Mathematics Curriculum Progression DP to MYP
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Curriculum mapping: Subtopics and Content details
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Name of the topicsNumber system

- Basic classification of numbers
- Order of operations (BODMAS)

Positive and Negative number/Integer
- The Number line
- Operation with numbers
- Combined operations using BODMAS

Properties of numbers
- Factors
- Multiplies
- H.C.F
- L.C.M
- Prime and Composite numbers

- Index notation (Positive indices) and different forms
- Squares and cubes, Square root & Cube roots
Integer and fractional exponents and surds

- Rational numbers
- Irrational numbers
- Absolute value
- Surds
- Laws of indices
- Index laws
- Expansion laws
- The negative index law
- The zero index law
Fractional Exponents
-Using the rules of indices to simplify numerical expressions involving radicals and exponents
- Scientific notations/Standard form
- Significant figures
Sets and Venn Diagrams

-Basic Vocabulary (element, subset, null set, and so on)
-Performing operations - Union, Intersection and complement of sets
-Properties of sets(commutative, associative, distributive)
-Drawing and interpreting Venn diagrams
-Using Venn diagrams to solve problems in real-life contexts
- Numbers in region
- Problem solving with Venn diagram
Fractional Exponents

-Using the rules of indices to simplify numerical expressions involving radicals and exponents

Functions

-Exponential,
-Trigonometric Functions (sine and cosine)
-Domain and range
-Transformations of functions
Functions

-Trigonometric Functions (Sine, Cosine and Tan)
-Logarithmic and rational functions
-Transformations of functions
-The modulus function
-Intersection of functions (where functions meet)
-Exponential functions (Growth, Decay, Compound Interest
Depreciation)

-Simplifying trigonometric expressions
-Trigonometric equations
-Negative and complemenatry angle formulae
-Compound angle formulae
NUMBERS & ALGEBRA

Numbers :
-Natural numbers, Rational numbers, Real numbers, Notations of different set of numbers
-Approximation : decimal places, significant figures
- Percentage errors
- Estimation
- Expressing numbers in the form a x 10^k , a belongs to the interval [1,10] and k is an integer. Operations with numbers in this form.
- SI and other basic units of measurements
- Currency conversions

Algebra :
-Use of GDC to solve pair of equations in two variables and quadratic equations
-Arithmetic and Geometric Sequences and Series and their applications. Formula to find nth terms of sequences and finding the sum upto nth terms
-Geometric sequences and series . Use the formulae for the nth term and the sum of the first n terms of the sequences
-Financial applications of the Geometric Sequences and Series for calculating compound interest and annual depreciation
ALGEBRA:

Sequences and Series:
Arithmetic sequences and series; sum of finite arithmetic series;
Geometric sequences and series; sum of finite and infinite geometric series.
Sigma Notations
Applications

Exponents and Logarithms:
Elementary treatment of exponents and logarithms,
Laws of exponents ; laws of logarithms.
Change of base

ALGEBRA :

SEQUENCES AND SERIES
Arithmetic sequences and series; sum of finite
arithmetic series; geometric sequences and
series; sum of finite and infinite geometric
series.
Sigma notation.
Applications. Examples include compound interest and
population growth.
Annuity - continuous investment must be taught with AP and GP

EXPONENTS AND LOGARITHMS
Exponents and logarithms.
Laws of exponents; laws of logarithms.
Change of base.
Prove all laws of logarithms in class.

BINOMIAL THEOREM :
Counting principles, including permutations
and combinations.
The binomial theorem: expansion of (a+b)^n

Not required:
Permutations where some objects are identical.
Circular arrangements.
Proof of binomial theorem.

MATHEMATICAL INDUCTION :
Proof by mathematical induction. Links to a wide variety of topics, for example,
complex numbers, differentiation, sums of
series and divisibility.

