Translate a situation into an algebraic expression. - ISEE Middle Level Quantitative Reasoning

SECTION 1

Where Did Algebra Come From?

Have you ever tried to describe a math problem to a friend using only words? It gets complicated fast! For thousands of years, people struggled with this exact issue. Ancient civilizations solved tricky problems, but they wrote everything out in long sentences. Eventually, mathematicians invented a shorthand — algebra — that lets us write math ideas quickly and clearly.

1800 BCE • Babylonian Word Problems

Ancient Babylonians solved problems about land and trade. They wrote solutions in words on clay tablets — no symbols at all!

820 CE • Al-Khwarizmi's Big Book

The mathematician al-Khwarizmi wrote a famous book about solving equations. The word "algebra" actually comes from the Arabic title of his book.

1637 • Descartes Uses x, y, z

French mathematician René Descartes started using letters like x, y, and z to stand for unknown numbers. This is the style we still use today.

Today • Algebra Everywhere

Video game designers, doctors, athletes, and engineers all use algebraic expressions every day to solve real-world problems.

On the ISEE, you'll see questions that describe a situation in words and ask you to pick the matching algebraic expression. The key skill is learning to translate English into math. Let's learn how!

SECTION 2

Core Principles of Translation

Translating a situation into algebra is like translating between two languages. English has its own words for math operations, and once you learn the "dictionary," you can convert any sentence into a neat expression. Here are the four big ideas you need.

Identify the Unknown

Look for the number you don't know yet. Use a variable (a letter like n, x, or y) to stand for that mystery number.

Spot the Operation Words

Words like "more than" mean addition. "Less than" means subtraction. "Times" means multiplication. "Divided by" means division.

Watch the Order

"5 less than a number" is n − 5, not 5 − n. The order of words matters — always read carefully!

Use Parentheses When Needed

Phrases like "twice the sum of a number and 3" need parentheses: 2(n + 3). The parentheses group what happens first.

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KEY TAKEAWAY

Think of a variable like a nickname. If you don't know someone's full name, you give them a nickname to talk about them. In algebra, when you don't know a number, you give it a letter — like n — so you can still write math about it.

SECTION 3

The Translation Dictionary

The diagram below is your cheat sheet. It shows common English words and the math operations they represent. Study it closely — you'll see these words again and again on the ISEE.

This chart shows the most common English phrases and the math operation each one signals. When you see one of these words in a problem, you know which symbol to write.

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ISEE Test Tip

Watch out for tricky phrases! "5 less than n" means n − 5, not 5 − n. The phrase "less than" flips the order. Similarly, "subtracted from" also reverses: "3 subtracted from x" means x − 3.

SECTION 4

Building Expressions Step by Step

An algebraic expression (a math phrase that uses numbers, variables, and operations) doesn't have an equals sign — that would make it an equation. Let's look at some common patterns.

ADDITION PATTERN

n + k

n = the unknown number, k = the number being added. Example: "a number increased by 7" → n + 7

SUBTRACTION PATTERN

n − k

n = the unknown number, k = the number being subtracted. Example: "8 less than a number" → n − 8

MULTIPLICATION PATTERN

k × n or kn

k = the multiplier, n = the unknown. In algebra, we usually write 3n instead of 3 × n. Example: "triple a number" → 3n

COMBINED PATTERN

k × n + c or kn + c

Many ISEE problems combine two operations. Example: "twice a number, increased by 5" → 2n + 5

Notice that we always pick a variable for the unknown first. Then we build the expression around it, one operation at a time. On the ISEE, always check: does the order match the words?

SECTION 5

Tricky Phrases & Common Mistakes

Some phrases on the ISEE are designed to trip you up. The diagram below shows the most common traps and how to avoid them. Pay special attention to the order-reversal phrases — they are the #1 source of mistakes.

The left column shows phrases where you read in normal order. The right column shows phrases where the order flips — the variable comes first in the expression even though it appears second in the sentence.

