Ridgefield Public Schools - Algebra 2/Precalculus Honors

High School - Algebra 2/Precalculus Honors

LONG TERM GOALS

Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.

Functions and Inverse Functions (early September)

BIG IDEAS AND ESSENTIAL QUESTIONS

Mathematicians developed specific notations so that any representation can be communicated in a succinct and uniform manner.
Functions are a subset of relations because they hold a specific property that every element in its domain can be mapped to only one element of its range.
Mathematicians can compare different functions, discuss their properties and characteristics, or operate on functions to make new functions.
A single function will not always model the behavior of a real-world context, resulting in the need for domain restrictions and/or the use of piecewise functions.
An inverse function undoes another function by mapping the original function's output to its input.
Mathematicians create and analyze different representations (table of values, algebraic, graphical, geometrical) to broaden their scope of modeling.

LEARNING OUTCOMES

Students will be skilled at . . .

distinguishing a relation from a function.
identifying the domain and range of a relation or function and using interval notation to denote such characteristics.
writing and evaluating functions using function notation.
analyzing, interpreting and representing key features of graphs, including (but not limited to): (1) intercepts, (2) extrema, (3) increasing, decreasing and constant intervals, (4) end behavior, (5) continuity, (6) x-axis, y-axis or origin symmetry, and (7) even or odd function classification.
identifying families of functions (constant, linear, quadratic, cubic, square root, cube root, rational, absolute value).
creating transformations of parent functions (i.e. vertical and horizontal shifts, vertical stretch and shrink, and reflection over the x-axis) as well as writing functions representing transformations given a graph, table, written description.
performing operations and compositions with multiple functions.
graphing a function and its inverse, including a restricted domain if necessary.
identifying whether or not two functions are inverses both algebraically (using composition of functions) and graphically (symmetry over the line y=x).
modeling a single function in a variety of ways (i.e. table, graph, mapping, equation).

CONTENT STANDARDS

Understand the concept of a function and use function notation.

HSF-IF.A.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

HSF-IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4. 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.

HSF-IF.B.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7b - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

HSF-IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that describes a relationship between two quantities.

HSF-BF.A.1b - Combine standard function types using arithmetic operations.

HSF-BF.A.1c - Compose functions.

Build new functions from existing functions.

HSF-BF.B.3 - 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.

HSF-BF.B.4 - Find inverse functions.

HSF-BF.B.4a - 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) =2x³ for x > 0 or f(x) = (x+1)/(x–1) for x ? 1.

HSF-BF.B.4b - Verify by composition that one function is the inverse of another.

HSF-BF.B.4c - Read values of an inverse function from a graph or a table, given that the function has an inverse.

HSF-BF.B.4d - Produce an invertible function from a non-invertible function by restricting the domain.

CONTENT

The students will know . . .

a function is a relation in which every element in the domain only pairs with one element in the range.
the domain of a function is the set of all of its x-values while the range of a function is the set of all of its y-values.
function notation is a way to represent and evaluate a relation as a function.
interval notation is a way in which mathematicians can succinctly represent intervals.
a piecewise function is a function which is defined by multiple functions, each of which is applied to a certain interval of the piecewise function's overall domain.
key features of graphs, including (but not limited to): (1) intercepts, (2) extrema, (3) increasing, decreasing and constant intervals, (4) end behavior, (5) continuity, (6) x-axis, y-axis or origin symmetry, and (7) even or odd function classification through the analysis of graphical and algebraic representations.
families of functions (constant, linear, quadratic, cubic, square root, cube root, rational, absolute value) can be categorized based on the characteristics of their graphs and modeled equations.
any parent function (constant, linear, quadratic, cubic, square root, cube root, rational, absolute value) can take on transformations (i.e. vertical and horizontal shifts, vertical stretch and shrink, and reflection over the x-axis) to create new functions.
restricting a function's domain may allow its inverse function to then also be a function.
a function and its inverse show symmetry over the line y=x.
to find a function's inverse, one can do so algebraically.

Applications of Linear Functions (early October)

BIG IDEAS AND ESSENTIAL QUESTIONS

Linear regression is a statistical technique that allows one to take a set of discreet data and use it to identify a linear pattern that best represent that set of data.
The solution set to any system of linear inequalities must be infinite or null based upon the number of ordered pairs that satisfy all inequalities in the system.
Linear programming is a method to achieve the best outcome in a mathematical model when there are multiple limitations or constraints which are represented by linear relationships.

LEARNING OUTCOMES

Students will model linear equations and inequalities given real world applications.
Students will apply linear equations and systems of linear equations to real world models
Students will apply systems of linear inequalities to real world models
Students will take a set of data and perform a linear regression to describe the general trend of the data set.
Students will be able to model the greatest integer function and other piece-wise defined linear functions given real world applications.

CONTENT STANDARDS

Reason quantitatively and use units to solve problems

HSN-Q.A.1 - Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

HSN-Q.A.2 - Define appropriate quantities for the purpose of descriptive modeling.

HSN-Q.A.3 - Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Interpret the structure of expressions

HSA-SSE.A.1 - Interpret expressions that represent a quantity in terms of its context.

HSA-SSE.A.1a - Interpret parts of an expression, such as terms, factors, and coefficients.

Create equations that describe numbers or relationships

HSA-CED.A.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.