COMPLEX NUMBERS :
-Complex numbers: the number i = sq.rt(-1) ; the terms real part, imaginary part, conjugate,
-modulus and argument.
-Cartesian form z = a + ib
-Sums, products and quotients of complex numbers.
-Modulus–argument (polar) form
-The complex plane.
-Powers of complex numbers: de Moivre’s theorem.
-nth roots of a complex number.
-Conjugate roots of polynomial equations with real coefficients.

SYSTEM OF LINEAR EQUATIONS :
-Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.
-These systems should be solved using both algebraic and technological methods, eg row reduction. Systems that have solution(s) may be referred to as consistent. When a system has an infinity of solutions, a general solution may be required.
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Geometry of Structures

Points and Lines
Measuring and classifying angles

Angle Properties (Angle Pairs of Parallel lines) + Basic Terminologies
- Angles on a straight line add up to 180 degree
- Angles around a point add up to 360 degree
- Vertically Opposite angles
- Corresponding angles
- Alternate angles (Interior and Exterior)
- Co-Interior angles
- Supplementary angles

Triangles (Classification and Properties)

- Concept of Pythagoras Theorem and its inverse
- Real life application of Pythagoras Theorem
- Pythagorean triplet
Ratio and Proportions and percentage

-Unitary method in percentage
-Percentage increase and decrease: Using both the methods- 1) With two steps 2) With one step using a multiplier
-Finding a percentage change
-Finding the original amount (Finding Cost price when Selling price and profit/loss% is given, Finding Marked price when Selling price and discount% is given)
-Simple Interest
-Compound Interest
-Equal Ratios and Equivalent Ratios
-Proportions
-Direct and inverse proportion
-Finding a constant of proportionality
-Using ratios to divide quantities (Word problems)
Number Sequences

-Predicting the next term in a number sequence
- General term of an arithmetic and geometric sequence
Arithmetic and Geometric series

-Developing, and justifying or proving, general rules/ formulae for sequences
-Finding the sum of the series, including infinite series
Transformation of Geometry

-Simple transformations, including isometric transformations.
Transformation of Geometry

-Transforming a figure by rotation, reflection, translation and enlarment
MATHEMATICAL MODELS :

FUNCTIONS :
-Concept of a function, domain, range and graph.
-Function notation, concept of a function as a mathematical model.
-Linear models, Linear functions and their graphs.
-Properties of a parabola, symmetry, vertex, intercepts on the x-axis and y-axis.
-Equation of the axis of symmetry.
-exponential models.
-Exponential functions and their graphs,
-Concept and equation of a horizontal asymptote.
-Polynomial functions of integral degree and their graphs, Y axis as a vertical asymptote.
-Drawing accurate graphs,
-Creating sketch from informations given.
-Transferring of graphs from GDC to paper,
-Reading, interpreting and making predictions using graphs.
-Addition and subtraction of all above mentioned functions.
- Use of a GDC to solve equations involving combinations of the functions above.
FUNCTIONS AND EQUATIONS:

Function Notation
Concept of function f : x |-> f (x) : domain,range; image (value).
Odd and even functions.
Composite functions f o g .
Identity function
Inverse function f ^(−1)

Features of graph
The graph of a function ; its equation y = f(x)
Function graphing skills
Investigation of key features of graph, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
Use of technology to graph a variety of functions, including ones not specifically mentioned.
The graphs of the function y = f ^ (-1) (x) as the reflection in the line y = x of the graph of y = f(x).
NOTE THE DIFFERENCE IN THE COMMAND TERMS """"DRAW"""" and """"SKETCH""""."""

Transformations of graphs
Translations : y = f(x) + b ; y = f(x-a)
Reflection (in both axes) : y = - f(x) ; y = f(-x)
vertical stretch with scale factor p : y = p f(x)
Stretch in the x-direction with scale factor (1/q) ; y = f (qx)
Composite transformation

The quadratic function x |----> ax^2 + bx + c : its graph, y-intercept (0, c) . Axis of symmetry.
The form x |---> a(x − p)(x − q) ,
x-intercepts ( p, 0) and (q, 0) .
The form x |---> a(x − h)^2 + k , vertex (h, k) .

