**Further Mathematics – SS1, SS2 & SS3 Curriculum; Scheme of Work; Assessment Tests**

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**ASSURE Educational Services** has put-together the scheme of work on **Further Mathematics**** **** – SS1, SS2 and SS3** based on the National Curriculum and the syllabus of WAEC and NECO. All the topics highlighted in the National Curriculum and the syllabus of WAEC and NECO have been accommodated in our Scheme of Work.

End of term Assessment Tests in SS1, SS2 and SS3 alongside Overall Assessment Tests are available. Schools may adopt our scheme of work because of its simplicity and ease of monitoring of academic activities.

The Scheme of Work stated below is inexhaustible. Schools are encouraged to input weeks and periods of completion of topics in accordance with the academic calendar, number of students and available classes.

**SS1 Further Mathematics – 1st Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS1 |
1st Term |
Pure Mathematics |
Sets |
(i) Idea of a set defined by a property, Set notations and their meanings. (ii) Disjoint sets, Universal set and complement of set (iii) Venn diagrams, Use of sets And Venn diagrams to solve problems. (iv) Commutative and Associative laws, Distributive properties over union and intersection. |

Surds |
Surds of the form a/√b, a√b and a+b√n where a is rational, b is a positive integer and n is not a perfect square. | |||

Binary Operations |
Properties: Closure, Commutativity, Associativity and Distributivity, Identity elements and inverses. Use of properties to solve related problems. |

**SS1 Further Mathematics – 2nd Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS1 |
2nd Term |
Pure Mathematics |
Logical Reasoning |
(i) Rule of syntax: true or false statements, rule of logic applied to arguments, implications and deductions. (ii) The truth table |

Functions |
(i) Domain and co-domain of a function. (ii) One-to-one, onto, identity and constant mapping; (iii) Inverse of a function. (iv) Composite of functions. |
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Polynomial Functions |
(i) Linear Functions, Equations and Inequality (ii) Quadratic Functions, Equations and Inequalities (ii) Cubic Functions and Equations |
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Rational Functions |
(i) Rational functions of the form Q(x) = (f(x))/(g(x)), g(x)≠ 0. where g(x) and f(x) are polynomials. e.g. f:x →(ax+b)/(px²+qx+r)(ii) Resolution of rational functions into partial fractions. |

**SS1 Further Mathematics – 3rd Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS1 |
3rd Term |
Pure Mathematics |
Indices and Logarithmic Functions |
(i) Indices Laws of indices. Application of the laws of indices to evaluating products, quotients, powers and nth root. Solve equations involving indices.(ii) Logarithms Laws of Logarithms. Application of logarithms in calculations involving product, quotients, power (log a ^{n}), nth roots (log√a , log a^{1/n}).Solve equations involving logarithms (including change of base).Reduction of a relation such as y = ax ^{b}, (a,b are constants) to a linear form:log _{10}y = b log_{10}x+log_{10}a.Consider other examples such as log ab ^{x} = log a + x log b;log (ab) ^{x} = x(log a + log b) = x log ab*Drawing and interpreting graphs of logarithmic functions e.g. y = ax ^{b}. Estimating the values of the constants a and b from the graph |

PermutationAnd Combinations. |
(i) Simple cases of arrangements| (ii) Simple cases of selection of objects.Knowledge of arrangement and selection is expected. The notations: ^{n}C_{r}, and ^{n}P_{r} for selection and arrangement respectively should be noted and used. e.g. arrangement of students in a row, drawing balls from a box with or without replacements.^{n}p_{r} = n!/(n-r)!^{n}C_{r}= n!/r!(n-r)! |
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Binomial Theorem |
Expansion of (a + b)^{n}.Use of (1+x) ^{n} ≈1+nx for any rational n, where x is sufficiently small. e.g (0.998)^{1/3}Use of the binomial theorem for positive integral index only.Proof of the theorem not required. |

**SS2 Further Mathematics – 1st Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS2 |
1st Term |
Pure Mathematics |
Sequences and Series |
(i) Finite and Infinite sequences.(ii) Linear sequence/Arithmetic Progression (A.P.) and Exponential sequence/Geometric Progression (G.P.) (iii) Finite and Infinite series. (iv) Linear series (sum of A.P.) and exponential series (sum of G.P.) |

