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 solve applied problems from a variety of fields including the Physical and Biological Sciences, Economics, and Business.

It is expected that students will:

• solve problems involving displacement, velocity, acceleration
• solve related rates problems
• solve optimization problems (applied maximum/minimum problems)

SUGGESTED INSTRUCTIONAL STRATEGIES

Calculus was developed to solve problems that had previously been difficult or impossible to solve. Such problems include related rate and optimization problems, which arise in a variety of fields that students may be studying (e.g., the physical and biological sciences, economics, and business).

• Have a student demonstrate a "student trip" in front of the class — constant speed, accelerate, stop, slow down, stop, back up. Have students sketch a graph of displacement against time for the movements.
• Discuss average velocity over a specified interval, instantaneous velocity at specified times. Have students in pairs create a graph and have their partner perform the motion.
• Have students brainstorm examples of the need for calculus in the real world:
  • population growths of bacteria
  • the optimum shape of a container
  • water draining out of a tank
  • the path that requires the least time to travel
  • marginal cost and profit
• Challenge students to create their own "new" problems, which they must then try to solve. These problems could be used to develop tests or unit reviews.
• Have students use technology (e.g., graphing calculators) to investigate problems and confirm their analytical solutions graphically.
• Discuss with students the merits of using versus not using a graphing calculator "to do" calculus.

SUGGESTED ASSESSMENT STRATEGIES

Students develop their knowledge of derivatives by solving problems involving rates of change, maximum, and minimum. When assessing student performance in relation to these problems, it is important to consider students’ abilities to make generalizations and predictions about how calculus is used in the real world.

Observe

• While students are working on problems involving derivatives, look for evidence they:
  • clearly understand the requirements of the problem
  • recognized when a strategy was not appropriate
  • can explain the process used to determine their answers
  • used a graphing calculator where appropriate to help visualize their solution
  • verified that their solutions where correct and reasonable
• Provide a number of problems where students are required to use average velocity or instantaneous velocity. Observe the extent to which students are able to:
  • determine whether the situation calls for the calculation of average velocity or instantaneous velocity
  • explain the differences between the two
  • provide other examples

Collect

• Assign a series of problems that require students to apply their knowledge of the dynamics of change: how, at certain time the speed of an object is related to its height, and how, at a certain time the speed of an object is related to the change in velocity.
• To discover how well students can recognize and explain the concepts of change and rapid change give them a problem such as:
  A bacterial colony grows at a rate . How large is the colony when it is growing most rapidly? In their analysis look for evidence that they understand the rate of change of "the rate of change."

Question

• While students are working on simple area problems based on real-life applications, ask them to explain the relationship between the graph’s units (on each axis) and the units of area under the curve.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  Ch. 4 (Sections 4.3, 5.4)
  pp. 97-98, 172, 180, 198, 202
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 182, 184, 195, 197, 209, 219-221, 247, 257
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 2 (Section 2.9)
  Ch. 3 (Section 3.11)
  Ch. 4 (Sections 4.2, 4.3, 4.9)

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 211, 290, 299, 363
• Calculus of a Single Variable, Sixth Edition
  Ch. 3 (Sections 3.2, 3.3, 3.8, 3.9)
  pp. 157, 171-175, 182-183

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 7, 2000

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