An introduction to Complex Numbers - amfidromie.nl€¦ · Lesson plan for lesson 1 8 Lesson 2: 10...

Teacher’s Guide

An introduction to

Complex Numbers

Rob Kosman Annelieke de Vos 2016

International School Twente: An introduction to Complex Numbers — Teacher’s Guide


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Content

Introduction 4 Educational & Didactical Justification 4 Intended Audience & Prior Knowledge 6 Lesson 1: 7

Learning goals for lesson 1 7 Educational & didactical justification for lesson 1 7 Lesson plan for lesson 1 8

Lesson 2: 10


Lesson 3: 13


Lesson 4: 16


References 20


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Introduction

This lesson material and the accompanying teacher’s guide concern 4 lessons that are intended to be the start of a larger lesson series introducing students to complex numbers. This lesson material was developed in the context of the academic Master’s course ‘History’, offered by MasterMath in the Netherlands, taught by J. Daems (University Utrecht, Netherlands) and S. Wepster (University Utrecht, Netherlands). This course was part of the programme ‘Master of Science Education and Communication’ of University Twente for both authors Rob Kosman and Annelieke de Vos. The lesson series was developed for Year 13 students at International School Twente, Netherlands, following the Cambridge A-level programme in mathematics (syllabus 9709). The authors have also been mindful of the transition International School Twente is making into the Diploma Program of the International Baccalaureat for the top two years. As such, this lesson material is also intended for use in the IB course Mathematics HL.

Educational & Didactical Justification

Most highschool textbooks introduce complex numbers by starting with the rule

2 1i and the use of Argand diagrams. A more formal construction of complex

numbers as a field extension of the real numbers usually falls well outside the scope of mathematics in a highschool curriculum. We believe there are serious downsides to this approach. First off, as is often the case in textbooks, a new concept is dropped on students (and a fairly exotic one at that) without there seeming to be any good reason to do so. An often quoted reason is that complex numbers allow us to factorise all quadratic expressions, sometimes even under the misconception that this is how complex numbers developed historically. But why would one want to factorise all quadratic expressions? It is undeniable that being able to do so is both beautiful and elegant. But it’s doubtful that many of us, or many of our students, just based on that, would feel the need to investigate such a possibility. Real linear factors of a quadratic expression correspond to real zeros of the corresponding function, and geometrically correspond to x-intercepts of the graph of that function. Since parabolas can happily live without x-intercepts, it seems perfectly natural that certain quadratic expressions don’t

factorise. There simply is no mystery here, no puzzle to solve, nothing that drives us to be discontent with the ‘old’ state of affairs. Furthermore, proficiency in mathematics at the highest level can only be achieved when we take a balanced approach in our education of mathematics. In Adding it up the authors argue that mathematical proficiency consists of 5 intertwined strands, all of which we need to pay attention to as educators (Kilpatrick et al., 2001).


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conceptual understanding — comprehension of mathematical concepts, operations, and relations

procedural fluency — skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

strategic competence — ability to formulate, represent, and solve mathematical problems

adaptive reasoning — capacity for logical thought, reflection, explanation, and justification

productive disposition — habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

