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ISEE Middle Level: Mathematics Achievement • Number Sense and Computation

Evaluate expressions using order of operations.

Master the rules that guarantee everyone gets the same answer from the same math expression.

Explore the lesson 11 min read
ISEE Middle Level emblem with a school, calculator, and reading passage
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Why Do We Need Rules for Math?

Imagine you and a friend both try to solve 3 + 4 × 2. You add first and get 14. Your friend multiplies first and gets 11. You used the same numbers and the same symbols, but you got different answers! That's a big problem, especially when math is used in science, engineering, and banking.

Mathematicians realized centuries ago that everyone needs to follow the same set of rules. Without a shared agreement, math becomes confusing and unreliable. The order of operations is that shared agreement. It tells you exactly which calculations to do first, second, and third.

1500s • Early Grouping Symbols

European mathematicians begin using parentheses and brackets to show which parts of a problem should be calculated first.

1600s • Exponents Appear

René Descartes introduces the modern exponent notation (like 3²), making it important to decide when to handle powers.

1800s • Rules Become Standard

Textbooks across Europe and America begin teaching a consistent order: parentheses first, then exponents, then multiplication and division, then addition and subtraction.

1900s • PEMDAS Enters Classrooms

The memory aid PEMDAS (Please Excuse My Dear Aunt Sally) becomes popular in American schools to help students remember the order.

Today, every calculator and computer on the planet follows these same rules. When you master the order of operations, you're speaking the universal language of math. Let's learn exactly how it works.

SECTION 2

The Core Rules: PEMDAS

The order of operations is a set of rules that tells you the exact sequence to follow when an expression has more than one operation. The memory trick PEMDAS helps you remember the order. Each letter stands for a step.

1

P — Parentheses

Always start inside parentheses (or brackets). Solve everything inside grouping symbols first, from the innermost set outward.
2

E — Exponents

Next, evaluate exponents (powers). For example, 3² means 3 × 3 = 9. Handle these before any other arithmetic.
3

MD — Multiply & Divide

Multiplication and division are equal partners. Work them from left to right, just like reading a sentence. Do NOT always multiply before dividing.
4

AS — Add & Subtract

Addition and subtraction are also equal partners. Work them from left to right. Do NOT always add before subtracting.
⚠

ISEE TIP

A very common mistake is thinking that multiplication always comes before division, or that addition always comes before subtraction. Remember: M and D are tied, and A and S are tied. For tied operations, always go left to right. Wrong answer choices on the ISEE are often built around this mistake!
✦

KEY TAKEAWAY

Think of PEMDAS like getting dressed in the morning. You have to put on your socks (parentheses) before your shoes (exponents), and your shirt (multiply/divide) before your jacket (add/subtract). Doing things out of order gives you a messy result!

SECTION 3

See the Steps in Action

The diagram below shows the PEMDAS staircase. You start at the top and work your way down. Each step must be completed before you move to the next one. Notice that multiplication and division share the same step, and addition and subtraction share the same step.

THE PEMDAS STAIRCASEStart at the top — work your way downPParenthesesDo everything inside ( ) first(3 + 5)EExponentsCalculate powers like 3² = 94² = 16MDMultiply & Divide (left → right)Equal rank — go left to right6 × 3 ÷ 2ASAdd & Subtract (left → right)Equal rank — go left to right10 + 3 − 4
The PEMDAS Staircase: Start with Parentheses at the top. Work down through Exponents, then Multiplication and Division (left to right), and finally Addition and Subtraction (left to right).

Notice how Multiply and Divide share one step, and Add and Subtract share another. When two operations are on the same step, you just read left to right. This is the key detail that many students miss, and it's exactly what the ISEE likes to test.

SECTION 4

The Rules Written Out

Let's write the order of operations in a clear, step-by-step way. When you see a long expression on the ISEE, follow these rules one at a time. Think of it as a checklist.

STEP 1 — PARENTHESES

Simplifyinsideallgroupingsymbols:(),[],Simplify inside all grouping symbols: ( ), [ ], { }

If there are nested parentheses, start with the innermost pair. For example, in 2 × [3 + (4 − 1)], you'd handle (4 − 1) first.

STEP 2 — EXPONENTS

Evaluateallpowers:32=9,52=25,23=8Evaluate all powers: 3² = 9, 5² = 25, 2³ = 8

On the ISEE Middle Level, you'll mostly see squares (like 4² = 16). Remember: the exponent tells you how many times to multiply the base by itself.

STEP 3 — MULTIPLY & DIVIDE

Workleft→right:12÷3×2=4×2=8Work left → right: 12 ÷ 3 × 2 = 4 × 2 = 8

Do NOT multiply first just because M comes before D in PEMDAS. They are equals. Whichever one you meet first reading left to right, do that one first.

STEP 4 — ADD & SUBTRACT

Workleft→right:10−3+5=7+5=12Work left → right: 10 − 3 + 5 = 7 + 5 = 12

Same idea as Step 3. Addition does NOT beat subtraction. Go left to right, handling whichever you see first.

