Generalising with repeated reasoning

Mastery Evidence

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

Assessment Prompt

“When [child] finds the rule for a number sequence — like "add 7 each time" or "multiply the position number by 3" — can they use that rule to find any term without having to list all the ones before it?”

Prerequisites4

Angles in triangles (age 10+)
softAges 10—11
Describing Rules & Patterns
softAges 8—9
Linear number sequences
softAges 10—11
Reasoning with Equivalences
hardAges 9—10

Show full prerequisite tree

Angles in triangles (age 10+) soft
Generalising polygon angle sums exemplifies repeated reasoning
- Types of angles (age 8+) hard
  Finding unknown angles in polygons requires reading diagrams where known angles are marked with arcs and tick marks
- Classifying shapes by properties hard
  Finding angles in polygons requires knowing shape properties
  - Regular and irregular polygons hard
    Hierarchy includes regular/irregular distinction from Y5
    - Understanding angles (age 8+) hard
      Shape classification supports distinguishing regular from irregular
      - 2-D shapes (age 6+) hard
        Identifying 2D shape properties is prerequisite to classifying by shared attributes
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
      - Angles in triangles (age 7+) hard
        Recognising shapes by attributes is prerequisite to quadrilateral hierarchy classification
        
        Angles in triangles (age 6+) hard
        Drawing shapes by attributes extends understanding defining vs non-defining attributes
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes (age 6+) hard
        Identifying pentagons, hexagons, quadrilaterals extends knowing 2-D shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
    - Estimating Angles soft
      Classifying regular/irregular polygons benefits from angle estimation skills
      - Degrees and turns hard
        Estimating/comparing angles requires understanding the degree system
        
        What Is an Angle? hard
        Degree measurement system requires understanding what an angle is
        
        Types of angles hard
        Angle definition builds on understanding right angles
        
        Right Angles & Turns hard
        Identifying right angles and greater/less than right angle is prerequisite to naming acute/obtuse
        
        Understanding angles hard
        Identifying right angles requires understanding what an angle is
        
        2-D shapes (age 6+) soft
        Understanding angles as shape properties requires knowing basic shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        Position, direction, and movement hard
        Recognising angles as turns extends Y2 work on quarter/half/three-quarter turns
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        Types of angles (age 8+) soft
        Identifying right angles and turns is supported by the convention of marking right angles with a small square
        
        Position, direction, and movement hard
        Right angles as quarter turns extends Y2 turn vocabulary
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Right Angles & Turns hard
        Angle definition builds on classifying acute/obtuse angles
        
        Understanding angles hard
        Identifying right angles requires understanding what an angle is
        
        2-D shapes (age 6+) soft
        Understanding angles as shape properties requires knowing basic shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        Position, direction, and movement hard
        Recognising angles as turns extends Y2 work on quarter/half/three-quarter turns
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Types of angles (age 8+) soft
        Identifying right angles and turns is supported by the convention of marking right angles with a small square
        
        Position, direction, and movement hard
        Right angles as quarter turns extends Y2 turn vocabulary
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Positional Language hard
        Describing movement and turns builds on positional language
  - Classifying shapes by line properties hard
    Property-based hierarchy extends Y5 classification by parallel/perpendicular lines
    - Lines, Rays & Angles hard
      Must draw/identify geometric elements before classifying shapes by them
      - Types of angles hard
        Y4 acute/obtuse angle identification is prerequisite to drawing and labelling angle types
        
        Right Angles & Turns hard
        Identifying right angles and greater/less than right angle is prerequisite to naming acute/obtuse
        
        Understanding angles hard
        Identifying right angles requires understanding what an angle is
        
        2-D shapes (age 6+) soft
        Understanding angles as shape properties requires knowing basic shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        Position, direction, and movement hard
        Recognising angles as turns extends Y2 work on quarter/half/three-quarter turns
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Types of angles (age 8+) soft
        Identifying right angles and turns is supported by the convention of marking right angles with a small square
        
        Position, direction, and movement hard
        Right angles as quarter turns extends Y2 turn vocabulary
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Positional Language hard
        Describing movement and turns builds on positional language
      - Parallel and perpendicular lines hard
        Y3 horizontal/vertical/perpendicular/parallel lines is prerequisite to drawing and identifying them formally
        
        Right Angles & Turns hard
        Perpendicular lines require understanding right angles
        
        Understanding angles hard
        Identifying right angles requires understanding what an angle is
        