But introducing complex numbers by starting off with 2 1i and the use of Argand

diagrams, in our opinion, does not do justice to all 5 strands. In such an approach there is little room for reflection, explanation and justification. And more importantly, it could genuinely diminish a student’s belief in diligence and one’s own efficacy. Presenting such exotic new material as a finished and polished product completely ignores the fact that some of the greatest mathematical minds in history have wrestled with this material for centuries, and that we only got to where we are now by centuries of diligent effort, making mistakes, and experiencing near misses. Being made aware of that human socio-cultural historical aspect of the development of mathematics can inspire students to want to be a part of that. Hence, we have chosen to introduce complex numbers to students from a historical perspective. We use a puzzle presented by Gerolamo Cardano in his Ars Magna as a ‘hook’ to make students curious and to draw them into the topic. Throughout the lesson material use is made of historical events and historical sources related to the subject of complex numbers. Students learn about the people involved in the creation of complex numbers and about their important contributions to the development of this field. Through the use of history students should become more motivated to work on the material and thus to understand the concept of complex numbers (Tzanakis & Arcavi, 2002). Despite the fact that the material currently only spans 4 lessons, and is intended to become part of a larger lesson series, we believe that this lesson material, as it stands, can already be very useful to supplement the presentation of complex numbers in most highschool textbooks in the form of a ‘prequel’. In our design of this lesson material we have tried to take a constructivist approach, and the material is set up as a kind of guided rediscovery, as proposed by Bruner (1966, 1986). The student material is also set up in a way that a student is able to do the lessons using self-study if this is necessary. This way, if a student has to miss class, they are still able to do the work and fully understand the material. Both primary and secondary sources are used to support the learning process of the students. Modern-day sources that we consulted are listed in the ‘References’ at the back of this teacher’s guide. We encourage teachers to consult these sources for additional information.

The choices made for certain material and exercises, as well as learning goals and lesson plans (based on lessons of 45 minutes), are detailed per lesson in the next section of this guide.


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Intended Audience & Prior Knowledge

This lesson material was designed with Year 13 students at International School Twente, Netherlands, in mind. These students follow the Cambridge A-level programme in mathematics (syllabus 9709). By the time they start on complex numbers, they should already have completes a Cambridge AS-level in mathematics. The list below details prior knowledge that is specific and essential to this lesson series:

Natural numbers, integers, rational numbers, real numbers.

Quadratic functions and equations, including:

o factorisation o completing the square o quadratic formula o role of the discriminant o geometrical interpretation of solutions to a quadratic equation

Cubic functions, including:

o graphs of cubic functions o geometrical interpretation of solutions to a cubic equation

Factor theorem

Polynomial long division

Systems of (non-linear) equations in two variables

Basic differential calculus, including:

o derivatives of polynomial functions o geometrical interpretation of a derivative


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Lesson 1

Learning goals for lesson 1 A student is able to:

express in his own words how his concept of number has expanded in the past;

summarize what he knows about quadratic functions and equations by means of a mind map;

summarize what he knows about cubic functions and equations by means of a mind map;

compare and contrast both mind maps and express in his own words how his knowledge with regards to cubic equations is incomplete.

Educational & didactical justification for lesson 1 The introduction is meant to show students that they have expanded their concept of number in the past, even when they thought they knew everything already. Through this introduction it is made clear that there are even more numbers, so it is clear that the new subject is about a new type of numbers. The students are faced with a problem (a puzzle by Gerolamo Cardano) which does not have any solutions using their knowledge up to this point. This problem is chosen as an introduction to the subject “complex numbers” and is intended to make students curious. This problem is chosen, because it is a real question that was posed in the 16th century. As stated by Tzanakis & Arcavi (2002) students understand the motivation for learning a new subject when given an example to reconstruct. The students get the original problem first, before it is translated to the notation which they know. This way students see the evolving nature of the representation of mathematical problems, which leads to the students understanding the advantages and disadvantages of modern forms of mathematics (Tzanakis & Arcavi, 2002). The next part of the lesson is activating students’ prior knowledge on quadratic and cubic equations. At first students are asked to make a mind map related to solving a quadratic equation. This mind map is a visual representation of their network of knowledge, or cognitive schema, with respect to quadratic equations. They have to do the same activity for solving cubic equations. By comparing these mind maps the students come to the realisation that their knowledge network related to quadratic equations is bigger and more complete than their network for cubic equations. So their knowledge is incomplete, may contain gaps, and there is still more to learn. Through these activities we make students activate their prior learning and the relevant neural networks involved. This way the new knowledge will be linked to their existing knowledge network which is already in their long-term memory, resulting in better integration and more adaptable use of new concepts (Van Streun, 2013).


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Lesson plan for lesson 1

Educational function

Subject material Activity teacher Activity students

Time

Lesson organisation

The following parts will be discussed this lesson:

Introduction

Puzzle

Quadratic equations

Cubic equations

The teacher should write down the plan for this lesson before class and walk students through this.

