5th Grade Math Checklist: What Your Child Should Know

Fractions

Your child is mastering advanced fraction operations — adding, subtracting, and multiplying fractions with different denominators, dividing with fractions, and solving real-world problems using visual models and mathematical reasoning.

• Equivalent fractions (age 9+)

  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 a fraction strip to show 2/3 = 4/6 = 6/9
  • Explain that multiplying numerator and denominator by the same number gives an equivalent fraction because the size of the whole is unchanged
  • Generate three fractions equivalent to 3/5 and verify with diagrams

• Decimals and fractions (age 10+)

  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

  • Convert 3/8 to 0.375 by dividing 3 ÷ 8
  • State that 1/5 = 0.2 = 20% from memory
  • Use fraction–decimal–percentage equivalences to compare 30%, 1/3, and 0.35 and put them in order

• Decimals for Tenths & 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)

  • Rewrite 0.62 as 62/100
  • Write 3/10 as 0.3 and locate on a number line
  • Read 0.07 and express as 7/100

• Converting tenths to hundredths

  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)

  • Rewrite 7/10 as 70/100
  • Calculate 3/10 + 4/100 = 34/100
  • Explain why 5/10 = 50/100 using a hundredths grid

• Fractions of a whole (age 10+)

  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

  • Explain that 3/4 means 3 ÷ 4 and verify by showing 3/4 × 4 = 3
  • Solve: '9 people share a 50-pound sack equally — how many pounds each?' and express the answer as 5 5/9
  • Use a visual model to show that sharing 3 wholes among 4 people gives each person 3/4

• Decimal & Percent Notation

  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

  • Read and write decimal numbers correctly, identifying the value of each digit (ones, tenths, hundredths)
  • Use the % symbol correctly and explain that per cent means 'out of 100'
  • Convert between simple fractions, decimals, and percentages (e.g. 1/2 = 0.5 = 50%) and explain why they are equal

• Tenths (age 9+)

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

  • Write 0.025 as 25/1000
  • Explain that 1 tenth = 100 thousandths
  • Place 0.345 on a number line between 0.34 and 0.35

• Simplifying Fractions

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

  • Simplify 18/24 to 3/4 by identifying the HCF of 18 and 24
  • Express 2/3 and 5/8 with a common denominator of 24
  • Explain why dividing numerator and denominator by a common factor produces an equivalent fraction

• Dividing fractions (unit fractions)

  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

  • Compute (1/3) ÷ 4 = 1/12 and explain using a visual model of splitting 1/3 into 4 equal parts
  • Create a story context for (1/6) ÷ 3 and solve it
  • Verify (1/3) ÷ 4 = 1/12 by showing (1/12) × 4 = 1/3

• Understanding fractions (age 9+)

  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)

  • Express 5/8 as a sum of five copies of 1/8
  • Show on a number line how 4/3 is built by iterating 1/3 four times
  • Explain why 7/4 = 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4

• Area with Fractions

  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

  • Tile a 2/3 × 3/4 rectangle with 1/12 squares and count to find area = 6/12 = 1/2
  • Explain why the number of unit-fraction tiles equals the product of the two fractions
  • Calculate the area of a rectangle with sides 1 1/2 and 2/3 using fraction multiplication

• Dividing unit fractions and whole numbers

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

  • Solve: '3 people share 1/2 lb of chocolate equally — how much each?' (1/6 lb)
  • Solve: 'How many 1/3-cup servings in 2 cups of raisins?' (6 servings)
  • Create a story context for a given unit-fraction division expression

• Multiplication as scaling

  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

  • Predict without calculating whether 3/4 × 7 is greater or less than 7
  • Explain why multiplying by 5/3 makes a number larger and multiplying by 2/5 makes it smaller
  • Relate multiplying a/b by n/n = 1 to the principle of fraction equivalence

• Dividing by Fractions

  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

  • Compute 4 ÷ (1/5) = 20 and explain: 'How many fifths fit in 4 wholes?'
  • Create a story context for 6 ÷ (1/4) and solve it
  • Verify 4 ÷ (1/5) = 20 by showing 20 × (1/5) = 4

• Real-world fraction multiplication

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

  • Solve: 'A recipe needs 2 1/3 cups of flour; you want to make 1 1/2 batches. How much flour?'
  • Draw a fraction model to represent and solve a multiplication word problem
  • Check the reasonableness of a fraction product in context using estimation

• Multiplying fractions (age 10+)

  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

  • Compute (2/3) × (4/5) = 8/15 and show with an area model
  • Use a visual model to demonstrate (2/3) × 4 = 8/3 = 2 2/3
  • Simplify the product 3/4 × 2/3 = 6/12 = 1/2

• Comparing Decimals

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

  • Compare 0.45 and 0.405 and explain which is greater
  • Order 3.142, 3.14, 3.2 from smallest to largest
  • Justify that 0.7 = 0.70 = 0.700 using place-value reasoning

• Multiplying fractions

  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)

  • Calculate 4 × 3/8 using repeated addition or the rule n × a/b = (n×a)/b
  • Multiply 2 1/3 × 3 and express as a mixed number
  • Use a visual model to show why 5 × 1/4 = 5/4

• Percentage and decimal equivalents

  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

  • State that 1/5 = 20% = 0.2 and use this to find 20% of 60
  • Convert 3/4 to 75% and to 0.75
  • A shop offers 25% off a £40 item — what is the sale price?