Matrices and Linear Transformation |
(i) Matrices Concept of a matrix – state the order of a matrix and indicate the type. Equal matrices – If two matrices are equal, then their corresponding elements are equal. Use of equality to find missing entries of given matrices Addition and subtraction of matrices (up to 3 x 3 matrices). Multiplication of a matrix by a scalar and by a matrix (up to 3 x 3 matrices)(ii) Determinants Evaluation of determinants of 2 x 2 matrices.(iii) Inverse of 2 x 2 Matrices Application of determinants to solution of simultaneous linear equations.(iv) Linear Transformation Finding the images of points under given linear transformation Determining the matrices of given linear transformation. Finding the inverse of a linear transformation (restrict to 2 x 2 matrices). Finding the composition of linear transformation. Recognizing the Identity transformation. |
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Trigonometry |
(i) Trigonometric Ratios and Rules Sine, Cosine and Tangent of general angles (0 ^{o}≤θ≤360^{o}).Identify trigonometric ratios of angles 30 ^{O}, 45^{O}, 60^{o} without use of tables.Use basic trigonometric ratios and reciprocals to prove given trigonometric identities. Evaluate sine, cosine and tangent of negative angles. Convert degrees into radians and vice versa. Application to real life situations such as heights and distances, perimeters, solution of triangles, angles of elevation and depression, bearing(negative and positive angles) including use of sine and cosine rules, etc, Simple cases only.(ii) Compound and Multiple Angles. sin (A±B), cos (A±B), tan(A±B). Use of compound angles in simple identities and solution of trigonometric ratios e.g. finding sin 75 ^{o}, cos 150^{o }etc, finding tan 45^{o} without using mathematical tables or calculators and leaving your answer as a surd, etc.Use of simple trigonometric identities to find trigonometric ratios of compound and multiple angles (up to 3A).(iii) Trigonometric Functions and EquationsRelate trigonometric ratios to Cartesian Coordinates of points (x, y) on the circle x ^{2} + y^{2} = r^{2}.f:x →sin x, g: x → a cos x + b sin x = c. Graphs of sine, cosine, tangent and functions of the form asinx + bcos x. Identifying maximum and minimum point, increasing and decreasing portions. Graphical solutions of simple trigonometric equations e.g. asin x + bcos x = k.Solve trigonometric equations up to quadratic equations e.g. 2sin ^{2}x – sin x – 3 =0, for 0^{o} ≤ x ≤ 360^{o}.Express f(x) = asin x + bcos x in the form Rcos (x ) or Rsin (x ) for 0 ^{o} ≤ ≤ 90^{o}and use the result to calculate the minimum and maximum points of a given functions. |

**SS2 Further Mathematics – 2nd Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS2 |
2nd Term |
Pure Mathematics |
Co-ordinate Geometry |
(i) Straight Lines Mid-point of a line segment Coordinates of points which divides a given line in a given ratio. Distance between two points; Gradient of a line; Equation of a line: (i) Intercept form; (ii) Gradient form; Conditions for parallel and perpendicular lines. Calculate the acute angle between two intersecting lines e.g. if m _{1} and m_{2} are the gradients of two intersecting lines, then tan θ = (m_{1}– m_{2})/(1+ m_{1 }m_{2} ). If m_{1}m_{2} = -1, then the lines are perpendicular.(ii) Conic SectionsLoci of variable points which move under given conditions Equation of a circle: (i) Equation in terms of centre, (a, b), and radius, r, (x – a) ^{2}+(y – b)^{2} = r^{2};(ii) The general form: x ^{2}+y^{2}+2gx+2fy+c = 0, where (-g, –f) is the centre and radius, r =√ a^{2}+b^{2 }– c.Tangents and normals to circlesEquations of parabola in rectangular Cartesian coordinates (y ^{2} = 4ax, include parametric equations (at^{2}, at)).Finding the equation of a tangent and normal to a parabola at a given point.*Sketch graphs of given parabola and find the equation of the axis of symmetry. |