Students listen.

2 min

Introduction

The number line is discussed with the students. They should start off with the natural numbers, followed by the integers, rational numbers and real numbers.

The teacher guides the class discussion and gets students to name the different types of numbers. Results on the board.

Students share their ideas in a class discussion.

6 min

Introducing puzzle

The puzzle by Gerolamo Cardano is introduced briefly. Next students are given time to read the problem and do exercise 1.

The teacher introduces the puzzle and invites students to try and solve it, either individually or in small groups.

Students listen and then go to work on solving the puzzle from exercise 1. Collaboration is allowed.

10 min

Activate prior knowledge

Student have worked with quadratic equations before. Their knowledge of solving these equations will be activated by making a mind map (exercise 2). The concepts that should at least be included in the mind map are listed in exercise 2b.

The teacher invites students to make a mind map about quadratic equations. After 5 minutes the results are discussed and a joint mind map is written on the board. The teacher encourages students to comment on the concepts listed in exercise 2b.

Students spend 5 minutes making a personal mind map about quadratic equations. Then they join in a class discussion and share their suggestions with the class.

12 min


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Time

Activate prior knowledge

Students have worked with cubic equations before only to some small extent. Their limited knowledge of solving these equations will be activated by making a mind map (exercise 3). The concepts that should at least be included in the mind map are listed in exercise 3b.

The teacher invites students to make a mind map about cubic equations. After 5 minutes the results are discussed and a joint mind map is written on the board. The teacher encourages students to comment on the concepts listed in exercise 3b.

Students spend 5 minutes making a personal mind map about cubic equations. Then they join in a class discussion and share their suggestions with the class.

12 min

Wrap up lesson

Students should finish the lesson material for lesson 1 before the start of next lesson.

The teacher briefly summarizes the lesson and assigns remaining exercises from lesson 1 as homework.

Students write down the homework.

3 min


10

Lesson 2


recount in his own words some socio-historical facts with respect to solving quadratic and cubic equations;

comment in his own words on the discovery of mathematics being a human endeavour that doesn’t happen in a social vacuum;

comment in his own words on the evolution of mathematical notation and its impact on the development of mathematics itself;

use Del Ferro’s formula from the lesson material (not from memory) to solve cubic equations of suitable type.

Educational & didactical justification for lesson 2 As in the previous lesson some attention is given to quadratic equations. The material offers information on how knowledge of solving quadratic equations has been around for a long time, in different cultures. This shows students that mathematics is not purely a product of Western culture. As stated by Tzanakis & Arcavi (2002): “In some cases, these cultural aspects may help teachers in their daily work with multi-ethnic classroom populations, in order to re-value local cultural heritage as a means of developing tolerance and respect among fellow students.” This notion is enhanced by showing the work of al-Khwarizmi and the exercise related to his work. Next, a timeline is given on the progress of solving cubic equations. Students will see that it took quite some time to find some sort of a solution to cubic equations. Struggling to find a solution to a problem is something students experience themselves. This helps them understand that it is normal to make mistakes and struggle, but that it pays off to keep trying (Tzanakis & Arcavi, 2002). The passage about “Keeping the Secret” intends to provide information on the social and cultural structure in the time of these mathematicians. As stated by Tzanakis & Arcavi (2002): “Students can be given the opportunity to appreciate that mathematics is driven not only by utilitarian reasons, but also developed for its own sake, motivated by aesthetic criteria, intellectual curiosity, challenge and pleasure, recreational purposes etc.” This makes mathematics a more “human” subject for students instead of something that simply is. The last part of this lesson focuses on the notation used in modern days versus the notation used by Cardano. This shows students the advantages and disadvantages of our modern notation (Tzanakis & Arcavi, 2002). Another important reason for the dictionary question (exercise 13) is given by Arcavi & Bruckheimer (1991): “This activity helps the students to become acquainted with unknown notation, symbols, names of concepts, or formulations in the source.” The students will need this dictionary in lesson 3.