The reciprocal function
x |---> (1/x) , x is not 0 ; its graph and self-inverse nature.
The rational function x | ---> (ax + b) / (cx + d) and its graph
vertical and horizontal asymptote

Exponential Function and graph
x |--> a^x , a > 0, x |--> e^x
Logarithmic functions and their graphs
x |--> log x with base a , x > 0 , x |--> ln x , x > 0
Relationships between these functions
a^x = e^(x ln a) ; log a^x with base a = x ; a^(Log x with base a) = x , x > 0.

Solving equations, both graphically and analytically.
Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
Solving ax^2 + bx +c = 0 ; a is not 0.
The discriminant delta = b^2 - 4ac and the nature of roots, that is , two distinct real roots, two equal roots, no real roots.
Solving exponential equations

Application of graphing skills and solving equations that relate to real-life situations"
FUNCTIONS AND EQUATIONS :

Concept of function
-domain,range; image (value).
-Odd and even functions.
-Composite functions
-Identity function.
-One-to-one and many-to-one functions.
-Inverse function , including domain restriction. Self-inverse functions.

The graph of a function

-its equation Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
-The graphs of the functions y = |f(x)| and t = f (|x|) .
The graph of y = 1/f(x) given the graph of y = f(x)

Transformations of graphs:
-translations;
-stretches;
-reflections in the axes.
-The graph of the inverse function as a reflection in y = x .

Other functions along with its graphs :
- The rational functions
-The reciprocal function
- The exponential functions
- The logarithmic functions
- The absolute function

Polynomial functions and their graphs.
-The factor and remainder theorems.
-The fundamental theorem of algebra.

-Use of the discriminant to determine the nature of the roots.
-Solving polynomial equations both graphically and algebraically.
-Sum and product of the roots of polynomial equations.
.-Solution of a^x = b using logarithms.
-Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

Inequalities
-Solutions of g(x) >= f (x) .
-Graphical or algebraic methods, for simple polynomials up to degree 3.
-Use of technology for these and other functions.

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Algebra expressions

- Writing algebraic expressions
- Key words in expression
- Equal algebraic expressions
- Collecting like terms
- Algebraic products
- Evaluating algebraic expressions
- Adding and subtracting algebraic fractions

Expressions, Equations and Formulae

Understanding Algebraic Expressions
Key words in algebra
Constructing expressions
Like and Unlike terms
Simplifying expressions
Understanding equations
Solving equations
Constructing and solving equations
Substitution into an expression and Formula
Changing the subject in the equation
Conversion between temperature scales (Fahrenheit and Celsius)
Writing a short situation based story from an expression
Investigating patterns
Algebraic fractions

-The distributive law
-The product (a + b) (c + d)
-Perfect square expansion: (a+b)^2 and (a-b)^2
-Difference of two squares expansion: (a+b) (a-b)
- Common Factors
- Factorising with common factors
- Perfect square factorization: a^2 + 2ab + b^2 and a^2 - 2ab + b^2
- Difference of two squares factorizing: a^2 - b^2

- Factorizing by grouping
- Factorizing quadratic exprexxions and equations
- Simplifying algebraic fractions
- Addition and Subtraction of algebraic fractions
- Simplifying complex algebraic fractions

-Simple Linear Equations
-Constructing Simple Equations
-Constructing complex
-Substition
-Rearranging Formulae / Changing the subject of Formulae
-Simultaneous Equations (Elimination method and Substitution method)
- Graphing of simulataneous equstions?????
- Problem solving with simultaneous Equation

Equations, Functions & Graphs

By completing the square method
By Qudratic formula method
- Problem solving

Functions
-Types of functions: linear and quadratic
Functions and Graphs

Relations, functions, functions notation
- Inverse and composite functions

-Axes intercepts, Axis of symmetry, Vertex, Quadratic optimisation
Statistics

-Graphical analysis and representation (scatter plots), data collection, constructing and interpreting graphs, drawing line of best fit.
-Population sampling (Selecting samples and making inferences about populations)

