Algebra I Scope and Sequence edited May 2014 Algebra I Overview Page # 2-4 11 Unit Foundations of Functions Linear Equations and Functions Linear Inequalities Abbreviated Name FOUNDFUN LINEAR First Semester TEKS Readiness Supporting A.1A, A.1B, A.1C, A.2C, A.1D, A.1E, A.2B, A.2D, A.4A A.4C A.2A, A.3A, A.3B, A.4B, A.2B, A.2D, A.4A, A.5C, A.6B, A.4C, A.5A, A.5B, A.6A, A.6C, A.6F, A.7B A.6D, A.6E, A.6G, A.7A, A.7C Time (Days) LININEQ A.7B A.1C, A.7A, A.7C 25 (inc final week) Second Semester 12 Systems SYS A.8B A.8A, A.8C 15 13-14 15 16 17 Quadratic Functions Polynomial Operations Quadratic Equations Exponential Functions and Equations 10 42 77 days QUADFUN A.2D, A.9D A.2A, A.9A, A.9B, A.9C 10 POLYOP A.4A A.4A, A.4B 15 QUADEQU A.10A A.10B 15 EXPO A.11A, A.11B, A.11C (includes inverse variation) STAAR Review Unit STAAR ALL TEKS 11 18 Algebra Methods ALGMET New TEKS: A.11A, A.11B, A.12C, A.12D, A.12E 23 (inc final week) 100 days 10
Foundations of Functions (10 days) Enduring Understandings The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student understands that data representing real-world situations can be collected, organized, and interpreted in order to solve problems. The student understands that functions can be used to model real-world situations. Vocabulary function, independent, dependent, discrete, continuous, domain, range, input, output, mapping, scatterplot The TEKS in this unit are not first-time instruction. 382 classes are covering this unit for a total of 30 days, including flex/assessment days. Bolded and underlined items will need more strengthening in high school. A.1A(S) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to describe independent and dependent quantities in functional relationships. The difference between the independent and dependent quantities. Identify the independent and dependent quantities in a function and/or situation. That not all relationships are functions (height vs eye color). Verify that a point is on a function by substituting (x, y) into the function equation. An ordered pair can represent a point Verbalize functional relationships. on the function. y depends on x y is a function of x x determines y y is a result of x Use related vocabulary for functional relationships. Input, output If, then Cause, effect A.1B(S) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to gather and record data and use data sets to determine functional relationships between quantities. Not all data sets represent functional Gather and represent data from situations (Ex: build a table of relationships. The characteristics of a functional relationship in its various forms (graph, table, set of ordered pairs). A function may or may not contain an infinite number of ordered pairs. data from a problem situation). Represent data in a graph, table, set of ordered pairs, mapping, and verbal description. Use data to determine if a relation is a function, including vertical line test. 2
A.1C(S) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to describe functional relationships for given problem situations and write equations [or inequalities] to answer questions arising from the situations. A.1D(R) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, [and inequalities]. Algebraic expressions can be used to represent generalizations. Symbols can be used to represent unknowns or variables. A table is a graphic organizer used to represent relationships among quantities. Patterns in a table create patterns in a graph. Verbally describe the relationship among quantities (Ex: As the term number increases by 1, the value decreases by 3). Create a graph from a given equation, table of data, concrete models, or a given situation. Complete a table of data from a given equation, a graph, concrete models, or a given situation. Use graphing calculator to verify table of data, equation, and/or graph. Recognize and connect the various representations of a situation (i.e. graph, table, equation, verbal description). Create a situation from a table of data, an equation, or a graph. Define what a variable(s) represents in a problem situation. Generate equation from scenario and from table. A.1E(R) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to interpret and make decisions, predictions, and critical judgments from functional relationships. A.2C(S) The student uses the properties and attributes of functions. The student is expected to interpret situations in terms of given graphs or creates situations that fit given graphs. Inferences and predictions can be made from a functional relationship. Make/interpret decisions, predictions, and critical judgments from functional relationships. A functional relationship can be connected to a situation. Answer related questions when given a real-world situation (Ex: term position and value of term). Create a situation that fits a given graph. Create a graph that fits a given situation. Interpret a situation in terms of a given graph (Ex: I know that Trisha was going faster during this part of the trip because the graph is steeper). 3
