Algebra 2, Quarter 1, Unit 1.2Complex Numbers (8 days)
Algebra 2, Quarter 1, Unit 1.2
Complex Number System
OverviewNumber of instructional days: / 8 / (1 day = 45 minutes)
Content to be learned
/Mathematical practices to be integrated
- Develop a conceptual understanding of the meaning of i.
- Perform arithmetic operations (addition, subtraction, and multiplication only) on the set of complex numbers to solve equations.
- Solve quadratic equations with real coefficients that have complex solutions.
Essential questions
- In what situations would you be required to use complex numbers?
- What do imaginary roots represent in a polynomial?
- What do complex numbers represent?What are the implications of complex numbers to solutions?
Written Curriculum
Common Core State Standards for Mathematical Content
Numbers and Operations
The Complex Number SystemN-CN
Perform arithmetic operations with complex numbers.
N-CN.1Know 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.2Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients]
N-CN.7Solve quadratic equations with real coefficients that have complex solutions.
Common Core State Standards for Mathematical Practice
6Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
8Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2+ x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
Eighth-grade students studied and solved equations with perfect squares and cube roots.In algebra 1, students recognized when the quadratic formula gives complex solutions and wrote them in the form a + bi.Also, algebra 1 students graphed quadratic functions that do not have real solutions, giving them a conceptual understanding of functions not crossing the x-axis.They also rewrote radicals in simplified form in algebra 1.
Current Learning
Algebra 2 students manipulate complex numbers using arithmetic operations, and they solve quadratic equations with complex solutions.Students learn the significance of the Fundamental Theorem of Algebra.
Future Learning
This learning is extended in precalculus by redefining the real and imaginary parts within the polar coordinate system and performing operations with them.
Additional Findings
Beyond Numeracy discusses that, given the expanded number system, we can prove the Fundamental Theorem of Algebra.With complex numbers, we soon extended the definition of trigonometric and exponential functions to the domain of complex numbers and generalized analysis to calculus and differential equations and related fields.These technical advances, including the geometrical interpretation of various operations on complex numbers, paved the way for their indispensable use in electrical theory and other physical sciences (pp. 116–117).
Principles and Standards for School Mathematics indicates that high school students should have substantial experience exploring the properties of different classes of functions. They should also learn that some quadratic equations do not have real roots and that this characteristic corresponds to the fact that their graphs do not cross the x-axis.Also, students should be able to identify the complex roots of such quadratics (p. 299).
Warwick Public Schools, in collaboration withC-1
the Charles A. Dana Center at the University of Texas at Austin