Mathematics › ⚡ Grade 6 › Deep Sea Diver: Integers and the Number Line

Deep Sea Diver: Integers and the Number Line

Sea level is zero — everything above it is positive, everything below it is negative, and the number line tells you exactly how far and which way.

Grade 6Pre-Algebra

Deep Sea Diver: Integers and the Number Line — illustration

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The big idea: Some quantities need more than a size — they need a direction too. A diver's depth, a thermometer's reading, a bank balance: each has a natural zero point, and numbers can sit above it or below it. Negative numbers are how we write “below zero,” and the number line lets you see at a glance which of two integers is bigger, smaller, or farther from zero.

🎯 By the end, you'll be able to

Explain how positive and negative numbers describe opposite directions from zero
Locate and order integers on a number line
Compare two integers correctly, including when both are negative
Find the absolute value of a number as its distance from zero

📎 You should already know

Whole numbers and place value
Plotting points on a number line

Below sea level

Picture a diver exploring a reef. The surface of the water is sea level — call that 0. Swim up toward the surface (or into the air above it) and your position is a positive number of metres. Swim down into the deep and your position is negative: −10 m means 10 metres below the surface.

Temperature works the same way. 0°C is freezing; a reading of −5°C is 5 degrees below freezing, not just “a small number.”

🔑 Negative numbers show the opposite direction

The integers are the whole numbers together with their opposites: …, −3, −2, −1, 0, 1, 2, 3, …. A negative sign does not mean “less than nothing” — it means “the same distance as the positive number, but in the opposite direction from zero.”

\[ -25 \;\longleftarrow\; 0 \;\longrightarrow\; +25 \]

−25 m (25 m below the surface) and +25 m (25 m above the surface) are the same distance from sea level, on opposite sides.

Reading the number line

On a number line, numbers increase as you move to the right and decrease as you move to the left. That rule still holds once you cross zero into negative territory: −3 is to the right of −7, so −3 is greater than −7 — even though 7 “looks bigger” than 3. Farther left always means smaller, no matter the sign.

🎮 Interactive: compare integers on the number line LIVE

Drag the Boundary (a) slider to set one depth and the Test point (x) slider to set another. Click “Change relation” to cycle through >, ≥, <, ≤ and watch the shaded ray light up every value that satisfies it. Try setting a = −3 and sliding x below and above it to confirm that anything to the left of −3 is less than −3.

✨ Which is colder?

Is −10°C or −3°C colder? Find them on the number line: −10 sits farther to the left, so −10 < −3, and −10°C is the colder temperature. The number with the larger digit is not always the larger integer once negative signs are involved.

Absolute value: distance, not direction

Sometimes you only care how far something is from zero, not which way. That distance is called the absolute value, written with two vertical bars. Distance is never negative, so absolute value is always zero or positive — a diver 18 m below the surface and a buoy 18 m above it are both “18 units from sea level.”

\[ |-7| = 7 \qquad |7| = 7 \]

A number and its opposite always have the same absolute value — both are the same distance from 0.

📝 Worked example: Which is deeper: a diver at −18 m or a diver at −25 m?

Locate both on the number line: −25 is farther to the left than −18.
Farther left means the smaller integer, so −25 < −18.
The smaller (more negative) integer is the deeper position.

✓ The diver at <strong>−25 m</strong> is deeper, since −25 < −18, even though 25 is the larger digit.

📝 Worked example: Order these depths from deepest to shallowest: −30 m, −5 m, −18 m.

Deepest means the most negative (smallest) integer; shallowest means the least negative (largest) integer.
Compare on the number line: −30 is farthest left, then −18, then −5 is farthest right.
So the order from deepest to shallowest follows −30, then −18, then −5.

✓ Deepest to shallowest: <strong>−30 m, −18 m, −5 m</strong>.

⚠️ “More negative” is smaller, not bigger

It is tempting to think −25 is “bigger” than −18 because 25 > 18. As integers it is the opposite: −25 < −18. Save comparisons like “25 is bigger than 18” for their absolute values (their distances from zero), not for the signed numbers themselves.

Check your understanding

1. Which number is greater: −7 or −3?

On the number line, −3 sits to the right of −7, so −3 > −7.

2. A diver is at −18 m. Is that above or below sea level, and what is |−18|?

A negative depth means below sea level. Absolute value strips the sign, so |−18| = 18, the distance from 0.

3. Which is colder: −12°C or −5°C?

−12 is farther left on the number line than −5, so −12 < −5, meaning −12°C is the colder reading.

4. Order these depths from deepest to shallowest: −30 m, −5 m, −18 m.

Deepest is the most negative integer. Since −30 < −18 < −5, the deepest-to-shallowest order is −30, −18, −5.

5. What is |−9|?

Absolute value is distance from zero, and distance is never negative. −9 is 9 units from 0, so |−9| = 9.

✅ Key takeaways

Positive and negative numbers show opposite directions from a zero point, like sea level or freezing.
On a number line, values increase to the right and decrease to the left — even below zero.
Comparing integers means comparing their positions on the number line, not just their digits: −25 < −18.
Absolute value is distance from zero and is always zero or positive, so |−7| = |7| = 7.
The most negative integer is the smallest, so 'deeper' or 'colder' readings are more negative, not larger digits.