|
|
DRAFT |
|
|
|
||||
|
|
NCC 2023 SLOs |
Guidance on NCC 2023 SLOs
Elaboration on the extent of depth of study required for the SLOs and assessment expectations |
Essential Questions |
Rationale |
Questions for Feedback from Stakeholders
(questions are numbered according to the corresponding SLO) |
|||
|
How are Broad Topics Conceptualised |
Complex Numbers Matrices and Determinants Vectors Sequences and Series Miscellaneous Series Permutation, Combination and Probability Mathematical Induction and Binomial Theorem Functions and Graphs Linear Programming Trigonometric Identities of Sum and Differences of Angles Application of Trigonometry Graphs of Trigonometric and Inverse Trigonometric Functions and Solution of Trigonometric Equations Introduction to Symbolic Package: MAPLE Functions and Limits Differentiation Higher Order Derivatives and applications Differentiation of Vector Functions Integration Plane Analytic Geometry – Straight Line Conics-I Conics-II Differential Equations
Partial Differentiation
Introduction to Numerical Methods |
Pure Mathematics 1 Quadratics Functions Coordinate Geometry Circular Measure Trigonometry Series Differentiation Integration Pure Mathematics 2 Algebra Logarithmic and exponential functions Trigonometry Differentiation Integration Numerical solution of equations Pure Mathematics 3 Algebra Logarithmic and exponential functions Trigonometry Differentiation Integration Numerical solution of equations Vectors Differential equations Complex numbers Mechanics Forces and equilibrium Kinematics of motion in a straight line Momentum Newton’s laws of motion Energy, work and power Probability and Statistics 1 Representation of data Permutations and combinations Probability Discrete random variables The normal distribution Probability & Statistics 2 The Poisson distribution Linear combinations of random variables Continuous random variables Sampling and estimation Hypothesis tests |
Number and Algebra · Sets of numbers; natural, integers, rational and real · Approximation, decimal places, significant figures, percentage error, estimation · Operations with scientific notation · Metric system · Arithmetic sequences and series · Geometric sequences and series · Solution of linear equations with GDC · Solutions of quadratic equations by factorization and GDC Functions · Concepts of functions as a mapping, domain, range, mapping diagrams · Linear functions and their graphs · Quadratic functions, axis of symmetry, vertex, intercepts · Exponential functions, growth and decay, asymptotic behavior · Sine and cosine functions, amplitude, period · Accurate graph drawing · Use of GDC to sketch and analyse new functions Sets, Logic and Probability · Basic concepts of set theory; subsets, intersection, union, complement · Venn diagrams and simple applications · Sample space and complementary events · Equally likely events, probability of an event A given by · probability of a · complementary event. · Venn diagrams, tree diagrams, table of outcomes · Laws of probability, combined events, mutually exclusive events, independent events, · conditional probability · Concepts of symbolic logic, definition of proposition, notation · Truth tables · Definition of implication, converse, inverse and contrapositive Geometry and Trigonometry · Coordinates in two dimensions, points, lines, planes, distance formula · Equations of line in two dimensions, gradient, intercepts parallel and perpendicular lines · Right-angle trigonometry, trigonometric ratios · Sine Rules, cosine rule, area of triangle, constructions · Three dimensional geometry, cubes, prisms, pyramids, cylinders, spheres, cones Statistics · Classification of data as discrete or continuous · Frequency tables and polygons · Histograms stem and leaf diagrams, boundaries · Cumulative frequency tables and graphs, box and whisker plots, percentiles, quartiles · Measures of central tendency, mean, median, mode · Measures of dispersion, range, interquartile range, standard deviation · Scatter plots, line of best fit, bivariate data, Pearson’s product-moment correlation coefficient, interpretation of correlations · The regression line for y on x, use of regression line for predictions · The Chi-Square test for independence, formulation of null and alternative hypothesis, · significance levels, contingency tables, expected frequencies, degrees of freedom Financial Mathematics · Currency conversions · Simple interest · Compound interest, depreciation · Construction and use of tables, loan and repayment schemes, investment and saving · schemes, inflation |
Grade 11 Matrices and Determinants Sequences and Series Permutation, Combination and Probability Mathematical Induction and Binomial Theorem Algebraic Manipulation Theory of Quadratic Equations and Inequalities Functions and Graphs Differenciation I Integration I Vectors in space Fundamental Laws of Trigonometry Trigonometric Functions and graphs
Grade 12 Complex Numbers Differencuation II Integration II Differntial Equations Vector Differenciation Mechanics Plane Analytic Geometry Conics Numerical Solutions of Non Linear Equations Inverse Trigonometric Functions and Graphs nverse Trigonometric Identities and solution of trigonometric equations Normal Distributions |
|
|
|
|
|
Nature of Mathematics |
N/A |
N/A |
Embedded as part of other SLOs and as questions to consider while teaching the concepts. |
1. Develop an appreciation for the beauty and elegance of mathematical ideas and the power of mathematics to explain and understand the world. [e.g., how the Fibonacci sequence was discovered through the mating behaviour of rabbits and the golden ratio exists throughout nature from flowers to shells to the human body.] [e.g. the beauty behind Euler’s formula]
2. Recognize the philosophical debates that arise in mathematics such as whether mathematics was invented or discovered. [e.g., consider the number e or logarithms–did they already exist before man defined them?]
3. Explain, that a paradox is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. It is a situation that defies logical explanation and challenges conventional thinking, often leading to new insights and discoveries in mathematics. [e.g., Zeno’s paradox of Achilles and the tortoise]
4. Construct mathematical models or use mathematical ideas to depict real-life situations and solve social problems, such as: - Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed. - Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player. - Designing the layout of the stalls in a school fair so as to raise as much money as possible. - Analyzing stopping distance for a car. - Modeling savings account balance, bacterial colony growth, or investment growth. - Analyzing risk in situations such as extreme sports, pandemics, and terrorism. - Relating population statistics to individual predictions.
5. Develop an understanding of the limitations of mathematical models and the need for mathematical rigor and caution when drawing conclusions from mathematical data. [e.g. problems caused by absence of representative samples such as in the US presidential elections in 1936, Literary Digest v George Gallup]. |
1. Students are not expected to write long answers to these questions, but students can be made to reflect on their views of mathematics through short answers. For example, students can be asked at the end of a unit, "What do you find beautiful in mathematics?" and "How is this beauty different or similar to the beauty experienced in other things such as when you view art or listen to music?"
2. Students are not expected to formally write their own opinions on some assessment but can have a class discussion on these ideas to help them think about the various aspects of mathematics.
3, 4 These should be regularly incorporated throughout the other math topics by selecting tasks that allow students to use their knowledge and tools to solve real-life problems. |
What is the place of beauty and elegance in mathematics?
Is mathematics invented or discovered?
What is a mathematical paradox?
How is mathematics useful?
|
These SLOs continue the unit on NOM introduced in Grade 9-10. After having learnt more mathematical ideas and concepts, students continue to explore the NOM through questions, debates and modeling. These will allow them to use their previous and current knowledge to understand the different sides of mathematics, in particular the beauty and usefulness of mathematics. |
|
|
Mathematical Proofs |
No mention of what a proof is, or how is it carried out but formal proofs of theorems are mentioned in the SLOs. For example, proving properties of union and intersection of sets, proving the Pythagoras theorem, the laws of logarithms, the remainder and factor theorem, theorems on circles, parallelograms and triangles, etc.
Proof by mathematical induction is part of Grade XII: Apply principle of Mathematical Induction to prove statements, identities and formulae.
|
N/A |
- Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.
- The symbols and notation for equality and identity.
- Proof by mathematical induction. Proof should be incorporated throughout the course where appropriate. Mathematical induction links specifically to a wide variety of topics, for example complex numbers, differentiation, sums of sequences and divisibility.
- Proof by contradiction. Examples: Irrationality of 3; irrationality of the cube root of 5; Euclid’s proof of an infinite number of prime numbers; if a is a rational number and b is an irrational number, then a + b is an irrational number.
- Use of a counterexample to show that a statement is not always true. Example: Consider the set P of numbers of the form n2 + 41n + 41, n ∈ ℕ, show that not all elements of P are prime. Example: Show that the following statement is not always true: there are no positive integer solutions to the equation x2 + y2 = 10. It is not sufficient to state the counterexample alone. Students must explain why their example is a counterexample |
1. Understand the different types of proofs such as direct proof, proof by counter example and proof by contradiction.
2. Apply the Principle of Mathematical Induction to prove statements, identities and formulae.
3. Determine the best method to show that a mathematical statement is true. |
1. Students are expected to prove only simple statements and common examples such as - The irrationality of the square root of 2 for proof by contradiction - Show that the following statement is not always true: there are no positive integer solutions to the equation x^2 + y^2 = 10, for proof by counterexample. It is not sufficient to state the counterexample alone. Students must explain why their example is a counterexample
2. The PMI should take examples from a wide variety of mathematical domains such as complex numbers, differentiation, and sums of sequences. For example, prove that the sum of the first n odd positive integers is n^2. |
What are the different kinds of proofs? |
This unit follows from the unit on proofs introduced in Grade 9-10. This builds on from the introduction of proofs to the different types of proofs. Again, the purpose is to help students be familiar with basic proof writing to help in their critical thinking skills and their transition to university level mathematics.
The 2006 National Curriculum includes Principle of Mathematical Induction but it is introduces it as a method without a sense of understanding of where it comes from and why is it useful. Placing it with other types of proofs will help students understand the power of each kind of proof and develop abilities to choose which method would be most suitable. |
|
|
COMPLEX NUMBERS |
Complex Numbers Recall complex number ⱬ represented by an expression of the form ⱬ = a + ib or of the form ( a,b ) where a and b are real numbers and i =√-1 ii) Recogtnize a as real part of ⱬ and b as imaginary part of ⱬ . iii) Know the condition for equality of complex numbers. iv) Carryout basic operations on complex numbers. v) Define as the complex conjugate of ⱬ = a + ib . vi) Define |ⱬ| = √(a+b) as the absolute value or modulus of a complex number ⱬ = a + ib . Properties of Complex Numbers Describe algebraic properties of complex numbers (e.g. commutative, associative and distributive) with respect to ‘+’ and ‘×’. ii) Know additive identity and multiplicative identity for the set of complex numbers. iii) Find additive inverse and multiplicative inverse of a complex number ⱬ. iv) Demonstrate the following properties v) Find real and imaginary parts of the following type of complex numbers Solution of equations i)Solve the simultaneous linear equations with complex coefficients. For example, ii) Write the polynomial P(z) as a product of linear factors. For example, iii) Solve quadratic equation of the form 0 2 pz + qz + r = by completing squares, where p, q, r are real numbers and z a complex number. For example: Solve 2 5 0 2 z − z + = . ⇒ (z −1− 2i)(z −1+ 2i) = 0 , ⇒ z = 1+ 2i, 1− 2i . |
"Candidates should be able to: • understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs represent complex numbers geometrically by means of an Argand diagram • carry out operations of multiplication and division of two complex numbers expressed in polar form r r cos si in ei i i + / i find the two square roots of a complex number understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram"
|
Not in IB |
"Complex Numbers Identify: complex numbers, complex conjugate, absolute value or modulus of a complex number, algebraic properties of complex no.and carryout basic operations on complex numbers Demonstrate additive identity and multiplicative identity for the set of complex numbers. Find additive inverse and multiplicative inverse of a complex number z . Demonstrate the following properties |z|=|-z|=|z ̅ |=|-z ̅ |, z ̿=z, z.z ̅=|z|^2, (z_1+z_2 ) ̅=(z_1 ) ̅+(z_2 ) ̅, (z_1 z_2 ) ̅=(z_1 ) ̅(z_2 ) ̅, ((z_1/z_2 ) ) ̅=(z_1 ) ̅/(z_2 ) ̅ ,z_2≠0 Solve the simultaneous linear equations with complex coefficients. Solve quadratic equation of the form pz^2+qz+r=0 by completing squares, where p, q, r are real numbers and z a complex number. Express complex numbers in polar form |
Grade 12 Complex Numbers Understand the following algebraic representation of complex numbers: Identification of complex numbers, Complex conjugate, Absolute value or modulus of a complex number, Applying algebraic properties and carryout basic operations on complex numbers, Additive identity and Multiplicative identity for the set of complex numbers, Additive inverse and Multiplicative inverse of a complex number, Demonstrating properties Finding real and imaginary parts of complex numbers of the type (x+iy)^n ∀n∈Z, Solving the simultaneous linear equations with complex coefficients, Solving quadratic equations of the form pz^2+qz+r=0 ∀p.q,r∈R and z∈∁. Understand the following geometric interpretation of the algebraic operations: Geometric interpretation of a complex number, Geometric interpretation of the modulus, Geometric interpretation of the algebraic operations.
