Spotting Lines: Identifying Linear Functions

Algebra 1 Grades High School 4:01 Video
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Lesson Description

Learn how to distinguish between linear and non-linear functions from a set of points, focusing on constant rates of change and graphical representation.

Video Resource

Recognizing linear functions | Linear equations and functions | 8th grade | Khan Academy

Khan Academy

Duration: 4:01
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Key Concepts

  • Linear Function
  • Non-Linear Function
  • Constant Rate of Change
  • Rate of Change

Learning Objectives

  • Students will be able to determine if a function represented by a set of points is linear or non-linear.
  • Students will be able to explain the importance of a constant rate of change in identifying linear functions.
  • Students will be able to visualize linear and non-linear functions by plotting points on a graph.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a function. Briefly discuss what a linear equation looks like in graph form and algebraic form. Ask students to provide examples of linear and non-linear equations they have seen before.
  • Video Explanation (10 mins)
    Play the Khan Academy video 'Recognizing linear functions'. Encourage students to take notes on the method of identifying linear functions using the constant change in x and y values.
  • Guided Practice (15 mins)
    Work through several examples together. Provide sets of points and guide students to calculate the change in x and the change in y. Emphasize the importance of a constant change in y for a constant change in x. Discuss examples of both linear and non-linear functions.
  • Independent Practice (15 mins)
    Provide students with a worksheet containing sets of points. Students should determine whether each set of points represents a linear or non-linear function. Students should also plot the points on a graph to visually confirm their answers.
  • Wrap-up and Assessment (5 mins)
    Review the key concepts of the lesson. Administer a short multiple-choice and fill-in-the-blank quiz to assess student understanding.

Interactive Exercises

  • Online Graphing Tool
    Use an online graphing tool (e.g., Desmos) to plot the points from the independent practice worksheet. Visually confirm whether the plotted points form a straight line, indicating a linear function. Experiment with different sets of points to observe the behavior of linear and non-linear functions.
  • Group Sorting Activity
    Divide students into small groups. Provide each group with a set of cards, some containing linear functions (represented by points) and others containing non-linear functions. Have students sort the cards into two piles: 'Linear' and 'Non-Linear'. Each group should justify their sorting decisions.

Discussion Questions

  • How can you tell if a function is linear just by looking at its graph?
  • Why is a constant rate of change important for a function to be linear?
  • Can you think of real-world examples of linear and non-linear relationships?

Skills Developed

  • Analytical Skills
  • Problem-Solving Skills
  • Graphical Interpretation
  • Pattern Recognition

Multiple Choice Questions

Question 1:

Which of the following is true about a linear function?

Correct Answer: It has a constant rate of change.

Question 2:

If a function has points (1,2), (2,4), (3,6), (4,8), is it linear or non-linear?

Correct Answer: Linear

Question 3:

What is the graph of a linear function?

Correct Answer: A straight line

Question 4:

In a non-linear function, what happens to the rate of change?

Correct Answer: It varies.

Question 5:

The points (1, 5), (2, 10), (3, 20), (4, 40) represent which type of function?

Correct Answer: Non-linear

Question 6:

Which of the following is NOT a characteristic of a linear function?

Correct Answer: Curved line graph

Question 7:

If the change in y divided by the change in x is always constant, the function is:

Correct Answer: Linear

Question 8:

What does it mean for a rate of change to be 'constant'?

Correct Answer: It stays the same

Question 9:

A table shows x values increasing by 2 and y values increasing by 5 each time. Is this function likely linear?

Correct Answer: Yes

Question 10:

Which equation represents a linear function?

Correct Answer: y = 2x + 3

Fill in the Blank Questions

Question 1:

A function with a __________ rate of change is a linear function.

Correct Answer: constant

Question 2:

The graph of a linear function is a __________.

Correct Answer: straight line

Question 3:

If the change in 'y' is not constant for a constant change in 'x', the function is ___________.

Correct Answer: non-linear

Question 4:

To determine if a function is linear from a table, check if the change in y divided by the change in x is always ___________.

Correct Answer: constant

Question 5:

A linear function grows by __________ intervals.

Correct Answer: equal

Question 6:

The constant rate of change in a linear function is also known as its __________.

Correct Answer: slope

Question 7:

In a linear function, for every change in x, the change in y is always the __________ value.

Correct Answer: same

Question 8:

If a set of points does not form a straight line when plotted, it represents a __________ function.

Correct Answer: non-linear

Question 9:

An exponential function grows by equal __________ over equal intervals.

Correct Answer: factors

Question 10:

The change in y over the change in x represents the __________ of the function.

Correct Answer: rate of change

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.A.4.1:Analyze, use and apply mathematical models and other data sets (e.g., graphs, equations, two points, a set of data points) to calculate and interpret slope and the x- and y-intercepts of a line.

Teaching Materials

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