Math // Addition & Subtraction

Addition and subtraction strategies (age 7+)

Explain why addition and subtraction strategies work, using place value and the properties of operations

Ages
7 to 8
Type
Thinking skill
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

If the learner works out '67 + 24' by splitting it into '60 + 20' and '7 + 4', can they explain to you in words *why* that approach gives the right answer?

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.
  • Explain why adding tens and ones separately gives the correct total
  • Describe why a compensation strategy works (e.g. 'I added 1 too many, so I subtract 1')
  • Use place-value language to justify a written method step by step

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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Addition and subtraction strategies (age 7+)

Ages 7 to 8 // Thinking skill

Leads to

1 topic

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

Children justify WHY addition and subtraction strategies work, using place value and the properties of operations — the step from executing a strategy to explaining it. Two justifications are central: partitioning ('add tens and ones separately, then recombine') is valid because a number is composed of tens and ones and addends may be split, reordered and regrouped without changing the total (commutativity/associativity); and compensation ('add 100 for 99, then subtract 1') is an equivalent transformation — you adjusted by a known amount and undo the surplus. The target is reasoning voiced in place-value language, not more correct answers. A dialogic 'prove-it' routine is well-suited as the primary vehicle because producing a justification requires the child to articulate their reasoning, which an open spoken exchange is well-suited to elicit; written annotation, manipulatives and spot-the-flaw tasks serve as complementary scaffolds and probes.

Common misconceptions

  • Restating the procedure or citing authority ('the teacher said') instead of giving a reason
  • Doubting that splitting/reordering preserves the total (fearing something is lost when a number is broken up)
  • Compensation direction error: adding 100 for 99 then adding 1 (not seeing 100 is one too many)
  • Explaining WHAT was done but not WHY it is valid

Useful representations

  • Base-ten blocks (whole conserved under partitioning)
  • Number line (compensation: overshoot then step back)
  • Part-whole diagram (the number as tens + ones)
  • Worked strategy with each step annotated by its reason
  • Bar model (equivalent transformations preserve the total)

Teaching practices

  • Ask 'why does that work?' routinely, and press for reasons beyond 'that's the rule' — model place-value language back to the child
  • Ground justification in materials first (base-ten shows the whole is conserved when partitioned) then move to verbal argument
  • Make the properties explicit: order and grouping of addends don't change the total; use this to justify partitioning
  • For compensation, track the adjustment quantitatively — 'I added one too many, so I subtract one' — and why the direction is forced
  • Use talk moves (revoice, 'do you agree and why?', counterexamples) so explanations are built collaboratively

Activities to try

  1. 1

    Talk with a 'prove-it' partner who never accepts a bare rule

    Why it works: A remembered procedure becomes an owned explanation: the number IS its tens and ones, so like parts can be added and recombined without loss, and this holds whichever way you split; adding 100 for 99 adds exactly one too many, so you subtract one to land on adding exactly 99.

    1. Solve 46 + 37 by adding tens and ones separately (40 + 30 = 70, 6 + 7 = 13, 70 + 13 = 83), then say the answer aloud.
    2. When your partner (teacher, peer, or an in-role character) asks 'but why doesn't breaking 46 into 40 and 6 lose anything?', explain in your own words why splitting and recombining keeps the same total.
    3. Try splitting 37 a different way — e.g. 35 + 2 instead of 30 + 7 — and explain whether the method still works and why, to show the reasoning transfers to a new split.
    4. Solve 46 + 99 by adding 100 and then adjusting; when asked 'why take one BACK rather than add one?', explain how much too much 100 is compared with 99 and why the surplus must be removed.
    5. Respond to a counterexample your partner offers (e.g. a friend who split 46 into 40 + 6 but only added the 40 and forgot the 6) and say whether that mistake means splitting itself is unreliable.
  2. 2

    Show the whole is conserved: partition with materials, compensate on a number line

    Why it works: Partitioning preserves the total because the parts still compose the same whole and any order/grouping of addends gives the same sum; compensation is 'overshoot then step back' by the known surplus.

    1. Using base-ten blocks, a part-whole diagram, or a bar model, split 46 into 40 + 6 and check that the pieces still make 46 — nothing is lost when you break it up.
    2. Rearrange the order and grouping of the parts (e.g. 6 + 40, or group the tens first) and confirm the total is unchanged.
    3. On a number line, add 46 + 99 by jumping 100 forward, then step back 1; highlight the extra +1 you overshot by.
    4. Say out loud why you step back rather than forward.
  3. 3

    Annotate a worked strategy with the reason for each step

    Why it works: Every step of a strategy has a place-value reason; a justification names those reasons, not just the actions.

    1. Take a written worked method for 46 + 37 (40 + 30 = 70; 6 + 7 = 13; 70 + 13 = 83).
    2. Label each step with WHY it is allowed, not just what was done (e.g. '46 is 40 and 6', 'the order of adding the parts doesn't change the total', 'recombining the parts gives the whole').
    3. Do the same for a compensation method (46 + 99 = 46 + 100 − 1) and name the surplus of one and its undoing.

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
2.NBT.9

Math // Addition & Subtraction

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