English Phrase	Common Mistake	Correct Expression
"10 less than a number"	10 − n	n − 10
"twice the sum of n and 4"	2n + 4	2(n + 4)
"a number divided into 20"	n ÷ 20	20 ÷ n

Common translation mistakes and their corrections

SECTION 6

Worked Example

Let's walk through a full ISEE-style problem together. We'll go step by step so you can see exactly how to think through it.

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SAMPLE QUESTION

Maria has some stickers. She gives away 4 stickers, then splits the remaining stickers equally among 3 friends. Which expression represents the number of stickers each friend receives?

Step-by-Step Solution

1
Step 1 — Identify the Unknown

We don't know how many stickers Maria starts with. Let's call that number s.

Unknown: s = starting number of stickers
2
Step 2 — Translate the First Action

"She gives away 4 stickers" means we subtract 4. After giving away stickers, Maria has s − 4 stickers left.

After giving away: s − 4
3
Step 3 — Translate the Second Action

"Splits equally among 3 friends" means we divide the remaining amount by 3. We need parentheses around (s − 4) because the entire leftover amount gets divided.

Each friend gets: (s − 4) ÷ 3
4
Step 4 — Check with a Real Number

Let's test it. If Maria starts with 16 stickers, she gives away 4, leaving 12. Split among 3 friends = 4 each. Now plug into our expression: (16 − 4) ÷ 3 = 12 ÷ 3 = 4. ✓ It matches!

Answer: (s − 4) ÷ 3

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PRO TIP: PLUG IN TO CHECK

On the ISEE, after you write your expression, plug in an easy number to make sure it works. This is like checking your answer on a video game before you hit submit — it only takes a few seconds and can save you from silly mistakes.

SECTION 7

ISEE Strategies: Expressions vs. Equations

Students sometimes confuse expressions and equations. The ISEE may test both, so it helps to know the difference. Here's a quick comparison.

Feature	Expression	Equation
Has an = sign?	No	Yes
Example	3n + 7	3n + 7 = 22
Can you solve for n?	Not by itself — you can evaluate it for a given n	Yes — you can find the exact value of n
Think of it like…	A recipe (a set of instructions)	A balanced scale (both sides match)

Expressions vs. Equations

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ISEE STRATEGY

When an ISEE question says "which expression represents…", your answer should never contain an equals sign. If you see an equals sign in your answer choice, it's an equation — not the expression the question asked for. Use process of elimination to cross it out!

SECTION 8

From Expressions to Equations and Beyond

Learning to translate situations into expressions is your first big step. Once you're comfortable with expressions, you'll be ready for more advanced topics. Here's where this skill leads.

This Lesson	Next Level
Write expressions: 2n + 5	Solve equations: 2n + 5 = 17
Use one variable: n	Use two variables: n and m
Translate a single situation	Write inequalities: 3n + 2 > 14
Simple operations (+, −, ×, ÷)	Exponents and roots: n² + 3

How this skill builds into future math

Every one of those advanced skills starts with the same first move: read the words, pick a variable, and build the expression. Master that now, and the rest will come much more easily later.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Remember: there's no penalty for guessing on the ISEE, so always pick an answer! Use process of elimination to cross out choices that don't make sense.

Question 1: CONCEPTUAL

Which expression represents "a number increased by 9"? (A) 9n (B) n − 9 (C) n + 9 (D) 9 − n

A. 9n
B. n − 9
C. n + 9
D. 9 − n

Show answer (C)

The correct answer is (C) n + 9. "Increased by" means addition, so we add 9 to the unknown number n. Choice (A) 9n is incorrect because multiplying is not the same as increasing. Choice (B) n − 9 is subtraction, not addition. Choice (D) 9 − n subtracts the number from 9, which reverses the meaning.

