Efficiency, Sankey diagrams, and work done

PROCEDURAL

Science›Energy|Ages 12—13|ID: mt_JyfLtl_nhw

Calculate energy efficiency as the ratio of useful output energy to total input energy, construct and interpret Sankey diagrams, and calculate work done using work = force × distance

Mastery Evidence

Calculates efficiency as a percentage from given input and useful output energy values
Draws a Sankey diagram to represent energy transfers in a device, with arrow widths proportional to energy amounts
Uses W = Fd to calculate work done when a force moves through a distance
Identifies which energy transformations in a Sankey diagram are useful and which are wasted

Assessment Prompt

“If [child] heard that a light bulb is 10% efficient and uses 60 J of energy, could they work out how much of that energy actually becomes light — and where the rest goes?”

Curriculum Standards1 alignment

KS3.Sci.Phys.Energy.3The national curriculum in England

Energy Efficiency and Work Done

calculating energy efficiency; the relationship between work done and energy transferred; the role of heat in changes of state

Science · KS3

Prerequisites3

Energy can't be created or destroyed
hardAges 11—12
Resultant Forces
softAges 11—12
Calculating Percentages
softAges 10—11

Show full prerequisite tree

Energy can't be created or destroyed hard
Efficiency calculations quantify how much energy is conserved vs dissipated — conservation and dissipation must be understood first
- Naming types of energy hard
  Explaining conservation of energy and dissipation requires energy store and transfer vocabulary
- Energy stores and transfers hard
  Conservation of energy is a property of energy stores and transfers — the stores and pathways model must be established first
  - Speed and energy hard
    KS3 kinetic energy store extends KS2 US understanding that faster-moving objects have more energy
    - Naming types of energy hard
      Relating an object's speed to its energy requires 'kinetic energy' vocabulary
    - Pushes & Pulls hard
      Must understand forces change motion before relating speed to energy
      - Drawing Force Diagrams soft
        Understanding pushes and pulls as forces is supported by the arrow representation of magnitude and direction
  - Naming types of energy hard
    Identifying energy stores and transfer pathways is a formalisation of the energy type vocabulary introduced earlier
  - Greenhouse Gas Science soft
    KS3 energy stores and radiation transfer provides the physics vocabulary for explaining how greenhouse gases re-emit infrared
    - Global Wind Patterns hard
      Greenhouse radiation physics provides the energy imbalance context that drives differential heating and global circulation
      - Climate Zones hard
        Global atmospheric circulation depends on climate zones
        
        Weather vs Climate hard
        Climate zones require distinguishing weather from climate
        
        Seasons & Weather Patterns soft
        Climate as long-term patterns benefits from knowing seasonal patterns
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Geography & Local Weather hard
        Climate concept requires knowing that different places have different typical weather
        
        Seasons & Weather Patterns hard
        Understanding global weather differences requires knowing seasons
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Types of Weather hard
        Comparing world weather requires weather vocabulary
        
        Types of Weather hard
        Distinguishing weather from climate requires understanding weather first
        
        Geography & Local Weather hard
        Climate zones formalize the concept of different weather around the world
        
        Seasons & Weather Patterns hard
        Understanding global weather differences requires knowing seasons
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Types of Weather hard
        Comparing world weather requires weather vocabulary
        
        Sun-Driven Weather Systems soft
        Climate zones benefit from understanding Sun drives weather systems
        
        Weather vs Climate soft
        Sun-driven weather benefits from weather vs climate distinction
        
        Seasons & Weather Patterns soft
        Climate as long-term patterns benefits from knowing seasonal patterns
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Geography & Local Weather hard
        Climate concept requires knowing that different places have different typical weather
        
        Seasons & Weather Patterns hard
        Understanding global weather differences requires knowing seasons
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Types of Weather hard
        Comparing world weather requires weather vocabulary
        
        Types of Weather hard
        Distinguishing weather from climate requires understanding weather first
        
        The Water Cycle hard
        Sun as weather engine requires understanding the water cycle it drives
        
        Cloud Types hard
        Water cycle requires understanding cloud formation (condensation)
        
