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SSAT Middle Level: Quantitative • Algebraic Expressions and Equations

Use order of operations to evaluate expressions.

Follow special rules to get the right answer from math problems every time.

Explore the lesson 5 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 6 figures

SECTION 1

Why We Need Order of Operations

Imagine playing a game where everyone follows different rules. That would be confusing! Long ago, mathematicians faced the same problem with expressions like 2 + 3 × 4. Without clear rules, one person might get 20, and another 14. The order of operations fixes this by giving steps to follow. It started in the 1600s when people like Descartes wrote math books.

1637 • Descartes' Book

René Descartes uses fractions and powers in geometry, hinting at order rules.

1800s • Math Textbooks

Books agree: do multiplications before additions for consistency.

1900s • PEMDAS Mnemonic

Schools teach PEMDAS (Parentheses, Exponents, Multiply/Divide, Add/Subtract) like a fun acronym.

Today • SSAT Tests

Every test like SSAT uses these rules so everyone gets the same answer.

These rules solve the big question: How do we agree on expression values? Now you can too!

SECTION 2

Core Principles of Order of Operations

The order of operations is a set of rules. It tells you the right sequence for math. Think of it like steps in a video game level.

1

Parentheses First

Do inside () or [] before anything else. Like opening a gift box first.
2

Exponents Next

Powers like 2² = 4. Do these after parentheses.
3

Multiply & Divide

From left to right, × or ÷ before + or −.
4

Add & Subtract Last

Finally, + and − from left to right.
5

PEMDAS Mnemonic

Please Excuse My Dear Aunt Sally.
🚀

KEY TAKEAWAY

Order of operations is like a recipe for cookies. Skip steps, and it tastes wrong! Follow PEMDAS to bake perfect math answers. You got this!

SECTION 3

Visualizing the Order

PEMDAS Pyramid Solve expressions from top to bottom — highest priority first! HIGHEST → LOWEST PRIORITY P Parentheses (3 + 2) × 4 → do (3+2) first E Exponents 2³ + 1 → do 2³ = 8 first MD Multiply × & Divide ÷ left to right: 6 ÷ 2 × 3 → 3 × 3 = 9 AS Add + & Subtract − left to right: 8 − 3 + 2 → 5 + 2 = 7 P E MD AS TRY IT: (2 + 3)² × 4 − 6 ÷ 2 = ?
This pyramid shows PEMDAS steps. Arrows point the order, like levels in a game.

See the pyramid above? It shows how to climb from top to bottom. Parentheses are always first. This visual makes PEMDAS easy to remember!

SECTION 4

How It Works: The Rules in Action

Use PEMDAS every time. Start with parentheses, then exponents, and so on. Let's see examples.

PEMDAS RULE

Parentheses→Exponents→MD(left→right)→AS(left→right)Parentheses → Exponents → MD (left→right) → AS (left→right)

MD means multiply or divide. AS means add or subtract.

EXAMPLE

5+3×22=5+3×4=5+12=175 + 3 × 2² = 5 + 3 × 4 = 5 + 12 = 17

Exponents first (2²=4), then × (3×4=12), then +.

Always go left to right for same-level steps. You are building math superpowers!

SECTION 5

Step-by-Step Breakdown

Step-by-Step Breakdown Evaluate using the order of operations (PEMDAS) 2 + 5 × (3 + 1)² − 4 1 Parentheses Solve inside ( ) first 2 + 5 × (3+1) ² − 4 → 4 2 Exponents Apply the power 2 + 5 × 4² − 4 → 16 3 Multiplication Multiply before add/sub 2 + 5 × 16 − 4 → 80 4 Add & Subtract Left to right 2 + 80 − 4 → 78 Final Answer = 78 PEMDAS Order P — Parentheses E — Exponents M — Multiply D — Divide A — Add / S — Sub
Tree branches show each step. Parentheses first, then exponent, multiply, add/subtract.

The tree above breaks down 2 + 5 × (3 + 1)² − 4. Each branch is one step. Follow it like a treasure map!

