Algebra 1, Quarter 2, Unit 2.1 Relations and Functions Overview Number of instructional days: 10 (2 assessments) (1 day = 45 60 minutes) Content to be learned Demonstrate conceptual understanding of linear functions and relations. Demonstrate the specific characteristics of functions from a graph including domain, range, increasing, decreasing, maximum, minimum, and intercepts. Use the concepts of constant and variable rate of change. Determine if a relationship is a function based on a graph, table, or set of ordered pairs. Evaluate functions given domain values. Essential questions What is the difference between a function and a relation? What real-world applications are represented by functions? Given a graph, how do you identify key features such as domain, range, intercepts, etc.? How do you determine if a relationship is linear or nonlinear? Mathematical practices to be integrated Model with mathematics. Use functions to model problem situations. Analyze the relationships of specific characteristics of functions. Look for and make use of structure. Demonstrate understanding of functions by using the structure of functions. Use the structure of functions to determine connections among graphs, tables, and equations. In what kind of real-world situations would the domain and range of a linear function be restricted? How do you determine if a relationship is a function based on a graph, table or set of ordered pairs? How do you evaluate functions for given domain values? 13
Algebra 1, Quarter 2, Unit 2.1 Relations and Functions (10 days) Written Curriculum Common Core State Standards for Mathematical Content Interpreting Functions F-IF Understand the concept of a function and use function notation [Learn as general principle; focus on linear and exponential and on arithmetic and geometric sequences] F-IF.1 F-IF.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). 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 [Linear, exponential, and quadratic] F-IF.4 F-IF.5 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. 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. Common Core Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 14
Algebra 1, Quarter 2, Unit 2.1 Relations and Functions (10 days) 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning Students have been working with linear relationships since fourth grade. They have identified, described, and compared situations that represent constant rates of change. In grade 8, students understood that a function is a rule that assigns to each input exactly one output. They compared properties of two functions, such as the rate of change of each function when the two functions were represented in different ways. By analyzing a graph, students described where a function was increasing or decreasing and if the function was linear or nonlinear. In middle school, students were introduced to problem-solving situations involving slope and constant vs. varying rates of change. In addition, students have had to distinguish between linear and nonlinear relationships. Current Learning This unit is a reinforcement of understanding linear functions and relations. It is an introduction to specific characteristics such as domain, range, intercepts, and increasing and decreasing intervals. Students work between and among different representations of functions and relations. Students determine if a relation is a function and determine the domain and range of a function. They use function notation to evaluate functions for inputs in the domain. They interpret statements that use function notation in terms of a context. Future Learning In grade 10, the average rate of change (slope) will be introduced, as well as applications of slope. This topic will also be revisited in grades 11, 12, and advanced math, where students will represent and analyze functions in several ways, analyze characteristics of functions (exponential, logarithmic, and trigonometric), and apply knowledge of functions to interpret situations. 15
Algebra 1, Quarter 2, Unit 2.1 Relations and Functions (10 days) Additional Findings A Research Companion to Principles and Standards for School Mathematics indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities (pp. 250 261). Principles and Standards for School Mathematics notes that high school students should be able to interpret functions in a variety of formats. Students should solve problems in which they use tables, graphs, words and symbolic expressions to represent and examine functions and patterns of change. Students should learn algebra both as a set of concepts and competencies tied to the representation of quantitative relationships and as a style of mathematical thinking for formalizing patterns, functions, and generalizations (p. 287). By the completion of algebra 1, high school students should have substantial experience in exploring the properties of different classes of functions. 16