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Time

Lesson organisation


History of quadratic equations

(Depressed) cubic equations


Students listen.

2 min

Historical background

A short history on the solving of quadratic equations is given involving the following:

Babylonian cuneiform tablet BM13901

Other cultures with the same knowledge (ancient India, ancient Greece, Arabic world)

al-Khwarizmi

A more detailed description is given in the student material “lesson 2: Solving the Quadratic”.

The teacher shows the students the picture of the Babylonian cuneiform tablet BM13901 and explains what is written on it. He emphasizes the historical and cultural aspects of the topic. The teacher shows the book (picture) of al-Khwarizmi with the transcribed version of the title, and asks students if they see a familiar word related to mathematics. The word the teacher is looking for is “algebra”. The teacher can call on a student to share his idea, or use the tool from www.answergarden.ch to visualize which answers are given by the entire group (the bigger a word, the more often it was given as an answer).

Students listen, ask questions, participate in the class discussion.

15 min


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Time

Depressed cubic equations

First a short historical overview of solving cubic equations is given as provided in “lesson 2: Solving the Cubic”.

Next it is explained what a depressed cubic equation is. Exercise 7.

The teacher presents a historical overview, followed by showing the depressed cubic equation that Del Ferro was able to solve:

3x px q , , 0p q .

The teacher invites students to work on exercise 7 with their neighbour for 5-10 minutes. Afterwards the exercise is discussed with the class. Students that run into difficulties can be supported by the teacher and guided on the way towards the right answer.

Students listen. Students work on exercise 7 in pairs for 5-10 minutes. They ask the teacher for help if they get stuck. Students take part in the class discussion and share their solutions with the rest of the class.

20 min

Exercise

Exercise 8.

The teacher invites students to work on exercise 8: try and find a solution for the depressed cubic equation

3x px q , , 0p q .

Students individually work on exercise 8.

5 min

Wrap up lesson


The teacher briefly summarizes the lesson and assigns the remaining material and exercises from lesson 2 as homework.


3 min


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Lesson 3


comment in his own words on the evolution of mathematical notation and its impact on the development of mathematics itself;

derive the formulae for the solutions to both cases of depressed cubics from each other;

reproduce Bombelli’s method of simplifying results from Cardano’s formulae, with sufficient scaffolding provided.

Educational & didactical justification for lesson 3 The first part of this lesson focuses on Bombelli’s work. Students are asked, with sufficient scaffolding provided, to rediscover the solution to the problem posed by Bombelli (Bruner, 1966, 1986). Working on a historical problem might spark the students’ interest, because it is not an artificial problem (Tzanakis & Arcavi, 2002). Also reproducing Bombelli’s work gives students the opportunity to debate the validity of his solution (Arcavi & Bruckheimer, 1991). First, students are made comfortable with the solution to a depressed cubic equation in which the radicand of the square root is always positive. Next, another type of depressed cubic is discussed, in the solution of which it is possible for a negative radicand to appear under the square root. Students first work on an example in which that is not yet the case. They have to decipher the specific example from two different provided primary historical sources, one by Cardano, and one by Bombelli. This way the students are again faced with differences in notation. This shows the students that mathematics has an evolving nature (Tzanakis & Arcavi, 2002). Students are also asked in exercise 23 to compare and contrast these different types of notation, which strengthens this insight. Then students are asked to reproduce Bombelli’s procedure for this example, this time with less scaffolding. Students are also asked to show that Cardano’s solution to this second type of depressed cubic can easily be derived from Del Ferro’s formula (exercise 19). This may raise the question for some students why these two forms of depressed cubics are regarded as separate cases in the first place. They wouldn’t be by us, who feel completely comfortable with negative parameters, but they were in the 16th century, when most would go to great lengths to be able to avoid negatives. The intention is to not draw any attention to this in lesson 3, but to revisit this exercise and this point in lesson 5 (which has yet to be designed) when a parallel is drawn between the slow acceptance of negative and complex numbers. Note that we have intentionally avoided anything that reeks of complex numbers up until this point. This way we can set the stage and first make students feel comfortable with the context. When the mystery is then introduced in lesson 4, the focus can be completely on that.


