The area A, in square meters, of a triangular garden plot can be represented by the expression \(\frac{1}{2}\mathrm{b}(\mathrm{b} + 8)\),...

1. TRANSLATE the problem information

• Given information:
  • Area formula: \(\mathrm{A = \frac{1}{2}b(b + 8)}\)
  • Base length: \(\mathrm{b}\) meters
  • Need to find: height in meters

2. INFER the approach needed

• Since we have an area expression and need height, we should use the triangle area formula
• Triangle area formula: \(\mathrm{A = \frac{1}{2} \times base \times height}\)
• We can set up an equation using both area expressions

3. SIMPLIFY by setting up and solving the equation

• Set the given area equal to the formula:
  \(\mathrm{\frac{1}{2}b(b + 8) = \frac{1}{2} \times b \times height}\)

• Divide both sides by \(\mathrm{\frac{1}{2}b}\):
  \(\mathrm{(b + 8) = height}\)

Answer: D. \(\mathrm{(b + 8)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to use the triangle area formula to find height. Instead, they might look at the given expression \(\mathrm{\frac{1}{2}b(b + 8)}\) and try to identify parts directly without setting up the proper equation.

They might think "since \(\mathrm{\frac{1}{2}}\) is there, and \(\mathrm{b}\) is the base, maybe \(\mathrm{\frac{1}{2}b}\) is the height?" This leads them to select Choice B (\(\mathrm{\frac{1}{2}b}\)).

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students misinterpret what each part of the expression represents. They might think the entire expression represents just one dimension, or confuse which variable represents which measurement.

This confusion about what \(\mathrm{b}\) represents versus what they're solving for can cause them to select Choice C (\(\mathrm{b}\)), thinking the base and height are equal.

The Bottom Line:

This problem requires students to connect a given algebraic area expression with the standard geometric area formula - a skill that bridges algebra and geometry understanding.