Integrated Algebra 2 Outline Opening: Summer Work Review P.0 Demonstrate mastery of algebra, geometry, trigonometric and statistics basic skills Skills and Concepts Assessed in Summer Work for Mastery: P.1 Evaluate and Simplify Algebraic Expressions with Real Numbers and Order of Operations with and without function notation Distributive Property, Laws of Exponents, FOILing, like and unlike terms fractions, decimals, integers, positive exponents square roots and cube roots absolute value scientific notation P.2 Solve Linear Equations and Inequalities with fractions and decimals with graphing calculator (might be missing in 2012-13 group) with units and in word problems simple literal equations, including working with Geometry formulas proportions P.3 Create Linear Equations from Conditions point- slope and y- intercept forms from graphs parallel/perpendicular relationships P.4 Solve Polynomials by Factoring factor and solve polynomials by setting the equation = 0 given a situation P.5 Work with Radicals estimate radicals combine and simplify like radicals P.6 Apply Geometry Concepts similar triangles Pythagorean Theorem word problems with area and volume unit analysis P. 7 Apply Simple Right Triangle Trigonometric Concepts (might be missing in 2012-13 group) P.8 Calculate Simple Probabilities (missing in 2012-13 group) UNIT 1: The Complex Number System 1. Simplifying and combining radicals 2. Converting terminating and non- terminating decimals to a/b to show rational relationship 3. Combining like terms Use properties of rational and irrational numbers 1.1 Subsets of a + bi and how to determine rational vs. irrational numbers (a/b conversions) (N.CN.1) 1.2 Simplification and evaluation of numerical expressions with real numbers (N.RN.3) 1.3 Radicals, including simplification, rationalization, square roots, cube roots Perform arithmetic operations with complex numbers 1.4 a. Definition of i (N.CN.1) b. Simplification and operations on complex numbers (N.CN.2) c. Conjugate of a complex number to rationalize (N.CN.3) Extend the properties of exponents to rational exponents 1.5 a. Definition of rational exponents and their relationship to radicals (N.RN.1) b. Use properties of exponents to rewrite expressions containing radicals (N.RN.2) c. Solve equations involving radicals and rational exponents (A.SSE.3) 1.6 Evaluate logs, rewrite logs in exponential form and apply log properties (A.SSE.3c) N.CN Number and Quantity- The Complex Number System
Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. 2. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. N. RN Number and Quantity- The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. A.SSE Algebra- Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. UNIT 2: Polynomial Relationships (A.REI.3) 1. Identify linear relationships (1 st degree equations) 2. Formulate point- slope linear equations and identify slope, points and intercepts 3. Graph linear equations in the form of two variables 4. Solve linear equations with real numbers 5. Use laws of exponents to simplify monomials 6. Factor perfect squares and trinomials with lead coefficients of 1 Perform arithmetic operations on polynomials. 2.1 Add and subtract polynomials (A.APR.1) 2.2 Multiply by distributive property (A.APR.1) 2.3 Dividing of polynomials: Long division and synthetic division (A.APR.MA1a) Understand the relationship between zeros and factors of polynomials 2.4 Know and apply the Remainder Theorem by choosing appropriate method(s) (A.APR.2, A.REI.4, N.CN.7) 2.4.a Factoring (2, 3, and 4 terms) 2.4.b Quadratic formula (including complex number solutions; using determinant) 2.4.c Solve by square rooting 2.4.d Completing the square 2.4.e Graphing calculator 2.4.f Synthetic division to find zeros A.APR Algebra- Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MA.1.a. Divide polynomials.
