fractions and convert between fractions, decimals and percentages.

Use place value (units, tenths, hundredths etc.) to change a simple decimal into a fraction.

Revise rounding numbers to the nearest 10, 100, 1000, etc., or to a set number of decimal places.

Explain carefully how to round a number to a given number of significant figures.

Use straightforward examples to determine upper and lower bounds for data. For example, a length, l, measured as 3cm to the nearest millimetre has lower bound 2.95cm and upper bound 3.05cm. Show how this information can be written using inequality signs e.g. 2.95cm ≤ l < 3.05cm.

Class activity: Investigate upper and lower bounds for quantities calculated from given formulae by specifying the accuracy of the input data.

Draw a graph to determine whether two quantities (y and x or y and x2, etc.) are in proportion.Solve problems involving direct or inverse proportion using the notation y ∝ x ⇒ y = kx and y ∝ 1/x ⇒ y = k/x , where k is a constant.

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 paper to that of the corresponding side of an A4 sheet of paper.Draw and use straight line graphs to convert between different units e.g. between metric and imperial units or between different currencies.

Revise: Work covered on percentages in Unit 1.Use simple examples to show how to calculate the original value of something before a percentage increase or decrease took place.

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 calculations using addition/subtraction and multiplication/division. (Percentage error will need to be discussed beforehand)

Use practical examples to illustrate how to convert between: millimetres, centimetres, metres and kilometres; grams, kilograms and tonnes; millilitres, centilitres and litres. Use standard form where appropriate.

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Topics /


Resources

Revise units for measuring time and use examples to convert between hours, minutes and seconds.Use television schedules and bus/train timetables 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 world map with times in various cities assuming it is noon where you live.

Solve straightforward problems involving exchange rates.Up to date information from a daily newspaper is useful

Solve simple problems using practical examples where possible, taking information from published tables or advertisements. (It is worth introducing a range of simple words and concepts here to describe different aspects of finance, e.g. tax, percentage profit, deposit, loan, etc.)Use the formula Ι = PRT to solve a variety of problems involving simple interest.

Class activity: Use newspapers to research the cost of borrowing money from different banks (or money lenders).

7 Weeks 2. ALGEBRA

2.1 Indices

2.2 Expansion and Simplification

2.3 Factorisation

2.4 Substitution

2.5 Changing the Subject of a Formula

2.6 Algebraic Fractions

2.7 Linear Equations

Use and interpret positive, negative and zero indices.

Use letters to express generalized numbers and express basic arithmetic processes algebraically

Substitute numbers for words and letters in formulae

Transform simple formula

Construct simple expressions and set up simple equations

Construct and transform more complicated formulae and equations

Manipulate directed numbers; use brackets and extract common factors

Expand products of algebraic expressions

Class activity: Revise writing an integer as a product of primes, writing answers using index notation.

Use simple examples to illustrate the rules of indices.

Introduce negative indices, e.g. 2 –1 = 2 (2 – 3)= 22

23= 12

and 20 = 2 (3–3) =

23

23 = 1

Introduce fractional indices by relating them to roots (of positive

integers), e.g. x12×x

12 = x1, so that x

12=√ x.

Use the rules of indices to show how values such as 16 ¾ can 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 √20 + √45 can be written as 5√5. (This is not on the syllabus but it will broaden candidates mathematical knowledge by introducing surds)

Solve simple exponential equations, e.g. 5x = 25,

Information and worksheets on many aspects of algebra at http://www.algebrahelp.com/worksheets.htm

Factorising quadratic expressions at http://www.bbc.co.uk/schools/gcsebitesize/maths/algebraih/index.shtml

Try the ‘Pyramid’ investigation at http://nrich.maths.org/public/leg.php

Information about inequalities and graphs at http://www.projectgcse.co.uk/

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http://www.projectgcse.co.uk/maths/inequalities.htm



Topics /


Resources2.8 Simultaneous Equations

2.9 Quadratic Equations

2.10 Inequalities

Factorize where possible expressions of the form: ax + bx + kay + kbya2x2 – b2y2

a2 + 2ab + b2

ax2 + bx + c

Manipulate algebraic fractions,

e.g., x3+ x−42

,

2x3

−3(x−5)2

, 3a4×5ab3

,

3a4

−9a10

, 1x−2

+ 2x−3

Factorize and simplify expressions

such as x2−2 xx2−2x+6

Solve simple linear equations in one unknown

Solve simultaneous linear equations in two unknowns

Solve quadratic equations by factorization and either by use of formula or by completing the square

Solve simple linear inequalities

3(x + 1) = 27, 2-x = 8, etc.

Revise simple algebraic notation, for example, ab and x2 .

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 of x .

Transform simple formulae, e.g., rearrange y = ax + b to make x the subject.

Revise : transforming simple formulae (use examples similar to those used in Unit 1).

Transform complex formulae, e.g. x2 + y2 = r2,

s = ut + ½at2, expressions involving square roots, etc.

Use examples to illustrate how to simplify algebraic fractions - build on the work with fractions in Unit 1. Transform formulae involving algebraic

fractions, e.g. 1f=1u+ 1v

Class activity: Revise transforming simple formulae (use examples similar to those used in Unit 1).

