Unit Overview. Content Area: Algebra 2 Unit Title: Preparing for Advanced Algebra Target Course/Grade Level Duration: 10 days


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1 Content Area: Algebra 2 Unit Title: Preparing for Advanced Algebra Target Course/Grade Level Duration: 10 days 11 th or 12 th graders Description This chapter 0 contains lessons on topics from previous courses. We use this chapter as a review of topics that students previously completed in other courses. We will review several concepts, skills, and vocabulary terms. Students are evaluated by a unit test along with other alternate assessments throughout the unit. Representing functions FOIL Factoring polynomials Counting techniques Adding probabilities Multiplying probabilities Congruent and similar figures The Pythagorean theorem Measures of center, spread and position CPI Codes FIF.HS.04 FIF.HS.05 GCO.HS.01 GCO.HS.07 GMG.HS.03 GSRT.HS.02 GSRT.HS.05 GSRT.HS.08 SMD.HS.06 SMD.HS st Century Themes and Skills See addendum What is the domain of a function? What is the range of a function? What does FOIL method mean? How to factor polynomials? How to find the total number of outcomes? How to find the probabilities of compound, independent, and dependent events? What are congruent figures? What are similar figures? What is the Pythagorean Theorem? How to find the measures of center, spread, and position? Identify the domain and range of functions. Use the FOIL method to multiply binomials. Use various techniques to factor polynomials. Find the total number of outcomes using of outcomes using a variety of methods. Compute theoretical and experimental probabilities. Compute probabilities of compound events. Find probabilities of independent and dependent events. Identify and use congruent and similar figures. Use the Pythagorean Theorem and its converse. Find measures of center, spread, and position.
2 Students will... Identify the domain and range of functions. Use the FOIL method to multiply binomials. Use various techniques to factor polynomials. Find the total number of outcomes using of a variety of methods. Compute theoretical and experimental probabilities. Compute probabilities of compound events. Find probabilities of independent and dependent events. Identify and use congruent and similar figures. Use the Pythagorean Theorem and its converse. Find measures of center, spread, and position. Locate coordinates Use a mapping illustration Use the Distributive Property Use factors and sums Use special products Fundamental counting principle Permutations of n objects taken r at a time Combinations of n objects taken r at a time Theoretical and experimental probability Probability of independent events Probability of dependent events Conditional probability Twoway frequency table Determine congruency Determine similarity Solve a problem involving similarity Find hypotenuse measures Find leg measures Identify a right triangle Measures of center Measures of spread Fivenumber summary Effect of an outlier Content Area: Mathematics Unit Title: Algebra 2 Equations and Inequalities. Target Course/Grade Level 11 th and/or 12 th graders Duration: 23 Days Description : Students begin this unit by using a number line to graph and order real numbers and by identifying the properties of real numbers in operations. After evaluating and simplifying algebraic expressions, students solve linear equations. They also rewrite equations with more than one variable, including formulas. To set up and solve reallife applications, students use a general fivestep problem solving plan, and implement various strategies such as drawing a model or looking for a pattern. Finally students use these skills to solve simple and compound inequalities as well as absolute value equations and inequalities. Students are evaluated by a unit test along with other alternate assessments throughout the unit.
3 Expressions and formulas Properties of real numbers Solving equations Solving absolute value equations Solving inequalities Solving compound and absolute value inequalities CPI Codes Use the order of operations to evaluate expressions. Use formulas. Classify real numbers. Use the properties of real numbers to evaluate expressions. Translate verbal expressions into algebraic expressions and equations and vice versa. Solve equations using the properties of equality. Evaluate expressions involving absolute values. Solve absolute value equations. Solve onestep inequalities. Solve multistep inequalities. Solve compound inequalities. Solve absolute value inequalities. MA.CC APR.HS.01 CED.HS.04 REI.HS.01 REI.HS st Century Themes and Skills See addendum What is the order of operations? How do we classify real numbers? What are the common properties to evaluate expressions? How do you evaluate expressions? How do we solve absolute value equations? How do we solve onestep inequalities? How do we solve compound inequalities? How do we solve absolute value inequalities? Students will... Use the order of operations to evaluate expressions. Use formulas. Classify real numbers. Use the properties of real numbers to evaluate expressions. Translate verbal expressions into algebraic expressions and equations and vice versa. Solve equations using the properties of equality. Evaluate expressions involving absolute values. Solve absolute value equations. Solve onestep inequalities. Solve multistep inequalities.
