
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) 

Envisioned Total Number of Teaching Hours 
280 hrs 

128 




Grade IXX Concepts 
Matrices and Determinants Real and Complex Numbers Logarithm Algebraic Expressions and Algebraic Formulas Factorization Algebraic Manipulation Linear Equations and Inequalities Quadratic equations Theory of Quadratic equations Variations Partial fractions Sets and Functions Basic Statistics Linear Graphs and their Applications Introduction to Coordinate Geometry Introduction to Trigonometry Congruent Triangles Parallelograms and Triangles Line Bisectors and Angle Bisectors Sides and Angles of a Triangle Ratio and Proportion Pythagoras Theorem Theorems related to Area Projection of a Side of a Triangle Chords of a Circle Tangent to a Circle Chords and Arcs Angle in a Segment of a Circle Practical Geometry – Triangles Practical Geometry – Circles 
Number Types of number Sets Powers and roots Fractions, decimals and percentages Ordering The four operations Indices I Standard form Estimation Limits of accuracy Ratio and proportion Rates Percentage Using a calculator Time Money Exponential growth and decay Surds Algebra and graphs Introduction to algebra Algebraic manipulation Algebraic fractions Indices II Equations Inequalities Sequences Proportion Graphs in practical situations Graphs of functions Sketching curves Functions Coordinate geometry Coordinates Drawing linear graphs Gradient of linear graphs Length and midpoint Equations of linear graphs Parallel lines Perpendicular lines Geometry Geometrical terms Geometrical constructions Scale drawings Similarity Symmetry Angles Circle theorems I Circle theorems II Mensuration Units of measure Area and perimeter Circles, arcs and sectors Surface area and volume Compound shapes and parts of shapes Trigonometry Pythagoras’ theorem Rightangled triangles Nonrightangled triangles Pythagoras’ theorem and trigonometry in 3D Transformations and vectors Transformations Vectors in two dimensions Magnitude of a vector Vector geometry Probability Introduction to probability Relative and expected frequencies Probability of combined events Statistics Classifying statistical data Interpreting statistical data Averages and measures of spread Statistical charts and diagrams Scatter diagrams Cumulative frequency diagrams Histograms 
Grade 9 1. Real Numbers 2. Sets and Functions 3. Radicals and Factorization 4. Linear Equations and Inequalities 5. Coordinate Geometry 6. Volume and Surface Area 7. Similar figures 8. Geometrical Properties of regular polygons, Triangles and Parallelograms 9. Loci 10. Trigonometry 11. Construction of Triangles 12. Frequency distribution and Probability Grade 10 1. Matrices and determinants 2. Algebraic fractions 3. Linear inequalities in two variables 4. Quadratic equations 5. Graphs in real life and scientific situations 6. Vectors in plane 7. Application of trigonometry 8. Chords and arcs of a circle 9. Angles and tangents on a circle 10. Practical geometry of circles 11. Cumulative frequency distribution and measure of dispersion 12. Probability of combined events





Introduction to Mathematics 
N/A 
N/A 
1. Describe mathematics as the study of pattern, structure and relationships.
2. Explain with examples how the modern world has introduced new branches in mathematics, derived from the old branches of algebra, number theory, geometry and arithmetic. Students should be able to distinguish in terms of the broad subject matter that is studied between the below fields:
 algebra  real analysis  topology  complex analysis  number theory  game theory  cryptography  graph theory  logic  statistics  probability  combinatorics  operations research  numerical analysis  computation  geometry  trigonometry
3. Understand how in each of these subdisciplines, applications parallel theory. (E.g. Even the most abstract fields of mathematics such number theory and logic, are now used routinely in applications in computer science and cryptography).
4. Recognise that mathematicians are people who specialize in the study of mathematics, use highlevel mathematics and technology to develop new mathematical principles, understand relationships between existing principles, and solve realworld problems.
5. Name a few female mathematics around the world and a few Pakistani mathematicians and their contributions in the field of math (E.g. Asghar Qadir, Arif Zaman, Maryam Mirzakhani). 
2. Students are not intended to write formal definitions of these fields. Rather they should be ready to suggest in MCQs and structured questions which field would most appropriately be matched to the activity being presented (e.g. analyzing and interpreting data would be best classified under statistics). 5. Students are not intended to write long paragraphs on specific mathematicians or match mathematicians with their contributions but should be able to name atleast a couple of mathematicians and mention their contribution in 12 lines. 
What is Mathematics? How porous or rigid are the boundaries between the subdisciplines of mathematics? What are some emerging fields in modern mathematics? 
In current curricula taught in Pakistan, students usually directly jump into studying one branch of mathematics after the other, without being provided a birdseye overview of the field as a whole. These introductory SLOs provide this context.
These SLOs also help provide students with an understanding of what are the big questions that mathematics today is aiming to address, and how these endeavors are often interdisciplinary.
These introductory SLOs are not intended to be comprehensive, and are complemented by the CrossCutting Theme SLOs that are embedded into each of the subsequent Units. This will help students also understand what kind of career trajectories are possible by studying further mathematics, beyond simply becoming a mathematician, engineer or a professor. They are also intended to help students see themselves as mathematicians and recognize that mathematics is not just meant for the white men in the society. 
1. Does the definition need any refinement, to help high school students understand the dynamic nature of mathematics?
2. Any recommendations for more fields, or should the fields be classified differently?
3. Any other recommendations for mathematicians or modern contributions in mathematics? 
Nature of Mathematics 
N/A 
N/A 
1. Explain, with examples, how mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its intrinsic interest.
2. Develop an understanding of the history and development of mathematics, including the contributions of different cultures and civilizations such as Arabic, Greek, Indian and Chinese. [e.g., the history of number from Sumerians and its development to the present Arabic system]
3. Identify the major figures and their contributions in the history of mathematics, such as Pythagoras, Euclid, Archimedes, Newton, Leibniz, Euler and Ramanujan.
4. Understand the role of technology in mathematics and how mathematics has influenced technological advances. [e.g., use of mathematical simulations to predict future climate change or spread of disease] [e.g., the mathematics of connections and logical chains, for example, has contributed greatly to the design of computer hardware and programming techniques]
5. Analyze the impact of mathematical ideas on society and culture, and understand the cultural and societal factors that have influenced the development of mathematics throughout history. [e.g. the discovery of the irrationality of square root of 2 by the Pythagoreans invalidated many of their geometric proofs, shattered their beliefs about the supremacy of whole numbers, and caused unrest in their brotherhood]
6. Understand the connection of mathematics to other fields and the ability to apply mathematical concepts and skills to other disciplines. [e.g. complex numbers are used in electrical engineering concepts such as apparent power and phase shifts].
7. Explain, with examples, how mathematical models and equations are often used to make predictions and test hypotheses in science. [e.g., In physics, mathematical equations are used to describe the motion of objects and the behavior of energy and matter. In chemistry, mathematical models are used to predict the behavior of chemical reactions and the properties of molecules. In biology, mathematical models are used to predict the growth and spread of populations and the spread of disease.]
8. Explain, with examples, how mathematical methods and techniques are used in the analysis and interpretation of data in science, engineering and technology. [e.g., Statistical methods are used to analyze data from experiments and observational studies, and to make inferences about the underlying processes.]
9. Explain, with examples, how mathematics plays a key role in the development of new scientific theories and technologies. [e.g., Mathematical models and simulations are used to design and optimize new materials and drugs, and to understand the behavior of complex systems such as the human brain.] 
2, 4, 5, 6  Students should not be assessed on these understandings. Rather, these SLOs should be imbedded throughout the other topics (such as mentioning the contribution of different cultures when talking about number systems) and emphasized on through teaching techniques.
Teachers should select tasks that will promote atleast one aspect of NOM, provide students with the opportunity to reflect on their views of NOM at the beginning of a unit, highlight any particular aspect that naturally comes up during the teaching activities, and then at the end of a unit, provide students the opportunity to reflect on and refine their views on the NOM. 
 How did Mathematics as a field emerge historically and what bearing do those histories continue to have on the field's development today?
 How is Mathematics interlinked with other disciplines?
 What are the main properties of mathematical activity or mathematical knowledge? 
The purpose of studying Mathematics at the introductory high school level is not only to prepare students for further study in mathematics. Most students will in fact not go on to study further math or STEM fields. The math that they learn in school may well remain their understanding of the subject for the rest of their lives. Hence an introductory math curriculum must consider what citizens in a democractic society ought to know about the nature of math.
“Nature of Mathematics” (NoM) means teaching about math's underlying assumptions, and its methodologies. This involves some broad ideas from the history and philosophy of mathematics. It is important to study NoM because it helps students develop a complete view of mathematics and move away from seeing mathematics as only a set of rules to memorize or a subject that has no use in their reallife. It also helps them develop critical thinking and problemsolving skills that will help them in analyzing the world around them.
Teaching NoM is also seen as an international trend. For example:
 The United States has developed Common Core Standards for Mathematical Practices that highlight some of the themes of NoM.
 The IB curriculum substantially incorporates NoM in all its MYP and DP curricula.
Teachers with math backgrounds can effectively teach introductory level modules on NOS with the support of teacher training, clear examples of assessment expectations and supportive online and textbook materials.The level of knowledege required up to Grade 12 on this topic is nicely elaborated on in the IB DP curriculum guidance documents and these can be adapted. 

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. 
N/A 
1. Differentiate between a mathematical statement and its proof.
2. Understand the difference between an axiom, conjecture and theorem.
3. Understand when the negation of a statement and conjunction and disjunctions of statements are true or false.
4. Differentiate between, and be able to find the inverse, converse, and contrapositive of conditional statements.
5. Differentiate between deductive and inductive reasoning.
6. Understand the symbols and notation for equality and identity.
7. Formulate simple deductive proofs [algebraic proofs that require showing the LHS to be equal to the RHS. E.g., showing (x3)^2 + 5=x^2  6x + 14].
8. Understand the role of mathematical proof in the justification of mathematical claims and the development of mathematical knowledge [E.g. Stating how Fermat proposed a theory that remained unsolved for years and attempting to prove Fermat's last theorem led to new developments in the field of number theory]. 
2. Students can be given the example of the parallel postulate, Collatz conjecture and Fermat's last theorem as an example of an axiom, conjecture and theorem. The chosen examples pave way for highlighting the importance of each in the history of mathematics. 
What is a proof? Why are proofs important? 
This unit has been added to the Math National Curriculum 2023 to further highlight the nature of mathematics. Proof in mathematics is an essential element in developing critical thinking. Engaging students in the process of proving a statement enables a deeper understanding of mathematical concepts. Writing proofs enables students to appreciate proof techniques and mathematical thought processes. Proofs also encourage students to reflect on mathematical rigour, efficiency and the elegance of showing that a statement is true.
Students who are not familiar with proofs at the high school level also tend to develop a distorted image of mathematics as only comprising of applied mathematics. This will help introduce students to the abstract and formal side of mathematics, easing their transition into university level mathematical ideas. 

