* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download McREL Technology Solutions (MTS) Lesson Plan Template
Survey
Document related concepts
Mathematical model wikipedia , lookup
Mathematics wikipedia , lookup
History of mathematics wikipedia , lookup
Mathematics and architecture wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Ethnomathematics wikipedia , lookup
Elementary mathematics wikipedia , lookup
Vector space wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Classical Hamiltonian quaternions wikipedia , lookup
Euclidean vector wikipedia , lookup
Bra–ket notation wikipedia , lookup
Transcript
Common Core Standards Template Domain: The Complex Number System Cluster: Perform arithmetic operations with complex numbers. Class Standards Precalculus (Q3) Notes N.CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. A moduli is “The positive square root of the sum of the squares of the real and imaginary parts of a complex number.” Or a 2 b2 . Student Friendly Language: “I can…” I can determine the absolute value of a complex number. I can divide complex numbers and simplify the quotient. I can determine the conjugate of a complex number. Know (factual) Understand (conceptual) a + bi is the standard form of a complex number, where a is the real part and b describes the imaginary part. |a + bi| = a b . a + bi and a – bi are conjugates. 2 2 The moduli of a complex number refers to the distance the value is away from the origin on the coordinate plane. To find the quotient of complex numbers, we must multiply by the conjugate of the denominator/divisor on both the numerator and denonminator. Do (procedural, application, extended thinking) Use the distance formula to determine the moduli of a point. Instructional Strategies, Applications and Resources: Connect the complex number system to vectors, magnitude, and finding conjugates of square roots by comparing and contrasting these topics. Use a graphic organizer for all of the new vocabulary that will be introduced this chapter. Include definitions, pictures, page reference numbers, an example, a place for side notes. Student Misconceptions Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Multiply by the conjugate of the numerator to simplify quotients of complex numbers. Critical Questions/Problems: Questions that define the standard and are representative of what students will need to be able to do to show evidence of understanding. Closed Problems or Questions: Open Problems or Questions: Divide, then write your answer in standard form: 3 4i Write the conjugate: 16 3 1 16 i b) a) 16 4 3 4i 3 1 3 4i i d) c) 16 4 16 6 10i . 2i Evaluate | 7 – 3i | a. Evaluate: |4 + 3i| 58 b. 4 c. 2 d. -58 Divide, then write your answer in standard form. 4 i 1 4i 8 8 i b. i c. i d. i a. 17 17 Divide, then write your answer in standard form. 3 2i 4 3i Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Domain: The Complex Number System Cluster: Represent complex numbers and their operations on the complex plane Class Precalculus (Q3) Standards Notes N.CN.4 Represent complex numbers on the complex plane in rectangular There are an infinite number of options for the polar form of a number given in rectangular form. 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 Student Friendly Language: “I can…” I can plot complex numbers in rectangular and polar form. I can convert a point in rectangular form into point in polar form. I can convert a point in polar form into a point in rectangular form. Know (factual) Understand (conceptual) Complex numbers in rectangular form are written a + bi, and plotted with the real part, a, on the x-axis, and the imaginary part, b, on the y-axis. The point would have coordinates (a, b). The polar form of a point is written r, , where r is the You can have an infinite number of polar coordinates for a rectangular point, simply by adding or subtracting 360 / 2 from . Do (procedural, application, extended thinking) To convert from polar to rectangular complex points, use x r cos y r sin To convert from rectangular to polar complex points, use moduli of the r 2 x2 y 2 number, and theta is the angle of the number. tan y x Instructional Strategies, Applications and Resources: Use a graphing calculator that has apps that can do conversion between the two forms. Meaningful student connections: explain why they use the variable “r” in polar form. Teach using “big math ideas.” Create flow chart for conversion of polar and rectangular points. Graphic organizer for new vocabulary. Student Misconceptions Using the wrong r value, not understanding what a negative r value really means and which positive or negative r values. goes with the They also don’t think of both forms as points rather than vectors, and sometimes draw a line from the origin to the point. Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Critical Questions/Problems: Questions that define the standard and are representative of what students will need to be able to do to show evidence of understanding. Closed Problems or Questions: In polar coordinates, which of the following is not a correct representation for the point 2, 6 b) 2, 11 6 c) 2, 11 d) 2, 7 6 6 2, 5 6 ? Open Problems or Questions: Convert from rectangular to polar coordinates: 6, 2 a) Convert from polar to rectangular coordinates: 8, 7 6 Plot -3 + 4i in the complex plane. 4 3, 4 b) 4, 4 3 c) 4, 4 3 d) 4, 4 3 a) Calculate | 5 + 12i |. Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Domain: Vector Quantities and Matrices Cluster: Represent and model with vector quantities. Standards Class Precalculus (Q3) Notes N.VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). Student Friendly Language: “I can…” I can find the magnitude of a vector. I can find the direction of a vector. I can draw a vector on the coordinate plane. Know (factual) Magnitude ( Understand (conceptual) v ) of a vector, a, b , is the length of the vector. The formula for finding magnitude is v a 2 b2 . Direction of a vector is found using the formula tan Vectors are equivalent if their direction and magnitude are equal, regardless of their position on the coordinate plane. There are 2 parts to a vector: direction and magnitude. Do (procedural, application, extended thinking) Determine the magnitude of a vector. Use a variety ways to represent a function. Compare two vectors to determine if they are equivalent. y . x Instructional Strategies, Applications and Resources: Student Misconceptions Two vectors are equivalent if they share the same direction or magnitude, not both. Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Critical Questions/Problems: Questions that define the standard and are representative of what students will need to be able to do to show evidence of understanding. Closed Problems or Questions: Find the direction of v : v = 7i – 2j a. 344o b. 16o c. 164o d. 196o Open Problems or Questions: Find the direction of v: v = 1, 4 . Convert the vector v into component form: v = 4i + 2j a. 2, 4 b. 4, 0 c. 0, 2 d. 4, 2 Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Domain: Vector Quantities and Matrices Cluster: Represent and model with vector quantities Standards Class Precalculus (Q3) Notes N.VM.2. Find the components of a vector by subtracting the coordinates of Head-minus tails rule. an initial point from the coordinates of a terminal point. Student Friendly Language: “I can…” I can identify the head of a vector. I can identify the tail of a vector. I can use the head-minus-tail rule to find a vector in component form. Know (factual) If the initial point of vector v is x1 , y1 and the terminal point of v is Understand (conceptual) The head of the vector is the end with the arrowhead on it. The tail does not have any marking on it. Do (procedural, application, extended thinking) Be able to determine if two vectors are equivalent vectors by performing the head-minus-tail rule. x2 , y2 , then the component form of v x2 x1 , y2 y1 . Instructional Strategies, Applications and Resources: 2-Column Notes used to show equivalent vectors. Student Misconceptions Two vectors aren’t the equivalent unless they have the same initial and terminal point. Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Critical Questions/Problems: Questions that define the standard and are representative of what students will need to be able to do to show evidence of understanding. Closed Problems or Questions: A vectorv has initial point (3, 7) and terminal point (3, -2). Find its component form. a. 0, 9 b. 9, 0 c. 0, 9 d. Open Problems or Questions: Find the component form of the vector below. 9, 0 Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Domain: Vector Quantities and Matrices Cluster: Represent and model with vector quantites. Standards Class Precalculus Notes N.VM.3. Solve problems involving velocity and other quantities that can be represented by vectors. Student Friendly Language: “I can…” I can determine the actual velocity of an airplane when factoring in the velocity of the wind. I can find a resultant vector. I can determine the speed and direction of any vector. Know (factual) Understand (conceptual) The sum of 2 vectors is called the resultant. A vector is composed of speed and direction. Speed is the magnitude of the vector, and direction is the direction angle of the vector when in standard position. The sum of two vectors represents the effect of 2 or more forces acting on an object. The graph of the sum of 2 or more vectors would represent the single vector that starts at the origin and ends at the head of the second vector. Do (procedural, application, extended thinking) Determine the speed and compass heading a pilot should set an airplane at in order to arrive at the correct destination when figuring in the force of wind. Determine the resultant force on an object if 2 or more forces are pushing or pulling on that object. Instructional Strategies, Applications and Resources: Use sequencing for multi-step problems. KWL process. Student Misconceptions Compass headings and direction angles are not the same. The calculator gives the direction angle, and must be converted into a compass heading. Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Template Critical Questions/Problems: Questions that define the standard and are representative of what students will need to be able to do to show evidence of understanding. Closed Problems or Questions: Two forces, one of 120 pounds and the other 200 pounds, act on the same object at angles 30o and -30o respectively, with the positive x-axis. Find the direction of the resultant of these two forces. a) 14o b) -8.2o c) 0o d) 18o Given v = 5, 2 and w = 6,1 , find the angle between v and w. a) 58.7o b) 148.7o c) 31.3o d) 121.3o Open Problems or Questions: What force is required to keep a 2000-pound vehicle from rolling down a ramp inclined at 30o from the horizontal? A storm front is moving east at 30.0 mph and north at 18.5 mph. Find the resultant velocity of the front. An airliner’s navigator determines that the jet is flying 475 mph with a heading of 42.5o north of west, but the jet is actually moving 465 mph in a direction 37.7o north of west. What is the velocity of the wind? Two forces, one of 150 pounds and the other of 200 pounds, act on the same object at angles of 20o and -30o, respectively, with the positive x-axis. Find the magnitude and direction of the resultant force. Standards of Mathematical Practices: 1. 2. 3. 4. Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 5. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.