Algebra I, Unit 2


Linear and Exponential Relationships
  • Representing Linear and Exponential Functions
  • Modeling Data with Linear and Exponential Functions
  • Systems of Equations and Inequalities

Overview (Big Ideas):
Representing Linear and Exponential Functions
Sequences are functions whose domains are subsets of integers representing the term numbers. The sixth term of a sequence is commonly defined as f(6) or a6 . Sequences can be defined either explicitly or recursively. An explicit formula allows the student to find a sequence term by simply substituting the term-number into the formula and evaluating. A recursive formula relates each term of a sequence to the term before. In order to find a sequence term, the student must find all prior terms in chronological order. Students will write sequences in both forms and translate between them.
The focus in Unit 2 will be on the comparison of arithmetic and geometric sequences. Connections should made between arithmetic sequences and linear functions and between geometric sequences and exponential functions. It is up to the students to determine which type of model is appropriate for a given situation. In cases of relationships with common differences, arithmetic models are appropriate. In cases of relationships with common ratios or percent rates, geometric models are appropriate.
Formulas for sequences can be constructed from a description of the pattern, a graph, or two or more input-output pairs. These pairs are often given in a table of values. Students should write formulas for sequences to represent given real-world situations. They will use this understanding to write linear and exponential functions. Students will examine graphs, data tables, equations, and verbal models to represent the set of all solutions. Students will be asked to create linear or exponential functions from a given table, graph, or verbal model. Students will need to determine which representation best fits the scenario and describe the limits to each representation, including when the representation does not make sense.
Students will identify common differences or common ratios of linear and exponential functions. All representations should be analyzed for increasing or decreasing behavior, which will be described by slope, common differences, rate of growth, or common ratios. Students must calculate the average rate of change over a given interval for exponential functions. In addition, students should be able to describe the domain and range and any necessary restrictions for a given function based upon the context of the given scenario. Students will identify intercepts, end behavior, maxima and minima, focusing on how these key features help make decisions regarding real-life scenarios.
Students will transform linear and exponential functions by adding or subtracting a constant and by scalar multiplication. Students will use function notation to build a new function from an existing relationship. For example, by taking f(x) + 3, all values in the range corresponding to the given x-values of the function f are increased by 3. Students will explore vertical translations of linear and exponential functions. Vertical translations should be related to the y-intercepts of given functions. Students will be required to create an equation, graph, table, or verbal description given key properties including increasing or decreasing behavior, sign of the leading coefficient, and end behavior.

Modeling Data with Linear and Exponential Functions
Students will be able to create a scatterplot given two sets of quantitative data. They will describe the features of scatterplots and determine if a linear or exponential model of best fit would be appropriate. Using the correlation coefficient, paired with an analysis of residuals, students will justify their choice. Residuals refer to the difference between the y-coordinate of an actual data point and its theoretical output based on a linear regression. For linear data, residuals should not reveal a pattern when the independent variable is plotted on a coordinate plane against the residual values. If a pattern is revealed by the residual plot, students should recognize that a linear model is unsuitable. The correlation coefficient, or r, describes the strength and direction of a linear relationship. Perfect linear relationships will have a correlation coefficient of 1 or -1. Conversely, relationships that have little to no linear correlation will have correlation coefficients close to 0. The correlation coefficient should not be used to describe non-linear data. Students will distinguish the difference between correlation, or association, and causation. When using a linear model, students will interpret the slope and y-intercept of the equation in the context of the problem. In addition, students will be able make predictions using either linear or exponential regression models. Calculations and regressions should be performed using technology.

Systems of Equations and Inequalities
From their work in Unit 1, students should be aware that the graph of a line or curve is a representation of every solution of the graphed equation. This includes ordered pairs of all real numbers, not only rational numbers. Systems of equations will expose students to determining solutions that simultaneously satisfy two or more equations. The solution to a system of linear equations is the point of intersection of two or more lines. If the lines are parallel, there will be no solution to the system. If the system of linear equations produces the same line, there are infinite solutions. Students will recognize through real-world application tasks that the solution, or point of intersection, will be common to the two given situations. Determining the system of linear equations from the context of a problem and then interpreting the solution will be used to justify solutions within the context of a real-world problem.
Students will build on their prior knowledge of graphing linear equations in order to graph a system of equations. They will predict solutions by estimating points of intersection. Graphing calculators may also be used as tools to find the intersection point. In addition to systems of equations, students will graph systems of inequalities. Students need to apply appropriate scales and intervals to a given system of equations or inequalities when using graphical representations. An appropriate window will be chosen to view the system of equations/inequalities when using graphing technology. Students will make connections between intersecting, parallel, and perpendicular lines and the meaning of the solutions they produce. Students will connect the real-world solution to the intersection point of the graph.
In addition to graphing, students will solve two-variable systems algebraically using the substitution and elimination methods. Regardless of the method used to solve a system of equations, the same solution will be produced. Students should be able to justify their method of solving and their solutions. Having multiple methods to solve systems allows students to determine the solution through a variety of pathways. Through practice and application, students will determine the most efficient or practical method for solving a system of equations.
Given two pairs of points, students will write linear equations and determine whether the lines will intersect. Students will recognize that a system has no solution by identifying the slope of the lines algebraically or by viewing a graphical representation of the system of equations. Students will determine that a system shares all solutions using graphic and algebraic methods to solve.



