Recognize a pattern in a sequence of numbers - CogAT Grades K 2

SECTION 1

Why Do We Look for Patterns?

People have been finding patterns in numbers for a very, very long time! A pattern is like a secret rule. When you figure out the rule, you know what comes next. This is a skill that helps us count, build, and solve problems every day.

Long Ago • Counting with Fingers

Early people counted 1, 2, 3, 4, 5 on their fingers. They noticed numbers go up by one each time!

Ancient Times • Patterns on Walls

People in Egypt and Greece drew number patterns on walls. They used dots and shapes to show how numbers grow.

In School • Skip Counting

Kids learn to count by 2s, 5s, and 10s. These are all number patterns! Each number follows a rule.

Today • Patterns Are Everywhere

We use patterns in math, music, games, and even on tests like the CogAT. Finding patterns makes you a great thinker!

So here is the big question: if you see the numbers 2, 4, 6, 8 — can you guess what comes next? That is what this lesson is all about. Let's learn how to find the hidden rule in a row of numbers!

SECTION 2

What Is a Number Pattern?

A number pattern is a list of numbers that follow a rule. The rule tells you how to get from one number to the next. When you know the rule, you can figure out any number in the list!

Sequence

A sequence is a row of numbers in a special order, like 1, 2, 3, 4.

Rule

The rule is what you do each time to get to the next number, like adding 2.

Growing Pattern

Numbers get bigger each time. Example: 3, 6, 9, 12. The numbers go up!

Shrinking Pattern

Numbers get smaller each time. Example: 10, 8, 6, 4. The numbers go down!

Repeating Pattern

The same group of numbers shows up again and again. Example: 1, 2, 1, 2, 1, 2.

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KEY TAKEAWAY

Think of a number pattern like a jump rope song. Every time you jump, you say the next number. If you always jump the same way (like adding 2), you know exactly which number comes next. The jump is the rule!

SECTION 3

See the Pattern!

Let's look at a picture that shows how a number pattern works. In this picture, you can see each number and the hop between them. Each hop is the same size. That is how you know it is a pattern!

Each circle shows a number. The curved arrows show the rule: add 2 every time. The last circle has a question mark. Can you figure out it is 10?

In the picture above, each hop says "+2." That means you add 2 to get to the next number. 2 plus 2 is 4, 4 plus 2 is 6, 6 plus 2 is 8, and 8 plus 2 is 10. The answer is 10!

SECTION 4

How to Find the Rule

Finding the rule is like being a number detective! You look at two numbers that sit next to each other. Then you ask: what did I do to the first number to get the second number? Let's learn the steps.

Step-by-Step Detective Work

Look at two neighbors. Pick the first two numbers. Are they going up or down?
Find the hop. Count how many you add or take away to get from one number to the next.
Check the hop again. Try the same hop between the next two numbers. Does it still work?
Use the rule! If the hop is the same every time, you found the rule. Use it to find the missing number.

THE PATTERN RULE

Next Number = This Number + Hop

The hop is the amount you add (or take away) each time. If the hop is +3, you add 3. If the hop is −1, you take away 1.

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Tip!

If numbers go up, you are adding. If numbers go down, you are taking away. Look at whether the numbers get bigger or smaller first!

SECTION 5

Different Kinds of Number Patterns

Not all patterns look the same! Some go up, some go down, and some repeat. Let's look at a chart that shows different types of patterns you might see.

This picture shows three types of patterns: growing (numbers go up), shrinking (numbers go down), and repeating (a group of numbers shows up again and again). It also shows skip counting examples at the bottom.

Pattern Type	Example	Rule
Add 1	1, 2, 3, 4, 5	Each number is 1 more
Add 2	2, 4, 6, 8, 10	Each number is 2 more
Add 5	5, 10, 15, 20, 25	Each number is 5 more
Subtract 1	9, 8, 7, 6, 5	Each number is 1 less
Repeating	3, 5, 3, 5, 3, 5	3 then 5, over and over

Common patterns you might see on the CogAT

SECTION 6

Let's Solve One Together!

Let's be number detectives! Here is a sequence: 5, 10, 15, 20, ?. What number goes where the question mark is?

