
ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary 
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Fish Farm I
A Fishy Family
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
 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.
Bonus: Explain why all possible answers in question 2 result in the same pie graph for Gar's pond.
Extension Problems (not to be submitted):
 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?
Teacher Support Page Students used several different approaches in thinking about how to distribute the fish into the three ponds.
 Many students starting by giving each pond the minimum number of fish necessary to meet the ratio (i.e., 1 male and 1 female in Angel's pond, 3 male and 1 female in Molly's pond, and 1 male and 2 females in Gar's pond). Then they did another "round" of giving fish to each pond in the same ratios. After this there were 5 females and 3 males left, and students then had to think about how to distribute the remaining fish. Many decided to concentrate on how to put fish into Gar's or Molly's pond so that an equal number of male and fish would be left to put into Angel's pond.
 Some students realized up front that they needed to put fish in Molly and Gar's ponds so that the sum of the number of males in Molly's and Gar's ponds would be equal to the sum of the number of females in Molly's and Gar's ponds. Then there would be an equal number of female and male fish left in the tank, and those could all be put in Angel's pond to satisfy the 1:1 ratio.
 A few students approached this problem using algebra, and set up equations that would model relation like those described in (2) above.
Many students found correct solutions to both question 1 and question 2, but gave limited justifications for why the ratios are equivalent. Some students made statements about ratios "reducing" to the same thing. In order to earn credit for a solution and justification, a student needed to give a reason why ratios are equivalent. For example, 4:8 is the same as 1:2 because in each case, the number of males is exactly half the number of females, or in each ratio there are twice as many females as males.
Several students expressed the ratio as a fraction (1:2 as 1/2). Although a ratio can be expressed in the form of a fraction, this representation led some students to reason that the pie graph for a 1:2 relation would be half yellow and half brown: when they tried to use a pie graph to represent 1:2, they tended to give it two equal parts instead of the correct three equal parts. Perhaps when people wrote the ratio as 1/2, they got confused about the number of equal parts because they were thinking of "one half" instead of counting the total number of parts, which in this case was 3 (1 and 2, rather than 1 divided by 2).
Finally, some people called the kids "twins." I wonder why they would say that. Check the problem description to see what the difficulty with that is...

QUESTIONS 1. How many male fish and female fish does each triplet get in their 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?)I began by putting 1 male and 1 female in angel's pond. Then I went straight down to Molly's pond and put in 3 males and 1 female. Then I went straight down again and put 1 male and 2 females in Gar's pond. Then I repeated the above process bringing me to a grand total of 2 males and 2 females in Angel's pond, 6 males and 2 females in Molly's pond, and 2 males and 4 females in Gar's pond. This left me with 8 fish left to place in the ponds divided among 3 males and 5 females. I chose to place 2 more males and 4 more females in Gar's pond and then 1 more male and 1 more female in Angel's pond. Angel's ratio had to be 1:1 which means that for every 1 male present there had to be 1 female present, or in other words, the number of males had to be equal to the number of females. My solution gave her 3males:3females for a total of 6 fish. Molly's ratio had to be 3males:1female. This ratio can be written as a fraction of 3/1. My solution gave her 6males:2females for a total of 8 fish. This ratio can be written as a fraction of 6/2. If I were to factor a 2 out of the top and bottom of my 6/2, it would become 2/2 * 3/1. The 2/2 = 1 and so my fraction of 6/2 can be simplified to 3/1 because 2/2*3/1=1*3/1 and 1*3/1= 3/1. These fractions, 6/2 and 3/1 are equivalent fractions. This shows that the two ratios of 3males:1female is equivalent to my solution of 6males:2females. Gar's ratio had to be 1male:2females which can be written as a fraction as 1/2. My solution gave him 4males:8females for a total of 12 fish. This ratio can be written as a fraction of 4/8. If I were to factor a 4 out of the top and bottom of my 4/8, it would become 4/4 * 1/2. The 4/4 = 1 and so my fraction of 4/8 can be simplified to 1/2 because 4/4*1/2=1*1/2 and 1*1/2= 1/2. This shows that the two ratios of 1male:2females is equivalent to my solution of 4males:8females. You can add the totals of fish in each pond to get 6+8+12=26 total fish 2. Given the 13 males and 13 females, what are ALL the possible amounts of male and female fish that would satisify 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. The possible amounts of fish in Gar's pond could have been 1male:2females, 2males:4females, 3males:6females, 4males:8females, 5males:10females, 6males:12females. This is without regard to the ratios set for the other two triplets! Each of these ratios when written as a fraction of males over females is equivalent to 1/2. A ratio can be written as a fraction and simplified. To simplify each fraction of 1/2, 2/4, 3/6, 4/8, 5/10, 6/12, you must factor out their greatest common factor. For example, 5/10=5/5*1/2. Then you can simplify and replace the 5/5 with its equivalent value of 1. That gives us 5/10=1*1/2. Since 1 is the multiplicative identity, then 5/10=1/2. The same process can be followed for each of the proposed fractions or ratios in this problem. are also equivalent if you write them as females over males to 2/1. Bonus: Explain why all possible answers in #2 result in the same pie graph for Gar's pond. The pie chart