Understand the relationship between zeros and factors of polynomials 2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). A. REI Algebra- Reasoning with Equations and Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MA.3.a. Solve linear equations and inequalities in one variable involving absolute value. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) 2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. MA.4.c Demonstrate an understanding of the equivalence of factoring, completing the square, or using the quadratic formula to solve quadratic equations. N.CN Number and Quantity- Complex Numbers Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions. UNIT 3: Functions 1. Graph linear functions 2. Graph functions by using a T- table 3. Graph functions by using a graphing calculator (Y= and Graph commands) 4. Find x- and y- intercepts 5. Find slope between two points Understand the concept of a function and use function notation 3.1 Domain and ranges of functions (F.IF.1) 3.2 Determine a function, i.e. Vertical Line Test, and utilize function notation by evaluating functions of all types. (F.IF.2) 3.2.a Vertical Line Test 3.2.b Evaluate with function notation 3.2.c Evaluate composition of functions 3.3 Identify types of functions and their characteristics including linear, polynomial (with degree), trigonometric, exponential, logarithmic, piece- wise, absolute value, radical, and rational. (F.IF.MA.10) Analyze functions using different representations 3.4 Identify and graph functions using the key characteristics (F.IF.7 a b c) 3.4.a By Hand: Linear and quadratic: Intercepts(quadratics- simple factoring and quadratic formulas), max/mins with vertex 3.4.b With GC: Square root, cube root, absolute value functions, rational (with asymptotes: domain and factoring denominators), exponential and logs. 3.4.c By Hand: Piece- wise defined (Course 233: linear/quadratic, Course 220: Include absolute value) 3.5 Understand and find relations, domain and range of a function, both with and without the graphing calculator. 3.6 Find inverse functions (F.BF.4.a, F.BF.5) 3.6.a Show graphically how inverses are related 3.6.b Write expressions for inverses by interchanging x and y s 3.6.c Understand inverse relationship between exponential and log functions Interpret functions in applications 3.7 Interpret key features of graphs and tables of functions to answer applications. (F.IF 4, F.IF.5) 3.7.a Intercepts 3.7.b Increasing, decreasing intervals
3.7.c Positive or negative position (above or below independent axis, i.e. projectile motion) 3.7.d Relative max/mins 3.8 Apply average rate of change of a function by calculating slope between two points on the curve. (F.IF.6) 3.9 Solve simple systems of equations and inequalities with linear and quadratic functions (A.REI.7) 3.9.a. Algebraically by substitution 3.9.b Graphically by finding points of intersection F.IF Functions- Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person- hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise- defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. MA.10. Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric. F.BF Functions- Building Functions Build a function that models a relationship between two quantities 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x + 1)/(x 1) for x 1. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. A.REI Algebra- Reasoning with equations and inequalities Solve systems of equations 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2 + y2 = 3. UNIT 4: Create Equations and Build Functions 1. Simplify proportional relationships 2. Label linear, area and volume quantities with correct units 3. Use problem solving strategies to formulate equations 4. Translate literal expressions into algebraic expressions 5. Solve literal equations for a given variable indicates Modeling standard.
Reason quantitatively and use units to solve problems 4.1 Use units as a way to understand problems and to guide the solution of multi- step problems, and define appropriate quantities for descriptive modeling (N.Q.1, N.Q.2) 4.1.a. Use dimensional analysis to solve problems (G.MG.MA.4) 4.1.b. Apply concepts of density based on area and volume (G.MG.2) 4.2 Choose a level of accuracy when reporting quantities (N.Q.3) Interpret the structure of expressions 4.3 Interpret expressions that represent a quantity in terms of its context (A.SSE.1) 4.4 Use structure of an expression to identify ways to rewrite it (A.SSE.2) Create equations that describe numbers or relationships 4.5 Create equations and inequalities in one variable to solve problems (A.CED.1) 4.5.a From linear and quadratic models 4.5.b From simple rational models 4.5.c From simple exponential models 4.6 Create equations in two or more variables to represent relationships between quantities, and graph them on a coordinate plane with labels and scales. (A.CED.2) 4.7 Represent constraints by equations or inequalities, graph as a system and interpret viable and non- viable options, i.e. linear programming. (A.CED.3) Write expressions in equivalent forms to solve problems. 4.8 Create expressions/equations to reveal or explain properties of the quantity, and solve (ASSE.3) 4.8.a Rearrange formulas to solve for a given variable or a given situation (A.CED.4) 4.8.b Quadratic expressions equal to zero (ASSE.3.a) 4.8.c Complete square to find max/min value (ASSE.3.b) 4.8.d Use properties of exponents to transform and solve exponential expressions (ASSE.3.c) 4.8.e Mortgage payment problems (sum of a finite geometric series) (ASSE.MA.4) N.Q Number and Quantity- Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi- step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. MA.3.a. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure. A- SSE Algebra- Seeing Structures in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.151/12)12t 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. indicates Modeling standard. (+) indicates standard beyond College and Career Ready.
4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ACED Algebra- Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. G.MG Geometry- Modeling with Geometry Apply geometric concepts in modeling situations 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). MA.4. Use dimensional analysis for unit conversions to confirm that expressions and equations make sense. UNIT 5: Right Triangle Trigonometry Grade 9- Integrated Geometry Course: Unit 7- Right Triangles/Trigonometry Define trigonometric ratios and solve problems involving right triangles 5.1 Side ratios in right triangles leading to all six definitions of trionometric ratios for acute angles 5.2 Relationship between sine and cosine of complementary angles 5.3 Apply trigonometric ratios to right triangle problem solving situations Translate between the geometric description and the equation for a conic section 5.4 Derive the equation of a circle given a center and a radius by using the Pythagorean Theorem and completing the square method 5.5 Derive the equation of a parabola given a focus and directrix G.SRT. Similarity, Right Triangles and Trigonometry 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.GPE. Expressing Geometric Properties with Equations 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix UNIT 6: Statistics and Probability Grade 9- Integrated Geometry Course: Unit 10- Probability Utilize Combinatorials 6.1 Correctly denote, formulate and calculate permutations and combinations in an experimental setting Conditional Probability 6.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities. Use this to prove they are independent. 6.3 Understand conditional probability and interpret independence of A and B 6.4 Denote and calculate the probability of two dependent events. Model conditional probability with two stages by using a probability tree.
S- CP. Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data. 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 4. Construct and interpret two- way frequency tables of data when two categories are associated with each object being classified. Use the two- way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. indicates Modeling standard. (+) indicates standard beyond College and Career Ready.