IGCSE Math Extended Exhaustive Syllabus

Integers: Integers are whole numbers (positive, negative, or zero) and are used to represent values such as gains and losses, temperatures, and elevations.

Prime numbers: Prime numbers are numbers that have only two factors, one and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, and so on.

Common factors and common multiples: Common factors are numbers that divide evenly into two or more numbers. Common multiples are numbers that are multiples of two or more numbers.

Rational and irrational numbers: Rational numbers are numbers that can be expressed as a ratio of two integers, while irrational numbers cannot. Examples of irrational numbers include pi (π) and the square root of 2.

Real numbers: Real numbers are all the numbers on the number line, including rational, irrational, positive, negative, and zero.

Reciprocals: Reciprocals are numbers that, when multiplied together, result in 1. For example, the reciprocal of 4 is 1/4, and the reciprocal of 1/2 is 2.

Definition of sets: A set is a collection of distinct objects or elements. Sets can be defined by listing the elements between braces or by using set-builder notation. For example, the set A = {1, 2, 3, 4} contains the elements 1, 2, 3, and 4, while the set B = {x | x is a positive integer less than 5} contains the same elements but is defined using set-builder notation. The set C = {a, b, c} contains the elements a, b, and c.

Square roots: The square root of a number is the value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5×5 = 25.

Cubes: Cubing a number means multiplying it by itself three times. For example, the cube of 3 is 3x3x3 = 27.

Cube roots: The cube root of a number is the value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because 3x3x3 = 27.

Other powers and roots: In addition to squares and cubes, numbers can be raised to any power (e.g. to the fourth power, to the fifth power, etc.) and can have any root (e.g. the fourth root, the fifth root, etc.). Calculating with powers and roots involves using the appropriate formulas or rules for each type of operation.

Recognise equivalence and convert between these forms.  

Decimals: A decimal is a way of expressing a number that is not a whole number. It is written with a decimal point and digits after the point. Decimals can be used to represent parts of a whole or to compare quantities.

Percentages: A percentage is a way of expressing a fraction or a decimal as a number out of 100. Percentages can be used to compare quantities, to express a part of a whole, or to show a change over time.

Conversion between fractions, decimals, and percentages: Understanding how to convert between fractions, decimals, and percentages is important for solving problems and working with data. For example, knowing that 50% is the same as 0.5 or 1/2 can be useful when calculating discounts or working with data in graphs.

Applications in real-life situations: The language and notation of fractions, decimals, and percentages are used in many real-life situations, such as cooking recipes, financial calculations, and data analysis. Being able to understand and use these concepts is important for making informed decisions and solving problems in everyday life.

The symbols used to compare quantities are:

• = (equal to): used to indicate that two quantities are the same
• ≠ (not equal to): used to indicate that two quantities are different
• < (less than): used to indicate that one quantity is smaller than another
• (greater than): used to indicate that one quantity is larger than another
• ≤ (less than or equal to): used to indicate that one quantity is smaller than or equal to another
• ≥ (greater than or equal to): used to indicate that one quantity is larger than or equal to another

Understanding how to order quantities by magnitude and use these symbols is important for many mathematical operations, including solving equations and inequalities.

Use the rules of indices: This subtopic involves using the rules of indices to simplify expressions involving powers and roots, and to solve equations involving indices. This includes understanding how to use the product, quotient, power, and root rules of indices, as well as understanding how to apply these rules in various contexts.

Use the standard form A × 10n where n is a positive or negative integer, and 1 ⩽ A < 10: This subtopic involves understanding how to write numbers in standard form and how to perform calculations involving numbers in standard form. This includes understanding the meaning of the terms “mantissa” and “exponent,” as well as understanding how to convert between standard form and ordinary decimal form.

Correct ordering of operations: To solve a mathematical problem that involves more than one operation, it is important to perform the operations in the correct order. The order of operations is usually remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Use of brackets: Brackets are used in mathematical expressions to indicate that the operations inside the brackets should be performed first. This can help to avoid confusion and ensure that the correct order of operations is followed.

Approximating to specified numbers of significant figures and decimal places: This involves rounding a number to a certain number of significant figures or decimal places. This is important in science, engineering and other technical fields where precise measurements are required.

Rounding off answers to reasonable accuracy: This involves rounding a final answer to a reasonable number of decimal places or significant figures depending on the context of the problem. This ensures that the answer is both accurate and practical to work with in the given context.

Obtain appropriate upper and lower bounds to solutions of simple problems given data to a specified accuracy.

Significant figures: Significant figures are the digits that carry meaning in a number. When giving an upper or lower bound, it is important to consider the number of significant figures and to round the answer accordingly.

Accuracy and precision: Accuracy refers to how close a measurement or calculation is to the true or accepted value. Precision refers to the degree of exactness or reproducibility of a measurement or calculation. When giving upper and lower bounds, it is important to consider both accuracy and precision in order to give a reasonable estimate of the true value.

Error bounds: Error bounds are the limits of the possible error or uncertainty in a measurement or calculation. These bounds are used to ensure that the result of a calculation or measurement is within an acceptable range of accuracy.

Average Speed: Understand the concept of speed and average speed, and solve problems involving average speed.

Common Measures of Rate: Understand the concept of rate and solve problems involving common measures of rate such as distance per unit time, work per unit time, and cost per unit weight.

Express one quantity as a percentage of another: This involves expressing a given quantity as a percentage of another quantity, which requires finding the ratio of the two quantities and multiplying by 100 to convert the ratio to a percentage.

Calculate percentage increase or decrease: This involves finding the percentage increase or decrease between two quantities, which requires finding the difference between the two quantities, dividing by the original quantity, and multiplying by 100 to convert the value to a percentage.

