Advanced Algebra Scope and Sequence First Semester. Second Semester

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1 Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs 030 SLOs Time (days) R., R.3, R.4 8-5, 8-6, , R.5, R.6, R.7 8-5, , 0-3, Unit 3a: Linear Equations. 8-, (of 30) (includes eam and review days) Total 78 Unit Name Unit 3b: Applications and Modeling with Linear Equations Unit 4: Quadratic Equations with Applications Unit 5: Other Equations and Inequalities with Applications Second Semester Sections in Book 0308 SLOs 030 SLOs Time (days). 8-, (of 30).3,.4, , ,.7,.8 8- Unit 6: Linear Functions., , 8-4 Unit 7: Polynomial and Rational Functions 0-3, 0-4, 0-5, , 0-6, Unit 8: Systems 5., , Unit 9: Eponential and Logarithmic Functions (optional) 4., 4., (includes eam and review days) Total 99

2 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 Unit : Review of Basic Concepts and Polynomials Tetbook Sections: R., R.3, R.4 Time Spent: 5 Days The student understands that mathematics is a structure with rules and definitions that can be applied to solve problems. The student understands the importance of the skills required to manipulate symbols in order to solve problems. The student understands that epressions can be mathematically manipulated to create equivalent forms of the epression. The student understands that polynomials can be written in a variety of ways and still be equivalent. Natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, simplify, evaluate, base, power, eponent, monomial, binomial, trinomial, polynomial, degree, term, coefficient, constant, leading coefficient, factor 8-5 Simplify epressions using definitions and laws of integer eponents. 8-6 Add, subtract, multiply, and divide polynomials. 8-7 Factor polynomials. 0- Define, represent, and perform operations on real and comple numbers. 0-3 Recognize and use algebraic (field) properties, concepts, procedures (including factoring), and algorithms to combine, transform, and evaluate absolute value, polynomial, radical, and rational epressions. The characteristics of the number lines and the coordinate plane for all real numbers. Distinguish between the number sets (Natural, Whole, Integers, Rational, Irrational, Real). Properties of real numbers can be used to simplify epressions. Identify the set to which a given number belongs (E: is an irrational number). Apply properties of real numbers and order of operations to evaluate and/or simplify epressions. 5 E: E: 5q p r E: y z when q = 3, r = 5 and p = E: 5y when = 4 and y = Use eponent rules to simplify monomial epressions. E: n n n 3 y E: 3 y Simplify epressions, including +,,,. 5 5 E: p 3 p p E: m5m 9 5 m Factor polynomials of various types. GCF (E: y 7 y 35 y )

3 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 Grouping (E: 0ab 6b 35a ) Trinomials (E: 6k 5kp 6p ) Perfect square trinomials (E: Difference of two squares (E: Sum/Difference of two cubes (E: 9m n mn 4 ) 4 36z 8y ) 3 3 8m 7n ) 4 Factoring with substitution: (E: 5 36 ) Unit : Rational and Radical Epressions Tetbook Sections: R.5, R.6, R.7 Time Spent: 33 Days The student understands that epressions can be mathematically manipulated to create equivalent forms of the epression. The student understands that concepts involving simple epressions (such as numerical fractions) can be applied to more comple epressions. The student understands that there are rules that define how an epression can be mathematically manipulated, including rules involving eponents, rational epressions, and radicals. rational epression, domain (of a rational epression), zero-factor Property, fundamental principle of fractions, least common denominator, comple fraction, negative eponent, rational eponent, product rule, quotient rule, power to a power, radical notation, radical sign, radicand, inde, n th root, principal n th root, like radicals, unlike radicals 8-5 Simplify epressions using definitions and laws of integer eponents. 8-0 Simplify, multiply and divide rational epressions. 0- Define, represent, and perform operations on real and comple numbers. 0-3 Recognize and use algebraic (field) properties, concepts, procedures (including factoring), and algorithms to combine, transform, and evaluate absolute value, polynomial, radical, and rational epressions. 0-6 Model, interpret and justify mathematical ideas and concepts using multiple representations. that there are some value(s) for which a rational epression may be undefined. the fundamental principle of fractions, ac a for b 0, c 0, and bc b how to apply it. the properties of eponents, particularly as they apply to negative and rational eponents. that the properties of determine the domain over which a rational epression is defined. 6 4 E: the domain of is, 4 4 apply the fundamental principle of fractions to multiplication, division, addition, subtraction, and simplification of rational epressions and comple fractions. p 7p 4 E: 5p 0p 3

