2012-2013 CHEMISTRY 1 PACING GUIDE Prepared by: Melvin G ...images.pcmac.org/Uploads/GadsdenCounty... · MACC.912.A-CED.1.2 – Create equations in two or more variables to represent

2013-2014 Algebra 2

1

All mathematics courses must incorporate the Eight Mathematical Practice Standards

MACC.K12.MP.1.1 MACC.K12.MP.2.1 MACC.K12.MP.3.1

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others

MACC.K12.MP.4.1 MACC.K12.MP.5.1 MACC.K12.MP.6.1

Model with mathematics Use appropriate tools strategically Attend to precision

MACC.K12.MP.7.1 MACC.K12.MP.

Look for and make use of structure

Look for and express regularity in repeated reasoning

Algebra 2 Honors incorporates the following LACC standards These standards can be meet by using essential questions as writing prompts for bellringers, assigning projects which have the students explore a concept and elaborate on it in detail, incorporating new vocabulary and mathematical symbols into

each lesson and having the students make connections to the words and the concepts, classroom discussions, etc

LACC.1112.RST.1.3 LACC.1112.RST.2.4 LACC.1112.RST3.7

LACC.1112.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks; analyze the specific results based on explanations in the text.

LACC.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11-12 texts and topics.

LACC.1112.RST.3.7: Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem.

2013-2014 Algebra 2

2

LACC.910.SL.1.1 LACC.910.SL.1.2 LACC.910.SL.1.3

LACC.910.RL.1.1: Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

LACC.910.RL.1.2: Determine a theme or central idea of a text and analyze in detail its development over the course of the text, including how it emerges and is shaped and refined by specific details; provide an objective summary of the text.

LACC.910.RL.1.3: Analyze how complex characters (e.g., those with multiple or conflicting motivations) develop over the course of a text, interact with other characters, and advance the plot or develop the theme.

LACC.910.SL.2.4 LACC.1112.WHST.1.1 LACC.1112.WHST.2.4

LACC.910.RL.2.4: Determine the meaning of words and phrases as they are used in the text, including figurative and connotative meanings; analyze the cumulative impact of specific word choices on meaning and tone (e.g., how the language evokes a sense of time and place; how it sets a formal or informal tone).

LACC.910.WHST.1.1: Write arguments focused on discipline-specific content.

LACC.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

LACC.1112.WHST.3.9

LACC.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.

2013-2014 Algebra 2

3

COURSE CODE: 1200330 COURSE NAME: Algebra 2

UNIT TITLE: Chapter 1 – Equations and Inequalities

UNIT ESSENTIAL QUESTION: Why is Algebra a universal Language? How does it facilitate problem solving? Jeopardy Review Game http://www.avon-schools.org/Page/7739

SEMESTER: 1 Grading Period: 1

CONCEPT CONCEPT CONCEPT

Lesson 1-1 Expressions and Formulas

Using the order of operations to evaluate expressions

Use formulas

Lesson 1-2 Properties of Real Numbers

Classify real numbers

Use the properties of real numbers to evaluate expressions

Lesson1-3 Solving Equations

Translate verbal expressions into algebraic expressions and equations, and vice versa

Solve equations using the Properties of equality

STANDARD(S) STANDARD(S) STANDARD(S)

MACC.912.A-SSE.1.1a – Interpret parts of an expression, such as terms, factors, and coefficients MACC.912.F-BF.1.1a – Determine an explicit expression, a recursive process, or steps for calculation from a context

MACC.912.N-RN.2.3 - Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

MACC.912.A-CED.1.1 – Create equations and inequalities in one variable and use them to solve problems MACC.912.A-CED.1.4 – Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations MACC.912.A-REI.1.1 – Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

http://www.avon-schools.org/Page/7739

2013-2014 Algebra 2

4

LESSON ESSENTIAL QUESTION LESSON ESSENTIAL QUESTION LESSON ESSENTIAL QUESTION

How would you apply the order of operations to evaluating a variety of problems? How would you use formulas to solve real world problems?

How do you classify real numbers and apply the properties of real numbers to evaluate expressions? How do you use real number properties to solve problem equations and inequalities?

How do you translate expressions into algebraic expressions and equations, vice versa? How do you solve equations using the properties equality?

VOCABULARY VOCABULARY VOCABULARY Variables Algebraic expressions Order of Operations Formula

Real numbers Rational numbers Irrational numbers Integers Whole numbers Natural numbers

Open Sentence Equation Solution

RESOURCES Math Worksheets

http://www.kutasoftware.com/

http://www.softschools.com/math/worksheets/

http://www.superteacherworksheets.com

http://freemathworksheets.net/

http://www.onlinemathlearning.com/ http://teachnology.com/worksheets/math

Math Lessons and Worksheets http://www.algebrahelp.com/ http://exchange.smarttech.com/index.html

#tab=0 http://www.nctm.org/resources/content.as

px?id=16385 http://www.onlinemathlearning.com/ http://www.sharemylesson.com/ http://teachers.henrico.k12.va.us/math/hcp

salgebra1/ http://worksheets.theteacherscorner.net/

Math Lessons and Worksheets http://www.crctlessons.com/algebra-

lessons.html http://people.hofstra.edu/stefan_Waner/r

ealworld/tut_alg_review/framesA_3B.html https://www.khanacademy.org/

Math Games http://www.math-play.com/ http://www.onlinemathlearning.com/ http://coolmath.com/ http://www.mathplayground.com/

2013-2014 Algebra 2

5

Additional Information

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.3.1 – Solve linear equations in one variable that include simplifying algebraic expressions.

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSS Benchmark MA.912.A.3.2 – Identify and apply the distributive, associative, and commutative properties and real numbers and the properties of equality. Also addresses MA.912.A.1.1 and MA.912.A.1.4

Cognitive Complexity (respectively):

Level 2: Basic Application of Skills & Concepts

Level 1: Recall

Level 3: Strategic Thinking & Complex Reasoning

NGSSS Benchmark MA.912.A.3.1 – Solve linear equations in one variable that include simplifying algebraic expressions MA.912.A.3.3 – Solve literal equations for a specified variable

2013-2014 Algebra 2

6


Lesson 1-4 Solving Absolute Value Equations

Evaluate expressions involving absolute values

Solve absolute value equations

Lesson 1-5 Solving Inequalities

Solve one-step inequalities

Solve multi-step inequalities

Lesson 1-6 Solving Compound and Absolute Value Inequalities

Solve compound inequalities

Solve absolute value inequalities


MACC.912.A-SSE.1.1b – Interpret parts of an expression, such as terms, factors, and coefficients MACC.912.A-CED.1.1 – Create equations and inequalities in one variable and use them to solve problems

MACC.912.A-CED.1.1 – Create equations and inequalities in one variable and use them to solve problems

MACC.912.A-SSE.1.1b – Interpret parts of an expression, such as terms, factors, and coefficients MACC.912.A-CED.1.1 – Create equations and inequalities in one variable and use them to solve problems


Construct how to evaluate expressions involving absolute values? Show how to solve absolute value equations?