Efficient Use of Calculator: Using the calculator efficiently by knowing which functions to use for a given problem, inputting the values correctly, and using brackets where necessary to avoid errors.

Checking Accuracy: Applying appropriate checks to ensure the accuracy of calculator calculations, such as estimating the answer before calculating, checking that the answer is reasonable, and performing the calculation using a different method to verify the answer.

Reading clocks and dials: Students learn to read and interpret clocks and dials, including analog and digital clocks, as well as other types of gauges and instruments that display numerical data.

Timetables: Students learn to read and interpret timetables, which show schedules for events such as public transportation, sporting events, or performances. This includes understanding the layout and format of timetables, as well as calculating time intervals and journey durations based on the information provided.

Currency conversion: The process of converting one currency to another using exchange rates and performing calculations with different currencies.

Buying and selling currencies: Understanding how to buy and sell currencies, including commission and transaction fees.

Profit and loss in currency exchange: Understanding how to calculate profit and loss when exchanging currencies, including exchange rate fluctuations.

Real-world applications: Applying currency conversion and exchange rate calculations to real-world situations, such as international travel, online shopping, and global business transactions.

• Earnings: Understanding earnings, income, salary, and wages. Calculating gross and net earnings. Understanding deductions and taxes.
• Simple Interest: Understanding the concept of simple interest, calculating simple interest on a principal amount for a given period at a given rate.
• Compound Interest: Understanding the concept of compound interest, calculating compound interest on a principal amount for a given period at a given rate. Understanding the difference between simple and compound interest.
• Extracting Data: Understanding how to read, interpret, and extract data from tables and charts. Analyzing data, identifying trends, and making predictions based on data.

Exponential Decay: Exponential decay is a pattern of decrease in which the rate of decrease is proportional to the amount present. In mathematics, exponential decay is represented by an exponential function.

Population Growth: Population growth is the increase in the number of individuals in a population over time. It is a type of exponential growth that can be influenced by factors such as birth rate, death rate, immigration, and emigration.

Population Decay: Population decay is the decrease in the number of individuals in a population over time. It is a type of exponential decay that can be influenced by factors such as death rate, emigration, and other factors that decrease the number of individuals in the population.

Finance: Exponential growth and decay are commonly used in finance to model the growth or decay of investments, loans, or other financial instruments. Understanding these concepts is important for financial planning and decision making.

Simplifying algebraic expressions: Simplifying algebraic expressions involves combining like terms, removing parentheses, and applying the distributive property. This allows for easier calculations and a clearer representation of the expression.

Substituting values: Substituting values involves replacing letters or variables in an algebraic expression with specific numbers. This allows for the evaluation of the expression and the calculation of a specific result.

Expanding brackets: Expanding brackets involves applying the distributive property to an expression with brackets. This involves multiplying each term inside the brackets by the term outside the brackets.

Factorising: Factorising involves breaking down an expression into its factors. This can help to simplify the expression or identify common factors.

Solving equations: Solving equations involves finding the value of the variable that makes the equation true. This can be done by using inverse operations to isolate the variable.

Forming equations: Forming equations involves translating a real-world problem into an algebraic equation. This requires an understanding of the problem and the ability to identify the relevant variables and operations.

Simultaneous equations: Simultaneous equations are equations that contain two or more variables. They can be solved by either substitution or elimination.

Quadratic equations: Quadratic equations are equations that contain a variable raised to the power of two. They can be solved by either factorisation or the quadratic formula.

Inequalities: Inequalities are expressions that compare two quantities using inequality symbols such as <, >, ≤, or ≥. They can be solved using similar methods to equations, but the solution is a range of values rather than a single value.

Expand products of algebraic expressions. Factorise possible expressions

• Directed numbers are positive or negative whole numbers including zero.
• Manipulating directed numbers includes adding, subtracting, multiplying, and dividing them.

Use brackets:

• Brackets are used to group terms in an expression.
• Brackets can be used to avoid confusion when performing arithmetic operations.
• Multiplying terms by brackets can also help to simplify expressions.

Extract common factors:

• Common factors are numbers or variables that can be divided from all terms in an expression.
• Extracting common factors involves finding the largest factor that can be divided from all terms in an expression and factoring it out.
• This process can simplify expressions and make them easier to solve.

Simplify rational expressions: Rational expressions can be simplified by canceling out common factors or by multiplying by the conjugate. In this subtopic, students will learn how to simplify rational expressions using these methods, and also how to simplify expressions with negative exponents and expressions with radicals. They will also learn how to solve equations involving rational expressions.

Use the rules of indices.

Use the rules of indices: This subtopic covers the rules of indices, including how to multiply and divide powers with the same base, raise a power to a power, and use the zero and negative index rules. Students will also learn how to simplify expressions with indices using these rules.

Derive and solve simultaneous linear equations in two unknowns: Simultaneous linear equations involve two or more linear equations with two or more variables that must be solved at the same time. Techniques include substitution, elimination, and graphing.

Derive and solve simultaneous equations involving one linear and one quadratic: This subtopic involves solving systems of equations where one equation is linear and the other is quadratic. Techniques include substitution, elimination, and graphing.

Derive and solve quadratic equations by factorisation, completing the square and by use of the formula: Quadratic equations involve a single variable raised to the power of two. This subtopic involves different methods of solving quadratic equations such as factorization, completing the square, and using the quadratic formula.

Derive and solve linear inequalities: This subtopic involves inequalities that contain linear expressions. Techniques include graphing and algebraic manipulation to solve for the variable that satisfies the inequality.

