Designing a Sample Learning Activity for Complex Numbers
A teacher plans a 70-minute lesson on complex numbers for a Unit 3 class. The intended outcome is that students can represent complex numbers in polar form, multiply and divide them using modulus-argument properties, and interpret the results geometrically. The teacher's draft plan is: 10 minutes of definitions on the board, 45 minutes of students working through 20 routine conversion exercises, and 15 minutes of going through answers. The head of department observes that the plan is procedurally heavy and asks for a redesign that develops reasoning, uses a concrete representation, and includes formative assessment. The class has access to graphing software and CAS.
1. Critique the draft plan against the intended outcome and propose a redesigned sequence with timings.
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Critique: the draft spends 45 of 70 minutes on routine conversion, which targets a lower-order skill (procedural fluency in converting forms) and does not develop geometric interpretation or reasoning about modulus-argument properties. The 15-minute answer review is teacher-centred and gives no formative information about student thinking during the lesson.
Redesigned sequence (70 minutes):
- 0-10: Activate prior knowledge with a quick diagnostic: three conversions and one question asking students to locate a complex number on the plane. Collect responses to identify misconceptions.
- 10-25: Guided exploration with graphing software. Students enter several complex numbers, observe modulus and argument, and predict where the product of two numbers will lie. Record predictions.
- 25-45: Paired task. Students multiply and divide pairs using both polar form and software, then write a general statement relating moduli and arguments. Teacher circulates and probes.
- 45-60: Whole-class consolidation. Two pairs present their general statements; the class tests them against a counterexample chosen by the teacher.
- 60-70: Exit ticket with one conversion and one geometric interpretation question, collected for formative use.
This shifts the balance toward reasoning and representation while retaining necessary procedural practice.
2. Write the pseudocode for a short activity in which students verify the modulus-argument rule for multiplication, and state what the output should show.
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INPUT z1 = a1 + b1*i, z2 = a2 + b2*i
SET r1 = sqrt(a1^2 + b1^2)
SET r2 = sqrt(a2^2 + b2^2)
SET theta1 = atan2(b1, a1)
SET theta2 = atan2(b2, a2)
SET z3 = z1 * z2
SET r3 = sqrt(Re(z3)^2 + Im(z3)^2)
SET theta3 = atan2(Im(z3), Re(z3))
PRINT r1*r2, r3
PRINT theta1 + theta2, theta3
Expected output: r1r2 equals r3 (to within rounding), and theta1 + theta2 equals theta3 modulo 2pi. Students should test with numbers in different quadrants, including cases where theta1 + theta2 exceeds pi, and observe that the argument comparison must be taken modulo 2*pi. This teaches both the rule and the subtlety of argument representation.
3. Identify two formative assessment opportunities in the redesigned lesson and explain what each reveals about student understanding.
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The opening diagnostic (0-10 minutes). It reveals whether students can convert between forms and locate numbers on the plane, exposing gaps in prerequisite knowledge before the main task. If several students cannot locate a number, the teacher should adjust the exploration phase rather than proceed.
The paired task predictions (10-25 minutes). Asking students to predict where the product lies before computing reveals whether they are reasoning geometrically about modulus and argument or merely applying a formula. Predictions that ignore the argument addition indicate a procedural view; predictions that anticipate rotation and scaling indicate conceptual understanding. The teacher can use this to decide which pairs present and what to emphasise in consolidation.
The exit ticket provides a third, summative-for-learning check, but the two in-lesson opportunities are the ones that allow real-time adjustment.