Foundation and Higher skills, split into teachable steps.

Integers, place value and directed number

Pearson groups integer work together; Corbett separates place value, negative numbers, operations and rounding, so these are split for lesson writing.

• Read and write whole numbers using place value
• Order and compare positive integers
• Use inequality symbols correctly
• Place negative numbers on a number line
• Order and compare positive and negative numbers
• Add and subtract with negative numbers
• Multiply and divide with negative numbers
• Round whole numbers to the nearest 10, 100 and 1000
• Use rounding to estimate answers

Decimals and calculation checks

Rounding is separated into whole number, decimal place and significant-figure skills because students often confuse the three.

• Understand decimal place value
• Order and compare decimals
• Add and subtract decimals
• Multiply decimals
• Divide decimals
• Round decimals to the nearest whole number
• Round to a given number of decimal places
• Round to a given number of significant figures
• Use one calculation to check another calculation

Indices, powers, roots and calculators

• Recognise square numbers and square roots
• Recognise cube numbers and cube roots
• Use index notation for positive powers
• Understand simple negative index notation
• Multiply numbers written with the same base
• Divide numbers written with the same base
• Use BIDMAS with powers and brackets
• Use a calculator accurately for powers, roots and brackets

Factors, multiples and primes

The broad Pearson row is split into factor, multiple, prime factor, HCF and LCM lessons to match how exam questions usually isolate the skill.

• List factors of a number
• List multiples of a number
• Identify prime numbers
• Write a number as a product of prime factors
• Use index form in prime factorisation
• Find common factors
• Find the highest common factor
• Find the lowest common multiple of two numbers
• Find the lowest common multiple of three numbers

Expressions and simplifying

• Write an expression from words
• Identify terms in an expression
• Collect like terms
• Simplify expressions with addition and subtraction
• Simplify expressions with multiplication and division
• Use simple index laws in algebra

Brackets and factorising

Corbett separates expanding, simplifying and factorising; the app should keep those as separate lessons.

• Expand a single bracket
• Expand and simplify expressions
• Simplify expressions containing squares and cubes
• Factorise by taking out a single common factor
• Choose between expanding and factorising in a question

Substitution and formulae

• Substitute positive values into expressions
• Substitute negative values into expressions
• Substitute values into expressions with brackets
• Substitute values into expressions with powers
• Substitute values into formulae
• Use word formulae in context

Tables and grouped information

• Sort data into categories
• Use tally charts
• Read data from lists and tables
• Read information from timetables
• Find the mode from a list
• Find the mode from a table

Charts and diagrams

Chart work is broad, so each chart type should become its own note or worksheet.

• Choose a suitable chart for a data set
• Draw and interpret bar charts
• Draw and interpret dual bar charts
• Draw and interpret composite bar charts
• Draw and interpret line graphs
• Draw and interpret time-series graphs
• Draw and interpret frequency polygons
• Draw and interpret pictograms
• Draw and interpret stem-and-leaf diagrams

Pie charts

• Draw pie charts from frequencies
• Read values from pie charts
• Find the modal category from a pie chart
• Find total frequency from a pie chart
• Compare two pie charts

Scatter graphs

• Plot a scatter graph
• Describe correlation
• Identify outliers
• Draw a line of best fit
• Use a line of best fit to estimate a value

Fractions

Corbett separates fraction operations, fraction of amount and mixed-number work; those should be separate app lessons.

• Find equivalent fractions
• Simplify fractions
• Compare fractions
• Write one amount as a fraction of another
• Convert mixed numbers to improper fractions
• Convert improper fractions to mixed numbers
• Add and subtract fractions
• Multiply fractions
• Divide fractions
• Find a fraction of an amount

Fractions, decimals and recurring decimals

• Convert common fractions to decimals
• Convert decimals to fractions
• Recognise terminating decimals
• Recognise recurring decimals
• Use recurring decimal notation

Percentages and multipliers

Percentage of an amount, percentage change and multiplier methods are split because they create different exam methods.

