Learning Units

recognise the tests of divisibility of 4, 6, 8 and 9

Students are required to recognise tests of divisibility of 4, 6, 8 and 9. "Tests of divisibility" means the methods of identifying whether a positive integer IS or IS NOT divisible by the specified positive integer. Students already know divisibility rules for 2, 3, 5, and 10 from KS2. The test for 6 is typically stated as: a number is divisible by 6 if and only if it passes both the tests for 2 and 3. Teachers should point out this holds not merely because 6 = 2 × 3, but because 2 and 3 share no common factor larger than 1. The analogous argument need not be proved in the curriculum. Counter-examples such as 4, 12, 20, 28, 36 (divisible by both 2 and 4 but not by 8) help students see that composite-number rules cannot always be formed by simple combination. Proofs or explanations of why the rules hold are optional enrichment beyond the curriculum requirement. This objective supports LO 1.3 (prime factorisation) and LO 1.4 (GCD/LCM).

understand the concept of power

The concept of power is not part of the primary Mathematics curriculum. This objective introduces it as a tool for numerical computation to support subsequent objectives. Students are required to: - compute a given power of any positive integer (e.g. $3^4 = 3 \times 3 \times 3 \times 3 = 81$); - express a result using power notation (e.g. write 81 as $3^4$). Computations combining powers (e.g. $7^2 \times 7^3 = 7^5$) are NOT required at this stage; they are covered in LU 10 "Laws of integral indices". Teachers may introduce the word "exponent" as an alternative to "index".

perform prime factorisation of positive integers

Students already know about prime and composite numbers from KS2 (4N3), including that 1 is neither prime nor composite. In this objective students are required to decompose any positive integer into a product of prime factors and represent the result using power (index) notation (e.g. $360 = 2^3 \times 3^2 \times 5$). Teachers may choose whether to introduce the term "index notation" here. This objective directly supports LO 1.4.

find the greatest common divisor and the least common multiple

Students extend their KS2 skills (listing, short division for two numbers) by applying prime factorisation from LO 1.3 and by extending short division to three or more numbers. Students are also required to recognise that H.C.F., gcd, G.C.D., etc. are all short forms of "greatest common divisor". This objective is not limited to two numbers. Teachers should use examples to illustrate that, when finding the LCM of more than two numbers by short division, the order of division does not affect the result. Over-complicated operations or exceedingly large numbers should be avoided, as the primary purpose is to support algebraic simplification in later stages (see connections to KS3.LU11 and KS4.CP_LO4.4).

perform mixed arithmetic operations of positive integers involving multiple levels of brackets

In KS2 (3N4), students used brackets in mixed operations but were not required to handle multiple levels of nesting. That restriction is lifted in KS3. Students are required to evaluate expressions with multiple levels of brackets, e.g. 12 + (7 – (5 – 2)) or ((35 – 20) – (5 + 7)) × 2. Teachers may introduce the three standard bracket notations: ( ), [ ], { }.

perform mixed arithmetic operations of fractions and decimals

In KS2, mixed arithmetic operations of three numbers (integers, fractions, decimals) were required, but the restriction of denominators ≤ 12 and no more than three operands applied. Both restrictions are lifted in KS3. Together with LO 1.5, students should be able to perform mixed arithmetic operations of integers, fractions, and decimals with multiple levels of brackets. Over-complicated computations should still be avoided.

understand the laws of positive integral indices

The five index laws for positive integral exponents are established: (1) a^m × a^n = a^(m+n), (2) a^m ÷ a^n = a^(m−n) (m > n), (3) (a^m)^n = a^(mn), (4) (ab)^n = a^n b^n, (5) (a/b)^n = a^n/b^n. Students derive these laws from the definition of a positive integral index and apply them to simplify expressions. The base a is assumed to be non-zero where division is involved.

understand the definitions of zero exponent and negative exponents

The zero exponent is defined as a^0 = 1 (a ≠ 0), motivated by the pattern of dividing by a or by extending law (2). A negative exponent is defined as a^(−n) = 1/a^n (a ≠ 0, n a positive integer). Students should be able to evaluate numerical expressions and simplify algebraic expressions using these definitions.

understand the laws of integral indices

The five index laws established in LO 10.1 are extended to all integral (positive, zero, and negative) exponents. Students simplify expressions involving combinations of integral index laws. Expressions with fractional exponents are not required at this stage.

understand scientific notations

Scientific notation expresses a number as a × 10^n where 1 ≤ a < 10 and n is an integer. Students convert between ordinary decimal notation and scientific notation for both very large and very small numbers. They perform multiplication and division of numbers in scientific notation using index laws. Rounding to a given number of significant figures is required when expressing results in scientific notation.

understand the binary number system and the denary number system

Non-foundation topic. Students understand that the denary (base-10) system uses digits 0–9 and that each place has value 10^n. The binary (base-2) system uses only digits 0 and 1, with each place having value 2^n. Students convert positive integers between binary and denary representations. Addition of binary numbers may be included; subtraction and other operations are not required.

understand other numeral systems, such as the hexadecimal number system

Enrichment topic. Students explore numeral systems other than binary and denary, with the hexadecimal (base-16) system as the primary example (digits 0–9 and A–F). Conversion between hexadecimal and denary (and possibly binary) is the main activity. Applications in computing (e.g. colour codes, memory addresses) may motivate the topic. Formal treatment of general base-n arithmetic is not required.

understand the concept of polynomials

Students learn the definitions of polynomial, term, coefficient, degree of a term, degree of a polynomial, and leading coefficient. They distinguish between monomials, binomials, trinomials, and general polynomials. Students recognise polynomials in one or two variables. Expressions involving negative or fractional exponents (not polynomials) are contrasted. Students also learn to arrange a polynomial in descending order of degree.

perform addition, subtraction, multiplication and their mixed operations of polynomials

Addition and subtraction involve collecting like terms. Multiplication of a polynomial by a monomial applies the distributive law. Multiplication of two polynomials (including two binomials and a binomial by a trinomial) uses repeated application of the distributive law, illustrated by the area model. Mixed operations may involve brackets and require correct order of operations. Division of a polynomial by a monomial is included; polynomial long division is not required at this stage.

factorise polynomials

Students factorise polynomials by the following methods: (1) Taking out the highest common factor (HCF) of all terms. (2) Grouping terms. (3) Using algebraic identities (cross-referencing LU12): difference of two squares $a^2 - b^2 = (a+b)(a-b)$, perfect square trinomials $a^2 \pm 2ab + b^2 = (a \pm b)^2$. (4) The cross method (factor method) for factorising trinomials $ax^2 + bx + c$. Students verify factorisation by expanding. Over-complicated expressions requiring multiple passes of grouping are not required.

understand the concept of identities

An identity is an equation that is true for every value of the variable(s), in contrast to a conditional equation which is true only for specific values. Students learn to verify a given identity by expanding both sides and checking they are equal, or by testing several values. The symbol ≡ (identical to) may be introduced informally. Students should be able to distinguish identities from equations.

use identities to expand algebraic expressions

The required identities for expansion are: (1) $(a + b)^2 = a^2 + 2ab + b^2$ (2) $(a - b)^2 = a^2 - 2ab + b^2$ (3) $(a + b)(a - b) = a^2 - b^2$ Students apply these to expand expressions where a and b may themselves be monomials or simple expressions (e.g. $(2x + 3)^2$, $(x^2 - 1)(x^2 + 1)$). Expansion of $(a + b)^3$ is not required.

