Klb Mathematics Book 2 Teachers Guide Pdf

KCSE Mathematics Syllabus

Mathematics General Objectives

By the end of the course, the learner should be able to:

1. develop a positive attitude towards learning Mathematics;

2. perform mathematical operations and manipulations with confidence, speed and accuracy;

3. think and reason precisely, logically and critically in any given situation;

4. develop investigative skills in Mathematics;

5. identify, concretise, symbolise and use Mathematical relationships in everyday life;

6. comprehend, analyse, synthesise, evaluate and make generalizations so as to solve Mathematical problems;

7. collect, organise, represent, analyse, interprete data and make conclusions and predictions from its results;

8. apply mathematical knowledge and skills to familiar and unfamiliar situations;

9. appreciate the role, value and use of Mathematics in society;

10. develop a willingness to work collaboratively;

11. acquire knowledge and skills for further education and training;

12. communicate mathematical ideas.

1.1.0 Symbols

1.1.1 SI Units will be used throughout this syllabus. Besides the usual operational symbols +, -, x, +, the combined ± will be used

1.1.2 Rational symbols = is equal to. * is not equal to. > is greater than. > is greater than or equal to. < is less than. < is less than or equal to. a:b ratio of a to b. cc varies as. m is congruent to or identical to. is congruent to or identical to. ....— approximately equal to. <=> is equivalent implies therefore

2.1.0 Numbers

2.1.1 Natural Numbers

i) Specific Objectives The learner should be able to:

(a) identify, read and write natural numbers in symbols and words;

(b) round off numbers to the nearest tens, hundreds, thousands, millions and billions;

(d) solve word problems involving natural numbers.

ii) Content

(a) Place values of numbers

(b) Rounding off numbers to the nearest tens, hundreds, thousands, millions and billions

(d) Even numbers

(e) Prime numbers

(f) Word problems involving natural numbers.

2.1.2 Factors

(i) Specific Objectives The learner should be able to:

(a) express composite numbers in factor form;

(b) express composite numbers as product of prime factors,

(ii) Content

(a) Factors of composite numbers

(b) Prime factors

2.1.3 Divisibility Tests

(i) Specific Objectives The learner should be able to test the divisibility of numbers by 2, 3, 4, 5, 6, 8, 9, 10 and 11.

(ii) Content Divisibility test of numbers by 2, 3, 4, 5, 6, 8, 9, 10 and 11

2.1.4 Greatest Common Divisor (GCD)/highest Common Factor (HCF)

(i) Specific Objectives The learner should be able to:

(a) find the GCD/HCF of a set of numbers;

(b) apply GCD to real life situations.

(ii) Content

(a) GCD of a set of numbers

(b) Application of GCD/HCF to real life situations

2.1.5 Least Common Multiple (LCM)

(i) Specific Objectives The learner should be able to:

(a) list multiples of numbers;

(b) find the LCM of a set of numbers;

(ii) Content

(a) Multiples of a number

(b) LCM of a set of numbers

2.1.6 Integers

(i) Specific Objectives The learner should be able to:

(a) define integers;

(b) identify integers on a number line;

(d) work out combined operations on integers in the correct order;

(e) apply knowledge of integers to real life situations.

(ii) content

(a) Integers

(b) The number line

(d) Order of operations

(e) Application to real life situations

2.1.7 Fractions

(i) Specific Objectives The learner should be able to:

(a) identify proper and improper fractions and mixed numbers;

(b) convert mixed numbers to improper fractions and vice versa;

(d) perform the four basic operations on fractions;

(e) carry out combined operations on fractions in the correct order;

(f) apply the knowledge of fractions to real life situations.

(ii) Content

(a) Fractions

(b) Proper, improper fractions and mixed numbers

(d) Comparing fractions

(e) Operations on fractions

(f) Order of operations on fractions

(g) Word problems involving fractions in real life situations.

2.1.8 Decimals

(i) Specific Objectives The learner should be able to:

(a) convert fractions into decimals and vice versa;

(b) identify recurring decimals;

(d) round off a decimal number to the required number of decimal places.

(e) write numbers in standard form;

(f) perform the four basic operations on decimals;

(g) carry our operations in the correct order,

(h) apply the knowledge of decimals to real life situations.

(ii) Content

(a) Fractions and decimals

(b) Recurring decimals

(d) Decimal places

(e) Standard form

(f) Operations on decimals

(g) Order of operations

(h) Real life problems involving decimals.

2.1.9 Squares and Square Roots

(i) Specific Objectives The learner should be able to:

(a) find squares of numbers by multiplication;

(b) find squares from tables;

(d) find square root from tables.

(ii) Content

(a) Squares by multiplication

(b) Squares from tables

(d) Square roots from tables.

