Functions, Equations and Graphs Linear Systems … you find the vertex for an absolute value equation When and how would you shade a two variable inequality? Key Terms Function, slope,

Curriculum Design Template

Content Area: Mathematics

Course Title: Algebra 2 Grade Level: 11

Functions, Equations and Graphs

Linear Systems

Quadratic Functions and Equations

Marking Period 1


Polynomials and Polynomial Functions

Probability and Statistics

Marking Period 2

Radical Functions and Rational Exponents

Exponential and Logarithmic Functions

Marking Period 3

Rational Functions

Sequence and Series

Periodic and Trig Functions

Marking Period 4

Date Created: August 2012

Board Approved on: August 27, 2012

Algebra 2

Unit Title Lessons to include Time Common Core Standards

Math

Expressions, Equations, and

Inequalities

1.2 Properties of Real Numbers Daily warm-

ups and mixed

reviews

N.RN.3

1.3 Algebraic Expressions A.SSE.1A

1.4 Solving Equations A.CED.1, A.CED.4

1.5 Solving Inequalities A.CED.1

1.6 Absolute Value Eqs and Inequalities A.SSE.1.b, A.CED.1

Functions, Equations, and

Graphs

2.1 Relations and Functions

3 weeks

F.IF.1, F.IF.2

2.2 Direct Variation A.CED.2, F.IF.1, F.BF.1

2.3 & 2.5 Linear Equations, Slope-Intercept Form, Linear Models

A.CED.2, F.IF.4, F.IF.6, F.IF.7, F.BF.1

2.6 Families of Functions F.IF.7, F.BF.3 2.7 Absolute value functs and graphs F.IF.7, F.IF.7.B, F.BF.3

2.8 Two-variable, inequalities A.CED.2, F.IF.7.B

Linear Systems

3.1 Solving Systems Using Tables/ Graphs

3 weeks

A.CED.2, A.CED.3, A.REI.6, A.REI.11

3.2 Solving Algebraically A.CED.2, A.CED.3, A.REI.5

3.3 Systems of Inequalities A.CED.3, A.REI.6, A.REI.12

3.4 Linear Programming A.CED.3

3.5 Systems with 3 variables Extends A.REI.6


4.1 Quadratic Functions and Transformations

5 weeks

A.CED.1, F.IF.4, F.IF.6, F.IF.7, F.BF.3

4.2 Standard Form of a Quad Function A.CED.2, F.IF.4, F.IF.6, F.IF.7, F.IF.8, F.IF.9, F.BF.1 4.3 Modeling data w/ Quad Functions F.IF.4, F.IF.5

