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Teacher Guide: Introduction to Negative Numbers

Learn what negative numbers are, where they appear in real life, and how to place them on a number line.

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

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Class quiz

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

For Teachers

Learning Objectives
  • Define negative numbers as values less than zero
  • Locate and plot negative numbers on a number line
  • Compare and order integers including negative numbers
  • Identify real-world situations involving negative numbers
  • Find the opposite of any integer
Prerequisites
  • • Understanding of counting numbers (1, 2, 3, ...)
  • • Familiarity with number lines
  • • Basic understanding of zero as a quantity
Discussion Starters
  • 1. If you had to explain negative numbers to someone who had never heard of them, what would you say?
  • 2. Can you think of situations where negative numbers would not make sense? (Hint: Can you have negative apples?)
  • 3. Why do you think it took mathematicians so long to accept negative numbers as real numbers?
  • 4. How would our world be different if negative numbers did not exist?
  • 5. Where have you seen negative numbers in your daily life?
Common Misconceptions

Larger digit means larger number (thinking −9>−2-9 > -2)

Remediation: Use temperature examples: Is −9°C-9°C warmer or colder than −2°C-2°C? Connect to the number line - right is always greater.

Negative numbers work the same as positive numbers

Remediation: Show that some familiar rules change: −3-3 is not simply 3 with a minus. Demonstrate with number line movements.

Zero is a negative number

Remediation: Emphasize zero as the neutral point. It is neither positive nor negative - it is the boundary between them.

Differentiation Ideas

For Struggling Students:

  • • Provide a physical number line students can touch and move along
  • • Use two-color counters (red for negative, yellow for positive)
  • • Focus only on single-digit negative numbers initially
  • • Connect every example to temperature, which is intuitive

For On-Level Students:

  • • Practice comparing negative numbers using number line visuals
  • • Work with real-world contexts like elevation and bank balances
  • • Introduce the concept of opposites

For Advanced Students:

  • • Explore negative decimals and fractions
  • • Research the history of negative numbers across cultures
  • • Create word problems involving negative numbers for classmates
Standards Alignment
  • 6.NS.C.5 (CCSS.MATH.CONTENT.6.NS.C.5)

    Understand that positive and negative numbers describe quantities having opposite directions or values

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

    Understand a rational number as a point on the number line

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

    Understand ordering and absolute value of rational numbers

Lesson Resources
  • visualInteractive Number Line

    Drag and drop numbers onto the correct positions on a number line from −10-10 to 1010

  • activityTemperature Investigation

    Students research and compare winter temperatures in different cities around the world

  • gameInteger Card Game

    Students play a card game where the greater integer wins, including negative numbers

Lesson Content

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

Definition

A negative number is any number less than zero. We write negative numbers with a minus sign in front, like −3-3 or −7-7. On a number line, negative numbers are found to the left of zero, while positive numbers are to the right.
The number zero is special - it is neither positive nor negative. It sits right in the middle of the number line, separating the positive numbers from the negative numbers.
The counting numbers 1, 2, 3, ..., their negatives −1-1, −2-2, −3-3, ..., and zero together make up the set of integers: {...,−3,−2,−1,0,1,2,3,...}\{..., -3, -2, -1, 0, 1, 2, 3, ...\}

Worked Examples

A thermometer shows −8-8 degrees Celsius. What does this mean?
1

Identify the sign

The minus sign (−-) tells us this is a negative number → Negative
2

Understand the meaning

Negative temperature means below zero → −8°C-8°C is 8 degrees below zero

Common Mistakes

Thinking −7-7 is greater than −3-3 because 7 is bigger than 3

Why it's wrong: With negative numbers, the number closer to zero is actually greater. Think of temperature: −3°C-3°C is warmer (greater) than −7°C-7°C.

Correct: −3>−7-3 > -7 because −3-3 is closer to zero and further right on the number line.

Forgetting that zero is neither positive nor negative

Why it's wrong: Zero is the dividing point between positive and negative numbers. It has no sign.

Correct: Zero is neutral - neither positive nor negative. It separates the two types of numbers.

Confusing the minus sign with subtraction

Why it's wrong: The minus sign in front of a number (like −5-5) means negative, not subtract. Context matters!

Correct: In −5-5, the minus means negative five. In 8−58 - 5, the minus means subtract.

Placing negative numbers to the right of zero on a number line

Why it's wrong: Negative numbers are less than zero, so they belong on the left side of the number line.

Correct: Negative numbers go LEFT of zero, positive numbers go RIGHT of zero.

