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### Fish Farm II - posted March 19, 2001

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Fish Farm II

### A Carp Family Picnic

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

1. 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].

2. What is your best estimate of the percentage of male fish in Lake Mark? Describe the strategy you used to determine this estimate.

3. 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.

### Comments

Teacher Support Page

For question 1, some students were unwilling even to evaluate the 1-to-1 ratio as a possibility if the scoops did not produce an exact 1-to-1 ratio. They were able to compare the other values but did not recognize that a 57% male population was closer to the 1:1 ratio than to 1:2. They seemed to have a better grasp of what 1-to-1 means both visually and mathematically, thus making it easy to discount based upon preliminary, although slightly inaccurate, data. Students generally picked a sample size and then evaluated it from 3 to 5 times. Not changing the sample size, and missing the leveling trend on male percentages, affected the following questions.

For question 3, many students did not base their confidence levels on the leveling effect of numerous scoops with large samples. Answers were generally vague, and stated that it would have been better if we knew how many fish were in the pond, or if they could all be counted. Students either used large samples from the start, or never even noticed that the percentages changed when larger samples were scooped. This prevented a high confidence rating. Not many students seemed to see the data trend change over sample size.

The bonus caused some confusion for a group of students from the Caroline Davis school. They did not know whether there were fish in the pond before more were added. The number of students that tried the bonus was significantly lower because of this. Others were able to complete the mathematical percentages expected or they figured that the larger number of scoops would be more accurate. Only one student received credit for the bonus for understanding the entire concept.

### Highlighted solutions:

 From: Katie B., age 20 School: North Carolina State University, Raleigh, NC

```QUESTIONS

1. Which triplet's backyard pond most closely matches the male:female
ratio in the lake?  Justify your answer with data you collected.
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].

Angel's backyard pond most closely matches the male:female ratio in
Lake Mark. When collecting my data, as my number of scoops approached
infinity, my ratio approached closer and closer to 1:1.  For example,
at 100 scoops, I had 58% males and 42% females. Then at 500 scoops, I
had 54% male and 46% female. At 10000 scoops, I had 55% male and 45%
female. So the more scoops I had, the closer my percentages of male
and female fish reached 50% males and 50% females. We know that our
1:1 ratio is the same as 50% male and 50% female, our 3:1 ratio is
equivalent to 75% male and 25% female, and our 1:2 ratio is the same
as 33% male and 67% female. Since the percentages I collected in my
data were closer to Angel's percentages than Molly's and Gar's, I knew
the ratio of fish in Lake Mark was closest to Angel's pond.

2. What is your best estimate of the percent of fish in Lake Mark
that are male? Describe the strategy you used to determine this
estimate.

55%; The strategy I used to determine this estimate was to do a bunch
of trials adding more scoops each time.  The more scoops I added, the
closer my male percent reached 55. I did ten trial runs; my final
being 10000 scoops.

3. How confident are you that your estimate is close to the actual
percent of fish that are male?  What makes you feel confident
about your estimate, and what would make you feel more confident?

I am confident that my estimate is close to the actual percent of male
fish. I feel pretty confident about my estimate since the last 6
trials I ran were all within 1% of 55% for the  male fish. I would be
more confident of my estimate if I ran more trials and added more
scoops to the trials.  If I run more trials and add more scoops, my
percent of male fish will be more acurate.

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.

I might get 4 males and 6 females if I took 10 scoops. I am not very
confident that this hypothesis will be correct since there are 1000
fish in the lake. I guessed this number because there are 13 female
fish for every 7 male fish. So I took half of 7 and rounded it up and
I took half of 13 and rounded it down.  This gave me 10 fish. If
I scooped 700 fish, I would predict that I would get 245 male fish and
455 female fish.  Again I not extremely confident about this
hypothesis for the same reason, but I am more confident that it will
be a closer estimate than the first one. 35% of the lake's fish is
male and 65% is female.  So I took 35% of 700 and got 245 and I took
65% of 700 and got 455. 700 is closer to 1000 than 10, so I figure my
second estimate is better than my first.  It is still all a gamble,
though.
```

