Coordinate Algebra: Unit 3 – Linear and Exponential Functions
|Coordinate Algebra: Unit 3 – Linear and Exponential Functions |
|3a (4 weeks) & 3b (3 weeks) |
|(7 weeks) |
|Unit Overview: 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. |
| |
|This unit lays the foundation for the entire course. The concept of a function is threaded throughout each unit in Coordinate Algebra and acts as a bridge to |
|future courses. Students will develop a critical understanding of the concept of a function by examining linear functions and comparing and contrasting them with |
|exponential functions. Note that exponential functions are restricted to those of the form: f(x) =bx + k, where b > 1, k is an integer and x is any real number. |
|Content Standards: |
|Represent and solve equations and inequalities graphically |
|MCC9-12.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). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.) |
|MCC9-12.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.★ |
| |
|Understand the concept of a function and use function notation |
|MCC9-12.F.IF.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). (Draw examples from linear and exponential functions.) |
|MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. |
|(Draw examples from linear and exponential functions.) |
|MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which |
|requires students to write arithmetic and geometric sequences.) |
| |
|Interpret functions that arise in applications in terms of the context |
|MCC9-12.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.★ (Focus on linear and exponential functions.) |
|MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★ (Focus on linear and exponential|
|functions.) |
|MCC9-12.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.★ (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.) |
| |
|Analyze functions using different representations |
|MCC9-12.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.★|
|(Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.) |
|MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★ |
|MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and |
|amplitude.★ |
|MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal |
|descriptions). (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.) |
| |
|Build a function that models a relationship between two quantities |
|MCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★ (Limit to linear and exponential functions.) |
|MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.) |
|MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Limit to linear and exponential functions.) |
|MCC9-12.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 |
|MCC9-12.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. (Focus on vertical translations of graphs of linear and exponential functions. Relate the |
|vertical translation of a linear function to its y-intercept.) |
| |
|Construct and compare linear, quadratic, and exponential models and solve problems |
|MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★ |
|MCC9-12.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.★ |
|MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.★ |
|MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.★ |
|MCC9-12.F.LE.2 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).★ |
|MCC9-12.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.★ |
| |
|Interpret expressions for functions in terms of the situation they model |
|MCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★ (Limit exponential functions to those of the form f(x) = bx + |
|k.) |
| |
|Standards for Mathematical Practice: |
|4 Model with mathematics. |
|8 Look for and express regularity in repeated reasoning. |
|Coordinate Algebra: Unit 3a – Linear and Exponential Functions |
|(4 weeks) |
|Content Standards: |
|Represent and solve equations and inequalities graphically |
|MCC9-12.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). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.) |
|MCC9-12.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.★ |
| |
|Understand the concept of a function and use function notation |
|MCC9-12.F.IF.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). (Draw examples from linear and exponential functions.) |
|MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. |
|(Draw examples from linear and exponential functions.) |
|MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which |
|requires students to write arithmetic and geometric sequences.) |
| |
|Interpret functions that arise in applications in terms of the context |
|MCC9-12.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.★ (Focus on linear and exponential functions.) |
|MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★ (Focus on linear and exponential|
|functions.) |
|MCC9-12.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.★ (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.) |
| |
|Analyze functions using different representations |
|MCC9-12.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.★|
|(Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.) |
|MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★ |
|MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and |
|amplitude.★ |
|MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal |
|descriptions). (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.) |
| |
|Standards for Mathematical Practice: |
|4 Model with mathematics. |
|8 Look for and express regularity in repeated reasoning. |
| |
|Standards for Mathematical Practice (4, 8) |
| |
|EQ: How do mathematically proficient students use mathematical models to solve problems? (MP4) How can recognizing repetition or regularity help solve problems |
|more efficiently? (MP8) |
| |
|Learning Targets: |
|I can … |
|apply the mathematics I know to solve problems arising in everyday life, society, and the workplace. (MP4) |
|write an equation to describe a situation. (MP4) |
|apply what I know to make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. (MP4) |
|identify important quantities in a practical situation. (MP4) |
|map quantity relationships using such tools as diagrams, tables, graphs, and formulas. (MP4) |
|analyze relationships mathematically to draw conclusions. (MP4) |
|interpret my mathematical results in the context of the situation. (MP4) |
|reflect on whether my results make sense, possibly improving the model if it has not served its purpose. (MP4) |
|notice if calculations are repeated, and look both for general methods and for shortcuts. (MP8) |
|maintain oversight of the problem solving process, while also attending to the details. (MP8) |
|continually evaluate the reasonableness of my intermediate results. (MP8) |
| |
|Concept Overview: |
|MP4 Model with mathematics. |
|Linear and exponential functions often serve as effective models for real life contexts. Teachers who are developing students’ capacity to "model with |
|mathematics" move explicitly between real-world scenarios and mathematical representations of those scenarios. Teachers might represent a comparison of different |
|DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted. |
|One strategy for developing this skill is to pose scenarios with no question, and ask student to complete the statements, “I notice …, I wonder…” For sample |
|scenarios, click here. |
| |
|MP8 Look for and express regularity in repeated reasoning. |
|In this unit, students will have the opportunity to explore linear and exponential functions using tables. In the Make a Table strategy (which should really be |
|called Make a Table and Look for Patterns) students have the opportunity to explore and talk through patterns they see in repeated calculations. Students are |
|encouraged to look for and describe patterns both horizontally and vertically, as well as to describe what’s happening “over and over again.” Even the simple |
