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# Mathematics series

Course Level

### Intermediate

5,000.00 4,000.00

8 Lessons

#### ( a ) Number bases

( i ) conversion of numbers from one base to another

( ii ) Basic operations on number bases

Conversion from one base to base 10 and vice versa. Conversion from one base to another base .

Addition, subtraction and multiplication of number bases.

#### (b) Modular Arithmetic

(i) Concept of Modulo Arithmetic.

(ii) Addition, subtraction and multiplication operations in modulo arithmetic.

(iii) Application to daily life

Interpretation of modulo arithmetic e.g. 6 + 4 = k(mod7), 3 x 5 = b(mod6), m = 2(mod 3), etc.

Relate to market days, clock,shift duty, etc.

#### ( c ) Fractions, Decimals and Approximations

(i) Basic operations on fractions and decimals. (ii) Approximations and significant figures.

Approximations should be realistic e.g. a road is not measured correct to the nearest cm

#### WAEC Syllabus For Mathematics (General) | ( d ) Indices

( i ) Laws of indices

( ii ) Numbers in standard form ( scientific notation)

e.g. a x x a y = a x + y , a x ÷ a y = a x – y , ( a x ) y = a xy , etc where x , y are real numbers and a ≠0. Include simple examples of

negative and fractional indices.

Expression of large and small numbers in standard form e.g. 375300000 = 3.753 x 108 0.00000035 = 3.5 x 10-7 Use of tables of squares, square roots and reciprocals is accepted

#### ( e) Logarithms

( i ) Relationship between indices and logarithms e.g. y = 10 k implies log10 y = k . ( ii ) Basic rules of logarithms e.g. log10( pq ) = log10 p + log10 q
log10( p / q ) = log10 p – log10 q
log10 p n = n log10 p . (iii) Use of tables of logarithms and antilogarithms.

Calculations involving multiplication, division, powers and roots

#### WAEC Syllabus For Mathematics (General) | ( f ) Sequence and Series

(i) Patterns of sequences.

(ii) Arithmetic progression (A.P.) Geometric Progression (G.P.)

Determine any term of a given sequence. The notation Un = the nth termof a sequence may be used.

Simple cases only, including word problems. (Include sum for A.P. and exclude sum for G.P

#### WAEC Syllabus For Mathematics (General) | ( g ) Sets

(i) Idea of sets, universal sets, finite and infinite sets, subsets, empty sets and disjoint sets. Idea of and notation for union, intersection and complement of sets.

(ii) Solution of practical problems involving classification using Venn diagrams

Notations: ℰ, ⊂, ∪, ∩, { }, ∅, P’( the compliment of P).

♦• properties e.g. commutative, associative and distributive

Use of Venn diagrams restricted to at most 3 sets

#### WAEC Syllabus For Mathematics (General) | ( h ) Logical Reasoning

Simple statements. True and false statements. Negation of statements, implications.

Use of symbols: ⟹,⇐, use of Venn diagrams

(i) Positive and negative integers, rational numbers

The four basic operations on rational numbers

Match Natural numbers with points on the number line

#### ( j ) Surds (Radicals

Simplification and rationalization of simple surds

Surds of the form √, a√ and a ±√where a is a rational number and b is a positive integer. Basic operations on surds (exclude surd of the form √).

#### ( k ) Matrices and Determinants

( i ) Identification of order, notation and types of matrices.

( ii ) Addition, subtraction, scalar multiplication and multiplication of matrices.

( iii ) Determinant of a matrix

Not more than 3 x 3 matrices. Idea of columns and rows.

Restrict to 2 x 2 matrices.

Application to solving simultaneous linear equations in two variables. Restrict to 2 x 2 matrices.

#### ( l ) Ratio, Proportions and Rates

Ratio between two similar quantities. Proportion between two or more similar quantities.

Financial partnerships, rates of work, costs, taxes, foreign exchange, density (e.g. population), mass, distance, time and speed

Relate to real life situations.

Include average rates, taxes e.g. VAT, Withholding tax, etc

#### WAEC Syllabus For Mathematics (General) | ( m ) Percentages

Simple interest, commission, discount, depreciation, profit and loss, compound interest, hire purchase and percentage error

Limit compound interest to a maximum of 3 years

∗( n) Financial Arithmetic

( i ) Depreciation/ Amortization.

( ii ) Annuities

(iii ) Capital Market Instruments

Definition/meaning, calculation of depreciation on fixed assets, computation of amortization on capitalized assets

Definition/meaning, solve simple problems on annuities.

Shares/stocks, debentures, bonds, simple problems on interest on bonds and debentures.

