A model predicts that a certain animal weighed 241 pounds when it was born and that the animal gained 3...

1. TRANSLATE the problem information

• Given information:
  • Animal weighed 241 pounds at birth
  • Animal gained 3 pounds per day in first year
  • Model equation: \(\mathrm{f(x) = a + bx}\)
  • \(\mathrm{f(x)}\) = predicted weight in pounds
  • \(\mathrm{x}\) = days after birth
  • Need to find: value of a

2. INFER what the parameters represent

• In the linear function \(\mathrm{f(x) = a + bx}\):
  • 'a' is the y-intercept (starting value when \(\mathrm{x = 0}\))
  • 'b' is the slope (rate of change per unit of x)

• Since x represents days after birth, when \(\mathrm{x = 0}\), we have the birth weight

3. INFER the value of parameter a

• Substitute \(\mathrm{x = 0}\) into the equation:

\(\mathrm{f(0) = a + b(0) = a}\)

• Since \(\mathrm{f(0)}\) represents the weight at birth (0 days after birth), and we know the animal weighed 241 pounds at birth:

\(\mathrm{a = 241}\)

Answer: 241

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Confusing which parameter represents what in the linear model

Students might think 'a' represents the rate of change (3 pounds per day) because it appears first in the equation, leading them to incorrectly conclude \(\mathrm{a = 3}\). This stems from not properly connecting the mathematical form \(\mathrm{f(x) = a + bx}\) to the real-world context where 'a' is the starting value.

This may lead them to guess or select an incorrect answer.

The Bottom Line:

This problem tests whether students understand that in a linear function \(\mathrm{f(x) = a + bx}\), the parameter 'a' represents the y-intercept or initial value when the independent variable equals zero. Success requires connecting the abstract mathematical form to the concrete real-world situation.