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Time

Lesson organisation


Discuss homework exercises 12 & 13

Continuation depressed cubic equations

Other type of depressed cubic equations


Students listen.

2 min

Go over previous lesson

The material discussed in “Lesson 2: Cardano’s Ars Magna” should be briefly discussed. The results from exercise 12 and exercise 13 should be discussed, because these results are of importance for the next set of homework.

The teacher discusses exercise 12 with the students to see if there were difficulties with this exercise.

For exercise 13 the teacher picks students at random for their answers. This to check if students deciphered Cardano’s piece correctly.

Students participate in discussing exercise 12. Students give their answers to exercise 13 when called on.

10 min


15


Subject material

Activity teacher Activity students

Time

Continuation of the depressed cubic and its solution

The information discussed in “Lesson 3: Bombelli to the rescue”. Exercises 16a and 16b.

The teacher discusses Bombelli’s idea that

3

10 108a b .

After this, the teacher lets students work on exercises 16a and 16b. The teacher should give students about 10 minutes to work on this. If they are not finished, they should finish it at home.

Students listen and ask questions in case they do not understand something. Students work on exercises 16a and 16b. Students are allowed to discuss with their neighbours.

15 min

Introducing another type of depressed cubic

The depressed cubic equation of the form

3x px q

with , 0p q

should be introduced as a new type. Its solution is provided in “Lesson 3: Another type of ‘depressed cubic’”. Exercise 18.

The teacher explains the new depressed cubic form and clearly notes the difference with the depressed cubic shown before this. Also, the solution to this depressed cubic equation should be shown to students. After that, the teacher lets the students work on exercise 18 in pairs for about 5-10 minutes. Afterwards, the teacher briefly discusses the answers found. The teacher compares these results to the results found in exercise 7 together with the students.

Students listen and ask questions in case they do not understand something. Students work on exercise 18 with their neighbours. Students actively participate in the discussion of the exercise.

15 min

Wrap up lesson




3 min


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Lesson 4


describe in his own words how negative radicands under square roots can appear in Cardano’s formula and why the geometry of the problem might move one to try and make sense of such roots instead of just dismissing them;

explain in his own words how the technique Bombelli used to simplify solutions from Cardano’s formulae before, provide him sufficient reason to at least try something similar in the case when square roots of negative numbers pop up;

use the rules for calculating with square roots of negative numbers, as provided in the lesson material;

express his current personal opinion on the ontological nature of complex numbers, based on the knowledge he has so far.

Educational & didactical justification for lesson 4 Up to this point students were only faced with positive radicands under square roots, to make them comfortable with the solutions found by Del Ferro and Cardano for these depressed cubic equations. Now that they’ve reached a certain level of comfort with Cardano’s formula, they are exposed to examples that lead to negative radicands under the square root. In the text given to the students, the following is stated: “No doubt Cardano was aware of this. But in his Ars Magna he carefully avoids using the above formula for such cases.” This demonstrates to students that Cardano was not quite ready to deal with square roots of negatives yet. This makes it acceptable for students to be sceptical about the idea as well (Tzanakis & Arcavi, 2002). Just like in lesson 3, students are asked to decipher a historic piece (exercise 24). This shows students again the evolving nature of mathematics, which could lead to appreciation of modern mathematics, as stated by Tzanakis & Arcavi (2002).

Students are not asked immediately whether they feel b , with 0b , makes any

sense. Instead they are first asked to just check if they could come to a solution, using the algebraic rules provided. This process of just trying something out, letting one’s imagination run wild, before taking a step back to reflect, is something Bombelli did himself. Again this demonstrates the evolving nature and the human aspect of mathematics to students. Most of the time students see mathematics in its reorganised, polished and finished state. Though this has some advantages, due to this reorganisation students often do not see the motivation for the development of a new mathematical idea (Tzanakis & Arcavi, 2002). As stated by Tzanakis & Arcavi (2002) a reconstruction will lead to students having better understanding of the subject matter.