- Probability
- Probability scale
- Events
- Sample space
Probability of an event - with and without replacement
Probability of independent, mutually exculusive and combined events
Probability of successive trials
Solving problems using tree and Venn diagrams

Statistics

-Standard deviation
-Variance
- Making inferences about the data given the mean and stadard deviation
GEOMETRY AND TRIGONOMETRY :

Co-ordinate Geometry on Cartesian Plane:
-Slope of straight lines,
-equations of straight lines in two dimension of the form y = mx+c and ax+by+d = 0,
-intercepts on axes,
-Lines with gradients m1 and m2, conditions of parallelism and perpendicularity of two lines.

Trigonometry
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-Use of sine, cosine and tangent ratios to find the sides and angles of right angled triangles.
-Angles of elevation and depression,
-Application of the sine rule and cosine rule, use of area of a triangle.
-Concepts of bearings and standard angles for construction of labelled diagrams from verbal statements, ambiguous cases of triangle

Geometry of 3D solids :
- Cuboid, right prism, right pyramid, right cone, cylinder, sphere, hemisphere, combinations of solids.
-Surface Area and Volume of three dimensional solids.
CIRCULAR FUNCTIONS AND TRIGONOMETRY:

The circle
radian measure of angles; lengths of an arc; area of a sector

Definition of cosθ and sinθ in terms of the unit circle.
Definition of tanθ as sin θ / cos θ
Exact values of trigonometric ratios of 0, π/6 , π/4 , π/3 , π/2 and their multiples.

The Pythagorean identity cos^2 θ + sin^2 θ =1.
Double angle identities for sine and cosine.
Relationship between trigonometric ratios

The circular functions sin x , cos x and tan x : their domains and ranges; amplitude, their periodic nature; and their graphs.
Composite functions of the form f (x) = asin (b(x + c) ) + d .
Transformations
Applications

Solving trigonometric equations in a finite interval, both graphically and analytically.
Equations leading to quadratic equations in sin x, cos x or tan x

Solutions of triangle
The cosine rule
The sine rule, including the ambiguous case
Area of a triangle (0.5 ab sin C)
Applications
CIRCULAR FUNCTIONS AND TRIGONOMETRY :

-The circle: radian measure of angles.
-Length of an arc; area of a sector.
-Trigonometric functions and identities
-Definition of cosθ , sinθ and tanθ in terms of the unit circle.
-Exact values of sin, cos and tan of 0,π/6, π/4 , π/3, π/2 and their multiples.
-Definition of the reciprocal trigonometric ratios secθ , cscθ and cotθ .
-Pythagorean identities: cos^2θ + sin^2θ =1; 1+ tan^2θ = sec^2θ ; 1+ cot^2θ = csc^2
-Trigonometric Identities
-Compound angle identities. Double angle identities.
-Derivation of double angle identities from compound angle identities.
-Finding possible values of trigonometric ratios without finding θ, for example, finding sin 2θ given sinθ
- Composite functions of the form f(x) = a sin (b(x+c)) + d and its application
-Trigonometric Equations
-The inverse functions x |--> arcsin x ,x |--> arccos x , x |--> arctan x ; their domains and ranges; their graphs.
-Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization.
-Sine and Cosine rule
-The sine rule including the ambiguous case.
-Area of triangle as (1/2) a b sin C.
-Application : Examples include navigation, problems in two and three dimensions, including angles of elevation and depression."