• Convert between fractions, decimals and percentages
• Order fractions, decimals and percentages
• Write one amount as a percentage of another
• Find a percentage of an amount
• Calculate percentage increase
• Calculate percentage decrease
• Use decimal multipliers
• Increase an amount by a percentage
• Decrease an amount by a percentage

Equations and function machines

• Use one-step function machines
• Use two-step function machines
• Solve one-step linear equations
• Solve two-step linear equations
• Solve equations with brackets
• Solve equations with unknowns on both sides
• Rearrange simple formulae
• Form and solve equations from problems

Inequalities and error intervals

Pearson places error intervals here, while Corbett treats them as accuracy work; the app should cross-link both algebra and number.

• Represent inequalities on a number line
• List integer values that satisfy an inequality
• Solve one-step inequalities
• Solve two-step inequalities
• Show an inequality solution on a number line
• Write error intervals from rounded values
• Write error intervals from truncated values

Sequences

• Continue number sequences
• Continue picture sequences
• Find a term-to-term rule
• Find the nth term of a linear sequence
• Use an nth-term rule to generate a sequence
• Use an nth-term rule to decide whether a number is in a sequence

Shape properties and angle facts

Angle facts are split by diagram type to make weak spots easy to diagnose.

• Measure and draw lines accurately
• Measure and draw angles accurately
• Recognise common 2D shapes
• Describe properties of 2D shapes
• Recognise simple 3D shapes
• Identify parallel lines
• Identify perpendicular lines
• Use angles at a point
• Use angles on a straight line
• Use vertically opposite angles
• Use angle facts in triangles and quadrilaterals
• Use alternate, corresponding and co-interior angles

Interior and exterior angles

• Find the sum of interior angles in a polygon
• Find missing interior angles in irregular polygons
• Find interior angles in regular polygons
• Use exterior angles of regular polygons
• Connect interior and exterior angles

Sampling and bias

• Understand what a sample is
• Identify biased questions
• Identify biased samples
• Suggest ways to reduce bias

Averages and spread

Averages from lists, discrete tables and grouped tables should be separated because the methods are different.

• Find the mean from a list
• Find the median from a list
• Find the mode from a list
• Find the range from a list
• Find averages from discrete frequency tables
• Find the median from a frequency table
• Find the modal class from grouped data
• Estimate the mean from grouped data
• Use charts and diagrams to compare averages

Measures and scale reading

• Convert between metric length units
• Convert between metric mass units
• Convert between metric capacity units
• Read scales accurately
• Solve time calculations

Perimeter, area and surface area

Corbett separates rectangle, triangle, trapezium, compound area and surface area, so lesson pages should do the same.

• Find the perimeter of 2D shapes
• Find the area of rectangles
• Find the area of triangles
• Find the area of parallelograms
• Find the area of trapezia
• Find the area of compound 2D shapes
• Find the surface area of prisms
• Find the surface area of simple compound solids

3D shapes and volume

• Name common 3D shapes
• Describe properties of 3D shapes
• Use faces, edges and vertices
• Find the volume of a cuboid
• Find the volume of a prism
• Find the volume of compound solids

Coordinates and real-life graphs

Real-life graphs should split into conversion, cost, travel and velocity/time because each context has its own language.

• Plot coordinates in all four quadrants
• Find the midpoint of a line segment
• Read and use conversion graphs
• Read fixed-cost and cost-per-unit graphs
• Interpret distance-time graphs
• Interpret velocity-time graphs

Straight-line graphs

• Draw straight-line graphs from tables
• Read values from straight-line graphs
• Interpret gradient
• Identify parallel lines
• Find the equation of a line from a graph
• Use y = mx + c

Translations, rotations and reflections

• Translate a shape using a vector
• Describe a translation
• Rotate a shape about a centre
• Describe a rotation
• Reflect a shape in a mirror line
• Describe a reflection

Enlargement and combined transformations

Enlargement should be split from other transformations because centre of enlargement and scale factor cause common exam errors.

• Enlarge a shape by a positive integer scale factor
• Describe an enlargement
• Find a centre of enlargement
• Use combinations of transformations
• Describe a sequence of transformations

Ratio

Ratio notation, sharing, finding missing parts and ratio-to-fraction conversion should be separate because students mix the methods.