Why It Matters

Negative numbers are everywhere in daily life, even if you do not always notice them!
Temperature is one of the most common places we see negative numbers. When the weather report says it is −5-5 degrees outside, that means the temperature is 5 degrees below zero. In cold places like Canada, Russia, or Scandinavia, winter temperatures often drop well below zero.
Money and banking also use negative numbers. If you spend more money than you have in your bank account, you might have a negative balance. For example, a balance of 20 euros in debt means you owe the bank 20 euros.
Elevation uses negative numbers too. Sea level is considered zero, so places below sea level have negative elevations. The Dead Sea, for instance, is about −430-430 meters - that is 430 meters below sea level!
Understanding negative numbers is essential for math you will learn later, including algebra, graphing, and working with data.

Real World Applications

Weather and Temperature

Meteorologists use negative numbers every day when reporting temperatures, especially in winter months. Understanding negative temperatures helps us prepare for cold weather and stay safe.

Example:

If today's high is −2°C-2°C and tonight's low will be −15°C-15°C, the temperature will drop 1313 degrees.

1Try It Yourself

The weather forecast says the morning temperature is −6°C-6°C and it will rise by 10 degrees by afternoon.

What will the afternoon temperature be?

Step 1: Write the mathematical expression

Start at −6-6 and add 10 degrees:

Bank Accounts and Debt

Banks use negative numbers to show when you owe money (debt). A negative balance means you have spent more than you had and need to deposit money to get back to zero or positive.

Example:

If your account shows −25-25 euros, you owe the bank 25 euros. You need to deposit at least 25 euros to have a balance of zero.

2Try It Yourself

Your bank account balance is −30-30 euros. You deposit 50 euros.

What is your new balance?

Step 1: Write the mathematical expression

Start with your debt and add the deposit:

Elevation and Geography

Geographers measure elevation relative to sea level, which is defined as zero. Mountains have positive elevations, while places below sea level, such as the shore of the Dead Sea or the ocean floor, have negative elevations.

Example:

Mount Everest is about 8,8498,849 meters above sea level. The Mariana Trench is about −10,994-10,994 meters (below sea level).

3Try It Yourself

A submarine is at −200-200 meters (200 meters below sea level). It rises 75 meters.

What is the submarine's new position relative to sea level?

Step 1: Write the mathematical expression

Start at the current position and add the rise:

Key Takeaways

  • 1Negative numbers are less than zero and written with a minus sign (e.g., −5-5)
  • 2On a number line, negative numbers are to the LEFT of zero
  • 3Zero is neither positive nor negative - it separates them
  • 4The further left on the number line, the smaller the number
  • 5Comparing negatives: −3>−7-3 > -7 because −3-3 is closer to zero
  • 6Every number has an opposite: the opposite of 55 is −5-5
  • 7Integers are the counting numbers 1, 2, 3, ..., their negatives, and zero

Frequently Asked Questions

What is the difference between a negative number and subtraction?

The minus sign can mean two different things depending on context. When it appears in front of a number by itself (like −5-5), it means the number is negative (5 units below zero). When it appears between two numbers (like 8−58 - 5), it means subtraction (take away). Context helps you know which meaning applies.

Why is −3-3 greater than −7-7?

Think of it like temperature: −3°C-3°C is warmer than −7°C-7°C. On a number line, −3-3 is to the right of −7-7, and numbers to the right are always greater. Another way to think about it: −3-3 is only 3 away from zero, while −7-7 is 7 away from zero. Being closer to zero makes a negative number greater.

Is zero a positive or negative number?

Zero is neither positive nor negative - it is neutral. Zero is the special point on the number line that separates positive numbers (on the right) from negative numbers (on the left). This is why we sometimes say a number is non-negative (zero or positive) or non-positive (zero or negative).

Where do negative numbers come from? Who invented them?

Negative numbers have a long history! Ancient Chinese mathematicians used red and black rods to represent positive and negative numbers over 2,000 years ago. Indian mathematicians like Brahmagupta formally defined rules for negative numbers around 628 CE. European mathematicians were slower to accept them - some called them absurd or fictitious numbers well into the 1700s!

What are integers?

Integers are the counting numbers (1, 2, 3, ...), their negatives (..., −3-3, −2-2, −1-1), and zero. The word integer comes from Latin and means whole or untouched. Numbers like 12\frac{1}{2} or 2.52.5 are not integers.

Glossary

Negative number
A number less than zero, written with a minus sign in front (e.g., −3-3, −100-100)
Positive number
A number greater than zero (e.g., 55, 4242)
Integer
One of the counting numbers 1, 2, 3, ..., their negatives −1-1, −2-2, −3-3, ..., or zero
Number line
A line showing numbers in order, with negative numbers on the left, zero in the middle, and positive numbers on the right
Opposite
Two numbers that are the same distance from zero but on different sides. The opposite of 77 is −7-7
Zero
The number that separates positive from negative; neither positive nor negative
Absolute value
The distance a number is from zero, so it is never negative. The absolute value of both 55 and −5-5 is 55

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