Progression maps: Using and applying
Problem Solving:
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Try different approaches to solve a problem.
Try different approaches and find ways of overcoming difficulties that arise when solving problems.
Use a range of strategies when solving problems.
Develop strategies for solving problems and use these strategies both in working within
mathematics and in applying mathematics to practical contexts.
Begin to structure an approach when exploring a simple task or solving a problem. Generate and check the
necessary information.
Identify the necessary information to carry through tasks and solve mathematical problems. Check
results and consider whether they are sensible.
Solve more complex problems by breaking them into smaller steps or tasks, choosing and using efficient
techniques for calculation, algebraic manipulation and graphical representation, and resources, including
ICT.
Solve substantial problems by breaking them into simpler tasks, using a range of efficient
techniques, methods and resources, including ICT.
Starting from given problems or contexts, progressively refine or extend the mathematics used to
generate fuller solutions.
Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics
across a range of contexts: number, algebra, shape, space and measures, and handling data.
Communicating:
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Discuss work using mathematical language. Represent work using symbols and simple diagrams.
Begin to organise work. Use and interpret mathematical symbols and diagrams.
Begin to refine ways of recording and use appropriate mathematical symbols correctly.
Present information and results in a clear and organised way. Present solutions/findings in the
context of the problem/task.
Present and interpret solutions/findings in the context of the problem/task. Begin to develop correct and
consistent use of notation, symbols and diagrams.
Show understanding of situations by describing them mathematically, making correct use of
symbols, words, diagrams, tables and graphs.
Choose and use correctly symbols, diagrams and graphs. Present and interpret solutions/findings in the
context of the original problem/task.
Interpret, discuss and synthesise information presented in a variety of mathematical forms. Begin
to explain reasons for selection and use of diagrams.
Represent problems and synthesise information in algebraic, geometric or graphical form; move from one
form of presentation to another to gain a different perspective on the problem/task.
Examine critically, improve and justify the choice of mathematical presentation, explaining features
selected.
Reasoning:
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Explain why an answer is correct.
Understand a general statement by finding particular examples that match it.
Try out ideas to find a pattern or solution.
Make general statements, based on evidence produced, and explain reasoning.
Solve problems and investigate in a range of contexts, explaining and justifying methods and conclusions;
begin to generalise and to understand the significance of a counter-example.
Draw simple conclusions and explain reasoning; suggest extensions to problems; conjecture and
generalise.
Use logical argument to establish the truth of a statement; begin to give mathematical justifications and test
by checking particular cases.
Present a concise reasoned argument, using symbols, diagrams, graphs and related explanatory texts.
Show some insight into mathematical structure by using pattern and symmetry to justify generalisations,
arguments or solutions.
Appreciate the difference between mathematical explanation and experimental evidence.
Progression maps: Number
Place Value:
Read, write and order whole numbers to at least 1000; know what each digit represents.
Read and write the vocabulary of comparing and ordering numbers. Use symbols correctly, including
less than (<), greater than (>), equals (=).
Round any positive integer less than 1000 to the nearest 10 or 100
Multiply and divide any positive integer up to 10,000 by 10 or 100 and understand the effect.
Round a number with one or two decimal places to the nearest integer.
Understand and use decimal notation and place value.
Order a given set of positive and negative integers.
Compare and order a mixed set of numbers or measurements with up to three decimal places.
Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and explain the effect.
Round positive whole numbers to the nearest 10, 100 or 1000 and decimals to the nearest whole number
or one decimal place.
Find the difference between a positive and negative integer or between two negative integers, in a context
such as temperature or the number line.
Round positive numbers to any given power of 10; round decimals to the nearest whole number or to one
or two decimal places.
Multiply and divide integers and decimals by 0.1, 0.01
Use rounding to make estimates; round numbers to the nearest whole number or to one or two decimal
places.
Begin to write numbers in standard form.
Round numbers to a given number of significant figures.
Understand upper and lower bounds.
Estimate calculations by rounding numbers to one significant figure and multiplying or dividing mentally.
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Mental Calculation:
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Know by heart all addition and subtraction facts for each number to 20.
Add and subtract mentally a 'near multiple of 10' to or from a two-digit number.
Understand division and recognise that division is the inverse of multiplication.
Know by heart facts for the 2, 5 and 10 multiplication tables.
Use known number facts and place value to add or subtract mentally, including any pair of twodigit whole numbers.
Know by heart facts for the 2, 3, 4, 5 and 10 multiplication tables.
Derive quickly division facts corresponding to the 2, 3, 4, 5 and 10 multiplication tables.
Find remainders after division.
Know by heart all multiplication facts up to 10 × 10.
Find differences by counting up through next multiple of 10, 100 or 1000, e.g. calculate mentally a
difference such as 8006 – 2993.