        2-D shapes (age 6+) soft
        Understanding angles as shape properties requires knowing basic shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        Position, direction, and movement hard
        Recognising angles as turns extends Y2 work on quarter/half/three-quarter turns
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Types of angles (age 8+) soft
        Identifying right angles and turns is supported by the convention of marking right angles with a small square
        
        Position, direction, and movement hard
        Right angles as quarter turns extends Y2 turn vocabulary
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Positional Language soft
        Horizontal/vertical builds on positional vocabulary
    - Understanding angles (age 8+) hard
      Y4 quadrilateral classification supports classifying by lines/angles
      - 2-D shapes (age 6+) hard
        Identifying 2D shape properties is prerequisite to classifying by shared attributes
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
      - Angles in triangles (age 7+) hard
        Recognising shapes by attributes is prerequisite to quadrilateral hierarchy classification
        
        Angles in triangles (age 6+) hard
        Drawing shapes by attributes extends understanding defining vs non-defining attributes
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes (age 6+) hard
        Identifying pentagons, hexagons, quadrilaterals extends knowing 2-D shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
- Angle Sum Rules hard
  Finding unknown angles builds on Y5 angle sum facts (180°, 360°)
  - Types of angles (age 8+) hard
    Angle sum rules (360° at a point, 180° on a line) are applied through reading angle diagrams with correct notation
  - Degrees and turns hard
    Angle facts (at a point = 360°) require degree system knowledge
    - What Is an Angle? hard
      Degree measurement system requires understanding what an angle is
      - Types of angles hard
        Angle definition builds on understanding right angles
        
        Right Angles & Turns hard
        Identifying right angles and greater/less than right angle is prerequisite to naming acute/obtuse
        
        Understanding angles hard
        Identifying right angles requires understanding what an angle is
        
        2-D shapes (age 6+) soft
        Understanding angles as shape properties requires knowing basic shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        Position, direction, and movement hard
        Recognising angles as turns extends Y2 work on quarter/half/three-quarter turns
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Types of angles (age 8+) soft
        Identifying right angles and turns is supported by the convention of marking right angles with a small square
        
        Position, direction, and movement hard
        Right angles as quarter turns extends Y2 turn vocabulary
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Positional Language hard
        Describing movement and turns builds on positional language
      - Right Angles & Turns hard
        Angle definition builds on classifying acute/obtuse angles
        
        Understanding angles hard
        Identifying right angles requires understanding what an angle is
        
        2-D shapes (age 6+) soft
        Understanding angles as shape properties requires knowing basic shape properties
        
        Angles in triangles (age 6+) soft
        Understanding defining attributes supports describing shape properties formally
        
        2-D shapes hard
        Distinguishing defining vs non-defining attributes requires knowing common 2-D shape names first
        
        3-D shapes (age 5+) hard
        Identifying defining attributes builds on informal analysis and comparison of shapes
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        2-D shapes hard
        Describing properties of 2-D shapes (sides, symmetry) requires knowing the shapes first
        
        3-D shapes (age 5+) hard
        Formal property description extends informal analysis of sides and vertices
        
        2-D shapes hard
        Analysing and comparing shapes requires being able to name them first
        
        3-D shapes hard
        Analysing 3-D shapes requires recognising and naming them
        
        Position, direction, and movement hard
        Recognising angles as turns extends Y2 work on quarter/half/three-quarter turns
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Positional Language hard
        Describing movement and turns builds on positional language
        
        Types of angles (age 8+) soft
        Identifying right angles and turns is supported by the convention of marking right angles with a small square
        
        Position, direction, and movement hard
        Right angles as quarter turns extends Y2 turn vocabulary
        
        Positional Language hard
        Position/direction vocabulary with right angles extends basic positional language
        
        Turns & Directions hard
        Right-angle turns (clockwise/anti-clockwise) build directly on whole/half/quarter turns from Year 1
        