• Write ratios in simplest form
• Write ratios from context
• Share an amount in a two-part ratio
• Share an amount in a three-part ratio
• Find one quantity when another is known
• Compare ratios
• Write a ratio in the form 1:n
• Write a ratio in the form n:1
• Convert between ratios and fractions

Proportion and best value

• Recognise direct proportion
• Recognise inverse proportion
• Use proportion graphs
• Compare best value offers
• Scale recipes up and down
• Use currency conversions

Pythagoras and trigonometry

Pythagoras and SOHCAHTOA should be separate lessons before mixed right-angled-triangle questions.

• Use Pythagoras' theorem to find a hypotenuse
• Use Pythagoras' theorem to find a shorter side
• Choose sin, cos or tan
• Find a missing side using trigonometry
• Find a missing angle using trigonometry
• Know exact values for 0, 30, 45, 60 and 90 degrees where required

Probability basics

• Use the probability scale from 0 to 1
• List all possible outcomes
• Complete two-way tables
• Use frequency trees
• Use the complement rule 1 - p

Diagrams and repeated events

Venn diagrams, tree diagrams and sample spaces are distinct exam methods, so they should not be one combined worksheet.

• Use relative frequency
• Use sample space diagrams
• Use Venn diagrams
• Use set notation
• Use probability tree diagrams
• Calculate probabilities for combined events

Compound measures

• Use speed, distance and time
• Use density, mass and volume
• Use pressure, force and area
• Convert units in compound-measure questions

Financial percentages

Reverse percentage, simple interest and compound interest are split because they use different reasoning.

• Calculate percentage profit
• Calculate percentage loss
• Find original amounts using reverse percentages
• Calculate simple interest
• Calculate compound interest
• Calculate growth using multipliers
• Calculate depreciation and decay
• Solve rates of pay questions

Plans and elevations

• Name 3D shapes from drawings
• Sketch 3D forms
• Draw a plan of a 3D shape
• Draw front and side elevations
• Build or identify a 3D form from its plan and elevations

Constructions, loci and bearings

Constructions, loci and bearings are often separate exam questions and should have separate practice sets.

• Construct perpendicular bisectors
• Construct angle bisectors
• Construct triangles accurately
• Draw loci from a point or line
• Find regions satisfying more than one locus
• Use maps and scale drawings
• Measure and draw bearings
• Solve bearing problems

Quadratic expressions and equations

Quadratics should be split into expand, factorise and solve before moving to graphs.

• Expand double brackets
• Factorise simple quadratic expressions
• Solve quadratic equations by factorising
• Check solutions to quadratic equations

Quadratic graphs

• Plot quadratic graphs from a table
• Find roots from a quadratic graph
• Find the y-intercept of a quadratic graph
• Find the turning point of a quadratic graph
• Use quadratic graphs to estimate solutions

Circle measures

• Name parts of a circle
• Find circumference
• Find area of a circle
• Find arc length
• Find area of a sector

Cylinders, cones, spheres and composite solids

• Find the volume of a cylinder
• Find the surface area of a cylinder
• Recognise spheres, pyramids and cones
• Solve volume questions with cones and spheres
• Solve surface area questions with curved solids
• Solve composite solid questions

Fractions and reciprocals

• Add and subtract mixed numbers
• Multiply mixed numbers
• Divide mixed numbers
• Find the reciprocal of an integer
• Find the reciprocal of a decimal
• Find the reciprocal of a fraction

Index laws and standard form

Standard form should split into conversion, operations and calculator use.

• Use index laws to simplify numerical expressions
• Calculate the value of expressions with powers
• Convert ordinary numbers to standard form
• Convert standard form to ordinary numbers
• Add and subtract in standard form
• Multiply and divide in standard form
• Use a calculator for standard form

Similarity and congruence

• Use SSS congruence
• Use SAS congruence
• Use ASA congruence
• Use RHS congruence
• Identify similar shapes
• Find scale factors
• Find missing lengths in similar shapes

Vectors

• Use column vector notation
• Draw column vectors
• Identify parallel column vectors
• Add and subtract column vectors
• Multiply vectors by a scalar

Formulae and algebra language

• Use the terms expression, equation, formula and identity
• Change the subject of a formula
• Answer simple show-that questions
• Check whether a statement is an identity

Non-linear graphs and simultaneous equations

Cubic graphs, reciprocal graphs and simultaneous equations should be split before mixed graph-solving tasks.