use identities to factorise polynomials

Students recognise and apply the identities in reverse to factorise: (1) $a^2 + 2ab + b^2 = (a + b)^2$ [perfect square trinomial] (2) $a^2 - 2ab + b^2 = (a - b)^2$ [perfect square trinomial] (3) $a^2 - b^2 = (a + b)(a - b)$ [difference of two squares] Students check that a given polynomial matches the identity pattern before applying it. Combined factorisation (e.g. first taking out HCF, then applying an identity) is included.

perform operations of algebraic fractions

Operations include simplification (cancelling common factors from numerator and denominator), multiplication, division, addition, and subtraction of algebraic fractions. For addition and subtraction, students find the lowest common denominator (LCD) — which may require factorising the denominators. The resulting expression should be simplified to lowest terms. Over-complicated expressions with high-degree or multi-variable denominators should be avoided.

use substitution to find the values of unknowns in the formulae

Students substitute given values into a formula to compute the value of the subject (the isolated variable on the left-hand side). Formulae from geometry (e.g. area of a circle $A = \pi r^2$), physics (e.g. speed v = d/t), and finance (e.g. simple interest I = PRT) are appropriate. Students should handle formulae with multiple variables and be comfortable substituting directed numbers or fractions.

change the subject of formulae not involving radical signs

Students rearrange a formula to express a different variable as the subject. For example, rearranging $v = u + at$ to give $t = \frac{v - u}{a}$. Formulae involving addition, subtraction, multiplication, division, and integer powers are included. Formulae involving square roots or other radical signs are not required at this stage (radical-sign formulae are treated in KS4). Students verify the rearrangement by substituting back into the original formula.

understand the concept of inequalities

Students understand the meaning of the four inequality symbols (<, >, ≤, ≥) and their relationship to the number line. They compare directed numbers using inequality notation and understand that a ≤ b means a < b or a = b. The difference between an equation (specific solution) and an inequality (set of solutions) is established.

recognise the basic properties of inequalities

The properties are: (1) Addition/subtraction: adding or subtracting the same quantity to both sides preserves the inequality direction. (2) Multiplication/division by a positive number: preserves the inequality direction. (3) Multiplication/division by a negative number: reverses the inequality direction. Students verify these properties with numerical examples before applying them algebraically. The reversal rule is a common source of error and must be explicitly practised.

solve linear inequalities in one unknown

Students solve linear inequalities of the form ax + b < cx + d (and with ≤, >, ≥). The solution is a set of real numbers, represented on a number line using open or closed circles and a ray or segment. Simultaneous inequalities (finding the intersection or union of two solution sets) may be included. Students should present solutions in inequality notation and as number line diagrams.

solve problems involving linear inequalities in one unknown

Students formulate a linear inequality from a word problem, solve it, and interpret the solution in context. Problems include minimum/maximum quantity problems (e.g. how many items can be bought within a budget) and constraint problems (e.g. ranges of measurements satisfying a condition). Answers are stated as inequality statements or as sets (e.g. "x ≥ 5, so at least 5 items").

recognise the concept of errors in measurement

All physical measurements involve a degree of approximation. The error of a measurement is the difference between the measured value and the true value. Since the true value is generally unknown, the maximum absolute error (half the smallest unit of measurement) serves as an upper bound on the error. Students distinguish between the measured value, the range of true values (true value lies within measured value ± maximum absolute error), and the error itself.

recognise the concepts of maximum absolute errors, relative errors and percentage errors

Maximum absolute error = half the smallest graduation unit of the measuring instrument. Relative error = maximum absolute error ÷ measured value. Percentage error = relative error × 100%. Students calculate these quantities for given measurements and understand that relative/percentage error provides a fairer comparison of accuracy across measurements of different sizes (e.g. 0.5 cm error in a 5 cm measurement is much larger relatively than in a 500 cm measurement).

solve problems related to errors

Problems involve calculating the maximum absolute error, relative error, and percentage error for given measurements, finding the range of true values, and choosing the more accurate of two measurements based on percentage error. Students may be asked to determine the required precision of a measurement to keep the percentage error below a given threshold. Over-complicated multi-step error propagation is not required.

design estimation strategies in measurement according to the contexts and judge the reasonableness of the results obtained

Enrichment topic. Students design measurement and estimation strategies that control the size of the error for a specific context (e.g. estimating the area of an irregular shape, or determining the height of a building indirectly). They evaluate the reasonableness of their results by comparing the estimated percentage error to an acceptable threshold. Formal error propagation formulae are not required.

understand the formula for arc lengths of circles

The arc length formula is derived by proportionality: an arc subtended by a central angle of $\theta^\circ$ is (θ/360) of the full circumference. Arc length = $\frac{\theta}{360} \times 2\pi r$. Students derive this formula from the full-circle circumference formula and apply it to calculate arc lengths given the radius and central angle. The distinction between minor arc and major arc is noted. Students should also be able to find the radius or angle given the arc length and the other quantity.

understand the formula for areas of sectors of circles

Similarly, the area of a sector is (θ/360) of the full circle area. Area of sector = $\frac{\theta}{360} \times \pi r^2$. Students derive this from the full-circle area formula. They calculate sector areas given radius and central angle, and find the radius or angle when given the sector area and the other quantity. The perimeter of a sector (arc length plus two radii) is a natural extension.

solve problems related to arc lengths and areas of sectors of circles

Problems include composite figures involving sectors and other shapes (e.g. finding the shaded area between a square and an inscribed quarter-circle). Students apply the formulae in context, giving answers in terms of π where appropriate or as decimal approximations to a stated accuracy.

recognise the Circle Dissection Algorithm of the ancient Chinese mathematician Liu Hui and further recognise Huilu and Tsulu (approximations of π)

Enrichment topic. Liu Hui (3rd century CE) approximated π by inscribing a regular polygon in a circle and repeatedly doubling the number of sides. As the number of sides increases, the polygon's perimeter approaches the circumference. Students explore how this method produces increasingly accurate approximations of π. Huilu (徽率) is Liu Hui's approximation $\pi \approx 3.14159$; Tsulu (祖率) is Zu Chongzhi's more precise approximation 355/113. This historical context reinforces the concept of approximation and the value of iterative methods.

recognise the concepts of right prisms, right circular cylinders, right pyramids, right circular cones, regular prisms, regular pyramids, polyhedra and spheres

A right prism has a uniform cross-section and lateral edges perpendicular to the bases; a regular prism has a regular polygon as its base. A right pyramid has its apex directly above the centroid of the base; a regular pyramid has a regular polygon base. A right circular cylinder is a prism with circular cross- section. A right circular cone has its apex above the centre of the circular base. A polyhedron is a solid bounded by flat polygonal faces; a regular polyhedron (Platonic solid) has congruent regular polygonal faces with the same number meeting at each vertex. Spheres are introduced as solids where all surface points are equidistant from the centre.

recognise the sections of prisms, circular cylinders, pyramids, circular cones, polyhedra and spheres

