Evidence Outcomes:

1. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (CCSS: HS.N-CN.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Perform arithmetic operations with complex numbers

  • Extend understanding of the numbers system to include complex numbers

  • Solve problems that previously appeared to have no solutions by performing operations with complex numbers

In order to meet these essential understandings, students must know...

• The format of a conjugate

In order to meet these essential understandings, students must be able to...

• Use conjugates to find quotients and moduli of complex numbers

Evidence Outcomes:

1. (+) 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. (CCSS: HS.N-CN.B.4)

2. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1+√3i)^3=8 because (-1+√3i) has modulus 2 and argument 〖120〗^∘. (CCSS: HS.N-CN.B.5)

3. (+) 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. (CCSS: HS.N-CN.B.6)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Attend to precision

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Represent complex numbers and their operations on the complex plane

  • Perform operations and graph complex numbers in polar and rectangular forms

In order to meet these essential understandings, students must know...

• The difference between rectangular and polar form

• Why rectangular and polar forms of a given complex number represent the same number

In order to meet these essential understandings, students must be able to...

• Calculate the distance between complex numbers

• Add, subtract, multiply, and conjugate complex numbers

• Represent complex numbers in rectangular and polar form

• Find modulus and argument

Evidence Outcomes:

1. (+) 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. (CCSS: HS.N-CN.B.4)

2. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1+√3i)^3=8 because (-1+√3i) has modulus 2 and argument 〖120〗^∘. (CCSS: HS.N-CN.B.5)

3. (+) 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. (CCSS: HS.N-CN.B.6)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Attend to precision

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Represent complex numbers and their operations on the complex plane

  • Perform operations and graph complex numbers in polar and rectangular forms

In order to meet these essential understandings, students must know...

• The difference between rectangular and polar form

• Why rectangular and polar forms of a given complex number represent the same number

In order to meet these essential understandings, students must be able to...

• Calculate the distance between complex numbers

• Add, subtract, multiply, and conjugate complex numbers

• Represent complex numbers in rectangular and polar form

• Find modulus and argument

Evidence Outcomes:

1. (+) 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 ). (CCSS: HS.N-VM.A.1)

2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. (CCSS: HS.N-VM.A.2)

3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. (CCSS: HS.N-VM.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Model with Mathematics

• Attend to Precision

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Represent and model with vector quantities

  • Model real-world scenarios with vectors

In order to meet these essential understandings, students must know...

• Vocabulary: magnitude, direction, vector, velocity, initial point, and terminal point

• Vector quantities have both magnitude and direction

• The symbols for vectors and their magnitudes

In order to meet these essential understandings, students must be able to...

• Solve problems involving velocity and other quantities that can be represented by vectors

• Represent magnitude and direction when graphing and describing vectors

Evidence Outcomes:

1. 4. (+) Add and subtract vectors. (CCSS: HS.N-VM.B.4)

   • 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. (CCSS: HS.N-VM.B.4.a)

   • Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. (CCSS: HS.N-VM.B.4.b)

   • Understand vector subtraction v-w as v+(-w), where -w is the additive inverse of w, with the same magnitude as and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. (CCSS: HS.N-VM.B.4.c)

2. (+) Multiply a vector by a scalar. (CCSS: HS.N-VM.B.5)

   • 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). (CCSS: HS.N-VM.B.5.a)

   • 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). (CCSS: HS.N-VM.B.5.b)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Attend to Precision

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Perform operations on vectors

  • Solve real-world problems using operations on vectors

  • Accurately represent magnitude and direction when using vector arithmetic

In order to meet these essential understandings, students must know...

• The magnitude of a sum of two vectors is typically not the sum of the magnitude of the two vectors

• Additive inverse of vectors

  • (v) - (w) = (v) + (-w)

• The effects of multiplying a vector by a scalar

• Parallelogram rule

In order to meet these essential understandings, students must be able to...

