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The Geometric Interpretation of Difference Quotient Lesson Plan

Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

In this lesson plan, your students will engage in learning to find the difference quotient of a function. Students will be given an opportunity to cement their understanding of the lesson concepts through an activity.

Learning Objectives

After this lesson, students will be able to:

  • Define the difference quotient of a function
  • Describe the difference quotient from a geometric standpoint
  • Name the type of line identified by the difference quotient

Length

60 minutes

Curriculum Standards

  • CCSS.MATH.CONTENT.8.F.A.1

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

  • CCSS.MATH.CONTENT.HSF.IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Materials

Warm-up

  • Review concepts important to this lesson:
    • What is a function?
    • What does the word difference mean in math? What about quotient?
    • What is the slope of a line?
    • What is a secant line?
  • Ask for volunteers to solve a few functions on the board or poster paper to remind your students how to work with functions. Choose simple functions such as:
    • f(x)= x - 2
    • f(2) = x - 2 = 2 - 2 = 0
    • f(x + 3)= x + 3 - 2 = x + 1

Instruction

  • Hand out transcripts of the video lesson.
  • Play the video lesson pausing at time marker 0:53. Discuss:
    • Can you explain the two points required for the difference quotient?
    • What is meant by point (x + h)?
  • Remind students that difference means subtraction and quotient means to divide, thus the difference quotient is simply a method of dividing a subtraction problem (subtraction on top of the fraction bar).
  • Continue the video pausing at time marker 2:43. Allow students time to ask any questions to clarify their understanding of the concepts to this point (prior to starting the example).
  • Continue the video pausing at time marker 3:43.
  • Work through two to three more examples to ensure that students understand the method for finding the difference quotient.
  • Play the remainder of the video.

Activities

Worksheet

Instructions

  • Instruct your students to create a 5 question worksheet covering the concepts of the lesson.
  • Remind students to create an answer key for their worksheet.
  • If working with a large group, instruct students to swap papers with another student and complete their new partner's worksheet. If working with an individual student, provide them with a few questions of your own to ensure thorough understanding.
  • Allow time for students to check their answers and ask questions if necessary.

Create a Circle

Materials

  • Index Cards (one per student)

Set-up

  • If working with a group of students, you will need exactly one card per student. If working with an individual student, you will need a set of cards for that student.
  • Prior to the activity, you will need to write specific functions and their difference quotients (DQ) on the index cards in this pattern:

Front of card Back of card
Card 1 DQ5 Function 1
Card 2 DQ1 Function 2
Card 3 DQ2 Function 3
Card 4 DQ3 Function 4
Card 5 DQ4 Function 5
  • Notice that the difference quotient for the function on the last card is shown on the front of the first card.
  • When placed so that each function faces its difference quotient, these cards form a circle because you must curve the line around so that the function printed on the last card can face its answer printed on the first card.


An example image of five possible cards in order.
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  • Consider including more complex functions than seen in the lesson to challenge your students.

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