        Rain & Puddles hard
        Cloud types build on knowing rain comes from clouds
        
        Types of Weather soft
        Rain concept benefits from knowing rain as a weather type
        
        Where water is found on Earth soft
        Water cycle benefits from knowing where water is found on Earth
        
        Shapes of land and water hard
        Must know land and water features before learning where water is found on Earth
        
        Seasonal changes soft
        Seasonal observation supports understanding land and water features
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Heating & Cooling Changes soft
        Understanding state changes supports knowing water as solid (ice) and liquid
        
        States of Matter Vocabulary hard
        Describing and measuring changes of state requires solid/liquid/gas vocabulary and the term 'change of state'
        
        Drawing Particle Diagrams hard
        Observing and describing change of state requires reading particle diagrams showing how arrangement changes on heating or cooling
        
        Heating & Cooling Changes soft
        Water cycle benefits from curriculum states of matter (heating/cooling changes state)
        
        States of Matter Vocabulary hard
        Describing and measuring changes of state requires solid/liquid/gas vocabulary and the term 'change of state'
        
        Drawing Particle Diagrams hard
        Observing and describing change of state requires reading particle diagrams showing how arrangement changes on heating or cooling
        
        Temperature & Thermometers soft
        Water cycle evaporation relates to temperature (heat drives evaporation)
        
        Rain & Puddles hard
        Full water cycle builds on basic rain/puddle/evaporation concept
        
        Types of Weather soft
        Rain concept benefits from knowing rain as a weather type
        
        Sunlight warms things up soft
        Sun drives weather builds on curriculum sunlight warms Earth's surface
        
        Seasonal changes soft
        Seasonal observation supports understanding sunlight warming effects
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        What Causes Wind hard
        Sun-driven weather systems extend the basic wind-cause concept to global scale
        
        What Is Wind? hard
        Understanding wind causes requires knowing what wind is
    - Sun-Driven Weather Systems hard
      Greenhouse radiation physics depends on sun as driver of weather
      - Weather vs Climate soft
        Sun-driven weather benefits from weather vs climate distinction
        
        Seasons & Weather Patterns soft
        Climate as long-term patterns benefits from knowing seasonal patterns
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Geography & Local Weather hard
        Climate concept requires knowing that different places have different typical weather
        
        Seasons & Weather Patterns hard
        Understanding global weather differences requires knowing seasons
        
        Seasonal changes soft
        Seasons/weather patterns parallels curriculum four seasons topic
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Types of Weather soft
        Seasons benefit from weather vocabulary
        
        Types of Weather hard
        Comparing world weather requires weather vocabulary
        
        Types of Weather hard
        Distinguishing weather from climate requires understanding weather first
      - The Water Cycle hard
        Sun as weather engine requires understanding the water cycle it drives
        
        Cloud Types hard
        Water cycle requires understanding cloud formation (condensation)
        
        Rain & Puddles hard
        Cloud types build on knowing rain comes from clouds
        
        Types of Weather soft
        Rain concept benefits from knowing rain as a weather type
        
        Where water is found on Earth soft
        Water cycle benefits from knowing where water is found on Earth
        
        Shapes of land and water hard
        Must know land and water features before learning where water is found on Earth
        
        Seasonal changes soft
        Seasonal observation supports understanding land and water features
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
        
        Heating & Cooling Changes soft
        Understanding state changes supports knowing water as solid (ice) and liquid
        
        States of Matter Vocabulary hard
        Describing and measuring changes of state requires solid/liquid/gas vocabulary and the term 'change of state'
        
        Drawing Particle Diagrams hard
        Observing and describing change of state requires reading particle diagrams showing how arrangement changes on heating or cooling
        
        Heating & Cooling Changes soft
        Water cycle benefits from curriculum states of matter (heating/cooling changes state)
        
        States of Matter Vocabulary hard
        Describing and measuring changes of state requires solid/liquid/gas vocabulary and the term 'change of state'
        
        Drawing Particle Diagrams hard
        Observing and describing change of state requires reading particle diagrams showing how arrangement changes on heating or cooling
        
