know that an integer is a whole number; that an integer can be greater or less than zero; and that zero is an integer; know the difference between odd and even numbers; understand that even numbers are in the two times table; understand that the position of a digit in a number gives value; read and write numbers using both words and figures;
Adding two integers with up to four digits; subtracting two integers of up to four digits; knowledge of multiplication tables of up to 12 x 12; carry out multiplication of one integer by another integer with a single digit; carry out multiplication of an integer by another integer that has more than one digit; divide one integer by a single digit; divide an integer by another integer which has more than one digit.
Recognise a negative number and know that it has a value of less than zero; add and subtract negative numbers; multiply and divide negative numbers: when one number is negative and one number is positive, or where both numbers are negative.
Know that multiplying or dividing by 10 moves each digit in a number up or down a place; know that a decimal number consists of an integer part and a decimal part; and the place value of digits in the decimal part of the number; add and subtract decimal numbers; multiply decimal numbers; divide decimals; place decimal numbers in order.
Know that a fraction is part of a whole number, and that it is shown as one integer over another integer; identify common denominators; add two fractions; multiply two fractions; divide two fractions; use mixed number; understand that a mixed number can be represented as vulgar or top-heavy fraction; convert between vulgar fractions and mixed number; carry out addition, subtraction, multiplication and division on mixed number; order fractions by using a common denominator.
Fractions and Decimals
Change terminating decimals to fractions and vice versa; recognise that a terminating decimal is a fraction; know that some fractions will yield a recurring number; understand that a rational number can be described as a fraction with integers as a numerator and a denominator, and that an irrational number cannot be shown in that way; use simplification across fractions; calculate exactly with fractions; know what a recurring decimal is; use the dot method to show a recurring number. Change a recurring decimal to a fraction and vice versa;
Order of Numbers
Order positive and negative integers, decimals and fractions; compare numbers using the equals, not equals, greater than, less than, greater or equal to and less than or equal to signs.
Order of Operations
Understand the priority of operations (Brackets, Indices, Division, Multiplication, Addition, Subtraction); understand the inverse of an operation; understand that brackets can control a sequence of operations; know that a number multiplied by its reciprocal yields an answer of 1 (multiplicative inverse); that the multiplicative identity is 1; and that zero does not have an inverse.
Understand the definition of sets; use set notation too define sets; understand the concept of a universal set; join sets together and define the result of the union and intersection of sets; use the notation for the number of elements in a set; display and manipulate sets using Venn diagrams; manipulate sets shown with populations or with totals; manipulate sets algebraically, including the use of subsets; use the complement of a set, elements of a set, subsets, proper subsets and empty sets; show that an element belongs to a set; definition of sets; define sets algebraically;; use specific sets of numerical types of data.
Understand that a factor is an integer that divides into another integer without leaving a remainder; determine the factors for a number; understand that a prime number is a number that can only be divided by itself and one other number; know that all numbers can be described as a product of its prime number factors; recognize the Unique Factorization theorem; determine the product of its primes for a number; show this product in index form (product notation); determine the Highest Common Factor of two numbers; understand that multiples of a number will share factors; determine the Lowest Common Multiple of two numbers;
Powers and Roots
Know that a square number is a number calculate by itself; know that a cube of a number is calculated from three instances of the same number multiplied together; understand that a square root is an inverse of squaring a number; determine a square root for a given number; understand that a cube root is an inverse of cubing a number; determine a cube root for a given number; raise an integer to a given power; use a calculator to raise an integer to a power or to determine a real root for a given positive number; recognise powers of 2, 3, 4, 5; know that 1000 = 10^3 and 1 million = 10^6; estimate powers and roots of any positive number; use Trial and Improvement to estimate answers.
Know the Laws of Indices and use these laws in context; calculate with integer indices including zero; calculate with negative indices; calculate roots using fractional indices with a unitary numerator; work with fractional and compound indices.
Understand why surds are used; adding and subtracting surds; simplify surd expressions; calculate with surds using the three laws of surds (multiplication, division, power); rationalise surd denominators.
