Where Do Number Patterns Come From?
People have loved number patterns for thousands of years! Long ago, mathematicians noticed that some numbers follow a special rule. When you add the same amount each time, you get what we call an arithmetic sequence (say it like: air-ith-MET-ick). That just means a list of numbers where you keep adding the same number.
On the ISEE, you will see a list of numbers with one missing. Your job is to figure out the pattern and fill in the blank. Let's learn how!
Core Ideas You Need to Know
Before we solve problems, let's learn four big ideas. These are your superpowers for finding missing numbers!
Sequence
Common Difference
The Rule
Find the Gap
See the Pattern!
Let's look at the sequence 4, 7, 10, ?, 16. The picture below shows how the numbers grow by jumping the same amount each time.
See how every jump is the same size? That's the secret of an arithmetic sequence. Once you find the jump, you can fill in any missing number!
The Three-Step Method
Here is a simple three-step method you can use every time. It works like a recipe!
Different Kinds of Patterns
On the ISEE, the missing number can be anywhere — at the beginning, in the middle, or at the end. The pattern might go up (adding) or go down (subtracting). Let's see all the types!
Notice something cool? For Type 3, you subtract to go backward. It's like rewinding! But the common difference is still the same number.
Let's Solve One Together
Here is a problem just like you might see on the ISEE. Let's solve it step by step!
Tips and Common Traps
Even great math students can make mistakes on pattern problems. Here are some tips to help you and traps to avoid.
| ✅ Do This | ❌ Avoid This |
|---|---|
| Check the difference between at least two pairs of neighbors. | Only checking one pair — you might pick the wrong difference. |
| Read the full sequence with your answer plugged in. | Picking an answer without double-checking. |
| Use subtraction to go backward when the blank is at the start. | Adding when you should subtract. |
| Use process of elimination — try each answer choice if stuck. | Leaving the question blank — there's no penalty for guessing! |
Patterns Beyond Adding
On the ISEE, most sequences use adding or subtracting. But as you grow as a math student, you will see fancier patterns too. Here's a sneak peek!
| Type | What Happens | Example |
|---|---|---|
| Adding (arithmetic) | Add the same number each time | 2, 5, 8, 11, 14 (+3) |
| Subtracting | Subtract the same number each time | 30, 25, 20, 15, 10 (−5) |
| Multiplying (future topic) | Multiply by the same number each time | 2, 4, 8, 16, 32 (×2) |
For now, focus on adding and subtracting patterns. That's what the ISEE Lower Level tests. You've got this!
Practice Problems
Time to practice! Try each problem on your own, then check the answer. Remember: find the common difference first, then use it to find the missing number.
Lesson Summary
An arithmetic sequence is a list of numbers where you add (or subtract) the same amount each time. That amount is called the common difference. To find a missing number, use the three-step method: first, find the common difference by subtracting two neighbors. Second, check it with another pair. Third, add or subtract to fill in the blank.
Remember that the blank can appear at the beginning, middle, or end of the sequence. If it's at the start, subtract the common difference from the next number. Always double-check by reading the whole sequence with your answer plugged in. On the ISEE, there is no penalty for guessing, so always answer every question. You've got this!