HSA-CED.A.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSA-CED.A.3 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Solve equations and inequalities in one variable

HSA-REI.B.3 - Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve systems of equations

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

Represent and solve equations and inequalities graphically

HSA-REI.D.10 - Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

HSA-REI.D.11 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

HSA-REI.D.12 - Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Interpret functions that arise in applications in terms of the context

HSF-IF.B.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations

HSF-IF.C.7a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

Construct and compare linear and exponential models and solve problems

HSF-LE.A.1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

HSF-LE.A.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

HSF-LE.A.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Interpret expressions for functions in terms of the situation they model

HSF-LE.B.5 - Interpret the parameters in a linear or exponential function in terms of a context.

Summarize, represent, and interpret data on two categorical and quantitative variables

HSS-ID.B.6a - Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

HSS-ID.B.6c - Fit a linear function for scatter plots that suggest a linear association.

Interpret linear models

HSS-ID.C.7 - Interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data.

HSS-ID.C.8 - Compute (using technology) and interpret the correlation coefficient of a linear fit.

CONTENT

The students will know . . .

the three forms of a linear equation (standard, point-slope, and slope-intercept) and how to apply it them to various application problems.
that a graphing calculator can find a line of best fit using linear regression.
how to describe the strength of the linear regression model using r^2.
the definition of the feasible region in a linear programming problem.
that the maximum or minimum value in a linear programming problem must come from one of the vertices of the feasible region.
the greatest integer function and how to apply it to various application problems.

Quadratic Functions (mid October)

BIG IDEAS AND ESSENTIAL QUESTIONS

Understanding the three forms (standard, intercept, vertex) of quadratic functions allows for efficiency in graphing, modeling and writing quadratic functions.
Becoming familiar with the five solving techniques (factoring, square roots, completing the square, quadratic formula, graphing) for quadratic functions allows a mathematician to approach a problem in the most efficient manner.
The complex number system allows one to find all possible solutions of a quadratic function by defining the square root of -1 as the imaginary number, i.
Quadratic regression is a statistical technique that allows one to take a set of discreet data and use it to identify a quadratic pattern that best represents that set of data.
Solving any quadratic inequality is found by graphically discovering the infinite region containing all possible values satisfying the inequality, therefore allowing one to conclude the equation has an infinite solution set.

LEARNING OUTCOMES

Students will be skilled at . . .

graphing, writing and modeling quadratic functions in three forms (vertex, standard and intercept forms).
identifying key features of quadratic functions, including all features from Unit 1 with the addition of a quadratic function's vertex and explaining what those characteristics represent if presented in a real world context.
using factoring, completing the square, graphing and/or the quadratic formula to solve a quadratic function with real and/or complex solutions.
identifying the solutions to a quadratic function on its graph.
understanding a complex number system exists outside of the real number system and how complex numbers are used in modeling and solving quadratic functions.
simplifying imaginary numbers for any given positive integer power.
adding, subtracting, multiplying and dividing complex numbers, including rationalizing the denominator.
modeling quadratic functions in all forms.
using the properties of quadratic functions to describe and analyze real-world applications, including quadratic regression, projectile motion and optimization of geometric area.

CONTENT STANDARDS

Understand the concept of a function and use function notation.

HSF-IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4 - 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.

HSF-IF.B.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7a - Graph linear and quadratic functions and show intercepts, maxima, and minima.

HSF-IF.C.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF-IF.C.8a -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.

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that describes a relationship between two quantities.

CONTENT

The students will know that . . .

graphing, writing and modeling quadratic functions can occur in three forms (vertex, standard and intercept forms).
zeroes, roots, and solutions are all synonyms for the x-intercepts of a quadratic function.
quadratic equations with real coefficients that have either real and/or complex solutions can be solved by (1) factoring, (2) taking square roots, (3) completing the square, (4) quadratic formula, and/or (5) graphing
a complex number system exists outside of the real number system and complex numbers are used in modeling and solving quadratic functions.
there is a complex number i such that i^2 = sqrt(-1) and every complex number takes on the form a plus/minus bi where a and b are real numbers.
there is a pattern to simplifying imaginary numbers for any given positive integer that can be derived using properties of exponents.
a complex number should be written in its simplified form by rationalizing its denominator when needed.
the properties of quadratic functions can be used to describe and analyze real-world applications including quadratic regression, projectile motion and optimization of geometric area.

Polynomial Functions (early November)

BIG IDEAS AND ESSENTIAL QUESTIONS

A binomial expression that is a sum/difference of perfect cubes can be written as an equivalent expression made up of two factors, one of which is linear and the other is an irreducible quadratic.
One can relate the multiplicity of the linear factors of a polynomial function to its number of turning points as well as its behavior at each root which allows one to sketch an accurate model of the interior of its graph as well as write an accurate algebraic model of the polynomial function.
One can relate the degree of a polynomial function to its end behavior which allows one to sketch an accurate model on both ends of the graph; as x approaches infinity and as x approaches negative infinity.\
Cubic and quartic regression are statistical techniques that allow one to take a set of discreet data and use it to identify a pattern that best represents that set of data.
When dividing a polynomial expression by a linear expression, one can determine if that linear expression is a factor of the given polynomial expression.
Any polynomial function has a finite number of possible rational zeros that can be found by taking the ratio of factors of the constant term to the factors of the leading coefficient.

LEARNING OUTCOMES

Students will be skilled at . . .

factoring the sum/difference of perfect cubes.
solving polynomial equations for all real and/or complex solutions by factoring, long/synthetic division and/or applying the Factor, Remainder and Rational Root Theorems.
proving the existence of real zeros of a polynomial function by utilizing the Intermediate Value Theorem.
identifying key features of polynomial functions, including all features from Unit 1 with the addition of a polynomial function's multiplicity at roots, degree, end behavior and maximum number of turning points and explaining what those characteristics represent if presented in a real world context.
using key attributes (i.e. roots, multiplicity at roots, degree, end behavior and maximum number of turning points) to sketch polynomial functions in both factored and standard forms.
modeling and writing polynomial functions given degree, roots (real and/or complex), multiplicity, and at least one point on the curve.
using the properties of polynomial functions to describe and analyze real-world applications, including cubic and quartic regression, and optimization of geometric area.