Use known number facts and place value to consolidate mental addition and subtraction
Derive quickly division facts corresponding to multiplication tables up to 10 x 10
Consolidate and extend mental methods of calculation to include decimals, fractions and
percentages, accompanied where appropriate by suitable jottings; solve simple word problems mentally
Consolidate and extend mental methods of calculation, working with decimals, fractions and percentages,
squares and square roots, cubes and cube roots; solve word problems mentally.
Make and justify estimates and approximations of calculations
Written Calculation:
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Use informal pencil and paper methods to support, record or explain HTU ± TU, HTU ± HTU. Develop
and refine written methods for column addition and subtraction for two whole numbers less then
1000, and for addition of more than two such numbers.
Extend efficient written methods to:
 column addition/subtraction of two integers less than 10,000;
 short multiplication of HTU or TU by U;
 long multiplication of TU by TU;
 short division of HTU by U (with integer remainder).
Use all four operations to solve simple word problems involving numbers and quantities including
time, explaining methods and reasoning.
Extend written methods to:
 column addition and subtraction of numbers involving decimals;
 long multiplication of a three-digit by a two-digit integer;
 short division of TU by U (mixed-number answer);
 division of HTU by TU (long division, whole-number answer);
 short division of numbers involving decimals.
Identify and use the appropriate operations (including combinations of operations) to solve word
problems involving numbers and quantities, and explain methods and reasoning.
Know and use the order of operations including brackets.
Check a result by considering whether it is of the right order of magnitude and by working the
problem backwards.
Use standard column procedures for multiplication and division of integers and decimals,
including by decimals such as 0.6 or 0.06; understand where to position the decimal point by
considering equivalent calculations.
Enter numbers in a calculator and interpret the display in different contexts.
Use a calculator efficiently and appropriately to perform complex calculations with numbers of any size,
knowing not to round during intermediate steps of a calculation.
Use a calculator efficiently and appropriately, including using the reciprocal key.
Integers and Powers:
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Recognise odd and even numbers up to 1000, and some of their properties, including the outcomes of
sums or differences of pairs of odd/even numbers.
Describe and extend number sequences: count on or back in tens or hundreds, starting from any twoor three-digit number.
Recognise negative numbers in contexts (e.g. on a number line, on a temperature scale).
Recognise multiples of 2, 3, 4, 5 and 10, up to the tenth multiple.
Recognise and extend number sequences formed by counting from any number in steps of constant size.
(See progression map for Sequences, Functions and Graphs.)
Recognise multiples of 6, 7, 8 and 9 up to the tenth multiple.
Know the squares of numbers up to 10 × 10.
Know all the pairs of factors of any number up to 100.
Know and apply tests of divisibility by 2, 4, 5, 10 or 100.
Factorise numbers to 100 into prime factors.
Recognise and use multiples, factors and primes (less than 100).
Know and apply simple tests of divisibility.
Recognise the first few triangular numbers, squares of numbers to at least 12 × 12 and the corresponding
roots.
Add and subtract positive and negative numbers in context.
Add, subtract, multiply and divide integers.
Use index notation for integer powers and simple instances of the index laws.
Know and use the index laws for multiplication and division of positive integer powers.
Fractions, Decimals, Percentages, Ratio and Proportion:
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Recognise unit fractions such as ½, ⅓, ¼, , ... and use them to find fractions of shapes and numbers.
Recognise simple fractions that are several parts of a whole, such as ⅔ or ⅝, and mixed numbers,
such as 5¾; recognise the equivalence of simple fractions (e.g. fractions equivalent to ½, ¼ or ¾).
Relate fractions to division and to their decimal representations.
Order a set of fractions such as 2, 2¾, 1¾, 2½, 1½ and position them on a number line.
Use decimal notation for tenths and hundredths.
Reduce a fraction to its simplest form by cancelling common factors.
Use a fraction as an 'operator' to find fractions of numbers or quantities, (e.g. ⅝ of 32,
of 40,
of 400 cm).
Understand percentage as the number of parts in every 100 and find simple percentages of small
whole-number quantities.
Solve simple problems involving ratio and proportion.
Begin to add and subtract simple fractions and those with common denominators.
Simplify fractions by cancelling all common factors and identify equivalent fractions.
Recognise the equivalence of percentages, fractions and decimals; calculate simple percentages and
use percentages to compare simple proportions.
Multiply and divide a fraction by an integer.
Use the equivalence of fractions, decimals and percentages to compare proportions; calculate
percentages and find the outcome of a given percentage increase or decrease.
Order fractions by writing them with a common denominator or by converting them into decimals.
Divide a quantity into two or more parts in a given ratio; use the unitary method to solve simple
word problems involving ratio and direct proportion.
Add and subtract fractions by writing them with a common denominator.
