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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...

 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.

 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...

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

 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 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 use matrices to represent data based on transportation networks. Students multiply a matrix by a scalar, add and subtract matrices of appropriate dimensions, and interpret...

 Student Outcomes Students understand the forces involved in constructing a stone arch. Students add and subtract vectors given in magnitude and direction form. Students solve problems that can be...

 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...

 Student Outcomes Students use matrix transformations to represent motion along a straight line.

 In Topic E students apply the knowledge developed in this module to understand how firstperson video games use matrix operations to project threedimensional objects onto twodimensional screens and...

 Student Outcomes Students use matrix transformations to model circular motion. Students use coordinate transformations to represent a combination of motions.

 Student Outcomes Students understand that an inverse transformation, when represented by a 2 × 2 matrix, exists precisely when the determinant of that matrix is nonzero.

 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 write the equation for a line in ℝ2 or ℝ3 using vectors. Students write the parametric equations for a line in ℝ2 or ℝ3 . Students convert between parametric equations and...

 Student Outcomes Students apply linear transformations and vectors to understand the conditions required for a sequence of transformations to preserve the solution set to the system of equations.