## "Fish Farm Part II" Teacher Support

Fish 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.

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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 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.

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.

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Alignment to the NCTM Standards - Grades 6-8

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

Possible Activities:

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?

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Related Lessons Online:

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Resources to help students:

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Extending the Fish Farm Activity into Ecology and Other Sampling Methods

- Making Predictions A hands on lesson for sampling fish with a large net (e.g., more than 1 fish sampled at a time), tagging the fish, and replacing them.
- How many fish in the sea? Using a Capture-Tag-Recapture method to estimate population

Fish image used with permission from: http://members.aol.com/sirchin/goldfish.htm

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