Identify a Rule for a Pattern - ISEE Middle Level Mathematics Achievement

SECTION 1

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!

~500 BC • Pythagorean Patterns

Greek mathematician Pythagoras studied triangular and square number patterns. He found that shapes and numbers are deeply connected.

~1200 • Fibonacci's Famous Sequence

Italian mathematician Fibonacci described the sequence 1, 1, 2, 3, 5, 8, … where each number is the sum of the two before it. This pattern appears in sunflowers, pinecones, and seashells!

~1600s • Algebra Meets Patterns

Mathematicians began writing pattern rules as formulas. This allowed them to jump to any term in a sequence without listing every number.

Today • Patterns Everywhere

Computer scientists, engineers, and scientists use pattern rules every day. The ISEE tests your ability to identify these rules — a skill you will use throughout your math career.

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!

SECTION 2

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

You add or subtract the same number each time. Example: 3, 7, 11, 15 (add 4 each time). This is the most common type on the ISEE.

Geometric Patterns

You multiply or divide by the same number each time. Example: 2, 6, 18, 54 (multiply by 3 each time). The numbers grow (or shrink) quickly.

Two-Operation Patterns

The rule uses two steps, such as "multiply by 2, then add 1." These are trickier because a single operation won't explain the whole pattern.

Position-Based Rules

The rule connects each number to its position in the list. Example: 1, 4, 9, 16 — the rule is "position × position" (1², 2², 3², 4²).

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KEY TAKEAWAY

Think of a pattern rule like a vending machine. You put in the position number (1st, 2nd, 3rd…), and the machine spits out the value. The rule is whatever the machine does on the inside — add 5, multiply by 3, or something fancier. Your job is to figure out what the machine is doing.

SECTION 3

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.

The diagram above compares two common patterns. On the left, the arithmetic pattern grows by adding 3 each time. On the right, the geometric pattern grows by multiplying by 3 each time. Notice how the geometric pattern's numbers get larger much faster.

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.

SECTION 4

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.

ARITHMETIC PATTERN RULE

Value = start + (n − 1) × d

start = the first number in the pattern; n = the position (1st, 2nd, 3rd…); d = the common difference (how much you add each time)

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.

GEOMETRIC PATTERN RULE

Value = start × r^(n − 1)

start = the first number; r = the common ratio (what you multiply by each time); n = the position

For the pattern 2, 6, 18, 54 … the start = 2 and r = 3. The 5th term would be 2 × 3⁴ = 2 × 81 = 162.

SIMPLE POSITION RULE

Value = (multiplier × n) + adjustment

Many ISEE problems give a pattern like 4, 7, 10, 13 and ask which rule works. You can test each answer choice by plugging in n = 1, n = 2, etc. Here, the rule is 3n + 1 because 3(1) + 1 = 4, 3(2) + 1 = 7, and so on.

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ISEE TEST TIP

When answer choices are formulas, don't panic! Just plug in n = 1 and see which formula gives you the first number in the pattern. Then check n = 2. If both match, that's almost certainly the answer. Plug in n = 3 to be sure.

SECTION 5

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)

Common pattern types on the ISEE

Follow this flowchart whenever you see a pattern question on the ISEE. Start at the top by finding the differences, then work your way down to identify the pattern type.

SECTION 6

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.

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SAMPLE PROBLEM

What is the rule for the pattern 7, 13, 19, 25, 31, …? (A) Add 5 (B) Add 6 (C) Multiply by 2 (D) Add 7

Finding the Rule for 7, 13, 19, 25, 31, …

1
Step 1 — Find the Differences

Subtract each number from the one after it: 13 − 7 = 6, 19 − 13 = 6, 25 − 19 = 6, 31 − 25 = 6.

Differences: 6, 6, 6, 6
2
Step 2 — Check: Are the Differences the Same?

Yes! Every difference is 6. This tells us the pattern is arithmetic.

Pattern type: Arithmetic (add a constant)
3
Step 3 — State the Rule

Since you add 6 to get from each term to the next, the rule is "add 6." Check: 7 + 6 = 13 ✓, 13 + 6 = 19 ✓, 19 + 6 = 25 ✓, 25 + 6 = 31 ✓.