4.4 Factoring Quadratic Expressions A.SSE.2

4.5 Quadratic Equations A.SSE.1.A, A.APR.3, A.CED.1

4.6 Completing the Square Extends A.REI.4.B

4.7 The Quadratic Formula Reviews A.REI.4.B

4.8 Complex Numbers

N.CN.1, N.CN.2, N.CN.7, N.CN.8

Quadratic Systems A.CED.3, A.REI.7

Polynomials and Polynomial functions

5.2 Polynomial & Linear Factors & Zeros

4 weeks

A.SSE.1.A, A.APR.3, F.IF.7, F.IF.7.C, F.BF.1

5.3 Solving Polynomial Equations A.SSE.2, A.REI.11

5.4 Dividing Polynomials A.APR.1, A.APR.2, A.APR.6

5.5 Thms About Roots of Polynomials N.CN.7, N.CN.8

5.6 The Fundamental Thm of Algebra N.CN.7, N.CN.8, N.CN.9, A.APR.3

5.8 Polynomial Models in Real World F.IF.4, F.IF.5, F.IF.6, F.IF.7

5.9 Transforming Polynomial Functions F.IF.7, F.IF.8, F.IF.9, F.BF.3


11.3 Probability of Multiple Events

2 weeks

S.CP.2, S.CP.5, S.CP.7

11.4 Conditional Probability S.CP.3, S.CP.4, S.CP.5, S.CP.6, S.CP.8

11.5 Analyzing Data S.IC.6

11.6 Standard Deviation S.ID.4, S.IC.6

11.7 Samples and Surveys S.IC.1, S.IC.3, S.IC.4, S.IC.6

Midterm Review & exam weeks

Chapters 2-5 & 11 1 weeks All

Radical functions and Rational

Exponents

6.1 Roots and Radical Expressions

5 weeks

A.SSE.2

6.2 Multiply Divide Radical Expressions

A.SSE.2 6.3 Binomial Radical Expressions ASS.E.2

6.4 Rational Exponents N.RN.1, N.RN.2

6.5 Solving Square Root and Other Radical Equations

A.CED.4, A.REI.2

6.6 Function Operations F.BF.1, F.BF.1.B

6.7 Inverse Relations and Functions F.BF.4.A, F.BF.4.C

6.8 Graphing Radical Functions

F.IF.7, F.IF.7.B, F.IF.8

Exponential and Logarithmic

functions

7.2 Properties of Exponential Functions

4 weeks

A.SSE.1.B, A.CED.2, F.IF.7 F.IF.7.E, F.IF.8, F.BF.1, F,BF.1.B

7.3 Logarithmic Functions as Inverses A.SSE.1.B, F.IF.7.E, F.IF.8, F.IF.9, F.BF.4.A

7.4 Properties of Logarithms F.LE.4

7.5 Exponential & Log Equations A.REI.11, F.LE.4

Rational Functions

8.1 Inverse Variation

5 weeks

A.CED.2, A.CED.4

8.2 The Reciprocal Function Family A.CED.2, F.BF.1, F.BF.3

8.3 Rational Functions and Graphs A.CED.2, F.IF.7, F.BF.1, F.BF.1.B

8.4 Simplify, Multiply, and Divide Rational Expressions

A.SSE.1, A.SSE.1.A, A.SSE.1B, A.SSE.2, A.APR.7

8.5 Adding/Subtract Rational Express A.APR.7

8.6 Solving Rational Equations A.APR,6, A.APR.7, A.CED.1, A.REI.2, A.REI.11

Sequence and Series

9.2 Arithmetic Sequences

2 weeks

F.IF.3

9.3 Geometric Sequences A.SSE.4

9.4 Arithmetic Series F.IF.3

9.5 Geometric Series A.SSE.4

Periodic Functions and Trigonometry

T-1 Rt Triangles and Trig Ratios

2 weeks

F.IF.4, F.IF.7.E, F.TF.2, F.TF.5

T-3 The Unit Circle F.IF.4, F.IF.7.E, F.TF.2, F.TF.5

T-4 Degrees and Radian Measure F.TF.1

Final review and Exam week

Chapters 6-9 & T 1 weeks All

Course Title: Algebra 2 Grade Level: 11th & 12th

Overarching Essential Questions

What are Functions?

What are linear systems and how are they used?

What are matrices and how are they used in life?

What is a quadratic equation and what is its function?

What are polynomials?

What are radical expressions and functions?

What are logarithmic and exponential functions?

Overarching Enduring Understanding

Students in Algebra 2 will learn about functions, linear systems, and matrices and

how they are used in everyday life. They will also become fluent with quadratics,

polynomials, radical expressions, logarithmic and exponential functions.

Course Description

Algebra 2 is a continuation of the skills learned in Algebra I. This course will cover such topics as linear and quadratic functions, quadratic relations, linear systems, and powers and roots. The course will also explore areas such as exponents, and logarithms. Graphing calculators will be used extensively in this course. This course is designed for the college bound student who intends to attend a 2-year college and/or a STEM career.

Functions, Equations and Graphs

Essential Questions

What is a function? How is slope used and can you put it into linear form? What is the difference between dependent and independent variables? What is point slope form? What types of graphs are produced by absolute value equations? Can you find the vertex for an absolute value equation When and how would you shade a two variable inequality?

Key Terms

Function, slope, point-slope, dependent and independent variables, vertex, shading

Objectives

Students will be able to:

Graph a function with and without a calculator

Find slope of a given line

Work with independent and dependent variables

Graph an absolute value inequality

Shade a 2 variable inequality

Standards associate with objectives

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

MA.F.IF.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).

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

MA.F.IF 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ ★

MS.F.IF.6 - 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.★

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

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

MA.F.BF.1 - Write a function that describes a relationship between two quantities.★ MA.F.BF.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.