• Understanding Percentages

  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

  • Write 35% as 35/100 and as 0.35
  • Shade 40% of a 10×10 grid and write the fraction 40/100
  • Explain that 100% means the whole, 50% means half

• Adding Fractions (Unlike Denominators)

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

  • Compute 2/3 + 5/4 by converting to twelfths: 8/12 + 15/12 = 23/12
  • Subtract 1 3/5 from 3 1/4 using a common denominator
  • Explain the strategy of finding a common denominator and verify using a visual model

• Adding fractions (different denominators)

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

  • Calculate 1/3 + 1/6 by converting to sixths: 2/6 + 1/6 = 3/6 = 1/2
  • Calculate 3/4 − 1/8 by converting to eighths
  • Add 2/5 + 3/10 and simplify the answer

• Fraction Word Problems

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

  • Solve: 'Tara ate 2/5 of a pizza and Sam ate 1/3. How much did they eat together?'
  • Recognise that 2/5 + 1/2 ≠ 3/7 because 3/7 < 1/2 but the sum should exceed 1/2
  • Estimate a fraction sum using benchmarks (0, 1/2, 1) before computing exactly

• Comparing fractions (age 9+)

  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

  • Compare 3/8 and 5/12 by finding a common denominator of 24
  • Order 2/3, 3/5, and 7/10 from smallest to largest
  • Use the benchmark 1/2 to decide that 5/8 > 3/7

• Fraction Addition Concepts

  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

  • Show that 3/8 = 1/8 + 1/8 + 1/8 and also 3/8 = 1/8 + 2/8
  • Write two different decompositions of 5/6
  • Use a visual model to justify a decomposition of 2 1/8 into whole-number and fraction parts

• Comparing fractions (age 10+)

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

  • Order 3/4, 7/8, 5/6, and 11/12 from smallest to largest
  • Compare 7/5 and 4/3 using common denominators
  • Place improper fractions and mixed numbers on a number line in correct order

• Decimals to three places

  Solve problems involving numbers with up to three decimal places

  • Calculate the total of three measurements: 1.234 m + 0.567 m + 2.199 m
  • A bottle holds 1.5 litres; 0.375 l is poured out — how much remains?
  • Find two numbers with 3 d.p. that add to make 1

• Mixed numbers and improper fractions

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

  • Convert 11/4 to 2 3/4
  • Convert 3 2/5 to 17/5
  • Write 6/5 + 4/5 = 10/5 = 2 as both improper fraction and whole number

• Adding and subtracting mixed numbers

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

  • Calculate 2 3/5 + 1 4/5 and express as a mixed number
  • Subtract 3 1/4 from 5 3/4
  • Solve 4 2/6 − 1 5/6 by regrouping the whole number

• Addition and subtraction word problems

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

  • A jug contains 3/4 litre of juice; 2/4 litre is poured out — how much remains?
  • Two pieces of ribbon are 2 3/8 and 1 5/8 inches — what is their total length?
  • Draw a fraction model to represent and solve a fraction word problem

• Decimal place value (age 9+)

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

  • Round 3.47 to the nearest whole number (3) and to 1 d.p. (3.5)
  • Round 12.95 to 1 d.p. (13.0) and explain the boundary case
  • Estimate 4.83 + 2.17 by rounding each to the nearest whole number

• Fractions of a whole (age 9+)

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

  • Each person eats 3/8 of a pizza and there are 5 people — how many pizzas are needed?
  • A ribbon is cut into pieces of 2/3 metre; how long are 4 pieces altogether?
  • Between what two whole numbers does 6 × 3/4 lie?

Geometry

Your child is developing advanced spatial skills — working with 3D shapes and nets, using coordinate grids with negative numbers, calculating angles, and understanding geometric transformations like reflection and translation.

• Types of angles (age 8+)

  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

  • Mark a right angle with a small square symbol in a diagram and explain what it means
  • Use tick marks to show equal lengths in a shape and double tick marks for a second pair of equal sides
  • Read and interpret angle notation (e.g. angle ABC) and identify the angle being referred to in a diagram

• Degrees and turns

  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

  • Explain that a full turn is 360° and a right angle is 90°
  • Describe what 'one degree' means in terms of a fraction of a circle
  • State that an angle of 45° has turned through 45 one-degree angles

• What Is an Angle?

  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

  • Identify the vertex and arms of an angle in a diagram
  • Find examples of angles in the classroom (e.g. open door, clock hands)
  • Explain why two rays meeting at a point form an angle

• Measuring angles

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

  • Measure an angle with a protractor and read 47°
  • Draw an angle of 135° using a protractor
  • Identify which scale on the protractor to use for an obtuse angle

• Lines, Rays & Angles

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

  • Draw a pair of perpendicular lines and a pair of parallel lines
  • Identify all pairs of parallel sides in a trapezoid
  • Mark a right angle, an acute angle, and an obtuse angle in a given figure

• Coordinates (age 10+)

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

  • Plot points with negative coordinates such as (−3, 4) and (2, −5) on a four-quadrant grid
  • Identify the quadrant in which a given point lies
  • Describe the position of a shape using coordinates in all four quadrants

• Angle Sum Rules

  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

  • State that two angles on a straight line add to 180°
  • Explain that four right angles at a point make 360°
  • Identify 270° as three right angles

• Translating and reflecting shapes

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

  • Translate a triangle 3 units right and 2 units down on a coordinate grid and state the new coordinates
  • Reflect a shape in the x-axis and list the coordinates of the reflected vertices
  • Explain how translation changes coordinates (add/subtract) while reflection changes the sign of one coordinate

• Regular and irregular polygons

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

  • Classify a set of polygons as regular or irregular
  • Explain that a regular pentagon has 5 equal sides and 5 equal angles
  • Identify that a rectangle is irregular (equal angles but not all equal sides) unless it is a square