A section (cross-section) is the 2-D figure obtained by cutting a solid with a plane. Students identify the shape of cross-sections of common solids: — prism: uniform cross-section parallel to the bases is congruent to the base; oblique cuts produce other shapes. — cylinder: circle (cut parallel to base), rectangle (cut through axis), ellipse (oblique). — pyramid/cone: similar polygon/circle (parallel to base), triangle (through apex). — sphere: always a circle (or point). Students describe cross-sections verbally and by sketch.

sketch the 2-D representations of 3-D figures

Students sketch common 3-D solids using conventions: visible edges as solid lines, hidden edges as dashed lines, curved surfaces suggested by ellipses. The two figures in the source PDF (right pyramid, right circular cylinder) illustrate the expected level of accuracy. Students should be able to sketch all solids covered in LO 17.1. Formal perspective drawing or isometric drawing skills are not required.

recognise the three orthographic views of 3-D figures

Enrichment topic. Orthographic projection represents a 3-D solid by three views: front view (elevation), side view (side elevation), and top view (plan). Students identify the three views of simple solids and, conversely, identify the solid given its three views. This is a useful skill in engineering and design contexts.

recognise Euler's formula and explore the number of regular polyhedra (Platonic solids)

Enrichment topic. Euler's formula states $V - E + F = 2$ for any convex polyhedron, where V, E, F are the numbers of vertices, edges, and faces respectively. Students verify the formula for several polyhedra by counting. Using Euler's formula together with the properties of regular polygons, students explore why there are exactly five regular (Platonic) solids: tetrahedron (4 triangular faces), cube (6 square faces), octahedron (8 triangular faces), dodecahedron (12 pentagonal faces), icosahedron (20 triangular faces). A combinatorial argument at an intuitive level is appropriate; a formal proof is not required.

recognise the formulae for volumes of prisms, circular cylinders, pyramids, right circular cones and spheres

The required volume formulae are: — Prism / cylinder: $V = \text{base area} \times \text{height}$ — Pyramid / cone: $V = \frac{1}{3} \times \text{base area} \times \text{height}$ — Sphere: $V = \frac{4}{3}\pi r^3$ Students should be able to recall and apply these formulae. Derivations are not required, but the relationship $V(\text{pyramid}) = \frac{1}{3} V(\text{prism with same base and height})$ may be demonstrated by practical activities or dissection arguments.

find the surface areas of right prisms, right circular cylinders, right pyramids, right circular cones and spheres

Surface area is computed by summing the areas of all faces (for polyhedra) or using formulae (for curved surfaces). Key results: — Right prism: sum of two base areas + lateral area (perimeter of base × height). — Right circular cylinder: $2\pi r^2 + 2\pi rh$ (total); or $2\pi rh$ (lateral only). — Right pyramid: base area + (1/2) × perimeter of base × slant height (for regular pyramid). — Right circular cone: $\pi r^2 + \pi rl$ where l is the slant height (total). — Sphere: $4\pi r^2$. Net diagrams (unfolded solids) are a helpful teaching tool. Students should be able to identify the relevant measurements from a net.

recognise the relations among lengths, areas and volumes of similar figures and solve related problems

For two similar figures with linear scale factor k: — Ratio of corresponding lengths = k. — Ratio of corresponding areas $= k^2$. — Ratio of corresponding volumes $= k^3$. Students apply these relationships to find unknown lengths, areas, or volumes when the scale factor or one pair of corresponding measurements is given. Problems may involve similar containers (e.g. two similar conical vessels) or map-scale problems extended to three dimensions.

solve problems involving volumes and surface areas

Composite-figure problems require combining the formulae from LO 18.1 and 18.2. Examples include a solid made up of a cylinder topped by a cone, or a hemisphere on top of a cuboid. Students also encounter practical contexts such as finding the capacity of a container, the amount of material needed to make a solid, or the cost of painting a surface. Answers should be given to a reasonable degree of accuracy (e.g. 3 significant figures or as specified).

explore ways to form a container with the greatest capacity by folding an A4-sized paper with squares cutting from its four corners

Enrichment topic. An open box is formed by cutting equal squares of side x from the four corners of an A4-sized sheet (297 mm × 210 mm) and folding up the sides. The volume $V = x(297 - 2x)(210 - 2x)$. Students compute V for several values of x, plot a table or graph, and estimate the value of x that maximises the volume. This is an introduction to optimisation; formal calculus methods are not required. The activity also reinforces substitution into formulae and unit conversion skills.

understand the concepts and properties of adjacent angles on a straight line, vertically opposite angles and angles at a point

Adjacent angles on a straight line (supplementary, sum = 180°), vertically opposite angles (equal), and angles at a point (sum = 360°) are the three basic angle relationships. Students state these properties, apply them to find unknown angles, and use them in simple proofs. The convention of labelling angles with letters (e.g. ∠AOB or ∠a) is established here.

understand the concepts of corresponding angles, alternate interior angles and interior angles

When a transversal crosses two straight lines (not necessarily parallel), it creates four pairs of corresponding angles (F-shape), two pairs of alternate interior angles (Z-shape), and two pairs of co-interior (same-side interior, C-shape) angles. Students identify and name these angle pairs from a diagram. The specific angle relationships (equal or supplementary) hold when the lines are parallel; this is addressed in LO 19.3 and 19.4.

recognise the conditions for two straight lines being parallel

The three tests for parallel lines (given a transversal): (1) Corresponding angles are equal → lines are parallel. (2) Alternate interior angles are equal → lines are parallel. (3) Co-interior (same-side interior) angles are supplementary (sum = 180°) → lines are parallel. Students apply these tests to determine whether two given lines are parallel from angle measurements.

recognise the angle properties associated with parallel lines

The converse of LO 19.3: if lines are parallel, then corresponding angles are equal, alternate interior angles are equal, and co-interior angles are supplementary. Students use these properties to find unknown angles in figures involving parallel lines and transversals, giving reasons for each step (e.g. "corresponding angles, AB ∥ CD").

understand the properties of the interior and exterior angles of triangles

Interior angle sum of a triangle = 180° (proved using the parallel-line property by drawing a line through one vertex parallel to the opposite side). An exterior angle of a triangle equals the sum of the two non-adjacent interior angles (exterior angle theorem). Students apply these results to find unknown angles in triangles and in composite figures. Simple angle-chasing proofs (stating reasons at each step) are expected.

understand the concept of directed numbers

Students are introduced to positive and negative numbers as directed numbers that represent quantities with both magnitude and direction (e.g. temperature above/below zero, altitude above/below sea level, credit/debit). The number line is used to order directed numbers. Students learn to compare directed numbers and to place them correctly on the number line. The absolute value (modulus) of a directed number is introduced as its distance from zero, without requiring formal absolute-value notation.

perform mixed arithmetic operations of directed numbers

Students perform addition, subtraction, multiplication, and division of directed numbers, including integers, fractions, and decimals. Mixed operations involving multiple operations and brackets are included. The multiplication table of directed integers (−3 to +3) is a useful reference for establishing sign rules. Students should be proficient in applying the rules of signs without relying solely on the number line for calculation.

solve problems involving directed numbers

Problems are drawn from everyday contexts such as temperature changes, sea depth, financial transactions, and score differences in games. Students formulate and solve arithmetic expressions using directed numbers to answer contextual questions. Over-complicated multi-step calculations should be avoided; the emphasis is on correct interpretation and sign handling.