CONTENT STANDARDS

Understand the concept of a function and use function notation.

HSF-IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4 - 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.

HSF-IF.B.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7c - Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

HSF-IF.C.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF-IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that describes a relationship between two quantities.

CONTENT

The students will know that . . .

(a³ ± b³) can be re-written as (a ± b)(a² ± ab + b²)
the maximum number of turning points in a polynomial function is one less than that of its degree.
when a linear factor has a multiplicity that is even, its respective root "bounces/touches and turns" at that point, but when a linear factor has a multiplicity that is odd, its respective root "crosses through" at that point.
a polynomial function of even degree will have the same end behavior at both ends of its graph whereas a polynomial function of odd degree will have different end behavior at each end.
cubic and quartic regression are statistical techniques that allow one to take a set of discreet data and use it to identify a pattern that best represents that set of data.
if a polynomial expression is divided by a linear expression (x - c) and results in no remainder, then that linear expression is a factor of the polynomial and (c, 0) is a point on the graph. (The Factor Theorem)
if a polynomial expression is divided by a linear expression (x - c) and results in a remainder, then the point (c, f(c)) exists where f(c) is the remainder that was found. (The Remainder Theorem)
the finite number of possible rational zeros of any polynomial function can be found by taking the ratio of factors of the constant term to the factors of the leading coefficient. (The Rational Roots Theorem)
a polynomial function can be solved by factoring, long/synthetic division and/or applying the Factor, Reminder and Rational Root Theorems.
the Intermediate Value Theorem can be applied to polynomial functions to prove the existence of real zeros.

Rational Expressions and Functions (late November)

BIG IDEAS AND ESSENTIAL QUESTIONS

Rational expressions can be added, subtracted, multiplied, or divided using the same strategy that one would use when performing these same operations on fractions containing numeric values only.
When solving or performing operations on rational expressions or equations, the manipulation used may force a change in the function, resulting in a different domain than what was originally intended and therefore creating the possibility of extraneous solutions or values.
Variable connected in direct proportionality imply a linear relationship while variables connected by inverse proportionality imply a mathematical model in rational form.
One can relate the domain restrictions of a rational function to its vertical asymptotes and holes which allows one to sketch an accurate model of the interior of its graph as well as write an accurate algebraic model of the rational function.
One can relate the end behavior model of a rational function to its horizontal or slant asymptote which allows one to sketch an accurate model on both ends of the graph; as x approaches infinity and as x approaches negative infinity.
Through the process of partial fraction decomposition, one can decompose a rational expression to return it to its expanded addition or subtraction form.

LEARNING OUTCOMES

Students will be skilled at . . .

adding, subtracting, multiplying and dividing rational expressions as well as simplifying complex fractions.
solving simple rational equations in one variable, solving rational equations that lead to linear or quadratic equations, as well as solving rational equations where extraneous solutions may also arise.
graphing, writing and analyzing rational functions.
identifying key features of rational functions, including all features from Unit 1 with the addition of a rational function's asymptotes, holes and end behavior and explaining what those characteristics represent if presented in a real world context.
identifying the limits as x approaches infinity and as x approaches negative infinity.
using the properties of rational functions to describe and analyze real-world applications, including direct, inverse and joint variation.
partial fraction decomposition including denominators with linear factors, denominators with repeating factors and denominators with irreducible quadratic factors.

CONTENT STANDARDS

Understand the concept of a function and use function notation.

HSF-IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4 - 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.

HSF-IF.B.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7d - Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

HSF-IF.C.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF-IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that describes a relationship between two quantities.

CONTENT

The students will know that . . .

(a³ ± b³) can be re-written as (a ± b)(a² ± ab + b²)
the maximum number of turning points in a polynomial function is one less than that of its degree.
when a linear factor has a multiplicity that is even, its respective root "bounces/touches and turns" at that point, but when a linear factor has a multiplicity that is odd, its respective root "crosses through" at that point.
a polynomial function of even degree will have the same end behavior at both ends of its graph whereas a polynomial function of odd degree will have different end behavior at each end.
cubic and quartic regression are statistical techniques that allow one to take a set of discreet data and use it to identify a pattern that best represents that set of data.
if a polynomial expression is divided by a linear expression (x - c) and results in no remainder, then that linear expression is a factor of the polynomial and (c, 0) is a point on the graph. (The Factor Theorem)
if a polynomial expression is divided by a linear expression (x - c) and results in a remainder, then the point (c, f(c)) exists where f(c) is the remainder that was found. (The Remainder Theorem)
the finite number of possible rational zeros of any polynomial function can be found by taking the ratio of factors of the constant term to the factors of the leading coefficient. (The Rational Roots Theorem)
a polynomial function can be solved by factoring, long/synthetic division and/or applying the Factor, Reminder and Rational Root Theorems.
the Intermediate Value Theorem can be applied to polynomial functions to prove the existence of real zeros.

Power Functions and Exponent Properties (early December)

BIG IDEAS AND ESSENTIAL QUESTIONS

A radical expression can be written as an equivalent expression involving rational exponents, allowing one to understand the inverse relationship that is then needed to solve equations involving radicals or rational exponents as well as understand the graphical representation of a power function with fractional powers.
A monomial that contains a negative exponent has an equivalent form found through the process of repeated division, allowing one to connect a power function to that of a rational function.
Power functions with negative whole number exponents are simple examples of rational functions and thus contain similar characteristics of rational functions.
Power functions with positive whole number exponents are simple examples of polynomial functions and thus contain similar characteristics of polynomial functions.

