Why Do Patterns Matter?
Humans have been fascinated by patterns since ancient times. Early farmers noticed patterns in the seasons to know when to plant crops. Ancient astronomers tracked patterns in the night sky to predict eclipses. Patterns are everywhere in nature, music, art, and especially math.
In mathematics, a pattern rule (also called a rule or function) is the instruction that tells you how to get from one number to the next. Finding this rule is like cracking a secret code. Once you know the rule, you can predict any term in the sequence.
The key question this lesson answers is: How do you figure out the rule that makes a pattern tick? Let's find out!
Core Principles of Pattern Rules
Before we dive into solving problems, let's learn the building blocks. There are a few key ideas you need to understand about patterns and their rules.
Sequence
Term
Common Difference
Common Ratio
Two-Step Rules
Seeing the Pattern
The best way to find a pattern rule is to look at the differences between consecutive terms. The diagram below shows how this works for the sequence 4, 7, 10, 13, 16.
Notice how we checked every gap between terms. On the ISEE, always check at least two or three gaps. Sometimes the first gap looks like one rule, but the next gap is different. That means you might be dealing with a multiplication pattern or a two-step rule.
The Math Behind Pattern Rules
There are three main types of pattern rules you'll see on the ISEE. Let's look at the formula (or shortcut) for each one.
Type 1: Additive Patterns (Add or Subtract)
Example: 20, 17, 14, 11, … → The common difference is −3 (you subtract 3 each time).
Type 2: Multiplicative Patterns (Multiply or Divide)
Example: 3, 12, 48, 192, … → Each term is multiplied by 4. The common ratio is 4.
Type 3: Two-Step Rules
Classifying Pattern Types
The diagram below shows a decision tree. It's a step-by-step guide to help you figure out which type of pattern you're dealing with. Follow the arrows based on what you see in the numbers.
| Pattern Type | Example Sequence | Rule | How to Spot It |
|---|---|---|---|
| Additive | 5, 9, 13, 17, 21 | Add 4 | Same difference every time |
| Subtractive | 30, 25, 20, 15, 10 | Subtract 5 | Same negative difference |
| Multiplicative | 2, 10, 50, 250 | Multiply by 5 | Same ratio between terms |
| Two-Step | 1, 5, 13, 29 | ×2, then +3 | Differences change; ratios aren't constant |
Worked Example: Cracking the Code
Let's walk through a full ISEE-style problem step by step.
ISEE Strategies & Common Traps
The ISEE tests pattern rules in both regular word problems and quantitative comparison questions. Here are strategies for both types, plus common mistakes to avoid.
| Strategy | Why It Works | Watch Out For |
|---|---|---|
| Check differences first | Most ISEE patterns are additive, so this catches the majority. | Don't assume additive if the differences aren't constant. |
| Check ratios second | Quickly identifies multiplication/division patterns. | Remember to divide term 2 by term 1, not the other way around. |
| Use answer choices | On the ISEE, you can work backward from the choices to test which rule fits. | Don't just test one choice — make sure it works for ALL terms. |
| Eliminate wrong answers | Even if you're unsure, crossing out impossible answers raises your odds. | There's no penalty for guessing on the ISEE — always pick an answer! |
Connecting to Bigger Ideas
Pattern rules aren't just an ISEE topic — they connect to important ideas you'll use in algebra and beyond. Understanding pattern rules now gives you a head start on more advanced math.
| What You're Learning Now | What It Becomes Later |
|---|---|
| Additive patterns (add the same number) | Arithmetic sequences & linear equations (y = mx + b) |
| Multiplicative patterns (multiply by the same number) | Geometric sequences & exponential growth |
| Finding the nth term | Writing formulas for sequences & functions |
| Testing a rule with multiple values | Verifying solutions in algebra & proofs |
The skill of looking for a rule, testing it, and applying it is the core of mathematical thinking. Every time you identify a pattern rule, you're training your brain to think like a mathematician. That's a superpower that will help you far beyond test day.
Practice Problems
Try these five problems. They get harder as you go. Remember to use the decision tree: check differences first, then ratios, then try two-step rules.
Lesson Summary
A pattern rule is the operation that turns one term into the next in a sequence. To find the rule, start by computing the differences between consecutive terms. If the differences are constant, the pattern is additive. If not, check the ratios — equal ratios mean a multiplicative pattern. If neither works, look for a two-step rule that combines operations.
On the ISEE, always verify your rule against every given term before selecting an answer. Use process of elimination on the answer choices, and remember: there is no penalty for guessing, so never leave a question blank. For quantitative comparisons, compute both columns' values and compare. You've got this!