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Co-ordinate Geometry

- The Cartesian Plane
- Number grids
- Positive and Negative Coordinates
- Plotting points from a table of values
- Linear Relationships
- Axes Intercepts
- Graphing Straight Lines
- Horizontal and Vertical Lines
- Equation of a line?????
- Midpoint
- Distance formula
- Perpendicular and parallel lines
- Finding equations from graph
Statistics

-Graphical analysis and representation (pie charts, histograms, line graphs,box- and - whisker plots), data collection, constructing and interpreting graphs
-Measures of central tendency/location (mean, median, mode, quartile, percentile)for discrete and continuous data
-Measures of dispersion/spread (range, interquartile range) for continuous and discrete data
-Cumulative Frequency data

Vectors

-Directed line segment representation
-Vector equality
-Vectors in component form
-Scalar multiplication of vectors, both algebraically and graphically
-Parallelism of vectors
-Dot product
-3D vectors
-Relative velocity
DESCRIPTIVE STATISTICS :

-Classification of data as discrete or continuous,
-Simple discrete data, frequency tables of Discrete,
-Grouped discrete and continuous data, mid-interval values, upper and lower boundaries,
-Frequency histograms.
-Cumulative frequency tables and application of GDC to generate lists and cumulative frequency curves, median and quartiles,
-Box-and-whisker diagrams,
- Measures of central tendency, estimation of mean, median and mode for discrete, grouped discrete and continuous data, model class,
-Measures of dispersion, range, IQR, Standard deviation, using GDC to find 5 number summary
VECTORS:

Vectors as displacements in the plane and in three dimensions
Components of a vector ; column representation;
v = v1 i + v2 j + v3k
Algebraic and geometric approaches to the following :
the sum and difference of two vectors; the zero vector, the vector −v ;
multiplication by a scalar, kv ; parallel vectors;
magnitude of a vector, | v |;
unit vectors; base vectors; i, j and k;
position vectors OA = a
→ → →
AB OB OA = b - a

The scalar product of two vectors.
Perpendicular vectors; parallel vectors
The angle between two vectors.

Vector equation of a line in two and three dimensions : r = a + tb
The angle between two lines

Distinguishing between coincident and parallel lines.
Finding the point of intersection of two lines
Determining whether two lines intersect.
VECTORS :

-Concept of a vector
-Representation of vectors using directed line segments.
-Unit vectors, base vector, component of a vector,
-Algebraic and geometric approaches to the following :
the sum and difference of two vectors; the zero vector, the vector −v ;
multiplication by a scalar, kv ; parallel vectors;
- magnitude of a vector, | v |;
- unit vectors; base vectors; i, j and k;
- position vectors OA = a
→ → →
AB = OB - OA = b - a

SCALAR PRODUCT :
-The definition of the scalar product of two vectors.
-Properties of the scalar product.
-The angle between two vectors.
-Perpendicular vectors, parallel vectors

VECTOR EQUATION OF A LINE :
-Vector equation of a line in two and three dimensions.
-Simple applications to kinematics.
-The angle between two lines.
-Coincident, parallel, intersecting and skew lines;
- distinguishing between these cases.
-Points of intersection.

VECTOR PRODUCT :
-The definition of the vector product of two vectors. Properties of the vector product
-Geometric interpretation of | v × w |.
-Areas of triangles and parallelograms.

VECTORS AND PLANES :
-Vector equation of a plane
-Use of normal vector to obtain the form
-Cartesian equation of a plane
-Intersections of: a line with a plane; two planes; three planes.
-Angle between: a line and a plane; two planes.
-Geometrical interpretation of solutions.
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Measurement and Mensuration

- Length,Mass and capacity
- Relationships between units
- Measurement of Time
- Area and Perimeter of 2D shapes(parallelogram, trapezium, circle)
- Surface area and Volume of 3D Shapes(cube, cuboid, cylinder)
- Connecting volume and capacity
Mensuration

- Area of compound figures
- Volume and surface area of cone, sphere and hemisphere
- Connecting volume and capacity in detail
- Connection between volume of cylinder and cone.
Inequalities

-Solving and graphing linear inequalities
-Linear Programming (Graphing)
-Number lines (Expressing the solution set of a linear inequality on the number line (as well as set notation)
Inequalities

-Solving nonlinear inequalities
Probability

-Probability of an event
-Independent, mutually exclusive and combined events
-Probability of successive trials
Probability

-Conditional Probability
STATISTICAL APPLICATIONS :

The normal distribution,
-The concept of a random variable, parameters of normal distribution ( mean and standard deviation or variance ) and their notations, bell shaped diagrams of distribution, symmetry about mean, diagrammatic representation,Calculation of Normal probability, Expected value and inverse normal with the help of GDC.

