An object hangs from a spring. The formula l = 30 + 2w relates the length l, in centimeters, of...

1. TRANSLATE the problem information

• Given formula: \(\mathrm{l = 30 + 2w}\)
• Need to determine: What does the 2 represent in this context?

2. INFER the structure and meaning

• This is a linear equation in the form \(\mathrm{y = b + mx}\) where:
  • \(\mathrm{l}\) (length) is the dependent variable
  • \(\mathrm{w}\) (weight) is the independent variable
  • 30 is the constant term (y-intercept)
  • 2 is the coefficient of \(\mathrm{w}\)
• Key insight: In linear relationships, the coefficient of the independent variable tells us the rate of change

3. INFER what the coefficient means in context

• Since 2 is multiplied by \(\mathrm{w}\) (weight), it represents how much \(\mathrm{l}\) (length) changes for each unit change in \(\mathrm{w}\)
• Specifically: For each 1-newton increase in weight, the length increases by 2 centimeters

4. Verify with examples

• When \(\mathrm{w = 0}\): \(\mathrm{l = 30 + 2(0) = 30}\) cm
• When \(\mathrm{w = 1}\): \(\mathrm{l = 30 + 2(1) = 32}\) cm
• When \(\mathrm{w = 2}\): \(\mathrm{l = 30 + 2(2) = 34}\) cm

Each 1-newton increase in weight causes a 2-cm increase in length.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the roles of different parts of the equation and think 30 represents the rate of change instead of 2.

They see that 30 is a larger, more prominent number and assume it must be the "important" part that describes how the spring changes. They don't recognize that coefficients (the numbers multiplied by variables) represent rates of change in linear relationships.

This may lead them to select Choice A (The length, in centimeters, of the spring with no weight attached)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "rate of change" means and think about the inverse relationship - how weight changes with length instead of how length changes with weight.

They might think: "If length increases by 1 cm, weight increases by some amount," rather than "If weight increases by 1 newton, length increases by 2 cm." This backwards thinking confuses the independent and dependent variables.

This may lead them to select Choice C (The increase in the weight, in newtons, of the object for each one-centimeter increase in the length of the spring)

The Bottom Line:

Understanding linear equations requires recognizing that coefficients represent rates of change, and being clear about which variable depends on which. The number multiplied by the input variable tells you how much the output changes per unit of input.