|activity provided in an extension of the Guess and Check activities (courtesy of Dr. Math), in which students do calculations without ever simplifying any steps, |
|help them see and exploit repeated reasoning. |
| |
|Resources: |
|MP4 Inside Mathematics Website |
|Make a Mathematical Model |
| Diagnostic: Prerequisite Assessment 3a |
| |
|Solving Equations Graphically |
| |
|Represent and solve equations and inequalities graphically |
|MCC9-12.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). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.) |
|MCC9-12.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.★ |
| |
|EQ: How can graphs of linear or exponential equations be used to solve problems? |
| |
|Learning Targets: |
|I can … |
|identify solutions and non-solutions of linear and exponential equations. (REI.10) |
|graph points that satisfy linear and exponential equations. (REI.10) |
|explain why a continuous curve or line contains an infinite number of points on the curve, each representing a solution to the equation modeled by the curve. |
|(REI.10) |
|approximate or find solutions of a system of two functions (linear and/or exponential) using graphing technology or a table of values. (REI.11) |
|explain what it means when two curves {y = f(x) and y = g(x)} intersect i.e. what is the meaning of x and what is the meaning of f(x) = g(x). (REI.11) |
|graph a system of linear equations, find or estimate the solution point, and explain the meaning of the solution in terms of the system. ♦ (REI.11) |
| |
|Concept Overview: Beginning with simple, real-world examples help students to recognize a graph as a set of solutions to an equation. For example, if the equation|
|y = 6x + 5 represents the amount of money paid to a babysitter (i.e., $5 for gas to drive to the job and $6/hour to do the work), then every point on the line |
|represents an amount of money paid, given the amount of time worked. |
| |
|Explore visual ways to solve an equation such as 2x + 3 = x – 7 by graphing the functions y = 2x + 3 and y = x – 7. Students should recognize that the intersection|
|point of the lines is at (–10, –17). They should be able to verbalize that the intersection point means that when x = -10 is substituted into both sides of the |
|equation, each side simplifies to a value of –17. Therefore, –10 is the solution to the equation. This same approach can be used whether the functions in the |
|original equation are linear, nonlinear or both. |
| |
|Using technology, have students graph a function and use the trace function to move the cursor along the curve. Discuss the meaning of the ordered pairs that |
|appear at the bottom of the calculator, emphasizing that every point on the curve represents a solution to the equation. Begin with simple linear equations and how|
|to solve them using the graphs and tables on a graphing calculator. Then, advance students to nonlinear situations so they can see that even complex equations that|
|might involve quadratics, absolute value, or rational functions can be solved fairly easily using this same strategy. While a standard graphing calculator does not|
|simply solve an equation for the user, it can be used as a tool to approximate solutions. |
| |
|Use the table function on a graphing calculator to solve equations. For example, to solve the equation x2 = x + 12, students can examine the equations y = x2 and y|
|= x + 12 and determine that they intersect when x = 4 and when x = –3 by examining the table to find where the y-values are the same. |
| |
|Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce |
|approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing |
|calculators or programs to generate tables of values, graph, or solve a variety of functions. |
| |
|Vocabulary: |
|x - coordinate – the first number in an ordered pair |
|Intersection – the ordered pair or set of elements common to both equations or inequalities |
|Solution – a replacement for the variable in an open sentence that results in a true sentence |
|Linear function – a function that can be written in the form y = mx + b, where x is the independent variable and m and b are real number. Its graph is a line. |
|Sample Problem(s): |
|Given a graph of the equation x + 3y = 6, find three solutions that will satisfy the equation. (REI10) REI10 Solution |
| |
|Given a graph representing the growth of a savings account over time with a given rate of return, determine the value of the account after 3 years, 5 years, 10 |
|years, 12 years and 6 months. (REI10) |
| |
|The cost of producing a soccer ball is modeled by C = 10x + 1000. The sales price of a soccer ball is $20. Explain why a company has to sell 100 soccer balls |
|before they will make a profit. (REI11) |
| |
|Use technology to graph and compare a beginning salary of $30 per day increased by $5 each day and a beginning salary of $0.01 per day, which doubles each day. |
|When are the salaries equal? How do you know? (REI11) |
|Standard |Topic |Resources |Teacher Notes |
| | |Textbook Section #’s: |Student Misconceptions: |
|MCC9-12.A.REI.10 |How graphs represent |(Online Textbook Codes) |Students may believe that the graph of a function is simply a line or curve |
| |solutions |Prentice Hall M3- 11-2, |“connecting the dots,” without recognizing that the graph represents all |
| | |11-5 |solutions to the equation. Additionally, students may believe that |
| | |Prentice Hall A1- 5-1, |two-variable inequalities have no application in the real world. Teachers |
| | |7-1,8-7 |can consider business related problems (e.g., linear programming |
| | |McDougal-Littell M1-1.2,1.3 |applications) to engage students in discussions of how the inequalities are |
| | |McDougal-Littell M2-4.4,4.5 |derived and how the feasible set includes all the points that satisfy the |
| | | |conditions stated in the inequalities. |
| | |Additional resources: | |
| | | |Probing questions: |
| | |Is this a Function | |
| | | |1. What do the points on a line or curve represent? |
| | | |2. How do you determine if a point or ordered pair is a solution to an |
| | | |equation? |
| | | | |
| | | |Differentiation Strategy: |
| | | |1. Have students create a table of value by hand or using a graphing |
| | | |calculator, use the TABLE feature to find points on the curve. |
| | | |2. Have students make a table of values for the given function before |
| | | |plotting points. |
| | | | |
| | | |Cooperative Learning Strategy: Sage and Scribe How graphs represent |
| | | |solutions |
| | |Textbook Section #’s: |Student Misconceptions: Students may also believe that graphing linear and |
|MCC9-12.A.REI.11 |Graphical approaches to |Prentice Hall A1-7-1 |other functions is an isolated skill, not realizing that multiple graphs can|
| |solving equations | |be drawn to solve equations involving those functions. |
| | |Additional resources: | |
| | |Matching with Graphs |Probing questions: |
| | |Solving Graphically |1. What is the difference between an independent system and a dependent |
| | |Notetaking Guide Solving Graphically |system? |
| | |Remediation Solving Graphically | |
| | | |Differentiation Strategy: |
| | | |1. Use graphing calculators to graph equations to help students understand |
| | | |that the solution, or intersection, does not have to be integer coordinates |
| | | | |
| | | |Cooperative Learning Strategy: Roundtable Graphical Approach to Solving |
| | | |Equations |
|Interpreting Functions |
| |
|Understand the concept of a function and use function notation |
|MCC9-12.F.IF.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). (Draw examples from linear and exponential functions.) |
|MCC9-12.F.IF.2 Use function notations, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. |
|(Draw examples from linear and exponential functions.) |
|MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which |
|requires students to write arithmetic and geometric sequences.) |
| |
|Interpret functions that arise in applications in terms of the context |
|MCC9-12.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.★ (Focus on linear and exponential functions.) |
|MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★ (Focus on linear and exponential|
|functions.) |
|MCC9-12.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.★ (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.) |
| |
|Analyze functions using different representations |
|MCC9-12.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.★|
|(Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.) |
|MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★ |
|MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and |
|amplitude.★ |
|MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal |
|descriptions). (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.) |
| |
|EQ: How can functions be used to represent relationships between quantities? |
| |
|Learning Targets: |
|I can … |
|define functions and explain what they are in my own words. (IF1) |
|use function notation, evaluate functions at any point in the domain, give general statements about how f(x) behaves at different regions in the domain (as x gets |
|very large or very negative, close to 0 etc.), and interpret statements that use function notation. (IF2) |
|explain the difference and relationship between domain and range. (IF2) |
|state, and/or find, the domain and range of a function from a function equation, table or graph. (IF2) |
|look at data (from a table, graph, or set of points) and determine if the data is a function and explain my conclusion. (IF3) |
|write a function from a sequence or a sequence from a function. (IF3) |
|explain how an arithmetic or geometric sequence is related to its algebraic function notation. (IF3) |
|write arithmetic and geometric sequences recursively. (IF3) |
|write an explicit formula for arithmetic and geometric sequences. (IF3) |
|connect arithmetic sequences to linear functions and geometric sequences to exponential functions and explain those connections. (IF3) |
|interpret x and y intercepts, where the function is increasing or decreasing, where it is positive or negative, its end behaviors, given the graph, table or |
|algebraic representation of a function in terms of the context of the function. (Only linear and exponential). ♦ (IF4) |
|find and/or interpret appropriate domains and ranges for linear and exponential functions. (IF5) |
|explain the relationship between the domain of a function and its graph in general and/or to the context of the function. (IF5) |
|calculate and interpret the average rate of change over a given interval of a function from a function equation, graph or table, and explain what that means in |
|terms of the context of the function. (IF6) |
|estimate the rate of change of a function from its graph at any point in its domain. (IF6) |
|accurately graph a linear function by hand by identifying key features of the function such as the x- and y-intercepts and slope. ♦ |
|graph a linear or exponential function using technology. (IF7a) |
|sketch the graph of an exponential function accurately identifying x- and y-intercepts and asymptotes. (IF7e) |
|describe the end behavior of an exponential function (what happens as x goes to positive or negative infinity). (IF7e) |
|discuss and compare two different functions (linear and/or exponential) represented in different ways (tables, graphs or equations). Discussion and comparisons |
|should include: identifying differences in rates of change, intercepts, and/or where each function is greater or less than the other. (IF9) |
| |
|Concept Overview: There are five important ideas to consider when thinking about functions. |
|Definition – A function is a rule that assigns each element of set A to a unique element of set B. |
|Thus, a function is a mapping of some element from a domain (set A) into a range (set B). While most of the time we think of the domain and range as being sets of |
|numerical values, this is not always the case. It is important that students understand that a function can operate on non-numerical values. |
|Example: An amusement park has a sign that displayed in front of the bumper car that says a person must be at least 4 feet tall to get on the ride. If John is 3 |
|feet 11 inches, the rule assigns him to the group of non-riders. Susan is 4 feet 2 inches, so according to the rule (function) Susan is assigned to the group of |
|riders. |
|Example: Applying rigid motion to a triangle. The triangle is the input or domain. The rotations, reflections, translations the triangle is put through are the |
|function “rules” and the final transformed triangle is the output or range. (See Unit 5 & 6 of this course) |
| |
|Covariance and rate of change: The independent and dependent variables of a function have a covariant relationship. Patterns in how the two variables change |
|together, let us know to which family of functions a particular function belongs. Examining the rate of change of a function gives us some important information. |
|(Difference tables are valuable tools for examining rates of change.) |
| |
|Families of Functions: Functions that share the same type of rate of change belong to the same family of functions. A linear function will have a constant rate of |
|change. An exponential function has a rate of change that is proportional to the function value. In this course, students will need to be able to discern between |
|linear and exponential functions. In later courses, students will explore quadratic, exponential and trigonometric functions. Note that sequences – both arithmetic|
|and geometric – can be considered to be functions where the domain is restricted to only the positive integers. |
| |
|Combining and Transforming Functions: Under certain conditions, it is possible to add, subtract, multiply or divide functions, as well as to compose functions |
|together. It is also possible to create transformations of functions in predictable ways. There are patterns in transformations of functions which are consistent |
|across all different families of functions. These are helpful when graphing functions. Under appropriate conditions, functions have inverses which “undo” them. |
| |
|Multiple Representations of Functions: Functions have multiple representations – Algebraic equations, Table, Graph, Verbal descriptions and Context. Students |
|should be comfortable with all representations and be fluent in moving between representations. They should understand that changing the representation does NOT |
|change the function. Each different representation can help students understand a different facet of a function. Certain representations are more useful in certain|
|contexts. Understanding the links between different representations is critical in gaining a deeper understanding of a function. For example: What does the rate |
|of change (slope) of a linear function look like in a table? On a graph? In an algebraic equation? In a verbal description? |
| |
|It is highly recommended that teachers use technology when teaching this unit – use graphing calculators, Excel or an online graphing utility. |
| |
|Provide applied contexts in which to explore functions. For example, examine the amount of money earned when given the number of hours worked on a job, and |
|contrast this with a situation in which a single fee is paid by the “carload” of people, regardless of whether 1, 2, or more people are in the car. |
| |
|Use diagrams to help students visualize the idea of a function machine. Students can examine several pairs of input and output values and try to determine a simple|
|rule for the function. |
| |