Find the Missing Number

1
Step 1 — Look at the First Two Numbers

The first number is 5. The second number is 10. Is 10 bigger or smaller than 5? It is bigger! So this is a growing pattern.
2
Step 2 — Find the Hop

How much bigger is 10 than 5? Let's count up from 5: 6, 7, 8, 9, 10. That is 5 hops! So the rule might be add 5.

Hop = +5
3
Step 3 — Check the Hop

Does the same hop work between 10 and 15? 10 + 5 = 15. Yes! Does it work between 15 and 20? 15 + 5 = 20. Yes! The rule is definitely add 5.

Rule confirmed: Add 5 every time ✓
4
Step 4 — Use the Rule

Now use the rule on the last number. 20 + 5 = 25. The missing number is 25!

Answer: 25

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REMEMBER

Always check the hop more than once! If it works between every pair of neighbors, you found the right rule.

SECTION 7

What Makes Patterns Easy or Tricky?

Some patterns are easy to spot, and some are a little harder. Let's look at what makes a pattern easy and what makes it tricky.

Easy Patterns 😊	Tricky Patterns 🤔
The hop is small, like +1 or +2	The hop is bigger, like +4 or +5
Numbers only go up	Numbers go up and down (repeating)
You can see many numbers	Only a few numbers are shown
The missing number is at the end	The missing number is in the middle

Easy patterns vs. tricky patterns

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KEY TAKEAWAY

Even tricky patterns follow a rule. It is like a treasure hunt. The treasure (the answer) is always there — you just need to follow the clues (the hops) to find it! Take your time and check each hop carefully.

SECTION 8

Getting Ready for Bigger Patterns

Right now you are learning patterns with small numbers. As you grow up, you will see even cooler patterns! Let's peek at what is coming.

What You Learn Now	What Comes Later
Add the same number each time (like +2)	Multiply by the same number each time (like ×2)
Small numbers (1 to 20)	Bigger numbers (100s and 1000s)
Find the next number	Write a rule using letters like n
Number patterns	Shape and picture patterns too

Now vs. later

Every big math idea starts with something simple. The patterns you learn today are the building blocks for all the amazing math you will do later. You are building your math superpower right now!

SECTION 9

Practice Problems

Now it is your turn! Try to find the pattern and the missing number in each problem. Take your time and remember to find the hop.

Question 1: CONCEPTUAL

Look at this sequence: 1, 2, 3, 4, ___. What number comes next?

Show answer

The answer is 5. The rule is add 1. Each number is 1 more than the one before it. 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, and 4 + 1 = 5. This is just regular counting!

Question 2: BASIC CALCULATION

What number is missing? 2, 4, 6, ___, 10

Show answer

The answer is 8. The rule is add 2. From 2 to 4 is +2. From 4 to 6 is +2. So the next hop is 6 + 2 = 8. Then 8 + 2 = 10, which matches!

Question 3: INTERMEDIATE

Find the missing number: 10, 8, 6, ___, 2

Show answer

The answer is 4. This is a shrinking pattern! The numbers go down. From 10 to 8 is −2. From 8 to 6 is −2. So 6 − 2 = 4. And then 4 − 2 = 2, which matches the last number. The rule is subtract 2.

Question 4: APPLIED

You are putting stickers in a book. On page 1 you put 3 stickers. On page 2 you put 6 stickers. On page 3 you put 9 stickers. How many stickers will you put on page 4?

Show answer

The answer is 12 stickers. The pattern is 3, 6, 9, ___. From 3 to 6 is +3. From 6 to 9 is +3. The rule is add 3. So 9 + 3 = 12. You will put 12 stickers on page 4!

Question 5: CRITICAL THINKING

Look at this pattern: 1, 2, 4, 1, 2, 4, 1, ___. What are the next two numbers?

Show answer

The next two numbers are 2 and 4. This is a repeating pattern! The group 1, 2, 4 repeats over and over. After the 1, the group starts again with 2, then 4. Repeating patterns do not add or subtract — they just loop the same group of numbers again and again.

SUMMARY

What We Learned

Summary

A number pattern is a row of numbers that follows a rule. The rule tells you the hop — how much you add or take away each time. Patterns can be growing (numbers go up), shrinking (numbers go down), or repeating (a group shows up again and again).

To find a missing number, be a number detective: look at two neighbors, find the hop, check that the hop works every time, and then use the rule. Always check your hop more than once to make sure you have the right answer!