Translating Words into Math - Sixth Grade Mathematics

Translating verbal descriptions into algebraic expressions converts everyday scenarios into precise mathematical formulas, requiring careful attention to operational keywords and phrase structure.

The Operational Keyword Lexicon

Every mathematical operation is signaled by specific words and phrases in the English language.

+-------------------------------------------------------------------------+
|                  THE ALGEBRAIC TRANSLATION LEXICON                      |
+---------------+---------------------------------------------------------+
| Operation     | Common English Keywords and Phrases                     |
+---------------+---------------------------------------------------------+
| Addition (+)  | Sum, plus, increased by, more than, total of, added to  |
+---------------+---------------------------------------------------------+
| Subtraction(-)| Difference, minus, decreased by, less than, subtracted  |
|               | from, reduced by, take away                             |
+---------------+---------------------------------------------------------+
| Multiplication| Product, times, multiplied by, of, double (2x),         |
| (*)           | triple (3x), twice                                      |
+---------------+---------------------------------------------------------+
| Division (/)  | Quotient, divided by, ratio of, split into, half (x/2)  |
+---------------+---------------------------------------------------------+

The Order-Reversal Trap: "Less Than" and "Subtracted From"

The most frequent error in translating verbal algebra is misinterpreting phrases that reverse the physical order of terms. Standard phrasing maintains word order: "Twelve decreased by a number n" translates directly to: 12 - n. Order-reversal phrasing inverts the sequence: "Five less than a number x" means you start with x and take 5 away: x - 5. Writing 5 - x is completely false!

+-------------------------------------------------------------------------+
|                  THE ORDER-REVERSAL PHRASE WATCHLIST                    |
+------------------------------------+------------------------------------+
| Verbal Phrase                      | Correct Algebraic Translation      |
+------------------------------------+------------------------------------+
| "8 less than y"                    | y - 8   (NOT 8 - y!)               |
| "4 subtracted from p"              | p - 4   (NOT 4 - p!)               |
| "6 fewer than twice a number k"    | 2k - 6  (NOT 6 - 2k!)              |
+------------------------------------+------------------------------------+

Translating Multi-Step Compound Expressions

Many real-world situations involve multiple operations combined with grouping symbols. Parentheses are signaled by phrases like "the sum of" or "the difference of" multiplied by another number.

Translating Compound Phrasing:
"Three times the sum of a number x and 4"
Notice: The multiplication applies to the ENTIRE sum!
Correct: 3 * (x + 4)  or  3(x + 4)
Incorrect: 3x + 4 (This multiplies x by 3 first, then adds 4!)

"The quotient of a number n and 5, increased by 8"
Correct: (n / 5) + 8

Real-World Narrative Modeling

Translating practical problems into expressions requires identifying fixed and changing quantities: A taxi ride charges a $3.00 pickup fee plus $2.50 per mile m: Expression = 2.50m + 3.00. A cell phone plan costs $40 monthly plus $0.10 for each gigabyte g over the limit: Expression = 40 + 0.10g.

Exercises

Exercise 1. Translate each verbal phrase into an algebraic expression: (a) A number k increased by 14; (b) The product of 9 and a number m; (c) Seven less than a number w; (d) The quotient of a number p and 6.

Exercise 2. Translate each complex phrase into an algebraic expression: (a) Four times the difference of a number x and 7; (b) Five more than twice a number y; (c) The sum of 12 and the square of a number n.

Exercise 3. An amusement park charges $25 for admission plus $3 for each ride ticket t. Write an algebraic expression for the total cost of visiting the park.

Solutions

Solution to Exercise 1. For (a) k increased by 14: k + 14. For (b) product of 9 and m: 9m. For (c) seven less than w: order is reversed; w - 7. For (d) quotient of p and 6: p / 6 (or p/6).

Solution to Exercise 2. For (a) four times the difference of x and 7: the difference is grouped; 4(x - 7). For (b) five more than twice y: twice y is 2y; adding 5 gives 2y + 5. For (c) sum of 12 and the square of n: square of n is n^2; sum is 12 + n^2.

Solution to Exercise 3. The admission fee of $25 is a fixed constant. The cost for rides depends on the number of tickets t at $3 each (3t). Combining gives the algebraic expression: 25 + 3t (or 3t + 25).