Use Common Pattern Types (Add/Subtract, Multiply/Divide, Alternating) - CogAT Grades 3 Plus

SECTION 1

Historical Context & Motivation

Humans have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that numbers can follow rules, and those rules help us predict what comes next. Today, pattern recognition is one of the most important skills tested on the CogAT (Cognitive Abilities Test). The Quantitative Battery measures how well you can find hidden rules in number sequences.

~300 BC • Euclid's Elements

The Greek mathematician Euclid studied number patterns. He explored sequences of even numbers, odd numbers, and prime numbers.

1202 • Fibonacci's Sequence

Leonardo Fibonacci introduced a famous number pattern (1, 1, 2, 3, 5, 8…) where each number is the sum of the two before it.

1654 • Pascal's Triangle

Blaise Pascal published a triangle of numbers full of addition patterns. Each row is built by adding pairs of numbers from the row above.

1968 • First CogAT Published

The Cognitive Abilities Test was created to measure reasoning skills, including pattern recognition in number sequences.

The big question is: when you see a list of numbers, how do you figure out the hidden rule? That's exactly what this lesson will teach you.

SECTION 2

Core Principles & Definitions

A number pattern (also called a number sequence) is a list of numbers that follows a specific rule. Your job on the CogAT is to find that rule and use it to figure out what number comes next—or which number is missing.

Add / Subtract Patterns

Each number is found by adding or subtracting the same amount every time. Example: 3, 7, 11, 15 (add 4 each time).

Multiply / Divide Patterns

Each number is found by multiplying or dividing by the same amount. Example: 2, 6, 18, 54 (multiply by 3).

Alternating Patterns

The rule switches back and forth between two operations. Example: 2, 6, 4, 8, 6 (add 4, then subtract 2, repeat).

Growing / Shrinking Steps

The amount added or subtracted changes each time. Example: 1, 2, 4, 7, 11 (add 1, add 2, add 3, add 4).

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KEY TAKEAWAY

Think of a number pattern like a recipe. When you bake cookies, you follow the same steps over and over for each batch. A number pattern follows the same math "recipe" over and over. Your job is to figure out what that recipe is!

SECTION 3

Visual Explanation

The diagram below shows how each of the three main pattern types works. Look at how the arrows between numbers show the hidden rule.

This diagram compares three pattern types. The add pattern (top) adds the same amount each time. The multiply pattern (middle) multiplies by the same amount. The alternating pattern (bottom) switches between two different rules.

Notice how the arrows between numbers are labeled with the operation. In the add pattern, every arrow says "+5." In the multiply pattern, every arrow says "×2." In the alternating pattern, the arrows switch between "+3" and "−1." When you take a CogAT test, always look at the jump from one number to the next. That jump is the key to cracking the code!

SECTION 4

How Each Pattern Type Works

Let's look at the math behind each pattern type. Don't worry—it's simpler than it looks! The key is learning how to find the difference (for add/subtract patterns) or the ratio (for multiply/divide patterns) between numbers.

ADD / SUBTRACT RULE

next number = current number + d

Here, d is the common difference. If d is positive, the numbers go up. If d is negative, the numbers go down.

MULTIPLY / DIVIDE RULE

next number = current number × r

Here, r is the common ratio. If r is greater than 1, the numbers grow. If r is a fraction (like ½), the numbers shrink.

ALTERNATING RULE

Step A → Step B → Step A → Step B → …

Alternating patterns use two different rules that take turns. For example, Step A might be "add 5" and Step B might be "subtract 2."

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Finding the Difference

To find the difference, subtract each number from the one after it. For the sequence 4, 9, 14, 19: do 9 − 4 = 5, then 14 − 9 = 5, then 19 − 14 = 5. The difference is always 5, so it's an add pattern!

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Finding the Ratio

To find the ratio, divide each number by the one before it. For 5, 15, 45, 135: do 15 ÷ 5 = 3, then 45 ÷ 15 = 3, then 135 ÷ 45 = 3. The ratio is always 3, so it's a multiply pattern!

SECTION 5

Strategy Guide: How to Identify the Pattern

When you see a number sequence on the CogAT, follow this step-by-step strategy. The flowchart below shows you exactly what to do.

Follow this flowchart when you encounter a number sequence. First, check for a common difference (add/subtract). If that doesn't work, check for a common ratio (multiply/divide). If neither works, look for an alternating pattern.

Step 1: Subtract each number from the next. If you always get the same answer, it's an add/subtract pattern.
Step 2: If the differences are different, try dividing each number by the one before it. If you always get the same answer, it's a multiply/divide pattern.
Step 3: If neither works, look at every other number. Check if the odd-position numbers follow one rule and the even-position numbers follow another. That's an alternating pattern!

SECTION 6

Worked Example

Let's walk through a CogAT-style problem step by step. The sequence is: 4, 12, 8, 24, 20, ?

Finding the Pattern in 4, 12, 8, 24, 20, ?

1
Step 1 — Find the Differences

Subtract each pair: 12 − 4 = 8, then 8 − 12 = −4, then 24 − 8 = 16, then 20 − 24 = −4. The differences are 8, −4, 16, −4. They are NOT all the same, so this is not a simple add/subtract pattern.

Differences: 8, −4, 16, −4 → NOT constant
2
Step 2 — Check for Ratios

Divide each pair: 12 ÷ 4 = 3, then 8 ÷ 12 ≈ 0.67, then 24 ÷ 8 = 3, then 20 ÷ 24 ≈ 0.83. The ratios are not all the same either. So this is not a simple multiply/divide pattern.

