Mathematics

Understand subtraction as taking away or separating from a group to find how many remain

Understand addition as combining or putting together two groups to find the total

Find the number that makes 10 when added to a given number from 1 to 9 (number bonds to 10)

Decompose numbers up to 10 into pairs in more than one way (part-part-whole)

Solve addition and subtraction word problems within 10 using objects or drawings

Represent addition and subtraction using objects, drawings, and mental images

Read, write, and interpret the symbols +, −, and = in number sentences

Fluently add and subtract within 5

Recall number bonds (addition and related subtraction facts) within 20

Add and subtract one-digit and two-digit numbers to 20, including zero

Solve one-step word problems involving addition and subtraction to 20, including missing-number problems

Add and subtract within 20 using strategies such as making ten, decomposing a number leading to ten, and using known facts

Fluently add and subtract within 10

Understand the meaning of the equal sign as 'is the same as' and determine if equations are true or false

Add within 100 using strategies based on place value, including adding a two-digit and one-digit number, and a two-digit and a multiple of 10

Understand and apply the commutative property of addition: addends can be added in any order

Understand subtraction as finding an unknown addend (e.g. 10 − 8 = ? is the same as 8 + ? = 10)

Recognise and use the inverse relationship between addition and subtraction to check calculations and solve missing-number problems

Solve addition and subtraction problems using mental and written methods, including problems involving numbers, quantities, and measures

Add and subtract two two-digit numbers using concrete objects, pictorial representations, and mental methods

Add and subtract a two-digit number and ones mentally and using concrete/pictorial representations

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers

Recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100

Add and subtract a two-digit number and tens mentally and using concrete/pictorial representations

Subtract multiples of 10 (10–90) from multiples of 10 using place value strategies

Add three one-digit numbers using strategies including looking for pairs that make 10

Understand and apply the associative property of addition: when adding three numbers, any two can be added first

Use counting on and counting back as strategies for addition and subtraction

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction

Fluently add and subtract within 20 using mental strategies; know from memory all sums of two one-digit numbers

Add and subtract within 1000 using concrete models, drawings, and strategies based on place value; understand composing and decomposing tens and hundreds

Add and subtract numbers with up to three digits using formal written methods of columnar addition and subtraction

Estimate the answer to a calculation and use inverse operations to check answers; apply to increasingly large numbers using rounding and inverse reasoning

Solve addition and subtraction problems including missing-number problems, using number facts, place value, and more complex methods

Solve one- and two-step word problems within 100 using addition and subtraction, with unknowns in all positions

Explain why addition and subtraction strategies work, using place value and the properties of operations

Represent whole numbers as lengths on a number line and represent sums and differences within 100 on a number line diagram

Mentally add and subtract a three-digit number and ones

Add up to four two-digit numbers using strategies based on place value and properties of operations

Mentally add and subtract a three-digit number and hundreds

Mentally add and subtract a three-digit number and tens

Solve two-step word problems using the four operations; represent problems using equations with a letter standing for the unknown quantity

Add and subtract numbers with up to four digits using formal written methods of columnar addition and subtraction

Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and the relationship between addition and subtraction

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

Use rounding to check answers to calculations and determine appropriate levels of accuracy in context

Fluently add and subtract whole numbers with more than four digits using the standard columnar algorithm

Add and subtract numbers mentally with increasingly large numbers, using place-value knowledge and derived facts

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why, with numbers up to 10,000,000 and decimals

Add and subtract decimals to hundredths using strategies based on place value, properties of operations, and the relationship between addition and subtraction; relate strategies to written methods and explain reasoning

Understand positive and negative numbers as describing quantities with opposite directions or values; use them in context such as temperature, floors in a building, and bank balances

Use simple formulae expressed in words or symbols to calculate values (e.g. perimeter = 2 × (length + width))

Express missing number problems algebraically using letters for unknowns; translate word problems into equations

Generate two numerical patterns using two given rules; identify relationships between corresponding terms; form ordered pairs and graph them on a coordinate plane

Find pairs of numbers that satisfy an equation with two unknowns (e.g. find pairs (a, b) where a + b = 10 or 2a + b = 15)

Enumerate possibilities of combinations of two variables systematically (e.g. all ways to choose from a set of options)

Generate and describe linear number sequences, including those with negative and decimal steps; identify the term-to-term rule

Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations

Plot and read coordinates in all four quadrants of the Cartesian plane, using positive and negative x- and y-values to describe positions precisely

Use and interpret algebraic notation including: ab for a × b, 3y for y + y + y, a² for a × a, a/b for a ÷ b, coefficients as fractions, and brackets for grouping; read and write algebraic expressions fluently

Use algebraic methods to solve linear equations in one variable, including equations that require rearrangement, expanding brackets, and collecting terms on both sides; solve equations with rational number coefficients

Simplify algebraic expressions by collecting like terms — combine terms with the same variable and power (e.g., 3a + 2b + 5a = 8a + 2b) while maintaining equivalence

Expand (multiply out) a single term over a bracket using the distributive property, e.g., 3(2x + 5) = 6x + 15; expand expressions involving negative multipliers

Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors; distinguish between an expression (no equals sign), an equation (equals sign), and an inequality (inequality sign)

Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs; move between word problems, algebraic representations, tables, and graphical representations

Substitute numerical values into formulae and expressions including scientific formulae; evaluate expressions by replacing variables with given values and computing the result using correct order of operations

Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither

Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-intercept in context

Plot linear graphs by generating a table of values, reduce a two-variable linear equation to the form y = mx + c, and calculate gradients from two points on a line

Understand and use standard mathematical formulae; rearrange formulae to change the subject, performing inverse operations to isolate a different variable

Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence

Factorise algebraic expressions by taking out common factors — identify the highest common factor of all terms and write the expression as a product, e.g., 6x + 9 = 3(2x + 3)

Use graphs of linear and quadratic functions to estimate output values for given inputs, find approximate solutions to equations, and interpret graphical information in real-world contexts

Recognise that quadratic functions produce curved (parabolic) graphs, distinguish them from linear graphs, and use plotted quadratic graphs to estimate values and find approximate solutions

Understand that two linear equations can be solved simultaneously by finding the point where their graphs intersect, and interpret this graphically and algebraically as the pair of values satisfying both equations

