SECTION 1
Why Order of Operations Exists
Imagine texting a friend the expression 3 + 4 × 2 and asking for the answer. Without a shared set of rules, one person might add first to get 14, while another multiplies first to get 11. Throughout the history of mathematics, scholars recognized that a single expression must produce a single, unambiguous result. The order of operations is the convention that eliminates this confusion.
1600s • Early Algebraic Notation
1700s • Exponent and Grouping Conventions
1800s • Multiplication Before Addition
1900s • PEMDAS Enters the Classroom
The central question that the order of operations answers is straightforward yet critical: when an expression contains several different operations, which one do you calculate first? On the ISEE, roughly two to four questions per Mathematics Achievement section depend on your ability to apply these rules accurately and efficiently. Mastering this topic is one of the most reliable ways to pick up points, because the rules never change.
SECTION 2
Core Principles of PEMDAS
The order of operations is often summarized by the acronym PEMDAS. Each letter stands for a level of priority. However, many students misinterpret PEMDAS by thinking multiplication always comes before division, or addition always before subtraction. In reality, multiplication and division share the same priority level, and addition and subtraction share another. Within each level, you work strictly from left to right.
Parentheses (P)
Exponents (E)
Multiplication & Division (MD)
Addition & Subtraction (AS)
SECTION 3
Visualizing the Priority Hierarchy
The diagram below shows the order of operations as a priority pyramid. Operations at the top of the pyramid must be completed before those below. Notice that multiplication and division sit on the same level, and addition and subtraction sit on the same level. The left-to-right arrow on each tier reminds you of the tiebreaker rule within each priority level.
SECTION 4
The Mathematical Framework
While PEMDAS is a mnemonic, the underlying mathematics is precise. Understanding each rule as a formal principle helps you apply it correctly in complex expressions, especially those involving negative numbers, nested grouping, or exponents combined with subtraction — all of which appear on the ISEE.
GROUPING SYMBOLS FIRST
Parentheses force the addition to happen before multiplication. For example, 3 × (2 + 5) = 3 × 7 = 21, but 3 × 2 + 5 = 6 + 5 = 11.
EXPONENT BEFORE MULTIPLICATION
The exponent applies only to its immediate base. So 2 × 3² = 2 × 9 = 18. If you mistakenly compute (2 × 3)² you get 36 — a common wrong answer on the ISEE.
NEGATIVE BASE CAUTION
Without parentheses, the negative sign is treated as subtraction applied after the exponent. With parentheses around −3, the entire negative number is squared, yielding a positive result.
LEFT-TO-RIGHT TIE RULE
Division and multiplication have equal priority, so you process them left to right. Dividing first gives 4, then multiplying by 2 gives 8. Multiplying 6 × 2 first would violate the left-to-right rule.
SECTION 5
Common ISEE Traps and How to Avoid Them
The ISEE deliberately designs wrong answer choices to match errors that students commonly make with order of operations. By knowing what these traps look like, you can recognize when you have accidentally fallen into one and correct your work before choosing an answer. The diagram below maps the most frequent mistakes to their root causes.
Notice that every trap stems from applying operations in the wrong sequence. Trap 1 violates the priority hierarchy by doing addition before multiplication. Traps 2 and 6 violate the left-to-right tiebreaker rule. Trap 3 misidentifies what the exponent applies to. Traps 4 and 5 overlook grouping symbols. When you practice, label each step with the PEMDAS level you are using — this habit makes errors visible before they cost you points.
SECTION 6
Worked Example: Full PEMDAS Walkthrough
Let's evaluate a multi-step expression that tests every level of PEMDAS. Follow each step carefully and notice how we process one priority level at a time, always moving left to right within a level.
