Why Do Patterns Matter?
People have loved patterns for thousands of years! A sequence is a list of numbers that follow a rule. Long ago, mathematicians noticed that patterns are everywhere — in nature, music, and buildings.
Think about counting by twos: 2, 4, 6, 8. That's a sequence! Once you know the rule, you can find any number in the list — even a missing one.
So here's the big question: if someone hides one number in a pattern, can you figure out what it is? That's exactly what we'll learn today!
Core Ideas About Sequences
Before we start finding missing numbers, let's learn a few important ideas. These will be your tools!
What Is a Sequence?
What Is a Rule?
Find the Pattern First
Check Your Work
See the Pattern!
Let's look at a picture that shows how a sequence works. This diagram shows the sequence 3, 6, 9, ?, 15 and how to find the missing term.
See how we subtract neighbors to find the rule? That trick works with most sequences on the ISEE. Always check that your answer fits both the number before AND the number after!
The Three Main Types of Sequences
On the ISEE, you'll mostly see three types of sequences. Let's learn each one!
Type 1: Add (or Subtract) the Same Number
Type 2: Multiply (or Divide) by the Same Number
Type 3: Two-Step Rules
Your Step-by-Step Strategy
Here's a simple plan you can follow every time you see a sequence question on the ISEE. Think of it as your detective toolkit!
If subtraction doesn't show a pattern, try dividing each term by the one before it. If each time you get the same answer, the rule is multiply! For example, 3, 6, 12, 24 → each term is the one before it × 2.
Worked Example
Let's work through a problem together, step by step. Ready? Here we go!
Great job following along! See how checking both sides at the end gives you confidence? On the real ISEE, that quick check can save you from picking the wrong answer.
Common Mistakes & How to Avoid Them
Even smart students make mistakes on sequence problems. Here are the most common ones — and how to dodge them!
| Common Mistake | Why It Happens | How to Fix It |
|---|---|---|
| Only checking one pair of numbers | You look at only the first two terms and assume you know the rule. | Always check at least two pairs before deciding the rule. |
| Forgetting to check after the blank | You apply the rule to get the missing term but don't see if the NEXT term also follows the rule. | Always check both sides of the blank. |
| Mixing up add and multiply | The numbers get bigger fast, so you think it's adding, but it's actually multiplying. | If the differences aren't the same, try dividing each number by the one before it. |
| Rushing and making an arithmetic error | You know the rule but mess up the addition or subtraction. | Slow down! Write out your work in the test booklet. |
Beyond Simple Sequences
Once you're comfortable with "add the same number" sequences, you might see trickier patterns. Here's how the basic idea connects to more advanced ones.
| What You Know Now | What Comes Next |
|---|---|
| Add the same number (like +5) | Add a changing amount (like +1, +2, +3, +4) |
| Multiply by the same number (like ×2) | Patterns that switch between add and multiply |
| Sequences of whole numbers | Sequences with fractions or decimals |
| Numbers going up | Numbers going down (subtracting or dividing) |
For now, focus on the patterns we learned today. They cover most of what you'll see on the ISEE Lower Level. As you get more practice, the trickier patterns will feel easy too!
Practice Problems
Time to try some on your own! Remember your detective plan: find the differences, spot the rule, fill the gap, and check. You've got this!
Putting It All Together
A sequence is a list of numbers that follow a rule. To find a missing term, use the four-step detective plan: find the differences, spot the rule, fill the gap, and check both sides. Most ISEE sequences use adding, subtracting, or multiplying by the same number.
If the missing number is at the beginning, work backwards using the opposite operation. Watch out for changing-difference patterns where the amount added goes up each time. And remember — on the ISEE, there's no penalty for guessing, so always answer every question. Use process of elimination to cross out wrong answers. You've got this!