A producer is creating a video with a length of 70 minutes. The video will consist of segments that are...

1. TRANSLATE the problem information

• Given information:
  • Total video length: 70 minutes
  • Video has 1-minute segments and 3-minute segments
  • \(\mathrm{x}\) = number of 1-minute segments
  • \(\mathrm{y}\) = number of 3-minute segments

2. INFER what each type of segment contributes to the total

• The key insight: Total time = time from 1-minute segments + time from 3-minute segments
• Time from 1-minute segments = \(\mathrm{x}\) segments × 1 minute each = \(\mathrm{x}\) minutes
• Time from 3-minute segments = \(\mathrm{y}\) segments × 3 minutes each = \(\mathrm{3y}\) minutes

3. TRANSLATE the relationship into an equation

• Total time = \(\mathrm{x + 3y}\)
• Since total time is 70 minutes: \(\mathrm{x + 3y = 70}\)

Answer: D. \(\mathrm{x + 3y = 70}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse which variable represents which type of segment, thinking \(\mathrm{x}\) represents 3-minute segments and \(\mathrm{y}\) represents 1-minute segments.

Following this incorrect interpretation, they write: \(\mathrm{3x + y = 70}\), thinking "\(\mathrm{x}\) 3-minute segments contribute \(\mathrm{3x}\) minutes, and \(\mathrm{y}\) 1-minute segments contribute \(\mathrm{y}\) minutes."

This may lead them to select Choice C (\(\mathrm{3x + y = 70}\))

Second Most Common Error:

Poor INFER reasoning about combining segments: Students don't understand how to properly combine different segment lengths, instead thinking they should multiply the variables together or treat all segments as the same length.

This conceptual confusion leads them to consider expressions like \(\mathrm{4xy}\) or \(\mathrm{4(x + y)}\), not recognizing that we need to add the time contributions separately.

This causes them to get stuck and guess between Choice A (\(\mathrm{4xy = 70}\)) or Choice B (\(\mathrm{4(x + y) = 70}\))

The Bottom Line:

Success requires carefully translating which variable represents which segment type, then understanding that total time comes from adding the separate contributions of each segment type.