• Measuring angles (age 9+)

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

  • Two angles on a straight line are 65° and x°; find x = 115°
  • An angle is decomposed into 35° and 40°; state the whole angle is 75°
  • Solve: angles at a point are 120°, 90°, and x°; find x = 150°

• Estimating Angles

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

  • Estimate an angle as approximately 130° and classify it as obtuse
  • Identify a reflex angle in a diagram and estimate it as about 270°
  • Order four angles from smallest to largest by estimation before measuring

• Classifying shapes by properties

  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

  • Explain why all squares are rectangles but not all rectangles are squares
  • Place quadrilaterals in a hierarchy diagram showing subset relationships
  • Identify properties shared by all parallelograms and explain why rhombuses and rectangles are special cases

• Numbers on a number line

  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

  • Identify the x-axis, y-axis, and origin on a coordinate grid
  • Explain that (3, 5) means go 3 along the x-axis and 5 up the y-axis
  • Distinguish (2, 4) from (4, 2) by explaining each coordinate's meaning

• Plotting points in the first quadrant

  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

  • Plot the point (4, 7) accurately on a first-quadrant grid
  • Graph a set of data pairs (e.g. time vs distance) as points on the coordinate plane
  • Read coordinates of a plotted point and explain what they represent in a given scenario

• Classifying shapes by line properties

  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

  • Sort a set of quadrilaterals into those with parallel sides and those without
  • Identify which triangles in a set are right triangles
  • Classify shapes that have both perpendicular and parallel sides

• Understanding angles (age 9+)

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

  • Given one side of a rectangle is 8 cm and the perimeter is 28 cm, find the other side
  • Explain that all angles in a rectangle are 90°
  • Find a missing side of a rectangle given its area and one side length

• Angles in triangles (age 10+)

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

  • Find a missing angle in a triangle given two angles (using angle sum = 180°)
  • Calculate a missing angle in a quadrilateral (angle sum = 360°)
  • Calculate the interior angle of a regular hexagon from the angle sum formula

• 2-D shapes (age 10+)

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

  • Draw a triangle with sides 5 cm, 7 cm and an included angle of 60°
  • Construct a rectangle with given length and width using a ruler and set square
  • Draw a regular pentagon given the side length and interior angle

• Understanding angles (age 10+)

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

  • Find a missing angle at a point given the other angles (total 360°)
  • Use the straight-line property (angles sum to 180°) to find an unknown angle
  • Identify vertically opposite angles and use the property that they are equal to find unknowns

• 3-D shapes (age 9+)

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

  • Identify a cuboid from its net
  • Name the 3-D shape shown in an isometric drawing
  • Match 2-D representations to the correct 3-D shape

• Transformations on a Grid

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

  • Reflect a triangle in a vertical mirror line on a coordinate grid and state the new coordinates
  • Translate a shape 3 units right and 2 units up and describe the movement
  • Explain that a reflected/translated shape is congruent to the original

• Transformations on a grid

  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

  • Reflect a shape in a given mirror line on a grid and label the new coordinates
  • Translate a shape by a given number of squares horizontally and vertically and describe the movement
  • Rotate a shape 90° or 180° about a given centre on a grid and check the image is congruent to the original

• Nets of 3-D Shapes

  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

  • Draw the net of a cube, cuboid, or triangular prism and fold it mentally to identify which faces connect
  • Build a 3-D shape from its net and check that all faces, edges, and vertices match
  • Identify which of several given nets will fold into a specific 3-D shape and explain why the others won't

• 2-D shapes (age 8+)

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

  • Find all lines of symmetry in a rectangle, square, and equilateral triangle
  • Determine whether a given shape has a line of symmetry when rotated
  • Identify which shapes in a set have exactly one line of symmetry

• 3-D shapes (age 10+)

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

  • Identify which net will fold into a given 3-D shape
  • Construct the net of a triangular prism and fold it to verify
  • Describe a 3-D shape by naming its faces, edges, and vertices

• Parts of a circle

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

  • Label the radius, diameter, and circumference on a circle diagram
  • Calculate the diameter given a radius of 4.5 cm (diameter = 9 cm)
  • Explain the relationship between radius and diameter in their own words

Multiplication & Division

Your child is advancing to sophisticated multiplication and division — using formal written methods for complex calculations, working with decimals, and applying the order of operations to solve multi-step problems.

• Long multiplication

  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

  • Calculate 2,347 × 6 using formal written layout
  • Calculate 34 × 27 using long multiplication
  • Draw an area model for 45 × 23 and connect to the written method

• Long multiplication (age 10+)

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

  • Correctly compute 2,463 × 37 using long multiplication
  • Explain each partial product in a long multiplication and why they are added
  • Multiply a four-digit number by a two-digit number without procedural errors

• Division with remainders (age 10+)

  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

  • Compute 4,752 ÷ 13 using long division and express the remainder as a fraction
  • Decide whether to round up or down a division remainder in a real-life context (e.g. buses needed)
  • Explain each step of the long division algorithm for a 4-digit ÷ 2-digit calculation

• Rounding Answers

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

  • Compute a division and round the result to one decimal place as specified
  • Determine the appropriate degree of accuracy for a measurement context (e.g. round to nearest penny)
  • Solve a problem where an unrounded decimal answer must be interpreted in context (e.g. whole containers needed)

• Division with remainders

  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

  • 52 children go on a trip in minibuses holding 9 each — how many minibuses are needed? (interpret remainder)
  • A shop sells packs of 6 pencils for £1.50 each — how much do 5 packs cost?
  • Represent a two-step problem with an equation using n for the unknown

