Why Do Patterns Matter?
Humans have been fascinated by number patterns for thousands of years. Ancient civilizations noticed repeating patterns in the stars, the seasons, and even in music. They realized that if you can find the rule behind a pattern, you can predict what comes next.
Mathematicians turned this idea into a powerful tool. A pattern rule is like a recipe that tells you how to get from one number to the next. Once you know the recipe, you can figure out any number in the sequence — even the 100th one!
On the ISEE, pattern questions ask you to figure out the rule that connects the numbers in a list. Let's learn exactly how to do that!
Core Principles of Pattern Rules
Before we start solving problems, you need to know the main types of patterns the ISEE will test. Every pattern follows a rule — a consistent operation (like adding, subtracting, multiplying, or dividing) that takes you from one term to the next.
Arithmetic Patterns
Geometric Patterns
Two-Operation Patterns
Position-Based Rules
Seeing the Pattern
The best first step for any pattern question is to look at the differences between the numbers. The diagram below shows how to find the differences for an arithmetic pattern and a geometric pattern side by side.
Here is the key idea: when you subtract each number from the one after it, if those differences are the same, you have an add or subtract pattern. If the differences keep changing, try dividing each number by the one before it. If those ratios are the same, you have a multiply or divide pattern.
The Math Behind Pattern Rules
On the ISEE, you might need to write a rule using the term's position number (which spot it is in the sequence). We often call the position n. Let's see how to turn a pattern into a formula.
For example, the pattern 5, 8, 11, 14 … has start = 5 and d = 3. The 10th term would be 5 + (10 − 1) × 3 = 5 + 27 = 32.
For the pattern 2, 6, 18, 54 … the start = 2 and r = 3. The 5th term would be 2 × 3⁴ = 2 × 81 = 162.
Classifying Patterns You'll See on the ISEE
Let's look at the different pattern types and how to quickly identify each one. The table below is your cheat sheet.
| Pattern Type | What to Look For | Example | Rule |
|---|---|---|---|
| Add a constant | Same difference between each pair | 10, 15, 20, 25 | Add 5 |
| Subtract a constant | Numbers decrease by same amount | 50, 43, 36, 29 | Subtract 7 |
| Multiply by a constant | Same ratio between each pair | 4, 12, 36, 108 | Multiply by 3 |
| Divide by a constant | Numbers shrink by same factor | 256, 64, 16, 4 | Divide by 4 |
| Two operations | Neither differences nor ratios are constant | 3, 7, 15, 31 | Multiply by 2, then add 1 |
| Position-based | Value relates to its position number (n) | 1, 4, 9, 16 | n × n (perfect squares) |
Worked Example: Finding the Rule Step by Step
Let's walk through a full ISEE-style problem together. Take it one step at a time.
ISEE Strategies: Strengths & Pitfalls
Knowing the math is important, but knowing how to handle tricky answer choices is just as valuable on test day. Let's look at what works and what can trip you up.
| Strategy | Why It Helps | Watch Out For… |
|---|---|---|
| Find differences first | Works on the most common ISEE pattern problems quickly | If differences aren't constant, don't force it — switch to ratios |
| Plug in n = 1, 2, 3 | Great when answer choices are formulas; you can test without solving | One value might match by coincidence — always test at least two |
| Process of elimination | Narrow down choices by testing each one against the first term | Don't spend too long — if two remain, test the second term to decide |
| Look at growth speed | Numbers growing fast → probably multiplication; growing slowly → probably addition | This is a first guess, not proof — always confirm with actual calculation |
Patterns Now and Later
The pattern skills you're learning right now lay the groundwork for bigger ideas in algebra and beyond. Here's a quick preview of how these concepts grow.
| What You Learn Now | What It Becomes Later |
|---|---|
| Arithmetic pattern: add the same number | Linear functions (y = mx + b) in Algebra |
| Geometric pattern: multiply by the same number | Exponential functions in Algebra 2 |
| Position-based rules (like n²) | Quadratic functions in advanced math |
| Finding the nth term | Writing and evaluating algebraic expressions |
Don't worry about those advanced topics yet. The point is that every pattern you solve today is training your brain for the math ahead. You're building a strong foundation!
Practice Problems
Try these five problems on your own. They start easier and get harder. Remember to find the differences first, then check ratios if needed. Good luck!
Lesson Summary
To identify a rule for a pattern, start by finding the differences between consecutive terms. If the differences are constant, you have an arithmetic pattern (add or subtract). If the differences change, find the ratios between terms. If the ratios are constant, you have a geometric pattern (multiply or divide). If neither works, look for a two-operation rule or a position-based rule.
When answer choices are formulas, use the plug-in strategy: substitute n = 1, then n = 2, and see which formula produces the correct output. Always verify with at least two values. Remember, there is no penalty for guessing on the ISEE, so never leave a question blank. Use process of elimination to cross off wrong choices and give yourself the best chance of picking the right answer.