In Grade 5 you wrote expressions from words using only numbers. Now the words can include a variable.
Key idea
Turn each phrase into an operation, in the order the words describe: "5 more than $n$" is $n + 5$, "5 less than $n$" is $n - 5$, "3 times the sum of $y$ and 4" is $3(y + 4)$.
| Words | Operation | Example |
|---|---|---|
| sum, more than, increased by, plus | $+$ | "$n$ increased by 2": $n + 2$ |
| difference, less than, decreased by, minus | $-$ | "7 less than $n$": $n - 7$ |
| product, times, twice, of | $\times$ | "twice $n$": $2n$ |
| quotient, divided by, half of | $\div$ | "half of $n$": $\frac{n}{2}$ |
Why it works
Test a phrase with a number. "5 less than 12" is 7, and $12 - 5 = 7$, while $5 - 12$ is not 7. The phrase "5 less than $n$" starts from $n$ and takes 5 away, so $n$ is written first even though 5 is said first. Words like "less than" and "more than" describe a change made to the number that comes after them.
Worked example: Two operations
Write "7 more than twice $x$".
- Twice $x$ is $2x$.
- 7 more than that: $2x + 7$.
Answer: $2x + 7$
Worked example: A sum as one quantity
Write "3 times the sum of $y$ and 4".
- The sum of $y$ and 4 is $y + 4$, and it is multiplied as a whole, so it goes in parentheses.
- $3(y + 4)$. Without parentheses, $3y + 4$ would mean something else.
Answer: $3(y + 4)$
Worked example: From a story
Ana had $d$ dollars and spent \$8. Write an expression for the money left.
- Spending takes money away.
- Start with $d$ and subtract 8.
Answer: $d - 8$
Common mistake
Writing "5 less than $n$" as $5 - n$, in the order the words are spoken. Quick check: try $n = 12$. Five less than 12 is 7. $12 - 5 = 7$ works; $5 - 12$ does not.