Identify a rule for a set of numbers. - ISEE Lower Level Mathematics Achievement

SECTION 1

Where Did Number Patterns Come From?

People have loved finding number patterns for thousands of years! Long ago, people noticed patterns in nature — like the petals on a flower or the stars in the sky. These patterns helped them count, trade, and build amazing things.

3000 BC • Ancient Egypt

Egyptian builders used number patterns to build the pyramids. They counted by twos, fives, and tens!

500 BC • Pythagoras & the Greeks

A mathematician named Pythagoras loved number patterns so much that he started a math club! His group studied odd numbers, even numbers, and more.

1200 AD • Fibonacci's Rabbits

An Italian mathematician named Fibonacci discovered a famous pattern: 1, 1, 2, 3, 5, 8… Each number is the sum of the two before it!

Today • Patterns Everywhere

We use number patterns in computers, video games, music, and even on the ISEE test! Knowing how to spot a rule is a super important skill.

On the ISEE, you will see a set of numbers and need to figure out the rule that connects them. Let's learn how to become a pattern detective!

SECTION 2

Core Ideas About Number Rules

A rule is like a recipe that tells you how to get from one number to the next. Once you know the rule, you can figure out any number in the pattern. Here are the big ideas you need to know.

Look at the Difference

Subtract each number from the next one. If the difference is always the same, you found the rule! For example: 3, 6, 9, 12 → the difference is always 3.

Check for Multiplication

Divide each number by the one before it. If the answer is always the same, the rule uses multiplication. For example: 2, 4, 8, 16 → each number is multiplied by 2.

Going Up or Going Down?

Numbers can grow bigger (add or multiply) or get smaller (subtract or divide). Always notice the direction first!

Test Your Rule

Once you think you found the rule, check it with EVERY number in the set. A good rule works for all of them, not just some.

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KEY TAKEAWAY

Think of a number rule like a machine at a factory. You put a number in, the machine does something to it (like adding 5), and a new number comes out. The same machine works on every number in the pattern!

SECTION 3

See the Pattern!

Let's look at a number pattern on a picture. Seeing the pattern drawn out makes it much easier to understand the rule.

Each circle shows a number in the pattern. The dashed arrows show what happens between each number. Notice how we always add 4 to get to the next number.

See how the difference between every pair of numbers is the same? That is the biggest clue! When the difference stays the same, the rule is add (or subtract) that number. You've got this!

SECTION 4

How to Find the Rule — Step by Step

Here are the most common types of rules you will see on the ISEE. Let's learn how each one works!

Adding Rules

ADD RULE

Next Number = Current Number + Same Amount

Example: 5, 10, 15, 20 → Each number is the current number + 5

Subtracting Rules

SUBTRACT RULE

Next Number = Current Number − Same Amount

Example: 20, 17, 14, 11 → Each number is the current number − 3

Multiplying Rules

MULTIPLY RULE

Next Number = Current Number × Same Amount

Example: 2, 6, 18, 54 → Each number is the current number × 3

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ISEE Test Tip!

On the ISEE, always try adding or subtracting first. Those are the most common rules! If the numbers are growing really fast, then check for multiplication.

SECTION 5

Types of Rules You'll See

Let's look at all the types of rules side by side. This chart will help you tell them apart quickly on test day.

This chart shows the four main types of rules. Notice how multiply rules make numbers grow much faster than add rules. That's a great clue!

Here is a quick trick. If the gaps between numbers stay the same, it's an add or subtract rule. If the gaps keep getting bigger, it might be a multiply rule. Always write down the differences to check!

SECTION 6

Let's Solve One Together!

Here is a problem just like one you might see on the ISEE. Let's work through it step by step!

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Sample Problem

What is the rule for this set of numbers? 6, 12, 18, 24, 30

Finding the Rule: 6, 12, 18, 24, 30

1
Step 1 — Check the Direction

Are the numbers going up or down? They go 6 → 12 → 18 → 24 → 30. The numbers are getting bigger, so the rule probably uses adding or multiplying.

Numbers are going UP
2
Step 2 — Find the Differences

Subtract each number from the next one. 12 − 6 = 6. Then 18 − 12 = 6. Then 24 − 18 = 6. Then 30 − 24 = 6. The difference is always 6!

Difference = 6 every time
3
Step 3 — Write the Rule

Since the difference is always 6, the rule is add 6. You could also say "each number is 6 more than the number before it."

Rule: Add 6
4
Step 4 — Check Your Answer

Let's make sure! Start at 6. Add 6 → 12. ✓ Add 6 → 18. ✓ Add 6 → 24. ✓ Add 6 → 30. ✓ Our rule works for every number. Great job!

✓ Confirmed: Rule is "Add 6"

SECTION 7

Test-Taking Strategies

Knowing the math is great, but knowing how to use it on the ISEE is even better! Here are some smart strategies.

