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Geometry Resources & Lesson Plans7 chapters | 199 lessons

Instructor:
*Maria Airth*

Maria has a Doctorate of Education and over 15 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 through collaboration. Students will be out of their seats actively moving around as they cement their understanding of the lesson concepts.

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

60 minutes

- 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.

- Printed copies of the transcript of the video lesson The Geometric Interpretation of Difference Quotient
- Hard copies of the lesson quiz
- Audio/video equipment
- Index cards

- 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 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

- f(

- Hand out transcripts of the video lesson.
- Play the video lesson pausing at time marker 0:53. Discuss:
- Can anyone 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 on the board as a class to ensure that students understand the method for finding the difference quotient.
- Play the remainder of the video.

- Divide your students into pairs.
- Instruct your students to create a 5 question worksheet covering the concepts of the lesson.
- Remind the pairs to create an answer key for their worksheet.
- Instruct students to swap papers and work in their pairs to complete the other's worksheet.
- Allow time for students to check their answers and ask questions if necessary.

- Index Cards (one per student)

- You will need exactly one card per student.
- Prior to class 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.

- Consider including more complex functions than seen in the lesson to challenge your students.

- Before the activity, ask students to move their desks away from the middle of the classroom.
- Randomly hand out one card per student.
- Explain to your students that their card has a function on one side and the answer to a different function on the back. Their goal, as a class, is to set the cards up in a circle so that each function faces its correct answer (the functions and their answers will be on different cards).
- Allow students free reign to work together to form a circle with the cards. They should line themselves up in the correct order to form a circle of functions and their corresponding difference quotients (if you have students missing, you will need to assist your class to get the extra cards placed in the right order).
- If students are struggling, remind then it is like a matching game in which all the answers are given, they just need to match up the correct questions with the correct answers to form the circle.
- You may want to consider including one letter in the corner of each card so that when the cards are in order they spell out a secret message to the class (such as 'Good job everyone.')

- Replay the 'Lesson Summary' section of the video lesson (starting at time marker 3:43).
- Allow students time to review their worksheets and ask any final questions before the quiz.
- Hand out the lesson quiz. Go over each question and answer with the class after they have finished it.

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