2.1.10 Cubes and Cube Roots

(i) Specific Objectives The learner should be able to: (a) find the cube of a number by multiplication;

(b) find the cube root of a number by factor method;

(d) evaluate expressions involving cubes and cube roots;

(e) apply the knowledge of cubes and cube roots in real life situations.

(ii) Content

(a) Cubes of numbers by multiplication

(b) Cubes from tables

(d) Evaluation of cube and cube root expressions

(e) Application of cubes and cube roots to real life situations.

2.1.11 Reciprocals

(i) Specific Objectives The learner should be able to:

(a) find reciprocals of numbers by division;

(b) find reciprocals of numbers from tables;

(ii) Content

(a) Reciprocals of numbers by division

(b) Reciprocals of numbers from tables

2.1.12 Rates, Ratios, Percentages and Proportion

(i) Specific Objectives The learner should be able to:

(a) define rates;

(b) solve problems involving rates;

(d) compare two or more quantities using ratios;

(e) change quantities in a given ratio;

(t) compare two or more ratios;

(g) represent and interpret proportional parts;

(h) recognise direct and inverse proportions;

(i) solve problems involving direct and inverse proportions;

(j) convert fractions and decimals to percentages and vice-versa;

(k) calculate percentage change in a given quantity;

(1) apply rates, ratios, percentages to real life situations and proportion.

(ii) Content

(a) Rates

(b) Solving problems involving rates

(d) Comparing quantities using ratio

(e) Increase and decrease in a given ratio

(f) Comparing ratios

(g) Proportion: direct and inverse

(h) Solve problems on direct and inverse proportions

(i) Fractions and decimals as percentages

(j) Percentage increase and decrease

(k) Application of rates, ratios, percentages and proportion to real life situations.

2.1.13 Compound Proportions and Rates of Work

(i) Specific Objectives By the end of the topic the learner should be able to:

(a) solve problems involving compound proportions using unitary and ratio methods;

(b) apply ratios and proportions to real life situations;

(ii) Content

(a) Proportional parts

(b) Compound proportions

(d) Proportions applied to mixtures.

2.1.14 Linear Motion

(i) Specific Objectives By the end of the topic the learner should be able to:

(a) define displacement, speed, velocity and acceleration;

(b) distinguish between

(i) distance and displacement,

(ii) speed and velocity;

(d) plot and draw graphs of linear motion (distance and velocity time graphs);

(e) interpret graphs of linear motion;

(f) define relative speed;

(g) solve problems involving relative speed.

(ii) Content

(a) Displacement, velocity, speed and acceleration

(b) Determining velocity and acceleration

(d) Distance — time graph

(e) Velocity time graph

(f) Interpretation of graphs of linear motion

(g) Solving problems involving relative speed.

2.1.15 Indices and Logarithms

(i) Specific Objectives By the end of the topic the learner should be able to:

(a) define indices (powers);

(b) state the laws of indices;

(d) relate the powers of 10 to common logarithms;

(e) use the tables of common logarithms and anti-logarithms in computation.

(ii) Content

(a) Indices (powers) and base

(b) Laws of indices (including positive integers, negative integers and fractional indices)

(d) Common logarithms:

i) characteristics and

ii) mantissa

(e) Logarithm tables

(f) Application of common logarithms in multiplication, division , powers and roots.

2.1.6 Further Logarithms

(i) Specific Objectives The learner should be able to:

(a) derive logarithmic relation from index form and vice versa;

(b) state the laws of logarithms;

(d) apply laws of logarithms for further computations.

(ii) Content

(a) Logarithmic notation (eg. an=b, log ab=n)

(b) The laws of logarithms: log (AB) = log A + log B, log(A÷B) = log A-log B and Log An= n x log A.

(d) Solution of logarithmic equations

(e) Further computation using logarithmic laws.

2.1.17 Approximations and Errors

(i) Specific Objectives The learner should be able to:

(a) Perform various computations using a calculator;

(b) make reasonable approximations and estimations of quantities in computations and measurements;

(d) define absolute, relative, percentage, round-off and truncation errors;

(e) determine possible errors made from computations;

(f) find maximum and minimum errors from operations.

(ii) Content

(a) Computing using calculators

(b) Estimations and approximations

(d) Absolute, relative, percentage, round-off (including significant figures) and truncation errors

(e) Propagation of errors from simple calculations

(f) Maximum and minimum errors.

2.2.0 Measurement (1)

2.2.1 Length

(i) Specific Objectives The learner should be able to:

(a) state the units of measuring length;

(b) convert units of length from one form to another;

(d) find the perimeter of a plane figure and circumference of a circle.

