Use arithmetic and algebraic patterns appropriate to level - CogAT Grades 3 Plus

SECTION 1

Why Patterns Matter

People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain numbers follow rules. These rules helped them build pyramids, track the stars, and trade goods. Today, pattern-finding is one of the most important skills you can develop — and it is exactly what the CogAT Quantitative Battery tests.

~2000 BCE • Babylonian Number Tables

Ancient Babylonians carved multiplication and square tables into clay tablets. They used patterns to solve everyday problems.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid studied patterns in prime numbers and even/odd numbers. His work still shapes math today.

1202 • Fibonacci's Sequence

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

1968 • CogAT Created

The Cognitive Abilities Test was developed. Its Quantitative Battery uses number patterns to measure reasoning ability.

So here is the big question: when you see a list of numbers, how do you figure out the hidden rule? That is what this lesson is all about.

SECTION 2

Core Principles of Pattern Recognition

Before you can solve pattern problems, you need to understand a few key ideas. Every number pattern is built on a rule (a set of steps that repeat). Once you find the rule, you can predict what comes next.

Arithmetic Patterns

You add (or subtract) the same number each time. Example: 3, 7, 11, 15 … (add 4).

Geometric / Multiplicative Patterns

You multiply (or divide) by the same number each time. Example: 2, 6, 18, 54 … (multiply by 3).

Alternating Patterns

Two different rules take turns. Example: 1, 10, 2, 20, 3, 30 … (count up by 1, then multiply by 10).

Growing-Difference Patterns

The amount you add changes in a pattern of its own. Example: 1, 2, 4, 7, 11 … (add 1, 2, 3, 4 …).

Algebraic Patterns

A formula connects the position number to the value. Example: the 'nth' term = 2n + 1 gives 3, 5, 7, 9 …

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KEY TAKEAWAY

Think of a number pattern like a playlist on repeat. Once you figure out the order of songs (the rule), you know exactly which song comes next. On the CogAT, your job is to find that 'playlist rule' hiding inside the numbers.

SECTION 3

Seeing the Pattern

A great way to understand patterns is to draw them out. The diagram below shows two common types side by side: an arithmetic pattern (add the same amount) and a growing-difference pattern (the amount you add keeps increasing).

On the left, the difference between neighbors is always 5 — a classic arithmetic pattern. On the right, the differences themselves grow by 1 each time, forming a growing-difference pattern.

Notice how the first step is always the same: subtract neighboring numbers to find the differences. This one trick helps you crack most CogAT number-pattern questions.

SECTION 4

The Math Behind Patterns

You can describe many patterns with simple formulas. A formula lets you jump straight to any term without listing every number before it.

ARITHMETIC PATTERN FORMULA

term = first value + (position − 1) × common difference

first value = the starting number; position = which term you want (1st, 2nd, 3rd …); common difference = the amount added each time.

For example, in the sequence 4, 9, 14, 19 … the first value is 4 and the common difference is 5. To find the 10th term: 4 + (10 − 1) × 5 = 4 + 45 = 49.

MULTIPLICATIVE PATTERN FORMULA

term = first value × ratio^(position − 1)

ratio = the number you multiply by each time. The ^ symbol means 'raised to the power of.' For 2, 6, 18, 54 … the ratio is 3.

ALGEBRAIC RULE (nth TERM)

value = a × n + b

n = the position number; a = the common difference; b = an adjustment. Example: if the rule is 2n + 1, the sequence is 3, 5, 7, 9 …

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Don't Panic About Formulas!

On the CogAT, you will not be asked to write a formula. But understanding how formulas work helps you think faster when choosing the right answer. Focus on the pattern of differences first, then check your answer.

SECTION 5

Classifying CogAT Pattern Questions

CogAT pattern questions come in several flavors. The table below shows each type, an example, and the strategy to solve it.

Type	Example	Strategy
Add / Subtract	5, 12, 19, 26, ?	Find the constant difference (+7). Add it to the last term: 26 + 7 = 33.
Multiply / Divide	4, 12, 36, 108, ?	Find the constant ratio (×3). Multiply the last term: 108 × 3 = 324.
Two-Operation	1, 3, 7, 15, ?	Try ×2 then +1: (1×2+1=3), (3×2+1=7), (7×2+1=15), so 15×2+1=31.
Alternating	2, 10, 4, 20, 6, ?	Split into odd and even positions. Odd: 2, 4, 6 (+2). Even: 10, 20, ? (×2). Answer: 30.
Growing Difference	2, 3, 5, 8, 12, ?	Differences: 1, 2, 3, 4. Next difference is 5. So 12 + 5 = 17.

Common CogAT Pattern Types

Follow this flowchart whenever you get stuck. Start by finding the differences, then follow the path. If you reach the bottom-right, try splitting the sequence into two interleaved lists or look for a two-step rule.

SECTION 6

Worked Example: Finding the Missing Number

Let's solve a CogAT-style problem together. The sequence is: 5, 8, 14, 23, 35, __. What number comes next?

Finding the Next Term in 5, 8, 14, 23, 35, __

1
Step 1 — Find the Differences

Subtract each number from the one after it: 8 − 5 = 3, 14 − 8 = 6, 23 − 14 = 9, 35 − 23 = 12.

