You already use numbers below zero — a weather app says -7°, an elevator button says B2, a bank app turns a balance red. Negative numbers live to the LEFT of zero, positives to the RIGHT, and together with zero they form the integers.
↔️The Number Line: Left Is Less. Always.
Zero is the agreed-on middle — neither positive nor negative. Numbers grow as you walk right and shrink as you walk left… even when their digits look big.
Integers are the whole numbers and their negative partners — …-3, -2, -1, 0, 1, 2, 3… — no fractions, no decimals, marching off forever in BOTH directions.
📏Absolute Value: Distance, Not Direction
Two straight bars ask one question: how far from zero? Direction is thrown away, so absolute value is never negative.
Two different numbers can be equally far from zero. |-15| and |15| are both 15 — distance is a length, and lengths don’t come in negative.
🚶Adding Integers = Taking Walks
No rule-memorizing required: start at the first number, walk the second one — right for positive, left for negative. And subtracting? Just add the opposite.
With counters instead: one + chip and one − chip make a zero pair and vanish. 7 + (-7) = 0 — which is exactly why opposites cancel.
🌍 Zero is an agreement, not a law of nature. Sea level counts as zero so a mountain gets +29,000 ft and a trench gets a negative number. Death Valley sits at -282 ft. Above ground floor or below it, warmer than freezing or colder — you pick the zero, and the number line does the rest.
🔑Key Terms
🔢IntegerAny whole number and its negative partner, including zero: …-3, -2, -1, 0, 1, 2, 3… No fractions or decimals allowed.
🧊Negative NumberA number less than zero, written with a minus sign, like -8. On a number line it sits to the LEFT of zero.
☀️Positive NumberA number greater than zero, like 8 — RIGHT of zero. The plus sign is usually invisible: 8 and +8 mean the same thing.
⚖️ZeroThe one integer that is neither positive nor negative — the agreed-on starting point and the boundary between directions.
📏Absolute ValueHow far a number is from zero, either direction: |-15| = 15 and |15| = 15. A distance, so never negative.
🪞OppositeTwo numbers the same distance from zero on opposite sides, like -6 and 6. A number plus its opposite always makes zero.
↔️Number LineA straight line of evenly spaced numbers showing order and distance at a glance — larger to the right, smaller to the left.
🏔️ElevationHeight measured from sea level, which counts as zero. Above is positive, below is negative — like Death Valley’s -282 feet.
🌊Sea LevelThe ocean’s average surface height, chosen as Earth’s zero point for elevation. An agreed-on reference, not a natural boundary.
🤝Zero PairOne positive counter matched with one negative counter. They cancel to zero — why 7 + (-7) = 0.
🌍Where You’ll See This in Real Life
🌊Ocean & Earth scienceScientists treat average sea level as zero so every height on the planet shares one ruler. Summit = positive, deep-sea trench = negative, and subtracting the two gives the true vertical distance. Without a zero in the middle, “how much higher?” has no clean answer.
💳Money & budgetingA banking app shows a balance below zero when you owe more than you have — and a -$45 balance is worse than -$5 for exactly the reason you learned: -45 sits farther left. Understanding that ordering is the difference between noticing a problem and being surprised by it.
🀄Ancient Chinese mathematicians used counting rods — RED for positive, BLACK for negative — the exact opposite of the modern saying “in the red,” which means you owe money.
🥶The coldest natural temperature ever measured on Earth: about -128.6°F (-89.2°C) at Vostok Station, Antarctica, in July 1983 — midwinter down there.
✈️Amsterdam’s Schiphol Airport sits roughly 11 feet below sea level — jets there touch down at an elevation of about -11 feet, lower than the nearby sea.
📌Remember This
1Farther LEFT is always smaller — even when the digits look bigger. That is why -20 < -4.
2Absolute value is distance from zero, so it’s never negative. |-15| and |15| are both 15 — two different numbers, equally far from zero.
3Subtracting is adding the opposite: 6 − 10 becomes 6 + (-10) = -4. One idea replaces two rules.
🤔 Think about it
Absolute value keeps the distance and throws away the direction. Name a real situation where the distance alone is genuinely enough — and another where losing the direction would be a disaster.
Temperature has a real floor (about -273.15°C, absolute zero), but the number line has no floor at all. Does that mean some negative numbers can never describe anything real — or just that we picked our zero in an arbitrary spot?
⭐Remember: every integer question is a picture question. Put the numbers on the line — smaller is left, distance is absolute value, subtracting is walking the opposite way — and the answers stop being rules and start being obvious.
✏️ ClickClass Anchor Chart · Integers: Life Below Zero