LEARNING OUTCOMES

Students will be able to . . .

to extend the properties of exponents to rational exponents.
rewrite expressions involving radicals and rational exponents using the properties of exponents.
solve simple equations involving rational exponents and radicals in one variable.
identify extraneous solutions to equations involving rational exponents and radicals in one variable.
analyze power functions based on their exponent value.
write power functions.
model power functions.
graph power functions.

CONTENT STANDARDS

Extend the properties of exponents to rational exponents.

HSN-RN.A.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.

HSN-RN.A.2 - Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Interpret the structure of expressions.

HSA-SSE.A.2 - Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems.

HSA-SSE.B.3 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Understand solving equations as a process of reasoning and explain the reasoning.

HSA-REI.A.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4 - 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.

HSF-IF.B.5- Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7b - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

CONTENT

The students will know . . .

the components of a rational power correspond to the power and root of a radical expression.
the properties of exponents (power rule, product rule, and quotient rule) apply to rational exponents.
the algebraic process for solving a radical equation.
the algebraic process for solving simple equations involving rational exponents.
the domain of a radical function and how that impacts the solutions one finds while solving either radical equations or equations with rational powers.
power functions that involve even valued powers are even functions
power functions that involve odd valued powers are odd functions
power functions that involve positive whole number powers take on the same behavior as polynomial functions.
power functions that involve negative powers take on the same behavior as rational functions.
power functions that involve powers greater than one grow faster than power functions that involve powers that are between 0 and 1.

Exponential and Logarithmic Functions (mid December)

BIG IDEAS AND ESSENTIAL QUESTIONS

An exponential function models the numerical pattern represented by repeated multiplication within a geometric sequence.
The domain of the logarithmic function of base b is the same as the range of the exponential function b^x and the range of the logarithmic function of base b is the same as the domain of the exponential function b^x, which means that the logarithmic and exponential functions are inverse of each other.
Becoming familiar with the logarithmic form of an exponential function allows a mathematician to approach any problem involving an exponential or logarithmic function in the most efficient manner.
Given that exponential functions and logarithmic functions hold an inverse relationship, one can use this inverse relationship to solve both exponential or logarithmic equations.
Solving logarithmic equations may require one to apply the properties of logarithms because, without condensing the expression, one would not be able to convert it to its exponential form, thus allowing for the solving to take place.
Exponential and logarithmic regression are statistical techniques that allows one to take a set of discreet data and use it to identify either an exponential or logarithmic pattern that best represents that set of data.

LEARNING OUTCOMES

Students will be skilled at...

graphing exponential and logarithmic functions.
identifying key features of exponential functions, including all features from Unit 1 with the addition of a exponential and logarithmic function's asymptote and end behavior and explaining what those characteristics represent if presented in a real world context.
solving exponential and logarithmic equations.
working through applications of exponential functions.
converting between logarithmic and exponential forms.
evaluating logarithms.
applying the properties of logarithms to simplify expressions and solve equations.

CONTENT STANDARDS

Understand the concept of a function and use function notation.

HSF-IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4 - 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.

HSF-IF.B.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7e - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

HSF-IF.C.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF-IF.C.8b - Use the properties of exponents to interpret expressions for exponential functions.

HSF-IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that describes a relationship between two quantities.

Build new functions from existing functions.

HSF-BF.B.4 - Find inverse functions.

HSF-BF.B.4b - Verify by composition that one function is the inverse of another.

HSF-BF.B.4c - Read values of an inverse function from a graph or a table, given that the function has an inverse.

HSF-BF.B.5 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Construct and compare linear and exponential models and solve problems.

HSF-LE.A.1 - Distinguish between situations that can be modeled with linear functions and with exponential functions.

HSF-LE.A.1a - Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

HSF-LE.A.1c - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

HSF-LE.A.2 - Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

HSF-LE.A.4 - For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret expressions for functions in terms of the situation they model.

HSF-LE.B.5 - Interpret the parameters in a linear or exponential function in terms of a context.

CONTENT

The students will know that . . .

The numerical pattern represented by repeated multiplication within a geometric sequence offers an exponential function.
Logarithmic functions are the inverse function to that of a exponential function.
Converting between logarithmic and exponential forms can be done using the definition: the logarithm of x to base b is the solution y to the equation b^y = x; which is sometimes necessary for evaluating, simplifying and solving.
The power, product, and quotient properties of logarithms; which are sometimes necessary for evaluating, simplifying and solving.
How to take a set of discreet data and use it to identify either an exponential or logarithmic pattern that best represents that set of data.
The properties of exponential and logarithmic functions can be used to describe and analyze real-world applications including exponential growth, exponential decay, compound interest and continuous compounding.

Statistics and Probability (early January)

BIG IDEAS AND ESSENTIAL QUESTIONS

Statisticians developed specific vocabulary so that any statistics can be communicated in a succinct and uniform manner.
The mean, median, and mode are three different ways to describe the central position within a set of data, allowing one to determine a possible most common data point, but if there are outliers or skewed distributions, some measures are more appropriate than others.
The range and standard deviation of a set of data allows a way to describe the spread of the data, which can be important information when making conclusions about the whole data set.
Knowing whether a data set is uniform, skewed, or normal allows one to understand how a data set is distributed and will allow one to know which statistical tests are most appropriate to apply to the data set.
The probability density underneath a normal curve holds a predictable pattern based on the number of standard deviations a region is away from the mean, allowing one to find the percent of data that falls within one (68%), two (95%), and three (99.7%) standard deviations from the mean.
There are a variety of sampling techniques that can be utilized within an study, each of which must be considered then chosen based on the question being posed and the topic being studied.
There are a variety of experimental designs that can be utilized within a study, each of which must be considered and one chosen based on the question being posed and the topic being studied.
Two way tables are a way to organize data involving two categorical variables, allowing one to easily compute conditional probabilities.