        Temperature & Thermometers soft
        Water cycle evaporation relates to temperature (heat drives evaporation)
        
        Rain & Puddles hard
        Full water cycle builds on basic rain/puddle/evaporation concept
        
        Types of Weather soft
        Rain concept benefits from knowing rain as a weather type
      - Sunlight warms things up soft
        Sun drives weather builds on curriculum sunlight warms Earth's surface
        
        Seasonal changes soft
        Seasonal observation supports understanding sunlight warming effects
        
        Days, Weeks, Months & Years soft
        Observing and describing seasonal changes requires basic date and time vocabulary (months, seasons, year)
        
        Ordering Events in Time hard
        Understanding days/months/years builds on sequencing events chronologically
      - What Causes Wind hard
        Sun-driven weather systems extend the basic wind-cause concept to global scale
        
        What Is Wind? hard
        Understanding wind causes requires knowing what wind is
  - How energy travels around hard
    KS3 energy stores model extends KS2 US observation that energy is transferred by sound, light, heat and electric currents
    - Light & Seeing in the Dark soft
      Light knowledge supports understanding energy transfer by light
      - Light & Sound Vocabulary hard
        Observing that objects need illumination to be seen requires the 'light source' vocabulary
    - Naming types of energy hard
      Describing energy transfer by sound, light, heat, and electric current requires energy type vocabulary
    - Building a simple circuit hard
      Must know circuits before understanding energy transfer by electric current
      - What uses electricity at home hard
        Must identify electrical appliances before building circuits
    - Vibrations & Sound soft
      Sound knowledge supports understanding energy transfer by sound
      - Light & Sound Vocabulary hard
        Understanding vibrating materials and sound requires 'vibration' vocabulary
      - Communication with Light & Sound hard
        Must understand how sound works before designing a communication device using it
        
        Light & Seeing in the Dark hard
        Must understand how light works before designing a communication device using it
        
        Light & Sound Vocabulary hard
        Observing that objects need illumination to be seen requires the 'light source' vocabulary
Resultant Forces soft
Work done = force × distance links energy to mechanics — understanding forces as vectors provides the context
- Pushes & Pulls hard
  KS3 forces as vectors extends KS2 introduction to pushes and pulls changing speed and direction
  - Drawing Force Diagrams soft
    Understanding pushes and pulls as forces is supported by the arrow representation of magnitude and direction
- Force & Motion Vocabulary hard
  Describing balanced and unbalanced forces as vector quantities requires resultant force, balanced forces vocabulary
- Drawing Force Diagrams hard
  Forces as vectors with magnitude and direction is the formal underpinning of the force arrow representation
- Contact & Non-Contact Forces hard
  KS3 resultant force and balanced forces extends KS2 distinction between contact and non-contact forces
  - Forces Vocabulary hard
    Distinguishing contact from non-contact forces requires these exact terms
  - Drawing Force Diagrams soft
    Distinguishing contact and non-contact forces is clarified by drawing force diagrams showing where arrows originate
  - Friction & Surfaces hard
    Must experience contact forces like friction before distinguishing contact vs non-contact forces
    - Pushes & Pulls hard
      Must understand forces change motion before comparing movement on different surfaces
      - Drawing Force Diagrams soft
        Understanding pushes and pulls as forces is supported by the arrow representation of magnitude and direction
    - Forces Vocabulary hard
      Comparing how things move on different surfaces requires friction vocabulary
Calculating Percentages soft
Calculating energy efficiency as a percentage requires percentage calculation skills from Math
- Decimals and fractions (age 10+) hard
  Calculating percentages requires fraction-decimal-percentage equivalence
  - Fractions of a whole (age 10+) hard
    Calculating decimal equivalents requires understanding fraction as division
    - Multiplying fractions hard
      Understanding fraction as division builds on fraction × whole number (inverse reasoning)
      - Understanding fractions (age 9+) hard
        a/b as sum of 1/b is prerequisite to understanding a/b as multiple of 1/b
        