Question 2: BASIC CALCULATION

Jake earns d dollars each hour. He works for 5 hours and then spends $12 on lunch. Which expression represents the amount of money Jake has left? (A) 5d − 12 (B) 5d + 12 (C) 5(d − 12) (D) 12 − 5d

A. 5d − 12
B. 5d + 12
C. 5(d − 12)
D. 12 − 5d

Show answer (A)

The correct answer is (A) 5d − 12. Jake earns d dollars per hour for 5 hours, giving him 5 × d = 5d dollars. Then he spends $12, so we subtract 12 from 5d. Choice (B) adds 12 instead of subtracting (he spent money, not gained it). Choice (C) subtracts 12 from d before multiplying, which would mean he loses $12 per hour — that's not what the problem says. Choice (D) reverses the subtraction, which would give a negative result if Jake earned more than $12.

Question 3: INTERMEDIATE

Quantitative Comparison: A number n is greater than 0. Column A: The value of "6 less than twice n" when n = 5 Column B: The value of "twice the result of n less 6" when n = 5 (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

A. Column A is greater.
B. Column B is greater.
C. The two quantities are equal.
D. Cannot be determined.

Show answer (A)

The correct answer is (A). First, translate both phrases. Column A: "6 less than twice n" = 2n − 6. When n = 5: 2(5) − 6 = 10 − 6 = 4. Column B: "twice the result of n less 6" = 2(n − 6). When n = 5: 2(5 − 6) = 2(−1) = −2. Since 4 > −2, Column A is greater. Notice how the placement of parentheses completely changes the answer! Choice (B) is wrong because −2 is not greater than 4. Choice (C) is wrong because 4 ≠ −2. Choice (D) is wrong because both columns use only n = 5, so we can determine the exact values.

Question 4: APPLIED

A movie theater charges $8 per ticket and $3 for each bucket of popcorn. A group of friends buys t tickets and p buckets of popcorn, then uses a $10 coupon. Which expression represents the total amount the group pays? (A) 8t + 3p + 10 (B) 8t + 3p − 10 (C) 11tp − 10 (D) 8t − 3p − 10

A. 8t + 3p + 10
B. 8t + 3p − 10
C. 11tp − 10
D. 8t − 3p − 10

Show answer (B)

The correct answer is (B) 8t + 3p − 10. Tickets cost $8 each, so t tickets cost 8t dollars. Popcorn costs $3 each, so p buckets cost 3p dollars. The total before the coupon is 8t + 3p. A $10 coupon reduces the price, so we subtract 10. Choice (A) adds 10 instead of subtracting (a coupon saves money). Choice (C) incorrectly multiplies t and p together and combines the prices. Choice (D) subtracts the popcorn cost, but popcorn is an additional expense that should be added.

Question 5: CRITICAL THINKING

Quantitative Comparison: n is a positive integer. Column A: The expression for "3 more than half of n" evaluated at n = 10 Column B: The expression for "half of 3 more than n" evaluated at n = 10 (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

A. Column A is greater.
B. Column B is greater.
C. The two quantities are equal.
D. Cannot be determined.

Show answer (A)

The correct answer is (A). Translate each phrase carefully. Column A: "3 more than half of n" = (n ÷ 2) + 3. At n = 10: (10 ÷ 2) + 3 = 5 + 3 = 8. Column B: "half of 3 more than n" = (n + 3) ÷ 2. At n = 10: (10 + 3) ÷ 2 = 13 ÷ 2 = 6.5. Since 8 > 6.5, Column A is greater. Choice (B) is wrong because 6.5 < 8. Choice (C) is wrong because the two values are different. Choice (D) is wrong because n = 10 is specified, giving exact values.

SUMMARY

Summary & Quick Review

Summary

To translate a situation into an algebraic expression, start by choosing a variable for the unknown quantity. Next, identify the operation words — words like "more than" (addition), "less than" (subtraction), "times" (multiplication), and "divided by" (division). Pay special attention to order-reversal phrases like "less than" and "subtracted from," which flip the position of the variable and the number.

Use parentheses when a phrase like "twice the sum" means you must group operations together. Always plug in a test number to check your expression — this quick strategy catches most errors. Remember: an expression has no equals sign, and on the ISEE, there is no penalty for guessing, so always answer every question!