Bivariate Data :
-the concept of correlation, scatter diagram, line of best fit ( hand calculation and with the help of GDC ), by eye, passing through mid point.
-Pearson's product-moment correlation coefficient, r.
-Interpretation of positive,zero and negative, strong or weak correlations.

Regression line for y on x.
-Use of the regression line for the prediction purpose.
-Chi square test for independence : formulation of null and alternative hypothesis, significance levels, contingency tables, expected frequencies, degree of freedom, p-values.
STATISTICS AND PROBABILITY:

Manipulation and presentation of statistical data
Concepts of population, sample, random sample and frequency distribution of discrete and continuous data
Presentation of data : frequency distributions (tables), frequency histograms with equal class intervals.
Box-Whisker plots; outliers
Grouped data : mid-interval values for calculations, interval width, upper and lower interval boundaries, modal class.

Statistical measures and their interpretations.
Central tendency: mean, median, mode.
Quartiles, percentiles.
Dispersion: range, interquartile range, variance, standard deviation.
Effect of constant changes to the original data.
Applications.

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.

Linear correlation of bivariate data
Pearson's product-moment correlation coefficient 'r'
Scatter diagrams ; lines of best fit.
Equation of the regression line of y on x.
Use of the equation for prediction purposes.
Mathematical and contextual interpretation.

Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
The probability of an event A is P(A) = n(A) / n(U)
The complementary events of A and A' (not A).
Use of Venn-diagrams, tree diagrams and tables of outcomes

Combined events , P(A U B)
Mutually exclusive events : P(A intersection B) = 0
Conditional Probability ; the definition
P( A | B) = P(A intersection B ) / P( A | B' )
Independent events; the definition
P(A | B)= P(A)= P(A |B')
Probabilities with and without replacement.

Concepts of discrete random variables and their probability distribution
Expected value (mean) , E (X) for discrete data
Applications

Binomial distribution
Mean and Variance of the binomial distribution

normal distributions and curve
Standardization of normal variables ( z- values, z - scores)
Properties of the normal distribution.
STATISTICS AND PROBABILITY :

Probability :
- Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
- Grouped data: mid-interval values, interval width, upper and lower interval boundaries.
- Mean, variance, standard deviation.
- Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
- The probability of an event A
- The complementary events A and A' (not A).
- Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems
- Combined events; the formula for P(AUB) .
- Mutually exclusive events.
- Conditional Probability, independent events
-Use of Bayes’ theorem for a maximum of three events

Random Variables :
-Concept of discrete and continuous random variables and their probability distributions.
-Definition and use of probability density functions.
-Expected value (mean), mode, median, variance and standard deviation.
-For a continuous random variable, a value at which the probability density function has a maximum value is called a mode.
- Applications.

Distribution :
-Binomial distribution, its mean and variance.
-Poisson distribution, its mean and variance.
-Normal Distribution. Properties of the normal distribution. Standardization of normal variables.
Properties of the normal distribution.
Standardization of normal variables.
-Probabilities and values of the variable must be found using technology.
-The standardized value (z) gives the number of standard deviations from the mean.
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Statistics

Discrete and Continuous Data
Data Collection
Displaying data - Two-way tables
Categorical data - Tally and Frequency tables and Displaying categorical data through bar graphs and pie charts
Numerical Data
Grouped Data - Displaying through Histograms and Stem & Leaf
Measuring the centre(averages) and spread - Mean, Median, Mode & Range from data sets
Interpreting data & graphs, Comparing graphs

Circle Geometry

Using circle theorems to find:
-lengths of chords and arcs
-measures of angles
- Area and perimeter of sector and segment
INTRODUCTION TO DIFFERENTIAL CALCULUS :