4 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 eponents may be combined to simplify certain types of epressions. that comple fractions may be simplified, in some cases, from the standpoint of negative eponents. the product, quotient, and power rules for radicals, and how to use them, in combination if necessary, to simplify radical epressions. that radical notation and rational eponents of an epression are related. y y y E: y y y y y 3 3 E: y y 4 m 4 3m E: 3m 4 4 3m 3 E: 6 m E: m 4 (increased the level of difficulty) m y 3 4 y y E: (increased the level of difficulty) y y y apply eponent rules to simplify epressions and solve problems, including those with rational and negative eponents r z E: 9 0r z E: 3 E: E: E: z z z z E: y y y E: y convert between radical notation and the rational eponent form of an epression E: p q p q E: simplify radical epressions E: E: 6 E: 3 3 m m E: 7 3 4

5 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 E: y z E: y y E: E: E: rationalize the denominator of a rational epression with a radical in the denominator. E: E: 5 y (increased the level of difficulty) Unit 3: Linear Equations with Applications Tetbook Sections:.,. Time Spent: 0+0 Days (split) The student understands the basic terminology of linear equations and how to solve them. The student understands that certain principles and methods can be applied to finding solutions to real-world problems. The student understands that there are many practical real-world applications that can be modeled with linear equations. equation, solution, root, solution set, equivalent equation, variable, linear equation, identity, conditional equation, contradiction, empty set, literal equation, independent variable, dependent variable, linear modeling 8- Solve linear equations and inequalities in one variable and compound inequalities in one variable. 8- Use linear equations to solve applications. 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. the basic terminology of equations and the definition of a linear equation in one variable. that there is a variety of methods than can be used to solve linear equations. that the process of solving equations can be broken down into steps. that there are different types of linear equations. solve linear equations on one variable E: remove fractions from an equation by multiplying both sides of the equation by the common denominator. 0 E: E: 0.3( + ) 0.5( + ) = identify different types of linear equations (i.e. identities, conditional equations, contradictions). E: 0 = 0 (Identity) E: = 3 (Conditional) E: -3 = 7 (Contradiction) 5

6 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 the addition and multiplication principles of equality. that linear equations can be used to model real-world situations. solve a literal equation for a specified variable. E: Solve S lw wh hl for h solve certain types of geometry, simple interest, motion, and miture problems that can be modeled with linear equations. E: If the length of each side of a square is increased by 3 cm, the perimeter of the new square is 40 cm more than twice the length of each side of the original square. Find the dimensions of the original square. E: Latoya borrowed $540 for new furniture. She will pay it off in months at an annual simple interest rate of 4.5%. How much interest will she pay? E: Russ and Janet are running in the Strawberry Hill Fun Run. Russ runs at 7 mph and Janet runs at 5 mph. If they start at the same time, how long will it be before they are.5 miles apart? E: Marin needs 0% hydrochloric acid for a chemistry eperiment. How much 5% acid should she mi with 60 ml of 0% acid to get a 0% solution? E: Linda won $00,000 in a state lottery. She first paid income ta of 30% on her winnings. Of the rest, she invested some at.5% and some at 4% earning $4,350 in interest per year. How much money did she invest at each rate? Unit 4: Quadratic Equations with Applications Tetbook Sections:.3,.4,.5 Time Spent: 0 Days The student understands that the number system used to solve problems can be etended to include nonreal numbers (i.e. comple numbers). The student understands the basic terminology of quadratic equations and how to solve them. The student understands that certain principles and methods can be applied to finding solutions to real-world problems. The student understands that there are many practical real-world applications that can be modeled with quadratic equations. imaginary unit, comple number, real number, pure imaginary, comple conjugate, standard form (a+bi), quadratic equation, second-degree equation, standard form ( a b c 0), zero-factor property, double root, square root property, completing the square, quadratic formula, cubic equation, discriminant, Pythagorean theorem 8-8 Solve quadratic equations using the factoring method. 0- Define, represent, and perform operations on real and comple numbers. 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. the definition and characteristics of the imaginary unit and comple numbers. that a quadratic equation may rewrite an epression with a negative radicand using the imaginary unit E: 6 4i perform operations (+, -,, ) on comple numbers, including using a comple conjugate for division and/or 6

