Compare and order integers on a number line.

SECTION 1

Historical Context & Motivation

Long ago, people only used positive numbers, like counting apples or steps in a game. Negative numbers came later to show debts or temperatures below zero. The number line helped everyone see and compare them easily, like a straight road.

628 AD

Brahmagupta in India writes rules for negative numbers, like subtracting debts.

1202

Fibonacci

Brings negative numbers to Europe through his book on math.

1637

Descartes

Invents coordinate system, linking numbers to a line.

1685

John Wallis

Draws the first clear number line to show all integers.

Before the number line, comparing negatives confused people, like who owes more money. It solved that by showing order visually. Now you can master it for the SSAT!

SECTION 2

Core Principles & Definitions

**Integers** are whole numbers like -5, 0, or 7, including negatives (less than zero) and positives (greater than zero). A number line is a straight line with zero in the middle, negatives to the left, and positives to the right. Numbers get bigger as you move right.

Zero in the Center

Zero splits positives (right) from negatives (left).

Direction Matters

Right = larger, left = smaller, like a race track.

Symbols for Comparison

< means left of, > means right of.

Ordering from Least to Greatest

Read left to right on the line.

🌡️ Think Like a Thermometer!

The number line is like a thermometer for temperatures. Zero is room temp, left is freezing (smaller), right is hot (bigger). Plot points to compare easily!

SECTION 3

Visual Explanation

See the points: -7 (leftmost, smallest) < -2 < 0 < 6 (rightmost, largest). Farther right means greater.

Look at the diagram above. The arrow points right for bigger numbers. Plot any integer to compare—you'll see the order right away. Great job visualizing!

SECTION 4

Mathematical Framework

To compare two integers a and b, check their positions. If a is right of b, then a > b. Use symbols: < (less than, opens left), > (greater than, opens right), = (same spot).

COMPARISON RULES

If a is to the right of b, then a > b Examples: −5 < 0, 3 > −2, −1 = −1

− means negative. Right on line = greater value.

ORDERING STEP

Least to greatest: plot → read left → right Ex: −4, 1, −6 → −6 < −4 < 1

These rules work every time. Practice plotting, and comparing feels like a game. You're building SSAT skills!

SECTION 5

Detailed Breakdown

Break it down by plotting sets of numbers. See how negatives are smaller than positives, and among negatives, closer to zero is bigger. This visual trick makes ordering simple.

Plot the set and read left to right: −6, −4, −1, 0, 3. Dashed arrow shows increasing order.

In the diagram, notice −1 > −4 because it's right of it, even both negative. Zero beats all negatives. This breakdown builds your confidence!

SECTION 6

Worked Example

Let's order these from least to greatest: 2, −8, 0, −3, 5. You'll see how to do it step by step. You got this!

Order: 2, −8, 0, −3, 5

Step 1 — Plot on Number LineImagine line: mark −8 far left, then −3, 0, 2, 5 right.

Positions: −8 < −3 < 0 < 2 < 5

Step 2 — Read Left to RightLeast (leftmost) first, greatest last. Check: −8 smallest, 5 biggest.

−8, −3, 0, 2, 5

Step 3 — Verify Comparisons−3 > −8 (right), 0 > −3, etc. All good!

Final order correct!

SECTION 7

Strengths & Limitations

Number line wins for SSAT integers!

Method	Strengths	Limitations
With Number Line	Visual—easy for negatives. Quick ordering.	Needs drawing or imagining.
Without Number Line	Fast for positives only.	Confusing for negatives, like −5 vs −2.

🏆 KEY TAKEAWAY

Number line is your superpower for integers, like a map in a video game. It turns confusion into wins!

SECTION 8

Connection to Advanced Theory

Integers are the start—next come fractions and decimals on the same line. This builds to algebra inequalities. You're on the path!

Level	What You Do	Example
Integers (Now)	Whole numbers only.	−3 < 1 < 4
Advanced (Fractions)	Split spots between integers.	−1 < −½ < 0

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Which number is between −2 and 0 on the number line? A. −3 B. −1 C. 1 D. 2 E. −4

PROBLEM 2 — BASIC CALCULATION

Order −1 and 3 from least to greatest. A. 3, −1 B. −1, 3 C. −1 = 3 D. 3 > −1 E. −1 > 3

PROBLEM 3 — INTERMEDIATE

Which is greatest: −5, 0, −2, 4, −1? A. −5 B. 0 C. −2 D. 4 E. −1

PROBLEM 4 — APPLIED

Temperatures: Miami 3°C, Chicago −6°C, NYC −1°C. Order coldest (least) to warmest. A. 3, −1, −6 B. −6, −1, 3 C. −1, −6, 3 D. 3, −6, −1 E. −6, 3, −1

PROBLEM 5 — CRITICAL THINKING

Order least to greatest: −7, 1, −2, 4, 0, −5. A. −7, −5, −2, 0, 1, 4 B. −2, −5, −7, 0, 1, 4 C. 4, 1, 0, −2, −5, −7 D. −7, −2, −5, 0, 1, 4 E. −5, −7, −2, 0, 4, 1

SUMMARY

Lesson Summary

Master comparing and ordering integers with the number line: zero center, left smaller (negatives), right bigger. Plot, compare > < =, read left to right for order.

Practice visuals build speed for SSAT. You can ace this!