• Brackets in Expressions

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

  • Evaluate 3 × (4 + 5) by computing inside the parentheses first
  • Evaluate {2 × [3 + (7 − 1)]} with nested grouping symbols
  • Insert parentheses into an expression to make it equal a target value

• Order of operations

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

  • Explain why 8 + 2 × 5 = 18 (not 50) by referencing multiplication before addition
  • Evaluate a multi-step expression like 12 ÷ 3 + 4 × 2 correctly as 12
  • State the correct order: brackets, then orders/exponents, then multiplication/division (L→R), then addition/subtraction (L→R)

• Arrays for multiplication (age 9+)

  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

  • Calculate 4,932 ÷ 6 using short division
  • Solve 125 ÷ 8 and interpret: 15 remainder 5 means 15 full bags with 5 left over
  • Use the multiplication–division relationship to check a division answer

• Writing Number Sentences

  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)

  • Write an expression for 'add 8 and 7, then multiply by 2' using parentheses
  • Explain what 4 × (365 − 12) represents without computing it
  • Compare two expressions and determine which is larger without evaluating

• Decimal place value

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

  • Correctly compute 4.56 × 7 using a written method
  • Use place-value reasoning to explain why 3.2 × 5 = 16.0
  • Model a decimal multiplication using an area diagram or expanded form

• Division with Decimals

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

  • Compute 14.76 ÷ 4 using a written method to get 3.69
  • Explain how to continue long division past the decimal point to obtain a decimal quotient
  • Solve a context problem that requires dividing a decimal amount equally (e.g. sharing £18.60 among 5 people)

• Multi-step problems: choosing operations

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

  • Solve a three-step word problem involving a mix of all four operations
  • Explain why particular operations were chosen for each step of a multi-step problem
  • Identify and correct an error in a multi-step solution that used the wrong operation

• Multiplying and dividing (age 10+)

  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

  • Compute 3.456 × 100 = 345.6 correctly
  • Compute 45.2 ÷ 1000 = 0.0452 correctly
  • Explain why multiplying by 10 shifts each digit one place to the left

• Factors, multiples, and primes

  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

  • List all factor pairs of 36: (1,36), (2,18), (3,12), (4,9), (6,6)
  • State the first five multiples of 7
  • Find the common factors of 24 and 36

• Estimation to check answers to calculations

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

  • Estimate 487 × 23 by rounding to 500 × 20 and use this to check a calculated answer
  • Determine whether an exact answer or an estimate is more appropriate in a given context
  • Spot an unreasonable answer by comparing with a quick mental estimate

• Multiplying and dividing

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

  • Calculate 3.45 × 100 = 345
  • Calculate 72 ÷ 1000 = 0.072
  • Explain that multiplying by 10 shifts each digit one place to the left

• Ratio (age 10+)

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

  • Mentally compute 45 × 8 by partitioning into 40 × 8 + 5 × 8
  • Use known facts to derive 6.5 × 4 mentally
  • Solve a multi-operation mental calculation such as 250 × 3 + 500 and explain the strategy used

• Dividing by two-digit numbers

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

  • Use short division to compute 3,648 ÷ 16 efficiently
  • Explain when short division is more appropriate than long division
  • Interpret a remainder as a decimal or fraction in a measurement context

• Multiplicative Comparison

  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

  • Explain that '4 times as many' means multiply by 4
  • Write an equation for: Sarah has 3 times as many stickers as Tom, who has 8 stickers
  • Distinguish multiplicative comparison from additive comparison in word problems

• Shape patterns

  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

  • Given 'start at 1, add 3', generate terms and notice they alternate odd/even
  • Given a shape pattern, predict the next three terms and describe the rule
  • Explain why starting at 2 and adding 4 always gives even numbers

• Understanding fractions

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

  • A recipe for 4 people needs 200 g of flour — how much for 6 people?
  • If 3 kg costs £12, how much does 5 kg cost?
  • Scale a shape by a factor of 1/2 and find the new dimensions

• Factors, multiples, and primes (age 9+)

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

  • Is 156 a multiple of 6? Explain how you know
  • Find two square numbers that add to make 100
  • Use factor pairs to simplify 24 × 25 as 6 × (4 × 25) = 6 × 100 = 600

• Mental multiplication and division (age 9+)

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

  • Calculate 40 × 60 mentally by using 4 × 6 = 24 then appending zeros
  • Derive 7 × 15 by calculating 7 × 10 + 7 × 5
  • Calculate 360 ÷ 9 mentally using 36 ÷ 9 = 4

• Prime numbers

  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

  • Explain that a prime number has exactly two factors: 1 and itself
  • Determine whether 51 is prime or composite and justify the answer
  • List all prime numbers up to 19 from memory

• Multiplicative Comparison

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

  • A blue ribbon is 3 times as long as a red ribbon of 7 cm — how long is the blue ribbon?
  • 36 is 4 times a number — what is the number?
  • Explain why 'Sam has 5 more' is additive but 'Sam has 5 times as many' is multiplicative

• Square and cube numbers

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

  • Identify 49 as 7² and explain that 7 × 7 = 49
  • Calculate 4³ = 64 and explain it means 4 × 4 × 4
  • List the first ten square numbers

Measurement

Your child is mastering advanced measurement skills — calculating volume and area using formulas, converting between different units of measurement, and solving real-world problems involving length, mass, and capacity.

