Place Value Problem-Solving

Mastery Evidence

Solve a problem requiring rounding, comparing, or ordering numbers beyond 1000
Apply place-value knowledge in a practical context (e.g. population figures, distances)
Explain the strategy used, referencing place-value understanding

Assessment Prompt

“If [child] is asked 'what is the largest 4-digit even number you can make from the digits 3, 7, 1, and 8?', can they work through it systematically and give you an answer?”

Curriculum Standards1 alignment

Ma/KS2/Y4/NPV/8The national curriculum in England

Solve number problems

solve number and practical problems that involve all of the above and with increasingly large positive numbers

Mathematics · Key Stage 2

Prerequisites2

Place value of each digit
hardAges 8—9
Rounding to 10, 100, 1000
softAges 8—9

Show full prerequisite tree

Place value of each digit hard
PV problems require four-digit place-value understanding
- 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
    - The two digits of a two-digit number hard
      Must understand two-digit place value before extending to hundreds
      - A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
  - 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
    - The teen numbers hard
      General two-digit place value extends from understanding teen number composition
      - How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
      - Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
Rounding to 10, 100, 1000 soft
Many PV problems involve rounding
- Ordering Numbers to 1000 soft
  Understanding which multiple is closer uses comparison skills
  - The three digits of a three-digit number hard
    Comparing three-digit numbers requires 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - The two digits of a two-digit number hard
        Must understand two-digit place value before extending to hundreds
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
    - 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
  - Comparing and ordering numbers hard
    Comparing to 1000 extends comparing two-digit numbers
    - The two digits of a two-digit number hard
      Comparing two-digit numbers using PV requires understanding 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
    - Two written numerals between 1 and 10 soft
      Comparing two-digit numbers extends from comparing single-digit written numerals
      - Comparing groups: more or fewer soft
        Comparing written numerals is the symbolic form of comparing quantities — conceptual comparison helps but isn't strictly required
        
        Counting objects to 20 soft
        Counting a set helps when comparing groups, but younger children (GB age 4) can compare using matching without formal counting to 20
        
        How Many in Total? hard
        Answering 'how many?' requires the cardinality principle
        
        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'
        
        One-to-one counting hard
        Counting objects to answer 'how many?' requires one-to-one correspondence
- Place value of each digit hard
  Rounding to nearest 1000 requires four-digit PV understanding
  - 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - The two digits of a two-digit number hard
        Must understand two-digit place value before extending to hundreds
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
    - 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
- Estimating by rounding soft
  Rounding supports estimation
  - Inverse: addition undoes subtraction hard
    Using inverse to check answers requires understanding the inverse relationship
    - Finding a missing number in addition hard
      Inverse relationship builds on understanding subtraction as unknown-addend
      - Addition as combining or putting together two hard
        Unknown-addend requires understanding both addition and subtraction
        
        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
        Subtraction as unknown-addend reframes subtraction conceptually
        
        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'
  - Addition and subtraction within 1000 soft
    Estimating and checking applies to three-digit calculations
    - The three digits of a three-digit number hard
      Three-digit operations require three-digit place-value understanding
      - 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The two digits of a two-digit number hard
        Must understand two-digit place value before extending to hundreds
        
        A Ten Is Ten Ones hard
        Understanding tens and ones place value requires the concept of 10 as a bundle
        
        The teen numbers hard
        Understanding 10 as a bundle builds on understanding teen numbers as 'a ten and some ones'
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
    - Fluent adding and subtracting within 100 hard
      Adding/subtracting within 1000 extends within-100 skills
      - Adding within 100 hard
        Fluency within 100 builds on strategy-based adding within 100
        
        Addition and subtraction within 20 hard
        Adding within 100 extends within-20 strategies to larger numbers
        
        Number bonds to 9 hard
        Making-ten strategy requires knowing bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fluent adding and subtracting within 10 hard
        Strategies for within-20 calculation build on fluent within-10 knowledge
        
        Number bonds to 9 hard
        Fluency within 10 requires knowing number bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        Fluent adding and subtracting within 5 hard
        Fluency within 10 extends from fluency within 5
        
        Addition as combining or putting together two hard
        Fluency with addition within 5 requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Fluency with subtraction within 5 requires understanding subtraction as taking away
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        The two digits of a two-digit number hard
        Adding within 100 using PV requires understanding 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
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
        
        The teen numbers hard
        General two-digit place value extends from understanding teen number composition
        
        How Many in Total? hard
        Understanding tens-and-ones composition requires cardinality — knowing numbers represent quantities
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Reading and writing numbers to 20 hard
        Composing/decomposing teen numbers requires reading and writing those numerals
        
        How Many in Total? hard
        Reading/writing numerals 0–20 requires understanding that numerals represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Writing digits 0-9 hard
        Writing numerals requires the motor skill of forming digits 0-9 (taught in English handwriting)
        
        Sitting and holding a pencil soft
        Same motor skills for digit formation
      - Fluent adding and subtracting within 20 hard
        Fluency within 100 requires fluent single-digit facts
        
        Addition and subtraction within 20 hard
        Fluency within 20 requires prior strategy-based adding/subtracting within 20
        
        Number bonds to 9 hard
        Making-ten strategy requires knowing bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fluent adding and subtracting within 10 hard
        Strategies for within-20 calculation build on fluent within-10 knowledge
        
        Number bonds to 9 hard
        Fluency within 10 requires knowing number bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        Fluent adding and subtracting within 5 hard
        Fluency within 10 extends from fluency within 5
        
        Addition as combining or putting together two hard
        Fluency with addition within 5 requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Fluency with subtraction within 5 requires understanding subtraction as taking away
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fluent adding and subtracting within 10 hard
        Fluency within 20 builds on fluency within 10
        
        Number bonds to 9 hard
        Fluency within 10 requires knowing number bonds to 10
        
        Numbers up to 10 into pairs hard
        Making 10 is a specific application of decomposing numbers into pairs
        
        Addition as combining or putting together two hard
        Decomposing numbers into pairs requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Fluent adding and subtracting within 5 hard
        Fluency within 10 extends from fluency within 5
        
        Addition as combining or putting together two hard
        Fluency with addition within 5 requires understanding addition as combining
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
        
        Subtraction as taking away or separating hard
        Fluency with subtraction within 5 requires understanding subtraction as taking away
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

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