At a certain time and day, the Washington Monument in Washington, DC, casts a shadow that is 300 feet long....

1. INFER the relationship between objects and shadows

• Key insight: At the same time of day, all objects have the same height-to-shadow ratio because the sun hits them at the same angle
• This creates similar right triangles (object-ground-shadow tip)
• Therefore: \(\frac{\mathrm{Monument\ height}}{\mathrm{Monument\ shadow}} = \frac{\mathrm{Tree\ height}}{\mathrm{Tree\ shadow}}\)

2. TRANSLATE the problem into a mathematical proportion

• Given information:
  • Washington Monument: 555 feet tall, 300 feet shadow
  • Cherry tree: unknown height, 16 feet shadow
• Set up the proportion: \(\frac{555}{300} = \frac{h}{16}\) (where h = tree height)

3. SIMPLIFY by cross multiplication

• Cross multiply: \(555 \times 16 = 300 \times h\)
• Calculate: \(8880 = 300h\) (use calculator)
• Solve for h: \(h = 8880 \div 300 = 29.6\) feet (use calculator)

4. APPLY CONSTRAINTS to select from answer choices

• Compare 29.6 to given choices: A. 10, B. 20, C. 30, D. 35
• 29.6 is closest to 30

Answer: C. 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a proportional relationship problem and instead try to find some other connection between the numbers (like subtraction or simple ratios).

For example, they might calculate 555 - 300 = 255, then try to relate this to 16 somehow, leading to confusion and random guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the proportion incorrectly, such as \(\frac{555}{16} = \frac{300}{h}\), mixing up which measurements correspond to which object.

This leads to \(h = \frac{300 \times 16}{555} = \frac{4800}{555} \approx 8.6\) feet, which might lead them to select Choice A (10) as the closest option.

The Bottom Line:

The key insight is recognizing that shadows create a proportional relationship - once you see this connection, the rest is straightforward algebra. Students who miss this foundational concept often get lost in the numbers without a clear strategy.