Mathematics · 7th Grade

Active learning ideas

Dividing Integers

Active learning works for dividing integers because students must repeatedly apply sign rules while manipulating concrete quantities. Moving beyond memorized steps to constructing and deconstructing expressions helps them internalize why the sign of a quotient behaves the way it does.

• Common Core State Standards
• CCSS.Math.Content.7.NS.A.2b

• 15–25 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Stations Rotation
• 25 min · Small Groups

Fact Family Investigation: Integer Triangles

Groups receive cards with three integers (e.g., -5, 4, -20) and write all four related multiplication and division equations. They must verify each equation and mark the sign patterns they observe. Groups then create a "family portrait" poster showing a fact family of their choice with a real-world story connecting the numbers.

Justify why the rules for dividing integers are similar to the rules for multiplying integers.

Facilitation Tip

During Fact Family Investigation: Integer Triangles, circulate and ask each pair to verbalize how the sign of the product predicts the sign of each quotient in the triangle.

What to look for

Provide students with three problems: 1) 24 / -6, 2) -35 / -7, 3) -50 / 10. Ask them to write the answer and briefly explain the sign rule used for each problem.

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

Activity 02

• Think-Pair-Share
• 15 min · Pairs

Think-Pair-Share: Division Scenarios

Present a real-world scenario: "A submarine descends 120 feet over 8 minutes at a constant rate. What is the change in depth per minute?" Students set up the division expression, justify the sign, solve, and interpret. Pairs compare their expressions and interpretations before the class discusses.

Analyze the relationship between multiplication and division of integers.

Facilitation Tip

During Think-Pair-Share: Division Scenarios, listen for correct sign explanations and gently correct any scenario that implies division by a negative always yields a negative result.

What to look for

Present students with the equation -4 * 5 = -20. Ask them to write two division equations that belong to the same fact family and explain how they relate to the original multiplication equation.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 03

• Stations Rotation
• 20 min · Small Groups

Collaborative Sort: Division Expression Sign

Groups receive 16 division expressions. Without computing exact values, they sort each into one of three bins: positive quotient, negative quotient, or zero. After sorting, they compute to check their predictions and record their error rate. The class discusses which sign cases caused the most errors.

Construct a scenario where dividing negative integers provides a meaningful solution.

Facilitation Tip

During Collaborative Sort: Division Expression Sign, challenge students who finish early to create an extra card that breaks the current rule.

What to look for

Pose the scenario: 'A group of 4 friends owes a total of $60 for a shared item. How much does each friend owe?' Ask students to solve this using integer division and explain why the answer is negative. Then, ask them to create a different scenario involving negative integers where division provides a meaningful solution.

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

A few notes on teaching this unit

Start with fact families so students see division as the inverse of multiplication. Use number lines or counters to model division with negatives so the sign change is tied to quantity rather than a rule. Avoid teaching “two negatives make a positive” as a separate mantra; instead, connect it to multiplication’s sign rules to prevent later confusion.

Students will confidently determine the sign of any integer quotient and justify it with a related multiplication fact or fact family. They will also explain the difference between sign changes and magnitude changes in the context of integer division.

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Watch Out for These Misconceptions

• During Fact Family Investigation: Integer Triangles, watch for students who assume the quotient’s sign is always the same as the dividend’s sign.

  Ask students to cover the product in the triangle and look only at the dividend and divisor. Have them predict the quotient’s sign before uncovering the multiplication fact that confirms it.

• During Collaborative Sort: Division Expression Sign, watch for students who believe the absolute value of the quotient changes when signs change.

  Have students calculate the quotients side by side on the same card (e.g., 20/-4 and -20/-4) and circle the difference in sign without altering the magnitude.

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Methods used in this brief

• Stations Rotation
• Think-Pair-Share