LEARNING OUTCOMES

Students will be able to . . .

determine the difference between a sample and a population.
identify if a quantity is a statistic or a parameter.
explain how randomization relates to each method of data collection.
recognize the purposes of and differences among sample surveys, experiments, and observational studies.
identify the type of sampling method used in a statistical design?
identify the possible bias caused in the sampling process.
describe the measures of central tendency and apply them to describe a data set.
describe the standard deviation and apply it to describe a data set.
identify the shape of a frequency distribution.
describe and predict the approximate location of the measures of central tendency when data sets contain an outlier or skewed distribution.
calculate simple probability.
use the Empirical Rule to find probabilities for normally distributed data sets.
find probabilities on the normal curve using calculator.
calculate conditional probability using two-way tables.

CONTENT STANDARDS

Summarize, represent, and interpret data on a single count or measurement variable.

HSS-ID.A.2 - Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

HSS-ID.A.3 - Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

HSS-ID.A.4 - Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve.

Summarize, represent, and interpret data on two categorical and quantitative variables.

HSS-ID.B.5 - Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.

Understand and evaluate random processes underlying statistical experiments.

HSS-IC.A.1 - Understand that statistics is a process for making inferences about population parameters based on a random sample from that population.

HSS-IC.A.2 - Decide if a specified model is consistent with results from a given data-generating process, e.g. using simulation.

Make inferences and justify conclusions from sample surveys, experiments and observational studies.

HSS-IC.B.3 - Recognize the purposes of and differences among sample surveys, experiments and observational studies; explain how randomization relates to each.

HSS-IC.B.4 - Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

HSS-IC.B.5 - Use data from a randomized experiment to compare two treatments; justify significant differences between parameters through the use of simulation models for random assignment.

HSS-IC.B.6 - Evaluate reports based on data.

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

HSS-CP.A.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”).

HSS-CP.A.2 - 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.

HSS-CP.A.3 - 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.

HSS-CP.A.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.

HSS-CP.A.5 - Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

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

HSS-CP.B.6 - 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.

CONTENT

The students will know . . .

a sample is a subset of a populations.
a quantity that describes a sample is called a statistic and a quantity that describes a population is a parameter.
some sampling techniques include: Simple Random Sample, Stratified, Cluster, Convenience, Systematic.
the mean, median, and mode are the three measures of central tendency that describe a data set.
a distribution can be skewed left, skewed right, uniform, normal, or inverse normal.
standard deviation describes the spread of a data set.
the mean is the measure of central tendency that i most effected by an outlier or skewed data set.
an outlier exists if the data point falls outside of 1.5 times the interquartile range.
the Empirical Rule describes the probability of data within 1 standard deviation of the mean (68%), 2 standard deviations of the mean (95%), or 3 standard deviations of the mean (99.7%) for a normally distributed set of data.
the graphing calculator's normalcdf feature allows one to find probabilities for normally distributed data sets when the probabilities one is trying to find does not fall perfectly within a standard deviation.
a two-way table is an organizational structure than helps to describe the same logical constructs as a venn diagram.

Trigonometry (late January)

BIG IDEAS AND ESSENTIAL QUESTIONS

The unit circle ties together Euclidean geometry (circles, points, lines, triangles, etc.), coordinate geometry (the x-y plane, coordinates on the plane, etc), and trigonometry (the sine and cosine ratios, etc) by describing x and y motion in the coordinate system using right triangles and then mapping that motion along a circle.
As a point rotates counterclockwise around a circular path (unit circle) in the complex plane the real component oscillates back and forth along the real axis creating a cosine (change in x motion) or sine (change in y motion) function.
Tangent and reciprocal trigonometric functions exist from the derivation of sine and cosine functions allowing one to model using the whole family of trigonometric functions.
Using the domain of the sine, cosine and tangent functions would produce graphical representations for inverses that are relations, but not functions, which requires the need for domain restrictions on the original functions in order to produce an inverse that is a function.
Becoming familiar with the trigonometric identities allows a mathematician to approach any problem involving trigonometric expressions in the most efficient manner.
Solving trigonometric equations may require one to apply trigonometric identities because, without them, one would not be able to isolate a given variable.
When given limited information about a triangle's sides and angle measures, the use of the Law of Sines and/or Law of Cosines become necessary to make the appropriate determination of whether one, two or no triangles exists.

LEARNING OUTCOMES

Students will be skilled at . . .

working with the unit circle.
graphing sine, cosine, and tangent functions, and their reciprocal functions.
identifying key features of trigonometric functions, including all features from Unit 1 and explaining what those characteristics represent if presented in a real world context.
recognizing graphs of inverse trigonometric functions.
trigonometric identities and proofs.
solving trigonometric equations.
applications with Trigonometry.
Law of Sines
Law of Cosines
area of triangle

CONTENT STANDARDS

Understand the concept of a function and use function notation.

HSF-IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4 - 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.

HSF-IF.B.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Analyze functions using different representations.

HSF-IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7e - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

HSF-IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that describes a relationship between two quantities.

Build new functions from existing functions.

HSF-BF.B.3 - 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.

HSF-BF.B.4c - Read values of an inverse function from a graph or a table, given that the function has an inverse.

