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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...

 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...

 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...

 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...

 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 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...

 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...

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

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

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

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

 Student Outcomes Students represent multiplication of complex numbers geometrically on the complex plane.

 Student Outcomes Students calculate distances between complex numbers as the modulus of the difference. Students calculate the midpoint of a segment as the average of the numbers at its endpoints.

 Student Outcomes Students apply distances between complex numbers and the midpoint of a segment. Students derive and apply a formula for finding the endpoint of a segment when given one endpoint and...

 Student Outcomes Students represent complex numbers in polar form and convert between rectangular and polar representations. Students explain why the rectangular and polar forms of a given complex...

 Student Outcomes Students determine the geometric effects of transformations of the form L(z) = az, L(z) = (bi)z, and L(z) = (a+bi)z for real numbers a and b.

 Student Outcomes Students understand why the geometric transformation effect of the linear transformation L(z) = wz is dilation by w and rotation by the argument of w.

 Student Outcomes Students create a sequence of transformations that produce the geometric effect of reflection across a given line through the origin.

 Student Outcomes Students apply their knowledge to understand that multiplication by the reciprocal provides the inverse geometric operation to a rotation and dilation. Students understand the...

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

 Student Outcomes Students understand how a formula for the nth roots of a complex number is related to powers of a complex number. Students calculate the nth roots of a complex number.