(ii) Content

(a) Units of length (mm, cm, m, km)

(b) Conversion of units of length from one form to another

(d) Perimeter

(e) Circumference (include length of arcs).

2.2.2 Area

(i) Specific Objectives The learner should be able to:

(a) state units of area;

(b) convert units of area from one form to another;

(d) estimate the area of irregular plane figures by counting squares;

(e) calculate the surface area of cubes, cuboids and cylinders.

(ii) Content

(a) Units of area (0112, m², km², Ares, ha)

(b) Conversion of units of area

(d) Area of irregular plane shapes

(e) Surface area of cubes, cuboids and cylinders.

2.2.3 Pythagoras Theorem

(i) Specific Objectives The learner should be able to:

(a) derive Pythagoras Theorem;

(b) solve problems using Pythagoras Theorem;

(ii) Content

(a) Pythagoras Theorem

(b) Solution of problems using Pythagoras Theorem

2.2.4 Volume and Capacity

(i) Specific Objectives The learner should be able to:

(a) state units of volume;

(b) convert units of volume from one form to another;

(d) state units of capacity;

(e) convert units of capacity from one form to another;

(f) relate volume to capacity;

(g) solve problems involving volume and capacity.

(ii) Content

(a) Units of volume

(b) Conversion of units of volume

(d) Units of capacity

(e) Conversion of units of capacity

(f) Relationship between volume and capacity

(g) Solving problems involving volume and capacity.

2.2.5 Mass, Density and Weight

(i) Specific Objectives The learner should be able to:

(a) define mass;

(b) state units of mass;

(d) define weight;

(e) state units of weight;

(f) distinguish mass and weight;

(g) relate volume, mass and density.

(ii) Content

(a) Mass and units of mass

(b) Weight and units of weight

(d) Problem solving involving real life experiences on mass, volume, density and weight.

2.2.6 Time

(i) Specific Objectives The learner should be able to:

(a) convert units of time from one form to another;

(b) relate the 12 hour and 24 hour clock systems;

(d) solve problems involving travel time tables.

(ii) Content

(a) Units of time

(b) 12 hour and 24 hour clock systems

(d) problems involving travel time tables.

2.3.0 Measurement (2)

2.3.1 Area of a Triangle

(i) Specific Objectives

The learner should be able to:

(a) Derive the formula; Area = V2 ab sin C;

(b) Solve problems involving area of triangles using the formula Area = 1/2 ab sin C;

(ii) Content

(a) Area of triangle A = ab sin C

(b) Area of a triangle A = .J√s(s — a)(s — b)(s —c)

2.3.2 Area of Quadrilaterals and Other Polygons

(i) Specific Objectives The learner should be able to:

(a) find the area of a quadrilateral;

(b) find the area of other polygons (regular and irregular).

(ii) Content

(a) Area of quadrilaterals

(b) Area of other polygons (regular and irregular).

2.3.3 Area of Part of a Circle

(i) Specific Objectives The learner should be able to:

(a) find the area of a sector;

(b) find the area of a segment;

(ii) Content

(a) Area of a sector

(b) Area of a segment

2.3.4 Surface Area of Solids

(i) Specific Objectives The learner should be able to:

(a) find the surface area of a prism;

(b) find the surface area of a pyramid;

(d) find the surface area of a frustum;

(e) find the surface area of a sphere and a hemisphere.

(ii) Content Surface area of prisms, pyramids, cones, frustums and spheres.

2.3.5 Volume of Solids

(i)Specific Objectives The learner should be able to:

(a) find the volume of a prism;

(b) find the volume of a pyramid;

(d) find the volume of a frustum;

(e) find the volume of a sphere and a hemisphere.

(ii) Content

Volumes of prisms, pyramids, cones, frustums and spheres.

2.4. 0 Algebra

2.4.1 Algebraic Expressions

(i) Specific Objectives The learner should be able to:

(a) use letters to represent numbers;

(b) write statements in algebraic form;

(d) factorise an algebraic expressions by grouping;

(e) remove brackets from algebraic expressions;

(f) evaluate algebraic expressions by substituting numerical values;

(g) apply algebra in real life situations.

(ii) Content

(a) Letters for numbers

(b) Algebraic fractions

(d) Factorisation by grouping

(e) Removal of brackets

(f) Substitution and evaluation

(g) Problem solving in real life situations.

2.4.2 Equation of Straight Lines

(i) Specific Objectives The learner should be able to:

(a) define gradient of a straight line;

(b) determine the gradient of a straight line through known points;

(d) express a straight line equation in the form y = mx + c;

(e) interpret the equation y = mx + c;

(f) find the x- and y- intercepts from an equation of a line;

(g) draw the graph of a straight line using gradient and x- and y- intercepts;

(h) state the relationship of gradients of perpendicular lines;

(i) state the relationship of gradients of parallel lines;

(j) apply the relationship of gradients of perpendicular and parallel lines to get equations of straight lines.