4 Solve compound inequalities. Solve absolute value inequalities. Review the graphing of equations on a plane. (y=mx + b) Apply method to reallife situations: wreaths for the holiday s problem. Discuss solutions as they relate to graphs and the kinds of solutions possible. (One, none, many). Use the linear combination method to solve a system when terms are lined up, when they are not lined up, when neither, one or both need to be multiplied in order to be solved. Determine the number of solutions to a given system and why. Use the substitution method to solve a system when it is in the y=mx+b form and when it has to be rewritten. Determine the number of solutions to a given system and why. Determine the bounds formed by constraints in order to find optimized values. Use optimization method on reallife situations. Evaluate functions of two variables. Plot points in three dimensions. Graph linear equations in three variables. Content Area: Algebra 2 Unit Title: Polynomials and Polynomial Functions Target Course/Grade Level 11 th and/or 12th Duration: 18 Days Description Students use properties of exponents and scientific notation to simplify algebraic expressions and to model reallife problems. They use synthetic substitution to evaluate polynomial expressions. They also graph polynomials and investigate their end behavior. They will add, subtract, multiply and divide polynomial functions and will use factoring, synthetic division, and rational zero theorem to find the zeros of polynomial functions. Students will see how the fundamental theorem of algebra can be used to determine the number of solutions of a polynomial equation and will use graphing calculators to approximate the real zeros of a polynomial function. They will use zeros to write polynomial functions and they will use xintercepts and turning points to graph polynomial functions. Students are evaluated by a unit test along with other alternative assessments throughout the unit. Operations with polynomials Dividing polynomials Polynomial functions Analyzing graphs of polynomial functions Solving polynomial equations The remainder and factor theorems Roots and zeros Rational zero theorem Multiply, divide and simplify monomials and expressions involving powers. Add, subtract, and multiply polynomials. Divide polynomials using long division. Divide polynomials using synthetic division. Evaluate polynomial functions. Identify general shapes of graphs of polynomial functions. Graph polynomial functions and locate their zeros. Find the relative maxima and minima of polynomial functions. Factor polynomials. Solve polynomial equations by factoring. Evaluate functions by using synthetic substitution. Determine whether a binomial is a factor of a polynomial by using synthetic substitution. Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function. Identify possible rational zeros of a polynomial function.
5 CPI Codes APR.HS.01 APR.HS.02 APR.HS.03. APR.HS.04 APR.HS.05 REI.HS.11 NQ.HS.03 Find all of the rational zeros of a polynomial function. 21 st Century Themes and Skills See addendum What is a power? What is a base? What is the product of powers property? What is the power of a power property? What is the negative exponent property? What is the zero exponent property? What is the quotient of powers property? What is the power of a quotient property? How do you simplify expressions using the properties of exponents? What is a polynomial function? What is the leading coefficient? What is the degree of a polynomial function? What is the standard form of a polynomial function? How do you identify a polynomial function? How do you evaluate a polynomial function? What is direct substitution? What is synthetic substitution? How do you graph a polynomial function? How do you add polynomials? How do you subtract polynomials? How do you multiply polynomials? How do you factor a quadratic expression? How do you factor a polynomial expression? How do you factor the sum or difference of cubes? How do you factor by grouping? How do you factor polynomials in quadratic form? How do you solve polynomials equations using factoring? What is the remainder theorem? What is the factor theorem? How does the remainder theorem relate to the factor theorem? How do you divide a polynomial using polynomial long division? How do you divide a polynomial using synthetic division? What is the rational zero theorem? How do you find the rational zeros of a polynomial function using the rational zero theorem? How are zeros, factors, solutions, and xintercepts closely related? How do you use xintercepts to graph a polynomial function? What are turning points of a polynomial function?
6 How do you find turning points? What is the local maximum and minimum of a polynomial function? What is a cubic function? What is a finite difference? How do you find finite differences? What are the properties of finite differences? How do you model with finite differences? How do you model with cubic regression? Students will... Multiply, divide and simplify monomials and expressions involving powers. Add, subtract, and multiply polynomials. Divide polynomials using long division. Divide polynomials using synthetic division. Evaluate polynomial functions. Identify general shapes of graphs of polynomial functions. Graph polynomial functions and locate their zeros. Find the relative maxima and minima of polynomial functions. Factor polynomials. Solve polynomial equations by factoring. Evaluate functions by using synthetic substitution. Determine whether a binomial is a factor of a polynomial by using synthetic substitution. Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function. Identify possible rational zeros of a polynomial function. Find all of the rational zeros of a polynomial function. Evaluate a polynomial function using direct substitution and synthetic substitution. Graph polynomials Use the properties of exponents and scientific notation in a reallife problem. Model polynomial subtraction and multiplication. Solve a polynomial equation by factoring. Divide a polynomial using polynomial long division and by using synthetic division. Find the rational zeros of a polynomial function using the rational zero theorem. Test the zeros using synthetic division. Use the fundamental theorem of Algebra to determine the number of zeros a polynomial function has. Determine the zeros of a polynomial function. Use zeros to write a polynomial function. Graph a polynomial function using xintercepts. Find the turning points of a polynomial function. Content Area: Algebra 2 Unit Title: Inverses and Radical Functions and Relations Target Course/Grade Level Duration: 15 Days 11 th and/or 12th Description Students learn how to evaluate nth roots of real numbers using both radical and exponential notation. They use properties of rational exponents to evaluate and simplify expressions, and they evaluate power functions and perform arithmetic operations with functions as well as composition of functions. They will find inverses of functions, and they observe the graphing of square root and cube root functions. Students will solve equations that have radicals or rational exponents. They use roots, rational exponents, power functions, function operations, and radical equations to solve reallife problems. Students are evaluated by a unit test along with other alternate assessments throughout the unit.