⚠

COMMON ISEE TRAP

Test-makers love to include wrong answers that come from doing operations in the wrong order. If you see your answer among the choices, double-check that you followed PEMDAS. The wrong-order answer is almost always there as a trap choice!

SECTION 5

Tracing Through a Full Expression

Let's trace through a medium-difficulty expression step by step. The diagram below shows how the expression 5 + 3 × (8 − 2)² ÷ 6 gets simplified one piece at a time. Each colored bar shows which operation we do at that step.

STEP-BY-STEP TRACEExpression: 5 + 3 × (8 − 2)² ÷ 6STEP 1: Parentheses5 + 3 × (8 − 2)² ÷ 6→ 8 − 2 = 6STEP 2: Exponents5 + 3 × 6² ÷ 6→ 6² = 36STEP 3: Multiply (left to right)5 + 3 × 36 ÷ 6→ 3 × 36 = 108STEP 4: Divide (left to right)5 + 108 ÷ 6→ 108 ÷ 6 = 18STEP 5: Add5 + 18→ 5 + 18 = 23
Each row shows one step. The highlighted part is the operation being performed. Notice that multiplication and division were both done in Step 3 and Step 4, reading left to right.

Take a close look at Steps 3 and 4. We multiplied before we divided, but only because multiplication came first from left to right. If the division had been on the left, we would have divided first. This left-to-right rule is the most important detail to remember on test day.

SECTION 6

Worked Example: Full Solution

Let's work through a problem just like one you might see on the ISEE. We'll solve it slowly, showing every step and explaining our reasoning.

Evaluate: 18 ÷ 3 + 2 × (7 − 4)²

  1. 1

    Step 1 — Parentheses first

    Look for grouping symbols. We see (7 − 4). Solve inside: 7 − 4 = 3. The expression becomes: 18 ÷ 3 + 2 × 3².
    18 ÷ 3 + 2 × 3²
  2. 2

    Step 2 — Exponents

    We see 3². That means 3 × 3 = 9. The expression becomes: 18 ÷ 3 + 2 × 9.
    18 ÷ 3 + 2 × 9
  3. 3

    Step 3 — Multiply and divide, left to right

    Reading left to right, the first MD operation is 18 ÷ 3 = 6. Next is 2 × 9 = 18. The expression becomes: 6 + 18.
    6 + 18
  4. 4

    Step 4 — Add and subtract, left to right

    Only addition is left. 6 + 18 = 24. That's our final answer!
    24
💡

TEST STRATEGY

On the ISEE, write each simplified version of the expression on your scratch paper. This keeps you organized and helps you avoid careless mistakes. If you skip steps in your head, it's easy to lose track.

SECTION 7

Common Mistakes & How to Avoid Them

The ISEE test-makers design wrong answer choices based on the most common student errors. If you know what mistakes people usually make, you can avoid falling into those traps. Here's a comparison of correct approaches versus common mistakes.

MistakeWhat Students Do WrongCorrect Approach
Left-to-right ignoreAlways multiply before dividing (e.g., in 12 ÷ 4 × 3, they do 4 × 3 first)Do ÷ and × left to right: 12 ÷ 4 = 3, then 3 × 3 = 9
Adding before multiplyingIn 2 + 5 × 3, they add 2 + 5 = 7, then multiply: 7 × 3 = 21Multiply first: 5 × 3 = 15, then add: 2 + 15 = 17
Exponent errorThink 3² means 3 × 2 = 63² means 3 × 3 = 9. The exponent tells you how many times to multiply the base.
Forgetting parenthesesIn (4 + 6)², they do 4 + 6² = 4 + 36 = 40Parentheses first: (4 + 6) = 10, then 10² = 100

Common order-of-operations mistakes and how to fix them

✦

KEY TAKEAWAY

Think of wrong answer choices as booby traps. The test-makers know the mistakes most students will make, and they put those wrong results right there in the answer choices. If you follow PEMDAS carefully and go left to right for tied operations, you'll step right over those traps.

SECTION 8

Order of Operations and Algebra

The order of operations isn't just for arithmetic. It's the foundation for everything you'll do in algebra and beyond. When you start working with variables (letters that stand for numbers), PEMDAS still applies in exactly the same way.

What You Know NowWhat's Coming Next
Evaluate 3 + 4 × 2 = 11Evaluate 3 + 4x when x = 2 → 3 + 4(2) = 11
Simplify (5 + 1)² = 36Simplify (a + 1)² when a = 5 → same PEMDAS rules
Follow left-to-right for × and ÷Solve multi-step equations using reverse order of operations
Use parentheses to group numbersUse the distributive property: 3(x + 2) = 3x + 6

How order of operations connects to algebra

On the ISEE, you may see problems that combine the order of operations with substituting a number into a variable expression. Don't worry — if you follow PEMDAS step by step, those problems work exactly the same way. You'll just replace the variable with the given number first, then follow the rules.

SECTION 9

Practice Problems

Try these five problems. They get harder as you go. Remember to follow PEMDAS and work left to right for multiplication/division and addition/subtraction. There is no penalty for guessing on the ISEE, so always pick an answer!

