Ordering Integers — Compare and order integers on a number...

SECTION 1

Historical Context & Motivation

For thousands of years, people only used counting numbers like 1, 2, and 3. But what happens when you owe someone money, or the temperature drops below zero? Ancient mathematicians realized they needed numbers that go in the opposite direction from counting numbers. That's where negative numbers come in. Together, the positive numbers, negative numbers, and zero form what we call integers.

200 BCE

China — First Negative Numbers

Chinese mathematicians used red rods for positive numbers and black rods for negative numbers to track debts and credits.

628 CE

India — Brahmagupta's Rules

The Indian mathematician Brahmagupta wrote the first rules for adding and subtracting with negative numbers and zero.

1600s

Europe — The Number Line Appears

European mathematicians began drawing number lines to show how negative and positive numbers relate to each other visually.

Today

SHSAT and Beyond

Ordering integers is a core skill on the SHSAT. You'll use it in algebra, graphing, and real-world problems like temperature and elevation.

Here's the big question this lesson answers: if you have a list of integers—some positive, some negative—how do you figure out which is the smallest and which is the biggest? A number line gives you a simple, visual way to do exactly that.

SECTION 2

Core Principles & Definitions

Before we start ordering, let's lock down the key ideas you'll need. These four principles are the building blocks of everything in this lesson.

What Are Integers?

Integers are whole numbers that can be positive, negative, or zero. Examples: −5, −1, 0, 3, 12. They do NOT include fractions or decimals.

The Number Line

A number line is a straight line where every integer has its own spot. Negative numbers sit to the left of 0, and positive numbers sit to the right.

Left Is Less

On a number line, any number to the left is always less than a number to the right. So −7 < −2 because −7 is farther to the left.

Comparison Symbols

Use < (less than), > (greater than), and = (equal to) to show how two integers relate. The open end of the symbol always points toward the bigger number.

✦ KEY TAKEAWAY

Think of the number line like a thermometer turned on its side. The farther left you go, the colder (smaller) the number. The farther right you go, the hotter (bigger) the number. Zero is the freezing point right in the middle!

SECTION 3

Visual Explanation — The Number Line

Let's look at a number line with several integers plotted on it. Pay close attention to where each number sits. Remember: farther left means smaller, and farther right means larger.

Five integers are plotted: −7 (red), −4 (orange), −1 (violet), 3 (cyan), and 6 (green). Notice that −7 is farthest left, so it is the smallest. 6 is farthest right, so it is the largest.

From the diagram, you can read the integers in order from least to greatest: −7 < −4 < −1 < 3 < 6. You just go from left to right along the line. That's really all there is to it! The tricky part is that with negative numbers, a "bigger looking" digit can actually be a smaller number. For example, −7 has a bigger digit than −4, but −7 is less than −4.

SECTION 4

Mathematical Framework — Comparing Integers

You don't always have a number line drawn for you on the SHSAT. Here are the rules that let you compare any two integers in your head.

RULE 1 — POSITIVE vs. NEGATIVE

Any positive integer > 0 > Any negative integer

Every positive number is greater than every negative number. Zero sits right in between.

RULE 2 — TWO POSITIVE INTEGERS

If a > b (both positive), then a is farther right on the number line.

This works just like normal counting. 8 > 5 because 8 is farther to the right.

RULE 3 — TWO NEGATIVE INTEGERS

If |a| > |b|, then −a < −b

The absolute value (the distance from zero, ignoring the sign) tells you which negative number is farther from zero. The negative number farther from zero is smaller. Example: |−9| = 9 and |−3| = 3. Since 9 > 3, we know −9 < −3.

ORDERING NOTATION

a < b means a is to the LEFT of b on the number line

The symbol < points to the smaller number. The symbol > points away from the smaller number. A helpful trick: the "mouth" of the symbol always opens toward the bigger number, like a hungry alligator eating the larger value.

💡 SHSAT TIP

When a question says "order from least to greatest," you're going from left to right on the number line. When it says "order from greatest to least," you go from right to left.

SECTION 5

Detailed Breakdown — Why Negatives Trick Us

The most common mistake students make on the SHSAT is thinking that a negative number with a bigger digit is "bigger." Let's break down why this happens and how to avoid it.

The top boxes show the common mistake (left, red) vs. correct reasoning (right, green). The number line below confirms that −8 is farther to the left than −3, so −8 < −3.

Here's the key rule for negative numbers: the closer a negative number is to zero, the greater it is. So −1 is greater than −100 because −1 is much closer to zero. Think of it like debt. If you owe $1, you're in better shape than if you owe $100!