A.2B(R) The student uses the properties and attributes of functions. The student is expected to identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete. A.4A(R) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to find specific function values, [simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations.] A.4C(S) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1. The domain represents the independent values (x-values) in a function. The range represents the dependent values (y-values) in a function. The meaning of the symbols >, <,,. The difference between continuous data and discrete data. The domain and range of a function may be different from the domain and range for a situation represented by that function. Identify domain and range from a graph, table, or set of ordered pairs. Identify domain/range values given range/domain of a function. - Find y given x, find x given y from all representations. - Find f(3) given the f(x) = x + 3 - Find x given an f(x) value. Solve 1 and 2-step equations in context of finding x and y values. Represent the domain and range using inequality notation, including formal set notation. Ex: {x 3<x<10} Identify domain and range of a continuous or discrete situation. - Ex 1: The cost of a food bill equals $2 per hamburger times the number of hamburgers ordered. For the function y = 2x, D={real #s} and R={real #s} and the graph of the function is a line (continuous). For the portion of the function relative to the situation, the D={positive whole numbers} and R={2, 4, 6, 8, } The graph shows points (1, 2), (2, 4), etc. (discrete). - Ex 2: The perimeter of a rectangle is 12. Its area is represented by equation A = x(6 x), where A is the area of the rectangle and x is the length of the rectangle. For the function A = x(6 x), the D={real #s} and R={all numbers less than or equal to 9} (continuous) or {y y 9}. For the portion of the function relative to the situation, D={all numbers between (but not including) 0 and 6} or {x 0<x<6}, and R={all numbers between (but not including) 0 and 10} or {y 0<y<10} (continuous). A.2D(R) The student uses the properties and attributes of functions. The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations. There are different ways to organize data (table, scatterplot, etc.). The attributes of a scatterplot (horizontal axis represents the independent variable and vertical axis represents the dependent variable). Information can be gathered and interpreted from a scatterplot. Predictions can be made based on trends in data. Collect data from problem situations. Create and/or interpret scatterplots from problem situations. Analyze patterns in scatterplots to make predictions, decisions, and critical judgments, including linear and nonlinear. Determine if a set of data has a positive, negative, or no correlation from the scatterplot, situation, or table. Identify independent and dependent quantities in a given situation. 4
Linear Equations and Functions (42 days) Enduring Understandings The student understands that linear functions have unique properties. The student understands that linear functions can be represented in a variety of ways. The student understands that slopes of linear functions can be found from various representations and that they have meaning based on the related linear situation. The student understands that changes in a function equation can result in specific changes to the graph of the function. The student understands that the zero of a linear function is connected to the solution of the related linear equation, f(x) = 0. The student understands the relationship between equations and functions. The student understands that equations can be solved in a variety of ways. Vocabulary correlation, linear, nonlinear, parent function, rate of change, coefficient, constant, arithmetic sequence, nth term, common difference, recursive, arithmetic rate of change scatterplot, slope, slope-intercept form, parallel, perpendicular, point-slope form, standard form, trend line, x- and y- intercept, zeros, x, y, solution, horizontal, vertical, literal equation, proportional change, coefficient, constant, direct variation, directly proportional, zero of a function, constant of proportionality A.5C(R) The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. A.3A(S) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to use symbols to represent unknowns and variables. A.3B(S) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to look for patterns and represent generalizations algebraically. A linear function can be Represent linear functions in a variety of ways (table, described in a variety of ways. graph, verbal description, equation). A linear function describes the Make connections between various representations of relationship between unknowns. An ordered pair on the line indicates x and y values which makes the equation of the line a true statement. The ordered pairs of a function have meaning in the context of a situation. linear functions. Analyze a situation involving linear functions. Create a situation from a table of data, an equation, or a graph. 5