7 CPI Codes Operations on functions Inverse functions and relations Square root functions and inequalities Nth roots Operations with radical expressions Rational exponents Solving radical equations and inequalities REI.HS.02 FBF.HS.04 FIF.HS.01 FIF.HS.02 FIF.HS.07 FIF.HS.08 N RN.HS.01 N RN.HS.02 N RN.HS.03 Find the sum, difference, product, and quotient of functions. Find the composition of functions. Find the inverse of a function or relation. Determine whether two functions or relations are inverses. Graph and analyze square root functions. Graph square root inequalities. Simplify radicals. Use a calculator to approximate radicals. Simplify radical expressions. Add, subtract, multiply, and divide radical expressions. Write expressions with rational exponents in radical form and vice versa. Simplify expressions in exponential or radical form. Solve equations containing radicals. Solve inequalities containing radicals. 21 st Century Themes and Skills See addendum How to evaluating Nth Roots? How to using Nth Roots in Real Life? How do you find the nth root of a number? How do you evaluate expressions with rational exponents? How do you solve equations using nth roots?
8 How do you use properties of rational exponents to simplify expressions How do you use write radicals in simplest form? How do you add and subtract roots and radicals? How to use properties of rational exponents in real life? How do you add and subtract functions? How do you multiply and divide functions? How do you find the composition of functions? Students will... Find the sum, difference, product, and quotient of functions. Find the composition of functions. Find the inverse of a function or relation. Determine whether two functions or relations are inverses. Graph and analyze square root functions. Graph square root inequalities. Simplify radicals. Use a calculator to approximate radicals. Simplify radical expressions. Add, subtract, multiply, and divide radical expressions. Write expressions with rational exponents in radical form and vice versa. Simplify expressions in exponential or radical form. Solve equations containing radicals. Solve inequalities containing radicals. Evaluating nth roots of real numbers using both radical notation and rational exponent notation. Using properties of rational exponents to evaluate and simplify expression Perform operations with functions including power functions. Find inverses of linear functions and find inverses of nonlinear functions. Graph square root and cube root functions. Solve equations that contain radicals or rational exponents. Use measures of central tendency and measures of dispersion to describe data sets and use boxandwhisker plots and histograms to represent data graphically. Content Area: Algebra 2 Unit Title: Exponential and Logarithmic Functions and Relations Target Course/Grade Level 11 th and/or 12th Duration: days Description The chapter begins by defining b to the x power, when x is an integer and extending the existing properties to include rational exponents. The meaning of b to the x power is extended to include irrational values of x. Students relate to the exponential functions. Students define composites and inverse functions, leading to the definition of the logarithmic function as the exponential inverse. The culmination of the topic ends in students solving equations of the type 2 to the x = 5. Students extend and apply these skills to reallife situations. Evaluation of student learning occurs via homework, quizzes and test upon completion of the chapter.
9 Graphing exponential functions Solving exponential equations and inequalities Logarithms and logarithmic functions Solving logarithmic equations and inequalities Properties of logarithms Common logarithms Base e and natural logarithms Using exponential and logarithmic functions CPI Codes FLE.HS.02 FLE.HS.03 FLE.HS.04 Graph exponential growth functions. Graph exponential decay functions. Solve exponential equations. Solve exponential inequalities. Evaluate logarithmic expressions. Graph logarithmic functions. Solve logarithmic equations. Solve logarithmic inequalities. Simplify and evaluate expressions using the properties of logarithms. Solve logarithmic equations using the properties of logarithms. Solve exponential equations and inequalities using common logarithms. Evaluate logarithmic expressions using the change of base formula. Evaluate expressions involving the natural base and natural logarithm. Solve exponential equations and inequalities using natural logarithms. Use logarithms to solve problems involving exponential growth and decay. Use logarithms to solve problems involving logistic growth. 21 st Century Themes and Skills See addendum What is the exponential form? What is the radical form? What is an exponent equation? What does the graph of an exponential function look like? What is the composite of the functions? What are the inverse functions? What is log? How do you write an equation in logarithmic form? How do you solve a logarithmic equation? What are the laws of logarithms? How can one use a calculator to find logarithms? How can solve an exponential equation using logarithms? What is exponential growth? What is exponential decay? What is the natural logarithm?
