Objectives

This lesson shows students how much more helpful the vertex form is than the standard form. Students will: - comprehend that a parabola has roots that are real, imaginary, and complex. - be able to apply the four basic operations to complex numbers. - be able to plot complex numbers on the complex plane.

Core Questions

- How can graphs and/or tables that show quadratic equations help us understand what's going on in the world?

Vocabulary

- Root: The values of x for which the equation f(x) = 0 are true.  - Zero of a Function: The roots of a function; the x-values at which the function crosses the x-axis. - Quadratic Formula: A formula that can be used to find the roots to any quadratic equation of the form y = ax² + bx + c; the quadratic equation is \( x = -b \pm { \sqrt{b^2-4ac} \over 2a} \). - Quadratic Function: A function of the form f(x) = ax² + bx + c, where a, b, and c are real numbers. - Complex Plane: The two-dimensional plane in which the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part of a complex number. - Conjugate: For a complex number of the form a + bi, the conjugate is the number a – bi; similarly for a complex number of the form a - bi, the conjugate is the number a + bi. - Fractal: An object or quantity that displays self-similarity; when “zooming in” on a geometric fractal, for instance, each smaller part of the fractal displays the same geometric type as the whole. - Irrational Conjugate Pairs: Two numbers, one of the form a + b and one of the form a – b, where at least one of a or b is an irrational number. - Discriminant: For a quadratic function of the form y = ax² + bx + c, the discriminant is equal to b² – 4ac. - Complex numbers: A number of the type a + bi, where i = \(\sqrt{-1}\) and a and b are real numbers. - Imaginary number: The representation of \(\sqrt{-1}\) as i.

Materials

- paper - pencils - rulers - 3 by 5-inch index cards - graph paper - graphing calculator - copies of Draw the Tree activity sheet (M-A2-2-3_Draw the Tree)  - copies of Isosceles Tree activity sheet (M-A2-2-3_Isosceles Tree) - Lesson 3 Exit Ticket and KEY (M-A2-2-3_Lesson 3 Exit Ticket and KEY)

Assignment

- Pair work on complicated numbers with partners must be assessed to ensure that foundational abilities are present. The discussion between partners should stay on topic, and both partners should listen to each other's questions and observations. Tell students that when you check their work, you should go backward from their answers to find where they went wrong. Group exercises are most effective when the teacher participates in the discussions and asks questions and challenges about student observations. In class discussion, when a student makes an incorrect statement, ask them leading questions. - The Lesson 3 Exit Ticket (M-A2-2-3_Lesson 3 Exit Ticket and KEY) tests how well students understand how to multiply and divide complicated numbers.

Supports

Active Engagement, Modeling, Explicit Instruction W: This lesson will teach students how to use imaginary numbers to solve quadratic problems that do not have real answers. Students will be able to draw a complex number on the complex plane and learn about the set of complex numbers, which is made up of real and imaginary numbers. After that, students will know how to use complex conjugates to divide complex numbers and simplify them in fraction form, as well as the four basic operations of math. You will be able to solve quadratic equations with no real answers after this lesson. You will also understand the properties of complex numbers and be able to use them to solve problems. Finally, you will be able to draw complex numbers.  H: To find the x-intercepts of y = (x - 2)² + 9, students will need to use both math and graphs to find answers. The mathematical answers will test how well they can change expressions and make substitutions. They will better understand the quadratic function after seeing it in a graph.  E: Students will have to come up with a new way to understand how a negative number times itself can be written as a positive number to find a useful interpretation of the square roots of –25 and –16. To get them to think in a new way, they need to understand what it means to write i as the square root of -1.  R: In Activity 5, the kids will show and tell what they know. The first thing they have to do is translate between standard form and vertex form. Then they have to solve for the imaginary roots and hand their work to the next student until all four parts of the task are done. Students can judge the work of their classmates and think again about what they know about quadratic functions through this game.  E: The Lesson 3 Exit Ticket has students draw a picture of the graph of the quadratic function to show how they understand the link between the a, b, and c terms of the function. Along with that, it tests how well they can use multiplication and division to show complex numbers.  T: Let students who are having trouble finding the vertex of the parabola of a quadratic function use a graphing computer to see how the location of the vertex changes over time. Start with y = x² and move on to y = x² - 3x, y = x² + 3x, and y = (x - 3)². In writing, have each student explain how the parabola changes with each change in the function. Use a, b, and c terms that are reasonable numbers written as fractions and decimals for smarter students.  O: This lesson connects new concepts to past knowledge. The first part of the lesson teaches students how to find the area under a curve. Then, complicated numbers are introduced by saying that students will need them to find the area under a curve more accurately at times. A parabola has x-intercepts even if it doesn't touch the x-axis. This is taught to the students. Even though the x-intercepts are made up, they let us use algebra to solve any quadratic problem. Science teachers show their students how to use imaginary numbers in certain areas. This shows students that there is a reason to learn this information.

