The table gives the average time t, in minutes, it takes Carly to travel a certain distance d, in kilometers....

1. TRANSLATE the problem information

• Given information:
  • Table showing distance d (km) vs time t (minutes)
  • Data points: \(\mathrm{(0.32, 8)}\), \(\mathrm{(0.56, 14)}\), \(\mathrm{(0.68, 17)}\)
  • Need to find equation representing this linear relationship
• What this tells us: We need to find the pattern connecting distance and time

2. INFER the approach

• Since it's a linear relationship, we're looking for an equation \(\mathrm{t = ad + b}\)
• Looking at answer choices, none have a constant term, so likely \(\mathrm{t = ad}\)
• Strategy: Calculate slope between data points to find coefficient a

3. SIMPLIFY to find the slope coefficient

• Using first two points \(\mathrm{(0.32, 8)}\) and \(\mathrm{(0.56, 14)}\):
  \(\mathrm{slope = \frac{14 - 8}{0.56 - 0.32}}\)
  \(\mathrm{= \frac{6}{0.24}}\)
  \(\mathrm{= 25}\) (use calculator)

• Verify with second pair \(\mathrm{(0.56, 14)}\) and \(\mathrm{(0.68, 17)}\):
  \(\mathrm{slope = \frac{17 - 14}{0.68 - 0.56}}\)
  \(\mathrm{= \frac{3}{0.12}}\)
  \(\mathrm{= 25}\) (use calculator)

4. INFER the complete equation

• Since slope = 25 consistently, we have \(\mathrm{a = 25}\)
• Check for y-intercept using point \(\mathrm{(0.32, 8)}\):
  \(\mathrm{8 = 25(0.32) + b}\)
  \(\mathrm{8 = 8 + b}\)
  \(\mathrm{b = 0}\)
• Therefore: \(\mathrm{t = 25d}\)

5. APPLY CONSTRAINTS to verify and select answer

• Test equation with all data points:
  • \(\mathrm{d = 0.32}\): \(\mathrm{t = 25(0.32) = 8}\) ✓
  • \(\mathrm{d = 0.56}\): \(\mathrm{t = 25(0.56) = 14}\) ✓
  • \(\mathrm{d = 0.68}\): \(\mathrm{t = 25(0.68) = 17}\) ✓

Answer: C (\(\mathrm{t = 25d}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when calculating \(\mathrm{6/0.24}\) or \(\mathrm{3/0.12}\), getting incorrect slope values like 4 instead of 25.

For example, they might incorrectly compute \(\mathrm{6/0.24}\) as 4 (perhaps confusing it with \(\mathrm{6/1.5}\)), leading them to think the coefficient is 4. This may lead them to select Choice A (\(\mathrm{t = 4d}\)).

Second Most Common Error:

Poor INFER reasoning about linear relationships: Students might think they need to find which answer "works" for one data point without checking consistency across all points.

They might substitute the first data point \(\mathrm{(0.32, 8)}\) into each choice and find that both A and C seem to work approximately, then guess incorrectly. This causes them to get stuck and guess between seemingly valid options.

The Bottom Line:

This problem requires systematic slope calculation and verification across multiple data points. Students who rush or make arithmetic errors with decimal division will select incorrect coefficients, while those who don't verify consistency may not recognize the true linear pattern.