13. Products and Quotients with Multiple Negatives

13. Products and Quotients with Multiple Negatives

Learning Targets

• Use negative-factor parity.
• Evaluate chained products and quotients.
• Handle zero factors.

Why This Matters

Long products become fast when sign is determined once before arithmetic.

Core Lesson

Count negative factors. An even number gives a positive result; an odd number gives a negative result. Each pair of negative factors contributes a positive product.

For \((-2)(-3)(-4)\), three negatives make the result negative and the magnitude is \(24\), giving \(-24\). For multiplication and division at equal priority, preserve left-to-right order. Any zero factor makes an entire product zero.

Detailed Concept Development

Each negative factor reverses the product's direction. Two reversals cancel, so negative factors can be paired. This creates the parity rule: even count gives positive; odd count gives negative.

Count only factors, not subtraction symbols that belong to later addition. In a multiplication/division chain, determine sign from all nonzero signed factors, then compute magnitude left to right where division order matters.

Pause here and explain the core relationship in your own words. Name what the sign means, which quantity is the reference, and why the operation preserves the situation being modeled. If the explanation relies only on a memorized phrase, connect it to a number line, grouping model, inverse operation, or estimate.

Explanatory SVG

Read the diagram from left to right. Cover the final panel and reproduce the middle transformation from memory. Then replace the displayed values with your own values while keeping the same mathematical relationship.

Five Fully Worked Examples

The examples move from direct computation to reasoning and verification. Before reading a solution, predict the sign and approximate size, write a plan, and attempt the first step independently.

Worked Example 1: Foundation

Problem: Find the sign of \((-2)(5)(-3)(-4)\).

1. Identify the skill. This example uses the session target: Use negative-factor parity.
2. Predict before calculating. Use direction, magnitude, units, or a benchmark to decide what kind of answer is reasonable.
3. Choose the method. Apply the relationship developed in the core lesson and keep every sign visible.
4. Calculate step by step. Three negative factors make the product negative.
5. Interpret the result. State what the signed value means rather than reporting an unexplained number.
6. Verify. Mark each negative factor, count, then calculate magnitude.

Worked Example 2: Foundation

Problem: Evaluate \((-48)\div(-6)\div(-2)\).

1. Identify the skill. This example uses the session target: Use negative-factor parity.
2. Predict before calculating. Use direction, magnitude, units, or a benchmark to decide what kind of answer is reasonable.
3. Choose the method. Apply the relationship developed in the core lesson and keep every sign visible.
4. Calculate step by step. Work left to right: \(8\div(-2)=-4\).
5. Interpret the result. State what the signed value means rather than reporting an unexplained number.
6. Verify. Mark each negative factor, count, then calculate magnitude.

Worked Example 3: Detailed Reasoning

Problem: Evaluate \((-2)(-5)(3)(-4)\).

1. Identify what is given and asked. There are three negative factors.
2. Select a plan. Odd negative count means negative; multiply magnitudes.
3. Show the mathematical work. \(2\cdot5\cdot3\cdot4=120\).
4. State the answer clearly. \(-120\)
5. Interpret the sign and units. Explain what positive, negative, or zero communicates in this problem.
6. Verify independently. Removing any pair of negatives leaves one negative factor.

Worked Example 4: Detailed Reasoning

Problem: Evaluate \((-72)\div(-3)\div4\).

1. Identify what is given and asked. Two negatives occur in the first quotient.
2. Select a plan. Work left to right.
3. Show the mathematical work. \((-72)\div(-3)=24\), then \(24\div4=6\).
4. State the answer clearly. \(6\)
5. Interpret the sign and units. Explain what positive, negative, or zero communicates in this problem.
6. Verify independently. \(6\cdot4\cdot(-3)=-72\).

Worked Example 5: Detailed Reasoning

Problem: Evaluate \((-7)(0)(-9)(5)\).

1. Identify what is given and asked. A zero factor appears.
2. Select a plan. Apply zero-product rule before other arithmetic.
3. Show the mathematical work. The entire product equals \(0\).
4. State the answer clearly. \(0\)
5. Interpret the sign and units. Explain what positive, negative, or zero communicates in this problem.
6. Verify independently. No sign count can override a zero factor.

Real-World Application

Scenario: A game applies three direction reversals to a movement of 8 units.

1. Define the reference and quantities. Each reversal acts like multiplication by \(-1\).
2. Translate words into mathematics. Model as \(8(-1)(-1)(-1)\).
3. Calculate in a visible sequence. Three negatives are odd, so the final movement is \(-8\).
4. Answer in context. The object moves 8 units in the opposite direction.
5. Check reasonableness. Two reversals cancel; the third creates one remaining reversal.
6. Reflect. Change one value in the scenario, predict how the result would change, and then recalculate to test the prediction.

Common Trap

Stopping the sign count after the first pair or reordering a division chain.

GED® Success Habit

Mark each negative factor, count, then calculate magnitude.

Session Check

Mastery Note

Complete a teach-back: explain one worked example without looking, solve a parallel problem with different values, and identify one tempting wrong answer. Verify the answer's sign, approximate size, and meaning. Record the first incorrect step—not only the final answer—whenever an error occurs.