10 Students will... Graph exponential growth functions. Graph exponential decay functions. Solve exponential equations. Solve exponential inequalities. Evaluate logarithmic expressions. Graph logarithmic functions. Solve logarithmic equations. Solve logarithmic inequalities. Simplify and evaluate expressions using the properties of logarithms. Solve logarithmic equations using the properties of logarithms. Solve exponential equations and inequalities using common logarithms. Evaluate logarithmic expressions using the change of base formula. Evaluate expressions involving the natural base and natural logarithm. Solve exponential equations and inequalities using natural logarithms. Use logarithms to solve problems involving exponential growth and decay. Use logarithms to solve problems involving logistic growth. Write an expression in exponential form Write an expression in simplest radical form Show the graph of exponential function. Solve an exponential equation. Find the composition of functions. Show the inverse of a function. Evaluate functions. Write an equation in exponential form. Write an equation in logarithmic form. Simplify logarithms. Content Area: Algebra 2 Unit Title: Linear Equations and Functions Target Course/Grade Level 11 th and/or 12th Duration: 27Days Description Students begin this unit by identifying and representing relations and functions, and by graphing and evaluating linear functions. They find the slope of a line, and identify parallel and perpendicular lines from their slopes. Students generalize slope as a rate of change. They graph linear equations using both slopeintercept and standard forms and identify and graph horizontal and vertical lines. They write equations of lines using the slope and intercept, a point and the slope, or two points. Students write direct variation equations. They explore positive and negative correlation using scatter plots, and approximate bestfitting lines. Students then complete the unit by graphing linear inequalities in two variables, piece wise functions, and absolute value functions, while using all of these to model reallife applications. Students are evaluated by a unit test along with other alternate assessments throughout the unit. Relations and functions Linear relations and functions Rate of change and slope Analyze relations and functions. Use equations of relations and functions. Identify linear relations and functions.
11 CPI Codes Garfield High School Writing linear equations Scatter plots and lines of regression Special functions Parent functions and transformations Graphing linear and absolute value inequalities CED.HS.02 REI.HS.10 REI.HS.11 FIF.HS.01 FIF.HS.02 FIF.HS.07 SID.HS.07 SID.HS st Century Themes and Skills See addendum What is a function? What forms of equations do you know? What is the slopeintercept form of a line? What forms of equations do you know? What is a scatter plot? What is the first step in graphing an inequality on a plane? What does piecewise mean? What is meant by absolute value? Students will... Analyze relations and functions. Use equations of relations and functions. Identify linear relations and functions. Write linear equations in standard form. 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. Write linear equations in standard form. 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. Use scatter plots and prediction equations. Model data using lines of regression. Write and graph piecewisedefined functions. Write and graph step and absolute value functions. Identify and use parent functions. Describe transformations of functions. Graph linear inequalities. Graph absolute value inequalities.
12 Use scatter plots and prediction equations. Model data using lines of regression. Write and graph piecewisedefined functions. Write and graph step and absolute value functions. Identify and use parent functions. Describe transformations of functions. Graph linear inequalities. Graph absolute value inequalities. Represent relations and functions as well as to graph and evaluate linear functions. Find slopes of lines and classify parallel and perpendicular lines as well as solving reallife problems. Use the slopeintercept form of a line to graph linear equations. Write linear equations in different forms given various types of data: slopes, points, intercepts, or graphs. Graph linear inequalities in two variables on a plane. Represent piecewise functions on a graph and to use piecewise functions to represent reallife situations. Represent and write absolute value functions. Content Area: Algebra 2 Unit Title: Quadratic Functions and Relations Target Course/Grade Level 11 th and/or 12th Duration: 23 Days Description Students are led through all major topics involving quadratic functions. First quadratic equations are graphed and then the standard, vertex and intercept forms are introduced. These forms are used throughout the unit and students learn to convert between them. Quadratic expressions are factored then quadratic equations are solved by factoring, finding square roots, completing the square or using the Quadratic Formula. Factoring is also used to find the zeroes of quadratic function. Students solve quadratic equations with complex solutions and perform operations with complex numbers. The discriminate is used to determine the number and nature of the solutions to a quadratic equation. Students are evaluated by a unit test along with other alternate assessments throughout the unit. Graphing quadratic functions Solving quadratic equations by graphing Solving quadratic equations by factoring Complex numbers Completing the square The quadratic formula and the discriminate Transformations of quadratic graphs Quadratic inequalities Graph quadratic functions. Find and interpret the maximum and minimum values of a quadratic function. Solve quadratic equations by graphing. Estimate solutions of quadratic equations by graphing. Write quadratic equations in standard form. Solve quadratic equations by factoring. Perform operations with pure imaginary numbers. Perform operations with complex numbers. Solve quadratic equations by using the Square Root Property. Solve quadratic equations by completing the square. Solve quadratic equations by using the Quadratic Formula.
13 CPI Codes Garfield High School Use the discriminate to determine the number and type of roots of a quadratic equation. Write a quadratic function in the form Transform graphs of quadratic functions of the form Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable. REI.HS.10 N CN.HS.01 N CN.HS.02 N CN.HS.03 N CN.HS.04 N CN.HS.05 N CN.HS.07 N CN.HS st Century Themes and Skills See addendum How do we graph a quadratic function? What are the maximum and the minimum values of a quadratic function? How do we estimate the solutions of a quadratic equation? How do we factor quadratic equations? What is an imaginary number? What is a complex number? How do we perform operations with imaginary and complex numbers? What is the Square Root Property? What is the quadratic formula? How can we use the discriminate to predict the type and number of solutions in a quadratic equation? What is the vertex formula of a quadratic equation? How do we solve quadratic inequalities with one or two variables? Students will... Graph quadratic functions. Find and interpret the maximum and minimum values of a quadratic function.