Suggested Lesson Activities

Work with real world problems with linear equations Use the graphing application or Geometer Sketchpad to show and calculate slope Show how we find the vertex of an absolute value equations Work with shading on an absolute value inequality

Differentiation /Customizing learning (strategies)

Allow students to work in groups WITHOUT a calculator and graph different absolute value equations. Some students may want to work with negative coefficients, while other may not be ready for this.

Have students work on a short essay to explain the differences in absolute value inequalities

Linear Systems

Essential Questions

Name and show how to use the methods for solving systems of equations What is the best way to check your solutions to a given set of equation? When graphing a set of lines, how close is it feasible to get to a given solution? When and how to use the substitution method and elimination method to solve

systems with two variables? How are systems used in life? How do you solve systems with three variables?

Key Terms

Coefficients, elimination, graphing, additive inverses, linear equations, non-linear

equations

Objectives


Solve sets of equations using both the substitution method, elimination method,

and graphing

Work with sets of equations using the graphing method WITHOUT a calculator

Use the graphing method to find the point of intersection given 3 equations with 3

unknowns

Use technology to solve a 3 x 4 equation set



MA.A.CED.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

MA.A.REI.5- Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions

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

MA.A.REI.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.

MA.A.REI.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.


Work with numerous examples to solve sets of equations Put sets of linear equations on the board and use 3 methods to solve for a solution Discuss inconsistent sets of equations Do higher level forms of linear equations with non whole number coefficients Use technology to solve for solutions to systems of equations graphically


Allow students of advanced abilities to work with non-whole number coefficients and use the word problems involving linear equation

Use group work as a tool to let students of varying abilities to learn in a cooperative environment

Allow students that are struggling to work on their algebraic skills by giving reinforcement problems for homework


Essential Questions

How do you model data using quadratic functions? Can you prove the quadratic equation? Explain the properties of parabolas and give examples? Which ways do parabolas transform? WITHOUT a calculator, explain how to find the vertex of a parabola? Do you know how to complete the square, show your work? When are complex numbers used? How do you solve a quadratic without the middle term? How do you solve quadratic systems?

Key Terms

Quadratic, parabola, vertex, transform, quadratic equation, complex numbers, root, nth

term, function, minimum, maximum

Objectives

Students will be able to

Model data using a quadratic

Know what a parabola is and find its vertex

Find the vertex of a parabola, with and without a calculator

Know the properties of a parabola

Work with parabolas of higher dimension coefficients

Transform a parabola (shrink, stretch)

Move a parabola or switch its direction

Factor all types of quadratic expressions

Use the quadratic equation to solve for solutions

Work with complex numbers

Solve for extraneous solutions


MA.N.CN.1 - 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.

MA.N.CN.2– Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

MA.N.CN.7 - Solve quadratic equations with real coefficients that have complex solutions.

MA.N.CN.8 - (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).

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

MA.A.SSE.2 – Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

MA.A.APR.3 – Identify zeros of polynomials when suitable factorizations are

available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MA.A.CED.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.


MA.A.CED.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

MA.A.REI.4.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.

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

MA.F.IF.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

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

MA.F.IF.6 - 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.

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

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

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

MA.F.BF.1 - Write a function that describes a relationship between two quantities MA.F.BF.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.


Work with examples quadratics to find their properties

Show properties of parabolas on the smartboard with vertex Explain maximum and minimum values for any given parabola Show how complex numbers work and give quadratic examples Factor quadratics and show examples with and without solutions Show the difference of two perfect squares Use square roots to solve quadratics if possible


Allow students of varying levels to work collaboratively on the proof of the quadratic equation

Show how completing the square will help to make the proof go much more smoothly

Students who struggle can work on basic quadratics with low coefficients and work their way up to more advanced ones

Polynomials and Polynomial Functions

Essential Questions

Can you write an equation in factored form? What are the zeroes of an equation? Discuss the relative maximum and minimums of a given function? Name one of the many ways to solve an equation of higher order? Explain how you would factor a perfect square trinomial? Discuss where you would find complex roots? Use the quadratic equation to work with complex numbers? What purposes does the fundamental theorem of algebra have?