Matrices and Determinants 
Introduction to Matrices Define • a matrix with real entries and relate its rectangular layout (formation) with real life, • rows and columns of a matrix, • the order of a matrix, • equality of two matrices. Types of Matrices Define and identify row matrix, column matrix, rectangular matrix, square matrix, zero/null matrix, identity matrix, scalar matrix, diagonal matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition and Subtraction of Matrices Know whether the given matrices are conformable for addition/subtraction. ii) Add and subtract matrices. iii) Multiply a matrix by a real number. iv) Verify commutative and associative laws under addition. v) Define additive identity of a matrix. vi) Find additive inverse of a matrix. Multiplication of Matrices i) Know whether the given matrices are conformable for multiplication. ii) Multiply two (or three) matrices. iii) Verify associative law under multiplication. iv) Verify distributive laws. v) Show with the help of an example that commutative law under multiplication does not hold in general (i.e. AB ≠ BA). vi) Define multiplicative identity of a matrix. vii) Verify the result ( ) t t t AB = B A Multiplicative Inverse of a Matrix Define the determinant of a square matrix. Evaluate determinant of a matrix. Define singular and nonsingular matrices. iv) Define adjoint of a matrix. v) Find multiplicative inverse of a nonsingular matrix A and verify that AA I A A −1 −1 = = where I is the identity matrix. vi) Use adjoint method to calculate inverse of a non[1]singular matrix. vii) Verify the result 1 1 1 ( ) − − − AB = B Solution of Simultaneous Linear Equations Solve a system of two linear equations and related reallife problems in two unknowns using • Matrix inversion method, • Cramer’s rule. 
Not in O levels 
Matrices and Determinants · Recognize matrices as a display of information in the form of rectangular arrays of any order · Describe the different types of matrices · Calculate the product of the scalar quantity and a matrix · Solve situations involving sum, difference, and product of two matrices · Calculate the determinant and inverse of 2 × 2 nonsingular matrices · Solve the simultaneous linear equations in two variables using matrix inversion method 
This concept is shifted to grade 10 to build a progression with grade 11 SLOs and remove the gap between the concepts 
What is a matrix, and how is it used to represent data in various fields, including mathematics, physics, and engineering? 
Kept in 2023 curiculum It’s not the part of O levels now but it is retained in new curriculum keeping in mind the importance and application of Matrices. Matrices have many applications in diverse fields of science, commerce, and social science. Matrices are used in: (i) Computer Graphics (ii) Optics (iii) Cryptography (iv) Economics (v) Chemistry (vi) Geology (vii) Robotics and animation (viii) Wireless communication and signal processing (ix) Finances (x) Mathematic 

REAL AND COMPLEX NUMBERS 
Real Numbers i) Recall the set of real numbers as a union of sets of rational and irrational numbers. ii) Depict real numbers on the number line. iii) Demonstrate a number with terminating and non[1]terminating recurring decimals on the number line. iv) Give decimal representation of rational and irrational numbers. Properties of Real Numbers Know the properties of real numbers. Radicals and Radicands i) Explain the concept of radicals and radicands. ii) Differentiate between radical form and exponential form of an expression. iii) Transform an expression given in radical form to an exponential form and vice versa. Complex Numbers i) Define complex number z represented by an expression of the form z = a + ib , where a and b are real numbers and i = −1 . ii) Recognize a as real part and b as imaginary part of z = a + ib iii) Define conjugate of a complex number. iv) Know the condition for equality of complex numbers. Basic Operations on Complex Numbers Carryout basic operations (i.e., addition, subtraction, multiplication and division) on complex numbers. 
Surds Understand and use surds, including simplifying expressions. Rationalise the denominator Indices I Understand and use indices (positive, zero, negative and fractional). Understand and use the rules of indices. Indices II Understand and use indices (positive, zero, negative and fractional). Understand and use the rules of indices. · Convert numbers into and out of standard form. · Calculate with values in standard form. 
Real Numbers • Describe the set of real numbers • Demonstrate irrational numbers by: • representing, identifying, and simplifying irrational numbers • ordering irrational numbers • Demonstrate the properties of real numbers 
Part of grade 9 concepts suggested to Sort a set of numbers into rational and irrational numbers. Determine an approximate value of an irrational number. Approximate the locations of irrational numbers on a horizontal or vertical number line, using a variety of strategies, and explain the reasoning. Order a set of irrational numbers on a horizontal or vertical number line. Express a radical as a mixed radical in simplest form (limited to numerical radicands). Express a mixed radical as an entire radical (limited to numerical radicands). Explain, using examples, the meaning of the index of a radical. Represent, using a graphic organizer, the relationship among the subsets of the real numbers (natural, whole, integer, rational, irrational). 
How are matrices added, subtracted, multiplied, and transposed, and what are the properties of these operations? 
Retained, to build a strong base of Real Numbers before moving onto Complex Numbers. Complex Numbers which were part of grade 9 were removed, as there was no continuity, along with the concept not available in parallel curriculum. 

Indices/Exponents 
Laws of Exponents/Indices i) Recall base, exponent and value. ii) Apply the laws of exponents to simplify expressions with real exponents. 
Understand and use indices (positive, zero, negative and fractional) Understand and use the rules of indices 
Demonstrate an understanding of powers with integral and rational exponents. 
concept of Grade 9 Explain, using patterns or exponent laws, Explain, using patterns, Apply the exponent laws to expressions with rational or variable bases and integral or rational exponents, and explain the reasoning Express powers with rational exponents as radicals and vice versa. Solve a problem that involves exponent laws or radicals. Identify and correct errors in the simplification of an expression that involves power 
How can matrices be used to solve systems of linear equations, and what is the relationship between the rank of a matrix and the solutions to a system of equations? 
Retained, to build a strong base of Indices and exponents however some practice of the concept done in grade 8 also 

LOGARITHMS 
Scientific Notation Express a number in standard form of scientific notation and vice versa. Logarithm i) Define logarithm of a number to the base a as the power to which a must be raised to give the number (i.e. a y x = ⇔ log y x, a = a > 0, y > 0 and a ≠ 1). ii) Define a common logarithm, characteristic and mantissa of log of a number. iii) Use tables to find the log of a number. iv) Give concept of antilog and use tables to find the antilog of a number. Common and Natural Logarithm Differentiate between common and natural logarithm. Laws of Logarithm Prove the following laws of logarithm. • loga (mn) = loga m + loga n , • a m a n nm a log ( ) = log − log , • m n a m n a log = log , • loga mlogm n = loga n . Application of Logarithm Apply laws of logarithm to convert lengthy processes 
Standard form Use the standard form A × 10n where n is a positive or negative integer and 1 ⩽ A < 10. 

Shifted to grade 11 
What is a determinant, and how is it computed for a given square matrix, and what is the relationship between determinants and the invertibility of a matrix? 
Not in O levels 

ALGEBRAIC EXPRESSIONS AND ALGEBRAIC FORMULAS 
Algebraic expressions i) Know that a rational expression behaves like a rational number. ii) Define a rational expression as the quotient p(x)/q(x) of two polynomials p(x) and q(x) where q(x) is not the zero polynomial. iii) Examine whether a given algebraic expression is a • polynomial or not, • rational expression or not. iv)Define p(x)/q(x) as a rational expression in its lowest terms if p(x) and q(x)a re polynomials with integral coefficients and having no common factor. v) Examine whether a given rational algebraic expression is in lowest form or not. vi) Reduce a given rational expression to its lowest terms. vii) Find the sum, difference and product of rational expressions. viii) Divide a rational expression with another and express the result in its lowest terms. ix) Find value of algebraic expression at some particular real number. Algebraic Formulae Know the formulas. Surds and their Application Recognize the surds and their application. ii) Explain the surds of second order. Use basic operations on surds of second order to rationalize the denominators and evaluate it. Rationalization Explain rationalization (with precise meaning) of real numbers of the types a+b x 1 , x + y 1 and their combinations where x and y are natural numbers and a and b are integers 
Equations Construct expressions, equations and formulas. Solve linear equations in one unknown. Solve fractional equations with numerical and linear algebraic denominators Solve simultaneous linear equations in two unknowns. Solve quadratic equations by factorisation, completing the square and by use of the quadratic formula. Change the subject of formula 
Rational Expressions and Functions Describe rational expressions Factorize and simplify rational expressions. Demonstrate manipulation of algebraic fractions. Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials, or trinomials) Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials, or trinomials). 
Concept of Grade 10 Determine a rational expression that is equivalent to a rational expression by multiplying the numerator and denominator by the same factor (limited to a monomial or a binomial), and state the nonpermissible values of the equivalent rational expression. Simplify a rational expression. Explain why the nonpermissible values of a rational expression and its simplified form are the same. Identify and correct errors in a simplification of a rational expression, and explain the reasoning Compare the strategies for performing an operation on rational expressions to the strategies for performing the same operation on rational numbers. Determine the nonpermissible values when performing operations on rational expressions. Determine, in simplified form, the sum or difference of rational expressions with the same denominator. Determine, in simplified form, the sum or difference of rational expressions in which the denominators are not the same and which may or may not contain common factors. Determine, in simplified form, the product or quotient of rational expressions. Simplify an expression that involves two or more operations on rational expressions. Determine the nonpermissible values for the variable in a rational equation. Determine the solution to a rational equation algebraically, and explain the process. Explain why a value obtained in solving a rational equation may not be a solution of the equation. Solve a problem by modelling a situation using a rational equation. 
How can determinants be used to find the eigenvalues and eigenvectors of a matrix, and how are these concepts used in various applications, including computer graphics, cryptography, and data analysis? 
The concept of Algebraic expressions and Algebraic formula is covered in grade 8 however rest of the SLOs are added under the concept of Algebraic Manipulation 

Factorization 
Factorization Recall factorization of expressions of the following types. Remainder Theorem and Factor Theorem State and prove Remainder theorem and explain through examples. ii) Find remainder (without dividing) when a polynomial is divided by a linear polynomial. iii) Define zeros of a polynomial. iv) State and prove factor theorem. Factorization of a Cubic Polynomial Use factor theorem to factorize a cubic polynomial. 
Introduction to algebra Know that letters can be used to represent generalised numbers. Substitute numbers into expressions and formulas. Algebraic manipulation Simplify expressions by collecting like terms. Expand products of algebraic expressions. Factorise by extracting common factors. Factorise expressions of the form: • ax + bx + kay + kby • a 2 x 2 − b2 y 2 • a 2 + 2ab + b2 • ax 2 + bx + c • ax 3 + bx2 + cx . Complete the square for expressions in the form ax 2 + bx + c . Algebraic fractions Manipulate algebraic fractions. Factorise and simplify rational expressions 
Factorization: Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially, and symbolically. Factorize algebraic expressions of the types: 〖a^4+a^2 b^2+b^(4 ) or a^4+4b〗^4 x^4+px+q ax^2+ bx+c (ax^2+ bx+c)( ax^2+ bx+d)+k (x+a)(x+b) (x+c)(x+d)+k (x+a)(x+b)(x+c)(x+d)+kx^2 a^3+3a^2b+3ab^2+b^3 a^33a^2b +3ab^2b^3 a^3±b^3 
Grade 9 concept Determine the common factors in the terms of a polynomial, and express the polynomial in factored form. Model the factoring of a trinomial, concretely or pictorially, and record the process symbolically. Factor a polynomial that is a difference of squares, and explain why it is a special case of factoring a trinomial of the form ax2 + bx + c = 0 where b = 0 and c < 0. Identify and explain errors in a polynomial factorization. Factor a polynomial, and verify by multiplying the factors. Explain, using examples, the relationship between multiplication and factoring of polynomials. Generalize and explain strategies used to factor a trinomial. Express a polynomial as a product of its factors. 
How do singular value decomposition (SVD) and LU decomposition factorize matrices, and what are their applications in linear algebra, statistics, and machine learning? 
Remainder and factor theorem are shifted to grade 11 to align with A levels Most of the factorization has already done in grade 8 so the remaining types of equations are added in grade 9 

ALGEBRAIC MANIPULATION 
Highest Common Factor and Least Common Multiple i) Find highest common factor and least common multiple of algebraic expressions. ii) Use factor or division method to determine highest common factor and least common multiple. iii) Know the relationship between HCF and LCM. iv) Solve real life problems related to HCF and LCM. Basic Operations on Algebraic Fractions Use highest common factor and least common multiple to reduce fractional expressions involving +,−,×,÷ . Square Root of Algebraic Expression Find square root of algebraic expression by factorization and division. 


