Pre-Algebra Class 11: Variables and Algebraic Expressions — Translating Words — 8 SOMATH Posters, Theory & 40 Practice Questions

A variable stands for an unknown number. A constant is a fixed number.

Every algebraic expression is built from two kinds of pieces:

• A variable is a letter (usually x, y, n, a, m, or similar) that stands for an unknown or changing number. Its value can change — that's why we use a letter instead of a specific number.
• A constant is a specific, fixed number whose value never changes: 5, 12, 100, 3.14, 1/2.

In the expression x + 5:

• The x is a variable — it could be 1, 100, or any other number.
• The 5 is a constant — it is always exactly 5.

Two more names you'll hear in the same breath:

• Coefficient — the number multiplied by a variable, like the 3 in 3x. If a variable has no visible coefficient (like plain x), the coefficient is 1.
• Term — a single piece of an expression separated by + or −. In 3x + 5 there are two terms: 3x and 5.

Words that mean + (addition)

When you see any of these words in a phrase, use a + sign in your expression:

Good news about addition: the order doesn't matter. x + 7 and 7 + x represent the same number, so both are correct translations of “7 more than x.” That's the commutative property of addition. Order only matters for subtraction and division (coming up).

Words that mean − (subtraction)

These words all mean subtract — but some of them flip the order:

The order-flip trap. The phrases less than, fewer than, and subtracted from put the numbers in the opposite order from how they appear in the sentence. Read carefully:

• “5 less than n” means start with n, and take 5 away. So it's n − 5, NOT 5 − n.
• “2 subtracted from z” means start with z, subtract 2. So it's z − 2, NOT 2 − z.

Trick: the word than or from tells you what to start with. Whatever comes right after than or from goes first in the expression.

Words that mean × (multiplication)

Algebra shorthand. Between a number and a variable, we drop the multiplication sign completely. We write 3x, not 3 × x or 3 · x. The number goes in front of the variable (called the coefficient).

Between two numbers, we cannot drop the sign (3 · 5 is not the same as 35). Between two variables, we can drop it (xy means x · y).

Watch: “the product of n and 6” is written 6n — the number goes first as the coefficient, even though it appeared second in the sentence. That's just a convention: x3 looks like a typo, so we always write 3x.

Words that mean ÷ (division)

Algebra shorthand. Instead of the ÷ sign, we write division as a fraction. So “x divided by 4” is written x/4 — the first thing goes on top, the second goes on the bottom.

Order matters. Division is not commutative: 12/3 = 4 but 3/12 = 1/4. So “the quotient of x and 4” is x/4, not 4/x. Whatever is named first goes on top.

Grouping. When a phrase inside the division has its own operation, wrap it in parentheses so the whole thing goes together: “x divided by (y + 3)” is x / (y + 3), meaning add y and 3 first, then divide x by that result.

Multi-step phrases — go one step at a time

Some phrases use more than one operation. The trick is to slow down and read one step at a time. Follow the order the words appear, but pause on the two things that flip:

1. “less than” and “subtracted from” still flip the order.
2. “the sum of … and …”, “the difference of … and …”, “the product of … and …”, “the quantity …” group two things — do that operation first, in parentheses if needed.

Examples from the poster:

• “Three times a number plus 5”: first “three times a number” = 3x, then “plus 5” = + 5. Answer: 3x + 5.
• “7 less than twice y”: first “twice y” = 2y, then “7 less than” flips the order — subtract 7 from 2y. Answer: 2y − 7 (NOT 7 − 2y).
• “The sum of a number and 4, then divided by 2”: “the sum of” groups x + 4 in parentheses, then divide by 2. Answer: (x + 4)/2.
• “5 more than n divided by 3”: careful — this means “n divided by 3” first (n/3), then add 5. Answer: n/3 + 5. (If the sentence said “5 more than n, divided by 3” with a comma, it would be different.)
• “The product of 2 and a number, minus 1”: “product of 2 and x” = 2x, then “minus 1” = − 1. Answer: 2x − 1.

Parentheses = “keep these together, do them first”

Parentheses are the algebra way to say “this whole thing is one group.” They keep parts of an expression together and tell you what to do first under the order of operations.

Words that signal parentheses:

• the sum of … and …
• the difference of … and …
• the product of … and …
• the quantity …
• the total of …
• the whole expression, group together

Examples from the poster:

Common mistake: writing “twice the sum of x and 4” as 2x + 4. That is WRONG. 2x + 4 doubles only the x. What the phrase actually asks for is to add first (x + 4), then double the whole thing — so 2(x + 4). Try x = 3: 2(3 + 4) = 14, but 2(3) + 4 = 10. Different answers.

Putting it all together

Now you have every tool you need. The 4-step method works for any word phrase, no matter how long:

1. Read the phrase. Look for the unknown — assign it a variable (usually x, n, or whatever makes sense).
2. Identify the operations. Underline every key word: sum, plus, more than, minus, less than, times, divided by, product of, quotient of, twice, half of, the quantity…
3. Watch for the two order-flippers — “less than” and “subtracted from” — and for grouping words that need parentheses.
4. Write the expression one step at a time, left to right, honoring the flips and the parentheses.

Examples from the poster:

Sanity-check trick. When you're done, plug in a small number for the variable and check that the expression matches what the words describe. If “5 less than x” is supposed to be x − 5, try x = 10: x − 5 = 5. That matches “5 less than 10”. If you had written 5 − x, plugging in 10 gives −5 — a red flag.

Class 12 (next class) puts these expressions to work: evaluating expressions and combining like terms. Class 13 introduces equations — setting two expressions equal to each other and solving for the variable.

How to attack a word problem

1. Name the unknown. Pick a variable and write exactly what it means (“let x = number of pencils”).
2. Translate the sentence into an expression. Use everything from Class 11: key words for operations, the order-flip on “less than”/“subtracted from”, parentheses for grouping.
3. Set up an equation only if the problem tells you what the expression equals. Otherwise, just evaluate.
4. Solve and check. Plug the answer back in and re-read the sentence to be sure it fits.