• Use inverse proportion graphs
• Recognise cubic graphs
• Sketch cubic graphs
• Recognise reciprocal graphs
• Sketch reciprocal graphs
• Solve simultaneous equations algebraically
• Solve simultaneous equations graphically

Calculation, checking and rounding

• Use all four operations with integers
• Use all four operations with decimals
• Use one calculation to find another
• Apply the product rule for counting
• Round to a suitable degree of accuracy
• Estimate answers using rounded values
• Check whether an answer is reasonable

Indices, roots, reciprocals and BIDMAS

Higher index work should split positive, negative and fractional powers before algebraic index laws.

• Use positive integer powers
• Use negative powers
• Use fractional powers
• Use roots and reciprocals
• Apply order of operations with powers
• Evaluate complex calculator expressions

Factors, multiples and primes

• Identify factors, multiples and primes
• Use prime factorisation
• Write prime factorisation in index form
• Find common factors and HCF
• Find LCM using listing
• Find LCM using prime factors

Standard form and surds

Corbett separates standard form, surds and rationalising; these should be separate Higher notes.

• Use index laws to simplify numerical expressions
• Convert between ordinary numbers and standard form
• Add and subtract in standard form
• Multiply and divide in standard form
• Use calculators with standard form
• Simplify surds
• Multiply and divide surds
• Expand brackets containing surds

Expressions, brackets and factorising

This Pearson section is large; Corbett splits it into collecting terms, expanding, factorising, quadratics and difference of two squares.

• Write algebraic expressions
• Collect like terms
• Simplify algebraic expressions
• Use index laws in algebra
• Expand single brackets
• Expand double brackets
• Factorise single brackets
• Factorise quadratic expressions
• Factorise using the difference of two squares

Equations, formulae and iteration

• Set up expressions from context
• Set up equations from context
• Substitute into expressions and formulae
• Solve linear equations
• Solve linear inequalities
• Change the subject of a formula
• Use iteration to approximate a solution
• Interpret iteration results

Sequences

Higher sequence work needs linear, quadratic and geometric routes rather than one combined lesson.

• Continue arithmetic sequences
• Find the nth term of a linear sequence
• Generate terms from an nth term
• Find the nth term of a quadratic sequence
• Distinguish arithmetic and geometric sequences
• Recognise simple geometric progressions
• Find term-to-term rules for geometric sequences
• Use negative, fractional and decimal terms in sequences

Averages and range

• Find averages from lists
• Find range from lists
• Use two-way tables
• Find averages from discrete frequency tables
• Find modal class from grouped data
• Estimate the mean from grouped data
• Draw stem-and-leaf diagrams
• Interpret stem-and-leaf diagrams

Representing and comparing data

Higher data work should separate chart choice, histograms and distribution comparison.

• Choose an appropriate chart or diagram
• Draw and interpret bar charts
• Draw and interpret line graphs
• Draw and interpret time-series graphs
• Draw and use pie charts
• Compare two pie charts
• Construct histograms
• Interpret histograms
• Compare distributions using averages and spread

Scatter graphs

• Plot scatter graphs
• Describe correlation
• Identify outliers
• Draw a line of best fit
• Use a line of best fit for interpolation and extrapolation

Fractions and recurring decimals

• Simplify and compare fractions
• Write one amount as a fraction of another
• Convert between mixed numbers and improper fractions
• Use all four operations with fractions
• Find fractions of amounts
• Convert recurring decimals to fractions
• Convert fractions to recurring decimals

Percentages

Higher percentage work should split comparison, change, multipliers and reverse percentages.

