by Thomas Price, Delaware Valley High School

Teacher Introduction

This was designed with the following rationale in mind.

Optimization is a topic in differential calculus, generally falling at the end of that division.  It is a practical application of curve sketching, or analyzing a graph using the first and second derivatives of the equation.  Before studying optimization problems, the student should have a solid foundation in drawing graphs.  Solving problems in optimization problems requires the student to use previously learned skills to find the maximum or minimum values for equations that model the specific situation. 

  

Optimization problems lie in the realms of both physical and natural sciences.  To solve such problems, the student is required first to find a function that models the problem, and determine a logical interval.  The student then needs to find the derivatives of that function.  Using that information will give the student the necessary knowledge to determine whether the function has a maximum or a minimum value in the predetermined interval and to develop a logical solution.

  

Calculus is typically a course taken by seniors in high school who are more than likely attending college the following year.  This WebQuest is written with those students in mind.  It is required that these students possess a strong background in Algebra skills.  To reach this height in high school mathematics, the typical calculus student most likely has a strong logical mathematical intelligence.   According to Howard Gardner, the developer of the multiple intelligences theory, this type of student has the ability to analyze problems, use appropriate mathematical operations and thoroughly investigate the issue at hand.  In addition, a calculus student possesses strong problem-solving skills and is strong in using higher level cognitive skills such as analysis, synthesis and evaluation. 

  

For the above-mentioned student, this web curriculum was designed.  Optimization requires the student to be accomplished in all of the above-mentioned skills.  Being confronted with a problem and being asked to determine the maximum or minimum value requires that the student first organize the information in a logical sequence.  The student must then determine an appropriate model, in this case an equation.   An interval must be chosen that fits both the model and the information given.  Once this is determined, the student must rely on previously learned knowledge of differentiation to calculate the maximum or minimum values, determine if they lie in the predetermined interval and test other possible values.  In conclusion, the student will display the correct answer to the problem.  This WebQuest offers multiple activities involving the solving of optimization problems which allow student to display their learning in an authentic manner.

  

Another reason for choosing this topic for a curriculum web is the authenticity of the topic.  It is not just a set of abstract problems taken from a textbook.  Optimization involves real life situations.  Given necessary data based on past revenue, a student can determine the ideal price that will produce maximum revenue.  Given the amount of material available, the student can determine the exact dimensions that will yield the maximum volume or area at the minimum cost.  These are just two of the possible scenarios of optimization.  One of the most common and most frustrating questions that teachers are asked by students is “When am I ever going to use this?”  Presenting them with real world situations will avoid this question, and allow the students to project their learning on a real world problem.

  

Lastly, this curriculum web allows students to gain proficiency in applicable mathematical standards set forth by the Department of Education of the Commonwealth of Pennsylvania .  Like other states, Pennsylvania has standards for both Problem solving and Calculus.  Those standards are listed in the Learner's Section

  

This WebQuest was designed as an interesting activity for students to use those concepts learned from graphical analysis and determining maximum and minimum values.  They will use those skills to solve word problems. 

In this WebQuest students become treasure hunters who have just found the gold from a Spanish galleon that was sunk off the coast during a hurricane.  The salvage boat has run out of gas, and the treasure must be transported in a rowboat.  The students must complete the following three tasks to get the treasure safely ashore.
  1. On board the boat there is a large piece of plywood.  Only the treasure that is able to fit in an open-topped box made from the plywood can be taken ashore in the rowboat.  Students must design a plywood box so that the box has the maximum volume.
  2. Given the speed of rowing the boat and the speed of carrying the treasure on land to the museum where the treasure will be displayed, the students must determine the route that will take the minimum amount of time.
  3. Given the current price of tickets to the museum, the students must determine a new price to charge for tickets which will produce the maximum revenue.