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Culpeper County Public Schools Curriculum Pacing Guide High School Semester Algebra I Revised June 2011 A YEAR AT A GLANCE ALGEBRA I High School Semester Algebra I (June 2011) Benchmark 1 Benchmark 2 Benchmark 3 Benchmark 4 UNIT 1 (1.5 weeks) UNIT 3 (1 week) UNIT 5 (2.5 weeks) UNIT 7 (3 days) Problem Solving - Strategies Functions – Exponent (A.2 a) Best Fit Curves (A.11) (A.4 f) *Analyze linear and quadratic function families Polynomial Operations SOL PREDICTOR TEST Expressions (A.1) (A.7 a-f) (A.2 b) (13 weeks) Multi-Step Equations (A.4 Direct and Inverse Variation Factoring UNIT 8 (1.5 weeks) d) (A.8) (A.2 c) Statistics – Analyzing Data Sets Properties (A.4 b) UNIT 4 (2 weeks) UNIT 6 (1.5 weeks)) (Box and Whisker Plots) (A.10) Multi-Step Inequalities with Systems of Linear Equations Radicals (A.3) one variable (A.5 a-c) (A.4 e-f) – (Include Real World Deviations and Z -Scores (A.9) Problems) Quadratics (A.4 c) Literal Equations (A.4 a) (Review A.7 with Quadratics) Systems of Linear Inequalities (A.5 d) SOL REVIEW UNIT 2 (2.5 weeks) COUNTY BENCHMARK II Linear Equations in Two ( 8 weeks) Variables (A.4 d) Slopes and Lines (A.6 a) Equations of Lines (A.6 b) Problem Solving with Linear Equations/Inequalities (A.4 f) COUNTY BENCHMARK I (4.5 weeks) CCPS Curriculum Guide Course Map June 2011 THIS COURSE: Algebra I is about the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures 1. How are algebraic expressions and equations used to solve real world problems? 2. What is the relationship between a line and its graph, slope, intercepts, and table of values? 3. How is a function, in all its forms, used to investigate relationships between quantitative data? 4. How is a linear system used to solve real world problems? 5. What are the different ways to simplify a polynomial expression? 6. What is the most effective way to solve a given quadratic equation? 7. How is a line or curve of best fit used to make predictions about a set of data? 8. How are measures of deviation and z-scores used to describe and analyze a set a data? CCPS Curriculum Guide Course Map June 2011 Critical Vocabulary and/or Concepts Solve Linear Equations Box and Whisker Plots Properties Quadratic Equations Systems Polynomials Rate of Change Absolute Value Inequalities Zeros Factoring Simplify Exponents Domain and Range Absolute Deviation Radicals Graphing Intercepts Standard Deviation Linear Inequalities Problem Solving Variations Z-Scores Slope Reciprocal Functions Learned in Basic Algebra these Concepts UNITS Statistics A.4 b,d,f A.9 and A.10 A.1 A.5 a-c Slopes and Best Fit Lines Curves A.4 d,f A.11 A.6 Functions Quadratics A.7 and A.8 A.3, A.4 c Systems Polynomials A.4 e,f A.2 A.5 d Benchmark I Unit 1 Standard A.4 a, b, d, f Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 1 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A solution to an equation is the mathematical communication, Unit 1 Organizer Interactive Achievement value or set of values that can be mathematical reasoning, connections, Unit Tests, Benchmark Tests, substituted to make the equation Introductory and representations to and Projects true. Function Problems Solve a literal equation (formula) (see Instr. Specialist The solution of an equation in one for a specified variable. Sue Jenkins) variable can be found by graphing the expression on each side of the Simplify expressions and solve Cube Train (see Toni equation separately and finding the equations, using the field Miller) x-coordinate of the point of properties of the real numbers and intersection. properties of equality to justify Algeblocks or simplification and solution. Algebra Tiles Real-world problems can be interpreted, represented, and solved Solve quadratic equations. Hands-On Equations using linear and quadratic equations. Identify the roots or zeros of a Property Match quadratic function over the real Game (see Toni The process of solving linear and number system as the solution(s) Miller) quadratic equations can be modeled to the quadratic equation that is in a variety of ways, using concrete, formed by setting the given Cover-Up Problem pictorial, and symbolic quadratic expression equal to zero. (VDOE ESS (2004) representations. Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 1 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) Properties of real numbers and Solve multistep linear equations in properties of equality can be used to one variable. Algeblocks and justify equation solutions and Equation Solving expression simplification. Confirm algebraic solutions to VDOE ESS (2004) linear and quadratic equations, using a graphing calculator. The zeros or the x-intercepts of the A Mystery to Solve quadratic function are the real Given a system of two linear VDOE ESS (2004) root(s) or solution(s) of the equations in two variables that has quadratic equation that is formed by a unique solution, solve the system Solving Linear setting the given quadratic by substitution or elimination to Equations VDOE expression equal to zero. find the ordered pair which ESS (2004) satisfies both equations. A system of linear equations with Video Power of exactly one solution is characterized Given a system of two linear Algebra by the graphs of two lines whose equations in two variables that has intersection is a single point, and a unique solution, solve the system Algebra/ Solving the coordinates of this point satisfy graphically by identifying the Equations Jeopardy both equations. point of intersection. Style A system of two linear equations (http://quia.com) Determine whether a system of with no solution is characterized by two linear equations has one the graphs of two lines that are Rags to Riches solution, no solution, or infinite (http://quia.com ) parallel. Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 1 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) A system of two linear equations solutions. having infinite solutions is characterized by two graphs that Write a system of two linear coincide (the graphs will appear to equations that models a real-world Algebra Tiles be the graph line), and the situation. (NCTM website) coordinates of all points on the line http://illuminations.n satisfy both equations. Interpret and determine the ctm.org/activitydetail. reasonableness of the algebraic or aspx?id=216 ) Systems of two linear equations can graphical solution of a system of be used to model two real-world two linear equations that models a conditions that must be satisfied real-world situation. Promethean Board Site simultaneously. Determine if a linear equation in Equations and systems of equations one variable has one, an infinite can be used as mathematical models number, or no solutions.† for real-world situations. Set builder notation may be used to represent solution sets of equations. † Revised March 2011 Benchmark I Unit 1 Standard A.1 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 1 Standard: A.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, Algebra is a tool for reasoning mathematical communication, Unit 1 Organizer Interactive Achievement about quantitative situations so that mathematical reasoning, connections, Algeblocks or Algebra relationships become apparent. and representations to Tiles Unit Tests, Benchmark Tests, and Projects Algebra is a tool for describing and Algebra Magic Translate verbal quantitative representing patterns and Packet (see Inst. situations into algebraic relationships. Specialist Sue expressions and vice versa. Jenkins) Mathematical modeling involves Model real-world situations with creating algebraic representations of Traffic Jam VDOE algebraic expressions in a variety quantitative real-world situations. ESS (2004) of representations (concrete, pictorial, symbolic, verbal). The numerical value of an Expression Bingo expression is dependent upon the Evaluate algebraic expressions for (see Toni Miller) values of the replacement set for the a given replacement set to include variables. Evaluating and rational numbers. Simplifying There are a variety of ways to Evaluate expressions that contain Expressions VDOE compute the value of a numerical ESS (2004) absolute value, square roots, and expression and evaluate an cube roots. algebraic expression. Promethean Board Site The operations and the magnitude of the numbers in an expression impact the choice of an appropriate computational technique. An appropriate computational technique could be mental mathematics, calculator, or paper Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 1 Standard: A.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) and pencil. Benchmark I Unit 1 Standard A.5 a-c Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 1 Standard: A.5 The student will solve multistep linear inequalities in two variables, including a) solving multistep linear inequalities algebraically and graphically; b) justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set of real numbers and its subsets; c) solving real-world problems involving inequalities; and d) solving systems of inequalities. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A solution to an inequality is the mathematical communication, Unit 1 Organizer Interactive Achievement value or set of values that can be mathematical reasoning, connections, substituted to make the inequality Inequalities VDOE and representations to ESS (2004) true. Solve multistep linear inequalities Real-world problems can be in one variable. Promethean Board Site modeled and solved using linear inequalities. Justify steps used in solving inequalities, using axioms of Properties of inequality and order inequality and properties of order can be used to solve inequalities. that are valid for the set of real numbers. Set builder notation may be used to represent solution sets of Solve real-world problems inequalities. involving inequalities. Solve systems of linear inequalities algebraically and graphically. Benchmark I Unit 2 Standard A.4 d, f Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 2 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) Unit 2 Organizer The student will use problem solving, A solution to an equation is the mathematical communication, Interactive Achievement value or set of values that can be Greetings VDOE mathematical reasoning, connections, ESS (2004) substituted to make the equation and representations to true. Solve a literal equation (formula) for Algeblocks: Solving The solution of an equation in one a specified variable. for Y VDOE ESS variable can be found by graphing (2004) the expression on each side of the Simplify expressions and solve equation separately and finding the equations, using the field properties Estimating Ages of x-coordinate of the point of of the real numbers and properties of Famous People intersection. equality to justify simplification and (mathbits.com) solution. Real-world problems can be interpreted, represented, and solved Solve quadratic equations. Promethean Board Site using linear