Procedures

Example 1: "How can we calculate the dimensions (base, height, and hypotenuse) of a right triangle if we know the area and how the dimensions relate to one another? For example, what are the dimensions of a right triangle with a height that is 7 units greater than its base and an area of 30 square units? Because we know that the height is 7 units longer than the base, let x = base length and x + 7 = height length. The area of the triangle is bh. Then x(x + 7) = 30." "Because the dimensions of a triangle must be positive, the base is 5 units, and the height is 12 units (5 + 7). According to the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the two sides." 5² + 12² = c² c² = 25 +144 = 169 c = 13 units "We can create an infinite number of right triangles with infinitely different bases, heights, and hypotenuses. Fractals are an interesting and crucial concept related to all of these right triangles." "Fractals are rough or fragmented geometric shapes that can be divided into parts, each of which is a smaller version of the total. They are generally easier to demonstrate than explain." Here is an example of one called a Sierpinski triangle: <a href="http://en.wikipedia.org/wiki/Sierpi%C5%84ski_Triangle">http://en.wikipedia.org/wiki/Sierpi%C5%84ski_Triangle</a>  "Observe how the smaller triangles repeat themselves inside the first triangle with equal sides." Activity 1: Drawing a Pythagoras Tree Distribute copies of the Draw the Tree activity sheet (M-A2-2-3_Draw the Tree). Students will require a pencil with a good eraser, a ruler, and a protractor. Note: Activities requiring students to hand draw can be incredibly beneficial in assisting students in creating meaningful representations of the mathematical processes at work in right triangles and fractals. Encourage students to retry and make new drawing attempts if they are having problems getting started. Better outcomes will come soon with a little practice. Show students sketches of different variations of the Pythagoras Tree: <a href="http://mathworld.wolfram.com/PythagorasTree.html">http://mathworld.wolfram.com/PythagorasTree.html</a>  "Three squares surround the right triangle ABC. The area of square ABKJ is the square of the hypotenuse AB; the area of square BCGH is the square of side CB; and the area of square ACED is the square of side AC." "To design the Pythagoras Tree using the right triangles that form the fractal, ensure that all triangles are similar to triangle ABC. To draw them correctly, all matching angles must be congruent." "Begin by finding a new point, F, outside the square ACED, between points D and E. Use your protractor to determine the angle CAB. Measure this angle at least twice to ensure an accurate measurement. Now, with the vertex of your protractor on point D, lightly mark the same angle CAB with a point. Draw a line from that point to point D. Next, use your protractor to determine the angle ABC, measuring at least twice to ensure accuracy. Place the vertex of your protractor at point E, and lightly indicate the line that will intersect with the line you drew from point D. Mark the intersection point F. Triangle DFE, if measured and drawn correctly, should resemble triangle ABC." "Now, repeat the procedure to find another point I outside square BCGH above side GH. Mark the same angle as angle CAB using your protractor's vertex at point G. Lightly extend the line from point G through your angle measurement mark. Next, with your protractor's vertex at point H, mark the same angle as angle ABC. Extend that line to cross with the line from point G, and label it as point I. Triangle GHI should resemble triangles ABC and DFE." "To continue the fractal pattern, draw two more squares, each adjacent to triangles DFE and GHI. Start by determining the length of side IH. Measure this length in millimeters using your ruler's centimeter scale. Measuring lengths to the nearest millimeter is significantly easier and more precise than using fractions of an inch. The length of IH determines the dimensions of the other three sides of the square next to triangle GHI. Place your protractor on side IH, with the vertex at point I, and mark 90° from point I. Repeat the 90° measurement from point H. Draw two perpendicular lines from points I and H. Use your ruler to mark the same lengths between points I and H as length IH. Repeat the process to draw the next square adjacent to side FE of triangle