Start. Draw a number line. A fictional score starts at −3, rises by 8 and then falls by 4. Model −3 + 8 − 4 = 1 with two movements. Compare “subtract 4” with “add −4”. Then show three changes of −2: 3 × (−2) = −6. Explain multiplication signs with a pattern: 2 × (−3) = −6, 1 × (−3) = −3, 0 × (−3) = 0, so continuing the constant change gives (−1) × (−3) = 3. Division reverses multiplication: −12 ÷ 3 = −4 and −12 ÷ (−3) = 4. These are numerical models, not directions to incur a debt or use equipment.
Route 26A · A log of changes
A fictional level indicator starts at −5, changes by +9, then −7, then +2. Mark each stage on a number line and find the final level. Reorder the signed additions and check whether the final total changes. Explain why the intermediate levels can differ even when the final sum does not.
Teacher next move: −5 → 4 → −3 → −1. Signed addends −5 + 9 − 7 + 2 total −1; ordering addition changes the path but not its sum. Do not extend this claim to arbitrary reordering of subtraction or division expressions.
Route 26B · Use a pattern to justify a sign
Complete a table for n × (−4) at n = 3, 2, 1, 0, −1, −2. Explain the change from one row to the next. Use multiplication to check −20 ÷ (−4) and 20 ÷ (−4). Then compare −2 × (3 − 7) with −2 × 3 − 7 and explain the role of the brackets.
Teacher next move: −12, −8, −4, 0, 4, 8; quotients 5 and −5. Bracketed expression 8; unbracketed −13. A recalled sign rule is useful only when it is applied to the intended expression.
Route 26C · Make a target, then diagnose
Use the integers −6, 3 and 2 once each to make 0 using addition, multiplication, division and optional brackets. Show one valid expression. Diagnose this false claim: “−6 ÷ 3 + 2 is −6 ÷ 5, so it is not zero.” State the order of operations and check your expression by substitution.
Teacher next move: −6 + 3 × 2 = 0 or −6 ÷ 3 + 2 = 0. The false combination changes the expression; −6 ÷ 3 + 2 = −2 + 2 = 0. Accept another valid use of all three integers with operations named. A learner's method must match the written expression.
Close: explain one sign or bracket choice using a model, inverse operation or pattern.
Check the operation you have not yet seen
After the chosen Day 26 route, note which signs and operations the learner actually explained. Route A mainly shows signed addition; it cannot establish all four operations. Choose a short ordinary follow-up if another operation is uncertain:
- Signed addition/subtraction: calculate 8 − 12 + 5 and show the changes on a number line. Key: 1.
- Multiplication: explain why (−4) × (−3) = 12 using a sign pattern or reversal of a negative change. A named rule alone is a starting point for a question about why it works.
- Division: calculate −28 ÷ 7 and −28 ÷ (−7), then check each with multiplication. Keys: −4 and 4; 7 × (−4) = −28 and (−7) × 4 = −28.
- Order and brackets: compare 6 − 2 × (−3) with (6 − 2) × (−3). Keys: 12 and −12. Ask what the brackets keep together.
Keep the chosen response and the support used. Offer another teaching example where needed. These are ordinary follow-ups with visible staff answers, not fresh checks or a requirement to complete every task. E and F do not establish full integer mastery.