How do you explain the steps to solve one step inequalities? How do you solve multi-step inequalities?

How can you apply principles for equivalent equations to solve equations and inequalities?

VOCABULARY VOCABULARY VOCABULARY

Absolute value Empty set Extraneous solution

Set-builder notation

Compound inequality Intersection Union

2013-2014 Algebra 2

7

RESOURCES

Math Resources http://mathforum.org/mbower/johnandbe

tty/frame.htm http://mathforum.org/ http://www.purplemath.com/modules/ http://mathwords.com/ http://people.hofstra.edu/stefan_Waner/r

ealworld/tut_alg_review/framesA_3B.html

Math Resources http://www.mathsisfun.com/ http://www.aaamath.com/ http://interactmath.com/Default.aspx http://algebrafunsheets.com/AlgebraII/ind

ex.php

Math Resources https://www.teachingchannel.org/videos http://illuminations.nctm.org/ http://krysstal.com/numbers.html http://www.ic-schools.net/tutorials


Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.3.6 – Solve and graph the solutions of absolute value equations and inequalities with one variable. Also addresses MA.912.A.1.4

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.3.4 – Solve and graph simple and compound inequalities in one variable and be able to justify each step in a solution

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.3.4 – Solve and graph simple and compound inequalities in one variable and be able to justify each step in a solution MA.912.A.3.6 – Solve and graph the solution of absolute value equations and inequalities with one variable

2013-2014 Algebra 2

8


UNIT TITLE: Chapter 2 – Linear Relations and Functions

UNIT ESSENTIAL QUESTION: How do linear equations, slopes, intercepts, and points on a line help us solve relevant problems and make predictions?



Lesson 2-1 Relations and Functions

Analyze relations and functions

Use equations of relations and functions

Lesson 2-1 Linear Relations and Functions

Identify linear relations and functions

Write linear equations in standard form

Lesson 2-3 Rate of Change and Slop

Find rate of change

Determine the slope of a line


MACC.912.F-IF.1.1 - 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). MACC.912.F-IF.1.2 – Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MACC.912.F-IF.2.5 – Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MACC.912.A-CED.1.2 – Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MACC.912.F-IF.2.4 – 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.

MACC.912.A-CED.1.2 – Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MACC.912.F-IF.2.4 – 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. MACC.912.F-IF.2.6: 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. MACC.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling


How would you analyze relations and functions? How would you use equations and functions to represent a relation?

How do you identify and construct linear relations and functions and write linear equations in standard form? How do you identify and sketch graphs of parent functions?

How do you find rate of change and determine the slope of a line? How do you identify and sketch graphs of parent functions, including quadratics?

2013-2014 Algebra 2

9

VOCABULARY VOCABULARY VOCABULARY One-to-one function Onto function Discrete relationships Continuous relationships Vertical line test Independent variable Dependent variable Function notation

Linear relation Nonlinear relation Linear equation Linear function Standard form y-intercept x-intercept

Rate of change Slope












lessons.html http://people.hofstra.edu/stefan_Waner/realworld/tut_alg_review/framesA_3B.html Math Games

http://www.math-play.com/ http://www.onlinemathlearning.com/ http://coolmath.com/ http://www.mathplayground.com/


Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.10.3 – Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities rational or radical expressions or logarithmic or exponential functions)

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.2.6 – Identify and graph common functions (including but not limited to linear, rational, quadratic, cubic, radical, absolute value) MA.912.A.10.3 – Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities rational or radical expressions or logarithmic or exponential functions)

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.10.3 – Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities rational or radical expressions or logarithmic or exponential functions)

2013-2014 Algebra 2

10


Lesson 2-4 Writing Linear Equations

Write an equation of a line given the slope and a point on the line

Write an equation of a line parallel or perpendicular to a given line

Lesson 2-5 Scatter Plots and Lines of Regression

Use scatter plots and predictions equations

Model data using lines of regression

Lesson 2-6 Special Functions

Write and graph piecewise-defined functions

Write and graph step and absolute value functions


MACC.912.F-IF.3.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MACC.912.F-IF.3.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

MACC.912.S-ID.2.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. MACC.912.F-IF.3.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MACC.912.F-IF.3.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

MACC.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MACC.912.F-IF.3.7b: Graph square root, cubic root, and piecewise-defined functions, including step functions and absolute value functions.


How do you write an equation of a line given the slope and a point on the line? How do you write an equation of a line parallel or perpendicular to a given line? How do you apply functions to problem situations?

How do you use scatter plots and prediction equations? How do you model data using lines of regression? How do you create models from data and use models to make decisions and critical judgments?

How do you write and graph piecewise defined functions? How do you write and graph step and absolute value functions? How do you identify and sketch graphs of parent functions, including quadratic functions?

2013-2014 Algebra 2

11


Slope-Intercept form Point-slope form Parallel Perpendicular

Bivariate data Scatter plot Dot plot Positive correlation Negative correlation Line of fit Prediction equation Regression Line Correlation coefficient

Piecewise-defined function Piecewise-linear function Step function Greatest integer function Absolute value function

RESOURCES





ex.php



Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.3.10 – Write an equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.3.11 – Write an equation of a line that models a data set and use the equation or the graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.2.5 – Graph absolute value equations and inequalities in two variables MA.912.A.2.9 – Recognize, intercept, and graph functions defined piecewise, with and without technology. Also addresses MA.912.A.2.6

2013-2014 Algebra 2

12

CONCEPT CONCEPT

Lesson 2-7 Parent Functions and Transformations

Identify and use parent functions

Describe transformations of functions

Lesson 2-8 Graphing Linear and Absolute Value Inequalities

Graph linear inequalities

Graph absolute value inequalities

STANDARD(S) STANDARD(S)

MACC.912.F-BF.2.3: 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.

MACC.912.F-BF.2.3: 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. MACC.912.F-IF.3.7b: Graph square root, cubic root, and piecewise-defined functions, including step functions and absolute value functions.

LESSON ESSENTIAL QUESTION LESSON ESSENTIAL QUESTION

How do you analyze and use parent functions and describe transformations of functions? How do you analyze a situation modeled by a rational function, formulate an equation or inequality, and solve the problem?

How do you graph linear and absolute value inequalities? How do you formulate systems of inequalities?

