Integers and the Number Line

🌡️ Integers and the Number Line

Part 1 of 5 — Meet the Integers

Topics in This Part

🔑 Key Concept: Integers are the whole numbers and their opposites: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ — no fractions, no decimals. They let us describe amounts below zero, like a temperature of $-5^\circ$ or a debt of $20.

What Is an Integer?

An integer is any number from this list, stretching forever in both directions:

$\ldots,\; -4,\; -3,\; -2,\; -1,\; 0,\; 1,\; 2,\; 3,\; 4,\; \ldots$

There are three kinds of integers:

⚠️ Not integers: $\frac{1}{2}$, $0.75$, and $-3.2$ are not integers because they fall between the whole numbers. Integers are always "whole" — they never have a fraction or decimal part.

Concept Check 🎯

Integers in the Real World

Negative integers describe everyday situations that go below a starting point:

💡 Signal words: below, loss, owe, withdraw, fell, descend usually mean negative. Above, gain, earn, deposit, rose, climb usually mean positive.

Match the Situation to Its Integer 🔽

Choose the integer that best represents each situation.

Write the Integer 🧮

Write the integer (with a $+$ or $-$ sign) that represents each situation. Type a leading minus for negatives; positives can be written as a plain number.

1) A withdrawal of $25 from a bank account → $\,?$ 2) A mountain climber rises $30$ feet → $\,?$ 3) A temperature of $7$ degrees below zero → $\,?$

What's Next

You can now spot an integer and decide whether a situation is positive or negative. But to really understand integers — to compare them and find distances — we need a picture.

That picture is the number line, and it's the star of Part 2.

🌡️ Integers and the Number Line

Part 2 of 5 — Plotting on the Number Line

🔑 The Idea: A number line is a straight line with evenly spaced marks. Zero sits in the center, positives go right, and negatives go left. Every integer has exactly one home on the line.

How the Number Line Works

$\overset{-5}{\bullet}\;-4\;-3\;-2\;-1\;\overset{0}{\bullet}\;1\;2\;3\;4\;\overset{5}{\bullet}$

Three rules tell you where any integer lives:

1. Find zero — it's the middle of the line.
2. Pick a direction — positive numbers go right, negative numbers go left.
3. Count the steps — move one mark for each unit.

Example: Plot $-3$

Start at $0$, move left (because it's negative) by 3 marks. You land on $-3$.

Example: Plot $+4$

Start at $0$, move right by 4 marks. You land on $4$.

💡 The marks are always equally spaced. The jump from $0$ to $1$ is the same size as the jump from $-7$ to $-6$ — exactly one unit.

Concept Check 🎯

Read the Number Line 🧮

A point sits at each described spot. Enter the integer it lands on.

1) Start at $0$ and move right $7$ units. The point is at $\,?$ 2) Start at $0$ and move left $5$ units. The point is at $\,?$ 3) Start at $2$ and move left $6$ units. The point is at $\,?$

Counting Across Zero

The trickiest moves cross zero. Take it slow and count one mark at a time.

Example: Start at $-2$ and move right $5$ units

$-2 \;\to\; -1 \;\to\; 0 \;\to\; 1 \;\to\; 2 \;\to\; 3$

That's $5$ marks, landing on $\mathbf{3}$. Notice you pass through zero on the way.

⚠️ Watch out: Don't skip zero! Many students count $-2 \to -1 \to 1$ and forget that $0$ is its own mark. Always pause on zero.

Cross Zero Carefully 🧮

Count one mark at a time. Enter the integer you land on.

1) Start at $-4$, move right $4$ units → $\,?$ 2) Start at $-1$, move right $3$ units → $\,?$ 3) Start at $3$, move left $7$ units → $\,?$

🌡️ Integers and the Number Line

Part 3 of 5 — Comparing and Ordering Integers

🔑 The Golden Rule: On a number line, the number farther to the right is greater. This single idea lets you compare any two integers — even two negatives.

Greater Than and Less Than

We use these symbols to compare:

💡 Trick: The symbol is like a hungry mouth — it always opens toward the bigger number. $5 > 2$ and $2 < 5$ both say the same thing.