Strategy	How It Helps	Example
Write the differences	Helps you SEE the pattern quickly	Write "+5" between each number
Try each answer choice	Eliminates wrong answers fast	If choice says "add 3" — test it!
Use the answer choices	Answers give you clues about the rule	If choices are "add 2, add 4, ×2, ×4" — try adding first
Never leave it blank	No penalty for wrong answers on the ISEE!	Guess if you run out of time

Strategies for Pattern Problems on the ISEE

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REMEMBER THIS!

Finding a rule is like being a detective. You look for clues (the differences), test your theory (try the rule), and prove it (check every number). If one answer choice doesn't work for every number, cross it out and try the next one!

SECTION 8

Patterns Beyond the ISEE

The skills you're learning now will help you in math class for years to come! Let's see how pattern rules connect to bigger ideas.

What You Know Now	What You'll Learn Later
Add the same number each time (like +3)	In middle school, this is called an "arithmetic sequence"
Multiply by the same number (like ×2)	In middle school, this is called a "geometric sequence"
Use position to find the value	In algebra, you'll write this as a formula using a letter like n
Find the missing number	In algebra, you'll solve equations to find unknown values

How Pattern Skills Grow Over Time

You're building a strong math brain right now! Every pattern you solve makes you a better problem-solver for the future.

SECTION 9

Practice Problems

Time to practice! Try these five problems. Remember: find the differences, figure out the rule, and check your answer. You've got this!

Question 1: CONCEPTUAL

What is the rule for this set of numbers? 2, 4, 6, 8, 10

A. Add 1
B. Add 2
C. Add 3
D. Multiply by 2

Show answer (B)

The correct answer is B, "Add 2." Find the differences: 4 − 2 = 2, 6 − 4 = 2, 8 − 6 = 2, 10 − 8 = 2. The difference is always 2, so the rule is add 2. Choice A (add 1) doesn't work because 2 + 1 = 3, not 4. Choice C (add 3) doesn't work because 2 + 3 = 5, not 4. Choice D (multiply by 2) works for the first two numbers (2 × 2 = 4), but 4 × 2 = 8, not 6, so it fails.

Question 2: BASIC CALCULATION

What is the rule for this set of numbers? 40, 35, 30, 25, 20

A. Subtract 3
B. Subtract 5
C. Subtract 10
D. Subtract 15

Show answer (B)

The correct answer is B, "Subtract 5." The numbers go down, so we're subtracting. Find the differences: 40 − 35 = 5, 35 − 30 = 5, 30 − 25 = 5, 25 − 20 = 5. The difference is always 5. Choice A (subtract 3) gives 40 − 3 = 37, not 35. Choice C (subtract 10) gives 40 − 10 = 30, not 35. Choice D (subtract 15) gives 40 − 15 = 25, not 35.

Question 3: INTERMEDIATE

The numbers below follow a rule. What number comes next? 7, 14, 21, 28, ___

A. 30
B. 32
C. 35
D. 56

Show answer (C)

The correct answer is C, 35. Find the differences: 14 − 7 = 7, 21 − 14 = 7, 28 − 21 = 7. The rule is add 7. So the next number is 28 + 7 = 35. Choice A (30) would mean adding only 2. Choice B (32) would mean adding 4. Choice D (56) would mean multiplying by 2, but 7 × 2 = 14 and 14 × 2 = 28, not 21, so multiplying doesn't work.

Question 4: APPLIED

Maria saves money each week. After week 1 she has $5, after week 2 she has $10, after week 3 she has $15, and after week 4 she has $20. If the pattern continues, how much will she have after week 6?

A. $24
B. $25
C. $30
D. $35

Show answer (C)

The correct answer is C, $30. The rule is add $5 each week (10 − 5 = 5, 15 − 10 = 5, 20 − 15 = 5). After week 4 she has $20. Week 5: $20 + $5 = $25. Week 6: $25 + $5 = $30. Choice A ($24) adds only $4. Choice B ($25) is what she has after week 5, not week 6. Choice D ($35) is what she would have after week 7.

Question 5: CRITICAL THINKING

Look at the number pattern below. 3, 6, 12, 24, ___ What number comes next?

A. 36
B. 42
C. 48
D. 96

Show answer (C)

The correct answer is C, 48. The rule is to multiply each term by 2 to get the next term. Check: 3 × 2 = 6 ✓. 6 × 2 = 12 ✓. 12 × 2 = 24 ✓. 24 × 2 = 48 ✓. Choice A (36) would mean adding 12 each time, but 3 + 12 = 15, not 6, so that rule does not work. Choice B (42) has no consistent rule connecting all the terms. Choice D (96) would mean multiplying by 4 each time, but 3 × 4 = 12, not 6, so that rule does not work either.

SUMMARY

Let's Wrap It Up!

Summary

To find a rule for a set of numbers, start by finding the difference between each pair of numbers. If the difference is always the same, the rule is add or subtract that amount. If the numbers grow very fast, check if each number is being multiplied by the same number.

Always test your rule with every number in the set to make sure it works. On the ISEE, try plugging in each answer choice to see which one fits. Remember, there's no penalty for guessing, so always pick an answer. You've learned an awesome skill — now go find those patterns!