Solving simple linear programming problems: Linear programming is a method used to optimize a linear objective function, subject to constraints represented by linear inequalities. In simple linear programming problems, the constraints and the objective function involve only one or two variables. The solution to a linear programming problem is the set of values that satisfies all the constraints and maximizes or minimizes the objective function.

Identifying feasible regions: The feasible region is the set of points that satisfy all the constraints in a linear programming problem. To identify the feasible region, we graph the constraints on a coordinate plane and shade the region that satisfies all the constraints.

Identifying optimal solutions: The optimal solution is the point in the feasible region that maximizes or minimizes the objective function in a linear programming problem. To identify the optimal solution, we evaluate the objective function at each corner point of the feasible region and choose the point that gives the highest or lowest value.

Recognise patterns in sequences including the term to term rule and relationships between different sequences: This involves identifying the underlying pattern in a given sequence of numbers and determining the rule for generating subsequent terms.

Find and use the nth term of sequences: This involves using the term-to-term rule to determine a formula for the nth term of a sequence, and then using this formula to find specific terms in the sequence.

Inverse Proportion: Inverse proportion is the relationship between two variables, where an increase in one variable results in a decrease in the other variable. This relationship can be represented algebraically as y = k/x, where k is the constant of proportionality.

Using Algebraic Expressions: Direct and inverse proportion can be expressed algebraically, allowing for the calculation of unknown quantities in proportional relationships. This involves the use of ratios and proportions, solving equations, and substitution.

Solving Problems: Expressing proportional relationships algebraically can be used to solve a variety of problems, such as calculating rates, ratios, and unit prices. Inverse proportion can be used to solve problems related to time, distance, speed, and density.

Real-World Applications: Direct and inverse proportion are commonly used in the real world to represent relationships in a wide range of fields, including finance, physics, engineering, and biology. Understanding and applying these concepts is important for making informed decisions and solving problems in various contexts.

Inverse Functions: An inverse function is a function that “undoes” the original function. For example, if f(x) = 2x + 3, then the inverse function would be f^-1(x) = (x – 3) / 2. Inverse functions have the property that f(f^-1(x)) = x and f^-1(f(x)) = x.

Composite Functions: A composite function is a function that results from combining two functions, such as g(f(x)). It can be thought of as performing one function and then the other, or “plugging in” one function as an input into the other. Composite functions can be simplified using function notation and the properties of functions.

Draw graphs from given data. Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and deceleration. Calculate distance travelled as area under a speed–time graph.

Travel graphs: A travel graph is a type of graph that shows the distance travelled against time. Students will learn how to read and interpret travel graphs, as well as how to use them to calculate average speed and distance.

Conversion graphs: A conversion graph is a type of graph that shows the relationship between two different units of measurement. Students will learn how to use conversion graphs to convert between different units, such as miles and kilometers, or pounds and kilograms.

Drawing graphs from given data: In this subtopic, students will learn how to draw graphs from given data, including line graphs, bar graphs, and scatter plots. They will also learn how to choose an appropriate scale for their graphs and how to label the axes and title of the graph.

• Tables of values can be constructed for functions of the form ax + b, ±x2 + ax + b, and xa (x ≠ 0), where a and b are integer constants. In these tables, values of x and corresponding values of the function can be calculated and listed.
• Graphs can be drawn based on the values obtained from the tables of values. For linear functions (ax + b), the graph is a straight line, while for quadratic functions (±x2 + ax + b), the graph is a curve. For functions of the form xa (x ≠ 0), the shape of the graph depends on the value of a.

Solving equations approximately:

• Roots of linear and quadratic equations can be found by graphical methods. To do this, a graph of the equation is plotted, and the x-coordinate of the point where the graph intersects the x-axis gives the root.

Recognizing, sketching, and interpreting graphs of functions:

• Different types of functions can have different shapes of graphs. For example, linear functions have straight line graphs, quadratic functions have parabolic graphs, and exponential functions have graphs that increase or decrease rapidly.
• Graphs can be used to interpret the behavior of a function, including its domain, range, intercepts, maximum and minimum values, and any asymptotes.

Simple Sums: In this subtopic, we learn how to differentiate simple sums of not more than three functions of the form axn. The power rule of differentiation is applied in this case.

Gradients and Turning Points: This subtopic involves applying differentiation to find the gradient of a curve at any point. It also involves finding the turning points or stationary points of a curve, which can be maxima or minima.

Discriminating Maxima and Minima: Discrimination between maxima and minima is a process of determining whether a turning point on a curve represents a maximum or a minimum. This can be done by using the second derivative test, by examining the sign of the second derivative at the turning point.

Plotting Points on the Coordinate Plane: To plot a point on the coordinate plane, we start at the origin and move horizontally (left or right) by the value of the x-coordinate, and then vertically (up or down) by the value of the y-coordinate. The point where these two movements intersect is the point on the plane.

Recognizing Quadrants: The plane is divided into four quadrants, numbered counterclockwise from quadrant I (upper right) to quadrant IV (lower right). Each quadrant has a unique combination of positive and negative x and y values.

Finding the Distance between Two Points: To find the distance between two points on the plane, we use the distance formula:

distance = √((x₂ – x₁)² + (y₂ – y₁)²)

where (x₁,y₁) and (x₂,y₂) are the coordinates of the two points.

Recognizing Lines and Slopes: A line is a set of points that extends infinitely in both directions. The slope of a line is a measure of how steep it is, and is defined as the change in y over the change in x between any two points on the line. The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).

Graphing Linear Equations: To graph a linear equation in two variables, we can find two or more points that satisfy the equation, plot them on the plane, and then draw a straight line through them. We can also use the slope-intercept form of the equation to identify the y-intercept and slope, and then graph the line using these values.

