The length, y, of a white whale was 162 centimeters (cm) when it was born and increased an average of...

1. TRANSLATE the problem information

• Given information:
  • Birth length: 162 cm
  • Growth rate: 4.8 cm per month
  • \(\mathrm{x}\) = months after birth
  • \(\mathrm{y}\) = current length in cm

2. INFER the mathematical structure

• This describes linear growth: starting value + (rate × time)
• We need the form \(\mathrm{y = mx + b}\) where:
  • \(\mathrm{m}\) = slope (rate of change per month)
  • \(\mathrm{b}\) = y-intercept (starting value when \(\mathrm{x = 0}\))

3. INFER which values go where

• At birth (\(\mathrm{x = 0}\)), the whale was 162 cm → y-intercept = 162
• Each month (\(\mathrm{x}\) increases by 1), length increases by 4.8 cm → slope = 4.8
• Therefore: \(\mathrm{y = 4.8x + 162}\)

4. Verify with the answer choices

• Choice D matches our equation exactly

Answer: D. \(\mathrm{y = 4.8x + 162}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students mix up which number represents the rate of change versus the starting amount.

They might think: "162 is bigger, so it must be what multiplies with x" or "4.8 comes with 'per month' so it goes with the constant." This leads them to write \(\mathrm{y = 162x + 4.8}\) or similar incorrect combinations.

This may lead them to select Choice B (\(\mathrm{y = 162x + 162}\)) - getting the y-intercept right but using 162 as the slope instead of 4.8.

Second Most Common Error:

Poor INFER reasoning about linear equation structure: Students understand the numbers but don't grasp that the y-intercept represents the value when \(\mathrm{x = 0}\) (at birth).

They might think: "The whale grows 4.8 cm per month and started with some length, so it's 4.8x plus something small." This leads them to use 4.8 as the y-intercept instead of 162.

This may lead them to select Choice C (\(\mathrm{y = 4.8x + 4.8}\)) - getting the slope right but using 4.8 instead of 162 for the starting length.

The Bottom Line:

Success requires clearly distinguishing between "how much it starts with" (y-intercept) and "how much it changes per unit time" (slope) when translating real-world linear growth into \(\mathrm{y = mx + b}\) form.