        Fractions of a whole hard
        Understanding a/b as a parts of size 1/b is prerequisite to understanding a/b as sum of 1/b
        
        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
    - Division with remainders (age 10+) soft
      Division leading to fractions connects to long division skills
      - Arrays for multiplication (age 9+) hard
        Long division by 2-digit extends Y5 short division by 1-digit
        
        Division as Unknown Factor hard
        Understanding division as unknown-factor supports short division strategy
        
        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
        
        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)
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        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
        
        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
        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'
        
        Fluent multiplication and division facts hard
        Fluent ×÷ within 100 is prerequisite to short division of larger numbers
        
        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
        
        Multiplication as repeated addition hard
        Using arrays requires understanding what multiplication means
        
        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)
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        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
        
        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)
        
        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)
        
        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
        
        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)
        
        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
      - Long multiplication (age 10+) soft
        Checking division with multiplication requires fluent multiplication
        
        Long multiplication hard
        Long multiplication 4×2 extends Y5 multiplication up to 4×1 and 2×2
        
        Written Multiplication hard
        2/3-digit × 1-digit written method is prerequisite to 4-digit × 1-digit and 2-digit × 2-digit
        
        Written Multiplication & Division hard
        Formal short multiplication extends Y3 written multiplication
        
        The three digits of a three-digit number soft
        Two-digit × one-digit uses place-value partitioning (e.g. 23 × 4 = 20 × 4 + 3 × 4)
        
        A Hundred Is Ten Tens hard
        Three-digit place value requires understanding 100 as a unit
        
        A Ten Is Ten Ones hard
        100 as ten tens extends understanding of 10 as ten ones
        
        The two digits of a two-digit number hard
        Must understand two-digit place value before extending to hundreds
        
        The two digits of a two-digit number hard
        Three-digit PV extends two-digit PV (tens and ones)
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        Times tables (age 7+) hard
        Written multiplication/division requires known table facts
        
        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
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        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
        
        Reading ×, ÷, and = Symbols hard
        Writing multiplication/division statements requires fluency with symbols
        
        Reading +, −, and = symbols soft
        Reading × and ÷ symbols parallels reading + and − symbols
        
        Reading and writing numbers to 20 hard
        Writing number sentences requires reading and writing numerals
        
        Addition as combining or putting together two hard
        Reading/writing the + symbol requires understanding what addition means
        
        Subtraction as taking away or separating hard
        Reading/writing the − symbol requires understanding what subtraction means
        
        Division as equal sharing hard
        Writing ÷ sentences requires understanding division
        
        Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        Multiplication as repeated addition hard
        Writing × sentences requires understanding multiplication
        
        Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        All times tables to 12×12 hard
        Written multiplication requires fluent table facts
        
        Counting in 6s soft
        Counting in 6s/7s/9s supports learning those table facts
        
        Skip Counting (4s, 8s, 50s, 100s) hard
        Counting in 6s/7s/9s/25s/1000s extends counting in 4s/8s/50s/100s
        
        Counting in 2s hard
        Counting in 4s and 8s extends prior skip counting in 2s, 5s, 10s
        
        Times tables (age 7+) hard
        All tables to 12×12 extends knowledge of 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
        
        Multiplication as repeated addition hard
        Recalling times table facts requires understanding multiplication as repeated addition/grouping
        
        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
        
        Area and the distributive property soft
        Area models for distributive property support understanding long multiplication layout
        
        Understanding angles (age 8+) hard
        Must multiply side lengths for area before using area models for distributive property
        
        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
  - Percentage and decimal equivalents hard
    Extends Y5 percentage/decimal equivalents to broader range of fractions
    - Fraction-Decimal Equivalents soft
      Decimal equivalents of 1/4, 1/2, 3/4 support percentage equivalence problems
      - Equivalent fractions on a number line soft
        Equivalent fractions understanding supports recognising decimal equivalents of common 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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        
        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
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        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
        
