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Mathematics (WAEC)

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Course Curriculum

Number bases
( i ) conversion of numbers from one base to another ( ii ) Basic operations on number of 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.

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

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

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 the 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 are accepted

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

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

Sets
(i) Idea of sets, universal sets, finite and infinite sets, subsets, empty sets, and disjoint sets. The 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

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

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 √).

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. The 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.

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. The 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.

Ratio, Proportions and Rates
The ratio between two similar quantities. The 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

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

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.

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

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

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

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.

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

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

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.

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

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

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=-

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

Mensuration (Length 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

Mensuration (Areas)
i ) Triangles and special quadrilaterals – rectangles, parallelograms, and trapeziums (ii) Circles, sectors, and segments of circles. (iii) Surface areas of cubes, cuboids, cylinders, pyramids, right triangular prisms, cones, and spheres Areas of similar figures. Include the 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.

Volumes
(i) Volumes of cubes, cuboids, cylinders, cones, right pyramids, and spheres. ( ii ) Volumes of similar solids Include volumes of compound shapes.

Plane Geometry (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

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 a proportional division of a line segment

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

Circles
(i) Chords. (ii) The angle which an arc of a circle subtends at the center 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 center 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 are subtended by chords in a circle and at the center. Perpendicular bisectors of chords. ∗the formal proofs of those underlined may be required

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

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

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

Trigonometry
(a) Sine, Cosine, and Tangent of an angle ( 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 = as in x + bcos x, etc

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.

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

Statistics
(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, the concept of class intervals, classmark, and class boundaries. 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 everyday 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 the range, variance, standard deviation, etc. ∗The standard deviation of grouped data

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 ahead when tossing a fair coin. With replacement. ∗without replacement Simple practical problems only. Interpretation of “and” and “or” in probability

Vectors and Transformation
(a) Vectors in a Plane (b) Transformation in the Cartesian Plane Vectors as a directed line segment. Cartesian components of a vector The magnitude of a vector, equal vectors, addition and subtraction of vectors, zero vector, parallel vectors, multiplication of a vector by a 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 center for a given scale factor. Use given scales to enlarge or reduce plane figures.

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