| Title: |
Unit 2: Linear and Exponential Relationships |
| Author & Email: |
Karen Mitchell mitchelk@marshall.edu
Modified by Vivian Jean Brown vbrown@access.k12.wv.us and Michael D. Brown mdbrown@access.k12.wv.us |
| Grade Level: |
High School Math I |
Unit Overview:
|
In this unit students will build on the work they have already done with linear relationships. While they will see and work with other function families, the emphasis will be on linear and exponential families. They will compare and contrast the properties of linear and exponential functions. They will spend time working to understand functions as an important mathematical structure by examining function characteristics, notation, representations, and operations of specific function families. Students will practice building functions that describe the relationship between two quantities from a context. They will look at multiple ways to apply functions to solve problems and represent situations. They will also work with sets of order pairs that do not represent functions. |
Next Generation Content Standards and Objectives:
|
Objectives Directly Taught or Learned
Through Inquiry/Discovery | Evidence of Student Mastery of Content | M.1HS.LER.1
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.) | Equation Analysis Sheet 11, Equation Analysis Sheet 12 (Lesson 4) | M.1HS.LER.2
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.* (Focus on cases where f(x) and g(x) are linear or exponential.) | Function Analysis Worksheet and presentation (Lesson 10) | M.1HS.LER.3
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. | Determine a feasible region of solution (Lesson 13) | M.1HS.LER.4
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). | Good Definition Handout (Lesson 1) | M.1HS.LER.5
use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context | Function Notation Practice (Lesson 2) | M.1HS.LER.6
recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1. | First three attachments from Rows of Numbers (Lesson 14) | M.1HS.LER.7
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.) | Functions Represented by Graphs Handout (Lesson 5) | M.1HS.LER.8
relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (Focus on linear and exponential functions.) | Function Notation Handout (Lesson 2) | M.1HS.LER.9
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. Mathematics II and III will address other function types). | Table Analysis Handout, Table Analysis Homework (Lesson 3) | M.1HS.LER.10
graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. - graph linear and quadratic functions and show intercepts, maxima, and minima.
- graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude.
| Solving Equations by Using Functions Handout (Lesson 9) | M.1HS.LER.11
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. | Recorder’s Sheet (Lesson 8) | M.1HS.LER.12
write a function that describes a relationship between two quantities. - Determine an explicit expression, a recursive process, or steps for calculation from a context.
- 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.
| Newspaper article (Lesson 15), Investigation Sheet (Lesson 12) | M.1HS.LER.13
write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.) | Summary in Attachments for Part 1(Lesson 14) | M.1HS.LER.14
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 | Transforming Equations, Transforming Graphs and Tables, Understanding Check (Lesson 11) | M.1HS.LER.15
distinguish between situations that can be modeled with linear functions and with exponential functions. - prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.
- recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
| Recording Sheet and presentation of provided situation (Lesson 6) | M.1HS.LER.16
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). | Poster presentation of work from Equations, Tables, and Graphs (Lesson 7) | M.1HS.LER.17
observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (Limit to comparisons between exponential and linear models.) | Function Notation Handout(Lesson 2) | M.1HS.LER.18
interpret the parameters in a linear or exponential function in terms of a context. | Analysis 1A and Analysis 1B (Lesson 16) | |
Mathematical Practices::
|
| Mathematical Practices | Evidence of Student Engagement in Mathematical Practices | | 1. Make sense of problems and persevere in solving them. | Lessons 7, 10, 13, 14, 15, 16 | | 2. Reason abstractly and quantitatively. | Lessons 1, 2, 11, 12 | | 3. Construct viable arguments and critique the reasoning of others. | Lessons 2, 3, 6,12 | | 4. Model with mathematics. | Lessons 7, 10, 14, 15, 16 | | 5. Use appropriate tools strategically. | Lessons 8, 9, 11 | | 6. Attend to precision. | Lessons 1, 5, 9 | | 7. Look for and make use of structure. | Lessons 2, 3, 4, 5, 6, 8, 11, 12, 14 | | 8. Look for and express regularity in repeated reasoning. | Lessons 3, 5, 11, 14 | |
Focus/Driving Question:
|
What roles do functions play in mathematics? How are different families of functions used to model situations and predict outcomes? |
Student will Know:
|
Vocabulary from all the individual lessons A formal definition of function Properties of specific families of functions Four ways to represent functions The five function operations The four components of a good definition Changes that transformations make in function representations |