#### WAEC Syllabus For Mathematics (General) | ( o ) Variation

Direct, inverse, partial and joint variations.

Expression of various types of variation in mathematical symbols e.g. direct (z ∝n ), inverse (z ∝ ), etc. Application to simple practical problems

### B. ALGEBRAIC PROCESSES

#### ( a ) Algebraic expressions

(i) Formulating algebraic expressions from given situations

( ii ) Evaluation of algebraic expressions

e.g. find an expression for the cost C Naira of 4 pens at x
Naira each and 3 oranges at y
naira each. Solution: C = 4 x + 3 y

e.g. If x =60 and y = 20, find C . C = 4(60) + 3(20) = 300 naira.

#### WAEC Syllabus For Mathematics (General) | ( b ) Simple operations on algebraic expressions

( i ) Expansion

(ii ) Factorization

(iii) Binary Operations

e.g. ( a + b )( c + d ), ( a + 3)( c – 4), etc.

factorization of expressions of the form ax + ay, a ( b + c ) + d ( b + c ), a 2 – b 2, ax 2 + bx + c where a , b , c
are integers. Application of difference of two squares e.g. 492 – 472 = (49 + 47)(49 – 47) = 96 x 2 = 192.

Carry out binary operations on real numbers such as: a*b = 2a +b-ab etc.

#### WAEC Syllabus For Mathematics (General) | ( c ) Solution of Linear Equations

( i ) Linear equations in one variable

( ii ) Simultaneous linear equations in two variables.

Solving/finding the truth set (solution set) for linear equations in one variable.

Solving/finding the truth set of simultaneous equations in two variables by elimination, substitution and graphical methods. Word problems involving one or two variables

#### ( d ) Change of Subject of a Formula/Relation

( i ) Change of subject of a formula/relation (ii) Substitution

e.g. if =
+ , find v. Finding the value of a variable e.g. evaluating v
given the values of u and f

#### WAEC Syllabus For Mathematics (General) | ( e ) Quadratic Equations

( i ) Solution of quadratic equations

(ii) Forming quadratic equation with given roots.

(iii) Application of solution of quadratic equation in practical problems

Using factorization i.e. ab = 0 ⇒ either a = 0 or b = 0.

By completing the square and use of formula

Simple rational roots only e.g. forming a quadratic equation whose roots are -3 and ⇒ ( x
+ 3)( x – ) = 0.

#### (f) Graphs of Linear and Quadratic functions

(i) Interpretation of graphs, coordinate of points, table of values, drawing quadratic graphs and obtaining roots from graphs.

( ii ) Graphical solution of a pair of equations of the form: y = ax2 + bx + c and y = mx + k

(iii) Drawing tangents to curves to determine the gradient at a given point

Finding: (i) the coordinates of maximum and minimum points on the graph. (ii) intercepts on the axes, identifying axis of symmetry, recognizing sketched graphs.

Use of quadratic graphs to solve related equations e.g. graph of y = x 2 + 5 x + 6 to solve x 2 + 5 x + 4 = 0. Determining the gradient by drawing relevant triangle

#### WAEC Syllabus For Mathematics (General) | ( g ) Linear Inequalities

(i) Solution of linear inequalities in one variable and representation on the number line.

∗(ii) Graphical solution of linear inequalities in two variables.

∗(iii) Graphical solution of simultaneous linear inequalities in two variables

Maximum and minimum values. Application to real life situations e.g. minimum cost, maximum profit, linear programming, etc

#### ( h ) Algebraic Fractions

Operations on algebraic fractions with: ( i ) Monomial denominators

( ii ) Binomial denominators

( ii ) Binomial denominators

Simple cases only e.g. +

= ( x≠0, y≠ 0).
Simple cases only e.g. + = ! ! where a and b
are constants and x ≠ a or b . Values for which a fraction is undefined e.g. “is not defined for x=-3

#### (i) Functions and Relations

Types of Functions

One-to-one, one-to-many, many-to-one, many-to-many. Functions as a mapping, determination of the rule of a given mapping/function

### C. MENSURATION

#### ( a ) Lengths and Perimeters

(i) Use of Pythagoras theorem, ∗sine and cosine rules to determine lengths and distances. (ii) Lengths of arcs of circles, perimeters of sectors and segments. (iii) Longitudes and Latitudes.

No formal proofs of the theorem and rules are required.

Distances along latitudes and Longitudes and their corresponding angles

#### ( b ) Areas

( i ) Triangles and special quadrilaterals – rectangles, parallelograms and trapeziums

(ii) Circles, sectors and segments of circles.

(iii) Surface areas of cubes, cuboids, cylinder, pyramids, right triangular prisms, cones andspheres.