Use straightforward examples (with both positive and negative numbers) to illustrate expanding brackets. Extend this technique to multiplying two brackets together - use a 2x2 grid to help understanding.

Use straightforward examples (with both positive and negative numbers) to illustrate factorizing simple expressions. Extend this technique to factorizing quadratic expressions, including spotting expressions which are the difference of two squares.

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

Class activity: Use algebra to show that the solution to the following problem is always 2. “Think of a number, add 7, multiply by 3, subtract 15, multiply by ⅓, take away the number you first thought of.” Investigate similar problems.

Revise how to solve linear equations (including expressions with brackets).

Use straightforward examples to illustrate how to solve simultaneous equations by elimination and by substitution.

maths/inequalities.htm

IGCSE Mathematics (2nd edition) by Ric Pimentel and Terry Wall Pg 174 – 184

IGCSE Mathematics by Karen MorrisonPg 49 - 81



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Topics /


ResourcesClass activity: Approximate the solution to simultaneous linear equations by graphical means.

Use straightforward examples to illustrate how to solve quadratic equations by factorisation, and either by using the quadratic formula or 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 or quadratic 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.

4 Weeks 3. GRAPHS I

3.1. Straight Line Graphs

3.1.1 Gradient

3.1.2 Length of a line segment

3.1.3 Mid-point of a line segment

3.1.4 Equation of straight line

3.2 Linear Programming

Calculate the gradient of the straight line

Calculate the length of a straight line segment from the coordinates of its end points

Calculate the mid-point of a straight line segment from the coordinates of its end points

Interpret and obtain the equation of a straight line in the form of y = mx + c

Determine the equation of a straight line parallel to a given line

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 regions will be expected)

Revise : drawing a graph of y = mx + c from a table of values

Starting with a straight line graph shows how its equation y= mx + c can be obtained.

Using examples which illustrate both positive and negative gradients, show how to calculate the gradient of a straight line given only the coordinates of two points on it.

Use straight forward examples to illustrate how to solve linear programming problems by graphical means.

Construct inequalities from constraints given and show that a number of possible solutions exist, indicated by the unshaded region on a graph.

IGCSE Mathematics (2nd

edition) by Ric Pimentel and Terry Wall Pg 139 – 149

Information about inequalities and graphs at http://www.projectgcse.co.uk/maths/inequalities.htm

IGCSE Mathematics by Karen Morrison Pg 99 - 103

IGCSE Mathematics (2nd

edition) by Ric Pimentel and Terry WallPg 185 – 190


1 Week 4. FUNCTIONS

4.1 Evaluation of Function

4.2 Inverse function

4.3 Composite function

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 of x. Evaluate simple functions for specific values, 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 inverse functions for specific values,

describing the functions using f-1(x) notation and mapping notation.

Using linear and/or quadratic functions, f(x) and g(x), form composite functions, gf(x), and evaluate them for specific values of x.

IGCSE Mathematics (2nd edition) by Ric Pimentel and Terry Wall Pg 153-158

IGCSE Mathematics by Karen Morrison Pg 116-120

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Topics /


Resources

3 Weeks 5. GRAPHS II

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;

Draw and interpret such graphs;

Solve linear and quadratic equations approximately by graphical methods.

Construct tables of values and draw graphs for functions of the

form axnwhere 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 axwhere ais a positive integer;

Estimate gradients of curves by drawing tangents; solve associated equations approximately by graphical methods.

Draw quadratic functions from a table of values.

Show how the solutions to a quadratic equation may be approximated using a graph. Extend this work to show how the solution(s) to pairs of

equations (e.g. y = x2- 2x - 3 and y = x ) can be estimated using a graph. Class activity: Computer packages such as Omnigraph or derive are useful here.

Draw functions of the form where 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 this to find the equation of the tangent at a point on a curve.

IGCSE Mathematics (2nd edition) by Ric Pimentel and Terry Wall Pg 165-172

IGCSE Mathematics by Karen Morrison Pg 87-115

2 – 3 Weeks

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 and conversion graphs, draw graphs from given data.

Apply the idea of rate of change to easy kinematics involving distance-time and speed-time graphs, acceleration and deceleration;

calculate distance travelled as area under a linear speed-time graph.

Revise coordinates in two dimensions.

Draw and use straight line graphs to convert between different units e.g. between metric and imperial units or between different currencies.

Solve straightforward problems using compound measures, e.g., problems involving rate of flow.

Draw and use distance-time graphs to calculate average speed (link to calculating gradients in Unit 1).

Interpret information shown in travel graphs.

Draw travel graphs from given data.

Class activity: Draw a travel graph for the journey to and from school. Answer a set of questions about the journey, e.g. what is the 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 time in hours).

Revise how to calculate the area of a rectangle and the area of a right angled triangle [Further work on this is completed in Unit 17].

Draw and use speed-time graphs to calculate acceleration and deceleration.

Information on speed, distance and time at http://www.mathforum.org/dr.math/faq/faq.distance.html


IGCSE Mathematics by Karen MorrisonPg 88 – 98Pg 211 - 217

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Topics /


Resources

Use straightforward examples to show that the area under a linear speed-time graph is equivalent to the distance travelled.

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24610552 maths igcse scheme of work 0580 2010year10

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