Multiplying Integers

Active learning helps students grasp multiplying integers because the topic relies on visual patterns and real-world reasoning rather than abstract memorization. When students build tables, debate scenarios, and move around the room, they connect the sign rules to concrete evidence they can see and test.

• 7th Grade
• Mathematics
• 3 activities
• 15 min–30 min

Learning Objectives

1. Explain the mathematical reasoning that leads to the product of two negative integers being positive.
2. Calculate the product of integers involving positive and negative numbers using established rules.
3. Analyze patterns in multiplication tables to predict the sign of products with multiple negative factors.
4. Compare the sign of a product when the number of negative factors is odd versus even.

Carousel Brainstorm

⏱ 30 min·Small Groups

Pattern Investigation: Building the Integer Multiplication Table

Small groups fill in a multiplication table that includes rows and columns for -3 through 3. They first complete the positive portion using known facts, then use the decreasing pattern in each row to extend into negatives. Groups record what they notice about the signs and share a rule they derived from the pattern.

Prepare & details

Why does multiplying two negative numbers result in a positive product?

Facilitation Tip: During Pattern Investigation, have students work in pairs to complete the table row by row, discussing each new product before moving forward to reinforce the pattern.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: The Negative Times Negative Debate

Students individually write an explanation for why (-3) x (-4) = 12 using the row-pattern argument or a real-world analogy. They share with a partner, combine the clearest reasoning, and then a few pairs present to the class. The class votes on the most convincing explanation.

Prepare & details

Explain the pattern of signs when multiplying multiple integers.

Facilitation Tip: For Think-Pair-Share, assign roles: one student explains the real-world scenario, another connects it to the math, and a third records the group’s conclusion.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

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

Gallery Walk

⏱ 25 min·Small Groups

Gallery Walk: Sign Rule Scenarios

Post six posters around the room, each showing a real-world multiplication scenario (e.g., losing per day for 4 days; reversing a loss of per day). Groups rotate, write the multiplication expression on a sticky note, place it on the poster, and check if the sign matches the context before moving on.

Prepare & details

Predict the sign of a product involving an odd or even number of negative factors.

Facilitation Tip: During the Gallery Walk, post only the completed sign rule scenarios around the room, and require students to add sticky notes with alternative examples or questions for each poster.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Experienced teachers approach this topic by starting with repeated addition and debts to ground the concept in familiar contexts. Avoid rushing to the rule—let students discover the patterns themselves through structured tables and debates. Research shows that students retain the concept longer when they construct the rule through exploration rather than being told it outright.

What to Expect

Successful learning looks like students explaining why two negatives make a positive using patterns or real-life examples, not just applying rules. They should confidently predict signs for expressions with multiple factors and justify their reasoning during discussions.

Watch Out for These Misconceptions

Common Misconception

During Pattern Investigation, watch for students who assume negative times negative should be negative because 'two negatives are bad.'

What to Teach Instead

Redirect their attention to the completed table rows, pointing out how each product increases as the second factor decreases, showing that the pattern leads to a positive product.

Common Misconception

During Think-Pair-Share, watch for students who struggle to predict the sign of products involving three or more negative factors.

What to Teach Instead

Have groups sort their expressions by the number of negative factors before computing, emphasizing that an even number of negatives flips the sign back to positive.

Assessment Ideas

Exit Ticket

After Pattern Investigation, provide students with three problems: 1) 5 x (-3), 2) (-7) x (-4), 3) (-2) x 3 x (-5). Ask them to calculate the product for each and write one sentence explaining the sign rule used for problem #2.

Discussion Prompt

After Think-Pair-Share, pose the question: 'Imagine you owe your friend $5. If you do this 3 times, your debt increases. But if you remove 3 of those $5 debts, what happens to your financial situation?' Guide students to connect this to why negative times negative is positive.

Quick Check

During Gallery Walk, present a partially completed multiplication table with rows and columns for positive and negative integers. Ask students to fill in the missing products, focusing on the pattern of signs. Circulate to observe their application of the rules.

Extensions & Scaffolding

• Challenge: Provide expressions with four or more negative factors (e.g., (-2) x (-3) x (-1) x (-4)) and ask students to predict the sign before calculating the product.
• Scaffolding: Give students a partially completed multiplication table with only positive integers filled in, and ask them to extend it to include negative integers step by step.
• Deeper exploration: Ask students to create their own real-world scenario for a negative times a negative situation and present it to the class.

Key Vocabulary

Integer	A whole number, including positive numbers, negative numbers, and zero.
Product	The result of multiplying two or more numbers together.
Factor	A number that divides into another number exactly. In multiplication, the numbers being multiplied are factors.
Sign Rule	A mathematical convention that determines whether the result of an operation (like multiplication) is positive or negative.