(ii) Content

(a) Gradient of a straight line

(b) Equation of a straight line

(d) The x and y intercepts of a line

(e) The graph of a straight line

(f) Perpendicular lines and their gradients

(g) Parallel lines and their gradients

(h) Equations of parallel and perpendicular lines.

2.4.3 Linear Equations

(i) Specific Objectives The learner should be able to:

(a) solve linear equations in one unknown;

(b) solve simultaneous linear equations by substitution and elimination;

(ii) Content

(a) Linear equations in one unknown

(b) Simultaneous linear equations

2.4.4 Quadratic Expressions and Equations (1)

(i) Specific Objectives The learner should be able to:

(a) expand algebraic expressions that form quadratic equations;

(b) derive the three quadratic identities;

(d) factorise quadratic expressions including the identities;

(e) solve quadratic equations by factorization;

(f) form and solve quadratic equations.

(ii) Content

(a) Expansion of algebraic expressions to form quadratic expressions of the form aX2+b X+c, where a, b and c are constants

(b) The three quadratic identities: (a+b)2 = a2 + 2ab+b2 (a-b)2 = a2 - 2ab+ b2 (a-b) (a+b) = a2-b2

(d) Factorisation of quadratic expressions

(e) Solve quadratic equations by factorization

(f) Form and solve quadratic equations.

2.4.5 Quadratic Expressions and Equations (2)

(i) Specific Objectives The learner should be able to:

(a) factorise quadratic expressions;

(b) identify perfect squares;

(d) solving quadratic equations by completing the square;

(e) derive the quadratic formula;

(f) solve quadratic equations using the formula;

(g) form and solve quadratic equations from roots and given situations;

(h) make tables of values from a quadratic relation;

(i) draw the graph of a quadratic relation;

(j) solve quadratic equations using graphs;

(k) solve simultaneous equations (one linear and one quadratic) analytically and graphically;

(1) apply the knowledge of quadratic equations to real life situations.

(ii) Content

(a) Factorisation of quadratic expressions

(b) Perfect squares

(d) Solution of quadratic equations by completing the square

(e) Quadratic formula x = -b ± ,-,tac.) 2a

(f) Solution of quadratic equations using the formula.

(g) Formation of quadratic equations and solving them

(h) Tables of values for a given quadratic relation

(i) Graphs of quadratic equations

(j) Simultaneous equation — one linear and one quadratic

(k) Application of quadratic equation to real life situation.

2.4.6 Linear Inequalities (1)

(i) Specific Objectives The learner should be able to:

(a) identify and use inequality symbols;

(b) illustrate inequalities on the number line;

(d) represent the linear inequalities graphically;

(e) solve the linear inequalities in two unknowns graphically; (f) form simple linear inequalities from inequality graphs.

(ii) Content

(a) Inequalities on a number line

(b) Simple and compound inequality statements e.g. z>aand 2, ZbaZ x Lb

(d) Graphical representation of linear inequalities

(e) Graphical solutions of simultaneous linear inequalities

(f) Simple linear inequalities from inequality graphs.

2.5.0 Linear Programming

2.5.1 Linear Inequalities (2)

(i) Specific Objectives The learner should be able to:

(a) form linear inequalities based on real life situations;

(b) represent the linear inequalities on a graph;

(d) apply linear programming to real life situations.

(ii) Content

(a) Formation of linear inequalities

(b) Analytical solutions of linear inequalities

(d) Optimisation (include objective function)

(e) Application of quadratic equations to real life situations.

2.6.0 Surds

(i) Specific Objectives The learner should be able to:

(a) define rational and irrational numbers;

(b) simplify expressions with surds;

(ii) Content

(a) Rational and irrational numbers

(b) Simplification of surds

2.7.0 Sequences and Series

(i) Specific Objectives The learner should be able to:

(a) identify simple number patterns;

(b) define a sequence;

(d) determine a term in a sequence;

(e) recognise arithmetic and geometric sequences;

f) define a series;

(g) recognise arithmetic and geometric series (Progression);

(h) derive the formula for partial sum of an arithmetic and geometric series (Progression);

(i) apply A.P and G.P to solve problems in real life situations.

(ii) Content

(a) Simple number patterns

(b) Sequences

(d) Geometric sequence

(e) Determining a term in a sequence

f) Arithmetic progression (A.P)

(g) Geometric progression (G.P)

(h) Sum of an A.P

(i) Sum of a G.P (exclude sum to infinity)

(j) Application of A.P and G.P to real life situations.

