RQ: Relationships between Quantities

(Framing Text): Extend the properties of exponents to rational exponents.

RQ.M.2HS.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (e.g., We define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5.)

Exponents and Power Rules

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

RQ.M.2HS.4: 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.

Points in the Complex Plane
Roots of a Quadratic

RQ.M.2HS.5: Use the relation i² = –1 and the commutative, associative and distributive properties to add, subtract and multiply complex numbers.

Points in the Complex Plane

(Framing Text): Perform arithmetic operations on polynomials.

RQ.M.2HS.6: Understand 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.

Addition and Subtraction of Functions
Addition of Polynomials
Modeling the Factorization of x²+bx+c

QF: Quadratic Functions and Modeling

(Framing Text): Interpret functions that arise in applications in terms of a context.

QF.M.2HS.7: 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 given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Absolute Value with Linear Functions
Cat and Mouse (Modeling with Linear Systems)
Exponential Functions
Function Machines 3 (Functions and Problem Solving)
General Form of a Rational Function
Graphs of Polynomial Functions
Logarithmic Functions
Points, Lines, and Equations
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Roots of a Quadratic
Slope-Intercept Form of a Line
Translating and Scaling Sine and Cosine Functions

QF.M.2HS.8: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., 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.)

General Form of a Rational Function
Introduction to Functions
Logarithmic Functions
Radical Functions
Rational Functions

QF.M.2HS.9: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Distance-Time Graphs
Distance-Time and Velocity-Time Graphs

(Framing Text): Analyze functions using different representations.

QF.M.2HS.10: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

QF.M.2HS.10.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

Linear Functions
Points, Lines, and Equations
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Slope-Intercept Form of a Line
Zap It! Game

QF.M.2HS.10.b: Graph square root, cube root and piecewise-defined functions, including step functions and absolute value functions.

Absolute Value with Linear Functions
Radical Functions
Translating and Scaling Functions

QF.M.2HS.11: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

QF.M.2HS.11.a: 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.

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Quadratics in Factored Form
Quadratics in Vertex Form
Roots of a Quadratic

QF.M.2HS.11.b: Use the properties of exponents to interpret expressions for exponential functions. (e.g., Identify percent rate of change in functions such as y = (1.02) to the 𝘵 power, 𝘺 = (0.97) to the 𝘵 power, 𝘺 = (1.01) to the 12𝘵 power, 𝘺 = (1.2) to the 𝘵/10 power, and classify them as representing exponential growth or decay.)

Compound Interest
Exponential Functions

QF.M.2HS.12: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum).

General Form of a Rational Function
Graphs of Polynomial Functions
Linear Functions
Logarithmic Functions
Quadratics in Polynomial Form
Quadratics in Vertex Form

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

QF.M.2HS.13: Write a function that describes a relationship between two quantities.

QF.M.2HS.13.a: Determine an explicit expression, a recursive process or steps for calculation from a context.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

QF.M.2HS.13.b: Combine standard function types using arithmetic operations. (e.g., 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.

Addition and Subtraction of Functions

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

QF.M.2HS.14: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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.

Absolute Value with Linear Functions
Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Quadratics in Vertex Form
Radical Functions
Rational Functions
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Translations
Zap It! Game

QF.M.2HS.15: Find inverse functions. 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.

Logarithmic Functions

(Framing Text): Construct and compare linear, quadratic, and exponential models and solve problems.

QF.M.2HS.16: Using graphs and tables, observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically; or (more generally) as a polynomial function.

Compound Interest
Introduction to Exponential Functions

EE: Expressions and Equations

(Framing Text): Interpret the structure of expressions.

EE.M.2HS.17: Interpret expressions that represent a quantity in terms of its context.

EE.M.2HS.17.a: Interpret parts of an expression, such as terms, factors, and coefficients.

Compound Interest
Exponential Growth and Decay
Unit Conversions

EE.M.2HS.17.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.

Compound Interest
Exponential Growth and Decay
Translating and Scaling Functions
Using Algebraic Expressions

EE.M.2HS.18: Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x⁴)⁴ – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).

Dividing Exponential Expressions
Equivalent Algebraic Expressions I
Equivalent Algebraic Expressions II
Exponents and Power Rules
Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Multiplying Exponential Expressions
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Simplifying Trigonometric Expressions
Solving Algebraic Equations II
Using Algebraic Expressions

(Framing Text): Write expressions in equivalent forms to solve problems.

EE.M.2HS.19: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

EE.M.2HS.19.a: Factor a quadratic expression to reveal the zeros of the function it defines.

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

EE.M.2HS.19.c: Use the properties of exponents to transform expressions for exponential functions.

Exponents and Power Rules

(Framing Text): Create equations that describe numbers or relationships.

