This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will understand that calculus was developed to help model dynamic situations.

It is expected that students will:

• distinguish between static situations and dynamic situations
• identify the two classical problems that were solved by the discovery of calculus:
  • the tangent problem
  • the area problem
• describe the two main branches of calculus:
  • differential calculus
  • integral calculus
• understand the limit process and that calculus centers around this concept

SUGGESTED INSTRUCTIONAL STRATEGIES

Students will quickly come to realize that calculus is very different from the mathematics they have previously studied. Of greatest importance is an understanding that calculus is concerned with change and motion. It is a mathematics of change that enables scientists, engineers, economists, and many others to model real-life, dynamic situations.

• The concepts of average and instantaneous velocity are a good place to start. These concepts could be related to the motion of a car. Ask students if the displacement of the car is given by , what is the average velocity of the car between times and , and the instantaneous velocity at ? Point out that calculus is not needed to determine the average velocity , but is needed to determine the velocity at (the speedometer reading).
• As a way of solving the tangent problem (i.e., the need for two points and the fact that only one point is available), create a secant line that can be moved toward the tangent line. Ask students if they can move the secant line towards the tangent line using algebraic techniques.
• Ask students to determine the area under a curve by estimating the answer using rectangles. Introduce the concept of greater accuracy when the rectangles are of smaller and smaller widths (the idea of limit).
• Present the following statement for discussion:
  
  Use geometry to illustrate the relationship between the left-hand and right-hand sides.

SUGGESTED ASSESSMENT STRATEGIES

To demonstrate their achievement of the outcomes for this organizer, students need opportunities to engage in open-ended activities that allow for a range of responses and representations.

Although formative assessment of students’ achievement of the outcomes related to this organizer should be ongoing, summative assessment can only be carried out effectively toward the end of the course, when students have sufficient understanding of the details of calculus to appreciate the "big picture."

Self-Assessment

• Work with the students to develop criteria and rating systems they can use to assess their own descriptions of the two main branches of calculus. They can represent their progress with symbols to indicate when they have addressed a particular criterion in their work. Appropriate criteria might include the extent to which they:
  • make connections to other branches of mathematics
  • demonstrate understanding of the classical problems
  • consider the different attempts to solve the classical problem(s)
  • explain the significance of limits with reference to specific examples
  • accurately and comprehensively describe the relationship between integral and differential calculus
    (methodology and purpose)

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  pp. 83-85, 95, 190, 262, 272-273,375
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 57 - 61, 63
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 2 (Section 2.2)
  pp. 3-4, 6-9, 85, 351

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 2, 112, 114, 375
• Calculus of a Single Variable, Sixth Edition
  Ch. 1 (Section 1.1)
  Ch. 4 (Section 4.2, 4.3)
  pp. 41-45, 91, 265

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 5, 2000

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