7 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 have up to solutions. the definition of a quadratic equation. the zero-factor property and how to apply it. the square root property and how to apply it. that there are various methods of solving a quadratic equation, including: factoring, taking the square root, completing the square, and the quadratic formula. the significance of the discriminant in a quadratic equation. that there are real-world applications problems that can be modeled with quadratic equations. simplification. E: E: i 8 3i E: E: 6 5i 6 5i E: 3 i 5 i simplify powers of i. E: i 40 solve quadratic equations using various methods (zerofactor property, square root property, completing the square, quadratic formula). E: E: E: E: etc i etc ( 6)( 3) etc. ( 6) solve cubic equations. 3 E: 8 0 solve literal quadratic equations for a specified variable. E: rt st k ( r 0) Solve for t. identify the discriminant of a quadratic equation and use it to identify the type of solutions for that equation. E: has a discriminant of 4(5)( 4) 84 so the equation has irrational solutions. solve application problems modeled by quadratic equations (area and volume, Pythagorean theorem, height of a projectile). E: The difference of the squares of two positive consecutive odd integers is 3. Find the integers. E: The length of each side of a square is 3 in. more than the length of each side of a smaller square. The sum of the areas of the squares is 49 in.. Find the lengths of the sides of the two squares. E: Tanner and Sheldon received walkie talkies for Christmas. If 7

8 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 they leave from the same point at the same time, Tanner walking north at.5 mph and Sheldon walking east at 3 mph, how long will they be able to talk to each other if the range of the walkie talkies is 4 mi.? Round your answer to the nearest minute. E: An astronaut on the moon throws a baseball upward. The astronaut is 6 ft, 6 in. tall, and the initial velocity of the ball is 30 ft per second. The height s of the ball in feet is given by the equation s.7t 30t 6.5 where t is the number of seconds after the ball was thrown. a) After how many seconds is the ball ft above the moon s surface? b) How many seconds will it take for the ball to return to the surface? Unit 5: Other Equations and Inequalities with Applications Tetbook Sections:.6,.7,.8 Time Spent: 0 Days The student understands show to use identity properties to solve equations and inequalities, including: square root, cube root, and rational equations and inequalities. The student understands that strategies used to solve basic linear equations can be applied to more comple equations. The student understands the basic terminology of radical equations and inequalities and how to solve them. The student understands the basic terminology of rational equations and inequalities and how to solve them. The student understands the basic terminology of inequalities and how to solve them. The student understands that certain principles and methods can be applied to finding solutions to real-world problems. The student understands that there are many practical real-world applications that can be modeled with inequalities. rational equation, linear inequality, interval notation, open interval, closed interval, quadratic inequality, rational inequality, absolute value equation, absolute value inequality, union, intersection 8- Solve linear equations and inequalities in one variable and compound inequalities in one variable. 0-3 Recognize and use algebraic (field) properties, concepts, procedures (including factoring), and algorithms to combine, transform, and evaluate absolute value, polynomial, radical, and rational epressions. 0-4 Identify and solve absolute value, polynomial, radical, and rational equations. 0-5 Identify and solve absolute value and linear inequalities. 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. that some solutions to rational and radical equations may not be valid. there are real-world situations solve rational equations and determine if the solutions are valid. 8

