The side length of a square is equal to the average of the numbers a and a + b, where...

1. TRANSLATE the problem information

• Given information:
  • Side length of square = average of \(\mathrm{a}\) and \(\mathrm{(a + b)}\)
  • Need to find an expression for the area

• What this tells us: We need to first find the side length, then square it for area

2. TRANSLATE 'average' into mathematical notation

• Average of two numbers \(\mathrm{a}\) and \(\mathrm{(a + b)}\):
  • Average = \(\frac{\mathrm{a + (a + b)}}{2}\)

3. SIMPLIFY the average expression

• \(\frac{\mathrm{a + (a + b)}}{2}\) = \(\frac{\mathrm{a + a + b}}{2}\) = \(\frac{\mathrm{2a + b}}{2}\) = \(\mathrm{a + \frac{b}{2}}\)

• So the side length is \(\mathrm{(a + \frac{b}{2})}\)

4. INFER the next step for finding area

• Since area of square = \(\mathrm{(side\ length)}^2\)
• We need: Area = \(\mathrm{(a + \frac{b}{2})}^2\)

5. SIMPLIFY by expanding the squared expression

• Using \(\mathrm{(x + y)}^2 = \mathrm{x}^2 + 2\mathrm{xy} + \mathrm{y}^2\):
• \(\mathrm{(a + \frac{b}{2})}^2\) = \(\mathrm{a}^2 + 2\mathrm{a}(\frac{\mathrm{b}}{2}) + (\frac{\mathrm{b}}{2})^2\)
• = \(\mathrm{a}^2 + \mathrm{ab} + \frac{\mathrm{b}^2}{4}\)

Answer: B. \(\mathrm{a}^2 + \mathrm{ab} + \frac{\mathrm{b}^2}{4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when expanding \(\mathrm{(a + \frac{b}{2})}^2\)

Many students correctly find the side length as \(\mathrm{(a + \frac{b}{2})}\), but then struggle with the expansion. They might:

• Forget the middle term: getting \(\mathrm{a}^2 + \frac{\mathrm{b}^2}{4}}\) instead of \(\mathrm{a}^2 + \mathrm{ab} + \frac{\mathrm{b}^2}{4}}\)
• Make fraction errors with \(\mathrm{(\frac{b}{2})}^2\): writing \(\frac{\mathrm{b}^2}{2}\) instead of \(\frac{\mathrm{b}^2}{4}\)
• Incorrectly compute the middle term: getting \(\mathrm{a}\cdot\mathrm{b}\) instead of \(\mathrm{ab}\)

This may lead them to select Choice A (\(\mathrm{a}^2 + \frac{\mathrm{b}^2}{4}}\)) if they miss the middle term entirely.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what 'average' means in this context

Some students might think the problem is asking for something more complex, or they might incorrectly set up the average calculation. This leads to confusion about what expression to square, causing them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem tests whether students can systematically work through a multi-step algebraic process: translating words to math, simplifying expressions, and carefully expanding squared binomials with fractions.