Key Terms

Zeroes, relative maximum, relative minimum, complex roots, polynomial functions,

fundamental theorem of algebra, imaginary roots

Objectives


Find extraneous solutions to given equations

Look for irrational roots and be able to decipher why they have irrational solutions

Find a 4th degree polynomial equation with integer coefficients

Find all roots for a polynomial equation

Explain how and why we use the fundamental theorem of algebra

Work with complex roots of given polynomial equations


MA.N.CN.7 – Solve quadratic equations with real coefficients that have complex solutions.

MA.N.CN.8 - (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).

MA.N.CN.9 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

MA.A.SSE.1 - Interpret expressions that represent a quantity in terms of its context. MA.A.SSE.2 – Use the structure of an expression to identify ways to rewrite it. For

example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

MA.A.APR.1 - 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.

MA.A.APR.2 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

MA.A.APR.3 – Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MA.A.APR.6 - Rewrite simple rational expressions in different forms; write a(x)/b(x)

in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

MA.A.REI.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

MA.F.IF.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

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

MA.F.IF.6 - 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.

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

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


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

MA.F.BF.1 - Write a function that describes a relationship between two quantities. MA.F.BF.3 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx),



Examples on the smart board, graph and find the zeroes of polynomial equations Work with relative maximum and minimum of higher order equations Visualize polynomial equations with Geometer Sketchpad Practice factoring sum and differences of 2 cubes with examples Work with the fundamental theorem of algebra to show complex roots and

solutions Differentiation /Customizing learning (strategies)

Allow students of differing abilities to work collaboratively to do complex root

solutions Review the quadratic equation and do practice problems from the book with

respect to complex solutions Use kinesthetic learning by bringing in a cereal box, use variables and see how close

the student can come to a decimal approximation


Essential Questions

How do you find probability of multiple events? How do you find conditional probabilities? How can you compare and describe sets of data? What are standards deviation and variance? How are measures of central tendency different from standard deviation? How can you collect unbiased data in a sample?

Key Terms

Dependent events, independent events, mutually exclusive events, conditional probability,

mean, median, mode, bimodal, outlier, range, quartile, percentile, variance, standard

deviation, population, sample, bias, controlled experiment, survey

Objectives


Find the probability of the event A and B

Find the probability of the event A or B

Find conditional probabilities

Use tables and tree diagrams to determine conditional probabilities

Calculate measures of central tendency

Draw and interpret box-and-whisker plots

Find the standards deviation and variance of a set of values

Apply standard deviation and variance

Identify sampling methods

Recognize bias in samples and surveys


MA.S.ID.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.

MA.S.IC.1 - Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

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

MA.S.IC.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.

MA.S.IC.6 - Evaluate reports based on data. MA.S.CP.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.

MA.S.CP.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.

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

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

MA.S.CP.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.

MA.S.CP.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.

MA.S.CP.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.


Have students perfrom an activity to model independent and dependet events – choosing marbles form a bag Use online “dynamic activity” to manioulate box and whisker plots and see the results of various manipulations Work with percentiles – and use standardized testing a to give real life relevance

Use practice HSPA open ended provblems that ask students to use measures of

central tendancy and outlier

Use graphing calculator to show mean and standard deviation

Use charts and line graphs to be visual showing mean, standard deviation, etc (see p.

734, problem 3)


Use smaller sample sizes to simplify concepts Use tangible activitities to help studnets understand independent and dependent Use visual representations to help with standard deviation

Radical Functions and Rational Exponents

Essential Questions

To simplify the nth root of an expression, what must be true about the expression? When you square each side of an equation, is the resulting equation equivalent to

the original? How are a function and its inverse related? Solve for x in an equation using powers in fractional form? Explain how to expand a binomial by the 3 step process of expansion? Can you explain when you can find extraneous solutions to equations? What type of equations have inverses and why? Do inverses have a tie to extraneous solutions, and if so, why? Give the domain and range of inverse functions and explain their roots?

Key Terms

Extraneous solution, inverse, radicand, binomial, square root equation, rational exponent,

like radicals, rationalize the denominator

Objectives


Discuss the differences between a square root and a principal square root

Expand a given binomial in different forms

Simplify radical expressions and use absolute values for given solutions

Rationalize the denominator of given expressions

Use conjugates to simplifying rational expressions

Determine when an expression is fully simplifying using conjugates

Put given expressions in different forms using a radicand

Solve square root equations and other fractional root equations

Solve and check for extraneous solutions


MA.N.RN.1 - 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.