A.2A(S) The student uses the properties and attributes of functions. The student is expected to identify and sketch the general forms of linear (y = x) [and quadratic (y = x 2 )] parent functions. A.5A(S) The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to determine whether or not given situations can be represented by linear functions. A.5C(R) The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. The graph of the parent function y = x is Identify and sketch the linear parent function. the set of ordered pairs in which the x- Create ordered pairs that would fall on the coordinate and the y-coordinate of each graph of the line y = x. ordered pair are equal. Determine if situations are represented by a The attributes of a linear function. The differences between linear and nonlinear functions when represented in various ways. A linear function has a constant arithmetic rate of change. (i.e. the change is an addition rather than multiplication) There is a connection between the mathematical patterns in a linear function and the idea of constant rate of change in a real-life situation. linear function. Make connections between a recursive linear pattern and a constant arithmetic rate of change. *Ex) In the sequence 2, 5, 8, 11 each term is three more than the term before (recursive pattern) For this sequence. Compare and contrast linear and non-linear situations based on the constant arithmetic rate of change as seen in tables, graphs, verbal descriptions, scenarios and equations. Apply vocabulary such as flat rate, initial fee, rate of change, service fee, etc. when given a situation. Define what a variable(s) represents in a problem situation. A.5B(S) The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to determine the domain and range for linear functions in given situations. The meaning of the Identify the domain/range, when given the range/domain for symbols >, <,,. The difference between continuous data and discrete data. The domain and range of a linear function may be different from the domain and range for a situation represented by that function. given linear functions in situations. Determine the domain and range values for a given linear situation using the graph, verbal description, table, or equation. Use appropriate notation for domain and range, based on the nature of the data (set notation, inequality, or verbal description, including formal set notation). Ex. {x x>4} Identify an appropriate and reasonable domain and range for a given situation, both continuous and discrete (Ex: The domain of y = 10x+7 is all real numbers. But the situation of a babysitter earning a flat fee of $7 plus $10/hour has a reasonable domain of {1, 2, 3, 4, 5, 6}). 6
A.6A(S) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations. A.6E(S) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to determine the intercepts of the graphs of linear functions [and zeros of linear functions] from graphs, tables, and algebraic representations. A.4C(S) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1. The vertical/horizontal change from point to point on a line remains constant. The slope or rate of change describes the relationship (ratio) between the change in the dependent variable and the change in the independent variable ( y ). x There are various ways to find the slope or rate of change. The y-intercept represents the value of the dependent variable (y) when the independent variable (x) is zero (0, y). The symbol f(x) can be used interchangeably with the dependent variable of a function, usually y. Find the vertical and horizontal pattern in the graph of a linear function, and connect that vertical and horizontal change to its corresponding table of values. Determine the slope or the rate of change. Graphs Tables Ordered pairs Algebraic representations Arithmetic Sequence (common difference) Formalize y = mx + b formula. Determine the y-intercepts of a linear function given a table, equation, graph, or verbal description. Connect equation notation with function notation. 7
A.6D(S) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept. A.3A(S) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to use symbols to represent unknowns and variables. A.3B(S) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to look for patterns and represent generalizations algebraically. A.7A(S) The student formulates equations [and inequalities] based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to analyze situations involving linear functions and formulate linear equations [or inequalities] to solve problems. Arithmetic The m and b in the formula y mx b can be used to determine the graph of a Graph a line from an equation in y = mx + b form. Graph a line. sequences line. Two points in a table (or as ordered pairs) A line can be graphed using various information, including slope, point(s), and intercepts. A point and the slope The slope and y-intercept Algebraic expressions can be used to represent generalizations. Symbols can be used to represent unknowns or variables. A table is a graphic organizer used to represent relationships among quantities. Patterns in a table represent patterns in a graph and vice versa. The connection between concrete and symbolic methods of solving equations. Expressions can be written in different ways and still be equivalent (Ex: 2x + 6 = 6 + 2x). Given linear data, write the function rule from a table, graph, model, or situation. Create a situation from a table of data, an equation, or a graph. Define what a variable(s) represents in a problem situation. Given a problem situation, write a linear equation to solve the problem. Identify the terms of an arithmetic sequence using a recursive process. 8