• Add and subtract vectors end to end, component-wise, parallelogram rule

• Determine the magnitude and direction of their sum, given two vectors

• Use vector subtraction with additive inverse and represent vector subtraction graphically

• Multiply a vector by a scalar

• Represent scalar multiplication graphically

• Compute the magnitude of a scalar multiple cv

• Compute the direction of cv

Evidence Outcomes:

1. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. (CCSS: HS.A-REI.C.5)

2. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (CCSS: HS.A-REI.C.6)

3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y=-3x and the circle 𝑥2+𝑦2=3. (CCSS: HS.A-REI.C.7)

4. (+) Represent a system of linear equations as a single matrix equation in a vector variable. (CCSS: HS.A-REI.C.8)

5. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater). (CCSS: HS.A-REI.C.9)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Reason abstractly and quantitatively

• Model with mathematics

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Solve systems of equations

  • Solve systems of linear equations algebraically and graphically

Advanced - All Functions

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• Represent a system of linear equations as a single matrix equation in a vector variable, which may model a real-world scenario

• Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices 3x3 and larger)

Evidence Outcomes:

1. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* (CCSS: HS.F-IF.C.7)

   • (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (CCSS: HS.F-IF.C.7.d)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Reason abstractly and quantitatively

• Use appropriate tools strategically

• Attend to precision

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Analyze functions using different representations (Modeling)

  • Reason abstractly and understand the connections between the symbolic representation, the table of values, and the key features of the graph of a function (Rule of 4)

  • Compare properties of functions written in different forms

Advanced - Logarithmic/Trigonometric - mismatch with CDE Colorado Academic Standards and Common Core state standards

In order to meet these essential understandings, students must know...

• Vocabulary: intercepts, max/min, zeros, (+) asymptotes, end behavior, period, midline, amplitude

In order to meet these essential understandings, students must be able to...

• Graph functions and show key features both by hand and using technology

  • Trigonometric functions: period, midline, amplitude

Evidence Outcomes:

1. Write a function that describes a relationship between two quantities.* (CCSS: HS.F-BF.A.1)

   • (+) Compose functions. For example, 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. (CCSS: HS.F-BF.A.1.c)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Model with mathematics

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Build a function that models a relationship between two quantities (Modeling)

Advanced - Compose Functions

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• Compose functions

Evidence Outcomes:

1. Find inverse functions. (CCSS: HS.F-BF.B.4)

   • (+) Verify by composition that one function is the inverse of another. (CCSS: HS.F-BF.B.4.b)

   • (+) Read values of an inverse function from a graph or table, given that the function has an inverse. (CCSS: HS.F-BF.B.4.c)

   • (+) Produce an invertible function from a non-invertible function by restricting the domain. (CCSS: HS.F-BF.B.4.d)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Construct viable arguments and critique the reasons of others

• Attend to precision

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Build new functions from existing functions

  • Finding inverse functions

Advanced - trig transformations

In order to meet these essential understandings, students must know...

• The inverse relationship between exponents and logarithms and use these relationships to solve problems involving logarithms and exponents

In order to meet these essential understandings, students must be able to...

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

• Read values of an inverse function from a graph or a table

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

Evidence Outcomes:

1. (+) Use special triangles to determine geometrically the values to sine, cosine, tangent for π/3, π/4, and π/6 and use the unit circle to express the values sine, cosine, and tangent for x, π+x, and 2π-x and in terms of their values for x where x is any real number. (CCSS: HS.F-TF.A.3)

2. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. (CCSS: HS.F-TF.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• ADVANCED

  • Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Extend the domain of trigonometric functions using the unit circle

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• Use special right triangles and reference angles to determine values for sine, cosine, and tangent(x) where x is any real number and x = pi - x, x = pi + x, x = 2pi - x

• apply knowledge of reference angles to other angles in the unit circle

• Use the unit circle to explain even and odd symmetry and periodicity of trig functions

Evidence Outcomes:

1. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (CCSS: HS.F-TF.B.6)

2. (+) Use inverse function to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.* (CCSS: HS.F-TF.B.7)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• ADVANCED

  • Reason abstractly and quantitatively

  • Model with mathematics

  • Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Model periodic phenomena with trigonometric functions (Modeling)

  • Inverse Trigonometric Functions

In order to meet these essential understandings, students must know...