and quadratic equations. Identify the roots or zeros of a quadratic function over the real The process of solving linear and number system as the solution(s) to quadratic equations can be modeled the quadratic equation that is formed in a variety of ways, using concrete, by setting the given quadratic pictorial, and symbolic expression equal to zero. representations. Solve multistep linear equations in Properties of real numbers and Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 2 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) properties of equality can be used to one variable. justify equation solutions and expression simplification. Confirm algebraic solutions to linear and quadratic equations, using a graphing calculator. The zeros or the x-intercepts of the quadratic function are the real Given a system of two linear root(s) or solution(s) of the equations in two variables that has quadratic equation that is formed by a unique solution, solve the system setting the given quadratic by substitution or elimination to expression equal to zero. find the ordered pair which satisfies both equations. A system of linear equations with exactly one solution is characterized Given a system of two linear by the graphs of two lines whose equations in two variables that has intersection is a single point, and a unique solution, solve the system the coordinates of this point satisfy graphically by identifying the both equations. point of intersection. A system of two linear equations Determine whether a system of with no solution is characterized by two linear equations has one the graphs of two lines that are solution, no solution, or infinite parallel. solutions. A system of two linear equations Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 2 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) having infinite solutions is Write a system of two linear characterized by two graphs that equations that models a real-world coincide (the graphs will appear to situation. be the graph of one line), and the coordinates of all points on the line Interpret and determine the satisfy both equations. reasonableness of the algebraic or graphical solution of a system of Systems of two linear equations can two linear equations that models a be used to model two real-world real-world situation. conditions that must be satisfied simultaneously. Determine if a linear equation in one variable has one, an infinite number, or no solutions.† Equations and systems of equations can be used as mathematical models for real-world situations. Set builder notation may be used to represent solution sets of equations. † Revised March 2011 Benchmark I Unit 2 Standard A.6 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 2 Standard: A.6 The student will graph linear equations and linear inequalities in two variables, including a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined; and b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, Changes in slope may be described mathematical communication, Unit 2 Organizer Interactive Achievement by dilations or reflections or both. mathematical reasoning, connections, Slippery Slope and representations to VDOE ESS (2004) Changes in the y-intercept may be described by translations. Graph linear equations and Sally Snail VDOE inequalities in two variables, Linear equations can be graphed ESS (2004) including those that arise from a using slope, x- and y-intercepts, variety of real-world situations. The Submarine and/or transformations of the parent function. VDOE ESS (2004) Use the parent function y = x and describe transformations defined The slope of a line represents a Transformationally constant rate of change in the by changes in the slope or y- Speaking VDOE dependent variable when the intercept. ESS (2004) independent variable changes by a Find the slope of the line, given Transformation constant amount. the equation of a linear function. Investigation) VDOE The equation of a line defines the ESS (2004) Find the slope of a line, given the relationship between two variables. coordinates of two points on the Slope 2 Slope VDOE line. The graph of a line represents the ESS (2004) set of points that satisfies the Find the slope of a line, given the equation of a line. Equations of Lines graph of a line. VDOE ESS (2004) A line can be represented by its Recognize and describe a line with graph or by an equation. a slope that is positive, negative, The graph of the solutions of a Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 2 Standard: A.6 The student will graph linear equations and linear inequalities in two variables, including a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined; and b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) linear inequality is a half-plane zero, or undefined. Equations of Attack bounded by the graph of its related (NCTM linear equation. Points on the Use transformational graphing to http://illuminations.n boundary are included unless it is a investigate effects of changes in ctm.org/lessondetail.a strict inequality. equation parameters on the graph spx?id=L782 ) of the equation. Parallel lines have equal slopes. Movie Lines (NCTM Write an equation of a line when http://illuminations.n The product of the slopes of given the graph of a line. ctm.org/lessondetail.a perpendicular lines is -1 unless one spx?id=L629 ) of the lines has an undefined slope. Write an equation of a line when The Line Runner given two points on the line whose (NCTM coordinates are