# Fish Farm 1 Design

The Fish Farm problem is designed to have students apply the concepts of ratio and equivalent ratio to distribute a set number of male and female fish into three ponds. Each pond has a target ratio of male-to-female fish. As the students move fish from the tank into the individual ponds, the numerical and graphical displays of the current numbers of male and female fish will change.

## Contents

## Problem Description

For their birthday, the Carp triplets received 26 tropical fish: 13 females and 13 males. They discussed ways to divide the fish among their three tiny backyard ponds. Angel said, "I want the same number of male and female fish in my pond." "Okay," said Molly. "I want three times as many males as females in my pond." "Then I want twice as many females as males in my pond," Gar replied. Is there a way to put all 26 fish into those three ponds, while giving each triplet what he or she wants? Use the applet to explore this question.

## Questions / Explorations

- How many male fish and female fish does each triplet get in his or her pond? Describe the work you did to find the solution. (Sample questions you can answer: Into which pond did you put fish first? How many fish of each kind went into that pond? Why? What was your next step? How were you sure a pond had the correct ratio?)
- Given the 13 males and 13 females, what are ALL the possible numbers of male and female fish that would satisfy the ratio of 1 male to 2 female fish in Gar's pond? Explain why these different amounts are equivalent to the ratio 1:2.
- Explain why all possible answers in question 2 result in the same pie graph for Gar's pond.
- Find a different way to distribute all 26 fish among the three ponds in the correct ratios.
- Distribute the fish into the three ponds such that only 1 female fish is left in the large tank. Is there more than one way to distribute the fish and achieve the target ratios?
- Distribute the fish into the three ponds such that only 1 male fish is left in the large tank. Is there more than one way to distribute the fish and achieve the target ratios?

## Possible Activities

### Pre-Activity

- Review the concept of ratios in the form a:b.
- Print out the matching tasks from the Web page All About Ratios by Cynthia Lanius. Have students work in small groups to discuss these tasks and make convincing arguments about their answers.
- Have students discuss how to represent the ratio 2:3 in a circle graph.

### Post-Activity

- Have students find a DIFFERENT way to distribute all 26 fish among the three ponds.
- Have students distribute the fish into the three ponds such that only 1 female fish is left in the large tank. Is there more than one way to distribute the fish and achieve the target ratios?
- Have students distribute the fish into the three ponds such that only 1 male fish is left in the large tank. Is there more than one way to distribute the fish and achieve the target ratios?
- Encourage students to develop a procedure and justification for determining whether two ratios are equivalent.

## Standards

The pre-activity should help students begin to think about the concept of equivalent ratios, and how a ratio might be represented in a circle graph. The post-activity is an extension of the original problem, and should give students more opportunities to experience creating equivalent ratios in both numerical and graphical form.

- Algebra
- understand patterns, relations, and functions
- use mathematical models to represent and understand quantitative relationships

- Number & Operations
- understand and use ratios and proportions to represent quantitative relationships;
- develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios

- Representation
- create and use representations to organize, record, and communicate mathematical ideas
- select, apply, and translate among mathematical representations to solve problems
- use representations to model and interpret physical, social, and mathematical phenomena

- Problem Solving
- solve problems that arise in mathematics and in other contexts
- monitor and reflect on the process of mathematical problem solving

- Communication
- communicate mathematical thinking coherently and clearly to peers, teachers, and others
- use the language of mathematics to express mathematical ideas precisely

## Lesson Plans

- ePOW (electronic Problem of the Week) activity hosted on the MathForum
- Teacher Support
- Lesson plans created by participants of the 2010 Summer Institute