Algebra 8 Unit 2 Overview

Transcription

1 HMH Unit 2 Linear and Exponential Relationships Before: 7 th Grade A study of functions begins with a study of co-variation. In Grade 6, students use variables to represent two quantities and study how they change in relation to each other. Grade 6 students also create an equation to express the dependent variable in terms of the independent variable and then use graphs and tables to represent and analyze the relationship. In Grade 7, students recognize and represent proportional relationships using equations, tables, diagrams, words and graphs. They learn that two varying quantities y and x are in a proportional relationship if there is a constant k such that y = kx, without being asked to think of y = kx as an example of a linear function where b = 0. During: 8 th Grade It is in Grade 8 that the CCSS makes its first explicit mention of functions. Here students understand that a function is a rule that assigns to each input exactly one output. The graph of a function is not thought of as the set of solutions but as the set of ordered pairs consisting of an input and the corresponding output. In this study, student focus is on functions in one variable even though y is used instead of f(x) because functional notation is not introduced until high school. A Grade 8 study of functions might begin with proportional relationships. In 8. EE 5 students are asked to graph proportional relationships, interpret the unit rate as the slope of the graph, and compare two different proportional relationships represented in different ways. Students represent functions in tables, graphs, equations, or verbal descriptions. In Grade 8 students are asked to create examples and non-examples of linear functions, and to determine the rate of change and initial value of a linear function from a description of a relationship or from two ordered pairs that might be read from a table or a graph. Finally, in Grade 8, students describe qualitatively a linear function given its graph or sketch a graph given the qualitative description of the linear function. In 8 th Grade Algebra students study exponential functions and compare them with linear functions. After: 9 th Grade In high school, work with functions becomes more sophisticated and technical. Students interpret, build, and compare functions. They model with functions. In high school students understand the concept of a function and use function notation, interpret functions that arise in applications in terms of the context, and analyze them using different representations. High school students build a function which models a relationship between two quantities and build new functions from existing functions. They construct, compare, and contrast linear, quadratic, and exponential models. Students also study trigonometric functions.

2 UNIT 2: Linear and Exponential Relationships Quarterly Planner TIME: 9 weeks UNIT NARRATIVE: In this unit, students will have a variety of experiences working with expressions and creating equations. In this unit, students continue this work by using quantities to model and analyze situations, to interpret expressions, and by creating equations to describe situations. By the end of accelerated 7 th grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation and to justify the process used in solving a system of equations. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. Students explore systems of equations and inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them. In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Textbook Correlations: HMH Go Math Unit 2: Modules 6, 7, 8, 9, 10, and 11 ESSENTIAL QUESTIONS: 1. Why do simultaneous linear equations have different types of solutions? 2. What is a function? 3. What are efficient strategies for comparing functions? 4. What quantitative information can be extracted from a linear function? 5. What is the relationship between a linear equation and a linear graph? 6. What strategy is used to solve systems of two equations? Additional Resources Personal Math Trainer, Math on the Spot, Formative Assessment Lessons, and Real Player Activator videos ACADEMIC VOCABULARY: elimination, intersection, intersecting, parallel lines, coefficient, distributive property, like terms, substitution, solution, solve, system of linear equations, functions, y-value, x-value, vertical line test, input, output, rate of change, linear function, nonlinear function, linear relationship, rate of change, slope, initial value, y-intercept, domain, range, function notation, intercepts, increasing intervals, decreasing intervals, positive intervals, negative intervals, relative maximum, relative minimum, exponential function,

3 7. How and when are systems of two linear equations solved precisely and graphically? 8. How does a graphed solution of equations and inequalities in two variables indicate the set of all its solutions? 9. How are functions interpreted when used in applications? 10. What essential information is indicated when graphing linear and exponential functions? 11. How can multiple representations of functions help reveal and explain different properties of the function? 12. What arithmetic and geometric sequences can be used to model situations? 13. How do I determine the situations that can be modeled with linear and exponential functions? 14. How do I prove the equal factor growth of linear and exponential functions? 15. What are the strategies and methods used to construct and compare linear, and exponential models and solve problems? CLUSTER HEADING & STANDARDS: Create equations that describe numbers or relationships. A.CED.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. Define, evaluate, and compare functions. 8. F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8. F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Use functions to model relationships between quantities. 8. F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from exponential growth, exponential decay, constant function, arithmetic sequence, geometric sequence MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit. MP 1 Make sense of problems and persevere when solving them. MP 2 Reason quantitatively and abstractly. MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. MP 5 Use appropriate tools strategically. MP 6 Attend to precision. MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning

4 a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8. F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Represent and solve equations and inequalities graphically. A.REI.12 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. Solve systems of equations 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. Represent and solve equations and inequalities graphically Interpret functions that arise in applications in terms of the context F.IF.4 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. F.IF.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. Analyze functions using different representations 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. F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

5 F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Build a function that models a relationship between two quantities F.BF.1 Write a function that describes a relationship between two quantities. F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Build new functions from existing functions 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. F.BF.4 Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Construct and compare linear, quadratic, and exponential models and solve problems F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. F.LE.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

6 F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). F.LE.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. 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. S.ID.6a. 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. S.ID.6b. Informally assess the fit of a function by plotting and analyzing residuals. S.ID.6c. Fit a linear function for a scatter plot that suggests a linear association. Learning Outcomes: Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m A. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation. Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations. Students know expressions and can define functions, and equivalent expressions define the same function. They know that two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling. Students should be able to interpret the intercepts; intervals where the function is increasing, decreasing, positive, or negative. They should be able to compare and graph characteristics of a function represented in a variety of ways. The should know that characteristics include domain, range, vertex, axis of symmetry, zeros, intercepts, intervals of increase and decrease, and rates of change. End of the Unit Assessment: AC Driven

7 Visual aids Balance scales and Algebra Tiles Balancing equation manipulatives Real-world situations where students can work problems through using reverse order of operations. Real-world situations where students can work problems through linear input/output data tables. Revisit 7 th /8 th grade Common Core Standards prior to moving into Algebra. Start with equations and move into functions. Develop lessons with graph vocabulary DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Technology ELD Standard Spiral Functions applied in the real world. Graphic organizers Student presentation of self-made Highlighting : cloze activities Applications. SIOP strategies Advanced Algebra Tile Process Real-world visuals Start with equations and move into Group collaboration functions Number talks Number talks Small group instruction One on one peer support Smaller size quantities Start with equations and move into functions Balance scales and Algebra Tiles Balancing equation manipulatives Real-world situations where students can work problems through linear input/output data tables. Revisit 7 th /8 th grade Common Core Standards prior to moving into Algebra.