|Rewrite sequences of numbers in tabular form, where the input represents the term number (the position or index) in the sequence, and the output represents the |
|number in the sequence. |
| |
|Help students to understand that the word “domain” implies the set of all possible input values and that the integers are a set of numbers made up of {…-2, -1, 0, |
|1, 2, …}. |
| |
|Distinguish between relationships that are not functions and those that are functions (e.g., present a table in which one of the input values results in multiple |
|outputs to contrast with a functional relationship). Examine graphs of functions and non-functions, recognizing that if a vertical line passes through at least two|
|points in the graph, then y (or the quantity on the vertical axis) is not a function of x (or the quantity on the horizontal axis). |
| |
|Flexibly move from examining a graph and describing its characteristics (e.g., intercepts, relative maximums, etc.) to using a set of given characteristics to |
|sketch the graph of a function. |
| |
|Examine a table of related quantities and identify features in the table, such as intervals on which the function increases, decreases, or exhibits periodic |
|behavior. |
|Recognize appropriate domains of functions in real-world settings. For example, when determining a weekly salary based on hours worked, the hours (input) could be |
|a rational number, such as 25.5. However, if a function relates the number of cans of soda sold in a machine to the money generated, the domain must consist of |
|whole numbers. |
| |
|Given a table of values, such as the height of a plant over time, students can estimate the rate of plant growth. Also, if the relationship between time and height|
|is expressed as a linear equation, students should explain the meaning of the slope of the line. Finally, if the relationship is illustrated as a linear or |
|non-linear graph, the student should select points on the graph and use them to estimate the growth rate over a given interval. |
| |
|Explore both linear and exponential function and help students to make connections in terms of general features. Examine multiple real-world examples of |
|exponential functions so that students recognize that a base between 0 and 1 (such as an equation describing depreciation of an automobile [ represents the value |
|of a $15,000 automobile that depreciates 20% per year over the course of x years]) results in an exponential decay, while a base greater than 1 (such as the value |
|of an investment over time [ represents the value of an investment of $5,000 when increasing in value by 7% per year for x years]) illustrates growth. |
|f(x)=15,000(0.8)x |
|f(x)=5,000(1.07)x |
| |
|Click here for a more detailed conceptual foundation for this topic. Please note that systems of inequalities is not addressed in this unit. |
| |
|Vocabulary: |
|Polynomial function – a function of the form f(x) = anxn + an-1xn-1 + …. + a1x + a0 where an ≠0, a0, a1, a2, . . . an are real numbers, and the exponents are all |
|whole numbers. |
|Rational function – a function in the form f(x) = p(x)/g(x) where p(x) and q(x) are polynomials and q(x) ≠ 0. |
|Absolute Value function – a function that contains an absolute value expression |
|Exponential function – a function that involves the expression bx where the base b is a positive number other than 1. |
| |
|Sample Problem(s): |
|Write a story that would generate a relation that is a function. Write a story that would generate a relation that is not a function. (IF1) |
| |
|Find a function from science, economics, or sports, write it in function notation and explain its meaning at several points in the domain. (IF2) |
| |
|Draw the next arrangement of blocks in the sequence and describe the sequence using symbols. (IF3) |
|[pic] |
| |
|Create a story that would generate a linear or exponential function and describe the meaning of key features of the graph as they relate to the story. (IF4) |
| |
|You are hoping to make a profit on the school play and have determined the function describing the profit to be f(t) = 8t – 2654 where t is the number of tickets |
|sold. What is a reasonable domain for this function? Explain. (IF5) |
| |
| |
|Create a function in context where the domain would be: (IF5) |
|All real numbers. |
|Integers. |
|Negative integers. |
|Rational Numbers. |
|(10, 40). |
| |
| |
|The graph models the speed of a car. Tell a story using the graph to describe what is happening in various intervals. (IF6) |
|[pic] |
| |
| |
|Graph the linear function that has a slope of 2 and crosses the y-axis at – 3. Write the equation for the function and identify the intercepts. (IF7a) IF7a |
|Solution |
| |
|The population of salmon in a lake triples each year. The current population is 472. Model the situation graphically. Include the last three years and the next |
|two. Model the situation with a function. (IF7e) |
| |
|Which has a greater slope? (IF9) |
|f(x) = 3x + 5 |
|A function representing the number of bottle caps in a shoebox where 5 are added each time. |
| |
|Commentary |
|Slope measures the rate of change in the dependent variable as the independent variable changes. The greater the slope the steeper the line. When the slope, m, is |
|positive, the line slants upward to the right. The more positive m is, the steeper the line will slant upward to the right |
| |
|Solution |
|This linear function: f(x) = mx + b; the slope of the line is m. |
| |
|f(x) = 3x + 5, the slope is 3 |
|f(x) = x + 5, is the function that represents the bottle caps in the shoebox, the slope is 1 |
| |
|The second function has a greater slope. |
| |
| |
|Create a graphic organizer to highlight your understanding of functions and their properties by comparing two functions using at least two different |
|representations. (IF9) |
|Standard |Topic |Resources |Teacher Notes |
| | |Textbook Section #’s: |Model Lesson: Function Notation (K) |
|MCC9-12.F.IF.1 |Functions and Function |(Online Textbook Codes) |Students will learn about the various representations of relations and |
|MCC9-12.F.IF.2 |Notation |Prentice Hall A1- 5-2,5-3,5-4, 8-7,8-8 |functions, including function notation |
|MCC9-12.F.IF.3 | |Prentice Hall M3- 11-1, 11-3 | |
| | |McDougal Littell M1- 3-13 |Student Misconceptions: |
| | | |Students may believe that all relationships having an input and an output |
| | |Additional resources: |are functions, and therefore, misuse the function terminology. Students may |
| | |Function Notation Intro (K) |also believe that the notation f(x) means to multiply some value f times |
| | |(organizer from model lesson) |another value x. The notation alone can be confusing and needs careful |
| | |Functions Practice (K) |development. For example, f(2) means the output value of the function f when|
| | |(HW from model lesson) |the input value is 2. |
| | |Functions: Graphs, Tables, Equations (S)| |
| | |Functions with Fiona |Probing questions: |
| | | |1. What is the difference between a relation and a function? |
| | | |2. All functions are relations, are all relations functions? Why? |
| | | |3. What does a mapping represent? |
| | | |4. What is the difference between finding the values of linear functions and|
| | | |non-linear functions? |
| | | |5. Sequences are functions with what type of numbers as the domain? |
| | | |6. How can a pattern in a sequence of numbers be represented? |
| | | |7. What can be asked to describe the relationship between each term of a |
| | | |sequence to the ones before it? |
| | | |8. Do you connect the points when graphing a sequence? |
| | | | |
| | | |Differentiation Strategy: |
| | | |1. Have students say ‘f or x’ when reading the symbol f(x) to help develop |
| | | |the understanding that the solution is a function of x, the input. |
| | | |2. Have students look up the definition of recur, and use the definition to|
| | | |understand the meaning of recursive. |
