Use operations and order of operations to complete quantitative puzzles - CogAT Grades 3 Plus

SECTION 1

Where Did Order of Operations Come From?

Imagine if everyone solved the problem 3 + 2 × 4 differently. Some people might get 20, and others might get 11. That would be chaos! Centuries ago, mathematicians realized they needed a set of rules so that every person who reads the same expression gets the same answer.

These rules grew over hundreds of years. Different cultures contributed ideas about arithmetic and how to write math clearly. Let's look at a few key moments in this story.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid wrote about how multiplication relates to addition, laying early groundwork for treating operations differently.

~800 CE • Al-Khwarizmi & Algebra

The Persian scholar Al-Khwarizmi wrote books on solving equations. His work helped people see that some operations should be done before others.

1500s • Symbols for +, −, ×, ÷

Mathematicians in Europe started using the symbols we know today. Once everyone wrote math the same way, they needed rules for reading it the same way too.

1800s–1900s • PEMDAS Becomes Standard

Textbooks around the world agreed on a single set of rules for the order of operations. Students began learning the acronym PEMDAS (or BODMAS in some countries).

The big question these rules answer is simple: When a math expression has several operations, which one do you do first? That question is at the heart of every quantitative puzzle you'll see on the CogAT.

SECTION 2

Core Principles: The Rules You Need

Before you can solve puzzles, you need to know the building blocks. There are four main operations (actions you do to numbers): addition, subtraction, multiplication, and division. The order of operations tells you which action to perform first when more than one appears in the same problem.

Parentheses First

Always do whatever is inside parentheses ( ) before anything else. Parentheses are like a VIP pass — they cut to the front of the line.

Exponents Second

An exponent (like 3²) means you multiply a number by itself. Handle exponents right after parentheses.

Multiply & Divide (Left to Right)

Multiplication and division are equal partners. Work through them from left to right, in the order they appear.

Add & Subtract (Left to Right)

Addition and subtraction are also equal partners. After all multiplication and division are done, handle these from left to right.

A popular way to remember the order is the acronym PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Some people remember it with the phrase "Please Excuse My Dear Aunt Sally."

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KEY TAKEAWAY

Think of order of operations like a recipe. You wouldn't frost a cake before you bake it! The steps have to happen in the right order, or you end up with a mess. PEMDAS is your recipe card for math expressions.

SECTION 3

Seeing the Order: A Visual Guide

The diagram below shows the PEMDAS pyramid. Operations at the top are done first. Operations at the bottom are done last. Notice that multiplication and division sit on the same level, and so do addition and subtraction.

The PEMDAS pyramid shows that parentheses come first (top) and addition/subtraction come last (bottom). Multiplication and division share a level, meaning you do whichever comes first reading left to right.

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Common Mistake Alert

Many students think multiplication always comes before division because the M is before D in PEMDAS. That's not true! Multiplication and division are equals. The same goes for addition and subtraction. Always work left to right within each level.

SECTION 4

The Mathematical Framework

On the CogAT, you won't just simplify expressions. You'll also figure out which operation is missing. That means you need to be comfortable going both directions — solving expressions and filling in blanks. Here are the key ideas written as rules.

PEMDAS ORDER

P → E → (M or D, left to right) → (A or S, left to right)

P = Parentheses, E = Exponents, M = Multiplication, D = Division, A = Addition, S = Subtraction. The arrow → means "then do."

FINDING A MISSING OPERATION

a ? b = c → Try +, −, ×, ÷ to find which makes it true

When you see a blank (?) between two numbers, test each operation. For example, if 6 ? 3 = 18, try each one: 6 + 3 = 9 (no), 6 − 3 = 3 (no), 6 × 3 = 18 (yes!). The answer is ×.

INVERSE OPERATIONS

Addition ↔ Subtraction Multiplication ↔ Division

Inverse means "undo." If adding 5 gives you 12, then subtracting 5 takes you back to 7. Knowing inverses helps you work backward in puzzles.

These three tools — the order of operations, testing for a missing operation, and using inverse operations — are everything you need for CogAT quantitative puzzles. Let's see how they look in a real problem.

SECTION 5

Types of Quantitative Puzzles

CogAT quantitative puzzles come in several flavors. The diagram below shows the most common types you'll run into. Understanding what each type asks will save you time on test day.

This diagram shows three puzzle types you'll see on the CogAT. Type 1 asks for a missing number, Type 2 asks for a missing operation, and Type 3 asks you to balance an equation. The strategy checklist at the bottom applies to all three.

Puzzle Type	What's Missing?	Your Strategy
Missing Number	A number in the expression	Use the inverse operation to undo and find the blank.
Missing Operation	The +, −, ×, or ÷ sign	Test all four operations and see which one makes the equation true.
Balance Both Sides	A number that makes two sides equal	Simplify the complete side first, then solve for the unknown.
Multi-Step Expression	The final answer after multiple operations	Apply PEMDAS carefully step by step, showing each stage.

Overview of common CogAT quantitative puzzle formats

SECTION 6

Worked Example: Step by Step

Let's work through a CogAT-style puzzle together. Follow along and notice how we apply PEMDAS at every step.

Solve: 4 + 3 × (10 − 2) ÷ 6 = ?

1
Step 1 — Parentheses First

Look for parentheses. We see (10 − 2). Calculate that first: 10 − 2 = 8. The expression now looks like: 4 + 3 × 8 ÷ 6.

Expression becomes: 4 + 3 × 8 ÷ 6
2
Step 2 — Check for Exponents

Are there any exponents? No. Move on to the next level.
3
Step 3 — Multiply and Divide Left to Right

We have 3 × 8 and then ÷ 6. Going left to right: first do 3 × 8 = 24. Now the expression is 4 + 24 ÷ 6. Next, do 24 ÷ 6 = 4. Now the expression is 4 + 4.