A.6C (R) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b. A.6F(R) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to interpret and predict the effects of changing slope and y-intercept in applied situations. Changing the slope and/or intercepts in a situation can affect the other attributes of the function. Changing m in the function y mx baffects the steepness of the graph. Changing b in the function y mx baffects the y-intercept of the graph. Interpret and predict the effects of changing slope and y-intercept in applied situations. Describe and predict the effects of changes in m and b on the graph of y = mx + b. Examples are listed below. Compare y = 3x to y = 3x + 5 Compare y = 2x to y = 4x Compare y = 2x + 3 to y = 3x + 7 A.2D(R) The student uses the properties and attributes of functions. The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations. There are different methods to Collect data and create and/or interpret scatterplots organize linear data (table, from linear problem situations. scatterplot. etc.). Determine if a set of data can be modeled by a linear Predictions can be made based on function. trends in data. An equation that approximates the trend line can be determined for a set of data. Identify independent and dependent quantities in a given situation. Transform the linear parent function to create an accurate trend line and its corresponding equation for given data (no linear regression). Use accurate trend line and/or its equation to make predictions. Strength of correlation coefficient Association vs causation Regression with technology 9
A.4A(R) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to find specific function values, simplify [polynomial] expressions, transform and solve equations, [and factor as necessary] in problem situations. A.4B(S) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to use the commutative, associative, and distributive properties to simplify algebraic expressions. A.4C(S) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1. A.7B(R) The student formulates equations [and inequalities] based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to investigate methods for solving linear equations [and inequalities] using concrete models, graphs, and the properties of equality, select a method, and solve the equations [and inequalities]. A.7C(S) The student formulates equations [and inequalities] based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to interpret and determine the reasonableness of solutions to linear equations [and inequalities]. Solve literal The connection between concrete and symbolic methods of solving equations. An equation can be solved using a variety of methods: concrete models, tables, algebraic methods. The concept of an equation means that the value of the expression on the left side is always equivalent to the value of the expression on the right side. To solve an equation means to determine the value of the variable which makes the number sentence a true statement. To keep an equation balanced, an operation performed on one side of the equation must also be performed on the other side of the equation. The sum of a number and its opposite is zero. Write and solve equations algebraically and with models. Combining like terms Distributive property Variables on both sides Introduce each type with a real-world scenario Verbal descriptions (Ex: x is five more than four times a number) Function notation When given an equation in a problem situation, substitute the appropriate value(s) in order to solve the problem. Find the indicated value in a function equation (i.e. find y given x and vice versa, such as (x, 3)). Algebraically (by substitution) Graphically Tabular (ordered pair in the table) Solve and justify solutions to equations algebraically and with graphing calculator. Home screen Y1 = and table Solve a literal linear equation of up to 4 variables for a specified variable. (Ex: Solve I=PRT for P) equations Translate verbal descriptions into expressions 10
A.6B(R) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs. A.6D(S) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept. Identify To write a linear equation Translate equations from standard form to slope-intercept form translations and in y = mx + b form, the and vice versa. graph from point-slope form slope and the y-intercept With algebra tiles Horizontal of the line must be known Algebraically transformations or determined. Determine the x- and y-intercept of a linear function using Equations of lines can be data, symbolic representations, or graphs. determined from various Write the equation of a line (point-slope, slope-intercept, information sets. standard) from a variety of information. The slope of a line has Graph of a line meaning based on the Two points in a table (or as ordered pairs) situation it represents. Two points