ALGEBRA 1 H2 COURSE SYLLABUS Primary Course Materials: Book: Big Ideas Math Algebra 1 Publisher: Big Ideas Learning LCC Authors: Larson and Boswell Additional Resources: Algebra 1Student Journals, Algebra 1 Chapter Resource Books, Online Algebra 1 Resources (www.bigideasmath.com), TI-nSpire, and Graphing Calculator Applications Course Description: This course is designed for ninth grade students who continue to demonstrate the necessary ability, maturity, and motivation to be successful in a rapidly paced algebra program. All students will be actively engaged in Problem Solving, Reasoning, Connecting, and Communicating as they study the following topics: solving linear equations, solving linear inequalities, graphing linear functions, writing linear functions, solving systems of linear equations, exponential functions and sequences, polynomial equations and factoring, graphing quadratic functions, solving quadratic equations, radical functions and equations, and data analysis and displays. Students will be required to keep a notebook, read and interpret the algebra text and complete independent, partner, and group work. Emphasis will be place on investigating and solving real world problems that will include open-ended and open response questions to assist in preparing students for standardized exams and future mathematics courses. Since the course will advocate and encourage the proper use of technology, the purchase of a TI-nSpire CX CAS calculator is highly recommended. Essential Questions: 1. Where and how are algebraic operations and concepts used to solve everyday problems? 2. How does Algebra help us understand and relate to other objects as well as to other areas of mathematics? 3. How are the function families used to analyze real world applications in a variety of disciplines such as science, business, and economics? 4. How are the functions families used to model real world systems such as moving objects, revenue costs, populations and cost & profit? 5. How are transformational properties of functions families related to the algebraic representations of a function? 6. How can technology be used to deepen understanding of function families and algebraic structres? Course Objectives: This course addresses the following common goals for Chelmsford High School: 21 st Century Learning Expectations: Thinking and Communicating 1) Read information critically to develop understanding of concepts, topics and issues. 2) Write clearly, factually, persuasively and creatively in Standard English. 3) Speak clearly, factually, persuasively and creatively in Standard English. 4) Use computers and other technologies to obtain, organize and communicate information and to solve problems. 5) Conduct research to interpret issues or solve complex problems using a variety of data and information sources. Gain and Apply Knowledge in and across the Disciplines 9/17/2015-1 -
6) Gain and Apply Knowledge in: a) Literature and Language b) Mathematics c) Science and Technology d) Social Studies, History and Geography e) Visual and Performing Arts f) Health and Physical Education Work and Contribute 7) Demonstrate personal responsibility for planning one s future academic and career options 8) Participate in a school or community service activity 9) Develop informed opinions about current economic, environmental, political and social issues affecting Massachusetts, the United States and the world and understand how citizens can participate in the political and legal system to affect improvements in these areas. Learning Standards from the Massachusetts Curriculum Framework: A chart is attached identifying which of the standards from the Massachusetts Curriculum Frameworks will be assessed in this course. Additional Learning Objectives Beyond the Curriculum Framework: 21st Century Skills: o Instructional practices support the achievement of 21st C. Learning Expectations by (Check those that apply to the Unit): personalizing instruction engaging students in cross disciplinary learning engaging students as active and self-directed learners emphasizing inquiry, problem solving and higher order thinking applying knowledge and skills in authentic tasks engaging students in self-assessment and reflection integrating technology Learning Standards from the Massachusetts Curriculum Framework: A chart is attached identifying which of the standards from the Massachusetts Curriculum Frameworks will e assessed in this course. Content Outline: Suggested Time Line I. Chapter 1: Solving Linear Equations 9 Days 1.1 Solving Simple Equations 1.2 Solving Multi-Step Equations 1.3 Solving Equations with Variable on Both Sides 1.4 Solving Absolute Value Equations 1.5 Rewriting Equations and Formulas II. Chapter 2: Solving Linear Inequalities 9 Days 2.1 Writing and Graphing Inequalities 2.2 Solving Inequalities Using Addition and Subtraction 2.3 Solving Inequalities Using Multiplying or Division 2.4 Solving Multi-Step Inequalities 9/17/2015-2 -
2.5 Solving Compound Inequalities 2.6 Solving Absolute Value Inequalities III. Chapter 3: Graphing Linear Functions 10 Days 3.1 Functions 3.2 Linear Functions 3.3 Function Notation 3.4 Graphing Linear Equations in Standard Form 3.5 Graphing Linear Equation in Slope-Intercept Form 3.7 Graphing Absolute Value Functions IV. Chapter 4: Writing Linear Functions 12 Days 4.1 Writing Equations in Slope-Intercept Form 4.2 Writing Equations in Point-Slope Form 4.3 Writing Equations of Parallel and Perpendicular Lines 4.4 Scatter Plots and Lines of Fit 4.5 Analyzing Lines of Fit (tone down) 4.6 Arithmetic Sequences (light treatment) 4.7 Piecewise Functions (light treatment) V. Chapter 5: Solving Systems of Linear Equations 12 Days 5.1 Solving Systems of Linear Equations by Graphing 5.2 Solving Systems of Linear Equations by Substitution 5.3 Solving Systems of Linear Equations by Elimination 5.4 Solving Special Systems of Linear Equations 5.6 Graphing Linear Inequalities in Two Variables 5.7 Systems of Linear Inequalities VI. Chapter 6: Exponential Functions and Sequences 13 Days 6.1 Properties of Exponents 6.2 Radicals and Rational Exponents 6.3 Exponential Functions 6.4 Exponential Growth and Decay 6.6 Geometric Sequences (light) VII. Chapter 7: Polynomial Equations and Factoring 16 Days 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Solving Polynomial Equations in Factor Form 7.5 Factoring x 2 + bx + c 7.6 Factoring ax 2 + bx + c 7.7 Factoring Special Products 7.8 Factoring Polynomials Completely VIII. Chapter 8: Graphing Quadratic Functions 14 Days 8.1 Graphing f(x) = ax 2 8.2 Graphing f(x) = ax 2 + c 8.3 Graphing f(x) = ax 2 + bx + c 8.4 Graphing f(x) = a(x-h) 2 + k 8.5 Using Intercept Form (basic make connection) 9/17/2015-3 -