remains the same each time because it is comparing the number of males and females to the whole pie or total number of fishes in Gar's pond. If Gar has a ratio of 1male:2females then the male fish will from a pie slice of 1/3 which associates 1 male fish to 3 total fish (1male and 2females) and the female fish will form a pie slice of 2/3 which associates 2 female fish to 3 total fish (1male and 2females). If we raise our ratio to 2 males:4 females then the resulting part to whole comparison remains the same. The number of males will be 2/6 or 1/3 and the number of females will be 4/6 or 2/3. This will continue for each of the possible ratios listed in #2. The reason why the pie slices never change size is because of the equivalent fractions that I discussed above. Lets return to the example above of 5males:10females. This tranlates to a fraction in our pie graph of 5/15 which represents 5 male fish with the 15 total fish (5male and 10female). The fraction of 5/15 can be simplified to 1/3 by using the same process that I outlined above by factoring out the greatest common factor. Since 5/15=5/5*1/3 then 5/15=1*1/3 and then 1/15=1/3. This shows that a pie graph ratio of 5males:10females yields 1/3 of the graph because 5/15=1/3. The same process can be used when discussing the pie grahp interpretation of female fish. The ratio is 5males:10females which yields 10/15 and compares the 10 female fish to the 15 total fish (5males and 10females). This fraction will alsp yield 2/3 for the female fish because 10/15=5/5*2/3 which gives 10/15=1*2/3 which yields 10/15=2/3. No matter what fraction is chosen from #2, the resulting areas on the pie chart will always be 1/3 for the males and 2/3 for the females because of the equivalent fractions. Extension Problems (not to be submitted): 1. Find a different way to distribute all 26 fish to the three ponds in the correct ratios. 2. 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? 3. 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

QUESTIONS 1. How many male fish and female fish does each triplet get in their 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?) Angel had 8 male fish and 8 female fish in her pond. Molly had 3 male fish and 1 female fish in her pond. Gar had 2 male fish and 4 female fish in her pond. I first put one male and one female in Angel's pond, 3 male and 1 female in Molly's pond, and 2 males and 4 females in Gar's pond. I thought I could put the rest into Angel's pond, but I noticed that there was unequal amounts of males and females. So to make it equal, I put one more male fish and 2 more female fish in in Gar's pond, that would still be the same as 1:2. That left me with the same amount of male fish and female fish, so they could all go into Angel's pond (that would still equal 1:1). Since I did everything slowly, I made sure that my amounts of fish were equal to the ratios. All I did was get the total amounts and then reduce them, and the reduced number should've equaled the ratio. I knew I got everything right when the bricks turned green. 2. Given the 13 males and 13 females, what are ALL the possible amounts of male and female fish that would satisify 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. All of the possible amounts of male and female fish that would satisfy the ratio of 1 male to 2 female fish are: 1:2 2:4 3:6 4:8 5:10 6:12 These work because when they are all reduced to lowest terms, they equal 1:2. To reduce something to its lowest terms is to divide both sides of the ratio by the same number. Then to make sure it's in its lowest term, keep on dividing the sides by a number until the different sides of the ratio can't be divided anymore. Another way to check is divide the first half of the fraction by the second half you'd get .5, and if you did this for every single ratio, then you compared them, you will see that they are all the same. If they're not the same, then you know that they are not equivalent to the ratio. This tells me that the ratios are equivalent because when the above ratios are all reduced, they equal the same as 1:2. Bonus: Explain why all possible answers in #2 result in the same pie graph for Gar's pond. All the ratios I got in question #2 result in the same pie graph for Gar's pond because they are the same thing. If you looked at (for example) 1:2 by ratios, you would notice that that the first number is one third of the total (the 3, the total, is goten from the first and second number added together). Also, you would notice that every single other number in problem #2 also have the first number 1/3 of the total of the ratio. If you look at the pie graph for Gar's pond, then you would also see that 1/3 of it is one color, and 2/3 is the other color, which is the same as the ratios above in problem #2.
Alex Acosta, age 15  Westside High School, Houston, TX
Joanne B., age 22  North Carolina State University, Raleigh, NC
Katie B., age 20  North Carolina State University, Raleigh, NC
Emily C., age 13  Taipei American School, Taipei, Taiwan
Kelly C., age 13  Taipei American School, Taipei, Taiwan
Jerson, Jr. Cometa, age 18  Westside High School, Houston, TX
Paul D., age 25+  Centennial Campus Middle School, Raleigh, NC
Jennifer F., age 23  North Carolina State University, Raleigh, NC
Sarah F., age 14  Taipei American School, Taipei, Taiwan
Nina H., age 14  Taipei American School, Taipei, Taiwan
Kristin K., age 21  North Carolina State University, Raleigh, NC
Mary K., age 25+  North Carolina State University, Raleigh, NC
Josefin L., age 14  Taipei American School, Taipei, Taiwan
Katie L., age 13  Taipei American School, Taipei, Taiwan
Eric R., age 16  McLean High School, McLean, VA
Klaas S., age 14  Taipei American School, Taipei, Taiwan
Mary S., age 25+  North Carolina State University, Raleigh, NC
Max S., age 14  Taipei American School, Taipei, Taiwan
Rebecca S., age 21  North Carolina State University, Raleigh, NC
Mark Vuong, age 18  Westside High School, Houston, TX
Grace Hsiang Wen Y., age 14  Taipei American School, Taipei, Taiwan
Kristina Y., age 21  North Carolina State University, Raleigh, NC