2.8.0 Binomial Expansions

(i) Specific Objectives The learner should be able to:

(a) expand binomial expressions up to the power of four by multiplication;

(b) building up — Pascal's Triangle up to the eleventh row;

(d) apply binomial expansion in numerical cases.

(ii) Content

(a) Binomial expansion up to power four

(b) Pascal's triangle

(d) Computation using binomial expansion

(e) Evaluation of numerical cases using binomial expansion.

2.9.0 Formulae and Variations

(i) Specific Objectives The learner should be able to:

(a) rewrite a given formula by changing the subject;

(b) define direct, inverse, partial and joint variations;

(d) form and solve equations involving variations;

(e) draw graphs to illustrate direct and inverse proportions;

(f) use variations to solve everyday life problems.

(ii) Content

(a) Change of the subject

(b) Direct, inverse, partial and joint variations

(d) Graphs of direct and inverse proportion (e) Equations on variation from real life situations.

2.10.0 Geometry

2.10.1 Angles and Plane Figures

(i) Specific Objectives The learner should be able to:

(a) name and identify types of angles;

(b) solve problems involving angles on a straight line;

(d) solve problems involving angles on a transversal cutting parallel lines,

(e) state angle properties of polygons;

(f) solve problems involving angle properties of polygons;

(g) apply the knowledge of angle properties to real life situations.

(ii) Content

(a) types of angles

(b) angles on a straight line

(d) angles on a transversal (corresponding, alternate and allied angles)

(e) angle properties of polygons

(f) application to real life situations.

2.10.2 Geometrical Constructions

(i) Specific Objectives The learner should be able to: (a) use a ruler and compasses only to:

i) construct a perpendicular bisector of a line;

ii) construct an angle bisector;

iii) construct a perpendicular to a line from a given point;

iv) construct a perpendicular to a line through a given point on the line;

v) construct angles whose values are multiples of 71/4

vi) construct parallel lines;

vii) divide a line proportionally;

(b) use a ruler and a set square to construct parallel lines, divide a line proportionally, and to construct perpendicular lines;

(d) construct irregular polygons using a ruler, compasses and protractor.

(ii) Content

(a) Construction of lines and angles using a ruler and compasses only

(b) Construction of perpendicular and parallel lines using a ruler and a set square only

(d) Construction of regular polygons (upto a hexagon)

(e) Construction of irregular polygons (upto a hexagon).

2.10.3 Loci

(i) Specific Objectives The learner should be able to:

(a) define Locus;

(b) describe common types of Loci;

i) loci involving inequalities;

ii) loci involving chords;

iii) loci involving points under given conditions;

iv) intersecting loci.

(ii) Content

(a) common types of Loci

(b) perpendicular bisector loci

(d) angle bisector loci

(e) other loci under given condition including intersecting loci

f) loci involving inequalities

(g) loci involving chords (constant angle loci).

2.10.4 Scale Drawing

(i) Specific Objectives The learner should be able to:

(a) interpret a given scale;

(b) choose and use an appropriate scale;

(d) state the bearing of one point from another;

(e) locate a point using bearing and distance;

f) determine angles of elevation and depression;

(g) solve problems involving bearings elevations and scale drawing;

(h) apply scale drawing in simple surveying.

(ii) Content

(a) Types of scales

(b) Choice of scales

(d) Bearings

(e) Bearings, distance and locating points

f) Angles of elevation and depression

(g) Problems involving bearings, scale drawing, angles of elevation and depression

(h) Simple surveying techniques.

2.10. 5 Common Solids

(i) Specific Objectives The learner should be able to:

(a) identify and sketch common solids;

(b) sketch and accurately draw nets of solids;

(d) calculate surface area of solids from nets;

(e) find distances between two points on a solid.

(ii) Content

(a) Common solids, eg cubes, cuboids, pyramids, prisms, cones, spheres, cylinders etc.

(b) Sketches of solids

(d) Models of solids from nets

(e) Surface area of solids

(f) Distance between two points on the surface of solid.

2.10.6 Three Dimensional Geometry

(i) Specific Objectives The learner should be able to:

(a) state the geometric properties of common solids;

(b) identify projection of a line onto a plane;

(d) calculate the length between two points in three dimensional geometry;

(e) identify and calculate the angle between

(i) two lines;

(ii) a line and a plane;

(ii) two planes.

(ii) Content

(a) Geometrical properties of common solids

(b) Skew lines and projection of a line onto a plane

(d) The angle between

i) a line and a line

ii) a line a plane

iii) a plane and a plane

iv) angles between skewlines.