        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
      - Decimal equivalents of tenths and hundredths hard
        General decimal equivalents prerequisite to specific 1/4, 1/2, 3/4 equivalents
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
    - Understanding Percentages hard
      Must understand % concept before solving % problems
      - Decimals for Tenths & Hundredths hard
        Decimal notation for fractions is prerequisite to understanding % as parts per 100
        
        Converting tenths to hundredths hard
        10ths as 100ths supports using decimal notation for both
        
        Tenths (age 8+) hard
        Understanding hundredths is prerequisite to working with 10ths and 100ths together
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        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
        
        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
        
        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
        
        Equivalent fractions (age 9+) hard
        Generating equivalent fractions supports converting 10ths to 100ths
        
        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
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        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
        
        Fractions of amounts hard
        Writing fractions and recognising equivalence requires knowing what the fractions mean
        
        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
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        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
        Extends partitioning from halves/fourths to include thirds
        
        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
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        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
        
        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
        
        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
        
        Understanding fractions hard
        Extends equivalence of 2/4 and 1/2 to broader equivalent fractions
        
        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
        
        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
        
        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 a whole hard
        Must understand unit fractions before placing them on number line
        
        Decimal equivalents of tenths and hundredths hard
        Y4 decimal equivalents of 10ths/100ths is prerequisite to formal decimal notation for fractions
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        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
        
        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
        
        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
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
        
        Decimal & Percent Notation hard
        Using decimal notation for fractions requires decimal, tenths, and hundredths vocabulary
      - Decimal & Percent Notation hard
        Understanding the % symbol and 'per cent means parts per hundred' is the LANGUAGE node content
  - Decimal & Percent Notation hard
    Recalling equivalences between fractions, decimals, and percentages requires all three sets of vocabulary
  - Understanding Percentages soft
    Builds on Y5 understanding of percentages
    - Decimals for Tenths & Hundredths hard
      Decimal notation for fractions is prerequisite to understanding % as parts per 100
      - Converting tenths to hundredths hard
        10ths as 100ths supports using decimal notation for both
        
        Tenths (age 8+) hard
        Understanding hundredths is prerequisite to working with 10ths and 100ths together
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        Equivalent fractions (age 9+) hard
        Generating equivalent fractions supports converting 10ths to 100ths
        
        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
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        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
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        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
        
        Addition as combining or putting together two hard
        Reading/writing the + symbol requires understanding what addition means
        
        Subtraction as taking away or separating hard
        Reading/writing the − symbol requires understanding what subtraction means
        
        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
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        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
        
        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
        
        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
        
        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
        
        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
        Partitioning shapes into halves extends from Y1 understanding of halves
        
        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
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        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
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        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
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        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
        Extends partitioning from halves/fourths to include thirds
        
        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
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        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
        
        Fractions of amounts hard
        Writing fractions and recognising equivalence requires knowing what the fractions mean
        
        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
        Working with fractions extends from Y1 understanding of halves
        
        Division as equal sharing soft
        Finding fractions of quantities uses equal sharing (division)
        
        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
        Extends partitioning from halves/fourths to include thirds
        
        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
        
        Tenths soft
        Counting in tenths supports placing fractions on a number line
        
        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
        
        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
      - Decimal equivalents of tenths and hundredths hard
        Y4 decimal equivalents of 10ths/100ths is prerequisite to formal decimal notation for fractions
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
      - Decimal & Percent Notation hard
        Using decimal notation for fractions requires decimal, tenths, and hundredths vocabulary
    - Decimal & Percent Notation hard
      Understanding the % symbol and 'per cent means parts per hundred' is the LANGUAGE node content
- Bar Models for Ratios soft
  Percentage-of-amount problems can be set up as bar models showing 100% divided into parts
- Percentages (age 9+) hard
  Calculating percentages of amounts requires 'percentage', 'proportion', and 'out of' vocabulary
- Decimal place value soft
  Finding percentages often involves decimal multiplication
  - Multiplying and dividing (age 10+) hard
    Understanding decimal place value and powers-of-10 scaling is essential for decimal multiplication
    - Multiplying and dividing hard
      ×÷ by 10/100/1000 to 3dp extends Y5 ×÷ by 10/100/1000
      - Dividing by 10 and 100 hard
        Dividing by 10/100 (Y4 fractions context) is prerequisite to ×÷ by 10/100/1000 with decimals
        
