The function g gives the product of a number, y, and a number that is 23 less than y. Which...

1. TRANSLATE the problem information

• Given information:
  • Function g gives the product of two numbers
  • First number: y
  • Second number: "a number that is 23 less than y"

• Key translation: "23 less than y" means \(\mathrm{y - 23}\)

2. TRANSLATE the product relationship

• The function g gives the product of these two numbers:
  \(\mathrm{g(y) = y \times (y - 23)}\)

3. SIMPLIFY the algebraic expression

• Use the distributive property to expand:
  \(\mathrm{g(y) = y(y - 23)}\)
  \(\mathrm{g(y) = y^2 - 23y}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "23 less than y" as \(\mathrm{23 - y}\) instead of \(\mathrm{y - 23}\).

This backwards translation comes from reading the phrase left to right rather than understanding the mathematical meaning. When we say "23 less than y," we mean we're taking 23 away from y, which gives us \(\mathrm{y - 23}\). Students who make this error set up \(\mathrm{g(y) = y \times (23 - y) = 23y - y^2}\), which doesn't match any of the given choices exactly, leading to confusion and guessing.

Second Most Common Error:

Conceptual confusion about language: Students might misunderstand "product" and think the problem is asking for addition rather than multiplication.

This leads them to set up \(\mathrm{g(y) = y + (y - 23) = 2y - 23}\), which again doesn't match the answer choices. This may lead them to select Choice B (\(\mathrm{y^2 - 23}\)) if they recognize they need a quadratic but guess at the structure.

The Bottom Line:

This problem tests whether students can accurately translate English phrases into mathematical expressions. The phrase "23 less than y" is a classic stumbling point because the natural left-to-right reading doesn't match the mathematical order of operations.