Making Connections

Connecting Functions and Their Derivatives

Cacey Wells, Michell Eike, Sherry Franklin, Cacey Wells | Published: October 14th, 2022 by K20 Center

• Subject Mathematics
• Course AP Calculus
• Time Frame 95-135 minutes
• Duration 2-3 class periods

Summary

The goal of this lesson is to help students understand the relationships between a function and its first and second derivatives. Students will analyze graphs recalling their knowledge of sketching graphs by hand using the first and second derivatives and apply their graphical knowledge verbally through a Leap Frog game.

Essential Question(s)

How are the graphs of a function and its derivatives related?

Snapshot

Engage

Students make observations about four functions and their corresponding derivative as the derivative of the function is graphed in real time.

Explore

Students use one graph, viewing it once as the first derivative and once as the second derivative, to draw conclusions about the function.

Explain

Students look at three unlabeled curves and label them as f, f ’, and f ’’, and as a class, formalize their understanding of how the function and its derivatives are related.

Extend

Students apply their understanding of the relationships between the function and its first and second derivatives through a Leap Frog game.

Evaluate

Students complete an Exit Ticket where they answer an AP exam-style question to demonstrate their understanding of derivative relationships.

Materials

• Lesson Slides (attached)

• Exploring Relationships handout (attached; one per pair; printed front and back)

• Which Graph is Which? handout (attached; one per student; printed front only)

• Leap Frog Student Cards handout (attached; one set per student; printed front only)

• Leap Frog Teacher Cards document (attached; for teacher use; one set; printed front only)

• Exit Ticket handout (attached; one half per student; printed front only)

• Pencils

• Coloring utensils (3 colors per student; highlighters, colored pencils/pens, etc.)

• Graphing calculators

• Student devices with internet access

• Card stock (optional)

Engage

15 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 to share the lesson's essential question with students. Move to slide 4 to share the lesson's learning objectives. Review each of these with students to the extent you feel necessary.

Ask students to find a partner or assign them. Display slide 5 and give each pair of students a copy of the attached Exploring Relationships handout. Introduce students to the I Notice, I Wonder strategy and direct them to draw a table on the back of their handout.

Show slide 6 and have students go to the Making Connections Wakelet collection of graphs. These graphs have an animated tangent line and first derivative graph. Direct students to click on any two of the graphs to create and complete an I Notice, I Wonder table. Give students some time to reflect and talk with their partner about their discoveries.

Ask for volunteers to share which graphs they selected and what they noticed and wondered about.

Explore

30 Minute(s)

Display slide 7 and introduce students to the Inverted Pyramid strategy. Direct students’ attention to the front of the Exploring Relationships handout. Have students graph the given function on their graphing calculator. This could be done by hand but the calculator will save time.

For Questions 1 and 2, have students imagine that the graph on their calculator is the first derivative of a function. Question 1 asks students to describe the function, using the graph of the derivative (the graph on their calculator). Question 2 asks students to describe the second derivative.

Have students work in pairs to answer Questions 1 and 2. After a few minutes, show slide 8 and have pairs of students find another pair of students (creating a group of four) to compare their results and reasoning.

Bring the class together for a whole group discussion. Have one student from each group share their responses and write the responses on the board for all to see. Before moving on, make sure students have represented everything for Questions 1 and 2.

Give students time to ask questions, and encourage students to justify their answers.

Show slide 11 and have students split back into their original pairs to answer the remaining questions. For Questions 3 and 4, students are looking at the same graph on their calculator, but now imagining that it is the second derivative of some function. Question 3 asks students to describe the function, and Question 4 asks them to describe the first derivative.

After a few minutes, display slide 12 and have pairs of students create new groups of four to share and compare their results and reasoning. Working with different peers fosters the development of academic vocabulary and encourages students to consider different approaches to a problem.

Bring the class together for a whole group discussion. Have one student from each group share their responses and write the responses on the board for all to see. Before moving on, make sure students have represented everything for Questions 3 and 4.

Give students time to ask questions, and encourage students to justify their answers.

Explain

20 Minute(s)

Show slide 15 and give each student a copy of the attached Which Graph is Which? handout and three coloring utensils.

Students are to use the coloring utensils to color coordinate their functions. In other words, students should use one color for the first derivative on the first and second graph, a second color on the second derivative of both graphs, and a third color for f(x) on both graphs. Therefore, students could trace each curve on the first graph now or after the class decides which graph is which.

After students have had some time to analyze the graphs and make notes, ask if anyone thinks they know which graph is the original function and why. If there are not any volunteers, ask what types of notes they have made to help the class see how using the properties of the graphs and using process of elimination they are able to make a decision.

Use slide 16 to share and review one approach to the question. Give students time to process and ask questions.

Show slide 17 and ask students to work with their partner to try the second example on their handout. As students work, circulate the room and listen to student discussions. Once a few pairs of students finish, display slide 18 so that students can check their work.

Once the class has finished, ask for volunteers to explain why the graph on the slide is labeled correctly.

Extend

25 Minute(s)

Display slide 19. Remind students to be kind and careful with the printed cards, then pass out a set of the Leap Frog Student Cards to each student and direct them to arrange the cards on their desk such that they could easily select one as the answer to a question. Give the warning that students should not spend too much time arranging their cards, as they will be at different desks throughout the game.

Display slide 20. Explain how the game is played and then use one of the attached Leap Frog Teacher Cards to model the procedure. This is an ideal time to select a card that everyone should get correct.

Explain to students that only one of their cards is correct even if multiple feel like they could be. The correct card is the one that creates a biconditional statement (if and only if). In other words, if the teacher card is true, then the student card is true AND if the student card is true, then the teacher card is true. So students need to think about that before selecting a card. For example, if the function has a relative maximum (teacher card), then the first derivative changes from positive to negative (student card). The first derivative would also be zero (student card), but it being zero does not determine if the function has a relative maximum (teacher card). Therefore, the correct card is the student card that states that the first derivative changes from positive to negative and not the card that says that the first derivative is zero.

Read the teacher card aloud, repeating it at least once, then have students select a student card from their desk and place it on their forehead, facing you. Scan the room to get an idea of who answered the question correctly but do not try to keep track of this information. Read the correct response aloud and direct students to place the card back on their desk. Have students who selected the correct card stand up and move along the path you explained earlier to the next empty seat. Students usually do a good job of honestly self-managing.

Repeat this until a student makes it back to his/her seat (or time is called).

If time allows, play another round by having students go back to their original seats and begin the game again.

Evaluate

5 Minute(s)

Display slide 21 and use the Exit Ticket strategy to assess what students have learned individually. Give each student a copy of the attached Exit Ticket handout. Students are asked to determine which graph would have a relative minimum given graphs of the derivative.