• Convert between fractions, decimals and percentages
• Order fractions, decimals and percentages
• Write one amount as a percentage of another
• Find percentages of amounts
• Calculate percentage increase and decrease
• Use decimal multipliers
• Increase or decrease by a percentage
• Use reverse percentages

Ratio and proportion

• Simplify ratios in context
• Share in two-part and three-part ratios
• Find a missing quantity from a ratio
• Compare ratios
• Write ratios in the form 1:n and n:1
• Convert between ratios and fractions
• Write a ratio as a linear function
• Use direct proportion
• Use inverse proportion
• Recognise proportion graphs
• Scale recipes
• Use currency conversions

Polygons, angles and parallel lines

• Measure and draw lines and angles
• Identify 2D and 3D shape properties
• Use parallel and perpendicular lines
• Use angle facts at a point and on a straight line
• Use vertically opposite angles
• Use angles in parallel lines
• Find interior angles of polygons
• Find exterior angles of regular polygons
• Use compound shape angle properties

Pythagoras and trigonometry

• Use Pythagoras' theorem
• Choose sin, cos or tan
• Find missing sides with trigonometry
• Find missing angles with trigonometry
• Use exact trig values

Coordinates and real-life graphs

• Use coordinates in all four quadrants
• Use conversion graphs
• Use cost graphs
• Interpret distance-time graphs
• Interpret velocity-time graphs
• Find the midpoint of a line segment
• Find the length of a line segment

Linear graphs and coordinate geometry

Higher linear graph work should split drawing, gradients, equations, parallel lines and perpendicular lines.

• Draw straight-line graphs
• Interpret gradient and intercept
• Find the equation of a line through two points
• Find the equation of a line from a graph
• Identify parallel lines
• Identify perpendicular lines
• Generate equations of parallel lines
• Generate equations of perpendicular lines

Quadratic, cubic, reciprocal and circle graphs

Corbett separates each non-linear graph type; app lessons should do the same before mixed graph recognition.

• Plot quadratic graphs
• Find roots and intercepts of quadratic graphs
• Find turning points of quadratic graphs
• Recognise and sketch cubic graphs
• Recognise and sketch reciprocal graphs
• Draw circles centred at the origin
• Use the circle equation x² + y² = r²

Perimeter, area and circles

• Convert metric measures
• Read scales
• Find perimeter of 2D shapes
• Find area of 2D and compound shapes
• Name parts of a circle
• Find circumference
• Find area of a circle
• Find arc length
• Find sector area

3D forms and curved solids

• Name and describe 3D shapes
• Find volume of cuboids
• Find volume of prisms
• Find volume of composite forms
• Find surface area of prisms
• Find surface area and volume of cylinders
• Solve problems with spheres, pyramids, cones and frustums

Accuracy and bounds

Bounds should split into single-value bounds, expression bounds and error intervals.

• Find upper and lower bounds of rounded numbers
• Find bounds for calculated expressions
• Use error intervals for rounded values
• Use error intervals for truncated values
• Decide whether bounds are inclusive or exclusive

Transformations

Higher transformation work should split negative enlargement, fractional enlargement and invariant points.

• Translate, rotate and reflect shapes
• Describe translations, rotations and reflections
• Enlarge by positive integer scale factors
• Enlarge by fractional scale factors
• Enlarge by negative scale factors
• Use combinations of transformations
• Describe combined transformations
• Identify invariant points and lines

Constructions, loci and bearings

• Draw plans and elevations
• Draw a 3D form from plans and elevations
• Use maps and scale drawings
• Use bearings
• Complete standard constructions
• Find regions satisfying loci
• Use loci in 3D contexts
• Use constructions to solve bearings and loci problems

Quadratic and simultaneous equations

Completing the square, formula and simultaneous-equation types should be separate Higher lessons.

• Set up quadratic equations
• Solve quadratic equations by factorising
• Solve quadratic equations by completing the square
• Solve quadratic equations with the quadratic formula
• Solve simultaneous linear equations algebraically
• Solve simultaneous linear equations graphically
• Solve linear and quadratic simultaneous equations
• Solve linear and circle simultaneous equations

Inequalities

• Represent inequalities on a number line
• List integer values satisfying inequalities
• Solve linear inequalities
• Show inequality solutions on a number line
• Represent inequalities graphically
• Interpret regions on inequality graphs

Probability methods

Corbett separates conditional probability and tree diagrams at Higher, so the app should keep multi-event probability granular.