Common integer comparisons and their results

Comparison	Number Line Position	Result
−2 vs. −9	−2 is closer to 0 (farther right)	−2 > −9
−5 vs. 1	−5 is left of 0; 1 is right of 0	−5 < 1
0 vs. −6	0 is to the right of −6	0 > −6
−12 vs. −3	−12 is farther left than −3	−12 < −3

SECTION 6

Worked Example

Let's walk through an SHSAT-style problem step by step.

Order from least to greatest: 4, −6, 0, −1, 7, −6, 3

Step 1 — Separate Negatives, Zero, and PositivesGroup the numbers into three categories. Negatives: −6, −1, −6. Zero: 0. Positives: 4, 7, 3.

Negatives → Zero → Positives

Step 2 — Order the Negative NumbersFor negative numbers, the one with the largest absolute value is the smallest. |−6| = 6, |−1| = 1. Since 6 > 1, we know −6 < −1. We have two copies of −6.

−6, −6, −1

Step 3 — Order the Positive NumbersFor positive numbers, just use normal counting order. 3 < 4 < 7.

3, 4, 7

Step 4 — Combine All GroupsPut the negatives first (smallest), then zero, then the positives (largest). This gives us the full list from least to greatest.

−6, −6, −1, 0, 3, 4, 7

✔ CHECK YOUR WORK

After ordering, quickly imagine placing each number on a number line from left to right. Does each number land to the right of the one before it? If yes, you're correct!

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students can slip up with integers. Here are the most common errors that show up on the SHSAT and strategies to avoid each one.

Top 4 SHSAT mistakes with integer ordering

Mistake	Why It Happens	How to Fix It
Thinking −8 > −3	You focus on the digit 8 being bigger than 3	Picture the number line. −8 is farther LEFT, so it's less.
Forgetting that 0 > any negative	You think 0 means "nothing" so it must be smallest	Zero is always to the RIGHT of all negatives on the line.
Mixing up < and >	The symbols look similar, especially under time pressure	The "mouth" opens toward the BIGGER number: 5 > 2 and 2 < 5.
Ordering greatest to least when asked for least to greatest	Reading the question too quickly	Circle the words "least to greatest" or "greatest to least" before solving.

✦ KEY TAKEAWAY

Think of negative numbers like floors in a building's basement. Basement level 8 (−8) is deeper underground than basement level 3 (−3). Deeper = lower = less. The lobby (0) is always above every basement floor.

SECTION 8

Connection to Advanced Topics

Once you master ordering integers, you've built a foundation for many harder topics you'll see in algebra and on advanced SHSAT questions. Let's see how this concept connects to what's coming next.

How integer ordering connects to future math

This Lesson (Integers)	What Comes Next
Ordering integers on a number line	Ordering rational numbers (fractions and decimals) on a number line
Comparing two integers with < and >	Solving inequalities like x > −3 and graphing solution sets
Understanding negative values	Plotting points in all four quadrants of the coordinate plane
Using absolute value to compare negatives	Solving absolute value equations and inequalities

On the SHSAT, you might also see integer ordering hidden inside word problems. For example, a question might describe temperatures of different cities or elevations of different locations and ask you to rank them. The strategy is the same: convert each description to an integer, then use the number line to order them.

SECTION 9

Practice Problems

Try these five problems. They start simple and get harder. For each one, imagine a number line to help guide your thinking.

PROBLEM 1 — CONCEPTUAL

True or false: −15 is greater than −4. Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

Place the correct symbol (<, >, or =) between each pair: (a) −7 ___ 2, (b) −3 ___ −10, (c) 0 ___ −5.

PROBLEM 3 — INTERMEDIATE

Order these integers from least to greatest: 5, −12, 8, −3, 0, −7, 2.

PROBLEM 4 — APPLIED

Five cities recorded these temperatures on a winter morning: Chicago −8°F, Miami 65°F, Denver −2°F, Anchorage −14°F, New York 12°F. List the cities from coldest to warmest.

PROBLEM 5 — CRITICAL THINKING

Integer A is 5 units to the left of −2 on a number line. Integer B is 3 units to the right of −7. Integer C is halfway between −10 and 4. Order A, B, and C from greatest to least.

SUMMARY

Lesson Summary

Integers are whole numbers that can be positive, negative, or zero. A number line is the tool you use to compare and order them. The most important rule is simple: numbers farther to the left are smaller, and numbers farther to the right are larger. Every positive integer is greater than zero, and zero is greater than every negative integer.

When comparing two negative numbers, the one with the smaller absolute value (closer to zero) is actually the greater number. Use the symbols < (less than) and > (greater than) to show the relationship, with the open end always pointing at the bigger number. On the SHSAT, always picture a number line before you answer!