2013-2014 Algebra 2

13

VOCABULARY VOCABULARY

Family of graphs Parent graph Parent function Constant function Identity function Quadratic function Translation Reflection Line of reflection Dilation

Linear Inequality Boundary

RESOURCES

Math Worksheets http://www.kutasoftware.com/

http://www.softschools.com/math/worksh

eets/ http://www.superteacherworksheets.com




#tab=0 http://www.nctm.org/resources/content.a

spx?id=16385 http://www.onlinemathlearning.com/ http://www.sharemylesson.com/ http://teachers.henrico.k12.va.us/math/hc

psalgebra1/ http://worksheets.theteacherscorner.net/


Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.2.6 – Identify and graph common functions (including but not limited to linear, rational, quadratic, cubic, radical, absolute value) MA.912.A.2.10 – Describe and graph transformation of functions. Also addresses MA.912.A.2.5

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.A.2.5 – Graph absolute value equations and inequalities in two variables

2013-2014 Algebra 2

14


UNIT TITLE: Chapter 3 Systems of Equations and Inequalities

UNIT ESSENTIAL QUESTION: How do you use algebraic, graphical, and substitution methods to solve systems of linear equations and inequalities?



Lesson 3-1 Solving systems of Equations by Graphing

Solve systems of linear equations by using tables and graphs

Classify systems of linear equations

Lesson 3-2 Solving Systems of Equations Algebraically

Solve systems of linear equations by using substitution

Solve systems of linear equations by using elimination

Lesson 3-3 Solving Systems of Inequalities by Graphing

Solve systems of inequalities by graphing

Determine the coordinates of the vertices of a region formed by the graph of a system of inequalities


MACC.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MACC.912.A-CED.1.3: 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. MACC.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. MACC.912.A-REI.3.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

MACC.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. MACC.912.A-REI.3.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. MACC.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MACC.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. MACC.912.A-CED.1.3: 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.

2013-2014 Algebra 2

15

MACC.912.A-REI.4.11: 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.


How do you solve systems of linear equations by using tables and graphs? How do you determine whether a system of linear equations is inconsistent, consistent and dependent, or consistent and independent? How do you use algebraic methods to solve systems of linear equations?

How do you solve systems of linear equations by using substitution? How do you solve systems of linear equations by using elimination? How do you solve systems of inequalities?

How do you solve systems of inequalities by graphing? How do you determine the coordinates of the vertices of a region formed by the graph of a system of inequalities? How do you solve real-world optimization problems using systems of inequalities?

VOCABULARY VOCABULARY VOCABULARY System of equations Consistent Independent Dependent Inconsistent

Substitution method Elimination method

System of Inequalities Region Bounded Unbounded Boundary line

2013-2014 Algebra 2

16

RESOURCES


http://www.softschools.com/math/works

heets/ http://www.superteacherworksheets.com











Cognitive Complexity:

Level 1: Recall (A-REI.3.6)

Level 2: Basic Application of Skills & Concepts (The other standards)

Level 3: Strategic Thinking & Complex Reasoning (A-CED.1.2)

NGSSS Benchmark MA.912.A.3.14 – Solve systems of linear equations and inequalities in two and three variables using graphical, substitution, and elimination methods MA.912.A.3.15 – Solve real-world problems involving systems of linear equations and inequalities in two and three variables



Level 2: Basic Application of Skills & Concepts (The other standards)




Level 3: Strategic Thinking & Complex Reasoning (A-CED.1.2)


2013-2014 Algebra 2

17


Lesson 3-4 Optimization with Linear Programming

Find the maximum and minimum values of a function over a region

Solve real-world optimization problems using linear programming

Lesson 3-5 Systems of Equations in Three Variables

Solve systems of linear equations in three variables

Solve real-world problems using systems of linear equations in three variables


MACC.912.A-CED.1.3: 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.

MACC.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.


How do you solve systems of linear equations by using tables and graphs? How do you determine whether a system of linear equations is inconsistent, consistent and dependent, or consistent and independent? How do you use algebraic methods to solve systems of linear equations?

How do you solve systems of linear equations by using substitution? How do you solve systems of linear equations by using elimination? How do you solve systems of inequalities?

2013-2014 Algebra 2

18


Linear programming Feasible region Unbounded Constraints Optimize

Ordered triple

RESOURCES





ex.php

2013-2014 Algebra 2

19






Level 1: Recall NGSSS Benchmark MA.912.A.3.14 – Solve systems of linear equations and inequalities in two and three variables using graphical, substitution, and elimination methods MA.912.A.3.15 – Solve real-world problems involving systems of linear equations and inequalities in two and three variables

2013-2014 Algebra 2

20


UNIT TITLE: Chapter 5 Quadratic Functions and Relations

UNIT ESSENTIAL QUESTION: How do you solve quadratic equations and inequalities using multiply methods and techniques? Jeopardy Review Game http://www.avon-schools.org/Page/7739 Common Core Does Not Require N-VM to be taught in Algebra 2, therefor, Chapter 4 is Optional



Lesson 5-1 Graph quadratic functions

Graph quadratic functions

Find and interpret the maximum and minimum values of a quadratic functions

Lesson 5-2 Solving Quadratic Equations by Graphing

Solve quadratic equations by graphing

Estimate solutions of quadratic equations by graphing

Lesson 5-3 Solving Quadratic Equations by Factoring

Write quadratic equations in intercept form

Solve quadratic equations by factoring


MACC.912.F-IF.3.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

MACC.912.A-CED.1.1: Create equations and inequalities in one variable and use them to solve problems. MACC.912.A-APR.2.3: 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. MACC.912.A-REI.2.4: Solve quadratic equations in one variable.

MACC.912.A-SSE.1.2: Use the structure of an expression to identify ways to rewrite it. MACC.912.A-SSE.2.3a: Factor a quadratic expression to reveal the zeros of the function it defines. MACC.912.A-REI.2.4: Solve quadratic equations in one variable.

http://www.avon-schools.org/Page/7739

2013-2014 Algebra 2

21


How do you graph quadratic functions? How would you find and interpret the maximum and minimum values of quadratic functions?

How do you solve quadratic equations by graphing? How do you estimate solutions of quadratic equations by graphing?

How do you write quadratic equations in intercept form? How do you solve quadratic equations by factoring?

VOCABULARY VOCABULARY VOCABULARY Quadratic function Quadratic linear Constant terms Parabola Axis of symmetry Vertex Maximum value Minimum value

Quadratics Equations Standard form Root Zero

Factored form FOIL method














2013-2014 Algebra 2

22




NGSSS Benchmark MA.912.A.2.6: Identify and graph common functions (including but not limited to linear, rational, quadratic, cubic, radical, absolute value. MA.912.A.7.6 Identify the axis of symmetry, vertex, domain, range and intercept(s) for a given parabola. Also addresses MA.912.A.10.5


Level 1: Recall (A-APR.2.3)


NGSSS Benchmark MA.912.A.7.6 Identify the axis of symmetry, vertex, domain, range and intercept(s) for a given parabola. MA.912.A.7.10 Use graphing technology to find approximate solutions of quadratic equations



NGSSS Benchmark MA.912.A.4.3 Factor polynomial expressions MA.912.A.10.3-Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities rational or radical expressions or logarithmic or exponential functions)

2013-2014 Algebra 2

23


Lesson 5-4 Complex Numbers

Perform operations with pure imaginary numbers

Perform operations with complex numbers

Lesson 5-5 Completing the Square

Solve quadratic equations by using Square Root Property

Solve quadratic equations by completing the square

Lesson 5-6 The Quadratic Formula and the Discriminant

Solve quadratic equations by using the Quadratic Formula

Use the discriminant to determine the number and type of roots of a quadratic equation


MACC.912.N-CN.1.1: 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. MACC.912.N-CN.1.2: Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. MACC.912.N-CN.3.7: Solve quadratic equations with real coefficients that have complex solutions.

