K-12 Math (US): 6-8 Number & Operations K-12 Math Integers Addsubtract integers

Study Guide: Adding and Subtracting Integers (Grade 6–8 Math)

1. The Driving Question

"If you owe your friend $5 and then borrow another $3, how do you write that as a single number—and why does it feel like you’re ‘losing’ money even though you’re adding? And what happens when you ‘subtract’ a negative number—does that mean you’re actually gaining something?"

This isn’t just about memorizing rules like "a negative plus a negative is a negative." It’s about figuring out how numbers can represent real things—like debt, temperature, or even video game scores—and how adding and subtracting them actually works in the real world.

2. The Core Idea — Built, Not Listed

Imagine you’re playing a video game where points can be gained (positive) or lost (negative). Your score starts at 0. If you: - Gain 4 points (+4), your score is 4.
- Lose 3 points (–3), your score drops to 1.
- Now, if you lose 5 more points (–5), your score goes to –4.

Here’s the key: Adding a negative number is the same as subtracting its positive version. Losing 5 points (–5) is the same as subtracting 5 from your score. So, 1 + (–5) is the same as 1 – 5, which is –4.

But what if you subtract a negative number? Imagine your score is –2, and the game says, "You just avoided a 3-point penalty!" That means you’re removing a loss of 3 points, so your score increases by 3 (–2 – (–3) = 1). Subtracting a negative is like undoing a debt—it makes your total go up.

Key Vocabulary:
- Integer: Any whole number (positive, negative, or zero), like –7, 0, 12.
Example: The temperature at the North Pole in winter might be –30°F—that’s an integer.
- Additive inverse: The number you add to another to get zero. The additive inverse of 5 is –5 because 5 + (–5) = 0.
Example: If you have $10 in your bank account but owe $10, your net worth is $0—the debt cancels the money.
(Grade 9–12 note: In algebra, this concept expands to solving equations like x + 3 = 0, where x = –3 is the additive inverse of 3.) - Absolute value: The distance a number is from zero on the number line, always positive. The absolute value of –6 is 6.
Example: If you walk 6 steps backward (–6) or 6 steps forward (+6), you’ve moved the same distance from your starting point.
- Number line: A visual tool where numbers increase to the right and decrease to the left.
Example: If you’re at –2 on a number line and move 3 steps right, you land on 1 (–2 + 3 = 1).

3. Assessment Translation

How this appears on state tests (Grade 6–8):
- Multiple choice: Questions like "What is –8 + 5?" with distractors that mix up signs (e.g., –13, 3, 13, –3).
Distractor pattern: Students often pick –13 (adding the numbers and keeping the negative) or 3 (ignoring the negative sign entirely).
- Short answer/constructed response: Problems like "Explain why –4 – (–7) is the same as –4 + 7. Use a number line or real-world example." Proficient response: Shows the step where subtracting a negative becomes addition (e.g., "Removing a $7 debt is like gaining $7") and includes a number line or example.
- Word problems: Scenarios like "A submarine is 200 feet below sea level (–200). It ascends 75 feet. What is its new depth?" Proficient response: Writes the equation –200 + 75 = –125 and labels the answer "125 feet below sea level."

Model Proficient Response (Short Answer):
Prompt: "Solve: –12 – (–5). Explain your steps using a number line or real-world example." Response:

"First, I know that subtracting a negative is the same as adding a positive. So –12 – (–5) becomes –12 + 5.
On a number line, I start at –12. Adding 5 means moving 5 spaces to the right. I land on –7.
Real-world example: If I owe $12 (–12) and a friend cancels a $5 debt I owe them (– (–5)), it’s like I gained $5. My new debt is $7 (–7)."

What teachers look for:
- Developing: Gets the answer right but can’t explain why (e.g., "I just remembered the rule"). Or mixes up signs (e.g., writes –12 + (–5)).
- Proficient: Shows the step where subtracting a negative becomes addition. Uses a number line, example, or clear reasoning.
- Advanced: Connects to other concepts (e.g., "This is like the additive inverse—subtracting –5 is the same as adding its opposite, 5").

4. Mistake Taxonomy

Mistake 1: Sign Confusion in Addition
Prompt: "Solve: –7 + (–4)." Common wrong answer: –3 (student adds the numbers but drops the negative sign).
Why it loses credit: The student ignores the rule that adding two negatives makes the number more negative. They treat it like 7 + 4 = 11 and then slap a negative sign on it.
Correct approach:

"Both numbers are negative, so I’m moving left on the number line. –7 + (–4) means I start at –7 and move 4 more spaces left. That lands me at –11."

Mistake 2: Misapplying "Subtracting a Negative"
Prompt: "Explain why –3 – (–6) is the same as –3 + 6." Common wrong answer: "Because two negatives make a positive." (This is a half-truth that doesn’t explain why.) Why it loses credit: The student memorized a slogan but can’t justify it. They might also write –3 – 6 = 3 (ignoring the parentheses).
Correct approach:

"Subtracting –6 is like removing a $6 debt, which is the same as gaining $6. So –3 – (–6) becomes –3 + 6. On a number line, I start at –3 and move 6 spaces right, landing on 3."

Mistake 3: Word Problem Misinterpretation
Prompt: "The temperature was –5°F at 6 AM. By noon, it had risen 8 degrees. What was the temperature at noon?" Common wrong answer: –13°F (student subtracts 8 from –5 instead of adding).
Why it loses credit: The student misreads "risen" as "dropped" or doesn’t connect "risen" to addition. They might also write 3°F but forget to label it as the final temperature.
Correct approach:

"‘Risen’ means the temperature went up, so I add 8 to –5. –5 + 8 = 3. The temperature at noon was 3°F."

5. Connection Layer

1. Within math: Integers → Coordinate planes
   Why it matters: The x- and y-axes on a coordinate plane are just number lines! Plotting points like (–3, 4) uses integers to describe location. Understanding how to add/subtract integers helps you move between quadrants (e.g., "If I start at (2, –1) and move left 5 spaces, where do I land?").

2. Across subjects: Integers → Physics (electric charge)
   Why it matters: In physics, protons have a +1 charge and electrons have a –1 charge. Adding/subtracting integers helps you calculate net charge (e.g., "If an atom has 3 protons and 5 electrons, what’s its net charge?").

3. Outside school: Integers → Sports statistics (plus/minus in hockey or basketball)
   Why it matters: A player’s "+/–" stat tracks how many goals their team scores vs. allows while they’re on the ice. A +5 means their team scored 5 more goals than the opponent; a –3 means they were outscored by 3. Adding/subtracting integers helps you compare players (e.g., "Player A is +7 and Player B is –2. How much better is Player A’s impact?").

6. The Stretch Question

"If you add a number to its additive inverse, you always get zero. But what if you multiply a number by its additive inverse? For example, 5 × (–5) = –25. Is there any integer where this product is positive? Why or why not?"

Pointer toward the answer:

"Think about the signs. A positive number’s additive inverse is negative, and vice versa. When you multiply a positive by a negative, the result is always negative. The only way the product could be positive is if both numbers were positive or both were negative—but an additive inverse pair always has one of each. Zero is its own additive inverse, but 0 × 0 = 0, which isn’t positive. So no, there’s no integer where this product is positive!"

(This sets up the idea of multiplicative inverses—like 5 and 1/5—which students will encounter in fractions and algebra.)