YOUNGSTOWN CITY SCHOOLS

MATH: PRECALCULUS

UNIT 8: COMPLEX NUMBERS AND POLAR COORDINATES (3 WEEKS) 2013-2014Synopsis: In this unit, students become familiar with complex numbers. They add, subtract, multiply, divide and simplify them, using the properties of real numbers. After they are proficient with complex numbers, they convert them to the polar form. Examining the relationship between graphical representation of the complex number and its polar form will aid in understanding the concept.

STANDARDSN.CN.1 Know there is a complex number I such that i2 = -1, and every complex number has the form a + bi with a and b real.

N.CN.2 Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

N.CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

N.CN.4(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

N.CN.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. (e.g., (-1 + 3i) has modulus 2 and argument 120°

N.CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints

MATH PRACTICES1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning

LITERACY STANDARDSL.2 Communicate using correct mathematical terminology L.3 Read, discuss and apply the mathematics found in literature, including looking at the author’s purpose L.7 Research mathematics topics or related problemsL.8 Read appropriate text, providing explanation for mathematical concepts, reason or proceduresL.9 Use information found in texts to support reasoning and develop a “works cited document” for research done to solve a problem.

MOTIVATION TEACHER NOTES1. Show students images of polar graphs such as the ones on page 6. Explain to them

upon completion of this unit, they will be able to change complex numbers to polar coordinates and graph polar equations. Images of polar graphs on website below:http://www.bing.com/images/search?q=polar+graphs&qpvt=polar+graphs&FORM=IGRE#a

2. Have students graph some polar equations on the TI or calculator software.http://mathbits.com/MathBits/TISection/PreCalculus/polargraphs.htm instructions on graphing polar equations on the TI

3. Preview expectations for end of Unit.

4. Have students set both personal and academic goals for this Unit.7/02/2013 YCS PRE-CALC UNIT 8: COMPLEX NUMBERS AND POLAR COORDINATES 2013-2014 1

TEACHING-LEARNING TEACHER NOTESVocabulary:Complex number Real part Imaginary part ConjugateModuli Polar form Pure imaginary argument

Web sites you may find helpful: http://www.math.hmc.edu/calculus/tutorials/complex/ explanation of all facets of complex numbers and polar coordinateshttp://www.mathworksheetsland.com/topics/complex.html worksheets on complex numbershttp://en.wikipedia.org/wiki/Complex_number http://www.mathworksheetsgo.com/sheets/algebra-2/complex-numbers/

1. Review with students in story form the number systems they have used to date: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. The number systems were created to solve particular problems. Natural numbers were the first to be used by the cavemen to count their possessions. Then whole numbers were created to represent the fact that perhaps one had nothing. As time progressed and scientific discoveries, such as the thermometer, were made, man needed negative numbers to represent a temperature colder than zero. Hence the integers were created to meet this need. During the Greek and Roman Empires, there was much building and when measuring for the construction, the measurements weren’t always whole numbers, so rational numbers had to be invented and measuring devices with fractions were made. Then along came Pythagoras with his theorem. Not all answers were rational. Sometimes the third side of the triangle was not a perfect square. So, you guessed it, the irrational number system was invented to accommodate this. Then you studied the quadratic formula and sometimes the number under the square root (discriminant) is a negative number. So the complex number system needed to be invented to accommodate this phenomenon. Astronomers have always referred to objects in space using a distance and direction. So as to simplify this, polar coordinates were invented.

To ensure students understand the different number systems, have them do a graphic organizer to illustrate all the number systems: natural, whole, integers, rational, irrational, and complex.(N.CN.1, MP.4, L-2)

2. Define a complex number as a+bi where a is the real part and b is the imaginary part.Consider a pure imaginary number in which a = 0. Discuss that , so squaring both sides gives: -1 = I. Using that definition, examine the powers of i and its repetitive nature (repeating every 4th one). Discuss adding, subtracting and multiplying complex numbers and have students identify the commutative, associative and distributive properties while they are performing those operations. Students should realize that the complex number system uses the same properties as the real number system. A worksheet is attached on page 7-10 (N.CN.1, N.CN.2, MP.2, MP.4, MP.8, L.2)

3. Ask students how they might simplify , that is write the fraction without a complex number in the denominator. Remind them of that a2 + b2 = (a-b)(a+b) and the fact that i2 = -1. When discussing this tell them the conjugate of a complex number is used to accomplish this. Let z = a +

bi then and . The modulus of z, . From this,

. Practice simplifying quotients of complex numbers. Additional problems are found on http://www.mathworksheetsgo.com/downloads/algebra-2/complex-numbers/dividing-complex-numbers-worksheet.pdf .(N.CN.3, MP.2, MP.4, MP.8, L.2)

4. Looking at a complex number on the coordinate system enhances the understanding of adding, subtracting, multiplying and conjugating complex numbers. Let the x axis be the real axis and the y axis be the imaginary axis. To plot (3+4i), from (0,0) go to the right 3 and up 4 units. Place a dot at the point (3,4). The complex number is represented by the vector from (0,0) to (3,4). The modulus

is the distance from (0,0) to (3,4) = . Have students practice graphing complex numbers and finding their moduli. .(N.CN.1, MP.2, MP.4, MP.8, L.2)

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TEACHING-LEARNING TEACHER NOTES5. Addition and subtraction can also be represented graphically. Consider (2+6i)+(3-i). Graphically plot

the first complex number (2+6i), i.e. from (0,0) go to the right 2 and up 6 units. From that point, (2,6), proceed up three units and down one unit. You are now at the point (5,5) which is (5+5i), so looking at it graphically, the sum is the resultant of the two vectors. When subtracting two complex numbers graphically such as (4 + 5i) – (6 + 2i), plot 4+5i then from the point (4,5) go to the left 6 units and down 2 units, arriving at (-2, 3) = -2 + 3i. The following link shows adding and subtracting of complex numbers: http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson(N.CN.5, MP.4, MP.5, MP.8, L.2)

6. To calculate the distance between two complex numbers, find the difference between their coordinates and find the modulus of that difference.

