Which expression is equivalent to \(3(2\mathrm{y}^2 - 4\mathrm{y}) + 8\mathrm{y}^2 + 5\mathrm{y}\)?

Part 1: Brief Solution

Concepts Tested: Combining like terms in algebraic expressions
Primary Process Skills: Simplify, Translate

Essential Steps:

• Identify like terms: \(\mathrm{9x^2}\) and \(\mathrm{7x^2}\) are like terms (same variable with same exponent)
• Combine the coefficients of like terms: \(\mathrm{9x^2 + 7x^2 = (9 + 7)x^2 = 16x^2}\)
• Keep the remaining term: \(\mathrm{-9x}\) stays as is
• Final simplified expression: \(\mathrm{16x^2 - 9x}\)

Answer: The mathematically correct answer is \(\mathrm{16x^2 - 9x}\), which most closely corresponds to choice D, though there appears to be a sign discrepancy.

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Incorrect Term Multiplication - Phase: Executing Approach → Choice A (\(\mathrm{63x^4 + 9x}\))
   • Process skill failure: Simplify
   • Students incorrectly multiply coefficients and add exponents instead of combining like terms (\(\mathrm{9 \times 7 = 63}\), \(\mathrm{x^2 \times x^2 = x^4}\)).
2. Adding All Coefficients Together - Phase: Executing Approach → Choice C (\(\mathrm{25x^5}\))
   • Process skill failure: Simplify
   • Students add all coefficients (\(\mathrm{9 + 7 + 9 = 25}\)) and incorrectly combine all variables into one term with wrong exponent.
3. Mishandling the x Term - Phase: Executing Approach → Choice B (\(\mathrm{9x^2 + 16x}\))
   • Process skill failure: Simplify
   • Students correctly add \(\mathrm{7x^2}\) to the x term (\(\mathrm{7 + 9 = 16x}\)) but fail to combine the \(\mathrm{x^2}\) terms properly.

Part 3: Detailed Solution

Understanding the Problem:
We need to simplify the algebraic expression \(\mathrm{9x^2 + 7x^2 - 9x}\) by combining like terms. Think of this like organizing similar items - if you have 9 apples and 7 apples, you have 16 apples total.

Process Skill: TRANSLATE - We need to identify what "like terms" means: terms that have exactly the same variable parts. Here, \(\mathrm{9x^2}\) and \(\mathrm{7x^2}\) are like terms because they both have \(\mathrm{x^2}\).

Step-by-Step Solution:

Step 1: Identify Like Terms

Starting expression: \(\mathrm{9x^2 + 7x^2 - 9x}\)

Like terms have identical variable parts:

• \(\mathrm{9x^2}\) and \(\mathrm{7x^2}\) are like terms (both have \(\mathrm{x^2}\))
• \(\mathrm{-9x}\) stands alone (has \(\mathrm{x^1}\), which is different from \(\mathrm{x^2}\))

Process Skill: SIMPLIFY - Now we combine the like terms by adding their coefficients.

Step 2: Combine Like Terms

For the \(\mathrm{x^2}\) terms: \(\mathrm{9x^2 + 7x^2}\)
\(\mathrm{= (9 + 7)x^2}\)
\(\mathrm{= 16x^2}\)

The x term remains unchanged: \(\mathrm{-9x}\)

Step 3: Write the Final Expression

Combining our results: \(\mathrm{16x^2 + (-9x)}\)
\(\mathrm{= 16x^2 - 9x}\)

Why This Works:
Combining like terms follows the distributive property in reverse. When we write \(\mathrm{9x^2 + 7x^2}\), we can factor out \(\mathrm{x^2}\): \(\mathrm{(9 + 7)x^2 = 16x^2}\). This is like saying "9 groups of \(\mathrm{x^2}\) plus 7 groups of \(\mathrm{x^2}\) equals 16 groups of \(\mathrm{x^2}\)."

Verification:
Let's check with \(\mathrm{x = 1}\):
Original: \(\mathrm{9(1)^2 + 7(1)^2 - 9(1) = 9 + 7 - 9 = 7}\)
Our answer: \(\mathrm{16(1)^2 - 9(1) = 16 - 9 = 7}\) ✓

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

Conceptual Confusion About Operations: Some students think they need to multiply terms instead of add them, leading to approaches like \(\mathrm{9x^2 \times 7x^2 = 63x^4}\). This represents a failure in the TRANSLATE process skill - misunderstanding what "equivalent expression" means in algebra.

Misunderstanding Like Terms: Students might not recognize that \(\mathrm{9x^2}\) and \(\mathrm{7x^2}\) are like terms, or conversely, might think that \(\mathrm{x^2}\) and x terms can be combined. This is a SIMPLIFY process skill failure where students don't properly categorize terms.

Errors while executing the approach:

Coefficient and Exponent Confusion: Students might multiply coefficients and add exponents (\(\mathrm{9 \times 7 = 63}\), \(\mathrm{x^2 \times x^2 = x^4}\)) instead of adding coefficients for like terms. This is a computational error in applying the rules for combining like terms.

Wrong Arithmetic: Simple addition errors like \(\mathrm{9 + 7 = 15}\) instead of 16, leading to \(\mathrm{15x^2 - 9x}\). This is a basic computational error.

Sign Errors: Students might lose track of the negative sign, writing \(\mathrm{16x^2 + 9x}\) instead of \(\mathrm{16x^2 - 9x}\), or might incorrectly combine the 9 from \(\mathrm{9x^2}\) with the \(\mathrm{-9x}\).

Errors while selecting the answer:

Partial Recognition: Students might correctly get \(\mathrm{16x^2}\) but then select choice B (\(\mathrm{9x^2 + 16x}\)) because it contains familiar numbers from their work, failing to recognize they haven't fully simplified.

Sign Confusion in Final Step: Students might have the correct structure (\(\mathrm{16x^2}\) and \(\mathrm{9x}\)) but choose the wrong sign, potentially selecting choice D if there was a transcription error in their work or the problem.

Pattern Matching Errors: Students might select choice A because they see familiar numbers (9 and coefficients that sum to larger numbers) without checking if their algebraic manipulation was correct.