Teacher Notes

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The fact that the HL syllabus content for complex numbers is in the first syllabus topic (Number & Algebra) is a perfect illustration of why the order in which content is listed in the syllabus does not necessarily correlate to the order in which it should be taught. Concepts and skills covered in the 3rd syllabus topic (Geometry & Trigonometry) are essential preparation for studying syllabus content in complex numbers such as: modulus-argument (trig) form of a complex number, and plotting complex numbers in the complex plane (Argand diagram). Also it is very useful (and a recommended teaching approach) to connect complex numbers and vectors (now only in HL in Geo & Trig topic) - in that a complex number can be represented as a vector in the complex plane having a magnitude (modulus) and direction (argument). While I would never consider teaching complex numbers before covering the Geometry & Trigonometry topic, I have some times taught complex numbers before covering vectors. When doing so, I use the complex numbers unit as a means to introduce some of the basic concepts of vectors.

Personally, I prefer to wait until after I have taught the Calculus topic (or at least the differential calculus part of it) before I teach complex numbers. Assuming that the binomial theorem has also been taught before complex numbers, then this provides a nice opportunity to derive Euler's identity, \({{\textrm{e}}^{{\textrm{i\theta }}}} = \cos {\theta } + {\textrm{i}}\sin {\theta }\) (not on the syllabus) Even though it is not in the syllabus, I feel that using differential calculus and the binomial theorem to derive Euler's identity can be a valuable experience for HL students - and serve as a useful introduction to power series, and specifically Maclaurin series. Furthermore, one can then present the special case of Euler's identity, \({{\textrm{e}}^{{\textrm{i\pi }}}} + 1 = 0\), which is often considered the most beautiful equation in mathematics. See the page on Euler's identity which shows a detailed derivation of the equation/identity.

♦ teaching materials

EXS_1-5-20v2_HL_complex_nmbrs_forms
Exercises & answers for expressing complex numbers in three different forms: Cartesian, modulus-argument & Euler's form
This is a corrected version (v2) with four answers corrected and corrected instructions for 5(a).

EXS_1-5-35v2_HL_complex_nmbrs
set of 6 miscellaneous exercises on complex numbers - includes answers

EXS_1-7-25v1_HL_deMoivres_thm 
Five exercises with worked solutions for using de Moivre's Theorem

Quiz_HL_complex_numbers1_v1 
Quiz covering complex numbers with four questions (and a bonus question) - total of 40 marks on quiz; no GDC allowed; solution key available below

Quiz_HL_complex_numbers1_v1_SOL_KEY
Worked solutions for complex numbers quiz above.

AA_HL_Test1_complex_numbers_v1 
Unit test on complex numbers. 5 Qs with no  GDC, and 4 Qs with GDC allowed. Also included a bonus question. Worked solutions available below.

AA_HL_Test1_complex_numbers_v1_SOL_KEY 
Worked solutions for Test1 on complex numbers above.

Test2_HL_complex_nmbrs_v1
Another unit test on complex numbers. 9 questions: 4 with no GDC, and 5 with GDC allowed.  Worked solutions available below.

Test2_HL_complex_nmbers_SOL_KEY_v1
Worked solutions for Test2 on complex numbers above.

Test3_HL_complex_nmbrs_v1
A unit test on complex numbers with 8 questions (50 marks) where a GDC is not allowed for any of the questions. Worked solutions available below.

Test3_HL_complex_nmbrs_SOL_KEY_v1 
Worked solutions for Test3 on complex numbers above.