Integrated Math II Unit 2 Packet - Radicals and Complex Numbers PDF

GRADE 10 INTEGRATED MATH II UNIT 2 PACKET: RADICALS AND COMPLEX NUMBERS MR. JANUARY (ROOM 200) AND MR. DAGTEKIN (ROOM 213) In this Unit, we will cover the following concepts: *Listing of Perfect Squares and Perfect Cubes *Simplify the powers of i *Listing of Perfect S...

GRADE 10 INTEGRATED MATH II UNIT 2 PACKET: RADICALS AND COMPLEX NUMBERS MR. JANUARY (ROOM 200) AND MR. DAGTEKIN (ROOM 213) In this Unit, we will cover the following concepts: *Listing of Perfect Squares and Perfect Cubes *Simplify the powers of i *Listing of Perfect Squares and Perfect Cube Roots *Operations with Complex Numbers *Classifying Rational and Irrational Numbers *Rationalizing denominators involving i *Properties and Operations of Rational and Irrational Numbers *Graphing complex numbers on a complex plane *Radicals and Rational Exponent Form Conversions *Comparison of Real versus Imaginary/Complex Numbers *Identifying Real, Imaginary, and Complex Numbers INTENTIONALLY LEFT BLANK  UNIT 2 INTEGRATED MATH II AERO COMMON CORE STANDARDS The following AERO Common Core Standards will be covered in this Unit:  UNIT 2 WEEKLY SCHEDULE OF TOPICS AND ASSOCIATED HOMEWORKS NOTE: Unit 2 corresponds with all material found within this packet and the corresponding Delta Math assignments. WEEK 1 TOPICS AND HOMEWORK *Perfect Squares, Perfect Cubes, Perfect Square Roots, Perfect Cube Roots *Classifying Rational and Irrational Numbers and Properties and Operations of Rational and Irrational Numbers (Day 1) [HWK: Unit 2 Homework 1 (2 pages) AND Unit 2 Homework 2 (3 pages)] *Properties and Operations of Rational and Irrational Numbers [HWK: Unit 2 Homework 2 (3 pages)] *Laws of Exponents Rule and Converting between Radical Form and Rational Form (Day 1) [HWK: Unit 2 Homework 3 (4 pages)] WEEK 2 TOPICS AND HOMEWORK *Unit 2 Homework Quiz 1 [NOTE: Unit 2 Homework Quiz 1 covers Unit 2 Homeworks 1,2, and 3] *Identifying between Real, Imaginary, and Complex Numbers [HWK: Unit 2 Homework 4 (1 page)] *Simplifying Powers of i (Day 1) [HWK: Unit 2 Homework 5 (2 pages)] *Simplifying Powers of i (Day 2) [HWK: Unit 2 Homework 5 (2 pages)] NOTE: Any changes to this schedule will be communicated to you in class and/or Schoology!!  UNIT 2 WEEKLY SCHEDULE OF TOPICS AND ASSOCIATED HOMEWORKS WEEK 3 TOPICS AND HOMEWORK *Addition and Subtraction of Complex Numbers [HWK: Unit 2 Homework 6 (1 page)] *Unit 2 Quiz 1 [NOTE: Unit 2 Quiz will cover ALL Unit 2 concepts up to Adding and Subtracting Complex Numbers] *Multiplication of Imaginary and Complex Numbers (Day 1) [HWK: Unit 2 Homework 7 (1 page)] *Division of Complex Numbers [HWK: Unit 2 Homework 8 (1 page)] WEEK 4 TOPICS AND HOMEWORK *Graphing Complex Numbers [HWK: Unit 2 Homework 9 (4 pages)] *Unit 2 Review Problems for Summative Assessment on Complex Numbers [HWK: Unit 2 Review Problems for Summative Assessment on Complex Numbers (6 pages)] *Unit 2 Test 1 Review [NOTE: Unit 2 will cover ANY and ALL concepts from Unit 2’s concepts.] *Begin Unit 3 Concepts NOTE: Any changes to this schedule will be communicated to you in class and/or Schoology!!  REVIEW TOPIC: LIST OF PERFECT SQUARES AND CUBES LIST THE PERFECT SQUARES FOR WHICH YOU WILL BE RESPONSIBLE: LIST THE PERFECT CUBES FOR WHICH YOU WILL BE RESPONSIBLE:  REVIEW TOPIC: LIST OF PERFECT SQUARE ROOTS AND CUBE ROOTS LIST THE PERFECT SQUARE ROOTS FOR WHICH YOU WILL BE RESPONSIBLE: LIST THE PERFECT CUBE ROOTS FOR WHICH YOU WILL BE RESPONSIBLE: CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS – CONCEPTUAL EXPLANATION CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS- VENN DIAGRAM AND CHART  CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS PRACTICE PROBLEMS Directions: Classify the given numbers by placing them in the appropriate areas in the Venn diagram. 1.)  CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS PRACTICE PROBLEMS (CON’T) Directions: Classify the given numbers by placing them in the appropriate areas in the Venn diagram.  CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS UNIT 2 HOMEWORK 1 (PAGE 1) Directions: Classify the given numbers by placing them in the appropriate areas in the Venn diagram.  CLASSIFYING RATIONAL AND IRRATIONAL NUMBERS UNIT 2 HOMEWORK 1 (PAGE 2) Directions: Classify the given numbers by placing them in the appropriate areas in the Venn diagram. Properties and Operations of Rational and Irrational Numbers – Conceptual Explanation  Properties and Operations of Rational and Irrational Numbers Practice Problems Directions: Given the operations below, simplify the expressions below.  Properties and Operations of Rational and Irrational Numbers Unit 2 Homework 2 (Page 1) Directions: Choose the best answer for each question below.  Properties and Operations of Rational and Irrational Numbers Unit 2 Homework 2 (Page 2) Directions: Choose the best answer and given an explanation where requested.  Properties and Operations of Rational and Irrational Numbers Unit 2 Homework 2 (Page 3) Directions: Choose the best answer and given an explanation where requested.  Laws of Exponents Conceptual Review This table is an overview of the different laws/rules of exponents. Radicals and Rational Exponents Conceptual Explanation Radicals and Rational Exponents Conceptual Explanation (Con’t)  Converting from Radical Form to Rational Form Practice Problems Directions: Convert the following radical expressions to rational form.  Converting from Rational Form to Radical Form Practice Problems Directions: Convert the rational exponent expressions to radical form.  Converting Between Radical Form and Rational Form Unit 2 Homework 3 (Page 1) Directions: Choose the best answer for the given questions.  Converting Between Radical Form and Rational Form Unit 2 Homework 3 (Page 2) Directions: Choose the best answer for the given questions.  Converting Between Radical Form and Rational Form Unit 2 Homework 3 (Page 3) Directions: Choose the best answer for the given questions.  Converting Between Radical Form and Rational Form Unit 2 Homework 3 (Page 4) Directions: Choose the best answer for the given questions. Comparison of Real vs. Complex Numbers Review Charts Real vs. Complex Numbers Conceptual Explanation Identifying Real and Imaginary Components of Complex Numbers Practice Problems Identifying Real and Imaginary Components of Complex Numbers Practice Problems (Con’t) Identifying Real and Imaginary Components of Complex Numbers Practice Problems (Con’t)  Identifying Real and Imaginary Components of Complex Numbers Unit 2 Homework 4 Directions: Simplify each problem below; then, identify the real and imaginary parts of your answers.  Simplifying Powers of i Conceptual Explanation Performing operations on complex numbers requires multiplying by i and simplifying powers of i. By definition, i = the square root of –1, so i 2 = –1. If you want i 3, you compute it by writing i 3 = i 2 x i = –1 x i = –i. Also, i 4 = i 2 x i 2 = (–1)(–1) = 1. And then the values of the powers start repeating themselves, because i 5 = i, i 6 = –1, i 7 = –i, and i 8 = 1. So, what do you do if you want a higher power, such as i 345, or something else pretty high up there? You don’t want to have to write out all the powers up to i 345 using the pattern. Instead, use the following rule. To compute the value of a power of i, determine whether the power is a multiple of 4, one more than a multiple of 4, two more than, or three more than a multiple of 4. Then apply the following: i 4n = 1 i 4n+1 = i i 4n+2 = –1 i 4n+3 = –i NOTE: i0 = 1 i1 = i i2 = −1 i3 = −i You may NOT leave powers of i in your answer. After i3, the pattern start repeating, meaning that i4 = i0, i5 = i1… etc. Simplifying Powers of i Conceptual Circle Simplifying Powers of i Practice Problems Simplifying Powers of i Practice Problems (Con’t) Simplifying Powers of i Unit 2 Homework 5 (Page 1) Simplifying Powers of i Unit 2 Homework 5 (Page 2)  Operations with Complex Numbers Conceptual Explanation All operations on complex numbers are exactly the same as you would do with variables… just make sure there is no power of i in your final answer. Your answer should be in a + bi form. Addition and Subtraction of Complex Numbers Conceptual Explanation Addition and Subtraction of Complex Numbers Practice Problems  Addition and Subtraction of Imaginary and Complex Numbers Unit 2 Homework 6 Directions: Simplify the expressions below. Be sure to show your work!! Multiplication of Complex Numbers Conceptual Explanation  Multiplication of Complex Numbers Practice Problems Directions: Simplify the expressions below. Be sure to show your work!!  