Understand the following polar representation of complex numbers: Polar coordinates in the plane, Polar representation of a complex number, Operations with complex numbers in polar representation, Solving simple equations and in-equations involving complex numbers in polar form. |
How do you add, subtract, multiply and divide complex numbers? Why are functions and relations represented by parametric equations? Why are functions and relations represented by polar equations? |
Shifted to grade 12 as Students are young enough to understand the concept and it is the part of A levels also. |
|
|
"MATRICES AND DETERMINANTS" |
"Matrices Recall the concept of • a matrix and its notation, • order of a matrix, • equality of two matrices. Define row matrix, column matrix, square matrix, rectangular matrix, zero/null matrix, identity matrix, scalar matrix, diagonal matrix, upper and lower triangular matrix, transpose of a matrix, symmetric matrix and skew-symmetric matrix. Algebra of Matrices Carryout scalar multiplication, addition/subtraction of matrices, multiplication of matrices with real and complex entries. ii) Show that commutative property • holds under addition. • does not hold under multiplication, in general. iii) Verify that ( ) t t t AB = B A . Determinants Describe determinant of a square matrix, minor and cofactor of an element of a matrix. ii) Evaluate determinant of a square matrix using cofactors. iii) Define singular and non-singular matrices. iv) Know the adjoint of a square matrix. v) Use adjoint method to calculate inverse of a square matrix. vi) Verify the result ( ) −1 −1 −1 Properties of Determinants State and prove the properties of determinants. ii) Evaluate the determinant without expansion (i.e. using properties of determinants). Row and Column Operations Know the row and column operations on matrices. ii) Define echelon and reduced echelon form of a matrix. iii) Reduce a matrix to its echelon and reduced echelon form. iv) Recognize the rank of a matrix Use row operations to find the inverse and the rank of a matrix. Solving System of Linear Equations i) Distinguish between homogeneous and nonhomogeneous linear equations in 2 and 3 unknowns. ii) Solve a system of three homogeneous linear equations in three unknowns. iii) Define a consistent and inconsistent system of linear equations and demonstrate through examples. iv) Solve a system of 3 by 3 non-homogeneous linear equations using: • matrix inversion method, • Gauss elimination method (echelon form), • Gauss-Jordan method (reduced echelon form), • Cramer’s rule. " |
Not in A Level |
Not in IB |
Matrices & Determinants Apply matrix operations (addition/subtraction and multiplication of matrices) with real and complex entries. Evaluate determinants of 3×3 matrix by using properties of determinant. find multiplicative inverse matrices. Solve a system of 3 by 3 non-homogeneous linear equations using matrix inversion method Solve a system of three non-homogeneous linear equations in three unknowns. Solve daily life problems involving matrices |
Grade 11 concept |
What are the similarities and differences between matrices and real numbers? • How do the commutative, associative, and distributive properties apply to matrices? • What methods do you have to solve systems of equations and what are the advantages of each method? • How do we find the inverse of a matrix and when does a matrix not have an inverse defined? • How can I use vector operations to model, solve, and interpret real-world problems? • How do geometric interpretations of addition, subtraction, and scalar multiplication of vectors help me perform computations efficiently? |
|
|
|
Algebraic Manipulation |
|
Algebra: understand the meaning of |x|, sketch the graph of y = |ax + b| and use relations such as |a| = |b| ⇔ a2 = b2 and |x – a| < b ⇔ a – b < x < a + b when solving equations and inequalities • divide a polynomial, of degree not exceeding 4,by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero) • use the factor theorem and the remainder theorem. recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than – (ax + b)(cx + d)(ex + f) – (ax + b)(cx + d)2 – (ax + b)(cx2 + d) Excluding cases where the degree of the numerator exceeds that of the denominator • use the expansion of (1 + x)n , where n is a rational number and x 1 1. |
|
Division of polynomial: Divide a polynomial of degree up to 4 by a linear and quadratic polynomial to identify quotient and remainder.
[Remainder Theorem and Factor Theorem]: Demonstrate and apply remainder theorem Analyze and apply factor theorem to factorize a cubic polynomial [Partial Fractions] Resolve an algebraic fraction into partial fractions when its denominator consists of •non-repeated linear factors, •repeated linear factors, |
Explain how long division of a polynomial expression by a binomial expression of the form x – a, a є I is related to synthetic division. Divide a polynomial expression by a binomial expression of the form x – a, a є I, using long division or synthetic division. Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function. Explain the relationship between the remainder when a polynomial expression is divided by x – a, a є I and the value of the polynomial expression at x = a (remainder theorem). Explain and apply the factor theorem to express a polynomial expression as a product of factors. |
How do variables help you model real-life situations? How can you use the properties of real numbers to simplify algebraic expressions? How do you solve an equation or inequality? |
Division of polynomial, Remainder Theorem and Factor Theorem and Partial fractions are the part of Grade 9 while this concept is the part of A levels therefore Shifted to grade 11 |
|
|
Solution of Quadratic Equations /Inequalities |
Not in Grade 11 and !2 |
carry out the process of completing the square for a quadratic polynomial ax 2 + bx + c and use a completed square form find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant solve quadratic equations, and quadratic inequalities, in one unknown solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic recognise and solve equations in x which are quadratic in some function of x. |
Solution of linear equations with GDC Solutions of quadratic equations by factorization and GDC |
Theory of Quadratic Equations and Inequalities Solve quadratic equations by any method Establish relationship between roots and coefficients of quadratic equations Form quadratic equation when roots are given Find discriminant of a given quadratic equation. Identify the nature of roots of a quadratic equation through discriminant. Solve a pair of linear and quadratic equations simultaneously Solve quadratic inequalities in one unknown. Solve quadratic equations, and quadratic inequalities, in one unknown Solve problems that involve quadratic equation |
Grade11 concept Explain, using examples, the relationship among the roots of a quadratic equation, the zeros of the corresponding quadratic function, and the x-intercepts of the graph of the quadratic function. Derive the quadratic formula, using deductive reasoning. Solve a quadratic equation of the form ax2 + bx + c = 0 by using strategies such as determining square roots factoring completing the square applying the quadratic formula graphing its corresponding function Select a method for solving a quadratic equation, justify the choice, and verify the solution. Explain, using examples, how the discriminant may be used to determine whether a quadratic equation has two, one, or no real roots; and relate the number of zeros to the graph of the corresponding quadratic function. Identify and correct errors in a solution to a quadratic equation. Solve a problem by determining or analyzing a quadratic equation |
How do quadratic equations model real world problems and situations? What do quadratic solutions mean in terms of the problem? What are the advantages of a quadratic function in vertex form and in standard form? |
The study of Quadratic Equations was moved from grade 10 to grade 11 due to its inclusion in the A levels syllabus.The concept of Quadratic Equations has been a part of the mathematics curriculum since 2006 and has continued to be a part of the mathematics curriculum in 2023.
Quadratic Equations are a fundamental concept in algebra that deals with the study of second-degree equations in one variable. It involves the use of mathematical formulas and techniques to solve equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
In 2006, the study of quadratic equations was included in the mathematics curriculum in detail however half of it is shifted to grade 11. It has continued to be a part of the curriculum in 2023, as it provides a foundation for more advanced mathematical concepts in higher education and in various fields such as physics, engineering, and computer science.
The study of quadratic equations is essential because it helps students develop problem-solving skills and logical reasoning. It also enables students to understand real-world problems and make informed decisions based on data and analysis. Therefore, the continued inclusion of the quadratic equations concept in the mathematics curriculum is essential for ensuring that students receive a well-rounded education and are prepared for success in their future careers. |
|
|
VECTORS IN PLANE |
"Vectors in Plane Define a scalar and a vector. ii) Give geometrical representation of a vector. iii) Give the following fundamental definitions using geometrical representation: • magnitude of a vector, • equal vectors, • negative of a vector• unit vector, • zero/null vector, • position vector, • parallel vectors, • addition and subtraction of vectors, • triangle, parallelogram and polygon laws of addition, • scalar multiplication. iv) Represent a vector in a Cartesian plane by defining fundamental unit vectors i and j . v) Recognize all above definitions using analytical representation. vi) Find a unit vector in the direction of another given vector. vii) Find the position vector of a point which divides the line segment joining two points in a given ratio. viii) Use vectors to prove simple theorems of descriptive geometry. |
"use standard notations for vectors, i.e. • carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms • calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a + tb, and find the equation of a line, given sufficient information determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points. " |
Vectors as displacements in the plane and in three dimensions Components of a vector; column representation ( Algebraic and geometric approaches to the following: o the sum and difference of two vectors; the zero vector, the vector −v ; o multiplication by a scalar, kv ; parallel vectors; o magnitude of a vector, v ; o unit vectors; base vectors; i, j and k; o position vectors OA = a; o 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: . |
Vectors in Plane Recognize rectangular coordinate system in plane. Represent vectors as directed line segments express a vector in terms of two non-zero and non-parallel coplanar vectors. Express a vector in terms of position vector Express translation by a vector Find magnitude of a vector. add and subtract vectors, multiply a vector by a scalar Solve geometrical problems involving the use of vectors
|
Shifted to Grade 10 |
Why are functions and relations represented by vectors? |
Shifted to Grade 10 as it is the part of O levels |
|
|
VECTORS IN SPACE |
Vectors in Space Recognize rectangular coordinate system in space. ii) Define unit vectors i , j and k . iii) Recognize components of a vector. iv) Give analytic representation of a vector. v) Find magnitude of a vector. vi) Repeat all fundamental definitions for vectors in space which, in the plane, have already been discussed. Properties of Vector Addition State and prove • commutative law for vector addition. • associative law for vector addition. ii) Prove that: • 0 as the identity for vector addition. • − A as the inverse for A Properties of Scalar Multiplication State and prove: • commutative law for scalar multiplication, • associative law for scalar multiplication, • distributive laws for scalar multiplication. Dot or Scalar Product Define dot or scalar product of two vectors and give its geometrical interpretation. ii) Prove that: • i ⋅i = j ⋅ j = k ⋅ k =1, • i ⋅ j = j ⋅ k = k ⋅i = 0 . iii) Express dot product in terms of components. iv) Find the condition for orthogonality of two vectors. v) Prove the commutative and distributive laws for dot product. vi) Explain direction cosines and direction ratios of a vector. vii) Prove that the sum of the squares of direction cosines is unity. viii) Use dot product to find the angle between two vectors. ix) Find the projection of a vector along another vector. x) Find the work done by a constant force in moving an object along a given vector. Cross or Vector Product Define cross or vector product of two vectors and give its geometrical interpretation. ii) Prove that: • i × i = j × j = k × k = 0 , • i × j = − j × i = k , • j × k = −k × j = i , • k × i = −i × k = j . iii) Express cross product in terms of components. iv) Prove that the magnitude of A× B represents the area of a parallelogram with adjacent sides A and B . v) Find the condition for parallelism of two non-zero vectors. vi) Prove that A× B = −B× A. vii) Prove the distributive laws for cross product. viii) Use cross product to find the angle between two vectors. ix) Find the vector moment of a given force about a given point. Scalar Triple Product Define scalar triple product of vectors. ii) Express scalar triple product of vectors in terms of components (determinantal form). iii) Prove that: • i ⋅ j × k = j ⋅ k × i = k ⋅i × j =1, • i ⋅k × j = j ⋅i ×k = k ⋅ j ×i = −1. iv) Prove that dot and cross are inter-changeable in scalar triple product. v) Find the volume of • a parallelepiped, • a tetrahedron, determined by three given vectors. vi) Define coplanar vectors and find the condition for coplanarity of three vectors. " |
use standard notations for vectors, • carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms e.g. ‘OABC is a parallelogram’ is equivalent to OB = OA + OC . The general form of the ratio theorem is not included, but understanding that the midpoint of AB has position vector 2 OA OB 1 _ + i is expected. • calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors In 2 or 3 dimensions. • understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a + tb, and find the equation of a line, given sufficient information e.g. finding the equation of a line given the position vector of a point on the line and a direction vector, or the position vectors of two points on the line. • determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists Calculation of the shortest distance between two skew lines is not required. Finding the equation of the common perpendicular to two skew lines is also not required. • use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points. |
|
Vectors in Space Recognize rectangular coordinate system in space. Recognize unit vectors i , j and k . Recognize components of a vector. Give analytic representation of a vector. Find magnitude of a vector. Repeat all fundamental mathematical operations for vectors in space which, in the plane, have already been discussed. Demonstrate and prove properties of vector addition Commutative law for vector addition. Associative law for vector addition. 0 as the identity for vector addition. -A as the inverse for A. Dot or Scalar Product Transform dot or scalar product of two vectors and give its geometrical interpretation. Express dot product in terms of components. Find the condition for orthogonality of two vectors. Prove the commutative and distributive laws for dot product. Explain direction cosines and direction ratios of a vector. Prove that the sum of the squares of direction cosines is unity. Use dot product to find the angle between two vectors. Find the projection of a vector along another vector. Find the work done by a constant force in moving an object along a given vector. Solve daily life problems based of vectors. Cross or Vector Product Transform cross or vector product of two vectors and give its geometrical interpretation. Express cross product in terms of components. Prove that the magnitude of ▁a×▁b represents the area of a parallelogram with adjacent sides ▁a and ▁b. Find the condition for parallelism of two non - zero vectors Verify that ▁a×▁b=-▁b×▁a Verify the distributive laws for cross product. Use cross product to find the angle between two vectors. Find the vector moment of a given force about a given point. Solve situations in daily life based on Cross orVector Product. Scalar Triple Product Transform scalar triple product of vectors. Express scalar triple product of vectors in terms of components (determinantal form). Prove that dot and cross are inter-changeable in scalar triple product. Find the volume of a parallelepiped and a tetrahedron, determined by three given vectors. Identify coplanar vectors and find the condition for coplanarity of three vectors.
