Operation Overhaul

Sum: \(1\frac{1}{8}\) or \(\frac{9}{8}\)

10 • SUBTRACTION \(3\frac{1}{4} - 1\frac{1}{2}\)

\(\text{LCD} = 4\) → \(3\frac{1}{4} - 1\frac{2}{4}\)
Regroup 1 whole: \(2\frac{5}{4} - 1\frac{2}{4}\)
\(2 - 1 = 1\); \(\frac{5-2}{4} = \frac{3}{4}\) → \(\mathbf{1\frac{3}{4}}\)

Difference: \(1\frac{3}{4}\) or \(\frac{7}{4}\)

11 • MULTIPLICATION \(1\frac{1}{2} \times \frac{4}{5}\)

Convert to improper: \(1\frac{1}{2} = \frac{3}{2}\)
\(\frac{3}{2} \times \frac{4}{5} = \frac{12}{10}\)
Simplify by dividing by 2 → \(\frac{6}{5} = \mathbf{1\frac{1}{5}}\)

Product: \(1\frac{1}{5}\) or \(\frac{6}{5}\)

12 • DIVISION \(\frac{3}{4} \div \frac{1}{2}\)

Multiply by reciprocal: \(\frac{3}{4} \times \frac{2}{1}\)
Cross-cancel 2 & 4: \(\frac{3 \times 1}{2 \times 1} = \frac{3}{2} = \mathbf{1\frac{1}{2}}\)

Quotient: \(1\frac{1}{2}\) or \(\frac{3}{2}\)

Stage 4: Real-World Applications (Solutions)

Problems 13–14 • 2 Points Each

13 • SOLUTION Multi-Step Operations: ×, +, & -

Step 1: Soccer Balls \(8 \times \$14.50 = \mathbf{\$116.00}\)

Step 2: Total Cost \(\$116.00 + \$18.00 = \mathbf{\$134.00}\)

Step 3: Change Paid: \(2 \times \$100 = \$200\)
\(\$200.00 - \$134.00 = \mathbf{\$66.00}\)

Change Returned: $66.00

14 • SOLUTION Multi-Step Operations: × and -

Step 1: Total Sliced Wedges \(4 \times 1\frac{1}{4}\text{ lbs} = 4 \times \frac{5}{4} = \mathbf{5\text{ pounds}}\)

Step 2: Remaining Block \(6\frac{1}{2} - 5 = \mathbf{1\frac{1}{2}\text{ pounds}}\) (or 1.5 lbs)

Remaining Weight: \(1\frac{1}{2}\) pounds (or 1.5 lbs)

Operation Overhaul • Complete Answer Key Page 2 of 4 (Turn over for Stages 5 & 6 Solutions) →

Teacher Key Stages 5 & 6 Worked Solutions

FLUENCY & PEMDAS • KEY

Order of operations progression and sprint solutions.

Enforce PEMDAS left-to-right rules

Stage 5: Order of Operations (Solutions)

Problems 15–18 • 4 Points

15 • PEMDAS \(24 - 3 \times (2^2 + 1) + 7\)

1. Inside ( ): \(2^2 + 1 = 4 + 1 = \mathbf{5}\)
2. Multiply: \(3 \times 5 = \mathbf{15}\)
3. Left to right: \(24 - 15 = 9 \implies 9 + 7 = \mathbf{16}\)

Answer: 16

16 • PEMDAS \(4.5 + 2 \times (6.8 - 2.8) - 10 \div 2\)

1. Inside ( ): \(6.8 - 2.8 = \mathbf{4.0}\)
2. Multiply/Divide: \(2 \times 4.0 = \mathbf{8.0}\); \(10 \div 2 = \mathbf{5}\)
3. Add/Sub: \(4.5 + 8.0 = 12.5 \implies 12.5 - 5 = \mathbf{7.5}\)

Answer: 7.5

17 • PEMDAS \(\left(\frac{3}{4} - \frac{1}{4}\right) \times (6 + 2)\)

1. Left ( ): \(\frac{3}{4} - \frac{1}{4} = \mathbf{\frac{2}{4} = \frac{1}{2}}\)
2. Right ( ): \(6 + 2 = \mathbf{8}\)
3. Multiply: \(\frac{1}{2} \times 8 = \mathbf{4}\)

Answer: 4

18 • PEMDAS \(30 \div (2 + 3) + 4.5\)

1. Parentheses: \(2 + 3 = \mathbf{5}\)
2. Divide: \(30 \div 5 = \mathbf{6}\)
3. Add: \(6 + 4.5 = \mathbf{10.5}\)

Answer: 10.5

Stage 6: Mixed Fluency Sprint (Solutions)