Enduring Understandings:
1. Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For example, sequences are functions.
2. For functions that map real numbers to real numbers, certain patterns of covariation, or patterns in how two variables change together, indicate membership in a particular family of functions and determine the type of formula that the function has.
3. Arithmetic sequences can be thought of as linear functions whose domains are the positive integers.
4. Geometric sequences can be thought of as exponential functions whose domains are the positive integers.
5. Functions can be represented in various ways, including through algebraic means, graphs, word descriptions, and tables.
6. Some representations of a function may be more useful than others, depending on the context.
7. Links between algebraic and graphical representations of functions are especially important in studying relationships and change.
8. A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the co-variation between two variables.
9. A function’s rate of change is one of the main characteristics that determines what kinds of real-world phenomena the function can model.
10. Members of a family of functions share the same type of rate of change.11. Linear functions are characterized by a constant rate of change. Reasoning about the similarity of “slope triangles” allows deducing that linear functions have a constant rate of change and a formula of the type constant m and b.
12. Exponential functions are characterized by a rate of change that is proportional to the value of the function. It is a property of exponential functions that whenever the input is increased by one unit, the output is multiplied by a constant factor. Exponential functions connect multiplication to addition through the equation.
13. For functions that map the real numbers, composing a function with “shifting” or “scaling” functions changes the formula and graph of the function in readily predictable ways.
14. Bi-variate data can be represented on scatter plots, which should be examined for form, direction, shape, and influential points.
15. Mathematical models can be used to capture key elements of the relationship between bivariate data and to make predictions.
16. A combination of the correlation coefficient and an analysis of the residuals can be used to assess the fit of a mathematical model to real-world data.

Source:
Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing Essential Understanding of Functions Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.
http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf



Essential Questions:
Representing Linear and Exponential Functions
  • How can the given explicit or recursive formula be used to find the value of a specific term?
  • What are the benefits of explicit and recursive formulas? Which would be more useful in a given situation?
  • Is the given relationship an example of an arithmetic or geometric sequence? How can this be justified?
  • How do arithmetic and geometric sequences differ?
  • How can a given set of data be modeled by a recursive or explicit formula?
  • What type of sequence can be modeled by a linear equation? An exponential equation model?
  • In what situations are linear functions practical to use? In what situations are exponential functions practical to use?
  • When is it appropriate to use inequalities?
  • What do the features of the graph reveal about the problem situation? Does this help in the decision-making process? How do these features affect the symbolic and numeric representations of the problem?
  • What is the effect of adding or subtracting a constant to a function? How does this affect the y-intercept?
  • What is the effect of multiplying a function by a scalar? What happens if the scalar is a negative number?

Modeling Data with Linear and Exponential Functions
  • How can we model two-variable data and use models to make predictions?
  • When are linear models appropriate for two-variable data? When are exponential models appropriate?
  • How can the correlation coefficient and a plot of the residuals be used to justify the strength of a linear model?
  • What is the difference between correlation and causation?

Systems of Equations and Inequalities
  • What types of real-world problems would systems of equations be used to solve?
  • What is the meaning of a solution to a system of linear equations in the context of a problem? How is this solution represented algebraically and graphically?
  • How are solutions to systems of linear equations different from solutions to systems of linear inequalities?
  • What is the benefit of having multiple methods to use to solve systems? When is it appropriate to use each method?
  • How can multiplication be used to eliminate a variable in a system of equations?
  • Why does changing the equations in a system by a multiple not change the solution?
  • Explain the meaning of having no solution or infinite solutions for a system of linear equations in the context of a real-world problem/scenario. What does these solutions mean about the graphs of the system?



Common Misconceptions:
Representing Linear and Exponential Functions
  • Arithmetic and geometric sequences do not follow the same formulas.
  • o Students may attempt to substitute a term number in order to evaluate a recursive formula. Conversely, they may attempt to substitute a prior term in order to evaluate an explicit formula.
  • o Subscripts are used to indicate the term-number of a given value. The notation an-1 indicates the term before the nth term.
  • o Students often use an incorrect initial value to create an equation, especially in the context of a problem situation. Often, they mistake the initial x-value for 1 instead of 0.
  • There is sometimes confusion regarding increasing and decreasing behavior, or positive versus negative slope. When given a table or graph, students sometimes confuse the order in which to add/subtract for constant differences, or multiply/divide for constant ratios. Encourage them to identify the pattern, visualize the graph, and make sense of the situation.
  • Students have difficulty recognizing increasing versus decreasing behavior graphically when all four quadrants are given.
  • Student struggle with shading and solid versus dotted lines when dealing with inequalities, often forgetting to think of the problem in terms of its context.

Modeling Data with Linear and Exponential Functions
  • Students may assume that a correlation coefficient close to 1 or -1 is the only information necessary to determine the fit of a linear model.
  • Students often follow steps to calculate residuals without a conceptual understanding of the meaning of the values.
  • Students may have difficulty applying the differences between correlation and causation to real-life situations. Many students use the term “cause” too often.
  • Students struggle with contextual interpretation of slope and y-intercept in terms of real-world data.
  • It may be difficult for students to understand that a regression model is only a predictor and may not accurately represent what they observe in their own experience.

Systems of Equations and Inequalities
  • Students may not understand that, while each method is a possible strategy, all methods should reveal the same solution set.
  • Linear systems of equations can only have multiple shared solutions if they have infinite solutions.
  • Changing equations by a multiple will not change the solution.
  • Algebraically, arriving at 0 = 0 indicates infinite solutions not that (0,0) is the solution. Additionally, arriving at -5 = 0 indicates no solution, not a solution of (-5, 0).
  • Systems with no solution will result in parallel lines and infinite solutions will result in the same line.
  • When using substitution, either variable can be substituted but must be substituted into the alternate equation.
  • A solution that satisfies one equation may not satisfy the other in a given system.
  • When graphing systems of inequalities, the solution set lies only where the solution sets of each inequality overlap.
  • Students may forget to consider whether the points on the boundary lines are included as solutions in a system of inequalities.