|
Concept of grade 11 |
What is the difference between Vector and Vector Space? Is zero a vector space? What are Equal Vectors? |
Previously the whole concept was covered in grade 11 however now vector in plane will be covered in Grade 10,11 and 12 |
|
|
SEQUENCES AND SERIES |
"Sequence Define a sequence (progression) and its terms. ii) Know that a sequence can be constructed from a formula or an inductive definition. iii) Recognize triangle, factorial and Pascal sequences. Arithmetic Sequence Define an arithmetic sequence. ii) Find the n th or general term of an arithmetic sequence. iii) Solve problems involving arithmetic sequence. Arithmetic Mean Know arithmetic mean between two numbers. ii) Insert n arithmetic means between two numbers. Arithmetic Series Define an arithmetic series. ii) Establish the formula to find the sum to n terms of an arithmetic series. iii) Show that sum of n arithmetic means between two numbers is equal to n times their arithmetic mean. iv) Solve real life problems involving arithmetic series. Geometric Sequence i) Define a geometric sequence. ii) Find the n th or general term of a geometric sequence. iii) Solve problems involving geometric sequence. Geometric Mean Know geometric mean between two numbers. ii) Insert n geometric means between two numbers Geometric Series Define a geometric series. ii) Find the sum of n terms of a geometric series. iii) Find the sum of an infinite geometric series. iv) Convert the recurring decimal into an equivalent common fraction. v) Solve real life problems involving geometric series. Harmonic Sequence i) Recognize a harmonic sequence. ii) Find nth term of harmonic sequence. Harmonic Mean Define a harmonic mean. ii) Insert n harmonic means between two numbers. " |
"use the expansion of (a + b)n, where n is a positive integer recognise arithmetic and geometric progressions use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression." |
Arithmetic sequences and series Geometric sequences and series |
Sequences and Series • Analyze arithmetic sequences and series to solve problems • Analyze geometric sequences and series to solve problems • Analyze harmonic sequences and series to solve problems |
Grade 11 concept Identify the assumption(s) made when defining an arithmetic sequence or series. Provide and justify an example of an arithmetic sequence. Derive a rule for determining the general term of an arithmetic sequence. Describe the relationship between arithmetic sequences and linear functions. Determine the first term, the common difference, the number of terms, or the value of a specific term in a problem involving an arithmetic sequence. Derive a rule for determining the sum of n terms of an arithmetic series. Determine the first term, the common difference, the number of terms, or the value of the sum of specific numbers of terms in a problem involving an arithmetic series. Solve a problem that involves an arithmetic sequence or series. Identify assumptions made when identifying a geometric sequence or series. Provide and justify an example of a geometric sequence. Derive a rule for determining the general term of a geometric sequence. Determine the first term, the common ratio, the number of terms, or the value of a specific term in a problem involving a geometric sequence. Derive a rule for determining the sum of n terms of a geometric series. Determine the first term, the common ratio, the number of terms, or the value of the sum of a specific number of terms in a problem involving a geometric series. Generalize, using inductive reasoning, a rule for determining the sum of an infinite geometric series. Explain why an infinite geometric series is convergent or divergent. Solve a problem that involves a geometric sequence or series. |
How can you write a rule for the nth term of a sequence? How can you recognize an arithmetic sequence from its graph? How can you recognize a geometric sequence from its graph? How can you find the sum of an infinite geometric series? |
Combined both Sequence and Series and Misceleneous Series under one heading |
|
|
MISCELLANEOUS SERIES |
Evaluation of 2 3 ∑n, ∑n and ∑n i) Recognize sigma ( ∑ ) notation. ii) Find sum of • the first n natural numbers ( ∑n ), • the squares of the first n natural numbers ( 2 ∑n ), • the cubes of the first n natural numbers ( 3 ∑n ). Arithmetico-Geometric Series i) Define arithmetico-geometric series. ii) Find sum to n terms of the arithmetico-geometric series Method of Differences Define method of differences. Use this method to find the sum of n terms of the series whose differences of the consecutive terms are either in arithmetic or in geometric sequence. Summation of Series using Partial Fractions Use partial fractions to find the sum to n terms and to infinity the series of the type a(a 1 +d ) + (a+d )( 1 a+2d ) +L |
use the expansion of (a + b)n , where n is a positive integer recognise arithmetic and geometric progressions • use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression. |
Not in IB |
|
|
|
|
|
|
Permutation and Combination |
Factorial of a Natural Number Know Kramp’s factorial notation to express the product of first n natural numbers by n!. Permutation i) Recognize the fundamental principle of counting and illustrate this principle using tree diagram. ii) Explain the meaning of permutation of n different objects taken r at a time and know the notation r n P . Prove that P = n(n −1)(n − 2) (n − r +1) r n L and hence deduce that iv) Apply r n P to solve relevant problems of finding the number of arrangements of n objects taken r at a time (when all n objects are different and when some of them are alike). v) Find the arrangement of different objects around a circle. Combination Define combination of n different objects taken r at a time. ii) Prove the formula !( )! ! = and deduce that iii) Solve problems involving combination. Probability i) Define the following: • statistical experiment, • sample space and an event, • mutually exclusive events, • equally likely events, • dependent and independent events, • simple and compound events. ii) Recognize the formula for probability of occurrence of an event E, that is iii) Apply the formula for finding probability in simple cases. iv) Use Venn diagrams and tree diagrams to find the probability for the occurrence of an event. v) Define the conditional probability. vi) Recognize the addition theorem (or law) of probability: P(A∪ B) = P(A)+ P(B)− P(A∩ B), where A and B are two events. Deduce that P(A∪ B) = P(A) + P(B) where A and B are mutually exclusive events. vii) Recognize the multiplication theorem (or law) of probability P(A∩ B) = P(A)P(B A) or P(A∩ B) = P(B)P(A B) where P(B A) and P(A B) are conditional probabilities. Deduce that P(A∩B) = P(A)P(B) where A and B are independent events. viii) Use theorems of addition and multiplication of probability to solve related problems. |
understand the terms permutation and combination, and solve simple problems involving selections solve problems about arrangements of objects in a line, including those involving – repetition (e.g. the number of ways of arranging the letters of the word ‘NEEDLESS’) – restriction (e.g. the number of ways several people can stand in a line if two particular people must, or must not, stand next to each other). evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events, or by calculation using permutations or combinations use addition and multiplication of probabilities, as appropriate, in simple cases understand the meaning of exclusive and independent events, including determination of whether events A and B are independent by comparing the values of P^A + Bh and P^Ah # P^Bh • calculate and use conditional probabilities in simple cases. sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians) use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles use the notations sin–1x, cos– 1x, tan–1x to denote the principal values of the inverse trigonometric relations use the identities cos sin / tan i i i and sin cos 1 find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included |
Not required: formal treatment of permutations and formula for nPr · Concepts of population, sample, random sample, discrete and continuous data. · Presentation of data: frequency distributions (tables); frequency histograms with equal class intervals; box-and-whisker plots; outliers. · Grouped data: use of mid-interval values for calculations; interval width; upper and lower interval boundaries; modal class. · Not required: frequency density histograms. · 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. · Not required: the coefficient of determination R2. · Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. · The probability of an event A is . · The complementary events A and A’(not A). · Use of Venn diagrams, tree diagrams and tables of outcomes. · Combined events, . · Mutually exclusive events: . · Conditional probability; the definition · Independent events; the definition P( A| B) = P(A) = P( A| B’) . · Probabilities with and without replacement. · Concept of discrete random variables and their probability distributions. · Expected value (mean), E(X ) for discrete data. · Applications. · Binomial distribution. · Mean and variance of the binomial distribution. · Not required: formal proof of mean and variance. · Normal distributions and curves. · Standardization of normal variables (z-values, z-scores). · Properties of the normal distribution. |
Permutation and Combination • Solve problems that involve the fundamental counting principle. • Solve problems that involve permutations. • Solve problems that involve combinations. |
Grade 11 concept Count the total number of items in the sample space, using graphic organizers such as lists and tree diagrams. Explain, using examples, why the total number of items is found by multiplying rather than adding the number of ways the individual choices can be made. Solve a simple counting problem by applying the fundamental counting principle Count, using graphic organizers such as lists and tree diagrams, the number of ways of arranging the elements of a set in a row. Determine, in factorial notation, the number of permutations of n different elements taken n at a time to solve a problem. Determine, using a variety of strategies, the number of permutations of n different elements taken r at a time to solve a problem. Explain why n must be greater than or equal to r in the notation nPr . Solve an equation that involves nPr notation. Explain, using examples, the effect on the total number of permutations when two or more elements are identical. Explain, using examples, the difference between a permutation and a combination. Determine the number of combinations of n different elements taken r at a time to solve a problem. Explain why n must be greater than or equal to r in the notation nCr or Explain, using examples, why nCr = nCn–r or Solve an equation that involves nCr or notation. |
How can you write a rule for the nth term of a sequence? How can you recognize an arithmetic sequence from its graph? How can you recognize a geometric sequence from its graph? How can you find the sum of an infinite geometric series? |
Already concept of grade 11 retained |
|
|
THE NORMAL DISTRIBUTION |
|
understand the use of a normal distribution to model a continuous random variable, and use normal distribution tables solve problems concerning a variable X, where X N , 2 + _n v i, including – finding the value of P X x > 1 _ i, or a related probability, given the values of x1, n, v. – finding a relationship between x1, n and v given the value of P X x > 1 _ i or a related probability For calculations involving standardisation, full details of the working should be shown. recall conditions under which the normal distribution can be used as an approximation to the binomial distribution, and use this approximation, with a continuity correction, in solving problems |
Normal distributions and curves. Standardization of normal variables (z-values, z-scores). Properties of the normal distribution |
The normal distribution Demonstrate an understanding of normal distribution, including • standard deviation • z-scores |
Grade 12 concept Explain, using examples, the meaning of standard deviation. Calculate, using technology, the population standard deviation of a data set. Explain, using examples, the properties of a normal curve, including the mean, median, mode, standard deviation, symmetry, and area under the curve. Determine whether a data set approximates a normal distribution, and explain the reasoning. Compare the properties of two or more normally distributed data sets. Explain, using examples that represent multiple perspectives, the application of standard deviation for making decisions in situations, such as warranties, insurance, or opinion polls. Solve a contextual problem that involves the interpretation of standard deviation. Determine, with or without technology, and explain the z-score for a value in a normally distributed data set. Solve a contextual problem that involves normal distribution |
How do you find percents of data and probabilities of events associated with normal distributions? How is the mean of a sampling distribution related to the population mean or proportion? How can you use shape, center and spread to characterize a data distribution? |