Concept of Grade 10 
What are the different types of matrices, including diagonal, triangular, symmetric, and orthogonal matrices, and how are they used in various applications? 
HCF and LCM are not included seperately as these are the part of addition and subtraction of rational fractions therefore instead of giving seperately it is the part of rational fractions. 

LINEAR EQUATIONS AND INEQUALITIES 
Linear Equations i) Recall linear equation in one variable. ii) Solve linear equation with rational coefficients. iii) Reduce equations, involving radicals, to simple linear form and find their solutions. Equation involving Absolute Value i) Define absolute value. ii) Solve the equation, involving absolute value, in one variable. Linear Inequalities i) Define inequalities (>, <) and (≥, ≤) . ii) Recognize properties of inequalities (i.e. trichotomy, transitive, additive and multiplicative). Solving Linear Inequalities Solve linear inequalities with rational coefficients. 
Equations Solve linear equations in one unknown. Solve fractional equations with numerical and linear algebraic denominators Inequalities 1 Represent and interpret inequalities, including on a number line. 2 Construct, solve and interpret linear inequalities. 3 Represent and interpret linear inequalities in two variables graphically. 4 List inequalities that define a given region. 
Linear Equations and Inequalities Solve linear equation and inequalities with rational coefficients. Solve equations with radicals reducible to simple linear form. Solve problems that involve systems of linear equations in two variables, graphically and algebraically. Solve Linear Equations and Inequality involving Absolute Value. 
Concept of Grade 9 Model a situation, using a system of linear equations. Relate a system of linear equations to the context of a problem. Determine and verify the solution of a system of linear equations graphically, with or without technology. Explain the meaning of the point of intersection of a system of linear equations. Determine and verify the solution of a system of linear equations algebraically. Explain, using examples, why a system of equations may have no solution, one solution, or an infinite number of solutions. Describe a strategy to solve a system of linear equations. Solve a contextual problem that involves a system of linear equations, with or without technology. 
How can matrices be used to represent linear transformations in Euclidean space, and how are concepts such as rotations, reflections, and scaling represented using matrices? 
In the 2023 curriculum, the concept of linear equations and inequalities is a continuation of what was covered in Grade 8. In Grade 8, students are typically introduced to the concept of linear equations, which involve finding the slope and yintercept of a line and using that information to graph the equation. They also learn how to solve linear equations with one variable.
In Grade 10, students build on this foundation by learning about linear inequalities in two variables, which involve graphing regions of the coordinate plane that satisfy the inequality. They also learn how to solve systems of linear inequalities and use them to model realworld situations.
Additionally, in Grade 10, students are introduced to absolute value functions, which involve finding the distance between a number and zero. They learn how to graph absolute value functions and use them to solve equations and inequalities.
Overall, the concepts of linear equations, linear inequalities, and absolute value functions in Grade 10 are a natural continuation of the concepts covered in Grade 8. By building on this foundation, students are better prepared for higherlevel math courses and are better equipped to apply these concepts to realworld problems. 

Linear Inequalities in two variables 
Not in Grade IX and X 
Construct, solve and interpret linear inequalities. Represent and interpret linear inequalities in two variables graphically 
Linear Inequalities in two variables Graph and solve problems that involve systems of linear inequalities in two variables Solve two linear inequalities with two unknowns simultaneously. Interpret and Identify regions in plane bounded by two linear inequalities in two unknowns. (Note: Problems from linear programing are not included). 
Concept of grade 10 Model a problem, using a system of linear inequalities in two variables. Graph the boundary line between two half planes for each inequality in a system of linear inequalities, and justify the choice of solid or broken lines. Determine and explain the solution region that satisfies a linear inequality, using a variety of strategies when given a boundary line. Determine, graphically, the solution region for a system of linear inequalities, and verify the solution. Explain, using examples, the significance of the shaded region in the graphical solution of a system of linear inequalities. Express the equation of a linear function in two variables, using function notation. Express an equation given in function notation as a linear function in two variables. Determine the related range value, given a domain value for a linear function. Determine the related domain value, given a range value for a linear function. Sketch the graph of a linear function expressed in function notation. 
How can matrices be used to solve optimization problems, including linear programming and quadratic programming, and how are these techniques used in various fields, including finance, economics, and operations research? 
The inclusion of linear inequalities in two variables in the Grade 10 curriculum of 2023 is important for several reasons:
Realworld applications: Linear inequalities are used to model many realworld situations, such as business decisions, resource allocation, and production planning. By introducing students to linear inequalities, they can learn to use mathematical tools to analyze and solve problems that have practical applications.
Development of critical thinking skills: Studying linear inequalities in two variables requires students to think critically and logically. They need to be able to identify patterns, use algebraic reasoning, and solve equations to find solutions. These skills are transferable to other subjects and are useful in many areas of life.
Preparation for higher education: Linear inequalities are a fundamental concept in algebra and are a prerequisite for more advanced mathematical topics, such as linear programming and optimization. By introducing linear inequalities in Grade 10, students are better prepared for higherlevel math courses in high school and college.
Alignment with standardized testing: Standardized tests, such as the Olevels, often include questions on linear inequalities in two variables. By teaching this topic in the Grade 10 curriculum, students are better prepared for these types of questions and are more likely to perform well on standardized tests.
Overall, the inclusion of linear inequalities in two variables in the Grade 10 curriculum of 2023 is important for building students' mathematical skills, preparing them for higher education, and ensuring that they are prepared for standardized testing 

QUADRATIC EQUATIONS 
Quadratic Equation Define quadratic equation. Solution of Quadratic Equations Solve a quadratic equation in one variable by: • factorization, • completing square. Quadratic Formula i) Use method of completing square to derive quadratic formula. ii) Use quadratic formula to solve quadratic equations. Equations Reducible to Quadratic Form i) Solve equations, reducible to quadratic form, of the type ax + bx + c = . ii) Solve the equations of the type a p x c p x b + = ( ) ( . ) iii) Solve reciprocal equations of the type iv) Solve exponential equations in which the variables occur in exponents. v) Solve equations of the type (x + a)(x + b)(x + c)(x + d) = k where a + b = c + d . Radical Equations Solve equations of the type: 
Solve quadratic equations by factorisation, completing the square and by use of the quadratic formula 
Quadratic Equations and their Solution Solve quadratic equations by using the methods of: factorization, completing squares, and quadratic formula to solve quadratic equation. Solve problems of “changing the subject of formula”. Solve fractional equations that can be reduced to quadratic equations. Solve real world situations by formulating a quadratic equation in one variable. 
Concept of grade 10 Explain the reasoning for the process of completing the square, as shown in an example. Write a quadratic function given in the form y = ax2 + bx + c as a quadratic function in the form y = a(x – p)2 – q by completing the square. Identify, explain, and correct errors in an example of completing the square. Determine the characteristics of a quadratic function given in the form y = ax2 + bx + c, and explain the strategy used. Sketch the graph of a quadratic function given in the form y = ax2 + bx + c. Verify, with or without technology, that a quadratic function in the form y = ax2 + bx + c represents the same function as the quadratic function in the form y = a(x – p)2 + q. Write a quadratic function that models a situation, and explain any assumptions made. Solve a problem, with or without technology, by analyzing a quadratic function 
How can matrices and determinants be used to model and analyze complex systems, including networks, circuits, and chemical reactions, and what are the advantages and limitations of these methods compared to other modeling techniques? 
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 seconddegree 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 problemsolving skills and logical reasoning. It also enables students to understand realworld 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 wellrounded education and are prepared for success in their future careers. 

THEORY OF QUADRATIC EQUATIONS 
Nature of the Roots of a Quadratic Equation i) Define discriminant (b 4ac 2 − ) of the quadratic expression ax + bx + c 2 . ii) Find discriminant of a given quadratic equation. iii) Discuss the nature of roots of a quadratic equation through discriminant. iv) Determine the nature of roots of a given quadratic equation and verify the result by solving the equation. v) Determine the value of an unknown involved in a given quadratic equation when the nature of its roots is given. Cube Roots of Unity and their properties i) Find cube roots of unity. i) Recognize complex cube roots of unity as ω and 2 ω . iii) Prove the properties of cube roots of unity. iv) Use properties of cube roots of unity to solve appropriate problems. Roots and Coefficients of a Quadratic Equation i) Find the relation between the roots and the coefficients of a quadratic equation. ii) Find the sum and product of roots of a given quadratic equation without solving it. iii) Find the value(s) of unknown(s) involved in a given quadratic equation when • sum of roots is equal to a multiple of the product of roots, • sum of the squares of roots is equal to a given number, • roots differ by a given number, • roots satisfy a given relation (e.g. the relation 2α + 5β = 7 where α and β are the roots of given equation), • both sum and product of roots are equal to a given number. Symmetric Functions of Roots of a Quadratic Equation i) Define symmetric functions of roots of a quadratic equation. ii) Evaluate a symmetric function of the roots of a quadratic equation in terms of its coefficients. Formation of Quadratic Equation i) Establish the formula, (Sum of roots) (Product of roots) 0 2x − x + = , to find a quadratic equation from the given roots. ii) Form the quadratic equation whose roots, for example, are of the type: • 2α +1, 2β +1, • 2 2 α , β , • α β1 1 , , • αββ α , , • α β α β 1 1 + , + , where α , β are the roots of a given quadratic equation. Synthetic Division i) Describe the method of synthetic division. ii) Use synthetic division to • find quotient and remainder when a given polynomial is divided by a linear polynomial, • find the value(s) of unknown(s) if the zeros of a polynomial are given, • find the value(s) of unknown(s) if the factors of a polynomial are given, • solve a cubic equation if one root of the equation is given, • solve a biquadratic (quartic) equation if two of the real roots of the equation are given. Simultaneous Equations Solve a system of two equations in two variables when • one equation is linear and the other is quadratic, • both the equations are quadratic. Solve the real life problems leading to quadratic equations. 
Not in O levels 

Shifted to grade XI as its the part of A levels 

Quadratic equations are a fundamental concept in mathematics that is typically introduced in secondary education. In the 2023 Mathematics curriculum, the theory of quadratic equations has been shifted to Grade 11 to align it with A levels.
The study of quadratic equations involves the analysis and solution of equations of the form ax^2 + bx + c = 0. It is an important topic in mathematics and is used extensively in various fields, including physics, engineering, and economics.
By shifting the study of quadratic equations to Grade 11, learners will have a more indepth understanding of this important mathematical concept. They will have a strong foundation in algebraic manipulation, factoring, and solving equations, which are essential skills for advanced mathematics.
Furthermore, the shift to Grade 11 aligns the curriculum with A levels, which provides learners with a seamless transition to higher education.
In conclusion, the theory of quadratic equations has been shifted to Grade 11 in the 2023 Mathematics curriculum to align it with A levels. This decision provides learners with a more indepth understanding of this important mathematical concept and prepares them for further studies in mathematics or related fields at the higher education level. 