        What Is a Half? soft
        Understanding half and quarter turns benefits from the concept of halves and quarters
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Positional Language hard
        Describing movement and turns builds on positional language
Describing Rules & Patterns soft
Describing algebraic rules and extending angle facts to any polygon is the mature maths form of the universal generalisation habit
- Spotting Patterns hard
  Generalising a rule requires first being able to spot the recurring pattern that the rule captures
  - Connecting New & Old Ideas soft
    Spotting patterns across domains is an extension of the habit of connecting new ideas to existing ones
    - Thinking Before Starting hard
      Making connections between new and old ideas requires the habit of activating prior knowledge first
      - Persisting When It's Hard hard
        Activating prior knowledge requires the foundational habit of persistent engagement with new material
Linear number sequences soft
Linear sequences are key Y6 generalisation context
- Shape patterns hard
  Linear sequences with negative/decimal steps extends Y5 pattern generation
  - Patterns in Times Tables hard
    Identifying arithmetic patterns feeds into generating and analysing patterns from rules
    - Properties of Operations hard
      Explaining patterns uses properties of operations
      - Commutative Multiplication hard
        Applying all three properties extends Y2 commutativity understanding
        
        Arrays for multiplication soft
        Arrays help demonstrate commutativity (rotate the array)
        
        Division as equal sharing hard
        Using arrays for division requires understanding division as grouping
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Commutativity of multiplication requires understanding multiplication
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - What Multiplication Means hard
        Using properties requires understanding what multiplication means
        
        Arrays for multiplication (age 7+) hard
        Extends array-based repeated addition to formal multiplication interpretation
        
        Arrays for multiplication hard
        Rectangular arrays with repeated addition extends array representation from Y2
        
        Division as equal sharing hard
        Using arrays for division requires understanding division as grouping
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Expressing array totals as sums of equal addends requires understanding multiplication as repeated addition
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Interpreting products formalises repeated addition/equal groups from Y1
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
    - Fluent multiplication and division facts hard
      Spotting patterns in tables requires knowing the facts
      - Division as Unknown Factor soft
        Using mult/div relationship as strategy supports fluency
        
        What Multiplication Means hard
        Connecting division to multiplication requires understanding products
        
        Arrays for multiplication (age 7+) hard
        Extends array-based repeated addition to formal multiplication interpretation
        
        Arrays for multiplication hard
        Rectangular arrays with repeated addition extends array representation from Y2
        
        Division as equal sharing hard
        Using arrays for division requires understanding division as grouping
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        Multiplication as repeated addition hard
        Expressing array totals as sums of equal addends requires understanding multiplication as repeated addition
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Interpreting products formalises repeated addition/equal groups from Y1
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        What Division Means hard
        Seeing division as unknown-factor requires understanding division first
        
        Division as equal sharing hard
        Interpreting quotients formalises equal sharing/grouping from Y1
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Times tables hard
        Fluency within 100 builds on knowing 2/5/10 tables
        
        Counting in 2s hard
        Times tables for 2, 5, 10 build on skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Times tables (age 7+) hard
        Fluency within 100 builds on knowing 3/4/8 tables
        
        Times tables hard
        3, 4, 8 times tables build on known 2, 5, 10 tables
        
        Counting in 2s hard
        Times tables for 2, 5, 10 build on skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Skip Counting (4s, 8s, 50s, 100s) soft
        Counting in 4s and 8s supports learning those table facts
        
        Counting in 2s hard
        Counting in 4s and 8s extends prior skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
- Measuring temperature soft
  Sequences crossing zero require negative number understanding
  - Negative numbers in context hard
    Calculating intervals across zero extends Y5 negative number context
    - Negative Numbers hard
      Counting through zero is prerequisite to interpreting negative numbers in context
      - Counting Within 1,000 hard
        Counting backwards through zero extends counting backwards within 1000
        
        Counting in 2s hard
        Counting to 1000 by 5s/10s/100s extends skip counting from Year 2
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        The multiples of 100 soft
        Understanding multiples of 100 supports skip counting by 100s
        
        A Hundred Is Ten Tens hard
        Multiples of 100 require understanding 100 as a unit
        
        A Ten Is Ten Ones hard
        100 as ten tens extends understanding of 10 as ten ones
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        The two digits of a two-digit number hard
        Must understand two-digit place value before extending to hundreds
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
Reasoning with Equivalences hard
Y6 generalisation extends Y5 repeated reasoning
- Shape patterns soft
  Generating patterns from rules exercises repeated reasoning and generalisation
  - Patterns in Times Tables hard
    Identifying arithmetic patterns feeds into generating and analysing patterns from rules
    - Properties of Operations hard
      Explaining patterns uses properties of operations
      - Commutative Multiplication hard
        Applying all three properties extends Y2 commutativity understanding
        
        Arrays for multiplication soft
        Arrays help demonstrate commutativity (rotate the array)
        