14 Solve quadratic equations by graphing. Estimate solutions of quadratic equations by graphing. Write quadratic equations in standard form. Solve quadratic equations by factoring. Perform operations with pure imaginary numbers. Perform operations with complex numbers. Solve quadratic equations by using the Square Root Property. Solve quadratic equations by completing the square. Solve quadratic equations by using the Quadratic Formula. Use the discriminate to determine the number and type of roots of a quadratic equation. Write a quadratic function in the form Transform graphs of quadratic functions of the form Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable. Graphing quadratic function and writing it in standard form Using the quadratic model Factoring a trinomial and monomial. Solving a quadratic equation by finding the zeros Using a quadratic equation as a model. Modeling a falling object s height with a quadratic faction. Adding, subtracting, multiplying, dividing and finding the absolute value of complex numbers. Solving a quadratic equation with two real numbers and two imaginary numbers. Solving vertical motion problems. Using a quadratic inequality as a model Writing a quadratic function in a vertex form. Finding a quadratic model for data set. Using a quadratic recession to find model. Content Area: Algebra 2 Unit Title: Rational Functions and Relations Target Course/Grade Level 11 th and/or 12th Duration: 15 days Description Students study rational algebraic expressions studied. The laws of exponents are reviewed and extended to include zero and negative exponents. Students use scientific notation by applying the law of exponents in practical applications. They will simplify, add, subtract, multiply, and divide rational expressions in word problems. Students then extend the skills to solve equations with fractional coefficients and fractional equations. They also extend and apply these skills to reallife situations. Variation is presented by way of problems in physics and other areas. Students apply direct variation and proportion with real life application. Inverse and joint variation follows. Long division for polynomials is presented and is related synthetic division introduced. Students extend and apply these skills to reallife situations. Evaluation of student learning occurs via homework, quizzes and test upon completion of the chapter. Multiplying and dividing rational expressions Adding and subtracting rational expressions. Graphing reciprocal functions Simplify rational expressions. Simplify complex fractions. Determine the LCM of polynomials.
15 CPI Codes Graphing rational functions Variation functions Solving rational equations and inequalities APR.HS.01 APR.HS.02 APR.HS.03 APR.HS.04 APR.HS.06 APR.HS.07 CED.HS.01 SSE.HS.02 APR.HS.01 APR.HS.07 SSE.HS.03 SSE.HS.04 FBF.HS.01 FBF.HS.03 Add and subtract rational expressions. Determine properties of reciprocal functions. Graph transformations of reciprocal functions. Recognize and solve direct and joint variation problems. Recognize and solve inverse and combined variation problems. Solve rational equations. Solve rational inequalities.
16 FIF.HS.08 FIF.HS.09 FLE.HS st Century Themes and Skills See addendum How do you simplify fractions with exponents? How do you simplify expressions with zero and negative exponents? How do you write a number in scientific notation? How do you simplify rational expression? What are the domain and the zeros of a given function? How do you find a significant digit? How do you simplify rational expression using addition and subtraction? How do you simplify complex fractions? How do you solve an equation with fractional coefficients? How do you solve an inequality with fractional coefficients? How do you solve a fractional equation? What is direct variation? How to find the constant in direct variation? What is Inverse Variation? What is Joint Variation? Students will... Simplify rational expressions. Simplify complex fractions. Determine the LCM of polynomials. Add and subtract rational expressions. Determine properties of reciprocal functions. Graph transformations of reciprocal functions. Recognize and solve direct and joint variation problems. Recognize and solve inverse and combined variation problems. Solve rational equations. Solve rational inequalities. Simplify fractions with exponents. Write expressions in simplest form without negative or zero exponents. Find a onesignificantdigit estimate of each given quotient. Find the domain of each given function and its zeros, if any. Simplify given rational expressions. Solve inequality and equations. Solve problems involving direct variation and find the constant of direct variation. Use synthetic division to divide a polynomial by a binomial. Divide one polynomial by another polynomial. Content Area: Algebra 2 Unit Title: Systems of Linear Equations and Inequalities Target Course/Grade Level 11 th and/or 12th
17 Duration: 25 Days Description In this unit students learn to solve systems of two linear equations in two variables algebraically and by graphing. Included are these systems with one solution, no solution, and many solutions. They also learn to graph the solutions of systems of linear inequalities. Systems of linear equations and inequalities are used to model and solve reallife problems. The work with linear equations is extended to linear programming problems, which are used to solve reallife optimization problems. Students learn to add, subtract and multiply matrices by a scalar and by another matrix. They then use these operations to solve realworld problems. Students find determinants of a 2 x 2 and a 3 x 3 matrix. Then, using Cramer's Rule, the determinants are used to solve systems of equations. Students are evaluated by a unit test along with other alternative assessments throughout the unit. The graphing calculator becomes an invaluable at this point, as the value of the determinant is calculated using this technology. Students are evaluated by a unit test along with other alternate assessments throughout the unit. Solving systems of equations Solving systems of inequalities by graphing Optimization with linear programming Systems of equations in three variables Operations with matrices Multiplying matrices Solving systems of equations using Cramer s Rule Solving systems of equations using inverse matrices CPI Codes REI.HS.01 REI.HS.05 REI.HS.06 REI.HS.10 Solve systems of linear equations graphically. Solve systems of linear equations algebraically. Solve systems of inequalities by graphing. Determine the coordinates of the vertices of a region formed by the graph of a system of inequalities. Find the maximum and minimum values of a function over a region. Solve realworld optimization problems using linear programming. Solve systems of linear equations in three variables. Solve realworld problems using systems of linear equations in three variables. Analyze data in matrices. Perform algebraic operations with matrices. Multiply matrices. Use the properties of matrix multiplication. Evaluate determinants. Solve systems of linear equations by using Cramer s Rule. Find the inverse of a 2 Write and solve matrix equations for a system of equations.