Expand products of two or more binomials, e.g., (x + 3)(x - 2) = x² + x - 6, using the grid method or FOIL; simplify the result by collecting like terms

One-to-one correspondence when counting objects: each object is paired with exactly one number name

Cardinality principle: the last number said when counting a set tells how many objects are in the set, regardless of arrangement or order counted

Compare two groups of objects to determine which has more, fewer, or whether they are equal, using matching and counting strategies

Represent numbers using objects, pictorial representations, and the number line

Each successive counting number represents a quantity that is one larger than the previous number

Rote count forwards and backwards from 0 to 100, beginning from 0, 1, or any given number, by ones

Count in multiples of 2, 5, and 10 (skip counting)

Count a set of objects to answer 'how many?' for sets up to 20 (arranged in lines, arrays, circles, or scattered)

Compare two written numerals between 1 and 10 to determine which is greater or less

Count forwards and backwards in tens from any number (not just multiples of 10)

Count forwards and backwards in steps of 3 from 0

Count from 0 in multiples of 4, 8, 50, and 100

Count within 1000, including skip-counting by 5s, 10s, and 100s

Count in multiples of 6, 7, 9, 25, and 1000

Classify objects into given categories, count the number in each category, and sort the categories by count

Organise and represent data with up to three categories by counting objects in each category and sorting categories by quantity

Read, write, and use the vocabulary of data collection and display — data, tally, tally chart, frequency, frequency table, survey, pictogram, bar chart, axis/axes, scale, label, category, discrete data, continuous data, line graph, pie chart — and apply these terms when collecting, organising, and presenting data

Interpret and construct simple pictograms, tally charts, block diagrams, and simple tables

Interpret categorical data by asking and answering questions about totals, how many in each category, and how many more or less one category has than another

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories; solve put-together, take-apart, and compare problems using information presented in a bar graph

Draw a scaled picture graph and a scaled bar graph to represent a data set; solve one- and two-step comparison, sum, and difference problems using bar charts, pictograms, and tables

Interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs

Complete, read, and interpret information in tables, including timetables

Solve comparison, sum, and difference problems using information presented in a bar graph

Read, write, and use the vocabulary of statistical analysis — mean, median, mode, range, frequency, data, sample, average, chart, table, graph, pie chart, scatter graph, correlation — with understanding of what each term describes

Interpret and construct pie charts and line graphs; use these to solve problems

Calculate and interpret the mean as an average of a data set

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8); use operations on fractions to solve problems involving data in line plots (e.g. redistribute total equally)

Describe, interpret, and compare distributions of a single variable using appropriate measures of central tendency (mean, median, mode) and spread (range), including the effect of outliers

Construct and interpret frequency tables, bar charts, pie charts, pictograms, and vertical line charts for both categorical and grouped numerical data, choosing appropriate representations for the data type

Plot bivariate data on a scatter graph with correctly labelled axes and appropriate scales; describe the correlation (positive, negative, none) and draw an estimated line of best fit where appropriate

Describe simple mathematical relationships between two variables using scatter graphs, identify positive, negative, or no correlation, and use a line of best fit to make predictions

Recognise, find, and name a half as one of two equal parts of an object, shape, or quantity

Recognise, find, and name a quarter as one of four equal parts of an object, shape, or quantity

Recognise, find, name, and write fractions 1/3, 1/4, 2/4, and 3/4 of a length, shape, set of objects, or quantity

Read, write, and use fraction notation correctly — fraction, numerator, denominator, unit fraction, non-unit fraction, proper fraction, improper fraction, mixed number, equivalent fraction, simplest form — and understand what each term describes, including the roles of the numerator and denominator in expressing parts of a whole

Understand that decomposing a shape into more equal shares creates smaller shares

Partition circles and rectangles into two and four equal shares and describe them using the words halves, fourths, and quarters

Write simple fractions (e.g. 1/2 of 6 = 3) and recognise the equivalence of 2/4 and 1/2

Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and from dividing one-digit numbers or quantities by 10

Recognise and use fractions as numbers: place unit fractions and non-unit fractions with small denominators on a number line

Partition circles and rectangles into two, three, or four equal shares; describe shares as halves, thirds, and fourths; recognise that equal shares of identical wholes need not have the same shape

Recognise and show, using diagrams, equivalent fractions with small denominators

Add and subtract fractions with the same denominator within one whole (e.g. 5/7 + 1/7 = 6/7)

Compare and order unit fractions, and fractions with the same denominator

Recognise, find, and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators

Solve problems involving counting in tenths, fractions of quantities, equivalence, fraction addition/subtraction, and fraction comparison

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a/b as a parts of size 1/b

Understand two fractions as equivalent if they are the same size or the same point on a number line; recognise and show families of common equivalent fractions using diagrams

Generate simple equivalent fractions and explain why they are equivalent using visual fraction models

Represent fractions on a number line: partition the interval 0 to 1 into b equal parts to locate 1/b, then mark off a lengths of 1/b from 0 to locate a/b

Recognise and write decimal equivalents of any number of tenths or hundredths (e.g. 3/10 = 0.3, 27/100 = 0.27)

Count up and down in hundredths; recognise that hundredths arise when dividing an object by 100 or dividing tenths by 10

Read, write, and use decimal and percentage notation correctly — decimal, decimal point, tenths, hundredths, thousandths, percentage, per cent, % symbol, convert, terminating decimal — and understand the relationships between fractions, decimals, and percentages as three ways of expressing the same value

Recognise and write decimal equivalents of 1/4, 1/2, and 3/4

Compare numbers with the same number of decimal places up to two decimal places

Find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits as ones, tenths, and hundredths

Add and subtract fractions with the same denominator, including results greater than one whole (e.g. 5/8 + 6/8 = 11/8)

Compare two fractions with the same numerator or the same denominator by reasoning about size; record comparisons with >, =, or < symbols

Express whole numbers as fractions (e.g. 3 = 3/1) and recognise fractions equivalent to whole numbers (e.g. 4/4 = 1, 6/1 = 6)

Solve problems involving increasingly harder fractions to calculate quantities, including non-unit fractions where the answer is a whole number

Solve simple measure and money problems involving fractions and decimals to two decimal places

