King William High School 2010-2011 Curriculum Pacing Guide Grade/Course ALGEBRA II Time Frame Unit/SOLs 7 days Recognize field properties AII. 1 (commutative, associative, identity, inverse, distributive) axioms of equality( reflexive, symmetric, transitive, substitution, addition, multiplication) Solve graph absolute value AII. 4 equations inequalities. SOL # Str Resources Assessments Additional Notes/ Adjustments Expressions Operations For all material: on each SOL Virginia SOL Teacher-made test Coach Book for Algebra II can serve as a resource for review questions. Also, ESS CF* (see at end of this guide) can be used as a resource. Chapter 1 (Sec. 2-6) For absolute value inequalities: Show students how to find the endpoints of the shaded region by replacing inequality sign with an = sign. Field properties Commutative Associative Identity Inverse Distributive Axioms of Equality Inequality Reflexive Symmetric Transitive Substitution Trichotomy Whole numbers Natural numbers Integers Rational numbers Irrational numbers Real numbers Absolute value equation inequalities
7 days Graph linear functions. Solve linear equations Graph step functions. Write line of best-fit (regression line) given set of points use to predict value for practical problems. Identify correct scatterplot for given situation involving two variables. 4 days Use systems of linear inequalities to solve linear programming problems. *This SOL will be moved to Precalculus in 2011-2012. AII. 8 AII. 19 AII. 13* Systems of Chapter 2 (Sec. 2 7) Chapter 3 (Sec. 3-4, then 3.2) on each SOL Teacher-made test. Collect data for regression line using toy cars. Record distance car travels down ramp vs. height of ramp. on each SOL Teacher-made test Hs-on activity of using dominoes wood blocks to simulate actual linearprogramming problem Review function definition vertical line test as go through sections. Stress slope-intercept form for graphing lines. Linear function Linear equation Identity function Step function Solution of an Equation Scatterplot Data Equation(s) Slope-intercept form Will apply properties from SOL 1 to matrices, stress how multiplication of matrices is not commutative. Sec. 3.2: Substitution linearcombination methods to solve systems can be reviewed after the matrix chapter, chapter 4. Point of intersection System of linear equations inequalities Linear programming model Constraints Corner points of a feasibility region Optimization process Maximum or minimum value
4 days Solve systems of two three equations using inverse matrix method. Use matrix multiplication to solve practical problems. *This SOL will be moved to Algebra I in 2011-2012. End of 1 st Nine Weeks AII. 12 AII. 11* Systems of Chapter 4 (Sec. 1, 3, 6, 7 (through example 2), 8 (Sec. 3.2) on each SOL Teacher-made test Project options: Visual aid for any SOL covered so far; data collection activity similar to car/ramp one in class; set up company problem to be solved using linear programming Benchmark Review 1 st Benchmark Test Teach Cramer s Rule for 2x2 systems only. Calculator matrix program inverse key will be used extensively. For matrix multiplication, stress how number of items across the top of first matrix must match the number of items going down the second matrix. Stress dimensions of matrices how inside numbers of the dimensions of two matrices must match in order for multiplication to be allowed. Matrix multiplication Matrix equation Inverse of a matrix System of linear equations
18 days Review properties of exponents (negative positive). Review adding like terms. Factor polynomials representing difference of squares, perfect square trinomials, sum difference of cubes, general trinomials. Add, subtract, multiply, divide simplify radical expressions. Convert from radical expressions to expressions with rational (fraction) exponents, viceversa. Solve radical equations equations with rational exponents. Simplify powers of i; perform operations with complex numbers AII. 5 AII. 3A AII.3B AII. 7 AII. 17 Expressions Operations Expressions Operations Chapter 5 (Sec. 1, 2, 3 (through Example 1),4, 5 (through Ex. 2), 6, 7, 8 (through Ex. 3), 9 zes teachermade tests Project options: Visual Aid for SOL covered since matrices; Make model representing x, y, z planes mark label three points (see page 136 of text); 300-word paper /or poster on famous mathematician or someone modern excelling in mathrelated field) Section 5.9 on complex numbers may be done after mid-term exam. Connect root number to roots underground to help students with rational exponents. Stress identifying as binomial or trinomial to aid in identifying type of factoring needed. Teacher-made hout (in color) of different types of factoring examples of each to be kept by student for entire year. Stress how inverse operation is needed to check answers for radical equations as can get extraneous solutions. Show how powers of i are cyclic why division of exponent by four is useful in simplifying powers of i. Complex numbers Pure imaginary numbers Monomials Binomials Trinomials Factored form Polynomial expression Radical expression Rational exponent Radical(s) Index number Radic Rationalizing the denominator Complete factorization Greatest monomial factor Difference of squares Sum difference of cubes Perfect square trinomial Conjugates i
2 days End of 2 nd Nine Weeks 3 days Operations on composition of functions (f(g(x)) Inverse AII. 9 Chapter 7, Sec. 7 8 Exam Review Midterm Exam (Includes Benchmark #2) Exam review h-out of review problems will be distributed to each student. Stress use of substitution principle order needed for f(g(x)). Explain symbols: f g (x) g f (x) Use charts to show how an equation its inverse undo each other; the original x values (input) become the y values (output) of the inverse equation, vice-versa. Inverse(s) Composition of functions