Example: Find the distance between 2+3i and 5+12i. The coordinates are (2,3) and (5,12). Find the differences between the coordinates: 5-2 = 3 and 13-3=9, the modulus

is so the distance between 2+3i and 5+12i is 6 units. To calculate the midpoint of the same segment, find the average of the numbers at its endpoints.

Example: Find the midpoint of the segment joining (2+3i) and (5+12i). Again working with (2,3) and (5,12), find the midpoint of the segment joining those two points which is (3.5, 7.5). (N.CN.6, MP.1, MP.2, MP.4, MP.6, MP.8, L.2)

TEACHER GENERATED ASSESSMENT

7. A complex number (a+bi) can be represented by an ordered pair (a,b). To represent a number in polar form a radius “r” and an angle “θ” is needed. So the polar form of a complex number is (r,θ). If you look at the graph of 3+4i which can be represented as the vector from (0,0) to (3,4), you see an angle in standard position and a length from (0,0) to (3,4). Draw the right triangle using the x axis as one leg and the other leg parallel to the y axis and ending at (3,4). The angle is found by using

the tangent function and the radius is found by using the Pythagorean theorem. So tan θ =

yielding θ = 530 and r = . The polar form of (3,4) is (5, 530). When representing the complex number as 3+ 4i, the polar form can be represented as

5cos 53 + 5i sin 53 = 5(cos 53 = i sin 53), since cos 53 = and sin 53 = . Notice that r is the modulus of the complex number. The angle is called the argument. Discuss why the two different forms represent the same number.

z = a + bi = r(cosθ + i sinθ), where r = Reinforce converting from complex to polar and polar to complex with additional problems. Refer to section 9-6 in the textbook. (N.CN.4, MP.4, MP.5, MP.8, L.2)

8. Now look at multiplication and division of complex numbers geometrically. Review some trig: Refer to pages 594-597 in the textbook(a+bi)(c+di) = r1(cos θ + i sinθ) * r2(cos β + i sin β)= r1* r2 (cos θ cos β + i cos β sinθ + i cos θ sin β + i2 sinθ sin β r1* r2 (cos θ cos β - sinθ sin β + i(cos β sinθ + cos θ sin β) note: i2 = -1 r1* r2(cos (θ + β) + I sin (θ + β) note: sum identity for cosine and sine

So if we would like to multiply (2+3i)(5+12i), r1 = = 5 and r2 = =13, θ = 56.3 and = 67.4, the product is 5*13((cos (56.3+67.4) + i (sin (56.3+67.4))=65(-0.5546 + 0.832i)Looking at this graphically, draw 2+3i and 5+12i. Look at r1 = 5, θ = 56.3, r2 = 13 and β = 67.4. The vector forming the product has angle 123.7 and radius 65Division is found on page 594 in the textbook and also on the link below. (N.CN.5, MP.1, MP.2, MP.4, MP.6, MP.8, L.2)http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-

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TEACHING-LEARNING TEACHER NOTESin-complex-plane-.lesson shows multiplication and division graphically

9. Using section 9-2 in the textbook as a guide for graphing polar equations. Have students try to graph a polar equation by plotting points such as r = 2 sin θ. After they have attempted to graph it, discuss the classic curves and their equations. Then reinforce with additional problems. (N.CN.4, MP.1, MP.2, MP.4, MP.6, MP.8, L.2)

TRADITIONAL ASSESSMENT TEACHER NOTES1. Unit Test: Multiple-Choice Questions

TEACHER CLASSROOM ASSESSMENT TEACHER NOTES1. Teacher Classroom Assessments: 2. 2- and 4- point questions

AUTHENTIC ASSESSMENT TEACHER NOTES1. Using the following website: http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=18

Students choose an occupation, and research how complex numbers are used in that occupation; the average wage for that occupation; the training and amount of time needed for that training; and list the duties that are required and explain them. With the information gathered, write a 500 word report and include a “works cited” page with at least three references. (N.CN.1, MP.2, MP.4, L.2, L.3, L.7, L.8, L.9)

2. Have students evaluate goals for the unit.

RUBRICELEMENTS OF THE

PROJECT 0 1 2

Write a 500 word report Did not write report Wrote report with only 300 words or less Wrote report with 500 words

“Works cited” page Did not include “works cited” page

“Works cited” page with only one reference

“Works cited” page with three references

Use of complex numbers Did not include use of complex numbers NA. Included use of complex

numbers

Average wage Did not include an average wage NA Included an average wage

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Type of training needed Did not include training NA Included training needed

Time it takes for the training Did not include time for training NA Included time for training

Duties Did not list the duties Listed the duties however no explanation

Listed the duties with an explanation

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MOTIVATION #1 - - POLAR GRAPHS

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T-L #2 http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Operations%20with%20Complex%20Numbers.pdf

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