Round decimals with one decimal place to the nearest whole number

Partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole

Explain why a fraction a/b is equivalent to (n×a)/(n×b) using visual models; use this principle to recognise and generate equivalent fractions, including tenths and hundredths

Use decimal notation for fractions with denominators 10 or 100; read and write decimal numbers as fractions (e.g. 0.62 = 62/100, 0.71 = 71/100)

Express a fraction with denominator 10 as an equivalent fraction with denominator 100 and use this to add fractions with denominators 10 and 100 (e.g. 3/10 + 4/100 = 34/100)

Recognise and use thousandths; relate them to tenths, hundredths, and their decimal equivalents (e.g. 1/1000 = 0.001, 35/1000 = 0.035)

Understand a fraction a/b with a > 1 as a sum of fractions 1/b (e.g. 3/5 = 1/5 + 1/5 + 1/5)

Compare two decimals to hundredths (or up to three decimal places) by reasoning about size using place-value understanding; record with >, =, <

Understand a/b as a multiple of 1/b; multiply proper fractions and mixed numbers by whole numbers, supported by visual models (e.g. 3 × 2/5 = 6/5 = 1 1/5)

Solve problems requiring knowledge of percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and fractions with denominators that are multiples of 10 or 25

Understand the per cent symbol (%); know that per cent means ‘number of parts per hundred’; write percentages as a fraction with denominator 100 and as a decimal

Add and subtract fractions with denominators that are multiples of the same number by finding a common denominator

Compare and order fractions with different numerators and denominators by creating common denominators/numerators or comparing to a benchmark such as 1/2; justify conclusions with visual models

Understand addition and subtraction of fractions as joining and separating parts; decompose a fraction into a sum of fractions with the same denominator in more than one way

Solve problems involving numbers with up to three decimal places

Recognise mixed numbers and improper fractions; convert from one form to the other (e.g. 2/5 + 4/5 = 6/5 = 1 1/5)

Add and subtract mixed numbers with like denominators, including by converting to improper fractions or using properties of operations

Solve word problems involving addition and subtraction of fractions with like denominators, using visual models and equations

Round decimals with two decimal places to the nearest whole number and to one decimal place

Solve word problems involving multiplication of a fraction by a whole number using visual models and equations

Associate a fraction with division and calculate decimal fraction equivalents for simple fractions (e.g. 3/8 = 0.375); recall and use equivalences between simple fractions, decimals, and percentages in different contexts

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b); solve word problems involving division of whole numbers leading to fractional or mixed-number answers

Use common factors to simplify fractions to their simplest form; use common multiples to express fractions with a common denominator

Interpret and compute division of a unit fraction by a non-zero whole number (e.g. 1/3 ÷ 4 = 1/12); use visual models and the relationship between multiplication and division to explain the result

Find the area of a rectangle with fractional side lengths by tiling with unit-fraction squares; show that the area equals the product of the side lengths

Solve real-world problems involving division of unit fractions by whole numbers and whole numbers by unit fractions, using visual models and equations

Interpret multiplication as scaling (resizing): compare the size of a product to a factor based on the size of the other factor without computing; explain the effect of multiplying by fractions greater than, equal to, or less than 1

Interpret and compute division of a whole number by a unit fraction (e.g. 4 ÷ 1/5 = 20); use visual models and the relationship between multiplication and division to explain why the quotient is larger than the dividend

Solve real-world problems involving multiplication of fractions and mixed numbers, using visual fraction models or equations

Multiply a fraction or whole number by a fraction, including proper fractions by proper fractions; interpret (a/b) × q as a parts of q partitioned into b equal parts; write answers in simplest form

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions to produce a common denominator

Solve word problems involving addition and subtraction of fractions with unlike denominators, using visual models and benchmark fractions to estimate and assess reasonableness

Compare and order fractions including fractions greater than 1, by converting to common denominators or using benchmarks

Interpret fractions and percentages as operators — find a fraction or percentage of an amount by multiplying, understanding that 'of' means multiply (e.g., 3/4 of 200 = 3/4 × 200 = 150)

Use the four operations with formal written methods applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8); convert fluently between the two forms

Divide a fraction by a fraction using the 'keep-change-flip' method and visual models; interpret and solve word problems involving division of fractions by fractions

Recognise and name common 3-D shapes (cubes, cuboids, pyramids, spheres, cylinders, cones)

Recognise and name common 2-D shapes (circles, triangles, rectangles including squares)

Describe the position of objects using terms such as above, below, beside, in front of, behind, next to

Analyse and compare 2-D and 3-D shapes using informal language to describe sides, vertices, and other attributes

Describe movement and direction, including whole, half, quarter, and three-quarter turns

Model shapes by building from components (e.g. sticks and clay balls) and by drawing

Compose simple shapes to form larger shapes (e.g. two triangles make a rectangle)

Distinguish two-dimensional (flat) shapes from three-dimensional (solid) shapes

Identify and describe properties of 2-D shapes including the number of sides and line symmetry in a vertical line

Distinguish defining attributes of shapes (e.g. triangles are closed and three-sided) from non-defining attributes (e.g. colour, orientation, overall size)

Use mathematical vocabulary to describe position, direction, and movement, including straight lines and distinguishing rotation as a turn in terms of right angles (quarter, half, three-quarter turns, clockwise and anti-clockwise)

Identify and describe properties of 3-D shapes including the number of edges, vertices, and faces

Compare and sort common 2-D and 3-D shapes and everyday objects by their properties

Compose three-dimensional shapes (cubes, right rectangular prisms, right circular cones, right circular cylinders) and create composite shapes; build new shapes from component shapes

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, quarter-circles) to create composite shapes, and compose new shapes from the composite shape

Identify 2-D shapes on the surface of 3-D shapes (e.g. a circle on a cylinder, a triangle on a pyramid)

Order and arrange combinations of mathematical objects in patterns and sequences

Identify right angles; recognise that two right angles make a half-turn, three make three-quarters, and four make a complete turn

Recognise angles as a property of shape or a description of a turn

Identify horizontal and vertical lines and pairs of perpendicular and parallel lines

Recognise and draw shapes having specified attributes (e.g. a given number of angles or equal faces); identify triangles, quadrilaterals, pentagons, hexagons, and cubes

Draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations and describe them

Identify acute and obtuse angles; compare and order angles up to two right angles by size

Use and interpret standard geometric diagram conventions: mark right angles with a small square, equal lengths with single or double tick marks, and equal angles with arc marks; label angles in three-letter notation (∠ABC) and individual angles with a single letter or number; draw diagrams showing angles at a point, angles on a straight line, and angles inside polygons with these conventions; read diagrams with these marks to identify given information and find unknown values

Understand that shapes in different categories may share attributes defining a larger category; classify quadrilaterals (rhombuses, rectangles, squares) and draw examples of quadrilaterals not in those subcategories

Describe positions on a 2-D grid as coordinates in the first quadrant

Plot specified points on a coordinate grid and draw sides to complete a given polygon

Represent and carry out geometric transformations on squared paper or a coordinate grid: reflections (in horizontal, vertical, and diagonal mirror lines, including the axes), translations (described as a vector or as left/right/up/down moves), and rotations (90° or 180° about a stated centre point); describe each transformation precisely using the correct language; identify which transformation maps one shape onto its image by comparing position, orientation, and size

Identify, draw, and interpret nets of common 3-D shapes — cubes, cuboids, triangular prisms, and square-based pyramids — by predicting which 3-D shape a given flat arrangement of faces will fold into, checking whether a net will close completely, and sketching a net from a description or 3-D model; understand the relationship between the number of faces and the structure of the net

Describe movements between positions as translations of a given unit to the left/right and up/down

Identify lines of symmetry in 2-D shapes presented in different orientations; recognise line-symmetric figures and draw lines of symmetry

Complete a simple symmetric figure with respect to a specific line of symmetry

Know that angles are measured in degrees, where one degree is 1/360 of a full turn; understand that an angle turning through n one-degree angles has a measure of n degrees

Understand that an angle is a geometric shape formed by two rays sharing a common endpoint (vertex); recognise angles in real-life contexts and 2-D shapes

Measure angles in whole-number degrees using a protractor; draw given angles and sketch angles of specified measure

Draw and identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines; identify these in two-dimensional figures

Know that angles at a point sum to 360° (one whole turn), angles on a straight line sum to 180°, and vertically opposite angles are equal; use these facts to find missing angles

Distinguish between regular and irregular polygons based on reasoning about equal sides and equal angles

Recognise angle measure as additive; find unknown angles by adding or subtracting on a diagram using equations with a symbol for the unknown

Estimate and compare acute, obtuse, and reflex angles in degrees; classify angles by type and order them by size

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or angles of a specified size; recognise right triangles as a category

Use the properties of rectangles to deduce related facts and find missing lengths and angles

Identify 3-D shapes, including cubes and other cuboids, from 2-D representations

Identify, describe, and represent the position of a shape following a reflection or translation using appropriate language; know that the shape has not changed

Describe positions on the full coordinate grid (all four quadrants); use coordinates with negative values

Draw and translate simple shapes on the coordinate plane; reflect shapes in the axes

Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties

Understand a coordinate system defined by two perpendicular number lines (axes) with an origin at (0,0); know that an ordered pair (x, y) specifies a unique point where the first number gives horizontal distance and the second gives vertical distance from the origin

Plot and read ordered pairs in the first quadrant of the coordinate plane; represent real-world and mathematical problems by graphing points and interpreting coordinate values in context

Find unknown angles in triangles, quadrilaterals, and regular polygons using angle sum properties

Draw 2-D shapes using given dimensions and angles, using a ruler and protractor accurately

Recognise angles where they meet at a point, are on a straight line, or are vertically opposite; find missing angles using these properties

Recognise, describe, and build simple 3-D shapes, including making nets

Illustrate and name parts of circles, including radius, diameter, and circumference; know that the diameter is twice the radius

Derive and apply formulae for the area of triangles, parallelograms, and trapezia, and for the volume of cuboids and other prisms (including cylinders), connecting each formula to its geometric reasoning

Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons

Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original

Use conventional geometric terms and notation to describe, sketch, and draw points, lines, parallel and perpendicular lines, right angles, regular polygons, and reflectively/rotationally symmetric polygons

Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths

Draw and measure line segments and angles accurately using ruler and protractor, and interpret scale drawings to extract real measurements

Apply the properties of angles at a point (360°), on a straight line (180°), and vertically opposite angles to find unknown angles in multi-step problems

Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections

Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids

Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts

Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent

Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle

Understand and use the relationship between parallel lines cut by a transversal: corresponding angles, alternate interior angles, and co-interior (same-side interior) angles; use these to find unknown angles

Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem

Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts

Show and tell how a mathematical answer was found using objects, drawings, and spoken words

Make sense of a problem by identifying what is being asked, choosing concrete objects or pictures to represent the situation, and explaining a pathway to the solution

Use objects, drawings, or simple number sentences to represent a real-world situation (early mathematical modelling)

Notice simple patterns and structures: spot that changing order doesn't change the total, and recognise how numbers relate to each other

Use mathematical words carefully when counting, comparing, and describing shapes and positions

Move between a real-world situation and a mathematical representation using concrete objects, drawings, diagrams, tables, number sentences, or bar models

Notice when a calculation or pattern repeats and use this to count more efficiently or predict results

Select and use familiar tools (concrete objects, fingers, ten frames) to help solve a mathematical problem

With teacher prompting, explain and justify mathematical reasoning using drawings, number sentences, or words

With teacher guidance, make sense of multi-step and more complex problems by planning a pathway to the solution, identifying relevant information, and choosing appropriate operations

Represent real-world problems with number sentences, bar models, or diagrams, and interpret the mathematical result back in context

Select and use appropriate tools and representations (number lines, hundred squares, rulers, part-whole models) to support problem-solving

Recognise and use repeated reasoning to generalise: spot calculation patterns, describe rules for sequences, and predict results using known mathematical facts

Communicate with mathematical precision: use correct vocabulary, specify units, and use symbols accurately

Move between real-world situations, drawings, and number sentences, explaining how each representation connects to the others (quantitative reasoning)

Look for and use mathematical structure: apply properties of operations, place-value patterns, and relationships between shapes to solve problems efficiently