Problems 19–22 • 4 Points

19 • 2-DIGIT DIVISOR \(435 \div 12\)

\(43 \div 12 = 3\) R7 \((43 - 36 = 7)\)
\(75 \div 12 = 6\) R3 \((75 - 72 = 3)\)
Quotient: 36, Remainder: 3 (or \(36\frac{1}{4} = 36.25\))

Answer: 36 R3 (or \(36\frac{1}{4}\))

20 • DECIMALS \(1.5 \times 0.4\)

Multiply whole numbers: \(15 \times 4 = \mathbf{60}\)
Decimal places: \(1 + 1 = \mathbf{2}\text{ places}\)
Move point 2 places left: \(\mathbf{0.60} = \mathbf{0.6}\)

Answer: 0.6 (or 0.60)

21 • MIXED DIVISION \(2\frac{1}{2} \div \frac{1}{2}\)

Improper: \(2\frac{1}{2} = \frac{5}{2}\)
Multiply reciprocal: \(\frac{5}{2} \times \frac{2}{1}\)
Cancel 2s: \(\frac{5 \times 1}{1 \times 1} = \mathbf{5}\)

Answer: 5

22 • ZERO REGROUP \(6,000 - 2,438\)

Borrow cascade: 5-9-9-10

  6,000
- 2,438

  3,562

Answer: 3,562

Operation Overhaul • Complete Answer Key Page 3 of 4 (Turn over for Stages 7 & 8 Solutions) →

Teacher Key Stages 7 & 8 Solutions

ERROR ANALYSIS & TASKS • KEY

Misconception explanations, rubrics, and extended performance task proofs.

Award full credit for sound explanations

Stage 7: Error Detective (Solutions)

Problems 23–24 • 2 Points Each

23 • SOLUTION \(45.2 - 18.35\)

Error Explanation:

Student failed to annex a placeholder zero to write \(45.20\). They subtracted bottom digit from top (\(5 - 0 = 5\)) instead of regrouping from tenths.

Correct: \(45.20 - 18.35 = \mathbf{26.85}\)

Difference: 26.85

24 • SOLUTION \(\frac{2}{3} \div \frac{1}{4}\)

Error Explanation:

Student multiplied across instead of multiplying by the reciprocal of the divisor (\(\frac{4}{1}\)).

Correct: \(\frac{2}{3} \times \frac{4}{1} = \frac{8}{3} = \mathbf{2\frac{2}{3}}\)

Quotient: \(2\frac{2}{3}\) or \(\frac{8}{3}\)

Stage 8: Real-World Performance Tasks (Solutions)

Problems 25–26 • 2 Points Each

25 • SOLUTION Multi-Step Fraction & Finance Application

Step 1: Sugar Used \(12 \times 1\frac{1}{2} = 12 \times \frac{3}{2} = \mathbf{18\text{ cups}}\)
Remaining: \(25 - 18 = \mathbf{7\text{ cups}}\)

Step 2: Total Revenue \(12 \times \$8.00 = \mathbf{\$96.00}\)

Step 3: Net Profit \(\$96.00 - \$35.00 = \mathbf{\$61.00}\)

Sugar Remaining: 7 cups

Net Profit: $61.00

26 • SOLUTION Multi-Step Decimal & Budget Calculations

Step 1: Tickets \(24 \times \$12.50 = \mathbf{\$300.00}\)

Step 2: Total Cost \(\$140.00 + \$300.00 = \mathbf{\$440.00}\)

Step 3: Balance Remaining \(\$500.00 - \$440.00 = \mathbf{\$60.00}\)

Trip Cost: $440.00

Balance: $60.00

Operation Overhaul • Complete Answer Key Total Possible Points: 26 Points (or 52 Points with process credit)

Enrichment & Extension

• Create an Error Bug: Have students write their own fictional student mistake for Problem 20 or 21 and have a partner diagnose it.
• Mental Math Challenge: Ask students to solve Stage 6 Fluency Sprint problems mentally using estimation strategies first.
• Variable Budgeting: Challenge students in Problem 26 to determine how many more students could attend before funds run out.

Scoring Matrix (26 Problems • 52 Total Points) 1 pt clear mathematical process + 1 pt accurate numerical value & units.

46–52: Mastery

38–45: Proficient

≤ 37: Intervention

Operation Overhaul • Comprehensive Teacher Reference Standards: CCSS.MATH.CONTENT.5.NBT / 5.NF / 6.NS / 6.EE