HSF-BF.B.4d - Produce an invertible function from a non-invertible function by restricting the domain.

Extend the domain of trigonometric functions using the unit circle.

HSF-TF.A.1 - Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

HSF-TF.A.2 - Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

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

HSF-TF.A.4 - Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions.

HSF-TF.B.5 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

HSF-TF.B.6 - Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

HSF-TF.B.7 - Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Prove and apply trigonometric identities.

HSF-TF.C.8 -Prove the Pythagorean identity sin²(?) + cos²(?) = 1 and use it to calculate trigonometric ratios.

HSF-TF.C.9 - Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

CONTENT

The students will know that . . .

the unit circle is a circle with a radius equal to one and its measures can be derived using right triangle trigonometry on a cartesian plane with the origin as its center.
the trigonometric measures represented along the unit circle (sine, cosine, tangent, cosecant, secant, cotangent) can be used to graph each of the ratios along the x-y plane as an oscillating function.
trigonometric functions can be used to represent sinusoidal phenomena.
tangent and reciprocal functions are derived from the sine and cosine functions.
the inverse relations of sine, cosine and tangent functions will require a restriction on their domain to yield inverse functions.
the trigonometric identities can be derived through prior knowledge of triangle relationships and then used to simplify expressions and solve equations.
solving trigonometric equations may require one to apply trigonometric identities.
solving trigonometric equations may result in one, more than one, or no solutions.
the Law of Sines and the Law of Cosines can be derived through prior knowledge of triangle relationships and becomes necessary to solve some more complicated triangles.
given information about a triangle may result in a case of one, two, or no triangles.
when given the measures of all sides of a triangle, the area of any triangle can be found using Heron's formula.

Parametric, Polar and Vectors (late March)

BIG IDEAS AND ESSENTIAL QUESTIONS

Mathematicians developed specific notations so that any representation can be communicated in a succinct and uniform manner.
A vector allows one to not only represent magnitude, but also direction allowing one to determine the exact position of one point in space relative to another.
The scalar product between vectors represents the angle between the vectors and magnitude of the vectors.
The scalar product determines perpendicular and parallel vectors by projecting component vectors.
Mathematicians can operate on vectors by adding, subtracting or finding the dot product in two dimensional space and thus creating new vectors.
A function can become parametrized when a function's x and y variables are each functions of a third variable allowing one to analyze the change in both the x and y directions.
The polar coordinate system provides a two-dimensional coordinate system in which each point on a plane can be determined by a distance from a reference point and an angle from a reference direction.
When the radius of a polar equation is equal to one, you are producing a unit circle, but you are creating a dilation of the unit circle when the radius is anything other than one.
Similar to one analyzing how the dependent variable is being effected by the independent variable in a rectangular coordinate system, one can analyze how the dependent variable is being effected by the independent variable in the polar coordinate system as the distance from a point to the origin changes in a counterclockwise motion.

LEARNING OUTCOMES

Students will be able to . . .

graph vectors and find a position vector.
add and subtract vectors.
find a scalar product and the magnitude of a vector.
find a unit vector.
find a vector from its direction and magnitude
work with objects in static equilibrium.
find the dot product of two vectors.
find the angle between two vectors.
determine whether two vectors are parallel or perpendicular (orthogonal).
decompose a vector into two orthogonal vectors.
compute work using vectors.
find the distance between two points in 3-D Space
plot points using polar coordinates.
convert from polar coordinates to rectangular coordinates and vice versa.
graph and identify polar equations by converting to rectangular equations.
graph polar equations by hand and use a graphing utility to check their work.
convert a complex number from rectangular form to polar form and be able to plot points in the complex plane.
find products and quotients of complex numbers in polar form.
use DeMoivre's Theorem to find power of complex numbers in polar form as well as find the distinct complex roots.
move between a parameterized form of an equation and the equation in cartesian form.
describe motion using parameterized equations.

CONTENT STANDARDS

Represent complex numbers and their operations on the complex plane.

HSN-CN.B.4 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent and model with vector quantities.

HSN-VM.A.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).

HSN-VM.A.2 - Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

HSN-VM.A.3 - Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

HSN-VM.B.4 - Add and subtract vectors.

HSN-VM.B.4a - 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.

HSN-VM.B.4b - Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

HSN-VM.B.4c - 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.

HSN-VM.B.5 - Multiply a vector by a scalar.

HSN-VM.B.5a - Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v?, v?) = (cv?, cv?).

HSN-VM.B.5b - 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).

CONTENT

The students will know . . .

a vector is made up of a magnitude and a direction.
vector addition is both commutative and associative.
scalars are quantities that have only magnitude.
a vector can be represented using brackets, paranthesis, or with i, j, k notations.
a unit vector will only have a magnitude of 1.
the dot product holds for both the commutative and the distributive properties.
one can use the dot product to calculate the angle between two vectors.
two vectors are parallel is the angle between them is 0 or Pi
two vectors are perpendicular (or orthogonal) if the angle between them is pi/2
polar coordinates use the independent variable of radius and the dependent variable of angle measured from polar axis.
polar coordinates with negative radii are measure 180 degrees from the given angle measurement of the given coordinate.
one can use the trigonometric ratios for sine and cosine within a right triangle to convert from polar coordinates to rectangular coordinates.
one can use the equation for the standard form of a circle as well as the trigonometric ratio for tangent inside of a right triangle to convert from rectangular form to polar form.
polar equations with r = constant is a circle.
polar equations with theta = constant is a line passing through the origin.
polar equations make the shape of circles, cardiods, limacons, kenbuscates, spirals and rose curves. Each are found based on the components of the given polar equations.