With teacher support, make sense of multi-step problems involving larger numbers or mixed operations by breaking them into parts, choosing strategies, and checking answers for reasonableness — children at this stage are developing the habit with guidance; independent strategy evaluation comes later

Communicate with mathematical precision: use correct place-value and fraction vocabulary, specify units in measurement answers, and use notation accurately

Construct and follow multi-step mathematical arguments; identify errors in reasoning and explain why a method works or does not work

Model real-world problems involving measurement, money, and time by choosing appropriate representations and interpreting results in context

Move fluently between real-world situations, diagrams, and symbolic equations involving three-digit numbers and fractions, explaining what each part represents

Select and use appropriate tools and representations strategically, including choosing between mental methods, jottings, formal algorithms, and calculators for arithmetic with multi-digit numbers, decimals, and fractions

Look for and use mathematical structure: apply place-value patterns to three-digit operations, use multiplication/division relationships, and exploit shape properties to classify

Recognise and use repeated reasoning to generalise: extend multiplication table patterns, derive unknown facts from known ones, and describe rules for sequences

Communicate with mathematical precision: use correct fraction/decimal vocabulary, name angle types accurately, specify units in measurement and money, and use notation (=, <, >, ÷, ×) correctly

Make sense of multi-step problems involving four operations, fractions, and area/volume by identifying sub-steps, choosing a strategy, and monitoring progress

Construct and present multi-step mathematical arguments; critique the reasoning of others and explain clearly why a method works or fails

Select and use appropriate tools and representations strategically: choose between mental, written, and diagrammatic methods; use calculators for checking; select fraction models suited to the task

Model real-world problems involving multiplication, area, fractions, and unit conversion by choosing appropriate representations and interpreting mathematical results in context

Recognise and use repeated reasoning to generalise: extend patterns in times tables and equivalent fractions, derive unknown facts from known facts efficiently, describe general rules

Move fluently between real-world situations, diagrams, number lines, and symbolic equations involving multiplication, fractions, and decimals, explaining what each representation shows

Look for and use mathematical structure: exploit place-value patterns for ×10/×100, use the distributive property to break apart multiplications, apply fraction equivalence to compare and compute, use shape properties to classify quadrilaterals

Make sense of complex multi-step problems involving large numbers, fractions, decimals, and percentages by analysing what is known and unknown, planning multi-step strategies, and evaluating reasonableness through estimation and inverse operations

Communicate with mathematical precision: use correct vocabulary for primes, factors, multiples, angle types, and polygon regularity; specify units including cm², m³, °; use notation for squares/cubes and percentages accurately

Construct and present logical mathematical arguments involving multiple steps; critique others' reasoning about fractions, angles, or calculations and clearly explain errors or alternative methods

Model real-world problems involving scaling, unit conversion, area/perimeter, and percentage by selecting appropriate mathematical representations and interpreting results in context

Select and use tools and representations strategically: choose between mental methods, formal written methods, protractors, fraction strips, and diagrams based on the demands of the problem

Move fluently between real-world situations, diagrams, number lines, bar models, and symbolic equations involving multi-digit multiplication, fractions, decimals, and percentages, explaining connections between representations

Recognise and use repeated reasoning to generalise: extend patterns in equivalent fractions and percentage conversions, derive unknown facts from known facts, describe general rules for sequences and predict terms

Look for and use mathematical structure: exploit the relationship between fractions, decimals, and percentages; use factor pairs to simplify multiplication; apply angle facts to find unknowns; use properties of regular polygons systematically

Make sense of complex multi-step problems involving ratio, proportion, algebra, negative numbers, and all four operations with fractions and decimals by analysing given and unknown quantities, planning solution strategies, and evaluating reasonableness using estimation and inverse operations

Move fluently between real-world situations, diagrams, coordinate grids, algebraic expressions, tables, and symbolic equations involving fractions, ratio, and algebra, explaining connections between representations

Model real-world problems involving ratio, scale, volume, unit conversion, and proportional reasoning with appropriate tools, diagrams, or equations

Communicate with mathematical precision: use correct vocabulary for ratio, proportion, algebra, volume, coordinate geometry, and circle parts; specify units including cm³, m³, and miles/km; use notation for algebraic expressions and order of operations accurately

Construct and present logical mathematical arguments involving multiple steps and formal reasoning; critique others' reasoning about fractions, algebra, ratio, or geometry and clearly explain errors or alternative approaches

Select and use tools and representations strategically: choose between mental methods, formal written methods, algebraic approaches, coordinate grids, and technology based on the demands of the problem

Look for and use mathematical structure: exploit the hierarchy of 2-D shapes to deduce properties; use order of operations and algebraic structure to simplify expressions; connect fraction–decimal–percentage equivalences; use ratio structure to solve proportion problems efficiently

Recognise and use repeated reasoning to generalise: describe algebraic rules for nth terms, use properties of operations to simplify, and verify generalisations with specific cases

Describe and identify measurable attributes of objects such as length, height, weight, and capacity; use comparative language (longer, shorter, heavier, lighter, more, less)

Compare two objects directly by length or height and describe the difference using language such as long, short, tall, longer, shorter, taller, double, half

Compare and describe capacity and volume using language such as full, empty, more than, less than, half full

Compare two objects directly by mass or weight and describe the difference using language such as heavy, light, heavier than, lighter than

Sequence events in chronological order using language such as before, after, next, first, today, yesterday, tomorrow, morning, afternoon, evening

Use comparative language for time: quicker, slower, earlier, later

Measure and begin to record lengths and heights using non-standard and standard units

Measure and begin to record capacity and volume using non-standard and standard units

Measure and begin to record mass/weight using non-standard and standard units

Recognise and use language relating to dates, including days of the week, weeks, months, and years

Measure and begin to record time in hours, minutes, and seconds

Tell the time to the hour and half past the hour, and draw clock hands to show these times

Recognise and know the value of different coins and notes

Choose and use appropriate standard units to measure length (m/cm), mass (kg/g), temperature (°C), and capacity (litres/ml) to the nearest appropriate unit

Measure the length of an object using same-size length units laid end to end with no gaps or overlaps

Order three objects by length and compare the lengths of two objects indirectly using a third object

Compare and order lengths, mass, and capacity and record results using >, <, and =

