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

Study Guide: Ordering Integers (Grade 6–8, Math)

1. The Driving Question

"If you owe your friend $5 and your sister owes you $3, who’s actually richer—you or your sister? And how do you write those amounts as numbers so a bank or a video game scoreboard can tell the difference between owing money and having money?"

2. The Core Idea — Built, Not Listed

Imagine a vertical thermometer outside your school in Minneapolis in January. At noon, it reads 5°F above zero, but by midnight, it drops to 12°F below zero. The numbers on the thermometer don’t just tell you how cold it is—they tell you direction: above or below a starting point (zero). Integers work the same way. They’re numbers with a sign (+ or –) that tells you whether you’re moving up (positive) or down (negative) from zero on a number line.

• Positive integers (like +3) mean you have something—money in your pocket, points in a game, or steps above ground level.
• Negative integers (like –5) mean you owe something—money to a friend, a penalty in a game, or steps below ground level.
• Zero is the neutral point—you neither have nor owe.

Key Vocabulary:
- Integer: Any whole number (positive, negative, or zero), like –7, 0, or +12.
Example: The elevator in a 20-story building uses integers—floor 3 is +3, the basement is –1, and the lobby is 0.
- Number line: A straight line where integers are placed in order, with zero in the middle.
Example: A football field’s yard lines are like a number line—midfield is 0, your team’s end zone is +50, and the opponent’s is –50.
- Absolute value: The distance a number is from zero, ignoring direction. Written as |–4| = 4.
Example: If you dig a 4-foot hole (–4) and climb a 4-foot ladder (+4), you’ve moved the same distance from the ground, even though you’re in different places.
Grade 9–12 note: In advanced math, absolute value becomes a function with its own graph and properties, not just a distance measure.
- Opposite: The integer with the same absolute value but opposite sign. The opposite of –9 is +9.
Example: If you lose 9 points in a game (–9), gaining 9 points (+9) cancels it out—you’re back to zero.

3. Assessment Translation

How this appears in class (Grade 6–8):
- Exit tickets: "Plot –3, 1, –5, and 4 on a number line. Which is the greatest? Explain in one sentence." - Short constructed response: "The temperature at 6 AM was –8°F. By noon, it rose 12 degrees. What was the temperature at noon? Show your work." - State standardized tests (e.g., SBAC, PARCC): - Multiple choice: "Which list orders the integers –2, 5, –7, 0 from least to greatest?"
Distractor patterns:
- Students ignore negative signs (e.g., picking –7 as greater than –2).
- Students confuse "least" with "smallest absolute value" (e.g., picking 0 as least).
- Evidence-based selected response: "A scuba diver descends 15 feet (–15), then ascends 8 feet. Which integer represents their new depth? Explain using a number line."
Proficient response: "The diver starts at –15, moves up 8, so –15 + 8 = –7. On a number line, this is 7 feet below the surface."

Model Proficient Response (Short Answer):
Prompt: "Order the integers –10, 3, –1, 0, –5 from least to greatest. Explain your reasoning." Response: "From least to greatest: –10, –5, –1, 0, 3. On a number line, numbers to the left are smaller. –10 is farthest left, then –5, then –1, then 0, and 3 is farthest right. Negative numbers are always less than positive numbers, and –10 is more negative than –5."

4. Mistake Taxonomy

Mistake 1: Ignoring the Negative Sign
- Prompt: "Which is greater: –4 or –9?" - Common wrong answer: "–9 is greater because 9 is bigger than 4." - Why it loses credit: The student compares absolute values but ignores the sign. On a number line, –4 is to the right of –9, so it’s greater.
- Correct approach: Draw a quick number line. –4 is closer to zero, so it’s greater than –9.

Mistake 2: Confusing "Least" with "Closest to Zero"
- Prompt: "Which integer is the least: –2, 5, –1, 0?" - Common wrong answer: "–1 is the least because it’s closest to zero." - Why it loses credit: "Least" means smallest value, not closest to zero. –2 is farther left on the number line, so it’s the least.
- Correct approach: List the numbers in order: –2, –1, 0, 5. The first one is the least.

Mistake 3: Misapplying Absolute Value
- Prompt: "True or false: |–6| > |3|. Explain." - Common wrong answer: "False, because –6 is less than 3." - Why it loses credit: The student compares the integers directly, not their absolute values. |–6| = 6, which is greater than |3| = 3.
- Correct approach: Calculate absolute values first: 6 > 3, so the statement is true.

5. Connection Layer

• Within math: Ordering integers → Comparing rational numbers — Once you can order integers, you can order fractions and decimals by converting them to a common form (e.g., –0.5 is –1/2, which is between –1 and 0).
• Across subjects: Integers on a number line → Timeline of historical events — BC/BCE years are like negative integers (e.g., 500 BCE is –500), and AD/CE years are positive. Ordering them helps you sequence events (e.g., –300 comes before –200).
• Outside school: Negative integers → Sports standings — In hockey, a team with "–5 goal differential" has let in 5 more goals than they’ve scored. The lower the number, the worse the team’s performance—just like on a number line!

6. The Stretch Question

"If you’re playing a video game where your score starts at 0, and you gain 10 points (+10) but then get a penalty of 15 points (–15), your score is –5. Now imagine the game has a ‘mirror mode’ where every positive score becomes negative and vice versa. What would your score be in mirror mode? Would you rather have –5 or its mirror version—and why?"

Pointer toward the answer: The mirror mode is like finding the opposite of your score. The opposite of –5 is +5. But which is "better"? In most games, +5 is better because it’s a higher score—but if the game penalizes positive scores (e.g., in a "reverse scoring" mode), the answer flips. The real question is: What does the sign mean in this context? The math is the same, but the meaning changes with the rules.