Use common pattern types (add/subtract, multiply/divide, alternating) - CogAT Grades K 2

SECTION 1

Why Do We Look for Patterns?

People have loved patterns for a very long time! Long ago, farmers looked at the sky and noticed patterns in the stars. They used those patterns to know when to plant seeds. Patterns help us guess what comes next.

Long Ago • Counting with Stones

People lined up stones in rows to count animals. They noticed that adding one more stone each time made a growing pattern!

Ancient Times • Number Patterns in Math

Mathematicians in Greece and Egypt found special patterns in numbers. They wrote rules to describe them.

School Today • Patterns on Tests

Today, tests like the CogAT ask kids to find patterns. This shows how well you can think and solve puzzles!

On the CogAT test, you will see rows of numbers. Your job is to find the secret rule that connects them. What rule makes the numbers change? Let's learn how!

SECTION 2

The Three Big Pattern Types

There are three main kinds of number patterns you will see. Each one has its own rule. Once you know the rule, you can figure out what number comes next!

Add or Subtract

The same number is added or taken away each time. Example: 2, 4, 6, 8 — you add 2 each time!

Multiply or Divide

Each number is multiplied (doubled, tripled) or divided. Example: 1, 2, 4, 8 — you multiply by 2!

Alternating

The rule switches back and forth. Example: 1, 3, 2, 4, 3 — you add 2, then subtract 1, over and over!

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KEY TAKEAWAY

Think of patterns like a recipe. If you bake cookies and add 2 chocolate chips to each cookie, that is an adding pattern. If you eat half your cookies each day, that is a dividing pattern. If you eat one then save one, eat one then save one, that is an alternating pattern!

SECTION 3

See the Patterns!

Let's look at a picture that shows all three pattern types. Each row of numbers follows a different rule. Can you see how the numbers change?

The top row adds 3 each time. The middle row multiplies by 2 each time. The bottom row switches between adding 3 and subtracting 1.

In the picture above, each row has a different rule. The add pattern goes up by 3, so the answer is 17. The multiply pattern doubles each time, so the answer is 32. The alternating pattern goes +3 then −1, and the next step is +3, so the answer is 8.

SECTION 4

How Each Pattern Rule Works

Let's look at each pattern type a little closer. We will learn a simple way to find the rule.

Add or Subtract Patterns

In an add pattern, you add the same number each time. The numbers go up. In a subtract pattern, you take away the same number each time. The numbers go down.

ADD PATTERN

next number = this number + same amount

Example: 3, 5, 7, 9 → each number is the one before it plus 2

SUBTRACT PATTERN

next number = this number − same amount

Example: 10, 8, 6, 4 → each number is the one before it minus 2

Multiply or Divide Patterns

In a multiply pattern, each number gets bigger fast! You multiply by the same number each time. In a divide pattern, each number gets smaller. You split it by the same number.

MULTIPLY PATTERN

next number = this number × same amount

Example: 2, 4, 8, 16 → each number is the one before it times 2

Alternating Patterns

An alternating pattern switches between two rules. It might add then subtract, or it might use two different numbers. It goes back and forth like a seesaw!

ALTERNATING PATTERN

Rule A, then Rule B, then Rule A, then Rule B …

Example: 1, 4, 2, 5, 3 → add 3 then subtract 2, over and over

SECTION 5

How to Be a Pattern Detective

Here is a trick to find patterns. Look at the space between each pair of numbers. We call this the jump. If every jump is the same, you found an add or subtract pattern. If the jumps get bigger and bigger, it might be a multiply pattern. If the jumps go back and forth, it is an alternating pattern!

Follow these four steps to discover the hidden rule in any number pattern. Start by writing down the jumps between each pair of numbers.

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Helpful Tip!

Use your fingers to count the jumps! Touch the first number, then count up or down to the second number. Write that number above the jump. Do this for every pair.

SECTION 6

Let's Solve One Together!

Here is a pattern puzzle: 3, 5, 4, 6, 5, ? What number comes next? Let's be pattern detectives!

Find the Missing Number: 3, 5, 4, 6, 5, ?

1
Step 1 — Write the Jumps

From 3 to 5 is +2. From 5 to 4 is −1. From 4 to 6 is +2. From 6 to 5 is −1.

Jumps: +2, −1, +2, −1
2
Step 2 — Are All Jumps the Same?

No! The jumps are not all the same. Some are +2 and some are −1. So this is NOT a simple add or subtract pattern.

Not an add/subtract pattern
3
Step 3 — Do Jumps Switch Back and Forth?

