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Teacher Guide: Comparing Integers

Learn how to compare positive and negative numbers using inequality symbols and the number line.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Negative Numbers. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Compare two integers using <<, >>, and == symbols correctly
  • Use a number line to determine which integer is greater or smaller
  • Order a set of integers from least to greatest and greatest to least
  • Apply integer comparison to real-world contexts like temperature and elevation
Prerequisites
  • • Understanding of positive and negative numbers
  • • Ability to locate integers on a number line
  • • Familiarity with the concept of zero as neither positive nor negative
Discussion Starters
  • 1. Why is −1-1 greater than −1000000-1000000? How can something that looks smaller be greater?
  • 2. If you owe someone 50 dollars and your friend owes 20 dollars, who is in a better financial situation? How does this relate to comparing negative numbers?
  • 3. Can you think of a situation where having a 'more negative' number is actually better? (Hint: think about golf scores)
  • 4. Why do you think mathematicians chose to put negative numbers to the left of zero on the number line?
Common Misconceptions

Larger digits mean larger numbers, even for negatives (thinking −50>−5-50 > -5)

Remediation: Use the thermometer analogy consistently. Ask: 'Which is colder, −50°C-50°C or −5°C-5°C?' Connect the physical experience of cold to the mathematical concept.

Zero is the smallest number

Remediation: Show the number line extending infinitely in both directions. Ask students to name numbers smaller than zero, then numbers smaller than those.

Mixing up << and >> symbols

Remediation: Teach the 'alligator mouth' or 'arrow pointing to smaller' mnemonic. Practice reading inequalities both ways: '5>35 > 3' as 'five is greater than three' and 'three is less than five.'

Differentiation Ideas

For Struggling Students:

  • • Provide a physical number line manipulative that students can touch
  • • Use only single-digit integers initially (−9-9 to 99)
  • • Connect every comparison to temperature: 'Which is colder?'
  • • Allow students to draw the number line for every problem

For On-Level Students:

  • • Compare two-digit integers including numbers like −25-25 and −52-52
  • • Order sets of 4-6 integers from least to greatest
  • • Solve word problems involving temperature and elevation comparisons
  • • Use both << and >> to write the same comparison two ways

For Advanced Students:

  • • Compare integers with absolute values: 'Which is further from zero?'
  • • Create their own real-world comparison problems
  • • Explore comparing negative fractions and decimals
  • • Count the integers between two given integers (for example, between −8-8 and 33)
Standards Alignment
  • 6.NS.C.7 (CCSS.MATH.CONTENT.6.NS.C.7)

    Understand ordering and absolute value of rational numbers

  • 6.NS.C.7.A (CCSS.MATH.CONTENT.6.NS.C.7.A)

    Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram

  • 6.NS.C.7.B (CCSS.MATH.CONTENT.6.NS.C.7.B)

    Write, interpret, and explain statements of order for rational numbers in real-world contexts

Lesson Resources
  • visualInteractive Number Line

    Students drag and compare integers on a dynamic number line

  • activityTemperature Sorting Game

    Order cities by temperature from coldest to warmest

  • worksheetInequality Practice

    Fill in <<, >>, or == between integer pairs

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Comparing integers means determining which number is greater, smaller, or if they are equal.
We use inequality symbols to compare:
  • >> means greater than (the number on the left is larger)
  • << means less than (the number on the left is smaller)
  • == means equal to (both numbers have the same value)
The Number Line Rule:
Numbers further RIGHT are GREATER\text{Numbers further RIGHT are GREATER}
Numbers further LEFT are SMALLER\text{Numbers further LEFT are SMALLER}
For example: −3>−7-3 > -7 because −3-3 is to the right of −7-7 on the number line.

Worked Examples

Compare −4-4 and −9-9. Which is greater?
1

Visualize on the number line

Place both numbers: −9-9 is at position −9-9, −4-4 is at position −4-4 → −4-4 is further right
2

Apply the rule

Numbers further right are greater → −4>−9-4 > -9
3

Verify with real-world context

−4°C-4°C is warmer than −9°C-9°C, so −4-4 is greater → Confirmed: −4>−9-4 > -9

Common Mistakes

Thinking −15>−5-15 > -5 because 15>515 > 5

Why it's wrong: With negative numbers, the further a number is from zero, the smaller it is. −15-15 is 15 units left of zero and −5-5 only 5 units, so −15-15 is actually smaller.

