Evidence Outcomes:

1. (+) Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. (CCSS: HS.N-RN.A.3)

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

• Construct viable arguments and critique the reasoning 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...

• Use properties of rational and irrational numbers

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

• The meaning of rational numbers

• The meaning of irrational numbers

• Properties of integer exponents extend to rational exponents

• the sum or product of two rational numbers is rational

• the sum of a rational number and an irrational number is irrational

• The product of a nonzero rational number and an irrational number is irrational

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

• Explain how it is possible that multiplying 2 irrational numbers gives a product that is not irrational. Why doesn’t this phenomenon apply to rational numbers?

• Communicate conclusions about rational and irrational numbers

Evidence Outcomes:

1. Define complex number i such that i^2=–1, and show that every complex number has the form a+bi where a and b are real numbers. (CCSS: HS.N-CN.A.1)

2. Use the relation i^2=–1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (CCSS: HS.N-CN.A.2)

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 definition of “i”

• The format of a complex number

• That operations of adding, subtracting, and multiplying and commutative, associative, and distributive properties extend to complex number system

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

• Add, subtract, and multiply complex numbers

Evidence Outcomes:

1. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ (CCSS: HS.A-SSE.B.3)

   • Factor a quadratic expression to reveal the zeros of the function it defines. (CCSS: HS.A-SSE.B.3.a)

   • Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (CCSS: HS.A-SSE.B.3.b)

   • Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.15 1/12) ^12t ~ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. (CCSS: HS.A-SSE.B.3.c)

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

• Look for and make use of structure

• 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...

• Write expressions in equivalent forms to solve problems (Modeling)

  • Move between multiple forms of functions and using them to find key characteristics of the function and to solve

  • Recognize the difference in the structure of linear, quadratic, and other equations, and apply strategies to solve

    • This standard is the procedural piece of this concept - the previous standard is the conceptual piece.

Math 2 - Quadratic/Exponential

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

• The zeros of a function are the x-intercepts of the function

• The key features found in each different form of a function (point-slope vs slope-intercept, standard vs vertex vs factored)

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

• Factor quadratic expressions to find the zeros of functions

• Complete the square in a quadratic expression to find maximum and minimum values

• Use properties of exponents to transform exponential functions

• Transform expressions to set up solution strategies

• Use the formula for the sum of a finite geometric series

Evidence Outcomes:

1. Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (CCSS: HS.A-APR.A.1)

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

• Look for and make use of structure

• 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...

• Perform arithmetic operations with polynomials

  • Understand that polynomials form a system analogous to the integers (closed under addition, subtraction, and multiplication)

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

• Understand how types of numbers and operations form a closed system

• How operations on polynomials yield polynomials

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

• Explain how polynomials are closed under operations of +, -, x

• Add, subtract, multiply polynomials

• Make hypotheses and draw conclusions about polynomials and operations on them

Evidence Outcomes:

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (CCSS: HS.A-CED.A.1)

2. Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. (CCSS: HS.A-CED.A.2)

3. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V=IR to highlight resistance R. (CCSS: HS.A-CED.A.4)

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

• Reason abstractly and quantitatively

• Model with mathematics

• Use appropriate tools strategically

• 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...

• Create equations that describe numbers or relationships (Modeling)

  • Model and solve problems arising in the real-world

Math 2 - Quadratic/Exponential

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

• When to use one-variable equations vs two-variable equations

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

• Create equations and inequalities in one variable and use them to solve problems

  • Linear, quadratic, and simple rational and exponentials

• Create equations in two or more variables and graph (label scales and axes)

• Use structure and operations to rearrange formulas to isolate a variable (ex. solve for ‘a’ given the formula: F = ma)

• Interpret mathematical results in the context of the situation and reflect on whether the results make sense and serve a purpose

• Use pencil and paper and technology to make sense of and solve mathematical equations

Evidence Outcomes:

1. Solve quadratic equations in one variable. (CCSS: HS.A-REI.B.4)

   • • Use the method of completing the square to transform any quadratic equation in x into an equation of the form (𝑥−𝑝)2=𝑞 that has the same solutions. Derive the quadratic formula from this form. (CCSS: HS.A-REI.B.4.a)

     • Solve quadratic equations (e.g., for x^2=49) by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a±bi for real numbers a and b . (CCSS: HS.A-REI.B.4.b)

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

• Use appropriate tools strategically

• 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...

