G3 Mathematics
Lessons in g3 mathematics. This course is still being developed; independent practice and assessment coverage is incomplete.
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G3 Mathematics
- Number and proportionSuggested next · 14 lessonsG3 Mathematics 2027 · Express Secondary 1 Mathematics 2020 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in number and proportion.
- Algebra and equationsSuggested next · 22 lessonsG3 Mathematics 2027 · Express Secondary 1 Mathematics 2020 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Explore expressions, linear equations and rational-expression restrictions. Other algebra methods are still being prepared.
- Functions and graphsSuggested next · 5 lessonsG3 Mathematics 2027 · Express Secondary 1 Mathematics 2020 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in functions and graphs.
- Sets and matricesSuggested next · 2 lessonsG3 Mathematics 2027 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in sets and matrices.
- Geometry and reasoningSuggested next · 10 lessonsG3 Mathematics 2027 · Express Secondary 1 Mathematics 2020 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in geometry and reasoning.
- Coordinates and vectorsSuggested next · 2 lessonsG3 Mathematics 2027 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in coordinates and vectors.
- Area, volume and mensurationSuggested next · 7 lessonsG3 Mathematics 2027 · Express Secondary 1 Mathematics 2020 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in area, volume and mensuration.
- TrigonometrySuggested next · 4 lessonsG3 Mathematics 2027 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in trigonometry.
- Statistics and probabilitySuggested next · 9 lessonsG3 Mathematics 2027 · Express Secondary 1 Mathematics 2020 · Express Secondary 2 Mathematics 2021 · Express Secondary 3–4 Mathematics 2022Build and connect the representations and methods in statistics and probability.
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Express Secondary 1 Mathematics 2020
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G3 Mathematics 2027
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Lessons in g3 mathematics. This course is still being developed; independent practice and assessment coverage is incomplete.
Learning goals
- Identify primes, factors and multiples
- Use prime factorisation to find HCF and LCM
- Calculate with integers and rational numbers
- Compare and order numbers and representations
- Round, estimate and judge numerical accuracy
- Use integer, zero, negative and fractional indices appropriately
- Represent and simplify ratios of quantities
- Share a quantity in a specified ratio
- Relate part-to-part ratios to part-to-whole fractions
- Model and solve direct or inverse proportional relationships
- Interpret a percentage relative to its base
- Calculate percentage changes and contextual amounts
- Recover a whole or original amount from a percentage part
- Use rate, speed, distance and time relationships
- Convert compatible measurement units
- Interpret algebraic notation and represented operations
- Evaluate an algebraic expression by substitution
- Represent a situation with algebraic expressions
- Express and use the general term of a sequence
- Simplify linear expressions by combining like terms
- Expand and factorise algebraic expressions
- Calculate and simplify algebraic fractions with restrictions
- Change the subject of a formula
- Solve linear equations and check the solution
- Solve simultaneous equations by justified methods
- Solve quadratic equations by suitable methods
- Solve inequalities and represent their solution sets
- Formulate, use and interpret a mathematical model
- Multiply and divide algebraic fractions with domain restrictions
- Add and subtract algebraic fractions with domain restrictions
- Interpret and draw graphs of functions
- Read and plot ordered coordinate pairs
- Find and interpret a graph gradient
- Represent sets and reason with set notation
- Represent data and calculate with matrices
- Classify shapes by their properties
- Deduce unknown angles from geometric relationships
- Construct geometric figures from stated conditions
- Reason about congruence, similarity and scale factors
- Classify angles by their size
- Reason and calculate using coordinate geometry
- Represent and calculate with plane vectors
- Calculate and justify area and perimeter
- Calculate and justify volume and surface area
- Calculate circle, arc, sector and segment measures
- Use and justify Pythagoras’ theorem
- Use trigonometric ratios in right-angled triangles
- Use sine and cosine rules for general triangles
- Read and interpret data displays
- Organise data and construct a suitable display
- Evaluate the purposes and limitations of data displays
