# Teacher Resources

## Exploring Other Function Models

On this page you will find lesson objectives, standards, instructional notes and UDL opportunities.

Estimated Time:  60 - 75 minutes

### Lesson Objectives:

• Students will review behaviors of linear, exponential and quadratic functions to describe when these functions may be used to model a data set.
• Students will be introduced to other function families that may be used to model a data set.
• Students will apply understanding of residuals and functional behavior to determine whether a function model is the best fit for a data set.

Key Common Core State Standards:

S.ID.6 Represent data on two quantitative variables on a scatter plot and describe how the variables are related.

1. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Supporting Common Core State Standards:
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.

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.

Standards for Mathematical Practice Emphasized:

MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.

## Instructional Notes:

In this lesson, students build on their understanding of modeling with linear, quadratic and exponential functions to begin to include other function families.

Students will be provided with a link to a Shodor Applet . This applet is limited to finding the line of best fit and does not allow other regression equations.  Students also need to use a graphing calculator to graph scatter plots, calculate the regression equation and display the residual plot.

The first scenario allows students to analyze known functions to find a model of best fit. Upon initial inspection, students should identify a quadratic function as the best fit of the functions they know.  This scenario provides an entry point into other function families as well.  In this case, students will learn that the cubic model is in fact the model of best fit.

The function interactive is intended to provide students with a very brief introduction to other function families.  As a lesson reflection, students will have the opportunity to research a function family in more depth.

The second scenario allows students the opportunity to review the list of additional function families to examine potential models of best fit.  Students may not yet identify a logarithmic function but should be able to identify criteria for eliminating other function models.

For the lesson reflection, students will explore another function family in depth.  With your class, make sure that all function families are selected.  Students will need time to explore other function families in depth before completing this module.

> Go to Exploring Other Function Models lesson

## Sample Responses

For sample responses to the Algebra II Journal questions, visit the Algebra II Journal in Teacher Resources.

## Exploring Other Function Models - Page 3

The first scenario allows students to analyze known functions to find a model of best fit. Upon initial inspection, students should identify a quadratic function as the best fit of the functions they know.  This scenario provides an entry point into other function families as well.  In this case, students will learn that the cubic model is in fact the model of best fit.

> Go to lesson, page 3

## Exploring Other Function Models - Page 7

This function interactive is intended to provide students with a very brief introduction to other function families.  As a lesson reflection, students will have the opportunity to research a function family in more depth.

## Exploring Other Function Models - Page 13

This scenario allows students the opportunity to review the list of additional function families to examine potential models of best fit.  Students may not yet identify a logarithmic function but should be able to identify criteria for eliminating other function models.

> Go to lesson, page 13

## Exploring Other Function Models - Page 14

With your class, make sure that all function families are selected.  Students will need time to explore other function families in depth before completing this module.

> Go to lesson, page 14

## UDL Opportunities:

Prior to this lesson assess the background knowledge of students informally or formally to determine readiness for this online module.  Students will need to understand key features and behaviors of linear, quadratic, and simple exponential functions.  They will need to be comfortable calculating and analyzing residuals and the correlation coefficient (for linear models).  A link is provided to Module 1: Linear, Quadratic and Exponential Regression for students that need additional support or review.  Within the lesson, students have opportunities to review and build upon prior learning.

Checkpoint 3.2: Highlight patterns, critical features, big ideas, and relationships.
By the end of this lesson, students should begin to identify critical features different function families as strategies to identify which model is best for a given data set.  Encourage student discussion about critical features and relationships between function families.

Checkpoint 4.1: Vary the methods for response and navigation.
Checkpoint 5.1: Use multiple media for communication.
Checkpoint 7.1: Optimize individual choice and autonomy.
Checkpoint 8.3: Foster collaboration and community.
Students have options to use an online applet and/or graphing calculator to analyze the data sets.  The end of lesson reflection allows students the opportunity to select a function family of interest and become the expert for that function family.  Allow alternatives for communicating that understanding, including using multimedia (PowerPoint, online tools such as http://edu.glogster.com/ , Google Docs, etc.) to share findings, graphic organizers or Frayer Models, posters or other visual representations of findings, or class presentations.  Consider options for collaboration to include group projects and/or modified jigsaws to share understanding of function families.

Checkpoint 7.2: Optimize relevance, value, and authenticity.
The scenarios in this lesson are based on real-world data in order to optimize relevance and authenticity.