VARIATIONS 
Ratio, Proportions and Variations i) Define ratio, proportions and variations (direct and inverse). ii) Find 3rd, 4th mean and continued proportion. Theorems on Proportions Apply theorems of invertendo, alternendo, componendo, dividendo and componendo & dividendo to find proportions. Joint Variation i) Define joint variation. ii) Solve problems related to joint variation. KMethod i) Use KMethod to prove conditional equalities involving proportions. ii) Solve real life problems based on variations. 
Variation express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities 

Already covered in grade 8 

The concepts of Ratio, Proportion, and Variation are fundamental concepts in mathematics and are typically introduced in the early years of secondary education. In the 2023 Mathematics curriculum, these concepts are introduced in Grade 8 and are covered in depth.
The concepts of Ratio, Proportion, and Variation are essential in many areas of mathematics, including algebra, geometry, and trigonometry. They are also used extensively in many realworld applications, such as finance, engineering, and physics.
In Grade 8, learners are introduced to the basic concepts of Ratio, Proportion, and Variation. They learn to solve problems using these concepts, develop their reasoning skills, and gain a deeper understanding of their realworld applications.
Since these concepts are already covered in depth in Grade 8, they are not included in the Grade 912 Mathematics curriculum. This allows learners to focus on more advanced concepts and applications in mathematics, building on the foundation established in earlier grades.
In conclusion, the concepts of Ratio, Proportion, and Variation are fundamental concepts in mathematics, and they are typically introduced in the early years of secondary education. In the 2023 Mathematics curriculum, these concepts are introduced in Grade 8 and are covered in depth. Since these concepts are already covered in depth in Grade 8, they are not included in the Grade 912 Mathematics curriculum. This allows learners to focus on more advanced concepts and applications in mathematics, building on the foundation established in earlier grades. 

PARTIAL FRACTIONS 
Proper, Improper and Rational Fraction Define proper, improper and rational fraction. Resolution of Fraction into Partial Fractions Resolve an algebraic fraction into partial fractions when its denominator consists of : • nonrepeated linear factors, • repeated linear factors, • nonrepeated quadratic factors, • repeated quadratic factors. 
Not in O levels 

Shifted to grade XI as it is the part of A levels 

Partial fractions is a topic that deals with the decomposition of a rational function into simpler fractions. It is an important concept in mathematics and is often used in calculus, differential equations, and other higherlevel mathematics courses.
In the 2023 Mathematics curriculum, the study of partial fractions has been shifted to Grade 11. This decision has been made based on the maturity level of learners in Grade 11, who are better equipped to understand the concept and relate it to further concepts.
Partial fractions involve several advanced mathematical concepts, such as algebraic manipulation, factoring, and solving equations. These concepts require a strong foundation in mathematics, which is typically developed in the earlier years of secondary education.
In Grade 11, learners have had more exposure to these concepts and are better prepared to handle the challenges of partial fractions. Furthermore, the study of partial fractions in Grade 11 helps learners build on their prior knowledge and deepen their understanding of more advanced mathematical concepts.
By shifting the study of partial fractions to Grade 11, learners can develop a more comprehensive understanding of this fundamental mathematical concept. They are better equipped to handle the advanced concepts involved in partial fractions, which helps them prepare for further study in higherlevel mathematics courses.
In conclusion, the study of partial fractions is an important concept in mathematics that has been shifted to Grade 11 in the 2023 Mathematics curriculum. This decision has been made based on the maturity level of learners in Grade 11, who are better equipped to understand the concept and relate it to further concepts. By studying partial fractions in Grade 11, learners can develop a more comprehensive understanding of this important mathematical concept and are better prepared for further study in higherlevel mathematics courses 

SETS 
Sets Properties of Union i) Recall the sets denoted by N, Z, W, E, O, P and Q. ii) Recognize operation on sets (∪, ∩, \,K). iii) Perform operation on sets • union, • intersection, • difference, • complement. Properties of Union and Intersection Give formal proofs of the following fundamental properties of union and intersection of two or three sets. • Commutative property of union, • Commutative property of intersection, • Associative property of union, • Associative property of intersection, • Distributive property of union over intersection, • Distributive property of intersection over union, • De Morgan’s laws. v) Verify the fundamental properties for given sets. Venn Diagram vi) Use Venn diagram to represent • union and intersection of sets, • complement of a set. vii) Use Venn diagram to verify • commutative law for union and intersection of sets, • De Morgan’s laws, • associative laws, • distributive laws. Ordered Pairs and Cartesian Product Recognize ordered pairs and Cartesian product. Binary relation Define binary relation and identify its domain and range. 
Understand and use set language, notation and Venn diagrams to describe sets and represent relationships between sets. Venn diagrams are limited to two or three sets. The following set notation will be used: • n(A) Number of elements in set A • ∈ ‘… is an element of …’ • ∉ ‘… is not an element of …’ • A′ Complement of set A • ∅ The empty set • Universal set • A ⊆ B A is a subset of B • A ⊈ B A is not a subset of B • A ∪ B Union of A and B • A ∩ B Intersection of A and B. Example definition of sets: A = {x: x is a natural number} B = {(x, y): y = mx + c} C = {x: a ⩽ x ⩽ b} D = {a, b, c, …}. 
Sets Demonstrate union and intersection of three sets (Subsets, overlapping sets and disjoint sets), Complement of a set using Venn diagrams Solve problems on classification and cataloguing by using Venn diagrams for Scenarios involving two sets and three sets. Further application of sets. Solve problems involving sets and linear inequalities, sets and coordinate geometry. Apply formulas for the union and intersection of three sets. 
Concet of grade 9 Explain how set theory is used in applications such as Internet searches, database queries, data analysis, games, and puzzles. Provide examples of the empty set, disjoint sets, subsets, and universal sets in context, and explain the reasoning. Organize information such as collected data and number properties, using graphic organizers, and explain the reasoning. Explain what a specified region in a Venn diagram represents. Determine the elements in the complement, the intersection, or the union of two sets. Identify and correct errors in a solution to a problem that involves sets. Solve a contextual problem that involves sets, and record the solution Functions Express the equation of a linear function in two variables, using function notation. Express an equation given in function notation as a linear function in two variables. Determine the related range value, given a domain value for a linear function. Determine the related domain value, given a range value for a linear function. Sketch the graph of a linear function expressed in function notation. 

The concept of sets is a fundamental part of mathematics, and it is introduced in Grade 8 of the 2023 Mathematics curriculum. In Grade 9, the study of sets is continued, building on the concepts learned in the previous grade. The inclusion of the concept of sets in the 2023 Mathematics curriculum helps students develop their understanding of this crucial mathematical concept.
In Grade 8, students learn about the basic properties of sets, including the elements of a set, the intersection and union of sets, and the complement of a set. In Grade 9, the study of sets is continued, and students learn more advanced concepts, such as Venn diagrams, the Cartesian product of sets, and set operations.
The study of sets in Grade 9 is essential as it helps students understand the relationships between different sets and how they can be combined and manipulated to solve problems. Additionally, the study of sets helps students develop their logical reasoning and problemsolving skills, which are essential in mathematics and many other fields.
By linking the concept of sets in Grade 9 to the concepts learned in Grade 8, students can build on their prior knowledge and deepen their understanding of sets. This continuity in the curriculum ensures that students have a strong foundation in this fundamental mathematical concept and are wellprepared for further study in higherlevel mathematics courses.
In conclusion, the concept of sets is an integral part of mathematics, and it is introduced in Grade 8 of the 2023 Mathematics curriculum. In Grade 9, the study of sets is continued, building on the concepts learned in the previous grade. By linking the concept of sets in Grade 9 to the concepts learned in Grade 8, students can build on their prior knowledge and deepen their understanding of sets. The study of sets is essential for developing students' logical reasoning and problemsolving skills, which are essential in mathematics and many other fields. 

FUNCTION 
Function i) Define function and identify its domain, codomain and range. ii) Demonstrate the following: • into function, • oneone function, • into and oneone function (injective function), • onto function (surjective function), • oneone and onto function (bijective function). iii) Examine whether a given relation is a function or not. iv) Differentiate between oneone correspondence and oneone function. v) Include sufficient exercises to clarify/differentiate between the above concepts. 