8.6 Comparing Linear, Exponential, and Quadratic Functions (light) IX. Chapter 9: Solving Quadratic Equations 14 Days 9.1 Properties of Radicals 9.2 Solving Quadratics by Graphing 9.3 Solving Quadratic Equations Using Square Roots 9.4 Solving Quadratic Equations by Completing the Square 9.5 Solving Quadratic Equations using the Quadratic Formula X. Chapter 11: Data Analysis and Displays 5 Days 11.1 Measures of Center and Variations (Eliminate SD) 11.2 Box-and-Whisker Plots/ Box Plots 11.5 Choosing a Data Display Major Evaluation Strategies: Name of Assessment Type of Assessment Common Goals Standards Assessed Test Performance Assessed Assessment Chapter Tests & Quizzes 2,3,4,5,6b,7 listed under course objectives for all assessments MCAS test Materials 4,5,6b,7 Activities 4,5,6b,7 Algebra Tiles Hiker Shoe Size and Height Bouncing Ball Maximize Area of Rectangle Tossing a basketball Calculator Regression M&M Paper Folding Paper Cutting Dice Roll Fast Talker Journals, Notebooks, Portfolios 1,2,6b,7 Common Assessments Equations and Inequalities (Who s in First?, In Stereo ) Linear (Rub a Dub Dub, Are we there yet? We ll Leave the light on!, African American Officials) Other Objectives Assessed Systems of Equations (Rent a Deal, Can you hear me now?) Quadratic Functions and Equations (Fore!, Cliffhanger, 9/17/2015-4 -
For the record, Fired in the Skies, Show me the money) Exponential Functions (Investment Decisions, Population Exploration ) Probability and Statistics (Bulls Eye, Plots of Fun) Homework/Class Participation Mid-Year Exam 80% Common Final Exam 80% Common 2,3,4,5,6b,7 2,3,4,5,6b,7 High School Content Standards Conceptual Category: Number and Quantity N-RN The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponent follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. N-Q N-CN A-SSE Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. The Complex Number System Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i 2 1, and every complex number has the form a + bi with a and b real. 2. Use the relation i 2 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Use complex numbers in polynomial identities and equations. 3. Solve quadratic equations with real coefficients that have complex solutions. Conceptual Category: Algebra Seeing Structure in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. (+) indicates standard beyond College and Career Ready. (+) indicates standard beyond College and Career Ready. 9/17/2015-5 -
a. Interpret parts of an expression, such as terms, factors, and coefficients. Write expressions in equivalent forms to solve problems. 2. 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. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-APR Arithmetic with Polynomials and Rational Expressions 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. Understand the relationship between zeros and factors of polynomials. Understand the relationship between zeros and factors of polynomials. 2. 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. Use polynomial identities to solve problems. 3. (+) Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. 1 Rewrite rational expressions. 4. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-CED Creating Equations Create equations that describe numbers or relationships. 1. 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. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. For example, represent inequalities describing nutritional and 1 The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. (+) indicates standard beyond College and Career Ready. 9/17/2015-6 -
cost constraints on combinations of different foods. 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. A-REI Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 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. Solve equations and inequalities in one variable. 2. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. MA.3.a. Solve linear equations and inequalities in one variable involving absolute value. 3. 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. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. MA.4.c. Demonstrate an understanding of the equivalence of factoring, completing the square, or using the quadratic formula to solve quadratic equations. Solve systems of equations. 4. 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. 5. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically. 6. 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). 7. 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. 8. Graph the solutions to a linear inequality in two 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. 9/17/2015-7 -
F-IF F-LE Conceptual Category: Functions Understand the concept of a function and use function notation. 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). 2. 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. 3. 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. 4. 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. Analyze functions using different representations. 5. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior (quadratic functions only).. e. Graph exponential and functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 6. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, and y = (1.2) t/10, and classify them as representing exponential growth or decay. MA.8.c. Translate among different representations of functions and relations: graphs, equations, point sets, and tables. Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric 9/17/2015-8 -
sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Interpret expressions for functions in terms of the situation they model. 4. Interpret the parameters in a linear or exponential function in terms of a context. G-GPE Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 5. 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). G-MG Modeling with Geometry Apply geometric concepts in modeling situations. MA.4. Use dimensional analysis for unit conversions to confirm that expressions and equations make sense. S-ID Conceptual Category: Statistics & Probability Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). 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. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Summarize, represent, and interpret data on two categorical and quantitative variables. 4. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Interpret linear models. 5. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 6. Compute (using technology) and interpret the correlation coefficient of a linear fit. S-CP Conditional Probability and the Rules of Probability 9/17/2015-9 -
Understand independence and conditional probability and use them to interpret data. 2 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 4. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model. 3 5. Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. 6. Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. 7. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. 8. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. 2 Link to data from simulations or experiments. (+) indicates standard beyond College and Career Ready. 3 Introductory only. 9/17/2015-10 -