Understanding Systems of Equations: A system of equations is a set of two or more equations with two or more variables. The solution to a system of equations is the set of values for the variables that make all of the equations true simultaneously. A system of equations can have one, none, or infinitely many solutions. We can solve a system of equations using substitution or elimination, and we can graph the system on the coordinate plane to visualize the solutions.

Method to find gradient: The gradient of a straight line can be found using the formula: gradient = (change in y-coordinate) / (change in x-coordinate) or simply by using the slope formula, which is (y2 – y1) / (x2 – x1).

Interpretation of gradient: The gradient can be interpreted as the rate of change of the dependent variable with respect to the independent variable. A positive gradient indicates an increasing relationship between the variables, while a negative gradient indicates a decreasing relationship.

Uses of gradient: The gradient is used in many applications of mathematics, such as finding the equation of a line, determining the rate of change in a physical system, and in calculus to calculate derivatives.

Midpoint Formula: The midpoint formula is used to find the midpoint of a line segment in a coordinate plane. It involves finding the average of the x-coordinates and the average of the y-coordinates of the two endpoints of the line segment.

Section Formula: The section formula is used to find the coordinates of a point that divides a line segment into two given ratios. It involves finding the weighted averages of the x-coordinates and y-coordinates of the two endpoints of the line segment.

Parallel and Perpendicular Lines: Two lines are parallel if they have the same slope, and they are perpendicular if their slopes are negative reciprocals of each other. Given the slope of one line and a point on the other line, we can find the equation of the line that is either parallel or perpendicular to the given line and passes through the given point.

Gradient: The gradient (or slope) of a line is a measure of how steep the line is. It is calculated by dividing the change in y-coordinates by the change in x-coordinates between two points on the line. The gradient can also be negative, zero, or undefined depending on the orientation of the line.

Finding the Equation of a Straight Line: Given two points on a line, we can find the equation of the line by calculating the gradient of the line and using one of the points to find the y-intercept. Alternatively, if we know the gradient of the line and the y-intercept, we can directly write the equation in the form y = mx + c.

Interpreting the Equation of a Straight Line: The equation of a straight line can provide information about the slope of the line, its intercept with the y-axis, and the direction of the line (whether it is rising or falling).

Graphing a Straight Line: Given an equation in the form y = mx + c, we can graph the line by plotting the y-intercept (c) and then using the gradient (m) to find additional points on the line. Alternatively, we can use two points on the line to graph it.

Applications of Straight Line Graphs: Straight line graphs can be used to model a variety of real-world phenomena, such as distance-time graphs for motion, cost-revenue graphs for business, and temperature-time graphs for weather. By interpreting the slope and intercept of the line, we can make predictions and draw conclusions about the situation being modeled.

Finding the Equation of a Line: To find the equation of a line, you need to know its gradient and y-intercept. This is usually done in the form y = mx + c, where m is the gradient and c is the y-intercept.

Point-Slope Form: Another way to find the equation of a line is by using the point-slope form, which uses a point on the line and the gradient to find the equation. The formula is y – y1 = m(x – x1), where (x1, y1) is the given point and m is the gradient.

Using Parallel Lines to Find Equations: If you have the equation of one line and need to find the equation of a parallel line, you can use the fact that parallel lines have the same gradient. Simply substitute the gradient of the first line into the equation y = mx + c and solve for the y-intercept.

Applications: The concept of parallel lines and finding equations of parallel lines is useful in a variety of real-life situations, such as designing roads, constructing buildings, and creating mathematical models.

Gradient of Perpendicular Lines: Perpendicular lines have negative reciprocal gradients, which means that their steepness or incline is opposite and reciprocal of each other. To find the gradient of perpendicular lines, you need to find the negative reciprocal of one of the lines.

Line: A line is a straight path that extends infinitely in both directions. It has no thickness and is represented by a straight line with two arrowheads.

Parallel: Two lines are parallel if they are always the same distance apart and never meet.

Bearing: A bearing is a measurement of an angle in degrees, measured clockwise from north.

Right angle: A right angle is an angle that measures exactly 90 degrees.

Acute angle: An acute angle is an angle that measures less than 90 degrees.

Obtuse angle: An obtuse angle is an angle that measures between 90 and 180 degrees.

Reflex angle: A reflex angle is an angle that measures between 180 and 360 degrees.

Perpendicular: Two lines are perpendicular if they meet at a right angle.

Similarity: Two figures are similar if they have the same shape but not necessarily the same size.

Congruence: Two figures are congruent if they have the same shape and size.

Triangle: A triangle is a closed two-dimensional shape with three straight sides and three angles.

Quadrilateral: A quadrilateral is a closed two-dimensional shape with four straight sides and four angles.

Circle: A circle is a closed curved shape where all points on the curve are the same distance from the center point.

Polygon: A polygon is a closed two-dimensional shape with straight sides.

Solid figure: A solid figure is a three-dimensional shape with length, width, and height.

Nets: Nets are two-dimensional shapes that can be folded to create three-dimensional figures.

Construct a triangle given the three sides: This subtopic covers how to use a ruler and pair of compasses to construct a triangle given the lengths of its three sides. The process involves drawing three circles with centers at the endpoints of the sides, and then finding the intersection of these circles to locate the vertices of the triangle.

Scale drawings: Scale drawings are drawings that show an object, building or area with accurate proportions relative to the actual size of the object. Scale drawings are often used in architecture, engineering, and design.

Representing objects on a smaller scale: When representing objects on a smaller scale, the dimensions of the object are reduced by the same factor, which is the scale factor.