        Division as equal sharing hard
        Using arrays for division requires understanding division as grouping
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Commutativity of multiplication requires understanding multiplication
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - What Multiplication Means hard
        Using properties requires understanding what multiplication means
        
        Arrays for multiplication (age 7+) hard
        Extends array-based repeated addition to formal multiplication interpretation
        
        Arrays for multiplication hard
        Rectangular arrays with repeated addition extends array representation from Y2
        
        Division as equal sharing hard
        Using arrays for division requires understanding division as grouping
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Expressing array totals as sums of equal addends requires understanding multiplication as repeated addition
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Interpreting products formalises repeated addition/equal groups from Y1
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
    - Fluent multiplication and division facts hard
      Spotting patterns in tables requires knowing the facts
      - Division as Unknown Factor soft
        Using mult/div relationship as strategy supports fluency
        
        What Multiplication Means hard
        Connecting division to multiplication requires understanding products
        
        Arrays for multiplication (age 7+) hard
        Extends array-based repeated addition to formal multiplication interpretation
        
        Arrays for multiplication hard
        Rectangular arrays with repeated addition extends array representation from Y2
        
        Division as equal sharing hard
        Using arrays for division requires understanding division as grouping
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        Multiplication as repeated addition hard
        Expressing array totals as sums of equal addends requires understanding multiplication as repeated addition
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Multiplication as repeated addition hard
        Interpreting products formalises repeated addition/equal groups from Y1
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        What Division Means hard
        Seeing division as unknown-factor requires understanding division first
        
        Division as equal sharing hard
        Interpreting quotients formalises equal sharing/grouping from Y1
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Times tables hard
        Fluency within 100 builds on knowing 2/5/10 tables
        
        Counting in 2s hard
        Times tables for 2, 5, 10 build on skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Times tables (age 7+) hard
        Fluency within 100 builds on knowing 3/4/8 tables
        
        Times tables hard
        3, 4, 8 times tables build on known 2, 5, 10 tables
        
        Counting in 2s hard
        Times tables for 2, 5, 10 build on skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Skip Counting (4s, 8s, 50s, 100s) soft
        Counting in 4s and 8s supports learning those table facts
        
        Counting in 2s hard
        Counting in 4s and 8s extends prior skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
- Times tables (age 8+) hard
  Age 8-9 repeated reasoning is prerequisite to age 9-10 level
  - Extending Table Patterns hard
    Age 7-8 repeated reasoning is prerequisite to age 8-9 level
    - Generalising Patterns hard
      Age 7-8 repeated reasoning builds on age 6-7
      - 10 More or 10 Less soft
        Mentally finding 10 more/less uses generalised repeated reasoning about place value
        
        One More Each Time soft
        10 more/less generalises the concept of one more/one less to tens
        
        How Many in Total? hard
        Understanding 'one more/one less' requires understanding that each number represents a quantity (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        The two digits of a two-digit number hard
        Mentally finding 10 more/less requires understanding that the tens digit changes
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - Addition and subtraction within 20 soft
        Deriving facts within 20 from known facts (near doubles, make ten) exercises generalising from patterns
        
        Number bonds to 9 hard
        Making-ten strategy requires knowing bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fluent adding and subtracting within 10 hard
        Strategies for within-20 calculation build on fluent within-10 knowledge
        
        Number bonds to 9 hard
        Fluency within 10 requires knowing number bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fluent adding and subtracting within 5 hard
        Fluency within 10 extends from fluency within 5
        
        Addition as combining or putting together two hard
        Fluency with addition within 5 requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Fluency with subtraction within 5 requires understanding subtraction as taking away
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Finding efficient methods hard
        Age 6-7 generalising from repeated reasoning builds on age 5-6 noticing repeated patterns
        
        Counting in 2s soft
        Skip counting exercises recognising repeating calculation patterns
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        One More Each Time soft
        The +1 pattern in counting is the earliest repeated reasoning
        
        How Many in Total? hard
        Understanding 'one more/one less' requires understanding that each number represents a quantity (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
    - Skip Counting (4s, 8s, 50s, 100s) soft
      Skip counting patterns exercise generalising from repetition
      - Counting in 2s hard
        Counting in 4s and 8s extends prior skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
    - Times tables (age 7+) soft
      Multiplication table patterns exercise repeated reasoning
      - Times tables hard
        3, 4, 8 times tables build on known 2, 5, 10 tables
        