on a graph The x-intercept represents A point and a slope the value of the A slope and y-intercept independent variable (x) x- and y- intercepts when the dependent Situations variable (y) is zero (x, 0). Models (e.g. algebra tiles, geometric figure, diagram, etc.) Identify the slope and a point from the point-slope form of a line. Graph a line. Two points in a table (or as ordered pairs) A point and the slope The slope and y-intercept The standard form equation The point-slope equation Interpret the meaning of the slope, x-intercept, and y-intercept in a problem situation. A.6C (R) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b. Parallel lines have the Determine if a pair of lines is parallel, perpendicular or neither. same slope, but different Table y-intercepts. Perpendicular lines have slopes which are oppositesigned reciprocals of each other. Graph Equation Write the equation of a line that contains a given point and is parallel to a given line. Write the equation of a line that contains a given point and is perpendicular to a given line. 11
A.6G(S) The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to relate direct variation to linear functions and solve problems involving proportional change. Remove The characteristics of direct Connect proportional change to a direct variation variation (proportional function of the form y = kx in table, equation, graph, relationships) given in a table, and problem situations. graph, or algebraic representation. The differences between proportional and non-proportional situations. Given the constant of proportionality and one ordered pair (x, y), additional ordered pairs can be determined. Find k, the constant of proportionality in a given situation from table, equation, graph, and problem situations. Compare and contrast proportional and nonproportional situations. Solve direct variation application problems. Using the value of k Using a proportion Determine if a situation is directly proportional. inverse variation (after next year may still be on 2014 STAAR) 12
Linear Inequalities (25 days including final week and benchmarks) Enduring Understandings The student understands that an inequality is the comparison of two expressions such that the values of the two expressions are not necessarily equal. The student understands that an inequality can be solved using a variety of methods. Vocabulary greater than or equal to, less than or equal to, inequality, solution set, between, inclusive, exclusive A.1C(S) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to describe functional relationships for given problem situations and write [equations or] inequalities to answer questions arising from the situations. A.7A(S) The student formulates [equations and] inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to analyze situations involving linear functions and formulate [linear equations or] inequalities to solve problems. A.7B(R) The student formulates [equations and] inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to investigate methods for solving linear [equations and] inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the [equations and] inequalities. A.7C(S) The student formulates [equations and] inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to interpret and determine the reasonableness of solutions to linear [equations and] inequalities. Linear inequalities can be formulated by using symbols to represent unknowns. A linear inequality describes a relationship between unknowns. A linear inequality can be used to generate an answer(s) to a problem situation. The solution to a linear inequality is represented by a set of values. The solution of a linear inequality must be reasonable to the situation it describes. The difference between > and, and between < and. The word between implies a solution set exclusive of the endpoints of its graph on a number line. Given one representation of an inequality, translate it into another representation (graph, slope-intercept form, standard form, verbal description). Formulate inequalities based on linear functions. Analyze an inequality situation involving linear functions. Write a one or two-variable inequality to solve application problems. Solve one and two-variable inequalities in problem situations using a variety of methods. Graph the solution of a one-variable inequality on a number line. (Do NOT include compound inequalities.) Graph the solution set of a two-variable inequality on a coordinate plane. Interpret the solution to one- and two-variable inequalities from a variety of representations. Graph Table Algebraic methods Determine the reasonableness of the solution to linear inequalities in terms of the situation. Determine if a situation is discrete or continuous. Solve inequalities involving distributive and variables on both sides. Solve inequalities involving distributive and variables on both sides. 13