Know the number of minutes in an hour and the number of hours in a day

Tell and write the time to five minutes, including quarter past and quarter to, and draw clock hands to show these times

Compare and sequence intervals of time

Solve simple money problems involving addition and subtraction, including giving change

Find different combinations of coins that equal the same amount of money

Recognise and use symbols for pounds (£) and pence (p) and combine amounts to make a particular value

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, metre sticks, and measuring tapes

Measure, compare, add, and subtract lengths (m/cm/mm), mass (kg/g), and volume/capacity (l/ml) using standard units

Measure the perimeter of simple 2-D shapes

Know the number of seconds in a minute and the number of days in each month, year, and leap year

Compare durations of events and calculate the time taken by particular events or tasks

Tell and write time from analogue and digital clocks to the nearest five minutes, using a.m., p.m., and 12-hour and 24-hour notation

Estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes, and hours

Solve word problems involving lengths within 100, using addition and subtraction with drawings and equations

Generate measurement data by measuring lengths to the nearest whole unit and display the data on a line plot

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit

Add and subtract amounts of money to give change, using both £ and p in practical contexts

Estimate lengths using units of inches, feet, centimetres, and metres

Measure the length of an object using two different length units and describe how the measurements relate to the size of the unit chosen

Measure areas by counting unit squares (square cm, square m, square in, square ft)

Understand that a unit square has one square unit of area and that the area of a plane figure is the number of unit squares that cover it without gaps or overlaps

Find the area of a rectangle by tiling it with unit squares and show that the result equals the product of the side lengths

Multiply side lengths to find areas of rectangles and represent whole-number products as rectangular areas

Use tiling to demonstrate the distributive property: the area of a rectangle with sides a and (b+c) equals a×b + a×c; use area models to represent the distributive property

Convert between different units of measure (e.g. kilometre to metre, hour to minute, minute to second, year to month, week to day)

Solve problems involving perimeters of polygons: find perimeter from side lengths, find an unknown side length, and explore rectangles with same perimeter but different areas (or vice versa)

Solve word problems involving elapsed time by adding and subtracting time intervals in minutes, including using a number line

Generate measurement data by measuring lengths to the nearest half and quarter inch; display the data on a line plot with a scale marked in whole numbers, halves, and quarters

Tell and write time to the nearest minute using analogue and digital clocks

Estimate, compare, and calculate different measures including money in pounds and pence

Measure and estimate liquid volumes and masses of objects using grams, kilograms, and litres; solve one-step word problems involving mass or volume

Recognise area as additive; find areas of rectilinear figures by decomposing into non-overlapping rectangles and summing their areas

Read, write, and convert time between analogue and digital 12-hour and 24-hour clocks

Solve word problems involving distances, time intervals, liquid volumes, masses, and money using the four operations with fractions or decimals; represent with diagrams including number lines

Solve problems involving converting between units of time (hours↔minutes, minutes↔seconds, years↔months, weeks↔days)

Know relative sizes of measurement units within one system (km/m/cm/mm, kg/g, l/ml, hr/min/sec); convert between different metric units and express measurements in terms of a smaller unit; record equivalents in conversion tables

Apply the area formula (l × w) and perimeter formula (2l + 2w) for rectangles including squares in real-world and mathematical problems; calculate and compare areas using standard units (cm², m²) and estimate areas of irregular shapes

Make a line plot to display measurement data in fractions of a unit (1/2, 1/4, 1/8); solve problems involving addition and subtraction of fractions using line plot data

Understand and use approximate equivalences between metric units and common imperial units (inches, pounds, pints)

Measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres

Estimate volume of cuboids using 1 cm³ blocks; estimate capacity of containers using water

Calculate the area of parallelograms and triangles using formulae (A = b × h for parallelograms, A = ½ × b × h for triangles)

Convert between miles and kilometres using the approximate relationship (5 miles ≈ 8 km)

Recognise volume as additive; find volumes of composite solid figures made of two or more non-overlapping right rectangular prisms

Convert among different-sized standard measurement units within a given system (e.g. 5 cm to 0.05 m) using decimal notation to up to three decimal places; convert between smaller and larger units of length, mass, volume, and time

Solve problems involving the calculation and conversion of units of measure, using decimal notation and multi-step reasoning in real-world contexts

Find the volume of right rectangular prisms by packing with unit cubes and show it equals l × w × h (or base area × height); apply V = l × w × h and V = B × h to solve real-world problems; calculate, estimate, and compare volumes of cubes and cuboids in standard units (cm³, m³)

Recognise that shapes with the same area can have different perimeters and vice versa; explore this relationship systematically

Measure volumes by counting unit cubes using cubic cm, cubic in, cubic ft, and other units

Recognise volume as an attribute of solid figures; understand that a unit cube (side length 1 unit) has 'one cubic unit' of volume and can be used to measure volume; a solid packed with n unit cubes has volume n cubic units

Understand division as sharing equally into groups or as grouping (how many groups of a given size can be made)

Understand multiplication as repeated addition and grouping equal sets

Use arrays to represent multiplication and division situations

Recall and use multiplication and division facts for the 2, 5, and 10 multiplication tables

Read, write, and interpret the symbols ×, ÷, and = in multiplication and division number sentences

Understand and apply the commutative property of multiplication and recognise that division is not commutative

Solve problems involving multiplication and division using arrays, repeated addition, mental methods, and known facts

Recognise odd and even numbers

Recall and use multiplication and division facts for the 3, 4, and 8 multiplication tables

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and 5 columns; write an equation to express the total as a sum of equal addends

Write and calculate mathematical statements for multiplication and division using known tables, including two-digit × one-digit, using mental and progressing to formal written methods

Solve problems involving multiplication and division, including scaling problems and correspondence problems where n objects are connected to m objects

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them

Interpret products of whole numbers (e.g. 5 × 7 as the total number of objects in 5 groups of 7)

Recall multiplication and division facts for multiplication tables up to 12 × 12

Multiply two-digit and three-digit numbers by a one-digit number using formal written layout

Apply properties of operations (commutative, associative, distributive) as strategies to multiply and divide

Fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division

Understand division as an unknown-factor problem (e.g. find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8)

Interpret whole-number quotients (e.g. 56 ÷ 8 as the number of objects in each share or the number of equal groups)