Yes! The jumps go +2, −1, +2, −1. That is switching back and forth! This is an alternating pattern.

Alternating: +2 then −1
4
Step 4 — Find the Next Number

The last jump was −1. The next jump should be +2. So we take 5 and add 2.

5 + 2 = 7

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GREAT JOB!

Always write out the jumps first. If the jumps go back and forth like a seesaw, you have an alternating pattern. Then just follow the pattern to find what's next!

SECTION 7

How Are the Pattern Types Different?

Each pattern type looks and acts differently. This table helps you tell them apart quickly!

Pattern Type	What You See	Example
Add	Numbers go up by the same amount	1, 3, 5, 7, 9 (+2)
Subtract	Numbers go down by the same amount	10, 8, 6, 4, 2 (−2)
Multiply	Numbers get bigger fast	2, 4, 8, 16 (×2)
Divide	Numbers get smaller fast	16, 8, 4, 2 (÷2)
Alternating	Jumps switch back and forth	2, 5, 3, 6, 4 (+3, −2)

How to tell the five pattern types apart

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REMEMBER THIS

If numbers go up like climbing stairs — steady and even — it's adding. If they grow super fast like a bouncing ball going higher and higher, try multiplying. If they go up-down-up-down like a seesaw, it's alternating!

SECTION 8

Growing Into Bigger Patterns

Right now you are learning simple patterns. As you get older, patterns get more exciting! Here is a peek at what comes later.

What You Know Now	What You'll Learn Later
Add 2 each time: 1, 3, 5, 7	Skip counting and times tables
Multiply by 2: 1, 2, 4, 8	Doubling and exponents (powers)
Alternating: +3 then −1	More complex sequences with two or three rules

Your pattern skills will keep growing!

Every time you solve a pattern puzzle, your brain gets stronger. The patterns you learn now are the building blocks for all of math. Keep practicing and you'll be a pattern superstar!

SECTION 9

Practice Problems

Now it's your turn! Try these five puzzles. Start easy and work your way up. Remember to find the jumps first!

Question 1: CONCEPTUAL

Look at this pattern: 2, 4, 6, 8. Is this an adding pattern, a multiplying pattern, or an alternating pattern? How do you know?

Show answer

This is an adding pattern. Each number goes up by 2. The jumps are all the same: +2, +2, +2. When every jump is the same and you are adding, it is an add pattern.

Question 2: BASIC CALCULATION

What number comes next? 10, 8, 6, 4, ?

Show answer

The answer is 2. This is a subtract pattern. Each time we take away 2. The jumps are −2, −2, −2. So 4 − 2 = 2.

Question 3: INTERMEDIATE

What number comes next? 1, 2, 4, 8, 16, ?

Show answer

The answer is 32. This is a multiply pattern. Each number is multiplied by 2. The jumps get bigger: +1, +2, +4, +8. That is a clue! Each number × 2 gives the next number. So 16 × 2 = 32.

Question 4: APPLIED

Look at this pattern: 2, 5, 4, 7, 6, ?. What number comes next? Tell what kind of pattern it is.

Show answer

The answer is 9. This is an alternating pattern. The jumps go +3, −1, +3, −1. The pattern switches between adding 3 and subtracting 1. The last jump was −1, so the next jump is +3. We take 6 + 3 = 9.

Question 5: CRITICAL THINKING

Here is a tricky one: 1, 3, 2, 6, 5, ?. What comes next? Hint: Look at the jumps carefully — there are TWO rules switching back and forth.

Show answer

The answer is 15. The jumps are: +2, −1, +4, −1. Wait — look again! From 1 to 3 the jump is +2, from 3 to 2 is −1, from 2 to 6 is +4, from 6 to 5 is −1. The subtract is always −1. But the add jumps are +2, +4 — those double! Actually, try another way: the odd-position numbers (1, 2, 5) show a pattern, and the even-position numbers (3, 6) show a pattern. One way to see it: the rule alternates between ×3 and −1. 1 × 3 = 3, 3 − 1 = 2, 2 × 3 = 6, 6 − 1 = 5, 5 × 3 = 15. The answer is 15!

SUMMARY

What We Learned

Summary

Number patterns follow secret rules. In an add or subtract pattern, the same number is added or taken away each time. In a multiply or divide pattern, each number is multiplied or divided by the same amount, and the numbers grow or shrink fast. In an alternating pattern, two different rules take turns, like a seesaw going back and forth.

To solve any pattern, be a pattern detective: write down the jumps between each pair of numbers, check if the jumps are the same, growing fast, or switching. Then use the rule to find the missing number. Practice makes you faster and smarter!