BR: Building Relationships among Complex Numbers, Vectors, and Matrices

(Framing Text): Perform arithmetic operations with complex numbers.

BR.M.4HSTP.1: Find the conjugate of a complex number; use conjugates to find moduli (magnitude) and quotients of complex numbers.

Points in the Complex Plane
Roots of a Quadratic

(Framing Text): Represent complex numbers and their operations on the complex plane.

BR.M.4HSTP.2: 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.

Points in the Complex Plane

BR.M.4HSTP.3: 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°.

Points in the Complex Plane

BR.M.4HSTP.4: 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.

Points in the Complex Plane

(Framing Text): Represent and model with vector quantities.

BR.M.4HSTP.5: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

Adding Vectors
Vectors

BR.M.4HSTP.6: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Vectors

BR.M.4HSTP.7: Solve problems involving velocity and other quantities that can be represented by vectors.

2D Collisions
Golf Range

(Framing Text): Perform operations on vectors.

BR.M.4HSTP.8: Add and subtract vectors.

BR.M.4HSTP.8.a: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Adding Vectors
Vectors

BR.M.4HSTP.8.b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Adding Vectors
Vectors

BR.M.4HSTP.8.c: Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order and perform vector subtraction component-wise.

Adding Vectors
Vectors

BR.M.4HSTP.9: Multiply a vector by a scalar.

BR.M.4HSTP.9.a: Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

Dilations
Vectors

BR.M.4HSTP.9.b: Compute the magnitude of a scalar multiple cv using ||cv ||= |c|.||v||. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Vectors

(Framing Text): Perform operations on matrices and use matrices in applications.

BR.M.4HSTP.11: Multiply matrices by scalars to produce new matrices (e.g., as when all of the payoffs in a game are doubled.

Dilations

BR.M.4HSTP.12: Add, subtract and multiply matrices of appropriate dimensions.

Translations

BR.M.4HSTP.16: Work with 2 × 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

Dilations
Translations

(Framing Text): Solve systems of equations.

BR.M.4HSTP.17: Represent a system of linear equations as a single matrix equation in a vector variable.

Solving Linear Systems (Matrices and Special Solutions)

ASF: Analysis and Synthesis of Functions

(Framing Text): Analyze functions using different representations.

ASF.M.4HSTP.19: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Absolute Value with Linear Functions
Exponential Functions
General Form of a Rational Function
Graphs of Polynomial Functions
Introduction to Exponential Functions
Logarithmic Functions
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Rational Functions

(Framing Text): Build a function that models a relationship between two quantities.

ASF.M.4HSTP.20: Write a function that describes a relationship between two quantities, including composition of functions.

Function Machines 1 (Functions and Tables)
Points, Lines, and Equations

(Framing Text): Build new functions from existing functions.

ASF.M.4HSTP.21: Find inverse functions.

ASF.M.4HSTP.21.a: Verify by composition that one function is the inverse of another.

Logarithmic Functions

ASF.M.4HSTP.21.b: Read values of an inverse function from a graph or a table, given that the function has an inverse.

Logarithmic Functions

ASF.M.4HSTP.22: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Logarithmic Functions

TIF: Trigonometric and Inverse Trigonometric Functions of Real Numbers

(Framing Text): Extend the domain of trigonometric functions using the unit circle.

TIF.M.4HSTP.23: 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.

Cosine Function
Sine Function
Sum and Difference Identities for Sine and Cosine
Tangent Function
Translating and Scaling Sine and Cosine Functions

(Framing Text): Prove and apply trigonometric identities.

TIF.M.4HSTP.28: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Simplifying Trigonometric Expressions
Sum and Difference Identities for Sine and Cosine

(Framing Text): Apply transformations of function to trigonometric functions.

TIF.M.4HSTP.29: Graph trigonometric functions showing key features, including phase shift.

Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions

DAG: Derivations in Analytic Geometry

(Framing Text): Translate between the geometric description and the equation for a conic section.

DAG.M.4HSTP.30: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Ellipses
Hyperbolas

MP: Modeling with Probability

(Framing Text): Calculate expected values and use them to solve problems.

MP.M.4HSTP.33: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Lucky Duck (Expected Value)

MP.M.4HSTP.34: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. (e.g., Find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.)

Binomial Probabilities
Geometric Probability
Independent and Dependent Events
Lucky Duck (Expected Value)
Probability Simulations
Theoretical and Experimental Probability

MP.M.4HSTP.35: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Geometric Probability
Independent and Dependent Events
Lucky Duck (Expected Value)
Probability Simulations
Theoretical and Experimental Probability

(Framing Text): Use probability to evaluate outcomes of decisions.

MP.M.4HSTP.36: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

MP.M.4HSTP.36.a: Find the expected payoff for a game of chance. (e.g., Find the expected winnings from a state lottery ticket or a game at a fast food restaurant.)

Lucky Duck (Expected Value)

MP.M.4HSTP.36.b: Evaluate and compare strategies on the basis of expected values. (e.g., Compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.)

Lucky Duck (Expected Value)