Functions Represent a linear function, using function notation. Demonstrate functions, domain and range Solve problems on functions involving linear expressions. Determine the inverse of a given functions Demonstrate an understanding of operations on, and compositions of, functions. 
FUNCTION Explain, using examples, why some relations are not functions but all functions are relations. Determine whether a set of ordered pairs represents a function. Sort a set of graphs as functions or nonfunctions. Generalize and explain rules for determining whether graphs and sets of ordered pairs represent functions. Determine and express in a variety of ways the domain and range of a relation Sketch the graph of a function that is the sum, difference, product, or quotient of two functions, given their graphs. Write the equation of a function that is the sum, difference, product, or quotient of two or more functions, given their equations. Determine the domain and range of a function that is the sum, difference, product, or quotient of two functions. Write a function f(x) as the sum, difference, product, or quotient of two or more functions. Determine the value of the composition of functions when evaluated at a point using the forms f(f(a)), f(g(a)), or g(f(a)). Determine, given the equations of two functions f(x) and g(x), the equation of the composite function of the forms f(f(x)), f(g(x)), or g(f(x)), and explain any restrictions. Sketch, given the equations of two functions f(x) and g(x), the graph of the composite function in the forms f(f(x)), f(g(x)), or g(f(x)). Sketch the graph of the function y = f(x) or given the graph of y = f(x), and explain the strategies used. Write a function f(x) as the composition of two or more functions. Write a function f(x) by combining two or more functions through operations on, or compositions of, functions 
How can you represent and describe functions? How can functions describe realworld situations, model predictions and solve problems? What is a function? Describe what it means for a situation to have a functional relationship. What is the relationship between the input and output of a function? In what ways can different types of functions be used and altered to model various situations that occur in life? What units, scales and labels must be applied to accurately represent a linear function in the context of a problem situation? Can students represent a function using real world contexts, algebraic equations, tables, and with words? What are the advantages of representing the relationship between quantities symbolically? Numerically? Graphically? Are students able to compare the properties of multiple functions, given a linear function, and determine which function has the greater rate of change? Can students construct a function to model a linear relationship between two quantities, and determine the rate of change and initial values of the functions? How can proportional relationships be used to represent authentic situations in life and solve actual problems? In what way(s) do proportional relationships relate to functions and functional relationships? Are students able to calculate the slope of a line graphically, apply direct variation, differentiate between zero slope and undefined slope, and understand that similar right triangles can be used to establish that slope is a constant for a nonvertical line? Do students have the knowledge to solve multistep equations using simple cases by inspection, one solution, infinitely many solutions, or no solution? Are students able to solve systems of linear equations numerically, graphically, or algebraically using substitution or elimination? Do students have the ability to discuss efficient solution methods when solving a system of equations? Can students use a system of equations to solve realworld problems and interpret the solution in the context of the problem? 
The concept of functions is an essential part of mathematics, and it is often introduced as a part of the concept of sets and functions. In the 2023 Mathematics curriculum, the study of functions is included to align our curriculum with O levels and to provide students with a comprehensive understanding of this fundamental mathematical concept.
Functions are used to describe the relationship between two variables, and they are a crucial tool in various fields, including science, engineering, and economics. By studying functions, students learn how to analyze the behavior of different variables and how to make predictions based on that behavior.
In the 2023 Mathematics curriculum, students will learn about the basic properties of functions, including domain, range, and inverse functions. They will also learn how to graph functions and how to use them to solve realworld problems.
By including the study of functions in our curriculum, students will have a solid foundation in this important mathematical concept, and they will be wellprepared for further study in higherlevel mathematics courses. Furthermore, the study of functions helps students develop their critical thinking and problemsolving skills, which are essential in many fields.
In conclusion, the concept of functions is an integral part of mathematics, and it is often introduced as a part of the concept of sets and functions. The addition of the study of functions in the 2023 Mathematics curriculum aligns our curriculum with O levels and provides students with a comprehensive understanding of this fundamental mathematical concept. By studying functions, students develop their critical thinking and problemsolving skills and are wellprepared for further study in higherlevel mathematics courses. 

VOLUME AND SURFACE AREA 

Carry out calculations and solve problems involving perimeters and areas of: • compound shapes • parts of shapes. Carry out calculations and solve problems involving surface areas and volumes of: • compound solids • parts of solids. 
Volume and Surface Area Solve problems involving the volume and surface area of solids including • right cones • right cylinders • right prisms • right pyramids • spheres Determine the surface area of composite 3D objects to solve problems 
Concept of Grade 9 Sketch a diagram to represent a problem that involves surface area or volume. Determine the surface area (may be expressed with scientific notation) of a right cone, right cylinder, right prism, right pyramid, or sphere, using an object or its labelled diagram. Determine the volume (may be expressed with scientific notation) of a right cone, right cylinder, right prism, right pyramid, or sphere, using an object or its labelled diagram. Determine an unknown dimension of a right cone, right cylinder, right prism, right pyramid, or sphere, given the object’s surface area or volume and the remaining dimensions. Solve a contextual problem that involves surface area or volume, given a diagram of a composite 3D object. Describe the relationship between the volumes of right cones and right cylinders with the same base and height right pyramids and right prisms with the same base and height 
How do you find the surface area and volume of a solid? How do the surface areas andvo lumes of similar solids compare? 
The study of volume and surface area of composite figures is a crucial part of the mathematics curriculum. In the 2023 curriculum, this topic is included, as it builds on the concepts covered in previous grades, especially in Grade 8. However, reinforcement of the concepts and the solving of advanced level problems in Grade 9 further enhance the students' understanding of the topic.
Composite figures are made up of simple shapes such as cylinders, cones, spheres, and prisms, and finding their volume and surface area requires an understanding of the properties of these shapes. In Grade 8, students learn about the basic properties of these shapes and how to find their volume and surface area. However, in Grade 9, they build on this knowledge and learn how to find the volume and surface area of more complex shapes made up of these simple shapes.
By studying the volume and surface area of composite figures in Grade 9, students can reinforce their understanding of the properties of simple shapes and how they can be combined to form more complex shapes. They also develop their critical thinking and problemsolving skills by applying these concepts to solve more advanced level problems.
Moreover, the study of the volume and surface area of composite figures is essential in various fields, including engineering, architecture, and construction. By studying this topic, students are better prepared for further study in these fields and are equipped with the necessary skills to solve realworld problems.
In conclusion, the study of the volume and surface area of composite figures is included in the 2023 mathematics curriculum as it is the part of the O levels syllabus, building on the concepts covered in Grade 8. Reinforcement of these concepts and the solving of advanced level problems in Grade 9 further enhance the students' understanding of the topic and develop their critical thinking and problemsolving skills. The study of this topic is also essential for further study in various fields and the solving of realworld problems 

SIMILAR FIGURES 

Similarity Calculate lengths of similar shapes. Use the relationships between lengths and areas of similar shapes and lengths, surface areas and volumes of similar solids. Solve problems and give simple explanations involving similarity 
Similar Figures Demonstrate an understanding of similarity of polygons Area and Volume of Similar Figures Solve problems using the relationship between areas of similar figures and volume of different solids 
Concept of Grade 9 Determine if the polygons in a presorted set are similar, and explain the reasoning. Draw a polygon similar to a given polygon, and explain why the two are similar. Solve a problem using the properties of similar polygons. 
How can you use proportions to find side lengths in similar polygons? How can you identify similar polygons and triangles? 
The basic concepts of similar and congruent figures are introduced in Grade 8 in the mathematics curriculum. Students learn about the properties of these figures and how to identify them. These concepts are essential in various fields, including science, engineering, architecture, and art.
However, in Grade 9, students are introduced to more advanced level problems related to similar and congruent figures. They learn how to apply these concepts to solve complex problems that involve multiple figures and require a deeper understanding of geometry.
By solving advanced level problems related to similar and congruent figures, students can enhance their understanding of these concepts and their ability to apply them in different situations. They develop critical thinking and problemsolving skills that are essential in various fields and prepare them for further study in higherlevel mathematics courses.
Furthermore, the study of similar and congruent figures helps students develop their spatial reasoning skills, which are essential in many fields, including science, engineering, and architecture. By studying these concepts, students learn how to think spatially and visualize geometric figures, which is an important skill in many realworld situations.
In conclusion, the basic concepts of similar and congruent figures are introduced in Grade 8, but solving advanced level problems in Grade 9 further enhances the students' understanding of these concepts. It helps them develop critical thinking and problemsolving skills, prepares them for further study in higherlevel mathematics courses, and develops their spatial reasoning skills. 

BASIC STATISTICS 
Frequency Distribution i) Construct grouped frequency table. ii) Construct histograms with equal and unequal class intervals. iii) Construct a frequency polygon. Cumulative Frequency Distribution i) Construct a cumulative frequency table. ii) Draw a cumulative frequency polygon. Measures of Central Tendency i) Calculate (for ungrouped and grouped data): • arithmetic mean by definition and using deviations from assumed mean, • median, mode, geometric mean, harmonic mean. ii) Recognize properties of arithmetic mean. 
Classifying statistical data Classify and tabulate statistical data. e.g. tally tables, twoway tables. Interpreting statistical data · Read, interpret and draw inferences from tables and statistical diagrams. · Compare sets of data using tables, graphs and statistical measures. · Appreciate restrictions on drawing conclusions from given data. e.g. compare averages and measures of spread between two data sets. Averages and measures of spread · Calculate the mean, median, mode and range for individual data and distinguish between the purposes for which these are used. · Calculate an estimate of the mean for grouped discrete or grouped continuous data. · Identify the modal class from a grouped frequency distribution. Statistical charts and diagrams Draw and interpret: (a) bar charts (b) pie charts (c) pictograms (d) simple frequency distributions 
Frequency Distribution • Construct a grouped frequency table, histogram(with unequal class intervals) and frequency polygon Measure of Central Tendency • Calculate the mean modal class and median of a grouped frequency distribution • Solve real life situations involving mean, range, median, and mode for given data 
Determine the possible graphs that can be used to represent a data set, and explain the advantages and disadvantages of each. Create, with or without technology, a graph to represent a data set. Describe the trends in the graph of a data set. Interpolate or extrapolate values from a graph. Explain, using examples, how the same graph can be used to justify more than one conclusion. Explain, using examples, how different graphic representations of the same data set can be used to emphasize a point of view. Solve a contextual problem that involves the interpretation of a graph. Explain, using examples, the advantages and disadvantages of each measure of central tendency. Determine the mean, median, and mode for a set of data. Identify and correct errors in a calculation of a measure of central tendency. Identify the outlier(s) in a set of data. Explain the effect of outliers on mean, median, and mode. Explain, using examples such as course marks, why some data in a set would be given a greater weighting in determining the mean. Determine the weighted mean of a set of data, and justify the different weightings. Explain, using examples from print or other media, how measures of central tendency and outliers are used to provide different interpretations of data. Solve a contextual problem that involves measures of central tendency 
When shoud you use mean, median and mode? 
The inclusion of cumulative frequency distribution and measures of central tendency in the mathematics curriculum is based on several rationales, including:
Realworld applications: Cumulative frequency distribution and measures of central tendency have numerous applications in various fields, including finance, science, engineering, and social sciences. Understanding these concepts is essential for analyzing and interpreting data in realworld situations.
Development of critical thinking and problemsolving skills: The study of cumulative frequency distribution and measures of central tendency requires students to use critical thinking and problemsolving skills to analyze and interpret data. It encourages students to think logically, critically, and creatively and to make informed decisions based on evidence.
Connection to other mathematical concepts: Cumulative frequency distribution and measures of central tendency are connected to many other mathematical concepts, including statistics, algebra, and geometry. By studying these concepts, students develop a deeper understanding of these concepts and how they can be applied in different situations.
Preparation for further study: Cumulative frequency distribution and measures of central tendency are essential topics in many fields, including science, engineering, and social sciences. By studying these topics in high school, students are better prepared for further study in these fields.
Continuation of concepts introduced in Grades 68: The introduction of cumulative frequency distribution and measures of central tendency in Grades 912 is a continuation of the concepts introduced in Grades 68. In earlier grades, students learn about basic concepts of data representation, such as frequency distribution tables and bar graphs. In Grades 912, students build on these concepts and learn more advanced topics, such as cumulative frequency distribution, mean, median, and mode.
In summary, the inclusion of cumulative frequency distribution and measures of central tendency in the mathematics curriculum is essential for the development of critical thinking and problemsolving skills, preparation for further study in various fields, connection to other mathematical concepts, realworld applications, and a continuation of concepts introduced in Grades 68. By studying these concepts, students develop a deeper understanding of mathematics and its applications and are better prepared for the challenges of the 21st century 

PROBABILITY 

Understand and use the probability scale from 0 to 1. Understand and use probability notation. Calculate the probability of a single event. Understand that the probability of an event not occurring = 1 – the probability of the event occurring. Relative and expected frequencies Understand relative frequency as an estimate of probability. Calculate expected frequencies. Probability of combined events Calculate the probability of combined events using,where appropriate: • sample space diagrams • Venn diagrams • tree diagrams. 
Probability • Calculate the probability of a single event and the probability of event not occurring Relative and expected frequencies • Calculate relative frequency as an estimate of probability. • Calculate expected frequencies. 
Provide an example from print and electronic media (e.g., newspapers, the Internet) where probability is used. Identify the assumptions associated with a given probability, and explain the limitations of each assumption. Explain how a single probability can be used to support opposing positions. Explain, using examples, how decisions based on probability may be a combination of theoretical probability, experimental probability, and subjective judgment. 

