Math // Addition & Subtraction

Grouping numbers to add

Understand and apply the associative property of addition: when adding three numbers, any two can be added first

Ages
6 to 7
Type
Concept
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

If the learner needs to add 3 + 7 + 6, can they spot that adding 3 and 7 first (to make 10) is easier — rather than always working left to right?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Solve 2 + 6 + 4 by first adding 6 + 4 = 10, then 2 + 10 = 12
  • Explain that grouping addends differently gives the same total
  • Choose which two numbers to add first to make the calculation easier

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics

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Grouping numbers to add

Ages 6 to 7 // Concept

Leads to

1 topic
  • Adding Three Small NumbersRequired

    Addition & Subtraction // Ages 6 to 7

    Adding three numbers uses the associative property to group pairs

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Teaches the associative property of addition for ages 6–7: when adding three or more numbers, any two can be combined first without changing the total, e.g. (2+8)+5 = 2+(8+5) = 15. The point is the freedom to choose which pair to combine first, used as a practical strategy (make a ten, add a double first) rather than as a stated law. Contrast with commutativity (reordering the numbers) and avoid over-generalising to subtraction, where (10−3)−2 = 5 but 10−(3−2) = 9. Includes hands-on manipulation, a reasoning/justification layer, consolidation checks, and a static side-by-side anchor, each with its pedagogical purpose and limits noted.

Common misconceptions

  • Believing the total changes if you add in a different grouping (failing to conserve the whole).
  • Conflating associativity with commutativity — thinking they are the same 'any order' rule.
  • Over-generalising to subtraction, e.g. assuming (10−3)−2 = 10−(3−2).
  • Losing track of an addend when regrouping (dropping or double-counting the third number).
  • Thinking brackets are just decoration rather than an instruction about what to combine first.

Useful representations

  • Three counter groups / cube towers that can be pushed into different pairs before merging
  • Bracket notation over three addends, (a+b)+c vs a+(b+c), with the first-combined pair highlighted
  • Ten-frame showing a make-ten pair filling first, then the remainder added
  • Bar/part-whole model with three parts and one conserved whole bar
  • Empty number line where the child chooses which jump to take first

Teaching practices

  • Introduce through the practical goal of making calculation easier (look for a pair that makes ten, or add a double first) rather than as an abstract named law.
  • Use concrete equal-status objects (counters, cubes) the child physically pushes into different pairs, so the conserved whole is seen while the grouping order varies.
  • Contrast explicitly with commutativity: reordering (which comes first) versus associating (which pair we combine first) — pupils confuse the two.
  • Have children compute the same triple by two different groupings and compare totals, making the invariance an observed result, not a told fact.
  • Connect forward to written column addition, where partial sums (ones, tens) are grouped and recombined, so the property is seen as load-bearing later.

Activities to try

  1. 1

    Grouping bench: choose which pair to combine first

    Why it works: You may choose which pair to combine first; a smart choice (make a ten, or add a double) makes the sum easier while the total stays identical because the three amounts themselves never change.

    1. Set out three equal-status objects or number tiles for a triple such as 2, 8, 5 (counters, cubes, or draggable tiles).
    2. Choose any two to combine first; merge them into a single running sum, then bring in the third.
    3. Try the same triple a second way (e.g. combine 8+5 first) and compare the final total.
    4. State out loud which pair you combined first and why that choice made the calculation easier.
    5. Repeat with a triple that rewards a make-ten pair (e.g. 2+8+5, where 2+8=10 then 10+5=15) and one that rewards adding a double first (e.g. 4+4+7, where 4+4=8 then 8+7=15).
  2. 2

    Why is that allowed? Justify your grouping

    Why it works: Grouping is a reasoning move you can justify: the total is conserved because the three quantities stay the same, no matter which pair you combine first — but this freedom holds for addition, not for subtraction.

    1. Given a triple, say which two numbers you would add first.
    2. Answer a questioner (a peer, tutor, or 'maths buddy') who asks 'Why is that allowed?' and 'Would we get the same total if we grouped the numbers differently? How do you know?'
    3. Explain in your own words that the three amounts are unchanged, so only which pair you combine first differs.
    4. Try a new triple such as 4+7+6 and justify which pair you would add first (here 4+6=10, then 10+7=17). Notice that 4 and 6 are not next to each other, so adding them first means changing the order as well as the grouping; the total is still 17.
    5. Test the contrast with subtraction: work out (10−3)−2 = 5 and 10−(3−2) = 9, notice the answers differ, and conclude that the grouping freedom belongs to addition, not subtraction.
  3. 3

    Consolidate: match, choose, and check equal totals

    Why it works: Recognising equivalent groupings and spotting the efficient first pair is intended to consolidate that regrouping preserves the total while making the calculation easier.

    1. Decide which pair you would add first to make a given triple easy (e.g. for 2+8+5, which pair makes a ten?).
    2. Match equivalent groupings, e.g. (2+8)+5 with 2+(8+5).
    3. Judge true/false statements about whether two groupings of the same triple share a total, including an item where one number has been changed so the total genuinely differs.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
1.OA.3

Math // Addition & Subtraction

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