- Calculate and interpret the arithmetic mean
- Find and interpret the mode and median
- Estimate the mean from grouped data
- Calculate measures of spread
- Compare variability using appropriate measures
- Calculate event probabilities from an explicit sample space
- Distinguish independence from mutual exclusivity
Syllabus statements covered
- primes and prime factorisation • finding highest common factor (HCF) and lowest common multiple (LCM), squares, cubes, square roots and cube roots by prime factorisation • negative numbers, integers, rational numbers, real numbers, and their four operations • calculations with calculator • representation and ordering of numbers on the number line • use of the symbols <, >, ⩽, ⩾ • approximation and estimation (including rounding off numbers to a required number of decimal places or significant figures and estimating the results of computation) • use of standard form A × 10^n, where n is an integer, and 1 ⩽ A < 10 • positive, negative, zero and fractional indices • laws of indices
- ratios involving rational numbers • writing a ratio in its simplest form • map scales (distance and area) • direct and inverse proportion
- expressing one quantity as a percentage of another • comparing two quantities by percentage • percentages greater than 100% • increasing/decreasing a quantity by a given percentage • reverse percentages
- average rate and average speed • conversion of units (e.g. km/h to m/s)
- using letters to represent numbers • interpreting notations: ab as a × b; a/b as a ÷ b or a × (1/b); a² as a × a, a³ as a × a × a, a²b as a × a × b, …; 3y as y + y + y or 3 × y; 3(x + y) as 3 × (x + y); (3 + y)/5 as (3 + y) ÷ 5 or (1/5) × (3 + y) • evaluation of algebraic expressions and formulae • translation of simple real-world situations into algebraic expressions • recognising and representing patterns/relationships by finding an algebraic expression for the nth term • addition and subtraction of linear expressions • simplification of linear expressions such as −2(3x − 5) + 4x and 2x/3 − 3(x − 5)/2 • use brackets and extract common factors • factorisation of linear expressions ax + bx + kay + kby • expansion of the product of algebraic expressions • changing the subject of a formula • finding the value of an unknown quantity in a given formula • use of (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², a² − b² = (a + b)(a − b) • factorisation of quadratic expressions ax² + bx + c • multiplication and division of simple algebraic fractions such as (3a/(4b²))(5ab/3); (3a/4) ÷ (9a²/10) • addition and subtraction of algebraic fractions with linear or quadratic denominator such as 1/(x − 2) + 2/(x − 3); 1/(x² − 9) + 2/(x − 3); 1/(x − 3) + 2/(x − 3)²
- solving linear equations in one variable • solving simple fractional equations that reduce to linear equations, such as x/3 + (x − 2)/4 = 3 and 3/(x − 2) = 6 • solving simultaneous linear equations in two variables by substitution and elimination or graphically • solving quadratic equations in one variable by factorisation, formula, completing the square for y = x² + px + q, or graphically • solving fractional equations that reduce to quadratics, such as 6/(x + 4) = x + 3 and 1/(x − 2) + 2/(x − 3) = 5 • formulating equations to solve problems • solving linear inequalities in one variable and representing the solution on the number line
- Cartesian coordinates in two dimensions • graph of a set of ordered pairs as a representation of a relationship between two variables • linear functions (y = ax + b) and quadratic functions (y = ax² + bx + c) • graphs of linear functions • the gradient of a linear graph as the ratio of the vertical change to the horizontal change (positive and negative gradients) • graphs of quadratic functions and their properties: - positive or negative coefficient of x^2 - maximum and minimum points - symmetry • sketching the graphs of quadratic functions given in the form: - y = (x − p)² + q - y = − (x − p)² + q - y = (x − a)(x − b) - y = − (x − a)(x − b) • graphs of power functions of the form y = ax^n, where n = −2, −1, 0, 1, 2, 3, and simple sums of not more than three of these • graphs of exponential functions y = ka^x, where a is a positive integer • estimation of the gradient of a curve by drawing a tangent
- use of set language and notation: A ∪ B, A ∩ B, n(A), ∈, ∉, A′, ∅, universal set ξ, A ⊆ B, A ⊈ B, A ⊂ B, A ⊄ B • union and intersection of two sets • Venn diagrams
- display of information in the form of a matrix of any order • interpreting the data in a given matrix • product of a scalar quantity and a matrix • problems involving the calculation of the sum and product (where appropriate) of two matrices
