Is It Linear? Spotting Linear Functions from Tables

Algebra 1 Grades High School 2:26 Video
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Lesson Description

Learn how to determine if a function represented by a table is linear by checking for a constant rate of change. This lesson uses real-world examples to make the concept clear.

Video Resource

Linear and nonlinear functions (example 1) | 8th grade | Khan Academy

Khan Academy

Duration: 2:26
Watch on YouTube

Key Concepts

  • Linear Function
  • Rate of Change
  • Constant Rate of Change
  • Change in x (Δx)
  • Change in y (Δy)

Learning Objectives

  • Students will be able to determine if a function represented in a table is linear.
  • Students will be able to calculate the rate of change from a table of values.
  • Students will be able to identify a constant rate of change.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a linear function and its graphical representation (a straight line). Briefly discuss different representations of functions (equations, graphs, tables).
  • Video Viewing (7 mins)
    Play the Khan Academy video 'Linear and nonlinear functions (example 1) | 8th grade | Khan Academy'. Instruct students to pay attention to how the rate of change is calculated and used to determine linearity.
  • Guided Practice (15 mins)
    Work through additional examples of tables, calculating the rate of change between different pairs of points. Emphasize the importance of a *constant* rate of change for a function to be linear. Guide students through the calculation of Δy/Δx.
  • Independent Practice (15 mins)
    Provide students with a worksheet containing several tables of values. Students should determine whether each table represents a linear function and justify their answer by showing their rate of change calculations.
  • Wrap-up and Assessment (8 mins)
    Review the key concepts and address any remaining questions. Administer the multiple-choice and fill-in-the-blank quizzes.

Interactive Exercises

  • Rate of Change Calculator
    Use an online rate of change calculator (or spreadsheet software) to input values from a table and visualize the rate of change between different points. Discuss how deviations in the rate of change indicate non-linearity.
  • Table Sort
    Create cards with different tables of values (some linear, some non-linear). Have students work in groups to sort the tables into linear and non-linear categories, justifying their choices.

Discussion Questions

  • What does a constant rate of change mean in the context of a linear function?
  • How can you quickly identify a non-linear function from a table?
  • Can a linear function have a rate of change of zero? What would the graph look like?

Skills Developed

  • Calculating rate of change
  • Identifying linear functions
  • Analyzing data from tables
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the most important characteristic of a linear function when examining a table of values?

Correct Answer: A constant rate of change

Question 2:

Which of the following represents 'change in y'?

Correct Answer: Δy

Question 3:

If the rate of change between two points in a table is 2, and the rate of change between another two points is 3, is the function linear?

Correct Answer: No

Question 4:

What is the formula for calculating the rate of change (slope) between two points (x1, y1) and (x2, y2)?

Correct Answer: (y2 - y1) / (x2 - x1)

Question 5:

A table shows x-values increasing by 1 each time. If the y-values also increase by a constant amount each time, what can you conclude?

Correct Answer: The function is linear

Question 6:

What does it mean if the rate of change (Δy/Δx) is negative?

Correct Answer: y is decreasing as x increases

Question 7:

A horizontal line on a graph indicates what type of rate of change?

Correct Answer: Zero

Question 8:

If Δx = 0, what can you conclude about the line?

Correct Answer: The line is vertical

Question 9:

A table shows x-values that are not consistently increasing. How does this affect your ability to determine linearity?

Correct Answer: You must still calculate the rate of change between each pair of points

Question 10:

The rate of change is also known as the ______ of a line.

Correct Answer: slope

Fill in the Blank Questions

Question 1:

A linear function has a ______ rate of change.

Correct Answer: constant

Question 2:

The change in y is represented by the symbol ______.

Correct Answer: Δy

Question 3:

To determine linearity from a table, you must calculate the ______ between pairs of points.

Correct Answer: rate of change

Question 4:

If the rate of change is different between two pairs of points, the function is ______.

Correct Answer: non-linear

Question 5:

The rate of change is calculated by dividing the change in y by the change in ______.

Correct Answer: x

Question 6:

If Δy/Δx is always the same, the table represents a ______ function.

Correct Answer: linear

Question 7:

A constant increase in x and a constant decrease in y still represents a ______ function.

Correct Answer: linear

Question 8:

A table is one way to represent a ______.

Correct Answer: function

Question 9:

The formula for the rate of change is often described as rise over ______.

Correct Answer: run

Question 10:

A linear function's graph is always a straight ______.

Correct Answer: line

Educational Standards

A1.F.2.1:Distinguish between linear and nonlinear (including exponential) functions. Understand that linear functions grow by equal intervals (arithmetic) and that exponential functions grow by equal factors over equal intervals (geometric). A1.F.3.1:Identify and generate equivalent representations of linear functions, graphs, tables, and real-world situations. A1.F.1.1:Distinguish between relations and functions.

Teaching Materials

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