understand the concept of regular polygons

A regular polygon has all sides equal in length and all interior angles equal. Students identify regular polygons (equilateral triangle, square, regular pentagon, hexagon, etc.) and know their names for 3 to 12 sides. They understand that a regular polygon is both equilateral and equiangular.

understand the formula for the sum of the interior angles of a polygon

By dividing an n-gon into (n − 2) triangles (using diagonals from one vertex), the sum of interior angles = $(n - 2) \times 180^\circ$. Students derive this formula, apply it to find unknown angles in polygons, and calculate the size of each interior angle of a regular n-gon: [$(n - 2) \times 180^\circ$] / n. Problems may require finding n given the angle sum or one interior angle.

understand the formula for the sum of the exterior angles of a convex polygon

The sum of the exterior angles of any convex polygon (one at each vertex) is always 360°. Students verify this for specific polygons and use it to find individual exterior angles of regular polygons (360°/n) and related interior angles. The relationship between interior and exterior angles (supplementary) is reinforced.

appreciate the triangles, quadrilaterals, and regular polygons that tessellate in the plane

Non-foundation topic. A tessellation is a tiling of the plane with no gaps or overlaps. Students explore which regular polygons can tessellate alone (equilateral triangle, square, regular hexagon) and why — because their interior angles must divide 360° evenly. They also investigate semi-regular tessellations (using two or more types of regular polygon). All triangles and all quadrilaterals tessellate; students explore why by examining angle sums. Artistic and real-life examples (e.g. honeycomb, floor tiles) motivate the topic.

construct equilateral triangles and regular hexagons with compasses and straightedge

Non-foundation topic. Students construct an equilateral triangle by drawing two arcs of the same radius from each endpoint of a line segment. A regular hexagon is constructed by marking off the radius around a circle six times (since the side of an inscribed regular hexagon equals the radius). These constructions require only compasses and an unmarked straightedge; protractors and rulers for measuring are not used. Students verify the constructions using angle and length properties.

explore ways to construct regular pentagons with compasses and straightedge

Enrichment topic. The interior angle of a regular pentagon is 108°. Students explore how to construct a regular pentagon using compasses and straightedge, connecting the construction to the golden ratio (φ = (1 + √5)/2) and the diagonal-to-side ratio of a regular pentagon. One classical construction involves bisecting a right angle to locate 54° and building from there. Full rigorous proof is not required; the emphasis is on exploration and verification.

understand the concept of congruent triangles

Two triangles are congruent if one can be mapped onto the other by a combination of translations, rotations, and reflections (isometries). Congruent triangles have all corresponding sides equal and all corresponding angles equal. Students learn to identify and name corresponding vertices, sides, and angles when two triangles are stated to be congruent (e.g. △ABC ≅ △DEF implies AB = DE, ∠A = ∠D, etc.).

recognise the conditions for congruent triangles

The five congruence conditions are: (1) SSS — three sides equal. (2) SAS — two sides and the included angle equal. (3) ASA — two angles and the included side equal. (4) AAS — two angles and a non-included side equal. (5) RHS — right angle, hypotenuse, and one other side equal (for right-angled triangles). Students apply these conditions to prove that two triangles are congruent and to deduce further equalities (sides and angles) from the congruence. The condition AAA is not sufficient (produces similar but not necessarily congruent triangles).

understand the property of isosceles triangles

The base angles of an isosceles triangle are equal (proved using triangle congruence: the triangle is congruent to its mirror image). Students apply this property to find unknown angles and side lengths in isosceles triangles and in composite figures containing them.

understand the condition for isosceles triangles

The converse: if two angles of a triangle are equal, then the sides opposite those angles are equal (i.e. the triangle is isosceles). Students prove this by the ASA or AAS congruence condition and apply it to identify isosceles triangles in geometric contexts.

construct angle bisectors, perpendicular bisectors, perpendicular lines, parallel lines, special angles and squares with compasses and straightedge

Non-foundation topic. Students perform the following compass-and-straightedge constructions and provide geometric justifications: — Bisect a given angle. — Construct the perpendicular bisector of a line segment. — Erect a perpendicular to a line at a given point on it, or from a given point not on the line. — Construct a line through a given point parallel to a given line. — Construct angles of 30°, 45°, 60°, 90°, and their combinations. — Construct a square given one side. Students should justify each construction step using congruence results. Measurement (ruler or protractor) is not permitted in constructions.

recognise the concept of congruent 2-D figures

Non-foundation topic. Congruence extends beyond triangles to any pair of 2-D figures: two figures are congruent if one can be mapped onto the other by isometries (translation, rotation, reflection, or a combination). Students identify congruent figures in familiar contexts and understand that congruent figures have equal corresponding lengths and angles. The concept lays the groundwork for transformation geometry in KS4.

explore the angles that can be constructed with compasses and straightedge

Enrichment topic. Starting from an angle of 60° (equilateral triangle) and using bisection, students generate angles of 60°, 30°, 15°, 45°, 90°, 120°, 135°, etc. They investigate which multiples of 1° can be constructed exactly (those expressible using combinations of bisection and addition of constructible angles). The impossibility of trisecting an arbitrary angle with only compasses and straightedge may be mentioned as an historical result; formal proof is not required.

understand the concept of similar triangles

Two triangles are similar (△ABC ~ △DEF) if their corresponding angles are equal and corresponding sides are proportional (the ratio of any pair of corresponding sides is constant — the scale factor). Congruence is the special case where the scale factor equals 1. Students identify corresponding vertices, write the ratio of corresponding sides, and find unknown lengths by proportion.

recognise the conditions for similar triangles

The three similarity conditions are: (1) AAA (AA) — two pairs of equal angles (since angle sum = 180°, the third is automatic). (2) SSS — three pairs of sides in the same ratio. (3) SAS — two pairs of sides in the same ratio with the included angle equal. Students apply these conditions to prove triangles similar and deduce unknown lengths or angles. Applications include indirect measurement (e.g. finding the height of a tree using shadow lengths) and map scales.

recognise the concept of similar 2-D figures

Two 2-D figures are similar if one can be obtained from the other by an enlargement (or reduction) combined with isometries. All corresponding lengths are in the same ratio (scale factor k) and all corresponding angles are equal. Students recognise similar figures in familiar contexts (maps, scale drawings, photographs). The area ratio is $k^2$ for similar figures with scale factor k.

explore shapes of fractals

Enrichment topic. A fractal is a self-similar pattern: each part resembles the whole when magnified. Students explore well-known examples such as the Sierpiński triangle (subdividing an equilateral triangle repeatedly) and the Koch snowflake (iterating on triangle sides). Activities include computing the perimeter and area of successive iterations to observe convergence and divergence. Formal definitions of fractal dimension are not required; the emphasis is on pattern recognition and the concept of self-similarity.

understand the properties of parallelograms

Properties of a parallelogram: opposite sides are parallel and equal; opposite angles are equal; consecutive angles are supplementary; diagonals bisect each other. Students prove these properties using triangle congruence (by drawing a diagonal to split the parallelogram into two triangles). They apply the properties to find unknown lengths and angles.

