#2. \( \frac{x+5}{4} = 6 \) \( x = 19 \)
• OO: Numerator bracketed; multiply denominator first
• EQ: Multiply by 4 (\( x + 5 = 24 \)), subtract 5
• CP: \( 6 \times 4 = 24 \) and \( 24 - 5 = 19 \)
#3. \( 5(x - 4) = 30 \) \( x = 10 \)
• OO: Distribute \( 5x - 20 = 30 \) (or divide by 5)
• EQ: Add 20 to 30 (\( 5x = 50 \)), divide by 5
• CP: \( 30 + 20 = 50 \) and \( 50 \div 5 = 10 \)
#4. \( 6x - (3 \times 4) = 24 \) \( x = 6 \)
• OO: Multiply parentheses first: \( 3 \times 4 = 12 \)
• EQ: \( 6x - 12 = 24 \implies 6x = 36 \implies x = 6 \)
• CP: \( 24 + 12 = 36 \) and \( 36 \div 6 = 6 \)
#5. \( \frac{x}{3} + 4^2 = 21 \) \( x = 15 \)
• OO: Evaluate power first: \( 4^2 = 16 \)
• EQ: Subtract 16 (\( \frac{x}{3} = 5 \)), multiply by 3
• CP: \( 21 - 16 = 5 \) and \( 5 \times 3 = 15 \)
#6. \( 3(x + 2) - 4 = 17 \) \( x = 5 \)
• OO: Expand: \( 3x + 6 - 4 \implies 3x + 2 = 17 \)
• EQ: Subtract 2 (\( 3x = 15 \)), divide by 3
• CP: \( 6 - 4 = 2 \), \( 17 - 2 = 15 \), \( 15 \div 3 = 5 \)
*Each item allows diagnosis of isolated skill breakdown: check corresponding checkbox on probe sheet. Guide Page 1 of 2
Scoring Guide
Mixed-Skill Error Analysis
Form C Step-by-Step Key (6 Items • 18 Pts) Parallel Item Equivalents
#1. \( 2x + 4^2 = 30 \)\( x = 7 \)
OO: \( 4^2=16 \)
EQ: \( 2x = 14 \implies x = 7 \)
CP: \( 30-16=14 \)
#2. \( \frac{x+8}{2} = 9 \)\( x = 10 \)
OO: Bar groups numerator
EQ: \( x+8 = 18 \implies x = 10 \)
CP: \( 9 \times 2 = 18 \)
#3. \( 3(x - 5) = 24 \)\( x = 13 \)
OO: Distribute: \( 3x - 15 \)
EQ: \( 3x = 39 \implies x = 13 \)
CP: \( 24+15=39 \)
#4. \( 4x - (2 \times 5) = 22 \)\( x = 8 \)
OO: Multiply: \( 2 \times 5 = 10 \)
EQ: \( 4x - 10 = 22 \implies 4x = 32 \)
CP: \( 32 \div 4 = 8 \)
#5. \( \frac{x}{4} + 3^2 = 16 \)\( x = 28 \)
OO: \( 3^2=9 \implies \frac{x}{4} + 9 = 16 \)
EQ: \( \frac{x}{4} = 7 \implies x = 28 \)
CP: \( 7 \times 4 = 28 \)
#6. \( 4(x + 1) - 6 = 18 \)\( x = 5 \)
OO: \( 4x+4-6 \implies 4x-2=18 \)
EQ: \( 4x = 20 \implies x = 5 \)
CP: \( 20 \div 4 = 5 \)
Mixed-Skill Error Diagnostic Matrix & Misconception Identifiers
Error Category 1: Order-of-Operations (OO) Failures Code: ERR-OO
• Exponent Multiplication Error: Multiplies base by exponent instead of evaluating power (e.g., calculates \( 2^3 = 6 \) instead of \( 8 \), or \( 3^2 = 6 \) instead of \( 9 \)).
• Grouping / Fraction Bar Trap: Operates on numerator terms before eliminating denominator (e.g., subtracting 7 from \( \frac{x+7}{3} \) prior to multiplying by 3).
• Distributive Incompletion: Multiplies coefficient by the variable only, omitting the second term (e.g., \( 4(x - 3) \implies 4x - 3 \)).
• Premature Term Combining: Combines across unlike terms or ignores operational grouping (e.g., in \( 5x - (4 \times 3) \), subtracting 4 first).
Error Category 2: Equation Solving / Inverse Operation (EQ) Failures Code: ERR-EQ
• Inverse Operation Reversal: Uses addition instead of subtraction or multiplication instead of division (e.g., adding 8 to 26 in \( 3x + 8 = 26 \)).
• Unbalanced Equality: Performs an operation on one side of the equation without applying the identical operation to the opposite side.
• Isolation Sequence Error: Attempts to divide by coefficient before clearing added or subtracted constant terms.
• Coefficient Inversion: When solving \( \frac{x}{2} = 4 \), divides 4 by 2 instead of multiplying by 2 (yielding \( x = 2 \) instead of \( 8 \)).
Error Category 3: Computation & Arithmetic (CP) Slips Code: ERR-CP
• Fact Retrieval Slip: Conceptual equation and GEMDAS path are entirely correct, but basic multiplication or division facts are miscalculated (e.g., \( 18 \div 3 = 5 \)).
• Sign / Integer Error: Loses or drops negative signs during subtraction or multi-digit arithmetic operations.
Instructional Decision Rules & Tiered Action Triggers IEP / MTSS Decision Thresholds
• If OO Score < 4/6: Pause two-step equation practice. Reteach GEMDAS hierarchy with color-coded grouping models and exponent flashcard drills.
• If EQ Score < 4/6: Use two-pan balance scale manipulatives and vertical "river line" equation templates to isolate variable terms before coefficients.
• If CP Score < 4/6: Provide a 100-chart or multiplication grid support to isolate whether algebra reasoning is intact while remediating fact fluency.
