• Pair work on complex numbers with partners has to be evaluated to insure that foundational skills are in place. [IS.20 - All Students] Discussion between partners should be on-topic and both partners need to be listening to each other’s questions and observations. When checking results, tell students to work backwards from their answers to find the source of their errors. Group activities are most effective when the teacher participates in the discussions and poses questions and challenges to student observations. In class discussions, ask leading questions when a student makes an incorrect statement.

• Lesson 3 Exit Ticket (M-A2-2-3_Lesson 3 Exit Ticket.doc) assesses students’ understanding of multiplying and dividing with complex numbers. [IS.21 - All Students]

[IS.19 - All Students]

Example 1:

“How can we find the dimensions (the base, height, and hypotenuse) of a right triangle if we know the area and how the dimensions relate to one another? [IS.5 - All Students] For example, what are the dimensions of a right triangle whose height is 7 units greater than its base and whose area is 30 square units? Since we know that the height exceeds the base by 7 units, [IS.6 - All Students] let x = length of the base and let x + 7 =length of the height. The area of the triangle is bh. Then, x(x + 7) = 30.”

“Since a triangle’s dimensions must be positive, 5 units is the measure of the base and 12 units (5 + 7) is the measure of the height. From the Pythagorean Theorem, the square of the hypotenuse equals the sum of the squares of the two sides, .”

5² + 12² = c²

c² = 25 +144 = 169

c = = 13 units

“We can generate an infinite variety of right triangles with infinitely many different bases, heights, and hypotenuses. One interesting and important concept associated with all of these right triangles is fractals.”

“Fractals are rough or fragmented geometric shapes that can be split into parts, [IS.7 - All Students] each of which is a reduced-size copy of the whole. They are often easier to show than explain.”

Here is an example of one called a Sierpinski triangle: http://en.wikipedia.org/wiki/Sierpi%C5%84ski_Triangle

“Notice how the smaller triangles repeat themselves inside the original equilateral triangle.”

Activity 1: Drawing a Pythagoras Tree

Hand out copies of the Draw the Tree activity sheet (M-A2-2-3_Draw the Tree.doc). Students will need a pencil with a good eraser, a ruler, and a protractor.

Note: Activities that require students to hand draw can be extremely useful in helping students make meaningful representations of the mathematical principles at work in right triangles and fractals. [IS.8 - All Students] Encourage students to retry and make new drawing attempts if they find difficulty in getting started. Better results come quickly with a little practice.

Show students some sketches of several versions of the Pythagoras Tree: http://mathworld.wolfram.com/PythagorasTree.html

“Right triangle ABC is surrounded by three squares. The area of square ABKJ is the square of hypotenuse AB; the area of square BCGH is the square of side CB; the area of square ACED is the square of side AC.”

“In order to draw the Pythagoras Tree with right triangles that make the fractal, it is necessary to make sure that all triangles are similar to triangle ABC. To draw them properly, all the corresponding angles must be congruent.”

“Begin by finding a new point, F, outside the square ACED between point D and point E. Use your protractor to find the measure of angle CAB. Measure this angle at least twice to make sure you have an accurate measurement. Now place the vertex of your protractor on point D and lightly mark the same measure as angle CAB with a point. Connect that point with a line to point D. Next use your protractor to find the measure of angle ABC, measuring at least twice to be accurate. Place the vertex of your protractor on point E and lightly mark that line to intersect with the line you drew from point D. Mark that intersection point F. If measured and drawn properly, triangle DFE should be similar to triangle ABC.”

“Now repeat the procedure to find another point I outside square BCGH above side GH. With the vertex of your protractor on point G, mark the same angle measure as angle CAB. Lightly extend that line from point G through your mark for the angle measure. Next, with the vertex of your protractor on point H, mark the same angle measure as angle ABC. Extend that line to intersect with the line from point G and mark it point I. Triangle GHI should be similar to triangles ABC and DFE.”

“To continue the fractal pattern, you will need to draw two more squares each adjacent to triangles DFE and GHI. Begin by measuring the length of side IH. Measure this length in millimeters, using the centimeter scale of your ruler. Measuring lengths to the nearest millimeter is much easier and more accurate than using fractions of one inch. The length of IH will determine the dimensions of the other three sides of the square adjacent to triangle GHI. Place your protractor on side IH with the vertex of your protractor on point I and mark 90^o from point I. Repeat the 90^o measurement from point H. Lightly draw the two perpendicular lines from point I and H. Using your ruler, mark the same distances from points I and H as length IH. Repeat the procedure to draw the next square adjacent to side FE of triangle DFE.”

