Valley Central School District 944 State Route 17K Montgomery, NY Telephone Number: (845) ext Fax Number: (845)

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1 Valley Central School District 944 State Route 17K Montgomery, NY Telephone Number: (845) ext Fax Number: (845) Algebra LR Approved by the Board of Education On November 13, 2017

2 Unit Title: Functions Unit 1 (39 Days) Standards: Focus Standards: F-IF.A.1 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). F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.B.5 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. F-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. F-IF.7. C Foundational Standards: 8.F.B.4 8.F.B.5 Progressions in Future Courses: This is required for both Geometry, Algebra II and Financial Algebra. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

3 Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior Key Concepts: Identifying functions from various representations and representing functions in various ways. Transformation of functions. Functions with set domains. Key features of functions. Rate of change. Topic A: Graphing linear functions and determining if something is a function (5 days) Students will be able to determine if a set of numbers is a function and be able to graph linear functions. intro to functions represent, name and evaluate functions of graphing lines quiz Topic B: Piecewise functions 5 days Students will be able to graph piecewise functions. Piecewise day 1 Piecewise day 2 step functions quiz Topic C: Quadratic Functions 6 days

4 Students will be able to identify, graph and transform quadratic functions. Graphing quadratic functions day 1 Symmetry in graphs of quad functions 17a practicing finding key features Transformations of quad functions quiz Topic D: Absolute Value Functions 5 days Students will be able to identify, graph and transform Absolute Value functions. Graphing Absolute Value Functions Finding the key features of absolute value functions Transformations of absolute value functions quiz Topic E: Exponential Functions 5 days Students will be able to identify and graph graphing exponential functions transformations of exponential functions comparing linear and exponential functions quiz Topic F: Other functions 5 days Students will be able to graph and transform square root, cube root and cubic functions. graph and transform square root functions graph and transform cubic functions graph and transform cube root functions quiz

5 Topic G: Compare and contrast all functions 8 days Students will be able to identify the various functions and their parts given graphs verbal situations and equations. the graphs of functions graphs of y=f(x) Piecewise day 4 Mixed practice of Piecewise functions modeling a context from a graph unit test Evidence of Learning/Assessments: Daily homework assignments Quizzes at the end of each topic Unit Test at the end of the unit Collected Do Now s and Exit Tickets

6 Unit Title: Representing functions in real world applications and patterns. (20 days) Standards: Focus Standards: F-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. Foundational Standards: N-RN.B.3a F-IF.A.2 Progressions in Future Courses: This standard is shared with Algebra II. Standards for Mathematical Practice: Which of the following are included in this unit? CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

7 Key Concepts: How to recognize that a sequence is a function whose domain is a subset of the integers. How to find and use the explicit rule of geometric and arithmetic sequences. Topic A: Introducing sequences 6 days The students will be able to recognize a pattern within a set of given numbers. Introducing sequences worksheet Arithmetic explicit worksheet Geometric explicit worksheet Applications of sequences worksheet quiz Topic B: recognizing patterns given a verbal situations 6 days The students will be able to create and use a model given a verbal representation. compound interest the power of exponential growth exponential growth and decay exponential growth and decay day 2 quiz Topic C: recognizing patterns given a table (8 days) The students will be able to recognize a functions or pattern based on a given table. Comparing linear and exponential functions from a table analyzing a data set analyzing a data set day 2 analyzing a verbal description day 1 modeling from a sequence unit test

8 . Evidence of Learning/Assessments: Daily homework assignments Collected Do Now s and Exit Tickets Quizzes/Tests

9 Unit Title: Quadratics (54 days) Standards: Focus Standards: N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients.2 b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, Foundational Standards: 8.NS.A.1 8.EE.A.1 A-REI.D.10 F-IF.A.2 Progressions in Future Courses: This topic is found in Geometry and Algebra II. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

10 thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2+y2).3 A-SSE.B.3 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.4 b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic

11 functions, and simple rational and exponential functions. 6 A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI.D.119 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. Key Concepts: Factoring

12 Radicals Solving Quadratics by multiple methods Quadratic formula Topic A: Factoring Review (8 days) Factor quadratics using all types of quadratics GCF D2PS Trinomials (+) Trinomials (-) Factor completely Factor completely quiz Topic B: Solving Quadratics (8 days) Students will be able to solve and find the roots for a quadratic Solving quadratics set equal to zero/solving cubic(linear times quad) Solving quadratics not set equal to zero Solving quadratics/cubic mixed review Easy verbal expressions find the numbers Area verbal expressions Consecutive integers expressions Review quiz Topic C: Solving quad/linear (7 days) Students will be able to solve equations using quadratic and linear functions.