Worked example
- 1
Step 1 — Identify Grouping Symbols
The expression contains one set of parentheses: (10 − 2²). We must evaluate everything inside these parentheses first. But within the parentheses, there is an exponent, so the exponent inside the group comes before the subtraction inside the group. - 2
Step 2 — Exponent Inside Parentheses
Evaluate 2² = 4. The expression inside the parentheses becomes (10 − 4).4 + 3 × (10 − 4) ÷ 6 - 3
Step 3 — Subtraction Inside Parentheses
Now complete the parentheses: 10 − 4 = 6. The parentheses are resolved.4 + 3 × 6 ÷ 6 - 4
Step 4 — Multiplication and Division (Left to Right)
No more exponents remain. Moving to the M/D level, scan left to right. First, 3 × 6 = 18. Then 18 ÷ 6 = 3.4 + 3 - 5
Step 5 — Addition
Finally, perform the addition: 4 + 3 = 7.Final answer: 7
SECTION 7
PEMDAS vs. Common Misinterpretations
Many students have memorized PEMDAS but still misapply it because they treat it as six sequential steps rather than four priority levels. The table below compares the correct interpretation with the most common misunderstanding, side by side.
| Aspect | Correct Interpretation | Common Misunderstanding |
|---|---|---|
| M vs. D | Equal priority — process left to right | Multiplication always comes before division |
| A vs. S | Equal priority — process left to right | Addition always comes before subtraction |
| Negative signs | −3² means −(3²) = −9 unless parentheses say (−3)² | The negative is part of the base, so −3² = 9 |
| Fraction bars | Act as grouping symbols — evaluate numerator and denominator fully before dividing | Divide term by term without grouping |
| Nested groups | Work from the innermost group outward | Evaluate the outermost group first |
Correct vs. incorrect interpretations of PEMDAS
Key takeaway
SECTION 8
Connecting to Algebra and Beyond
Order of operations is not just an arithmetic skill — it is the foundation of every algebraic manipulation you will ever perform. When you simplify an expression like 2x² + 3x − 5 for a given value of x, you are applying PEMDAS. When you distribute in 3(x + 4), the reason you multiply 3 by both x and 4 is rooted in how parentheses override addition's lower priority. Mastering PEMDAS now prepares you for equation solving, function evaluation, and even calculus.
| Concept | Arithmetic Example | Algebraic Extension |
|---|---|---|
| Evaluating expressions | 3 + 4 × 2 = 11 | If x = 2, then 3 + 4x = 3 + 8 = 11 |
| Exponent rules | 2 × 3² = 2 × 9 = 18 | 2x² when x = 3 means 2(9) = 18 |
| Distributive property | 3 × (4 + 5) = 3 × 9 = 27 | 3(x + 5) = 3x + 15 |
| Substitution into functions | f(x) = x² − 4x + 1; f(3) = 9 − 12 + 1 = −2 | Requires PEMDAS at every substitution step |
How order of operations extends into algebra
On the ISEE, you may encounter problems that combine order of operations with variable substitution. For example, a question might ask you to find the value of 2a² − 3b + c when a = 4, b = 5, and c = 7. The key is to substitute carefully and then apply PEMDAS to the resulting numerical expression. The exponent applies to 4 (not to 2 × 4), you multiply before subtracting, and you process the full expression left to right at each priority level.
SECTION 9
Practice Problems
These five problems mirror the style and difficulty of actual ISEE Mathematics Achievement questions. Work through each one step by step, applying PEMDAS carefully. Remember: there is no penalty for wrong answers on the ISEE, so always select an answer — but use process of elimination to improve your odds when you are unsure.
Question 1: Foundational
- A. 16
- B. 23
- C. 40
- D. 55
Show answer (B)
Question 2: Conceptual
- A. 11
- B. 15
- C. 18
- D. 21
Show answer (B)
Question 3: Applied
- A. 2
- B. 9
- C. 18
- D. 162
Show answer (C)
Question 4: Applied
- A. 24
- B. 27
- C. 32
- D. 37
Show answer (C)
Question 5: Challenge
- A. 19
- B. 23
- C. 27
- D. 35