Algebra 1, Quarter 2, Unit 2.2 Linear Functions Overview Number of instructional days: 15 (2 assessments) (1 day = 45 60 minutes) Content to be learned Calculate and interpret the average rate of change from a table or graph (slope formula). Prove that the slope is constant over any interval of the linear function. Create a graph based on an equation, table, or verbal description. Graph a linear equation and identify the x- and y- intercepts. Construct a linear function (slope-intercept form, standard form, and point-slope form) based on a graph, verbal description, or 2 input/output pairs. Describe the effect on the graph when the slope and/or y-intercept change. Understand the relationship between the slopes of parallel and perpendicular lines. Find the equation of parallel and perpendicular lines. Use functions to model problem situations including data. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative) and find the value of k given the graphs. Essential questions How do you find the slope of a line from a graph or table? How can it be shown that the slope of a line is constant over different intervals of the line? Mathematical practices to be integrated Model with mathematics. Use functions to model problem situations. Analyze the relationships of specific characteristics of functions. Look for and make use of structure. Demonstrate understanding of functions by using the structure of functions. Use the structure of functions to determine connections among graphs, tables, and equations. Why do different functions create unique graphs? How does changing the slope and y- intercept affect the graph of a line? 17
Algebra 1, Quarter 2, Unit 2.2 Linear Functions (15 days) How do you create a graph based on an equation, table, or verbal description? What information is needed to write the equation of a line in different forms? How can functions model problem situations including data? How can the slopes be used to determine if two lines are parallel or perpendicular? How do you find the equation of a parallel or perpendicular line through a given point? Written Curriculum Common Core State Standards for Mathematical Content Interpreting Functions F-IF.6 F-IF 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 [Linear, exponential, quadratic, absolute value, step, and piecewise-defined] F-IF.7 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. Creating Equations A-CED Create equations that describe numbers or relationships [Linear, quadratic, and exponential (integer inputs only); for A.CED.3 linear only] A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.. Building Functions F-BF Build a function that models a relationship between two quantities [For F.BF.1, 2, linear, exponential, and quadratic] F-BF.1 Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Build new functions from existing functions [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only] F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 18
Algebra 1, Quarter 2, Unit 2.2 Linear Functions (15 days) Reasoning with Equations and Inequalities A-REI Represent and solve equations and inequalities graphically [Linear and exponential; learn as general principle] 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). Linear, Quadratic, and Exponential Models F-LE Construct and compare linear, quadratic, and exponential models and solve problems F-LE.1 F-LE.2 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. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret expressions for functions in terms of the situation they model [Linear and exponential of form f(x) = b x + k] F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. Expressing Geometric Properties with Equations G-GPE Use coordinates to prove simple geometric theorems algebraically G-GPE.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). Quantities N-Q Reason quantitatively and use units to solve problems. N-Q.2 Define appropriate quantities for the purpose of descriptive modeling. 19
Algebra 1, Quarter 2, Unit 2.2 Linear Functions (15 days) Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on two categorical and quantitative variables S-ID-6 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 S-ID.7 S-ID.8 S-ID.9 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Compute (using technology) and interpret the correlation coefficient of a linear fit. Distinguish between correlation and causation. Common Core Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 20
Algebra 1, Quarter 2, Unit 2.2 Linear Functions (15 days) 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning Students have been working with linear relationships since fourth grade. They have identified, described, and compared situations that represent constant rates of change. In grade 6, students used variables to represent two quantities and identified independent and dependent relationships in graphs, tables, and equations. In grade 7, students analyzed proportional relationships and identified constants of proportionality in tables, graphs, equations, diagrams, and verbal descriptions. Students explained what a point on the graph meant in terms of a situation. In grade 8, students understood that a function is a rule that assigns to each input exactly one output. They compared properties of two functions, such as the rate of change of each function when the two functions were represented in different ways. By analyzing a graph, students described where a function was increasing or decreasing and if the function was linear or nonlinear. Also in grade 8, students identified equations of the form y = mx + b as linear functions whose graphs are a straight line, and they identified functions that are not linear (e.g., A = s2). Students defined a function (not using function notation) and created function tables to generate ordered pairs as a means of graphing a function. Students determined rate of change and represented it in multiple ways. Students graphed proportional relationships and interpreted the unit rate as the slope of the graph. They used similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line and derived the equation y = mx + b. In middle school, students were introduced to problem-solving situations involving slope and constant vs. varying rates of change. In addition, students have had to distinguish between linear and nonlinear relationships. Current Learning Students study characteristics of linear functions and use them in solving contextual problems. They calculate the slope of linear functions and prove that the slope is constant over any interval. Students graph linear functions from an equation, table, or verbal description, and they identify the x- and y- intercepts. They identify transformations of linear functions. Students construct linear functions using various forms and representations. They explore the effects of changing the slope and/or y-intercept of a linear function on the graph of the function and they explore slope relationships and equations of parallel and perpendicular lines. Students model problem situations, including data, with linear functions. 21