The inclusion of probability in the mathematics curriculum is based on several rationales, including:
Realworld applications: Probability has numerous applications in various fields, including finance, science, engineering, and social sciences. Understanding probability is essential for making informed decisions in realworld situations, such as calculating the risk of a medical procedure or evaluating the potential return on an investment.
Development of critical thinking and problemsolving skills: The study of probability requires students to use critical thinking and problemsolving skills to analyze and interpret data. It encourages students to think logically, critically, and creatively and to make informed decisions based on evidence.
Connection to other mathematical concepts: Probability is connected to many other mathematical concepts, including statistics, algebra, and geometry. By studying probability, students develop a deeper understanding of these concepts and how they can be applied in different situations.
Preparation for further study: Probability is an essential topic in many fields, including science, engineering, and social sciences. By studying probability in high school, students are better prepared for further study in these fields.
Continuation of concepts introduced in Grades 68: The introduction of probability in Grades 912 is a continuation of the concepts introduced in Grades 68. In earlier grades, students learn about basic probability concepts, such as finding the probability of an event using fractions or decimals. In Grades 912, students build on these concepts and learn more advanced probability topics, such as conditional probability and probability distributions.
In summary, the inclusion of probability in the mathematics curriculum is essential for the development of critical thinking and problemsolving skills, preparation for further study in various fields, connection to other mathematical concepts, realworld applications, and a continuation of concepts introduced in Grades 68. By studying probability, students develop a deeper understanding of mathematics and its applications and are better prepared for the challenges of the 21st century. 

LINEAR GRAPHS AND THEIR APPLICATION 
Cartesian Plane and Linear Graphs i) Identify pair of real numbers as an ordered pair. ii) Recognize an ordered pair through different examples; for instance an ordered pair (2,3) to represent a seat, located in an examination hall, at the 2nd row and 3rd column. iii) Describe rectangular or Cartesian plane consisting of two number lines intersecting at right angles at the point O. iv) Identify origin (O) and coordinate axes (horizontal and vertical axes or xaxis and yaxis) in the rectangular plane. v) Locate an ordered pair (a, b) as a point in the rectangular plane and recognize: • a as the xcoordinate (or abscissa), • b as the ycoordinate (or ordinate). vi) Draw different geometrical shapes (e.g., line segment, triangle and rectangle etc) by joining a set of given points. vii) Construct a table for pairs of values satisfying a linear equation in two variables. viii) Plot the pairs of points to obtain the graph of a given expression. ix) Choose an appropriate scale to draw a graph. x) Draw the graph of • an equation of the form y = c . • an equation of the form x = a . • an equation of the form y = mx . • an equation of the form y = mx + c . xi) Draw a graph from a given table of (discrete) values. xii) Solve appropriate reallife problems. Conversion Graphs i) Interpret conversion graph as a linear graph relating to two quantities which are in direct proportion. ii) Read a given graph to know one quantity corresponding to another. iii) Read the graph for conversions of the form: • miles and kilometers, • acres and hectares, • degrees Celsius and degrees Fahrenheit, • Pakistani currency and another currency, etc. Graphic Solution of Equations in two Variables Solve simultaneous linear equations in two variables using graphical method. 
Graphs in practical situations 1 Use and interpret graphs in practical situations including travel graphs and conversion graphs. 2 Draw graphs from given data. 3 Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and deceleration. 4 Calculate distance travelled as area under a speed–time graph Graphs of functions Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms: • axn (includes sums of no more than three of these) • abx + c where n = –2, –1, − 1/2, 0, 1/2 , 1, 2, 3; a and c are rational numbers; and b is a positive integer. Solve associated equations graphically, including finding and nterpreting roots by graphical methods. Draw and interpret graphs representing exponential growth and decay problems. 4 Estimate gradients of curves by drawing tangents. 
Graphs in real life and scientific situations Make up graph of a data collected from a reallife situation. Interpret the reallife situation from graphical representation of data. Interpret graphs in practical situations such as travel graphs and conversion graphs. Interpret graphs in scientific situations such as kinematics: distancetime graphs, speedtime graph and acceleration/deceleration. Identify the distance travelled as the area under a speedtime graph. Plotting and Interpreting the Graphs Select functions of the form: ax^n (including the sums of few of these and taking n as a rational number), ab^x+c (taking a, b and c as positive integers) Solve linear and quadratic equations graphically and interpret their solution. Solve simple systems of linear or one linear and one quadratic graphically and interpret their solution. Discover exponential growth/decay of a practical phenomenon through its graph. Determine the gradients of curves through tangents. Curves sketching Identify, sketch and interpret graphs of the functions: linear functions (e.g. y=ax +b ), nonlinear functions (e.g. y = x 2 ). Sketch the graph of the function y = xn where n is (a + ve integer, a  ve integer ( x 0 ), a rational number for x > 0). Sketch graph of quadratic function of the form y ax2 + bx + c , a( 0) , b, c are integers. Cubic Reciprocal Exponential Logarithmic 
Concept of Grade 10 Identify and describe the characteristics of a linear relation represented in a graph, table of values, number pattern, or equation. Sort a set of graphs, tables of values, number patterns, or equations into linear and nonlinear relations. Write an equation for a context involving direct or partial variation. Create a table of values for an equation of a linear relation. Sketch the graph for a table of values. Explain why the points should or should not be connected on the graph for a context. Create, with or without technology, a graph to represent a data set. Describe the trends in the graph of a data set. Sort a set of scatterplots according to the trends represented (linear, nonlinear, or no trend). Solve a contextual problem that requires interpolation or extrapolation. Relate slope and rate of change to linear relations. Match contexts with their corresponding graphs, and explain the reasoning. Solve a contextual problem involving the application of a formula for a linear relation. 

Linear graphs are a fundamental topic in mathematics and have numerous applications in various fields. The inclusion of linear graphs in the mathematics curriculum is based on several rationales, including:
Developing essential mathematical skills: Linear graphs require students to understand basic concepts such as plotting points on a coordinate plane, finding slope, and identifying intercepts. These skills are essential for many other mathematical topics and are important for students to develop a strong foundation in mathematics.
Problemsolving and analytical skills: The study of linear graphs requires students to analyze realworld situations and problems and determine the most effective way to represent the data. This process develops students' problemsolving and analytical skills and encourages them to think critically.
Applications in Science, Technology, Engineering, and Mathematics (STEM): Linear graphs have numerous applications in various fields, including physics, engineering, economics, and computer science. By studying linear graphs, students develop an understanding of the mathematical principles that underlie many realworld problems and gain insight into how these problems can be solved.
Communication and representation of data: Linear graphs provide an efficient and concise way to represent and communicate data. This skill is essential for many professions, including business, finance, and science.
Continuation of graph concepts taught in Grades 68: Linear graphs are a continuation of the graph concepts introduced in Grades 68. By building on these concepts, students develop a deeper understanding of the relationship between variables and how they can be represented graphically.
In summary, the inclusion of linear graphs in the mathematics curriculum is essential for the development of essential mathematical skills, problemsolving and analytical skills, applications in STEM, communication and representation of data, and a continuation of the graph concepts taught in Grades 68. By studying linear graphs, students develop a strong foundation in mathematics and gain insight into how mathematics can be applied to realworld problems 

INTRODUCTION TO COORDINATE GEOMETRY 
Distance Formula i) Define coordinate geometry. ii) Derive distance formula to calculate distance between two points given in Cartesian plane. iii) Use distance formula to find distance between two given points. Collinear Points i) Define collinear points. Distinguish between collinear and noncollinear points. ii) Use distance formula to show that given three (or more) points are collinear. iii) Use distance formula to show that the given three noncollinear points form: • an equilateral triangle, • an isosceles triangle, • a right angled triangle, • a scalene triangle. iv) Use distance formula to show that given four noncollinear points form: • a square, • a rectangle, • a parallelogram. Midpoint Formula i) Recognize the formula to find the midpoint of the line joining two given points. ii) Apply distance and mid point formulae to solve/verify different standard results related to geometry. 
Coordinates Use and interpret Cartesian coordinates in two dimensions. Drawing linear graphs Draw straightline graphs for linear equations. Gradient of linear graphs Find the gradient of a straight line. Calculate the gradient of a straight line from the coordinates of two points on it. Length and midpoint Calculate the length of a line segment. Find the coordinates of the midpoint of a line segment. Equations of linear graphs Interpret and obtain the equation of a straightline graph Parallel lines Find the gradient and equation of a straight line parallel to a given line. Perpendicular lines Find the gradient and equation of a straight line perpendicular to a given line. 
Coordinate geometry Derive distance formula by locating the position of two points in coordinate plane Calculate the midpoint of a line segment Find the gradient of a straight line when coordinates of two points are given Find the equation of a straight line in the form y =mx +c Find the gradient of parallel and perpendicular lines Apply distance and midpoint formulas to solve real life situations 
Concept of grade 9 Determine the distance between two points on a Cartesian plane, using a variety of strategies. Determine the midpoint of a line segment, given the endpoints of the segment, using a variety of strategies. Determine an endpoint of a line segment, given the other endpoint and the midpoint, using a variety of strategies. Solve a contextual problem involving distance between two points or midpoint of a line segment 

Coordinate geometry is a branch of mathematics that uses algebraic techniques to study geometry. It is a very powerful tool for solving problems related to geometry, and it has numerous applications in various fields, including engineering, physics, computer science, and economics. The concept of coordinate geometry has been a part of the mathematics curriculum for many years, and it is still included in the 2023 curriculum for several reasons.
Firstly, coordinate geometry provides a way to represent geometric figures and problems in a systematic and algebraic way. This representation enables us to use algebraic techniques to solve geometric problems that may be difficult or impossible to solve using traditional geometric methods. For example, we can use the Pythagorean theorem to find the distance between two points in the coordinate plane, or we can use the equation of a line to find the point of intersection of two lines.
Secondly, coordinate geometry provides a way to visualize and analyze geometric objects in higher dimensions. By extending the concept of coordinates beyond the twodimensional plane, we can represent and analyze objects in three, four, or more dimensions. This ability to work in higher dimensions is crucial for many fields, including physics, computer graphics, and machine learning.
Thirdly, coordinate geometry is an essential tool for studying calculus. The connection between calculus and coordinate geometry is fundamental, as it allows us to study rates of change and optimization problems in a geometric context. In calculus, we often use the tools of coordinate geometry to find the derivatives and integrals of functions that describe geometric objects.
Finally, the inclusion of coordinate geometry in the mathematics curriculum helps students develop their problemsolving skills and logical reasoning abilities. Through the study of coordinate geometry, students learn to think abstractly, logically, and critically. They develop an understanding of the relationship between algebra and geometry, which is essential for success in many fields.
In summary, the concept of coordinate geometry has been included in the mathematics curriculum for many years and continues to be an essential part of the curriculum in 2023. This is because it provides a powerful tool for solving problems related to geometry, allows us to visualize and analyze objects in higher dimensions, is fundamental for studying calculus, and helps students develop their problemsolving skills and logical reasoning abilities. 