EE.M.2HS.20: Create equations and inequalities in one variable and use them to solve problems.

Absolute Value Equations and Inequalities
Arithmetic Sequences
Compound Interest
Exploring Linear Inequalities in One Variable
Exponential Growth and Decay
Geometric Sequences
Modeling and Solving Two-Step Equations
Quadratic Inequalities
Solving Linear Inequalities in One Variable
Solving Two-Step Equations

EE.M.2HS.21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

2D Collisions
Air Track
Compound Interest
Determining a Spring Constant
Golf Range
Points, Lines, and Equations
Slope-Intercept Form of a Line

EE.M.2HS.22: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)

Area of Triangles
Solving Formulas for any Variable

(Framing Text): Solve equations and inequalities in one variable.

EE.M.2HS.23: Solve quadratic equations in one variable.

EE.M.2HS.23.a: 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.

Roots of a Quadratic

EE.M.2HS.23.b: Solve quadratic equations by inspection (e.g., for x² = 49), 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.

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Points in the Complex Plane
Roots of a Quadratic

(Framing Text): Use complex numbers in polynomial identities and equations.

EE.M.2HS.24: Solve quadratic equations with real coefficients that have complex solutions.

Roots of a Quadratic

AP: Applications of Probability

(Framing Text): Understand independence and conditional probability and use them to interpret data.

AP.M.2HS.28: 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”).

Independent and Dependent Events
Probability Simulations
Theoretical and Experimental Probability

AP.M.2HS.29: Understand 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.

Independent and Dependent Events

AP.M.2HS.30: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret 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.

Independent and Dependent Events

AP.M.2HS.31: 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. (e.g., 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 tenth grade. Do the same for other subjects and compare the results.)

Histograms

AP.M.2HS.32: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. (e.g., Compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.)

Independent and Dependent Events

(Framing Text): Use the rules of probability to compute probabilities of compound events in a uniform probability model.

AP.M.2HS.33: 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.

Independent and Dependent Events

AP.M.2HS.35: 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.

Independent and Dependent Events

AP.M.2HS.36: Use permutations and combinations to compute probabilities of compound events and solve problems.

Binomial Probabilities
Permutations and Combinations

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

AP.M.2HS.37: Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).

Lucky Duck (Expected Value)
Probability Simulations
Theoretical and Experimental Probability

AP.M.2HS.38: 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).

Estimating Population Size
Probability Simulations
Theoretical and Experimental Probability

SRT: Similarity, Right Triangle Trigonometry, and Proof

(Framing Text): Understand similarity in terms of similarity transformations.

SRT.M.2HS.39: Verify experimentally the properties of dilations given by a center and a scale factor.

SRT.M.2HS.39.a: 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.

Dilations

SRT.M.2HS.39.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations
Similar Figures

SRT.M.2HS.40: 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.

Similar Figures

(Framing Text): Prove geometric theorems.

SRT.M.2HS.42: 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. Implementation may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.2HS.C.3.

Investigating Angle Theorems

SRT.M.2HS.43: 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.

Isosceles and Equilateral Triangles
Proving Triangles Congruent
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Triangle Angle Sum
Triangle Inequalities

SRT.M.2HS.44: 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.

Parallelogram Conditions
Special Parallelograms

(Framing Text): Prove theorems involving similarity.

SRT.M.2HS.45: 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.

Isosceles and Equilateral Triangles
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures
Triangle Angle Sum
Triangle Inequalities

SRT.M.2HS.46: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Dilations
Perimeters and Areas of Similar Figures
Similarity in Right Triangles

(Framing Text): Define trigonometric ratios and solve problems involving right triangles.

SRT.M.2HS.48: Understand 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.

Sine, Cosine, and Tangent Ratios

SRT.M.2HS.50: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

(Framing Text): Prove and apply trigonometric identities.

SRT.M.2HS.51: Prove the Pythagorean identity sin²(θ) + cos²(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.

Simplifying Trigonometric Expressions
Sine, Cosine, and Tangent Ratios

CWC: Circles With and Without Coordinates

(Framing Text): Understand and apply theorems about circles.

CWC.M.2HS.53: 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.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

CWC.M.2HS.54: Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

(Framing Text): Find arc lengths and areas of sectors of circles.

CWC.M.2HS.56: 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.

Chords and Arcs

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

CWC.M.2HS.57: 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.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

CWC.M.2HS.58: Derive the equation of a parabola given the focus and directrix.

Parabolas

(Framing Text): Explain volume formulas and use them to solve problems.

CWC.M.2HS.60: 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.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

CWC.M.2HS.61: Use volume formulas for cylinders, pyramids, cones and spheres to solve problems. Volumes of solid figures scale by k3 under a similarity transformation with scale factor k.

Prisms and Cylinders
Pyramids and Cones