Preliminary Topics Absolute Value 9N2 Compare, order and determine equivalent forms for rational and irrational numbers. Factoring Numbers 9N4 Demonstrate fluency in computations using real numbers. Fractions 9N4 Demonstrate fluency in computations using real numbers. Geometric Formulas 9M5 Solve problems involving unit conversion for situations involving distances, areas, volumes and rates within the same measurement system. Demonstrate fluency in computations using real numbers. LCM and GCF 9N4 Negative Numbers 9N4 Demonstrate fluency in computations using real numbers. Number Bases (binary, hexidecimal, etc.) Identify and justify whether properties (closure, identity, inverse, commutative and associative) hold for a given set and operations; e.g., even integers and multiplication. Number Properties (Distributive, etc.) 9N1 Number Types (natural, integer, real, etc.) 9N2 Compare, order and determine equivalent forms for rational and irrational numbers. Converting between Percents, Decimals, and Fractions 9N2 Compare, order and determine equivalent forms for rational and irrational numbers. 9N4 Demonstrate fluency in computations using real numbers. Rounding (and significant digits) 9N2 Compare, order and determine equivalent forms for rational and irrational numbers. Beginning Algebra Topics Canceling Units (Simplifying) 9N4 Demonstrate fluency in computations using real numbers. Distance Formula 9N4 Demonstrate fluency in computations using real numbers. 9G1 Define the basic trigonometric ratios in right triangles: sine, cosine and tangent. Evaluation Exponents: rules, negative, fractional, variable Factoring 9G2 11M4 9N4 9N4 9P7 9P12 9P11 Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle measures in similar figures. Calculate distances, areas, surface areas and volumes of composite three-dimensional objects to a specified number of significant digits. Demonstrate fluency in computations using real numbers. Demonstrate fluency in computations using real numbers. Use formulas to solve problems involving exponential growth and decay. Simplify rational expressions by eliminating common factors and applying properties of integer exponents. Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only). Adopted From Prentice Hall Courses Trumbull County Educational Service Center 1
Graphing: overview, absolute value, linear, radical Linear Inequalities: solving, graphing Intercepts 9P12 9G3 9P1 9P2 9P3 9P15 9P9 10P11 9P4 Simplify rational expressions by eliminating common factors and applying properties of integer exponents. Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. range element. freely translate among tabular, graphical and symbolic representations. Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations. Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs. Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology. Solve real-world problems that can be modeled, using systems of linear equations and inequalities. Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words. Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, rate of change maximum or minimum 9P5 Midpoint Formula 9N4 Demonstrate fluency in computations using real numbers. Order of Operations Polynomials Probability Explain the effects of operations such as multiplication or division, and of computing powers 9P11 Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only). 9D1 Classify data as univariate (single variable) or bivariate (two variables) and as quantitative (measurement) or qualitative (categorical) data. 9D2 Create a scatterplot for a set of bivariate data, sketch the line of best fit, and interpret the slope of the line of best fit. 9D3 Analyze and interpret frequency distributions based on spread, symmetry, skewness, clusters and outliers. 9D4 Describe and compare various types of studies (survey, observation, experiment), and identify possible misuses of statistical data. Describe characteristics and limitations of sampling methods, and analyze the effects of random versus biased sampling; e.g., determine and justify whether the sample is likely to 9D5 be representative of the population Make inferences about relationships in bivariant data, and recognize the difference 9D6 between evidence of relationship (correlation) and causation. Use counting techniques and the Fundamental Counting principle to determine the total 9D7 number of possible outcomes for mathematical situations. 9D8 Describe, create and analyze a sample space and use it to calculate probability. 9D9 Identify situations involving independent and dependent events, and explain differences between, and common misconceptions about, probabilities associated with those events. Adopted From Prentice Hall Courses Trumbull County Educational Service Center 2
9D10 10D2 10D4 10D5 10D6 Use theoretical and experimental probability, including simulations or random numbers, to estimate probabilities and to solve problems dealing with uncertainty; e.g., compound events independent events simple dependent events Represent and analyze bivariate data using appropriate graphical displays (scatterplots, parallel box-and-whisker plots, histograms with more than one set of data, tables, charts, spreadsheets) with and without technology Identify outliers on a data display; e.g., use interquartile range to identify outliers on a boxand-whisker plot. Provide examples and explain how a statistic may or may not be an attribute of the entire population; e.g., intentional or unintentional bias may be present. Interpret the relationship between two variables using multiple graphical displays and statistical measures; e.g., scatterplots, parallel box-and-whisker plots, and measures of center and spread Differentiate and explain the relationship between the probability of an event and the odds of an event, and compute one given the other. 10D8 Radicals (square roots, cube roots, etc.) 9N5 Estimate the solutions for problem situations involving square and cube roots. Ratio & Proportion 9M1 9M2 9M3 9M4 9M5 9P13 Convert rates within the same measurement system; e.g., miles per hour to feet per second; kilometers per hour to meters per second. Use unit analysis to check computations involving measurement. Use the ratio of lengths in similar two-dimensional figures or three-dimensional objects to calculate the ratio of their areas or volumes respectively. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. Solve problems involving unit conversion for situations involving distances, areas, volumes and rates within the same measurement system. Model and solve problems involving direct and inverse variation using proportional reasoning. Solve real-world problems involving area, surface area, volume and density to a specified degree of precision. 11M5 Scientific Notation 9N2 Compare, order and determine equivalent forms for rational and irrational numbers. 9N4 Demonstrate fluency in computations using real numbers. 11M1 Determine the number of significant digits in a measurement. Slope of a straight line Slope and Graphing Slope and y-intercept 9P8 9P14 9P8 9G3 9P4 Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point Describe the relationship between slope and the graph of a direct variation and inverse variation. Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words. Adopted From Prentice Hall Courses Trumbull County Educational Service Center 3
Solving: Absolute Value, Linear, Literal, Radical Equations 9N4 10P3 Explain the effects of operations such as multiplication or division, and of computing powers Demonstrate fluency in computations using real numbers. Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height. Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology. Solving Systems of Equations and Inequalities 9P9 Variables 9N4 Demonstrate fluency in computations using real numbers. x,y-plane (plotting points, etc.) Intermediate Algebra Topics Completing the Square: circles, ellipses, solving quadratics, and finding the vertex Descartes' Rule of Signs Domain and Range 9D6 9G3 9P1 10P4 10P8 11P10 10P2 9P1 10P2 Make inferences about relationships in bivariant data, and recognize the difference between evidence of relationship (correlation) and causation. Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. range element. Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. Use algebraic representations and functions to describe and generalize geometric properties and relationships. Graph the quadratic relationship that defines circles. Describe the characteristics of the graphs of conic sections. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; e.g., general shape, possible number of roots domain and range range element. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; e.g., general shape, possible number of roots domain and range Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e g Even and Odd Functions 10G6 Identify the reflection and rotation symmetries of two- and three-dimensional figures. Factor Theorem 11P5 Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e g Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis, x-axis or y = x. Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e g Adopted From Prentice Hall Courses Trumbull County Educational Service Center 4
Factoring Quadratics Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. Function Notation 10P1 Define function formally and with f(x) notation. Functions Function Transformations Graphing: Quadratic Equations, deriving the vertex formula Graphing Inequalities: Systems, Absolute Value, Quadratic Piecewise Functions Polynomials 9P1 range element. 9P2 freely translate among tabular, graphical and symbolic representations. Demonstrate the relationship among zeros of a function, roots of equations, and solutions 9P4 of equations graphically and in words. Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, rate of 9P5 change maximum or minimum 10P1 Define function formally and with f(x) notation. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; e.g., general shape, possible 10P2 number of roots domain and range Use algebraic representations and functions to describe and generalize geometric 10P4 properties and relationships. Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e g 11P6 Represent the inverse of a function symbolically and graphically as a reflection about y = Describe how a change in the value of a constant in a linear or quadratic equation affects 9P15 the related graphs. Identify families of functions with graphs that have rotation symmetry or reflection 11P5 symmetry about the y-axis, x-axis or y = x. Use algebraic representations and functions to describe and generalize geometric 10P4 properties and relationships. 10P7 Solve systems of linear inequalities. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; e.g., general shape, possible 10P2 number of roots domain and range Identify the maximum and minimum points of polynomial, rational and trigonometric 11P4 functions graphically and with technology. Adopted From Prentice Hall Courses Trumbull County Educational Service Center 5
Rational Roots Test Remainder Theorem Quadratic Formula Deriving the Quadratic Formula Solving: Quadratic Equations Special Factoring Formulas (difference of squares, cubes, etc.) Synthetic Division Systems of Linear Equations Variation (direct, inverse, & joint) Vertical Line Test Advanced Algebra Topics Asymptotes 9P5 9P11 9P9 10P11 11P9 9P13 9P14 9P1 9P2 10P1 11P5 Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, rate of change maximum or minimum Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only). Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology. Solve real-world problems that can be modeled, using systems of linear equations and inequalities. Solve 3 by 3 systems of linear equations by elimination and using technology, and interpret graphically what the solution means (a point, line, plane, or no solution). Model and solve problems involving direct and inverse variation using proportional reasoning. Describe the relationship between slope and the graph of a direct variation and inverse variation. range element. freely translate among tabular, graphical and symbolic representations. Define function formally and with f(x) notation. Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis, x-axis or y = x. Adopted From Prentice Hall Courses Trumbull County Educational Service Center 6
Binomial Theorem NEW 12P3 11P1 Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest. Apply combinations as a method to create coefficients for the Binomial Theorem, and make connections to everyday and workplace problem situations. 12N2 Complex Numbers 11N3 Represent complex numbers on the complex plane. 11N7 11P8 12N1 Composition of Functions 10P10 Compute sums, differences, products and quotients of complex numbers. Solve equations involving radical expressions and complex roots. Determine what properties (closure, identity, inverse, commutative and associative) hold for operations with complex numbers. Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions. 11P6 Represent the inverse of a function symbolically and graphically as a reflection about y = x. Conic Sections 11G5 Identify, sketch and classify the cross sections of three-dimensional objects. 11P10 Describe the characteristics of the graphs of conic sections. Data Analysis 11D1 11D2 11D3 11D4 11D5 11D6 11D7 11D8 11D9 11D10 Design a statistical experiment, survey or study for a problem; collect data for the problem; and interpret the data with appropriate graphical displays, descriptive statistics, concepts of variability causation correlation and standard deviation Describe the role of randomization in a well-designed study, especially as compared to a convenience sample, and the generalization of results from each. Describe how a linear transformation of univariate data affects range, mean, mode and median. Create a scatterplot of bivariate data, identify trends, and find a function to model the data. Use technology to find the Least Squares Regression Line, the regression coefficient, and the correlation coefficient for bivariate data with a linear trend, and interpret each of these statistics in the context of the problem situation Use technology to compute the standard deviation for a set of data, and interpret standard deviation in relation to the context or problem situation. Describe the standard normal curve and its general properties, and answer questions dealing with data assumed to be normal. Analyze and interpret univariate and bivariate data to identify patterns, note trends, draw conclusions, and make predictions. Evaluate validity of results of a study based on characteristics of the study design, including sampling method, summary statistics and data analysis techniques. Understand and use the concept of random variable, and compute and interpret the expected value for a random variable in simple cases. Examine statements and decisions involving risk; e.g., insurance rates and medical decisions. 11D11 Determinants and Cramer's Rule NEW 11N4 Use matrices to represent given information in a problem situation. Adopted From Prentice Hall Courses Trumbull County Educational Service Center 7
Exponentials and Logarithms Geometric Formulas 11P7 Model and solve problems with matrices and vectors. 11P9 Solve 3 by 3 systems of linear equations by elimination and using technology, and interpret graphically what the solution means (a point, line, plane, or no solution). 12P5 Set up and solve systems of equations using matrices and graphs, with and without technology. 9P2 freely translate among tabular, graphical and symbolic representations. 9P3 Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations. Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, 9P5 rate of change maximum or minimum 9P7 Use formulas to solve problems involving exponential growth and decay. 11P1 Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest. Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity 11P2 after 7 years Identify families of functions with graphs that have rotation symmetry or reflection 11P5 symmetry about the y-axis, x-axis or y = x. 11P6 Represent the inverse of a function symbolically and graphically as a reflection about y = x. 11P11 Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation. 12D2 Transform bivariate data so it can be modeled by a function; e.g., use logarithms to allow nonlinear relationship to be modeled by linear function. 11M3 Derive a formula for the surface area of a cone as a function of its slant height and the circumference of its base. 11M4 Calculate distances, areas, surface areas and volumes of composite three-dimensional objects to a specified number of significant digits. 11M5 Solve real-world problems involving area, surface area, volume and density to a specified degree of precision. Induction Proofs 12P6 Make arguments about mathematical properties using mathematical induction. Inverse Functions 11P6 Represent the inverse of a function symbolically and graphically as a reflection about y = x. 12P4 Represent the inverse of a transcendental function symbolically. Matrix Operations Determine what properties hold for matrix addition and matrix multiplication; e.g., use 11N1 examples to show addition is commutative and when multiplication is not commutative. Determine what properties hold for vector addition and multiplication, and for scalar 11N2 multiplication. 11N4 Use matrices to represent given information in a problem situation. 11N5 Model, using the coordinate plane, vector addition and scalar multiplication. Compute sums, differences and products of matrices using paper and pencil calculations 11N6 for simple cases, and technology for more complicated cases. Use vector addition and scalar multiplication to solve problems. 11N9 Adopted From Prentice Hall Courses Trumbull County Educational Service Center 8
Next Number in a Sequence Radical Expressions and Equations Rational Expressions and Equations Trigonometry 11G3 11P7 12G1 12P5 9P2 11P2 12P2 10N1 10P5 10P10 11N8 11P8 11P11 9P4 9P12 10P6 11M2 Describe multiplication of a vector and a scalar graphically and algebraically, and apply to problem situations. Model and solve problems with matrices and vectors. Use matrices to represent translations, reflections, rotations, dilations and their compositions. Set up and solve systems of equations using matrices and graphs, with and without technology. freely translate among tabular, graphical and symbolic representations. Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity after 7 years Translate between the numeric and symbolic form of a sequence or series. Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. Connect physical, verbal and symbolic representations of irrational numbers; e.g., construct square root of 2 as a hypotenuse or on a number line. Solve simple linear and nonlinear equations and inequalities having square roots as coefficients and solutions. Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions. Use fractional and negative exponents as optional ways of representing and finding solutions for problem situations; e.g., 272/3 = (27 /3)2 = Solve equations involving radical expressions and complex roots. Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation. Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words. Simplify rational expressions by eliminating common factors and applying properties of integer exponents. Solve equations and inequalities having rational expressions as coefficients and solutions. Use radian and degree angle measures to solve problems and perform conversions as needed. Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines. 11G4 Vectors 11G1 Use polar coordinates to specify locations on a plane. 11G2 Represent translations using vectors. Adopted From Prentice Hall Courses Trumbull County Educational Service Center 9