MACC.912.A-REI.2.4: Solve quadratic equations in one variable. MACC.912.A-SSE.2.3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

MACC.912.A-REI.2.4: Solve quadratic equations in one variable.


How do you perform operations with pure imaginary numbers? How do you perform operations with complex numbers?

How do you solve quadratic equations by using the Square root property? How do you solve quadratic equations by completing the square?

How do you write quadratic equations in intercept form? How do you solve quadratic equations by factoring?

2013-2014 Algebra 2

24


Imaginary unit Pure imaginary number Complex number Complex conjugates

Completing the Square

Quadratic Formula Discriminant

RESOURCES





ex.php




Level 1: Recall NGSSS Benchmark MA.912.A.1.6-Identify the real and imaginary parts of complex numbers and perform basic operations. Also addresses MA.912.A.1.1 and MA.912.A.1.7



NGSSS Benchmark MA.912.A.7.3-Solve quadratic equations over the real numbers by completing the square. MA.912.A.7.5 Solve quadratic equations over the complex number system.



NGSSS Benchmark MA.912.A.7.4-Use the discriminant to determine the nature of the roots of a quadratic equation. MA.912.A.7.5 Solve quadratic equations over the complex number system. Also assesses MA.912.A.10.3

2013-2014 Algebra 2

25


Lesson 5-7 Transformations with Quadratic Functions

Write a quadratic function in the form y = a(x – h)2 + k

Transform graphs of quadratic functions of the form y = a(x – h)2 + k

Lesson 5-8 Quadratic Inequalities

Graph quadratic inequalities in two variables

Solve quadratic inequalities in one variable


MACC.912.F-BF.2.3: 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.

MACC.912.A-APR.2.3: 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.


How do you write a quadratic function in the form y = a (x-h) 2 + K? How do you transform graphs of quadratic functions of the form y = a (x-h)2 + k?

How do you solve quadratic equations by graphing? How do you estimate solutions of quadratic equations by graphing?

2013-2014 Algebra 2

26


Vertex form Quadratic Inequality RESOURCES





ex.php




NGSSS Benchmark MA.912.A.2.10-Describe and graph transformation of functions.


Level 1: Recall NGSSS Benchmark MA.912.A.4.11- Solve a polynomial inequality by examining the graph with and without the use of technology. MA.912.A.10.3-Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities rational or radical expressions or logarithmic or exponential functions)

2013-2014 Algebra 2

27


UNIT TITLE: Chapter 6 Polynomials and Polynomial Functions

UNIT ESSENTIAL QUESTION: How do you use mathematical tools including factoring to transform and solve polynomial equations?



Lesson 6-1 Operations with polynomials

Multiply, divide, and simplify monomials and expressions involving powers

Add, subtract, and multiply polynomials

Lesson 6-2 Dividing Polynomials

Divide polynomials using long division

Divide polynomials using synthetic division

Lesson 6-3 Polynomial Function

Evaluate polynomial functions

Identify general shapes of graphs of polynomial functions


MACC.912.A-APR.1.1: 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.

MACC.912.A-APR.1.1: 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. MACC.912.A-APR.2.2: 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). MACC.912.A-APR.4.6: 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.

MACC.912.A-SSE.1.1a: Interpret parts of an expression, such as terms, factors, and coefficients. MACC.912.F-IF.3.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

2013-2014 Algebra 2

28


How do you multiply, divide and simplify monomials and expressions involving powers? How do you add, subtract and multiply polynomials?

How do you divide polynomials using long division? How do you divide polynomials using synthetic division?

How do you evaluate polynomial functions? How do you identify general shapes of graphs of polynomial functions?

VOCABULARY VOCABULARY VOCABULARY Simplify Degree of a polynomial

Synthetic division Polynomial in one variable Leading coefficient Polynomial function Power function End behavior Quartic function Quantic function












lessons.html http://people.hofstra.edu/stefan_Waner/r

ealworld/tut_alg_review/framesA_3B.html Math Games




Level 1: Recall NGSSS Benchmark MA.912.4.2-Add, subtract and multiply polynomials Also assess MA.912.A.1.3 and MA.912.A.1.4


Level 1: Recall

Level 2: Basic Application of Skills & Concepts (A-APR.4.6)

NGSSS Benchmark MA.912.A.4.4- Divide polynomials by monomials and polynomials with various techniques, including synthetic division



NGSSS Benchmark MA.912.A.4.5 Graph polynomial functions with and without technology and describe the end behavior.

http://www.crctlessons.com/algebra-lessons.html

http://www.crctlessons.com/algebra-lessons.html

2013-2014 Algebra 2

29


Lesson 6-4 Analyzing Graphs of Polynomial Functions

Graph polynomial functions and locate their zeros

Find the relative maxima and minima of polynomial functions

Lesson 6-5 Solving polynomial Equations

Factor polynomials

Solve polynomial equations by factoring

Lesson 6-6 The Remainder Factor Theorems

Evaluate functions by using synthetic substitution

Determine whether a binomial is a factor of a polynomial by using synthetic substitution


MACC.912.F-IF.3.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MACC.912.A-REI.4.12: Graph the solutions to a linear inequality in two variables as a halfplane (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. MACC.912.A-APR.3.4: Prove polynomial identities and use them to describe numerical relationships. MACC.912.A-SSE.1.2: Use the structure of an expression to identify ways to rewrite it.

MACC.912.N-CN.3.7: Solve quadratic equations with real coefficients that have complex solutions.


How do you multiply, divide and simplify monomials and expressions involving powers? How do you add, subtract and multiply polynomials?

How do you factor polynomials? How do you solve polynomial equations by factoring?

How do you evaluate functions by using synthetic substitution? How do you determine whether a binomial is a factor of a polynomial by using synthetic substitution?


Relative maximum Relative minimum Extrema Turning point

Prime polynomials Quadratic form

Synthetic substitution Depressed polynomial

2013-2014 Algebra 2

30

RESOURCES





ex.php





NGSSS Benchmark MA.912.2.6 –Identify and graph common functions (including, but not limited to linear, rational, quadratic, cubic, radical, absolute value). MA.912.A.4.5-Graph polynomial functions with and without technology and describe the end behavior


Level 1: Recall (A-APR.3.4)


NGSSS Benchmark MA.912.A.4.3-Factor polynomial expressions MA.912.A.4.10-Use polynomial equations to solve real-world problems.


Level 1: Recall NGSSS Benchmark MA.912.A.4.6-Use theorems of polynomial behavior to find the zeros of a polynomial function. MA.912.A.4.8-Describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression, with and without technology. Also addresses MA.912.A.4.3.

2013-2014 Algebra 2

31


Lesson 6-7 Roots and Zeros

Determine the number and type of roots for a polynomial equation.

Find the zeros of polynomial function

Lesson 6-8 Rational Zero Theorem

Identify possible rational zeros of a polynomial function

Find all of the rational zeros of a polynomial function


MACC.912.N-CN.3.7: Solve quadratic equations with real coefficients that have complex solutions. MACC.912.N-CN.3.8: Extend polynomial identities to the complex numbers. MACC.912.F-IF.3.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MACC.912.N-CN.3.7: Solve quadratic equations with real coefficients that have complex solutions.


How do you determine the number and type of roots for a polynomial equation? How do you find the zeros of a polynomial function?

How do you identify possible rational zeros of a polynomial function? How do you find all of the rational zeros of a polynomial function?


Fundamental Theorem of Algebra Zeros Factors Roots Complex Conjugate Theorem Descartes’s Rule of Signs

Rational Zero Theorem Corollary to the Rational Zero Theorem

2013-2014 Algebra 2

32

RESOURCES





ex.php



Level 1: Recall

Level 2: Basic Application of Skills & Concepts (F-IF.3.7c)

NGSSS Benchmark MA.912.A.4.6-Use theorems of polynomial behavior to find the zeros of a polynomial function. MA.912.A.4.8-Describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression, with and without technology. Also addresses MA.912.A.4.3.


Level 1: Recall NGSSS Benchmark MA.912.A.4.6-Use theorems of polynomial behavior to find the zeros of a polynomial function. MA.912.A.4.8-Describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression, with and without technology. Also addresses MA.912.A.4.3.

2013-2014 Algebra 2

33


UNIT TITLE: Chapter 7 Inverses and Radical Functions and Relations

UNIT ESSENTIAL QUESTION: How do you describe and analyze the relationship between a function and its inverse?



Lesson 7-1 Operations on Functions

Find the sum, difference, product, and quotient of functions

Find the composition of functions

Lesson 7-2 Inverse Functions and Relation

Find the inverse of a function or relation

Determine whether two functions or relations are inverses

Lesson 7-3 Square Root Functions and Inequalities

Graph and analyze square root functions

Graph square root inequalities


MACC.912.F-BF.1.1b: Combine standard function types using arithmetic operations. MACC.912.F-BF.1.1c: Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

MACC.912.F-BF.2.4a: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.

MACC.912.A-CED.1.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. MACC.912.A-REI.1.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.


How do you find the sum, difference, product and quotient of functions? How do you find the compositions of functions?

How do you find the inverse of a function or relation? How do you determine whether two functions or relations are inverses?

How do you graph and analyze square root functions? How do you graph square root inequalities?

VOCABULARY VOCABULARY VOCABULARY Composition of functions

Inverse relation Inverse functions

Square root functions Radical functions Square root inequality

2013-2014 Algebra 2

34

















NGSSS Benchmark MA.912.A.2.7 Perform operations of functions algebraically, numerically and graphically. MA.912.A.2.8 Determine the composition of functions



NGSSS Benchmark MA.912.A.2.11 Solve problems involving functions and their inverses.


Level 1: Recall


NGSSS Benchmark MA.912.A.2.6 Identify and graph common functions (including, but not limited to linear, rational, quadratic, cubic, radical, absolute value).

2013-2014 Algebra 2

35


Lesson 7-4 nth Roots

Simplify radicals

Use a calculator to approximate radicals

Lesson 7-5 Operations with Radical Expressions

Simplify radical expressions

Add, subtract, multiply, and divide radical expressions

Lesson 7-6 Rational Exponents

Write expressions with rational exponents in radical form and vice versa

Simplify expressions in exponential or radical form


MACC.912.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. MACC.912.N-RN.1.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.

MACC.912.F-BF.1.1b: Combine standard function types using arithmetic operations.

MACC.912.N-RN.1.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.


How do you simplify radicals? How do you use calculator to approximate radicals?

How do you simplify radical expressions? How do you add, subtract, multiply, and divide radical expressions?

How do you write rational exponents with radical form, and vice versa? How do you simplify expressions in exponential or radical form?


Nth root Radical sign Index Radicand Principal root

Rationalize the denominator Like radical expressions conjugate

Rational Exponents (concept box) Expressions with Rational Exponents (rule box)

2013-2014 Algebra 2

36

RESOURCES





ex.php





Level 1: Recall NGSSS Benchmark MA.912.A.10.3: Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities, rational or radical expressions or logarithmic or exponential function.



NGSSS Benchmark MA.912.A.6.2- Add, Subtract, multiply, divide radical expressions (square roots and higher). Also assesses MA.912.A.1.4


Level 1: Recall NGSSS Benchmark MA.912.A.6.3 – Simplify expressions using properties of rational exponents MA.912.A.6.4 – Convert between rational exponent and radical forms of expressions

2013-2014 Algebra 2

37


Lesson 7-7 Solving Radical Equations and Inequalities

Solve equations containing radicals

Solve inequalities containing radicals


MACC.912.A-REI.1.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. MACC.912.A-CED.1.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.


How do you solve equations containing radicals? How do you solve inequalities containing radicals?


Radical equation Extraneous solution Radical inequality

2013-2014 Algebra 2

38

RESOURCES







Level 1: Recall NGSSS Benchmark MA.912.A.6.5 Solve equations that contain radical expressions MA.912.A.10.3: Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities, rational or radical expressions or logarithmic or exponential function.

2013-2014 Algebra 2

39


UNIT TITLE: Chapter 8 Exponential and Logarithmic

UNIT ESSENTIAL QUESTION: How do you solve and graph exponential and logarithmic equations and inequalities?



Lesson 8-1 Graphing Exponential Functions

Graph exponential growth functions

Graph exponential decay functions

Lesson 8-2 Solving Exponential Equations and Inequalities

Solve exponential equations

Solve exponential inequalities

Lesson 8-3 Logarithms and Logarithmic Functions

Evaluate logarithmic expressions

Graph logarithmic


MACC.912.F-IF.3.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. MACC.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

MACC.912.F-BF.2.4a: 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. MACC.912.A-SSE.2.3c: Use the properties of exponents to transform expressions for exponential functions. MACC.912.F-LE.1.4: 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.

MACC.912.F-IF.3.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.


How do you graph exponential growth functions? How do you graph exponential decay functions?

How do you solve exponential equations? How do you solve exponential inequalities?

How do you write rational exponents with radical form, and vice versa? How do you simplify expressions in exponential or radical form?