Areas of similar figures. Include area of triangle = ½ base x height and ½absinC. Areas of compound shapes

Relationship between the sector of a circle and the surface area of a cone.

#### ( c ) Volumes

(i) Volumes of cubes, cuboids, cylinders, cones, right pyramids and spheres.

( ii ) Volumes of similar solids

Include volumes of compound shapes.

### D. PLANE GEOMETRY

#### (a) Angles

(i) Angles at a point add up to 360o. (ii) Adjacent angles on a straight line are supplementary. (iii) Vertically opposite angles are equal.

The degree as a unit of measure. Consider acute, obtuse, reflex angles, etc

#### (b) Angles and intercepts on parallel lines.

(i) Alternate angles are equal. ( ii )Corresponding angles are equal. ( iii )Interior opposite angles are supplementary ∗♣♠(iv) Intercept theorem

Application to proportional division of a line segment

#### (c) Triangles and Polygons

(i) The sum of the angles of a triangle is 2 right angles.

(ii) The exterior angle of a triangle equals the sum of the two interior opposite angles.

(iii) Congruent triangles.

( iv ) Properties of special triangles – Isosceles, equilateral, right-angled, etc

(v) Properties of special

quadrilaterals – parallelogram, rhombus, square, rectangle, trapezium.

( vi )Properties of similar triangles.

( vii ) The sum of the angles of a polygon

(viii) Property of exterior angles of a polygon.

(ix) Parallelograms on the same base and between the same parallels are equal in area

∗The formal proofs of those underlined may be required.

Conditions to be known but proofs not required e.g. SSS, SAS, etc.

Use symmetry where applicable.

Equiangular properties and ratio of sides and areas.

Sum of interior angles = (n – 2)180o or (2n – 4)right angles, where n is the number of sides

#### ( d ) Circles

(i) Chords.

(ii) The angle which an arc of a circle subtends at the centre of the circle is twice that which it subtends at any point on the remaining part of the circumference.

(iii) Any angle subtended at the circumference by a diameter is a right angle.

(iv) Angles in the same segment are equal. (v) Angles in opposite segments are supplementary.

( vi )Perpendicularity of tangent and radius.

(vii )If a tangent is drawn to a circle and from the point of contact a chord is drawn, each angle which this chord makes with the tangent is

(ii) The angle which an arc of a circle subtends at the centre of the circle is twice that which it subtends at any point on the remaining part of the circumference.

(iii) Any angle subtended at the circumference by a diameter is a right angle.

(iv) Angles in the same segment are equal. (v) Angles in opposite segments are supplementary.

( vi )Perpendicularity of tangent and radius.

(vii )If a tangent is drawn to a circle and from the point of contact a chord is drawn, each angle which this chord makes with the tangent is equal to the angle in the alternate segment.
Angles subtended by chords in a circle and at the centre. Perpendicular bisectors of chords.

∗the formal proofs of those underlined may be required

#### ( e ) Construction

( i ) Bisectors of angles and line segments (ii) Line parallel or perpendicular to a given line. ( iii )Angles e.g. 90o, 60o, 45o, 30o, and an angle equal to a given angle. (iv) Triangles and quadrilaterals from sufficient data

Include combination of these angles e.g. 75o, 105o,135o, etc

#### ♠( f ) Loci

Knowledge of the loci listed below and their intersections in 2 dimensions. (i) Points at a given distance from a given point. (ii) Points equidistant from two given points. ( iii)Points equidistant from two given straight lines. (iv)Points at a given distance from a given straight line

Consider parallel and intersecting lines. Application to real life situations

### E. COORDINATE GEOMETRY OF STRAIGHT LINES

(i) Concept of the x-y plane.

(ii) Coordinates of points on the x-y plane

Midpoint of two points, distance between two points i.e. |PQ| = # \$− \$!+ ’− ’!, where P(x1,y1) and Q(x2, y2), gradient (slope) of a line m= ( ) ( ), equation of a line in the form y = mx + c and y – y1 = m(x – x1), where m is the gradient (slope) and c is a constant

### F. TRIGONOMETRY

#### ( b ) Angles of elevation and depression

(i) Sine, Cosine and Tangent of acute angles.

(ii) Use of tables of trigonometric ratios.

(iii) Trigonometric ratios of 30o 45o and 60o.

(iv) Sine, cosine and tangent of angles from 0o to 360o.

( v )Graphs of sine and cosine.

(vi)Graphs of trigonometric ratios.