understand the properties of rectangles, rhombuses and squares

Each is a special parallelogram with additional properties: — Rectangle: all angles 90°; diagonals equal. — Rhombus: all sides equal; diagonals perpendicular bisectors of each other; diagonals bisect the vertex angles. — Square: has all properties of both rectangle and rhombus. Students prove these additional properties using triangle congruence or the parallelogram properties from LO 23.1. They apply them to calculate angles and lengths in given figures.

understand the conditions for parallelograms

Non-foundation topic. The five conditions sufficient to guarantee a quadrilateral is a parallelogram: (1) Both pairs of opposite sides are parallel. (2) Both pairs of opposite sides are equal. (3) One pair of sides is both equal and parallel. (4) Diagonals bisect each other. (5) Both pairs of opposite angles are equal. Students prove that each condition is sufficient and apply them in problems to determine whether a given quadrilateral is a parallelogram.

apply the above properties or conditions to perform simple geometric proofs

Non-foundation topic. Students write structured, two-column or flowing-prose geometric proofs using the properties and conditions of parallelograms, rectangles, rhombuses, and squares. Each step must cite the geometric reason (e.g. "diagonals of a parallelogram bisect each other"). Proofs should be concise and logically sequenced; over-complex multi-stage proofs are not required.

understand the mid-point theorem and the intercept theorem

Non-foundation topic. Mid-point theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length. Intercept theorem (also called the basic proportionality theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Students prove these theorems using similar triangles or congruence and apply them to find unknown lengths in triangles and quadrilaterals.

explore the conditions for congruent quadrilaterals

Enrichment topic. Unlike triangles, a quadrilateral is not determined by its four sides alone (SSS does not hold for quadrilaterals). Students explore what additional information (e.g. one diagonal, one angle, or a set of angles) is needed to make two quadrilaterals congruent. The investigation develops logical reasoning and deepens understanding of the difference between triangles and more general polygons.

understand the properties of angle bisectors and perpendicular bisectors

The angle bisector locus: every point on the angle bisector of an angle is equidistant from the two sides of the angle. Students prove this using congruent triangles (the perpendicular distances from a point on the bisector to the two sides are equal by AAS or RHS congruence). The perpendicular bisector locus: every point on the perpendicular bisector of a segment is equidistant from the two endpoints. Students prove this similarly. These locus properties are the basis for the concurrence theorems.

understand the concurrence of angle bisectors and the concurrence of perpendicular bisectors of a triangle

Non-foundation topic. Incentre: the three angle bisectors of a triangle meet at a single point (the incentre), which is equidistant from all three sides and is the centre of the inscribed circle (incircle). Students prove concurrence using the angle bisector locus property. Circumcentre: the three perpendicular bisectors of the sides of a triangle meet at a single point (the circumcentre), which is equidistant from all three vertices and is the centre of the circumscribed circle (circumcircle). Students prove concurrence using the perpendicular bisector locus property. Both constructions are achievable with compasses and straightedge.

recognise the concurrence of medians and the concurrence of altitudes of a triangle

Non-foundation topic. Centroid: the three medians (each joining a vertex to the midpoint of the opposite side) are concurrent at the centroid. The centroid divides each median in the ratio 2:1 from the vertex. Students verify this property numerically or by construction and may explore it using coordinates. The centroid is the centre of mass of a uniform triangular lamina. Orthocentre: the three altitudes (each from a vertex perpendicular to the opposite side) are concurrent at the orthocentre. Students verify this by construction. Formal proofs of these concurrence results using vectors or coordinate methods are not required at KS3.

understand the Pythagoras' theorem

For a right-angled triangle with legs a and b and hypotenuse c: $a^2 + b^2 = c^2$. Students see at least one proof (e.g. the area-dissection proof rearranging four congruent right triangles into two squares). They state the theorem correctly, identifying the hypotenuse as the side opposite the right angle, and apply it to find the unknown side given the other two. Results involving irrational lengths (e.g. √2, √5) should be left in surd form or evaluated to appropriate accuracy.

recognise the converse of Pythagoras' theorem

The converse states: if $a^2 + b^2 = c^2$ for the three sides of a triangle, then the triangle is right-angled (with the right angle opposite the longest side). Students verify the converse with examples (e.g. 3-4-5, 5-12-13) and apply it to determine whether a triangle with given side lengths is right-angled, acute, or obtuse (by comparing $a^2 + b^2$ with $c^2$).

solve problems related to Pythagoras' theorem and its converse

Applications include finding distances in 2-D figures (e.g. diagonal of a rectangle, altitude of an equilateral triangle), real-life problems (e.g. the length of a ladder against a wall, the shortest path across a field), and composite geometric figures. Students check whether a given set of three lengths forms a right-angled triangle using the converse. Answers involving surds should be simplified.

explore Pythagorean triples

Enrichment topic. A Pythagorean triple is a set of three positive integers (a, b, c) satisfying $a^2 + b^2 = c^2$. The primitive triples (where gcd(a,b,c)=1) include (3,4,5), (5,12,13), (8,15,17), and (7,24,25). Students investigate the parametric formula $a = m^2 - n^2$, $b = 2mn$, $c = m^2 + n^2$ (m > n > 0, integers) for generating Pythagorean triples. They verify that this formula always produces a valid triple and generate several examples. Historical context (Babylonian clay tablet Plimpton 322) may be mentioned.

recognise the rectangular coordinate system

Students learn the Cartesian coordinate system: origin, x-axis, y-axis, four quadrants, and the coordinate pair (x, y). They plot points given their coordinates and read coordinates of plotted points. They recognise that the x-coordinate gives horizontal position and the y-coordinate gives vertical position.

find the distance between two points on a horizontal line and the distance between two points on a vertical line

For two points on a horizontal line (same y-coordinate): distance = $|x_1 - x_2|$. For two points on a vertical line (same x-coordinate): distance = $|y_1 - y_2|$. This prepares for the general distance formula (LO 26.5) and is used in area calculations (LO 26.3).

find areas of polygons in the rectangular coordinate plane

Students find areas of triangles and other polygons whose vertices are given as coordinate pairs. Methods include: decomposing into simpler shapes (right triangles and rectangles), using the bounding-box subtraction method, and applying the area formula $\frac{1}{2} |\text{base} \times \text{height}|$ where base and height are read from coordinates. The Shoelace formula (Gauss's area formula) may be introduced for general polygons but is not required.

recognise the effect of transformations on a point in the rectangular coordinate plane

Students identify the image of a point (x, y) under the four basic transformations: — Reflection in the x-axis: $(x, y) \rightarrow (x, -y)$. — Reflection in the y-axis: $(x, y) \rightarrow (-x, y)$. — Reflection in the origin (rotation 180°): $(x, y) \rightarrow (-x, -y)$. — Translation by (a, b): $(x, y) \rightarrow (x+a, y+b)$. Rotation by other angles and general enlargements are not required at this stage.

understand the distance formula

For two points P(x₁, y₁) and Q(x₂, y₂): $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Students derive this formula by constructing a right triangle with horizontal and vertical legs and applying Pythagoras' theorem. They apply the formula to calculate distances and to verify geometric properties (e.g. that a triangle is equilateral by comparing the three side lengths).

understand the mid-point formula and the formula for the internal point of division

Mid-point formula (core): the midpoint of P(x₁, y₁) and Q(x₂, y₂) is $M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$. Students derive this as the average of the coordinates and apply it to find midpoints and to verify that a point is a midpoint. Internal point of division (non-foundation, underlined in curriculum): the point that divides PQ internally in the ratio m:n has coordinates $(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n})$. Students apply this formula to find points dividing a line segment in a given ratio.

