A collectible item decreases in value from $10,000 to $5,000 at a constant rate of $500 per year. Which of...

1. TRANSLATE the problem information

• Given information:
  • Initial value: \(\$10,000\)
  • Decreases at constant rate of \(\$500\) per year
  • Need to identify function type

• What "constant rate of \(\$500\) per year" means: Each year, exactly \(\$500\) is subtracted from the previous year's value

2. INFER the mathematical pattern

• Constant rate of change = same amount added/subtracted each time period
• This is the key characteristic of linear functions
• Let's write out a few values to see the pattern:
  • Year 0: \(\$10,000\)
  • Year 1: \(\$10,000 - \$500(1) = \$9,500\)
  • Year 2: \(\$10,000 - \$500(2) = \$9,000\)

3. INFER the function form

• The pattern gives us: \(\mathrm{V(t)} = \$10,000 - \$500\mathrm{t}\)
• This matches the linear form: \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\)
  • Where \(\mathrm{m} = -500\) (slope) and \(\mathrm{b} = \$10,000\) (y-intercept)
• Since the slope is negative (\(-500\)), the function is decreasing

4. INFER why it's not exponential

• Exponential functions involve multiplying by a constant factor each period
• This problem describes subtracting a constant amount each period
• Exponential decay would look like: \(\mathrm{V(t)} = \$10,000 \times (0.95)^\mathrm{t}\) (losing 5% each year)

Answer: B (Decreasing linear)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see "decreases in value" and immediately think "exponential decay" without carefully analyzing what "constant rate" means.

They might reason: "Things that lose value over time are usually exponential, like depreciation." This misses the crucial distinction between constant dollar amounts vs. constant percentages.

This may lead them to select Choice A (Decreasing exponential).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "constant rate of \(\$500\) per year" as meaning the item loses value faster and faster, or they confuse rate with percentage.

They might think: "If it's losing \(\$500\) per year, that must be getting worse over time, so it's accelerating downward." This shows confusion about what "constant" means in this context.

This leads to confusion and potentially guessing between the exponential options.

The Bottom Line:

The key insight is distinguishing between constant amounts (linear) and constant percentages or ratios (exponential). Many students default to exponential thinking for real-world decrease problems without carefully analyzing the specific type of change described.