Comparing and Ordering Integers on a Number Line

Grade: 6 | Topic: Arithmetic

What You Will Learn

After working through this page you will be able to compare any two integers using the symbols $<$, $>$, and $=$, and you will know how to arrange a set of integers in ascending or descending order. You will also understand why negative numbers that seem "bigger" (like $-100$) are actually smaller than those that seem "tiny" (like $-1$).

Theory

The number line is your best tool

A number line is a straight line where integers are placed in order. Numbers increase as you move to the right and decrease as you move to the left:

$\longleftarrow \;-6 \;\; -5 \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\;\; 0 \;\;\; 1 \;\;\; 2 \;\;\; 3 \;\;\; 4 \;\;\; 5 \;\;\; 6\; \longrightarrow$

Key rule: An integer is greater than any integer to its left and less than any integer to its right.

Comparing two integers

Use the inequality symbols:

Symbol	Meaning
$<$	less than
$>$	greater than
$=$	equal to

Comparing rules at a glance:

1. Every positive integer is greater than every negative integer: $1 > -100$.
2. Every positive integer is greater than zero, and every negative integer is less than zero.
3. For two positive integers, the one with the larger absolute value is greater: $8 > 3$.
4. For two negative integers, the one closer to zero (smaller absolute value) is greater: $-2 > -7$.

Rule 4 is the one most students find tricky. Think of it this way: $-2$ is only 2 steps below zero, while $-7$ is 7 steps below zero, so $-2$ is higher up on the number line.

Absolute value and comparison

The absolute value of an integer is its distance from zero on the number line:

$|{-5}| = 5 \qquad |5| = 5$

Absolute value is always non-negative. It tells you the magnitude, but not the direction. When comparing two negative integers, the one with the smaller absolute value is the greater integer:

$|-3| = 3 \quad\text{and}\quad |-8| = 8$

Since $3 < 8$, we know $-3 > -8$.

Ordering integers

Ascending order (least to greatest) means arranging from leftmost to rightmost on the number line.

Descending order (greatest to least) goes from rightmost to leftmost.

Strategy for ordering a set of integers:

1. Separate the numbers into negative, zero, and positive groups.
2. Order the negatives: the one with the largest absolute value is the smallest (farthest left).
3. Place zero next (if present).
4. Order the positives normally.
5. Combine the groups.

Example: Order $\{3, -7, 0, -2, 5, -7\}$ from least to greatest.

• Negatives (largest absolute value first): $-7, -7, -2$
• Zero: $0$
• Positives: $3, 5$
• Result: $-7, -7, -2, 0, 3, 5$

Comparing integers in real-world contexts

Comparing integers is not just abstract math. Consider:

• Temperature: $-15°\text{C}$ is colder than $-3°\text{C}$ because $-15 < -3$.
• Elevation: A valley at $-50$ m is lower than a shore at $0$ m because $-50 < 0$.
• Bank balance: A debt of $200 ($-200$) is worse than a debt of $50 ($-50$) because $-200 < -50$.

Worked Examples

Example 1: Comparing two negative integers (easy)

Problem: Insert $<$, $>$, or $=$ between $-4$ and $-9$.

Step 1: Find the absolute values. $|-4| = 4 \qquad |-9| = 9$

Step 2: Both are negative. The one with the smaller absolute value is greater. $4 < 9 \implies -4 > -9$

Answer: $-4 \;\mathbf{>}\; -9$

Example 2: Comparing a negative and a positive integer (easy)

Problem: Insert $<$, $>$, or $=$ between $-12$ and $3$.

Step 1: One is negative and the other is positive. Every positive integer is greater than every negative integer.

Answer: $-12 \;\mathbf{<}\; 3$

Example 3: Ordering five integers in ascending order (medium)

Problem: Arrange $8, -3, -11, 0, 5$ from least to greatest.

Step 1: Identify the negatives, zero, and positives.

• Negatives: $-11, -3$
• Zero: $0$
• Positives: $5, 8$

Step 2: Order the negatives (largest absolute value = smallest value). $-11 < -3$

Step 3: Combine all groups.