• Area of Triangles & Parallelograms

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

  • Calculate the area of a parallelogram with base 8 cm and height 5 cm
  • Calculate the area of a triangle with base 12 cm and height 7 cm
  • Explain why the area of a triangle is half the area of a related parallelogram

• Miles & Kilometres

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

  • Convert 40 miles to approximately 64 km
  • Explain why the conversion factor between miles and km is approximately 1.6
  • Use the miles↔km relationship to compare a 10 km race with a 6-mile run

• Volume as additive

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

  • Decompose an L-shaped solid into two cuboids and calculate total volume
  • Solve a real-world problem requiring the volume of a composite figure (e.g. a step-shaped structure)
  • Explain why splitting a composite solid into rectangular prisms allows calculation of total volume

• Decimal place value

  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

  • Convert 3,250 g to 3.25 kg using decimal notation
  • Convert 0.45 km to 450 m
  • Explain the relationship between mm, cm, m, and km using powers of 10

• Measurement Conversions

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

  • Solve: 'A recipe uses 0.75 kg of flour. How many grams are needed for 3 batches?'
  • Compare 2.5 litres and 2,450 ml by converting to the same unit
  • Solve a multi-step problem involving time, distance, and unit conversion

• Fractions on a number line

  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

  • A 2.5 kg bag of flour is split equally into 4 portions — what does each weigh?
  • A journey takes 1 hr 45 min; what time do you arrive if you leave at 09:20?
  • Three ribbons of 0.75 m, 1.2 m, and 0.95 m — what is the total length?

• Telling time to the minute (age 9+)

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

  • Convert 3 hours 25 minutes to 205 minutes
  • A programme lasts 150 seconds — express in minutes and seconds
  • How many days are in 8 weeks and 3 days?

• Converting measurement units (age 9+)

  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

  • Convert 3.5 km to 3,500 m
  • Complete a conversion table for cm and mm: (1,10), (2,20), (3,30)...
  • Express 2 kg 350 g as 2,350 g

• Estimating answers (age 9+)

  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

  • Find the width of a room given area = 48 m² and length = 8 m
  • Calculate the area of a square with side 7.5 cm
  • Estimate the area of an irregular pond drawn on a cm² grid

• Estimating answers (age 10+)

  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)

  • Use V = l × w × h to find the volume of a cuboid 4 cm × 3 cm × 5 cm = 60 cm³
  • Explain why packing a prism with unit cubes gives the same result as multiplying edge lengths
  • Compare volumes of two cuboids and identify which has greater capacity

• Measurement Line Plots

  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

  • Create a line plot of seed growth measurements in 1/8-inch increments
  • Use a line plot to find the difference between the longest and shortest specimens
  • Calculate the total length of all items in a line plot by adding the fractional measurements

• Metric & Imperial Conversion

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

  • State that 1 inch ≈ 2.5 cm and use this to estimate length in inches
  • Know that 1 kg ≈ 2.2 pounds and estimate a person's weight in pounds
  • Convert approximately between litres and pints (1 litre ≈ 1.75 pints)

• Perimeter of Compound Shapes

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

  • Find the perimeter of an L-shaped figure by identifying all side lengths
  • Calculate the perimeter of a composite shape where some sides must be deduced
  • Draw a composite rectilinear shape with a perimeter of 40 cm

• Perimeter (age 10+)

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

  • Draw two rectangles with area 24 cm² but different perimeters
  • Find two shapes with perimeter 20 cm but different areas
  • Explain why a long thin rectangle and a square can have the same area but different perimeters

• Estimating volume

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

  • Build a cuboid from 1 cm³ blocks and state its volume
  • Estimate how many cm³ blocks would fill a given box
  • Estimate the capacity of a jug by comparing to known litre measures

• Counting Unit Cubes

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

  • Count unit cubes in a 3D diagram to determine volume in cm³
  • Measure the volume of a small container by filling it with centimetre cubes
  • Express a measured volume using the correct cubic unit notation

• Measuring length (age 10+)

  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

  • Explain what 'volume' means for a 3D shape and how it differs from area
  • Identify a unit cube and explain that it represents one cubic unit of volume
  • Determine the volume of a small solid by counting the unit cubes that fill it

Mathematical Thinking

Your child is developing advanced mathematical reasoning skills — learning to construct logical arguments, make connections between different mathematical concepts, and solve complex real-world problems involving fractions, algebra, and ratio.

• Advanced Multi-Step Problems

  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

  • Break down a three-step ratio problem into sub-problems and plan a solution pathway
  • Identify which quantities are known/unknown in an algebraic word problem and set up equations
  • Use estimation to check whether a decimal division answer is reasonable before finalising

• Understanding fractions (age 10+)

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

  • Translate a word problem into an algebraic equation and also represent it on a bar model
  • Plot data from a table onto a coordinate grid and interpret the relationship
  • Explain how a pie chart, a fraction, and a percentage all represent the same proportion

• Real-World Mathematical Modelling

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

  • Choose a bar model or double number line to represent a ratio problem and solve it
  • Model a volume problem with a labelled diagram, apply the formula, and interpret the result in context
  • Determine whether a measurement answer should be rounded and to what degree of accuracy

• Advanced Maths Vocabulary

  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

  • Use 'radius', 'diameter', and 'circumference' correctly when describing a circle
  • Distinguish between an expression, an equation, and a formula in mathematical writing
  • Specify units correctly when presenting volume calculations (e.g. 60 cm³ not just 60)

• Constructing mathematical arguments

  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

  • Prove that a given angle must be 60° by chaining angle facts in a logical sequence
  • Find and explain the error in a peer's fraction division calculation
  • Construct a counter-example to disprove a false conjecture (e.g. 'multiplying always makes bigger')

