Summary
In this lesson, students will use their knowledge of derivatives and pattern-recognition skills to find antiderivatives. Students will begin with the power rule for integration, evaluate the indefinite integral of basic trigonometric expressions, then learn how to integrate with u-substitution. Students are introduced to the idea of integration and its notation then apply their knowledge of antidifferentiation in a digital escape-room style activity. This lesson is intended to be used after completing derivatives and before the introduction of area and definite integrals. However, it can be adapted to be used later in the course.
Essential Question(s)
How can we find antiderivatives using our knowledge of derivatives?
Snapshot
Engage
Students use a table of derivatives and antiderivatives and use pattern recognition to complete the table, finding their first antiderivative.
Explore
Students recall the chain rule for derivatives, then using pattern recognition, attempt to “undo” the chain rule.
Explain
Students formalize their notation and understanding of antiderivatives and indefinite integrals.
Extend
Students practice finding antiderivatives and learn a little calculus history within an escape-room style Desmos Classroom activity.
Evaluate
Students demonstrate their understanding by creating their own composition function, finding its derivative, then trading to find the antiderivative of their peer’s problem.
Materials
Lesson Slides (attached)
Backward Beginnings (attached; one per student; printed front only)
Breaking the Chain (attached; one per student; printed front only)
Guided Notes handout (attached; one per student; printed front/back)
Guided Notes (Model Notes) document (attached; for teacher use)
Great Calculus Mystery handout (attached; one per student; printed front only)
Great Calculus Mystery (Teacher Guide) document (attached; for teacher use)
Index cards (one per student)
Pencils
Paper
Student devices with internet access
Engage
10 Minute(s)
Introduce the lesson using the attached Lesson Slides. Share the lesson’s essential question on slide 3 and the learning objectives on slide 4. Review each of these with your class to the extent you feel necessary.
Have students find a partner or assign partners, then display slide 5. Give each student a copy of the attached Backward Beginnings handout. Let students know that today they will be learning about antiderivatives, but do not yet tell students what antiderivatives are, as they will be using their pattern-recognition skills to try to answer this question themselves. Direct students’ attention to the I Notice section of their handout. Have pairs work together to complete the table and find the missing antiderivative, using the pattern(s) they notice from the other rows of the table. As students work, monitor progress. For students who seem stuck, ask guiding questions, like the ones below, rather than directly telling them how to find the missing information.
What are you sure/unsure of?
What do we know about derivatives?
What have you tried so far?
Instead of trying to figure out how to go from the first column to the second column, what if we thought about it backwards and tried to figure out how to go from the second column to the first?
As pairs complete their tables, move to slide 6 and ask for volunteers to share what they wrote for their last antiderivative. Then introduce the I Think / We Think strategy. Direct students' attention to the I Think portion of their handout. Ask students to quietly think about how they could describe the reasoning they used to complete the table and independently record their thinking on their handout.
After a couple of minutes, have each student share with their partner their thinking then show slide 7. Direct students’ attention to the We Think portion of their handout. Have pairs write a general rule describing how they completed the table. Let students know that the idea of a general rule (or generalizing) is that the written rule needs to work for all of the rows of the table, not just the last row of the table. After a couple of minutes, ask for volunteers to share their general rules. Use student responses to see if there are misconceptions. If the misconceptions are of derivatives, pause the lesson to address those prior-knowledge misunderstandings. If the misconceptions are of antiderivatives, be sure to address those during the Explain portion of the lesson.
Explore
10 Minute(s)
Display slide 8 and have pairs of students form small groups of 4. Give each student a copy of the attached Breaking the Chain handout and have students determine who is “Student 1,” “Student 2,” etc. This numbering simply determines the order of the tasks, but does not change anyone’s difficulty of work. Introduce the class to the Pass the Problem strategy.