Ratios: 3, 0.67, 3, 0.83 → NOT constant
3
Step 3 — Look for Alternating Rules

Let's look at what happens from one number to the next more carefully. From 4 to 12, we multiply by 3. From 12 to 8, we subtract 4. From 8 to 24, we multiply by 3. From 24 to 20, we subtract 4. A-ha! The pattern alternates: ×3, then −4, then ×3, then −4.

Rule: ×3, −4, ×3, −4, …
4
Step 4 — Apply the Rule to Find the Missing Number

The last operation was −4 (from 24 to 20). So the next operation should be ×3. We calculate: 20 × 3 = 60.

The missing number is 60.

SECTION 7

Comparing Pattern Types

Each pattern type has its own look and feel. The table below helps you quickly tell them apart.

Pattern Type	What to Look For	Example	Common Trap
Add / Subtract	Numbers go up or down by the same amount	10, 17, 24, 31 (+7)	Confusing with multiply when the difference is large
Multiply / Divide	Numbers grow or shrink by the same factor	4, 12, 36, 108 (×3)	Numbers grow fast; check ratios, not differences
Alternating	Numbers seem to zigzag or switch direction	1, 4, 2, 5, 3 (+3, −2)	Thinking there's no pattern at all
Growing Steps	Differences between numbers keep increasing	2, 3, 5, 8, 12 (+1, +2, +3, +4)	Forgetting to check the differences of the differences

Quick comparison of common number pattern types

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KEY TAKEAWAY

Think of patterns like music rhythms. An add pattern is like a steady drum beat—the same beat repeats. A multiply pattern is like the music getting louder and louder. An alternating pattern is like a song that switches between a verse and a chorus. Once you hear the rhythm, you know what comes next!

SECTION 8

Connection to Advanced Patterns

The pattern types you've learned here are the building blocks for more advanced math concepts. As you move into high school, you'll see these same ideas show up again with fancier names.

What You Know Now	What It Becomes Later
Add/Subtract patterns (constant difference)	Arithmetic sequences — studied in Algebra 1 and 2
Multiply/Divide patterns (constant ratio)	Geometric sequences — important in Algebra 2 and finance
Growing steps (differences increase)	Quadratic sequences — related to parabolas
Alternating patterns	Recursive sequences — used in computer science

How CogAT pattern skills connect to future math

Mastering pattern recognition now doesn't just help you on the CogAT. It builds a foundation for algebra, data science, and even coding. Every time you spot a rule in a sequence, you're thinking like a mathematician!

SECTION 9

Practice Problems

Try these five problems. For each one, find the pattern and figure out the missing number. The problems get harder as you go!

Question 1: CONCEPTUAL

What type of pattern is this: 5, 10, 15, 20, 25? Is it add/subtract, multiply/divide, or alternating? How do you know?

Show answer

This is an add pattern. Each number is 5 more than the one before it (10 − 5 = 5, 15 − 10 = 5, 20 − 15 = 5, 25 − 20 = 5). The difference between every pair of neighbors is always 5, which tells us the rule is "add 5."

Question 2: BASIC CALCULATION

Find the missing number: 2, 10, 50, 250, ?

Show answer

This is a multiply pattern. Divide each number by the one before it: 10 ÷ 2 = 5, 50 ÷ 10 = 5, 250 ÷ 50 = 5. The ratio is always 5. So the next number is 250 × 5 = 1,250.

Question 3: INTERMEDIATE

Find the missing number: 3, 9, 6, 12, 9, 15, ?

Show answer

This is an alternating pattern. Check the jumps: 3 → 9 (+6), 9 → 6 (−3), 6 → 12 (+6), 12 → 9 (−3), 9 → 15 (+6). The rule alternates between +6 and −3. The last step was +6, so the next step is −3. The answer is 15 − 3 = 12.

Question 4: APPLIED

A baker makes cookies each day. On Monday she makes 6, Tuesday 10, Wednesday 14, Thursday 18. If this pattern continues, how many cookies will she make on Saturday?

Show answer

First, find the pattern: 10 − 6 = 4, 14 − 10 = 4, 18 − 14 = 4. This is an add pattern with a common difference of +4. Friday would be 18 + 4 = 22 cookies. Saturday would be 22 + 4 = 26 cookies. Notice that you had to apply the rule twice to get from Thursday to Saturday.

Question 5: CRITICAL THINKING

Look at this sequence: 1, 2, 4, 7, 11, 16, ?. The differences between numbers are NOT the same. Can you find a pattern in the differences themselves? What is the next number?

Show answer

The differences are: 2 − 1 = 1, 4 − 2 = 2, 7 − 4 = 3, 11 − 7 = 4, 16 − 11 = 5. The differences themselves form an add pattern (+1 each time)! This is called a growing steps pattern. The next difference should be 6, so the answer is 16 + 6 = 22. When the regular differences don't match, always try checking the "differences of the differences"—that's the key to cracking these tricky patterns.

SUMMARY

Lesson Summary

Summary

Number patterns on the CogAT follow a hidden rule. The three most common types are add/subtract patterns (where the same number is added or subtracted each time), multiply/divide patterns (where each number is multiplied or divided by the same factor), and alternating patterns (where two different rules take turns).

To identify the pattern, start by finding the differences between neighbors. If those aren't constant, check the ratios. If neither works, look for alternating rules or growing steps. Practice this strategy, and you'll be cracking CogAT number puzzles in no time!