Correct: −5>−15-5 > -15 because −5-5 is closer to zero (further right on the number line).

Confusing << and >> symbols

Why it's wrong: It's easy to mix up which symbol points which way.

Correct: The symbol always 'eats' the bigger number. Think of it as a hungry alligator that always wants the larger meal: 5>35 > 3 and 3<53 < 5.

Forgetting that zero is greater than all negatives

Why it's wrong: Zero might seem like 'nothing,' but it's actually greater than every negative number.

Correct: 0>−1>−1000 > -1 > -100. Zero is the boundary between positive and negative.

Why It Matters

Comparing integers is essential for understanding real-world situations:
  • Weather forecasting: Is −5°C-5°C colder than −2°C-2°C? Yes, because −5<−2-5 < -2
  • Elevation: Which is lower: −200-200 meters (below sea level) or −50-50 meters? −200<−50-200 < -50
  • Finance: A balance of −100-100 dollars is worse than −20-20 dollars because −100<−20-100 < -20
  • Sports: In golf, −5-5 (5 under par) is better than −2-2 (2 under par) because lower is better
Understanding how to compare integers helps you make sense of temperatures, debts, elevations, and many other measurements that can be positive or negative.

Real World Applications

Temperature Comparison

Weather forecasters compare temperatures to determine which days are colder or warmer.

Example:

Moscow recorded −18°C-18°C and Helsinki recorded −12°C-12°C. Since −12>−18-12 > -18, Helsinki was warmer.

1Try It Yourself

Three cities have the following temperatures: City A: −7°C-7°C, City B: −15°C-15°C, City C: −3°C-3°C

Order the cities from coldest to warmest.

Step 1: Choose the right answer

Which order goes from coldest to warmest?

Elevation Comparison

Geographers use negative numbers for elevations below sea level.

Example:

The Dead Sea is at −430-430 meters and Death Valley is at −86-86 meters. Since −430<−86-430 < -86, the Dead Sea is lower.

2Try It Yourself

A submarine is at −250-250 meters. A diver is at −30-30 meters. The ocean floor is at −400-400 meters.

Order these from deepest to shallowest.

Step 1: Choose the right answer

Which order goes from deepest to shallowest?

Bank Balance Comparison

Banks show negative balances when you owe money (overdraft).

Example:

Account A shows −75-75 dollars. Account B shows −20-20 dollars. Since −20>−75-20 > -75, Account B is in better standing (owes less).

3Try It Yourself

Three friends check their accounts: Alex has −50-50 dollars, Ben has 2525 dollars, Carla has −10-10 dollars.

Order the three balances from least to most money.

Step 1: Choose the right answer

Which order goes from least to most money?

Key Takeaways

  • 1Use << (less than) and >> (greater than) to compare integers
  • 2On the number line: RIGHT = GREATER, LEFT = SMALLER
  • 3Any positive number is greater than any negative number
  • 4Zero is greater than all negative numbers but less than all positive numbers
  • 5For negative numbers: closer to zero means greater (−2>−10-2 > -10)

Frequently Asked Questions

Why is −3-3 greater than −10-10?

Think of temperature: −3°C-3°C is warmer than −10°C-10°C. On the number line, −3-3 is closer to zero (further right), so it's greater. The rule is: for negatives, closer to zero = greater.

Is zero positive or negative?

Zero is neither positive nor negative. It's the boundary point. But zero IS greater than all negative numbers and less than all positive numbers.

How do I remember which way << and >> point?

The symbol always opens toward the bigger number, like a hungry mouth eating the larger value. Or remember: the small end points to the small number.

Why do we say −100-100 is less than −1-1?

Even though 100 is bigger than 1, we're talking about negative numbers. −100-100 is 100 units left of zero, while −1-1 is only 1 unit left. Further left = smaller, so −100<−1-100 < -1.

Glossary

Integer
A whole number that can be positive, negative, or zero (e.g., −5-5, 00, 1212)
Greater than (>>)
A symbol meaning the number on the left is larger than the number on the right
Less than (<<)
A symbol meaning the number on the left is smaller than the number on the right
Inequality
A mathematical statement that compares two values using <<, >>, ≤\leq, or ≥\geq
Number line
A visual representation of numbers arranged from least to greatest, left to right

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