Move to slide 9 and direct students to use the given composition function, f(g(h(x))) to write f( ) in the second column of their row: Student 1 writes in row 1, Student 2 writes in row 2, etc. Then students pass their paper to the next person: Student 1 passes to Student 2; …; and Student 4 passes to Student 1. On the paper they just received, have students write g( ) in the third column of the row that has been started.
Show slide 10 and have students pass their papers again, in the same manner. On the paper they just received, have students write h(x) in the fourth column of the row that has been started. Then have them pass their papers again and complete the row by finding the derivative of the composition function (using the chain rule).
Display slide 11 and direct students to pass their paper back to the original owner. Have students repeat these steps again for the next row, while skipping the fifth row: Student 1 starts row 2; …; and Student 4 starts row 1.
Once students have their original papers back a second time, transition to slide 12. Have students complete the remaining two rows, still skipping the fifth row. Once students are finished, have students check their work within their group.
As students are checking their work, transition through slides 13–14 so groups can check their work. Give students time to ask clarifying questions about the chain rule for derivatives.
Move to slide 15 and tell students that their challenge is to use the given derivative in the last row to complete the table. Have students work quietly and independently for a few minutes before allowing them to discuss with their group.
Show slide 16 and share the composition function that goes with the given derivative. Ask for volunteers to share how they figured out the original composition function. Then ask if they know what word they might use to describe the function in column 1 if the derivative is in the last column. Use this to transition to the Explain portion of the lesson where students learn more about antiderivatives.
Explain
20 Minute(s)
Show slide 17 and give each student a copy of the attached Guided Notes handout. Use the slide to explain to students the meaning of the notation for antiderivatives/integrals. Be sure to emphasize the use of antidifferentiation being the same as integration and that general antiderivatives are the same as indefinite integrals.
Move to slide 18, then complete the Guided Notes handout with the class. Ask students to recall the activity from the Engage portion of the lesson and have them tell you how to write the power rule for integrals. Similarly, instead of telling students the antiderivative of the sine and cosine functions, ask them to tell you what they should be. Be sure for each example to emphasize the importance of the constant of integration (+C). Use the attached Guided Notes (Model Notes) document as needed.
Extend
35 Minute(s)
Show slide 19 and provide students with your session code. Then, have students go to student.desmos.com and enter the session code.
Give each student a copy of the attached Great Calculus Mystery handout and have them get a piece of notebook paper and pencil. Introduce the activity using screen 1, which shares how this escape-room style activity will function. Remind students that they will need to show their work on their notebook paper and be expected to turn in this scratch work when they are finished.
Have students work independently in pairs or individually through the activity. Here students are asked to find general antiderivatives of basic functions, identify u and du, rewrite integrals using u-substitution, and lastly evaluate indefinite integrals. Use the attached Great Calculus Mystery (Teacher Guide) document for more details and support of this activity.
Evaluate
10 Minute(s)
Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Display slide 20 and have each student create a composition function, writing it on the back of their Great Calculus Mystery handout. Then have students find the derivative of their own function. Encourage students to not create composition functions that will require the product or quotient rule to find the derivative, and if they did, tell them to pick a different composition function. As they work, give each student an index card. Have students write the simplified derivative on their index card, labeling it h'(x).
Transition to slide 21. Have students trade index cards and evaluate the indefinite integral they have been given, labeling their antiderivative h(x).
Once they are done, show slide 22 and have students trade their index cards back for their classmate to check their work, comparing it to their original composition function. If h(x) does not match the original composition function, have students work together, sharing their work and thinking for both the derivative and antiderivative to find the error.
Resources
K20 Center. (n.d.). Bell ringers and exit tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Desmos classroom. Tech Tools. https://learn.k20center.ou.edu/tech-tool/1081
K20 Center. (n.d.). I think / we think. Strategies. https://learn.k20center.ou.edu/strategy/141
K20 Center. (n.d.). Pass the problem. Strategies. https://learn.k20center.ou.edu/strategy/151