Systems (15 days) Enduring Understandings The student understands that real-world situations can be represented by systems of linear equations. The student understands that a system can be solved in a variety of ways. The student understands that the solution to a system of linear equations is related to the situation it represents. Vocabulary system, solution, intersection, linear combination (elimination), substitution A.8A(S) The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to analyze situations and formulate systems of linear equations in two unknowns to solve problems. A system of equations is a set of Identify the unknowns in a given situation. equations that describes the Assign variables to represent unknowns in a relationship between unknowns. given situation. In order to solve for two unknowns, there must be two equations. Each equation in a system of equations represents different information in the situation. Formulate a system of linear equations (two unknowns) from a given situation. A.8B(R) The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to solve systems of linear equations using concrete models, graphs, tables, and algebraic methods. A system of linear equations can be solved in a variety of ways. Models Graphs Tables Algebraic methods The solution to a system of equations represents the values that make both equations true. A system may have 1 solution, no solution, or infinitely many solutions. Solve two-variable systems of equations using a variety of methods. Concrete models (algebra tiles) Tables Graphs Algebraic methods Substitution (including y1 = y2) Linear combination (elimination) Choose an appropriate method for solving a system of equations. Recognize and write equations in standard form or slope-intercept as appropriate to solve the system. Determine if an ordered pair is the solution to a system of equations. A.8C(S) The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to interpret and determine the reasonableness of solutions to systems of linear equations. The solution to a system of equations should be reasonable for the situation it describes. Ordered pairs of either equation (other than the point of intersection) have meaning in context of a situation. Interpret the solution to a system of equations in terms of the situation (Ex: The ordered pair (2, 5) represents 2 hamburgers and 5 tacos.). Determine the reasonableness of the solution to a system of equations in terms of the situation. Determine the relative meaning of other ordered pairs besides the solution. (e.g. When should I use Plan A vs Plan B? How do I know?) 14
Quadratic Functions (10 days) Enduring Understandings The student understands that quadratic functions have unique properties and attributes that are represented in a variety of ways. The student understands that the graphs of quadratic functions are affected by the parameters of the function. Vocabulary quadratic function, parabola, axis of symmetry, symmetric pairs, symmetry, vertex, maximum/minimum, second differences, best fit curve A.2A(S) The student uses the properties and attributes of functions. The student is expected to identify and sketch the general forms of [linear (y = x) and] quadratic (y = x 2 ) parent functions. The quadratic parent function is y=x 2. Identify and sketch the quadratic parent The graph of y = x 2 is a parabola. function. Parabolas are symmetric with exactly Create ordered pairs that would fall on the one axis of symmetry. graph of y = x 2. Connect the graph of y = x 2 to patterns in the table and the function rule. Identify the attributes of the quadratic parent function, y = x 2, from a table, graph, or equation (symmetric points, vertex, and axis of symmetry). Recognize the attributes of any quadratic function from a table or graph. A.9A(S) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to determine the domain and range for quadratic functions in given situations. The domain represents the independent values (x-values) in a function. The range represents the dependent values (y-values) in a function. The domain and range for a function are not always the same as the domain and range for a situation represented by the function. Determine reasonable domain and range values for given quadratic situations from the graph, verbal description, table, or equation. Determine if the domain and range are discrete or continuous. Use appropriate notation for domain and range, based on the nature of the data: set notation, inequality, verbal description. Identify the maximum/ minimum value of the function and connect to domain and range. 15
A.9B(S) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to investigate, describe, and predict the effects of changes in a on the graph of y = ax 2 + c. A.9C(S) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to investigate, describe, and predict the effects of changes in c on the graph of y = ax 2 + c. Horizontal Changing a in the function y = ax 2 + c will vertically expand or compress the graph of the function. Changing c in the function y = ax 2 + c will vertically translate the graph of the function. The graph of y = -ax 2 + c is a reflection of the graph of y = ax 2 + c over the horizontal axis. Describe and predict the effect of changes in a and c on the graph of y = ax 2 + c in the graph, table, equation, or problem situation. Graph quadratic functions using transformations (changes in a and c). Determine the axis of symmetry, symmetric pairs of points, and