- right, acute, obtuse and reflex angles • vertically opposite angles, angles on a straight line and angles at a point • angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles • properties of triangles, special quadrilaterals and regular polygons (pentagon, hexagon, octagon and decagon), including symmetry properties • classifying special quadrilaterals on the basis of their properties • angle sum of interior and exterior angles of any convex polygon • construction of simple geometrical figures from given data using compasses, ruler, set squares and protractors, where appropriate
- congruent figures and similar figures • properties of similar triangles and polygons: - corresponding angles are equal - corresponding sides are proportional • enlargement and reduction of a plane figure • scale drawings • properties and construction of perpendicular bisectors of line segments and angle bisectors • determining whether two triangles are - congruent - similar • ratio of areas of similar plane figures • ratio of volumes of similar solids • solving simple problems involving similarity and congruence
- symmetry properties of circles: - 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 - the line joining an external point to the centre of the circle bisects the angle between the tangents • angle properties of circles: - angle in a semicircle is a right angle - angle between tangent and radius of a circle is a right angle - angle at the centre is twice the angle at the circumference - angles in the same segment are equal - angles in opposite segments are supplementary
- finding the gradient of a straight line given the coordinates of two points on it • finding the length of a line segment given the coordinates of its end points • interpreting and finding the equation of a straight line graph in the form y = mx + c • geometric problems involving the use of coordinates
- use of column-vector notation (x, y), vectors AB and a, and magnitudes |AB| and |a| • vector as a directed line segment • translation by a vector • position vectors • magnitude √(x² + y²) • sum and difference of vectors to express given vectors in terms of two coplanar vectors • scalar multiplication • geometric problems using vectors
- area of parallelogram and trapezium • problems involving perimeter and area of composite plane figures • volume and surface area of cube, cuboid, prism, cylinder, pyramid, cone and sphere • conversion between cm^2 and m^2 , and between cm^3 and m^3 • problems involving volume and surface area of composite solids • arc length, sector area and area of a segment of a circle • use of radian measure of angle (including conversion between radians and degrees)
- use of Pythagoras’ theorem • determining whether a triangle is right-angled given the lengths of three sides • use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angled triangles • extending sine and cosine to obtuse angles • use of the formula ½ab sin C for the area of a triangle • use of sine rule and cosine rule for any triangle • problems in two and three dimensions including those involving angles of elevation and depression and bearings
- simple concepts in collecting, classifying and tabulating data • analysis and interpretation of: - tables - bar graphs - pictograms - line graphs - pie charts - dot diagrams - histograms with equal class intervals - stem-and-leaf diagrams - cumulative frequency diagrams - box-and-whisker plots • purposes and uses, advantages and disadvantages of the different forms of statistical representations • drawing simple inference from statistical diagrams • explaining why a given statistical diagram leads to misinterpretation of data • mean, mode and median as measures of central tendency for a set of data • purposes and use of mean, mode and median • calculation of the mean for grouped data • quartiles and percentiles • range, interquartile range and standard deviation as measures of spread for a set of data • calculation of the standard deviation for a set of data (grouped and ungrouped) • using the mean and standard deviation to compare two sets of data
- probability as a measure of chance • probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability) • probability of simple combined events (including using possibility diagrams and tree diagrams, where appropriate) • addition and multiplication of probabilities (mutually exclusive events and independent events)
- primes and prime factorisation
- finding highest common factor (HCF) and lowest common multiple (LCM), squares, cubes, square roots and cube roots by prime factorisation
- negative numbers, integers, rational numbers, real numbers and their four operations
- calculations with calculator
- representation and ordering of numbers on the number line
- use of <, >, ≤, ≥
- approximation and estimation (including rounding off numbers to a required number of decimal places or significant figures, and estimating the results of computation)