Added in Grade 12 as statistics is missing in Curriculum 2006 |
|
|
MATHEMATICAL INDUCTION AND BINOMIAL THEOREM |
Mathematical Induction i) Describe principle of mathematical induction. ii) Apply the principle to prove the statements, identities or formulae Binomial Theorem i) Use Pascal’s triangle to find the expansion of ( ) n x + y where n is a small positive integer. ii) State and prove binomial theorem for positive integral index. iii) Expand ( ) n x + y using binomial theorem and find its general term. iv) Find the specified term in the expansion of ( )n x + y . Binomial Series i) Expand ( ) n 1 where + x n is a positive integer and extend this result for all rational values of n. ii) Expand ( ) n 1 in ascending powers of + x x and explain its validity/convergence for x < 1 where n is a rational number. iii) Determine the approximate values of the binomial expansions having indices as –ve integers or fractions. |
use formulae for probabilities for the binomial and geometric distributions, and recognise practical situations where these distributions are suitable models • use formulae for the expectation and variance of the binomial distribution and for the expectation of the geometric distribution. recall conditions under which the normal distribution can be used as an approximation to the binomial distribution, and use this approximation, with a continuity correction, in solving problems. use the Poisson distribution as an approximation to the binomial distribution where appropriate formulate hypotheses and carry out a hypothesis test in the context of a single observation from a population which has a binomial or Poisson distribution, using – direct evaluation of probabilities – a normal approximation to the binomial or the Poisson distribution, where appropriate calculate the probabilities of making Type I and Type II errors in specific situations involving tests based on a normal distribution or direct evaluation of binomial or Poisson probabilities |
The binomial theorem: expansion of N. Calculation of binomial coefficients using Pascal’s triangle |
Mathematical Induction • Describe a mathematical argument, Identify the base case, induction of hypothesis and a precise conclusion. • Apply the principle of mathematical induction to prove statements, identities, divisibility of numbers and summation formulae. • Evaluate and justify conclusions, communicating a position clearly in an appropriate mathematical form in daily life. Binomial Theorem • State the binomial theorem using sigma notation and combination notation (and/or factorials), • Apply the binomial theorem to expand a binomial of any nth power, • Describe Binomial Theorem as expansion of binomial powers restricted to the set of natural numbers, • Calculate binomial coefficients using Pascal’s triangle. • Understand the properties of binomial theorem. • Manipulate or evaluate expressions and equations involving binomials and combinations (and/or factorials). Applications of Binomial Theorem • find the remainder when a number to some large exponent is divided by a number • find the digit of a number • test the divisibility by a number • compare two large numbers |
Grade 11 Explain the patterns found in the expanded form of (x + y)n, n ≤ 4, by multiplying n factors of (x + y). Explain how to determine the subsequent row in Pascal’s triangle, given any row. Relate the coefficients of the terms in the expansion of (x + y)n to the (n + 1)st row in Pascal’s triangle. Explain, using examples, how the coefficients of the terms in the expansion of (x + y)n are determined by combinations. Expand, using the binomial theorem (x + y)n. Determine a specific term in the expansion of (x + y)n |
How do we find binomial probabilities and test hypotheses? |
Shifted to Grade 11 from Grade 12 as to build foundation for the next concepts |
|
|
FUNCTIONS AND GRAPHS |
Functions i) Recall • function as a rule of correspondence, • domain, co-domain and range of a function, • one to one and onto functions. ii) Know linear, quadratic and square root functions. Inverse Function Define inverse functions and demonstrate their domain and range with examples. Graphical Representation of Functions i) Sketch graphs of • linear functions (e.g. y = ax + b ), • non-linear functions (e.g. 2 y = x ). ii) Sketch the graph of the function n y = x where n is • a + ve integer, • a − ve integer ( x ≠ 0 ), • a rational number for x >0. iii) Sketch graph of quadratic function of the form y = ax + bx + c 2 , a(≠ 0) , b, c are integers. iv) Sketch graph using factors. v) Predict functions from their graphs (use the factor form to predict the equation of a function of the type f x = ax + bx + c 2 ( , if two points where the graph ) crosses x-axis and third point on the curve, are given). Intersecting Graphs i) Find the intersecting point graphically when intersection occurs between • a linear function and coordinate axes, • two linear functions, • a linear and a quadratic function. ii) Solve, graphically, appropriate problems from daily life. |
understand the terms function, domain, range, one-one function, inverse function and composition of functions identify the range of a given function in simple cases, and find the composition of two given functions determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases illustrate in graphical terms the relation between a one-one function and its inverse understand and use the transformations of the graph of y = f(x) given by y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and simple combinations of these |
Concept of function Domain, range; image (value). Composite functions. Not required: Domain restriction. Identity function. Inverse function The graph of a function; its equation y = f (x). Function graphing skills. Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range. Use of technology to graph a variety of functions, including ones not specifically mentioned. |
Functions and Graphs Classify the functions as algebraic and transcendental functions Describe various transcendental functions, such as: Trigonometric functions Inverse trigonometric functions Logarithmic function Exponential function Identify the domain and range of fundamental transcendental functions. Logarithmic function Demonstrate an understanding of logarithms. Derive and apply product, quotient, and power laws of logarithms. Exponential function Graph and analyze exponential and logarithmic function Solve problems that involve exponential and logarithmic equations Graphical Representation: Display graphically: y = f (x) , where f (x)= e^x,a^x,logx,lnx Identify through graphs domain, co-domain and range of a function. Draw the graph of modulus function Interpret the relation between a one-one function and its inverse through graph. Demonstrate the transformations of a graph of the form: horizontal shift,vertical shift, scaling Limit of a Function Demonstrate and find the limit of a sequence. State and apply theorems on limit of sum, difference, product and quotient of functions to algebraic, exponential and trigonometric functions. Continuous and Discontinuous Functions Demonstrate and test Continuity, discontinuity of a function at a point and in an interval. |
Concept of Grade 11 Sketch, with or without technology, a graph of an exponential function of the form y = ax, a > . Identify the characteristics of the graph of an exponential function of the form y = ax, a > 0, including the domain, range, horizontal asymptote, and intercepts, and explain the significance of the horizontal asymptote. Sketch the graph of an exponential function by applying a set of transformations to the graph f y = ax, a > 0, and state the characteristics of the graph. Sketch, with or without technology, the graph of a logarithmic function of the form y = logb x, > 1. Identify the characteristics of the graph of a logarithmic function of the form y = logb x, b > 1, including the domain, range, vertical asymptote, and intercepts, and explain the significance of the vertical asymptote. Sketch the graph of a logarithmic function by applying a set of transformations to the graph of y = logb x, b > 1, and state the characteristics of the graph. Demonstrate, graphically, that a logarithmic function and an exponential function with the same base are inverses of each other. |
|
The concept was already included in the 2006 mathematics curriculum. However, it has been further upgraded to meet the requirements of the A-levels and International Baccalaureate (IB) programs. |
|
|
Logarithmic Function |
Transcendental Functions Recognize algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic (and their identities), explicit and implicit functions, and parametric representation of functions. |
understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base) • understand the definition and properties of ex and lnx, including their relationship as inverse functions and their graphs use logarithms to solve equations and inequalities in which the unknown appears in indices e.g. 2 5 x 1 , 3 2 < 5 3 1 x # − , 3 4 x x 1 2 1 = + − . • use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. |
Exponential functions and their graphs: Logarithmic functions and their graphs: . Relationships between these functions: Solving equations, both graphically and analytically. Solving exponential equations. |
Demonstrate an understanding of logarithms. Derive and apply product, quotient, and power laws of logarithms. |
Grade 11 Explain the relationship between logarithms and exponents. Express a logarithmic expression as an exponential expression, and vice versa. Determine, without technology, the exact value of a logarithm. Estimate the value of a logarithm, using benchmarks, and explain the reasoning. Develop and generalize the laws for logarithms, using numeric examples and exponent laws. Prove each law of logarithms. Determine, using the laws of logarithms, an equivalent expression for a logarithmic expression. Determine, with technology, the approximate value of a logarithmic expression |
|
Part of Functions and Graph concept discussed earlier |
|
|
Exponential Function |
Transcendental Functions Recognize algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic (and their identities), explicit and implicit functions, and parametric representation of functions. |
understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base) • use logarithms to solve equations and inequalities in which the unknown appears in indices e.g. 2 5 x 1 , 3 2 < 5 3 1 x # − , 3 4 x x 1 2 1 = + − . • use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. |
Exponential functions and their graphs: Logarithmic functions and their graphs: . Relationships between these functions: Solving equations, both graphically and analytically Solving exponential equations. |
Graph and analyze exponential and logarithmic function Solve problems that involve exponential and logarithmic equations |
Grade 11 Determine the solution of an exponential equation in which the bases are powers of one nother. Determine the solution of an exponential equation in which the bases are not powers of one another, using a variety of strategies. Determine the solution of a logarithmic equation, and verify the solution. Explain why a value obtained in solving a logarithmic equation may be extraneous. Solve a problem that involves exponential growth or decay. Solve a problem that involves the application of exponential equations to loans, mortgages, or investments. Solve a problem that involves logarithmic scales, such as the Richter scale or the pH scale. Solve a problem by modelling a situation with an exponential or a logarithmic equation |
|
Part of Functions and Graph concept discussed earlier |
|
|
LINEAR PROGRAMMING |
Define linear programming (LP) as planning of allocation of limited resources to obtain an optimal result. Linear Inequalities i) Find algebraic solutions of linear inequalities in one variable and represent them on number line. ii) Interpret graphically the linear inequalities in two variables. iii) Determine graphically the region bounded by up to 3 simultaneous linear inequalities of non-negative variables and shade the region bounded by them. Feasible Region Define • linear programming problem, • objective function, • problem constraints, • decision variables. ii) Define and show graphically the feasible region (or solution space) of an LP problem. iii) Identify the feasible region of simple LP problems. Optimal Solution i) Define optimal solution of an LP problem. ii) Find optimal solution (graphical) through the following systematic procedure: • establish the mathematical formulation of LP problem, • construct the graph, • identify the feasible region, • locate the solution points, • evaluate the objective function, • select the optimal solution, • verify the optimal solution by actually substituting values of variables from the feasible region. iii) Solve real life simple LP problems. |
Not in A Level |
Not in IB |
|
Leaner programming will not be included in the NCC 2023 curriculum. The reason for this is that it is considered to be a topic that can be taught at a higher level, and it is not included in the A-levels and IB programs either. This decision may have been made to make room for other topics that are deemed more important at the NCC level.
