Subject:
Mathematics
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
College Algebra, Domain, Functions, Linear Functions, Range, Tmm0012
License:
Creative Commons Attribution-NonCommercial 4.0
Language:
English

Education Standards (2)

Linear Functions

Linear Functions

Module Overview

This material covers Chapter 4: Linear Functions chapter of the OpenStax College Algebra Text. This module contains an overview of learning objectives mapped to the OTM state standards, worksheets that correspond to chapter sections, interactive Desmos Activities that pair with the chapter, and a list of supplemental videos that correspond to the chapter content.

 
Cover Photo by Natalia García on Unsplash

Section 1: Chapter 4: Introduction and Outcomes

Linear Functions

Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. 1 In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.

Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data.

(From OpenStax College Algebra)

Chapter Sections:

  • 4.1 Linear Functions
  • 4.2 Modeling with Linear Functions
  • 4.3 Fitting Linear Models to Data

OTM Outcomes

1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.). In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

1b. Convert between different representations of a function.*


1c. Perform operations with functions including addition, subtraction, multiplication, division, composition, and inversion; connect properties of constituent functions to properties of the resultant function; and resolve a function into a sum, difference, product, quotient, and/or composite of functions.*

4a. Interpret the function correspondence and behavior of a given model in terms of the context of the model.*

4b. Create linear models from data and interpret slope as a rate of change.*

Chapter Section Objectives and related OTM standards:

4.1 Linear Functions

  • Represent a linear function. (1a)
  • Determine whether a linear function is increasing, decreasing, or constant. (1a)
  • Interpret slope as a rate of change (1a).
  • Write and interpret an equation for a linear function. (1a)
  • Graph linear functions. (1a)
  • Determine whether lines are parallel or perpendicular. (1a)
  • Write the equation of a line parallel or perpendicular to a given line. (1a)

4.2 Modeling with Linear Functions

  • Build linear models from verbal descriptions. (4a, 4b)
  • Model a set of data with a linear function. (4a, 4b)

4.3 Fitting Linear Models to Data

  • Draw and interpret scatter diagrams.
  • Use a graphing utility to find the line of best fit.
  • Distinguish between linear and nonlinear relations.
  • Fit a regression line to a set of data and use the linear model to make predictions.

Section 2: Chapter 4 Worksheets

The files below are worksheets created to support practice and learning for chapter 4 of the OpenStax College Algebra text.

Section 3: Chapter 4 Desmos Activities

Please note, the Line of best Fit Activity does have discussions of residual. This is not presented in the OpenStax College Algebra text, so this activity will either need to be edited or the OpenStax material will need to be supplemented with concepts of residual. The content contributors feel that residual is a topic that should be presented with linear modeling, so it is recommended to supplement this content.

The links below will direct to pre-built Desmos activities. These can be easily copied, remixed, or compiled into activities for teachers to customize to their class and objectives. For more on how educators can integrate the Desmos activities into their class please click here to link to the Desmos teacher support site:

Below is a short explanation of the Desmos activates linked

Linear Bundle:

The linear bundle contains a large collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Polygraph: Lines
In this polygraph, students will work in pairs, one student will pick a line, it will be the job of the other person to eliminate lines that do not match that description. This will progress until the student has narrowed it down to one line. It requires the use of vocabulary and formulating a series of yes/no questions to correctly identify the line.

Polygraph: Lines Part 2
This is a follow up that explores steepness, positive and negative slope, as well as the concept of increasing and decreasing

Put the point on the Line
This activity's focus is slope. The goal is to sharpen students’ focus on slope. In particular, the activity asks students to estimate first, then to calculate, then to notice proportionality as they place points on an imaginary line. Use student ideas here to define slope as a ratio of change in y-coordinates to change in x-coordinates. By the time students get to the end of the activity, they should have a number of ways of talking about this, but it’s unlikely they’ll write a fraction with ∆y in the numerator and ∆x in the denominator. They’ll be ready for you to introduce this idea.

Match My Line
In this activity, students work through a series of scaffolded linear graphing challenges to develop their proficiency with direct variation, slope-intercept, point-slope, and other linear function forms.

Land the Plane
In this activity, students practice finding equations of lines in order to land a plane on a runway. Most of the challenges are well-suited to slope-intercept form, but depending on the goals of an individual class or student they are easily adapted to other forms of linear equations.

Card Sort: Linear Functions
This activity asks students to notice and use properties of linear functions to make groups of three. Different properties will lead to different groupings by different students. Later we ask students to make conjectures about different groupings – why might another student have grouped the cards in a particular way?

Two Truths and a Lie: Lines
Students will practice their understanding of the features and vocabulary of linear equations by creating a line, writing two true and one false statement about it, and inviting their peers to separate truth from lies.

Marbleslides: Lines
In this delightful and challenging activity, students will transform lines so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Charge!

In this activity, students model a situation with a linear function.
This first activity begins innocently enough: battery percentage as a function of time plugged in appears to grow at a steady, linear rate. And it does, at least in the beginning. So when students build a model, make a prediction, and check their answers, they find that they’re off—usually by an hour or more!

In addition to providing an opportunity to review key features of linear functions, Charge! invites students to consider their assumptions more carefully in future modeling problems, and provides a useful object lesson on the reliability of interpolation — making predictions between known data points — and extrapolation — making predictions beyond known data points.

Line of Best Fit

In this activity, students visualize a line to fit a data set, then graph that line with sliders, and use it to make a prediction. Teachers can use the final screen to introduce the concept of the residual.

Alligator Investigation

An enormous alligator lurks in the swamp. Can scatterplots and least-squares regression tell you if you have enough animal tranquilizer to stay safe?

 

Section 4: Chapter 4 Supplemental Videos

Below is a document that links to supplemental videos for this chapter. Note that these videos are not created by the publisher of the text, so some verbiage or problem solving strategies may vary from what is presented in the text.