understand the slope formula and solve related problems

The slope (gradient) of a straight line through P(x₁, y₁) and Q(x₂, y₂) is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Slope measures the steepness and direction of a line: positive slope rises left-to-right; negative slope falls; zero slope is horizontal; undefined slope is vertical. Students calculate slopes and interpret them in context (e.g. gradient of a road). The connection between slope and the graph of $y = mx + c$ is noted.

recognise the relation between the slopes of parallel lines and the relation between the slopes of perpendicular lines, and solve related problems

Parallel lines: $m_1 = m_2$ (equal slopes). Perpendicular lines: $m_1 \times m_2 = -1$ (negative reciprocal slopes), provided neither line is vertical. Students prove these relationships using the geometry of similar triangles (for parallel) and the Pythagorean condition (for perpendicular), and apply them to determine whether lines are parallel/perpendicular and to find the equation or slope of a line given these conditions.

use coordinate geometry to perform simple geometric proofs

Students use coordinate methods to prove geometric results about polygons, such as: proving that a quadrilateral is a parallelogram (by showing opposite sides have equal slope and length), proving that the diagonals of a square are perpendicular (using the slope condition $m_1 \times m_2 = -1$), or verifying that a triangle is isosceles (by computing side lengths). Proofs should include a clearly stated strategy, coordinate assignments, and a geometric conclusion.

explore the formula for the external point of division

Enrichment topic. The point that divides PQ externally in the ratio m:n (m ≠ n) lies on the line through P and Q but outside the segment PQ. Its coordinates are $(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n})$. Students derive this formula by extending the internal division result and verify it for specific cases. Applications include harmonic division and connections to projective geometry at an informal level.

understand sine, cosine and tangent of angles between 0° and 90°

For an acute angle θ in a right-angled triangle: $\sin \theta$ = opposite / hypotenuse $\cos \theta$ = adjacent / hypotenuse $\tan \theta$ = opposite / adjacent Students use a calculator to find trig ratios for any acute angle and to find the angle given a trig ratio (inverse trig). They solve for unknown sides and angles in right-angled triangles using these ratios. The mnemonic SOH-CAH-TOA is commonly used. Angles are given in degrees; radian measure is not required.

understand the properties of trigonometric ratios

Key properties students should know: — $\sin \theta = \cos(90^\circ - \theta)$ and $\cos \theta = \sin(90^\circ - \theta)$ (complementary angle relationships). — $\sin^2 \theta + \cos^2 \theta = 1$ (Pythagorean identity, derived from Pythagoras' theorem). — $\tan \theta = \frac{\sin \theta}{\cos \theta}$. — As θ increases from 0° to 90°: $\sin \theta$ increases from 0 to 1; $\cos \theta$ decreases from 1 to 0; $\tan \theta$ increases from 0 to ∞ (undefined at 90°). Students use these properties to simplify expressions and solve equations.

understand the exact values of trigonometric ratios of 30°, 45° and 60°

The exact values are derived from the 30-60-90 triangle (half an equilateral triangle with side 2) and the 45-45-90 triangle (isosceles right triangle with legs 1): — sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3 — sin 45° = cos 45° = 1/√2 = √2/2, tan 45° = 1 — sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3 Students should memorise or be able to quickly derive these values and use them to compute exact answers without a calculator.

solve problems related to plane figures

Students apply trigonometric ratios to 2-D problems involving right-angled triangles embedded in more complex plane figures (e.g. finding the height of an isosceles triangle, the diagonal of a rectangle at an angle, or the area of a triangle using the formula ½ab sin C). Problems should be drawn from realistic contexts; over-complicated multi-stage calculations should be avoided.

solve problems involving gradients, angles of elevation, angles of depression and bearings

Gradient of a slope = tan(angle of inclination). Angle of elevation: the angle measured upward from the horizontal to a line of sight to an object above. Angle of depression: the angle measured downward from the horizontal to a line of sight to an object below. Bearing: a three-figure bearing measured clockwise from north (e.g. 045°, 270°). Students solve real-life problems (e.g. height of a building, distance between ships, navigation problems) using these concepts and right-triangle trigonometry. Problems involving non-right-angled triangles (sine rule, cosine rule) are deferred to KS4.

recognise the concepts of discrete data and continuous data

Discrete data can take only specific, separate values (e.g. number of students, shoe size). Continuous data can take any value within a range (e.g. height, temperature, time). Students classify given data sets as discrete or continuous and understand why the distinction matters for how data is organised and displayed. Practical examples from everyday life should be used.

recognise organisation of data without grouping

Students construct frequency tables for ungrouped data (each distinct value has its own tally and frequency). They use tallying to count occurrences and verify that frequencies sum to the total number of data values. Ungrouped organisation is appropriate when the number of distinct values is small (e.g. a die-rolling experiment, grades A–F). Relative frequency (frequency/total) may be included.

recognise organisation of data in groups

When data spans a wide range or is continuous, grouping into class intervals is appropriate. Students construct grouped frequency tables, choosing a suitable class width and writing class boundaries correctly (e.g. 10 ≤ x < 20). Key decisions include the number of classes (typically 5–10) and whether class intervals are equal in width. Cumulative frequency is introduced as a column in the grouped table. Students should understand that grouping loses information about individual values.

recognise stem-and-leaf diagrams and histograms

Stem-and-leaf diagrams: the stem represents the leading digit(s) and each leaf represents the final digit. A back-to-back stem-and-leaf diagram compares two data sets. Students construct these from raw data. Histograms: bars represent frequency (or frequency density) for class intervals; there are no gaps between bars. Students construct histograms from grouped frequency tables with equal class widths. The distinction between a bar chart (discrete data, gaps allowed) and a histogram (continuous data, no gaps) should be noted.

interpret stem-and-leaf diagrams and histograms

Students read values, compare distributions, identify the shape (symmetric, skewed), and determine the mode and approximate median from stem-and-leaf diagrams. From histograms, they estimate frequencies for specific intervals, identify the modal class, and describe the overall distribution shape. They should be able to reconstruct a frequency table from a histogram.

interpret statistical charts representing two different sets of data in daily life

Students interpret comparative charts (e.g. back-to-back stem-and-leaf diagrams, dual-axis charts, side-by-side bar charts, comparative pie charts) that display two data sets simultaneously. They extract information, make comparisons, and draw conclusions. Real-life sources (government statistics, media reports) are appropriate contexts. Students should be aware of the potential for misleading comparisons.

recognise frequency polygons, frequency curves, cumulative frequency polygons and cumulative frequency curves

Frequency polygon: mid-points of histogram bar tops are connected by straight line segments. Frequency curve: a smooth curve through the mid-points. Students construct these from a grouped frequency table. Cumulative frequency polygon (ogive): cumulative frequency is plotted against the upper class boundary and points are connected by line segments. Cumulative frequency curve: a smooth S-shaped curve through these points. Students construct these from a cumulative frequency table.

interpret frequency polygons, frequency curves, cumulative frequency polygons and cumulative frequency curves

