Key stage 4 mathematics lesson plans (national curriculum)

ENG-NC-MA-KS4-001

consolidate their numerical and mathematical capability from key stage 3 and extend their understanding of the number system to include powers, roots {and fractional indices}

ENG-NC-MA-KS4-002

select and use appropriate calculation strategies to solve increasingly complex problems, including exact calculations involving multiples of π {and surds}, use of standard form and application and interpretation of limits of accuracy

ENG-NC-MA-KS4-003

consolidate their algebraic capability from key stage 3 and extend their understanding of algebraic simplification and manipulation to include quadratic expressions, {and expressions involving surds and algebraic fractions}

ENG-NC-MA-KS4-004

extend fluency with expressions and equations from key stage 3, to include quadratic equations, simultaneous equations and inequalities

ENG-NC-MA-KS4-005

move freely between different numerical, algebraic, graphical and diagrammatic representations, including of linear, quadratic, reciprocal, {exponential and trigonometric} functions

ENG-NC-MA-KS4-006

use mathematical language and properties precisely

ENG-NC-MA-KS4-007

extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically

ENG-NC-MA-KS4-008

extend their ability to identify variables and express relations between variables algebraically and graphically

ENG-NC-MA-KS4-009

make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments {and proofs}

ENG-NC-MA-KS4-010

reason deductively in geometry, number and algebra, including using geometrical constructions

ENG-NC-MA-KS4-011

interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning

ENG-NC-MA-KS4-012

explore what can and cannot be inferred in statistical and probabilistic settings, and express their arguments formally

ENG-NC-MA-KS4-013

assess the validity of an argument and the accuracy of a given way of presenting information

ENG-NC-MA-KS4-014

develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems

ENG-NC-MA-KS4-015

develop their use of formal mathematical knowledge to interpret and solve problems, including in financial contexts

ENG-NC-MA-KS4-016

make and use connections between different parts of mathematics to solve problems

ENG-NC-MA-KS4-017

model situations mathematically and express the results using a range of formal mathematical representations, reflecting on how their solutions may have been affected by any modelling assumptions

ENG-NC-MA-KS4-018

select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret their solution in the context of the given problem

ENG-NC-MA-KS4-019

apply systematic listing strategies, {including use of the product rule for counting}

ENG-NC-MA-KS4-020

{estimate powers and roots of any given positive number}

ENG-NC-MA-KS4-021

calculate with roots, and with integer {and fractional} indices

ENG-NC-MA-KS4-022

calculate exactly with fractions, {surds} and multiples of π {simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators}

ENG-NC-MA-KS4-023

calculate with numbers in standard form A × 10n, where 1 ≤⃒ A < 10 and n is an integer

ENG-NC-MA-KS4-024

{change recurring decimals into their corresponding fractions and vice versa}

ENG-NC-MA-KS4-025

identify and work with fractions in ratio problems

ENG-NC-MA-KS4-026

apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}

ENG-NC-MA-KS4-027

simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:

ENG-NC-MA-KS4-028

know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}

ENG-NC-MA-KS4-029

where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the ‘inverse function’; interpret the succession of 2 functions as a ‘composite function’}

ENG-NC-MA-KS4-030

use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through 2 given points, or through 1 point with a given gradient

ENG-NC-MA-KS4-031

identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}

ENG-NC-MA-KS4-032

recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = [Fraction:1/x] with x ≠ 0, {the exponential function y = kx for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size}

ENG-NC-MA-KS4-033

{sketch translations and reflections of the graph of a given function}

ENG-NC-MA-KS4-034

plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

ENG-NC-MA-KS4-035

{calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts}

ENG-NC-MA-KS4-036

{recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point}

ENG-NC-MA-KS4-037

solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph

ENG-NC-MA-KS4-038

solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph

ENG-NC-MA-KS4-039

{find approximate solutions to equations numerically using iteration}

ENG-NC-MA-KS4-040

translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution

ENG-NC-MA-KS4-041

solve linear inequalities in 1 {or 2} variable {s}, {and quadratic inequalities in 1 variable}; represent the solution set on a number line, {using set notation and on a graph}

ENG-NC-MA-KS4-042

recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a positive rational number {or a surd}) {and other sequences}

ENG-NC-MA-KS4-043

deduce expressions to calculate the nth term of linear {and quadratic} sequences.

ENG-NC-MA-KS4-044

compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)

ENG-NC-MA-KS4-045

convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

ENG-NC-MA-KS4-046

understand that X is inversely proportional to Y is equivalent to X is proportional to [Fraction:1/Y]; {construct and} interpret equations that describe direct and inverse proportion

ENG-NC-MA-KS4-047

interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion

ENG-NC-MA-KS4-048

{interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts}

ENG-NC-MA-KS4-049

set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}

ENG-NC-MA-KS4-050

interpret and use fractional {and negative} scale factors for enlargements

ENG-NC-MA-KS4-051

{describe the changes and invariance achieved by combinations of rotations, reflections and translations}

ENG-NC-MA-KS4-052

identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

ENG-NC-MA-KS4-053

{apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results}

ENG-NC-MA-KS4-054

construct and interpret plans and elevations of 3D shapes

ENG-NC-MA-KS4-055

interpret and use bearings

ENG-NC-MA-KS4-056

calculate arc lengths, angles and areas of sectors of circles

ENG-NC-MA-KS4-057

calculate surface areas and volumes of spheres, pyramids, cones and composite solids

ENG-NC-MA-KS4-058

apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures

ENG-NC-MA-KS4-059

apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in 2 {and 3} dimensional figures

ENG-NC-MA-KS4-060

know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45°, 60°

ENG-NC-MA-KS4-061

{know and apply the sine rule, [Fraction:a/sinA] = [Fraction:b/sinB] = [Fraction:c/sinC], and cosine rule, a2 = b2 + c2 - 2bc cos A, to find unknown lengths and angles}

ENG-NC-MA-KS4-062

{know and apply Area = [Fraction:1/2] ab sin C to calculate the area, sides or angles of any triangle}

ENG-NC-MA-KS4-063

describe translations as 2D vectors

ENG-NC-MA-KS4-064

apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}

ENG-NC-MA-KS4-065

apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1

ENG-NC-MA-KS4-066

use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

ENG-NC-MA-KS4-067

calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions

ENG-NC-MA-KS4-068

{calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams}

ENG-NC-MA-KS4-069

infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling

ENG-NC-MA-KS4-070

interpret and construct tables and line graphs for time series data

ENG-NC-MA-KS4-071

{construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use}

ENG-NC-MA-KS4-072

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:

ENG-NC-MA-KS4-073

apply statistics to describe a population

ENG-NC-MA-KS4-074

use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.