Answer: $\mathbf{-11, -3, 0, 5, 8}$

Example 4: Ordering with duplicates in descending order (medium)

Problem: Arrange $-5, 2, -5, 7, 0, -1$ from greatest to least.

Step 1: In ascending order first: $-5, -5, -1, 0, 2, 7$.

Step 2: Reverse for descending order.

Answer: $\mathbf{7, 2, 0, -1, -5, -5}$

Example 5: Real-world comparison (challenging)

Problem: During a science experiment, four sensors recorded temperatures at dawn: Sensor A: $-8°\text{C}$, Sensor B: $-3°\text{C}$, Sensor C: $2°\text{C}$, Sensor D: $-8°\text{C}$. Rank the sensors from coldest to warmest, and identify which sensors recorded the same temperature.

Step 1: Order the temperatures from least (coldest) to greatest (warmest). $-8 < -8 < -3 < 2$

Step 2: Map back to sensors.

• Coldest (tied): Sensor A and Sensor D (both $-8°\text{C}$)
• Next: Sensor B ($-3°\text{C}$)
• Warmest: Sensor C ($2°\text{C}$)

Answer: Coldest to warmest: A = D, B, C. Sensors A and D recorded the same temperature.

Common Mistakes

Mistake 1: Thinking a "bigger" negative number is greater

❌ $-10 > -2$ (because 10 is bigger than 2)

✅ $-10 < -2$

Why this matters: With negative numbers, a larger absolute value means the number is farther from zero in the negative direction -- so it is actually smaller. Picture the number line: $-10$ is far to the left of $-2$.

Mistake 2: Confusing the direction of $<$ and $>$

❌ Writing $5 < 3$ when you mean 5 is greater

✅ $5 > 3$ (the open side of the symbol faces the larger number)

Why this matters: A helpful memory trick is that the symbol looks like an arrow pointing to the smaller number, or that the wide-open side always faces the bigger number (like a hungry alligator eating the bigger one).

Mistake 3: Forgetting zero when ordering

❌ Ordering $\{-4, 3, 0, -1, 7\}$ as $-4, -1, 3, 7$ (zero is missing)

✅ $-4, -1, 0, 3, 7$

Why this matters: Zero is an integer and must be included in any ordered list. It sits between the negatives and positives on the number line.

Practice Problems

Try these on your own before checking the answers:

1. Insert $<$, $>$, or $=$: $\;-6 \;\square\; -1$
2. Insert $<$, $>$, or $=$: $\;-15 \;\square\; 0$
3. Order from least to greatest: $4, -9, 1, -3, 0, -9$
4. Order from greatest to least: $-20, 15, -7, 0, 8, -20$
5. During a week, a town recorded overnight lows of $-2°\text{C}$, $3°\text{C}$, $-5°\text{C}$, $0°\text{C}$, and $-2°\text{C}$. Which night was coldest? Which was warmest?

Click to see answers

1. $-6 \;\mathbf{<}\; -1$ ($-6$ has a larger absolute value, so it is smaller)
2. $-15 \;\mathbf{<}\; 0$ (all negative integers are less than zero)
3. $\mathbf{-9, -9, -3, 0, 1, 4}$
4. $\mathbf{15, 8, 0, -7, -20, -20}$
5. Coldest: $-5°\text{C}$ (the smallest integer). Warmest: $3°\text{C}$ (the largest integer).

Summary

• On a number line, numbers increase to the right and decrease to the left.
• Every positive integer is greater than every negative integer, and both are compared to zero.
• For two negative integers, the one closer to zero (smaller absolute value) is greater: $-2 > -7$.
• Ascending order = least to greatest (left to right on the number line).
• Descending order = greatest to least (right to left on the number line).

Related Topics

• Integers — Operations, Number Line, and Word Problems
• Adding and Subtracting Integers — Rules and Examples
• Integer Word Problems with Step-by-Step Solutions

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