A block of a certain metal has a mass of 468 grams. The density of the metal is 7.2 grams...

1. TRANSLATE the problem information

• Given information:
  • Mass = 468 grams
  • Density = 7.2 grams per cubic centimeter
  • Formula: Density = mass ÷ volume (\(\mathrm{D = m/V}\))

• What we need to find: Volume in cubic centimeters

2. INFER the solution approach

• Since we have density and mass, but need volume, we must rearrange the given formula
• The formula \(\mathrm{D = m/V}\) needs to become \(\mathrm{V = m/D}\)
• This means: Volume = mass ÷ density

3. SIMPLIFY by substituting and calculating

• Substitute the known values: \(\mathrm{V = 468 \div 7.2}\)
• Calculate the division (use calculator): \(\mathrm{V = 65}\)

Answer: 65

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which quantity is which, especially confusing mass and volume or setting up the wrong relationship.

They might write something like \(\mathrm{V = D \times m}\) or \(\mathrm{m = V/D}\), leading to incorrect calculations like \(\mathrm{V = 7.2 \times 468 = 3,369.6}\). This causes them to get stuck and guess since this large number doesn't seem reasonable for volume.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{V = 468 \div 7.2}\) but make arithmetic errors in the division.

Common mistakes include getting \(\mathrm{6.5}\) (off by a factor of 10) or other calculation errors. This leads to confusion when their answer doesn't match any reasonable expectation for the problem.

The Bottom Line:

This problem tests whether students can work backwards from a given relationship. The key insight is recognizing that when you know two of the three quantities in \(\mathrm{D = m/V}\), you can always solve for the third by rearranging the formula.