18 REI.HS.12 N VM.HS.06 N VM.HS.07 N VM.HS.08 N VM.HS.09 N VM.HS.10 N VM.HS.11 N VM.HS st Century Themes and Skills See addendum What is the order of a matrix? How do you know if you can perform operations on two matrices? What are equal matrices? What are corresponding entries? How do you add or subtract two matrices? What is a scalar? How do you multiply a matrix by a scalar? How do you determine if you two matrices can be multiplied together? How do you determine the order of the product of two matrices? How do you calculate the product of two matrices? What is a square matrix? What is a determinant? How do you find the determinant of a 2x2 matrix? How do you find the determinant of a 3x3 matrix? What is Cramer's Rule? How do you solve a linear system of equations using Cramer's rule? What is and identity matrix? What does a 2x2 identity matrix look like? What does a 3x3 identity matrix look like? What are the relationship between and identity and inverse matrix? How do you determine the inverse of a 2x2 matrix? How do you determine the inverse of a 2x2 matrix? How do you solve linear systems of equations using inverse matrices?
19 Students will... Solve systems of linear equations graphically. Solve systems of linear equations algebraically. Solve systems of inequalities by graphing. Determine the coordinates of the vertices of a region formed by the graph of a system of inequalities. Find the maximum and minimum values of a function over a region. Solve realworld optimization problems using linear programming. Solve systems of linear equations in three variables. Solve realworld problems using systems of linear equations in three variables. Analyze data in matrices. Perform algebraic operations with matrices. Multiply matrices. Use the properties of matrix multiplication. Evaluate determinants. Solve systems of linear equations by using Cramer s Rule. Find the inverse of a 2 Write and solve matrix equations for a system of equations. Review the graphing of equations on a plane. Apply method to reallife situations. Discuss solutions as they relate to graphs and the kinds of solutions possible. Use the linear combination method to solve a system when terms are lined up, when they are not lined up, when neither, one or both need to be multiplied in order to be solved. Determine the number of solutions to a given system and why. Graph systems of inequalities on a plane Determine the bounds formed by constraints in order to find optimized values Determine the optimized value for an unbounded region. Use o Identify and compare the orders of two matrices. Add and subtract two matrices. Multiply a matrix by a scalar. Determine if two matrices can be multiplied. Describe the order of matrix products Calculate the products of two matrices. Find the determinant of a 2x2 matrix Find the determinant of a3x3 matrix. Solve a linear system of equations using Cramer's Rule. Recognize 2x2 and 3x3 identity matrices Determine the inverse of a 2x2 matrix. Optimization method on reallife situations. Plot points in three dimensions.
20 Content Area: Algebra 2 and Trigonometry Unit Title: Products and Factors of Polynomials Target Course/Grade Level: Duration: 3 weeks Description : This chapter consists of working with polynomials, factors of polynomials and applications of factoring. The terminology of polynomials is reviewed and the laws of exponents are used to develop the concepts used to find the products of polynomials. Greatest common factors and least common multiples are used to develop methods of factoring polynomials. Students develop methods to illustrate how to solve polynomial equations, inequalities and word problems. These skills are applied to reallife situations. Student learning is evaluated via homework, quizzes and test upon completion of the chapter. Laws of Exponents Factoring Polynomials Applications of Factoring to Problem Solving RealLife Extension of Factoring to Problem Extending Problem Solving Methods to Inequalities CPI Codes APR.HS. 01 APR.HS. 02 APR.HS. 03 APR.HS. 04 APR.HS. The laws of exponents can be applied to simplify complex polynomial expressions. Rewriting sum and differences into products is essential in algebra. Operations with polynomials must utilize laws of exponents. Factoring quadratic polynomials using GCF, difference of two squares, perfect trinomial squares, general method, sum/difference of two cubes and grouping is essential to problem solving. Applying factoring methods to general quadratic equations is a basic concept. RealLife problem solving by applying methods of factoring gives validity to studying factoring. The extension of solving equations can be extended and broadened to inequalities.