| | | |3. Use a mapping diagram to allow students to draw arrows to show how each |
| | | |corresponding element from the domain, is paired with each corresponding |
| | | |element from the range. |
| | | |4. After graphing, encourage students to use the vertical line test to |
| | | |determine if a relation is a function. |
| | | | |
| | | |Cooperative Learning Strategy: Think- Pair-Share Functions? |
| | | |Student Misconceptions: |
|MCC9-12.F.IF.4 |Functions in context |Additional resources: |Students may believe that it is reasonable to input any x-value into a |
|MCC9-12.F.IF.5 | |Functioning Well |function, so they will need to examine multiple situations in which there |
|MCC9-12.F.IF.6 | |Comparing Linear and Exponential |are various limitations to the domains. Students may also believe that the |
| | |Functions |slope of a linear function is merely a number used to sketch the graph of |
| | |Notes on Linear and Exponential |the line. In reality, slopes have real-world meaning, and the idea of a rate|
| | |Functions |of change is fundamental to understanding major concepts from geometry to |
| | | |calculus. |
| | | | |
| | | |Probing questions: |
| | | |1. How can you find the domain and range of a function from a graph? |
| | | |2. What is always true about the domain and range of exponential functions? |
| | | |3. What do the x-intercept and y-intercept represent in a given scenario? |
| | | |4. What is the relationship between y and f(x) in a function rule? |
| | | |5. How do you write slope in terms of f(x)? |
| | | |6. How can the relationship between the domain and range be represented? |
| | | | |
| | | |Differentiation Strategy: |
| | | |1. Have students use graphs of ordered pairs to visually identify the type |
| | | |of function formed and then find the equation using the pattern. |
| | | |2. Allow students to use a graphing calculator to visually identify key |
| | | |features of graphs |
| | | | |
| | | |Cooperative Learning Strategy: Who Am I? Functions in Context |
| | |Textbook Section #’s: |Model Lesson: Linear & Exponential Functions Matching (M) |
|MCC9-12.F.IF.7a |Graphing and comparing |Prentice Hall M3- 11-2, |Students will find sets of cards that represent the same relationship in a |
|MCC9-12.F.IF.7e |linear and exponential |11-5 |variety of ways (equation, graph, table, verbal description) |
|MCC9-12.F.IF.9 |functions |Prentice Hall A1- 5-1, 8-7,8-8 | |
| | |McDougal-Littell M1-1.2,1.3 |Student Misconceptions: |
| | |McDougal-Littell M2 4.4,4.5 |Students may believe that each family of functions (e.g., quadratic, square |
| | | |root, etc.) is independent of the others, so they may not recognize |
| | |Additional resources: |commonalities among all functions and their graphs. Students may also |
| | |Linear & Expon’l Summary and Practice WS|believe that skills such as factoring a trinomial or completing the square |
| | |(S) |are isolated within a unit on polynomials, and that they will come to |
| | |(HW from model lesson) |understand the usefulness of these skills in the context of examining |
| | |Patterns and Functions: |characteristics of functions. |
| | |Linear and Exponential (U) |Additionally, student may believe that the process of rewriting equations |
| | |(An activity for students to explore |into various forms is simply an algebra symbol manipulation exercise, rather|
| | |fractals and other patterns) |than serving a purpose of allowing different features of the function to be |
| | |Graphing Linear and Exponential |exhibited. |
| | |Functions Learning Task | |
| | |Comparing Linear and Exponential |Probing questions: |
| | |Functions |What kind of questions can teachers ask that require higher order thinking? |
| | |Notes on Linear and Exponential |What kind of questions can help students when they get stuck? (U) |
| | |Functions | |
| | | |Differentiation Strategy: |
| | | |List strategies for addressing varying student needs, including readiness |
| | | |levels, learning styles, interests, and/or personal goals. (S) |
| | | | |
| | | |Cooperative Learning Strategy: Mix ‘n’ Match Graphing and comparing linear |
| | | |and exponential functions |
|Unit # Summative Assessment 3A (3B is below) |
[pic]
|Coordinate Algebra: Unit 3b – Linear and Exponential Functions |
|(3 weeks) |
|Content Standards: |
|Build a function that models a relationship between two quantities |
|MCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★ (Limit to linear and exponential functions.) |
|MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.) |
|MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Limit to linear and exponential functions.) |
|MCC9-12.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 |
|MCC9-12.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. (Focus on vertical translations of graphs of linear and exponential functions. Relate the |
|vertical translation of a linear function to its y-intercept.) |
| |
|Construct and compare linear, quadratic, and exponential models and solve problems |
|MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★ |
|MCC9-12.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.★ |
|MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.★ |
|MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.★ |
|MCC9-12.F.LE.2 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).★ |
|MCC9-12.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.★ |
| |
|Interpret expressions for functions in terms of the situation they model |
|MCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★ (Limit exponential functions to those of the form f(x) = bx + |
|k.) |
| |
|Standards for Mathematical Practice: |
|4 Model with mathematics. |
|8 Look for and express regularity in repeated reasoning. |
| |
|Standards for Mathematical Practice (4, 8) |
| |
|EQ: How do mathematically proficient students use mathematical models to solve problems? (MP4) How can recognizing repetition or regularity help solve problems more|
|efficiently? (MP8) |
| |
|Learning Targets: |
|I can … |
|apply the mathematics I know to solve problems arising in everyday life, society, and the workplace. (MP4) |
|write an equation to describe a situation. (MP4) |
|apply what I know to make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. (MP4) |
|identify important quantities in a practical situation. (MP4) |
|map quantity relationships using such tools as diagrams, tables, graphs, and formulas. (MP4) |
|analyze relationships mathematically to draw conclusions. (MP4) |
|interpret my mathematical results in the context of the situation. (MP4) |
|reflect on whether my results make sense, possibly improving the model if it has not served its purpose. (MP4) |
|notice if calculations are repeated, and look both for general methods and for shortcuts. (MP8) |
|maintain oversight of the problem solving process, while also attending to the details. (MP8) |
|continually evaluate the reasonableness of my intermediate results. (MP8) |
| |
| |
| |
| |
|Concept Overview: |
|MP4 Model with mathematics. |
|Linear and exponential functions often serve as effective models for real life contexts. Teachers who are developing students’ capacity to "model with mathematics"|
|move explicitly between real-world scenarios and mathematical representations of those scenarios. Teachers might represent a comparison of different DVD rental |
|plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted. One strategy |
|for developing this skill is to pose scenarios with no question, and ask student to complete the statements, “I notice …, I wonder…” For sample scenarios, click |
|here. |
| |
|MP8 Look for and express regularity in repeated reasoning. |
|In this unit, students will have the opportunity to explore linear and exponential functions using tables. In the Make a Table strategy (which should really be |
