Writing Numerical Expressions from Words: Fifth Grade Mathematics

Translating written scenarios into mathematical expressions requires mapping keywords to operations and placing grouping symbols to preserve operational meaning.

Translating Words to Operations

Certain mathematical keywords signal specific arithmetic operations:

+--------------------------------------------------------------------------+
|                  MATHEMATICAL TRANSLATION KEYWORDS                       |
+-------------+------------------------------------------+-----------------+
| Operation   | Signal Words and Phrases                 | Example Words   |
+-------------+------------------------------------------+-----------------+
| Addition    | sum, plus, increased by, more than,      | "sum of 8 and 4"|
| (+)         | combined, total of                       | ---> 8 + 4      |
| Subtraction | difference, minus, decreased by,         | "6 less than 15"|
| (-)         | less than, take away, subtracted from    | ---> 15 - 6     |
| Multiplication| product, times, multiplied by,         | "twice the sum" |
| (*)         | double, twice, triple, of                | ---> 2 * (...)  |
| Division    | quotient, divided by, split equally,     | "quotient of    |
| (/)         | shared among, separated into             | 20 and 4" -> 20/4|
+-------------+------------------------------------------+-----------------+

The "Turnaround" Words in Subtraction

Phrases like "less than" and "subtracted from" reverse the written order of numbers:
- "5 less than 12" means 12 - 5 (NOT 5 - 12).

- "8 subtracted from 20" means 20 - 8 (NOT 8 - 20).

Recognizing When Grouping Symbols Are Required

Whenever an operation that normally has lower precedence (addition or subtraction) must occur before multiplication or division, you must enclose it in grouping symbols.

Phrases Requiring Parentheses:
  "Multiply 6 by the sum of 8 and 3"       --->  6 * (8 + 3)
  (Without parentheses, 6 * 8 + 3 would multiply 6 * 8 first!)

  "Twice the difference of 15 and 9"       --->  2 * (15 - 9)

  "Divide the sum of 14 and 10 by 3"       --->  (14 + 10) / 3

Exercises and Solutions

Exercise 1: Translating Word Phrases to Expressions. Write a numerical expression for each of the following verbal descriptions: Description A: "Add 15 and 9, then multiply by 4." Description B: "Subtract 7 from 25, then divide by 3." Description C: "Three times the sum of 12 and 8, decreased by 5."

Solution 1: For Description A: The words "then multiply" mean the addition must happen first: (15 + 9) * 4. For Description B: Subtract 7 from 25 is (25 - 7), and dividing that result by 3 gives (25 - 7) / 3. For Description C: The sum of 12 and 8 is grouped: (12 + 8). Three times that sum is 3 * (12 + 8). Decreasing that result by 5 gives 3 * (12 + 8) - 5.

Exercise 2: Real-World Scenario Translation. At a school book fair, Liam buys 3 graphic novels for $8 each and 2 bookmarks for $1.50 each. He pays with a $50 bill. Write a single numerical expression using grouping symbols that represents Liam's change.

Solution 2: First, identify the cost of the graphic novels: (3 * 8). Identify the cost of the bookmarks: (2 * 1.50). Combine total spending in brackets: [ (3 * 8) + (2 * 1.50) ]. Subtract total spending from $50: 50 - [ (3 * 8) + (2 * 1.50) ]. This single expression represents Liam's change.