|
|
Linear inequalities in one and two varibles is included however linear p[rograming is not included in A level and IB also |
|
|
TRIGONOMETRIC IDENTITIES OF SUM AND DIFFERENCE OF ANGLES |
Fundamental Law of Trigonometry Use distance formula to establish fundamental law of trigonometry: • cos(α − β) = cosα cosβ + sinα sin β , and deduce that • cos(α + β ) = cosα cosβ − sinα sin β , • sin(α ± β) = sinα cosβ ± cosα sin β , • Trigonometric Ratios of Allied Angles i) Define allied angles. ii) Use fundamental law and its deductions to derive trigonometric ratios of allied angles. iii) Express asinθ + bcosθ in the form ) rsin(θ +φ where a = r cosφ and b = rsinφ . Double, Half and Triple Angle Identities Derive double angle, half angle and triple angle identities from fundamental law and its deductions. Sum, Difference and Product of sine and cosine i) Express the product (of sines and cosines) as sums or differences (of sines and cosines). ii) Express the sums or differences (of sines and cosines) as products (of sines and cosines). |
sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians) use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles use the notations sin–1 x, cos–1 x, tan–1 x to denote the principal values of the inverse trigonometric relations use the identities cos sin / tan i i i and sin cos 1 find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included). understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of – sec t 1 an 2 2 i i / + and cosec c 1 ot 2 2 i i / + – the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) – the formulae for sin 2A, cos 2A and tan 2A – the expression of a b sin c i i + os in the use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of – sec t 1 an 2 2 i i / + and cosec c 1 ot 2 2 i i / + – the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) – the formulae for sin 2A, cos 2A and tan 2A – the expression of a b sin c i i + os in the forms Rsin^ h i a ! and Rcos^ h i a ! . use trigonometrical relationships in carrying out integration use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of – sec t 1 an 2 2 i i / + and cosec c 1 ot 2 2 i i / + – the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) – the formulae for sin 2A, cos 2A and tan2A – the expression of a b sin c i i + os in the forms Rsin^ h i ! a and Rcos^ h i ! a . use trigonometrical relationships in carrying out integration |
Right-angle trigonometry, trigonometric ratios Sine Rules, cosine rule, area of triangle, construction The circle: radian measure of angles; length of an arc; area of a sector. Definition of cos and sin in terms of the unit circle. Definition of tan as sin/cos Exact values of trigonometric ratios of and their multiples. The Pythagorean identity . Double angle identities for sine and cosine. Relationship between trigonometric ratios. The circular functions , and : their domains and ranges; amplitude, their periodic nature; and their graphs |
Fundamental Law of Trigonometry Establish fundamental law of trigonometry: Use fundamental law and its deductions to derive: trigonometric ratios of allied angles. double angle and half angle identities Express the product (of sines and cosines) as sums or differences (of sines and cosines). |
Concept of Grade 11 Explain the difference between a trigonometric identity and a trigonometric equation. Verify a trigonometric identity numerically for a given value in either degrees or radians. Explain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid. Determine, graphically, the potential validity of a trigonometric identity, using technology. Determine the non-permissible values of a trigonometric identity. Prove a trigonometric identity algebraically. Determine, using the sum, difference, or double-angle identities, the exact value of a trigonometric ratio. |
How do you sketch the graphs of basic sine and cosine functions? How do you use amplitude and period to help sketch graphs of sine and cosine functions? How do you sketch translations of sine and cosine functions? How do you use sine and cosine functions to model real-life data? |
Kept at Grade 11 as 2006 curriculum only name of the concept is change as Fundamental Laws of Trigonometry |
|
|
TRIGONOMETRIC FUNCTIONS |
Solving Triangles i) Solve right angled triangle when measures of • two sides are given, • one side and one angle are given. ii) Define an oblique triangle and prove • the law of cosines, • the law of sines, • the law of tangents, and deduce respective half angle formulae. iii) Apply above laws to solve oblique triangles. Area of a Triangle Derive the formulae to find the area of a triangle in terms of the measures of • two sides and their included angle, • one side and two angles, three sides (Hero’s formula) Circles Connected with Triangle i) Define circum-circle, in-circle and escribed-circle. ii) Derive the formulae to find • circum-radius, • in-radius, • escribed-radii, and apply them to deduce different identities. |
understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of – sec 1 tan 2 2 i/ + iand cosec 1 cot 2 2 i/ + i – the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) – the formulae for sin 2A, cos 2A and tan 2A – the expression of a sin i + b cos i in the forms Rsin^i ! ah and Rcos^i ! ah. |
· The circle: radian measure of angles; length of an arc; area of a sector. · Definition of and in terms of the unit circle. · Definition of as . · Exact values of trigonometric ratios of and their multiples. · The Pythagorean identity . · Double angle identities for sine and cosine. · Relationship between trigonometric ratios. |
Trigonometric Functions Find the domain and range of the trigonometric functions Discuss even and odd functions Discuss the periodicity of trigonometric functions Find the maximum and minimum value of a given function of the type: a + bsinθ , a + b cosθ , a + bsin(cθ + d ) , a + b cos(cθ + d ) , the reciprocals of above, where a, b, c and d are real numbers. Graphs of Trigonometric Functions Graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems, Explain periodic, even/odd and translation properties of the graphs of sinθ ,cosθ and tanθ. |
Concept of Grade 11 Sketch, with or without technology, the graph of y = sin x, y = cos x, or y = tan x. Determine the characteristics (amplitude, asymptotes, domain, period, range, and zeros) of the graph of y = sin x, y = cos x, or y = tan x. Determine how varying the value of a affects the graphs of y = a sin x or y = a cos x. Determine how varying the value of d affects the graphs of y = sin x + d or y = cos x + d. Determine how varying the value of c affects the graphs of y = sin (x – c) or y = cos (x – c). Determine how varying the value of b affects the graphs of y = sin bx or y = cos bx. Sketch, without technology, graphs of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d, using transformations, and explain the strategies. Determine the characteristics (amplitude, asymptotes, domain, period, phase shift, range, and zeros) of the graph of a trigonometric function of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d. Determine the values of a, b, c, and d for functions of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d that correspond to a graph, and write the equation of the function. Determine a trigonometric function that models a context to solve a problem. Explain how the characteristics of the graph of a trigonometric function relate to the conditions in a problem context. Solve a problem by analyzing the graph of a trigonometric function |
n Determine, with technology, the approximate value of a trigonometric ratio for any angle with a measure expressed in either degrees or radians. n Determine, using the unit circle or reference triangle, the exact value of a trigonometric ratio for angles expressed in degrees that are multiples of 0º, 30º, 45º, 60º, or 90º, or for angles expressed in radians that are multiples of 0, and explain the strategy. n Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified domain, given the value of a trigonometric ratio. n Explain how to determine the exact values of the six trigonometric ratios, given the coordinates of a point on the terminal arm of an angle in standard position. n Determine the measures of the angles in a specified domain in degrees or radians, given a point on the terminal arm of an angle in standard position. n Determine the exact values of the other trigonometric ratios, given the value of one trigonometric ratio in a specified domain. n Sketch a diagram to represent a problem that involves trigonometric ratios. n Solve a problem, using trigonometric ratios |
The concept was already included in the 2006 mathematics curriculum. However, it has been divided over the two years .In Grade 11 Students will cover Trigomometric functions and graphs. |
|
|
INVERSE TRIGONOMETRIC FUNCTIONS AND SOLUTION OF TRIGONOMETRIC EQUATIONS |
Period of Trigonometric Functions i) Find the domain and range of the trigonometric functions. ii) Define even and odd functions. iii) Discuss the periodicity of trigonometric functions. iv) Find the maximum and minimum value of a given function of the type: • a + bsinθ , • a + bcosθ , • a + bsin(cθ + d) , • a + bcos(cθ + d) , • the reciprocals of above, where a, b, c and d are real numbers. Graphs of Trigonometric Functions i) Recognize the shapes of the graphs of sine, cosine and tangent for all angles. ii) Draw the graphs of the six basic trigonometric functions within the domain from –2π to 2π. iii) Guess the graphs of 2 2 sin 2θ , cos2θ , sin θ , cosθ etc. without actually drawing them. iv) Define periodic, even/odd and translation properties of the graphs of sinθ , cosθ and tanθ , i.e., sinθ has: • periodic property sin(θ ± 2π ) = sinθ , • odd property sin(−θ ) = −sinθ , • translation property v) Deduce sin(θ + 2kπ ) = sinθ where k is an integer. Solving Trigonometric Equations Graphically i) Solve trigonometric equations of the type sinθ = k , cosθ = k and tanθ = k , using periodic, even/odd and translation properties. ii) Solve graphically the trigonometric equations of the type: • 2 sinθ = θ , • cosθ =θ , • tanθ = 2θ when 2 2 − π ≤θ ≤ π . Inverse Trigonometric Functions i) Define the inverse trigonometric functions and their domain and range. ii) Find domains and ranges of • principal trigonometric functions, • inverse trigonometric functions. iii) Draw the graphs of inverse trigonometric functions. iv) Prove the addition and subtraction formulae of inverse trigonometric functions. v) Apply addition and subtraction formulae of inverse trigonometric functions to verify related identities. Solving General Trigonometric Equations i) Solve trigonometric equations and check their roots by substitution in the given trigonometric equations so as to discard extraneous roots. ii) Use the periods of trigonometric functions to find the solution of general trigonometric equations. |
use the notations sin–1x, cos– 1x, tan–1x to denote the principal values of the inverse trigonometric relations sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians) understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude |
The aims of this topic are to explore the circular functions and to solve problems using trigonometry. On examination papers, radian measure should be assumed unless otherwise indicated. · The circle: radian measure of angles; length of an arc; area of a sector. · Definition of and in terms of the unit circle. · Definition of as . · Exact values of trigonometric ratios of and their multiples. · The Pythagorean identity . · Double angle identities for sine and cosine. · Relationship between trigonometric ratios. · The circular functions , and : their domains and ranges; amplitude, their periodic nature; and their graphs. · Composite functions of the form . · 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 . · Not required: the general solution of trigonometric equations. · Solution of triangles. · The cosine rule. · The sine rule, including the ambiguous case. · Area of a triangle, . · Applications. |
Inverse Trigonometric Functions Find domains and ranges of principal trigonometric functions, inverse trigonometric functions. Graphs of Inverse Trigonometric Functions Draw the graphs of the inverse trigonometric functions of cosine, sine, tangent, secant, cosecant and cotangent within the domain from –2π to 2π. Inverse trigonometric identities and solution of trigonometric equations State, prove and apply the addition and subtraction formulae of inverse trigonometric functions. Solve trigonometric equations of the type sinθ=k ,cosθ=k and tanθ = k ,using periodic, even/odd and translation properties. Solve graphically the trigonometric equations of the type: sinθ=θ/2, cosθ=θ, tanθ = 2θ where -π/2<θ<π/2 Use the periods of trigonometric functions to find the general solution of the trigonometric equations. |
Grade 12 concept Determine, with technology, the approximate value of a trigonometric ratio for any angle with a measure expressed in either degrees or radians. Determine, using the unit circle or reference triangle, the exact value of a trigonometric ratio for angles expressed in degrees that are multiples of 0º, 30º, 45º, 60º, r 90º, or for angles expressed in radians that are multiples of 0, and explain the strategy. Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified domain, given the value of a trigonometric ratio. Explain how to determine the exact values of the six trigonometric ratios, given the coordinates of a point on the terminal arm of an angle in standard position. Determine the measures of the angles in a specified domain in degrees or radians, given a point on the terminal arm of an angle in standard position. Determine the exact values of the other trigonometric ratios, given the value of one trigonometric ratio in a specified domain. Sketch a diagram to represent a problem that involves trigonometric ratios. Solve a problem, using trigonometric ratios |
|
The concept was already included in the 2006 mathematics curriculum. However, it has been divided over the two years .In Grade 11 Students will cover Trigomometric functions and graphs.and in Grade 12 students will cover Inverse Trigonometric Functions and Graphs |
|
|
Mechanics |
|
Kinematics of motion in a straight line Candidates should be able to: • understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities • sketch and interpret displacement–time graphs and velocity–time graphs, and in particular appreciate that – the area under a velocity –time graph represents displacement, – the gradient of a displacement –time graph represents velocity, – the gradient of a velocity –time graph represents acceleration • use differentiation and integration with respect to time to solve simple problems concerning displacement, velocity and acceleration Calculus required is restricted to techniques from the content for |
|
Mechanics Kinematics of motion in a straight line • understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities. • sketch and interpret displacement–time graphs and velocity–time graphs, and in particular appreciate that – the area under a velocity • –time graph represents displacement, – the gradient of a displacement • –time graph represents velocity, – the gradient of a velocity–time graph represents acceleration |
Concept of Grade12 |
|
The inclusion of Mechanics in the Grade 12 curriculum, particularly with respect to kinematics, will help students understand the fundamental principles of motion and how to describe and analyze the motion of objects. Kinematics is the branch of Mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves describing the position, velocity, and acceleration of an object over time. By introducing kinematics in Grade 12, students will learn how to apply mathematical formulas to analyze and understand the motion of objects. Students will learn about concepts such as distance, displacement, speed, velocity, and acceleration. They will also learn how to use equations of motion to calculate these quantities for different types of motion, including uniform motion, uniformly accelerated motion, and motion with constant acceleration. In addition, the inclusion of kinematics in Mechanics will also help students understand the relationship between position, velocity, and acceleration graphs. They will learn how to interpret these graphs to describe and analyze the motion of an object. including Mechanics with a focus on kinematics in the Grade 12 curriculum will provide students with a strong foundation in the principles of motion and help them develop analytical and problem-solving skills that are essential for future study and career opportunities in the field of science and engineering. |
|
|
INTRODUCTION TO SYMBOLIC PACKAGE: MAPLE |
Introduction i) Recognize MAPLE environment. ii) Recognize basic MAPLE commands. iii) Use MAPLE as a calculator. iv) Use online MAPLE help. Polynomials Use MAPLE commands for • factoring a polynomial, • expanding an expression, • simplifying an expression, • simplifying a rational expression, • substituting into an expression. Graphics i) Plot a two-dimensional graph. ii) Demonstrate domain and range of a plot. iii) Sketch parametric equations. iv) Know plotting options. Matrices i) Recognize matrix and vector entry arrangement. ii) Apply matrix operations. iii) Compute inverse and transpose of a matrix. |
Not in A levels |
Not in IB |
Not using now |
|
|
Maple is not included in the curriculum 2023 Performance: Maple can be slow for some complex calculations or operations, especially when compared to specialized packages like Matlab or Mathematica. Cost: Maple is a commercial software and may be expensive for some individuals or organizations. Complexity: Maple has a steep learning curve and its interface may be overwhelming for some users. Alternatives: There are many open-source alternatives like Sympy and Maxima that provide similar functionality and may be more accessible to users. Performance: Maple can be slow for some complex calculations or operations, especially when compared to specialized packages like Matlab or Mathematica.
