Content Area: Algebra 2 Unit Title: Preparing for Advanced Algebra Target Course/Grade Level Duration: 10 days 11 th or 12 th graders Unit Overview 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. Concepts 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 Concepts & Understandings Learning Targets F-IF.HS.04 F-IF.HS.05 G-CO.HS.01 G-CO.HS.07 G-MG.HS.03 G-SRT.HS.02 GSRT.HS.05 GSRT.HS.08 S-MD.HS.06 S-MD.HS.07 21 st Century Themes and Skills See addendum Guiding Questions 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? Understandings 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.
Students will... Unit Results 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. Suggested Activities The following activities can be incorporated into the daily lessons: 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 Two-way 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 Five-number 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 Unit Overview 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 real-life applications, students use a general five-step 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. Concepts & Understandings
Concepts Expressions and formulas Properties of real numbers Solving equations Solving absolute value equations Solving inequalities Solving compound and absolute value inequalities CPI Codes Learning Targets Understandings 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 one-step inequalities. Solve multi-step inequalities. Solve compound inequalities. Solve absolute value inequalities. MA.CC APR.HS.01 CED.HS.04 REI.HS.01 REI.HS.03 21 st Century Themes and Skills See addendum Guiding Questions 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 one-step inequalities? How do we solve compound inequalities? How do we solve absolute value inequalities? Unit Results 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 one-step inequalities. Solve multi-step inequalities.
Solve compound inequalities. Solve absolute value inequalities. Suggested Activities The following activities can be incorporated into the daily lessons: Review the graphing of equations on a plane. (y=mx + b) Apply method to real-life 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 real-life 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 Unit Overview Description Students use properties of exponents and scientific notation to simplify algebraic expressions and to model real-life 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 x-intercepts and turning points to graph polynomial functions. Students are evaluated by a unit test along with other alternative assessments throughout the unit. Concepts 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 Concepts & Understandings Understandings 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.
CPI Codes APR.HS.01 APR.HS.02 APR.HS.03. APR.HS.04 APR.HS.05 REI.HS.11 Q.HS.03 Learning Targets Find all of the rational zeros of a polynomial function. 21 st Century Themes and Skills See addendum Guiding Questions 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 x-intercepts closely related? How do you use x-intercepts to graph a polynomial function? What are turning points of a polynomial function?
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? Unit Results 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. Suggested Activities The following activities can be incorporated into the daily lessons: Evaluate a polynomial function using direct substitution and synthetic substitution. Graph polynomials Use the properties of exponents and scientific notation in a real-life 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 x-intercepts. Find the turning points of a polynomial function. Content Area: Algebra 2 Unit Overview 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. Concepts Concepts & Understandings Understandings
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 F-BF.HS.04 F-IF.HS.01 F-IF.HS.02 F-IF.HS.07 F-IF.HS.08 RN.HS.01 RN.HS.02 RN.HS.03 Learning Targets 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 Guiding Questions 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?
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? Unit Results 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. Suggested Activities The following activities can be incorporated into the daily lessons: 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 box-and-whisker plots and histograms to represent data graphically. Content Area: Algebra 2 Unit Overview Unit Title: Exponential and Logarithmic Functions and Relations Target Course/Grade Level 11 th and/or 12th Duration: 10-12 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 real-life situations. Evaluation of student learning occurs via homework, quizzes and test upon completion of the chapter. Concepts & Understandings
Concepts 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 F-LE.HS.02 F-LE.HS.03 F-LE.HS.04 Learning Targets Understandings 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 Guiding Questions 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? Unit Results
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. Suggested Activities The following activities can be incorporated into the daily lessons: 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 Unit Overview 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 slope-intercept 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 best-fitting 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 real-life applications. Students are evaluated by a unit test along with other alternate assessments throughout the unit. Concepts Relations and functions Linear relations and functions Rate of change and slope Concepts & Understandings Understandings Analyze relations and functions. Use equations of relations and functions. Identify linear relations and functions.
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 Learning Targets CED.HS.02 REI.HS.10 REI.HS.11 F-IF.HS.01 F-IF.HS.02 F-IF.HS.07 S-ID.HS.07 S-ID.HS.08 21 st Century Themes and Skills See addendum Guiding Questions What is a function? What forms of equations do you know? What is the slope-intercept 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? Unit Results 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 piecewise-defined 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.
Use scatter plots and prediction equations. Model data using lines of regression. Write and graph piecewise-defined 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. Suggested Activities The following activities can be incorporated into the daily lessons: 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 real-life problems. Use the slope-intercept 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 real-life 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 Unit Overview 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. Concepts 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 Concepts & Understandings Understandings 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.
CPI Codes Garfield High School Learning Targets 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 CN.HS.01 CN.HS.02 CN.HS.03 CN.HS.04 CN.HS.05 CN.HS.07 CN.HS.08 21 st Century Themes and Skills See addendum Guiding Questions 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? Unit Results Students will... 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. 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. Suggested Activities The following activities can be incorporated into the daily lessons: 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 Unit Overview 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 real-life 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 real-life situations. Evaluation of student learning occurs via homework, quizzes and test upon completion of the chapter. Concepts Multiplying and dividing rational expressions Adding and subtracting rational expressions. Graphing reciprocal functions Concepts & Understandings Understandings Simplify rational expressions. Simplify complex fractions. Determine the LCM of polynomials.
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 F-BF.HS.01 F-BF.HS.03 Learning Targets 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.
F-IF.HS.08 F-IF.HS.09 F-LE.HS.03 21 st Century Themes and Skills See addendum Guiding Questions 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? Unit Results 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. Suggested Activities The following activities can be incorporated into the daily lessons: Simplify fractions with exponents. Write expressions in simplest form without negative or zero exponents. Find a one-significant-digit 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. Unit Overview Content Area: Algebra 2 Unit Title: Systems of Linear Equations and Inequalities Target Course/Grade Level 11 th and/or 12th
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 real-life problems. The work with linear equations is extended to linear programming problems, which are used to solve real-life optimization problems. Students learn to add, subtract and multiply matrices by a scalar and by another matrix. They then use these operations to solve real-world 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. Concepts 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 Concepts & Understandings Learning Targets Understandings 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 real-world optimization problems using linear programming. Solve systems of linear equations in three variables. Solve real-world 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.
REI.HS.12 VM.HS.06 VM.HS.07 VM.HS.08 VM.HS.09 VM.HS.10 VM.HS.11 VM.HS.12 21 st Century Themes and Skills See addendum Guiding Questions 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? Unit Results
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 real-world optimization problems using linear programming. Solve systems of linear equations in three variables. Solve real-world 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. Suggested Activities The following activities can be incorporated into the daily lessons: Review the graphing of equations on a plane. Apply method to real-life 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 real-life situations. Plot points in three dimensions.