21 05 APR.HS. 06 APR.HS st Century Themes and Skills See Addendum What is the degree of a polynomial? How do you add polynomials? How do you subtract polynomials? What does simplify a polynomial mean? How do you simplify a product of powers? When do you use a power of a product? How do you simplify a power of a power? How do you multiply two binomials? How do you multiply a binomial with a polynomial? What is the prime factorization? How do you find the GCF of integers and monomials? How do you find the LCM of integers and monomials? What is a perfect square trinomial? What does a difference of squares mean? How do you factor the sum and the difference of cubes? Why does regrouping a polynomial is important? How to factor a trinomial? How to factor a trinomial with a coefficient that is negative or positive? How to determine whether a polynomial is prime? How to set an equation to zero? How to factor a polynomial equation? How to find the zeros of an equation or function? How do you know how to set up a polynomial equation for a certain word problem? How do you find and graph the solution set of a polynomial inequality? Students will... Simplify, add, and subtract polynomials. Use the laws of exponents to multiply a polynomial by a monomial. Multiply polynomials. Find the Greatest Common Factor and the Least Common Multiple of integers and monomials. Factor polynomials by using the GCF, by recognizing special products, and by grouping terms. To factor quadratic polynomials. Solve polynomial equations. Solve problems in reallife situations using polynomial equations. Solve polynomial inequalities. Write the degree of a polynomial. Add, subtract and simplify polynomials. Simplify the product of a power and the power of a product. Multiply two binomials. Multiply and simplify a binomial and a polynomial.
22 Prime factorization of an integer. GCF and LCM of 2 integers. Find the GCF and LCM of monomials. Factor a trinomial that is the sum or difference of squares. Factor a sum or a difference of 2 cubes. Polynomial by regrouping. trinomial where the coefficient of x^2 is 1 trinomial where the coefficient of x^2 is Determine that a polynomial is prime. Set an equation to zero. Factor a polynomial equation. Find the zeros of an equation or a function. Solve reallife situations using polynomial equations. Content Area: Algebra 2 and Trigonometry Unit Title: Inequalities Target Course/Grade Level: Duration: 2 weeks Description : Students are presented with the basic techniques of solving equations and this procedure is extended to inequalities. Methods for solving inequalities involving conjunction and disjunction and methods for solving inequalities involving absolute value are also introduced. Students solve absolute value problems h algebraically and graphically. These skills are extended to reallife situations. Student learning is evaluated via homework, quizzes and test upon completion of the chapter. Inequalities in One Variable. Combined Inequalities. Problem Solving with Inequalities. Absolute Value in Open Sentences CPI Codes REI.HS.03 REI.HS.04 REI.HS.10 REI.HS.11 Use of the number line to see that there are an infinite number of solutions to inequalities. Disjunction and conjunction illustrate union and intersection of sets to solve combined inequalities. Applications of algebraic solutions to inequalities are studied. Absolute value problems are solved both graphically and algebraically.
23 REI.HS st Century Themes and Skills See Addendum How do you solve and graph a linear inequality? How do you solve and graph an inequality with the variable on both sides? How do you graph the solution set of a conjunction? How do you graph the solution set of a compound inequality? How do you graph the solution set of a disjunction? What does the expression "at least" mean? How can I find consecutive integers whose sum is a certain number? How do you solve an absolute value equation? How do you solve an absolute value inequality using <? How do you solve an absolute value inequality using >? What is the distance between the graphs of each pair of numbers? How do you solve graphically an absolute value equation? How do you solve graphically an absolute value inequality using < or >? Students will... Solve simple inequalities in one variable. Solve conjunctions and disjunctions. Solve word problems by using inequalities in one variable. Solve open sentences involving absolute value. Use number lines to obtain quick solutions to certain equations and inequalities involving absolute value. Solve & graph a various linear inequality and inequalities with the variable on both sides. Graph the solution set of a conjunction, disjunction and compound inequality. Solve a word problem with consecutive integers in an inequality. To solve an absolute value inequality using <,> and an absolute value equation. To solve using a graph an absolute value equation. To solve using a graph an absolute value inequality with > and <. Content Area: Algebra 2 and Trigonometry Unit Title: Target Course/Grade Level: Duration: 3 Weeks The Language of Algebra Description : Students review the basic concepts and skills of algebra studied in previous courses. This includes real numbers and expressions, operations with real numbers and problem solving in realworld situations so that students can see practical applications of algebra being taught. Emphasis is placed on dealing with real numbers symbolically and in context of word problems. Student learning is evaluated with quizzes and a chapter test upon completion of the chapter. Real Numbers All real numbers can be represented on a number
24 Order of Operations Simplifying Expressions. Field Properties of Real Numbers. Solving Equations. line which will reinforce the relationships between rational and irrational numbers and help in understanding the order. The need to use the order of operations is required to simplify mathematical expression. Algebraic expressions are converted to numerical expressions and the order of operations is used to simply. Applications of the field properties of equality, closure, associative, commutative, inverse, identity and distributive laws are required to solve equations. The plan for problem solving is necessary so that word problems can be set up and solved. CPI Codes CED.HS.01 CED.HS.03 REI.HS.03 REI.HS.04 SSE.HS.01 SSE.HS.03 N RN.HS.01 N RN.HS st Century Themes and Skills See Addendum 1. How to find coordinates of a point 2. How to write the inequality statement 3. How to find the absolute value of an expression 4. how to use equation and inequality 5. how to simplify expressions 6. how to evaluate algebraic expressions 7. How to choose the right property/ 8. How to simplify an algebraic expression using steps? 9. How to choose the right property?