• Complex Multi-Step Problems

  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

  • Break a three-step problem involving unit conversion and multiplication into sub-problems and solve systematically
  • Estimate 4,832 × 7 as roughly 5,000 × 7 = 35,000 to check a calculated answer of 33,824
  • Identify that a percentage answer over 100% doesn't make sense in context

• Choosing Maths Tools

  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

  • Explain why long division is more appropriate than mental methods for 4,752 ÷ 13
  • Choose a coordinate grid approach to verify a translation rather than computing from a description alone
  • Select a formula-based approach rather than counting cubes to find volume efficiently

• Order of operations (age 10+)

  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

  • Use the property that all rectangles are parallelograms to deduce missing angle facts
  • Recognise that 3 × (n + 5) = 3n + 15 by applying distributive structure
  • Use the equivalence 1/8 = 0.125 = 12.5% flexibly to solve a comparison problem

• Generalising with repeated reasoning

  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

  • Explain that the interior angle sum of an n-sided polygon is (n−2) × 180° based on the pattern for triangles, quadrilaterals, pentagons
  • Predict the 20th term of a linear sequence by identifying and applying the general rule
  • Generalise that dividing by n always gives a denominator of n in the fraction, for any whole numbers

• Precise Maths Vocabulary

  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

  • Distinguish 'factor' from 'multiple' in a written explanation
  • Write an area answer as 48 cm² (not just 48) and a volume estimate as approximately 60 cm³
  • Use 5² notation correctly and read it as 'five squared'

• Understanding fractions (age 9+)

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

  • Prove that 3/4 > 2/3 using a common denominator argument and a visual model
  • Find and explain an error in a long multiplication where a partial product was misaligned
  • Present a chain of reasoning to show that angles in a triangle sum to 180° by tearing and arranging

• Real-World Maths Modelling

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

  • Model a recipe-scaling problem with multiplication and fractions, then interpret the answer in grams
  • Represent a room-carpet problem by drawing a scale diagram and applying the area formula
  • Use a percentage bar model to find a sale price and explain the answer in £

• Choosing representations strategically

  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

  • Choose mental multiplication for 25 × 40 but long multiplication for 347 × 26
  • Select a protractor to verify an estimated angle rather than relying on visual inspection alone
  • Choose a common-denominator approach vs. benchmark comparison for ordering fractions, and explain why

• Fractions on a number line (age 9+)

  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

  • Represent a scaling problem as both a bar model and a multiplication equation
  • Show how 0.35, 35/100, and 35% all represent the same quantity on a hundredths grid
  • Translate a line-graph reading into a subtraction equation to find the difference

• Reasoning with Equivalences

  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

  • Notice that multiplying any number by 25 can be done by multiplying by 100 then dividing by 4, and explain why
  • Describe the general rule for a sequence and predict the 20th term
  • Generalise: to find 10% divide by 10, to find 5% halve 10%, and use this to find 35% of any number

• Fractions, Decimals & Percentages

  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

  • Use the structure 25 × 16 = 25 × 4 × 4 = 100 × 4 = 400 by exploiting factor pairs
  • Recognise that 0.75 = 3/4 = 75% and use whichever form is most efficient for the problem
  • Use the fact that angles on a straight line sum to 180° as a structural tool to find missing angles

Number Representation & Place Value

Your child is mastering large numbers and decimals — reading, writing, and comparing numbers up to 10 million, understanding how decimal places work, and using negative numbers in real-world contexts like temperature and money.

• Place Value × 10 and ÷ 10

  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

  • Explain that the 4 in 0.04 is 1/10 of the 4 in 0.4
  • State that a digit moving one place left is ×10 and one place right is ÷10
  • Compare the value of the 6 in 6,000 and in 0.006

• Reading and writing numbers (age 10+)

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

  • Write 347.392 in expanded form: 3×100 + 4×10 + 7×1 + 3×(1/10) + 9×(1/100) + 2×(1/1000)
  • Compare 0.372 and 0.38 using place-value reasoning
  • Write 'five and sixty-two thousandths' as 5.062

• Place Value × 10 Pattern

  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)

  • Explain that the 3 in 3,000 is ten times the 3 in 300
  • Complete: 700 ÷ 70 = __ and explain using place-value reasoning
  • State how many times greater the value of the 5 in 50,000 is than the 5 in 5,000

• Numbers to Ten Million

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

  • A country's population is 8,274,500 — round to the nearest million for a headline
  • Order the depths of three ocean trenches given in metres including negative values
  • Estimate the total attendance at three events by rounding each to the nearest 100,000

• Reading and writing numbers to 10,000,000

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

  • Write 4,302,561 in words
  • Order four seven-digit numbers from smallest to largest
  • State the value of the 7 in 7,045,200 as seven million

• Reading Decimal Places

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

  • State the value of each digit in 2.635 (2 ones, 6 tenths, 3 hundredths, 5 thousandths)
  • Explain the relationship between adjacent decimal places (each place is ten times smaller)
  • Write a number given digit values (e.g. 4 ones, 0 tenths, 7 hundredths, 3 thousandths = 4.073)

• Decimal place value

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

  • Round 3.4567 to 2 decimal places (3.46)
  • Round 7,654,321 to the nearest 100,000 (7,700,000)
  • Use rounding to estimate 4.83 × 2.17 ≈ 5 × 2 = 10

• Reading and writing numbers (age 9+)

  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

  • Write 403,072 in words and in expanded form
  • Compare 548,301 and 543,801 using < and explain the reasoning
  • Order four six-digit numbers from smallest to largest

• Negative numbers in context

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

  • Place –3, –1, 0, 2, 5 on a number line
  • The temperature is –4°C and rises by 7 degrees — what is the new temperature?
  • Count backwards from 3 in ones: 3, 2, 1, 0, –1, –2