• Use the probability scale
• List outcomes systematically
• Complete two-way tables
• Use frequency trees
• Use the complement rule 1 - p
• Use relative frequency
• Use sample space diagrams
• Use Venn diagrams and set notation
• Use probability tree diagrams
• Solve conditional probability questions

Rates, compound measures and finance

• Compare best value
• Use speed, density and pressure
• Calculate percentage profit and loss
• Use reverse percentages
• Calculate simple interest
• Calculate compound interest
• Use growth and decay multipliers
• Calculate depreciation
• Solve rates of pay questions

Similarity, congruence and scale factors

Higher similarity should split length scale factor, area scale factor, volume scale factor and frustum work.

• Use triangle congruence criteria
• Use formal proof with congruence
• Use formal proof with similarity
• Identify similar shapes
• Find missing lengths in similar shapes
• Use length scale factors
• Use area scale factors
• Use volume scale factors
• Find area and surface area of frustums

Trigonometric graphs

• Recognise sine, cosine and tangent graphs
• Sketch trigonometric graphs
• Interpret trigonometric graphs
• Use exact trigonometric values
• Transform graphical functions

Further trigonometry

Sine rule, cosine rule, triangle area and 3D Pythagoras/trig should be separate Higher lessons.

• Use the formula for area of a triangle
• Use the sine rule in 2D
• Use the sine rule in 3D contexts
• Use the cosine rule in 2D
• Use the cosine rule in 3D contexts
• Use Pythagoras' theorem in 3D
• Choose the correct trigonometry method in mixed problems

Collecting data

• Identify types of data
• Understand primary and secondary data
• Recognise bias
• Improve sampling methods
• Design unbiased questions

Cumulative frequency, box plots and histograms

These are three major Higher statistics methods and should be separate notes with linked comparison practice.

• Construct cumulative frequency tables
• Draw cumulative frequency graphs
• Find median and quartiles from cumulative frequency
• Find interquartile range
• Construct box plots
• Interpret and compare box plots
• Construct histograms
• Estimate the mean from a histogram
• Estimate the median from a histogram

Quadratics and graph solving

• Sketch quadratic graphs
• Identify roots of quadratic graphs
• Identify turning points and intercepts
• Complete the square
• Expand more than two linear brackets
• Sketch cubic graphs
• Solve simultaneous equations graphically
• Solve quadratic inequalities
• Represent quadratic inequalities

Circle theorems

Each circle theorem should become its own micro-lesson before mixed proof questions.

• Name parts of a circle
• Use angles at the centre and circumference
• Use angles in the same segment
• Use angle in a semicircle
• Use opposite angles in a cyclic quadrilateral
• Use tangent-radius facts
• Use alternate segment theorem
• Prove circle theorem results

Circle geometry

• Recognise circle graphs
• Construct the graph of a circle
• Find the equation of a circle
• Find the equation of a tangent to a circle

Algebraic fractions, surds and proof

This Pearson heading contains several Higher-only topics, so it should be split aggressively.

• Rationalise denominators involving surds
• Simplify algebraic fractions
• Multiply algebraic fractions
• Divide algebraic fractions
• Solve equations involving algebraic fractions
• Change the subject of complex formulae
• Use algebraic proof

Functions

• Use function notation
• Find outputs from functions
• Find inverse functions
• Use composite functions
• Solve equations using functions

Vectors and proof

• Use vector notation
• Use column vectors
• Find the length of a vector
• Calculate a resultant vector
• Use vectors divided in a given ratio
• Prove points are collinear
• Prove vectors or lines are parallel

Graphs, gradients and area under curves

Corbett includes exponential graphs, rates of change and area under curves; these should be added to the Higher graph path.

• Recognise reciprocal graphs
• Sketch reciprocal graphs
• Recognise exponential graphs
• Interpret exponential graphs
• Estimate area under a curve
• Interpret area under a curve
• Find the gradient of a tangent
• Interpret instantaneous rate of change
• Interpret average rate of change

Direct and inverse proportion

• Recognise direct proportion graphs
• Recognise inverse proportion graphs
• Set up direct proportion formulae
• Set up inverse proportion formulae
• Use proportion formulae to solve problems