Themes

for conceptual learning

Challenging questions usually arise from the required problem-solving involved, rather than the level of content knowledge. An ideal question is one that nearly every participant would understand the solution of, even though they may not have seen how to solve it in the first instance.

These lists aim to offer all potential participants a feel for the level of mathematical knowledge expected, and hence help them make a reasoned decision on their suitability in becoming a participant.

Senior (Grade 12 and below) Themes for conceptual learning comprises of the Junior level content as well as

Modular arithmetic

Working in different bases

Natural logarithms and e

Counting principles

Geometric averages

Sum and product of zeros

Partial fractions

Complex numbers in rectangular and polar form

Fermat's little theorem

Polynomial functions

Exponential and Logarithmic functions

Algebraic division

Rational and piecewise functions

Rational root theorem

Remainder and factor theorems

Completing the rectangle

Parametric equations

Descartes rule of signs

Factors of simple cubics e.g. x3 - 1

Addition, subtraction and scalar multiplication

Matrix multiplication

Inverse of 2x2 matrices

Elementary row operations

Gaussian elimination

Use cofactors and minors to calculate determinants

Use Cramer's rule

Use eigenvalues and eigenvectors to solve linear recurrence relations

Recursive sequences

Binomial expansions

Telescoping sequences

Linear regression

Discrete random variables

Normal distributions

Probability density functions

Radians

Circle theorems

Perpendicular lines

Triangles: Finding area, lengths and angles

Trigonometric graphs

Trigonometric functions and identities

Inverse trigonometric functions

Equations of circles, parabolas and ellipses

Polar coordinates

3D vectors

Heron's Formula

Chain, product and quotient rule differentiation of functions: Polynomials, Trigonometric, Exponential, Logarithmic

Anti-differentiation of known differentiation

Areas between curves and axes and volumes of revolution (in only 2-variables)

Kinematics

Optimisation

Participants are expected to be able to apply all of the above concepts in unfamiliar contexts

Combinatorics








Junior (Grade 9 and below) Themes for conceptual learning comprises of the Prime Plus level content as well as

Prime factors, multiples

Standard form

All real exponents

Surds

Rationalising denominators

Acceleration, speed, distance, time, including their graphical representations

Direct and inverse proportion

Binary; rudimentary knowledge of other bases

Pascals triangle

Factorisation

Zeros/Roots

Fractions

Null factor law

Domain and range

Composite and inverse functions

Quadratic functions

Polynomials, including reciprocal functions

Vertical and horizontal asymptotes

Difference of two squares

Simple proofs

Gradients/slopes

Straight lines

Addition, subtraction and scalar multiplication

Matrix multiplication

Inverse of 2x2 matrices

Solve simple systems of equations

Arithmetic and geometric

Converging to a limit

Common series

Infinite series

Simple recurrence relations

Iterative sequences

Arithmetic averages

Common charts and graphs

Measures of centre and measures of dispersion (including standard deviation)

Factorials, Combinations and permutations

Cumulative frequency, Box and whisker plots

Percentiles

Experiments Vs theory

Independent, Mutually exclusive and conditional probability

Venn diagrams

Simple correlation

Angles involving polygons and parallel lines

Bearings/elevation/depression

Similarity

Circle theorems

Right-angled triangles: Finding area, lengths and angles

Sine and Cosine rules

Area of any triangle

Common perimeters, areas and volumes

Differentiate simple polynomials

Simple limits (on continuous and reciprocal functions)

Find tangents and normals at a point on a function

Understand rate of change as the gradient (slope) of a function and its concavity describes the change in that rate of change

Participants are expected to be able to apply all of the above concepts in unfamiliar contexts

Combinatorics








Secondary (Grade 7 and below) Themes for conceptual learning

Prime factors, multiples

Standard form

Metric units

Rational exponents

Simple surds

Simple factorials

Simple speed, distance, time

Ratios of amounts

Direct proportion

Pascal's triangle

Simple divisibility rules

Factorisation

Fractions

Inverse functions

Simple composite functions

Simple quadratic functions

Difference of two squares identity

Null factor law (for quadratics)

Simple proofs

Straight lines

Simultaneous equations

Addition, subtraction and scalar multiplication

Matrix multiplication

Arithmetic

Common series

Arithmetic averages

Common charts and graphs

Tree diagrams

Venn diagrams

Percentiles

Experiments vs theory

Simple correlation

Angles involving polygons and parallel lines

Similarity

Simple parabolas

Common perimeters, areas and volumes

Not required

Participants are expected to be able to apply all of the above concepts in real-life contexts


Primary ROCKS (Grade 5 and below) Themes for conceptual learning

Time: Read from an analog clock

Measures: Have a good idea for estimating the magnitude of things

Order of operations: Understand how multiplication is short-hand for repeated addition and use it accordingly

Divisibility: Identify the divisibility by 10, 5 and 2 of any number

Quotients: Understand how division is short-hand for repeated subtraction and find exact quotients without remainders

Notation: Articulate place value of number digits within words

Exponents: Appreciate the difference between numbers arranged in a square compared to in an oblong

Finance: Appreciate and recognise 'value for money' involving decimals and percentages

Error: Identify obvious mis-calculations

Ratio and Proportion: Identify proportional relationships when comparing quantities

Sequences and Series: Recite up to the 15 times tables as a fundamental axiom of number patterns

Cryptography: Communicate the phonetic alphabet with precision

Direction: Provide directions using North, East, South and West

Mappings and Equations: Visualise bijective functions through a mapping and vice versa

Coordinates: Know and use first quadrant cartesian coordinates in context

Graphs: Appreciate the graphical representation of a constant input with independent output and vice versa

Triangles: Identify and label angles smaller or larger than a quarter turn and larger than a half turn

Perimeter and Area: Understand perimeter as distances and rectangular areas as a 2D geometrical representation of a times table and perform calculations for rectilinear figures

Factorising: Generate and identify the prime factor buiding blocks of a number and combine them appropriately

Theorems: Recognise the fundamental axioms of geometry: Amount of turn round a point, vertically opposite and corresponding angles

Transformations: Recognise the spacial effect of a translation slide movement and a mirror image reflection

Tesselations: Maintain a visual appreciation for shapes that create tiling patterns

Location and Dispersion: Location and Dispersion

Location and Dispersion: Place objects into given overlapping groups and interpret the result

Location and Dispersion: Identify frequencies through pictorial representations and interpret in context

History and Recreational: Identify and complete magic squares

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