• Measuring temperature

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

  • The temperature falls from 3°C to −5°C — what is the drop? (8 degrees)
  • Calculate the difference between a bank balance of −£120 and £350
  • Order −7, −3, 0, 2, 5 on a number line and find the interval from −7 to 5

• Patterns with Powers of Ten

  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)

  • Write 10⁴ = 10,000 and explain the exponent means four factors of 10
  • Explain why 3.4 × 10² = 340 by describing the decimal shift
  • Predict 2.56 × 10³ without calculating and explain the pattern

• Rounding Large Numbers

  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

  • Round 456,782 to the nearest 10,000 (460,000)
  • Round 950,500 to the nearest 100,000 and explain the boundary case
  • Use rounding to estimate the sum of 387,412 + 214,560

• Working with Large Numbers

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

  • What is the largest six-digit number with digits summing to 15?
  • A stadium holds 67,450 people. Round to the nearest thousand for a news report
  • Order the populations of five cities and find the difference between the largest and smallest

• Counting forwards and backwards (age 9+)

  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

  • Count on in 10,000s from 462,000
  • Count back in 100,000s from 800,000
  • Identify the next three terms: 375,000; 475,000; 575,000; ...

• Roman numerals to 1000

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

  • Read MCMXCIX as 1999
  • Write 2024 in Roman numerals (MMXXIV)
  • Explain the subtractive principle: IV = 4 not IIII

Probability

• Calculating Simple Probability

  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

  • Calculate the probability of drawing a blue marble from a bag of 3 blue and 7 red as 3/10
  • Use the formula P(event) = favourable outcomes ÷ total outcomes to solve at least three different problems
  • Explain why increasing the number of favourable outcomes increases the probability

• The 0-to-1 Probability Scale

  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

  • Place events on a number line from 0 (impossible) to 1 (certain), expressing positions as fractions or decimals
  • Explain that a probability of 0.5 means 'even chance' and connect this to the informal word 'likely'
  • Convert between informal language ('very unlikely') and a numerical position on the 0-to-1 scale

• Probabilities Sum to One

  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

  • List all outcomes of spinning a 4-colour spinner and verify their probabilities add up to 1
  • Calculate P(not rolling a 3) as 1 − 1/6 = 5/6 using the complement rule
  • Spot an error in a probability table where the values don't sum to 1 and explain what's wrong

• Experimental vs Theoretical

  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

  • Roll a die 60 times and compare experimental frequencies with the expected 10 per number
  • Explain why experimental results don't exactly match theoretical predictions but get closer with more trials
  • Predict what would happen if the experiment were repeated 600 times instead of 60

• Probability as a Fraction

  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

  • State that the probability of rolling a 6 on a fair die is 1/6 and explain why
  • Express the probability of picking a red card from a standard deck as 26/52 or 1/2
  • Write the probability of a simple event as a fraction: favourable outcomes over total outcomes

• Simple Chance Experiments

  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

  • Flip a coin 20 times, record heads and tails in a tally chart, and describe what they notice about the results
  • Roll a die 30 times and compare how often each number came up with what they expected
  • Pull counters from a bag, record results, and explain whether the outcomes matched their prediction

• Likelihood Language

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

  • Place five everyday events (e.g. 'the sun will rise tomorrow', 'it will snow in July', 'I'll flip heads') on a scale from impossible to certain
  • Use 'likely', 'unlikely', 'certain', 'impossible', and 'even chance' correctly to describe different events
  • Explain why pulling a red ball from a bag of mostly red balls is 'likely' but not 'certain'

• Equally Likely Outcomes

  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)

  • Explain that a fair coin has equally likely outcomes because heads and tails each have the same chance
  • Identify whether a spinner with unequal sections has equally likely outcomes or not, and explain why
  • Give an example of a situation with equally likely outcomes and one without, explaining the difference

• Ordering Likelihoods

  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

  • Order four or more events from least likely to most likely and justify each placement
  • Compare likelihoods when outcomes are not equally likely — e.g. 'Drawing red from a bag with 7 red and 3 blue is more likely than drawing blue'
  • Explain why some events are closer to 'even chance' and others are closer to 'certain' or 'impossible'

Addition & Subtraction

Your child is mastering complex addition and subtraction — solving multi-step problems with large numbers and decimals, and choosing the best strategies to work out challenging calculations.

• Adding and subtracting (age 10+)

  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

  • Solve a two-step word problem involving addition and subtraction of numbers beyond 1,000,000
  • Choose between mental, written, and calculator methods for a multi-step problem and justify the choice
  • Interpret a real-life context to identify which addition/subtraction operations are needed across multiple steps

• Addition and subtraction strategies (age 10+)

  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

  • Correctly compute 3.45 + 2.78 and explain regrouping across decimal places
  • Use a number line or base-ten blocks to model 5.03 − 2.67
  • Explain why the standard algorithm works for decimal addition by connecting to place-value understanding

• Adding and subtracting (age 9+)

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

  • A school collects 12,450 bottles in Week 1 and 8,372 in Week 2; their target is 25,000 — how many more do they need?
  • Choose between mental and written methods for each step and explain why
  • Solve a three-step problem involving addition and subtraction of five-digit numbers

• Checking Answers by Rounding

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

  • Check 47,832 + 23,156 ≈ 48,000 + 23,000 = 71,000 to verify the exact answer 70,988
  • Decide whether to round to the nearest 100 or 1,000 for a given estimation context
  • Identify that a calculated answer of 3,421 cannot be correct because the estimate gives approximately 50,000

• Fluent addition and subtraction (age 9+)

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

  • Calculate 34,567 + 28,945 using columnar addition
  • Calculate 500,000 − 234,178 using columnar subtraction with multiple exchanges
  • Add three five-digit numbers in a single column layout

• Mental addition and subtraction (age 9+)

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

  • Mentally calculate 45,000 + 8,000
  • Mentally subtract 3,200 from 10,000 by counting up
  • Use near-doubles: 2,500 + 2,600 = 5,100

Algebra

Your child is beginning to work with algebra — using letters to represent unknown numbers, creating and following rules for number patterns, and translating word problems into mathematical equations.

