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In everyday language, the phrases **"more than"** and **"less than"** act as cues for addition and subtraction, respectively. The phrase "$$b$$ more than $$a$$" means $$b$$ is added to $$a$$, giving $$a + b$$. For instance, "eight more than your current age" translates to your age $$+ 8$$. Conversely, "$$b$$ less than $$a$$" means $$b$$ is subtracted from $$a$$, giving $$a - b$$. For instance, "seven less than your current age" translates to your age $$- 7$$. A common mistake is to reverse the order—writing $$8 - \text{age}$$ instead of $$\text{age} + 8$$—so it is important to note that the quantity after "more than" or "less than" is the starting value from which you add or subtract.

'More Than' and 'Less Than' as Addition and Subtraction Cues

Translate each English phrase into an algebraic expression:

**(a)** "Seventeen more than $y$": The keyword *more than* signals addition and means "added to." Rewrite the phrase as "seventeen added to $y$," which translates to $y + 17$.

**(b)** "Nine less than $9x^2$": The keyword *less than* signals subtraction and means "subtracted from." Rewrite the phrase as "nine subtracted from $9x^2$," which translates to $9x^2 - 9$.

In both cases, the quantity that follows "more than" or "less than" is the starting value, and the number before the keyword is the amount being added or subtracted.

Translating 'Seventeen more than $y$' and 'Nine less than $9x^2$'

Word problems often describe one quantity in terms of another using phrases that must be translated into algebra. For example, translating the phrase "$$14$$ less than the width" to an algebraic expression for the length of a rectangle:

If we let $$w$$ represent the width of the rectangle, we can substitute $$w$$ for "the width" to get "$$14$$ less than $$w$$." Rewriting "less than" as "subtracted from" gives "$$14$$ subtracted from $$w$$." Translating this into algebra results in the expression $$w - 14$$.

Translating a Rectangle's Length as '$$14$$ less than the width'

When translating professional verbal cues into algebraic expressions, the phrases 'more than' and 'less than' dictate a specific order of terms. State the general rule for identifying which quantity in the phrase acts as the 'starting value' and explain where that starting value must be placed in the final algebraic expression.

The Rule for Identifying Starting Values

In professional documentation, verbal instructions must be accurately converted into mathematical formulas to ensure consistency. Describe the standard rule for identifying the 'starting value' in phrases containing the cues 'more than' or 'less than.' In your explanation, specify which part of the English phrase indicates the starting value and state the resulting algebraic structure for the phrase '15 units less than the baseline (b).'

Standard Rule for Translating Verbal Comparison Cues

Arrange the steps to correctly translate the phrase "800 less than $$e$$" into an algebraic expression.

In the phrase "$$8$$ more than $$x$$", which part represents the starting value?

Match each verbal phrase with its correct algebraic expression.

What is a common mistake when translating the phrase "$$10$$ less than $$x$$" into an algebraic expression?

When translating the phrase "$$12$$ less than $$q$$" into an algebraic expression, the starting value is _____.

In the phrase "$$5$$ less than $$d$$", $$d$$ is the starting value from which you subtract.

Case context:
A student translates 'nine less than a number $$x$$' as $$9 - x$$.

Question:
Explain the student's mistake and write the correct expression.

Sample answer:
The student subtracted $$x$$ from $$9$$ instead of $$9$$ from $$x$$. The correct expression is $$x - 9$$.

Key points:
- The phrase 'less than' means subtracting from the value that comes after it.
- The starting value is $$x$$, not $$9$$.
- The correct expression is $$x - 9$$.

Rubric:
Identifies that the subtraction order is reversed and provides the correct expression: $$x - 9$$.

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