|called Make a Table and Look for Patterns) students have the opportunity to explore and talk through patterns they see in repeated calculations. Students are |
|encouraged to look for and describe patterns both horizontally and vertically, as well as to describe what’s happening “over and over again.” Even the simple |
|activity provided in an extension of the Guess and Check activities (courtesy of Dr. Math), in which students do calculations without ever simplifying any steps, |
|help them see and exploit repeated reasoning. |
| |
|Resources: |
|MP4 Inside Mathematics Website |
|Make a Mathematical Model |
| |
|Diagnostic: Prerequisite Assessment 3b |
|Building Functions |
| |
|Build a function that models a relationship between two quantities |
|MCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★ (Limit to linear and exponential functions.) |
|MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.) |
|MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Limit to linear and exponential functions.) |
|MCC9-12.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 |
|MCC9-12.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. (Focus on vertical translations of graphs of linear and exponential functions. Relate the |
|vertical translation of a linear function to its y-intercept.) |
| |
|EQ: In what ways can functions be combined to create new functions? |
| |
|Learning Targets: |
|I can … |
|write a function that describes a linear or exponential relationship between two quantities.(BF1a) |
|combine different functions using addition, subtraction, multiplication, division and composition of functions to create a new function. (BF1b) |
|write arithmetic and geometric sequences recursively. (BF2) |
|write an explicit formula for arithmetic and geometric sequences. (BF2) |
|connect arithmetic sequences to linear functions and geometric sequences to exponential functions and explain those connections. (BF2) |
|identify and explain (in words, pictures or with tables) the effect “k” on a graph of f(x) i.e f(x) + k, kf(x), f(kx), and f(x + k). |
|find the value of “k” given the graphs. (BF3) |
|recognize even and odd functions from their graphs and algebraic expressions. (BF3) |
| |
|Concept Overview: |
|Provide a real-world example (e.g., a table showing how far a car has driven after a given number of minutes, traveling at a uniform speed), and examine the table |
|by looking “down” the table to describe a recursive relationship, as well as “across” the table to determine an explicit formula to find the distance traveled if |
|the number of minutes is known. |
| |
|Write out terms in a table in an expanded form to help students see what is happening. For example, if the y-values are 2, 4, 8, 16, they could be written as 2, |
|2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used multiple times as a factor. |
| |
|Focus on one representation and its related language – recursive or explicit – at a time so that students are not confusing the formats. |
| |
|Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function for the cost |
|of each (given the number of miles driven) is known. |
| |
|Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual models to generate sequences of numbers that can be explored and |
|described with both recursive and explicit formulas. Emphasize that there are times when one form to describe the function is preferred over the other. |
| |
|Use graphing calculators or computers to explore the effects of a constant in the graph of a function. For example, students should be able to distinguish between |
|the graphs of y = 2x, y = 2∙2x, y = (½)∙2x, y = 2x + 2, and y = 2(x + 2). This can be ½ accomplished by allowing students to work with a single parent function and |
|examine numerous parameter changes to make generalizations. |
| |
|Distinguish between even and odd functions by providing several examples and helping students to recognize that a function is even if f(–x) = f(x) and is odd if |
|f(–x) = –f(x). Visual approaches to identifying the graphs of even and odd functions can be used as well. |
| |
|Click here for a more detailed conceptual foundation on this topic. |
| |
| |
|Vocabulary: |
|Logarithmic function – the inverse of the exponential function y = bx, denoted by y = logb x |
|System of Equations – a set of two equations that can be written in the form of Ax + By = C and Dx + Ey = F where x and y are variables, A and B are not both zero, |
|and D and E are not both zero. |
|Substitution property – If a = b, then a may be replaced by b. |
| |
|Sample Problem(s): |
|Anne is shopping and finds a $30 sweater on sale for 20% off. When she buys the sweater, she must also pay 6% sales tax. Write an expression for the final price of |
|the sweater in such a way that the original price is still evident. (Extension: If the clerk just adds 14% will the price be correct?) (BF1) |
| |
|Find an expression, process or calculation to determine the number of squares needed to make the next three patterns in the series. (BF1) |
|[pic] |
| |
|You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Express the amount remaining to be paid off |
|as a function of the number of months, using a recursion equation. (BF1) |
| |
|A cup of coffee is initially at a temperature of 93º F. The difference between its temperature and the room temperature of 68º F decreases by 9% each minute. Write |
|a function describing the temperature of the coffee as a function of time. (BF1) |
| |
|The radius of a circular oil slick after t hours is given in feet by [pic], for 0 ≤ t ≤ 10. Find the area of the oil slick as a function of time. (BF1) |
| |
|Write two formulas that model the pattern: 3, 9, 27, 81…… (BF2) |
| |
|Commentary |
| |
|Functions can be defined explicitly, by a formula in terms of the variable. We can also define functions recursively, in terms of the same function of a smaller |
|variable. In this way, a recursive function "builds" on itself. A recursive formula may list the first two (or more) terms as starting values, depending upon the |
|nature of the sequence. In such cases, the an portion of the formula is dependent upon the previous two (or more) terms. |
| |
|Solution |
| |
|Explicit formula: |
|[pic] |
|Recursive formula: |
|[pic] |
|Certain sequences, such as this geometric sequence, can be represented in more than one manner. This sequence can be represented as either an explicit formula or a |
|recursive formula. |
| |
| |
| |
| |
|Continue the pattern for two more iterations graphically and then find a recursive or explicit formula to model the situation. (BF2) |
| |
|[pic] |
| |
| |
|There are 2,500 fish in a pond. Each year the population decreases by 25 percent, but 1,000 fish are added to the pond at the end of the year. Find the population |
|in five years. Also, find the long-term population. (BF2) |
| |
|Graph the following on a single set of axes: (BF3) BF3 Solution |
| |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
| |
|Compare and contrast the characteristics of these graphs. |
| |
|Describe the effect of varying the parameters a, h, and k on the shape and position of the graph f(x) = ab(x + h) + k., orally or in written format. What effect do |
|values between 0 and 1 have? What effect do negative values have? (BF3) |
|Standard |Topic |Resources |Teacher Notes |
| | |Textbook Section #’s: |Student Misconceptions: Students may believe that the best (or only) way to |
|MCC9-12.F.BF.1 |Writing equations for |(Online Textbook Codes) |generalize a table of data is by using a recursive formula. Students |
|MCC9-12.F.BF.1a |linear and exponential |Prentice Hall M3- 11-1, |naturally tend to look “down” a table to find the pattern but need to |
|MCC9-12.F.BF.1b |functions |11-6 |realize that finding the 100th term requires knowing the 99th term unless an|
|MCC9-12.F.BF.2 | |Prentice Hall A1- 5-4, 5-5, 5-7, 8-6, |explicit formula is developed. Students may also believe that arithmetic |
