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CogAT (Grades K-2) • Quantitative Battery

Use basic arithmetic patterns to recognize numeric relationships

Learn how numbers follow special patterns that help you figure out what comes next!

Explore the lesson 8 min read
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Why Do We Look for Patterns in Numbers?

People have loved number patterns for a very long time! Long ago, people counted things like sheep and apples. They noticed that numbers follow rules. When you add the same number again and again, you get a pattern. These patterns help us understand the world around us.

Long Ago • People Start Counting

People used fingers and stones to count things like animals and food. They noticed that numbers go in order: 1, 2, 3, 4, 5!

Ancient Times • Patterns in the Sky

People watched the moon and stars. They saw patterns that repeated, like the moon getting bigger and smaller every 30 days.

In School • Kids Learn Patterns

Children learn to skip count by 2s, 5s, and 10s. This helps them add and subtract faster!

Today • Patterns Are Everywhere

We use number patterns every day! When you count coins, tell time, or play number games, you are using patterns.

So here is the big question: If you see the numbers 2, 4, 6, 8 — can you guess what comes next? That is what finding patterns is all about! Let's learn how.

SECTION 2

What Are Number Patterns?

A pattern is something that repeats in a way you can predict. In numbers, a pattern means the numbers follow a rule. When you figure out the rule, you can find the next number!

1

Adding Pattern

Numbers go up by the same amount each time. Like 1, 3, 5, 7 — we add 2 every time!
2

Subtracting Pattern

Numbers go down by the same amount each time. Like 10, 8, 6, 4 — we take away 2 every time!
3

Skip Counting

When you count by 2s, 5s, or 10s, you are using an adding pattern! Like 5, 10, 15, 20.
4

Equal Pairs

Sometimes two groups of numbers are related. If one side has 3 + 2, the other side has 5. They are equal!
✦

KEY TAKEAWAY

Think of a number pattern like a staircase. Each step is the same height. If you climb up 2 steps every time, you know exactly where the next step will be! The rule is the height of each step.

SECTION 3

See the Pattern!

Let's look at a picture that shows how an adding pattern works. Watch how the numbers go up by the same amount each time!

Adding Pattern: +3 Each Time36912151st2nd3rd4th5th+3+3+3+3Each bar grows by 3. The rule is: add 3!
Each colored bar shows a number in the pattern 3, 6, 9, 12, 15. The bars get taller because we add 3 every time. The dashed arrows show the jump of +3 between each number.

See how each bar is a little taller than the one before it? That is because we keep adding 3. The rule is +3. Once you know the rule, you can keep the pattern going forever!

SECTION 4

How Patterns Work with Adding and Subtracting

Every number pattern has a rule. The rule tells you what to do to get from one number to the next. Let's look at some simple rules!

ADDING RULE

Number+Rule=NextNumberNumber + Rule = Next Number

If the rule is +2 and you start at 4, then: 4 + 2 = 6, then 6 + 2 = 8, then 8 + 2 = 10.

SUBTRACTING RULE

Number−Rule=NextNumberNumber − Rule = Next Number

If the rule is −3 and you start at 15, then: 15 − 3 = 12, then 12 − 3 = 9, then 9 − 3 = 6.

FINDING THE RULE

SecondNumber−FirstNumber=RuleSecond Number − First Number = Rule

Look at two numbers that are next to each other. Subtract the smaller one from the bigger one. That tells you the rule! For example: 10 − 5 = 5, so the rule is +5.

💡

REMEMBER THIS!

To find the rule, look at two numbers that are right next to each other. Ask yourself: "How do I get from this number to the next one?" If the numbers go up, you are adding. If they go down, you are subtracting.

SECTION 5

Different Kinds of Number Patterns

There are many kinds of number patterns! Let's look at a picture that shows three different ones. Each pattern follows its own rule.

Three Number PatternsCount by 2s (Rule: +2)2→4→6→8→?Answer: 10!Count by 5s (Rule: +5)5→10→15→20→?Answer: 25!Count Down by 1 (Rule: −1)9→8→7→6→?Answer: 5!
Three patterns are shown: counting by 2s (cyan), counting by 5s (yellow), and counting down by 1 (pink). Each row has a missing number with a dashed circle. Can you see the rule for each one?
PatternRuleExample
Count by 1sAdd 11, 2, 3, 4, 5
Count by 2sAdd 22, 4, 6, 8, 10
Count by 5sAdd 55, 10, 15, 20, 25
Count by 10sAdd 1010, 20, 30, 40, 50
Count down by 2sSubtract 210, 8, 6, 4, 2

Common number patterns and their rules

SECTION 6

Let's Solve One Together!

Let's find the missing number in a pattern, step by step. Here is the pattern: 4, 7, 10, 13, __. What number goes in the blank?

