Notice the cross-cancellation of 5s in the multiplication step.
8. \(\left(-\frac{1}{2}\right)^3 + \frac{5}{8} \div \left(-\frac{5}{4}\right)\) Ans: \(-\frac{5}{8}\) (or \(-0.625\))
• Exponent: \(\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}\)
• Divide: \(\frac{5}{8} \times \left(-\frac{4}{5}\right) = -\frac{4}{8} = -\frac{1}{2}\)
• Add: \(-\frac{1}{8} + \left(-\frac{4}{8}\right) = -\frac{5}{8}\)
Watch out: Odd power of negative base stays negative: \((-1/2)^3 = -1/8\).
Turn to Page 2 for Solutions to Nested Grouping, Error Analysis & Puzzles Answer Key • Page 1 of 3
Teacher Resource • Answer Key
Solutions for Level 3 nested grouping, Jordan's error analysis, and parentheses placement.
Scoring: 9-12 (2 pts each) • 13 (4 pts) • 14 (2 pts each)
C
9. \(-3 \cdot [4.5 - (-\frac{3}{2})] + (-2)^3\) Ans: \(-26\)
• Inner brackets: \(4.5 - (-1.5) = 4.5 + 1.5 = 6.0\)
• Multiply: \(-3 \cdot 6.0 = -18\)
• Exponent: \((-2)^3 = -8\)
• Combine: \(-18 + (-8) = -26\)
10. \(\frac{-15.6 + 3.6}{4 - (-2)} - (-\frac{2}{3}) \times 6\) Ans: \(2\)
• Fraction bar numerator: \(-15.6 + 3.6 = -12.0\)
• Denominator: \(4 - (-2) = 6 \implies \frac{-12}{6} = -2\)
• Multiply: \((-\frac{2}{3}) \times 6 = -4\)
• Subtract: \(-2 - (-4) = -2 + 4 = 2\)
11. \(|-8.4 + 2.4| \div (-0.5)^2 + 3(-\frac{4}{5})\) Ans: \(21.6\) or \(21\frac{3}{5}\)
• Inside absolute value: \(|-6.0| = 6.0\)
• Exponent: \((-0.5)^2 = 0.25\)
• Divide: \(6.0 \div 0.25 = 24\)
• Multiply: \(3 \times (-0.8) = -2.4 \implies 24 - 2.4 = 21.6\)
12. \([(-\frac{1}{3}) \times 9 + 4.8] \div (-0.6)\) Ans: \(-3\)
• Inside brackets: \((-\frac{1}{3}) \times 9 = -3\)
• Add: \(-3 + 4.8 = 1.8\)
• Divide: \(1.8 \div (-0.6) = -3\)
D
13a. Sample Student Explanation (2 pts):
"Jordan incorrectly subtracted \(12 - 4\) first. According to the order of operations, exponents and multiplication must be performed before subtraction. Jordan should have calculated \((-\frac{3}{2})^2 = \frac{9}{4}\) first, then multiplied by \(4\), and finally subtracted that from \(12\)."
13b. Correct Evaluation (2 pts): Ans: \(3\)
• Exponent: \((-\frac{3}{2})^2 = \frac{9}{4}\)
• Multiply: \(4 \times \frac{9}{4} = 9\)
• Subtract: \(12 - 9 = 3\)
E
14a. Target: \(-18\) Verified
(-5.5 + 2.5) \(\times 4 - 6 = -18\)
Check: \((-3.0) \times 4 - 6 = -12 - 6 = -18\).
14b. Target: \(-20\) Verified
\(15 \div\) (-2.5 + 5.5) \(\times (-4) = -20\)
Check: \(15 \div 3.0 \times (-4) = 5 \times (-4) = -20\).
Turn to Page 3 for Real-World Word Problem Solutions & Scoring Rubrics Answer Key • Page 2 of 3
Teacher Resource • Answer Key
Real-world modeling solutions, expression formulation, and full arithmetic verifications.
Scoring Rubric (4 pts each): 1 pt expression • 2 pts work • 1 pt answer & unit
15. Ocean Trench Exploration (Marine Science) Ans: \(-207.3\text{ meters}\)
Valid Expression: \(-180.5 - 6 \times 12\frac{1}{2} + 48.2\) (or \(-180.5 + 6(-12.5) + 48.2\))
Step 1: Descent product: \(6 \times 12.5 = 75\text{ m}\)
Step 2: Elevation after descent: \(-180.5 - 75 = -255.5\text{ m}\)
Step 3: Ascent addition: \(-255.5 + 48.2 = -207.3\text{ m}\)
Accept also: "207.3 meters below sea level".
16. Student Checking Account (Personal Finance) Ans: \(\$89.15\)
Valid Expression: \(52.40 - 3(14.75) + 85.50 - 4.50\)
Step 1: Total subscriptions: \(3 \times 14.75 = 44.25\)
Step 2: Balance after subscriptions: \(52.40 - 44.25 = 8.15\)
Step 3: Deposit addition: \(8.15 + 85.50 = 93.65\)
Step 4: Maintenance fee: \(93.65 - 4.50 = 89.15\)
Students must include the dollar sign (\(\$\)).
17. Polar Cold Front (Atmospheric Science) Ans: \(-11.7^\circ\text{F}\) or \(-11\frac{7}{10}^\circ\text{F}\)
Valid Expression: \(-6.5 - 4 \times 2\frac{3}{4} + 5.8\) (or \(-6.5 + 4(-2.75) + 5.8\))
Step 1: Total drop: \(4 \times 2.75 = 11.0^\circ\text{F}\) (or \(4 \times \frac{11}{4} = 11\))
Step 2: Temperature after drop: \(-6.5 - 11 = -17.5^\circ\text{F}\)
Step 3: Midday rise: \(-17.5 + 5.8 = -11.7^\circ\text{F}\)
Must include degree symbol and Fahrenheit unit (\(^\circ\text{F}\)).
18. Custom Bookcase Framing (Carpentry & Design) Ans: \(11\frac{5}{8}\text{ in}\) or \(11.625\text{ in}\)
Valid Expression: \(84 - [3(16\frac{1}{4}) + 2(11.5) + 5(\frac{1}{8})]\)
Step 1: Shelves: \(3 \times 16.25 = 48.75\text{ in}\)
Step 2: Slats: \(2 \times 11.5 = 23.0\text{ in}\); Kerf sawdust: \(5 \times 0.125 = 0.625\text{ in}\)
Step 3: Total wood used: \(48.75 + 23.0 + 0.625 = 72.375\text{ in}\) (\(72\frac{3}{8}\))
Step 4: Remaining: \(84 - 72.375 = 11.625\text{ in}\) (\(11\frac{5}{8}\text{ in}\))
Award full credit for either fraction (\(11\frac{5}{8}\text{ in}\)) or decimal (\(11.625\text{ in}\)).
Order of Operations with Rational Numbers • Teacher Reference Guide Answer Key • Page 3 of 3