Solve problems involving multiplying and adding, including using the distributive law, integer scaling problems, and harder correspondence problems

Recognise and use factor pairs and commutativity in mental calculations

Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations

Use place value, known and derived facts to multiply and divide mentally, including multiplying by 0 and 1, dividing by 1, and multiplying together three numbers

Use multiplication and division within 100 to solve word problems involving equal groups, arrays, and measurement quantities

Determine the unknown whole number in a multiplication or division equation relating three whole numbers (e.g. 8 × ? = 48, ? × 6 = 42)

Multiply a whole number of up to four digits by a one-digit number, and multiply two two-digit numbers, using formal written methods including long multiplication; illustrate with area models

Solve multi-step word problems using the four operations with whole numbers, including interpreting remainders in context; represent with equations using a letter for the unknown; check with estimation

Divide numbers up to four digits by a one-digit number using short division (and place-value/array strategies); interpret remainders appropriately for the context

Find all factor pairs for a whole number in the range 1–100; identify common factors and common multiples of two numbers; use these concepts to solve problems

Multiply and divide whole numbers and those involving decimals by 10, 100, and 1000

Interpret a multiplication equation as a comparison (e.g. 35 = 5 × 7 means 35 is 5 times as many as 7); represent verbal statements of multiplicative comparisons as equations

Generate a number or shape pattern that follows a given rule; identify apparent features of the pattern not explicit in the rule and explain informally why they occur

Solve problems involving scaling by simple fractions and problems involving simple rates

Solve problems involving multiplication and division using knowledge of factors, multiples, squares, and cubes

Multiply and divide numbers mentally drawing upon known facts, including related facts and place-value adjustments

Know and use the vocabulary of prime numbers, prime factors, and composite numbers; establish whether a number up to 100 is prime; recall prime numbers up to 19

Solve word problems involving multiplicative comparison using drawings and equations with a symbol for the unknown number

Recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)

Fluently multiply multi-digit whole numbers (up to 4 digits by 2 digits) using the formal written method of long multiplication

Divide numbers up to 4 digits by a two-digit divisor using formal written long division, interpreting remainders as whole numbers, fractions, or by rounding as appropriate

Solve problems which require answers to be rounded to specified degrees of accuracy

Use parentheses, brackets, or braces in numerical expressions and evaluate expressions containing these grouping symbols

Understand and apply the conventional order of operations (PEMDAS/BODMAS) to carry out calculations involving the four operations

Write simple numerical expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g. recognise that 3 × (18932 + 921) is three times as large as 18932 + 921)

Multiply one-digit numbers with up to two decimal places by whole numbers (e.g. 3.47 × 6)

Use written division methods in cases where the answer has up to two decimal places; divide decimals to hundredths by whole numbers

Solve problems involving addition, subtraction, multiplication, and division, deciding which operations and methods to use and why; solve multi-step problems in contexts

Multiply and divide numbers by 10, 100, and 1000 giving answers up to three decimal places, understanding that digits shift position in the place-value chart

Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Perform mental calculations including with mixed operations and large numbers, using strategies such as partitioning, compensation, and derived facts

Divide numbers up to 4 digits by a two-digit number using formal written short division where appropriate, interpreting remainders according to context

Recognise and use relationships between operations including inverse operations; use these relationships to check answers and simplify calculations

Use conventional notation for the priority of operations including brackets, powers, roots, and reciprocals; apply BIDMAS/BODMAS consistently to evaluate complex numerical expressions

Use the concepts and vocabulary of prime numbers, factors, multiples, common factors, common multiples, highest common factor (HCF), lowest common multiple (LCM), and prime factorisation including product notation and the unique factorisation property

Multiply and divide with positive and negative integers and rational numbers, understanding the rules for the sign of the product or quotient

Read and write numerals from 0 to 20

Understand that the teen numbers (11–19) are composed of ten ones and some further ones (early place value)

Read and write number words from one to twenty

Read and write numerals from 0 to 100

Understand that 10 can be thought of as a bundle of ten ones — called a 'ten'

Understand that the two digits of a two-digit number represent amounts of tens and ones

Compare and order two-digit numbers using the symbols >, =, and <, based on place value understanding

Identify, represent, and estimate numbers using different representations including the number line

Mentally find 10 more or 10 less than a given two-digit number without counting

Read and write numbers to at least 100 in words

Use place value understanding and number facts to solve problems

Understand that the multiples of 10 (10, 20, 30 … 90) represent one to nine tens and 0 ones

Count to 120 starting at any number less than 120; read and write numerals to 120

Understand that 100 can be thought of as a bundle of ten tens — called a 'hundred'

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones

Compare and order numbers up to 1000 using >, =, and < symbols, based on place-value understanding

Understand that the multiples of 100 (100–900) each represent a number of hundreds with 0 tens and 0 ones

Read and write numbers to 1000 in numerals, number names, and expanded form

Find 10 or 100 more or less than a given number up to 1000

Solve number problems and practical problems involving place value of numbers up to 1000

Determine whether a group of objects (up to 20) has an odd or even number of members

Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and ones)

Order and compare numbers beyond 1000

Count backwards through zero to include negative numbers

Round any number to the nearest 10, 100, or 1000

Identify, represent, and estimate numbers up to 10,000 using different representations

Solve number and practical problems involving place value with increasingly large positive numbers

Find 1000 more or less than a given number

Read Roman numerals to 100 (I to C) and understand that the numeral system changed over time to include zero and place value

Recognise that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (e.g. 700 ÷ 70 = 10)

Read, write, order, and compare whole numbers up to at least 1,000,000 using base-ten numerals, number names, expanded form, and place-value understanding

Interpret negative numbers in context (temperature, sea level, bank balance); count forwards and backwards with positive and negative whole numbers, including through zero

Round any whole number up to 1,000,000 to the nearest 10, 100, 1,000, 10,000, or 100,000 using place-value understanding

Solve number and practical problems involving reading, writing, ordering, comparing, and rounding whole numbers up to 1,000,000

Count forwards and backwards in steps of powers of 10 (10, 100, 1000, 10,000, 100,000) for any given number up to 1,000,000

