Question:A wheel makes 1.25 complete clockwise rotations from its starting position. When measured as an angle from the standard position...

1. TRANSLATE the problem information

• Given information:
  • Wheel makes 1.25 complete clockwise rotations
  • Angle measured from standard position (counterclockwise positive)
  • Answer form: -aπ where a is positive rational

• What this tells us: We need to convert rotations to radians and account for direction

2. INFER the conversion approach

• Key insight: Each complete rotation equals \(2\pi\) radians
• Strategy: Convert rotations to radians first, then apply direction

3. SIMPLIFY the rotation conversion

• 1.25 rotations × \(2\pi\) radians/rotation = \(2.5\pi\) radians
• Converting to fraction: \(1.25 = \frac{5}{4}\), so we get \(\frac{5}{4} \times 2\pi = \frac{5}{2}\pi\) radians

4. INFER the direction effect

• Standard position uses counterclockwise as positive
• Since rotation is clockwise, the angle is negative: \(-\frac{5}{2}\pi\)

5. TRANSLATE to match the required form

• We have: \(-\frac{5}{2}\pi\)
• Required form: \(-\mathrm{a}\pi\)
• Therefore: \(\mathrm{a} = \frac{5}{2}\)

Answer: 5/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often miss that clockwise rotation produces a negative angle in standard position. They calculate \(1.25 \times 2\pi = 2.5\pi\) correctly but forget to make it negative, leading them to think \(\mathrm{a} = \frac{5}{2}\) corresponds to \(+\mathrm{a}\pi\) instead of \(-\mathrm{a}\pi\). This leads to confusion about the sign convention and may cause them to second-guess their arithmetic or abandon systematic solution and guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students struggle with converting 1.25 to fraction form, either leaving it as a decimal (writing \(\mathrm{a} = 2.5\) instead of 5/2) or making arithmetic errors when computing \(1.25 \times 2\pi\). Since the problem specifically asks for a fraction in lowest terms, decimal answers don't match the required format, leading to uncertainty and potential guessing.

The Bottom Line:

This problem tests whether students truly understand the sign conventions in angle measurement and can work fluently between decimal and fractional representations. The conceptual leap that clockwise equals negative is crucial but easily overlooked.