        Counting in 2s hard
        Times tables for 2, 5, 10 build on skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Skip Counting (4s, 8s, 50s, 100s) soft
        Counting in 4s and 8s supports learning those table facts
        
        Counting in 2s hard
        Counting in 4s and 8s extends prior skip counting in 2s, 5s, 10s
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
  - Equivalent fractions on a number line soft
    Equivalent fraction patterns exercise repeated reasoning
    - Equivalent fractions hard
      Diagram-based equivalent fractions is prerequisite to formal equivalence understanding
      - Fractions on a number line soft
        Number line supports seeing equivalent fractions as the same point
        
        Fractions of amounts hard
        Placing fractions on number line requires knowing what fractions are
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
      - Understanding fractions hard
        Extends equivalence of 2/4 and 1/2 to broader equivalent fractions
        
        Reading +, −, and = symbols soft
        Writing fraction sentences (1/2 of 6 = 3) requires understanding the = sign
        
        Reading and writing numbers to 20 hard
        Writing number sentences requires reading and writing numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        Addition as combining or putting together two hard
        Reading/writing the + symbol requires understanding what addition means
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Reading/writing the − symbol requires understanding what subtraction means
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fractions of amounts hard
        Writing fractions and recognising equivalence requires knowing what the fractions mean
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Writing fractions and recognising equivalence requires 'equivalent fraction' vocabulary
      - Fraction Notation hard
        Recognising and showing equivalent fractions requires that vocabulary
    - Fractions of a whole hard
      Must understand unit fractions to reason about equivalence
      - Fractions of amounts hard
        Recognising fractions of shapes/quantities is prerequisite to formal unit fraction understanding
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
      - Fraction Notation hard
        Understanding a/b as a parts of 1/b requires numerator, denominator, and unit fraction vocabulary
      - Splitting shapes into equal parts (age 7+) hard
        Partition into equal shares is prerequisite to understanding unit fractions
        
        Decomposing a shape into more equal shares hard
        Understanding equal shares of different shapes requires concept of more shares = smaller
        
        Halves & Quarters of Shapes hard
        Comparing share sizes requires experience partitioning into halves and quarters
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Halves & Quarters of Shapes hard
        Extends partitioning from halves/fourths to include thirds
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
    - Fractions on a number line (age 8+) hard
      Equivalent fractions as the same point on a number line directly uses the fraction number-line representation
      - Fractions on a number line hard
        Prior number-line fraction experience feeds into formal unit-fraction placement
        
        Fractions of amounts hard
        Placing fractions on number line requires knowing what fractions are
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
      - Fractions of a whole hard
        Must understand unit fractions before placing them on number line
        
        Fractions of amounts hard
        Recognising fractions of shapes/quantities is prerequisite to formal unit fraction understanding
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Understanding a/b as a parts of 1/b requires numerator, denominator, and unit fraction vocabulary
        
        Splitting shapes into equal parts (age 7+) hard
        Partition into equal shares is prerequisite to understanding unit fractions
        
        Decomposing a shape into more equal shares hard
        Understanding equal shares of different shapes requires concept of more shares = smaller
        
        Halves & Quarters of Shapes hard
        Comparing share sizes requires experience partitioning into halves and quarters
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Halves & Quarters of Shapes hard
        Extends partitioning from halves/fourths to include thirds
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
  - Understanding angles (age 8+) soft
    Area calculation patterns (doubling side doubles area) exercise generalisation
    - Area by Tiling hard
      Must see tiling→multiplication connection before computing area via side lengths
      - Area (age 8+) hard
        Counting unit squares is prerequisite to connecting tiling to multiplication
        
        Understanding Area hard
        Must understand unit squares and area concept before measuring by counting
        
        Measuring length (age 7+) soft
        Length measurement experience supports understanding area as a 2D measurement
        
        Measuring length (age 6+) hard
        Using standard measurement tools extends measuring with non-standard units
        
        Measuring length and height (age 5+) hard
        Measuring with iterated units extends Y1 beginning to measure length
        
        Comparing Lengths & Heights hard
        Measuring length with units requires first being able to compare lengths directly
        
        Measurable Attributes of Objects hard
        Comparing lengths/heights requires first identifying length as a measurable attribute
        
        Measuring length soft
        Measuring with units benefits from ability to order and compare lengths
        
        Comparing Lengths & Heights hard
        Ordering 3 objects by length and indirect comparison extends direct length comparison
        
        Measurable Attributes of Objects hard
        Comparing lengths/heights requires first identifying length as a measurable attribute
        