        Tenths (age 8+) hard
        Must understand hundredths to identify digit values when dividing by 10/100
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        Decimal equivalents of tenths and hundredths hard
        Must know decimal notation to express results of dividing by 10/100
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        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
        
        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
        
        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
        
        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
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
      - Place Value × 10 Pattern hard
        ×10 place-value relationship is prerequisite to ×÷ by powers of 10
        
        Place value of each digit hard
        Four-digit place value is prerequisite to understanding ×10 relationship between places
        
        The three digits of a three-digit number hard
        Four-digit place value extends three-digit place value
        
        A Hundred Is Ten Tens hard
        Three-digit place value requires 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
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        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'
        
        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
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        The two digits of a two-digit number hard
        Three-digit PV extends two-digit PV (tens and ones)
        
        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)
    - Reading Decimal Places hard
      Understanding digit shifting requires knowing what each decimal place represents
      - Reading and writing numbers (age 10+) hard
        Identifying digit value in 3dp numbers requires reading decimals to thousandths
        
        Comparing Decimals hard
        Compare decimals to thousandths extends Y5 decimal comparison
        
        Decimal place value (age 8+) hard
        Comparing decimals to 2dp (Y4) is prerequisite to comparing to 3dp
        
        Decimal equivalents of tenths and hundredths hard
        Must understand decimal notation to compare decimals
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
        
        Tenths (age 9+) soft
        Thousandths understanding supports comparing decimals to 3dp
        
        Decimals for Tenths & Hundredths hard
        Decimal notation for 10ths/100ths is prerequisite to extending to thousandths
        
        Converting tenths to hundredths hard
        10ths as 100ths supports using decimal notation for both
        
        Tenths (age 8+) hard
        Understanding hundredths is prerequisite to working with 10ths and 100ths together
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Equivalent fractions (age 9+) hard
        Generating equivalent fractions supports converting 10ths to 100ths
        
        Equivalent fractions on a number line hard
        Understanding equivalence conceptually is prerequisite to explaining algebraically
        
        Equivalent fractions (age 8+) hard
        Generating equivalent fractions with visual models is prerequisite to algebraic explanation of equivalence
        
        Decimal equivalents of tenths and hundredths hard
        Y4 decimal equivalents of 10ths/100ths is prerequisite to formal decimal notation for fractions
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
        
        Decimal & Percent Notation hard
        Using decimal notation for fractions requires decimal, tenths, and hundredths vocabulary
        
        Place Value × 10 and ÷ 10 soft
        Place-value understanding supports decimal comparison reasoning
        
        Tenths (age 9+) soft
        Understanding 1/10 of requires familiarity with thousandths
        
        Decimals for Tenths & Hundredths hard
        Decimal notation for 10ths/100ths is prerequisite to extending to thousandths
        
        Converting tenths to hundredths hard
        10ths as 100ths supports using decimal notation for both
        
        Tenths (age 8+) hard
        Understanding hundredths is prerequisite to working with 10ths and 100ths together
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Equivalent fractions (age 9+) hard
        Generating equivalent fractions supports converting 10ths to 100ths
        
        Equivalent fractions on a number line hard
        Understanding equivalence conceptually is prerequisite to explaining algebraically
        
        Equivalent fractions (age 8+) hard
        Generating equivalent fractions with visual models is prerequisite to algebraic explanation of equivalence
        
        Decimal equivalents of tenths and hundredths hard
        Y4 decimal equivalents of 10ths/100ths is prerequisite to formal decimal notation for fractions
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
        
        Decimal & Percent Notation hard
        Using decimal notation for fractions requires decimal, tenths, and hundredths vocabulary
        
        Place Value × 10 Pattern hard
        ×10 both directions extends Y5 ×10 relationship
        
        Place value of each digit hard
        Four-digit place value is prerequisite to understanding ×10 relationship between places
        