2013-2014 Algebra 2

40

VOCABULARY VOCABULARY VOCABULARY Exponential functions Exponential growth Asymptote Growth factor Exponential decay Decay factor

Exponential equation Compound interest Exponential inequality

Logarithm Logarithmic function

















NGSSS Benchmark MA.912.A.8.3 Graph exponential and logarithmic functions MA.912.A.8.7 Solve applications of exponential growth and decay. Also addresses LA.912.4.2.1, MA.912.A.2.10, MA.912. A.8.1, and MA.912.A.10.3



NGSSS Benchmark MA.912.A.8.5 Solve logarithmic and exponential equations MA.912.A.10.3: Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities, rational or radical expressions or logarithmic or exponential function



NGSSS Benchmark MA.912.A.8.1 Define exponential and logarithmic functions and determine their relationship. MA.912.A.8.2 Define and use the properties of logarithms to simplify logarithmic to simplify logarithmic expressions and to find their approximate values. Also addresses MA.912.A.2.10 and MA.912.A.8.3

2013-2014 Algebra 2

41


Lesson 8-4 Solving Logarithmic Equations and Inequalities

Solve logarithmic equations

Solve logarithmic inequalities

Lesson 8-5 Properties of Logarithms

Simplify and evaluate expressions using the properties of logarithms

Solve logarithmic equations using the properties of logarithms

Lesson 8-6 Common Logarithms

Solve exponential equations and inequalities using common logarithms

Evaluate logarithmic expressions using the Change of Base Formula


MACC.912.F-LE.1.4: 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.




How do you solve logarithmic equations? How do you solve logarithmic inequalities?

Lesson Essential Question: How do you simplify and evaluate expressions using the properties of logarithms? How do you solve logarithmic equations

How do you solve exponential equations and inequalities using common logarithms? How do you evaluate logarithmic expressions using the Change of Base Formula?


Logarithmic equation Logarithmic inequality

Product Property of Logarithms Quotient Property of Logarithms Power Property of Logarithms

Common Logarithm Change of Base Formula

2013-2014 Algebra 2

42

RESOURCES





ex.php





NGSSS Benchmark MA.912.A.8.5 Solve logarithmic and exponential equations MA.912.A.8.2 Define and use the properties of logarithms to simplify logarithmic to simplify logarithmic expressions and to find their approximate values. Also addresses MA.912.A.2.10 and MA.912.A.8.3



NGSSS Benchmark MA.912.A.8.2 Define and use the properties of logarithms to simplify logarithmic to simplify logarithmic expressions and to find their approximate values. Also addresses MA.912.A.2.10 and MA.912.A.8.3



NGSSS Benchmark MA.912.A.8.2 Define and use the properties of logarithms to simplify logarithmic to simplify logarithmic expressions and to find their approximate values. Also addresses MA.912.A.2.10 and MA.912.A.8.3 MA.912.A.8.6 Use the change of base formula. Also addresses MA.912.A.8.5

2013-2014 Algebra 2

43


Lesson 8-7 Base e and Natural Logarithms

Evaluate expressions involving the natural base and natural logarithm

Solve exponential equations and inequalities using natural logarithms

Lesson 8-8 Using Exponential and Logarithmic Functions

Use logarithms to solve problems involving exponential growth and decay

Use logarithms to solve problems involving logistic growth



MACC.912.F-LE.2.5: Interpret the parameters in a linear or exponential function in terms of a context. MACC.912.F-LE.1.2: 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).


How do you evaluate expressions involving the nature base and natural logarithm? How do you solve exponential equations and inequalities using natural logarithms?

How do you use logarithms to solve problems involving exponential growth and decay? How do you use logarithms to solve problems involving logistic growth?


Natural base, e Natural base exponential function Natural logarithm

Rate of continuous growth Rate of continuous decay Logistic growth model

2013-2014 Algebra 2

44

RESOURCES





ex.php




NGSSS Benchmark MA.912.A.8.2 Define and use the properties of logarithms to simplify logarithmic to simplify logarithmic expressions and to find their approximate values. Also addresses MA.912.A.2.10 and MA.912.A.8.3 MA.912.A.8.7 Solve applications of exponential growth and decay. Also assesses MA.912.A.8.5 and MA.912.A.10.3



NGSSS Benchmark MA.912.A.8.5 Solve logarithmic and exponential equations MA.912.A.8.7 Solve applications of exponential growth and decay. Also assesses MA.912.A.8.5 and MA.912.A.10.3

2013-2014 Algebra 2

45


UNIT TITLE: Chapter 9 Rational Functions and Relations

UNIT ESSENTIAL QUESTION: How would you use rational equations and inequalities to model and make predictions in problem situations involving direct and inverse variations? SEMESTER: 2 Grading Period: 3


Lesson 9-1 Multiplying and Dividing rational Expressions

Simplify rational expressions

Simplify complex fractions

Lesson 9-2 Adding and Subtracting Rational Expressions

Determine the LCM of polynomials

Add and subtract rational expressions

Lesson 9-3 Graphing Reciprocal Functions

Determine properties of reciprocal functions

Graph transformations of reciprocal functions


MACC.912.A-APR.4.7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.


MACC.912.F-BF.2.3: 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. MACC.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.


How do you simplify rational expressions? How do you simplify complex fractions?

How do you determine the LCM of polynomials? How do you add, subtract rational expressions?

How do you determine properties of reciprocal functions? How do you graph transformations for reciprocal functions?

2013-2014 Algebra 2

46

VOCABULARY VOCABULARY VOCABULARY Rational expression Complex number

Adding Rational Expressions Subtracting Rational Expressions

Reciprocal functions Hyperbola

















NGSSS Benchmark MA.912.A.5.2 Add, Subtract, multiply and divide rational expressions MA.912.A.5.3 Simplify complex fractions. Also addresses MA.912.A.10.3



NGSSS Benchmark MA.912.A.5.2 Add, Subtract, multiply and divide rational expressions



NGSSS Benchmark MA.912.A.5.6 Identify removable and non-removable discontinuities and vertical, horizontal, and oblique asymptotes of a graph of a rational function, find the zeros, and graph the function. Also assesses MA.912.A.2.6

2013-2014 Algebra 2

47


Lesson 9-4 Graphing Rational Functions

Graph rational functions with vertical and horizontal asymptotes

Graph rational functions with oblique asymptotes and point discontinuity

Lesson 9-5 Variation Functions

Recognize and solve direct and joint variation problems

Recognize and solve inverse and combined variation problems

Lesson 9-6 Solving Rational Equations and Inequalities

Solve rational equations

Solve rational inequalities


MACC.912.A-APR.2.3: 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. MACC.912.F-BF.1.1b: Determine an explicit expression, a recursive process, or steps for calculation from a context.

MACC.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.



How do you graph rational functions with vertical and horizontal asymptotes? How do you graph rational functions with oblique asymptotes and point discontinuity?

