Words to Expressions — Translate a word phrase into an algebraic expression. - SHSAT Math

SECTION 1

Why Do We Turn Words into Math?

People have been solving word problems for thousands of years. Long before anyone used letters like x or n, ancient mathematicians wrote out every problem in full sentences. That made even simple problems very long and hard to follow. Over time, clever thinkers invented shortcuts — symbols and letters — to make math easier to read and solve.

1800 BCE • Babylonian Word Problems

Ancient Babylonians carved word problems into clay tablets. They described unknowns with phrases like "the thing" instead of using a letter.

250 CE • Diophantus Uses Abbreviations

The Greek mathematician Diophantus started using short abbreviations for unknown quantities, saving space and reducing confusion.

820 CE • Al-Khwarizmi Writes the First Algebra Book

Al-Khwarizmi published a book that gave us the word "algebra." He described rules for solving equations using words and sentences.

1637 • Descartes Introduces x, y, z

French mathematician René Descartes popularized the use of letters like x, y, and z for unknowns. This is the system we still use today!

On the SHSAT, you will see questions that give you a phrase in plain English and ask you to write it as an algebraic expression (a math statement that uses numbers, variables, and operations). Being able to translate words into algebra quickly and correctly is one of the most important skills you can build.

SECTION 2

Core Principles of Translating Words to Expressions

Before you translate anything, you need to know the basic building blocks. An algebraic expression is a combination of numbers, variables (letters that stand for unknown values), and operations (like addition or multiplication). Unlike an equation, an expression does not have an equals sign.

Identify the Unknown

Look for the value you don't know. Words like "a number," "some amount," or "a quantity" tell you to use a variable (such as x or n).

Spot the Operation

Key words hint at which math operation to use. "Sum" means add. "Difference" means subtract. "Product" means multiply. "Quotient" means divide.

Watch the Order

Order matters for subtraction and division! "Five less than a number" is n − 5, not 5 − n. Always check what comes first.

Combine the Pieces

Put the variable, the number, and the operation together. Use parentheses when a phrase says "the sum of" or "the difference of" something being multiplied or divided.

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KEY TAKEAWAY

Think of translating words to algebra like translating English to Spanish. Each English math word has a matching math symbol. "Sum" becomes +, "product" becomes ×, and "a number" becomes a letter like x. Once you learn the vocabulary, translation becomes automatic!

SECTION 3

Visual Guide: Key Words and Their Operations

The diagram below is your cheat sheet. It groups the most common English phrases by the math operation they represent. Study it carefully — you will see these words over and over on the SHSAT.

This diagram groups key English words by the math operation they signal. Blue = addition, violet = subtraction, pink = multiplication, amber = division. Save this map and refer to it as you practice.

Notice how some words can be tricky. The phrase "less than" flips the order — "five less than a number" means n − 5, not 5 − n. Similarly, "subtracted from" reverses the order. Always ask yourself: what is being taken away, and from what?

SECTION 4

The Translation Framework

Let's build a simple three-step framework you can use every single time you see a word-to-expression problem.

STEP 1 — ASSIGN THE VARIABLE

Let n = the unknown number

Pick any letter you like. Most SHSAT problems use n, x, or the first letter of the item described.

STEP 2 — IDENTIFY THE OPERATION

Key word → +, −, ×, or ÷

Use the Key Word Map from Section 3. Match each English word to its operation symbol.

STEP 3 — BUILD THE EXPRESSION

(number or variable) operation (number or variable)

Put the pieces together in the correct order. Remember: for subtraction and division, the order you write the pieces matters a lot!

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SHSAT Tip: Order Traps

The SHSAT loves to test whether you flip the order. Memorize these two rules: (1) "A less than B" means B − A. (2) "A subtracted from B" means B − A. In both cases, the number after "than" or "from" comes first in the expression.

Here are some common patterns. "Twice a number" means 2 × n, which we write as 2n. "A number squared" means n². "Three more than twice a number" combines multiplication and addition: 2n + 3.

SECTION 5

Common Phrase-to-Expression Translations

Below is a reference table of phrases you are very likely to see on the SHSAT. Study the "Expression" column and make sure the order feels natural to you.

English Phrase	Operation	Expression
a number increased by 9	Addition	n + 9
7 more than a number	Addition	n + 7
the sum of a number and 4	Addition	n + 4
a number decreased by 6	Subtraction	n − 6
5 less than a number	Subtraction	n − 5 (not 5 − n)
3 subtracted from a number	Subtraction	n − 3 (not 3 − n)
the product of 8 and a number	Multiplication	8n
twice a number	Multiplication	2n
a number divided by 4	Division	n ÷ 4 or n/4
the quotient of a number and 10	Division	n ÷ 10 or n/10
three more than twice a number	Multiply, then Add	2n + 3
four times the sum of a number and 5	Add, then Multiply	4(n + 5)

Common English-to-algebra translations for the SHSAT

This flowchart shows how to handle a two-operation phrase. First, translate the inner operation ("the sum of a number and 5" → n + 5). Then apply the outer operation ("four times" → multiply by 4). The parentheses keep the addition grouped together.

SECTION 6

Worked Example: From Words to Algebra

Let's walk through a full SHSAT-style problem together, step by step.

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PROBLEM

Translate the following phrase into an algebraic expression: "Seven less than three times a number."

Solution: "Seven less than three times a number"

1
Step 1 — Identify the Unknown

The phrase says "a number." We don't know what that number is, so we assign a variable. Let n = the unknown number.

Variable: n
2
Step 2 — Translate "three times a number"

The word "times" tells us to multiply. "Three times a number" means 3 × n, which we write as 3n.