Student will Do:
|
Convert from one function representation to any of the other three Determine the particular function that best models a situation Use functions to predict values and solve problems Solve equations using functions Model systems of linear inequalities Find and describe patterns and sequences as functions Use a graphing calculator to complete regression models that correspond to a given set of data Use function operations to create new functions Use transformations to create new functions Write an equivalent form of a good definition |
Materials/Resources/Websites:
|
The Function Journal is used in every lesson. Lesson 1: Good Definition Handout, Global Warming and Functions Handout, Criteria for Evaluating a Web Site, Record of Internet Research Lesson 2: Function Notation Practice, Function Notation Lesson 3: Table Analysis Handout, Table Analysis Homework Lesson 4: Equation Worksheets 11, 12, 21, 22, 31, 32 Lesson 5: Functions Represented by Graphs Handout Lesson 6: Launch Introduction Activity, Garden Plots Revisited Handout, Go with the Flow Handout, Population Growth Handout, Timbers for Square Garden Plots Handout Lesson 7: Chart paper, markers, Equations, Tables, and Graphs Lesson 8: graphing calculator, Cards, Recorder’s Score Sheet, Function Journal, Teacher’s Instructions for the Function Card Game Lesson 9: graph paper, rulers, graphing calculators, colored pencils (optional), Solving Equations Using Functions Handout Lesson 10: Graphing Worksheet (Attachment C), Function Analysis Instruction Sheet (Attachment A-Part 1), Copies of the Exploration (Attachment A-Part 2), Blank Tables (Attachment B), Graphing Calculator Lesson 11: graphing calculator, Transforming Equations, Understanding Check, Transforming Graphs and Tables Lesson 12: Investigation Sheet, Environmental Protection Agency Article, Summary Activity Lesson 13: Graphing Calculator, the Exploration Worksheet (Attachment A) and 5 copies of the Graphing Worksheet (Attachment B) for each student Lesson 14: Copies of the following for each student: Attachments for Part 1 (Attachment 1A, Attachment 1B, Attachment 1C), index cards, and centimeter cubes available for each student who wants to use them (optional). Lesson 15: Rulers, Graph paper, Graphing calculators, News Article Notes, News Article Criteria, News Article Rubric, Function Journal Lesson 16: Analysis Instruction Sheet (Attachment A), U.S. Population for 1900 to 1990 Data Table (Attachment B), Graphing Worksheet (Attachment C), Graphing Calculator. |
Assessment Plan:
|
Each lesson contains the possibility for both formative and summative assessment. The lessons have been written with a concern for providing teachers with the flexibility to make decisions that will best meet their students’ needs. See evidence of student mastery. For the unit assessment students are asked to find two functions that are important to a specified topic and discuss its use and representations in both a paper and presentation. |
Major Products:
|
Lesson 6 Presentation (Group) Lesson 10 Presentation (Group) Lesson 15 Newspaper Article (Individual) Unit Function Presentation and Paper (Individual) |
Unit Reflection:
|
Teachers need to consider the level of understanding students have achieved relative to the cluster topics. Can students represent and solve equations and inequalities graphically? Do students understand the concept of function and the use of function notation? Can students interpret functions that arise in the context of an application? Can students analyze functions using different representations? Can students build a function that models a relationship between two quantities? Can students build new functions from existing functions? Can student construct and compare linear and exponential models and solve related problems? Can students interpret expressions for functions in terms of the situation they model? |
| Tagged Next Generation Content Standards and Objectives |
| NxG ID |
NxG Objectives |
| M.1HS.LER.1 |
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.) (CCSS.Math.Content.HSA-REI.D.10) |
| M.1HS.LER.2 |
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.* (Focus on cases where f(x) and g(x) are linear or exponential.) (CCSS.Math.Content.HSA-REI.D.11) |
| M.1HS.LER.3 |
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. (CCSS.Math.Content.HSA-REI.D.12) |
| M.1HS.LER.4 |
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). (CCSS.Math.Content.HSF-IF.A.1) |
| M.1HS.LER.5 |
use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. (CCSS.Math.Content.HSF-IF.A.1) |
| M.1HS.LER.6 |
recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n = 1. (CCSS.Math.Content.HSF-IF.A.1) |
| M.1HS.LER.7 |
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.) (CCSS.Math.Content.HSF-IF.B.4) |
| M.1HS.LER.8 |
relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (Focus on linear and exponential functions.) (CCSS.Math.Content.HSF-IF.B.5) |
| M.1HS.LER.9 |
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. Mathematics II and III will address other function types). (CCSS.Math.Content.HSF-IF.B.6) |
| M.1HS.LER.10 |
graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
- graph linear and quadratic functions and show intercepts, maxima, and minima.
- graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude.
(CCSS.Math.Content.HSF-IF.C.7) |
| M.1HS.LER.11 |
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. (CCSS.Math.Content.HSF-IF.C.9) |
| M.1HS.LER.12 |
write a function that describes a relationship between two quantities.
- Determine an explicit expression, a recursive process, or steps for calculation from a context.
- 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.
(CCSS.Math.Content.HSF-BF.A.1) |
| M.1HS.LER.13 |
write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.) (CCSS.Math.Content.HSF-BF.A.2) |
| M.1HS.LER.14 |
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. (CCSS.Math.Content.HSF-BF.B.3) |
| M.1HS.LER.15 |
distinguish between situations that can be modeled with linear functions and with exponential functions.
- prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.
- recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
(CCSS.Math.Content.HSF-LE.A.1) |
| M.1HS.LER.16 |
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). (CCSS.Math.Content.HSF-LE.A.2) |
| M.1HS.LER.17 |
observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (Limit to comparisons between exponential and linear models.) (CCSS.Math.Content.HSF-LE.A.3) |
| M.1HS.LER.18 |
interpret the parameters in a linear or exponential function in terms of a context. (CCSS.Math.Content.HSF-LE.B.5) |
|
| Files Uploaded |
|
| Date Created: |
June 13, 2012 |
| Date Modified: |
August 06, 2012 |
Unit Plan Outline
(Lesson Plans) |
Lesson 1: Introduction to Functions
Lesson 2: Function Notation in a Context
Lesson 3: Functions Represented by Tables
Lesson 4: Functions Represented by Equations
Lesson 5: Functions Represented by Graphs
Lesson 6: Functions Represented by Situations
Lesson 7: Modeling a Situation Using Different Representations
Lesson 8: Recognizing the Connections among Function Representations
Lesson 9: Solving Equations by Using Functions
Lesson 10: Too Hot To Handle – Comparing Linear and Exponential Functions
Lesson 11: Geometric Transformations Applied to Function Families
Lesson 12: Function Operations
Lesson 13: Modeling Systems of Linear Inequalities
Lesson 14: Sequences Represented as Functions
Lesson 15: Interpreting Tables and Graphs
Lesson 16: Linear and Exponential Models
Lesson 17: Function Presentation Practice (Unit Assessment)
Lesson 18: Function Presentations (Unit Assessment)
|
Career Connections:
|
Lesson 1: purchasing agent, researcher, financial planner
Lesson 2: essential information common to many professions
Lesson 3: statistician, surveyor, inventory strategist
Lesson 4: Safety engineer, scientists, event planner, travel agent, bankers, accountants, floor layers
Lesson 5: Photographer, scientist, day trader
Lesson 6: Statisticians, Purchasing Agents, Safety Engineers, Scientists, Researchers
Lesson 7: Statistician, Product Manager, Safety Controller, Sales Representative
Lesson 8: Attorney, Physician, Computer Software Engineer, Medical Professionals
Lesson 9: This strategy may be employed by any professional who models situations.
Lesson 10: Forest rangers, fire fighters, helicopter pilots, air traffic controllers, meteorologists, conservationists, and civil engineers. |
Lesson 11: Software engineer, Robotics, Traffic controller, Designer, Artist
Lesson 12: Economist, Scientist
Lesson 13: Administrators of schools, businesses and hospitals will use this information, accountants, food workers, investors, advertising executives and sales executives
Lesson 14: Computer and information systems managers, financial managers, gaming programmers, biomedical engineers, nuclear engineers
Lesson 15: Journalist, Physician, Librarian
Lesson 16: statistics, social work, international communications, or business management |
| Key Word Search Fields |
functions, linear relationships, exponential relationships
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