Two-step addition and subtraction problems
PROCEDURALSolve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why
Mastery Evidence
- Identify the two steps needed to solve a contextual problem
- Choose between mental and written methods for each step based on the numbers
- Explain why the chosen operations and methods are appropriate
Assessment Prompt
“If [child] reads a word problem with two steps — like working out change from a shopping trip with two items — can they plan which calculations to do and in what order?”
Curriculum Standards1 alignment
Ma/KS2/Y4/AS/3The national curriculum in Englandsolve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why
Prerequisites2
- Adding and subtracting (age 8+)hardAges 8—9
- Missing number problems (age 7+)hardAges 7—8
Show full prerequisite tree
- The three digits of a three-digit number hard
Four-digit place value extends three-digit place value
- 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)
- 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)
- 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)
- 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)
- 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)
- Adding and subtracting (age 7+) hard
Four-digit columnar methods extend three-digit columnar methods
- Fluent adding and subtracting within 100 hard
Columnar methods require fluent within-100 addition/subtraction
- Addition and subtraction within 20 hard
Adding within 100 extends within-20 strategies to larger numbers
- 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
- 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'
- 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)
- 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)
- Addition and subtraction within 20 hard
Fluency within 20 requires prior strategy-based adding/subtracting within 20
- 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
- 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'
- 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'
- 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'
- 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'
- Addition and subtraction within 1000 hard
Formal columnar methods build on conceptual understanding of composing/decomposing
- The three digits of a three-digit number hard
Three-digit operations require three-digit place-value understanding
- 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)
- 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)
- 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)
- 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)
- 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)
- Fluent adding and subtracting within 100 hard
Adding/subtracting within 1000 extends within-100 skills
- Addition and subtraction within 20 hard
Adding within 100 extends within-20 strategies to larger numbers
- 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
- 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)
- 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)
- 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)
- Addition and subtraction within 20 hard
Fluency within 20 requires prior strategy-based adding/subtracting within 20
- 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
- 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)
- 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'
- 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'
- 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 100 hard
Columnar methods require fluent within-100 addition/subtraction
- Addition and subtraction within 20 hard
Adding within 100 extends within-20 strategies to larger numbers
- 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
- 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'
- 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)
- 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)
- Addition and subtraction within 20 hard
Fluency within 20 requires prior strategy-based adding/subtracting within 20
- 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
- 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'
- 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'
- 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'
- 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'
- Addition and subtraction within 1000 hard
Formal columnar methods build on conceptual understanding of composing/decomposing
- The three digits of a three-digit number hard
Three-digit operations require three-digit place-value understanding
- 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)
- 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)
- 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)
- 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)
- 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)
- Fluent adding and subtracting within 100 hard
Adding/subtracting within 1000 extends within-100 skills
- Addition and subtraction within 20 hard
Adding within 100 extends within-20 strategies to larger numbers
- 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
- 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)
- 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)
- 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)
- Addition and subtraction within 20 hard
Fluency within 20 requires prior strategy-based adding/subtracting within 20
- 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
- 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)
- 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'
- 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'
- 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 100 hard
Solving word problems within 100 requires fluent computation within 100
- Addition and subtraction within 20 hard
Adding within 100 extends within-20 strategies to larger numbers
- 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
- 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'
- 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)
- 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)
- Addition and subtraction within 20 hard
Fluency within 20 requires prior strategy-based adding/subtracting within 20
- 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
- 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'
- 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'
- 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'
- 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'
- Adding and subtracting hard
Word problems to 20 require the procedural ability to add/subtract to 20
- 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'
- 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'
- Addition and subtraction word problems soft
Word problems to 20 extend from word problems within 10 — same problem structures at a higher range
- Representing Addition and Subtraction hard
Solving word problems within 10 requires ability to represent the operations with objects/drawings
- Addition as combining or putting together two hard
Representing addition with objects/drawings 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
Representing subtraction with objects/drawings 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'
Unlocks1
- Adding and subtracting (age 9+)hardAges 9—10