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ISEE Lower Level: Mathematics Achievement • Operations and Computation

Evaluate expressions using order of operations.

Learn the special rules that tell you which math step to do first so you always get the right answer.

Explore the lesson 10 min read
ISEE Lower Level emblem with a school, foundational shapes, and open book
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Why Do We Need Rules for Math?

Imagine you and a friend are solving the same math problem. You get 10, but your friend gets 14. Who is right? Without a set of rules, people could get different answers to the same problem!

That is exactly the problem that mathematicians faced hundreds of years ago. As math grew more complicated, people needed to agree on which step to do first. Over time, they created a special set of rules called the order of operations.

1400s • Early Math Rules

Mathematicians in Europe began writing rules for the order of steps in a problem. They agreed that multiplication should come before addition.

1600s • Parentheses Appear

Mathematicians started using parentheses — the ( ) symbols — to show which part of a problem should be done first.

1800s • Schools Teach the Rules

Textbooks around the world began teaching the same order of operations so that every student would get the same answer.

Today • PEMDAS Is Standard

Students everywhere learn PEMDAS (or similar memory tricks) so everyone speaks the same math language.

So here is the big question: when a math problem has more than one operation (like adding AND multiplying), which do you do first? That is exactly what this lesson will teach you. Let's find out!

SECTION 2

The Core Rules: PEMDAS

The order of operations is a set of rules that tells you which step to do first in a math expression. An expression is a math phrase with numbers and operations (like 3 + 4 × 2). We use the memory helper PEMDAS to remember the order.

1

P — Parentheses

Always do what is inside parentheses ( ) first. They are like a VIP pass — they cut to the front of the line!
2

E — Exponents

Exponents (tiny numbers above a number) come next. On the ISEE Lower Level, you will rarely see these, so don't worry too much!
3

M & D — Multiply & Divide

Multiplication and division are partners. Do them from left to right, like reading a book. Neither one goes first — just go left to right.
4

A & S — Add & Subtract

Addition and subtraction are also partners. After you finish multiplying and dividing, do these from left to right.
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ISEE TIP

A fun way to remember PEMDAS is: "Please Excuse My Dear Aunt Sally." Each first letter stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
✦

KEY TAKEAWAY

Think of order of operations like getting dressed in the morning. You have to put on your socks BEFORE your shoes. If you do it in the wrong order, things go badly! Math works the same way — the order matters.

SECTION 3

See the Order in Action

Let's look at a picture that shows the order of operations as a staircase. You always start at the top step and work your way down!

The PEMDAS StaircaseStart at the top — work your way down!STEP 1P — Parentheses ( )Do what's inside the ( ) first!🥇STEP 2E — ExponentsRare on the ISEE — skip if none!🥈STEP 3M / D — × and ÷Left to right, like reading a book🥉STEP 4A / S — + and −Left to right, last step!🏁DOFIRSTDOLAST
The staircase shows four steps. Start at the top with Parentheses, then move down to Exponents, then Multiply/Divide, and finally Add/Subtract. Remember: multiply/divide and add/subtract are done left to right!

Notice how multiplication and division share the same step. That means neither one is "more important." You just go from left to right. The same is true for addition and subtraction — they also share a step.

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COMMON MISTAKE

Some students think multiplication ALWAYS comes before division because M comes before D in PEMDAS. That is not true! Multiplication and division are equal partners — just go left to right. The same rule applies to addition and subtraction.

SECTION 4

How the Rules Work Step by Step

Let's see how PEMDAS works with real numbers. We will solve the expression 3 + 4 × 2 two ways to prove that the order matters.

THE WRONG WAY (NO RULES)

3+4×2→7×2→14✗3 + 4 × 2 → 7 × 2 → 14 ✗

If you just go left to right without rules, you add first and get 14. This is incorrect!

THE RIGHT WAY (USING PEMDAS)

3+4×2→3+8→11✓3 + 4 × 2 → 3 + 8 → 11 ✓

PEMDAS says multiply before you add. So 4 × 2 = 8 first, then 3 + 8 = 11. That is the correct answer!

Now let's try one with parentheses: (5 + 1) × 3.