Representing objects on a larger scale: When representing objects on a larger scale, the dimensions of the object are increased by the same factor, which is the scale factor.

Using scales on maps: Maps often use scales to represent the distances between locations accurately. The scale on a map may be represented as a ratio or as a graphical scale bar.

Using scales in technical drawings: Technical drawings such as blueprints or engineering plans use scales to accurately represent the dimensions of the object being designed or constructed.

Reading and interpreting scale drawings: To read a scale drawing, you need to understand the scale factor and the units of measurement used. You can then use the scale to calculate measurements and dimensions of the object being represented.

Making scale drawings: To make a scale drawing, you need to choose an appropriate scale factor and accurately measure the dimensions of the object you are representing. You can then use the scale factor to reduce or enlarge the dimensions of the object and draw the scale drawing with accurate proportions.

• Definition of scale factor: The ratio of corresponding lengths of two similar figures.
• Finding unknown side lengths: Given two similar figures, you can use the scale factor to find the length of an unknown side. Multiply the length of a known side by the scale factor to find the length of the corresponding side in the other figure.
• Solving problems involving similar figures: You can use the concept of similar figures to solve problems involving indirect measurement, such as finding the height of a tree or the width of a river.
• Applications: Similar figures are used in a variety of real-life applications, including architecture, engineering, and mapmaking. For example, architects may use scale models of buildings to visualize and test design concepts, and cartographers may use scale drawings to create accurate maps of geographical areas.

Identifying congruent shapes: This involves recognising the characteristics of congruent shapes, such as equal side lengths, equal angles, and identical shapes. It may also involve identifying congruent parts of a shape, such as sides or angles.

Using congruence to solve problems: This involves using the concept of congruence to solve various types of problems, such as finding missing side lengths or angles in congruent triangles. It may also involve using congruence to prove the properties of shapes or to solve more complex problems involving multiple shapes.

Applying congruence in real-world situations: This involves applying the concept of congruence to real-world situations, such as in architecture or engineering, where precise measurements and identical shapes are important for construction and design.

• equal chords are equidistant from the centre 

• the perpendicular bisector of a chord passes through the centre

• tangents from an external point are equal in length.

Line Symmetry: A shape has line symmetry if it can be divided into two identical parts by a line, also known as a line of symmetry.

Rotational Symmetry: A shape has rotational symmetry if it can be rotated by a certain angle and still look the same. The order of rotational symmetry is the number of times a shape can be rotated to coincide with its original position.

Examples: A circle has infinite rotational symmetry, a square has 4 lines of symmetry and rotational symmetry of order 4, a rectangle has 2 lines of symmetry but no rotational symmetry, and a regular hexagon has 6 lines of symmetry and rotational symmetry of order 6.

• angles at a point 

• angles at a point on a straight line and intersecting straight lines 

• angles formed within parallel lines 

• angle properties of triangles and quadrilaterals 

• angle properties of regular polygons 

• angle in a semicircle 

• angle between tangent and radius of a circle • angle properties of irregular polygons 

• angle at the centre of a circle is twice the angle at the circumference 

• angles in the same segment are equal 

• angles in opposite segments are supplementary; cyclic quadrilaterals 

• alternate segment theorem.

Angles at a point on a straight line and intersecting straight lines: The sum of the angles on a straight line is 180 degrees. When two straight lines intersect, opposite angles are equal and the adjacent angles are supplementary (add up to 180 degrees).

Angles formed within parallel lines: When two parallel lines are intersected by a third line (a transversal), corresponding angles are equal, alternate angles are equal, and the interior angles on the same side of the transversal add up to 180 degrees.

Angle properties of triangles and quadrilaterals: The sum of the angles in a triangle is 180 degrees. The sum of the angles in a quadrilateral is 360 degrees. This can be used to find missing angles in these shapes.

Angle properties of regular polygons: In a regular polygon with n sides, each interior angle is (n-2)180/n degrees and each exterior angle is 360/n degrees. This can be used to find missing angles in regular polygons.

Angle in a semicircle: An angle in a semicircle is a right angle (90 degrees). This can be used to find missing angles in a circle.

Angle between tangent and radius of a circle: The angle between a tangent and a radius of a circle is 90 degrees. This can be used to find missing angles in a circle.

Angle properties of irregular polygons: The sum of the angles in an irregular polygon can be found by dividing it into triangles or quadrilaterals and using the angle properties of these shapes.

Angle at the centre of a circle is twice the angle at the circumference: The angle at the centre of a circle that subtends an arc is twice the angle at the circumference that subtends the same arc. This can be used to find missing angles in a circle.

Angles in the same segment are equal: Angles in the same segment of a circle are equal. This can be used to find missing angles in a circle.

Angles in opposite segments are supplementary; cyclic quadrilaterals: Angles in opposite segments of a circle that intersect at a point outside the circle are supplementary (add up to 180 degrees). A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. The opposite angles of a cyclic quadrilateral are supplementary.

Alternate segment theorem: The angle between a tangent and a chord at the point of contact is equal to the angle subtended by the chord in the alternate segment. This can be used to find missing angles in a circle.

Units of length: Understanding units of length such as millimeters, centimeters, meters, and kilometers, and converting between them in practical situations.

Units of area: Understanding units of area such as square centimeters, square meters, and hectares, and converting between them in practical situations.

Units of volume: Understanding units of volume such as cubic centimeters, cubic meters, and liters, and converting between them in practical situations.

Units of capacity: Understanding units of capacity such as milliliters and liters, and converting between them in practical situations.