Solve equations and inequalities in one variable

Math 2 - Quadratics

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

• What conclusions to draw from the solutions of quadratic equations

• The structures of a quadratic equation to determine the most efficient solution strategy

• Which form of quadratic functions lends itself to a tool

  • Completing the square

  • Quadratic formula

  • Square roots

  • Factoring

  • Technology

• When the quadratic formula yields complex solutions

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

• Solve quadratic equations in one variable by

  • Completing the square

  • Quadratic formula

  • Square roots

  • Factoring

• Derive the quadratic formula from completing the square

• Write out complex solutions

Evidence Outcomes:

1. 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)

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

  • Solve systems with one linear equation and one quadratic equation algebraically and graphically

Math 2 - Linear-Quadratic Systems

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

• When and how to use the elimination method to solve a system of equations

• When and how to use substitution to solve a system of equations

• How to use graphs to approximate solutions of systems of equations.

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

• Solve a simple system consisting of a linear equation and a quadratic equation in two variables (could also be a circle)

Evidence Outcomes:

1. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (CCSS: HS.F-IF.B.4)

2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* (CCSS: HS.F-IF.B.5)

3. Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.* (CCSS: HS.F-IF.B.6)

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

• Model with mathematics

• Use appropriate tools strategically

• 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...

• Interpret functions that arise in applications in terms of the context (Modeling)

  • Explain how real-world context influences the domain of the function

Math 1 - Linear/ Exponential

Math 2 - Quadratic

Math 3 - Rational/Square Root/Cube Root

Advanced - Log/Trig Functions

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

• Key features and understand how various functions behave in different representations

• Similarities and differences between linear, exponential, quadratic, polynomial, rational, radical, and trigonometric functions

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

• Interpret key features of graphs and tables and sketch graph showing key features

  • Intercepts

  • Intervals of inc, dec, positive, negative

  • Relative mins and maxs

  • Symmetries

  • End behavior

  • Periodicity

• Describe an appropriate domain for a function, especially functions that model real-world situations

• Use functions and their graphs to model, interpret and generalize about real-world situations

• Calculate and interpret the average rate of change presented symbolically or as a table

  • Estimate the rate of change from a graph

• Use technology as a tool to visualize and understand how various functions behave

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 linear and quadratic functions and show intercepts, maxima, and minima. (CCSS: HS.F-IF.C.7.a)

   • Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (CCSS: HS.F-IF.C.7.b)

2. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (CCSS: HS.F-IF.C.8)

   • Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (CCSS: HS.F-IF.C.8.a)

   • Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (​0.97)^t, y = (1.01)12^t, and y = (1.2)^t /10, and classify them as representing exponential growth or decay. (CCSS: HS.F-IF.C.8.b)

3. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (CCSS: HS.F-IF.C.9)

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

Math 2 - Quadratic/Exponential/Absolute value/Step/Piecewise

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

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

• Which forms of functions are used to find specific features of that function

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

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

  • Linear and quadratic: intercepts, maxima, minima

  • Square root, cube root, piece-wise (step), and absolute value

• Write and compare functions in equivalent forms to reveal properties

• Use models of functions (equations, graphs, tables, scenarios) to compare key features of multiple, different functions

Evidence Outcomes:

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

   • Determine an explicit expression, a recursive process, or steps for calculation from a context. (CCSS: HS.F-BF.A.1.a)

   • Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (CCSS: HS.F-BF.A.1.b)

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)

  • Use both explicit and recursive formulas

Math 2 - Quadratic/Exponential

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

• Vocabulary: explicit expressions, recursive process, arithmetic sequences, geometric sequences

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

• Write an explicit expression, recursive process, or steps for calculation from a context

