Operations with Integers: Multiplication and Division

Active learning works for multiplication and division of integers because these operations rely on concrete understanding of grouping, sharing, and patterns. Students need to move beyond rote rules and see the logic behind signs and remainders, which hands-on tasks make visible.

• 4th Class
• Mastering Mathematical Thinking: 4th Class
• 3 activities
• 25 min–45 min

Learning Objectives

1. Calculate the product of two integers, applying the rules for signs.
2. Calculate the quotient of two integers, applying the rules for signs.
3. Explain the rule for multiplying a negative number by a negative number using a pattern.
4. Predict the sign of the result when multiplying or dividing multiple integers.
5. Analyze the sign of a product or quotient based on the number of negative factors or divisors.

Role Play

⏱ 35 min·Small Groups

Role Play: The Party Planner

Students are given a set number of 'guests' (counters) and 'tables' (paper plates). They must divide the guests and decide what to do with the remainders. Does the 'leftover' guest need a whole new table, or do they just miss out?

Prepare & details

Explain the rules for multiplying and dividing integers with different signs.

Facilitation Tip: During The Party Planner, circulate with counters and explicitly ask students to show how many more groups could be made when they have a remainder.

Setup: Open space or rearranged desks for scenario staging

Materials: Character cards with backstory and goals, Scenario briefing sheet

• Apply
• Analyze
• Evaluate
• Social Awareness
• Self-Awareness

Inquiry Circle

⏱ 25 min·Pairs

Inquiry Circle: The Remainder Race

Give pairs a set of division problems. They must sort them into three categories: 'Ignore the remainder,' 'Round up the answer,' and 'The remainder is the answer.' They must justify their choices to another pair.

Prepare & details

Predict the sign of the product or quotient of multiple integers.

Facilitation Tip: In The Remainder Race, have pairs verbalize each step of their sharing or grouping process aloud to catch misconceptions in real time.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Division Strategies

Set up stations for different methods: one for repeated subtraction on a number line, one for 'chunking' using multiplication facts, and one for physical grouping with cubes. Students rotate to find which method they find most reliable.

Prepare & details

Construct a pattern that demonstrates why a negative number multiplied by a negative number results in a positive number.

Facilitation Tip: Set a timer for Station Rotation so students rotate before losing focus, and post expected strategies at each station for quick reference.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach integer multiplication by starting with patterns on a number line or grid, then connect to real-world debts and temperature changes. For division, use physical objects to contrast sharing (equal distribution) with grouping (how many equal sets fit). Encourage students to verbalize the meaning of each number in the problem rather than just computing.

What to Expect

Students will confidently explain why negative times negative makes a positive, and interpret remainders in context. They will use sharing and grouping language to describe division, and connect division to multiplication as inverse operations.

Watch Out for These Misconceptions

Common Misconception

During The Party Planner, watch for students who ignore the remainder or treat it as part of the quotient without checking if another group can be formed.

What to Teach Instead

Have students physically place counters into cups or plates labeled with the divisor, and prompt with 'Can you make one more equal group with what’s left? What does that extra counter mean in your story?'

Common Misconception

During The Remainder Race, watch for students who treat the remainder as a separate answer without connecting it to the context of the problem.

What to Teach Instead

Ask each pair to explain their final result using the race scenario: 'Does the remainder mean another lap? A partial lap? How do you decide?'

Assessment Ideas

Quick Check

After the Station Rotation, present students with three multiplication problems: 5 x -3, -7 x -2, and -4 x 6. Ask them to write the answer and circle the sign they applied. Review answers as a class, focusing on the sign rule used for each.

Exit Ticket

During The Party Planner, give each student a card with a division problem involving integers, such as -24 ÷ 8 or -36 ÷ -6. Ask them to solve the problem and write one sentence explaining how they determined the sign of their answer.

Discussion Prompt

During the number line or grid activity, pose the question: 'If we know that 3 x 4 = 12, and we also know that 3 x 0 = 0, how can we use this pattern to figure out what -3 x -4 should be?' Facilitate a discussion where students explore the pattern of decreasing the first factor by 1 and observe the corresponding change in the product.

Extensions & Scaffolding

• Challenge: Provide a mixed set of integer division problems including negative divisors and ask students to create their own real-world scenarios that match the answers.
• Scaffolding: Offer a template with blanks for the parts of a division problem (dividend, divisor, quotient, remainder) and ask students to fill in words that describe the context.
• Deeper exploration: Have students research and present how division with remainders is handled in different cultures or historical contexts.

Key Vocabulary

Integer	A whole 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 3 is 4.
Sign Rules	The established conventions for determining the sign (positive or negative) of a product or quotient when working with integers.