Expressing quantities in terms of larger or smaller units: Using prefixes such as kilo-, mega-, milli-, and micro- to express quantities in larger or smaller units, and converting between them in practical situations.

Practical applications: Applying knowledge of units of measurement and conversions to practical situations, such as measuring ingredients for cooking, calculating distances for travel, and estimating volumes of containers.

Area of a rectangle: The area of a rectangle is the amount of space inside the shape. It is calculated by multiplying the length by the width.

Perimeter of a triangle: The perimeter of a triangle is the total distance around its sides. It is calculated by adding the lengths of all the sides.

Area of a triangle: The area of a triangle is the amount of space inside the shape. It is calculated by multiplying half the base by the height.

Perimeter of a parallelogram: The perimeter of a parallelogram is the total distance around its sides. It is calculated by adding the lengths of all the sides.

Area of a parallelogram: The area of a parallelogram is the amount of space inside the shape. It is calculated by multiplying the base by the height.

Perimeter of a trapezium: The perimeter of a trapezium is the total distance around its sides. It is calculated by adding the lengths of all the sides.

Area of a trapezium: The area of a trapezium is the amount of space inside the shape. It is calculated by multiplying the average of the two parallel sides by the height.

Compound shapes derived from rectangles, triangles, parallelograms, and trapeziums: Compound shapes are made up of two or more simple shapes. The perimeter and area of compound shapes can be calculated by adding or subtracting the perimeters and areas of the individual shapes.

Solve problems involving the arc length and sector area as fractions of the circumference and area of a circle.

Area of a circle: The area of a circle is the amount of space inside the circle.

Arc length of a circle: The arc length of a circle is the length of a portion of the circumference of a circle.

Sector of a circle: A sector of a circle is a region bounded by two radii and an arc, with a central angle measured in radians.

Sector area of a circle: The sector area of a circle is the area of a sector of a circle, which is a fraction of the total area of the circle.

Surface Area and Volume of a Prism: Learn how to calculate the surface area and volume of a prism, which is a 3D figure with two identical and parallel faces that are polygonal in shape.

Surface Area and Volume of a Cylinder: Learn how to calculate the surface area and volume of a cylinder, which is a 3D figure with a circular base and a curved surface that is perpendicular to the base.

Surface Area and Volume of a Sphere: Learn how to calculate the surface area and volume of a sphere, which is a 3D figure with all points on its surface equidistant from its center.

Surface Area and Volume of a Pyramid: Learn how to calculate the surface area and volume of a pyramid, which is a 3D figure with a polygonal base and triangular faces that meet at a common vertex.

Surface Area and Volume of a Cone: Learn how to calculate the surface area and volume of a cone, which is a 3D figure with a circular base and a curved surface that tapers to a point (apex) at the top.

Decomposing compound shapes: Decomposing compound shapes involves breaking them down into simpler shapes so that their area or volume can be calculated more easily.

Area of compound shapes: To find the area of a compound shape, you can decompose it into simpler shapes, calculate the area of each shape, and then add them together.

Volume of compound shapes: To find the volume of a compound shape, you can decompose it into simpler shapes, calculate the volume of each shape, and then add them together.

A bearing is an angle measured clockwise from north, written as a three-digit number, such as 030°.

The first two digits represent the angle measured in degrees, and the third digit represents the angle measured in minutes.

Bearings can be used to describe the direction from one point to another, such as the direction of a ship from a lighthouse.

Bearings can be used to calculate the distance between two points using trigonometry.

Solve trigonometric problems in two dimensions involving angles of elevation and depression. Know that the perpendicular distance from a point to a line is the shortest distance to the line.

Sine, Cosine, and Tangent Ratios: These are trigonometric ratios that relate the angles and sides of a right-angled triangle. Sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse, cosine is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, and tangent is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. These ratios can be used to find the length of a side or the measure of an angle in a right-angled triangle if one length and one angle are known.

Calculation of a Side or Angle: These concepts can be used to find the length of a side or the measure of an angle in a right-angled triangle when some information is given. The process involves identifying the right angle, labeling the sides with their opposite angles and trigonometric ratios, and applying the relevant formula to find the unknown quantity.

Graphing trigonometric functions: Learning how to graph trigonometric functions by plotting points or using transformations, and understanding the properties of these graphs, such as their amplitude, period, phase shift, and range.

Interpreting trigonometric graphs: Developing the ability to interpret the features of trigonometric graphs, such as their intercepts, maximums, minimums, and asymptotes, and using these interpretations to solve problems.

Simple trigonometric equations: Learning how to solve basic trigonometric equations, such as sin x = 0.5 or cos 2x = -1, by finding the values of x between 0° and 360° that satisfy the equation.

Using trigonometry in real-world contexts: Applying trigonometry to solve problems in various fields, such as physics, engineering, architecture, and navigation. Examples include finding the height of a building, the distance between two ships, or the angle of elevation of a plane.

Cosine Rule: The cosine rule is used to find the unknown sides or angles of a triangle. It relates the lengths of the sides of a triangle to the cosine of one of its angles.

Area of a Triangle: The formula for the area of a triangle is 1/2 times the base times the height. In addition, the area of a triangle can also be found using the formula 1/2 times a times b times the sine of the included angle C.

Angle between a line and a plane: The angle between a line and a plane is the acute angle formed by the line and the perpendicular to the plane. This concept is used in various fields such as engineering, physics, and astronomy. Trigonometric functions can be used to calculate this angle.

Solving trigonometric problems in three dimensions: Trigonometric functions can be used to solve problems involving three-dimensional figures such as pyramids, cones, and spheres. These problems can involve finding missing angles or side lengths of these figures using trigonometric functions.