• Using Simple Formulae

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

  • Substitute values into P = 2(l + w) to find the perimeter of a rectangle
  • Use the formula for area of a triangle (A = 1/2 × b × h) given base and height values
  • Interpret a formula expressed in words and use it to compute an output

• Writing Algebraic Equations

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

  • Write 'I think of a number, double it, and add 5 to get 17' as 2n + 5 = 17
  • Solve a one-step equation such as 3x = 24 and explain the reasoning
  • Express a word problem as an algebraic equation and find the unknown value

• Number Pattern Relationships

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

  • Generate sequences starting at 0 with rules 'add 3' and 'add 6'; observe terms in one are twice the other
  • Form ordered pairs from corresponding terms and plot them on a coordinate grid
  • Explain informally why the relationship between the two sequences holds

• Equations with Two Unknowns

  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)

  • List all whole-number pairs (a, b) where a + b = 12 with a, b > 0
  • Find three pairs that satisfy 2x + y = 20
  • Explain systematically how to generate all integer solutions to a two-variable equation within a given range

• Systematic Listing

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

  • List all possible meal combinations from 3 starters and 4 mains
  • Organise combinations into a systematic table to ensure none are missed
  • Explain why the total number of combinations equals the product of the options in each category

• Linear number sequences

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

  • Continue the sequence 2.5, 4.0, 5.5, ... and state the rule as 'add 1.5'
  • Generate terms of a sequence that crosses zero (e.g. 3, 1, −1, −3, ...)
  • Describe the term-to-term rule for a given linear sequence and predict the 10th term

Data & Statistics

Your child is learning to interpret and create different types of graphs and charts, and beginning to calculate averages to understand what data tells us about real-world situations.

• Line graphs (age 10+)

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

  • Read values from a line graph showing temperature over a day and identify trends
  • Construct a pie chart from given data by calculating sector angles
  • Use a pie chart to determine the actual quantity represented by each sector given the total

• Calculating the Mean

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

  • Calculate the mean of five test scores: 72, 85, 90, 68, 95
  • Explain what the mean represents and how it differs from individual data values
  • Find a missing data value given the mean and all other values

• Understanding fractions

  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)

  • Create a line plot showing lengths measured to the nearest 1/8 inch
  • Use the line plot to find the total of all measurements by adding fractions
  • Solve: 'If the total liquid were redistributed equally among all beakers, how much would each have?'

• Reading tables

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

  • Read a bus timetable to find the departure time for a particular stop
  • Complete a two-way table from given data about favourite sports by gender
  • Calculate how long a train journey takes using a timetable

• Reading and Comparing Bar Graphs

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

  • Read a line graph of temperature over a week and identify the warmest day
  • Find the difference in rainfall between two months from a line graph
  • Explain what a rising/falling line means on a time-series graph

• Statistical Analysis Vocabulary

  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

  • Correctly define and calculate the mean, median, mode, and range of a small data set
  • Use 'outlier' correctly to identify a value that doesn't fit the pattern, and explain its effect on the mean
  • Use 'correlation' correctly when describing the relationship shown in a scatter graph

Ratio & Proportion

Your child is learning to compare quantities and solve problems involving ratios, proportions, percentages, and scale — essential skills for understanding relationships between numbers in real-world contexts.

• Calculating Percentages

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

  • Calculate 15% of 360 by finding 10% and 5% and combining
  • Compare two discounts given as percentages of different original prices
  • Explain a strategy for finding any percentage of an amount using known percentage facts

• Scale and similar shapes

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

  • Find the missing side of a similar rectangle given a scale factor of 3
  • Determine the scale factor between two similar triangles from given side lengths
  • Use a scale factor to enlarge or reduce a shape and verify that all sides are in the same ratio

• Ratio Problems

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

  • If the ratio of red to blue beads is 3:5 and there are 15 blue beads, find the number of red beads
  • Use multiplication facts to find the missing value: 'For every 2 apples there are 5 oranges; if there are 20 oranges, how many apples?'
  • Explain the multiplicative relationship between two quantities in a ratio context

• Understanding fractions

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

  • Share 40 sweets between two children in the ratio 3:5
  • Solve: 'Tom gets twice as many as Sam and Sam gets three times as many as Jo. If there are 30 altogether, how many does each get?'
  • Use fraction knowledge to explain why sharing in ratio 2:3 means one person gets 2/5 of the total

• Bar Models for Ratios

  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

  • Draw a bar model to represent a ratio problem — e.g. sharing £20 in the ratio 3:2 by drawing 5 equal blocks
  • Use a bar model to solve a proportion problem and explain each step
  • Compare bar models with other representations (tables, double number lines) and explain when each is most useful

• Percentages (age 9+)

  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)

  • Explain the difference between a 'ratio' and a 'proportion' using a concrete example like mixing paint
  • Use 'per cent' correctly and convert between fractions, decimals, and percentages in context
  • Define 'scale factor' and use it to describe how a shape has been enlarged or reduced