7 days Graph quadratic functions (parabolas); give domain range. Solve quadratic equations by graphing, factoring, using the quadratic formula. Write equations that translate parabolas; identify vertex. AII. 8 15 AII. 5, 6, 10 AII. 15 Chapter 6 (Sec. 1, 2, 3, Graphing Teacher-made test Calculator Use of graphing Investigation on calculator will be used page 320-321, 6.6 as part of assessment. (through Ex. 1), 7, 5 Project: Students memorize the quadratic formula say it verbally to 5 faculty members; they wear the quadratic formula for two days to each class to emphasize how important this formula is. Quadratic formula patches are provided by the teacher. When graphing parabolas, include several examples of parabolas on their side - simple ones with y 2 in the equation; also include several involving shading (y x 2, y x 2 ) but be brief. Show students how to do shading with graphing calculator. May want to have students graph square root function at this time find domain algebraically. Quadratic function Quadratic equation Quadratic [(h,k)] form Domain Range Vertex (maximum or minimum) Zeros Solutions X-intercepts Quadratic formula Discriminant Real solutions Imaginary solutions Roots (real imaginary) Algebraic solution Translations
4 days Graph polynomial functions recognize basic shape from degree. Factor solve polynomial equations by graphing. Investigate describe the relationship between solutions, zeros, roots, x-intercepts, factors. AII. 8, 10 AII. 8, 10 Analytical Geometry Chapter 7 (Sec. 1, 2, 5 (through Ex. 1), 9 (through Ex. 2) Teacher-made test Use of graphing calculator extensively Benchmark Test #3 Sec. 7.2 Use graphing calculators to graph find roots max/min; do not make charts by h. Use table function on graphing calculator. Stress difference between factors solutions/zeros/roots. Stress: 2 nd degree functions make a parabola. 3 rd degree make a hill valley. 4 th degree make a W shape. Fundamental Theorem of Algebra Turning point End behavior n th degree of polynomial functions n roots (solutions) Polynomial equation Zeros (roots) of a function
6 days Graph conic sections given equation written in h, k graph form. Identify type of conic section from an equation written in general form. Solve systems involving linearquadratic quadraticquadratic equations. *This SOL will be moved to Geometry in 2011-2012. End of 3 rd Nine Weeks AII. 18* AII. 18* AII. 14 Analytical Geometry Chapter 8 (Sec. 1, 3, 4, 5, 6, 7) Test Project Options: 1. Constuct ellipse model (see page 432) 2. Visual aid for an SOL taught during this grading period; 3. Poster depicting the quadratic formula in a creative way.. Distance formula in Sec. 1 should be treated as a review from Geometry Show students how to get circle ellipse equations in form that allows them to graph them on graphing calculator. Show how can solve systems algebraically May have to do Section 8.7 in 4 th nine-week period. Conic sections Ellipse Hyperbola Parabola Circle Asymptotes Vertex Center Transformations Non-linear systems of equations Coordinates of points of intersection Linear-quadratic systems Quadratic-quadratic systems
8 days Graph rational functions (using graphing calculator only) find domain from the equation (what values x cannot be). Identify, create solve practical AII. 20 problems involving direct inverse variation, joint variation, a combination of direct inverse variation. Add, subtract, multiply, divide, simplify rational expressions, including complex fractions. Solve rational equations. AII. 9 AII. 2 AII. 7 Expressions Operations Chapter 9 (Sec. 3, 4, 1, 2, 6) May want to use Sec. 5 as a review Test Stress that what makes a value undefined is division by zero. Review briefly adding subtracting fractions with just numbers before venturing into those with variables in the denominator. Stress checking solutions with rational equations as that will help students on the multiple-choice questions of this type on the SOL Test. Rational function Domain Complex fraction Common denominator Cross-product Means extremes Directly proportional Inversely proportional Varies jointly Constant of proportionality (k) Direct Variation Inverse Variation Asymptotes Undefined
1 day Identify basic shape of exponential logarithmic functions. Find curve of best fit for data modeling quadratic equation AII. 8, 15 AII. 19 Chapter 10 (Sec. 1,2) Graphing Calculator investigation (page 300 359) Project as an introduction to exponential growth: Students will make a calendar of their birth month with large squares for the dates. Beginning with.01 on the first day,.02 on the 2 nd day,.04 on the 3 rd day,.08 on the 4 th day, continuing this pattern, students will record on the calendar the amount of money received each day then the total amount received for all days of their birth month. Show students all curves of best fit that graphing calculator will do utilizing STAT button. If time, do at least one example of exponential growth. Come back to this chapter after SOL test (or do before if you have time) to do exponential growth decay word problems use of logarithms to sove exponential equations. May also want to review trigonometry covered in Geometry after SOL test. Exponential Logarithm Exponential growth Exponential decay Model Scatterplot
2 days Use arithmetic geometric sequences series appropriate notation; use summation notation. AII. 16 Chapter 11 (Sec. 1,2,3,4,5) Tell story of how Frederick Gauss added up the first 100 positive whole numbers by adding the first last term multiplying by 50 ( the number of terms divided by 2 as he had 50 pairs). This is how the formula for finding the sum of an arithmetic sequence developed. 4 days SOL Test Review ALL ALL Released tests from state department 4 days SOL Test Exam Review ALL 2 nd Sem SOLs ALL Teacher-made exam review Test after SOL test 10-problem quizzes covering SOLs (multiple-choice) SOL Test Final Exam Will need to hit highlights only as time will be running out. Stress use of a n notation. Arithmetic sequence series Geometric sequence series Element Summation symbol Σ n th term a n S n Infinite series Patterns ESS refers to the VDOE Enhanced Scope & Sequence guide, CF refers to the VDOE Curriculum Framework