Limited Interoperability: Maple may not be well-integrated with other software packages and tools, making it harder to use in a broader context |
|
|
FUNCTIONS AND LIMITS |
Functions i) Identify through graph the domain and range of a function. ii) Draw the graph of modulus function (i.e. y = x ) and identify its domain and range. Composition of Functions i) Recognize the composition of functions. ii) Find the composition of two given functions. Inverse of Composition of Functions Describe the inverse of composition of two given functions. Transcendental Functions Recognize algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic (and their identities), explicit and implicit functions, and parametric representation of functions. Graphical Representations i) Display graphically: • the explicitly defined functions like y = f (x) , where f (x) = ex , ax , x x a e log , log . • the implicitly defined functions such as x 2 + y 2 = a2 and 1 2 2 2 2 + = b y a x and distinguish between graph of a function and of an equation. • the parametric equations of functions such as x = at 2 , y = 2at ; x = a secθ , y = b tanθ . • the discontinuous functions of the type ii) Use MAPLE graphic commands for two-dimensional plot of: • an expression (or a function), • parameterized form of a function, • implicit function, by restricting domain and range. iii) Use MAPLE package plots for plotting different types of functions. Limit of a Function i) Identify a real number by a point on the number line. ii) Define and represent • open interval, • closed interval, • half open and half closed intervals, on the number line. iii) Explain the meaning of phrase: • x tends to zero ( x→0), • x tends to a ( x→a ), • x tends to infinity ( x→∞). iv) Define limit of a sequence. v) Find the limit of a sequence whose nth term is given. vi) Define limit of a function. vii) State the theorems on limits of sum, difference, product and quotient of functions and demonstrate through examples. Important Limits i) Evaluate the limits of functions of the following types: ii) Evaluate limits of different algebraic, exponential and trigonometric functions. iii) Use MAPLE command limit to evaluate limit of a function. Continuous and Discontinuous Functions i) Recognize left hand and right hand limits and demonstrate through examples. ii) Define continuity of a function at a point and in an interval. iii) Test continuity and discontinuity of a function at a point and in an interval. iv) Use MAPLE command iscont to test continuity of a function at a point and in a given interval. |
understand the terms function, domain, range, one-one function, inverse function and composition of functions identify the range of a given function in simple cases, and find the composition of two given functions determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases illustrate in graphical terms the relation between a one-one function and its inverse understand and use the transformations of the graph of y = f(x) given by y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and simple combinations of these. understand and use the transformations of the graph of y = f(x) given by y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and simple combinations of these. |
Concept of function · Domain, range; image (value). · Composite functions. · Not required: Domain restriction. · Identity function. Inverse function . · The graph of a function; its equation y = f (x). · Function graphing skills. · Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range. · Use of technology to graph a variety of functions, including ones not specifically mentioned. · The graph of as the reflection in the line of the graph of · Transformations of graphs. · Translations: ; . · Reflections (in both axes): ; · Vertical stretch with scale factor p: · Stretch in the x-direction with scale factor · Composite transformations. The quadratic function : its graph, y-intercept (0, c) . Axis of symmetry. · The form , x-intercepts ( p, 0) and (q, 0). · The form , vertex (h, k) . The reciprocal function , x ≠ 0 : its graph and self-inverse nature. · The rational function and its graph. · Vertical and horizontal asymptotes. · Exponential functions and their graphs: · Logarithmic functions and their graphs: . · Relationships between these functions: · . · 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 . The quadratic formula. · The discriminant and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots. · Solving exponential equations. · Applications of graphing skills and solving equations that relate to real-life situations. Not required: analytic methods of calculating limits. |
Limit of a Function •Demonstrate and find the limit of a sequence. •State and apply theorems on limit of sum, difference, product and quotient of functions to algebraic, exponential and trigonometric functions. Continuous and Discontinuous Functions •Demonstrate and test Continuity, discontinuity of a function at a point and in an interval. |
Concept of Grade 11 |
|
Previously it was in Grade 12 now added in Grade 11 Functions and Graphs to make the base for Differentiation and Integration |
|
|
DIFFERENTIATION I |
Derivative of a Function i) Distinguish between independent and dependent variables. ii) Estimate corresponding change in the dependent variable when independent variable is incremented (or decremented). iii) Explain the concept of a rate of change. iv) Define derivative of a function as an instantaneous rate of change of a variable with respect to another variable. v) Define derivative or differential coefficient of afunction. vi) Differentiate y = xn , where n∈Z (the set of integers), from first principles (the derivation of power rule). vii) Differentiate y = (ax + b)n , where q n = p and p, q are integers such that q ≠ 0 , from first principles. Theorems on Differentiation Prove the following theorems for differentiation. • The derivative of a constant is zero. • The derivative of any constant multiple of a function is equal to the product of that constant and the derivative of the function. • The derivative of a sum (or difference) of two functions is equal to the sum (or difference) of their derivatives. • The derivative of a product of two functions is equal to (the first function)×(derivative of the second function) plus (derivative of the first function)× (the second function). • The derivative of a quotient of two functions is equal to denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Application of Theorems on Differentiation Differentiate: • constant multiple of xn , • sum (or difference) of functions, • polynomials, • product of functions, • quotient of two functions. Chain Rule i) Prove that dx du du dy dx dy = ⋅ when y = f (u) and u = g(x) . ii) Show that dy dx dx dy = 1 . iii) Use chain rule to show that [ f (x)]n n[ f (x)]n 1 f (x) dx d = − ′ . iv) Find derivative of implicit function. Differentiation of Trigonometric and Inverse Trigonometric Functions Differentiate: • trigonometric functions ( sin x, cos x, tan x, cosecx, sec x, and cot x) from first principles. • inverse trigonometric functions ( arcsin x, arccos x, arctan x, arccosecx, arcsecx, and arccotx) using differentiation formulae. Differentiation of Exponential and Logarithmic Functions i) Find the derivative of ex and ax from first principles. ii) Find the derivative of ln x and x loga from first principles. iii) Use logarithmic differentiation to find derivative of algebraic expressions involving product, quotient and power. Differentiation of Hyperbolic and Inverse Hyperbolic Functions Differentiate: • hyperbolic functions ( sinhx, cosh x, tanh x, cosech x, sech x and coth x) . • inverse hyperbolic functions ( sinh−1 x, cosh−1 x, tanh−1 x, cosech−1x, sech−1x, and coth−1 x ). |
understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations f ′(x), f ″(x), x for first and second derivatives use the derivative of xn (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change locate stationary points and determine their nature, and use information about stationary points in sketching graphs. use the derivatives of ex, ln x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites differentiate products and quotients find and use the first derivative of a function which is defined parametrically or implicitly. use the derivatives of ex, ln x, sin x, cos x, tan x, tan–1 x, together with constant multiples, sums, differences and composites differentiate products and quotients find and use the first derivative of a function which is defined parametrically or implicitly. |
Informal ideas of limit and convergence. Limit notation. Definition of derivative from first principles as f"(x)=limh-0 Derivative interpreted as gradient function and as rate of change. Tangents and normals, and their equations. Not required: analytic methods of calculating limits. Derivative of xn(n belongs to Q), sinx, cos x, tanx, ex, 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 |
Differenciation I Gradient of a curve • Recognize the meaning of the tangent to a curve at a point. • Calculate the gradient of a curve at a point. • Identify the derivative as the limit of a difference quotient. • Calculate the derivative of a given function at a point. • Estimate the derivative as rate of change of velocity, temperature and profit. Derivative of a Function • Recognize the derivative function. • Find the derivative of a square root function. • Find the derivative of a quadratic function. • State the connection between derivatives and continuity. Differentiation Rules • State, prove and apply the following differentiation rules: the constant rule the coefficient rule the power rule the sum and difference rule the product rule the quotient rule • Extend the power rule to functions with negative exponents. • Combine the differentiation rules to find the derivative of a polynomial or rational function. • Apply rates of change to displacement velocity acceleration of an object moving along a straight line. Further on Differentiation • Find the derivative of trigonometric and inverse trigonometric functions. • Find the derivative of exponential functions. • Find the derivative of logarithmic functions. • Apply differentiation to state the increasing and decreasing functions. |
Concept of Grade 11 Application of Theorems on Differentiation: Differentiate: constant multiple of xn . sum (or difference) of functions, polynomials, product of functions, quotient of two functions.
Chain Rule: Apply the rule dy/dx=dy/du.du/dx when y=f(u) and u=g(x). Find derivative of an implicit and parametric function.
Utilization of derivatives: Apply differentiation to gradients, tangents and normal, increasing and decreasing functions and rates of change. Find and classify the stationary points and use them for sketching graphs.
Further on Differentiation:
Differentiate: trigonometric functions from first principles. arctan x of e^x and e^ax and ax. of ln x and loga x.
Higher Order Derivatives: Find higher order derivatives of algebraic, trigonometric, exponential and logarithmic functions. Find the second derivative of implicit, inverse trigonometric and parametric functions.
Maxima and Minima:
Demonstrate the second derivative rule to find the extreme values of a function at a point.
Utilize second derivative rule to examine a given function for extreme values.
Solve real life problems related to extreme values. |
|
Dividing differentiation over two years can be a great strategy to build a strong base of the concept. |
|
|
HIGHER ORDER DERIVATIVES AND APPLICATIONS |
Higher Order Derivatives i) Find higher order derivatives of algebraic, trigonometric, exponential and logarithmic functions. ii) Find the second derivative of implicit, inverse trigonometric and parametric functions. iii) Use MAPLE command diff repeatedly to find higher order derivative of a function. Maclaurin’s and Taylor’s Expansions i) State Maclaurin’s and Taylor’s theorems (without remainder terms). Use these theorems to expand sin x,cos x, tan x , a , e , log (1 x) a x x + and ln(1 + x). ii) Use MAPLE command taylor to find Taylor’s expansion for a given function. Application of Derivatives i) Give geometrical interpretation of derivative. ii) Find the equation of tangent and normal to the curve at a given point. iii) Find the angle of intersection of the two curves. iv) Find the point on a curve where the tangent is parallel to the given line. Maxima and Minima i) Define increasing and decreasing functions. ii) Prove that if f (x) is a differentiable function on the open interval (a,b) then • f (x) is increasing on (a,b) if f ′(x) > 0, ∀x∈(a,b) , • f (x) is decreasing on (a,b) if f ′(x) < 0, ∀x∈(a,b) . iii) Examine a given function for extreme values. iv) State the second derivative rule to find the extreme values of a function at a point. v) Use second derivative rule to examine a given function for extreme values. vi) Solve real life problems related to extreme values. vii) Use MAPLE command maximize (minimize) to compute maximum (minimum) value of a function. |
use the derivatives of ex, ln x, sin x, cos x, tan x, tan–1 x, together with constant multiples, sums, differences and composites tan x, together with constant multiples, sums, differences and composites |
Not in IB |
Differenciation II Higher-Order Derivatives •Find higher order derivatives of algebraic, implicit, parametric, trigonometric, inverse-trigonometric, exponential and logarithmic functions. Applications of Derivatives •The ability to approximate functions locally by linear functions. (Linear approximations of square root functions, trigonometric functions) •Draw a graph that illustrates the use of differentials to approximate the change in a quantity. •Calculate the relative error and percentage error in using a differential approximation. (Volume of a cube and sphere) Extreme Values •Illustrate Global extrema (absolute extrema) and local extrema (relative extrema) •Find the extreme values by applying the second derivative test. •Explain how to find the critical points of a function over a closed interval. •Describe how to use critical points to locate absolute extrema over a closed interval. •Apply derivatives to real-world problems to find the maximum and the minimum values of a function under certain conditions.