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In this process of just trying something out, and letting one’s imagination run wild, students are basically asked to work with objects that seem to make no sense as of yet, to even do aritmetic and algebra with these objects, violating at least one algebraic rule they had believed to be universally true. So at this point, students probably feel like they have been thrown in at the deep end. This is why the sections “Time to Take a Step Back” and “A Question of Ontology” have been included. Here students have to analyse this new information using a fragment form Bombelli’s L’Algebra. Such a reconstruction will lead to students having a better understanding of the subject matter (Tzanakis & Arcavi, 2002). In addition, seeing that Cardano and Bombelli themselves felt uneasy with these new ‘numbers’, validates the feelings the students probably still have at this point. It shows students it is alright to have difficulty accepting this new concept (Tzanakis & Arcavi, 2002). Armed with their new tentative knowledge, students are then invited to try and solve the initial puzzle from lesson 1. Returning to this ‘hook’ helps them to reflect on the journey so far and to feel motivated to continue.


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Time

Lesson organisation


Depressed cubic equation

Bombelli’s ideas


Students listen.

2 min

Debate validity

The depressed cubic

equation 3x px q ,

with , 0p q is revisited,

including Cardano’s formula for its solution. Example:

3 15 4x x .

It should be debated in class whether the solution according to Cardano’s formula is valid (square root with a negative radicand). It should be mentioned that Cardano avoided cases in which this happens, but Bombelli does not. A contrast should be made between the seeming breakdown of Cardano’s formula and the existence of a positive real root.

The teacher should start off by giving a lecture about the subject material as described left. After showing all this information the teacher should open up a debate with the students about what to do next, knowing there exists a positive real solution, but at the same time knowing that Cardano’s solution seems not valid according to the students’ knowledge.

Students listen to the teacher taking in the information provided. After that, students join in the debate about what to do with Cardano’s solution in the example given.

25 min


19


Subject material

Activity teacher Activity students Time

Introducing square roots of negative numbers

The information given in “Lesson 4: Bombelli’s Wild Thought” should be discussed together with exercise 26a.

The teacher invites students to suspend all disbelief for a moment, and just treat square roots of negative numbers as valid objects. The teacher discusses which algebraic rules would then be reasonable to assume

2

b b ,

and invites students to explore where this leads. The teacher should give students the opportunity to work on exercise 26a for 5-10 minutes. The results should be discussed with the students after.

Students listen to the teacher until being instructed to work on exercise 26a. After this students will discuss their results with the rest of the class.

15 min

Wrap up lesson




3 min


20

References

Arcavi, A. & Bruckheimer, M. (1991). Reading Bombelli’s x-purgated algebra.

The College Mathematics Journal, 22(3), 212-219.

Bruner, J.S. (1966). Toward a theory of instruction. Cambridge MA: Belknap.

Bruner, J.S. (1986). Actual minds, possible worlds. Cambridge MA: Harvard

University Press.

Kilpatrick, J., et al. (2001). Adding it up: Helping children learn mathematics.

Washington, DC: National Academy Press.

Tzanakis, C. & Arcavi, A. (2002). Integrating history of mathematics in the classroom:

an analytic survey. In J. Fauvel & J. van Maanen (ed.), History in Mathematics

Education (pp. 201-240). Groningen: Springer Netherlands.

Van Streun, A. (2013). Leren en onderwijzen van wiskunde. In B. Zwaneveld (red.),

Handboek wiskundedidactiek (pp. 3-52). Amsterdam: Epsilon Uitgaven.

The following recommended sources are not directly referenced in this teacher’s guide,

but were consulted in designing the lesson material:

Katz, V.J. (2014). History of mathematics. (3rd edition). Harlow, UK: Pearson Education

Limited.

Nahin, P.J. (1998). An imaginary tale: The story of 1 . Princeton: Princeton

University Press.

Needham, T. (1997). Visual complex analysis. Oxford: Oxford University Press.

Stillwell, J. (2010). Mathematics and its history. (3rd edition). New York: Springer.

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