All West Virginia teachers are responsible for classroom instruction that integrates content standards and objectives and mathematical habits of mind.  This course is designed for students who have completed the Math III (LA) course and subsequently decided they are interested in pursuing a STEM career.  It includes standards that would have been covered in Mathematics III (STEM) but not in Mathematics III (LA) (i.e. standards that are marked with a “+” ), selected topics from the Mathematics IV course, and topics drawing from standards covered in Mathematics I and Mathematics II as needed for coherence. Mathematical habits of mind, which should be integrated in these content areas, include:  making sense of problems and persevering in solving them, reasoning abstractly and quantitatively; constructing viable arguments and critiquing the reasoning of others; modeling with mathematics; using appropriate tools strategically; attending to precision, looking for and making use of structure; and looking for and expressing regularity in repeated reasoning.  Students will continue developing mathematical proficiency in a developmentally-appropriate progressions of standards. Continuing the skill progressions from previous courses, the following chart represents the mathematical understandings that will be developed:

### Arithmetic and Algebra of Complex Numbers

#### Standards

M.SRM.1

Know there is a complex number i such that i² = – 1, and every complex number has the form a + bi with a and b real.

M.SRM.2

Use the relation i² = -1 and the commutative, associative and distributive properties to add, subtract and multiply complex numbers.

M.SRM.3

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

#### Alignment

M.SRM.1

Know there is a complex number i such that i² = – 1, and every complex number has the form a + bi with a and b real.

M.SRM.2

Use the relation i² = -1 and the commutative, associative and distributive properties to add, subtract and multiply complex numbers.

M.SRM.3

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

#### Standards

M.SRM.4

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers) and explain why the rectangular and polar forms of a given complex number represent the same number.

M.SRM.5

Represent addition, subtraction, multiplication and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. (e.g., (-1 + √3 i)³ = 8 because (-1 + √3 i) has modulus 2 and argument 120°.)

M.SRM.6

Calculate the distance between numbers in the complex plane as the modulus of the difference and the midpoint of a segment as the average of the numbers at its endpoints.

#### Alignment

M.SRM.4

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers) and explain why the rectangular and polar forms of a given complex number represent the same number.

M.SRM.5

Represent addition, subtraction, multiplication and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. (e.g., (-1 + √3 i)³ = 8 because (-1 + √3 i) has modulus 2 and argument 120°.)

M.SRM.6

Calculate the distance between numbers in the complex plane as the modulus of the difference and the midpoint of a segment as the average of the numbers at its endpoints.

#### Standards

M.SRM.7

Solve quadratic equations with real coefficients that have complex solutions.

M.SRM.8

Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).

M.SRM.9

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

#### Alignment

M.SRM.7

Solve quadratic equations with real coefficients that have complex solutions.

M.SRM.8

Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).

M.SRM.9

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

### Polynomials, Rational, and Radical Relationships

#### Standards

M.SRM.10

Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

#### Alignment

M.SRM.10

Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

#### Standards

M.SRM.11

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions.

#### Alignment

M.SRM.11

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions.

### Probability for Decisions

#### Standards

M.SRM.12

Use probabilities to make fair decisions (e.g. drawing by lot or using a random number generator).

M.SRM.13

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).

#### Alignment

M.SRM.12

Use probabilities to make fair decisions (e.g. drawing by lot or using a random number generator).

M.SRM.13

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).

### Trigonometry of General Triangles

#### Standards

M.SRM.14

Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

M.SRM.15

Prove the Laws of Sines and Cosines and use them to solve problems.

M.SRM.16

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems or resultant forces).

#### Alignment

M.SRM.14

Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

M.SRM.15

Prove the Laws of Sines and Cosines and use them to solve problems.

M.SRM.16

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems or resultant forces).

### Functions and Modeling

#### Standards

M.SRM.17

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

M.SRM.18

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.

M.SRM.19

Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline, and amplitude.

#### Alignment

M.SRM.17

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

M.SRM.18

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.

M.SRM.19

Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline, and amplitude.

#### Standards

M.SRM.20

Write a function that describes a relationship between two quantities.

M.SRM.21

Compose functions. (e.g., If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.)

#### Alignment

M.SRM.20

Write a function that describes a relationship between two quantities.

M.SRM.21

Compose functions. (e.g., If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.)

#### Standards

M.SRM.22

Find inverse functions.

M.SRM.23

Verify by composition that one function is the inverse of another.

M.SRM.24

Read values of an inverse function from a graph or a table, given that the function has an inverse.

M.SRM.25

Produce an invertible function from a non-invertible function by restricting the domain.

M.SRM.26

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

#### Alignment

M.SRM.22

Find inverse functions.

M.SRM.23

Verify by composition that one function is the inverse of another.

M.SRM.24

Read values of an inverse function from a graph or a table, given that the function has an inverse.

M.SRM.25

Produce an invertible function from a non-invertible function by restricting the domain.

M.SRM.26

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

#### Standards

M.SRM.27

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

M.SRM.28

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

#### Alignment

M.SRM.27

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

M.SRM.28

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

#### Standards

M.SRM.29

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

M.SRM.30

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

#### Alignment

M.SRM.29

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

M.SRM.30

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

#### Standards

M.SRM.31

Prove the addition and subtraction formulas for sine, cosine and tangent and use them to solve problems.