|
Concept of Grade 12 |
|
By dividing differentiation over two years, students will have ample time to fully understand the concepts and build a solid foundation for more advanced applications of calculus. |
|
|
Vectors |
Scalar and Vector Functions i) Define scalar and vector function. ii) Explain domain and range of a vector function. Limit and Continuity i) Define limit of a vector function and employ the usual technique for algebra of limits of scalar function to demonstrate the following properties of limits of a vector function. • The limit of the sum (difference) of two vector functions is the sum (difference) of their limits. • The limit of the dot product of two vector functions is the dot product of their limits. • The limit of the cross product of two vector functions is the cross product of their limits. • The limit of the product of a scalar function and a vector function is the product of their limits. ii) Define continuity of a vector function and demonstrate through examples. Derivative of Vector Function Define derivative of a vector function of a single variable and elaborate the result: if f ( ) ( )i ( ) j ( )k 1 2 3 t = f t + f t + f t , where ( ), ( ), ( ) 1 2 3 f t f t f t are differentiable functions of a scalar variable t, then Vector Differentiation i) Prove the following formulae of differentiation: where a is a constant vector function, f and g are vector functions, and φ is a scalar function of t. ii) Apply vector differentiation to calculate velocity and acceleration of a position vector r(t) = x(t)i + y(t)j + z(t)k . |
use standard notations for vectors, i.e. carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a + tb, and find the equation of a line, given sufficient information determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points. |
· Vectors as displacements in the plane and in three dimensions · Components of a vector; column representation ( ) . · Algebraic and geometric approaches to the following: o the sum and difference of two vectors; the zero vector, the vector −v ; o multiplication by a scalar, kv ; parallel vectors; o magnitude of a vector, v ; o unit vectors; base vectors; i, j and k; o position vectors · 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: . · The angle between two lines. · Distinguishing between coincident and parallel lines. · Finding the point of intersection of two lines. · Determining whether two lines intersect. · Expected value. · Inverse normal calculations. · Not required: o Transformation of any normal variable to the standardized normal. · Bivariate data: the concept of correlation. · Scatter diagrams; line of best fit, by eye, passing through the mean point. |
Vector Valued Function]: Demonstrate the need of vector valued function. Construct vector valued function. Identify domain and range of vector valued function. Identify difference between scalar and vector valued functions Vector differentiation Find the limit of a vector function and employ the usual technique for algebra of limits of scalar function to demonstrate the following properties of limits of a vector function. • The limit of the sum (difference) of two vector functions is the sum (difference) of their limits. • The limit of the dot product of two vector functions is the dot product of their limits. • The limit of the cross product of two vector functions is the cross product of their limits. • The limit of the product of a scalar function and a vector function is the product of their limits. Apply continuity of a vector function Derivative of Vector Function Find derivative of a vector function of a single variable and elaborate the result: If f (t)= f_1 (t)i+f_2 (t)j+ f_3 (t)k Where f_1 (t),f_2 (t),f_3 (t) are differentiable functions of a scalar variable t, then df/dt=(df_1)/dt i+(df_2)/dt j+(df_3)/dt k Apply vector differentiation to calculate velocity and acceleration of a position vector f(t)=x(t)i+y(t)j+z(t)k |
Concept Grade 12 |
|
A vector-valued function is a mathematical function that takes one or more input variables and produces a vector as its output. Vector-valued functions are important in a variety of mathematical fields, including calculus, linear algebra, differential equations, and more. In calculus, vector-valued functions are used to describe curves and surfaces in three-dimensional space. They are also used in the study of functions of several variables and in the analysis of fields in physics and engineering. In linear algebra, vector-valued functions are used to represent linear transformations and to solve systems of linear equations. They are also used in the study of eigenvalues and eigenvectors. In differential equations, vector-valued functions are used to model systems that involve multiple interacting components. They are also used to study dynamical systems, chaos theory, and other areas of mathematical physics. Overall, vector-valued functions are an important part of modern mathematics and have a wide range of applications in various fields. Therefore included in the curriculum of mathematics 2023 |
|
|
INTEGRATION I |
|
understand integration as the reverse process of differentiation, and integrate (ax + b)n (for any rational n except –1), together with constant multiples, sums and differences solve problems involving the evaluation of a constant of integration evaluate definite integrals use definite integration to find – the area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line or between two curves – a volume of revolution about one of the axes. extend the idea of ‘reverse differentiation’ to include the integration of eax + b, ax b 1 + , sin(ax + b), cos(ax + b) and sec2(ax + b) use trigonometrical relationships in carrying out integration understand and use the trapezium rule to estimate the value of a definite integral. extend the idea of ‘reverse differentiation’ to include the integration of eax + b, ax b 1 + , sin(ax + b), cos(ax + b), sec2(ax + b) and x a 1 2 2 use trigonometrical relationships in carrying out integration integrate rational functions by means of decomposition into partial fractions recognise an integrand of the form x k x f f l ^ ^ h h , and integrate such functions ecognise when an integrand can usefully be regarded as a product, and use integration by parts use a given substitution to simplify and evaluate either a definite or an indefinite integral. |
|
Integration • Find the general antiderivative of a given function. • Recognize and use the terms and notations for antiderivatives. • State the power rule for integrals. • State and apply the properties of indefinite integrals. • State the definition of the definite integral. • Explain the terms integrand, limits of integration, and variable of integration. • State and apply the properties of definite integrals. • State and apply Fundamental Theorem of Calculus to evaluate the definite integrals. • Describe the relationship between the definite integral and net area. • Area of a region bounded by a curve and lines parallel to axes, or between a curve and a line, or between two curves. • Volume of revolution about one of the axes. • Demonstrate trapezium rule to estimate the value of a definite integral.
|
Grade 11 concept |
|
By dividing integration over two years, students will have ample time to fully understand the concepts and build a solid foundation for more advanced applications of calculus |
|
|
INTEGRATION II |
Introduction i) Demonstrate the concept of the integral as an accumulator. ii) Know integration as inverse process of differentiation. iii) Explain constant of integration. iv) Know simple standard integrals which directly follow from standard differentiation formulae. Rules of Integration i) Recognize the following rules of integration. • ∫ f x dx = ∫ f x dx = f x + c dx • where c is a constant of integration. • The integral of the product of a constant and a function is the product of the constant and the integral of the function. • The integral of the sum of a finite number of functions is equal to the sum of their integrals. ii) Use standard differentiation formulae to prove the results for the following integrals: Integration by Substitution i) Explain the method of integration by substitution. ii) Apply method of substitution to evaluate indefinite integrals. iii) Apply method of substitution to evaluate integrals of the following types: Integration by Parts i) Recognize the formula for integration by parts. ii) Apply method of integration by parts to evaluate integrals of the following types: • ∫ a2 − x2dx , ∫ a2 + x2 dx , ∫ x 2 − a2 dx iii) Evaluate integrals using integration by parts. Integration using Partial Fractions Use partial fractions to find ∫ dx g x f x ( ) ( ) , where f (x) and g(x) are algebraic functions such that g(x) ≠0. Definite Integrals i) Define definite integral as the limit of a sum. ii) Describe the fundamental theorem of integral calculus and recognize the following basic properties: iii) Extend techniques of integration using properties to evaluate definite integrals. iv) Represent definite integral as the area under the curve. v) Apply definite integrals to calculate area under the curve. vi) Use MAPLE command int to evaluate definite and indefinite integrals. |
extend the idea of ‘reverse differentiation’ to include the integration of e ax + b, ax b1+ , sin(ax + b), cos(ax + b) and sec2(ax + b) use trigonometrical relationships in carrying out integration e.g. use of double-angle formulae to integrate sin2 x or cos2 (2x). • understand and use the trapezium rule to estimate the value of a definite integral |
Not in IB |
Techniques of Integration • Utilize trigonometric relationships to evaluate integrals. • Integrate functions involving the exponential and logarithmic functions. • Recognize when to use integration by parts. • Use the integration-by-parts formula to solve integration problems. • Use the integration-by-parts formula for definite integrals. • Solve integration problems involving trigonometric substitution • Integrate a rational function using the method of partial fractions. • Recognize simple linear factors in a rational function. • Recognize repeated linear factors in a rational function. Recognize quadratic factors in a rational function.
|
Grade 12 concept |
|
By dividing integration over two years, students will have ample time to fully understand the concepts and build a solid foundation for more advanced applications of calculus |
|
|
PLANE ANALYTIC GEOMETRY – STRAIGHT LINE |
Division of a Line Segment i) Recall distance formula to calculate distance between two points given in Cartesian plane. ii) Find coordinates of a point that divides the line segment in given ratio (internally and externally). Show that the medians and angle bisectors of a triangle are concurrent. Slope of a Straight Line i) Define the slope of a line. ii) Derive the formula to find the slope of a line passing through two points. iii) Find the condition that two straight lines with given slopes may be • parallel to each other, • perpendicular to each other. Equation of a Straight Line Parallel to Co-ordinate Axes Find the equation of a straight line parallel to • y-axis and at a distance a from it, • x-axis and at a distance b from it. Standard Form of Equation of a Straight Line i) Define intercepts of a straight line. Derive equation of a straight line in • slope-intercept form, • point-slope form, • two-point form, • intercepts form, • symmetric form, • normal form. ii) Show that a linear equation in two variables represents a straight line. iii) Reduce the general form of the equation of a straight line to the other standard forms. Distance of a Point From a Line i) Recognize a point with respect to position of a line. ii) Find the perpendicular distance from a point to the given straight line. Angle Between Lines i) Find the angle between two coplanar intersecting straight lines. ii) Find the equation of family of lines passing through the point of intersection of two given lines. iii) Calculate angles of the triangle when the slopes of the sides are given. Concurrency of Straight Lines i) Find the condition of concurrency of three straight lines. ii) Find the equation of median, altitude and right bisector of a triangle. iii) Show that • three right bisectors, • three medians, • three altitudes, of a triangle are concurrent. Area of a Triangular Region Find area of a triangular region whose vertices are given. Homogenous Equation i) Recognize homogeneous linear and quadratic equations in two variables. ii) Investigate that the 2nd degree homogeneous equation in two variables x and y represents a pair of straight lines through the origin and find acute angle between them. |
find the equation of a straight line given sufficient information interpret and use any of the forms y = mx + c, y – y1 = m(x – x1), ax + by + c = 0 in solving problems understand that the equation (x – a)2 + (y – b)2 = r 2 represents the circle with centre (a, b) and radius r use algebraic methods to solve problems involving lines and circles understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. use appropriate formulae for motion with constant acceleration in a straight line. |
Not in IB |
Analytical Geometry Derive equation of a straight line in • slope-intercept form, • point-slope form, • two-point form, • intercepts form, • symmetric form, • normal form. Show that a linear equation in two variables represents a straight line. Reduce the general form of the equation of a straight line to the other standard forms. Distance of a Point from a Line Find the perpendicular distance from a point to the given straight line. Angle Between Lines Find the angle between two coplanar intersecting straight lines. Find the equation of family of lines passing through the point of intersection of two given lines. |
Concept of Grade 12 find the equation of a straight line given sufficient information interpret and use any of the forms y = mx + c, y – y1 = m(x – x1), ax + by + c = 0 in solving problems Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines. understand that the equation (x – a)2 + (y – b)2 = r2 represents the circle with centre (a, b) and radius r Including use of the expanded form x2 + y2 + 2gx + 2fy + c = 0. use algebraic methods to solve problems involving lines and circles Including use of elementary geometrical properties of circles, e.g. tangent perpendicular to radius, angle in a semicircle, symmetry. Implicit differentiation is not included. understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations |
|
Analytical Geometry is an essential branch of Mathematics that deals with the study of geometry using algebraic methods. This subject is crucial for students who wish to pursue higher studies in fields such as engineering, physics, computer science, and economics. Therefore, it is essential that students have a good understanding of Analytical Geometry.