| | |8-7, 8-8 |and geometric sequences are the same. Students need experiences with both |
| | |McDougal-Littell M2-4.2,4.4,4.5 |types of sequences to be able to recognize the difference and more readily |
| | | |develop formulas to describe them. Additionally, advanced students who study|
| | |Additional resources: |composition of functions may misunderstand function notation to represent |
| | |Talk is Cheap |multiplication (e.g., f(g(x)) means to multiply the f and g function |
| | |(from state frameworks) |values). |
| | |Arithmetic Sequence | |
| | |Recursive Sequence |Probing questions: |
| | |Writing Linear Functions |1. What do you know about the difference between consecutive terms in an |
| | |Writing Exponential Functions |arithmetic sequence? |
| | |Application |2. How can you use the first term in a sequence and the common difference |
| | |Writing Exponential Functions |to write a rule, or determine an explicit expression? |
| | |Arithmetic Sequence WS |3. Explain the differences between a recursive rule of a function and an |
| | | |explicit rule of a function. |
| | | | |
| | | |Differentiation Strategy: |
| | | |1. Have students use graphs of ordered pairs to visually identify the type |
| | | |of function formed and then find the equation using the pattern. |
| | | |2. Use graphing calculators or a spreadsheet application to help students |
| | | |see how the terms will change without performing calculations. |
| | | |3. Use the general form of an explicit and recursive formula and let |
| | | |students substitute needed parameters form the given scenario. |
| | | | |
| | | |Cooperative Learning Strategy: Sage and Scribe Writing equations of linear |
| | | |and exponential functions |
| | | | |
| | | |Literacy Strategy: Vocabulary Cut and Paste. (K) |
| | |Textbook Section #’s: |Student Misconceptions: |
|MCC9-12.F.BF.3 |Effect of transformations |McDougal-Littell M2- 4.4,4.5 |Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 |
| |on functions | |shifted 4 units to the left (due to the subtraction symbol). Examples should|
| | |Additional resources: |be explored by hand and on a graphing calculator to overcome this |
| | |High Functioning |misconception. Students may also believe that even and odd functions refer |
| | |(from state frameworks) |to the exponent of the variable, rather than the sketch of the graph and the|
| | |Transformations of Exponential Functions|behavior of the function. |
| | |Transformation of Linear Functions | |
| | | |Probing questions: |
| | | |1. How are the horizontal asymptotes and the y-intercepts affected during |
| | | |transformations> |
| | | |2. How does using the ZOOM feature on a graphing calculator to enlarge a |
| | | |part of the graph effect your answer? |
| | | | |
| | | |Differentiation Strategy: |
| | | |1. Have students make a table of values for each function along with the |
| | | |parent function. Have students find an ordered pair for the parent function |
| | | |that have the same y-value when first determining horizontal shift |
| | | |directions |
| | | | |
| | | |Cooperative Learning Strategy: Timed Pair Share Effect of transformations |
| | | |on functions |
| | | | |
| | | |Literacy Strategy: Vocabulary Jeopardy Game , Vocabulary Jeopardy Answer |
| | | |Key (K) |
|Linear and Exponential Models |
| |
|Construct and compare linear, quadratic, and exponential models and solve problems |
|MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★ |
|MCC9-12.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.★ |
|MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.★ |
|MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.★ |
|MCC9-12.F.LE.2 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).★ |
|MCC9-12.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.★ |
| |
|Interpret expressions for functions in terms of the situation they model |
|MCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★ (Limit exponential functions to those of the form f(x) = bx + |
|k.) |
|EQ: How can linear and exponential functions be used to represent problems in real life contexts? |
| |
|Learning Targets: |
|I can … |
|distinguish between situations that can be modeled with linear and exponential functions. (LE1) |
|prove (using words, pictures, numbers and/or difference tables) that linear functions grow by equal differences over equal intervals. (LE1a) |
|prove (using words, pictures, numbers and/or difference table) that exponential functions grow by equal factors over equal intervals. (LE1a) |
|recognize situations with a constant rate of change. (LE1b) |
|recognize situations that can be modeled linearly or exponentially and describe the rate of change per unit as constant or the growth factor as a constant percent.|
|(LE1c) |
|recognize situations in which a quantity either grows or decays by a constant percent rate. (LE1c) |
|construct a linear function given an arithmetic sequence, a graph, a description of a relationship or a table of input-output pairs. (LE2) |
|construct an exponential function given a geometric sequence, a graph, a description of a relationship or a table of input-output pairs. (LE2) |
|explain why a quantity increasing exponentially will eventually exceed a quantity increasing linearly. (LE3) |
|interpret and explain the parameters in an exponential function in terms of a given context (authentic situation, |
|graph, symbolic representation.) (LE5) |
| |
|Concept Overview: Compare tabular representations of a variety of functions to show that linear functions have a first common difference (i.e., equal differences |
|over equal intervals), while exponential functions do not (instead function values grow by equal factors over equal x-intervals). |
|Apply linear and exponential functions to real-world situations. For example, a person earning $10 per hour experiences a constant rate of change in salary given |
|the number of hours worked, while the number of bacteria on a dish that doubles every hour will have equal factors over equal intervals. |
|Provide examples of arithmetic and geometric sequences in graphic, verbal, or tabular forms, and have students generate formulas and equations that describe the |
|patterns. |
|Use a graphing calculator or computer program to compare tabular and graphic representations of exponential and polynomial functions to show how the y (output) |
|values of the exponential function eventually exceed those of polynomial functions. |
|Have students draw the graphs of exponential and other polynomial functions on a graphing calculator or computer utility and examine the fact that the exponential |
|curve will eventually get higher than the polynomial function’s graph. A simple example would be to compare the graphs (and tables) of the functions y = x2 and y =|
|2x and to find that the y values are greater for the exponential function when x > 4. |
| |
|Students can investigate functions and graphs modeling different situations involving simple and compound interest. Students can compare interest rates with |
|different periods of compounding (monthly, daily) and compare them with the corresponding annual percentage rate. Spreadsheets and applets can be used to explore |
|and model different interest rates and loan terms. |
| |
|Students can use graphing calculators or programs, spreadsheets, or computer algebra systems to construct linear and exponential functions. |
| |
|Use real-world contexts to help students understand how the parameters of linear and exponential functions depend on the context. For example, a plumber who |
|charges $50 for a house call and $85 per hour would be expressed as the function y = 85x + 50, and if the rate were raised to $90 per hour, the function would |
|become y = 90x + 50. On the other hand, an equation of y = 8,000(1.04)x could model the rising population of a city with 8,000 residents when the annual growth |