Conics (late April)

BIG IDEAS AND ESSENTIAL QUESTIONS

Understanding how the components of the graphical representations of conic sections relate to their respective algebraic models allows for efficiency in graphing, modeling, identifying and writing equations of conic sections.

LEARNING OUTCOMES

Students will be skilled at . . .

classifying conic sections based on their given equations.
graphing circles, parabolas, hyperbolas and ellipses.
identifying key characteristics of each conic section given a graph and/or equation.
writing equations for circles, parabolas, hyperbolas and ellipses given various information.
modeling conic sections in a real world context.

CONTENT STANDARDS

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

HSG-GPE.A.2 - Derive the equation of a parabola given a focus and directrix.

HSG-GPE.A.3 - Derive the equations of ellipses and hyperbolas given two foci for the ellipse, and two directrices of a hyperbola.

CONTENT

The students will know that . . .

conic sections, with the exception of y = x²and its family of quadratic functions, appear as relations rather than functions.
there are identifying characteristics in each conic section's algebraic model to make them identifiable.
there are identifying characteristics in each conic section's graphical model to make them identifiable.

Discrete Mathematics (early May)

BIG IDEAS AND ESSENTIAL QUESTIONS

Combinatorics allows one to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that one may be certain that no missed collections or arrangements of these things occurs.
Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways.
An arithmetic sequence represents the same numeric pattern as a linear function, allowing one to understand that both representations grow at a constant rate.
A geometric sequence represents the same numeric pattern as an exponential function, allowing one to understand that both representations grow at an exponential rate.
The results of an experiment over a long period of time do produce regular patterns that enable us to predict using probability with remarkable accuracy.

LEARNING OUTCOMES

Students will be able to . . .

find all the subsets of a set.
find the intersection and union of sets.
find the complement of a set.
count the number of elements in a set.
solve counting problems using the multiplication principle, permutations, combinations, and permutations when there are nondistinct objects.
know when to apply the various counting method based on the type of situation.
construct a probability model and compute probabilities of equally likely outcomes.
use the addition rule and the complement rule to find probabilities.
recognize, write, and interpret an arithmetic sequence.
find the sum of an arithmetic sequence.
recognize, write, and interpret a geometric sequence.
find the sum of a geometric sequence and geometric series.
write the first several terms and the nth term of a sequence.
write the nth term of a sequence involving factorial patterns.
write the terms of a sequence defined by a recursive formula.
use and write using summation notation.
find the sum of a sequence either algebraically or with a graphing utility.
use the Binomial Theorem and Pascal's Triangle to expand binomials of high powers.

CONTENT STANDARDS

Interpret the structure of expressions.

HSA-SSE.A.1 - Interpret expressions that represent a quantity in terms of its context.

HSA-SSE.A.1a - Interpret parts of an expression, such as terms, factors, and coefficients.

HSA-SSE.A.2 - Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems.

HSA-SSE.B.3 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

HSA-SSE.B.4 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Understand the concept of a function and use function notation.

HSF-IF.A.3 - Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Build a function that models a relationship between two quantities.

HSF-BF.A.1 - Write a function that models a relationship between two quantities.

HSF-BF.A.1a - Determine an explicit expression, a recursive process, or steps for calculation from a context.

HSF-BF.A.2 - Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Understand independence and conditional probability and use them to interpret data

HSS-CP.A.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”).

HSS-CP.A.2 - 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.

HSS-CP.A.3 - 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.

HSS-CP.A.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.

HSS-CP.A.5 - Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

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

HSS-CP.B.6 - 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.

HSS-CP.B.7 - 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.

HSS-CP.B.8 - 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.

HSS-CP.B.9 - Use permutations and combinations to compute probabilities of compound events and solve problems.

Use probability to evaluate outcomes of decisions

HSS-MD.B.5 - Weight the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

HSS-MD.B.5a - Find the expected payoff for a game of chance.

HSS-MD.B.5b - Evaluate and compare strategies on the basis of expected values.

CONTENT

The students will know that . . .

a set is a well-defined collection of distinct objects, which are called elements.
the null set or empty set is a set that has no elements.
the intersection of sets A and B includes all of the elements belonging to both sets A and B.
the union of sets A and B includes all of the elements belonging to either set or A set B.
the complement of set A is the set that includes all of the elements in the universal set that are not in A.
n( A U B) = n(A) + n(B) - n(intersection of A and B).
if a task contains a sequence of choices where there are r selections for the first choice, s selections for the second choice, t selections for the third choice, and so on, then the task of making these selections can be done in (r)(s)(t)... different ways.
a permutation is an ordered arrangement of r objects chosen from n objects.
for permutations when distinct objects can be repeated, the number of ordered arrangments of r objects from n objects is n^r.
for permutations when distinct objects are not repeated, the number of ordered arrangments are found by the formula n!/(n-r)!
the number of permutations involving n nondistinct objects can be found by the formula n!/[(n₁)!(n₂)!...(n_k)!
a combination is an arrangement without regard to order when r objects are chosen from n distinct objects.
for combinations, the number of arrangements can be found using the formula n!/[(n-r)!r!].
P(E) = number of ways E can occur / number of all logical possibilities.
P(E and F) = P(E) + P(F) - P(E or F).
P(E of F) = P(E) + P(F) when E and F are mutually exclusive.
P(complement of E) = 1 - P(E).
a general sequence can be determined from a pattern using algebraic rules (which include factorials).
for an arithmetic sequence, a_n = a_n-1 +d or a_n=a+(n-1)d.
for an arithmetic sequence, S_n=(n/2)(a+a_n).
for a geometric sequence, a_n=r(a_n-1) or a_n=a(r^n-1).
for a geometric sequence, S_n=a [(1-rⁿ)/(1-r)].
for an infinite geometric sequence whose absolute value of the common ratio is less than 1, the sequence sums to a/(1-r).
the binomial theorem.
Pascale's triangle.