Definition of Standard Form; converting in and out of standard form; converting into and out of standard form for small numbers; calculate (add, subtract, multiply) and interpret using standard form; knowledge of prefixes used for large and small numbers.
Use a systematic listing strategy; use the handshake listing strategy to solve one-way and two-way listing problems; use a strategy to solve 'menu list' problems; use and apply the product rule for counting; use and apply the first-fit rule; use the first-fit descending rule.
Know the standard units of time; use both the 12-hour and 24-hour clock; Use the standard units of time on a timetable; use a distance chart to determine travel time and distance between points;
round numbers to the nearest 10, 100 or 1000; round numbers to a give number of significant figures;; round numbers to a given number of decimal places; use inequality notation to show error intervals due to truncation or rounding.
Estimate answers to calculations; use a process of estimating to check calculations carried out by computers and calculators; apply and interpret limits of accuracy; determine upper and lower bounds based on limits of accuracy.
Understand that symbols may be used to represent numbers in equations or variables in expressions and formulae; use and interpret algebraic notation; understand that algebraic expressions follow the generalised rules of arithmetic; understand and use algebraic vocabulary; writing coefficients as fractions, rather than as decimals; using brackets in algebra to order calculations.
Use contexts expressed in words or diagram form and convert it to an algebraic form; understand the concept of a term; like and unlike terms; simplify algebraic expressions by collecting like terms; simplify expressions using multiplication and division; manipulate expressions with power terms; multiplying and dividing power terms in expressions; laws of indices; use index notation in an algebraic context and which involves fractional, negative and zero powers; laws of indices; multiply algebraic terms over brackets; identify common factors in algebraic expressions.
Evaluate expressions by substituting numerical values for variables; expand the product of two binomials; understand the concept of a quadratic expression; factorise quadratic expressions; identify quadratic expressions where the form is the difference of two squares; manipulate algebraic expressions involving surds.
expand the product of more than two binomials; factorise quadratic expressions when the coefficient on the squared term is an integer not equal to 1; use the method completing the square for a given quadratic expression to identify turning points on graphs; simplify and manipulate expressions that include algebraic fractions where the numerator and/or denominator can be numeric, linear or quadratic; divide polynomials algebraically; use the Factor Theorem to determine the roots of an equation.
Use formulae expressed in words for one- or two-step operations; Substitute numerical values into formulae and expressions; Rearrange formulae to change the subject (Higher: this may include instances where the subject may appear twice, or as a power term);
Know the difference between an equation and an identity; show that algebraic expressions are equivalent; use algebra to construct arguments; use algebra to support proofs.
Describe a function in algebraic terms; Understand and use the f(x) notation; a function as a variable; domains and ranges; exclusion of values from a domain; the 'Z' integer, 'Q' rational, 'N' natural numbers, 'R' real numbers, 'P' primes and complex numbers (complex numbers not in syllabus); interpret the reverse process of a function as an inverse function; interpret the succession of two functions as a composite function.
Graphs and Coordinates
Use graph scales; use axes and coordinates in the first quadrant; Work with graphs in all four coordinates;
Plot graphs of straight lines; use the form y=mx+c; identify and interpret gradients and interception points on a straight line graph, both graphically and algebraically; find the equation of a line through two given points; or through one point with a given gradient; use the form y=mx+c to plot parallel lines; understand how to determine a perpendicular line when using the form y=mx+c; calculate the midpoint of a line segment; use geometric information to identify points on a graph.
Identify the symmetrical property of a quadratic; interpret, graphically, roots and intercepts of a quadratic function; identify, graphically, turning points of a quadratic function; determine turning points by completing the square; deduce roots of a quadratic function algebraically; identify the axis of symmetry for a quadratic.
Graphs of Functions
Plotting a graph using a table, or intercept and gradient (linear); and arrangement of function; recognise, sketch and interpret graphs of linear functions; recognise, sketch and interpret graphs of quadratic functions; recognise, sketch and interpret graphs of simple cubic functions; recognise, sketch and interpret graphs of reciprocal functions of the form y = 1/x where x is not equal to zero; recognise, sketch and interpret graphs of exponential functions of the form y=k^x for positive values of k; recognise, sketch and interpret graphs of trigonometric functions for y=sin(x), y=cos(x) and y=tan(x) where x is in degrees for any angle.