-Concepts of Derivative as a rate of change
-Tangent to a curve
-Power rule of derivative
-The Derivative of Polynomial functions where all exponents are integer only
-Gradients of curve for given value of x ( analytical and GDC )
-Values of x where derivative is given
-Equation of the tangent at a given point,
-Equation of the line perpendicular to the tangent at a given point ( normal )
-Increasing and decreasing functions.
-Graphical representation of derivative value negative, positive or equal to zero.
-Values of x where the gradient of a curve is zero.
-Solutions of derivative = 0
-Stationary points
-Local maxima and minima.
-Optimization problems
CALCULUS:

Informal ideas of limit and convergence.
Limit Notation
Definition of derivative from first principles as
f ' (x) = limit where h |--> 0 ( f(x + h) - f(x) / h )
Derivative interpreted as gradient function and as rate of change
tangents and normals ; and their equations

Derivative of x^n (n∈Q) , sin x , cos x , tan x , e^x and ln x .
Differentiation of a sum and a real multiple of these functions.
The chain rule for composite functions.
The product and quotient rules.
The second derivative
Extension to higher derivatives

Local maximum and minimum points. Testing for maximum or minimum.
Points of inflexion with zero and nonzero gradients.
Graphical behaviour of functions,including the relationship between the graphs of f , f ′ and f ′′
Optimization.
Applications

Indefinite integration as anti-differentiation
Indefinite integral of x^n ( n belongs to Q), sin x, cos x, (1/x) , e^x
The composites of any of these with the linear function ax + b
Integration by inspection , or substitution of the form ∫ f (g(x)) g '(x) dx .

Anti-differentiation with a boundary condition to determine the constant term.
Definite integrals, both analytically and using technology.
Areas under curves (between the curve and the x-axis).
Areas between curves.
Volumes of revolution about the x-axis.

Kinematic problems involving displacement s, velocity v, and acceleration a.
Total distance travelled.
CALCULUS :

Limits and Differential calculus :
-Informal ideas of limit, continuity and convergence.
-Definition of derivative from first principles
-The derivative interpreted as a gradient function and as a rate of change.
-Finding equations of tangents and normals.
-Identifying increasing and decreasing functions..
-The second derivative.
-Higher derivatives.

-Derivatives of x^n , sin x , cos x , tan x , e^x and ln x .
-Differentiation of sums and multiples of functions.
-The product and quotient rules.
-The chain rule for composite functions.
-Related rates of change.
-Implicit differentiation.
-Derivatives of sec x , csc x , cot x , a^x , log x to the base a ,arcsin x , arccos x and arctan x .

-Local maximum and minimum values.
-Optimization problems.
-Points of inflexion with zero and nonzero gradients.
-Graphical behaviour of functions, including the relationship between the graphs of f , f' and f'' .

Integral Calculus :

-Indefinite integration as anti-differentiation.
-Indefinite integral of xn , sin x , cos x and ex .
-The composites of any of these with a linear function.
-Anti-differentiation with a boundary condition to determine the constant of integration.
-Definite integrals.
-Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves.The value of some definite integrals can only be found using technology.
-Volumes of revolution about the x-axis or y-axis.
-Integration by parts
- Integration by substitution

Application of calculus :
Kinematic problems involving displacement s, velocity v and acceleration a.
Total distance travelled.

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Trigonometry & Bearings

-Relating angles and sides of right-angled triangles using sine, cosine and tangent
-Solving problems in right-angled triangle using trigonometric ratios
-Using the sine and cosine rules to solve problems
- Bearings
- Angle of elevation and depression
-3D trigo
Trigonometry & Bearings

Using simple trigonometric identities to simplify expressions and solve equations where 0<= theta<= 360
Simple trigonometric identities

"Trigonometric Identities
Supplementary angles
-using simple trigonometric identities to simplify expressions and solve equations.
-Converting angles between degrees and radians
-Using radians to solve problems, where appropriate
-Finding the exact value of trigonometric functions of special angles
The unit circle
The relation between sin theta and cos theta
The multiples of 30 degree and 45 degree.

