Math // Addition & Subtraction

Two-Step Equations

Solve two-step word problems using the four operations; represent problems using equations with a letter standing for the unknown quantity

Ages
8 to 9
Type
Method
Builds on
3 topics
Leads to
0 topics
Standards
1 code

Quick check

Can the learner write an equation like 'n + 25 = 60' to represent a word problem, and then solve it to find what n equals?

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 a two-step problem that combines addition/subtraction with multiplication/division
  • Write an equation using a letter for the unknown (e.g. 3 × n + 5 = 26)
  • Assess the reasonableness of the answer using estimation and mental computation

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

3 topics

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Two-Step Equations

Ages 8 to 9 // Method

Leads to

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Nothing in the catalogue builds on this topic yet. Browse Addition & Subtraction

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

Learners bridge from a two-step word problem to algebraic notation: solve problems that mix addition/subtraction with multiplication/division, write the problem as an equation using a letter for the unknown (e.g. 3 × n + 5 = 26), solve by inverse operations in the correct order, and check by substitution/estimation. Two conceptual hurdles sit alongside the arithmetic: the letter must be understood as a NUMBER we do not yet know (not a label for an object), and '=' must be read RELATIONALLY ('is the same value as', so both sides stay balanced while undoing) rather than as 'the answer comes next'.

Common misconceptions

  • Letter-as-object / label — 'n means pencils' rather than a number (Kuchemann's 'letter as object')
  • Operator view of '=' — 'the answer comes next' — so the equation cannot be run backwards
  • Ignoring order of operations — adds the 5 before multiplying, or reads left to right
  • Undoing the two steps in the wrong order (dividing before subtracting)
  • Accepting an answer without substituting back to check

Useful representations

  • Bar model with an unknown box
  • Balance scale (both sides equal)
  • The equation itself (3n + 5 = 26)
  • Empty number line for the inverse steps

Teaching practices

  • Move bar model -> equation: the model gives the structure, the letter names the unknown box
  • Insist the letter stands for a NUMBER (how many in a pack), never for the object or its label
  • Teach solving as undoing the build in reverse order (last operation undone first), anchored on a balance
  • Read '=' as 'is equal to / the same as' and keep both sides balanced when undoing
  • Substitute the answer back and estimate to judge reasonableness

Activities to try

  1. 1

    Balance-scale equation bench

    Why it works: The letter box held a NUMBER (7) all along, and it could only be found by doing the SAME thing to both sides, so '=' means genuinely balanced. The +5 is undone before the ×3 because the +5 is the last operation built — undo the last operation first — and removing it from both pans keeps them level.

    1. Model the story 'Maya has 3 packs of pencils plus 5 loose pencils, 26 pencils in total' by placing 3 identical unknown boxes (each labelled n) and 5 single cubes on the left pan, against 26 cubes on the right pan.
    2. To find what is inside one box, keep the scale level: remove 5 cubes from BOTH pans, leaving the 3 boxes on the left against 21 cubes on the right.
    3. Share the remaining 21 equally among the 3 boxes, and open a box to reveal 7 inside each.
    4. Write the matching equation 3 × n + 5 = 26 next to the balance, and state in words what n stands for.
    5. Substitute n = 7 back: check that 3 × 7 + 5 = 26.
  2. 2

    From bar model to equation, then undo in reverse

    Why it works: The structure of the problem (equal groups plus an extra part) determines the equation and the ORDER of undoing. Solving reverses the build: the last operation done is undone first.

    1. Draw a bar model for the story: three equal boxes (each n) joined to a small part of 5, with the whole bar equal to 26.
    2. Use the bar structure to write the equation 3 × n + 5 = 26, naming the unknown box with the letter.
    3. List the undo steps in order on an empty number line or in writing: 26 − 5 = 21, then 21 ÷ 3 = 7.
    4. Say aloud why the −5 comes before the ÷3 (the +5 was the last thing built, so it is undone first).
    5. Check 3 × 7 + 5 = 26 and judge whether 7 is a sensible pack size.
  3. 3

    Match the story to the right equation, then solve in order

    Why it works: Translating a story into the correct equation forces attention to operation order, and verifying by substitution confirms the solution rather than trusting the arithmetic blindly.

    1. Read a short scenario and choose the equation that correctly represents it, rejecting distractors such as 3 + n × 5 = 26 that misplace the operations.
    2. Order the solving steps correctly (subtract first, then divide), filling in each result.
    3. Substitute the found value back into the chosen equation to confirm it holds.
    4. Explain in one sentence what the letter stands for.

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
3.OA.8

Math // Addition & Subtraction

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