8. Adding Integers with Different Signs

8. Adding Integers with Different Signs

Learning Targets

• Add unlike-signed values.
• Use magnitude to determine sign.
• Recognize cancellation.

Why This Matters

Unlike-sign addition appears constantly in net-change problems.

Core Lesson

Different signs represent opposite directions. Subtract the smaller absolute value from the larger, then keep the sign of the value with the larger absolute value:
\[ -11+4=-(11-4)=-7. \]
Equal magnitudes cancel to zero. Before calculating \(-48+52\), notice that the positive magnitude is slightly larger, so the result must be small and positive.

Detailed Concept Development

Unlike-signed addition models competition between directions. The magnitudes cancel in equal pairs. The uncancelled magnitude determines size, and the direction with more magnitude determines sign.

This method is equivalent to subtracting absolute values. When magnitudes are close, the result should be near zero. When one magnitude is much larger, the result should have that value's sign and a size near the difference.

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: Compute \(-9+14\).

1. Identify the skill. This example uses the session target: Add unlike-signed values.
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. Subtract \(14-9=5\). The larger magnitude is positive, so the result is \(5\).
5. Interpret the result. State what the signed value means rather than reporting an unexplained number.
6. Verify. Circle the larger absolute value before subtracting.

Worked Example 2: Foundation

Problem: Compute \(27+(-35)\).

1. Identify the skill. This example uses the session target: Add unlike-signed values.
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. Subtract \(35-27=8\). The larger magnitude is negative, so the result is \(-8\).
5. Interpret the result. State what the signed value means rather than reporting an unexplained number.
6. Verify. Circle the larger absolute value before subtracting.

Worked Example 3: Detailed Reasoning

Problem: Compute \(-62+45\).

1. Identify what is given and asked. The signs differ and 62 has greater magnitude.
2. Select a plan. Subtract magnitudes and keep the negative direction.
3. Show the mathematical work. \(62-45=17\), so the result is \(-17\).
4. State the answer clearly. \(-17\)
5. Interpret the sign and units. Explain what positive, negative, or zero communicates in this problem.
6. Verify independently. A larger negative movement than positive movement must finish below zero.

Worked Example 4: Detailed Reasoning

Problem: Compute \(91+(-37)\).

1. Identify what is given and asked. The signs differ and 91 has greater magnitude.
2. Select a plan. Subtract 37 from 91 and keep positive.
3. Show the mathematical work. \(91-37=54\).
4. State the answer clearly. \(54\)
5. Interpret the sign and units. Explain what positive, negative, or zero communicates in this problem.
6. Verify independently. Estimate \(90-40pprox50\), close to 54.

Worked Example 5: Detailed Reasoning

Problem: Compute \(-125+125\).

1. Identify what is given and asked. The values are exact opposites.
2. Select a plan. Pair equal magnitudes.
3. Show the mathematical work. \(-125+125=0\).
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. Opposite movements cancel completely.

Real-World Application

Scenario: An account is overdrawn by \(\$68\), then receives a deposit of \(\$105\).

1. Define the reference and quantities. Overdraft is negative and deposit is positive.
2. Translate words into mathematics. Model the balance as \(-68+105\).
3. Calculate in a visible sequence. The signs differ: \(105-68=37\), with positive winning magnitude.
4. Answer in context. The new balance is \(\$37\).
5. Check reasonableness. The deposit first clears 68 dollars of debt and leaves 37 dollars.
6. Reflect. Change one value in the scenario, predict how the result would change, and then recalculate to test the prediction.

Common Trap

Keeping the sign of the first addend rather than the larger magnitude.

GED® Success Habit

Circle the larger absolute value before subtracting.

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.