Generalization, model comparison, proof, and technology-assisted validation.
Directions: Show exact work before decimals, label domains or intervals, and explain calculator use.
1. {'id': 'u08-l06', 'number': 6, 'title': "De Moivre's Theorem and Powers", 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Write 3+3i in polar form.
2. {'id': 'u08-l06', 'number': 6, 'title': "De Moivre's Theorem and Powers", 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Multiply 2cis(π/6) by 3cis(π/4).
3. {'id': 'u08-l06', 'number': 6, 'title': "De Moivre's Theorem and Powers", 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Divide 8cis(2π/3) by 2cis(π/6).
4. {'id': 'u08-l06', 'number': 6, 'title': "De Moivre's Theorem and Powers", 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Use De Moivre to square 2cis(π/3).
5. {'id': 'u08-l06', 'number': 6, 'title': "De Moivre's Theorem and Powers", 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
How many distinct complex fourth roots does a nonzero number have?
6. {'id': 'u08-l06', 'number': 6, 'title': "De Moivre's Theorem and Powers", 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
What geometric action is multiplication by i?
7. {'id': 'u08-l07', 'number': 7, 'title': 'Roots of Complex Numbers & Geometric Structure', 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Write 3+3i in polar form.
8. {'id': 'u08-l07', 'number': 7, 'title': 'Roots of Complex Numbers & Geometric Structure', 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Multiply 2cis(π/6) by 3cis(π/4).
9. {'id': 'u08-l07', 'number': 7, 'title': 'Roots of Complex Numbers & Geometric Structure', 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Divide 8cis(2π/3) by 2cis(π/6).
10. {'id': 'u08-l07', 'number': 7, 'title': 'Roots of Complex Numbers & Geometric Structure', 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Use De Moivre to square 2cis(π/3).
11. {'id': 'u08-l07', 'number': 7, 'title': 'Roots of Complex Numbers & Geometric Structure', 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
How many distinct complex fourth roots does a nonzero number have?
12. {'id': 'u08-l07', 'number': 7, 'title': 'Roots of Complex Numbers & Geometric Structure', 'family': 'complex', 'week': 25, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
What geometric action is multiplication by i?