Students read approximate frequencies and cumulative frequencies for given values from these diagrams. They use the cumulative frequency curve to estimate the median (value at the 50th percentile), quartiles, and the number/percentage of data values below or above a given threshold. Percentiles and inter-quartile range may be introduced here as extensions; formal treatment is in KS4.

choose appropriate statistical charts to present data

Students select the most appropriate chart type for a given purpose and data type: — Discrete data (few categories): bar chart or pie chart. — Continuous data: histogram or frequency polygon. — Comparing two data sets: back-to-back stem-and-leaf, dual bar chart. — Showing cumulative frequencies: cumulative frequency polygon or curve. — Showing change over time: line graph. Justifications for the choice should refer to the properties of the data and the intended message.

recognise the uses and abuses of statistical charts in daily life

Students identify misleading features in charts, including: — Truncated y-axis (not starting at zero, exaggerating differences). — Inconsistent scale intervals. — 3-D effects that distort proportions. — Omission of sample size or source. — Cherry-picking time periods. Students should be critical readers of statistical displays in media and public discourse, and should be able to redraw a misleading chart in a more honest form.

recognise the concept of approximate values

Extends KS2 rounding skills (to a given place for whole numbers; to nearest tenth or hundredth for decimals). Students are required to round numbers to any specified degree of accuracy, including significant figures.

understand the estimation strategies

Students understand and apply strategies such as front-end estimation, rounding, and compatible numbers. They are not required to prove why a strategy produces a valid estimate.

solve related real-life problems

Problems should be drawn from everyday contexts. Students apply the estimation strategies from LO 3.2 to arrive at reasonable answers without requiring exact computation.

design numerical estimation strategies according to the contexts and judge the reasonableness of the results obtained

Enrichment topic. Teachers may, according to students' abilities and interests, guide students to design their own estimation strategies suited to specific contexts and to evaluate whether a given result is reasonable.

understand the concepts of mean, median and mode/modal class

Mean: the arithmetic average; sum of all values divided by the number of values. Median: the middle value when data is arranged in order; for an even number of values, the median is the mean of the two middle values. Mode: the most frequently occurring value; a data set may have no mode, one mode, or multiple modes. Modal class: the class interval with the highest frequency in grouped data. Students understand each measure conceptually before calculating.

calculate mean, median and mode of ungrouped data

Students compute the mean from a list or a frequency table ($\text{mean} = \frac{\Sigma fx}{\Sigma f}$). For the median, data must be ordered; the position of the median is $\frac{n+1}{2}$ for n data values. Mode is found by inspection of the frequency table. Students should recognise that the mean is affected by outliers while the median is resistant to them.

calculate mean, median and modal class of grouped data

For grouped data: the mean is estimated using class mid-points ($\text{mean} \approx \frac{\Sigma f \cdot x_m}{\Sigma f}$ where xₘ is the class mid-point). The median is estimated from the cumulative frequency curve (50th percentile) or by linear interpolation within the median class. The modal class is the class with the highest frequency. Students recognise that these are estimates because individual values within each class are unknown.

recognise the uses and abuses of mean, median and mode/modal class in daily life

Students evaluate which measure best represents a data set in a given context: — Mean is appropriate for symmetric distributions without outliers. — Median is better when the data is skewed or contains extreme values (e.g. household incomes). — Mode is used for categorical data or when the most popular item matters (e.g. best-selling shoe size). Students identify cases where statistics are used misleadingly (e.g. claiming an "average" salary that is actually only the mean, inflated by a few high earners).

understand the effects of the following operations on the mean, median and mode: (i) adding a common constant to each item of the set of data; (ii) multiplying each item of the set of data by a common constant

Non-foundation topic. (i) Adding a constant c to every data value: mean increases by c, median increases by c, mode increases by c. Measures of spread are unchanged. (ii) Multiplying every data value by a constant k: mean is multiplied by k, median is multiplied by k, mode is multiplied by k. Students verify these rules with numerical examples and apply them to simplify calculations (e.g. coding data by subtracting the mean and dividing by a constant to work with simpler numbers).

recognise the concept of weighted mean

The weighted mean is computed as $\frac{\Sigma (w_i x_i)}{\Sigma w_i}$ where wᵢ are the weights. It generalises the ordinary mean (which assigns equal weight to every value). Students understand that the weighted mean gives more importance to values with higher weights. Applications include grade-point averages (where subjects have different credit values) and index numbers.

solve problems involving weighted mean

Students compute the weighted mean given data values and their weights, and solve reverse problems (e.g. find the weight of one item given the weighted mean and the weights of all other items). Real-life contexts include overall exam scores (with different weightings for different components) and average prices weighted by quantity.

recognise the concepts of certain events, impossible events and random events

A certain event always occurs (probability = 1); an impossible event never occurs (probability = 0); a random event may or may not occur. Students distinguish between these three types with examples from everyday life (e.g. "the sun will rise tomorrow" is certain; "a rolled die shows 7" is impossible; "a rolled die shows an even number" is random). The probability of any event lies between 0 and 1 inclusive.

recognise the concept of probability

Classical (theoretical) probability: $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$. The sample space is the set of all possible outcomes of an experiment. An event is a subset of the sample space. Students understand that classical probability requires equally likely outcomes. The relationship between theoretical probability and relative frequency (empirical probability) is noted: as the number of trials increases, the relative frequency approaches the theoretical probability (law of large numbers, informally stated).

calculate probabilities of events by listing the sample space and counting

Students systematically list the sample space using methods such as listing, tables, and tree diagrams. Once the sample space is established, the probability of an event is the number of outcomes in the event divided by the total number of outcomes in the sample space. Problems involve one-stage experiments (e.g. drawing a card from a shuffled deck) and two-stage experiments (e.g. tossing a coin and rolling a die). Combinatorics formulae (permutations and combinations) are not required; complete enumeration of the sample space is sufficient.

solve problems involving probability

Students apply probability to realistic problems involving games, surveys, and quality control. They compute the probability of complementary events $P(\text{not } A) = 1 - P(A)$, and combine probabilities for mutually exclusive events $P(A \text{ or } B) = P(A) + P(B)$. Problems requiring independent events or conditional probability are deferred to KS4. Answers should be expressed as fractions, decimals, or percentages as appropriate.

recognise the concept of expectation

Non-foundation topic. The expectation (expected value) of a random variable X is $E(X) = \Sigma x_i P(x_i)$, the probability-weighted sum of all possible values. Students interpret the expectation as the long-run average value of the random variable over many repetitions. Examples include the expected score on a die, the expected winnings in a simple game of chance, and the expected number of heads in several coin tosses.

solve problems involving expectation

Non-foundation topic. Students compute the expected value for random variables arising from simple experiments and use it to make decisions (e.g. whether a game is fair, or which option has the higher expected return). Problems should involve small sample spaces where the probability distribution can be explicitly listed. Students should understand that expectation is a theoretical concept and does not predict the outcome of any single trial.

through various learning activities, discover and construct knowledge, further improve the ability to inquire, communicate, reason and conceptualise mathematical concepts