Read Roman numerals to 1000 (M) and recognise years written in Roman numerals

Recognise that in a multi-digit number, a digit in one place represents 10 times as much as in the place to its right and 1/10 of what it represents in the place to its left

Read, write, and compare decimals to thousandths using base-ten numerals, number names, and expanded form; compare using >, =, < based on place-value meaning

Solve number and practical problems involving reading, writing, ordering, comparing, rounding, and negative numbers up to 10,000,000

Read, write, order, and compare numbers up to 10,000,000 and determine the value of each digit

Identify the value of each digit in numbers given to three decimal places (e.g. in 4.378, the 7 represents 7 hundredths)

Round decimals to any place using place-value understanding; round any whole number to a required degree of accuracy

Use negative numbers in context (temperature, finance, sea level); calculate intervals across zero

Explain patterns in zeros when multiplying by powers of 10 and in decimal-point placement when multiplying/dividing by a power of 10; use whole-number exponents to denote powers of 10 (e.g. 10³ = 1000)

Order positive and negative integers, decimals, and fractions on a number line; use the symbols =, ≠, <, >, ≤, ≥ to compare values including negative numbers and mixed representations

Understand and use place value for decimals, measures, and integers of any size; extend the number system to include all positive and negative integers, decimals, and fractions on a single number line

Round numbers and measures to an appropriate degree of accuracy including to a specified number of decimal places or significant figures

Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations

Understand the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as magnitude in real-world contexts; distinguish absolute value comparisons from ordering statements

Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b; understand upper and lower bounds of rounded values

Interpret and compare numbers in standard form A × 10ⁿ where 1 ≤ A < 10 and n is an integer; convert between ordinary numbers and standard form

Appreciate the infinite nature of the sets of integers, real numbers, and rational numbers; position integers on a number line and distinguish between rational and irrational numbers

Describe the probability of simple equally-likely outcomes using unit fractions: the probability of rolling a 6 on a fair die is 1/6, flipping heads is 1/2, picking one specific colour from three equally represented colours is 1/3; place these fractional probabilities on a 0-to-1 probability scale

Conduct simple probability experiments — flipping a coin, rolling a die, pulling coloured counters from a bag — record results, and compare experimental outcomes with expected theoretical outcomes

Use probability language to describe and compare the likelihood of everyday events using words such as certain, likely, even chance, unlikely, impossible

Understand that 'equally likely' means every outcome has exactly the same chance of occurring; identify whether a given situation has equally likely outcomes (a fair coin, a fair die, a spinner with equal sections) or unequally likely outcomes (a bag with more of one colour, a spinner with unequal sections)

Compare the likelihood of different events and order them from least to most likely — including situations with unequal outcomes such as a bag with more of one colour than another, or a spinner with sections of different sizes — and explain reasoning using informal language

Calculate the probability of a simple event with equally likely outcomes using the formula: probability = number of favourable outcomes ÷ total number of possible outcomes; express the result as a fraction in its simplest form; apply to rolling dice, drawing from bags, and other simple chance situations

Understand probability as a measure expressed as a number between 0 (impossible) and 1 (certain); place events on the probability scale; express probabilities as fractions, decimals, and percentages

Understand that when all possible outcomes of a trial are listed, their probabilities must add up to 1; use this to find the probability of an event NOT happening: P(not A) = 1 − P(A); apply this shortcut to avoid counting all unfavourable outcomes directly

Run repeated probability experiments and compare experimental (relative frequency) results with theoretical predictions; understand and demonstrate that as the number of trials increases, the experimental probability tends towards the theoretical probability — and that short runs can give very different results

Understand and apply the rule that probabilities of all mutually exclusive outcomes sum to one; use this to find the probability of a complementary event (P(not A) = 1 − P(A))

Understand probability as a measure on a scale from 0 (impossible) to 1 (certain); use the language of probability including likely, unlikely, certain, and impossible

Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability

Generate theoretical sample spaces for single and combined events using listing, tables, and tree diagrams, and calculate theoretical probabilities as the number of favourable outcomes divided by the total number of equally likely outcomes

Enumerate sets and their unions and intersections systematically using tables, grids, and Venn diagrams to organise and count outcomes

Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events

Represent ratio and proportion problems using bar models (rectangular strips divided into equal parts labelled with quantities) and tape diagrams (segmented strips showing part-to-part and part-to-whole relationships); use these visual models to set up and solve unequal sharing, scaling, and percentage problems — drawing the diagram first, then reading off the answer

Know and use the vocabulary of ratio and proportion — ratio, proportion, percentage, scale, equivalent, unequal, relative size, part-to-part, part-to-whole, and out of — and understand the difference between ratio (comparing parts to parts) and proportion (comparing a part to the whole)

Solve problems involving the calculation of percentages of amounts (e.g. 15% of 360) and the use of percentages for comparison

Solve problems involving similar shapes where the scale factor is known or can be found

Solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts

Solve problems involving unequal sharing and grouping using knowledge of fractions and multiples

Use compound units such as speed (distance ÷ time), unit pricing (cost ÷ quantity), and density (mass ÷ volume); solve problems involving compound units

Use scale factors to interpret and create scale diagrams and maps, calculating real-life distances from map measurements and vice versa

Express one quantity as a fraction of another where the result may be less than 1 or greater than 1, and interpret the meaning in context

Define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal; express one quantity as a percentage of another; compare quantities using percentages; work with percentages greater than 100%

Convert freely between related standard units of measurement (time, length, area, volume/capacity, mass) using decimal notation to up to three decimal places where appropriate

Use ratio notation to describe the relationship between two or more quantities, simplify ratios to their simplest form, and convert between ratio and fraction representations

Divide a given quantity into two parts in a given part:part or part:whole ratio, and express the division as a fraction of the whole

Know and use advanced vocabulary of multiplicative reasoning — direct proportion, inverse proportion, ratio, rate, unit rate, compound unit, scale factor — accurately in problem-solving contexts

Represent proportional relationships using double number lines (two parallel number lines aligned at 0) and ratio tables; recognise that equivalent ratios generate straight lines through the origin when graphed

Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations

Understand that a multiplicative relationship between two quantities can be expressed as a ratio; use ratio notation; simplify ratios

Solve problems involving percentage increase, percentage decrease, finding the original value after a percentage change, and calculating simple interest