Application of trigonometry in physics: Trigonometry plays an important role in physics, especially in the study of motion and forces. Trigonometric functions can be used to calculate the components of vectors, the angles between them, and the magnitude of their resultant vector.

Navigation and trigonometry: Trigonometry is used extensively in navigation, including air and sea navigation. Trigonometric functions can be used to calculate distances, angles, and directions between two points, and to determine the location of a ship or airplane relative to a known location.

Vector: A quantity that has both magnitude and direction. Vectors are commonly represented by arrows, and the length of the arrow corresponds to the magnitude of the vector while the direction of the arrow indicates its direction.

Vector notation: The notation used to represent vectors, typically written in the form <x, y>, where x and y are the horizontal and vertical components of the vector, respectively.

Adding vectors: The process of combining two or more vectors to form a single vector.

Subtracting vectors: The process of finding the difference between two vectors by adding the opposite of the second vector to the first vector.

Scalar multiplication: The process of multiplying a vector by a scalar (a real number). This results in a new vector with the same direction as the original vector but a different magnitude.

Rotate simple plane figures about the origin, vertices, or midpoints of edges of the figures through multiples of 90°: This involves rotating a figure about a specific point on the coordinate plane (e.g. the origin, a vertex, or a midpoint of an edge) through 90, 180, or 270 degrees.

Construct given translations and enlargements of simple plane figures: This involves translating or enlarging a figure based on given instructions.

Recognize and describe reflections, rotations, translations, and enlargements: This involves identifying and describing the type of transformation that has been applied to a figure, such as a reflection, rotation, translation, or enlargement.

Representation of Vectors by Directed Line Segments: Vectors can be represented by directed line segments, where the length of the line segment represents the magnitude of the vector, and the direction of the line segment represents the direction of the vector.

Sum and Difference of Two Vectors: The sum and difference of two vectors can be used to express given vectors in terms of two coplanar vectors. The sum of two vectors can be found by adding their corresponding components, while the difference of two vectors can be found by subtracting their corresponding components.

Position Vectors: Position vectors are vectors that describe the position of a point in space with respect to a fixed point or origin. They can be used to represent vectors that point from one point to another.

Calculating probability: To calculate the probability of a single event, we need to determine the number of favourable outcomes and the total number of possible outcomes. The probability is then calculated by dividing the number of favourable outcomes by the total number of possible outcomes. The result can be expressed as a fraction, decimal or percentage.

Events and outcomes: An event is a specific outcome or set of outcomes of an experiment. Outcomes are the possible results of an experiment. For example, when tossing a coin, the possible outcomes are heads and tails.

Experimental and theoretical probability: Experimental probability is the probability of an event based on actual observations or experiments. Theoretical probability is the probability of an event calculated using mathematical formulas or models.

Sample space: The sample space is the set of all possible outcomes of an experiment. It can be represented using a list, table or diagram.

Mutually exclusive events: Mutually exclusive events are events that cannot occur at the same time. For example, when rolling a dice, the events of getting an even number and getting an odd number are mutually exclusive.

Independent events: Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. For example, when flipping a coin twice, the outcome of the first flip does not affect the outcome of the second flip.

and 1 represents a certain event.

Probability of an event: Calculating the probability of an event using the probability scale and expressing it as a fraction, decimal, or percentage.

Theoretical probability: Understanding and calculating the theoretical probability of an event using the formula: 

P(event) = Number of favorable outcomes / Total number of possible outcomes.

Experimental probability: Understanding and calculating the experimental probability of an event using the formula: P(event) = Number of times event occurred / Total number of trials.

Mutually exclusive events: Understanding and identifying mutually exclusive events, where the occurrence of one event excludes the occurrence of another event.

Independent events: Understanding and identifying independent events, where the occurrence of one event does not affect the probability of the occurrence of another event.

Dependent events: Understanding and identifying dependent events, where the occurrence of one event affects the probability of the occurrence of another event.

Expected frequency of occurrences: Expected frequency is the anticipated number of times an event is expected to occur over a certain period based on probability calculations. It is calculated by multiplying the total number of observations by the probability of the event occurring. For example, if the probability of a certain event occurring is 0.25 and there are 100 observations, the expected frequency would be 25.

Tree diagrams: Tree diagrams are diagrams that show the possible outcomes of a sequence of events. They are useful in calculating the probability of complex events.

Venn diagrams: Venn diagrams are diagrams that show the relationships between different sets of data. They are useful in calculating the probability of events that involve more than one set of data.

Calculating the probability of simple combined events: Simple combined events are events that involve more than one possible outcome. The probability of these events can be calculated using formulas and diagrams.

Using possibility diagrams to calculate probability: Possibility diagrams can be used to calculate the probability of simple combined events by counting the number of possible outcomes and dividing by the total number of outcomes.

Using tree diagrams to calculate probability: Tree diagrams can be used to calculate the probability of complex events by multiplying the probabilities of each event in the sequence.

Using Venn diagrams to calculate probability: Venn diagrams can be used to calculate the probability of events that involve more than one set of data by analyzing the overlaps between the sets.

Expected value of occurrences: Expected value is a measure of the average outcome of a given event over a large number of trials. It can be calculated by multiplying the probability of each outcome by its corresponding value and summing the products.

Venn diagrams: Venn diagrams are graphical representations of sets that use circles or ovals to show the relationship between different sets. They can be used to calculate conditional probabilities by shading the appropriate areas and calculating the probabilities of the shaded regions.

Tree diagrams: Tree diagrams are diagrams that show the branching of different possible outcomes of an event. They are useful in calculating conditional probabilities because they allow you to see all the possible outcomes of a given set of events and their probabilities.