Differences: 3, 6, 9, 12
2
Step 2 — Look at the Differences

Are the differences the same? No — but they form their own pattern! The differences go up by 3 each time (3, 6, 9, 12 …). This is a growing-difference pattern.

The next difference should be 12 + 3 = 15.
3
Step 3 — Apply the Rule

Add the next difference (15) to the last term: 35 + 15 = 50.

Answer: 50
4
Step 4 — Double-Check

Read the full sequence with the answer plugged in: 5, 8, 14, 23, 35, 50. Differences: 3, 6, 9, 12, 15. Each difference grows by 3. It checks out!

SECTION 7

Strengths and Pitfalls of Each Strategy

Not every strategy works perfectly every time. The table below compares your main tools.

Strategy	Strengths	Pitfalls
Find Differences	Works for most sequences. Quick and reliable.	Won't help if the rule involves multiplication or division.
Find Ratios	Perfect for sequences that grow very fast.	Doesn't help if numbers grow slowly or involve addition.
Split Into Two Lists	Great for alternating patterns that mix two rules.	You need to notice the alternating structure first.
Guess and Check	Works when nothing else does. Good backup plan.	Slow. You might run out of time on the test.

Strategy Comparison

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KEY TAKEAWAY

Think of these strategies like tools in a toolbox. A hammer is great for nails, but not for screws. On the CogAT, always try differences first. If the differences are messy, switch to ratios. If those don't work, try splitting the sequence. Use the right tool for the right job.

SECTION 8

Connecting Patterns to Algebra and Beyond

The pattern skills you are learning now connect directly to topics you will study in algebra and beyond. Here is a preview of how they relate.

CogAT Skill	Algebra Connection
Arithmetic pattern (constant difference)	Linear equations: y = mx + b, where m is the common difference.
Multiplicative pattern (constant ratio)	Exponential growth: y = a × rⁿ. Used in science, finance, and more.
Growing-difference pattern	Quadratic equations: y = ax² + bx + c. Creates curved graphs.
Two-operation rules	Recursive formulas: define each term using the previous one.

From CogAT to Algebra

When you master patterns now, you are building a strong foundation for high school and college math. Every time you spot a rule in a sequence, you are doing the same thinking that scientists and engineers use every day.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A sequence has the numbers 10, 15, 20, 25, 30. Is this an arithmetic pattern, a multiplicative pattern, or a growing-difference pattern? Explain how you know.

Show answer

This is an arithmetic pattern. When you subtract each term from the next, you get 5 every time (15 − 10 = 5, 20 − 15 = 5, etc.). Because the difference is constant, the pattern is arithmetic. A multiplicative pattern would require the same ratio between terms, and a growing-difference pattern would have differences that change.

Question 2: BASIC CALCULATION

What comes next in the sequence 3, 9, 27, 81, __?

Show answer

Divide each term by the one before it: 9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3. The ratio is a constant 3, so this is a multiplicative pattern. Multiply the last term by 3: 81 × 3 = 243. You can also recognize these as powers of 3: 3¹, 3², 3³, 3⁴, 3⁵ = 243.

Question 3: INTERMEDIATE

Find the missing number: 2, 5, 11, 20, 32, __.

Show answer

First, find the differences: 5 − 2 = 3, 11 − 5 = 6, 20 − 11 = 9, 32 − 20 = 12. The differences are 3, 6, 9, 12 — they increase by 3 each time. So the next difference is 12 + 3 = 15. Add it to the last term: 32 + 15 = 47. This is a growing-difference pattern where the second differences are all 3.

Question 4: APPLIED

A student saves money over five weeks: $2, $6, $18, $54, $162. If she continues this pattern, how much will she save in week 7?

Show answer

Find the ratio: 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3. The ratio is 3. Week 6: $162 × 3 = $486. Week 7: $486 × 3 = $1,458. This is a multiplicative (geometric) pattern. You can also use the formula: term = 2 ×

3^{n−1}

. For n = 7: 2 × 3⁶ = 2 × 729 = 1,458. Notice how quickly multiplicative patterns grow!

Question 5: CRITICAL THINKING

Look at this sequence: 1, 4, 2, 8, 3, 12, 4, __. What is the next number? Explain the rule.

Show answer

This is an alternating pattern with two interleaved sequences. The odd-position terms (1st, 3rd, 5th, 7th) are: 1, 2, 3, 4 — they increase by 1 each time. The even-position terms (2nd, 4th, 6th) are: 4, 8, 12 — they increase by 4 each time. The 8th term is an even position, so it continues the second list: 12 + 4 = 16. To solve alternating patterns, separate the sequence into two lists based on position. Each list follows its own simpler rule.

SUMMARY

Lesson Summary

Summary

In this lesson, you learned how to recognize and solve arithmetic patterns (constant difference), multiplicative patterns (constant ratio), growing-difference patterns (increasing differences), alternating patterns (two interleaved rules), and two-operation rules. Your first move should always be to find the differences between neighboring terms. If those differences are constant, you have an arithmetic pattern. If the differences grow, look for a pattern inside the differences themselves.

If differences don't help, try ratios (divide each term by the one before it) for multiplicative patterns. For tricky sequences, split the sequence into odd-position and even-position lists. These skills are tested on the CogAT Quantitative Battery and connect directly to linear equations, quadratic equations, and exponential growth that you will study in algebra and beyond.