DFE." "As you continue to draw identical triangles and adjacent squares, you will notice the fractal pattern of the Pythagoras Tree develop. You can keep drawing the pattern for as long as you have space on your paper." To complete the procedure as quickly as possible, try using large sheets of wrapping paper or butcher paper. Drawing the pattern on paper will take approximately five or six repetitions to see the squares and right triangles budding into a tree. Visit this website that allows the user to manipulate the sizes of the right triangles: <a href="http://www.ies.co.jp/math/java/geo/pytree/pytree.html">http://www.ies.co.jp/math/java/geo/pytree/pytree.html</a>  Take a look at this website to see some Pythagoras Trees of different sizes in triangles: <a href="http://www.wolframalpha.com/input/?i=Pythagoras+tree">http://www.wolframalpha.com/input/?i=Pythagoras+tree</a>  Ask students to participate in 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 This activity allows students to instantly view the solution to the follow-up question.  Distribute copies of the Isosceles Tree activity sheet (M-A2-2-3_Isosceles Tree). Ensure that each student has one 3-by-5-inch index card, many pieces of plain paper, and a ruler with a centimeter scale. Students will fold the index card into a simple T-square to help them draw correct and 45° angles. Demonstrate to students while explaining each step. "To make the fold, hold the index card in portrait position (3 inches horizontal and 5 inches vertical). Fold the upper right corner across and to the left so that the top edge is exactly aligned with the left edge. Try to produce a precise alignment and a sharp crease from the upper left corner to the fold's edge." Folding along the crease in both directions (front and rear) allows students to use the 45° angle from either side. Instruct students to orient their sketching paper in the landscape direction (8-inch side vertical and 11 ½-inch horizontal). Ask them to draw a 5-cm square in the center of the of the bottom of their paper. To do this, have them draw one right angle and then use a ruler to measure precisely 5 cm along the vertical and horizontal lines. "Mark each length in pencil with a little endpoint. Place the index card's right angle corner on one of the endpoints now, draw another right angle, and measure 5 centimeters to create the third side. Complete the square by drawing the final right angle vertex, then measure the sides to ensure they are all 5 centimeters long and each angle is right." If the square is not drawn correctly, tell the students that they should start over and repeat the instructions. The drawing will turn out better if the initial square is more exact. Students will now build a 45-45-90 triangle on top of their square, using the top border of the square as the hypotenuse. Each student should fold their 45-degree index card so that the corner and edge of the 45° angle match the corner and top edge of the square. Then, have them trace down the edge of the index card to form a 45° angle. Repeat the process on the square's other side, resulting in two 45° angles extending from the top. Make sure the students extend the 45° angles far enough so that the lines connect. Students should delete any overextending lines to create a tidy 45-45-90 triangle atop the original square. The following phase will be to build a smaller square that adjoins the new triangle. The four sides of this new square will be congruent with the two sides opposite the hypotenuse of the triangle. From there, students will use the ruler to measure one of the two sides. Its length is about 3.5 cm. Students should begin by inserting the right angle of an index card at the upper right vertex of the original 5-centimeter square. Then they'll draw the right angle from this point, measure 3.5 centimeters, and mark the spot. Repeat by drawing the right angle of the index card on the new point and measuring the next side of the 3.5-centimeter square. Students should draw a line from the fourth side of the square to the triangle's apex. Repeat the cycle, drawing a new isosceles right triangle on top of each new square. “What are the x-intercepts of y = (x - 2)² + 9?” Divide the class in half. Have one half solve for the x-intercepts algebraically, while the other half solve for the x-intercepts graphically. When solving algebraically, students should substitute 0 for y and solve for x. They should substract 9 from both sides and then recognize that they cannot take the square root of a negative integer. Students solving it graphically will notice that it does not intersect the x-axis.  