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Fundamental Theorem of Calculus

Calculus AB: Develop conceptual understanding, procedural fluency, modeling, and proof in Fundamental Theorem of Calculus.

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Beginner 15–20 minutes Calculus
Calculus Page 10 of 24
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At a Glance

Fundamental Theorem of Calculus
Binder SectionCalculus
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
Essential Question

How can the structures in Fundamental Theorem of Calculus be represented, justified, and applied to unfamiliar problems?

Visual Model
4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

  • I can define and represent the central ideas in Fundamental Theorem of Calculus.
  • I can select and execute an appropriate mathematical method.
  • I can justify each transformation and evaluate the reasonableness of a result.
  • I can transfer the idea to a new model or proof.
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Interactive Vocabulary

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Worked Examples

Example 1

Analyze a Fundamental Theorem of Calculus problem

1

State the given information, restrictions, and goal.

Example 2

Connect representations

1

Translate the result into a second form.

Notice and correct

Common Misconceptions

Ignoring restrictions or domain

Record restrictions before manipulating expressions or figures.

Using a theorem or rule without its conditions

Name the conditions and verify that they hold.

Stopping at a symbolic answer

Interpret, check, and communicate the result in context.

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Calculus Basics: Limits, Derivatives, and Integrals

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Quick Start

Warm-Up

Warm-up 1

Directions: Activate prior knowledge

State one prerequisite idea for Fundamental Theorem of Calculus, give an example, and explain the connection.

Picture your planUse the picture to organize what you know before you solve.
What do Iknow?Which rulefits?What must Ifind?

Choose how you want to work or answer.

2
We Do

Guided Practice

Guided 1

Directions: Develop the method

Solve a teacher-selected Fundamental Theorem of Calculus problem, annotating why each step is valid.

Picture your planUse the picture as you work through each step.
What do Iknow?Which rulefits?What must Ifind?

Choose how you want to work or answer.

3
You Do

Independent Practice

Independent 1

Directions: Demonstrate transfer

Create and solve a nonroutine Fundamental Theorem of Calculus problem with different parameters.

Picture equivalent unitsUse the picture to organize what you know before you solve.
1 meter1 m100 centimeters100 cm

Choose how you want to work or answer.

4
Stretch Your Thinking

Challenge & Real-World Practice

🧩 Challenge

Synthesis challenge: Combine Fundamental Theorem of Calculus with an earlier Calculus AB topic. State assumptions, solve using two approaches, compare efficiency, and defend the conclusion.

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Success Criteria

Before you begin, I can…

  • I can define and represent Fundamental Theorem of Calculus precisely.
  • I can select a valid and efficient method.
  • I can justify transformations, theorems, or modeling choices.
  • I can verify, interpret, and communicate a conclusion.
  • I can transfer the idea to an unfamiliar problem.
1
Question 1

What belongs at the start of an advanced problem?

2
Question 2

What makes a transformation valid?

3
Question 3

Which response is strongest?

4
Question 4

What is an independent verification?

5
Question 5

What demonstrates mastery?

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