• Combine function types (subtract a constant function from an exponential function)

• Apply understanding of functions to real-world contexts

Evidence Outcomes:

1. Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k both positive and negative; find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (CCSS: HS.F-BF.B.3)

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

   • Solve an equation of the form f(x)=c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x)=2x^3 or f(x)=(x+1)/(x-1) for x≠1. (CCSS: HS.F-BF.B.4.a)

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

  • Use function notation to write transformations

  • Finding inverse functions

Math 2 - Quadratic/Exponential/Absolute Value

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

• Vocabulary: even/odd functions, function family, inverse functions, transformation

• How transformations in function notation affect graphs

  • f(x)+k, f(x+k), kf(x), f(kx)

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

• Show how transformations by k appear on graphs of functions

• Find transformations from graph and write in function notation

• Use technology to create, describe and analyze related functions

• Communicate explanations of generalities across function families using accurate terms, definitions and mathematical symbols

• Find the inverse function for a simple function f that has an inverse

Evidence Outcomes:

1. Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (CCSS: HS.F-LE.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

• Model with mathematics

• 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...

• Construct and compare linear, quadratic, and exponential models and solve problems (Modeling)

  • Apply models to real-world scenarios

Math 2 - Quadratic/Exponential

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

• The difference between situations that can be modeled by linear and exponential functions

• The connection between table, graph, and function notation to better understand the function

• Exponential functions will always surpass all other functions in the long run

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

• Prove that by equal factors

• Identify situations where one quantity changes at a constant percent rate per unit interval

• Construct exponential functions given any form from the rule of 4 including arithmetic and geometric sequences

• Use graphs and tables to describe that an exponential function surpasses linear and polynomial functions in the long run

• Compare and contrast any two functions

Evidence Outcomes:

1. (+) Prove the Pythagorean identity sin^2 (θ)+cos^2 (θ)=1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. (CCSS: HS.F-TF.C.8)

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

  • Pythagorean Identity for sine, cosine, and tangent

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

• The Pythagorean Identity

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

• Prove the Pythagorean identity and use it to find sin(x), cos(x) or tan(x) given sin(x), cos(x) or tan(x) and the quadrant of the angle

Evidence Outcomes:

1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). (CCSS: HS.S-CP.A.1)

2. Explain that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. (CCSS: HS.S-CP.A.2)

3. Using the conditional probability of A given B as P(AandB)/P(B), interpret the independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. (CCSS: HS.S-CP.A.3)

4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in 10th grade. Do the same for other subjects and compare the results. (CCSS: HS.S-CP.A.4)

5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. (CCSS: HS.S-CP.A.5)

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

• Attend to precision

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

• Understand independence and conditional probability and use them to interpret data

  • Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations

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

• Vocabulary: sample space, unions, intersections, complements, independent, conditional probability

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

• Explain whether or not two events are independent (using the independence rule)

• Use conditional probability

• Construct and interpret a two-way table using conditional probability and addressing independence

Evidence Outcomes:

1. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. (CCSS: HS.S-CP.B.6)

2. Apply the Addition Rule, P(A or B)=P(A)+P(B)–P(A and B), and interpret the answer in terms of the model. (CCSS: HS.S-CP.B.7)

3. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B)=P(A)P(B∣A)=P(B)P(A∣B), and interpret the answer in terms of the model. (CCSS: HS.S-CP.B.8)

4. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. (CCSS: HS.S-CP.B.9)

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

• Model with mathematics

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

• Use the rules of probability to compute probabilities of compound events in a uniform probability model

  • Understand and use probability in the real world

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

• Vocabulary: conditional probability, (+)permutations, (+)combinations

• Addition rule: P(A or B) = P(A) + P(B) - P(A and B)

• (+)Multiplication rule: P(A and B) = P(A)P(B|A) = P(B)P(A|B)

• There are multiple representations for representing and understanding probabilities

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

• Find conditional probability and interpret the answer

• Use the addition rule and interpret the answer

• (+)Use the multiplication rule and interpret the answer

• (+)Use permutations and combinations to compute probabilities of compound events and solve problems.