25 10. How to simplify an algebraic expression using steps? 11. How to simplify and name the property in each step of an expression? 12. How to solve an equation for a given variable? 13. How to translate a phrase into an algebraic expression? 14. How to pick a variable to represent an unknown? Students will... Graph real numbers on a number line, to compare numbers, and to find their absolute values. Review the methods used to simplify numerical expressions and to evaluate algebraic expressions. Review properties of equality of real numbers and properties for adding and multiplying real numbers. Review the rules for adding and subtracting real numbers. Review rules for multiplying and dividing real numbers. Solve certain equations in one variable. Translate word phrases into algebraic expressions and word sentences into equations. Graph the given two numbers, and then write an inequality statement comparing them. Find the coordinate of the point one third of the way from C to D Find the value of each absolute value expression. Use the inequality and equality symbols to make a true statement. Simplify the given expressions using the order of operations and grouping symbols. Evaluate the algebraic expressions when the values of the variables are given. Name the properties of equality illustrated in each given statement. Choose a variable to represent an unknown number, and then write and solve an equation to describe the given situation. Content Area: Algebra 2 and Trigonometry Unit Title: Rational Expressions Target Course/Grade Level: Duration: 3 Weeks Description : Students study rational algebraic expressions studied. The laws of exponents are reviewed and extended to include zero and negative exponents. Students use scientific notation by applying the law of exponents in practical applications. They will be able to simplify, add, subtract, multiply, and divide rational expressions in word problems. Students then extend the skills to solve equations with fractional coefficients and fractional equations. They also extend and apply these skills to reallife situations. Evaluation of student learning occurs via homework, quizzes and test upon completion of the chapter. Quotients of Monomials Fractional Coefficients Fractional Equations Zero and Negative Exponents Rational Algebraic Expressions Products, Quotients, Sums and Differences of Rational Expressions. Complex Fractions The basic arithmetic operations of fractions are extended to algebraic expressions. The use of laws of exponents is combined with scientific notation to solve reallife problems. Factoring of polynomials is utilized to operate with rational expressions. The use of greatest common factor and least common multiple is applied to add and subtract
26 CPI Codes Garfield High School APR.HS.01 APR.HS.02 APR.HS.03 APR.HS.04 APR.HS.06 APR.HS.07 CED.HS.01 SSE.HS st Century Themes and Skills See Addendum How do you simplify fractions with exponents? How do you simplify expressions with zero and negative exponents? How do you write a number in scientific notation? How do you simplify rational expression? What are the domain and the zeros of a given function? How do you find a significant digit? How do you simplify rational expression using addition and subtraction? How do you simplify complex fractions? How do you solve an equation with fractional coefficients? How do you solve an inequality with fractional coefficients? How do you solve a fractional equation? rational algebraic problems. The use of greatest common factor is extended to provide tools to solve rational equations. Students will... Simplify fractions with and without variables. Review the first three laws of exponents and learn laws 4 and 5. Simplify expressions involving zero and negative exponents. Write very small or very large numbers in scientific notation. Analyze the problems to see how many digits the least accurate approximation has. Apply these skills in physics and chemistryrelated problems. Simplify rational algebraic expressions and how to find the domain and zeros of a rational function.
27 multiply and divide rational expressions using the multiplication and division rules for fraction Explain, in words, the procedures for multiplying and dividing rational expressions. Use the rules for adding and subtracting fractions to add and subtract rational expressions. Apply two methods for simplifying complex fractions. Solve equations and inequalities having fractional coefficients by using equivalent equations with integral coefficients. solve fractional equations and to check for extraneous roots Simplify fractions with exponents. Write expressions in simplest form without negative or zero exponents. Find a onesignificantdigit estimate of each given quotient. Find the domain of each given function and its zeros, if any. Simplify given rational expressions. solve inequality and equations Content Area: Algebra 2 and Trigonometry Unit Title: Analytic Geometry Target Course/Grade Level: Duration: 2 Weeks Description : Distance and midpoint formulas are the first topics introduced. Students discuss and study conic sections in depth as related to the distance formula. Students define circles, parabolas, ellipses and hyperbolas in terms of the distance formula and present them visually. They study their graphs and equations. They solve systems of three equations in three variables using graphing calculators and determinants serve to introduce technology to the students. Students extend and apply these skills to reallife situations. Evaluation of student learning occurs via homework, quizzes and test upon completion of the chapter. Distance and Midpoint Formulas Circles Parabolas Ellipses Hyperbola Central Conics Solving Quadratic Systems in Three Variables using Matrices and Graphing Calculator Ordered pairs are used to find the distance and midpoint of line segments in the coordinate plane. The distance formula is used to define an ellipse and its parts, i.e., the center, foci and intercepts The relationship between the center and radius of a circle and its equation is examined are discussed The use of focus, directrix, vertex and axis of symmetry completes the study of the parabola The distance formula is used to define a hyperbola and its parts.i.e, foci, intercepts and asymptotes The relationship between the foci, intercepts, and asymptotes of a hyperbola Conic sections with centers other than (0,0) are examined. Geometric solutions and algebraic solutions of systems involving conic
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