Therefore the 2023 Mathematics curriculum will include Analytical Geometry, with some theorems related to circles being removed from the syllabus due to their previous coverage in Grade 10 |
|
|
CONICS – I |
Introduction Define conics and demonstrate members of its family i.e. circle, parabola, ellipse and hyperbola. Circle Equation of a Circle General Form of an Equation of a Circle i) Define circle and derive its equation in standard form i.e. (x − h)2 + (y − k )2 = r 2 . ii) Recognize general equation of a circle x2 + y2 + 2gx + 2 fy + c = 0 and find its centre and radius. Equation of Circle determined by a given condition Find the equation of a circle passing through • three non-collinear points, • two points and having its centre on a given line, • two points and equation of tangent at one of these points is known, • two points and touching a given line. Tangent and Normal i) Find the condition when a line intersects the circle. ii) Find the condition when a line touches the circle. iii) Find the equation of a tangent to a circle in slope form. iv) Find the equations of a tangent and a normal to a circle at a point. v) Find the length of tangent to a circle from a given external point. vi) Prove that two tangents drawn to a circle from an external point are equal in length. Properties of Circle Prove analytically the following properties of a circle. • Perpendicular from the centre of a circle on a chord bisects the chord. • Perpendicular bisector of any chord of a circle passes through the centre of the circle. • Line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. • Congruent chords of a circle are equidistant from its centre and its converse. • Measure of the central angle of a minor arc is double the measure of the angle subtended by the corresponding major arc. • An angle in a semi-circle is a right angle. • The perpendicular at the outer end of a radial segment is tangent to the circle. • The tangent to a circle at any point of the circle is perpendicular to the radial segment at that point. |
Not |
Not in IB |
CONICS Demonstrate conics and members of its family i.e. circle, parabola, ellipse and hyperbola. Circle, Equation of a Circle Derive and apply equation of a circle in standard form i.e. 〖(x - h )〗^2+ 〖(y - k)〗^2 = r^2 Find the equation of a circle passing through: • three non-collinear points, • two points and having its centre on a given line, • two points and equation of tangent at one of these points is known, • two points and touching a given line. Tangent and Normal Find the condition when: a line intersects the circle. a line touches the circle. Find the equation of a tangent: to a circle in slope form. and a normal to a circle at a point. Find the length of tangent to a circle from a given external point. Parabola Derive and apply the standard equation of a parabola. sketch their graphs and find their elements. Find the equation of a parabola with the following given elements: • focus and vertex, • focus and directrix, • vertex and directrix. Equations of Tangent and Normal Find the condition when a line is tangent to a parabola at a point and hence write the equation of a tangent line in slope form. Find the equation of a tangent and a normal to a parabola at a point. Solve suspension and reflection problems related to parabola. Ellipse Derive and apply the standard form of equation of an ellipse and identify its elements. Convert a given equation to the standard form of equation of an ellipse, find its elements and draw the graph. Equations of Tangent and Normal Find points of intersection of an ellipse with a line including the condition of tangency. Find the equation of a tangent in slope form. Find the equation of a tangent and a normal to an ellipse at a point. Standard Form of Equation of Hyperbola Derive and apply the standard form of equation of a hyperbola and identify its elements. Find the equation of a hyperbola with the following given elements: • transverse and conjugate axes with centre at origin,two points, • eccentricity, latera recta and transverse axes, • focus, eccentricity and centre, • focus, centre and directrix. |
Concept of Grade 12 |
|
The 2023 Mathematics Curriculum includes the study of Conics. The study of Conics involves the properties and graphs of conic sections such as circles, parabolas, ellipses, and hyperbolas. However, in the updated curriculum, Conics I and II have been combined, as most of the properties of circles have already been covered in Grade 10.
In Grade 12, learners will focus on the standard form of equations for each type of conic section, as well as the equations of tangents and normals to these curves. This will enable them to have a better understanding of the properties of conic sections and their applications in various fields such as physics, engineering, and computer science.
It is essential for students to have a good understanding of Conics, as it is a fundamental concept in Mathematics and is used in many real-world applications. By combining Conics I and II, learners will have a more streamlined approach to learning about these important topics, and will be better equipped to apply their knowledge to practical situations |
|
|
CONICS – II |
Parabola i) Define parabola and its elements (i.e. focus, directrix, eccentricity, vertex, axis, focal chord and latus rectum). General Form of Equation of a Parabola ii) Derive the general form of an equation of a parabola. Standard Form of Equation of Parabola iii) Derive the standard equations of parabola, sketch their graphs and find their elements. iv) Find the equation of a parabola with the following given elements: • focus and vertex, • focus and directrix, • vertex and directrix. Equations of Tangent and Normal v) Recognize tangent and normal to a parabola. vi) Find the condition when a line is tangent to a parabola at a point and hence write the equation of a tangent line in slope form. vii) Find the equation of a tangent and a normal to a parabola at a point. Application of Parabola viii) Solve suspension and reflection problems related to parabola. 9.2 Ellipse 9.2.1 Standard Form of Equation of an Ellipse 9.2.2 Equations of Tangent and Normal i) Define ellipse and its elements (i.e. centre, foci, vertices, covertices, directrices, major and minor axes, eccentricity, focal chord and latera recta). ii) Explain that circle is a special case of an ellipse. iii) Derive the standard form of equation of an ellipse and identify its elements. iv) Find the equation of an ellipse with the following given elements • major and minor axes, • two points, • foci, vertices or lengths of a latera recta, • foci, minor axes or length of a latus rectum. v) Convert a given equation to the standard form of equation of an ellipse, find its elements and draw the graph. vi) Recognize tangent and normal to an ellipse. vii) Find points of intersection of an ellipse with a line including the condition of tangency. viii) Find the equation of a tangent in slope form. ix) Find the equation of a tangent and a normal to an ellipse at a point. 9.3 Hyperbola 9.3.1 Standard Form of Equation of Hyperbola i) Define hyperbola and its elements (i.e. centre, foci, vertices, directrices, transverse and conjugate axes, eccentricity, focal chord and latera recta). ii) Derive the standard form of equation of a hyperbola and identify its elements. iii) Find the equation of a hyperbola with the following given elements: • transverse and conjugate axes with centre at origin, 9.3.2 Equation of Tangent and Normal • two points, • eccentricity, latera recta and transverse axes, • focus, eccentricity and centre, • focus, centre and directrix. iv) Convert a given equation to the standard form of equation of a hyperbola, find its elements and sketch the graph. v) Recognize tangent and normal to a hyperbola. vi) Find • points of intersection of a hyperbola with a line B15including the condition of tangency, • the equation of a tangent in slope form. vii) Find the equation of a tangent and a normal to a hyperbola at a point. 9.4 Translation and Rotation of Axes i) Define translation and rotation of axes and demonstrate through examples. ii) Find the equations of transformation for • translation of axes, • rotation of axes. iii) Find the transformed equation by using translation or rotation of axes. iv) Find new origin and new axes referred to old origin and old axes. v) Find the angle through which the axes be rotated about the origin so that the product term xy is removed from the transformed equation. |
Not in A levels |
Not in IB |
|
merged in conics I |
|
The 2023 Mathematics Curriculum includes the study of Conics. The study of Conics involves the properties and graphs of conic sections such as circles, parabolas, ellipses, and hyperbolas. However, in the updated curriculum, Conics I and II have been combined, as most of the properties of circles have already been covered in Grade 10.
In Grade 12, learners will focus on the standard form of equations for each type of conic section, as well as the equations of tangents and normals to these curves. This will enable them to have a better understanding of the properties of conic sections and their applications in various fields such as physics, engineering, and computer science.
It is essential for students to have a good understanding of Conics, as it is a fundamental concept in Mathematics and is used in many real-world applications. By combining Conics I and II, learners will have a more streamlined approach to learning about these important topics, and will be better equipped to apply their knowledge to practical situations |
|
|
DIFFERENTIAL EQUATIONS |
Define ordinary differential equation (DE), order of a DE, degree of a DE, solution of a DE – general solution and particular solution. Demonstrate the concept of formation of a differential equation. Solution of Differential Equation i) Solve differential equations of first order and first degree of the form: • separable variables, • homogeneous equations, • equations reducible to homogeneous form. ii) Solve real life problems related to differential equations. Orthogonal Trajectories i) Find orthogonal trajectories (rectangular coordinates) of the given family of curves. ii) Use MAPLE graphic commands to view the graphs of given family of curves and its orthogonal trajectories. |
formulate a simple statement involving a rate of change as a differential equation find by integration a general form of solution for a first order differential equation in which the variables are separable use an initial condition to find a particular solution interpret the solution of a differential equation in the context of a problem being modelled by the equation. |
Not in IB |
Differential Equations Identify and construct first order differential equations from practical situations. Solution of Differential Equation Solve separable differential equations of first order and first degree of separable variable equations Homogeneous equations Solve real life problems related to first order differential equations.
|
Concept of Grade 12 |
|
In the 2023 Mathematics curriculum, the concept of Differential Equations will continue to be an essential topic. Students will be introduced to differential equations, including first-order differential equations and their solutions. This topic will provide students with an understanding of the relationship between rates of change and the variables that are changing.
The study of differential equations will also allow students to apply mathematical concepts to real-world problems, as many physical phenomena can be modeled using differential equations. Additionally, students will learn techniques for solving differential equations, including separation of variables and integrating factors.
By including the study of differential equations in the 2023 Mathematics curriculum, students will have the opportunity to develop valuable problem-solving skills and a deeper understanding of the role of mathematics in the natural sciences and engineering |
|
|
PARTIAL DIFFERENTIATION |
Differentiation of Function of Two Variables i) Define a function of two variables. ii) Define partial derivative. iii) Find partial derivatives of a function of two variables. Euler’s Theorem i) Define a homogeneous function of degree n. ii) State and prove Euler’s theorem on homogeneous functions. iii) Verify Euler’s theorem for homogeneous functions of different degrees (simple cases). iv) Use MAPLE command diff to find partial derivatives. |
Not |
Not in IB |
Not included in 2023 curriculum |
Not included on 2023 curriculum |
|
Partial differentiation is a mathematical concept that involves calculating the partial derivative of a function with respect to one of its variables, while holding all other variables constant. It is typically introduced in advanced undergraduate or graduate-level courses in mathematics, physics, and engineering, and requires a strong foundation in calculus and multivariable calculus. Based on its level of complexity, it is unlikely that the concept of partial differentiation would be included in a Grade 9-12 Mathematics curriculum 2023. |
|
|
INTRODUCTION TO NUMERICAL METHODS |
Numerical Solution of Non-linear Equations i) Describe importance of numerical methods. ii) Explain the basic principles of solving a non-linear equation in one variable. iii) Calculate real roots of a non-linear equation in one variable by • bisection method, • regula-falsi method, • Newton-Raphson method. iv) Use MAPLE command fsolve to find numerical solution of an equation and demonstrate through examples. Numerical Quadrature i) Define numerical quadrature. Use • Trapezoidal rule, • Simpson’s rule, to compute the approximate value of definite integrals without error terms. ii) Use MAPLE command trapezoid for trapezoidal rule and simpson for Simpson’s rule and demonstrate through examples. |
locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation understand how a given simple iterative formula of the form xn + 1 = F(xn) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy. locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation understand how a given simple iterative formula of the form xn + 1 = F(xn) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy. |
Not in IB |
Numerical Solution of Non-Linear Equations Analyze the searching of roots of an equation by graphical means and/or searching for the sign change. Demonstrate the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation. Apply a simple iterative formula of a form x_(n+1)=F(x_n ) to find roots of an equation to a prescribed degree of accuracy. Demonstrate how the iterative formula relates to the equation being |
Grade 12 concept |
|
In the updated 2023 curriculum, the traditional concept of "Introduction to Numerical Methods" has been replaced by the more specialized and focused topic of "Numerical Solution of Nonlinear Equations". This updated concept will provide students with a deeper understanding of the specific numerical methods and techniques used to solve nonlinear equations, which are a fundamental component of many fields of study such as engineering, physics, and economics.
The concept will cover a range of topics, including root-finding algorithms, fixed-point iteration, bisection methods, and Newton's method. Students will also learn about convergence analysis and error estimation, as well as practical applications of these numerical methods. By the end of the course, students will have gained a thorough understanding of how to effectively apply numerical techniques to solve complex nonlinear equations.
Overall, this change in the curriculum reflects the evolving needs of students and the demands of modern industries, and ensures that they are equipped with the skills and knowledge necessary to tackle real-world problems in their chosen fields. |
|