PARENTHESES EXAMPLE

(5+1)×3→6×3→18(5 + 1) × 3 → 6 × 3 → 18

The parentheses tell you to add 5 + 1 first, which gives 6. Then multiply 6 × 3 = 18.

LEFT-TO-RIGHT RULE

12÷3×2→4×2→812 ÷ 3 × 2 → 4 × 2 → 8

Division and multiplication are equal. Go left to right: 12 ÷ 3 = 4, then 4 × 2 = 8.

SECTION 5

Breaking Down a Harder Problem

Some ISEE problems mix several operations together. Let's trace through a bigger expression and watch each step carefully. We will solve: 2 × (3 + 5) − 4 ÷ 2.

Solving 2 × (3 + 5) − 4 ÷ 2 Step by Step2 × (3 + 5) − 4 ÷ 2Step 1: Parentheses3 + 5 = 82 × 8 − 4 ÷ 2Step 2: Multiply & DivideLeft to right: 2 × 8 = 16, then 4 ÷ 2 = 216 − 2Step 3: Subtract16 − 2 = 14= 14 ✓Each colored arrow shows one step of PEMDAS in action.
Follow the arrows from top to bottom. Cyan shows parentheses, pink shows multiply and divide, and amber shows subtraction. The final answer is 14.

See how each step simplifies the expression a little more? By the end, you are left with just one number. That is your answer! On the ISEE, circle or underline each step as you go — it helps you stay organized.

SECTION 6

Worked Example: Full PEMDAS Problem

Let's walk through a problem just like one you might see on the ISEE. Follow along step by step!

Evaluate: 10 + 2 × (6 − 3)

  1. 1

    Step 1 — Look for Parentheses

    We see (6 − 3). Do this first! 6 − 3 = 3. Now the expression becomes 10 + 2 × 3.
    10 + 2 × 3
  2. 2

    Step 2 — Look for Exponents

    Are there any exponents? Nope! Move on to the next step.
    No exponents — skip!
  3. 3

    Step 3 — Multiply and Divide (left to right)

    We see 2 × 3. Multiply! 2 × 3 = 6. Now the expression becomes 10 + 6.
    10 + 6
  4. 4

    Step 4 — Add and Subtract (left to right)

    Finally, add! 10 + 6 = 16. That is our answer!
    16
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ISEE TEST STRATEGY

On the ISEE, there is NO penalty for wrong answers. So always pick an answer, even if you have to guess! But first, use process of elimination: cross out any choices that look way too big or too small. This gives you a better chance of getting it right.

SECTION 7

Common Mistakes and How to Avoid Them

Many students make the same mistakes when using order of operations. The good news? Once you know about them, you can avoid them! Let's look at the most common ones.

MistakeWhat Really HappensHow to Fix It
Going left to right without checking for × or ÷ first3 + 4 × 2 is NOT 14. The answer is 11.Scan the whole expression for × and ÷ before you start adding.
Thinking M always comes before D12 ÷ 3 × 2 = 8, not 2. Go left to right.Remember: M and D are partners. Do whichever comes first from left to right.
Forgetting to do what is inside parentheses first(2 + 3) × 4 = 20, not 14.Circle any parentheses before you start. Solve them first every time.
Doing only one × or ÷ and skipping othersIn 2 × 3 + 8 ÷ 4, do BOTH × and ÷ before adding.Underline ALL × and ÷ parts. Solve them all before moving to + and −.

Common Order of Operations Mistakes

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KEY TAKEAWAY

Think of it like a traffic light. Before you "go" (add and subtract), you need to "slow down" (check for multiplication and division). And before that, check for the "red light" (parentheses) — they always stop you first!

SECTION 8

From Simple to More Advanced

Right now, you are learning order of operations with basic numbers. As you grow in math, these same rules will help you with bigger problems. Here is a peek at what comes next!

What You Know NowWhat Comes Later
Parentheses: (3 + 5)Brackets and braces: [2 × (3 + 5)]
Whole numbers: 2 × 4 + 1Fractions and decimals: 0.5 × 4 + 1
Missing number: 3 + __ = 10Variables: 3 + x = 10
Simple expressionsLong expressions with many steps

How Order of Operations Grows With You

The great news is that PEMDAS stays the same forever! Once you master it now, you will use the same rules in middle school, high school, and even college. You've got this!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own before looking at the answer. Remember: always answer every question on the ISEE — there is no penalty for guessing.