Matrices (mid May)

BIG IDEAS AND ESSENTIAL QUESTIONS

The mathematical concept of a matrix refers to a set of numbers, variables or functions ordered in rows and columns where that set can then be defined as a distinct entity, and further manipulated or operated on as a whole according to some basic mathematical rules.
An inverse matrix exists if the product of two matrices results in the identity matrix.

LEARNING OUTCOMES

Students will be able to . . .

write a matrix and identify its order.
perform elementary row operations on matrices.
use matrices and Gaussian elimination to solve systems of linear equations.
use matrices and Gauss-Jordan elimination to solve systems of linear equations.
use technology to solve systems of equations written as a matrix.
determine whether two matrices are equal.
add and subtract matrices as well as determine when it is possible to do so.
determine when it is possible to multiply two matrices and be able to multiply two matrices.
verify that two matrices are inverses of each other.
find the inverse of square matrices.
use inverse matrices to solves systems of linear equations.
find the determinant of square matrices.

CONTENT STANDARDS

Perform operations on matrices and use matrices in applications.

HSN-VM.C.6 - Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

HSN-VM.C.8 - Add, subtract, and multiply matrices of appropriate dimensions.

HSN-VM.C.9 - Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

HSN-VM.C.10 - Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

CONTENT

The students will know that . . .

an m x n matrix is a rectangular array with m rows and n columns.
an entry within a matrix is denoted by a_mn within an m x n matrix.
a matrix derived from a system of linear equations written in standard form is called an augmented matrix.
a matrix derived from the coefficients of the system (not including the constant term) is called the coefficient matrix of the system.
elementary row operations that can be performed on a matrix include interchanging two rows, multiplying a row by a nonzero constant, and adding a multiple of a row to another row.
a matrix in which all rows consisting entirely of zeros occur at the bottom of the matrix, each row that does not consist entirely of zeros has a first nonzero entry of 1, and when two successive nonzero rows has a leading 1 toward the right of the 1 in the row before it, the matrix is in row-echelon form.
a matrix is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1.
a system of linear equations corresponding to a matrix in row-echelon form can be used along with back-substitution to find the solution.
a system of linear equations corresponding to a matrix in reduced row-echelon form can be used to directly find the solution.
matrices can be added or subtracted only when the matrices are of the same order by adding or subtracting their corresponding entries.
a matrix can be multiplied by a scalar quantity by multiplying each entry in the matrix by the scalar quantity.
when an m x n matrix is multiplied by an n x p matrix, the result is an m x p matrix.
multiplying two matrices involves a row-by-column multiplication, where each entry in the "i"th row and "j"th column of the product AB is found by multiplying the entries in the "i"th row of A by the corresponding entired in the "j"th row of B and adding the results.
AA^-1=I_n determines that two square matrices are inverses of one another.
the determinant of a 2x2 matrix is found by finding ad-bc.

Limits and Continuity (early June)

BIG IDEAS AND ESSENTIAL QUESTIONS

Mathematicians developed specific notations so that any representation can be communicated in a succinct and uniform manner.
In order to help further describe the behavior of a function both within the interior of the function as well as the end behavior of the function, limits of functions can be evaluated numerically, graphically, and algebraically.
A function's continuity can be described either over an interval or at a distinct point.

LEARNING OUTCOMES

Students will be able to . . .

find a limit using a table.
find a limit using a graph.
find the limit of a sum, difference, or product.
find the limit of a polynomial, power/root, quotient, and average rate of change.
determine the one-sided limits of a function.
determine whether a function is continuous.
name the three-part definition for continuity at a point.
locate and name the type of discontinuity present on a graph.

CONTENT STANDARDS

Limits And Continuity

C.LC.1 - Understand the concept of limit and estimate limits from graphs and tables of values.

C.LC.2 - Find limits by substitution.

C.LC.3 - Find limits of sums, differences, products, and quotients.

C.LC.4 - Find limits of rational functions that are undefined at a point.

C.LC.5 - Find limits at infinity.

C.LC.6 - Decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior. Find special limits

C.LC.7 - Find one-sided limits.

C.LC.8 - Understand continuity in terms of limits.

C.LC.9 - Decide if a function is continuous at a point.

C.LC.10 - Find the types of discontinuities of a function.

CONTENT

The students will know that . . .

a limit is used in mathematics to describe a quantity that gets arbitrarily close to a value, without actually reaching that value.
the limit as x approaches x of f(x) is N if the limit exists.
the limit of sinx/x as x approaches 0 is 1.
the limit of f(x) as x approaches c of a constant function b is b.
the limit of f(x) as x approaches c of the function y=x is c.
the limit of f(x) +/- g(x) as x approaches c can be found by finding the limit of f(x) and g(x) individually and then adding or subtracting the results.
the limit of f(x) times g(x) as x approaches c can be found by finding the limit of f(x) and g(x) individually and then multiplying the results.
a one-sided limit will name the intended height of a function as x approaches c from one side.
a limit exists when x approaches c from the left and right hand-sides and approach the same intended height.
a function is continuous at a point c if f is defined at c, the limit as x approaches x exists, and f(c) is equal to the limit of f(x) as x approaches c.
discontinuity can be described as either removable or non-removable. Non-removable discontinuity can include vertical asymptotes, infinite oscillations, or breaks.