# BUILT. for. Regents Pathways

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1 BUILT for NY

2 Algebra 1 Regents Pathway Think Through Math s Algebra 1 Regents pathway is designed to equip students with the skills and conceptual understandings of high school level mathematics necessary for success on the Algebra 1 Regents Exam. The pathway includes only lessons aligned to major content of the Algebra 1 course.

3 Algebra I Regents Pathway Arithmetic with Polynomials and Rational Expressions Creating Equations Adding and Subtracting Polynomials Multiplying Polynomials Writing and Solving Linear Equations in One Variable HSA-APR.A.1 HSA-APR.A.1 HSA-CED.A.1 HSA-CED.A.3 Perform arithmetic operations with polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Perform arithmetic operations with polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Reasoning with Equations and Inequalities Solving Linear Equations in One Variable as a Reasoning Process HSA-REI.A.1 HSA-REI.B.3 Understand solving equations as a process of reasoning and explain the reasoning. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 3

4 Algebra I Regents Pathway Creating HSA-CED.A.2 Equations Writing and Graphing Linear Equations in Two or More Variables Writing Linear Inequalities in One Variable HSA-CED.A.1 HSA-CED.A.3 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve equations and inequalities in one variable. HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Represent and solve equations and inequalities graphically. Reasoning with Equations and Inequalities Solving Linear Equations Graphically HSA-REI.D.10 HSA-REI.D.11 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Represent and solve equations and inequalities graphically. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 4

5 Algebra I Regents Pathway Reasoning with Equations and Inequalities Solving Exponential Equations Graphically HSA-REI.D.10 HSA-REI.D.11 Represent and solve equations and inequalities graphically. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Represent and solve equations and inequalities graphically. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Graphing Linear Inequalities and Systems of Linear Inequalities in Real- World Situations HSA-CED.A.3 HSA-REI.D.12 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Creating Equations Solving Literal Equations HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. 5

6 Algebra I Regents Pathway Interpreting Correlation HSS-ID.C.7 Interpret linear models. Categorical and Quantitative Data Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. HSS-ID.C.9 Interpret linear models. Distinguish between correlation and causation. Understand the concept of a function and use function notation. Interpreting Functions Sequences as Functions HSF-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. Creating Equations Modeling Exponential Relationships with Equations, Inequalities, and Graphs HSA-CED.A.1 HSA-CED.A.2 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Interpreting Functions Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Function Notation I HSF-IF.A.1 Understand the concept of a function and use function notation. HSF-IF.A.2 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 6

7 Algebra I Regents Pathway Interpreting Functions Function Notation II HSF-IF.A.1 Understand the concept of a function and use function notation. Interpreting Graphs of Linear and Exponential Functions in Context Sketching Graphs of Linear and Exponential Functions from a Context Rate of Change for Linear and Exponential Functions HSF-IF.A.2 HSF-IF.B.4 HSF-IF.B.4 HSF-IF.B.6 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret functions that arise in applications in terms of the context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 7

8 Algebra I Regents Pathway Seeing Structure HSA-SSE.A.1a Interpret the structure of expressions. in Expressions Interpreting the Structure of Linear and Exponential Expressions Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. Factoring Quadratic Expressions HSA-SSE.A.2 HSA-SSE.B.3a Interpret the structure of expressions. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. Creating Equations Interpreting the Structure of Quadratic Expressions and Expressions with Rational Exponents Modeling Quadratic Relationships with Equations, Inequalities, and Graphs HSA-SSE.A.1a HSA-CED.A.1 Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. 8

9 Algebra I Regents Pathway Interpreting Functions Sketching Graphs of Quadratic Functions in Context HSF-IF.B.4 HSF-IF.B.5 Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Interpret functions that arise in applications in terms of the context. HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Solve equations and inequalities in one variable. Reasoning with Equations and Inequalities Solving Quadratics - Completing the Square HSA-REI.B.4a Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form. Building Functions Reasoning with Equations and Inequalities Writing Quadratic Functions From Their Graphs Problem Solving with Quadratic Functions HSA-CED.A.2 HSA-REI.B.4b Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve equations and inequalities in one variable. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form. 9

10 Geometry Regents Pathway Think Through Math s Geometry Regents pathway is designed to equip students with the skills and conceptual understandings of high school level mathematics necessary for success on the Geometry Regents Exam. The pathway includes only lessons aligned to major content of the Geometry course.

11 Geometry Regents Pathway Congruence What Is Proof? HSG-CO.B.7 Perform arithmetic operations with polynomials. Proving Theorems About Lines and Angles Rigid Motion and Congruence HSG-CO.C.9 HSG-CO.C.9 HSG-CO.A.5 HSG-CO.B.6 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. Experiment with transformations in the plane. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 11

12 Geometry Regents Pathway Congruence Proving Theorems About Congruent Triangles HSG-CO.B.8 Perform arithmetic operations with polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. HSG-CO.C.10 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove geometric theorems. Similarity, Right Triangles, and Trigonometry Problem Solving with Congruent Triangles HSG-CO.C.10 HSG-SRT.B.5 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove geometric theorems. Congruence Proving Theorems About Relationships in Triangles Proving Theorems About Parallelograms HSG-CO.C.10 HSG-CO.C.11 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove geometric theorems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 12

13 Geometry Regents Pathway Modeling with Geometry Similarity, Right Triangles, and Trigonometry Modeling Objects with Geometric Figures Using Geometric Relationships to Solve Design Problems Properties of Dilations I HSG-MG.A.1 HSG-MG.A.3 HSG-SRT.A.1b Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Apply geometric concepts in modeling situations. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: Properties of Dilations II Transformations and Similarity Problem Solving with Transformations and Similarity HSG-SRT.A.1a HSG-SRT.A.2 HSG-SRT.A.3 HSG-SRT.B.5 b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand similarity in terms of similarity transformations. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Understand similarity in terms of similarity transformations. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 13

14 Geometry Regents Pathway Geometry Pythagorean Theorem - Hypotenuse HSG-SRT.C.8 Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Pythagorean Theorem - Legs HSG-SRT.C.8 Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Pythagorean Theorem - Mixed Problems HSG-SRT.C.8 Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Similarity, Right Triangles, and Trigonometry Proving Theorems About Similar Triangles Similarity and Trigonometric Ratios HSG-SRT.B.4 HSG-SRT.C.6 Prove theorems involving similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Define trigonometric ratios and solve problems involving right triangles. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Modeling with Geometry Problem Solving with Similarity and Trigonometric Ratios Sine and Cosine of Complementary Angles Rates with Area and Volume HSG-SRT.C.8 HSG-SRT.C.7 HSG-MG.A.2 Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Define trigonometric ratios and solve problems involving right triangles. Explain and use the relationship between the sine and cosine of complementary angles. Apply geometric concepts in modeling situations. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 14

15 Geometry Regents Pathway Expressing Geometric Properties with Equations Coordinates of Parallel and Perpendicular Lines HSG-GPE.B.4 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. HSG-GPE.B.5 Use coordinates to prove simple geometric theorems algebraically. Problem Solving with Coordinates of Parallel and Perpendicular Lines Dividing a Segment Proportionally Using Coordinates to Find Perimeters and Areas Problem Solving with the Equation of a Circle HSG-GPE.B.4 HSG-GPE.B.6 HSG -GPE.B.7 HSG-GPE.A.1 HSG-GPE.B.4 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Use coordinates to prove simple geometric theorems algebraically. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Translate between the geometric description and the equation for a conic section. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). 15

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