Authentic Lessons for 21st Century Learning

Lesson 5E Lesson
Cover Image

I Sub, U-Sub, We All Sub Again

Integration: U-Subsitution, Part 2

Cacey Wells | Published: August 4th, 2022 by K20 Center

  • Grade Level Grade Level 12th
  • Subject Subject Mathematics
  • Course Course AP Calculus
  • Time Frame Time Frame 75 minutes
  • Duration More 2-3 class periods

Summary

This lesson is a follow-up to I Sub, U-Sub, We All Sub. In the previous lesson, students worked with the teacher to create an anchor chart of what "worked" and what "didn't work" for anti-differentiating derivatives of functions. The goal of this lesson is to build on that knowledge in order for students to begin seeing that what "worked" worked for a specific reason. Coupling this notion with the connection to what "didn't work" will allow students to see how U-substitution is a manipulated form of what did work in Part 1.

Essential Question(s)

How can we undo the chain rule? In other words, how can we find the antiderivative of a function that has been differentiated using the chain rule?

Snapshot

Engage

Using the anchor chart from the I Sub, U-Sub, We All Sub lesson, teachers and students identify what worked and what didn't work. Students label part of the equation as u and the other part as du.

Explore

Students explore the idea that part of the derivative of the chain rule looks similar to the derivative of another part of the function. Students then explore how to identify these parts as u and du.

Explain

Clarify what u represents and what du represents. Students then articulate that the integral of u du, looks like the integral of x dx.

Extend

Practice with these new types of problems in a scavenger hunt, identifying u and du, and then finding the antiderivative.

Evaluate

What? So What? Now What?

Materials

  • Paper and pencil

Engage

Display the anchor chart from the I Sub, U-Sub, We All Sub lesson.

Have students articulate which parts of the equation look like the derivative of the other part (u and u').

Pose the question, "What if we replace u' with du? Would that change anything?"

On the anchor chart, replace u' with du in each of the equations that "didn't work."

Explore

Distribute the Exploring U-Sub handout to each student. Ask students to work in pairs to go through the problems.

As students progress through the activity, check in with each pair to help clarify questions that they might have.

Explain

After students have finished both parts of the Explore activity, move on to the Explain activity, which is intended to clarify misconceptions and make sure students all understand the concepts.

Project or print the questions in the Explain Handout and ask students to explain each question. Give students ample time to work with their partner to explain their reasoning.

Extend

Have students complete the scavenger hunt, either individually or in pairs. Be sure to make yourself available if students have questions.

Evaluate

Display the attached Evaluate document to have students reflect on their learning and try a challenge problem.

Give students two or three minutes each to complete the first two questions. Allow students to explore the final question and use it as a starting point for conversations about manipulating the substitution.

And...we're done!

Resources

Publisher

K20 Center

View »

Downloads

Download PDF

Standards

Oklahoma Academic Standards for Mathematics (Process Standards)
PK.S: Mathematical Actions and Processes
M.1: Develop a Deep and Flexible Conceptual Understanding: Demonstrate a deep and flexible conceptual understanding of mathematical concepts, operations, and relations while making mathematical and real-world connections. Students will develop an understanding of how and when to apply and use the mathematics they know to solve problems.
M.2: Develop Accurate and Appropriate Procedural Fluency: Learn efficient procedures and algorithms for computations and repeated processes based on a strong sense of numbers. Develop fluency in addition, subtraction, multiplication, and division of numbers and expressions. Students will generate a sophisticated understanding of the development and application of algorithms and procedures.
M.3: Develop Strategies for Problem Solving: Analyze the parts of complex mathematical tasks and identify entry points to begin the search for a solution. Students will select from a variety of problem solving strategies and use corresponding multiple representations (verbal, physical, symbolic, pictorial, graphical, tabular) when appropriate. They will pursue solutions to various tasks from real-world situations and applications that are often interdisciplinary in nature. They will find methods to verify their answers in context and will always question the reasonableness of solutions.
M.4: Develop Mathematical Reasoning: Explore and communicate a variety of reasoning strategies to think through problems. Students will apply their logic to critique the thinking and strategies of others to develop and evaluate mathematical arguments, including making arguments and counterarguments and making connections to other contexts.
M.5: Develop a Productive Mathematical Disposition: Hold the belief that mathematics is sensible, useful and worthwhile. Students will develop the habit of looking for and making use of patterns and mathematical structures. They will persevere and become resilient, effective problem solvers.
M.6: Develop the Ability to Make Conjectures, Model, and Generalize: Make predictions and conjectures and draw conclusions throughout the problem solving process based on patterns and the repeated structures in mathematics. Students will create, identify, and extend patterns as a strategy for solving and making sense of problems.
K.PS2: Develop the Ability to Communicate Mathematically: Students will discuss, write, read, interpret and translate ideas and concepts mathematically. As they progress, students’ ability to communicate mathematically will include their increased use of mathematical language and terms and analysis of mathematical definitions.

Learner Handouts

Click on a file type icon to download an attachment.

  • Evaluate
  • Explain Handout
  • Exploring U-Sub
  • Scavenger Hunt

This work is licensed under a Creative Commons CC BY-SA 4.0 License.

Report copyright infringement »