13 Quad/linear graphically day 1 (set equal to zero) Quad linear graphically day 2 (have to set equal to zero) Quad linear algebraically day 1 (set equal to y already) Quad linear algebraically day 2 (have to set equal to y) Quad linear algebraically day 3 (substitution) Review Quiz Topic D: Adding/subtracting Radicals (6 days) Students will be able to add/subtract radicals and simplify. Intro to Radicals Simplify Radicals Simplifying mixed review Adding and Subtracting Radicals Mixed practice of Radicals Review Quiz Topic E: Multiplying, Dividing and Rationalizing Radicals (6 days) Students will be able to multiply, divide and rationalize radicals. Multiplying Radicals Dividing Radicals Rationalizing Radicals Mixed review Review of all radicals Test on all radicals

14 Topic F: completing the square (5 days) Students will be able to solve quadratics using completing the square. Completing the square day 1 Completing the square day 2 Completing the square day 3 Review quiz Topic G: Quadratic Formula (5 days) Solve quadratics using the quadratic formula. Quadratic Formula day 1 Quadratic formula day 2 Quadratic formula day 3 Review Quiz Topic H: Finding the key features of a quadratic (9 days) Students will be able to find the important key features of a quadratic Review of key features graphically Key features algebraically day 1 Key Features algebraically day 2 17 a applications of functions in standard form day 1 Applications of functions in standard form day 2 Vertex Form Day 1

15 Vertex Form Day 2 Review Quiz Evidence of Learning/Assessments: Daily homework assignments Collected Do Now s and Exit Tickets Quizzes/Tests

16 Unit 4: Statistics 15 days Standards: Focus Standards: S.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). S.ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). S.ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S.ID.B.6 Represent bivariate data on a scatter plot, and describe how the variables values are related. S.ID.B.6a Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data. S.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Foundational Standards: 6.SP.A.1 6.SP.A.2 6.SP.A.3 Progressions in Future Courses: Statistics is an important progression for students entering into Algebra II and/or statistics and probability. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

17 S.ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. S.ID.C.9 Distinguish between correlation and causation. Key Concepts: The three measure of central tendency. frequency and cumulative frequency histograms. qualitative and quantitative data. univariate and bivariate data. Box Plot Scatter plots Dot plots Standard Deviation Relative Frequency Residuals Correlation coefficient Topic A: Statistics (6 days) Students will be able to find the measure of central tendencies, create a box plot and histogram and understand outliers. Measures of central tendency, Standard deviation and dot plots Histograms Box plots Box plots with outliers quiz Topic B: Categorical data and two way frequency tables. (9 days)

18 The students will be able to define categorical data, complete frequency tables, complete scatter plots, find their correlations and find line of best fit. Categorical data and two way frequency tables relative frequency tables and conditional relative frequency tables Setting up verbal frequency tables scatter plots and their correlations coefficients Line of best fit exponential regression Topic exam Evidence of Learning/Assessments: Daily Homework Assignments Collected Do Now s and Exit Tickets Quizzes Topic Exam

19 Unit 5: Basic Algebra Skills 22 days Standards: Focus Standards: 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. A.APR.1 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. A.APR.1 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. A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Foundational Standards: 3.OA.8 6.EE.A.3 6.EE.A.4 6.EE.B.5 6.EE.B.6 6.EE.B.7 7.EE.A.1 6.EE.A.3 6.EE.A.4 6.EE.B.5 6.EE.B.6 6.EE.B.8 Progressions in Future Courses: It is important to understand input and output for all future coursework. Rational and irrational numbers are also in the Algebra II curriculum. Understanding the properties of polynomials is found in geometry and Algebra II. Progressions in Future Courses: The students will have to solve equations in Geometry and Algebra II. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

20 A.REI.1 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. A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. A.REI.10 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). A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.REI.D.12 Graph the solutions to a linear inequality in two

21 variables as a half- plane (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. A.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Key Concepts: Properties polynomials solving equations graphing linear functions graphing inequalities solving linear functions algebraically dimensional analysis Translating verbal inequalities and equations. Topic A: Properties (1 day) Students will be able to understand and define the algebraic properties. Defining properties

22 Topic B: Polynomials (2 days) Students will be able to perform the four basic operations with polynomials. Adding and subtracting polynomials multiplying and dividing polynomials Topic C: Equations (7 days) Students will be able to solve for a specific variable using the four basic operations. Solving Equations Solving proportions Dimensional analysis day 1 Dimensional analysis day 2 Literal equations Review Quiz Topic D: Equations of a line (2 days) Students will be able to write equations of lines with different variations of information given. Write the equation of a line given a slope and y-intercept, a graph and slope and a point. Find the equation given two points Topic E: Systems of equations (4 days) Students will be able to graph lines and systems of equations and solve equations algebraically. Graphing lines review Graphing systems Solving systems algebraically (elimination and substitution) Solving systems algebraically through verbal problems. Topic F: Graphing inequalities (6 days) Students will be able to graph inequalities, solve inequalities graphically and algebraically.

23 Graphing inequalities on a number line Graphing inequalities systems Verbal inequalities day 1 Verbal inequalities day 2 Review Quiz Evidence of Learning/Assessments: Daily Homework Assignments Collected Do Now s and Exit Tickets Quizzes

24 January - Midterm/Regents Review 10 days June - Final/Regents Review 10 days