Sketch, graphically, translations of a function; sketch, graphically, reflections of a function; sketch, graphically, enlargements of a function.
Graphs in Context
Plot and interpret graphs of non-standard graphs in real contexts such as problems involving kinematic problems; find approximate solutions to problems involving distance, speed and acceleration; plot and interpret reciprocal graphs; plot and interpret exponential graphs.
Gradients and Areas under Graphs
Calculate or estimate areas under graphs, including quadratic and other non-linear graphs; estimate an area under a graph using trapezia; interpret gradients and areas under graphs, including distance-time, velocity-time and graphs in financial contexts; understand the graphical properties of an equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point;
Solve linear equations, using algebra, with one unknown, including use of brackets; find approximate solutions to linear equations using a graph; solve linear equations algebraically with unknowns on both side of the equation; translate simple situations into algebraic expressions or formulae to derive an equation, and solve the equation; Solve geometrical problems using equations; find approximate solutions to equations using iteration; find approximate solutions to recursive formulae using a suffix notation.
Solving Quadratic Equations
Solve quadratic equations algebraically by factorising; find approximate solutions to a quadratic equation by using a graph; solve quadratic equations by completing the square, including equations that need re-arranging; solve quadratic equations by using the quadratic formula including equations that need re-arranging; use the discriminant to identify the number of possible roots in a quadratic.
Solve simultaneous equations with two variables (linear/linear) by using the addition method; Solve simultaneous equations with two variables (linear/linear) by substitution; Find approximate solutions to simultaneous equations using a graph, Solve simultaneous equations with two variables (linear/quadratic); Translate simple situations into algebraic expressions or formulae to derive a pair of simultaneous equations, and solve these equations; Solve geometrical problems using equations;
Represent inequalities on a number line, including the convention of open and closed circles; solve linear inequalities in one variable; represent inequalities using set notation; represent inequalities on a graph, including the convention of solid and dashed lines; solve linear inequalities with two variables; solve quadratic inequalities in one variable;
Generate terms of a sequence using either a term-to-term or position-to-term rule, including from patterns and diagrams; recognise and use sequences of square numbers; recognise and use sequences of cube numbers; recognise and use sequences of triangular numbers; recognise and use sequences based on Fibonacci sequences.
Recognise and use arithmetic progressions; recognise and use quadratic sequences; recognise and use geometric sequences of the form r^n where n is an integer and r is a rational number > 0; recognise and use sequences using surds and other sequences.
Deduce expressions to calculate the nth term of linear sequences; build sequences using a formula method; deduce expressions to calculate the nth term of quadratic sequences; calculate the sum of the first n terms.
Understand that rates of change can vary; differentiate powers of x; Obtain gradients, stationary points and turning points of a curve by differentiation; use differentiation when working with kinematics and similar problems.
Know the standard units of time, length, area, volume, mass and capacity; Change freely between standard units; understand and use compound measures; Change freely between compound measures; use standard units of measure and related concepts for area and volume; use units in an algebraic context.
Use scale factors; use scale factors on maps; interpret maps; use scale factors on drawings; interpret scale drawings.
Change freely between units of speed in numerical and algebraic context; use speed in problems; change freely between units of rates of pay in numerical and algebraic context; use rates of pay in problems; change freely between units using unit pricing in numerical and algebraic context; Use unit pricing in problems; use units of density and pressure; use density in problems; use pressure in problems.
Use ratio notation, including reduction to its simplest form; show ratio as either part: part or part:whole; divide a given quantity into two parts in a given part:part ratio; Apply ratio to real contexts and problems; express the division of a quantity into two parts as a ratio, including comparison and mixing.