integers. http://illuminations.n ctm.org/lessondetail.a Write an equation of a line when spx?id=L851 ) given the slope and a point on the line whose coordinates are School has this integers. software : Green Globs and Graphing Write an equation of a vertical line Equations (very as x = a. good) Write the equation of a horizontal line as y = c. Promethean Board Site Benchmark II Unit 3 Standard A.7 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 3 Standard: A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros of a function; d) x- and y-intercepts; e) finding the values of a function for elements in its domain; and f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A set of data may be characterized mathematical communication, Unit 3 Organizer Interactive Achievement by patterns, and those patterns can mathematical reasoning, connections, be represented in multiple ways. Functions VDOE and representations to ESS (2004) Graphs can be used as visual Determine whether a relation, Square Patio VDOE representations to investigate represented by a set of ordered ESS (2004) relationships between quantitative pairs, a table, or a graph is a data. Functions 2 VDOE function. ESS (2004) Inductive reasoning may be used to Identify the domain, range, zeros, make conjectures about Domain and intercepts of a function characteristics of function families. Representation presented algebraically or graphically. (NCTM Each element in the domain of a http://illuminations.n relation is the abscissa of a point of For each x in the domain of f, find ctm.org/lessondetail.a the graph of the relation. spx?id=L621 ) f(x). Each element in the range of a Roller Coasting Represent relations and functions relation is the ordinate of a point of through Functions using concrete, verbal, numeric, the graph of the relation. (NCTM graphic, and algebraic forms. Given one representation, students http://illuminations.n A relation is a function if and only ctm.org/lessondetail.a will be able to represent the if each element in the domain is Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 3 Standard: A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros of a function; d) x- and y-intercepts; e) finding the values of a function for elements in its domain; and f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) paired with a unique element of the relation in another form. spx?id=L839 ) range. Detect patterns in data and The values of f(x) are the ordinates represent arithmetic and geometric of the points of the graph of f. patterns algebraically. Promethean Board Site The object f(x) is the unique object in the range of the function f that is associated with the object x in the domain of f. For each x in the domain of f, x is a member of the input of the function f, f(x) is a member of the output of f, and the ordered pair [x, f(x)] is a member of f. An object x in the domain of f is an x-intercept or a zero of a function f if and only if f(x) = 0. Set builder notation may be used to represent domain and range of a relation. Benchmark II Unit 3 Standard A.8 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 3 Standard: A.8 The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, The constant of proportionality in a mathematical communication, Unit 3 Organizer Interactive Achievement direct variation is represented by the mathematical reasoning, connections, ratio of the dependent variable to Direct Variation and representations to VDOE ESS (2004) the independent variable. Given a situation, including a real- The constant of proportionality in Do I have to Mow the world situation, determine whether Whole Thing? an inverse variation is represented a direct variation exists. (NCTM by the product of the dependent variable and the independent http://illuminations.n Given a situation, including a real- ctm.org/lessondetail.a variable. world situation, determine whether spx?id=L729 ) an inverse variation exists. A direct variation can be represented by a line passing Write an equation for a direct Promethean Board Site through the origin. variation, given a set of data. Real-world problems may be Write an equation for an inverse modeled using direct and/or inverse variation, given a set of data. variations. Graph an equation representing a direct variation, given a set of data. Benchmark II Unit 4 Standard A.4 e, f Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 4 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A solution to an equation is the mathematical communication, Unit 4 Organizer Interactive Achievement value or set of values that can be mathematical reasoning, connections, substituted to make the equation Road Trip VDOE and representations to ESS (2004) true. Solve a literal equation (formula) The solution of an equation in one for a specified variable. What’s Your Call? variable can be found by graphing VDOE ESS (2004) the expression on each side of the Simplify expressions and solve equation separately and finding the equations, using the field Spring Fling VDOE x-coordinate of the point of properties of the real numbers and ESS (2004) intersection. properties of equality to justify simplification and solution. The Exercise Ring Real-world problems can be VDOE ESS (2004) interpreted, represented, and solved Solve quadratic equations. using linear and quadratic Talk or Text (NCTM equations. Identify the roots or zeros of a Equations of Attack quadratic function over the real (NCTM The process of solving linear and number system as the