MA.N.RN.2 - Rewrite expressions involving radicals and rational exponents using the properties of exponents.

MA.A.SSE.2 – Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

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

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

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


MA.F.BF.1 - Write a function that describes a relationship between two quantities. MA.F.BF.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 or f(x) = (x+1)/(x–1) for x ≠ 1.

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


Use negative exponent examples so students will understand their affect Expansion of binomial problems using Pascal’s Triangle Use smart board to show how absolute value solutions are used when simplifying

radical expressions Using conjugates to simplify expressions of fractional values Show rationalizing of a denominator and using conjugates Work with the 4 main methods of simplifying numbers with rational exponents (4

problems) Show examples on how to solve equations with 2 rational exponents


Allow more advanced students to simplify IRRATIONAL expressions and rational the denominator

Work with solving equations to a more advanced level such as solving solutions with conjugates and then looking for extraneous solutions

Students who are struggling may wish to work on more examples of principal square roots and determining when to use absolute value

Use on-line examples from the book reinforcing more basic skills when dealing with rational exponents , both positive and negative

Allow the exploring of different inverses of equations and introduce the ideas of calculus and slope tangent

Exponential and Logarithmic Functions

Essential Questions

What is a logarithm and how are they used to solve equations? What types of bases are used for logarithm and explain the value of “e” How do we go about using logs in everyday life word problems and show 3

examples of their uses Predict the graph to a logarithmic function by using values Find the asymptote of a logarithmic graph. Simplify logarithms of like bases Use the change of base formula to solve word problems of varying degrees. Solve logarithm problems by graphing and finding the intersection of the graphs Interpolate logarithmic values of equations using logarithms

Key Terms

Logarithm, Asymptote, logarithmic functions as inverses, compound interest and its uses,

graphing using tables

Objectives


Label asymptotes of a given logarithmic graph

Solve an equation with base e

Translate a logarithmic graph with and without a calculator

Use logarithmic functions as inverses and label the key parts of the graph

Simplify logarithmic with like bases to represent as a single function

Explain the properties of logarithms and use them to simplify and solve

Expand logarithms without like bases and solve

Use the properties of logarithms to evaluate expressions

Use a table to show the values of logarithms at given points on a graph


MA.A.SSE.1.b – Interpret expressions that represent a quantity in terms of its context. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

MA.A.REI.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



MA.F.IF.7e - Graph exponential and logarithmic functions, showing intercepts and end

behavior, and trigonometric functions, showing period, midline, and amplitude. MA.F.IF.8 - Write a function defined by an expression in different but equivalent forms

to reveal and explain different properties of the function. MA.F.IF.9 - Compare properties of two functions each represented in a different way

(algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum

MA.F.BF.1 - Write a function that describes a relationship between two quantities. MA.F.BF.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 or f(x) = (x+1)/(x–1) for x ≠ 1.

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


Use example to work with compound interest and discuss Graph logarithmic functions and their parent functions Have students show solutions on smart board of logarithmic functions Find the domain and range of given logarithmic functions Find the asymptotes of given logarithmic functions without a calculator Show simplifying logarithms Expansion of logarithms Solve logarithmic functions Use change of base formula to solve logarithmic functions


Allow students to use the ph balance of given chemicals to calculate how much needs to be mixed into a given solution to find another value

Use logarithmic functions of base”e” to discuss exponential growth and decay. Varying word problems for examples

Allow students who are struggling with the material of exponents to review how positive and negative exponents are solved in equation form

Use the books and videos on the smart board to reinforce logarithmic properties

Rational Functions

Essential Questions

What is inverse variation? What are reciprocal functions? What is a rational function? How is multiplying and dividing rational expressions similar to multiplying and

dividing fractions? How do you add and subtract rational expressions? How is solving rational expressions similar to solving a polynomial equation?

Key Terms

Inverse variation, combined variation, joint variation, reciprocal function, rational function,

rational expression, rational equation

Objectives


Recognize and use inverse variation

Use joint and other variations

Graph reciprocal functions

Graph translations of reciprocal functions

Identify properties of rational functions

Graph rational functions

Simplify rational expressions

Multiply and divide rational expressions

Add and subtract rational expressions

Solve rational equations

Use rational equations to solve problems


MA.A.APR.6 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

MA.A.CED.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.


