Identify a Pattern Rule - ISEE Middle Level Quantitative Reasoning

SECTION 1

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.

~2000 BCE • Babylonian Number Tables

Babylonian scribes carved tables of squares and cubes into clay tablets. They used patterns in these tables to solve practical problems like dividing land.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid studied number patterns like prime numbers and even/odd sequences. His work laid the foundation for number theory.

1202 • Fibonacci's Famous Sequence

Leonardo Fibonacci introduced a special pattern — 1, 1, 2, 3, 5, 8, 13 — where each number is the sum of the two before it. This pattern appears in flowers, shells, and galaxies!

Today • Patterns on the ISEE

The ISEE Middle Level test asks you to identify pattern rules in number sequences. This skill tests your ability to think logically and find relationships between numbers.

The key question this lesson answers is: How do you figure out the rule that makes a pattern tick? Let's find out!

SECTION 2

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

A sequence is an ordered list of numbers that follows a rule. Example: 3, 6, 9, 12, …

Term

Each number in a sequence is called a term. The 1st term is the starting number. The 2nd term is next, and so on.

Common Difference

If you add (or subtract) the same number each time, that number is the common difference. Example: In 5, 8, 11, 14, the common difference is +3.

Common Ratio

If you multiply (or divide) by the same number each time, that number is the common ratio. Example: In 2, 6, 18, 54, the common ratio is ×3.

Two-Step Rules

Some patterns use two operations, like "multiply by 2, then add 1." These are trickier, but the same approach works — look at how the numbers change.

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KEY TAKEAWAY

Think of a pattern rule like a recipe. If someone gives you cookies that taste the same every time, there must be a recipe. The recipe is the pattern rule. Your job is to figure out the recipe by tasting (looking at) the cookies (numbers)!

SECTION 3

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.

The curved arrows show the difference between each pair of terms. Since every difference is +3, the pattern rule is "add 3."

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.

SECTION 4

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)

ADDITIVE PATTERN

Next Term = Current Term + d

Where d is the common difference. If d is positive, you're adding. If d is negative, you're subtracting.

Example: 20, 17, 14, 11, … → The common difference is −3 (you subtract 3 each time).

Type 2: Multiplicative Patterns (Multiply or Divide)

MULTIPLICATIVE PATTERN

Next Term = Current Term × r

Where r is the common ratio. If the numbers get bigger, r is greater than 1. If they get smaller, r is a fraction (or you're dividing).

Example: 3, 12, 48, 192, … → Each term is multiplied by 4. The common ratio is 4.

Type 3: Two-Step Rules

TWO-STEP PATTERN

Next Term = (Current Term × r) + c

Some patterns combine multiplication and addition. For example, "multiply by 2, then add 1" turns 3 into 7 (3 × 2 + 1 = 7), then 7 into 15 (7 × 2 + 1 = 15).

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ISEE Test Tip

When the differences between terms are NOT the same, try dividing each term by the one before it. If you get the same answer every time, it's a multiplication pattern. If neither works, try a two-step rule.

SECTION 5

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.

Follow this decision tree whenever you encounter a pattern problem on the ISEE. Start by finding differences. If those are constant, it's additive. If not, check ratios for a multiplicative pattern. Otherwise, try a two-step rule.

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

Common pattern types on the ISEE

SECTION 6

Worked Example: Cracking the Code

Let's walk through a full ISEE-style problem step by step.

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Sample Problem

What is the next number in the pattern: 3, 6, 12, 24, …?

Step-by-Step Solution

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Step 1 — Find the Differences

Subtract each term from the next: 6 − 3 = 3, 12 − 6 = 6, 24 − 12 = 12. The differences are 3, 6, 12 — they're not the same. So this is NOT a simple additive pattern.

Differences: 3, 6, 12 → Not constant
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Step 2 — Check for a Common Ratio

Divide each term by the previous one: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. The ratio is 2 every time!

Common ratio = 2 ✓
3
Step 3 — State the Rule

The pattern rule is: multiply by 2. Each term is twice the previous term.

Rule: ×2
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Step 4 — Find the Next Term

Apply the rule to the last term: 24 × 2 = 48.