Expression becomes: 4 + 4
4
Step 4 — Add

Finally, add: 4 + 4 = 8.

Answer: 8

Find the Missing Operation: 12 ○ 4 = 3

1
Step 1 — Test Addition

12 + 4 = 16. That's not 3, so addition is wrong.
2
Step 2 — Test Subtraction

12 − 4 = 8. That's not 3 either. Keep going.
3
Step 3 — Test Multiplication

12 × 4 = 48. Way too big. Not the right operation.
4
Step 4 — Test Division

12 ÷ 4 = 3. That matches!

The missing operation is ÷ (division).

SECTION 7

Strengths, Pitfalls, and Tips

Using order of operations is a powerful tool, but there are common traps that trip students up. The table below highlights what works well and what to watch out for.

Strength	Common Pitfall	How to Avoid It
PEMDAS gives a clear, universal order.	Thinking M always comes before D.	Remember: M and D are equal — go left to right.
Parentheses let you control which part is done first.	Forgetting to solve everything inside parentheses.	Circle every set of parentheses before you start.
Inverse operations let you work backward.	Using the wrong inverse (e.g., dividing when you should subtract).	Ask: what was done to this number? Then do the opposite.
Testing all four operations finds missing signs.	Stopping after the first operation that looks close.	Always test all four and pick the exact match.

Strengths and common pitfalls when using order of operations

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KEY TAKEAWAY

Think of PEMDAS like traffic rules. Everyone on the road follows the same rules so there are no crashes. If you ignore a rule (like running a red light), you'll get the wrong answer. Following the rules every single time keeps you safe — and correct.

SECTION 8

Connecting to Algebra and Beyond

The skills you're building here are the exact same skills you'll use in algebra, geometry, and even calculus someday. When you learn to solve equations like 2x + 5 = 13, you'll use inverse operations to isolate x — just like finding a missing number in a puzzle!

What You Do Now (CogAT Puzzles)	What You'll Do Later (Algebra & Beyond)
Find a missing number: 5 + ? = 12	Solve for x: 5 + x = 12
Use PEMDAS to simplify: 3 + 2 × 4	Simplify algebraic expressions: 3 + 2y
Balance both sides: 3 + 5 = ? + 2	Solve equations: 3x + 5 = 2x + 8
Use inverse operations to undo	Use inverse functions (square root undoes squaring)

How CogAT puzzle skills connect to future math

The CogAT Quantitative Battery is measuring how well you think with numbers. Every time you practice these puzzles, you're building number sense — a deep, flexible understanding of how numbers work together. That skill will help you in every math class ahead.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. After each one, check the answer to see if you're on track.

Question 1: CONCEPTUAL

In the expression 6 + 4 × 2, which operation should you do first — the addition or the multiplication? Why?

Show answer

You should do the multiplication first. According to PEMDAS, multiplication comes before addition. So you calculate 4 × 2 = 8 first, then add 6 + 8 = 14. If you added first, you'd get 10 × 2 = 20, which is incorrect because it ignores the order of operations.

Question 2: BASIC CALCULATION

Simplify: 18 − 2 × (3 + 1)

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Start with the parentheses: 3 + 1 = 4. Now the expression is 18 − 2 × 4. Next, do the multiplication: 2 × 4 = 8. Finally, subtract: 18 − 8 = 10. The answer is 10. Remember, always handle parentheses first, then multiplication before subtraction.

Question 3: INTERMEDIATE

Find the missing operation: 15 ○ 3 ○ 2 = 7. Each circle represents an operation (+, −, ×, or ÷). Both circles can be different operations.

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You need to find two operations that make 15 ○ 3 ○ 2 = 7. Try 15 ÷ 3 first: that gives 5. Then 5 + 2 = 7. That works! So the operations are ÷ and +, giving 15 ÷ 3 + 2 = 5 + 2 = 7. You should also check that PEMDAS is respected: division comes before addition, so the left-to-right order gives us the correct result.

Question 4: APPLIED

A puzzle shows two sides of a balance: the left side says 2 × 6 + 4, and the right side says 4 × ? . What number replaces the question mark to make both sides equal?

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First, simplify the left side using PEMDAS. Do multiplication first: 2 × 6 = 12. Then add: 12 + 4 = 16. So the left side equals 16. Now set up the right side: 4 × ? = 16. To find the missing number, use the inverse of multiplication, which is division: 16 ÷ 4 = 4. The question mark is 4, because 4 × 4 = 16, which matches the left side.

Question 5: CRITICAL THINKING

Using only the numbers 2, 3, 4, and 6 (each used exactly once) and the operations +, −, ×, or ÷, create an expression that equals 13. You may use parentheses.

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One solution is 6 × 2 + 4 − 3 = 13. Here's why: by PEMDAS, do 6 × 2 = 12 first. Then work left to right: 12 + 4 = 16, and 16 − 3 = 13. Another possible solution is (4 − 2) × 6 + 3 − 2... but wait, that reuses 2. The clean answer is 6 × 2 + 4 − 3. This problem tests your ability to combine operations creatively while still respecting the order of operations. Trying different arrangements is a great strategy.

SUMMARY

Putting It All Together

Summary

CogAT quantitative puzzles test your ability to use addition, subtraction, multiplication, and division together. The order of operations (PEMDAS) tells you which step to take first: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right). Remember that multiplication and division are equal partners, and so are addition and subtraction.

When a puzzle has a missing number, use inverse operations to work backward. When a missing operation is the blank, test all four operations to find the one that works. For balance puzzles, simplify the complete side first, then solve for the unknown. These strategies will carry you through the CogAT and into algebra and beyond!