Evidence Outcomes:

1. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (CCSS: HS.S-MD.B.6)

2. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). (CCSS: HS.S-MD.B.7)

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

  • Analyze decisions and strategies using probability concepts

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

• (+)What “fair” means in the context of statistics

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

• (+)Use a random process to make decisions (RNG, Random number tables, big ol’ hat and mix it up)

Evidence Outcomes:

1. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. (CCSS: HS.G-CO.C.9)

2. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 〖180〗^∘; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. (CCSS: HS.G-CO.C.10)

3. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (CCSS: HS.G-CO.C.11)

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

• Construct viable arguments and critique the reasoning of others

• Attend to precision

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

• Prove geometric theorems

  • Using theorems about lines, angles, triangles, and parallelograms

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

• Theorems about lines and angles

  • Vertical angles, alternate interior angles, corresponding angles,

  • Points on a perpendicular bisector are equidistant from the segment’s endpoints

• Theorems about triangles

  • Measures of an interior angles of a triangle sum to 180 degrees

  • Base angles of isosceles triangles are congruent

  • The segment joining midpoints of 2 sides of a triangle is parallel to the third side and ½ the length

  • The medians of a triangle meet at a point

• Theorems about parallelograms

  • Opposite sides and angles are congruent

  • Diagonals bisect each other

  • Rectangles are parallelograms with congruent diagonals

• How to structure a proof

  • 2-column proof

  • Paragraph proof

  • Flow proof

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

• Justify/prove theorems about lines and angles

• Justify/prove theorems about triangles

• Justify/prove theorems about parallelograms

• Use theorems to determine missing parts of angles/triangles/parallelograms

Evidence Outcomes:

1. Verify experimentally the properties of dilations given by a center and a scale factor. (CCSS: HS.G-SRT.A.1)

   • Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (CCSS: HS.G-SRT.A.1.a)

   • Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. (CCSS: HS.G-SRT.A.1.b)

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (CCSS: HS.G-SRT.A.2)

3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (CCSS: HS.G-SRT.A.3)

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

• Reason abstractly and quantitatively

• Use appropriate tools strategically

• Look for and make use of structure

• 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...

• Understand similarity in terms of similarity transformations

  • Using dilations to show similarity

  • Proportional reasoning to connect algebraic and geometric content

  • AA criterion for similar triangles

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

• Properties of dilations

• Definition of similarity

• AA criterion of similarity

• Indirect measurement

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

• Show the properties of dilations given a center and a scale factor

• Use auxiliary lines not part of the original figure

• Use geometric tools and technology in properties of dilations

• Use the properties of similarity to determine if two figures are similar

• Explain that two similar triangles have congruent corresponding angles and proportional corresponding sides

• Use AA criterion to establish similar triangles

Evidence Outcomes:

1. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (CCSS: HS.G-SRT.B.4)

2. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (CCSS: HS.G-SRT.B.5)

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

• Construct viable arguments and critique the reasoning of others

• Attend to precision

• 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...

• Prove theorems involving similarity

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

• Theorems about triangles

  • Parallel lines divide triangles proportionally (and its converse)

  • Pythagorean Theorem proved using triangle similarity

• Congruence and similarity criteria of triangles

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

• Create proofs of theorems about triangles

• Solve problems and prove relationships in geometric figures relating to similarity

Evidence Outcomes:

1. Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (CCSS: HS.G-SRT.C.6)

2. Explain and use the relationship between the sine and cosine of complementary angles. (CCSS: HS.G-SRT.C.7)

3. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* (CCSS: HS.G-SRT.C.8)

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

• Reason abstractly and quantitatively

• Model with mathematics

• 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...

