Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number 432/433 Title Algebra II (A & B) H Grade 10-11 # of Days 120 Course Description This intensive course provides an accelerated and more rigorous treatment of the logical development of algebra. Part A topics include systems of equations and inequalities, matrices, and linear, quadratic and polynomial functions. Part B topics include powers, roots and radicals, exponential, logarithmic and rational functions, and sequences and series. Student owned graphing calculators are necessary for this course. This course is recommended for students who have demonstrated exceptional interest and ability in Honors Algebra I and Geometry Honors. All students who plan to take AP Calculus should take this course. This course addresses Whitman-Hanson Student Learning Expectations 1-6. Instructional Strategies Instructional Strategies include but may not be limited to the following: 1. Whole class instruction 2. Individual work: homework, classwork, assessments 3. Group work: activities, problem solving 4. Experiments, demonstrations, investigations 5. Video presentations 6. Use of technology (graphing calculators and computers) 7. Projects Student Learning Expectations 1. Read, write and communicate effectively. 2. Utilize technologies appropriately and effectively. 3. Apply critical thinking skills. 4. Explore and express ideas creatively. 5. Participate in learning both individually and collaboratively. 6. Demonstrate personal, social, and civic responsibility. 1
Unit of Study Prerequisite Topics MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: 18 Days Chapters 1-3 Essential Questions Concepts, Content: 1. How can interval notation describe inequalities? Replace x 4, -2 < x < 8, x > 5, x -6 with (, 4], (-2, 8), (5, ), and [-6, ) On a graph, replace closed circles with square brackets, open circles with parentheses and arrows with and parentheses 2. What are the uses for piecewise functions? Use piecewise functions when a function has distinct sections that are best described with different rules, or with step functions, and absolute value functions 3. How can a line of best fit (regression line) be found on a graphing calculator? What is correlation? Enter data through STAT Edit, find the linear regression line through STAT Calc Lin Reg Correlation gives a numeric value to the relationship between the x and y values on a scatterplot 4. How can linear systems be solved? How many solutions can a linear system have? Solve linear systems by graphing, elimination, or substitution. There can be no solution (parallel lines), one solution (intersecting lines) or infinitely many solutions (the same line twice) 5. How do you use linear programming to find the optimum value? Find the objective function, then use the constraints to graph the feasible region. All possible optimum values occur at the vertices of the feasible region. Substitute coordinates of the vertices into the objective function to find the maximum or minimum value. 6. How do you plot and name points in three dimensions? How do you graph a linear equation in three variables in three dimensions? The (x, y, z) points are plotted in 3-d space using the parallelogram or vector method. A linear equation in three variables can be graphed by finding the x, y and z intercepts and connecting them to form a triangular region representing a plane. Targeted Skill(s): 1. Use interval notation to describe inequalities or graph inequalities from interval notation 2. Create step, absolute value and other piecewise functions 2. Find f(x) when f is a piecewise function 2. Write the rules for f(x) and each restricted domain given the graph of a piecewise function 3. Find the linear regression equation on a graphing calculator 2
3. Find r, correlation on a graphing calculator 4. Solve a system of 2 or 3 linear equations by graphing, elimination and/or substitution 4. Based on the algebraic solution, determine the number of points of intersection 5. Determine the objective function from a verbal description 5. Determine the constraints from a verbal description 5. Graph the constraints (inequalities) to determine a feasible region 5. Determine when a feasible region is bounded or unbounded 5. Use the vertices of the feasible region to determine the optimum value 6. Draw a 3-d display of intersecting planes 6. Plot points on a 3-d graph using the parallelogram method or vectors 6. Find the x, y, and z intercepts of a linear equation in 3 variables: (x, 0, 0) (0, y, 0) and (0, 0, z) 6. Use intercepts to graph a linear equation in 3 variables Writing: Assessment Practices: Assessment questions such as: Describe and advantages and disadvantages to the methods of solving systems of equations. or Compare the similarities and differences between graphing in two and three dimensions. Answering word problems with labeled numeric answers and/or sentences Quizzes and tests that include vocabulary, graphs, and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. 3
MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: 11 days Chapter 4 Matrices and Determinants 1. What are matrices and why are they used? A matrix is a rectangular array of numbers, names by the number of rows x the number of columns. It is used to organize a large amount of data. 2. How do you compute with matrices? For addition and subtraction the dimensions of the matrices must be equal. Then the corresponding entries are combined. Scalar multiplication multiplies each entry by a number outside the matrix. With multiplication, the number of columns of the first matrix must equal the number of rows of the second or multiplication is undefined. Matrix multiplication is not commutative. 3. What is the determinant of a matrix and what is it used for? The determinant of A, A, is a real number and can be hand calculated in a 2 x 2 or 3 x 3 matrix. Determinants can be used to find the area of a triangle from its vertices, and in Cramer s Rule to solve systems of equations. 4. What is the Identity matrix and how is it used? The Identity matrix has 1 s on the main diagonal and 0 s everywhere else. Where I = the identity matrix: I x A =A, and A x I = A 5. What is the inverse of a matrix and how is it used? The inverse of matrix A, in symbol is A -1. If A and B are inverses then AB = I and BA = I 6. How are matrices used in the real world? Matrices are used to organize large amounts of data, to code and decode messages and in solving systems of linear equations. Targeted Skill(s): 1. Determine the dimensions of a matrix 1. Name the entries in a matrix 1. Write a matrix with rows and columns labeled with words, that describes a situation presented verbally 2. Add, subtract, and multiply two matrices and multiply a matrix by a scalar. 3. Find the determinant of a 2 x 2 or a 3 x 3 matrix by hand or with a graphing calculator 3. Use determinants to find the area of a triangle from its vertices 3. Use Cramer s Rule to solve a system of linear equations with determinants 4. Write the Identity Matrix 4. Multiply with the Identity Matrix 5. Given A is a 2 x 2 matrix, find A -1 (if it exists) by hand 5. Find the inverse of a matrix (if it exists) with a graphing calculator 6. Write a matrix to organize data presented in a world problem 6. Use matrices to solve problems (especially coding and encoding, and verbal problems solved with linear systems 4
Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. Students write small messages, put them into code, exchange them and decode the messages. 5
MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: Targeted Skill(s): 15 days Chapter 5 Quadratic Functions 1. What are the important aspects of the graph of a quadratic function? The shape is a parabola with a vertex, axis of symmetry, y intercept and sometimes 1 or 2 x intercepts. The equation can be written in standard form, vertex form or intercept form. 2. How can you solve a quadratic equation? Solve quadratic equations by graphing, factoring, by finding square roots, the quadratic formula and completing the square The discriminant indicates the number and the nature of the solutions 3. What are complex numbers and how are they used? Numbers in the form a + bi include the imaginary unit i which is the square root of negative 1 Computation with imaginary numbers differs from real number computation only with powers of i and imaginary components in the denominator of fractions (division) Complex numbers are used in fractals and electric circuits 4. How do you solve quadratic inequalities? The graph of the quadratic equation can be modified to include a solid or dotted line and shading above or below the line 5. How can you create a quadratic model from actual data? Given the vertex substitute into the vertex form, given the x intercepts use the intercept form, and given 3 points use the standard form and a system of 3 equations 1. Graph a parabola including: the axis of symmetry, vertex, any x and y intercepts for a minimum of 5 points 1. Use standard form, vertex form or intercept form to graph a parabola 2. Solve quadratic equations by graphing factoring, taking the square root of both sides, completing the square and the quadratic formula 2. Use the discriminant to determine the number and nature of the solutions 3. Compute with complex numbers, simplify as necessary 3. Plot points on the complex plane 4. Graph quadratic inequalities < and > with a dotted line parabola and shading, and with a solid line parabola and shading 5. Write the equation of a parabola using substitution given special points (vertex or x intercepts) 5. Write the equation of a parabola in standard form using 3 points on the parabola and a system of 3 equations Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. 6
MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: 10 days Polynomials and Polynomial Functions Chapter 6 1. What are the Properties of Exponents, and how can they be used to simplify computation? The properties are: the Product of Powers, the Power of a Power, the Power of a Product, the Negative Exponent, the Zero Exponent, the Quotient of Powers, and the Power of a Quotient. Numerically, scientific notation is an example of using properties in computation. These are also used in algebraic expressions. 2. How do you name a polynomial by knowing its degree? The polynomials of degree 0-5 have special names: constant, linear, quadratic, cubic, quartic and quintic. 3. How do you evaluate a polynomial? You can evaluate a polynomial directly with substitution or indirectly with synthetic substitution. 4. What attributes of a polynomial should be included (with or without technology) to show a complete graph? A complete graph of a polynomial includes the end behavior, the relative maximum and minimum points, the y-intercept, and any x intercepts. 5. What methods can be used to add, subtract or multiply polynomials? There is a horizontal and a vertical method for addition, subtraction and multiplication. Additionally with multiplication, there is the area model and special product patterns (the sum and difference of two binomials, the square of a binomial and the cube of a binomial). 6. What methods are used to factor higher degree polynomials? Students should factor any greatest common factor first, and use then other factoring techniques including: the sum of two cubes, the difference of two cubes, factoring by grouping, or factoring a polynomial in quadratic form. 7. How do you solve a polynomial equation? If a polynomial equation can be factored, students can use the zero product property to find solutions. All potential solutions should be verified in the original equation. Application problems may only have real number solutions. 8. How can you divide polynomials? Methods for polynomial division include long division and synthetic division (if the divisor is in the form (x k)). The remainder theorem will give the quotient and remainder, and the factor theorem will indicate if the divisor is factor of the polynomial. Using the zero product property, the factor theorem can produce zeros of the polynomial function. You can find rational zeros using the rational zero theorem and synthetic division. 9. What does the Fundamental Theorem of Algebra tell us? An nth-degree polynomial equation has exactly n solutions, and an nth-degree polynomial function has exactly n zeros. Complex zeros occur in conjugate pairs, if a + bi is a zero, then a bi is a zero. 10. Why is knowing the zeros of a function important? Given the zeros of a function, the function s degree and the leading coefficient; the function itself can be determined. 11. What are the connections between the algebraic attributes of a function and its graph? The zero of a function appears as an x-intercept of the graph of the function, the maximum and minimum points of the graph are turning points of the function. 7
12. How are the finite differences of a polynomial function related to the function s degree? When the nth-order differences are non-zero and constant, the function is an nth-degree function. 13. How are finite differences used? Finite differences are used to generate polynomials from data tables. Targeted Skill(s): 1. Simplify algebraic and numeric expressions using the properties of positive and negative exponents. 1. Use the properties of exponents and scientific notation to solve application problems. 2. Identify polynomials by name according to degree. 3. Use substitution of x into P(x) and synthetic substitution of x into P(x) to get the same correct answer. 4. Graph a polynomial using the end behavior, the relative maximum and minimum points, the y-intercept, and any x intercepts. 5. Correctly add, subtract and multiply polynomials. 5. Memorize the pattern of the product of the sum and difference of two binomials, the square of a binomial and the cube of a binomial to correctly multiply. 6. Completely factor higher degree polynomials. 7. Solve polynomials equations for all appropriate solutions. 8. Correctly divide polynomials. 8. Use the remainder and factor theorems to solve polynomial equations. 8. Use synthetic division to find rational zeros. 9. Find all real zeros of a polynomial function. 10. Given zeros of a function, its leading coefficient and its minimum degree; find the function. 11. Find all the necessary attributes of a given polynomial function to graph it by hand. 11. Graph a polynomial function on a graphing calculator in an appropriate viewing window. 12. Find the degree of the function by finding the nth-order differences from a table of data. 13. Use finite differences and a table of values to determine a function. Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. Find a picture of a parabola and use a superimposed grid system to determine coordinates that can be used to determine the equation of the parabola. 8