 From: Katie B., age 20 School: North Carolina State University, Raleigh, NC

```QUESTIONS

1. Which triplet's backyard pond most closely matches the male:female
ratio in the lake?  Justify your answer with data you collected.
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].
Angel's pond with a ratio of 1:1 most closely matches that of the
lake. In the data that I collected there were only a few more males
than females, resulting in having percentages like 55:45, 54:46, and
56:44, all three of which came up twice each. This data came from an
overall total of 700 scoops(200,200,100,100,50,50). With this data,
there is definitely not a 3:1 ratio of males to females. And since
there are more males coming up than females, I do not think that it
could possibly be a 1:2 ratio of males to females. Thus, the only
logical answer, as it seems to me, would be for a ratio closest to
1:1.

2. What is your best estimate of the percent of fish in Lake Mark
that are male? Describe the strategy you used to determine this
estimate.
My best estimate of percentage of male fish would be around
55%. I determined this by taking the average of the 55,55,54,54,56,56
different percentages that came from the data.

3. How confident are you that your estimate is close to the actual
percent of fish that are male?  What makes you feel confident
about your estimate, and what would make you feel more confident?
I am fairly certain, aoround 90%, that my estimate is close to the
actual percent of fish in the lake. I think that having the data show
so close results every time, that the data has to be fairly close to
accurate. I would feel more confident with more scooping to further
my hypothesis.

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.
I think that with only 10 scoops the prediction would not be
justified enough because of such little sampling, so I think that the
scooping would show an almost even trend, with a few more female fish
than males. With 700 scoops, I think that the true ratio would be
shown, or close to it, showing nearly twice as many females and
males. I think that my prediction is fairly accurate, around maybe
80%. I definitely think that there is a possibility of the outcome
being different from that of my prediction.
```

 From: Kristina Y., age 21 School: North Carolina State University, Raleigh, NC

```QUESTIONS

1. Which triplet's backyard pond most closely matches the male:female
ratio in the lake?  Justify your answer with data you collected.
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].

The triplet's pond that the lake matches best is Angel's pond.  When I
did 1000 scoops, the outcome was 55:45 ratio.  This is virtually a 1:1
ratio.  It is not quite that ratio, but it is the closest ratio out of
the three.

2. What is your best estimate of the percent of fish in Lake Mark
that are male? Describe the strategy you used to determine this
estimate.

I found that 55% are male and 45% are female.  I scooped 1000 times
to figure this out because the more scoops that I made, the closer the
percents will be to the correct percent.

3. How confident are you that your estimate is close to the actual
percent of fish that are male?  What makes you feel confident
about your estimate, and what would make you feel more confident?

I am fairly confident about this estimate because I scooped so many
times.  Each time I scooped I gained greater knowledge about the ratio
of fish in the pond.  By keeping track of each scoop, I could see how
the percentages became closer and closer to 55% and 45%.  I would feel
more confident if I could take all of the fish out of the pond and
count them.  However, I believe this is the best method for finding
the percentages without taking all of the fish out.

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.

350 males out of 1000 total fish is 35% and 650 females out of 1000
total fish is 65%.  So, to find the number pulled, you need to find
35% of 10 and 65% of 10.  However, you can't have ½ of a fish, so more
than likely there will be about 7 females and 3 males.  I subtracted
the 1 from the male side because there are so many more females, so
you are more likely to pull female fishes in general.  This is going
to be a less accurate statement then saying that about 245 males and
455 females will be pulled from 700.  This is because the more pulls
you make, the closer you come to the correct ratio of male to female
fish.  To better estimate the number of fish, you must pull millions
of fish to figure out an even closer ratio.
```

