The theme of Module 1, Topic C is to highlight the effectiveness of changing notations and the power provided by certain notations such as matrices. Lessons 18 and 19 exploit the connection to trigonometry, as students see how much complex arithmetic is simplified (N-CN.B.4, N-CN.B.5). Students use the connection to trigonometry to solve problems such as find the three cube roots. In Lesson 20, complex numbers are regarded as points in the Cartesian plane. Students begin to write analytic formulas for translations, rotations, and dilations in the plane and revisit the ideas of Geometry (G-CO.A.2, G-CO.A.4, G-CO.A.5) in this light. In Lesson 21, students discover a better notation (i.e., matrices) and develop the matrix notation for planar transformations represented by complex number arithmetic. This work leads to Lessons 22 and 23 as students discover how geometry software and video games efficiently perform rigid motion calculations. Students discover the flexibility of matrix notation in Lessons 24 and 25 as they add matrices and multiply by the identity matrix and the zero matrices (N-VM.C.8, N-VM.C.11). Students understand that multiplying matrix by the identity matrix results in the matrix and connect the multiplicative identity matrix to the role of 1, the multiplicative identity, in the real number system. This is extended as students see that the identity matrix does not transform the unit square. Students then add matrices and conclude that the zero matrices added to the matrix result in the matrix and are similar to in the real number system. They extend this concept to transformations on the unit square and see that adding the zero matrices has no effect, but multiplying by the zero matrix collapses the unit square to zero. This allows for the study of additional matrix transformations (shears, for example) in Lessons 26 and 27, multiplying matrices, and the meaning of the determinant of a matrix (N-VM.C.10, N-VM.C.12). Lessons 28–30 conclude Topic C and Module 1 as students discover the inverse matrix and determine when matrices do not have inverses. Students begin to think and reason abstractly about the geometric effects of the operations of complex numbers (MP.2) as they see the connection to trigonometry and the Cartesian plane.
The study of vectors and matrices is only introduced in Module 1 through a coherent connection to transformations and complex numbers. The further and more formal study of multiplication of matrices occurs in Module 2. N-M.C.8 is assessed secondarily, in the context of other standards, but not directly on Mid- and End-of-Module Assessments until Module 2.
Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to the Algebra II and Algebra II with Trigonometry Alabama Courses of Study.
Module 2, Topic B explores the geometric context for higher-dimensional matrices. The geometric effect of matrix operations—matrix product, matrix sum, and scalar multiplication—are examined, and students come to see, geometrically, that matrix multiplication for square matrices is not a commutative operation, but that it still satisfies the associative and distributive properties. The geometric and arithmetic roles of the zero matrices and identity matrix are discussed, and students see that a multiplicative inverse to a square matrix exists precisely when the determinant of the matrix is non-zero.
Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to the Algebra II and Algebra II with Trigonometry Alabama Course of Study.
Module 2, 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 frameworks for these operations. Students solve problems involving velocity and other quantities that can be represented by vectors. Parametric equations are introduced allowing students to connect their prior work with functions to vectors.
In Module 2, Topic E students apply the knowledge developed in this module to understand how first-person video games use matrix operations to project three-dimensional objects onto two-dimensional screens and animate those images to give the illusion of motion.