Comparing Using Ratio
Compare lengths, areas and volumes using ratio notation; use scale factors; Make links to similarity (including trigonometric ratios);
Ratios and Fractions
Express one quantity as a fraction of another. where the fraction is less than or greater than 1; show a quantity as a ratio or a fraction; identify and work with fractions when solving problems involving ratio; express a multiplicative relationship between two quantities as a ratio or as a fraction; relate ratios to linear functions.
Define percentage as number of parts per hundred; move between significant values written in percentage, fraction or decimal terms; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems including using percentage increases and percentage decreases; solve percentage problems to determine an original value; use repeated percentage change: one percentage change followed by another.
Fractions and Percentages
Interpret percentage changes as a decimal; Interpret percentage change as a fraction and vice versa; Interpret a fraction as an operator; Use and identify operators derived from a percentage;
Calculate simple interest using percentages; using percentages, apply a price increase to an amount; understand that Value Added Tax is a sales tax, and know when and how it is applied; know that discounts will reduce the cost of items, and can be seen as a percentage, fraction or fixed amount; know that income is money received, and when and how it is received by an individual in the workplace; understand how basic pay, overtime, bonus, commission may be calculated; know the difference between gross and net pay and understand how benefits, allowances, pensions, tax and National Insurance will change the amount received in a wage; know that central and local government requires tax to be paid on income, and how that tax is applied; set up, solve and interpret answers relating to compound interest; solve compound interest problems; solve compound interest problems relating to credit cards and loans; understand the general makeup of a utility bill, being a fixed amount and a variable amount based on a rate; determine how to create a budget using expected income and expenditure; carry out calculations involving the conversion of currencies; knowledge and understand of terms used in household and business finance: profit, loss, cost price, selling price, asset, liability, depreciation, balance, inflation;
Direct and Inverse Proportion
Understand and use proportion as equality of ratios; Solve problems using direct proportion; Use the process of proportionality to evaluate unknown quantities; Use ratio to determine best-buy or better value problems; Solve problems using inverse proportion; Solve problems involving direct and inverse proportion using graphical representations;
Solve problems involving direct and inverse proportion using algebraic representations; Interpret equations that describe direct and inverse proportion; Understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; Construct equations that describe direct and invers proportions; construct inverse proportion equations; and equations with power terms; use graphs to convert values across different units.
Graphs and Rates of Change
Interpret the gradient of a straight line as a rate of change; use a distance-time graph, and understand that the gradient gives velocity; use a velocity-time graph, and understand that the gradient gives acceleration; recognise and interpret graphs that illustrate direct and inverse proportion; apply the concepts of average and instantaneous rate of change (chords and tangent) in numerical, algebraic and graphical context;
Compound Growth and Decay
Set up, solve and interpret the answers in growth and decay problems; Work with general iterative processes;
Geometrical Terms and Conventions
Use conventional terms and notations: points, lines, right angles; use conventional terms: polygons; regular polygons; use conventional terms for 3D shapes: vertices, edges and planes; identify polygons with reflective and/or rotational symmetries; use the standard conventions for labelling and referring to the sides and angles of a triangle; draw a diagram from a written description; recognise both parallel and perpendicular lines.
