A total of 364 paper straws of equal length were used to construct two types of polygons: triangles and rectangles....

1. TRANSLATE the ordered pair notation

• Given: \(\mathrm{(x, y) = (24, 73)}\)
• This means: \(\mathrm{x = 24}\) and \(\mathrm{y = 73}\)

2. TRANSLATE what the variables represent in context

• From the problem setup:
  • \(\mathrm{x}\) = number of triangles constructed
  • \(\mathrm{y}\) = number of rectangles constructed
• Therefore: \(\mathrm{x = 24}\) means 24 triangles, \(\mathrm{y = 73}\) means 73 rectangles

3. INFER the complete interpretation

• If \(\mathrm{(x, y) = (24, 73)}\), then we constructed:
  • 24 triangles AND
  • 73 rectangles
• This matches Choice A exactly

4. Verify the solution (optional)

• Check: \(\mathrm{3(24) + 4(73) = 72 + 292 = 364}\) ✓

Answer: A. If 24 triangles were constructed, then 73 rectangles were constructed.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students correctly identify that \(\mathrm{(x, y) = (24, 73)}\) means \(\mathrm{x = 24}\) and \(\mathrm{y = 73}\), but then mix up which variable represents which quantity. They might think x represents rectangles and y represents triangles, reversing the relationship.

This leads them to incorrectly conclude that 24 rectangles and 73 triangles were constructed, causing them to select Choice C (If 73 triangles were constructed, then 24 rectangles were constructed).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what \(\mathrm{y = 73}\) represents in the context. Instead of recognizing that y is the number of rectangles, they think y represents the number of paper straws used when 24 triangles are constructed.

This confusion about variable meaning leads them to select Choice B (If 24 triangles were constructed, then 73 paper straws were used).

The Bottom Line:

This problem tests whether students can accurately connect mathematical notation to real-world context. The key is carefully tracking what each variable represents throughout the interpretation process.