2.10.7 Angle Properties of a Circle

(i) Specific Objectives The learner should be able to:

(a) identify an arc, chord and segment;

(b) relate and compute angle subtended by an arc at the circumference;

(d) state the angle in the semi-circle;

(e) state the angle properties of a cyclic quadrilateral;

f) find and compute angles of a cyclic quadrilateral.

(i) Content

(a) Arc, chord and segment

(b) Angle subtended by the same arc at the circumference

(d) Angle in a semi-circle

(e) Angle properties of a cyclic quadrilateral

f) Finding angles of a cyclic quadrilateral.

2.10.8 Circles: Chords and Tangents

(i) Specific Objectives The learner should be able to:

(a) calculate length of an arc and a chord;

(b) calculate lengths of tangents and intersecting chords;

(d) construct tangent to a circle;

(e) construct direct and transverse common tangents to two circles;

(f) relate angles in alternate segment;

(g) construct circumscribed, inscribed and escribed circles;

(h) locate centroid and orthocentre of a triangle;

(i) apply knowledge of circles, tangents and chords to real life situations.

(ii) Content

(a) Arcs, chords and tangents

(b) Lengths of tangents and intersecting chords

(d) Construction of tangents to a circle

(e) Direct and transverse common tangents to two circles

(t) Angles in alternate segment

(g) Circumscribed, inscribed and escribed circles

(h) Centroid and orthocentre

(i) Application of knowledge of tangents and chords to real life situations.

2.1 1.0 Graphs

2.1 1.1 Co-ordinates and Graphs

(i) Specific Objectives The learner should be able to:

(a) draw and label the complete cartesian plane;

(b) locate and plot points on the cartesian plane;

(d) make a table of values for a given linear relation,

(e) use values to draw a linear graph;

(t) solve simultaneous linear equations graphically:

(g) draw, read and interprete graphs.

(ii) Content

(a) Cartesian plane

(b) Cartesian co-ordinate

(d) Choice of appropriate scale

(e) Table of values for a given linear relation

(f) Linear graphs

(g) Graphical solutions of simultaneous linear equations

(h) Interpretation of graphs.

2.11.2 Graphical Methods

(i) Specific Objectives The learner should be able to:

(a) makes a table of values from given relations;

(b) use the table of values to draw the graphs of the relations;

(d) interpret information from graphs;

(e) draw and interpret graphs from empirical data;

f) solve cubic equations graphically;

(g) draw the line of best fit;

(h) identify the equation of a circle;

(i) find the equation of a circle given the centre and the radius;

(j) determine the centre and radius of a circle and draw the circle on a cartesian plane.

(ii) Content

(a) Tables and graphs of given relations

(b) Graphs of cubic equations

(d) Average rate of change

(e) Instantaneous rate of change

f) \empirical data and their graphs

(g) the line of best fit

(h) equation of a circle

finding of the equation of a circle

(j) determining of the centre and radius of a circle.

2.12.0 Trigonometry

2.12.1 Trigonometry (1)

(i) Specific Objectives The learner should be able to:

(a) define tangent, sine and cosine ratios from a right angled triangle;

(b) read and use tables of trigonometric ratios;

(c) use sine, cosine and tangent in calculating lengths and angles;

(d) establish and use the relationship of sine and cosine of complimentary angles;

(e) relate the three trigonometric ratios;

(f) determine the trigonometric ratios of special angles 30°, 45°, 60° and 90° without using tables;

(g) read and use tables of logarithms of sine, cosine and tangent;

(h) apply the knowledge of trigonometry to real life situations.

(ii) Content

(a) Tangent, sine and cosine of angles

(b) Trigonometric tables

(c) Angles and sides of a right angled triangle

(d) Sine and cosine of complimentary angles

(e) Relationship between tangent, sine and cosine

(f) Trigonometric ratios of special angles 30°, 45°, 60° and 90°

(g) Logarithms of sines, cosines and tangents

(h) Application of trigonometry to real life situations.

2.12.2 Trigonometry (2)

(i) Specific Objectives The learner should be able to:

(a) define and draw the unit circle;

(b) use the unit circle to find trigonometric ratios in terms of co-ordinates of points for 0 < 0 < 360';

(c) find trigonometric ratios of negative angles;

(d) find trigonometric ratios of angles greater than the unit circle,

(e) use mathematical tables and Whits to find trigonometric ratios of angles in the range 0 < 0 60

(f) define radian measure;

(g) draw graphs of trigonometric functions; y = sin x, y = cos x and y = tan x using degrees and radians;

(h) derive the sine rule;

(i) derive the cosine rule;

(j) apply the sine and cosine rule to solve triangles (sides, angles and area);

(k) apply the knowledge of sine and cosine rules in real life situations.

