🎯⭐ INTERACTIVE LESSON

Understanding Integers

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Understanding Integers - Complete Interactive Lesson

Part 1: 🌡️ Understanding Integers

🌡️ Understanding Integers

Part 1 of 5 — Concept Introduction

Have you ever seen a temperature like $-10°$ on a freezing winter morning? Or heard that a submarine dove to $-200$ feet below sea level? Those numbers with a little "minus" in front are part of a special set of numbers called integers.

Integers are the whole numbers, their opposites, and zero:

$..., -3, -2, -1, 0, 1, 2, 3, ...$

The three little dots ( $...$ ) mean the list keeps going forever in both directions!

The Three Kinds of Integers

Type	Examples	Where it lives
Positive integers	$1, 2, 3, 4, ...$	To the right of zero
Negative integers	$-1, -2, -3, -4, ...$

💡 Important: Zero is special — it is neither positive nor negative. It is the dividing line between the two!

Notice that fractions and decimals like $\frac{1}{2}$ or $2.5$ are not integers. Integers are only whole numbers and their opposites.

The Number Line 📏

The easiest way to picture integers is on a number line:

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

Why Integers Matter 🌍

Negative numbers help us describe things that go below a starting point. Here are some real places you see them:

Temperature: $-10°$ means 10 degrees below zero (brrr! ❄️).
Elevation: $-50$ feet means 50 feet below sea level.
Money: $-20$ , i.e. owing $20, means you owe $20 (a debt).

In each case, the positive version means the opposite direction: $+10°$ is above zero, feet is above sea level, and , i.e. having $20, means you $20.

Concept Check 🎯

Let's make sure the basics are clear before moving on.

Part 2: 📝 Worked Examples

📝 Worked Examples

Part 2 of 5 — Absolute Value Step by Step

The absolute value of a number is its distance from zero on the number line. Distance is never negative, so absolute value is always positive (or zero).

We write absolute value with two straight bars: $|\;\;|$ .

Example 1: Find $|-5|$

Picture $-5$ on the number line. It sits 5 steps to the left of zero.

Part 3: 🧭 Guided Practice

🧭 Guided Practice

Part 3 of 5 — Comparing Integers

Remember the golden rule: on a number line, the number farther to the right is greater, and the number farther to the left is less.

Fill In the Comparison 🔍

Choose the correct word to complete each sentence about the number line.

Part 4: 🌍 Application & Word Problems

🌍 Application & Word Problems

Part 4 of 5 — Integers in the Real World

Integers show up all around you. The trick to word problems is deciding which direction is positive and which is negative.

Situation	Positive ( $+$ ) means	Negative ( $-$ ) means
Temperature	Above zero	Below zero
Elevation	Above sea level	Below sea level
Money	You have / earn	You owe / spend
Football	Yards gained	Yards lost

Example 🏔️

Part 5: Review & Challenge

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You have learned a lot about integers! Here is a quick summary of the big ideas:

Idea	What it means	Example
Integers	Whole numbers, their opposites, and zero	$..., -2, -1, 0, 1, 2, ...$

The number line follows two simple rules:

As you move right, the numbers get bigger (increase).
As you move left, the numbers get smaller (decrease).

Every integer has an opposite — the number that is the same distance from zero but on the other side.

The opposite of $3$ is $-3$ .
The opposite of $-7$ is $7$ .
The opposite of $0$ is just $0$ itself.

Think of opposites as a mirror image, with zero acting as the mirror!

So integers let us talk about both directions — up and down, above and below, having and owing — all with one set of numbers.

Distance is just how far, not which direction.

Example 2: Find $|5|$

The number $5$ sits 5 steps to the right of zero.
That is also a distance of $5$ .
So $|5| = 5$ . ✅

💡 Big idea: A number and its opposite have the same absolute value, because they are the same distance from zero. Both $-5$ and $5$ are $5$ units away.

Example 3: Find $|0|$

Zero is $0$ steps away from itself, so $|0| = 0$ .

Putting It Together

Let's read this number line and answer a few quick questions.

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

What is the opposite of $-3$ ? Go the same distance on the other side of zero → $3$ .
What is $|-4|$ ? The number $-4$ is $4$ units from zero → .

Now it is your turn to try a few!

Your Turn 🧮

Use what you learned about opposites and absolute value. Type a single number for each answer.

What is $|-8|$ ?
What is the opposite of $6$ ? (Include the sign.)
What is $|7|$ ?

A hiker starts at sea level ( $0$ feet). She climbs to $+300$ feet, then a diver goes down to $-150$ feet in a nearby lake.

The hiker's elevation $+300$ is above sea level.
The diver's elevation $-150$ is below sea level.
Since $+300$ is to the right of $-150$ on a number line, the hiker is higher: $300 > -150$ .

Once you label the directions, comparing integers tells you who is higher, colder, or richer!

Word Problem Practice 🧮

On a freezing morning, a thermometer reads 8 degrees below zero. By noon it has warmed up, and a bank account shows a debt (money owed) of $15.

Type a single number for each answer, including a minus sign when needed.

Write the morning temperature as an integer.
Write the bank account debt of $15 as an integer.
What is the absolute value of that temperature, $|-8|$ ? (This tells how many degrees below zero it is.)

Word Problem Check 📋

💡 Remember: When comparing two negative numbers, the one closer to zero is the greater number. For example, $-2 > -8$ .

Now take on the challenge questions below to show what you know!

Challenge Questions 🎯

These mix together everything from this lesson. Take your time!

What is $|2|$ ? The number

2

2

units from zero →

2