3n
3
Step 3 — Translate "seven less than"

"Less than" means subtraction, and it reverses the order. "Seven less than [something]" means you start with that something and subtract 7. So we subtract 7 from 3n.

3n − 7
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Step 4 — Write the Final Expression

Combining everything, the algebraic expression is:

3n − 7
5
Step 5 — Check Your Work

Read the expression back as words. 3n − 7 means "three times a number, minus seven" — that matches "seven less than three times a number." ✓

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ALWAYS CHECK BY READING BACK

After writing your expression, read it back in English. If it matches the original phrase, you're correct. It's like proofreading a text message before you hit send — take two seconds to make sure it says what you mean!

SECTION 7

Tricky Phrases and Common Mistakes

Some phrases are designed to trip you up on the SHSAT. The table below shows the most common traps and how to avoid them.

Tricky Phrase	Common Wrong Answer	Correct Expression
10 less than n	10 − n	n − 10
n subtracted from 20	n − 20	20 − n
the quotient of 12 and n	n ÷ 12	12 ÷ n
5 times the difference of n and 2	5n − 2	5(n − 2)
the sum of n and 3, divided by 4	n + 3 ÷ 4	(n + 3) ÷ 4

Common SHSAT traps with subtraction, division, and grouping

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THE "LESS THAN" / "SUBTRACTED FROM" RULE

Think of it like a video game scoreboard. If you have 50 points and lose 8, your new score is 50 − 8, not 8 − 50. The thing you start with goes first, and the thing removed goes after the minus sign. "8 less than 50" = 50 − 8. "8 less than n" = n − 8.

SECTION 8

From Expressions to Equations and Beyond

Translating words to expressions is the foundation for bigger skills. Once you can write an expression, you can write full equations (expressions connected by an equals sign) and even inequalities (expressions connected by <, >, ≤, or ≥). The SHSAT tests all three.

Concept	What It Looks Like	Example from Words
Expression (this lesson)	No equals sign: 2n + 5	"five more than twice a number"
Equation (next step)	Has an equals sign: 2n + 5 = 17	"five more than twice a number is 17"
Inequality (advanced)	Has <, >, ≤, or ≥: 2n + 5 > 17	"five more than twice a number is greater than 17"

Progression from expressions to equations to inequalities

Notice how the expression is always the first building block. If you master this lesson, the next steps — solving equations and working with inequalities — will feel much easier. Think of an expression as a single sentence and an equation as a complete thought that makes a claim.

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LOOKING AHEAD

On the actual SHSAT, many word problems ask you to set up an equation and solve for the variable. You'll need the translation skills from this lesson plus the ability to isolate the variable. Practice translating first — the solving part comes next!

SECTION 9

Practice Problems

Try each problem on your own before checking the answer. Use the three-step framework: (1) assign the variable, (2) identify the operation, and (3) build the expression.

Question 1: CONCEPTUAL

Which math operation does the word "product" represent: addition, subtraction, multiplication, or division?

Show answer

The word "product" means multiplication. When a problem says "the product of 6 and a number," it means 6 × n, which we write as 6n. Remember: "sum" = add, "difference" = subtract, "product" = multiply, "quotient" = divide. Memorize these four words — they appear on nearly every SHSAT.

Question 2: BASIC CALCULATION

Translate the phrase into an algebraic expression: "a number increased by 12."

Show answer

The answer is n + 12. "Increased by" means addition. Step 1: Let n = the number. Step 2: The operation is addition (+). Step 3: Combine them as n + 12. You could also write 12 + n, since addition is commutative (order doesn't matter).

Question 3: INTERMEDIATE

Translate: "Eight less than four times a number."

Show answer

The answer is 4n − 8. First, translate "four times a number" as 4n (multiplication). Next, "eight less than" means subtract 8 from whatever came before it. So we get 4n − 8. A common mistake is writing 8 − 4n, but remember: "less than" flips the order. The item after "than" (4n) goes first, and the item before "than" (8) gets subtracted.

Question 4: APPLIED

A movie ticket costs $d. Your family buys 5 tickets and also spends $8 on popcorn. Write an expression for the total amount your family spends.

Show answer

The answer is 5d + 8. Each ticket costs d dollars, and your family buys 5 of them, so the ticket cost is 5 × d = 5d. The popcorn adds $8 more, so you add 8 to get 5d + 8. This is exactly how the SHSAT presents real-world word problems — you identify the variable (d), figure out the operations (multiply and add), and combine them.

Question 5: CRITICAL THINKING

Translate: "Twice the difference of a number and 3, increased by 7." Then explain why parentheses are needed.

Show answer

The answer is 2(n − 3) + 7. First, "the difference of a number and 3" translates to n − 3. Next, "twice" that difference means 2 × (n − 3). Finally, "increased by 7" means we add 7, giving us 2(n − 3) + 7. The parentheses are essential because without them, 2n − 3 + 7 would mean something different — it would only double n, not the whole difference. The phrase says "twice the difference," so the entire subtraction must be grouped inside parentheses before multiplying.

SUMMARY

Lesson Summary

Summary

Translating word phrases into algebraic expressions is a core SHSAT skill. You learned to assign a variable for the unknown, match key words to operations (sum → +, difference → −, product → ×, quotient → ÷), and watch the order — especially with "less than," "subtracted from," and "divided into," which flip the position of the numbers.

For multi-step phrases, translate the inner operation first and use parentheses to group it, then apply the outer operation. Always read your expression back in English to confirm it matches the original phrase. Master this skill, and you'll be ready to tackle equations and inequalities next.