(ii) Content

(a) The unit circles

(b) Trigonometric rations from the unit circle

(c) Trigonometric ratios of angles greater than 360° and negative angles

(d) Use of trigonometric tables and calculations

(e) Radian measure

(f) Simple trigonometric graphs

(g) Derivation of sine and cosine rule

(h) Solution of triangles

(i) Application of sine and cosine rule to real situation.

2.12.3 Trigonometry (3)

(i) Specific Objectives The learner should be able to:

(a) recall and define trigonometric ratios;

(b) derive trigonometric identity sin2x+cos2x = 1;

(c) draw graphs of trigonometric functions;

(d) solve simple trigonometric equations analytically and graphically;

(e) deduce from the graph amplitude, period, wavelength and phase angles.

(ii) Content

(a) Trigonometric ratios

(b) Deriving the relation sin2x+cos2x =1

(c) Graphs of trigonometric functions of the form y=sinx y=cosx, y=tanx y=asinx, y=acosx, y=atanx y = a sin bx, y = a cos bx y=atanbx y = a sin(bx ± 0) y = a cos(bx ± 0) y= a tan(bx ± 0)

(d) simple trigonometric equation

(e) amplitude, period, wavelength and phase angle of trigonometric functions.

2.13.0 Commercial Arithmetic

2.13.1 Commercial Arithmetic

(1) (i) Specific Objectives The learner should be able to:

(a) state the currencies of different countries;

(b) convert currency from one form into another given the exchange rates;

(c) calculate profit and loss;

(d) express profit and loss as percentages;

(e) calculate discount and commission;

(f) express discount and commission as percentage.

(ii) Content

(a) Currency

(b) Current currency exchange rates

(c) Currency conversion

(d) Profit and loss

(e) Percentage profit and loss

(f) Discounts and commissions

(g) Percentage discounts and commissions.

2.13.2 Commercial Arithmetic

(2) (i) Specific Objectives The learner should be able to:

(a) define principal, rate and time in relation to interest;

(b) calculate simple interest using simple interest formula;

(c) calculate compound interest using step by step method;

(d) derive the compound interest formula;

(e) apply the compound interest formula for calculating interest;

(f) define appreciation and depreciation;

(g) use compound interest formula to calculate appreciation and depreciation;

(h) calculate hire purchase;

(i) calculate income tax given the income tax bands.

(ii) Content

(a) Principal rate and time

(b) Simple interest

(c) Compound interest using step by step method

(d) Derivation of compound interest formula

(e) Calculations using the compound interest formula

(f) Appreciation and depreciation

(g) Calculation of appreciation and depreciation using the compound interest formula

(h) Hire purchase

(i) Income tax.

2.14.0 Statistics and Probability

2.14.1 Statistics (1)

(i) Specific Objectives The learner should be able to:

(a) define statistics;

(b) collect and organise data;

(c) draw a frequency distribution table;

(d) group data into reasonable classes;

(e) calculate measures of central tendency;

(f) represent data in form of linegraphs, bargraphs, pie-charts, pictogram, histogram and frequency polygons;

(g) interpret data from real life situations.

(ii) Content

(a) Definition of statistics

(b) Collection and organization of data

(c) Frequency distribution tables (for grouped and ungrouped data)

(d) Grouping data

(e) Mean, mode and median for ungrouped and grouped data

f) Representation of data: Line graph, Bar graph, Pie chart, Pictogram, Histogram, Frequency polygon, Interpretation of data..

2.14.2 Statistics (2)

(i) Specific Objectives The learner should be able to:

(a) state the measures of central tendency;

(b) calculate the mean using the assumed mean method;

(c) make cumulative frequency table;

(d) estimate the median and the quartiles by

(i) calculation and

(ii) using ogive;

(e) define and calculate the measures of dispersion: range, quartiles, interquartile range, quartile deviation, variance and standard deviation

(f) interpret measures of dispersion.

(ii) Content

(a) Mean from assumed mean;

(b) Cumulative frequency table

(c) Ogive

(d) Meadian

(e) Quartiles

(f) Range

(g) Interquartile range

(h) Quartile deviation

(i) Variance

(j) Standard deviation

2.14.3 Probability

(i) Specific Objectives The learner should be able to:

(a) define probability;

(b) determine probability from experiments and real life situations;

(c) construct a probability space;

(d) determine theoretical probability;

(e) differentiate between discrete and continuous probability;

f)differentiate mutually exclusive and independent events;

(g) state and apply laws of probability;

(h) use a tree diagram to determine probabilities.