- ratios involving rational numbers
- writing a ratio in its simplest form
- problems involving ratio
- expressing one quantity as a percentage of another
- comparing two quantities by percentage
- percentages greater than 100%
- increasing/decreasing a quantity by a given percentage (including concept of percentage point)
- reverse percentages
- problems involving percentages
- concepts of average rate, speed, constant speed and average speed
- conversion of units (e.g. km/h to m/s)
- problems involving rate and speed
- using letters to represent numbers
- interpreting notations: ab as a × b; a/b as a ÷ b or a × (1/b); a² as a × a, a³ as a × a × a, a²b as a × a × b, …; 3y as y + y + y or 3 × y; 3(x + y) as 3 × (x + y); (3 + y)/5 as (3 + y) ÷ 5 or (1/5) × (3 + y)
- evaluation of algebraic expressions and formulae
- translation of simple real-world situations into algebraic expressions
- recognising and representing patterns/relationships by finding an algebraic expression for the nth term
- addition and subtraction of linear expressions
- simplification of linear expressions such as −2(3x − 5) + 4x; 2x/3 − 3(x − 5)/2
- use brackets and extract common factors
- concept of equation
- solving linear equations in one variable
- solving simple fractional equations that can be reduced to linear equations such as x/3 + (x − 2)/4 = 3; 3/(x − 2) = 6
- formulating a linear equation in one variable to solve problems
- Cartesian coordinates in two dimensions
- graph of a set of ordered pairs as a representation of a relationship between two variables
- linear functions y = ax + b
- graphs of linear functions
- the gradient of a linear graph as the ratio of the vertical change to the horizontal change (positive and negative gradients)
- right, acute, obtuse and reflex angles
- vertically opposite angles, angles on a straight line, angles at a point
- angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles
- properties of triangles, special quadrilaterals and regular polygons (pentagon, hexagon, octagon and decagon), including symmetry properties
- classifying special quadrilaterals on the basis of their properties
- angle sum of interior and exterior angles of any convex polygon
- construction of simple geometrical figures from given data using compasses, ruler, set squares and protractors, where appropriate
- area of parallelogram and trapezium
- problems involving perimeter and area of composite plane figures
- volume and surface area of prism and cylinder
- conversion between cm² and m² , and between cm³ and m³
- problems involving volume and surface area of composite solids
- simple concepts in collecting, classifying and tabulating data
- analysis and interpretation of: • tables • bar graphs • pictograms • line graphs • pie charts
- purposes and uses, advantages and disadvantages of the different forms of statistical representations
- explaining why a given statistical diagram leads to misinterpretation of data
- map scales (distance and area)
- direct and inverse proportion
- expansion of the product of algebraic expressions
- changing the subject of a formula
- finding the value of an unknown quantity in a given formula
- use of: • (a + b)² = a^2 + 2ab + b^2 • (a − b)² = a^2 − 2ab + b^2 • a^2 − b^2 = (a + b)(a − b)
- factorisation of linear expressions of the form ax + bx + kay + kby
- factorisation of quadratic expressions ax^2 + bx + c
- multiplication and division of simple algebraic fractions such as (3a/(4b²))(5ab/3); (3a/4) ÷ (9a²/10)
- addition and subtraction of algebraic fractions with linear or quadratic denominator such as 1/(x − 2) + 2/(x − 3); 1/(x² − 9) + 2/(x − 3); 1/(x − 3) + 2/(x − 3)²
- concept of equation and inequality
- solving simple inequalities in the form ax + b ≤ c and ax + b < c and representing the solutions on the number line
- graphs of linear equations in two variables (ax + by = c)
- solving simultaneous linear equations in two variables by: • substitution and elimination methods • graphical method
- solving quadratic equations in one variable by factorisation
- formulating a pair of linear equations in two variables to solve problems
- quadratic functions y = ax^2 + bx + c
- graphs of quadratic functions and their properties: • positive or negative coefficient of x^2 • maximum and minimum points • symmetry
- congruent figures
- similar figures
- properties of similar triangles and polygons: • corresponding angles are equal • corresponding sides are proportional
- enlargement and reduction of a plane figure
- solving simple problems involving congruence and similarity
- volume and surface area of pyramid, cone and sphere
- use of Pythagoras’ theorem
- determining whether a triangle is right-angled given the lengths of three sides