9 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 that can be modeled with rational or radical equations. the concepts and terminology of inequalities and their solution sets. there are various methods for solving linear, three-part, quadratic, absolute value, and rational inequalities. that the solution to an inequality can be epressed in various ways, including setbuilder notation and interval notation. that the solution set for an inequality will often contain an infinite number of solutions. the definition and properties of absolute value epressions. E: solve radical equations and determine if the solutions are valid. E: 4 0 E: 6 8 solve inequalities and state the solution in interval notation. E: E: E: E: 4 use absolute value to describe distances. E: k is less than 5 units from 8 becomes k 8 5 solve absolute value equations and inequalities and state the solution set in interval notation using union and intersection of sets as necessary. 3 E: 4 E: 7 E: 3 4 Unit 6: Linear Functions Tetbook Sections:., Time Spent: 0 Days The student understands the concept of slope as a rate of change and can apply it to purely mathematical and real-world situations. The student understands the definition of a function as a relation and can represent a function in various ways, including graphical, tabular, verbal, and symbolic. The student understands the attributes of functions, including domain, range, continuity, and increasing/decreasing and can make connections of these attributes between various representations. The student understands the relationship between the algebraic and geometric representations of a circle and parts of a circle. The student understands that the graphs of parent functions have characteristics such as continuity and symmetry which set them apart from other functions. The student understands that functions can be combined through performing operations (+, -,,) or by composition. ordered pair, rectangular coordinate system, distance formula, midpoint formula, - ais, y-ais, Cartesian coordinate system, coordinate plan, quadrants, coordinates, graph, -intercept, y-intercept, circle, center-radius form, general form, relation, function, domain, range, function notation, increasing function, decreasing function, 9

10 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 constant function, linear function, standard form, slope, point-slope form, parallel, perpendicular, horizontal, vertical, continuous function, identity, square, square root, cube, cube root, greatest integer function, epansion, compression, reflection, symmetry, even function, odd function, translations, composite function 8-3 Sketch graphs of linear relations and determine a linear equation in two variables given pertinent information. 8-4 Find the slope and - and y- intercepts of a linear relation. 0- Recognize, understand, and analyze features of a function. 0-6 Model, interpret and justify mathematical ideas and concepts using multiple representations. 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. the distance and midpoint formulas. that an ordered pair is a solution of an equation if its values make the equation true when substituted in for the variables. the definition of a relation. that a function is a particular kind of relation. how to interpret functional notation. the concept of slope in a linear function. a linear function can be written in different forms. that the domain of a function represents the independent () values and the range represents the dependent (y) values of the function. the unique equations for a vertical line (=a) and a horizontal line (y=b) through a point (a, b). that transforming the graph of a parent function will create infinitely many new functions. the definition and characteristics of even and odd functions. a linear function can be written in different forms. the concept of continuity. functions can be added, subtracted, multiplied, or divided to create a new calculate the distance and midpoint between two points. P 5, 7 ; Q 3,, Given PQ E: Given 7, 8 3 and 5 7, 3 P Q, find PQ. state the domain and range for a function. E: y 7 determine if three given points can be the vertices of a right triangle. E: Are 6, 4, 0, and 0, 8 vertices of a right triangle? find the midpoint of a segment defined by two points or an endpoint when given the midpoint and the other endpoint. E: Given midpoint 5, 8 M and endpoint find the coordinates of the other endpoint. determine if a given relation is a function. E: {(5, ), (3, ), (4, 9), (7, 8)} 3, 0, E: E: E: E: + y < 3 7 E: y 5 identify the domain and range and intervals of increasing and decreasing of a function. 5 E: y find the values of functions. E: Find E: Find f ( ) for f 9 f ( ) for f 4, 0,, 6, 0, 8 0

