K-12 Math (US): 6-8 Algebra K-12 Math Expressions Translate words to expressions

Grade 6–8 Algebra Study Guide: Translating Words to Expressions

1. The Driving Question

"If your friend texts you, ‘I have twice as many stickers as you, plus five,’ how do you turn that sentence into a math problem you can actually solve? And why does the word ‘than’ change everything—like, why does ‘three less than a number’ not mean ‘3 – x’?"

2. The Core Idea — Built, Not Listed

Imagine you’re at a school bake sale. Your friend says: "I sold three more cookies than you did, and then I sold twice as many brownies as cookies."

To figure out how many brownies they sold, you can’t just guess—you need a math expression. Words like "more than," "less than," "twice as many," and "total" are clues that map directly to operations (+, –, ×, ÷). The trick is that English phrases don’t always follow the order you’d write the math. For example, "five less than a number" isn’t 5 – x—it’s x – 5, because "less than" flips the order.

Think of it like translating a secret code: the words are the cipher, and the expression is the decoded message. Once you crack the pattern, you can turn any word problem into an equation.

Key Vocabulary:
- Variable – A letter (like x or n) that stands for an unknown number.
Example: If your little brother is n years old, "three times his age" is 3n—not 3 × brother.
Note (HS+): In advanced math, variables can represent functions, matrices, or even sets, not just single numbers.

• Operation – A math action (+, –, ×, ÷) triggered by words like "sum," "difference," "product," or "quotient." Example: "The quotient of a number and 7" means x ÷ 7, not 7 ÷ x (because "quotient of A and B" always means A ÷ B).
  Note (HS+): In calculus, operations expand to include limits, derivatives, and integrals—each with their own "word triggers."

• Expression – A math phrase with numbers, variables, and operations (no equals sign).
  Example: "Half of a number, decreased by 4" = n/2 – 4—this isn’t an equation yet because there’s no "is" or "equals." Note (HS+): In algebra II, expressions can include exponents, roots, and absolute values, making the translation trickier.

• Key Phrase – A word or group of words that signals a specific operation or order.
  Example: "Seven fewer than twice a number" = 2x – 7—"fewer than" flips the order, and "twice" means 2 ×.
  Note (HS+): In word problems, key phrases can get buried in complex sentences (e.g., "the difference between the square of a number and its cube" = x² – x³).

3. Assessment Translation

How This Appears on State Tests (Grades 6–8):
- Multiple Choice: Questions like "Which expression represents ‘eight more than the product of 5 and a number’?" with options like 5x + 8, 8 + 5 + x, or 5(x + 8).
Distractor Patterns: - Order errors: Choosing 8 + 5x instead of 5x + 8 (same value, but the question specifies order).
- Operation errors: Picking 5x – 8 for "eight less than the product" (confusing "less than" with subtraction).
- Grouping errors: Selecting 5(x + 8) for "eight more than the product" (misplacing parentheses).

• Short Answer: "Write an expression for ‘three times the sum of a number and 9.’" or "Explain why ‘10 less than a number’ is written as x – 10, not 10 – x." What Proficient Looks Like:
• Correct expression (3(n + 9) or x – 10).
• Explanation that "less than" reverses the order (e.g., "If the number is 15, ‘10 less than 15’ is 5, which is 15 – 10, not 10 – 15.").

Model Proficient Response (Short Answer):
Prompt: "Write an expression for ‘the quotient of 12 and a number, decreased by 3.’ Then explain how you knew which operation to use for ‘quotient.’" Response: "The expression is 12 ÷ x – 3. I knew ‘quotient’ means division, and ‘decreased by’ means subtraction. The phrase ‘quotient of 12 and a number’ tells me to divide 12 by x, not the other way around, because ‘quotient of A and B’ always means A ÷ B."

4. Mistake Taxonomy

Mistake 1: Misordering "Less Than" or "More Than"
- Question: "Write an expression for ‘four less than a number.’" - Common Wrong Answer: 4 – x - Why It Loses Credit: The phrase "less than" reverses the order. 4 – x would mean "a number less than 4," not "four less than a number." - Correct Approach: "Four less than a number" = x – 4. Test with a number: if x = 10, "four less than 10" is 6, which is 10 – 4, not 4 – 10.

Mistake 2: Ignoring Parentheses for "Sum" or "Difference"
- Question: "Write an expression for ‘twice the sum of a number and 5.’" - Common Wrong Answer: 2x + 5 - Why It Loses Credit: "Twice the sum" means you multiply 2 by the entire sum (x + 5), not just x. 2x + 5 would mean "twice a number, plus 5." - Correct Approach: "Sum of a number and 5" = x + 5. "Twice the sum" = 2(x + 5).

Mistake 3: Confusing "Product" and "Sum" in Word Problems
- Question: "Which expression represents ‘the product of 6 and a number, increased by 2’?" (Options: 6 + x + 2, 6x + 2, 6(x + 2)) - Common Wrong Answer: 6 + x + 2 - Why It Loses Credit: "Product" means multiplication, not addition. 6 + x + 2 would mean "6 plus a number plus 2." - Correct Approach: "Product of 6 and a number" = 6x. "Increased by 2" = 6x + 2.

5. Connection Layer

• Within Math: Translating words to expressions → Solving equations Why it matters: If you can’t turn "three more than twice a number" into 2x + 3, you’ll never solve "Three more than twice a number is 11." The expression is the first step to the equation.

• Across Subjects: Key phrases in math → Grammar in ELA (prepositional phrases) Why it matters: Words like "less than" or "sum of" act like prepositional phrases—they change the order of the sentence. In ELA, "the book on the table" isn’t the same as "the table on the book," just like "5 less than x" isn’t 5 – x.

• Outside School: Word-to-math translation → Reading nutrition labels Why it matters: A label might say "25% less sugar than the leading brand." To compare, you’d write: leading brand sugar – 0.25 × leading brand sugar. The "less than" flips the order, just like in math.

6. The Stretch Question

"If ‘the square of a number’ is x², and ‘a number squared’ is also x², why isn’t ‘the cube of a number’ the same as ‘a number cubed’ in some contexts? Can you invent a word problem where the order actually changes the meaning?"

Pointer Toward the Answer: In math, "the cube of a number" and "a number cubed" are the same (x³), but in English, word order can imply different operations. For example: - "The difference of a number and its cube" = x – x³ - "A number and its cube’s difference" = x – (x³) (same here, but the phrasing is clunky).
The real fun starts when you mix operations: "Twice the square of a number" (2x²) vs. "The square of twice a number" ((2x)² = 4x²). The order changes the math—can you write a word problem where swapping two words changes the answer?