The answer is (B) Add 6.
4
Step 4 — Eliminate Wrong Answers

(A) Add 5 would give 7, 12, 17, 22 — that doesn't match. (C) Multiply by 2 would give 7, 14, 28, 56 — way too big. (D) Add 7 would give 7, 14, 21, 28 — close but doesn't match 13 as the second term.

✓

ALWAYS VERIFY

After you pick an answer, test it on at least two terms. If it works for the first pair but not the second, it's wrong. This takes only a few seconds and can save you from silly mistakes!

SECTION 7

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

ISEE Pattern Strategies

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REMEMBER: NO PENALTY FOR GUESSING

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

SECTION 8

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

How patterns connect to future math

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!

SECTION 9

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!

Question 1: CONCEPTUAL

What is the rule for the pattern 4, 8, 12, 16, 20, …?

A. Add 2
B. Add 4
C. Multiply by 2
D. Multiply by 4

Show answer (B)

The correct answer is (B) Add 4. The differences are: 8 − 4 = 4, 12 − 8 = 4, 16 − 12 = 4, 20 − 16 = 4. Each term increases by 4, so the rule is "add 4." (A) Add 2 is wrong because 4 + 2 = 6, not 8. (C) Multiply by 2 works for the first pair (4 × 2 = 8) but fails for the second (8 × 2 = 16, not 12). (D) Multiply by 4 gives 4 × 4 = 16, not 8, so it fails immediately.

Question 2: BASIC CALCULATION

What number comes next in the pattern 3, 9, 27, 81, …?

A. 108
B. 162
C. 243
D. 324

Show answer (C)

The correct answer is (C) 243. Check the ratios: 9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3. The rule is "multiply by 3." So the next term is 81 × 3 = 243. (A) 108 is wrong — that would mean adding 27, which is not a constant addition. (B) 162 is 81 × 2, not 81 × 3. (D) 324 is 81 × 4, using the wrong multiplier.

Question 3: INTERMEDIATE

Which rule describes the pattern 5, 11, 17, 23, 29, …?

A. 6n − 1
B. 6n + 1
C. 5n + 6
D. 7n − 2

Show answer (A)

The correct answer is (A) 6n − 1. Plug in n = 1: 6(1) − 1 = 5 ✓. Plug in n = 2: 6(2) − 1 = 11 ✓. Plug in n = 3: 6(3) − 1 = 17 ✓. It works! (B) 6n + 1 gives 7 when n = 1, not 5. (C) 5n + 6 gives 11 when n = 1, not 5. (D) 7n − 2 gives 5 when n = 1 (seems right!) but 12 when n = 2, not 11. This shows why you must always check at least two values.

Question 4: APPLIED

A baker makes 2 cupcakes on Day 1, 5 cupcakes on Day 2, 8 cupcakes on Day 3, and 11 cupcakes on Day 4. If the pattern continues, how many cupcakes will the baker make on Day 7?

A. 17
B. 18
C. 20
D. 23

Show answer (C)

The correct answer is (C) 20. First find the differences: 5 − 2 = 3, 8 − 5 = 3, 11 − 8 = 3. The rule is "add 3." Continue the pattern: Day 5 = 14, Day 6 = 17, Day 7 = 20. You can also use the formula: Value = 2 + (n − 1) × 3 = 2 + (7 − 1) × 3 = 2 + 18 = 20. (A) 17 is the Day 6 count, not Day 7. (B) 18 would result from adding only 1 extra at Day 6. (D) 23 is the Day 8 count, going one day too far.

Question 5: CRITICAL THINKING

Look at the input-output table: Input: 1, 2, 3, 4, 5 → Output: 3, 8, 15, 24, 35. Which rule matches this table?

A. n² + 2
B. n² + 2n
C. 3n²
D. n² + 3n

Show answer (B)

The correct answer is (B) n² + 2n. Test n = 1: 1 + 2 = 3 ✓. Test n = 2: 4 + 4 = 8 ✓. Test n = 3: 9 + 6 = 15 ✓. All match! (A) n² + 2 gives 3 when n = 1 (works!) but 6 when n = 2 (not 8), so it fails. (C) 3n² gives 3 when n = 1 but 12 when n = 2 (not 8). (D) n² + 3n gives 4 when n = 1, which doesn't match the first output of 3. The trick is to check at least two inputs to be certain. Notice the differences between outputs are 5, 7, 9, 11 — they increase by 2, which is a hint that the rule involves n² (a squared term).

SUMMARY

Lesson Summary

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.