# MA.8.F.1.2

Given a function defined by a graph or an equation, determine whether the function is a linear function. Given an input-output table, determine whether it could represent a linear function.

### Clarifications

Clarification 1: Instruction includes recognizing that a table may not determine a function.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Functions
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Function
• Linear Function

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students determined whether a relationship was proportional, given a table, equation or written description. In grade 8, students determine whether the function defined by a graph or an equation is a linear function. In Algebra 1, students will classify the function type given an equation or graph and compare key features of linear and nonlinear functions.
• Instruction includes determining if the function has a constant rate of change between the $x$- and $y$-values.
• For example, students can depict the rate of change between the values in a table like below.

• Instruction includes a focus on the connection between the equation and the graph.
• Students should develop an understanding of a linear function by using examples and non-examples.

### Common Misconceptions or Errors

• Students may not understand the connection from a table to the visual of a graph of the same function. To address this misconception, provide opportunities for students to make connections and see the graph and table side by side.

### Strategies to Support Tiered Instruction

• Teacher models how to get from a set of points displayed on a table to the points graphed on a coordinate plane, and how points from a coordinate plan can be written in a table. Then, teacher provides opportunities to notice any patterns in the graph or table that will help identify if the function is linear.
• Teacher co-constructs a graph from a table with students, as well as a table from a graph to increase understanding of the relationship between the two. Once students become comfortable moving between graphs and tables, students can begin inspecting tables that represent functions. Teachers can review proportional and linear relationships and work with students to dissect tables to find if they contain a proportional relationship, meaning they are linear. Students should note that not all linear relationships are proportional, but all proportional relationships are linear.
• Teacher provides opportunities for students to make connections and see the graph and table of the same function side by side.

The area, $A$, of an isosceles right triangle is a function of the length of its legs, $s$, and is represented by the equation $A$ = 0.5$s$².
• Part A. Create a table of values to represent this function.
• Part B. Plot the points on a coordinate plane.
• Part C. What is the domain and range of the function?
• Part D. Is this function linear or nonlinear? Explain and justify your answer.

Part A. Create a table that could represent a linear function.
Part B. Create a table that could represent a non-linear function.
Part C. Compare your table from Part B with a partner.

### Instructional Items

Instructional Item 1
Taro and Jiro climbed a mountain and hiked back down. At the summit and at every station along the way back down, they recorded their altitude and the amount of time they had been travelling. Can the data in the table represent a linear function?

Instructional Item 2
Does the graph below represent a linear function? If so, justify your answer.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.F.1.AP.2: Given a function displayed on a graph or an equation, identify whether the function is a linear function.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

What Am I?:

Students are asked to describe a linear function, its graph, and the meaning of its parameters.

Type: Formative Assessment

Linear or Nonlinear?:

Students are asked to identify a function as either linear or nonlinear and to justify their decision.

Type: Formative Assessment

Explaining Linear Functions:

Students areĀ asked to describe defining properties of linear functions.

Type: Formative Assessment

## Lesson Plans

The Linear Function Connection:

The students will compare two linear functions that have been represented in different ways (equation, table, graph, verbal description). They will be able to find and compare the rate of change, or slope, of the function from any of the representations.

Type: Lesson Plan

Functions: Are They Linear or Non-Linear?:

In this lesson, students will investigate 5 different functions to see if they are linear or non-linear. They will then analyze the functions in groups. After that they will present their results and reasoning.

Type: Lesson Plan

## Original Student Tutorials

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

Summer of FUNctions:

Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial.

Type: Original Student Tutorial

## Virtual Manipulative

Functions and Vertical Line Test:

This lesson is designed to introduce students to the vertical line test for functions as well as practice plotting points and drawing simple functions. The lesson provides links to discussions and activities related to the vertical line test and functions as well as suggested ways to integrate them into the lesson.

Type: Virtual Manipulative

## MFAS Formative Assessments

Explaining Linear Functions:

Students areĀ asked to describe defining properties of linear functions.

Linear or Nonlinear?:

Students are asked to identify a function as either linear or nonlinear and to justify their decision.

What Am I?:

Students are asked to describe a linear function, its graph, and the meaning of its parameters.

## Original Student Tutorials Mathematics - Grades 6-8

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

## Original Student Tutorials Mathematics - Grades 9-12

Summer of FUNctions:

Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

Summer of FUNctions:

Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial.

Type: Original Student Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.