The Surprising Part: Negatives

With negative numbers, the bigger-looking number can actually be smaller:

$-2 > -7$

Why? On the number line, $-2$ is to the right of $-7$, so $-2$ is greater. Think of temperature: $-2^\circ$ is warmer than $-7^\circ$.

Concept Check 🎯

Pick the Right Symbol 🔽

Choose $<$, $>$, or $=$ to make each statement true.

Ordering a List of Integers

To order integers from least to greatest, imagine walking the number line from left to right and listing them as you pass.

Example: Order $3,\; -5,\; 0,\; -1,\; 4$ from least to greatest

Walk left → right:

$-5,\;\; -1,\;\; 0,\;\; 3,\;\; 4$

The most negative number ($-5$) comes first, and the largest positive ($4$) comes last.

⚠️ Common trap: $-5$ is the least even though $5$ "looks big." For negatives, the larger the digit, the smaller the value.

Order Them 🧮

Order each list from least to greatest. Enter only the single integer that belongs in the requested spot.

1) List: $\;2,\; -3,\; -8,\; 5\;$ — the least (first) value is $\,?$ 2) Same list — the greatest (last) value is $\,?$ 3) List: $\;-1,\; -6,\; 0,\; -4\;$ — the least value is $\,?$

🌡️ Integers and the Number Line

Part 4 of 5 — Opposites and Absolute Value

🔑 Big Payoff: Every integer has an opposite (same distance from zero, other side) and an absolute value (its distance from zero, always positive). These two ideas show up everywhere in middle-school math.

Opposites

The opposite of a number is the same distance from zero, but on the other side.

Both $5$ and $-5$ are exactly 5 units from zero — they're mirror images across zero.

💡 Zero is its own opposite. It sits right on the mirror, so flipping it changes nothing.

⚠️ The opposite of a negative number is positive. The opposite of $-3$ is $+3$, not a "more negative" number.

Absolute Value

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 it with two straight bars:

$\lvert -7\rvert = 7 \qquad \lvert 4\rvert = 4 \qquad \lvert 0\rvert = 0$

Both $-7$ and $7$ are 7 units from zero, so $\lvert -7\rvert = \lvert 7\rvert = 7$.

⚠️ Don't confuse them: the opposite of $-7$ is $+7$ (also $7$ here), but the absolute value of $-7$ is $7$ because that's its distance from zero. Absolute value answers "how far?", not "which side?"

Concept Check 🎯

Opposites and Absolute Value 🧮

Enter each value.

1) The opposite of $11$ is $\,?$ 2) $\lvert -20\rvert = \,?$ 3) The opposite of $\lvert -6\rvert$ is $\,?$ (find $\lvert -6\rvert$ first, then take its opposite)

Bonus: Distance Between Two Integers

Because absolute value measures distance, you can use it to find how far apart two integers are. Just count the units between them on the number line.

Example: How far is $-3$ from $4$?

Count from $-3$ up to $4$:

$-3 \to -2 \to -1 \to 0 \to 1 \to 2 \to 3 \to 4 \quad = \quad 7 \text{ units}$

So $-3$ and $4$ are 7 units apart.

💡 A shortcut you'll learn soon: the distance equals $\lvert \text{first} - \text{second}\rvert$. Here $\lvert -3 - 4\rvert = \lvert -7\rvert = 7$. For now, counting on the number line works perfectly.

Opposite or Absolute Value? 🔽

Pick the correct value for each blank.

🌡️ Integers and the Number Line

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) identify integers, (2) plot them on the number line, (3) compare and order them, and (4) find opposites and absolute values. Let's put it all together.

Quick Reference

⚠️ Remember the two big traps: a bigger digit makes a negative smaller (so $-9 < -2$), and absolute value is never negative.

Mixed Practice 🔽

Fill in each blank to make a true statement.

Mixed Practice 🧮

Work each one carefully.

1) Start at $-2$ on the number line and move right $6$ units. You land on $\,?$ 2) How many units apart are $-5$ and $3$ ? $\,?$ 3) $\lvert -12\rvert = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.