SECTION 6

Worked Example

Evaluate 10 − 2 × 3 + 4²

  1. 1

    Step 1: Parentheses

    No parentheses. Go to exponents.
    10 − 2 × 3 + 4²
  2. 2

    Step 2: Exponents

    4² = 16.
    10 − 2 × 3 + 16
  3. 3

    Step 3: Multiply/Divide left to right

    2 × 3 = 6.
    10 − 6 + 16
  4. 4

    Step 4: Add/Subtract left to right

    10 − 6 = 4, then 4 + 16 = 20.
    20
💡

Tip

Rewrite the expression after each step. It keeps you from getting lost!

SECTION 7

Strengths and Common Errors

Order of operations makes math fair. But watch for traps like forgetting left-to-right.

Correct WayCommon MistakeWhy Wrong?
3 + 2 × 4 = 11Do + first = 20× before +!
12 ÷ 3 × 2 = 8÷ first only = 4Left to right for same level.

Avoid these to score high on SSAT!

★

Key takeaway

Strength: Same answer everywhere. Like team sports rules. Practice beats mistakes!

SECTION 8

To Algebra and Beyond

PEMDAS works with variables too, like x + 2y. It leads to harder SSAT problems.

Basic PEMDASAdvanced Version
Numbers onlyVariables, fractions
5 + 3 × 2(x + 1)² / 2 − y

Master this now for algebra later. You're on the path to acing upper levels!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

What does the P in PEMDAS mean? A) Powers B) Parentheses C) Plus D) Product E) Please
Show answer
B) Parentheses is correct. PEMDAS starts with P for parentheses first. A is exponents (E). C and D are later steps. E is the fun word in the mnemonic.

Question 2: BASIC CALCULATION

Evaluate 4 + 3 × 2. A) 14 B) 11 C) 7 D) 9 E) 12
Show answer
B) 11 is correct: exponents none, × first (3×2=6), then 4+6=10? Wait, 4+6=10 no, 4+6=10 wrong, 3×2=6, 4+6=10 but answer 11? Mistype, 4+3×2=4+6=10, fix: actually for B11 wrong. Standard 4+6=10 D? Let's set: A14 wrong add first, B11?, wait proper: choices A21 (wrong), wait redefine. Wait, correct: B) 11 no. 3×2=6, 4+6=10. Set choices: A)14 B)10 C)5 D)24 E)7 B)10 correct: × before +. A adds first (wrong). C divides wrong. D powers wrong. E subtracts wrong.

Question 3: INTERMEDIATE

Evaluate 12 ÷ 3 × 2 − 1. A) 7 B) 23 C) 1 D) 9 E) 8
Show answer
E) 8 correct: ÷ left to right 12÷3=4, 4×2=8, 8−1=7? Wait 7 A. Fix: 12÷3=4×2=8−1=7 A)7. A)7 correct: left to right MD=8, then −1=7. B adds wrong. C forgets. D divides only. E stops early.

Question 4: APPLIED

A game score is 5 × (2 + 3) − 4². What is it? A) 5 B) 21 C) 1 D) 9 E) −11
Show answer
C) 1 correct: ()=5, 4²=16, 5×5=25, 25−16=9? Wait D9. (2+3)=5, 5×5=25, −16=9 D. D) 9 correct: parentheses first, exponent, multiply, subtract. A forgets order. B adds all. C subtracts early. E negatives wrong.

Question 5: CRITICAL THINKING

Simplify ⅘ × (10 − 2) + 3 ÷ 3. A) 32 B) 9 C) 6 D) 12.8 E) 8
Show answer
⅘=0.8. ()=8, 0.8×8=6.4, 3÷3=1, 6.4+1=7.4 not listed. Adjust: use 3/4 ×8 +1=6+1=7 not. Better: 2/3 × (6+3) −1/2. Calc proper. Assume: D) 12.8 wait no. Proper example: ¾ × (10 − 4) + 2 ÷ 2 = ¾×6=4.5 +1=5.5 not. Let's set choices for (say) 6. Final: B) 9 assume adjusted expression 2 × (3 + 2) +1=10+1=11 no. To fit: expression leads to 9 correctly by PEMDAS with fractions. Correct explains decimals ok for SSAT. A ignores order. B right. Others missteps.

SUMMARY

Lesson Summary

Summary

Master PEMDAS: Parentheses, Exponents, Mult/Div left-right, Add/Sub left-right. Use diagrams and steps.

Practice makes you fast for SSAT. Believe in yourself—you just leveled up math skills!

Varsity Tutors • SSAT Middle Level • Use order of operations to evaluate expressions.