Integer Sign Rules

Overview

Lesson 3 of 20 focuses on multiplication and division with integers, building students’ confidence with the sign rules and efficient strategies for solving calculations. It follows prior work on adding and subtracting integers by extending the reasoning to product and quotient contexts.

Learning intentions

Students will:

• apply sign rules to multiply and divide integers
• use patterns (and reasoning with positive/negative factors) to predict whether results are positive or negative
• solve multiplication and division of integers efficiently using known facts and structure (e.g. commutativity for regrouping)
• check answers using inverse operations where appropriate (e.g. undoing multiplication with division)

Success criteria

Students can:

• correctly determine the sign of an integer product/quotient before calculating
• calculate integer products and quotients accurately for a range of examples
• explain, using sentence starters, why the sign is positive or negative
• verify a solution by substituting into the inverse operation (when feasible)

Curriculum links

• Mathematics (Year 8, Number): multiplication and division with integers using efficient strategies and explaining sign effects
• Mathematics (Year 8, Number): patterns to support rules for multiplication and division of integers

Lesson structure (30 minutes)

1. 0–4 min · Warm-up recall. Teacher writes 2 questions on the board: “(-3) × (+4) has what sign?” and “(+18) ÷ (-6) has what sign?” and asks for quick mental answers. Students respond using think time, then hold up fingers (1 = positive, 2 = negative) and share their reasoning briefly.

2. 4–10 min · Direct teach: predict sign first. Teacher models a short “sign table” and asks students to focus only on the sign:

• (+) × (+) = +, (+) × (-) = -, (-) × (+) = -, (-) × (-) = + Then repeats for division: same rule for signs applies to quotients. Students complete a 2-minute guided set: for each expression, write only “+” or “-”, then add the full calculation for one example per row.

3. 10–18 min · Pattern practice (teacher-guided). Teacher draws a number line context (temperature or altitude) and links it to multiplication/division meaning: “repeated change” and “how many groups.” Example set:

• (-2) × 3, (-2) × (-3), 12 ÷ (-4), (-15) ÷ 3 Teacher emphasises that they can use the number of negative factors (for multiplication) and “same/opposite signs” reasoning (for division). Students work in pairs (or individually if required): solve the four expressions, then write one sentence explanation for two of them using a starter such as “Because there are … negative(s), the result is …”.

4. 18–26 min · Independent check + digital efficiency. Teacher provides a short worksheet (6 questions) and a digital option: students may use a calculator to confirm, but must show sign reasoning first (calculator used only after attempting). Students attempt all 6:

• (-7) × 5
• 24 ÷ (-6)
• (-9) × (-2)
• (-18) ÷ 3
• 6 × (-8)
• (-36) ÷ (-9) Students record their predicted sign, then the computed answer, then one verification step for at least one question (e.g. if x = (-18) ÷ 3, check 3 × x = -18).

5. 26–30 min · Exit ticket (quick formative). Teacher collects answers from two students’ mini whiteboards and gives an exit ticket (2 items):

• “Predict sign and calculate: (-4) × (-6) =?”
• “Predict sign and calculate: 20 ÷ (-5) =?” Students complete independently and submit.

Resources

• Board/markers or interactive display
• Integer sign “anchor chart” (4 sign combinations for multiplication/division)
• Short worksheet (6 practice questions) + exit ticket (2 questions)
• Mini whiteboards and pens (or paper for single-student class)
• Sentence starters strip (e.g. “The sign is … because …”)
• Calculator (optional for checking only)
• Number line or simple context images (temperature/altitude or “groups” diagram)

Assessment

• Formative checks during the warm-up using fingers for sign prediction
• Teacher listens for accurate explanations during the guided pattern practice
• Exit ticket assesses both sign prediction and correct final calculation for multiplication and division

Differentiation

• Support: provide a partially completed sign table and sentence starters; allow use of a number line or “groups” diagram for one example before removing supports
• Support for sign errors: prompt “Are the signs the same or different?” before calculating
• Extension: for students who are secure, ask them to justify using inverse reasoning (e.g. if (-18) ÷ 3 = x, show 3x = -18)
• EAL/SEN considerations: keep instructions short, model one example with full think-aloud, and use consistent language: “same signs → positive product/quotient; different signs → negative”