How do you recognize and solve direct and joint variation problems? How do you recognize and solve inverse and combined variation problems?

How do you solve rational equations? How do you solve rational inequalities?


Rational function Point discontinuity

Direct variation Constant of variation

Rational equations Rational inequalities Weighted average

2013-2014 Algebra 2

48

RESOURCES





ex.php






NGSSS Benchmark MA.912.A.2.10 – Describe and graph transformations of functions MA.912.A.5.6 – Identify removable and non-removable discontinuities, and vertical, horizontal, and oblique asymptotes of a graph of a rational function, find the zeros, and graph the functions



NGSSS Benchmark MA.912.A.2.12 Solve problems using direct, inverse, and joint variations



NGSSS Benchmark MA.912.A.5.5 Solve rational equations MA.912.A.5.7 Solve real-world problems involving rational equations

2013-2014 Algebra 2

49


UNIT TITLE: Chapter 10 Cone Sections

UNIT ESSENTIAL QUESTION: How do you derive the equation of a parabola given a focus and directrix? Common Core Only Require G-GPE.1.2 be taught in Algebra 2 All other G-GPE standards are optional



Lesson 10-2 Parabolas

Write equations of parabolas in standard form

Graph parabolas

Lesson 10-6 Identify Conic Sections

Write equations of conic sections in standard form

Identify conic sections from their equations


MACC.912.G-GPE.1.2: Derive the equation of a parabola given a focus and directrix.

MACC.912.G-GPE.1.2: Derive the equation of a parabola given a focus and directrix.


How do you write equations of parabolas in standard form? How do you graph parabolas?

How do you write equations of conic sections in standard form? How do you identify conic sections from their equations?

VOCABULARY VOCABULARY VOCABULARY Parabola Focus Directrix Latus rectum Standard form General form

Standard forms of Conic Sections

2013-2014 Algebra 2

50

RESOURCES Math Resources

http://mathforum.org/mbower/johnandbetty/frame.htm

http://mathforum.org/ http://www.purplemath.com/modules/ http://mathwords.com/ http://people.hofstra.edu/stefan_Waner/r


Math Resources http://www.mathsisfun.com/ http://www.aaamath.com/ http://interactmath.com/Default.aspx http://algebrafunsheets.com/AlgebraII/inde

x.php




NGSSS Benchmark MA.912.A.9.1 Write the equations of conic sections in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.) MA.912.A.9.2 Graph conic sections with



NGSSS Benchmark MA.912.A.9.1 Write the equations of conic sections in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.) MA.912.A.9.2 Graph conic sections with

2013-2014 Algebra 2

51


UNIT TITLE: Chapter 11 Sequences and Series

UNIT ESSENTIAL QUESTION: How do you relate arithmetic sequences to linear functions? How do you relate geometric sequences to exponential functions? SEMESTER: 2 Grading Period: 4


Lesson 11-1 Sequences as Functions

Relate arithmetic sequences to linear functions

Relate geometric sequences to exponential functions

Lesson 11-2 & 11-3 Arithmetic and Geometric Sequences and Series

Use arithmetic sequences

Use geometric Sequences

Find sums of arithmetic series

Find sums of geometric series

Lesson 11-4 Infinite Geometric Series

Find sums of infinite geometric series

Write repeating decimals as fraction


MACC.912.A-SSE.2.4: 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.

MACC.912.A-SSE.2.4: 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. MACC.912.F-IF.1.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.



How to you relate arithmetic sequences to linear functions?

How do you use arithmetic sequences? How do you find sums of arithmetic series? How do you use geometric sequences? How do you find the sums of geometric series?

How do you find sums and infinite geometric series? How do you write repeating decimals as fractions?

VOCABULARY VOCABULARY VOCABULARY Sequence Term Finite sequence Infinite Sequence Arithmetic sequence Common difference Geometric sequence Common ratio

Arithmetic means Geometric means Series Arithmetic series Geometric series Arithmetic series Partial sum Sigma notation

Convergent Series Divergent series Infinity

2013-2014 Algebra 2

52

RESOURCES


http://www.softschools.com/math/works

heets/ http://www.superteacherworksheets.com












Level 3:Strategic Thinking & Complex Reasoning

NGSSS Benchmark MA.912.D.11.1 Define arithmetic and geometric sequences and series. MA.912.D.11.3 Find specified terms of arithmetic and geometric sequences.




NGSSS Benchmark MA.912.D.11.3 Find specified terms of arithmetic and geometric sequences. MA.912.D.11.4 Find Partial sums of arithmetic and geometric series, and find sums of infinite convergent geometric series. Use sigma notation where applicable. Also addresses MA.912.D.11.1 and MA.912.D.11.2




NGSSS Benchmark MA.912.D.11.2 Use sigma notation to describe series MA.912.D.11.4 Find Partial sums of arithmetic and geometric series, and find sums of infinite convergent geometric series. Use sigma notation where applicable. Also addresses MA.912.D.11.1 and MA.912.D.11.2.

2013-2014 Algebra 2

53


Lesson 11-5 Recursion and Iteration

Recognize and use special sequences

Iterate functions

Lesson 11-6 The Binomial Theorem

Use Pascal's Triangle to expand powers of binomials

Use the Binomial Theorem to expand powers of binomials

Lesson 11-7 Proof by Mathematical

Prove statements by using mathematical induction

Disprove statements by finding a counterexample






How do you recognize and use special sequences? How do you iterate functions?

How do you use Pascal’s triangle to expand powers of binomials? How do you use the binomial theorem to expand powers of binomials

How do you find sums and infinite geometric series? How do you write repeating decimals as fractions?

2013-2014 Algebra 2

54


Fibonacci sequence Recursive sequence Explicit formula Recursive formula Iteration

Pascal's triangle

Mathematical induction Induction hypothesis

RESOURCES





ex.php






NGSSS Benchmark MA.912.D.11.1 Define arithmetic and geometric sequences and series




NGSSS Benchmark MA.912.A.4.12 Apply the Binomial Theorem




NGSSS Benchmark MA.912.D.1.3 Use mathematical induction to prove various concepts in number theory ( such as sums of infinite integer series, divisibility statements, and parity statements) , recurrence relations, and applications

2013-2014 Algebra 2

55


UNIT TITLE: Chapter 12 Probability and Statistics

UNIT ESSENTIAL QUESTION: How do you use independent and conditional probability to interpret data?



Lesson 0-5 Permutations and Combinations

Solve problems involving permutations and combinations

Lesson 12-1 Experiments, Surveys, and Observational Studies

Evaluate surveys, studies, and experiments

Distinguish between correlation and causation

Lesson 12-2 Statistical Analysis

Use measures of central tendency and variations to compare sets of data

Explore measures of variation


MACC.912.S-CP.1.2: 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.