Multiplication of Complex Numbers Practice Problems (Part 2) Directions: Simplify the expressions below. Be sure to show your work!!  Multiplication of Complex Numbers Unit 2 Homework 7 Directions: Simplify the expressions below. Be sure to show your work!! Division of Complex Numbers Conceptual Explanation  Division of Complex Numbers by Rationalizing Denominators Conceptual Explanation Rationalizing Denominators containing i Just as you are not allowed to leave a radical in a denominator of a fraction, you are not allowed to leave an i in the denominator. This is because i is a radical!!  Division of Complex Numbers Practice Problems Directions: Simplify the expressions below. Be sure to show your work!!  Division of Complex Numbers Practice Problems (Part 2) Directions: Simplify the expressions below. Be sure to show your work!!  Division of Complex Numbers Unit 2 Homework 8 Directions: Simplify the expressions below. Be sure to show your work!!  Graphing Complex Numbers Conceptual Explanation Due to their unique nature, complex numbers cannot be represented on a normal set of coordinate axes. In 1806, J. R. Argand developed a method for displaying complex numbers graphically as a point in a coordinate plane. His method, called the Argand diagram, establishes a relationship between the x-axis (real axis) with real numbers and the y-axis (imaginary axis) with imaginary numbers. In the Argand diagram, a complex number a + bi is the point (a,b) or the vector from the origin to the point (a,b). Graph the complex numbers: 1. 3 + 4i 2. 2 - 3i 3. -4 + 2i 4. 3 (which is really means ) 5. 4i (which is really means ) The Parallelogram Rule for Complex Addition The parallelogram rule for complex addition says that if you are adding two complex numbers, then the sum of can be represented by the diagonal of the parallelogram that can be drawn using the two original vectors as adjacent sides. (1+4i)+(5+i)=6+5i Graphing Complex Numbers Conceptual Graphic  Graphing Complex Numbers Practice Problems Add 3 + 3i and -4 + 2i graphically. Subtract 3 + 4i from -2 + 2i Graphing Complex Numbers Practice Problems (Part 2)  Graphing Complex Numbers Practice Problems (Part 3) Find the Complex Number for each graph shown.  Graphing Complex Numbers Unit 2 Homework 9 Directions: Graph the Complex Numbers on the Complex Coordinate Plane.  Graphing Complex Numbers Unit 2 Homework 9 (Part 2) Directions: Graph the Complex Numbers on the Complex Coordinate Plane.  Graphing Complex Numbers Unit 2 Homework 9 (Part 3) Directions: Add or Subtract the Complex Numbers Graphically. Represent the solution on the graph as well and be sure to draw the resulting parallelogram. 1. (3 + 4i) + (5 – 3i) = 2. (2+ 3i) – (4 – 2i) = 3. (2 – 4i) – (-2 + 3i) = 4. (5 + 6i) + (4 – 2i) = 5. (3+ 8i) – (12 – 10i) = 6. (1 – 11i) – (-4 + 9i) =  Graphing Complex Numbers Unit 2 Homework 9 (Part 4) Directions: Find the Complex Number for each graph shown. Unit 2 Review Graphics on Complex Numbers Unit 2 Review Problems for Summative Assessment on Complex Numbers Unit 2 Review Problems for Summative Assessment on Complex Numbers (Part 2)  Unit 2 Review Problems for Summative Assessment on Complex Numbers (Part 3) 2. 3. gfgdg  Unit 2 Review Problems for Summative Assessment on Complex Numbers (Part 4) 5. 6. 7. fdfdf 8. 9. 10. 11. 12. 13. 14. Unit 2 Review Problems for Summative Assessment on Complex Numbers (Part 5) 15. 16. 17. 18. 19. 20. 21.22. Unit 2 Review Problems for Summative Assessment on Complex Numbers (Part 6) 23. 24.