Algebra 1, Quarter 2, Unit 2.2 Linear Functions (15 days) Future Learning In grade 10, the average rate of change (slope) will be introduced, as well as applications of slope. This topic will also be revisited in grades 11, 12, and advanced math, where students will represent and analyze functions in several ways, analyze characteristics of functions (exponential, logarithmic, and trigonometric), and apply knowledge of functions to interpret situations. Additional Findings In middle grades, students should work more frequently with algebraic symbols than in lower grades. It is essential that they become comfortable in relating symbolic expressions containing variables to verbal, tabular, and graphical representations of numerical and quantitative relations (Principles and Standards for School Mathematics, p. 223). According to the PARCC Model Content Frameworks for Mathematics, Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables). A Research Companion to Principles and Standards for School Mathematics indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities (pp. 250 261). 22
Algebra 1, Quarter 2, Unit 2.3 Solving Systems of Linear Equations (15 days) Overview Number of instructional days: 15 (2 assessments) (1 day = 45 60 minutes) Content to be learned Solve systems of linear equations by graphing, substitution, and linear combinations. Show that a system of equations could have one solution, no solution, or infinite solutions. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Use systems of equations to plan a solution pathway for a problem situation. Monitor and evaluate progress in solving a problem and change course if necessary. Use appropriate tools strategically. Determine appropriate tools to use for solving systems of linear equations. Detect possible errors by strategically using estimation and other mathematical knowledge. Essential questions What is the connection between the graph and the solution of a system of equation? How can you determine the number of solutions of the system from the equation or graph? Why is the graphical solution to a system of linear equations the intersection of the lines? 21
Algebra 1, Quarter 2, Unit 2.3 Solving Systems of Linear Equations (15 days) Written Curriculum Common Core State Standards for Mathematical Content Reasoning with Equations and Inequalities A-REI Solve systems of equations A-REI.5 A-REI.6 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. 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 [Linear and exponential; learn as general principle] A-REI.11 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. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 22
5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning In grade 8, students used systems of linear equations to represent, analyze, and solve a variety of problems. They understood that a solution to a system of two linear equations in two variables corresponds to a point of intersection of a graph. They also solved simple systems of two linear equations in two variables algebraically and by estimating solutions by graphing the equations. In the previous unit of this course, students graphed linear equations and inequalities in two variables. They graphed a single linear inequality in two variables in the coordinate plane by graphing its boundary line and then shading the half-plane. Current Learning Students solve systems of linear equations by graphing, substitution, and linear combinations. Translating problem situations into equations is reinforced. Students provide graphical interpretations of solution(s) in problem-solving situations and solve problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. Future Learning Students will use methods of solving systems of equations and inequalities in algebra 2 when studying linear programming. Solving systems is a skill that students will continue to use in subsequent course work as they solve systems of nonlinear equations and inequalities. 23
Algebra 1, Quarter 2, Unit 2.3 Solving Systems of Linear Equations (15 days) Additional Research Findings Principles and Standards for School Mathematics notes that in high school, students should build on their prior knowledge, learning more varied and more sophisticated problem-solving techniques. In addition, improving fluency with algebraic symbolism helps students represent and solve problems in many areas of the curriculum. Students should be able to operate fluently on algebraic expressions, combining them and re-expressing them in alternative forms (p. 288). According to A Research Companion to Principles and Standards for School Mathematics, Stasis and change presents a conceptually rich theme across the grades K 12 curriculum. It has the potential to tie together patterns, functions, and algebra (pp. 136 149). 24