|rate is 4%. Students can examine what would happen to the population over 25 years if the rate were 6% instead of 4% or the effect on the equation and function of |
|the city’s population were 12,000 instead of 8,000. |
| |
|Graphs and tables can be used to examine the behaviors of functions as parameters are changed, including the comparison of two functions such as what would happen |
|to a population if it grew by 500 people per year, versus rising an average of 8% per year over the course of 10 years. |
| |
|Sample Problem(s): |
|An accountant has two ways of depreciating equipment. One way is to depreciate by a fixed amount each year. The other way is to depreciate by a fixed percentage |
|each year. Which depreciation method is linear? Which depreciation method is exponential? (LE1) |
| |
|A cell phone company has three plans. Graph the equation for each plan, and analyze the change as the number of minutes used increases. When is it beneficial to |
|enroll in Plan 1? Plan 2? Plan 3? (LE1) |
|$59.95/month for 700 minutes and $0.25 for each additional minute, |
|$39.95/month for 400 minutes and $0.15 for each additional minute, and |
|$89.95/month for 1,400 minutes and $0.05 for each additional minute. |
| |
|A computer store sells about 200 computers at the price of $1,000 per computer. For each $50 increase in price, about ten fewer computers are sold. How much should|
|the computer store charge per computer in order to maximize their profit? (LE1) |
| |
|A couple wants to buy a house in five years. They need to save a down payment of $8,000. They deposit $1,000 in a bank account earning 3.25% interest, compounded |
|quarterly. How much will they need to save each month in order to meet their goal? (LE1) |
| |
|Sketch and analyze the graphs of the following two situations. What information can you conclude about the types of growth each type of interest has? (LE1) |
|Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a year, but she does not compound the interest. |
|Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest compounded annually. |
| |
|Create a story that demonstrates one quantity changing at a constant rate per unit interval relative to another. (LE1) |
| |
|Write a function that models the population of Smithville, a town that in 2003 was estimated to have 35,000 people that increases by 2.4% every year. Describe a |
|reasonable way to use your function to predict future population in Smithville. (LE2) |
| |
|Determine an exponential function of the form f(x) = abx using data points from the table. Graph the function and identify the key characteristics of the graph. |
|(LE2) LE2 Solution |
| |
|x |
|f(x) |
| |
|0 |
|1 |
| |
|1 |
|3 |
| |
|3 |
|27 |
| |
| |
|Sara’s starting salary is $32,500. Each year she receives a $700 raise. Write a sequence in explicit form to describe the situation. (LE2) |
| |
|Commentary |
|Functions can be defined explicitly, by a formula in terms of the variable. |
| |
|Solution |
|The explicit formula that represents the situation is , where her annual salary is a function of the number of years she works. Substituting the number of years |
|x, starting with 1, the sequence is 33250, 34000, 34750, 35500, 36250, 37000… |
| |
|What’s the better deal, earning $1000 a day for the rest of your life or earning $.01 the first day, and doubling it every day for the rest of your life? How do |
|you know? Do you think an 80-year-old would make the same choice? Should she? (LE3) |
| |
|Contrast the growth of the f(x)=x3 and f(x)=3x. (LE3) |
| |
|Annie is picking apples with her sister. The number of apples in her basket is described by n = 22t + 12, where t is the number of minutes Annie spends picking |
|apples. What do the numbers 22 and 12 tell you about Annie’s apple picking? (LE5) |
| |
|Commentary |
|A practical application of slope, m, is a rate. A rate describes how much one variable changes with respect to another. Rates are often used to describe |
|relationships between time and an action. This equation is called the slope-intercept form for a line. The changes occur as a function of time. The slope is m and |
|the y-intercept is b. The point where the graph crosses the y-axis is called the y-intercept. The y-intercept represents an initial condition, or what action is |
|occurring when the time is zero. |
| |
|Solution |
|This linear function n = 22t + 12, is written in the general form y = mx + b. The slope of the line is m. In the formula, 22 is the slope and represents the number|
|of apples Annie picks in 1 minute. In the formula, 12 represents the initial amount of apples in the basket. As she adds 22 apples to the basket each minute, that|
|amount is automatically increased by the 12 apples in the basket before she began. |
| |
|A function of the form f(n) = P(1 + r)n is used to model the amount of money in a savings account that earns 5% interest, compounded annually, where n is the |
|number of years since the initial deposit. What is the value of r? What is the meaning of the constant P in terms of the savings account? Explain either orally or |
|in written format. (LE5) |
|Standard |Topic |Resources |Teacher Notes |
| | | |Student Misconceptions: Students may believe that all functions have a first|
|MCC9-12.F.LE.1 |Comparing linear and |Additional resources: |common difference and need to explore to realize that, for example, a |
|MCC9-12.F.LE.1a |exponential models |Building and Combining Functions |quadratic function will have equal second common differences in a table. |
|MCC9-12.F.LE.1b | |(from state frameworks) |Students may also believe that the end behavior of all functions depends on |
|MCC9-12.F.LE.1c | |Community Service Sequences and |the situation and not the fact that exponential function values will |
|MCC9-12.F.LE.2 | |Functions |eventually get larger than those of any other polynomial functions. |
|MCC9-12.F.LE.3 | |(from state frameworks) | |
| | |PPT Exponential Models |Probing questions: |
| | |Exponential Worksheet |What kind of questions can teachers ask that require higher order thinking? |
| | | |What kind of questions can help students when they get stuck? (U) |
| | | | |
| | | |Differentiation Strategy: |
| | | |List strategies for addressing varying student needs, including readiness |
| | | |levels, learning styles, interests, and/or personal goals. (S) |
| | | | |
| | | |Cooperative Learning Strategy: Flash Card Game Comparing Linear and |
| | | |exponential models |
| | | |Student Misconceptions: Students may believe that changing the slope of a |
|MCC9- |Interpreting function |Additional resources: |linear function from “2” to “3” makes the graph steeper without realizing |
|MCC9-12.F.LE.5 |expressions in context |You’re Toast Dude |that there is a real-world context and reason for examining the slopes of |
| | |(from state frameworks) |lines. Similarly, an exponential function can appear to be abstract until |
| | |Linear In Context |applying it to a real-world situation involving population, cost, |
| | |PPT Exponential Models |investments, etc. |
| | | | |
| | | |Probing questions: |
| | | |What kind of questions can teachers ask that require higher order thinking? |
| | | |What kind of questions can help students when they get stuck? (U) |
| | | | |
| | | | |
| | | |Differentiation Strategy: |
| | | |1. Have students enter original function and translated functions in |
| | | |graphing calculator to observe shifts, stretches, and shrinks. |
| | | |2. Have students use tracing paper to trace the graph of original function |
| | | |and use the traced graph to model shifts. |
| | | | |
| | | |Cooperative Learning Strategy: Timed Pair Share Interpreting function |
| | | |expressions in context |
| | | | |
| | | |Literacy Strategy: Vocabulary Who Has I Have . (K) |
|Unit # Summative Assessment 3B (3A is above) |
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