Multiplying and Dividing Integers

Active learning helps students recognize the sign patterns in integer multiplication and division through concrete models. These hands-on activities let students test rules, see patterns emerge, and correct mistakes in real time, which builds lasting understanding better than passive rule recall. Moving from counters to number lines to real-world contexts lets students connect abstract ideas to tangible experiences.

• Grade 7
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Calculate the product of two integers, including those with different signs, by applying established multiplication rules.
2. Determine the quotient of two integers, including those with different signs, by applying established division rules.
3. Explain the pattern of signs in the product or quotient of integers based on the signs of the factors or dividend and divisor.
4. Analyze real-world scenarios involving financial transactions or temperature changes to solve problems requiring multiplication or division of integers.

Problem-Based Learning

⏱ 35 min·Pairs

Two-Color Counters: Sign Patterns

Give pairs red and yellow counters for negatives and positives. Students model products like (-3)x2 by grouping, then remove zero pairs for two negatives. Record signs in a table and test divisions by splitting groups. Discuss patterns as a class.

Prepare & details

Predict the sign of a product or quotient involving negative numbers.

Facilitation Tip: During Two-Color Counters, circulate and ask each group to explain how their zero pairs connect to the final sign.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 25 min·Small Groups

Relay Race: Predict and Justify

Divide into small groups for a relay. Each student predicts the sign of a product or quotient on a card, runs to board to justify with a quick sketch, then tags next teammate. Review all as whole class.

Prepare & details

Explain the patterns observed when multiplying or dividing integers with different signs.

Facilitation Tip: For Relay Race, pause after each round to have the next runner summarize the rule the team just used.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 45 min·Small Groups

Context Stations: Real-World Problems

Set up four stations with scenarios: bank debt, stock changes, temperature drops, sea depths. Small groups solve two problems per station using rules, rotate, and share solutions. Emphasize sign predictions.

Prepare & details

Analyze real-world scenarios where multiplying or dividing negative numbers is necessary.

Facilitation Tip: In Context Stations, listen for students who connect the math to the scenario when justifying their answers.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 30 min·Pairs

Number Line Walks: Division Models

Pairs use large floor number lines. Students walk multiples for products, then reverse for divisions with negatives. Note sign changes and record observations. Pairs present one model to class.

Prepare & details

Predict the sign of a product or quotient involving negative numbers.

Facilitation Tip: With Number Line Walks, check that students mark each step clearly and label both direction and value.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Teaching This Topic

Start with Two-Color Counters to ground the rules in physical evidence, then move to contexts that show why the rules matter. Avoid rushing to mnemonics; instead, build the rules from patterns students observe. Research shows that students who generate the rules through models retain them longer than those who memorize first. Emphasize peer explanation so students practice articulating the logic behind the signs.

What to Expect

Students will explain the sign rules for multiplying and dividing integers with clear examples. They will solve problems accurately and justify their answers using models or language from the activities. Small-group discussions should show students using correct vocabulary and reasoning to resolve disagreements.

Watch Out for These Misconceptions

Common Misconception

During Two-Color Counters, watch for students who collect negative chips and assume the product must be negative. Redirect them by asking, 'How many zero pairs did you form?' and 'What remains after removing the zero pairs?'

What to Teach Instead

During Two-Color Counters, if students still claim two negatives make a negative, ask them to model (-2) x (-3) with chips and count the positive tiles left after pairing negatives, then have them compare their model to their rule.

Common Misconception

During Relay Race, listen for teams that treat division sign rules as different from multiplication. Pause the race and ask them to model the division problem with chips to see the shared pattern.

What to Teach Instead

During Relay Race, if a team says the sign rules for division are different, challenge them to model -12 / 3 using the same chip method as multiplication, then compare the steps aloud.

Common Misconception

During Context Stations, notice students who write correct answers but do not connect them to the scenario when explaining. Ask them to read their answer back to the scenario and explain how the sign matches the context.

What to Teach Instead

During Context Stations, if a student says '4 x -5 is -20' without linking it to debt, prompt them to restate the problem as 'owing $4 five times' and explain how the sign fits the situation.

Assessment Ideas

Exit Ticket

After Two-Color Counters and Relay Race, provide three problems: 1) 8 x (-3), 2) -45 / 5, and 3) a scenario: 'A diver descends 20 meters in 4 equal stages. What is the change in depth per stage?' Ask students to calculate the answer and write one sentence explaining the sign rule they used for each problem.

Quick Check

After Context Stations, write on the board: 'When multiplying or dividing integers, what determines the sign of the answer?' Have students write the two conditions for a positive answer and the two conditions for a negative answer on a small whiteboard or paper.

Discussion Prompt

During Context Stations, pose the question: 'Imagine you have $100 and you spend $10 each day. How many days until you have $0? Now, imagine you owe $100 and you earn $10 each day. How many days until you owe $0?' Guide students to see the connection between division by a negative number and the context of owing money by recording their reasoning on chart paper.

Extensions & Scaffolding

• Challenge: Ask students to create a word problem involving both multiplication and division of integers, then trade with a partner to solve and justify the signs.
• Scaffolding: Provide a half-completed chart with some sign combinations filled in and ask students to finish the missing products or quotients.
• Deeper exploration: Have students design a mini-lesson for a younger student using one of the activities to teach the sign rules.

Key Vocabulary

Integer	A whole number (not a fractional number) that can be positive, negative, or zero. Examples include -3, 0, and 5.
Product	The result of multiplying two or more numbers. For example, the product of 3 and 4 is 12.
Quotient	The result of dividing one number by another. For example, the quotient of 12 divided by 4 is 3.
Sign Rule	A rule that determines whether the product or quotient of two integers will be positive or negative, based on the signs of the original numbers.