        Choosing measurement units soft
        Choosing appropriate tools builds on choosing appropriate units
        
        Capacity and volume hard
        Using standard units for capacity extends from beginning to measure capacity
        
        Comparing Capacity hard
        Measuring capacity with units requires first being able to compare capacities
        
        Measurable Attributes of Objects hard
        Comparing capacity requires understanding capacity as a measurable attribute
        
        Measuring length and height (age 5+) hard
        Using standard units for length extends from beginning to measure length
        
        Comparing Lengths & Heights hard
        Measuring length with units requires first being able to compare lengths directly
        
        Measurable Attributes of Objects hard
        Comparing lengths/heights requires first identifying length as a measurable attribute
        
        Measuring mass and weight hard
        Using standard units for mass extends from beginning to measure mass
        
        Measuring mass and weight (age 4+) hard
        Measuring mass with units requires first being able to compare masses directly
        
        Measurable Attributes of Objects hard
        Comparing mass/weight requires first identifying mass as a measurable attribute
  - Describing Rules & Patterns soft
    Generalising fraction and times-table patterns is the maths-specific application of the universal generalisation habit
    - Spotting Patterns hard
      Generalising a rule requires first being able to spot the recurring pattern that the rule captures
      - Connecting New & Old Ideas soft
        Spotting patterns across domains is an extension of the habit of connecting new ideas to existing ones
        
        Thinking Before Starting hard
        Making connections between new and old ideas requires the habit of activating prior knowledge first
        
        Persisting When It's Hard hard
        Activating prior knowledge requires the foundational habit of persistent engagement with new material
- Equivalent fractions (age 9+) soft
  Equivalent fraction patterns exercise generalisation
  - Equivalent fractions on a number line hard
    Understanding equivalence conceptually is prerequisite to explaining algebraically
    - Equivalent fractions hard
      Diagram-based equivalent fractions is prerequisite to formal equivalence understanding
      - Fractions on a number line soft
        Number line supports seeing equivalent fractions as the same point
        
        Fractions of amounts hard
        Placing fractions on number line requires knowing what fractions are
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
      - Understanding fractions hard
        Extends equivalence of 2/4 and 1/2 to broader equivalent fractions
        
        Reading +, −, and = symbols soft
        Writing fraction sentences (1/2 of 6 = 3) requires understanding the = sign
        
        Reading and writing numbers to 20 hard
        Writing number sentences requires reading and writing numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        Addition as combining or putting together two hard
        Reading/writing the + symbol requires understanding what addition means
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Reading/writing the − symbol requires understanding what subtraction means
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fractions of amounts hard
        Writing fractions and recognising equivalence requires knowing what the fractions mean
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Writing fractions and recognising equivalence requires 'equivalent fraction' vocabulary
      - Fraction Notation hard
        Recognising and showing equivalent fractions requires that vocabulary
    - Fractions of a whole hard
      Must understand unit fractions to reason about equivalence
      - Fractions of amounts hard
        Recognising fractions of shapes/quantities is prerequisite to formal unit fraction understanding
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
      - Fraction Notation hard
        Understanding a/b as a parts of 1/b requires numerator, denominator, and unit fraction vocabulary
      - Splitting shapes into equal parts (age 7+) hard
        Partition into equal shares is prerequisite to understanding unit fractions
        
        Decomposing a shape into more equal shares hard
        Understanding equal shares of different shapes requires concept of more shares = smaller
        
        Halves & Quarters of Shapes hard
        Comparing share sizes requires experience partitioning into halves and quarters
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Halves & Quarters of Shapes hard
        Extends partitioning from halves/fourths to include thirds
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
    - Fractions on a number line (age 8+) hard
      Equivalent fractions as the same point on a number line directly uses the fraction number-line representation
      - Fractions on a number line hard
        Prior number-line fraction experience feeds into formal unit-fraction placement
        
        Fractions of amounts hard
        Placing fractions on number line requires knowing what fractions are
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
      - Fractions of a whole hard
        Must understand unit fractions before placing them on number line
        
        Fractions of amounts hard
        Recognising fractions of shapes/quantities is prerequisite to formal unit fraction understanding
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Understanding a/b as a parts of 1/b requires numerator, denominator, and unit fraction vocabulary
        
        Splitting shapes into equal parts (age 7+) hard
        Partition into equal shares is prerequisite to understanding unit fractions
        