Finding the Missing Number

  1. 1

    Step 1 — Look at the First Two Numbers

    The first number is 4 and the second number is 7. Ask yourself: how do I get from 4 to 7? You count up: 4 … 5, 6, 7. That is 3 more. So maybe the rule is +3!
    7 − 4 = 3
  2. 2

    Step 2 — Check the Next Pair

    Let's make sure! The second number is 7 and the third number is 10. Count up: 7 … 8, 9, 10. That is 3 more again! The rule is +3.
    10 − 7 = 3 ✓
  3. 3

    Step 3 — Check One More Time

    The third number is 10 and the fourth number is 13. Count up: 10 … 11, 12, 13. That is 3 more. Great — the rule works every time!
    13 − 10 = 3 ✓
  4. 4

    Step 4 — Use the Rule to Find the Answer

    Now add 3 to the last number: 13 + 3 = 16. The missing number is 16!
    13 + 3 = 16
💡

Tip!

Always check the rule between at least two pairs of numbers. If the rule works for all pairs, you found it! If it does not work the same way every time, try a different rule.

SECTION 7

Tips and Tricks for Pattern Problems

On the CogAT test, you will see number pattern questions. Here are some helpful tips to remember!

What You SeeWhat to DoExample
Numbers going upFind what you add each time3, 6, 9 → add 3
Numbers going downFind what you subtract each time20, 15, 10 → subtract 5
Numbers that look equalCheck if both sides add up the same3 + 4 = 2 + ?
Missing number in the middleUse the numbers before AND after2, ?, 6 → must be 4

Quick tips for pattern problems

✦

KEY TAKEAWAY

Think of number patterns like clapping a beat. If you clap every 2 seconds — clap, clap, clap — you know when the next clap comes. Numbers work the same way! Once you find the beat (the rule), you can predict what comes next.

SECTION 8

Getting Ready for Bigger Patterns

The patterns you are learning now will help you with bigger math later! Let's see how these simple patterns connect to what you will learn when you are older.

What You Know NowWhat You Will Learn Later
Count by 2s: 2, 4, 6, 8Multiplying by 2 (2 × 1, 2 × 2, 2 × 3)
Find the rule (+3, +5)Equations like n + 3 = ?
Balance two sides (3 + 2 = 5)Solving equations (x + 2 = 5)
Number patterns going upGraphing lines on a chart

How today's patterns lead to tomorrow's math

Everything you learn about patterns now is like building blocks. Each block helps you build something bigger. The adding and subtracting patterns you practice today will make multiplication, division, and even algebra easier someday!

SECTION 9

Practice Problems

Now it is your turn! Try these problems. Think about the rule for each pattern. Take your time and check your work!

Question 1: CONCEPTUAL

Look at this pattern: 1, 2, 3, 4, __. What number comes next? How do you know?
Show answer
The answer is 5. The pattern counts up by 1 each time. We go 1, 2, 3, 4, and the next number after 4 is 5. The rule is +1. Each number is one more than the number before it.

Question 2: BASIC CALCULATION

What is the missing number? 2, 4, 6, __, 10. What is the rule?
Show answer
The missing number is 8. The rule is +2. We can check: 2 + 2 = 4, 4 + 2 = 6, 6 + 2 = 8, and 8 + 2 = 10. Every number goes up by 2. This is the same as counting by 2s!

Question 3: INTERMEDIATE

Find the missing number: 15, 12, __, 6, 3. What is the rule and what number goes in the blank?
Show answer
The missing number is 9. The rule is −3 (subtract 3). We check: 15 − 3 = 12, 12 − 3 = 9, 9 − 3 = 6, and 6 − 3 = 3. The numbers go down by 3 every time. This is counting backwards by 3s!

Question 4: APPLIED

Sam has 3 stickers on Monday. Every day, he gets 2 more stickers. How many stickers does he have on Thursday? Use the pattern: 3, 5, 7, __.
Show answer
Sam has 9 stickers on Thursday. The rule is +2. Monday: 3 stickers. Tuesday: 3 + 2 = 5 stickers. Wednesday: 5 + 2 = 7 stickers. Thursday: 7 + 2 = 9 stickers. Each day adds 2 more to the count.

Question 5: CRITICAL THINKING

Look at this pattern: 1, 4, 7, 10, 13. What is the rule? What would the 7th number in this pattern be?
Show answer
The rule is +3. We check: 4 − 1 = 3, 7 − 4 = 3, 10 − 7 = 3. The 6th number is 13 + 3 = 16. The 7th number is 16 + 3 = 19. To find a number far ahead in a pattern, just keep using the rule over and over!

SUMMARY

What We Learned

Summary

A number pattern is a list of numbers that follows a rule. The rule tells you what to do to get from one number to the next. Adding patterns make numbers go up, like 5, 10, 15, 20. Subtracting patterns make numbers go down, like 20, 18, 16, 14.

To find the rule, subtract two numbers that are next to each other. Always check the rule with more than one pair. Once you know the rule, you can find any missing number in the pattern. These skills help you on the CogAT test and make you a strong math thinker!

Varsity Tutors • CogAT (Grades K-2) • Use basic arithmetic patterns to recognize numeric relationships