Use proportional reasoning to solve a problem, choosing the correct numbers to take as 100%, or
as a whole.
Use efficient methods to add, subtract, multiply and divide fractions.
Understand and use proportionality and calculate the result of any proportional change using
multiplicative methods.
Progression maps: Algebra
Equations, Formulae and Identities:
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Understand division and recognise that division is the inverse of multiplication.
Use symbols correctly including less than (<), greater than (>) and equals (=).
Understand the principles of the commutative, associative and distributive laws as they apply to
multiplication.
Make general statements about odd and even numbers.
Explain a generalised relationship (formula) in words.
Understand and use the relationships between the four operations, and principles of the arithmetic laws.
Use brackets.
Use letter symbols to represent unknown numbers or variables.
Know and use the order of operations and understand that algebraic operations follow the same
conventions and order as arithmetical operations.
Construct and solve linear equations with positive integer coefficients (unknown on one side only)
using appropriate methods.
Construct and solve linear equations with integer coefficients (unknown on either or both sides,
without and with brackets) using appropriate methods (e.g. inverse operations, transforming both sides in
same way).
Simplify or transform linear expressions by collecting like terms; multiply a single term over a
bracket.
Substitute integers into simple formulae.
Construct and solve linear equations with integer coefficients (with and without brackets, with negative
signs anywhere in the equation, and with a positive or negative solution) using an appropriate method.
Use systematic trial and improvement methods and ICT tools to find approximate solutions to
equations such as x³ + x = 20.
Transform algebraic expressions by factorising to produce a single term multiplied by terms in a bracket.
Square a linear expression, expand the product of two linear expressions of the form x ± n and simplify the
corresponding quadratic expression.
Solve a pair of simultaneous linear equations by eliminating one variable; link a graphical
representation of an equation or pair of equations to the algebraic solution.
Sequences, Functions and Graphs:
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Describe and extend number sequences: count on or back in tens or hundreds, starting from any twoor three-digit number.
Investigate a general statement about familiar numbers by finding examples that satisfy it.
Recognise and extend number sequences formed from counting from any number in steps of constant size,
extending beyond zero when counting back.
Read and plot coordinates in the first quadrant.
Recognise and extend number sequences, such as the sequence of square numbers, or the sequence of
triangular numbers.
Read and plot coordinates in all four quadrants.
Generate coordinate pairs that satisfy a simple linear rule; plot the graphs of simple linear functions.
Begin to use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by
referring to the activity or practical context from which it was generated.
Express simple functions in symbols; represent mappings expressed algebraically.
Plot the graphs of linear functions, where y is given explicitly in terms of x; recognise that
equations of the form y = mx + c correspond to straight-line graphs.
Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence,
on paper and using ICT.
Construct functions arising from real-life problems and plot their corresponding graphs; interpret
graphs arising from real situations.
Given values for m and c, find the gradient of lines given by equations of the form y = mx + c.
Find the next term and the nth term of quadratic sequences and explore their properties.
Know simple properties of quadratic functions.
Plot the graphs of simple quadratic and cubic functions, e.g. y = x², y = 3x², y = x³.
Progression maps: Shape and Space
Shape and Space:
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Identify lines of symmetry in simple shapes and recognise shapes with no lines of symmetry.
Classify polygons, using criteria such as number of right angles, whether or not they are regular,
and symmetry properties.
Recognise perpendicular and parallel lines, and properties of rectangles.
Recognise and visualise the transformation and symmetry of 2-D shapes, including reflection in
given mirror lines and line symmetry.
Construct 3-D models by linking given faces or edges.
Identify parallel and perpendicular lines; know the sum of angles at a point, on a straight line, and in
a triangle, and recognise vertically opposite angles.
Use a ruler and protractor to measure and draw lines to the nearest millimetre and angles, including
reflex angles, to the nearest degree.
Recognise and visualise the transformation and symmetry of a 2-D shape:
 reflection in given mirror lines and line symmetry;
 rotation about a given point and rotational symmetry.
Transform 2-D shapes by simple combinations of rotations, reflections and translations, on paper
and using ICT; identify all the symmetries of 2-D shapes.
Use a straight edge and compasses to construct:
 the midpoint and perpendicular bisector of a line segment;
 the bisector of an angle;
 the perpendicular from a point to a line;
 the perpendicular from a point on a line.
Construct a triangle, given three sides (SSS); use ICT to explore these constructions.
Identify alternate and corresponding angles; understand a proof that the sum of the angles of a
triangle is 180° and of a quadrilateral is 360°.
Classify quadrilaterals by their geometric properties.
Enlarge 2-D shapes, given a centre of enlargement and a positive whole-number scale factor.
Know that translations, rotations and reflections preserve length and angle, and map objects on to
congruent images.
