Tanya earns $13.50 per hour at her part-time job. When she works z hours, she earns 13.50z dollars. Which of...

1. TRANSLATE the problem information

• Given information:
  • Tanya earns \(\$13.50\) per hour (constant rate)
  • For \(\mathrm{z}\) hours of work, she earns \(13.50\mathrm{z}\) dollars
  • We need the expression for earnings when working \(3\mathrm{z}\) hours
• What this tells us: We have a proportional relationship between hours and earnings

2. INFER the relationship between different amounts of work

• Key insight: Since the hourly rate stays constant, earnings scale directly with hours
• If she works \(3\mathrm{z}\) hours instead of \(\mathrm{z}\) hours, she's working 3 times as many hours
• Therefore, she should earn 3 times as much money

3. Apply the scaling relationship

• Earnings for \(\mathrm{z}\) hours = \(13.50\mathrm{z}\) dollars
• Earnings for \(3\mathrm{z}\) hours = \(3 \times\) (earnings for \(\mathrm{z}\) hours)
• Earnings for \(3\mathrm{z}\) hours = \(3 \times (13.50\mathrm{z}) = 3(13.50\mathrm{z})\) dollars

Answer: A. \(3(13.50\mathrm{z})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what "\(3\mathrm{z}\) hours" means mathematically and confuse it with adding 3 to something, rather than multiplying by 3.

They might think "\(3\mathrm{z}\) hours" means "\(\mathrm{z}\) hours plus 3 more hours" and incorrectly add 3 to the original earnings expression. This may lead them to select Choice B (\(3 + 13.50\mathrm{z}\)).

Second Most Common Error:

Conceptual confusion about variable expressions: Students might misunderstand how to combine the "\(3\mathrm{z}\)" with the earnings formula, treating it as if both terms should be added together.

They might think they need to add "\(3\mathrm{z}\)" as a separate term to "\(13.50\mathrm{z}\)," leading to the expression "\(3\mathrm{z} + 13.50\mathrm{z}\)." This may lead them to select Choice C (\(3\mathrm{z} + 13.50\mathrm{z}\)).

The Bottom Line:

This problem tests whether students understand that proportional relationships scale by the same factor—if you triple the input (hours), you triple the output (earnings). The key is recognizing that "\(3\mathrm{z}\) hours" means "3 times as many hours," not "3 plus \(\mathrm{z}\) hours."