 From: Matt P., age 21 School: North Carolina State University, Raleigh, NC

```QUESTIONS

1. Which triplet's backyard pond most closely matches the male:female
ratio in the lake?  Justify your answer with data you collected.
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].
The ratio in the lake most closely resembles Angel's ratio of 1:1.
The data I collected showed the amount to be 55% male and 45% female
in the lake.  This is closest to Angel's ratio which would be 50% male
and 50% female.  The closest other pond would be Gar's which was
approximatly 33% male and 66% female.  The data I collected was first
with 10 scoops.  It was 5 male and 5 female.  Then I tried 1000 scoops
and the lake had 54.3% male and 45.7% female.  I then tried 5000
scoops to make sure my percentages were correct.  when I tried 5000 I
got 55.3% male and 44.7% female.  Now I was sure I had the right
amount.  To compare the ratios i multiplied all the ponds so the total
of the ratio was 100.  Angel's ratio became 50:50, Gar's became 33:66
and Mollie's became 75:25.  Then I compared my ratio of 55:45 and saw
that Angel's ratio was the closest.

2. What is your best estimate of the percent of fish in Lake Mark
that are male? Describe the strategy you used to determine this
estimate.
As I discussed in #1, I used the ratios totalling 100 to find the
percent.  The lake was approx. 55% male and 45% female.  I used the
ratios I found and the percent given in the table to come up with this
estimate.

3. How confident are you that your estimate is close to the actual
percent of fish that are male?  What makes you feel confident
about your estimate, and what would make you feel more confident?

I am very confident because I scooped so many fish.  When I scooped
1000 and then 5000, this gave me a very broad look at the lake.  I am
confident because with this many trials, the chance of having data
that is falsly represented decreases.  In other words, the more scoops
I make the closer to the actual percent I will get.  The only way to
be more certain is to take even more scoops, perhaps 50,000.

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.

If we scooped ten fish, we could expect 3 male and 7 female or 4 male
and 6 female.  however, with such a small number, the amounts could be
unrepresentative of the amount.  In other words, we could get 8 male
and 2 female. On the other hand if we scooped 700 fish, we would be
more likely to get closer to the ratio that is represented in the
pond.  I would expect approximately 245 males and 455 females.  This
is because this ratio is equivalent to the 350:650 that was placed in
the lake.  Therefore, we can use this ratio to estimate the number of
each sex that will be taken out with any number of scoops.  Therefore,
given any number of scoops, you could be confident that your
prediction was fairly close to the correct amount because you had
equivalent ratios.  The reason these ratios are equivalent is that the
male amount is the same part of the total fish.  In other words, the
percent of male and female fish is the same in each ratio and the
ratio of males to females is also the same.
```

### 22 students received credit this week.

Danny B., age 22 - North Carolina State University, Raleigh, NC
Katie B., age 20 - North Carolina State University, Raleigh, NC
Raul B., age 14 - Caroline Davis Intermediate School, San Jose, CA
Jae C., age 12 - Caroline Davis Intermediate School, San Jose, CA
Nick G., age 13 - Caroline Davis Intermediate School, San Jose, CA
Jessica H., age 12 - Caroline Davis Intermediate School, San Jose, CA
Deanna J., age 20 - North Carolina State University, Raleigh, NC
Asim K., age 13 - Caroline Davis Intermediate School, San Jose, CA
David K., age 13 - Caroline Davis Intermediate School, San Jose, CA
Terry L., age 14 - Caroline Davis Intermediate School, San Jose, CA
Chip M., age 21 - North Carolina State University, Raleigh, NC
Sergio M., age 14 - Caroline Davis Intermediate School, San Jose, CA
Chau N., age 12 - Caroline Davis Intermediate School, San Jose, CA
Grace P., age 12 - Caroline Davis Intermediate School, San Jose, CA
Matt P., age 21 - North Carolina State University, Raleigh, NC
Gilbert S., age 13 - Caroline Davis Intermediate School, San Jose, CA
Will S., age 15 - McLean High School, McLean, VA
George V., age 21 - North Carolina State University, Raleigh, NC
Marie V., age 17 - DeLand High School, DeLand, FL
Grace Hsiang Wen Y., age 14 - Taipei American School, Taipei, Taiwan
Kristina Y., age 21 - North Carolina State University, Raleigh, NC

#### View most of the solutions submitted by the students above

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