Visualise and use 2-D representations of 3-D objects; analyse 3-D shapes through 2-D projections,
including plans and elevations.
Explain how to find, calculate and use the interior and exterior angles of regular polygons.
Enlarge 2-D shapes by a positive whole-number or fractional scale factor.
Find the locus of a point that moves according to a given rule, both by reasoning and by using ICT.
Solve problems using properties of angles, of parallel and intersecting lines, and of triangles and other
polygons, justifying inferences and explaining reasoning with diagrams and text.
Measures:
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Use units of time and know the relationships between them (second, minute, hour, day, week,
month, year).
Know and use the relationship between familiar units of length, mass and capacity.
Understand an area measured in square centimetres (cm²).
Understand and use the formula in words 'length × breadth' for the area of a rectangle.
Calculate the perimeter and area of simple compound shapes that can be split into rectangles.
Convert one metric unit to another (e.g. grams to kilograms); read and interpret scales on a range of
measuring instruments.
Use units of measure to estimate, calculate and solve problems in everyday contexts involving length, area,
volume, capacity, mass, time, angle and bearings; know rough metric equivalents of imperial measures in
daily use (feet, miles, pounds, pints, gallons).
Know and use the formula for the volume of a cuboid; calculate volumes and surface areas of
cuboids and shapes made from cuboids.
Deduce and use formulae for the area of a triangle, a parallelogram and a trapezium; calculate areas
of compound shapes made from rectangles and triangles.
Know and use the formulae for the area and circumference of a circle.
Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half of the
unit in either direction.
Calculate the surface areas and volumes of right prisms; calculate lengths, areas and volumes in right
prisms, including cylinders.
Understand and use measures of speed (and other compound measures such as density or
pressure) to solve problems.
Understand and apply Pythagoras' theorem when solving problems in two dimensions.
Progression maps: Handling Data
Handling Data:
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Solve a given problem by organising and interpreting numerical data in simple lists, tables and
graphs; for example:
 simple frequency tables;
 pictograms – symbol representing two units;
 bar charts – intervals labelled in ones and then twos;
 Venn and Carroll diagrams (one criterion).
Solve a problem by collecting, organising, representing and interpreting data in tables, charts,
graphs and diagrams, including those generated by a computer; for example:
 tally charts and frequency tables;
 pictograms – symbol representing two, five, ten or twenty units;
 bar charts – intervals labelled in twos, fives, tens or twenties;
 Venn and Carroll diagrams (two criteria).
Find the mode and range of a set of data.
Solve a problem by representing, extracting and interpreting data in tables, graphs and charts.
Calculate statistics for small sets of discrete data:
 find the mode, median and range;
 calculate the mean in simple cases.
Understand the effect on the mean and median of altering the data. Interpret diagrams and graphs (including
pie charts) and draw simple conclusions based on the shape of graphs and simple statistics for a single
distribution.
Compare two simple distributions using the range and one of the mode, median or mean.
Construct, on paper and using ICT:
 pie charts for categorical data;
 bar charts and frequency diagrams for discrete and continuous data;
 simple line graphs for time series;
 simple scatter graphs.
Identify which are most useful in the context of the problem.
Design a survey or experiment to capture the necessary data from one or more sources; determine
the sample size and degree of accuracy needed; design, trial and, if necessary, refine data collection
sheets;design and use two-way tables.
Communicate interpretations and results of a statistical enquiry using selected tables, graphs and
diagrams in support, using ICT as appropriate; construct tables for large, discrete and continuous sets of
raw data, choosing suitable class intervals.
Select, construct and modify, on paper and using ICT, suitable graphical representations to progress an
enquiry, including scatter graphs to develop further understanding of correlation.
Examine critically the results of a statistical enquiry, and justify the choice of statistical
representation in written presentations.
Identify possible sources of bias and plan how to minimise it.
Probability:
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Discuss the chance or likelihood of particular events.
Use the language associated with probability to discuss events, including those with equally likely outcomes.
Understand and use the probability scale from 0 to 1.
Find and justify probabilities based on equally likely outcomes in simple contexts.
Know that if the probability of an event occurring is p, then the probability of it not occurring is 1 – p.
Estimate probabilities from experimental data; understand that:
 if an experiment is repeated, there will be different outcomes;
 increasing the number of times an experiment is repeated generally leads to better estimates of
probability.
Find and record all possible mutually exclusive outcomes for single events and two successive events in a
systematic way.
Know that the sum of probabilities of all mutually exclusive outcomes is 1, and use this when
solving problems.
Use a single probability to find an estimate of frequency.
Find an estimate of the frequency for up to two events.
Understand relative frequency as an estimate of probability and use this to compare outcomes of
experiments.
Find the probability of two or more mutually exclusive events occurring.
Find the probability of two independent events occurring.