"Fish Farm Part II" Teacher SupportFish Farm Part II Archived PoW  Student Version Fish Farm, Part II is no longer the current ESCOT Problem of the Week. The student version allows teachers to use the problem with their students without giving the students access to the archived answers. Teachers can use the link to the archived problem to get ideas of student thinking. 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 maletofemale 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 I 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.
 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
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 34 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. PostActivity 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 handson activity in the classroom with sixsided number cubes. Does each side of the cube have the same chance of landing faceup? 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? [top] Related Lessons Online:
Simulating Probability Situations Using Box Models an interactive eexample from the NCTM Principles and Standards  Statistics and Probability Several lessons from PBSTeacherSource  Ideas that Lead to Probability  several experiments designed by the Shodor Foundation  Recommended Probability Investigations from the Intermath Project at the University of Georgia Lessons on Probability and Pie Graphs  Pie Chart This interactive pie chart from the Shodor Foundation allows you to manipulative the size of a sector and view its percent of the circle, or vice versa.  Spinner This interactive resizable spinner from the Shodor foundation will compare experimental to theoretical probabilities for regions with varied sizes. [top] Resources to help students:
Probability Central  a ThinkQuest site What are Your Chances? a site from NCES that simulates rolling two dice and explains the probability of each possible sum Ask Dr. Math Archives  Ratios as fractions  Figuring ratios  What is the second number in a ratio?  Dr. Math's Introduction to Probability  Defining Probability  Odds and Probability  Randomly Selecting a Card Previous MidPOWs related to probability  Label Shuffle  posted August 30, 1999  Rock Paper Scissors  a four part ESCOT ePOW from January 2000 [top] Extending the Fish Farm Activity into Ecology and Other Sampling Methods
 How many fish in the sea? Using a CaptureTagRecapture method to estimate population