vertex of a quadratic function from the graph or table. (include horizontal shifts, but not in equations) shifts on graph A.9D(R) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to analyze graphs of quadratic functions and draw conclusions. The graphs of quadratic functions can be used to draw conclusions about the situation represented by the graph. Each point on the graph of a quadratic function has specific meaning in relation to the situation represented by the function. Interpret points on the graph in terms of the situation. Draw conclusions from the graph of a quadratic function in a given situation. A.2D(R) The student uses the properties and attributes of functions. The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations. There are different ways to organize data (table, scatterplot, etc.). The attributes of a scatterplot: horizontal axis represents the independent variable and vertical axis represents the dependent variable. Information can be gathered from a scatterplot. Predictions can be made based on trends in data. An equation that approximates the trend line can be determined from given information. Collect data from problem situations. Create and/or interpret scatterplots from problem situations. Determine if a set of data is quadratic from table, looking at second differences. Identify independent and dependent quantities in a given situation. Transform the quadratic parent function to create an accurate best fit curve and its corresponding equation for given data (no quadratic regression). Change a in y = ax 2 + c Change c in y = ax 2 + c Use accurate trend curve and/or its equation to make predictions. 16
Polynomial Operations (15 days) Enduring Understandings The student understands the importance of the skills required to manipulate symbols in order to solve problems. The student understands that polynomials can be rewritten in a variety of ways and still be equivalent. Vocabulary base, exponent, term, like terms, degree, expression, polynomial, monomial, binomial, trinomial, factor, GCF, product, area, dimensions A.4A(R) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations. A.4B(S) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations [and inequalities] in problem situations. The student is expected to use the commutative, associative, and distributive properties to simplify algebraic expressions. The difference between simplifying, multiplying, and factoring polynomials and when each is appropriate. Polynomials can be simplified, multiplied, and factored using various strategies. Polynomials may have different, but equivalent forms. Not all polynomials have factors other than 1 (some are prime). The product of two expressions can be modeled by the area of a rectangle. Simplify polynomials with concrete models and algebraically. Multiply polynomials (monomial polynomial and binomial binomial). With concrete models Box method Algebraically Verify on the graphing calculator that the original expression is equivalent to the resulting expression. Multiply polynomials as appropriate in problem situations. Factor quadratic polynomials using various methods. With concrete models Box method Algebraically Verify on graphing calculator Include higher degree if GCF removed leaves a quadratic (introduce product rule with single variable) Have discussion with Alg 2 about box method 17
Quadratic Equations (15 days) Enduring Understandings The student understands that quadratic equations can be solved in a variety of ways. Vocabulary quadratic formula, zeros, roots, solution A.10A(R) The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student is expected to solve quadratic equations using concrete models, tables, graphs, and algebraic methods. A.10B(S) The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student is expected to make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. Quadratic equations may have 2, 1, or no real solutions. There is a relationship between the graph of a quadratic function and its zeros. The x-intercept(s) of a graph is the ordered pair(s) that describes the point(s) where the graph crosses the x-axis. The zeros of a function are the value(s) of the independent variable, x, that makes the function equal to zero. There are various ways to determine the zeros of a quadratic function. Solve quadratic equations. In problem situations With and without concrete models From the graph of the related function From the table By factoring Using the quadratic formula Connect the roots (solutions) of the equation, the zeros of the related function, and the x- intercepts of the graph of the function. Find the solutions of an equation setting a function rule equal to a non-zero number. (Ex. f(x) = 6 has solutions where y = 6) Determine the zeros of a quadratic function from the graph, table, or algebraic representation. horizontal intercept means the point where a parabola crosses a horizontal line, not necessarily the x-axis. 18