Use of right angled triangles

Without the use of tables

Relate to the unit circle. 0o≤ x ≤ 360o.

e.g. y = a sin x , y = b cos x

Graphs of simultaneous linear and trigonometric equations. e.g. y = asin x + bcos x, etc

#### ( c ) Bearings

(i) Bearing of one point from another.

(ii) Calculation of distances and angles

Notation e.g. 035o, N35oE

Simple problems only. Use of diagram is required.∗♣♠Sine and cosine rules may be used.

### G. INTRODUCTORY CALCULUS

(i) Differentiation of algebraic functions

(ii) Integration of simple Algebraic functions.

### CALCULUS

(i) Differentiation of algebraic functions.

(ii) Integration of simple Algebraic functions.
Concept/meaning of differentiation/derived function, +, +- , relationship between gradient of a curve at a point and the differential coefficient of the equation of the curve at that point. Standard derivatives of some basic function e.g. if y = x2, +, +- = 2x. If s = 2t3 + 4, +. +/ = v = 6t2, where s = distance, t = time and v = velocity. Application to real life situation such as maximum and minimum values, rates of change etc.

Meaning/ concept of integration, evaluation of simple definite algebraic equations

### H. STATISTICS AND PROBABILITY

(i) Frequency distribution

( ii ) Pie charts, bar charts, histograms and frequency polygons

(iii) Mean, median and mode for both discrete and grouped data.

(iv) Cumulative frequency curve (Ogive).

(v) Measures of Dispersion: range, semi inter-quartile/interquartile range, variance, mean deviation and standard deviation

Construction of frequency distribution tables, concept of class intervals, class mark and class boundary.

Reading and drawing simple inferences from graphs, interpretation of data in histograms. Exclude unequal class interval. Use of an assumed mean is acceptable but not required.

For grouped data, the mode should be estimated from the histogram while the median, quartiles and percentiles are estimated from the cumulative frequency curve.

Application of the cumulative frequency curve to every day life.

Definition of range, variance, standard deviation, interquartile range. Note that mean deviation is the mean of the absolute deviations from the mean and variance is the square of the standard deviation. Problems on range, variance, standard deviation etc. ∗♣♠Standard deviation of grouped data

#### ( b ) Probability

(i) Experimental and theoretical probability.

(ii) Addition of probabilities for mutually exclusive and independent events

(iii) Multiplication of probabilities for independent events

Include equally likely events e.g. probability of throwing a six with a fair die or a head when tossing a fair coin. With replacement. ∗without replacement

Simple practical problems only. Interpretation of “and” and “or” in probability

### I. VECTORS AND TRANSFORMATION

#### (b) Transformation in the Cartesian Plane

Vectors as a directed line segment.

Cartesian components of a vector

Magnitude of a vector, equal vectors, addition and subtraction of vectors, zero vector, parallel vectors, multiplication of a vector by scalar.

Reflection of points and shapes in the Cartesian Plane.

Rotation of points and shapes in the Cartesian Plane.

Translation of points and shapes in the Cartesian Plane.

Enlargement

(5, 060o)
e.g. 0.12345 673458.

Knowledge of graphical representation is necessary.

Restrict Plane to the x and y
axes and in the lines x = k, y
= x and y = k x , where k is an integer. Determination of mirror lines (symmetry).

Rotation about the origin and a point other than the origin. Determination of the angle of rotation (restrict angles of rotation to -180o to 180o).

Translation using a translation vector.

Draw the images of plane figures under enlargement with a given centre for a given scale factor.Use given scales to enlarge or reduce plane figures.

### 3. UNITS

Candidates should be familiar with the following units and their symbols.

( 1 ) Length 1000 millimetres (mm) = 100 centimetres (cm) = 1 metre(m). 1000 metres = 1 kilometre (km)

( 2 ) Area 10,000 square metres (m2) = 1 hectare (ha)

( 3 ) Capacity 1000 cubic centimeters (cm3) = 1 litre (l)

( 4 ) Mass 1000 milligrammes (mg) = 1 gramme (g)
1000 grammes (g) = 1 kilogramme( kg )
1000 ogrammes (kg) = 1 tonne.
( 5) Currencies
The Gambia – 100 bututs (b) = 1 Dalasi (D)
Ghana – 100 Ghana pesewas (Gp) = 1 Ghana Cedi ( GH¢)
Liberia – 100 cents (c) = 1 Liberian Dollar (LD) Nigeria – 100 kobo (k) = 1 Naira (N) Sierra Leone – 100 cents (c) = 1 Leone (Le) UK – 100 pence (p) = 1 pound (£) USA – 100 cents (c) = 1 dollar (\$) French Speaking territories: 100 centimes (c) = 1 Franc (fr) Any other units used will be defined.

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### Kenneth Emmanuel

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