- use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angled triangles
- analysis and interpretation of: • dot diagrams • histograms • stem-and-leaf diagrams
- purposes and uses, advantages and disadvantages of the different forms of statistical representations
- explaining why a given statistical diagram leads to misinterpretation of data
- mean, mode and median as measures of central tendency for a set of data
- purposes and use of mean, mode and median
- calculation of the mean for grouped data
- probability as a measure of chance
- probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability)
- use of standard form A×10^n , where n is an integer, and 1 ≤ A < 10
- positive, negative, zero and fractional indices
- laws of indices
- solving quadratic equations in one variable by: • use of formula • completing the square for y = x^2 + px + q • graphical method
- solving fractional equations that can be reduced to quadratic equations such as 6/(x + 4) = x + 3; 1/(x − 2) + 2/(x − 3) = 5
- solving linear inequalities in one variable (including simultaneous inequalities) and representing the solution on the number line
- formulating a quadratic equation in one variable to solve problems
- sketching the graphs of quadratic functions given in the form: • y = (x − p)² + q • y = −(x − p)² + q • y = (x − a)(x − b) • y = −(x − a)(x − b)
- graphs of power functions y = ax^n , where n = −2, −1, 0, 1, 2, 3, and simple sums of not more than three of these
- graphs of exponential functions y = ka^x , where a is a positive integer, a > 1
- estimation of the gradient of a curve by drawing a tangent
- use of set language and the following notation: Union of A and B A∪B Intersection of A and B A∩B Number of elements in set A n(A) “... is an element of ...” ∈ “... is not an element of ...” ∉ Complement of set A A′ The empty set ∅ Universal set ξ A is a subset of B A⊆B A is a not a subset of B A⊈B A is a (proper) subset of B A⊂B A is a not a (proper) subset of B A⊄B
- union and intersection of two sets
- Venn diagrams
- display of information in the form of a matrix of any order
- interpreting the data in a given matrix
- product of a scalar quantity and a matrix
- problems involving addition, subtraction and multiplication of matrices
- scale drawings
- properties and construction of perpendicular bisectors of line segments and angle bisectors
- determining whether two triangles are: • congruent • similar
- ratio of areas of similar plane figures
- ratio of volumes of similar solids
- symmetry properties of circles: • 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 • the line joining an external point to the centre of the circle bisects the angle between the tangents
- angle properties of circles: • angle in a semicircle is a right angle • angle between tangent and radius of a circle is a right angle • angle at the centre is twice the angle at the circumference • angles in the same segment are equal • angles in opposite segments are supplementary
- finding the gradient of a straight line given the coordinates of two points on it
- finding the length of a line segment given the coordinates of its end points
- interpreting and finding the equation of a straight line graph in the form y = mx + c
- geometric problems involving the use of coordinates
- use of notations: column vector (x, y), vector AB, vector a, magnitude |AB| and |a|
- representing a vector as a directed line segment
- translation by a vector
- position vectors
- magnitude of a column vector (x, y) as √(x² + y²)
- use of sum and difference of two vectors to express given vectors in terms of two coplanar vectors
- multiplication of a vector by a scalar
- geometric problems involving the use of vectors
- arc length, sector area and area of a segment of a circle
- use of radian measure of angle (including conversion between radians and degrees)
- extending sine and cosine to obtuse angles
- use of the formula ½ab sin C for the area of a triangle
- use of sine rule and cosine rule for any triangle
- problems in two and three dimensions including those involving angles of elevation and depression and bearings
- quartiles and percentiles
- range, interquartile range and standard deviation as measures of spread for a set of data
- analysis and interpretation of: • cumulative frequency diagrams • box-and-whisker plots
- purposes and uses, advantages and disadvantages of the different forms of statistical representations
- calculation of the standard deviation for a set of data (grouped and ungrouped)
- using the mean and standard deviation to compare two sets of data
- probability of simple combined events (including using possibility diagrams and tree diagrams, where appropriate)
- addition and multiplication of probabilities (mutually exclusive events and independent events)