“As you continue to draw the similar triangles and adjacent squares, you will begin to see the fractal pattern of the Pythagoras Tree develop. You can continue drawing the pattern for as long as you have room on your paper.”

To carry out the procedure as far as possible, try using large sheets of wrapping paper or butcher paper. Drawing the pattern on paper will require about five or six repetitions to see the arrangement of squares and right triangles sprouting into a tree.

For a demonstration that allows the user to manipulate the sizes of the right triangles, look at http://www.ies.co.jp/math/java/geo/pytree/pytree.html

To see some Pythagoras Trees of different size triangles, look at http://www.wolframalpha.com/input/?i=Pythagoras+tree

Ask students this Drawing Activity Follow-up Question, “Describe the shape of the Pythagoras Tree if all the right triangles are isosceles?” (bilaterally symmetrical)

Drawing Activity 2: Pythagoras Tree with Isosceles Triangles

In this activity, students can see the answer to the follow-up question directly. [IS.9 - All Students]

Hand out copies of the Isosceles Tree activity sheet (M-A2-2-3_Isosceles Tree.doc). Make sure each student has one 3 by 5-inch index card, several sheets of plain paper, and a ruler with a centimeter scale. Students will use the index card as a simple T-square by folding it to assist them in drawing right angles and 45-degree angles. Demonstrate to the students while instructing them on each step.

“To make the fold, hold the index card in the portrait orientation (3-inch side horizontal and 5-inch side vertical). [IS.10 - Struggling Learners] Fold the upper right corner across and to the left side so the top edge aligns exactly along the left edge. Try to make the alignment as exact as possible and make a sharp crease from the upper left corner to the edge of the fold.” Folding along the crease in both directions, front and back will allow students to use the 45° angle from either the left or right side.

Tell students to orient their drawing paper in the landscape direction (8-inch side in the vertical direction and 11 ½- inch side in the horizontal direction. [IS.11 - All Students] Instruct the students to draw a 5-cm square near the center bottom of their paper. To do this first have them draw one right angle and then use the ruler to measure exactly 5 centimeters along the vertical line and 5 centimeters along the horizontal line. “Mark each length in pencil with a small end point. Now place a right angle corner of the index card on one of the end points and draw another right angle, then measure 5 centimeters to make the third side. Complete the square by drawing the final right angle vertex and then measure the sides to make sure they are all 5 centimeters and each angle is a right angle.” Tell students that it is worth starting over and repeating the steps if the square is not accurately drawn. The more accurate the initial square, the better the results of the drawing.

Now students will construct a 45-45-90 triangle on top of their square, with the top edge of the square forming the hypotenuse. Have each student place their folded, 45-degree index card so the corner and edge of the 45-degree angle align with the corner and top edge of the square. Then, have them trace along the edge of the index card to create the 45-degree angle. Have them repeat the process on the other side of the top of the square, creating two 45-degree angles extending from the top. Make sure students extend the 45-degree angles far enough so the lines intersect. Have students erase any over-extending lines so they have a clean 45-45-90 triangle atop their original square.

The next step will be the construction of a smaller square adjoining the new triangle. The four sides of this new square will be congruent to the two sides opposite the hypotenuse of the triangle. From that point, students will measure one of those two sides with the ruler. Its length is approximately 3.5 centimeters. Have students begin by placing a right angle of the index card at the upper right vertex of the original 5-centimeter square. Then they will draw the right angle from this point, measure 3.5 centimeters, and mark the point. Repeat by placing the right angle of the index card on the new point, draw the right angle, and measure the next side of the 3.5 centimeter square. Have students connect the fourth side of the square by drawing the line to the apex of the triangle. Repeat the cycle in the same way by drawing a new isosceles right triangle atop each new square.

“What are the x-intercepts of y = (x − 2)² + 9?”

