# Fish Farm 2 Design

Fish Farm II extends the concepts of ratio and equivalent ratio to a situation in which the ratio of male to female fish in a large lake is unknown. Students can explore the ratio of male to female fish in the lake by "scooping" one fish at a time from the lake and then returning the unmarked fish to the lake before doing another "scoop." They can choose to do as many trial "scoops" as they wish to make a justifiable estimate of the male-to-female ratio in the lake. As students run many trial "scoops," the data will be updated in a table and a pie graph.

Students are challenged to figure out, based on their scooping data, which ratio from the Fish Farm 1 ponds (1:1, 3:1, or 2:1) most closely matches the lake's ratio of male to female fish. The number of repeated trials that a student runs should affect the reasonableness of his or her estimate. Students should note that the ratio of male to female fish is unstable with a small number of trials, and tends toward a more stable ratio with a large number of trials.

## Problem Description

The day after their birthday, the Carp triplets visit Lake Mark Park. Seeing some fish swimming in the lake, Angel scoops one out with a small net, then throws it back. It's a male . Angel then scoops out and throws back another fish, this time a female .

Excitedly, Angel says, "I think this lake has equal numbers of male and female fish, just like my pond at home!" (See Fish Farm I.)

Molly and Gar wonder whether the ratio of fish in the lake might be closer to the ratios in their own backyard ponds. (Recall that Molly's pond contains three times as many males as females, and Gar's pond has twice as many females as males. The triplets think they will need to do more "scooping" before reaching a conclusion about the lake's male-to-female fish ratio.

To Do: Use the applet to simulate scooping out fish from the lake. With a small net, you can only scoop out and throw back one fish at a time; however, in the applet, you can change the number of scoops you do in a row. Try changing the number of scoops to 10, 20, 50, or any other number of scoops you feel are needed to get a good estimate of the ratio of male to female fish. Be sure to save all your data to the table to use in answering the questions below.

## Questions / Explorations

• Which triplet's back-yard pond most closely matches the male:female ratio in the lake? Justify your answer with data you collect. In your explanation, tell us which display(s) of data you used to help you and how it helped. Remember, Angel's pond has a 1:1 ratio, Molly's pond has a 3:1 ratio [three times as many males as females], and Gar's pond has a 1:2 ratio [twice as many females as males].
• What is your best estimate of the percentage of male fish in Lake Mark? Describe the strategy you used to determine this estimate.
• How confident are you that your estimate is close to the actual percentage of male fish? What makes you feel confident about your estimate, and what would make you feel more confident?
• Bonus: Suppose we knew that Lake Mark had just been filled with 350 male and 650 female fish. About how many male and female fish might be caught if you scooped a fish 10 times? What if you scooped a fish 700 times? How confident are you that your prediction would be close to an actual sample? Explain your reasoning.

## Possible Activities

The pre-activity is designed to introduce students to sampling with replacement, the difficulty of making predictions based on small samples, and how to convert data to percentages and represent those percentages in a pie graph. Two different ideas are listed as possible post-activities.

The first activity idea extends the sampling with replacement procedure to a procedure using a capture-recapture method. The second activity idea extends the concepts of ratio and percent in a pie representation, as well as explicitly introducing students to theoretical and experimental probability. The third activity idea will help students generalize the notion of sample size to a situation with six possible outcomes.

### Pre-Activity

• Setup: Create several sets of three brown bags that contain snack goldfish (e.g., parmesan and cheddar) with the following contents: A) 5 parmesan and 15 cheddar; B) 10 parmesan and 10 cheddar; C) 12 parmesan and 6 cheddar. Do not tell the students the contents of the individual bags; however tell them that only one of the bags has the same number of parmesan fish as cheddar goldfish. They will do a sampling experiment to try to determine which bag has equal amounts of the two types of fish.
• Experiment: Place students in groups of 3-4 and give each group a set of the three bags. Each group can only sample 10 fish, one at a time with replacement, from each bag. Record the group's findings in a table on the board.
• Percentages and Pie Graphs: Discuss how to convert all the data to percentages (e.g., group 1 had 4 parmesan and 6 cheddar fish for Bag A. This would be converted to 40% parmesan and 60% cheddar). Have each group use the interactive pie chart to model the percentage of parmesan and percentage of cheddar in each of their bags. (This could be done as a whole class if only one computer and a display are used.) Groups can either print out their pie graphs or draw rough sketches of the representation.
• Prediction Discussion: Ask each group to analyze the class data, including the pie graphs, and to predict which bag they think has an equal number of parmesan and cheddar goldfish. There should be enough variability in the data that the students do not feel very confident in their predictions, since they only sampled 10 fish. This should raise issues of sample size. Ask students to say why a larger sample size might be more helpful. They will have an opportunity to explore larger samples in the Fish Farm II activity.
• Transitioning from the Fish Farm I Problem: Have students use the Shodor Organization's interactive pie chart to model the ratios of male to female fish in each of the ponds from Fish Farm I.
• Angel had the same number of male and female fish in her pond.
• Molly had three times as many males as females in her pond.
• Gar had twice as many females as males in his pond.

Discuss the appearance of the pie graphs and the associated percentages. Justify why the pie graph is an accurate representation of the three ratios.

Before students use the java applet to solve the Fish Farm II problem, have them read and act out the triplets' visit to the lake to reinforce the "scoop" sampling procedure used in the applet.

### Post-Activity

• Idea 1: Have the students work in pairs and use the Shodor Organization's Spinner applet to build a spinner with three regions representing the ratio 1:1:2. What are the corresponding fractions (parts of the pie) and percentages that match this ratio? These fractions and percentages represent the theoretical probability of landing on each region. Run several experiments with a small number of spins and a large number of spins. The results of an experiment are used to calculate the experimental probability. Use the "view results frame" to look at a pie graph of the experimental results. How does the experimental probability compare with the theoretical probability? How many spins do you need to do to have the experimental probabilities closely approximate the theoretical probability?
• Idea 2: Do a hands-on activity in the classroom with six-sided number cubes. Does each side of the cube have the same chance of landing face-up? Have each pair of students roll a number cube 20 times and record the data on a class chart. Discuss the wide variability in results with the different sets of 20 trials. Then have students add up all the class data and discuss the distribution of results with a large number of rolls. Does it appear that each side of the cube has the same chance?

## Standards

• Number and Operations
• understand and use ratios and proportions to represent quantitative relations
• work flexibly with fractions, decimals, and percents to solve problems
• Data Analysis and Probability
• use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken
• use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
• Representation
• create and use representations to organize, record, and communicate mathematical ideas
• select, apply, and translate among mathematical representations to solve problems
• 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