MACC.912.S-IC.2.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. MACC.912.S-IC.2.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. MACC.912.S-IC.2.6: Evaluate reports based on data.

MACC.912.S-IC.2.4: 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. MACC.912.S-ID.1.4: 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.

2013-2014 Algebra 2

56


How would you use permutations and combinations to solve real world problems?

How do you evaluate surveys, studies, and experiments? How do you distinguish between correlation and causation?

How do you use measures of central tendency and variation to compare sets of data? How do you explore measures of variation?

VOCABULARY VOCABULARY VOCABULARY Permutation Linear permutation Combination

Survey population Census sample Biased unbiased observational study experiment treated group control group correlation causation

Univeriate data Measure of central Tendency parameter statistic margin of sampling error measure of variation variance standard deviation














2013-2014 Algebra 2

57






NGSSS Benchmark MA.912.S.2.1- Compare the difference between surveys, experiments, and observational studies. MA.912.S.2.3 Identify sources of bias, including sampling and non-sampling errors



NGSSS Benchmark MA.912.S.3.3 Calculate and interpret measures of the center of a set of data, including mean, median and weighted mean, and use these measures to make comparisons among sets of data. MA.912.S.3.4 Calculate and interpret measures of variance and standard deviation. Use these measures to make comparisons among sets of data.


Lesson 12-3 Conditional Probability

Find probabilities of events given the occurrence of other events

Use contingency tables to find conditional probabilities

Lesson 12-4 Probability and Probability Distributions

Find probabilities by using combinations and permutations

Create and use graphs of probability distributions

Lesson 12-5 The Normal Distribution

Determine whether a set of data appears to be normally distributed or skewed

Use the Empirical Rule to find probabilities

2013-2014 Algebra 2

58


MACC.912.S-CP.1.3: 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. MACC.912.S-CP.1.4: 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. MACC.912.S-CP.1.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. MACC.912.S-CP.2.6: 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.

MACC.912.S-CP.2.9: Use permutations and combinations to compute probabilities of compound events and solve problems. MACC.912.S-IC.1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

MACC.912.S-IC.1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.


How do you find probabilities of events given the occurrence of other events? How do you use contingency tables to find conditional probabilities?

How do you find probabilities by using combinations and permutations? How do you create and use graphs of probability distributions?

How do you determine whether a set of data appears to be normally distributed or skewed? How do you use the empirical rule to find probabilities?

2013-2014 Algebra 2

59


Conditional probability Contingency table relative frequency

Success failure sample space random variable probability distribution uniform distribution relative-frequency histogram discrete probability expected value

RESOURCES





ex.php





NGSSS Benchmark MA.912.P.2.3: Understand and use the concept of conditional probability, including: understanding how conditioning affects the probability of events and finding conditional probabilities from a two-way frequency table



Level 1: Recall NGSSS Benchmark MA.912.P.1.2- Use formulas for permutations and combinations to count outcomes and determine probabilities of events. MA.912.P.3.1 Determine probabilities of events from distributions, including: discrete: Uniform, binomial, normal and exponential


Level 1: Recall NGSSS Benchmark MA.912.P.3.1 Determine probabilities of events from distributions MA.912.P.3.2 Determine the mean and variance of distributions; (including discrete, uniform, binomial, normal and exponential MA.912.P.3.3 Apply properties of the normal distribution

2013-2014 Algebra 2

60


Lesson 12-6 Hypothesis Testing

Compare sample statistics and population parameters

Design experiments to test hypotheses

Lesson 12-7 Binomial Distributions

Find probabilities for binomial experiments

Find probabilities by using binomial distributions and expansions

Use unions, intersections, or complements to describe the set of outcomes

Apply the Addition Rule


MACC.912.S-IC.1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. MACC.912.S-IC.1.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

MACC.912.S-IC.1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

MACC.912.S-CP.1.1: 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”). MACC.912.S-CP.2.7: 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.


How do you compare sample statistics and population parameters? How do you design experiments to test hypotheses?

How do you find probabilities for binomial experiments? How do you find probabilities by using binomial distributions of expansions?

How would you describe a set of outcomes using unions, intersections, or complements? How do you apply the Addition Rule?

2013-2014 Algebra 2

61


How do you compare sample statistics and population parameters? How do you design experiments to test hypotheses?

How do you find probabilities for binomial experiments? How do you find probabilities by using binomial distributions of expansions?

RESOURCES





ex.php




Level 1: Recall


NGSSS Benchmark MA.912.S.5.2 Apply the general principals of hypotheses testing MA.912.S.5.3: Explain and identify the following: null hypothesis, alternative hypotheses. Type 1 error, and Type II error


Level 1: Recall NGSSS Benchmark MA.912.P.3.1 Determine probabilities of events from distributions MA.912.P.3.2 determine the mean and variance of distributions; (including discrete, uniform, binomial, normal and exponential


Level 1: Recall


2013-2014 Algebra 2

62


UNIT TITLE: Trigonometric Functions and Identities

UNIT ESSENTIAL QUESTION: How do you use trigonometric functions and identities to solve real-world problems?



Lesson 13-2 Angles and Angle Measures

Draw and find angles in standard positions

Convert between degree measures and radian measures

Lesson 13-7 Graphing Trigonometric Functions

Describe and graph the sine, cosine, and tangent functions

Describe and graph other trigonometric functions

Lesson 14-1 Trigonometric Identities

Use trigonometric identities to find trigonometric values

Use trigonometric identities to simplify expressions


MACC.912.F-TF.1.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. MACC.912.F-TF.1.2: 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.

MACC.912.F-TF.2.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

MACC.912.F-TF.3.8: Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

2013-2014 Algebra 2

63


How do you convert between degree and measures?

How to describe and graph the sine, cosine, and tangent functions? How do you describe and graph other trigonometric functions?

How do you use trigonometric identities to find trigonometric values? How do you use trigonometric identities to simplify expressions?

VOCABULARY VOCABULARY VOCABULARY standard position initial side terminal side co-terminal side radian central angle arc length

Amplitude frequency

Trigonometric identities














2013-2014 Algebra 2

64



Level 1: Recall


NGSSS Benchmark MA.912.T.1.1 Convert between degree and radian measures



NGSSS Benchmark MA.912.T.1.5 Make connections between right triangle ratios, trigonometric functions, and circular functions MA.912.T.1.6 Define and graph trigonometric functions using domain, range intercepts, period, amplitude, phase shift, vertical shift, and asymptotes with and without the use of graphing technology



NGSSS Benchmark MA.912.T.3.1 Verify the basic Pythagorean identities e.g. sin2x + cos2x = 1 , and show they are equivalent to the Pythagorean Theorem.

Documents

2012-2013 CHEMISTRY 1 PACING GUIDE Prepared by: Melvin G ...images.pcmac.org/Uploads/GadsdenCounty... · MACC.912.A-CED.1.2 – Create equations in two or more variables to represent