solution(s) http://illuminations.n quadratic equations can be modeled to the quadratic equation that is ctm.org/lessondetail.a in a variety of ways, using concrete, formed by setting the given spx?id=L780 ) pictorial, and symbolic quadratic expression equal to zero. representations. Solve multistep linear equations in Properties of real numbers and Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 4 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) properties of equality can be used to one variable. There Has to be a justify equation solutions and System for this Sweet expression simplification. Confirm algebraic solutions to Problem (NCTM linear and quadratic equations, http://illuminations.n using a graphing calculator. ctm.org/lessondetail.a The zeros or the x-intercepts of the spx?id=L766 ) quadratic function are the real Given a system of two linear root(s) or solution(s) of the equations in two variables that has quadratic equation that is formed by a unique solution, solve the system Promethean Board Site setting the given quadratic by substitution or elimination to expression equal to zero. find the ordered pair which satisfies both equations. A system of linear equations with exactly one solution is characterized Given a system of two linear by the graphs of two lines whose equations in two variables that has intersection is a single point, and a unique solution, solve the system the coordinates of this point satisfy graphically by identifying the both equations. point of intersection. A system of two linear equations Determine whether a system of with no solution is characterized by two linear equations has one the graphs of two lines that are solution, no solution, or infinite parallel. solutions. A system of two linear equations Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 4 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) having infinite solutions is Write a system of two linear characterized by two graphs that equations that models a real-world coincide (the graphs will appear to situation. be the graph of one line), and the coordinates of all points on the line Interpret and determine the satisfy both equations. reasonableness of the algebraic or graphical solution of a system of Systems of two linear equations can two linear equations that models a be used to model two real-world real-world situation. conditions that must be satisfied simultaneously. Determine if a linear equation in one variable has one, an infinite number, or no solutions. Equations and systems of equations can be used as mathematical models for real-world situations. Set builder notation may be used to represent solution sets of equations. † Revised March 2011 Benchmark II Unit 4 Standard a,d Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 4 Standard: A.5 The student will solve multistep linear inequalities in two variables, including a) solving multistep linear inequalities algebraically and graphically; b) justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set of real numbers and its subsets; c) solving real-world problems involving inequalities; and d) solving systems of inequalities. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A solution to an inequality is the mathematical communication, Unit 4 Organizer Interactive Achievement value or set of values that can be mathematical reasoning, connections, substituted to make the inequality and representations to Promethean Board Site true. Solve multistep linear inequalities Real-world problems can be in one variable. modeled and solved using linear inequalities. Justify steps used in solving inequalities, using axioms of Properties of inequality and order inequality and properties of order can be used to solve inequalities. that are valid for the set of real numbers. Set builder notation may be used to represent solution sets of Solve real-world problems inequalities. involving inequalities. Solve systems of linear inequalities algebraically and graphically. Predictor Test Unit 5 Standard A.2 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 5 Standard: A.2 The student will perform operations on polynomials, including a) applying the laws of exponents to perform operations on expressions; b) adding, subtracting, multiplying, and dividing polynomials; and c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, The laws of exponents can be mathematical communication, Unit 5 Organizer Interactive Achievement investigated using inductive mathematical reasoning, connections, A.2a) reasoning. and representations to Exponents VDOE A relationship exists between the ESS (2004) laws of exponents and scientific Simplify monomial expressions notation. and ratios of monomial A.2b) expressions in which the Mo and Po Nomial Operations with polynomials can be exponents are integers, using the VDOE ESS (2004) represented concretely, pictorially, laws of exponents. and symbolically. Old Polly VDOE Model sums, differences, products, ESS (2004) Polynomial expressions can be used and quotients of polynomials with to model real-world situations. concrete objects and their related Polynomial Puzzler pictorial representations. The distributive property is the (NCTM unifying concept for polynomial Relate concrete and pictorial http://illuminations.n operations. manipulations that model ctm.org/lessondetail.a polynomial operations to their spx?id=L798 ) Factoring reverses polynomial corresponding symbolic multiplication. Battleship for representations. Polynomials Some polynomials are prime Find sums