 # Standard #: MAFS.912.F-LE.1.2

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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).

### General Information

Subject Area: Mathematics
Domain-Subdomain: Functions: Linear, Quadratic, & Exponential Models
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems. (Algebra 1 - Supporting Cluster) (Algebra 2 - Supporting Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes

### Test Item Specifications

Also assesses:

MAFS.912.F-BF.1.1

MAFS.912.F-IF.1.3

Assessment Limits :
In items where the student must write a function using arithmetic
operations or by composing functions, the student should have to
generate the new function only.

In items where the student constructs an exponential function, a
geometric sequence, or a recursive definition from input-output
pairs, at least two sets of pairs must have consecutive inputs.

In items that require the student to construct arithmetic or geometric
sequences, the real-world context should be discrete.

In items that require the student to construct a linear or exponential
function, the real-world context should be continuous.

Calculator :

Neutral

Clarification :
Students will write a linear function, an arithmetic sequence, an
exponential function, or a geometric sequence when given a graph
that models a real-world context.

Students will write a linear function, an arithmetic sequence, an
exponential function, or a geometric sequence when given a verbal
description of a real-world context.

Students will write a linear function, an arithmetic sequence, an
exponential function, or a geometric sequence when given a table of
values or a set of ordered pairs that model a real-world context.

Students will write an explicit function, define a recursive process, or
complete a table of calculations that can be used to mathematically
define a real-world context.

Students will write a function that combines functions using
arithmetic operations and relate the result to the context of the
problem.

Students will write a function to model a real-world context by
composing functions and the information within the context.

Students will write a recursive definition for a sequence that is
presented as a sequence, a graph, or a table.

Stimulus Attributes :
For F-LE.1.2 and F-BF.1.1, items should be set in a real-world context.

For F-IF.1.3, items may be set in a mathematical or real-world
context.

Items must use function notation.

In items where the student builds a function using arithmetic
operations or by composition, the functions may be given using
verbal descriptions, function notation or as equations.

Response Attributes :
For F-BF.1.1b and c, the student may be asked to find a value.

For F-LE.1.2 and F-BF.1.1, items may require the student to apply the
basic modeling cycle.

In items where the student writes a recursive formula, the student
may be expected to give both parts of the formula.

The student may be required to determine equivalent recursive
formulas or functions.

Items may require the student to choose an appropriate level of
accuracy.

Items may require the student to choose and interpret the scale in a
graph.

Items may require the student to choose and interpret units.

### Sample Test Items (1)

 Test Item # Question Difficulty Type Sample Item 1 A study estimates that the cost of tuition at a university will increase by 2.8% each year. The cost of tuition at the university in 2015 was \$33,741.The function, B(x), models the estimated tuition cost, where x is the number of years since 2015.Click on the blank to enter an expression that completes the function B(x). N/A EE: Equation Editor

#### Related Courses

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#### Related Access Points

 Access Point Number Access Point Title MAFS.912.F-LE.1.AP.2a Select the graph, the description of a relationship or two input-output pairs of linear functions.

#### Assessments

 Name Description Sample 4 - High School Algebra 1 State Interim Assessment This is a State Interim Assessment for 9th-12th grades. Sample 2 - High School Algebra 1 State Interim Assessment This is a State Interim Assessment for 9th-12th grades.

#### Formative Assessments

 Name Description What Is the Function Rule? Students are asked to write function rules for sequences given tables of values. Writing an Exponential Function From a Table Students are asked to write an exponential function represented by a table of values. Writing an Exponential Function From a Description Students are asked to write an exponential function from a written description of an exponential relationship. Writing an Exponential Function From Its Graph Students are asked to write an exponential function given its graph. Functions From Graphs Students are asked to write a function given its graph. Writing a Function From Ordered Pairs Students are given a table of values and are asked to write a linear function. The Cost of Water Students are asked to write a function to model the relationship between two variables described in a real-world context.

#### Lesson Plans

 Name Description The Towers of Hanoi: Experiential Recursive Thinking This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones.  The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers.   The lesson is experiential, in that each student will be working with her/his own Towers of Hanoi manipulative, inexpensively obtained.  There is no formal prerequisite, although some familiarity with set theory and functions is helpful.  The last three sections of the lesson involve some more formal concepts with recursive equations and proof by induction, so the students who work on those sections should probably be level 11 or 12 in a K-12 educational system.  The lesson has a Stop Point for 50-minute classes, followed by three more segments that may require a half to full additional class time.  So the teacher may use only those segments up to the Stop Point, or if two class sessions are to be devoted to the lesson, the entire set of segments.  Supplies are modest, and may be a set of coins or some washers from a hardware store to assemble small piles of disks in front of each student, each set of disks representing a Towers of Hanoi manipulative.  Or the students may assemble before the class a more complete Towers of Hanoi at home, as demonstrated in the video.  The classroom activities involve attempting to solve with hand and mind the Towers of Hanoi problem and discussing with fellow students patterns in the process and strategies for solution. Functions and Everyday Situations This lesson unit is intended to help you assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, interpret algebraic functions in terms of the contexts in which they arise and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous. Piles of Paper Piles of Paper is a student activity that demonstrates linear and exponential growth using heights of flat and folded paper. Data tables are created and then algebraic models are developed. Real world types of linear and exponential growth are also introduced. Determining the Hubble Constant Students will graph distance/velocity data of real galaxies to arrive at their own value of the Hubble constant (H). Once they have calculated their own value of H, they will use it to determine distances to real galaxies with known recessional velocities.

#### Original Student Tutorial

 Name Description Creating Exponential Functions By the end of this tutorial, you should be able construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world.

#### Unit/Lesson Sequences

Name Description
Sample Algebra 1 Curriculum Plan Using CMAP

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

### Using this CMAP

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Direct and Inverse Variation "Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

#### Virtual Manipulatives

 Name Description Loan Calculator This virtual manipulative allows the user to explore scenarios of a loan repayment by manipulating the amount of the loan, interest rate, payment amount, frequency of payments, and length of the loan in years. Linear Function Machine In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet. Introduction to Functions This lesson is designed to introduce students to functions as rules and independent and dependent variables. The lesson provides links to discussions and activities that motivate the idea of a function as a machine as well as proper terminology when discussing functions. Finally, the lesson provides links to follow-up lessons designed for use in succession to the introduction of functions. Data Flyer Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

#### Original Student Tutorial

 Name Description Creating Exponential Functions: By the end of this tutorial, you should be able construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world.

#### Virtual Manipulatives

 Name Description Loan Calculator: This virtual manipulative allows the user to explore scenarios of a loan repayment by manipulating the amount of the loan, interest rate, payment amount, frequency of payments, and length of the loan in years. Linear Function Machine: In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet. Data Flyer: Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.