construct a perpendicular to a line using the standard ruler and compass method; construct a perpendicular to a line from a given point using the standard ruler and compass method; Know that the perpendicular distance from a point to a line is the shortest distance to the line; construct a perpendicular to a line from a given point on the line using the standard ruler and compass method; bisect an angle using a ruler and compass; construct an angle of 60 degrees; use perpendicular and bisection methods to solve loci problems; interpret plans and elevations of 3d shapes; construct plans and elevations for 3D shapes; identify nets that belong to shapes; draw a net for a particular shape;
identify the names of angle types; measure angles in geometric figures; draw angles to one degree of accuracy; apply the properties of angles at a point; and on a straight line, and vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle; derive and use the sum of the angles in any polygon; derive properties of regular polygons;
Properties of Shapes
Derive and apply the properties and definitions of squares, rectangles, parallelograms, trapezia, kites and rhombi; Derive and apply the properties of triangles; know the names for the different types of polygon; Know the names and properties of isosceles, equilateral, scalene, right-angled and obtuse-angled triangles; Identify and apply circle definitions and properties, including centre, radius, diameter, circumference and chord; Identify and apply circle definitions and properties, including tangent, arc, sector and segment; Solve geometric problems on coordinate axes; Identify properties of faces, surfaces, edges and vertices for cubes, cuboids, prisms, cylinders, cones and spheres;
use standard methods of measurement for length, area, mass, volume and capacity; carry out calculations using standard units of mass, length, area. volume and capacity for metric units; choose the correct units for measurement; know and use imperial units for measurement for length, mass and capacity; use imperial/metric conversion factors; measure line segments in geometric figures; read measured values from different types of scales; understand the use of units in a compound measurement; Interpret and draw bearings, including reverse bearings; Know that bearings are always given as three figures; Know the eight compass point bearings;
know and apply formulae to calculate the area of rectangles; Know and apply formulae to calculate the area of triangles; know and apply formulae to calculate the area of parallelograms; know and apply formulae to calculate the area of trapezia;
know and apply formulae to calculate the volume of a cuboid; Know and apply formulae to calculate the volume of right prism; know and apply formulae to calculate the volume of a cylinder; know and apply formulae for the volume of a sphere; Know and apply formulae for the volume of a cone; know and apply formulae for the volume of a pyramid;
know and apply strategies to calculate the surface area of a cuboid; know and apply strategies to calculate the surface area of a right prism; know and apply formulae to calculate the surface area of a cylinder; know and apply formulae for the surface area of a sphere; know and apply formulae for the surface area of a cone; know and apply strategies to calculate the surface area of a pyramid;
identify, describe and construct congruent and similar shapes, including on coordinate axis, by considering rotation; identify, describe and construct congruent and similar shapes, including on coordinate axis, by considering reflection; identify, describe and construct congruent and similar shapes, including on coordinate axis, by considering translation; Use column vector notation for translations; identify, describe and construct congruent and similar shapes, including on coordinate axis, by considering enlargement; identify, describe and construct congruent and similar shapes, including on coordinate axis, by considering enlargement using fractional scale factors; identify, describe and construct congruent and similar shapes, including on coordinate axis, by considering negative scale factors; describe the changes and invariance achieved by combinations of rotations, reflections and translations;
Circles and Shapes
know and apply the formula for the circumference of a circle; know and apply the formula for the area of a circle; calculate arc lengths, angles and areas of sectors of circles; calculate the perimeters of 2D shapes, including circles; calculate the areas of compound shapes; resolve answers using multiples of pi;
know the formula for Pythagoras' theorem; Apply Pythagoras's theorem to find lengths in right angles triangles; Know that the base angles of an isosceles triangle are equal; Use Pythagoras' Theorem and the trigonometric ratios to find angles and lengths in three dimensional figures; use the results of knowledge about angle facts, congruence, similarity, properties of quadrilaterals to obtain simple proofs;
apply and prove the standard circle theorem for a chord; apply and prove the standard circle theorem for a tangent; apply and prove the standard circle theorem for a subtended angle; apply and prove the standard circle theorem for a semicricle; apply and prove the standard circle theorem for a segmented angle; apply and prove the standard circle theorem for a cyclic quadrilateral; apply and prove the standard circle theorem for an alternate segment; use circle theorems to prove related results;
Congruence and Similarity
Use the basic congruency criteria for triangles; Use congruency to derive results about angles and sides; Apply the concepts of congruence and similarity to determine the relationship between lengths in similar figures. Apply the concepts of congruence and similarity, including the relationships between areas and volumes;