Question 1: CONCEPTUAL

In the expression 5 + 3 × 2, which operation should you do first? (A) Addition (B) Subtraction (C) Multiplication (D) Division
Show answer
The correct answer is C — Multiplication. According to PEMDAS, multiplication comes before addition. You should do 3 × 2 = 6 first, then add 5 + 6 = 11. Choice A is wrong because addition comes after multiplication in PEMDAS. Choice B is wrong because there is no subtraction in this expression. Choice D is wrong because there is no division in this expression.

Question 2: BASIC CALCULATION

What is the value of 8 + 2 × 3? (A) 11 (B) 14 (C) 24 (D) 30
Show answer
The correct answer is B — 14. Using PEMDAS, multiply first: 2 × 3 = 6. Then add: 8 + 6 = 14. Choice A (11) might come from making an arithmetic error when adding. Choice C (24) might come from multiplying 8 × 3 = 24 and ignoring the 2 entirely. Choice D (30) comes from adding first (8 + 2 = 10) then multiplying by 3, which violates the order of operations.

Question 3: INTERMEDIATE

What is the value of (4 + 6) × 3 − 5? (A) 13 (B) 17 (C) 25 (D) 35
Show answer
The correct answer is C — 25. First, do the parentheses: 4 + 6 = 10. Next, multiply: 10 × 3 = 30. Finally, subtract: 30 − 5 = 25. Choice A (13) comes from ignoring the parentheses and only multiplying 6 × 3 = 18, then subtracting 5: 18 − 5 = 13, leaving the 4 out of the calculation entirely. Choice B (17) comes from ignoring the parentheses and correctly applying multiplication before addition: 4 + (6 × 3) − 5 = 4 + 18 − 5 = 17. Choice D (35) comes from adding 5 instead of subtracting it: 10 × 3 = 30, then 30 + 5 = 35.

Question 4: APPLIED

Emma buys 3 books for $4 each and 1 bookmark for $2. She uses the expression 3 × 4 + 2 to find the total cost. What is the total? (A) $9 (B) $12 (C) $14 (D) $20
Show answer
The correct answer is C — $14. Using order of operations (PEMDAS), multiply first: 3 × 4 = 12 (the cost of 3 books). Then add: 12 + 2 = 14 (adding the bookmark). Choice A ($9) comes from adding all the numbers instead of multiplying: 3 + 4 + 2 = 9, which ignores the multiplication in the expression. Choice B ($12) comes from a student who multiplies correctly — 3 × 4 = 12 — but forgets to add the bookmark cost at all, leaving out the $2. Choice D ($20) comes from a student who misreads the problem and thinks each of the 4 books also comes with a $2 bookmark, calculating 3 × 4 = 12 and then adding 2 for each book: 12 + 8 = 20, instead of adding just one $2 bookmark.

Question 5: CRITICAL THINKING

What is the value of 20 − 2 × (1 + 4) + 6? (A) 10 (B) 16 (C) 28 (D) 96
Show answer
The correct answer is B — 16. First, parentheses: 1 + 4 = 5. Next, multiply: 2 × 5 = 10. Now the expression is 20 − 10 + 6. Go left to right: 20 − 10 = 10, then 10 + 6 = 16. Choice A (10) comes from forgetting to add 6 at the end, stopping at 20 − 10 = 10. Choice C (28) comes from skipping the multiplication step entirely and just adding and subtracting left to right: 20 − 2 + 4 + 6 = 28. Choice D (96) comes from multiplying 20 − 2 = 18 first, then doing 18 × (1 + 4) = 90, then 90 + 6 = 96, which applies the operations in the wrong order.

SUMMARY

Order of Operations — Quick Review

Summary

The order of operations is a set of rules that tells you which math step to do first. Use PEMDAS to remember: Parentheses first, then Exponents, then Multiplication and Division from left to right, and finally Addition and Subtraction from left to right. Remember: "Please Excuse My Dear Aunt Sally."

Key reminders for the ISEE: Multiplication and division are equal partners — just go left to right. The same is true for addition and subtraction. Always do what is inside parentheses before anything else. There is no penalty for wrong answers on the ISEE, so always answer every question. You've got this!

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