INTRODUCTION TO TRIGONOMETRY 
Measurement of an Angle i) Measure an angle in sexagesimal system (degree, minute and second). ii) Convert an angle given in D M ′S′′ o form into a decimal form and vice versa. iii) Define a radian (measure of an angle in circular system) and prove the relationship between radians and degrees. Sector of a Circle i) Establish the rule l = rθ , where r is the radius of the circle, l the length of circular arc and θ the central angle measured in radians. ii) Prove that the area of a sector of a circle is θ. Trigonometric Ratios i) Define and identify: • general angle (coterminal angles), • angle in standard position. ii) Recognize quadrants and quadrantal angles. iii) Define trigonometric ratios and their reciprocals with the help of a unit circle. iv) Recall the values of trigonometric ratios for 45 , 30 ,60 . v) Recognize signs of trigonometric ratios in different quadrants. vi) Find the values of remaining trigonometric ratios if one trigonometric ratio is given. vii) Calculate the values of trigonometric ratios for 0 , 90 , 180 , 270 , 360 . Trigonometric Identities Prove the trigonometric identities and apply them to show different trigonometric relations. Angle of Elevation and Depression. i) Find angle of elevation and depression. ii) Solve real life problems involving angle of elevation and depression. 
Know and use Pythagoras’ theorem Know and use the sine, cosine and tangent ratios for acute angles in calculations involving sides and angles of a rightangled triangle. Solve problems in two dimensions using Pythagoras’ theorem and trigonometry. Know that the perpendicular distance from a point to a line is the shortest distance to the line. Carry out calculations involving angles of elevation and depression 
Trigonometry Demonstrate an understanding of angles in standard position, expressed in degrees and radians Apply Pythagoras’ theorem and the sine, cosine and tangent ratios for acute angles to find a side or of an angle of a rightangled triangle. Solve trigonometrical problems in two dimensions involving angles of elevation and depression Apply the trigonometric identities to show different trigonometric relations 
Concept of Grade 9 Sketch an angle in standard position, given the measure of the angle. Determine the reference angle for an angle in standard position. Explain, using examples, how to determine the angles from 0° to 360° that have the same reference angle as a given angle. Illustrate, using examples, that any angle from 90° to 360° is the reflection in the xaxis and/or the yaxis of its reference angle. Determine the quadrant in which an angle in standard position terminates. Draw an angle in standard position given any point P (x, y) on the terminal arm of the angle. Illustrate, using examples, that the points P (x, y), P (–x, y), P (–x, –y), and P (x, –y) are points on the terminal sides of angles in standard position that have the same reference angle. Explain, using illustrations, why the Pythagorean theorem only applies to right triangles. Describe historical and contemporary applications of the Pythagorean theorem. Determine if a triangle is a right triangle or if an angle is 90° using the Pythagorean theorem. Solve a problem using the Pythagorean theorem. Explain the primary trigonometric ratios. Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given acute angle in the triangle. Solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem. Solve a problem that involves direct and indirect measurement, using measurement instruments such as a clinometer or metre stick, the trigonometric ratios, or the Pythagorean theorem Identify situations where the trigonometric ratios are used for indirect measurement of angles and lengths. Solve a contextual problem that involves right triangles, using the primary trigonometric ratios. Explain or verify that a solution to a problem involving primary trigonometric ratios is reasonable. 
How do you describe angles? How do you use degree measure? How do you use radian measure? How do you convert between degree and radian measures? How do you use angles to model and solve real life problems? How do you evaluate trigonometric functions of acute angles? How do you use the fundamental trigonometric identities? How do you use a calculator to evaluate trigonometric functions? How do you use trigonometric functions to model and solve real life problems? 
Trigonometry is an essential branch of mathematics that deals with the study of triangles, their sides, and angles. It has numerous practical applications, including in engineering, physics, astronomy, and navigation. The concept of trigonometry was introduced in the 2006 curriculum to provide students with a strong foundation in this subject area and to prepare them for future studies in science and mathematics. Over the years, the concept of trigonometry has become increasingly important, as new technologies and scientific discoveries have expanded its applications. As a result, it remains a fundamental topic in the study of mathematics and science, and it is critical that students continue to learn and understand the principles of trigonometry. The 2023 curriculum builds upon the concepts introduced in the 2006 curriculum and includes more advanced topics such as inverse trigonometric functions, solving trigonometric equations, and the applications of trigonometry in reallife situations. By continuing to teach and emphasize the importance of trigonometry in the 2023 curriculum, we are providing students with a strong foundation in this subject area and preparing them for future success. The introduction to trigonometry was already a part of the 2006 curriculum, and it remains an essential component of the 2023 curriculum. By providing students with a solid understanding of the principles of trigonometry, we are preparing them for future studies in science and mathematics and giving them the tools they need to succeed in their future academic and professional pursuits. 

BEARING 
not in 2006 
interpret and use threefigure bearings 
interpret and use threefigure bearings 
concept of grade 9 

Bearing is a fundamental concept in mathematics, geography, and navigation. It is the direction or angle between two points, measured in degrees, with reference to the north, south, east, or west. The concept of bearing is used extensively in the field of geography, in surveying, and in navigation. Introducing learners to the concept of bearing at this age provides them with a solid foundation in this topic, which will be helpful in their future studies and in practical situations. In summary, introducing the concept of bearing to learners, with a focus on O level students who are already familiar with this topic, provides a strong foundation in an important mathematical and geographical concept. It also enables learners to develop critical thinking skills and problemsolving abilities that are essential for success in their future studies and practical applications. 

APPLICATION TO TRIGONOMETRY 

Use the sine and cosine rules in calculations involving lengths and angles for any triangle. Use the formula area of triangle = 1/2 ab sinC . Carry out calculations and solve problems in three dimensions using Pythagoras’ theorem and trigonometry, including calculating the angle between a line and a plane 
Application of Trigonometry interpret and use threefigure bearings • apply Pythagoras’ theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or of an angle of a rightangled triangle • solve trigonometrical problems in two dimensions involving angles of elevation and depression • extend sine and cosine functions to angles between 90° and 180° • solve problems using the sine and cosine rules for any triangle and the formula area of triangle = 1/2 ab sin C • solve simple trigonometrical problems in three dimensions 
Concept of grade 10 Sketch a diagram to represent a problem that involves a triangle without a right angle. Solve a nonright triangle using right triangle methods. Explain the steps in a given proof of the sine law or cosine law. Sketch a diagram and solve a contextual problem, using the cosine law. Sketch a diagram and solve a contextual problem, using the sine law. Describe and explain ambiguous case problems that may have no solution, one solution, or two solutions. 

The concept of Application to Trigonometry has been moved to Grade 10 from grade 11 as it is included in the Cambridge O Level Mathematics curriculum. This means that students who are studying O Level Mathematics will now be introduced to trigonometry and its practical applications in Grade 10. Trigonometry is an important branch of mathematics that deals with the relationships between angles and the sides of triangles, and has many practical applications in fields such as engineering, physics, and navigation. By studying trigonometry, students can develop important skills in problemsolving, critical thinking, and mathematical reasoning, which can be useful in a wide range of academic and professional contexts. 

CONGRUENT TRIANGLES 
Congruent Triangles Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) In any correspondence of two triangles, if one side and any two angles of one triangle are congruent to the corresponding side and angles of the other, the two triangles are congruent. ii) If two angles of a triangle are congruent then the sides opposite to them are also congruent. iii) In a correspondence of two triangles, if three sides of one triangle are congruent to the corresponding three sides of the other, the two triangles are congruent. iv) If in the correspondence of two rightangled triangles, the hypotenuse and one side of one are congruent to the hypotenuse and the corresponding side of the other, then the triangles are congruent. 
solve problems and give simple explanations involving similarity and congruence 

Already covered in grade 8 

Already covered in Grade 8 

PARALLELOGRAMS AND TRIANGLES 
Parallelograms and Triangles Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) In a parallelogram: • the opposite sides are congruent, • the opposite angles are congruent, • the diagonals bisect each other. ii) If two opposite sides of a quadrilateral are congruent and parallel, it is a parallelogram. iii) The line segment, joining the midpoints of two sides of a triangle, is parallel to the third side and is equal to one half of its length. iv) The medians of a triangle are concurrent and their point of concurrency is the point of trisection of each median. v) If three or more parallel lines make congruent intercepts on a transversal they also intercept congruent segments on any other line that cuts them. 
1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • cointerior (supplementary) angles sum to 180°. 3 Know and use angle properties of regular and irregular polygons 
Geometrical Properties of Parallelograms ,Triangles and Regular Polygons Solve problems that involve the properties of parallelogram,triangles and regular polygons 
Grade 9 concept Describe and illustrate properties of triangles, including isosceles or equilateral. Describe and illustrate properties of quadrilaterals in terms of angle measures, side lengths, diagonals, or angles of intersection. Describe and illustrate properties of regular polygons. Explain, using examples, why a given property does or does not apply to certain polygons. Identify and explain an application of the properties of polygons in construction, industrial, commercial, domestic, or artistic contexts. Solve a contextual problem that involves the application of the properties of polygons 
What is the name of regular triangle and of a regular quadrilateral? How can you find the sum of the measures of polygon angles? How can you classify quadrilaterals? How can you use coordinate geometry to prove general relationships? 
The Grade 9 mathematics curriculum now includes the application of properties of triangles and quadrilaterals, in addition to traditional theorem proof methods. In previous years, the focus of the curriculum was primarily on teaching students to prove theorems using various geometric properties. While this is an important skill, we recognize that there are other equally important mathematical concepts that should be introduced to students as well. The application of properties of triangles and quadrilaterals is a key area of study in geometry that can greatly benefit students in a variety of ways. For example, it can help them to understand and solve practical problems related to reallife situations, such as calculating the area and perimeter of irregular shapes. Moreover, the application of properties can also help students to develop their problemsolving and critical thinking skills, as they work to identify and apply the relevant properties to solve a given problem. 