Separate the class in half. Have one half solve for the x-intercepts algebraically and have the other half solve for the x-intercepts graphically. [IS.12 - All Students] Students solving it algebraically should plug in 0 for y and solve for x. They should subtract 9 from both sides and then realize that they cannot take the square root of a negative number. Students solving it graphically will realize that it doesn’t hit the x-axis.
The terms solution, root, x-intercept, and answer all refer to the same thing.

“Why can we not take the square root of a negative number?”

Review what it means to square a number. “If we multiply a positive number by itself, the product is positive. If we multiply a negative number by itself, the product is still positive.”

Show students the graphs of y = (x − 1)² and y = (x + 3)² − 4. “How many x-intercepts do each of these graphs have?” The first parabola has 1 and the second parabola has 2.

“Parabolas can hit the x-axis twice, once, or not at all. But even though the graph doesn’t hit the x-axis, the parabola of y = (x − 2)² + 9 does have two x-intercepts, but they are imaginary.”

Start with a simple quadratic equation such as x² = −16 and ask students what the solution is. Stress that it’s not that there are no solutions, but that there are no real number solutions. [IS.13 - All Students] There are actually two imaginary number solutions. Introduce the imaginary number i. Have students write down the following in their notes.

i is the imaginary number defined to be equal to .

For any real number k > 0, it follows that .

Example:

Ask students to solve x² = −16.

Have students simplify the following and write the answers in their notes. [IS.14 - All Students]

Review the quadratic formula with students:

Practice solving for the x-intercepts of the following:

1. x² + 6x + 5

2. x² + x – 6

3. Our original parabola (x – 2)² + 9 = x² – 4x + 13

Have students add to their notes: A complex number is any number in the form a + bi, where a and b are real numbers and i is the imaginary number. [IS.15 - Struggling Learners] Complex numbers include the sets of real and imaginary numbers, and that if b = 0, a complex number is real and if b0, a complex number is imaginary (if a = 0 and b0, the complex number bi is often called a pure imaginary number). (Definitions taken from Holt Algebra 2, 2004.) In Advanced Algebra (2nd Ed.) by University of Chicago School Mathematics Project (Scott Foresman,1996), they define a + bi, where b 0, as a nonreal number and a + bi, where a = 0 and b0, as an imaginary number.

Explain that all powers of i can be simplified. Have students work with partners to simplify the powers of i up to the 12th power.

i¹ = i

i² =

i³ = (i)(i²) =(i)(−1) = −i

i⁴ = (i²)(i²) = (−1)(−1) = 1

i⁵ = (i)(i⁴) = (i)(1) = i

i⁶ = (i⁴)(i²) = (1)(−1) = −1

i⁷ = (i)(i⁶) = (i)(−1) = −i

i⁸ = (i⁴)(i⁴) = (1)(1) = 1

i⁹ = (i⁵)(i⁴) = (i)(1) = i

i¹⁰ = (i⁶)(i⁴) = (−1)(1) = −1

i¹¹ = (i¹⁰)(i) = (−1)(i) = −i

i¹² = (i¹⁰)(i²) = (−1)(−1) = 1

Partners will discuss the strategies and patterns they find and then share with the class. After a class discussion, students will be able to simplify i to any power by finding the nearest multiple of four.

Activity 3: Review of Combining Like Terms (Pairs)

“With your partner, simplify the expressions that are written on the board.”

1. 8 + 4x + 5 + 2x                             13 + 6x

2. 8 + 4x − (5 + 2x)                            3 + 2x

3. (8 + 4x)(5 + 2x)                            40 + 36x + 8x²

“Adding and subtracting complex numbers are similar to combining like terms. You add or subtract the real part and you add or subtract the imaginary part. [IS.16 - All Students] In the three problems you just did, change the x to an i and perform the same operations.”

1. 8 + 4i + 5 + 2i                                13 + 6i

2. 8 + 4i − (5 + 2i)                              3 + 2i

3. (8 + 4i)(5 + 2i)                               40 + 36i + 8i² = 40 + 36i + 8(−1) = 32 + 36i

Activity 4: Pair Work

Have students work individually on the following problems and then have them pair up to check their work. [IS.17 - All Students]

“Add, subtract, and multiply the following pairs of complex numbers. Write your answer in the form a + bi.”