• Must restrict a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed

In order to meet these essential understandings, students must be able to...

• Use inverse functions to solve trigonometric equations that arise in modeling context

• Restrict domains to create inverse functions

• Use inverse functions to solve

Evidence Outcomes:

1. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. (CCSS: HS.F-TF.C.9)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• ADVANCED

  • Reason abstractly and quantitatively

  • Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Prove and apply trigonometric identities

  • Sum and Difference formulas for sine, cosine, and tangent

In order to meet these essential understandings, students must know...

• The addition and subtraction formulas for sine, cosine, and tangent

• Trigonometric expressions can oftentimes be manipulated like algebraic expressions (factoring, distributing, steps of solving, etc.)

• Trigonometric identities can be written as multiple, equivalent forms (use when simplifying trigonometric expressions)

In order to meet these essential understandings, students must be able to...

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

• Understand trigonometric quantities as expressions and use their relationships in problem situations

Evidence Outcomes:

1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. (CCSS: HS.S-MD.A.1)

2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. (CCSS: HS.S-MD.A.2)

3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. (CCSS: HS.S-MD.A.3)

4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? (CCSS: HS.S-MD.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• ADVANCED

  • Reason abstractly and quantitatively

  • Model with mathematics

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Calculate expected values and use them to solve problems

  • Develop probability distributions to find expected values

In order to meet these essential understandings, students must know...

• Vocabulary: random variable, expected value, probability distribution

In order to meet these essential understandings, students must be able to...

• Define a random variable

• Graph probability distributions using the same displays as for data distributions

  • Include distributions where the theoretical probabilities can be calculated.

  • Include distributions in which empirical probabilities can be included

• Calculate the expected value of a random variable

  • Interpret as the mean of the probability distribution

Evidence Outcomes:

1. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. (CCSS: HS.S-MD.B.5)

   • Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or game at a fast-food restaurant. (CCSS: HS.S-MD.B.5.a)

   • Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or major accident. (CCSS: HS.S-MD.B.5.b)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• ADVANCED

  • Model with mathematics

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Use probability to evaluate outcomes of decisions

In order to meet these essential understandings, students must know...

• Vocabulary: payoff values, expected values

• What “fair” means in the context of statistics

In order to meet these essential understandings, students must be able to...

• Find the expected payoff for a game of chance by finding expected values

• Evaluate and compare different situations on the basis of expected value (two different insurance policies)

Evidence Outcomes:

1. (+) 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. (CCSS: HS.G-GPE.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Make sense of problems and persevere in solving them

• Reason abstractly and quantitatively

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Translate between the geometric description and the equation for a conic section

  • Parabolas, ellipses, hyperbolas

In order to meet these essential understandings, students must know...

• Vocabulary: parabola, focus, directrix, ellipse, hyperbola

• In ellipses and hyperbolas, the sum or difference of distances from the foci is constant

In order to meet these essential understandings, students must be able to...

• Derive the equation of a parabola using the focus and directrix

• Derive the equations of ellipses and hyperbolas given the foci

Evidence Outcomes:

1. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. (CCSS: HS.G-GMD.A.2)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Construct viable arguments and critique the reasoning of others

• Model with mathematics

• Use appropriate tools strategically

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Explain volume formulas and use them to solve problems

In order to meet these essential understandings, students must know...

• Cavalieri’s principle

In order to meet these essential understandings, students must be able to...

• Give an informal argument, using Cavalieri’s principle, for the volume of a sphere and other figures