This unit has a single composite objective that encompasses the full range of mathematical process skills: inquiry, communication, reasoning, and conceptualisation. Unlike the content units (LU1–LU31), there is no fixed list of topics. Teachers design or select activities that require students to: — Pose and investigate mathematical questions. — Collect, organise, and analyse data or patterns. — Formulate and test conjectures. — Communicate findings in oral and written form. — Reflect on the reasonableness and generality of conclusions. Suitable activities include mathematical investigations (e.g. exploring number patterns, optimisation problems), statistical projects (designing and carrying out a survey), geometric constructions and explorations, and cross-curricular projects. The enrichment and non-foundation objectives from other units provide natural sources of investigative material. Assessment for this unit is through observation, project work, and presentations rather than conventional written tests.

recognise the concept of nth root

Students learn that the nth root of a positive number x is the positive number whose nth power equals x. Emphasis is on square roots and cube roots. Students should be able to evaluate simple nth roots exactly (e.g. √16 = 4, ∛8 = 2) and use a calculator for others. Negative numbers do not have real even-indexed roots; this boundary should be noted but not explored formally.

recognise the concepts of rational and irrational numbers

A rational number is any number expressible as p/q where p and q are integers and q ≠ 0. Irrational numbers (e.g. √2, π) cannot be expressed in this form. Together, rationals and irrationals form the real numbers. Students should be able to classify given numbers as rational or irrational and to place them approximately on the number line. Formal proof of irrationality (e.g. proof that √2 is irrational) is not required.

perform mixed arithmetic operations of simple quadratic surds of the form a√b

Non-foundation topic. Students simplify and perform addition, subtraction, multiplication, and division of surds of the form a√b where a is rational and b is a positive integer with no perfect-square factor other than 1. Operations include combining like surds (e.g. 3√2 + 5√2 = 8√2), multiplying surds (e.g. √3 × √5 = √15), and rationalising denominators of the form 1/√b. Over-complicated operations and surds involving cube roots are not required.

explore the relation between constructible numbers and rational and irrational numbers

Enrichment topic. Students investigate which lengths can be constructed with compasses and straightedge starting from a unit length, and how this relates to rationality and irrationality. For example, √2 is constructible (as the diagonal of a unit square) even though it is irrational. This activity connects geometry and number theory at an exploratory level; formal proofs are not required.

understand the concept of percentage changes

Percentage change = (change ÷ original value) × 100%. Students distinguish between percentage increase and percentage decrease and interpret these in context. The bar diagram model (showing original, change, and new value) is a useful representation. Students should understand that a 20% increase followed by a 20% decrease does not return to the original value.

solve related real-life problems

Real-life problems include profit and loss (profit/loss percentage relative to cost price), discount (percentage off marked price), taxation (goods and services tax applied to a price), and simple and compound interest. For compound interest, students apply the formula A = P(1 + r)^n; over-complicated compounding scenarios (e.g. continuous compounding) are not required. Problems may involve finding the original value given the new value and the percentage change (reverse percentage).

understand the concepts of rates, ratios and proportions

Students understand rate as a comparison of two quantities measured in different units (e.g. km/h, HK$/kg). Ratio compares two quantities of the same kind and is written in the form a : b. Proportion describes the equality of two ratios. Direct proportion: y ∝ x means y/x = k (constant); the graph is a straight line through the origin. Inverse proportion: y ∝ 1/x means xy = k; the graph is a hyperbola. Students should be able to identify whether a given table of values represents a direct, inverse, or neither type of proportion, using the counterexample table as a reference for non-proportional relationships.

solve problems involving rates, ratios and proportions

Problems include speed-distance-time, density-mass-volume, currency exchange, map scales, and dividing quantities in a given ratio. Students set up and solve proportion equations. Problems requiring the unitary method (find the value for one unit, then scale) are appropriate. Complex multi-step problems involving several different rates simultaneously should be avoided.

represent word phrases by algebraic expressions

Students translate verbal descriptions into algebraic expressions, including expressions with one or more variables, constants, and operations. Examples include "five more than twice a number" → 2x + 5, and "the product of two consecutive integers" → n(n+1). Students should handle expressions involving fractions and negative coefficients. Simplification of expressions by collecting like terms is included.

represent algebraic expressions by word phrases

Students interpret and read algebraic expressions back as word phrases in natural language. This develops fluency with algebraic notation and the ability to communicate mathematical relationships verbally. Students recognise the meaning of terms such as "coefficient", "constant term", and "variable".

recognise the concept of sequences of numbers

Students identify and extend number sequences, including arithmetic sequences (constant difference) and simple geometric sequences. They find the general term of simple arithmetic sequences (e.g. T(n) = 3n + 1) and use it to calculate any specified term. Sequences serve as a concrete bridge to the idea of a variable input producing a specific output.

recognise the preliminary idea of functions

Students are given an informal introduction to the concept of a function as a rule that assigns exactly one output value to each input value. Simple examples (e.g. f(x) = 2x + 1, or a temperature conversion formula) illustrate the input-output idea. Formal notation, domain, codomain, and composition of functions are not required at this stage; these are treated in KS4.

solve linear equations in one unknown

Students solve linear equations of the form ax + b = cx + d, including cases where coefficients and constants are integers, fractions, or decimals. Methods include transposing terms (moving terms across the equality sign) and dividing both sides by the coefficient of the unknown. Equations with the unknown in the denominator (after clearing fractions) are included. Students should check solutions by substitution.

formulate linear equations in one unknown from a problem situation

Students identify the unknown, assign a variable, translate the problem conditions into a linear equation, and verify that the equation correctly models the situation. Problems involve age relationships, geometric measurements, number puzzles, and simple mixture/distance-rate-time contexts. Over-complicated or multi-stage formulation problems should be avoided.

solve problems involving linear equations in one unknown

Students combine LO 8.1 and 8.2 in complete problem-solving cycles: formulate → solve → interpret the answer in context → check reasonableness. Final answers must be interpreted in the original units (e.g. "the number of students is 24"). Fractional or decimal solutions should be rounded or left exact as appropriate to the context.

understand the concept of linear equations in two unknowns and their graphs

A linear equation in two unknowns ax + by = c has infinitely many solutions, each represented as an ordered pair (x, y). When plotted, these form a straight line in the coordinate plane. Students learn to construct a table of values and plot the corresponding line. The concept that a pair of simultaneous equations corresponds to two such lines, and their solution is the intersection point, is established here.

solve simultaneous linear equations in two unknowns by the graphical method

Students plot both lines on the same coordinate plane and read off the coordinates of the intersection to obtain the approximate solution. They recognise that parallel lines indicate no solution and coincident lines indicate infinitely many solutions. Accuracy of the graphical method depends on the scale and care of plotting; exact solutions are obtained algebraically.

solve simultaneous linear equations in two unknowns by the algebraic methods

Both the substitution method and the elimination method are required. In the substitution method, one variable is expressed in terms of the other from one equation and substituted into the second. In the elimination method, equations are multiplied by suitable constants and added or subtracted to eliminate one variable. Students should be able to choose the more efficient method for a given system and should check their solutions by substitution.

formulate simultaneous linear equations in two unknowns from a problem situation

Students identify two unknowns, assign variables, and write two equations from the problem conditions. Contexts include mixture problems (e.g. two types of items with different prices), age problems, and geometric problems. Each equation must independently reflect a distinct constraint given in the problem.

solve problems involving simultaneous linear equations in two unknowns

Students complete the full problem-solving cycle: formulate the system from the context, solve algebraically or graphically, interpret the solution in the original context, and check that the answers satisfy both conditions. Answers must be stated with appropriate units and context.