The phrases solution, root, x-intercept, and answer all mean the same thing. "Why is it impossible to calculate a negative number's square root?" Go over what it means to square a number. "A positive number multiplied by itself results in a positive product. Multiplying a negative number by itself produces a positive result."  Present the graphs for y = (x - 1)² and y = (x + 3)² − 4 to the students. "How many x-intercepts are there in each of these graphs?" There is one in the first parabola and two in the second. "Parabolas have three possible x-axis hits: twice, once, or not at all. However, the parabola of y = (x - 2)² + 9 does have two imaginary x-intercepts, even though the graph does not intersect the x-axis." Start with a simple quadratic equation, like x² = -16 and ask students to find the solution of the equation. Stress that it’s not that there are no solutions, but that there are no real number solutions. In actuality, there are two imaginary number solutions. Let's introduce the imaginary number i. Ask students to record the following information in their notes: i is the imaginary number defined to be equal to \(\sqrt{-1}\). For any real number k > 0, it follows that \(\sqrt{-k}\) = \(\sqrt{k}\)\(\sqrt{-1}\) = ±\(\sqrt{k}\)(i). Example: \(\sqrt{-25}\) = \(\sqrt{25}\)\(\sqrt{-1}\) = ± 5i. Ask students to solve x² = −16. x = \(\sqrt{-16}\) x = \(\sqrt{16}\)\(\sqrt{-1}\) x = ±4i Make the following simpler for the students, and have them write the solutions in their notes.  1. \(\sqrt{-4}\)          2. \(\sqrt{-81}\)          3. \(\sqrt{-36}\)           4. \(\sqrt{-100}\) 5. \(\sqrt{-64}\)        6. \(\sqrt{-9}\)            7. \(\sqrt{-49}\)           8. \(\sqrt{-121}\) Key: 1. ±2i   2. ±9i   3. ±6i   4. ±10i   5. ±8i   6. ±3i   7. ±7i   8. ±11i Go over the quadratic formula with the students: \( x = {-b \pm \sqrt{b^2-4ac} \over 2a} \) Practice finding the x-intercepts for the following: 1. x² + 6x + 5 \( x = {-6 \pm \sqrt{6^2-4(1)(5)} \over 2(1)} \) = \({-6 \pm \sqrt{16} \over 2} \) = \({-6  \pm 4  \over 2} \) = -5, -1 2. x² + x – 6 \( x = {-1 \pm \sqrt{1^2-4(1)(-6)} \over 2(1)} \) = \({-1 \pm \sqrt{25} \over 2} \) = \({-1  \pm 5  \over 2} \) = 2, -3 3. Our original parabola (x - 2)² + 9 = x² – 4x + 13 \( x = {4 \pm \sqrt{(-4)^2-4(1)(13)}\over 2(1)} \) = \({4 \pm \sqrt{-36}\over 2} \) = \({4 \pm \sqrt{36} \sqrt{-1}\over 2} \) = \({4  \pm 6i  \over 2} \) = \(4 \over 2 \) ± \(6i \over 2 \) = 2 ± 3i Instruct students to add to their notes: Any integer in the pattern a + bi, where i is the imaginary number and a, and b are real numbers, is a complex number. Complex numbers are sets of real and imaginary numbers, and they are real if b = 0 and imaginary if b0 . The complex number bi is frequently referred to as a pure imaginary number when a = 0 and b0. (The following definitions are from Holt Algebra 2, 2004.) University of Chicago School Mathematics Project, Advanced Algebra (2nd Ed., Scott Foresman, 1996) defines a + bi as an imaginary number when a =0 and b0 as a nonreal number when b0 . Justify the simplification of all powers of i. Students should simplify the powers of i up to the 12th power in pairs. \(i^1\) = i \(i^2\) = √-1√-1 = -1 \(i^3\) = (i)(\(i^2\)) =(i)(−1) = −i \(i^4\) = (\(i^2\))(\(i^2\)) = (−1)(−1) = 1 \(i^5\) = (i)(\(i^4\)) = (i)(1) = i \(i^6\) = (\(i^4\))(\(i^2\)) = (1)(−1) = −1 \(i^7\) = (i)(\(i^6\)) = (i)(−1) = −i \(i^8\) = (\(i^4\))(\(i^4\)) = (1)(1) = 1 \(i^9\) = (\(i^5\))(\(i^4\)) = (i)(1) = i \(i^10\) = (\(i^6\))(\(i^4\)) = (−1)(1) = −1 \(i^11\) = (\(i^10\))(i) = (−1)(i) = −i \(i^12\) = (\(i^10\))(\(i^2\)) = (−1)(−1) = 1 After discussing the strategies and patterns they discover in pairs, partners will present their findings to the class. Students will be able to simplify i to any power by finding the nearest multiple of four following a class discussion. Activity 3: Review of Combining Like Terms (Pairs) "Simplify the expressions written on the board with your partner." 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 can add or subtract both the real and imaginary parts. In the three problems you just completed, change the x to an i and repeat the actions." 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 Students should work separately on the following problems before pairing up to double-check their answers.  "Add, subtract, and multiply the following pairs of complex numbers. Write your response in the format 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 may wonder why they have not divided complex numbers. Discuss with students what it means to rationalize the denominator when a fraction contains a square root. For instance, we do not leave the fraction \(2 \over {\sqrt{3}}\) unchanged. To justify the denominator, multiply the top and bottom by the square root of three. \(2 \over {\sqrt{3}}\)•\({\sqrt{3}} \over {\sqrt{3}}\) = \({2\sqrt{3}} \over 3\) Explain how dividing complex numbers is equivalent to reasoning about the denominator of a fraction. "We will rationalize the complex denominator by multiplying the top and bottom by their conjugate. The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate always yields a real number." “For example: (3 + 5i)(3 − 5i) = 9 − 15i + 15i − 25i² = 9 − 25(−1) = 9 + 25 = 34.” Explain how the property of multiplying conjugates can be used to rationalize the denominator of the following complex number fraction (write on board):<figure class="image"><img style="aspect-ratio:448/47;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_41.png" width="448" height="47"></figure>Activity 5: Additional Pair Work “Work with your partner to solve the following two problems: Remember to write your answer in a + bi form.”<figure class="image"><img style="aspect-ratio:448/191;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_42.png" width="448" height="191"></figure> Explain that complex numbers have several practical applications in domains such as electronics, engineering, and physics. The formula V = ZI describes the relationship between voltage (V) in volts, impedance or resistance (Z) in ohms, and current (I) in amps in an alternating-current (AC) circuit. (This real-world application can be found in Advanced Algebra (2nd ed.) by the University of Chicago School Mathematics Project (Scott Foresman, 1996) and on the website <a href="http://www.regentsprep.org/rEGENTS/mathb/2C3/electricalresouce.htm">http://www.regentsprep.org/rEGENTS/mathb/2C3/electricalresouce.htm</a>.) 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 plotted on a complex plane. The horizontal axis, or x-axis for real numbers, is known as the real axis, and the vertical axis, or y-axis for real numbers, is known as the imaginary axis. A complex number, a + bi, is plotted at the point (a, b). Model a few examples, and then ask them to 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 Additionally, the students must express the following complex plane points 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 Group students into fours. They will convert a problem from standard to vertex form (by completing the square), and then solve for the imaginary roots.  When everyone has finished their problem, they will pass it clockwise (holding on to their own work). This pattern will continue until each student has solved all four issues. If a student is having difficulty with a problem, he or she might ask the person who assigned it to them. Give each student one of the functions listed below. 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 circumstances, determining the function by using an equation's imaginary roots might be advantageous. For example, if we know the roots of a quadratic equation are +i and -i, we can write (x -(i))(x-(-i)) = 0. Multiply the two binomials: (x - i) (x + i) = 0 and x² + 1 = 0. To find the quadratic equation, use the following imaginary roots: (i), (–i)                                      [x² + 3 = 0] (2 + 3i), (2 –3i)                         [x² –4x +13 = 0] Disperse the Lesson 3 Exit Ticket (M-A2-2-3_Lesson 3 Exit Ticket and KEY) to assess the students' comprehension.

Layla Nguyen

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Complex Numbers and the Quadratic Formula (M-A2-2-3)

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Layla Nguyen

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Description

This lesson shows students how much more helpful the vertex form is than the standard form. Students will:
- comprehend that a parabola has roots that are real, imaginary, and complex.
- be able to apply the four basic operations to complex numbers.
- be able to plot complex numbers on the complex plane.

Grade

Grade 11 Grade 12