11 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 function. the concept of composition of functions. the graphs and characteristics of the following parent functions: identity (y=) square (y= ) square root ( y ) cube (y= 3 ) absolute value (y= ) inverse ( y ) inverse squared ( y ) greatest integer (y= ) E: Find f ( ) for write a linear function in various forms (i.e. slopeintercept, point-slope, standard) and convert between the forms. E: Write the function y 3 4 in slopeintercept form and in standard form. recognize graphs of basic functions and identify discontinuities, if they eist. represent intervals of increasing and decreasing in a function. E: Identify the intervals of increasing and decreasing for the function shown. graph functions by hand and on a graphing calculator. E: through (-, 3) with slope 3 E: through, 4, m 0 find the slope of a linear function. E: through (, -) and (-3, -3) E: vertical line through (4, -7) E: 4+3y= write the equation of a line in standard and slopeintercept form. E: through (-, -3) and slope is 3 4 E: through (-, 3) and (3, 4) E: slope is 5 and y-intercept is 5 E: vertical line through (-6, 4) E: horizontal line through (-7, 4) E: through (-, 4) and parallel to 3y 5 E: through (-5, 6) and perpendicular to E: find k so that the line through (4, -) and (k, ) is perpendicular to the line y 5 apply linear functions to real-life situations. E: The pressure p of water on a diver s body is a linear function of the diver s depth,. At the water s surface, the pressure is atmosphere. At a depth of 00 ft, the pressure is about 3.9 atmospheres. Find the linear function that relates p to and compute the pressure at a depth of 0 fathoms (60 ft). E: See p. 58, #53 in tetbook determine the slope and y-intercept of each line. E: +y=-4 graph piece-wise functions with domain restrictions.

12 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 (optional) if 5 E: f ( ) if 0 if 0 write a piece-wise function given a graph. (optional) E: transform the graphs of functions, including translations, reflections, and dilations and write the equations of transformed graphs. E: Graph and describe the transformations on the parent function for the following: f f 3 Compare the graph of y g to y g, and g. y g, determine symmetry of functions. E: Describe the symmetry of each of the following: y 4 y 3 y 6 y 5 determine algebraically whether a function is even, odd, or neither. 5 3 g 3 E: E: 3 E: 6 9 perform operations on functions and the composition of functions. E: For the functions f and g, find the following: f g and its domain g f and its domain Unit 7: Polynomial and Rational Functions Tetbook Sections: Time Spent: 5 Days The student understands the basic terminology and characteristics of polynomial functions and how to apply them. The student understands the basic terminology and characteristics of rational functions and how to apply them.

13 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 polynomial function, leading coefficient, zero (of a function), quadratic function, parabola, ais, verte, synthetic division, remainder theorem, factor theorem, rational zeros theorem, fundamental theorem of algebra, multiplicity of the zero, conjugate zeros theorem, turning point, end behavior, reciprocal function, asymptotes (vertical and horizontal), discontinuous, rational function, 8-0 Simplify, multiply and divide rational epressions. 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. the definition and characteristics of a polynomial function. that linear and quadratic functions are polynomial functions. the division algorithm. that synthetic division is a shortcut for long division of polynomials. the remainder theorem. the definition and characteristics of a rational function. some relationships between quantities can be epressed as a variation. graph and state the verte, ais, domain, and range of a quadratic function. E: f 3 write the equation of a quadratic function from its graph. E: transform the general form of a quadratic function to the verte (graphing) form using completing the square. E: f 4 5 ( ) 5 ( ) 5 3 analyze the quadratic models of real-world situations such as projectiles E: See p. 37, #6 in tetbook divide polynomials using long division or synthetic division. (no comple roots) 4 3 E: Given f 3 5 0, determine if 4 k 3 is a zero of the function. find the real zeros of polynomial function. (no comple) E: f use the factor theorem to factor polynomials. f 5 6 E: Factor 3 identify the characteristics of the graph of a polynomial function, including zeros, turning points, and end behavior. f E: 4 3 n graph and analyze the function f a where n {natural numbers}, and its transformations graph and analyze the reciprocal (inverse) functions 3