1. 5 + 3i and 4 − 2i

Add: 9 + i
Subtract: 1 + 5i
Multiply: 26 + 2i

2. −6 + 2i and 3 + 4i

Add: −3 + 6i
Subtract: −9 − 2i
Multiply: −26 − 18i

3. 2 − 5i and −7 − 3i

Add: −5 − 8i
Subtract: 9 − 2i
Multiply: −29 + 29i

Students might be wondering why they have not done division of complex numbers. Review with students what it means to rationalize the denominator when there is a square root on the bottom of a fraction. For example, we do not leave the fraction as is. We rationalize the denominator by multiplying the top and bottom by the square root of 3.

Explain that the division of complex numbers is similar to rationalizing the denominator of a fraction.

“We are going to rationalize the denominator that has a complex number in it by multiplying the top and bottom by the denominator’s conjugate. The conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate always produces a real number.”

“For example: (3 + 5i)(3 − 5i) = 9 − 15i + 15i − 25i² = 9 − 25(−1) = 9 + 25 = 34.”

Explain that this property of multiplying conjugates can be used to rationalize the denominator of a fraction made up of complex numbers as follows (write on board):

Activity 5: Additional Pair Work

“With your partner, solve the following two problems. Remember to write your answer in a + bi form.”

Explain that complex numbers have many real-world uses in fields such as electronics, engineering, and physics. An example is the relationship between voltage (V) in volts, impedance or resistance (Z) in ohms, and current (I) in amps in an alternating-current (AC) circuit, expressed in the formula V = ZI. (This real-world application was found in Advanced Algebra (2^nd ed.) by University. of Chicago School Mathematics Project (Scott Foresman, 1996) and the Web page http://www.regentsprep.org/rEGENTS/mathb/2C3/electricalresouce.htm .)

Students will use the formula above to find V when I = 6 + 5i amps and Z = 3 − 2i ohms.

V = ZI

= (3 − 2i)(6 + 5i)

= 18 + 15i − 12i − 10i²

= 18 + 3i − 10(−1)

= 28 + 3i volts

Show students how complex numbers can be graphed on a complex plane. The horizontal axis, which is the x-axis for real numbers, is called the real axis, and the vertical axis, which is the y-axis for real numbers, is called the imaginary axis. A complex number a + bi is plotted at the point (a, b). Model a couple of examples and then have students plot the following complex numbers on a complex plane:

1. 4 + 3i

2. −5 + 2i

3. 4 − 5i

4. −1 − 4i

5. 6i

6. 2 + 0i

Students will also be asked to express the following points from the complex plane as complex numbers:

1. (7, −2)                                             7 − 2i

2. (−3, 8)                                            −3 + 8i

3. (6, 0)                                               6 or 6 + 0i

4. (0, −12)                                            −12i

Activity 6: Solve, Show, and Tell

Place students into groups of four. They are going to convert a problem from standard form to vertex form (by completing the square) and then they need to solve for the imaginary roots. [IS.18 - All Students] When everyone is finished with their problem, they are going to pass that problem clockwise (they hold on to their own work). This pattern will continue until each student has completed all four problems. If a student is stuck on a problem, s/he can ask questions of the person who passed them the problem. Give each student one of the following functions.

1. x² − 6x + 10                        vertex form: (x − 3)² + 1; roots: 3 + i, 3 − i

2. x² − 4x + 20                       vertex form: (x − 2)² + 16; roots: 2 + 4i, 2 − 4i

3. x² + 2x + 5                          vertex form: (x + 1)² + 4; roots: −1 +2i, −1 − 2i

4. x² + 25                                 vertex form: x² + 25; roots: 5i, −5i

Extension:

• In some cases, it is useful to use the imaginary roots of an equation to determine the function. For example, if we know that the roots of a quadratic equation are +i and –i, then (x –(i))(x–(–i)) = 0. Multiply the two binomials, (x –i) (x + i) = 0 and x² + 1 = 0. Use these imaginary roots to find the quadratic equation:

• (i), (–i)                                  [x² + 3 = 0]
• (2 + 3i), (2 –3i)                     [x² –4x +13 = 0]

• Hand out the Lesson 3 Exit Ticket (M-A2-2-3_Lesson 3 Exit Ticket.doc and M-A2-2-3_Lesson 3 Exit Ticket KEY.doc) to evaluate students’ understanding.