(ii) Content

(a) Probability

(b) Experimental probability

(c) Range of probability measure 0 P (x) < 1

(d) Probability space

(e) Theoretical probability

(f) Discrete and continuous probability (simple cases only)

(g) Combined events (mutually exclusive and independent events)

(h) Laws of probability

(i) The tree diagrams.

2.15.0 Vectors

2.15.1 Vectors (1)

(i) Specific Objectives The learner should be able to:

(a) define vector and scalar;

(b) use vector notation;

(c) represent vectors both single and combined geometrically;

(d) identify equivalent vectors;

(e) add vectors;

(f) multiply vectors by scalars;

(g) define position vector and column vector;

(h) find magnitude of a vector;

(i) find mid-point of a vector;

(j) define translation as a transformation.

2.17.2 Matrices and Transformations

(i) Specific Objectives The learner should be able to:

(a) relate image and object under a given transformation on the cartesian plane;

(b) determine the matrix of a transformation;

(c) perform successive transformations;

(d) determine and identify a single matrix for successive transformation;

(e) relate identity matrix and transformation;

(f) determine the inverse of a transformation;

(g) establish and use the relationship between area scale factor and determinant of a matrix;

(h) determine shear and stretch transformations;

(i) define and distinguish isometric and non-isometric transformation;

(j) apply transformation to real life situations.

(ii) Content

(a) transformation on the Cartesian plane

(b) identification of transformation matrix

(c) successive transformations

(d) single matrix of transformation for successive transformations

(e) identity matrix and transformation

f) inverse of a transformations

(g) area scale factor and determinant of a matrix

(h) shear and stretch (include their matrices)

(i) isometric and non-isometric transformations

(j) application of transformation to real life situations.

2.18.0 Navigation

2.18.1 Longitude and Latitudes

(i) Specific Objectives The learner should be able to:

(a) define the great and small circles in relation to a sphere (including the earth);

(b) establish the relationship between the radii of small and great circles;

(c) locate a place on the earth's surface in terms of latitude and longitude;

(d) calculate the distance between two points along the great circles and small circles (longitude and latitude) in nautical miles (nm) an kilometers (km);

(e) calculate time in relation to longitudes;

(f) calculate speed in knots and kilometres per hour.

(ii) Content

(a) Latitude and longitude (great and small circles)

(b) The Equator and Greenwich Meridian

(c) Radii of small and great circles

(d) Position of a place on the surface of the earth

(e) Distance between two points along the small and great circles in nautical miles and kilometres

(f) Distance in nautical miles and kilometres along a circle of latitude

(g) Time and longitude

(h) Speed in knots and Kilometres per hour.

2.19.0 Area Approximation

(i) Specific Objectives The learner should be able to:

(a) approximate the area of irregular shapes by counting techniques;

(b) derive the trapezium rule;

(c) apply trapezium rule to approximate areas of irregular shapes;

(d) apply trapezium rule to estimate areas under curves;

(e) derive the mid-ordinate rule;

(f) apply mid-ordinate rule to approximate area under curves.

(ii) Content

(a) Area by counting techniques

(b) Trapezium rule

(c) Area using trapezium rule

(d) Mid-ordinate

(e) Area by the mid-ordinate rule.

2.20 0 Elementary Calculus

2.20.1 Differentiation

(i) Specific Objectives The learner should be able to:

(a) find average rates of change and instantaneous rates of change;

(b) find the gradient of a curve at a point using tangent;

(c) relate the delta notation to rates of change;

(d) find the gradient function of a function of the form y = xn (n is a positive integer);

(e) define derivative of a function, derived function of a polynomial and differentiation;

(f) determine the derivative of a polynomial;

(g) find equations of tangents and normals to the curves;

(h) sketch a curve;

i) apply differentiation in calculating distance, velocity and acceleration;

(j) apply differentiation in finding maxima and minima of a function.

(ii) Content

(a) Average and instantaneous rates of change

(b) Gradient of a curve at a point

(c) Gradient of y= x (where n is a positive integer)

(d) Delta notation ( A) or 8

(e) Derivative of a polynomial

(f) Equations of tangents and normals to the curve

(g) Stationery points

(h) Curve sketching

(i) Application of differentiation in calculation of distance, velocity and acceleration

(j) Maxima and minima.

2.20.2 Integration

(i) Specific Objectives The learner should be able to:

(a) carry out the process of differentiation;

(b) interpret integration as a reverse process of differentiation;

(c) relate integration notation to sum of areas of trapezia under a curve;

(d) integrate a polynomial;

(e) apply integration in finding the area under a curve;

(f) apply integration in kinematics.

(ii) Content

(a) Differentiation

(b) Reverse differentiation

(c) Integration notation and sum of areas of trapezia

(d) Indefinite and definite integrals

(e) Area under a curve by integration

(f) Application in kinematics.

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