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Teaching Math: A Video Library, 9-12

Fish Derby

Process Standards
Problem Solving, Communication, Reasoning, Connections
School: Alhambra High School; 1,015 students
Location: Martinez, California
Teacher: Carol Cho
Years Teaching: 27
Students in Classroom: 28
Grade: 10 – 12
Fish Derby
Video Overview
Prior to this lesson, students estimated the number of bass and carp in a fictional pond. In this lesson's linear programming problem, students work in groups using the data they gathered and information on the fishes' breeding and feeding habits to determine optimal fish populations. Students use graphing calculators to solve the problem. When students encounter difficulties, Ms. Cho reconvenes the class to discuss the problem. Students then return to their groups to complete their work.

The Class Challenge
Students use the following information to answer the questions below: The pond's bottom has 10,000 square meters of habitable breeding area. Each carp requires 4 square meters in order to breed, and each bass requires 1 square meter to breed.
In the area close to the pond's shore, there are 20,000 square meters of feeding area. Each carp eats in an area of 4 square meters and each bass eats in an area of 5 square meters.

Find two inequalities based on the breeding and feeding information.
Decide how to solve this problem graphically.
Determine the optimal combination of carp and bass for the pond.

Course and Curriculum
This class is required for students who take only two other mathematics courses. It is the third course in the College Preparatory Mathematics (CPM) curriculum. In this course, students study ratio and proportion, linear regression, estimation, functions, linear programming, logarithms, probability, sampling, interpreting data, and fractals and chaos. The goal of the lesson was to review many of these concepts.
Prior to this lesson, students worked in groups on several linear programming problems and then took a group quiz. In this lesson, students used trial and error and were asked to show support for their conclusions. They did not do a formal proof. At the end of the lesson, each group wrote a letter to a fictional board of supervisors with recommendations for reaching optimal fish populations by reducing the number of carp. Following the lesson, the groups sampled the fictional pond again after carp have been fished out. Then they determined why the bass and carp populations changed, predicted population changes if carp continued to be fished out over five years, and wrote a second letter of recommendations to the board of supervisors.
Ms. Cho assessed students by observing their questions and recorded data. She also asked each group to write a report that included graphs and the two letters. Students received a group grade for their report and an individual grade for their participation. Ms. Cho usually organizes student groups randomly, although sometimes students choose them. She changes the groups at the end of each unit.
Projects are included in each unit of the CPM curriculum and vary in length, skill focus, and number of concepts addressed. Ms. Cho chooses projects based on skills she wants to develop in each class.

A Pre–Viewing Exploration for Teacher Workshops
Linear Programming: Applying Systems of Linear Inequalities
Objective: To solve a linear programming problem and to think about the importance of linear programming in the algebra curriculum.
Materials: Graph paper or a graphing calculator for each teacher.
Activity
Solve this problem in individually.
Anna Thompson is the manager of a discount store. After the July 4th weekend, the store has 12 boxes of noisemakers and 32 boxes of sparklers left in stock. Anna can package these items in two ways for quick sales. Each type A package contains two boxes of noisemakers and 8 boxes of sparklers and sells for $6.50. Each type B package contains three boxes of noisemakers and 4 boxes of sparklers and sells for $5.75. Assuming all packages will be sold, how many of each type should be made in orderto maximize the store's income?

Write the statement to be maximized. Let P be the total income, in dollars, for A and B packaged sparklers and noisemakers. Let x be the number of type A packages, and let y be the number of type B packages.
Write the constraints. Constraint inequalities represent the conditions under which the sparklers and noisemakers will be sold.
Sketch a graph of the region bounded by the constraints. Find the vertices of the region. The maximum will occur at the vertices as the theorem states: If there is such a maximum or minimum value of ax + by, then it occurs when x and y are the coordinates of a vertex of the region.
Find the maximum value for P by evaluating P = 6.50x + 5.75y for the x and y values in each vertex.

Exploring Content

Is linear programming an important topic in algebra? Why or why not?
How would you help students to understand what the shaded region in the graph represents?
Could you solve the first problem using trial and error after you have established the constraint inequalities? Why or why not?
What is the difference between a solution and a maximum? (A solution is any point in the shaded region.)

Exploring Teaching Issues

Should a mathematical concept be presented in a real–world context first, or should students learn the algorithm before trying to apply it? (An algorithm is a specific procedure for solving a problem in a finite number of steps.) How can you teach for both process and algorithmic understanding?
How can you help students generalize key mathematical concepts in order to apply them in other contexts?
What applications can students create that would use this problem–solving strategy?

Exploration Answers
Activity

P = 6.50x + 5.75y The income from both types of packages
x >0, y >0 Some sparklers and noisemakers will be sold.
2x + 3y <= 12 In package A, there are 2 boxes of noisemakers (2x), and in package B, there are 3 boxes (3y). The combination of boxes must be less than or equal to 12 because there are only 12 boxes of noisemakers.
8x + 4y <= 32 In package A, there are 8 boxes of sparklers (8x) and in package B, there are 4 boxes (4y). The combination of boxes must be less than or equal to 32 because there are only 32 boxes of sparklers.

The vertices of the region are (0,4) (4,0) (3,2), (0,0).
Maximum value of P at (3,2): three A packages and two B packages.

Exploring Content

Answers will vary. One possible answer is that the process helps students to understand multiple constraints in problem situations.
Go back to solving simple inequalities and the relationship of the shaded region to the solution of the problem. The solution must fall somewhere in this shaded region because all other points will not solve the inequalities.
Yes. It is possible to substitute numbers to find the right combinations.
The maximum profit is the solution that gives the greatest profit. The maximum is a solution, but every solution is not a maximum. The maximum profit will always be one of the vertices.

Topics for Discussion
How did Ms. Cho respond to students' varying interpretations of graphs?

How did Ms. Cho respond to students' confusion about the feasibility range? What are some other ways to respond? How can you plan ahead of time to minimize confusion?
How would you help students to understand and complete the problem correctly without providing them with too much information?
Some of the students' contributions were incorrect or tangential to the lesson. How would you acknowledge contributions while still moving on to other students' statements or your own statements?
How can you help students to value and discuss other perspectives with a focus on building group consensus and understanding? With a focus on mathematical evidence?
How might you encourage students to think about the meaning and usefulness of algebraic representations?

How did Ms. Cho encourage students to think about the meaning and usefulness of algebraic representations?
How would you relate this lesson to other real–life experiences?
List reasons why it might be difficult for students to recognize the meaning and usefulness of algebraic representations.
How did Ms. Cho assess student progress during this lesson?

According to NCTM's Assessment Standards for School Mathematics, one of the purposes of assessment is to inform instruction. Identify instances in which Ms. Cho conducted informal assessments then changed the course of her lesson based on her observations.
How would you assess students' work in this project? How much weight would you give to correctness of answers, accuracy of data, learning and discovery process, cooperation, presentations, and letters to the board of supervisors? How would you assess these aspects? What else should be assessed?
How would you assign grades in this project?
What are some ways you can help students take responsibility for group discussion and presentations?
What kind of homework would you give to follow up the lesson?