Circle Geometry

- Unit circle - Finding the exact value of trigonometric functions of special angles
LOGIC, SETS AND PROBABILITY

LOGIC :
- Basic concepts of symbolic logic, Definition of a proposition, Symbolic notations of proposition,
-Compound statements,implications, =>, Equivalence, <=>, negation, conjunction,disjunction,exclusive disjunction, translation between verbal statements and symbolic form,
-Truth Tables, concepts of logical contradiction and tautology,
-Converse, inverse, contrapositive, logical equivalence, Testing the validity of simple arguments through the use of truth tables.

SETS :
-Basic concepts of set theory, elements, subsets, intersection, union, complement symbols and notations,
-Venn Diagrams and simple applications up to 3 sets

PROBABILITY:
-Sample space, Event space, complimentary event, Probability of an event, expected value,
-probability of combined events, mutually exclusive events, independent events, tree diagrams and its application, sample space diagrams and Venn diagrams. Table of outcomes, probability using ""with replacement"" and ""without replacement"". Conditional Probability.
OPTION TOPICS for paper 3

OPTION 7 : STATISTICS AND PROBABILITY

-Cumulative distribution functions for both discrete and continuous distributions.
-Geometric distribution.
-Negative binomial distribution.

-Probability generating functions for discrete random variables.
-Using probability generating functions to find mean, variance and the distribution of the sum of n independent random variables.
-Linear transformation of a single random variable.
-Mean of linear combinations of n random variables.
-Variance of linear combinations of n independent random variables.
- Expectation of the product of independent random variables.
-Unbiased estimators and estimates.
-Comparison of unbiased estimators based on variances.
- A linear combination of independent normal random variables is normally distributed.
- The central limit theorem.

- Confidence intervals for the mean of a normal population.

- Null and alternative hypotheses, H0 and H1
- Significance level.
-Critical regions, critical values, p-values, one tailed and two-tailed tests.
-Type I and II errors, including calculations of their probabilities.
-Testing hypotheses for the mean of a normal population.

- Introduction to bivariate distributions.
- Covariance and (population) product moment correlation coefficient
- Definition of the (sample) product moment correlation coefficient R in terms of n paired observations on X and Y. Its application to the estimation of product moment correlation coefficient

- Informal interpretation of r, the observed value of R. Scatter diagrams.
-Assumption of bivariate normality
- Knowledge of the facts that the regression of X on Y ans Y on X are linear.
- Least-squares estimates of these regression lines (proof not required).
- The use of these regression lines to predict the value of one of the variables given the value of the other.

OPTION 9 - CALCULUS

- Infinite sequences of real numbers and their convergence or divergence.

- Convergence of infinite series.
-Tests for convergence: comparison test; limit comparison test; ratio test; integral test., p-series test
-Series that converge absolutely.
-Series that converge conditionally.
-Conditions for convergence.
-Alternating series.
-Power series: radius of convergence and interval of convergence.
-Determination of the radius of convergence by the ratio test.

- Continuity and differentiability of a function at a point.
- Continuous functions and differentiable functions.

- The integral as a limit of a sum; lower and upper Riemann sums.
- Fundamental theorem of calculus.
- Improper integrals

- First-order differential equations.
- Geometric interpretation using slope fields, including identification of isoclines.
- Euler’s method.
-Variables separable.
-Homogeneous differential equation using the substitution y = vx.
- Solution of y' + P(x)y = Q(x), using the integrating factor

- Rolle’s theorem.
- Mean value theorem.
- Taylor polynomials; the Lagrange form of the error term
- Maclaurin series
- Use of substitution, products, integration and differentiation to obtain other series.
-Taylor series developed from differential equations.

- The evaluation of limits of the form f(x) / g(x) where limit tends to infinity or a constant value 'a' using l’Hôpital’s rule or the Taylor series.

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Triangle Properties

-Properties of similar triangles
- Properties of congruent triangles
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