[BUG: Whole Number Multiplication Blowout] STATION 2

The Issue: Multiplying a whole number by both numerator and denominator (e.g., \( 3 \cdot \frac{1}{4} = \frac{3}{12} \)).

Socratic Reset: "If you have 3 separate packages that each contain 1/4 of a gallon, do you have 3/12 of a gallon? That simplifies back to 1/4. Did you gain nothing? We must write 3 as \( \frac{3}{1} \) first."

[BUG: Reciprocal Omission on Division] STATION 2 & 3

The Issue: Dividing across or failing to invert the divisor before multiplying.

Socratic Reset: "Division is multiplication by a reciprocal. Locate the second component. Switch the division gear to multiplication, flip the fraction upside down, and run the calculation."

Technical Inquiry Prompts

To Guide Setup

"What is the mathematical structure of this components system? Are we adding, subtracting, multiplying, or dividing? Check the master system schematics on Page 1. Which Mechanic applies here?"

To Push Verification

"How can you prove this output value matches the physical tolerance of the blueprint? If you draw a model of this measurement on a ruler, does the length look mathematically reasonable?"

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MASTER DIAGNOSTIC KEY

TEACHER PROTOCOL

COMPLETE ANSWERS

STATION 1: ADD & SUBTRACT

• Card 1.1: \( 5 + \frac{3}{8} = \frac{40}{8} + \frac{3}{8} = \mathbf{\frac{43}{8}} \) or \( \mathbf{5 \frac{3}{8}} \)
• Card 1.2: \( \frac{7}{10} + \frac{4}{5} = \frac{7}{10} + \frac{8}{10} = \mathbf{\frac{15}{10}} = \mathbf{\frac{3}{2}} \) or \( \mathbf{1 \frac{1}{2}} \)
• Card 1.3: \( 4 - \frac{5}{6} = 3\frac{6}{6} - \frac{5}{6} = \mathbf{3\frac{1}{6}} \) or \( \mathbf{\frac{19}{6}} \)
• Card 1.4: \( 3 \frac{1}{4} - \frac{7}{8} = \frac{13}{4} - \frac{7}{8} = \frac{26}{8} - \frac{7}{8} = \mathbf{\frac{19}{8}} \) or \( \mathbf{2 \frac{3}{8}} \)

STATION 2: MULT & DIV

• Card 2.1: \( 8 \cdot \frac{3}{10} = \frac{24}{10} = \mathbf{\frac{12}{5}} \) or \( \mathbf{2 \frac{2}{5}} \)
• Card 2.2: \( \frac{2}{3} \cdot \frac{9}{4} = \frac{18}{12} = \mathbf{\frac{3}{2}} \) or \( \mathbf{1 \frac{1}{2}} \)
• Card 2.3: \( 6 \div \frac{2}{3} = \frac{6}{1} \cdot \frac{3}{2} = \frac{18}{2} = \mathbf{9} \) units
• Card 2.4: \( \frac{3}{4} \div 6 = \frac{3}{4} \cdot \frac{1}{6} = \frac{3}{24} = \mathbf{\frac{1}{8}} \) feet

STATION 3: PRACTICAL APPLICATIONS (FIELD SPECS)

• Card 3.1: \( \frac{3}{4} + \frac{5}{8} + 1 = \frac{6}{8} + \frac{5}{8} + \frac{8}{8} = \mathbf{\frac{19}{8}} \) or \( \mathbf{2 \frac{3}{8}} \) inches
• Card 3.2: \( 15 - 6 \frac{2}{3} = 14 \frac{3}{3} - 6 \frac{2}{3} = \mathbf{8 \frac{1}{3}} \) meters
• Card 3.3: \( \frac{7}{8} \cdot 3 = \frac{21}{8} = \mathbf{2 \frac{5}{8}} \) inches
• Card 3.4: \( 4 \div \frac{4}{5} = \frac{4}{1} \cdot \frac{5}{4} = \frac{20}{4} = \mathbf{5} \) complete kits

STATION 4: SYSTEM DEFECT DIAGNOSIS KEY

Card 4.1 Defect: Step 1.
Error: Wrote whole number \( 6 \) as \( \frac{6}{3} \) (which equals 2).
Correct: \( 6 - \frac{2}{3} = 5\frac{3}{3} - \frac{2}{3} = \mathbf{5\frac{1}{3}} \) or \( \mathbf{\frac{16}{3}} \)

Card 4.2 Defect: Step 1 & 2.
Error: Divided by \( \frac{1}{2} \) without using the reciprocal.
Correct: \( 4 \div \frac{1}{2} = 4 \cdot \frac{2}{1} = \mathbf{8} \)

Card 4.3 Defect: Step 2.
Error: Added denominators together directly (common error).
Correct: \( 2 \frac{1}{3} + \frac{2}{3} = 2 \frac{3}{3} = \mathbf{3} \)

Card 4.4 Defect: Step 1.
Error: Scaled both numerator and denominator by 5, which scales nothing (it's multiplying by 1).
Correct: \( \frac{3}{4} \cdot \frac{5}{1} = \mathbf{\frac{15}{4}} \) or \( \mathbf{3 \frac{3}{4}} \)

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