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 = ba. 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: . 
Shifted to Grade 10 Vectors in Plane Recognize rectangular coordinate system in plane. Represent vectors as directed line segments express a vector in terms of two nonzero and nonparallel 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

Why are functions and relations represented by vectors? 
Shifted to Grade 10 as it is the part of O levels 

LINE BISECTORS AND ANGLE BISECTORS 
Line Bisectors and Angle Bisectors Prove the following theorems along with corollaries and apply them to solve appropriate problems. ) Any point on the right bisector of a line segment is equidistant from its end points. ii) Any point equidistant from the points of a line segment is on the right bisector of it. iii) The right bisectors of the sides of a triangle are concurrent. iv) Any point on the bisector of an angle is equidistant from its arms. v) Any point inside an angle, equidistant from its arms, is on the bisector of it. vi) The bisectors of the angles of a triangle are concurrent. 


Concept of grade 8 

Line and angle bisector has been covered in grade 8 

SIDES AND ANGLES OF A TRIANGLE 
Sides and Angles of a Triangle Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) If two sides of a triangle are unequal in length, the longer side has an angle of greater measure opposite to it. ii) If two angles of a triangle are unequal in measure, the side opposite to the greater angle is longer than the side opposite to the smaller angle. iii) The sum of the lengths of any two sides of a triangle is greater than the length of the third side. iv) From a point, outside a line, the perpendicular is the shortest distance from the point to the line. 

Solve problems that involve the properties of parallelogram and triangles In a parallelogram: • the opposite sides are congruent, • the opposite angles are congruent, • the diagonals bisect each other. If two opposite sides of a quadrilateral are congruent and parallel, it is a parallelogram. If two sides of a triangle are unequal in length, if and only if the longer side has an angle of greater measure opposite to it. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. From a point, outside a line,the perpendicular is the shortest distance from the point to the line. A line parallel to one side of a triangle, intersecting the other two sides, divides them proportionally. If a line segment intersects the two sides of a triangle in the same ratio then it is parallel to the third side. The internal bisector of an angle of a triangle divides the side opposite to it in the ratio of the lengths of the sides containing the angle. 
Concept of Grade 9 Describe and illustrate properties of triangles, including isosceles or equilateral. Describe and illustrate properties of quadrilaterals in terms of angle measures, side lengths, diagonals, or angles of intersection. Describe and illustrate properties of regular polygons. Explain, using examples, why a given property does or does not apply to certain polygons. Identify and explain an application of the properties of polygons in construction, industrial, commercial, domestic, or artistic contexts. Solve a contextual problem that involves the application of the properties of polygons. 

Included as application of theorems and properties 

RATIO AND PROPORTION 
Ratio and Proportion Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) A line parallel to one side of a triangle, intersecting the other two sides, divides them proportionally. ii) If a line segment intersects the two sides of a triangle in the same ratio then it is parallel to the third side. iii) The internal bisector of an angle of a triangle divides the side opposite to it in the ratio of the lengths of the sides containing the angle. iv) If two triangles are similar, the measures of their corresponding sides are proportional. 


Coverd in grade 8 

Already covered in Grade 8 

PYTHAGORAS’ THEOREM 
Pythagoras’ Theorem Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) In a rightangled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides. (Pythagoras’ theorem). ii) If the square of one side of a triangle is equal to the sum of the squares of the other two sides then the triangle is a right angled triangle (converse to Pythagoras’ theorem). 
Know and use Pythagoras’ theorem. 

Covered in grade 8 

included as part of trigonometry 

THEOREMS RELATED WITH AREA 
Theorems Related with Area Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) Parallelograms on the same base and lying between the same parallel lines (or of the same altitude) are equal in area. ii) Parallelograms on equal bases and having the same altitude are equal in area. iii) Triangles on the same base and of the same altitude are equal in area. iv) Triangles on equal bases and of the same altitude are equal in area. 
Not in O levels 



Not included 

PROJECTION OF A SIDE OF A TRIANGLE 
Projection of a Side of a Triangle Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) In an obtuseangled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing the obtuse angle together with twice the rectangle contained by one of the sides, and the projection on it of the other. ii) In any triangle, the square on the side opposite to an acute angle is equal to the sum of the squares on the sides containing that acute angle diminished by twice the rectangle contained by one of those sides and the projection on it of the other. iii) In any triangle, the sum of the squares on any two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side (Apollonius’ Theorem). 
Not in O levels 



Not included separately will be the part of Application of Trigonometry sine rule cosine rule etc 

CHORDS OF A CIRCLE 
Chords of a Circle Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) One and only one circle can pass through three non collinear points. ii) A straight line, drawn from the centre of a circle to bisect a chord (which is not a diameter) is perpendicular to the chord. iii) Perpendicular from the centre of a circle on a chord bisects it. iv) If two chords of a circle are congruent then they will be equidistant from the centre. v) Two chords of a circle which are equidistant from the centre are congruent. 
Circle theorems I Calculate unknown angles and give explanations using the following geometrical properties of circles: • angle in a semicircle = 90° • angle between tangent and radius = 90° • angle at the centre is twice the angle at the circumference • angles in the same segment are equal • opposite angles of a cyclic quadrilateral sum to 180° (supplementary) • alternate segment theorem. 
Solve problems by using the following properties of circle: One and only one circle can pass through three non collinear points. A straight line, drawn from the centre of a circle to bisect a chord (which is not a diameter) is perpendicular to the chord. Perpendicular from the centre of a circle on a chord bisects it. If two chords of a circle are congruent then they will be equidistant from the centre. Two chords of a circle which are equidistant from the centre are congruent. If two arcs of a circle (or of congruent circles) are congruent then the corresponding chords are equal. If two chords of a circle (or of congruent circles) are equal, then their corresponding arcs (minor, major or semicircular) are congruent. Equal chords of a circle (or of congruent circles) subtend equal angles at the centre (at the corresponding centres). If the angles subtended by two chords of a circle (or congruent circles) at the centre (corresponding centres) are equal, the chords are equal. The measure of a central angle of a minor arc of a circle, is double that of the angle subtended by the corresponding major arc. Any two angles in the same segment of a circle are equal. The angle • in a semicircle is a right angle, • in a segment greater than a semicircle is less than a right angle, • in a segment less than a semicircle is greater than a right angle. The opposite angles of any quadrilateral inscribed in a circle are supplementary. 
Concept of grade 10 
How can you prove relationships between angles and arcs in a circle? When lines intersect a circle,or within a circle, how do you find the measures of resulting angles, arcs, and segments? How do you find the equation of a circle in the coordinate plane? 
included in 2023 curriculum 

TANGENT TO A CIRCLE 
Tangent to a Circle Prove the following theorems along with corollaries and apply them to solve appropriate problems. i) If a line is drawn perpendicular to a radial segment of a circle at its outer end point, it is tangent to the circle at that point. ii) The tangent to a circle and the radial segment joining the point of contact and the centre are perpendicular to each other. iii) The two tangents drawn to a circle from a point outside it, are equal in length. iv) If two circles touch externally or internally, the distance between their centres is respectively equal to the sum or difference of their radii.


Tangent to a Circle Solve problems by applying the following properties of circle: If a line is drawn perpendicular to a radial segment of a circle at its outer end point, it is tangent to the circle at that point. The tangent to a circle and the radial segment joining the point of contact and the centre are perpendicular to each other. The two tangents drawn to a circle from a point outside it, are equal in length. If two circles touch externally or internally, the distance between their centres is respectively equal to the sum or difference of their radii. 
Concept of grade 10 

included in 2023 curriculum 

PRACTICAL GEOMETRY – TRIANGLES 
Construction of Triangle i) Construct a triangle having given: • two sides and the included angle, • one side and two of the angles, • two of its sides and the angle opposite to one of them (with all the three possibilities). ii) Draw: • angle bisectors, • altitudes, • perpendicular bisectors, • medians, of a given triangle and verify their concurrency. Figures with Equal Areas i) Construct a triangle equal in area to a given quadrilateral. ii) Construct a rectangle equal in area to a given triangle. iii) Construct a square equal in area to a given rectangle. iv) Construct a triangle of equivalent area on a base of given length. 

Construct a triangle having given: • two sides and the included angle, • one side and two of the angles, • two of its sides and the angle opposite to one of them (with all the three possibilities). Draw: • angle bisectors, • altitudes, • perpendicular bisectors, • medians, of a given triangle and verify their concurrency. 
Concept of grade 9 

With some little changes included in grade 9 of curriculum 2023 

PRACTICAL GEOMETRY – CIRCLES 
Construction of Circle i) Locate the centre of a given circle. ii) Draw a circle passing through three given non_x0002_collinear points. iii) Complete the circle: • by finding the centre, • without finding the centre, when a part of its circumference is given. Construction of Circle i) Locate the centre of a given circle. ii) Draw a circle passing through three given non_x0002_collinear points. iii) Complete the circle: • by finding the centre, • without finding the centre, when a part of its circumference is given. ii) Draw a circle passing through three given non_x0002_collinear points. iii) Complete the circle: • by finding the centre, • without finding the centre, when a part of its circumference is given. Circles attached to Polygon i) Circumscribe a circle about a given triangle. ii) Inscribe a circle in a given triangle. iii) Escribe a circle to a given triangle. iv) Circumscribe an equilateral triangle about a given circle. v) Inscribe an equilateral triangle in a given circle. vi) Circumscribe a square about a given circle. vii) Inscribe a square in a given circle. viii) Circumscribe a regular hexagon about a given circle. ix) Inscribe a regular hexagon in a given circle. Tangent to the Circle i) Draw a tangent to a given arc, without using the centre, through a given point P when P is • the middle point of the arc, • at the end of the arc, • outside the arc. ii) Draw a tangent to a given circle from a point P when P lies • on the circumference, • outside the circle. iii) Draw two tangents to a circle meeting each other at a given angle. iv) Draw • direct common tangent or external tangent, • transverse common tangent or internal tangent to two equal circles. v) Draw • direct common tangent or external tangent, • transverse common tangent or internal tangent to two unequal circles. vi) Draw a tangent to • two unequal touching circles, • two unequal intersecting circles. vii) Draw a circle which touches • both the arms of a given angle, • two converging lines and passes through a given point between them, • three converging lines 

Construction of Circle Locate the centre of a given circle. Draw a circle passing through three given noncollinear points. Complete the circle: • by finding the centre, • without finding the centre, when a part of its circumference is given. Tangent to the Circle Draw a tangent to a given arc, without using the centre, through a given point P when P is • the middle point of the arc, • at the end of the arc, • outside the arc. Draw a tangent to a given circle from a point P when P lies • on the circumference, • outside the circle. Draw two tangents to a circle meeting each other at a given angle. 
Concept of grade 10 
Why is it important to understand geometric constructions? How are the geometric properties of circles, their angles and their arcs used to model and describereal world phenomena? 
With some little changes, included in grade 9 of curriculum 2023 