        The three digits of a three-digit number hard
        Four-digit place value extends three-digit place value
        
        A Hundred Is Ten Tens hard
        Three-digit place value requires understanding 100 as a unit
        
        A Ten Is Ten Ones hard
        100 as ten tens extends understanding of 10 as ten ones
        
        The two digits of a two-digit number hard
        Must understand two-digit place value before extending to hundreds
        
        The two digits of a two-digit number hard
        Three-digit PV extends two-digit PV (tens and ones)
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        Tenths (age 9+) hard
        Reading/writing decimals to thousandths requires understanding thousandths place
        
        Decimals for Tenths & Hundredths hard
        Decimal notation for 10ths/100ths is prerequisite to extending to thousandths
        
        Converting tenths to hundredths hard
        10ths as 100ths supports using decimal notation for both
        
        Tenths (age 8+) hard
        Understanding hundredths is prerequisite to working with 10ths and 100ths together
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Equivalent fractions (age 9+) hard
        Generating equivalent fractions supports converting 10ths to 100ths
        
        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 of a whole hard
        Must understand unit fractions to reason about equivalence
        
        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
        
        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
        
        Decimal equivalents of tenths and hundredths hard
        Y4 decimal equivalents of 10ths/100ths is prerequisite to formal decimal notation for fractions
        
        Tenths (age 8+) hard
        Must understand hundredths before writing decimal equivalents
        
        Tenths hard
        Count in tenths is prerequisite to extending to hundredths
        
        Fractions of amounts hard
        Tenths extend fraction understanding from halves, thirds, quarters
        
        Counting in 2s soft
        Skip counting supports counting in tenths
        
        Decimal & Percent Notation hard
        Writing decimal equivalents of tenths and hundredths requires decimal point and place-value vocabulary
        
        Decimal & Percent Notation hard
        Using decimal notation for fractions requires decimal, tenths, and hundredths vocabulary
  - Long multiplication (age 10+) hard
    Multiplying decimals by whole numbers builds on whole-number multiplication
    - Long multiplication hard
      Long multiplication 4×2 extends Y5 multiplication up to 4×1 and 2×2
      - Written Multiplication hard
        2/3-digit × 1-digit written method is prerequisite to 4-digit × 1-digit and 2-digit × 2-digit
        
        Written Multiplication & Division hard
        Formal short multiplication extends Y3 written multiplication
        
        The three digits of a three-digit number soft
        Two-digit × one-digit uses place-value partitioning (e.g. 23 × 4 = 20 × 4 + 3 × 4)
        
        A Hundred Is Ten Tens hard
        Three-digit place value requires 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
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        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'
        
        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
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        The two digits of a two-digit number hard
        Three-digit PV extends two-digit PV (tens and ones)
        
        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)
        
        Times tables (age 7+) hard
        Written multiplication/division requires known table facts
        
        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
        
        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)
        
        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
        
        Reading ×, ÷, and = Symbols hard
        Writing multiplication/division statements requires fluency with symbols
        
        Reading +, −, and = symbols soft
        Reading × and ÷ symbols parallels reading + and − symbols
        
        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'
        
        Division as equal sharing hard
        Writing ÷ sentences requires understanding 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'
        
        Multiplication as repeated addition hard
        Writing × sentences 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'
        
        All times tables to 12×12 hard
        Written multiplication requires fluent table facts
        
        Counting in 6s soft
        Counting in 6s/7s/9s supports learning those table facts
        
        Skip Counting (4s, 8s, 50s, 100s) hard
        Counting in 6s/7s/9s/25s/1000s extends counting in 4s/8s/50s/100s
        
        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+) hard
        All tables to 12×12 extends knowledge of 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
        
        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)
        
        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
      - Area and the distributive property soft
        Area models for distributive property support understanding long multiplication layout
        
        Understanding angles (age 8+) hard
        Must multiply side lengths for area before using area models for distributive property
        
        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
        
        Choosing measurement units soft
        Choosing appropriate tools builds on choosing appropriate units

Unlocks1

Heating experiments and Q = mcΔT
hardAges 12—13