Question 1: CONCEPTUAL

What is the value of 6 + 3 × 4?
  • A. 14
  • B. 18
  • C. 36
  • D. 72
Show answer (B)
The correct answer is B (18). By PEMDAS, multiply before adding: 3 × 4 = 12, then 6 + 12 = 18. Choice A (14) comes from dividing instead of multiplying. Choice C (36) comes from adding first (6 + 3 = 9, then 9 × 4 = 36), which is the classic mistake of ignoring order of operations. Choice D (72) comes from multiplying all three numbers together.

Question 2: BASIC CALCULATION

What is the value of 20 − 8 ÷ 2 + 3?
  • A. 9
  • B. 15
  • C. 19
  • D. 27
Show answer (C)
The correct answer is C (19). First, divide: 8 ÷ 2 = 4. The expression becomes 20 − 4 + 3. Now work left to right: 20 − 4 = 16, then 16 + 3 = 19. Choice A (9) comes from incorrectly grouping the subtraction before the division: (20 − 8) ÷ 2 + 3 = 12 ÷ 2 + 3 = 6 + 3 = 9. Choice B (15) comes from skipping the division entirely and treating 8 ÷ 2 as just 8: 20 − 8 + 3 = 15. Choice D (27) comes from adding the result of 8 ÷ 2 instead of subtracting it: 20 + 4 + 3 = 27.

Question 3: INTERMEDIATE

What is the value of 4 × (3 + 5) − 2²?
  • A. 24
  • B. 28
  • C. 30
  • D. 36
Show answer (B)
The correct answer is B (28). First, evaluate the parentheses: 3 + 5 = 8. Next, evaluate the exponent: 2² = 4. Then multiply: 4 × 8 = 32. Finally, subtract: 32 − 4 = 28. Choice A (24) comes from a student who subtracts 8 instead of 4 at the last step, perhaps reusing the parentheses result: 32 − 8 = 24. Choice C (30) comes from a student who ignores the exponent and treats 2² as simply 2, computing 4 × 8 − 2 = 32 − 2 = 30. Choice D (36) comes from a student who adds instead of subtracts at the final step, computing 4 × 8 + 4 = 32 + 4 = 36.

Question 4: APPLIED

Sarah bought 3 packs of pencils at $2 each and 1 notebook for $5. She paid with a $20 bill. Which expression correctly shows her change, and what is the value?
  • A. $7
  • B. $9
  • C. $13
  • D. $15
Show answer (B)
The correct answer is B ($9). The expression is 20 − 3 × 2 − 5. Using order of operations, multiply first: 3 × 2 = 6. Then work left to right: 20 − 6 = 14, then 14 − 5 = 9. Choice A ($7) comes from misreading 3 packs as 4 packs: 20 − 4 × 2 − 5 = 20 − 8 − 5 = 7. Choice C ($13) comes from forgetting to subtract the cost of the pencils entirely and only subtracting the notebook and one pack price: 20 − 2 − 5 = 13. Choice D ($15) comes from only subtracting the notebook: 20 − 5 = 15.

Question 5: CRITICAL THINKING

What is the value of 2 × (3² − 1) + 48 ÷ (10 − 4)?
  • A. 18
  • B. 22
  • C. 24
  • D. 28
Show answer (C)
The correct answer is C (24). Start with both sets of parentheses. In (3² − 1): apply the exponent first, 3² = 9, then 9 − 1 = 8. In (10 − 4): 10 − 4 = 6. The expression becomes 2 × 8 + 48 ÷ 6. Next, multiply and divide left to right: 2 × 8 = 16, and 48 ÷ 6 = 8. Finally, add: 16 + 8 = 24. Choice A (18) comes from treating 3² as 3 × 2 = 6 instead of squaring: (6 − 1) = 5, then 2 × 5 = 10, and 48 ÷ 6 = 8, giving 10 + 8 = 18. Choice B (22) comes from misapplying the exponent and subtraction inside the first parentheses: a student might compute 3² − 1 as 9 − 2 = 7 (subtracting 2 instead of 1), giving 2 × 7 = 14, then correctly computing 48 ÷ 6 = 8, and adding 14 + 8 = 22. Choice D (28) comes from ignoring the parentheses around the divisor and computing 48 ÷ 4 = 12 (using only the 4 from 10 − 4), then adding 16 + 12 = 28.

SUMMARY

Order of Operations: Your Quick Review

Summary

The order of operations is a universal set of rules for evaluating math expressions. Use the memory aid PEMDAS: first solve inside Parentheses, then calculate Exponents, then handle Multiplication and Division from left to right, and finally Addition and Subtraction from left to right.

The biggest trap on the ISEE is the left-to-right rule: multiplication does NOT always come before division, and addition does NOT always come before subtraction. When operations are at the same level, go left to right. Write out each step on scratch paper, check your work against PEMDAS, and you'll avoid the trap answers that trip up other students. You've got this!

Varsity Tutors • ISEE Middle Level • Evaluate expressions using order of operations.