Exponential Functions and Equations (10 days) Enduring Understandings The student understands that expressions involving exponents can be written in a variety of ways that are equivalent. The student understands that there are situations modeled by functions that are neither linear nor quadratic. Vocabulary exponential, exponent, base, common ratio, rate of growth/decay, geometric sequence, geometric rate of change A.11A (S) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to use patterns to generate the laws of exponents and apply them in problem-solving situations. The properties needed to simplify Develop and use the following laws of Negative exponents expressions containing exponents. exponents to simplify expressions. The laws (rules) of exponents are Product rule governed by the properties of Power rule exponents. Quotient rule Power of zero rule Negative exponents NO FRACTIONAL EXPONENTS HERE (ALGMET UNIT) Apply exponent laws in problem-solving situations. Distinguish between laws of exponents and combining like terms. A.11B(S) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods. Independent (x) values and Determine if a situation is inversely proportional. dependent (y) values vary inversely if Represent situations involving inverse variation. k Concrete models y for x Tables Graphs Algebraically Solve inverse variation application problems. Even though inverse functions Identify the graph of the inverse variation parent are not exponential, this section 1 function: y. was moved from the Linear x Equations unit to improve the flow of instruction. Inverse variation is not covered in the new TEKS so this is the last year to teach this. 19
A.11C(S) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods. Geometric sequence Exponential functions can be Represent situations involving exponential represented in a variety of ways. growth and decay. When x = 1, exponential function values change by a common ratio (i.e. geometric rate of change). Concrete models Tables Graphs A sequence in which each new term is the product of the previous term and a constant (common ratio) is called a geometric sequence. A geometric sequence is an exponential function in which the domain is the set of natural numbers and the range consists of the terms of the sequence. Algebraic methods Distinguish between a graph representing exponential growth and a graph representing exponential decay. Solve problems involving exponential growth and decay (finance, population, etc.). Determine if a set of data can be modeled by an exponential function. Introduce pattern of geometric sequence and relate to exponential function. Identify the terms of a geometric sequence using a recursive process. 20
Algebra Methods (23 days) Enduring Understandings The student understands that expressions and equations can be mathematically manipulated to create equivalent forms of the expression or equation. The student understands that certain patterns exist in nature and can be represented mathematically. Vocabulary radical, radicand, index, exponent, base, rational number, sequence (arithmetic or geometric), term, term value, nth term, function form, recursive form, common difference, common ratio, literal equation New TEKS A.11A The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to simplify numerical radical expressions involving square roots. A.11B. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents. A.12C The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes form. A.12D. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to write a formula for the nth term of arithmetic and geometric sequences, given the value of several of their terms. A.12E. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to solve arithmetic and scientific formulas, and other literal equations, for a specified variable. Radical expressions can be rewritten in simplified form while maintaining its exact value. A rational exponent may be rewritten as a radical. The unique characteristics of arithmetic and geometric sequences. A sequence in which each new term is the sum of the previous term and a constant (common difference) is called an arithmetic sequence. An arithmetic sequence is a linear function in which the domain is the set of natural numbers and the range consists of the terms of the sequence. A sequence in which each new term is the product of the previous term and a constant (common ratio) is called a geometric sequence. A geometric sequence is an exponential function in which the domain is the set of natural numbers and the range consists of the terms of the sequence. Simplify radicals (Ex. 50 252 5 2 ). Simplify exponential expressions containing rational 3 3 6 6 3 3 9 exponents (Ex. 2 4x 4x 2x 8x ). Determine whether a sequence is arithmetic, geometric, or neither. Determine the common difference of an arithmetic sequence. Determine the common ratio of a geometric sequence. Find specific term values when give the term number and vice versa given the function rule for an arithmetic or geometric sequence. Write several terms of the sequence given the recursion formula for an arithmetic or geometric sequence (Ex Given a1 4 and an 1 2an, the first 5 terms of the sequence are 4, 8, 16, 32, 64). Write the formula for the nth term of an arithmetic or geometric sequence given several terms of the sequence. Solve literal equations for a specified variable (Ex. Solve 2 V r h for r). 21