and differences of (http://quia.com ) polynomials and cannot be factored polynomials. over the set of real numbers. A.2c) Polynomial expressions can be used Find products of polynomials. The Functionality VDOE to define functions and these factors will have no more than five ESS (2004) functions can be represented total terms (i.e. (4x+2)(3x+5) Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 5 Standard: A.2 The student will perform operations on polynomials, including a) applying the laws of exponents to perform operations on expressions; b) adding, subtracting, multiplying, and dividing polynomials; and c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) graphically. represents four terms and (x+1)(2x2 +x+3) represents five Promethean Board Site There is a relationship between the terms). factors of any polynomial and the x- intercepts of the graph of its related Find the quotient of polynomials, function. using a monomial or binomial divisor, or a completely factored divisor. Factor completely first- and second-degree polynomials with integral coefficients. Identify prime polynomials. Use the x-intercepts from the graphical representation of the polynomial to determine and confirm its factors. Predictor Test Unit 6 Standard A.3 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 6 Standard: A.3 The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A square root in simplest form is mathematical communication, Unit 6 Organizer Interactive Achievement one in which the radicand mathematical reasoning, connections, (argument) has no perfect square Estimating Square and representations to Roots VDOE ESS factors other than one. (2004) Express square roots of a whole A cube root in simplest form is one number in simplest form. Simplifying Square in which the argument has no perfect cube factors other than one. Roots VDOE ESS Express the cube root of a whole (2004) number in simplest form. The cube root of a perfect cube is an integer. Express the principal square root Promethean Board Site of a monomial algebraic The cube root of a nonperfect cube expression in simplest form where lies between two consecutive variables are assumed to have integers. positive values. The inverse of cubing a number is determining the cube root. In the real number system, the argument of a square root must be nonnegative while the argument of a cube root may be any real number. Predictor Test Unit 6 Standard A.4c Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 6 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, A solution to an equation is the mathematical communication, Unit 6 Organizer Interactive Achievement value or set of values that can be mathematical reasoning, connections, substituted to make the equation Factoring for Zeros and representations to VDOE ESS (2004) true. Solve a literal equation (formula) The solution of an equation in one for a specified variable. Egg Launch variable can be found by graphing (Equations of Attack the expression on each side of the Simplify expressions and solve (NCTM equation separately and finding the equations, using the field http://illuminations.n x-coordinate of the point of properties of the real numbers and ctm.org/lessondetail.a intersection. properties of equality to justify spx?id=L738 ) simplification and solution. Real-world problems can be Battleship for interpreted, represented, and solved Solve quadratic equations. Polynomials using linear and quadratic (http://quia.com ) equations. Identify the roots or zeros of a quadratic function over the real The process of solving linear and number system as the solution(s) Promethean Board Site quadratic equations can be modeled to the quadratic equation that is in a variety of ways, using concrete, formed by setting the given pictorial, and symbolic quadratic expression equal to zero. representations. Solve multistep linear equations in Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 6 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) Properties of real numbers and one variable. properties of equality can be used to justify equation solutions and Confirm algebraic solutions to expression simplification. linear and quadratic equations, using a graphing calculator. The zeros or the x-intercepts of the Given a system of two linear quadratic function are the real equations in two variables that has root(s) or solution(s) of the a unique solution, solve the system quadratic equation that is formed by by substitution or elimination to setting the given quadratic find the ordered pair which expression equal to zero. satisfies both equations. A system of linear equations with Given a system of two linear exactly one solution is characterized equations in two variables that has by the graphs of two lines whose a unique solution, solve the system intersection is a single point, and graphically by identifying the the coordinates of this point satisfy point of intersection. both equations. Determine whether a system of A system of two linear equations two linear equations has one with no solution is characterized by solution, no solution, or infinite the graphs of two lines that are solutions. parallel. Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 6 Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including a) solving literal equations (formulas) for a given variable; b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; c) solving quadratic equations algebraically and graphically; d) solving multistep linear equations algebraically and graphically; e) solving systems of two linear equations in two variables algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) A system of two linear equations Write a system of two linear having infinite solutions is equations that models a real-world characterized by two graphs that situation. coincide (the graphs will appear to be the graph of one line), and the Interpret and determine the coordinates of all points on the line reasonableness of the algebraic or satisfy both equations. graphical solution of a system of two linear equations that models a Systems of two linear equations can real-world situation. be used to model two real-world conditions that must be satisfied Determine if a linear equation in simultaneously. one variable has one, an infinite number, or no solutions.† Equations and systems of equations can be used as mathematical models for real-world situations. Set builder notation may be used to represent solution sets of equations. † Revised March 2011 Predictor Test Unit 7 Standard A.11 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 7 Standard: A.11 The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, The graphing calculator can be used mathematical communication, Unit 7 Organizer Interactive Achievement to determine the equation of a curve mathematical reasoning, connections, of best fit for a set of data. Estimating Ages of and representations to Famous People The curve of best fit for the (mathbits.com) Write an equation for a curve of relationship among a set of data best fit, given a set of no more than Illuminations: Line of points can be used to make twenty data points in a table, a Best Fit (NCTM predictions where appropriate. graph, or real-world situation. Illuminations Many problems can be solved by http://illuminations.n Make predictions about unknown ctm.org/ActivityDetai using a mathematical model as an outcomes, using the equation of l.aspx?ID=146 interpretation of a real-world the curve of best fit. situation. The solution must then refer to the original real-world Design experiments and collect Barbee Bungee situation. data to address specific, real-world Jumping (NCTM questions. Considerations such as sample size, Illuminations randomness, and bias should affect http://illuminations.n Evaluate the reasonableness of a ctm.org/LessonDetail experimental design. mathematical model of a real- .aspx?id=L646 world situation. Line of Best Fit VDOE ESS (2004) SOL TEST Unit 8 Standard A.10 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 8 Standard: A.10 The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, Statistical techniques can be used to mathematical communication, Unit 8 Organizer Interactive Achievement organize, display, and compare sets mathematical reasoning, connections, of data. Box-and-Whisker and representations to Plots VDOE ESS Box-and-whisker plots can be used (2004) Compare, contrast, and analyze to analyze data. data, including data from real- Vashon-Maury Island world situations displayed in box- Soil Study VDOE and-whisker plots. ESS (2004) Promethean Board Site SOL TEST Unit 8 Standard A.9 Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 8 Standard: A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) The student will use problem solving, Descriptive statistics may include Unit 8 Organizer Interactive Achievement mathematical communication, measures of center and dispersion. mathematical reasoning, connections, Good Assorted Variance, standard deviation, and and representations to Problem Set mean absolute deviation measure developed by Jeff the dispersion of the data. Analyze descriptive statistics to Holt of UVA (see determine the implications for the Instr. Specialist Sue The sum of the deviations of data real-world situations from which Jenkins) points from the mean of a data set is the data derive. 0. Given data, including data in a Promethean Board Site Standard deviation is expressed in real-world context, calculate and the original units of measurement of interpret the mean absolute the data. deviation of a data set. Standard deviation addresses the Given data, including data in a dispersion of data about the mean. real-world context, calculate variance and standard deviation of Standard deviation is calculated by a data set and interpret the taking the square root of the standard deviation. variance. Given data, including data in a The greater the value of the real-world context, calculate and standard deviation, the further the interpret z-scores for a data set. data tend to be dispersed from the mean. Explain ways in which standard deviation addresses dispersion by For a data distribution with outliers, examining the formula for the mean absolute deviation may be standard deviation. a better measure of dispersion than Subject/Grade Level: Algebra I Suggested Pacing for Benchmarking Period: UNIT 8 Standard: A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores. Essential Knowledge Resources and Suggested Essential Understandings and Skills Activities Assessment(s) the standard deviation or variance. Compare and contrast mean absolute deviation and standard deviation in a real-world context. A z-score (standard score) is a measure of position derived from the mean and standard deviation of data. A z-score derived from a particular data value tells how many standard deviations that data value is above or below the mean of the data set. It is positive if the data value lies above the mean and negative if the data value lies below the mean.