14 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 f and f E: g and their transformations. 4 5 identify the characteristics of the graph of a rational P function, f where P() and Q() are Q polynomial functions: including intercepts, asymptotes (vertical and horizontal), removable discontinuities, and end behavior. 3 E: f 0 identify a variation problem and set up an equation to represent the situation, including direct variation, inverse variation, joint variation, and combined variation. E: Simple interest varies jointly as principal and time. If $000 left in an account for years earned $0, find the amount of interest earned by $5000 for 5 years. Unit 8: Systems Tetbook Sections: 5. Time Spent: 7 Days The student understands that linear systems can be solved by different methods to achieve the same solution. The student understands that there are many practical real-world applications that can be modeled with systems of equations and inequalities. linear systems, independent, dependent, inconsistent 8-9 Solve systems of linear equations in two variables, including applications. 0-6 Model, interpret and justify mathematical ideas and concepts using multiple representations. 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. the definition and characteristics of systems of equations. that some real-life situations can be represented by a system of equations. A system of linear equations in two variables can have 0,, or infinitely many solutions. Different methods of solving systems can be most efficient depending on the format of solve systems of linear equations by a variety of methods: substitution method, elimination method, or matrices. E: Solve by substitution 3 y y 3 E: Solve by elimination 6 7y 0 E: Solve by elimination 7 6y 6 0 y y 3 3 4

15 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 the system given. E: A system of linear equations in two variables given in standard form may be easiest to solve by the elimination method. E: 8 3y 6z 4 9y 4z 8 3y 8z determine is a system is one of the following: inconsistent (with no solution) independent (with one solution) dependent (with infinitely many solutions) solve linear inequalities y > 4 5 y < - y > Unit 9: Eponential and Logarithmic Functions (optional) Tetbook Sections: 4., 4., 4.3 Time Spent: 7 Days The student understands the basic terminology and characteristics of eponential functions and how to apply them. The student understands the basic terminology and characteristics of logarithmic functions and how to apply them. The student understands the inverse relationship between eponential and logarithmic functions and the concept of one-to-one correspondence. The student understands the equivalence relationship between eponential and logarithmic epressions and can utilize this relationship to solve problems. inverse operation, one-to-one function, horizontal line test, inverse function, eponential function, compound interest, e, compounding continuously, logarithm, base, argument, logarithmic function, properties of logarithms 0-7 Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. the definition of the inverse of a function. that an eponent can have a real irrational value the definition and characteristics of an eponential function. the number e and its value in application problems. the definition and characteristics of a logarithmic function. the properties of logarithms determine if a function is one-to-one by using the horizontal line test and/or algebraically. determine if two given functions are inverses of each other. E: f 4 and g 4 E: f +3; D: 0, and g 3, D: 3, determine the inverse of a function and if the inverse is a function. E: Given f 6, for 6 E: y graph and analyze eponential functions f a a., 0 graph and analyze eponential function transformations. 5

16 Advanced Algebra Level Scope and Sequence 0-3 Last update: April 03 f 5 36 E: use the properties of eponents to solve eponential equations. 4 E: 4 3 E: r 4 use eponential functions to calculate values involving compound interest, including continuously compounded interest. (optional) r E: A P n nt rt E: A Pe identify the logarithmic function as the inverse of a given eponential function and vice versa. apply properties of logarithms evaluate and/or simplify epressions and solve equations. log y 5 log y 3 E: Simplify 7 E: Solve log4 b b 6

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