Question:A fundraising campaign started the day having already raised some money by 8:00 a.m. From 8:00 a.m. onward, the campaign...

1. TRANSLATE the problem information

• Given information:
  • Started with some unknown amount at 8:00 a.m.
  • Collects \(\$85\) per hour from 8:00 a.m. onward
  • At 11:00 a.m. (3 hours later), total was \(\$1,505\)
  • Model: \(\mathrm{F(h) = A + Bh}\) where \(\mathrm{h}\) = hours after 8:00 a.m.
• What this tells us: A is the starting amount, B is the hourly rate, and we need to find A

2. INFER what the constants represent

• B must equal the hourly rate: \(\mathrm{B = \$85}\)
• We know one point on the line: at \(\mathrm{h = 3}\), \(\mathrm{F(3) = \$1,505}\)
• This gives us enough information to solve for A

3. SIMPLIFY by substituting and solving

• Substitute into \(\mathrm{F(3) = A + B(3)}\):
  \(\mathrm{1,505 = A + 85(3)}\)
  \(\mathrm{1,505 = A + 255}\)
• Solve for A:
  \(\mathrm{A = 1,505 - 255 = 1,250}\)

Answer: 1250

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what A and B represent in the context. They might think A represents the total amount collected (rather than the starting amount) or misunderstand the timing.

This leads them to set up incorrect equations like \(\mathrm{A = 1,505}\) or think B represents something other than the hourly rate, causing confusion and potentially guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{B = 85}\) and set up \(\mathrm{F(3) = A + 85(3) = 1,505}\), but make arithmetic errors when calculating \(\mathrm{85 × 3 = 255}\) or when solving \(\mathrm{A + 255 = 1,505}\).

This may lead them to get an incorrect value for A, such as 1,420 (if they subtract incorrectly) or other wrong calculations.

The Bottom Line:

This problem requires careful attention to what the linear function represents in context - distinguishing between the starting amount (A) and the rate of change (B), then using algebra to find the unknown initial value.