Tables: Tables can be used to calculate conditional probabilities by organizing the different events and their probabilities in a tabular form. This method is particularly useful when there are a large number of events and outcomes to consider.

Types of data: Data can be classified as qualitative or quantitative. Qualitative data describes qualities or characteristics, while quantitative data measures quantities or amounts.

Organizing data: Data can be organized in tables, charts, graphs, or frequency distributions. This helps to identify patterns, trends, and outliers in the data.

Measures of central tendency: Mean, median, and mode are statistical measures used to describe the center of a dataset.

Measures of dispersion: Range, variance, and standard deviation are statistical measures used to describe the spread of a dataset.

Frequency distributions: A frequency distribution is a table that shows how many times each value or range of values occurs in a dataset.

Bar charts and histograms: Bar charts and histograms are graphical representations of frequency distributions.

Scatter plots: Scatter plots are used to show the relationship between two variables. They are useful for identifying trends or patterns in the data.

Stem-and-leaf plots: Stem-and-leaf plots are a method for organizing and displaying numerical data in a way that shows the distribution of the data.

Compare sets of data using tables, graphs, and statistical measures: This subtopic focuses on comparing different sets of data using statistical measures such as the mean, median, and mode. Students will also learn how to create and interpret different types of graphs to compare data, such as scatter plots and box-and-whisker plots.

Appreciate restrictions on drawing conclusions from given data: This subtopic will help students understand the limitations of statistical data and the need to be cautious when drawing conclusions. They will learn how to identify potential biases and errors in data and how to address them. Students will also learn about the importance of sample size and how it affects the reliability of statistical data.

Pie charts: A pie chart is a circular graph that shows the distribution of data across different categories. The size of each slice of the pie represents the proportion of the data that falls into that category.

Pictograms: A pictogram is a chart that uses pictures or symbols to represent data. The size of the picture or symbol corresponds to the value of the data it represents.

Stem-and-leaf diagrams: A stem-and-leaf diagram is a way to display data using digits. The stems represent the tens or hundreds digit, and the leaves represent the ones digit. This allows for easy comparison of data values.

Simple frequency distributions: A simple frequency distribution is a table that shows the number of times each data value appears in a dataset.

Histograms with equal intervals: A histogram is a graph that displays data using rectangular bars. The bars are arranged side-by-side, with each bar representing a range of data values. The height of each bar corresponds to the frequency of data values within that range.

Scatter diagrams: A scatter diagram is a graph that shows the relationship between two variables. The variables are plotted on the x and y axes, and each point on the graph represents a data value for both variables.

Median: The median is the middle value in a set of data when the values are arranged in numerical order. If there are an even number of values, the median is the average of the two middle values. The median is used to describe the middle of a set of data, and is less affected by extreme values than the mean.

Mode: The mode is the value that appears most frequently in a set of data. A data set may have more than one mode, or no mode at all. The mode is used to describe the most common value or values in a set of data.

Range: The range is the difference between the largest and smallest values in a set of data. The range is used to describe the spread or variability of a set of data.

• Estimating the mean is a method of finding an approximate value of the mean for large datasets with limited information.
• Grouped data refers to data that has been organized into intervals or classes, while continuous data is data that can take on any value within a range.
• Methods for estimating the mean include the mid-point method, the assumed mean method, and the step deviation method.

Identify the modal class from a grouped frequency distribution:

• The modal class is the interval or class with the highest frequency in a grouped frequency distribution.
• A grouped frequency distribution is a way of organizing and displaying data in intervals or classes, along with their respective frequencies.
• Identifying the modal class is important for determining the most common value or interval in the dataset.

Estimate and interpret the median: The median is the middle value in a set of data, when the data is arranged in order. It is a measure of central tendency that is not affected by extreme values or outliers. Estimating the median involves finding the midpoint of the data set, which is useful when the data is continuous.

Estimate and interpret percentiles: Percentiles divide a data set into 100 equal parts, with each part representing 1% of the data. For example, the 75th percentile represents the value that is greater than 75% of the data. Percentiles are useful for comparing data between different groups.

Estimate and interpret quartiles: Quartiles divide a data set into four equal parts. The first quartile (Q1) is the value below which 25% of the data falls, the second quartile (Q2) is the median, and the third quartile (Q3) is the value below which 75% of the data falls. The interquartile range is the difference between the third and first quartiles.

Construct and interpret box-and-whisker plots: Box-and-whisker plots are used to display the distribution of a set of data. The box represents the middle 50% of the data, with the line inside the box representing the median. The whiskers extend from the box to the minimum and maximum values of the data, excluding outliers. Box-and-whisker plots provide a quick visual summary of the distribution of the data, and allow for easy comparison of multiple data sets.

Positive correlation: When the values of two variables increase or decrease together, it is known as a positive correlation. In a scatter diagram, the points on the graph will be arranged in a rising trend from left to right.

Negative correlation: When one variable increases while the other decreases, it is known as negative correlation. In a scatter diagram, the points on the graph will be arranged in a falling trend from left to right.

Zero correlation: When there is no apparent relationship between two variables, it is known as zero correlation. In a scatter diagram, the points will be randomly scattered across the graph without any trend or pattern.

Interpreting lines of best fit: Once a line of best fit has been drawn, you can use it to make predictions or estimates about other data points that are not on the scatter plot. The slope of the line of best fit gives an indication of the direction and strength of the relationship between the variables.

Using lines of best fit: Lines of best fit can be used to make predictions or estimates about other data points that are not on the scatter plot. For example, if you have data on the ages and heights of a group of people, you could use a line of best fit to predict the height of someone based on their age.