MA.A.CED.4 – Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V =IR to highlight resistance R.

MA.A.SSE.1.a – Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients.

MA.A.SSE.1.b – Interpret expressions that represent a quantity in terms of its context. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending

on P. MA.A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For

example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

MA.F.BF.1 - Write a function that describes a relationship between two quantities. MA.F.BF.3 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx),




Use online “Dynamic Activity” to compare and contrast graphs of direct and inverse variations.

Graph inverse relations by hand and with graphing calculator. Have students graph functions in f(x) = 1/x and discover the reciprocal parent

function Have students vary the function f(x) = a/(x-h) + k to see how it translates – use

technology to see the translations – draw conclusions from the activity Use the online “Dynamic Activity” to let students explore rational functions by

varying the polynomials in the numerator and denominator. Use this as an intro to the characteristics of rational function graphs

Review multiplying and dividing fractions and simplifying polynomial expressions as a warm up to rational expressions

Review adding and subtracting fractions and common denominators as a warm up to adding and subtracting rational expressions

Show multiple methods to solve rational equations (algebraically – eliminating the denominator or with a graphing calc)


Provide graph paper and easier example to show direct and inverse variations Use Sketchpad and graph in groups Be very visual with asymptotes, etc. Vary the complexity of the rational expressions and equations

Sequence and Series

Essential Questions

What is an arithmetic sequence? What is a geometric sequence? What is an arithmetic series? What is a geometric series?

Key Terms

Arithmetic sequence, common difference, arithmetic mean, geometric sequence, common

ratio, geometric mean, series, finite series, infinite series, arithmetic series, limits,

geometric series, converge, diverge

Objectives


Define, identify and apply arithmetic sequences

Define, identify and apply geometric sequences

Define arithmetic series and find their sums

Define geometric series and find their sums


MA.F.IF.3 - Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

MA.A.SSE.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. For example, calculate mortgage payments.★


Show examples of multiple sequences and help students determine the arithmetic sequences

Have students derive formulas for the sequence – and then show them the recursive definition

Show examples of multiple sequences and help students determine the geometric sequences

Have students derive formulas for the sequence – and then show them the recursive definition

Have students find the sum of a small finite arithmetic series and gradually increase the number of terms before showing them the formula

Have students find the sum of a small finite geometric series and gradually increase the number of terms before showing them the formula


Use calculators or technology to assist Visually represent the difference, etc.

Periodic Functions and Trigonometry

Essential Questions

What are the trigonometric ratios of a right triangle? How is the unit circle related to trigonometric ratios? What is a radian?

Key Terms

Trigonometric ratios, standard position, initial side, terminal side, reference angle, unit

circle, central angle, intercepted arc, radian

Objectives


Find the lengths of sides in a right triangle

Find measures of angles of a right triangle

Work with angles in standard position

Find coordinates of points on the unit circle

Use radian measures for angles

Find the length of an arc of a circle


MA.F.IF.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

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

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

MA.F.TF.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.

MA.F.TF.5 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★


Using geometer sketchpad 4.0, show examples of graphs of sine functions and show visually their number of cycles

Work with examples on page 721 to find exact values of special trigonometric angles

Show different examples on smart board calculator like page 725 (1-15) to show radian measure

Use geometer sketchpad to measure arc length, radian measure, and radius of any given circle

Page 731 Examples (31-42) to determine the sign of a given trigonometric function Show how to interpolate values of a sine function at given points on a graph Write an equation of a function using a given graph Show using the unit circle where tangent functions are undefined, and find its

asymptotes


Allow students to work with pictures of graphs of non whole coefficients and determine the properties of a trigonometric function

Have students review the unit circle and see if they can interpolate values for non special angles

Work with real-world word problems to show how trigonometric functions are used and how they will have more than one answer

Students who are struggling should review the unit circle and learn to use the TI-84 to aid in their knowledge of trigonometry

Reinforce oscillations using geometer sketchpad and show the period and amplitude of basic trigonometric functions.

Documents

Functions, Equations and Graphs Linear Systems … you find the vertex for an absolute value equation When and how would you shade a two variable inequality? Key Terms Function, slope,