Differentiation |
(i) The idea of a limit (i) Intuitive treatment of limit. Relate to the gradient of a curve. e.g. f1(x) = lim┬(h→0)〖(f(x+h)- f(x))/h〗.(ii) The derivative of a function(iii)Differentiation of polynomials(iv) Differentiation of trigonometric Functions(v) Product and quotient rules.(vi) Differentiation of implicit functions such as ax ^{2} + by^{2} = c(vii) Second order derivatives and Rates of change and small changes (x), Concept of Maxima and Minima |
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Integration |
(i) Indefinite Integral (ii) Definite Integral (iii) Applications of the Definite IntegralSimple problems on integration by substitution. Integration of simple trigonometric functions of the form ∫ _{a}^{b }sin x dx.(i) Plane areas and Rate ofChange. Include linear kinematics. Relate to the area under a curve. (ii)Volume of solid of revolution (iii) Approximation restricted to trapezium rule. |

**SS2 Further Mathematics – 3rd Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS2 |
3rd Term |
Statistics and Probability |
Statistics |
(i) Tabulation and Graphical representation of data Frequency tables. Cumulative frequency tables. Histogram (including unequal class intervals). Cumulative frequency curve (Ogive) for grouped data.(ii) Measures of location Central tendency: mean, median, mode, quartiles and percentiles. Mode and modal group for grouped data from a histogram. Median from grouped data. Mean for grouped data (use of an assumed mean required).(iii) Measures of Dispersion. Determination of: (i) Range, Inter- Quartile and Semi inter-quartile range from an Ogive. (ii) Mean deviation, variance and standard deviation for grouped and ungrouped data. Using an assumed mean or true mean.(iv) Correlation Scatter diagrams, use of line of best fit to predict one variable from another, meaning of correlation; positive, negative and zero correlations. Spearman’s Rank coefficient. Use data without ties. Equation of line of best fit by least square method. (Line of regression of y on x). |

Probability |
(i) Meaning of probability. Tossing 2 dice once; drawing from a box with or without replacement.(ii) Relative frequency. (iii) Calculation of Probability using simple sample spaces. Equally likely events, mutually exclusive, independent and conditional events. (iv) Addition and multiplication of probabilities. Include the probability of an event considered as the probability of a set. (v) Probability distributions. (i) Binomial distribution P(x=r)= ^{n}C_{r}p^{r}q^{n-r} , where probability of success = p, Probability of failure = q, p + q = 1 and n is the number of trials. Simple problems only. |

**SS3 Further Mathematics – 1st Term Scheme of Work**

Class |
Term |
Main Topics |
Sub-Topics |
Contents |

SS3 |
1st Term |
Vectors and Mechanics |
Vectors |
(i) Definitions of scalar and vector Quantities. (ii) Representation of Vectors.(iii) Algebra of Vectors. (iv) Commutative, Associative and Distributive Properties. (v) Unit vectors. (vi) Position Vectors. (vii) Resolution and Composition of Vectors. (viii) Scalar (dot) product and its application. (ix) Vector (cross) product and its application. Using the dot product to establish such trigonometric formulae as (i) Cos (a ± b) = cos a cos b ± sin a sin b (ii) sin (a ± b)= sin a cos b ± sin b cos a(iii) c^{2} = a^{2} + b^{2} – 2ab cos C(iv) |

Statics |
(i) Definition of a force. (ii) Representation of forces. (iii) Composition and resolution of coplanar forces acting at a point. (iv) Composition and resolution of general coplanar forces on rigid bodies. (v) Equilibrium of Bodies. (vi) Determination of Resultant.(vii) Moments of forces. (viii) Friction. |
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Dynamics |
(i) The concepts of motion The definitions of displacement, velocity, acceleration and speed. Composition of velocities and accelerations.(ii) Equations of Motion Rectilinear motion. Newton’s laws of motion. Application of Newton’s Laws Motion along inclined planes (resolving a force upon a plane into normal and frictional forces). Motion under gravity (ignore air resistance). Application of the equations of motions: V = u + at, S = ut + ½ at ^{2};v ^{2} = u^{2} + 2as.(iii) The impulse and momentum equations: Conservation of Linear Momentum(exclude coefficient of restitution). Distinguish between momentum and impulse. (iv) Projectiles. Objects projected at an angle to the horizontal. |

**SS3 Further Mathematics – 2nd Term Scheme of Work – Revision**

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