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 Precalculus Module 1: Complex Numbers and Transformations Module 1 sets the stage for expanding students' understanding of transformations by exploring the notion of linearity. This leads to the...

 Students revisit the fundamental theorem of algebra as they explore complex roots of polynomial functions. They use polynomial identities, the binomial theorem, and Pascal’s Triangle to find roots...

 Module 2 extends the concept of matrices introduced in Module 1. Students look at incidence relationships in networks and encode information about them via highdimensional matrices. Matrix...

 Student Outcomes Students solve quadratic equations with complex solutions. Students understand the geometric origins of the imaginary unit i in terms of 90degree rotations. Students use this...

 Student Outcomes Students learn when ideal linearity properties do and do not hold for classes of functions studied in previous years. Students develop familiarity with linearity conditions.

 Which Real Number Functions Define a Linear Transformation? Students develop facility with the properties that characterize linear transformations. Students learn that a mapping L:ℝ→ℝ is a linear...

 Topic A begins the study of linearity looking at common misconceptions made by math students. This study leads to complex solutions which launches the study of products, quotients of complex numbers...

 In Topic B, students develop an understanding that when complex numbers are considered points in the Cartesian plane, complex number multiplication has the geometric effect of a rotation followed by...

 Topic C highlights the effectiveness of changing notations and the power provided by certain notations such as matrices. The study of vectors and matrices is introduced through a coherent connection...

 Student Outcomes Students learn when ideal linearity properties do and do not hold for classes of functions studied in previous years. Students develop familiarity with linearity conditions.

 Student Outcomes Students derive the formula for zn = rn(cos(nθ) + i sin(nθ)) and use it to calculate powers of a complex number.

 Topic B explores the geometric context for higherdimensional matrices. The geometric effect of matrix operations—matrix product, matrix sum, and scalar multiplication—are examined, and students...

 Topic A brings students back to the study of complex roots of polynomial functions. Students briefly review quadratic and cubic functions and then extend familiar polynomial identities to both...

 Student Outcomes Students represent addition, subtraction, and conjugation of complex numbers geometrically on the complex plane.

 In Topic A, students look at incidence relationships in networks and encode information about them via highdimensional matrices. Questions on counting routes, the results of combining networks,...

 Student Outcomes Students use matrices to represent and manipulate data from network diagrams.

 Student Outcomes Students represent complex numbers as vectors. Students represent complex number addition and subtraction geometrically using vectors.

 Student Outcomes Students determine all solutions of polynomial equations over the set of complex numbers and understand the implications of the fundamental theorem of algebra, especially for...

 Student Outcomes Students describe complex numbers and represent them as points in the complex plane. Students perform arithmetic with complex numbers, including addition, subtraction, scalar...

 Topic D opens with a formal definition of a vector. The arithmetical work for vector addition, subtraction, scalar multiplication, and vector magnitude is explored along with the geometrical...

 Student Outcomes Students determine the multiplicative inverse of a complex number. Students determine the conjugate of a complex number.

 Student Outcomes Students add and subtract vectors and understand those operations geometrically and componentwise. Students understand scalar multiplication graphically and perform it component...

 Student Outcomes Students find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Students add vectors endtoend using the...

 Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers.