Mathematics Scope and Sequence Louisburg USD #416 - Revisions Algebra 1 Geometry Algebra 2 Advanced Math/PreCalculus

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1 Standard 1: Number and Computation Benchmark 1: Number Sense Whole numbers, parts of a whole Whole numbers, parts of a whole Reasonableness of numbers ematics Scope and Sequence 1.1.K.1 Knows, explains, and uses equivalent representations for the same real number and/or algebraic expression including integers, decimals, fractions, percents, ratios, scientific notation, absolute value and numbers with integer exponents. 1.1.A.1 Uses appropriate representations of real numbers and algebraic expressions (including integers, decimals, fractions, percents, ratios, scientific notation, absolute value and numbers with integer exponents) to formulate and solve real-world problems in Algebra I. 1.1.A.2 Compares and orders real numbers and/or algebraic expressions. 1.1.A2 Determines reasonableness of solutions to Geometry problems involving real numbers and/or algebraic expressions. 1.1.K1 Knows, explains, and uses equivalent representations of numeric and algebraic expressions. 1.1.A1 Uses appropriate representations of real numbers and/or algebraic expression to formulate and solve problems. 1.1.A2 Determines reasonableness of solutions to real-world problems involving real numbers and/or algebraic expressions. Applies properties of absolute value to compare and order real numbers. Calculus Personal Consumer Computes the costs associated with: Owning a home (mortgage, insurance, maintenance) Health and life insurance premiums Renting car for short period of time Uses numerical and computational concepts and procedures in a variety of ways. Determines reasonableness of solutions to Consumer problems. 1

2 Equivalent representation Equivalent representation of algebraic expressions ematics Scope and Sequence 1.1.K.3 Knows and explains what happens to the product or quotient when a real number is multiplied or divided by various rational numbers. 1.1.K.2 Determines reasonableness of solutions to Algebra I problems involving real numbers and/or algebraic expressions. Benchmark 2: Number Systems Classifies numbers by type Properties 1.2..K.1-2 Identifies the relationship between the subsets of the real number system. 1.2.K3 Names, uses, and describes the following properties: a. commutative, associative, distributive, and substitution properties b. identity and inverse properties of addition and multiplication c. Transitive, reflexive, and symmetric properties & 1.2.K4 1.1.K1-3 Knows, explains, and uses equivalent representations for the same real number and/or algebraic expression including integers, decimals, fractions, percents, ratios, scientific notation, absolute value and numbers with exponents. 1.2.K1-2 Uses appropriate representations of real numbers and/or algebraic expressions to formulate and solve problems in Geometry. 1.2.K3 Names, uses, and describes the following properties: commutative, associative, distributive, substitution, identity and inverse properties of addition and multiplication, transitive, reflexive, symmetric, addition and multiplication properties of equalities and inequalities, and the zero product property. 1.1.K3 Knows and explains what happens to the product or quotient when a real number is multiplied or divided by various rational numbers. 1.2.K1;1.2.K2 Identifies the relationship between the subsets of the real number system. 1.2.K3a-e Names and describes the properties of algebra, and uses properties of the real number system to formulate and solve realworld problems, both linear and quadratic, in Algebra 2. Converts numbers written in standard form to scientific notation using integral exponents. Calculus Personal Consumer Uses equivalent representations for fractions, decimals, and percents. 2

3 ematics Scope and Sequence d. addition and multiplication properties of equalities and inequalities e. the zero product property Calculus Personal Consumer Problem solving and reasoning with numbers 1.2.A2 Analyzes and evaluates the correct form of a real number to use when solving a real world problem. Benchmark #3: Estimation Estimation techniques 1.3.K1-4; 1.3.A1 Uses and applies estimation to check reasonableness of results and make predictions in real-world situations involving real numbers and algebraic expressions. Benchmark #4: Computation Computational Methods Division of whole numbers 1.4.K1; 1.4.K2a Explains and performs computational procedures emphasizing the order of operations without using a calculator. 1.4.K2e Simplifies expressions containing integral exponents. 1.2.A2 Analyzes and evaluates the correct form of a real number to use when solving a real world problem. 1.3.K1-4; 1.3.A1 Uses and applies estimation to check reasonableness of results and make predictions in real-world situations involving real numbers and algebraic expressions. 1.4.K1; 1.4.K2 Explains and performs computational procedures emphasizing the order of operations without using a calculator. 1.3.A1; 1.3.K1-4 Uses estimation to check reasonableness of results and make predictions in situations involving real numbers and algebraic expressions. 1.4.K1; 1.4.K2a Explains and performs computational procedures emphasizing the order of operations without using a calculator. Compares/contrasts differences of purchasing a new car or a leasing a car. Estimates solutions to real-world problems using a variety of methods. Fractions Decimals Factors and multiples Applies patterns for differences of squares and perfect trinomial 1.4.K3 Factors sums and differences of two cubes 3

4 ematics Scope and Sequence squares to factor binomials and trinomials. 1.4.K2g,h Uses matrix addition and multiplication with spreadsheet applications. and trinomials with leading coefficients greater than one. Calculus Personal Consumer 1.4.K3 Finds prime factors, greatest common factor, multiples, and the least common multiple of algebraic expressions. Order of operation Exponents Roots and powers 1.4.K2d Simplifies square root radicals 1.4.K2e,f Simplifies expressions containing variable exponents. 1.4.K2d Simplifies radical expressions and converts quantities with rational exponents into equivalent quantities in radical form, and vise versa. Ratio and percent Simplifies polynomials, rational expressions, and radicals using arithmetic Percent applications Computation and problem solving 1.4.A1a,b,d Uses arithmetic operations and inverse relationships to formulate and solve real-world problems (with applications to business 1.4.A1d Solves multi-step problems using percents. 1.4.A1a Uses arithmetic operations and inverse relationships to formulate and solve real-world problems. 1.4.K2g,h Uses matrix addition and multiplication with spreadsheet applications. 1.4.A1a-e Uses arithmetic operations and inverse relationships to operations. Uses arithmetic operations and inverse relationships to formulate and solve real-world problems involving real numbers and algebraic expressions, including Computes federal, state, and city income taxes. Finds a percent of a given number. Uses knowledge of percents to solve a variety of problems, including interest rates and taxes. Uses arithmetic operations to solve problems related to personal finances, such as employment, taxes, car and housing costs, and budgeting. 4

5 Computation and problem solving Volume and Surface ematics Scope and Sequence and science) involving real numbers and algebraic expressions. Standard 2: Algebra Benchmark #1: Patterns Identifies patterns 2.1.K1-4 Recognizes and explains the general rule of a pattern in a variety of situations, including the n th term of an arithmetic or geometric sequence. 1.4.A1b Calculates volume and surface area of rectangular solids and cylinders. 2.1.K1-4 Recognizes and explains the general rule of a pattern in a variety of situations, including the n th term of an arithmetic or geometric sequence. formulate and solve realworld problems involving real numbers and algebraic expressions, including volume and surface area, exponential growth and decay, applications of percents, and mixture problems. 2.1.K1-4 Recognizes and defines the generalization of arithmetic, geometric, and special patterns, using symbolic notation to represent the nth term in explicit form. volume and surface area, exponential growth and decay, and mixture problems. Creates patterns Factoring patterns Factoring Factors trinomials with Generalization of patterns Benchmark #2: Variables, equations and inequalities Equations and inequalities 2.2.A2a; 2.2.K3c Uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of 2.2.K1-2 Formulates and solves equations and inequalities using symbols and algebraic expressions (including variables as non-integral solutions. Computes arithmetic and geometric means and progressions. Sets up and solves advanced trigonometric equations by a variety of methods, including factoring. Calculus Personal Consumer Analyzes the costs that accompany financial decisions (such as car purchase, credit cards, home interest costs). 5

6 Solves equations and inequalities Solves equations and inequalities Solves equations and inequalities Solves equations and inequalities ematics Scope and Sequence 2.2.K1-2 Formulates and solves equations and inequalities using symbols, real numbers, and algebraic expressions (including variables as parameters for certain types of equations). 2.2.K3; 2.2.A2 Solves linear equations and inequalities graphically and analytically, quadratic equations, systems of linear equations, radical equations, absolute value equations and inequalities, and exponential equations. situations, including systems of linear equations. parameters for certain types of equations) within the complex number system. 2.2.K3b Applies a variety of solution methods to quadratic equations, including factoring, completing the square and the quadratic formula. 2.2.K3; 2.2.A2 Solves linear equations and inequalities graphically and analytically, quadratic equations, systems of linear equations, radical equations, absolute value equations and inequalities, and exponential equations. Sets up, solves, and graphs linear and quadratic equations, inequalities, and radical equations. Solves systems of equations (linear and non-linear) in two and three variables. Solves questions using synthetic division, remainder theorem, DesCartes Rule of Signs, and rational root theorem. Solves systems of equations by using matrices and determinants. Calculus Personal Consumer Explains/computes differences between: unit price of an item sale price of discounted item amount of markdown use of coupons & rebates Uses several methods to compute the interest, interest rate and maturity value of different types of loans. Develops & uses models to replicate the use of checkbooks & savings accounts. Solves linear and absolute value Inequalities inequalities. Proportions Prepares monthly budget based on previous expenditures. 6

7 Evaluation/ substitution Variation Growth and decay Benchmark #3: Functions Number line, coordinate grid T-tables, multiple representations T-tables, multiple representations Ordered pairs ematics Scope and Sequence 2.3.K1 Translates between verbal, symbolic, numerical, and graphical representations of using a variety of methods. 2.3.K2 Matches equations and graphs of linear and quadratic. 2.3.K3 Determines whether a graph, a list of ordered pairs, a table of values, or rule represents a function (including constant ) 2.3.K4-5; 2.3.A2 Determines x- and y- intercepts, maximum and minimum values, and domain and range of linear and quadratic. 2.3.K6 2.3.A2 Recognizes changes in constants and/or slope in linear and interprets the change in the appearance of the graph. Finds inverse and direct variation. Solves exponential growth and decay problems. Calculus Personal Consumer Evaluates given formulas for specific values of variables in the formula (such as I = prt, mpg = distance/total gallons). Functional notation 2.3.K7 Explains and uses functional notation. 2.3.K1; 2.3.K7 Evaluates (including composite ) using various methods, including appropriate technology and explains and uses Uses and explains functional notation. 7

8 Dependent/indepe ndent, domain/range, ematics Scope and Sequence 2.3.K6 Recognizes how changes in constants and/or slope within a linear function changes the appearance of a graph. functional notation. 2.3.K8-9 Identifies independent and dependent variables, relates them to domain and range, and evaluates a function given a specific domain. Limits Explains and applies the concept of a limit. Limits Find the value of a function when the limit is infinity or limit is Continuity Identify Composition of Composition of 2.3.K2-3 Recognizes, identifies, or matches the graphs of the following : linear, quadratic, absolute value, and constant. 2.3.A3 Analyzes how changes in constants and/or coefficients with quadratic and absolute value affect the appearance of the graph. Graphs and solves linear and quadratic. Finds the inverse of a function. Calculus Personal Consumer undefined. Determine continuity of a function using left and right-hand limits. Parameter changes 2.3.K8-9 Identifies independent and dependent variables, relates them to domain and range, and evaluates a function given a specific domain. 2.3.K6 2.3.A2 Recognizes changes in constants and/or slope in linear and interprets the change in the appearance of the graph. 2.3.K6 Recognizes how changes in constants and/or slope within a linear function changes the appearance of a graph. 8

9 Quadratic Logarithms Logarithms and exponents Trigonometric Trigonometric Trigonometric Derivatives of Derivatives of Derivatives of ematics Scope and Sequence 2.3.K4-5; 2.3.A2 Determines x- and y- intercepts, maximum and minimum values, and domain and range of linear and quadratic. Applies knowledge of logarithms to solution of equations and inequalities. Solves and graphs logarithmic and exponential. Uses basic trigonometric relationships to develop trigonometric identities. Knows trigonometric identities and applies them to the solution of problems. Applies the Law of Sines and Law of Cosines to the solution of problems. Calculus Personal Consumer Finds the first and second derivatives of by a variety of methods (by the definition of derivatives; quotient, and product rules; and implicit differentiation. Finds derivatives using the chain rule and generalized power rule. Identifies that have no derivatives. Finds derivatives of logarithmic and 9

10 Derivatives of Derivatives of ematics Scope and Sequence Calculus Personal Consumer exponential. Finds higher order derivatives. Integration Finds anti-derivatives of a function. Integration Finds the integral of a function by using the anti-derivative. Integration Evaluates integrals using a variety of methods, including exponents, logarithms, and substitution. Integration Evaluates definite integrals and logarithms. Integration Finds average value and area between curves. Integration Evaluates of two variables. Integration Evaluates partial derivatives. Integration Evaluates multiple integrals. Integration Uses Lagrange multipliers. Benchmark #4: Models Model relationships and procedures 2.4.K1 Uses or creates mathematical models to represent and explain mathematical concepts, relationships, and procedures. 2.4.K1 Uses or creates mathematical models to represent and explain mathematical concepts, relationships, and procedures. 2.4.K1 Uses or creates mathematical models such as graphs on the coordinate plane & algebraic expressions, equations, & inequalities to represent and explain mathematical concepts, relationships, and procedures 10

11 ematics Scope and Sequence 2.4.A1h,i Recognizes that various mathematical models (geometric models, process models, and frequency tables, and graphs) can be used to represent the same problem situation. 2.4.A1h,i Recognizes that various mathematical models (geometric models, process models, and frequency tables, and graphs) can be used to represent the same problem situation. Standard 3: Geometry and Measurement Benchmark #1: Geometric Figures and Their Properties 2-D Shapes 3-D Shapes Symmetry Symmetry Similar and congruent 3.1.K Recognizes and compared properties of two-and threedimensional figures, including regular polygons. 3.1.K1 Recognizes and compares properties of two-and three-dimensional figures using models. 3.1.K2 Recognizes properties of regular polygons, including angle measures and diagonals. 3.1.K4 Recognizes that similar figures have congruent angles, and their corresponding sides are proportional. 3.1.K6a Recognizes and describes congruence of triangles using SSS, ASA, SAS, Calculus Personal Consumer encountered in Algebra K1-2 Describes properties and critical attributes of rectangles, triangles, trapezoids, and parallelograms. 3.1.K3 Identifies symmetries of two- and three-dimensional figures. Uses or creates mathematical models such as graphs on the coordinate plane and algebraic expressions, equations, and inequalities to represent and explain mathematical concepts, relationships, and procedures encountered in Precalculus/College Algebra. Identifies points and lines of symmetry in trigonometric. Uses or creates mathematical models such as graphs on the coordinate plane and algebraic expressions, equations, and inequalities to represent and explain mathematical concepts, relationships, and procedures encountered in Calculus Describes the 3-D shape that results when a 2-D shape is rotated about an axis. 11

12 Lines, rays, angles Angle classification Parallel and perpendicular lines Pythagorean Theorem ematics Scope and Sequence and AAS. 3.1.K6b Recognizes and describes the ratios of sides in special right triangles. 3.1.K8 Identifies parts of a circle, including arcs, chords, sectors, secant and tangent lines, and central and inscribed angles. 3.1.K7 Recognizes and compares the relationships of angles formed when parallel lines are cut by a transversal. 3.1.A1b; 3.1.A2; 3.1.K5 Applies the Pythagorean theorem to determine right triangle relationships. 3.1.K6,8 Uses appropriate symbols and correct definitions for geometry terms and concepts such as lines, rays, angles, circles, and parallel and perpendicular lines. 3.1.K4 Identifies that similar figures have congruent angles, and their corresponding sides are proportional. 3.1.K7 Recognizes, explains, and compares parallel line properties including angle relationships created by a transversal. 3.1.A1b; 3.1.K5 Applies the Pythagorean theorem to determine right triangle relationships. Computes triangle inequalities as related to the Pythagorean theorem. Calculus Personal Consumer Properties of circles, intersecting lines Proof Proof Problem solving 3.1.A2-3 Uses definitions, axioms, theorems and deductive reasoning to understand and develop proofs. Uses deductive reasoning, and contrapositive, converse and inverse statements to test the validity of arguments. 12

13 ematics Scope and Sequence Calculus Personal Consumer Investigation methods 3.1.A1a,b; 3.1.K5; 3.4.K5 Formulates and solves real-world problems by applying measurements and measurement formulas; area, perimeter, surface area, and volume; distance; Pythagorean theorem; conversions within systems. Benchmark #2: Measurement and Estimation Measurement to desired unit Measurement tools Perimeter and area Volume and surface 3.2.K7 Knows, explains, and uses ratios and proportions to describe rates of change. 3.2.K1-2 Uses various estimation techniques to determine real number approximations of length, volume, surface area, area, perimeter, weight, capacity, temperatures, distance, time, and angle measurements. 3.2.K1 Approximates measurements using both standard and nonstandard units. 3.2.K2 Uses measurement tools with a level of precision appropriate to the given situation. 3.2.K4 Recognizes and applies formulas for perimeter and area of polygons, circumference and area of circles, and volume of rectangular solids and cylinders 3.1.A1 Formulates and solves realworld problems by applying properties of geometric figures, including similarity and congruence. 3.2.K1-2 Uses various estimation techniques to determine real number approximations of length, volume, surface area, area, perimeter, weight, capacity, temperatures, distance, time, and angle measurements. Applies knowledge of area and perimeter to the solution of realworld problems, such as carpentry and landscaping. Applies knowledge of volume and surface area to real-world situations, such as 13

14 Problem solving with measurement Conversions between systems ematics Scope and Sequence 3.2.K6 Recognizes and applies properties of corresponding parts of similar and congruent figures to find measurements of missing sides. 3.2.K3 Converts between the customary and metric measurement systems given the conversion units or formulas. 3.2.K6 Applies properties of corresponding parts of similar and congruent figures to find measurements of missing sides and angles. 3.2.K7 Uses ratios and proportions to describe rates of change. 3.2.K7 Knows, explains, and uses ratios and proportions to describe rates of change. 3.2.K3 Converts between the customary and metric measurement systems given the conversion units or formulas. Calculus Personal Consumer construction and home decoration. Benchmark #3: Transformational Geometry Spatial relations /transformations Impact of Transformations 3.3.K1 Recognizes, explains and performs up to two transformations (reflection, rotation, translation) on a twodimensional figures. 3.3.K1; 3.3.A1 Performs single and multiple transformations, including reflection, rotation, translation, reduction/enlargement on two- and threedimensional figures, and analyzes the impact on perimeter, area, and volume. 3.3.K3; 3.3.A3 Generates and analyzes a two-dimensional representation of a threedimensional figure. Benchmark #4: Geometry from an Algebraic Perspective Slope 3.4.K3 Calculates the slope of a line from a list of ordered pairs on the line and explains how the graph of the line is related to its scope. 3.4.K6 Recognizes an equation of a line in any form and transforms the equation into slope-intercept form to graph the line. 3.3K1-4 Recognizes and describes the impact of multiple transformations (rotations in 90º intervals, reflections, dilations) on two- and three-dimensional figures. 3.3.A1 Analyzes the impact of transformations on areas and perimeter of triangles and on volume of cylinders and rectangular prisms. 3.4.K6 Recognizes an equation of a line in any form and transforms the equation into slope-intercept form to graph the line. Computes slope and rate of change using derivatives. 14

15 ematics Scope and Sequence Calculus Personal Consumer Slope 3.4.K4 Finds or explains the relationship between the slopes of perpendicular and parallel lines. 3.4.K4 Finds or explains the relationship between the slopes of perpendicular and parallel lines. Coordinate Plane 3.4.K1 Recognizes and examines two- and three-dimensional figures and their attributes, including the graphs of on a coordinate Distance Linear equations/inequali ties Parabolas Conic sections Graphs of trigonometric 3.4.K6 Recognizes an equation of a line in any form and transforms the equation into slope-intercept form to graph the line. 3.4.K7 Recognizes the equation of a parabola and its characteristics (vertex, minimum, maximum, line of symmetry) and sketch the graph of a parabola. 3.4.K5 Calculates distance between two points using the Pythagorean Theorem or the distance formula. 3.4.K4 Recognizes and explains the relationship between slopes of parallel and perpendicular lines. 3.4.K8 Explains and examines the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their plane. 3.4.K5 Calculates distance between two points using the distance formula. 3.4.K7 Recognizes the equation of a parabola and its characteristics (vertex, minimum, maximum, line of symmetry) and sketch the graph of a parabola. 3.4.K8 Explains the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their corresponding graphs. Computes distance using the distance formula. Solves and graphs parabolas and circles Graphs conic sections with and without transformations. Graphs all six trigonometric, with and without transformations. Solves linear inequalities using interval notation. Uses the locus definition of conic sections (parabola, ellipse, hyperbola) to represent them both symbolically and graphically. 15

16 Graphs of trigonometric Paths on coordinate grid ematics Scope and Sequence 3.4.K1 Recognizes and examines two-and threedimensional figures and their attributes, including the graphs of on a coordinate plane. 3.4.K8 Explains the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their corresponding graphs. Data Analysis and Probability Benchmark #1: Probability Lists outcomes 4.1.K1 Finds the probability of two independent events in an experiment, simulation, or situation corresponding graphs. 3.4.K1 Examines two- and threedimensional figures and their attributes including the graphs of on a coordinate plane. Graphs sinusoids. Applies knowledge of computations and permutations to solution of problems. Calculus Personal Consumer Likelihood of results (probability) Probability vs. odds Conducts experiments, records results 4.1.K.2 Calculates conditional probability involving two dependent events. 4.1.K.3 Explains the relationship between probability and odds, and computes one given the other. 4.1.A2 Uses theoretical or empirical probability (simple events; compound events composed of two or more simple independent events) to make predictions about realworld situations. 4.1.K3 Explains the relationship between probability and odds, and computes one given the other. 4.1.K1-3 Uses theoretical or experimental probability or odds, including geometric probability, to make predictions about realworld situations. Calculates probability of independent events with and without replacement. 16

17 ematics Scope and Sequence Calculus Personal Consumer Benchmark #2: Statistics Displays data in tables or graphs 4.2.A1 Uses data analysis to make reasonable inferences, decisions, and predictions; and to develop convincing arguments from data displayed in a variety of formats 4.2.A1 Uses data analysis to make reasonable inferences, decisions, and predictions; and to develop convincing arguments from data displayed in a variety of formats. 4.2.A1 Uses data analysis to make reasonable inferences, decisions, and predictions; and to develop convincing arguments from data displayed in a variety of formats. Reads and interprets charts, tables, and graphs in a variety of formats. Sampling techniques. Statistical measures Predictions from data 4.2.K2 Explains how the reader s bias, measurement errors, and display distortion can affect the interpretation of data. 4.2.K3 Determines statistical measures including measures of central tendency, range, quartiles, and interquartile range for a real number data set. 4.2.K4 Explains the effects of outliers on the measures of central tendency and range and interquartile range of a real number data set. 4.2.K6 Compares and contrasts two given sets of data. 4.2.K4 Explains the effects of outliers on the measures of central tendency and range and interquartile range of a real number data set. 4.2.K5 Approximates a line of best fit given a scatter plot, and makes predictions using the equation of that line. 17

18 Scale or interval changes Data distortion ematics Scope and Sequence 4.2.A3 Recognizes or explains the effects of scale and/or interval changes on graphs of data sets. 4.2.K5 Approximates a line of best fit given a scatter plot, and makes predictions using the equation of that line. 4.2.K4 Explains the effects of outliers on the measures of central tendency and range and interquartile range of a real number data set. 4.2.K2 Explains how the reader s bias, measurement errors, and display distortion can affect the interpretation of data. Calculus Personal Consumer 52 objectives 39 objectives 47 objectives 32 objectives FSCC=Ft. Scott Community College 24 objectives FSCC=Ft. Scott Community College 10 objectives 11 objectives 18