Next term = 48
5
Step 5 — Verify

Double-check: 3 × 2 = 6 ✓, 6 × 2 = 12 ✓, 12 × 2 = 24 ✓, 24 × 2 = 48 ✓. The rule works for all terms.

Answer confirmed: 48

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STRATEGY REMINDER

Always verify your rule against ALL the given terms, not just the first pair. Think of it like solving a puzzle — if one piece doesn't fit, the whole picture is off.

SECTION 7

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!

Pattern Problem Strategies for the ISEE

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NO PENALTY FOR GUESSING

The ISEE does not take away points for wrong answers. If you're stuck, eliminate what you can and guess from what's left. Never leave a question blank!

SECTION 8

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

From Patterns to Algebra

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.

SECTION 9

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.

Question 1: CONCEPTUAL

What is the rule for the pattern 10, 15, 20, 25, 30, …? (A) Multiply by 5 (B) Add 5 (C) Add 10 (D) Multiply by 2

Show answer

The correct answer is (B) Add 5. Find the differences: 15 − 10 = 5, 20 − 15 = 5, 25 − 20 = 5, 30 − 25 = 5. The common difference is 5, so the rule is "add 5." Choice (A) is wrong because 10 × 5 = 50, not 15. Choice (C) is wrong because 10 + 10 = 20, not 15. Choice (D) is wrong because 10 × 2 = 20, not 15.

Question 2: BASIC CALCULATION

What is the next number in the sequence 4, 12, 36, 108, …? (A) 144 (B) 216 (C) 324 (D) 432

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The correct answer is (C) 324. Check the ratios: 12 ÷ 4 = 3, 36 ÷ 12 = 3, 108 ÷ 36 = 3. The common ratio is 3. The next term is 108 × 3 = 324. Choice (A) 144 would come from adding 36, which is not the rule. Choice (B) 216 would come from multiplying by 2, which is incorrect. Choice (D) 432 would come from multiplying by 4, which is also incorrect.

Question 3: INTERMEDIATE

Quantitative Comparison: The pattern rule for the sequence 2, 5, 11, 23, 47, … is "multiply by 2, then add 1." Column A: The 6th term of the sequence Column B: 95 (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) Cannot be determined

Show answer

The correct answer is (C) The two quantities are equal. Apply the rule to the 5th term: 47 × 2 + 1 = 95. So the 6th term is 95. Column A = 95 and Column B = 95, making them equal. We can verify the rule: 2 × 2 + 1 = 5 ✓, 5 × 2 + 1 = 11 ✓, 11 × 2 + 1 = 23 ✓, 23 × 2 + 1 = 47 ✓. Since both columns contain only numbers (no variables), answer (D) is not possible here.

Question 4: APPLIED

A gym membership starts at $20 in the first month. Each month after that, the cost increases by $15. How much does the membership cost in the 8th month? (A) $120 (B) $125 (C) $135 (D) $140

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The correct answer is (B) $125. This is an additive pattern with a starting value of $20 and a common difference of $15. The cost in the nth month is: 20 + 15 × (n − 1). For the 8th month: 20 + 15 × 7 = 20 + 105 = $125. You can also list the terms: 20, 35, 50, 65, 80, 95, 110, 125. Choice (A) $120 comes from miscalculating 15 × 8 − 15 incorrectly. Choice (C) $135 comes from using 20 + 15 × (8 − 1) + 10 erroneously. Choice (D) $140 comes from calculating 20 + 15 × 8, which counts 8 increases instead of 7.

Question 5: CRITICAL THINKING

Quantitative Comparison: Sequence P: 1, 4, 9, 16, 25, … Sequence Q: 3, 6, 12, 24, 48, … Column A: The 6th term of Sequence P Column B: The 6th term of Sequence Q (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

Show answer

The correct answer is (B) Column B is greater. Sequence P follows the pattern of perfect squares: 1², 2², 3², 4², 5², … so the 6th term is 6² = 36. Sequence Q is a multiplicative pattern where each term is doubled: the common ratio is 2. The 6th term of Q is 48 × 2 = 96. Since 36 < 96, Column B is greater. Answer (A) is wrong because 36 is less than 96. Answer (C) is wrong because 36 ≠ 96. Answer (D) is wrong because both columns are fixed numbers with no variables.

SUMMARY

Lesson Summary

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!