• Define trigonometric ratios and solve problems involving right triangles

  • Apply trig ratios and Pythagorean Theorem to model and solve real-world problems

Inverse Trig is covered in F-TF.B.6

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

• The ratios of corresponding sides of similar triangles are the same based on corresponding angle measures leading to definitions of trigonometric ratios Mo

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

• Explain and use sine and cosine of complementary angles

• Use trigonometric ratios and the Pythagorean Theorem to solve right triangles, including application problems

• Relate triangle similarity and the trigonometric ratios for acute angles (right triangles)

Evidence Outcomes:

1. Prove that all circles are similar. (CCSS: HS.G-C.A.1)

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (CCSS: HS.G-C.A.2)

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. (CCSS: HS.G-C.A.3)

4. (+) Construct a tangent line from a point outside a given circle to the circle. (CCSS: HS.G-C.A.4)

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

• Construct viable arguments and critique the reasoning of others

• Use appropriate tools strategically

• 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...

• Understand and apply theorems about circles

  • Use geometric tools and technology in circle-related constructions

  • Observe relationships among angles in circles and use in a variety of scenarios

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

• Vocabulary: inscribed angles, circumscribed angles, central angles, radii, chords, tangent lines

• Relationships among inscribed angles, radii, and chords

  • Between central, inscribed, and circumscribed angles

  • Inscribed angles on a diameter are right angles

  • Radius of a circle is perpendicular to the tangent where the radius intersects the circle

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

• Prove that all circles are similar

• Construct the inscribed and circumscribed circles of a triangle

• Prove properties of angles for a quadrilateral inscribed in a circle

• (+) Construct a tangent line from a point outside a given circle to the circle

Evidence Outcomes:

1. (+) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (CCSS: HS.G-C.B.5)

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

ADVANCED

• Construct viable arguments and critique the reasoning 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...

• (+)Find arc lengths and areas of sectors of circles

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

• (+)The radian measure of the angle is the constant of proportionality

• (+)Why it is more convenient to use radian measure for a central angle in a circle, rather than degree measure

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

• Derive using similarity that the length of an arc intercepted by an angle is proportional to the radius

• Derive the formula for area of a sector

Evidence Outcomes:

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. (CCSS: HS.G-GPE.A.1)

2. (+) Derive the equation of a parabola given a focus and directrix. (CCSS: HS.G-GPE.A.2)

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

  • Circles

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

• Vocabulary: Pythagorean Theorem

• Complete the square

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

• Derive the equation of a circle (not just centered at the origin) using the Pythagorean theorem

• Complete the square to find the center and radius of a circle

Evidence Outcomes:

1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0,2). (CCSS: HS.G-GPE.B.4)

2. (Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (CCSS: HS.G-GPE.B.6)

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

• Reason abstractly and quantitatively

• Construct viable arguments and critique the reasoning of others

• 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...

• Use coordinates to prove simple geometric theorems algebraically (Modeling)

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

• Distance formula

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

• Use coordinates to prove theorems (ex. prove or disprove that a figure in the coordinate plane is a rectangle)

• Find a point on a directed line segment that partitions the line given a certain ratio

Evidence Outcomes:

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. (CCSS: HS.G-GMD.A.1)

2. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* (CCSS: HS.G-GMD.A.3)

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

  • Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems in the real-world

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

• Formulas:

  • Circumference of a circle

  • Area of a circle

  • Volume of a cylinder

  • Volume of a pyramid

  • Volume of a cone

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

• Give an informal argument to explain where the formulas for circumference, area of circle, volume of cylinder, volume of pyramid, volume of cone come from (ex. Explain why the formula for volume of a cylinder makes sense)

• Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems

Evidence Outcomes:

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* (CCSS: HS.G-MG.A.1)

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* (CCSS: HS.G-MG.A.2)

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* (CCSS: HS.G-MG.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

• 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...

• Apply geometric concepts in modeling situations (Modeling)

  • Use geometric shapes, their measures, and their properties to describe objects and solve problems in the real-world

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

• Vocabulary: Density

• Common area and volume formulas

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

• Use geometric shapes, their measures, and their properties to describe objects

• Apply concepts of density based on area and volume (ex. population density)

• Apply geometric methods to solve design problems (ex. Minimize cost)