Mathematics · Grade 7 · Number Sense and Proportional Thinking · Term 1

Multiplying and Dividing Integers

Developing rules for multiplying and dividing integers and applying them to solve contextual problems.

• Ontario Curriculum Expectations
• 7.NS.A.2

About This Topic

Multiplying and dividing integers requires students to identify patterns in signs: the product or quotient is positive when there are zero or two negatives, negative with one or three. They predict outcomes, explain rules for different sign combinations, and solve contextual problems like calculating debt increases or elevation gains below sea level. These skills align with Ontario Grade 7 Number Sense standards, emphasizing operational fluency with integers.

This topic supports the unit on Number Sense and Proportional Thinking by strengthening pattern recognition and application to real scenarios, such as financial losses or temperature shifts. Students build confidence in handling negative values, a foundation for algebra and ratios. Collaborative exploration reveals why rules hold across operations, fostering deeper number sense.

Active learning benefits this topic greatly because students model operations with manipulatives like two-color counters or number lines. They physically pair positives and negatives to discover patterns firsthand. Group discussions then solidify explanations, making abstract sign rules concrete and memorable while encouraging peer teaching.

Key Questions

1. Predict the sign of a product or quotient involving negative numbers.
2. Explain the patterns observed when multiplying or dividing integers with different signs.
3. Analyze real-world scenarios where multiplying or dividing negative numbers is necessary.

Learning Objectives

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

Before You Start

Multiplying and Dividing Whole Numbers

Why: Students must have a solid understanding of the basic operations of multiplication and division with positive numbers before introducing negative integers.

Introduction to Integers

Why: Students need to be familiar with the concept of negative numbers and how to represent them on a number line or in context before performing operations with them.

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.

Watch Out for These Misconceptions

Common Misconception

The product of two negative numbers is negative.

What to Teach Instead

Students often extend addition rules incorrectly. Using two-color counters, they pair negatives and see zero pairs yield positive results, matching the pattern. Peer discussions compare models to reveal the true rule, building confidence through shared correction.

Common Misconception

Division sign rules differ from multiplication.

What to Teach Instead

Many think division ignores sign patterns. Modeling division as inverse multiplication with chips shows identical rules. Group explorations confirm consistency, helping students unify operations via hands-on evidence.

Common Misconception

Negative numbers lack real-world meaning in multiplication.

What to Teach Instead

Students dismiss negatives as abstract. Contextual stations with debt or losses make signs relevant. Collaborative solving connects rules to scenarios, shifting views through discussion and application.

Active Learning Ideas

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.

35 min·Pairs

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.

25 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.

45 min·Small Groups

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.

30 min·Pairs

Real-World Connections

• Accountants use integer multiplication and division to track financial losses or gains over time. For example, calculating a consistent weekly loss of $50 over 4 weeks results in a total loss of -$200.
• Meteorologists use integer operations to describe temperature changes. If the temperature drops by 3 degrees Celsius each day for 5 days, the total change is -15 degrees Celsius, which can be calculated as 5 x -3.
• Engineers working on underwater projects might use integer division to calculate how many days it will take to descend to a certain depth. If a submarine needs to descend 300 meters and descends 50 meters per day, it will take 6 days (300 / 50 = 6), or if it ascends 50 meters per day, it will take -6 days to reach the surface from that depth.

Assessment Ideas

Exit Ticket

Provide students with 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

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

Discussion Prompt

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.

Frequently Asked Questions

How do students discover rules for multiplying integers?

Start with pattern tables: compute 2x3, -2x3, 2x-3, -2x-3, then extend to more negatives. Students chart signs, spot even/odd patterns, and test predictions. This inductive approach, paired with manipulatives, leads to self-derived rules before formal teaching, enhancing retention over direct instruction.

What are common errors in dividing integers?

Errors include ignoring signs or treating division like subtraction. Address by linking to multiplication: if (-6) / 2 = -3 checks as (-3)x2=-6. Practice with number lines reinforces inverse relationships. Regular low-stakes quizzes with peer review catch issues early.

What real-world examples illustrate integer operations?

Use debt: owing $5 more daily is x(-1) on balance. Temperatures: -4C warming 3 degrees is +3. Elevations: diving 20m from -10m sea level. Financial gains/losses or coordinate changes provide contexts. Students solve mixed-sign problems to see practical value.

How can active learning strategies improve integer multiplication and division?

Active methods like chip models and relays engage kinesthetic learners, making sign patterns visible. Pairs discuss predictions, correcting errors collaboratively. Stations apply rules to contexts, boosting relevance. These approaches increase engagement, reduce anxiety with negatives, and improve fluency through repeated, varied practice over passive lecture.