know the formula for the sin ratio and apply it to find angles and lengths in right angled triangles; know the formula for the cosine ratio and apply it to find angles and lengths in right angled triangles; Know the formula for the tan ratio and apply it to find angles and lengths in right angled triangles; Know the exact values of sin and cos for the angles 0, 30, 45, 60 and 90; Know the exact values of tan for the angles 0, 30, 45 and 60; use Pythagoras' Theorem and trigonometrical ratios to find missing angles and lengths for more general triangles; know and apply the sine rule to any triangle to find unknown lengths and angles; know and apply the cosine rule to any triangle to find unknown lengths and angles; know and apply the formula for the area of any triangle to find missing lengths, angles and areas for any triangle; relate sin and cos as a fraction to equal tan; and know that sin and cos can be related using Pythagoras; know that trigonometrical ratios can solve problems relating to elevation and depression; use trigonometrical ratios where the angle involves a plane;
describe translations as 2D vectors; apply addition and subtraction of vectors to determine a resultant vector; represent vectors both diagrammatically and using a column representation; apply multiplication of vectors by a scalar to determine a resultant vector; calculate the magnitude of a vector; know that parallel vectors are similar vectors multiplied by a scalar; know that points on the same vector are colinear; use vectors in a 3D context; Use vectors to construct geometric arguments and proofs;
record, describe and analyse the frequency of outcomes of probability experiments using frequency tables; record, describe and analyse the frequency of outcomes of probability experiments using one-way tables; understand and use two-way tables; record, describe and analyse the frequency of outcomes of probability experiments using two-way tables; Record, describe and analyse the frequency of outcomes of probability experiments using frequency trees; understand that relative frequency relates to experimental rather than theoretical probability; and how to improve the accuracy of outcomes;
use appropriate probability language and use the 0 - 1 probability scale; apply the ideas of randomness, fairness and equally likely events to calculate the expected outcome of future experiments; relate relative expected frequencies to theoretical probability; understand that empirical unbiased samples tend towards theoretical probability distributions with increasing sample size; apply the probability that an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one; calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions; Know when to add and when to multiply probabilities;
enumerate sets and combinations of sets using tables and grids; enumerate sets and combinations of sets using Venn diagrams; enumerate sets and combinations of sets using tree diagrams; construct theoretical possibility spaces for single and combined experiments with equally likely outcomes, and use these to calculate theoretical probabilities;
calculate and interpret conditional probabilities through representation using expected frequencies with two way tables; calculate and interpret conditional probabilities through representation using expected frequencies with tree diagrams; Calculate and interpret conditional probabilities through representation using expected frequencies with Venn diagrams;
interpret and construct bar charts for categorical data and know their appropriate use; interpret and construct stem-and-leaf diagrams for data and know their appropriate use; interpret and construct pie charts for categorical data and know their appropriate use; interpret and construct pictograms for categorical data and know their appropriate use; interpret and construct vertical line charts for ungrouped discrete numerical data and know their appropriate use; interpret and construct vertical line charts for time series data and know their appropriate use;
construct and interpret histograms with equal intervals for grouped discrete data and continuous data; construct and interpret histograms with unequal intervals for grouped discrete data and continuous data; construct and interpret cumulative frequency graphs; Interpret spread of discrete, continuous data using quartiles and inter-quartile ranges; interpret, analyse and compare the distributions of data sets involving discrete, continuous and grouped data using box plots;
Averages and Graphs
interpret, analyse and compare the central tendency of data sets involving discrete, continuous and grouped data using the median; interpret, analyse and compare the central tendency of data sets involving discrete, continuous and grouped data using the mean; interpret, analyse and compare the central tendency of data sets involving discrete, continuous and grouped data using the mean; interpret and analyse and compare the measure of central density for a modal class; interpret spread of data using the range; consider outliers; Interpret, analyse and compare distributions of data using appropriate graphical representation; Interpret and analyse data using quartile measures; calculate and interpret standard deviation of data;
infer properties of populations or distributions from a sample; know the limitations of sampling; apply statistics to describe a population; Understand the terms primary data, secondary data, discrete data and continuous data; use and interpret scatter graphs of bivariate data; recognise correlation; know and understand the terms positive correlation, negative correlation, no correlation, weak correlation and strong correlation; know that a correlation does not indicate causation; Draw estimated lines of best fit; Make predications; Interpolate and Extrapolate data whilst knowing the dangers of doing so;
The wtMaths App is available for the iPhone and iPad on the App Store and covers Maths for GCSE (Higher and Foundation). The app is loaded with exam-style questions: in-app purchases are required to unlock all of the questions.