        Decomposing a shape into more equal shares hard
        Understanding equal shares of different shapes requires concept of more shares = smaller
        
        Halves & Quarters of Shapes hard
        Comparing share sizes requires experience partitioning into halves and quarters
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Halves & Quarters of Shapes hard
        Extends partitioning from halves/fourths to include thirds
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
  - Equivalent fractions (age 8+) hard
    Generating equivalent fractions with visual models is prerequisite to algebraic explanation of equivalence
    - Equivalent fractions on a number line hard
      Must understand equivalence before generating equivalent fractions
      - Equivalent fractions hard
        Diagram-based equivalent fractions is prerequisite to formal equivalence understanding
        
        Fractions on a number line soft
        Number line supports seeing equivalent fractions as the same point
        
        Fractions of amounts hard
        Placing fractions on number line requires knowing what fractions are
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Understanding fractions hard
        Extends equivalence of 2/4 and 1/2 to broader equivalent fractions
        
        Reading +, −, and = symbols soft
        Writing fraction sentences (1/2 of 6 = 3) requires understanding the = sign
        
        Reading and writing numbers to 20 hard
        Writing number sentences requires reading and writing numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        Addition as combining or putting together two hard
        Reading/writing the + symbol requires understanding what addition means
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Reading/writing the − symbol requires understanding what subtraction means
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fractions of amounts hard
        Writing fractions and recognising equivalence requires knowing what the fractions mean
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Writing fractions and recognising equivalence requires 'equivalent fraction' vocabulary
        
        Fraction Notation hard
        Recognising and showing equivalent fractions requires that vocabulary
      - Fractions of a whole hard
        Must understand unit fractions to reason about equivalence
        
        Fractions of amounts hard
        Recognising fractions of shapes/quantities is prerequisite to formal unit fraction understanding
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Understanding a/b as a parts of 1/b requires numerator, denominator, and unit fraction vocabulary
        
        Splitting shapes into equal parts (age 7+) hard
        Partition into equal shares is prerequisite to understanding unit fractions
        
        Decomposing a shape into more equal shares hard
        Understanding equal shares of different shapes requires concept of more shares = smaller
        
        Halves & Quarters of Shapes hard
        Comparing share sizes requires experience partitioning into halves and quarters
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Halves & Quarters of Shapes hard
        Extends partitioning from halves/fourths to include thirds
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
      - Fractions on a number line (age 8+) hard
        Equivalent fractions as the same point on a number line directly uses the fraction number-line representation
        
        Fractions on a number line hard
        Prior number-line fraction experience feeds into formal unit-fraction placement
        
        Fractions of amounts hard
        Placing fractions on number line requires knowing what fractions are
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
        
        Fractions of a whole hard
        Must understand unit fractions before placing them on number line
        
        Fractions of amounts hard
        Recognising fractions of shapes/quantities is prerequisite to formal unit fraction understanding
        
        Finding halves and quarters (age 5+) hard
        Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        What Is a Half? hard
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fraction Notation hard
        Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator
        
        Fraction Notation hard
        Understanding a/b as a parts of 1/b requires numerator, denominator, and unit fraction vocabulary
        
        Splitting shapes into equal parts (age 7+) hard
        Partition into equal shares is prerequisite to understanding unit fractions
        
        Decomposing a shape into more equal shares hard
        Understanding equal shares of different shapes requires concept of more shares = smaller
        
        Halves & Quarters of Shapes hard
        Comparing share sizes requires experience partitioning into halves and quarters
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Halves & Quarters of Shapes hard
        Extends partitioning from halves/fourths to include thirds
        
        Finding halves and quarters (age 5+) hard
        Partitioning into fourths/quarters extends from Y1 understanding of quarters
        
        What Is a Half? hard
        Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        What Is a Half? hard
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        Division as equal sharing hard
        Finding a half requires equal sharing into 2 groups — a division concept
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
- Describing Rules & Patterns soft
  Describing general rules for sequences and predicting terms applies the universal generalisation habit in a mathematical context
  - Spotting Patterns hard
    Generalising a rule requires first being able to spot the recurring pattern that the rule captures
    - Connecting New & Old Ideas soft
      Spotting patterns across domains is an extension of the habit of connecting new ideas to existing ones
      - Thinking Before Starting hard
        Making connections between new and old ideas requires the habit of activating prior knowledge first
        
        Persisting When It's Hard hard
        Activating prior knowledge requires the foundational habit of persistent engagement with new material

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