As I read submissions to So Many Salmon, the recent Math Fundamentals PoW, I was reminded of how “we” (meaning math educators) often talk about “open-ended” problems when in fact we mean “constructed response” problems.  Any problem where students are expected to write words, sentences, and even paragraphs to support their solution to a (possibly) complex problem is often called “open-ended”, when in fact there might be one or two paths to the same numerical answer.  I would call these types of problems “constructed response”.

So Many Salmon is closer to an actual opened ended problem.  We’re told how fast Armida, a worker at a salmon hatchery, can tag fish.  Then we’re told how many taggers work at the hatchery in one eight-hour shift, how many shifts there are, and how many salmon are tagged in a typical day.  We’re asked to decide whether the average worker is faster or slower than Armida. (That right there would make a good “scenario” – since there are no numbers, kids would really have to focus on the quantities and their relationships, not the values of those quantities!)

Most students answered the question by calculating how many salmon Armida and the average worker could each tag in a shift.  Here are a couple ways students figured that out.

by Erin, Troy Intermediate School

Answer: The average worker works slower than Armida in a shift. The new worker’s percentage of Armida’s daily total is 40%.

Explanation: I know that Armida can tag 1 fish every 4 seconds in an 8 hour shift. First I did 60 divided by 4. I got 15 that tells me how many she tagged in one minute. Then I figured out how many she tagged in one hour. I multiplied 15 by 60 and got 900 fish in one hour. Then I did 900 x 8 to know how many she tagged in her 8 hour shift .That was 7,200 fish. I knew that the Big Eagle tags 70,000 salmon a day. Next I figured out that 12 people worked total at the hatchery per day. I figured that out by multiplying 2 x 6. (shifts x workers) So if the hatchery tags 70,000 salmon a day, and 12 people do that work, then 70,000 divided by twelve tell me the number of fish tagged by one person. I did that and I got 5833.3. So the average worker works slower than Armida.

by Dayna, Birch Wathen Lenox School
Answer: The average worker is slower than Armida.
Explanation: First, I made a list of the important information. I knew that Armida tags 1 fish per 4 seconds. I also know that there are 60 seconds in an hour. So I did 60x60x8 to find the number of seconds in 8 hours, which was 28,800. Then I divided that by 4, to get the number of fish Armida tags in 8 hours. This was 7,200. So Armida tags 7,200 fish per shift. Then, I knew that all the workers tag 70,000 fish per day (2 shifts). There are 6 workers per shift, so there are 12 workers in 2 shifts. So I did 70,000 divided by 2 to find out how many fish per shift. It was 35,000. Than I did that divided by 6 to see how much an average worker does. So I got 5,833. That is less then Armida’s 7,200. So Armida is faster.

A second closely-related method was to figure out how many salmon each could tag in an hour.

by Alyssa, Sacajawea Middle School

Answer: Armida is faster than the average worker by 171. Extra: The percentage of Armida’s salmon with the new persons tag is 40%.

Explanation: I first wanted to find out the number of fish per hour for Armida and an average worker. I first needed to find out how many hours 6 people do in 2 shifts. I did 6 x 16=96 hours per day by the 6 workers. So to get the per hour, I knew that they tagged 70,000 fish a day so I did 70,000 / 96= 729.16. They can’t tag 0.16 fish so I rounded it to 729 fish per hour for an average worker.

I then needed to find Armida’s fish per hour. I knew she could do 1 fish in 4 seconds, so to find the fish per minutes, I knew 4 fit in to 60 15 times so that is the number of fish per minute. To get fish per hour I did 15 x 60= 900 fish per hour.
So an average workers fish per minute is 729 and Armida’s is 900 fish per minute so Armidia is faster than an average worker.

This was the method that I used, too. I’m not exactly sure why, but I think that as I wrote down “the facts”, and then calculated how many hours were worked each day, figuring out salmon per hour just seemed natural. Below I’ve recreated what I scribbled on the back of a scrap of paper.

Annie's Work

Some students figured out how many seconds it takes the average worker to tag a salmon, since we’re told that Armida can tag a salmon in about 4 seconds.

by Ethan, Rosemont School of the Holy Child

Answer: Amanda is faster than average.

Explanation: 2 8 hr shifts per day = 16 hours
6 workers=16*6=96 total hours per day
96 hours=5760 minutes=345,600 seconds
345,600seconds/70,000salmon= 4.9 second/salmon

Amanda takes 4 seconds and is faster than average

Some students didn’t directly calculate and compare any rates. Instead, they figured out how many salmon would get tagged at the hatchery in one day if everyone worked at the same rate as Armida.

by Samantha, Birch Wathen Lenox School

Answer: If the everyone worked at Armida’s pace, two six-person shifts would have produced 86,400 tagged fish at the end of the day. Therefore, the average worker is slower than Armida if the total production for the day is 70,000 tagged fish. Extra: In one day, the new person will tag 40% of Armida’s salmon.

Explanation: First I figured out how many seconds are in an hour. Then I divided that (3,600) by 4 seconds, and I got 900 fish per hour. I then multiplied 900 by 8 hours. My answer was 7,200 per 8 hours, for Armida. If 6 workers worked at Armida’s pace it would have totaled 43,200 fish per shift, or 86,400 for two shifts at Armida’s pace. If the Big Eagle produced 70,000 salmon at the end of the day using two six-person shifts, the average worker is slower than Armida.

Along those same lines, this student figured out how many workers would be needed to tag 70,000 salmon if they all worked at the same rate as Armida:

by Mia, Sacajawea Middle School

Answer: armida tags 7,200 fish a day. the average worker is slower than armida.

Explanation: 60 min. x 60 min.= 3,600 x 8= 28,800/4=7200 so armida tags 7200 fish a day.
70,000/7200= 9.7= how many people it takes to tag 70,000 fish if they were working at the same rate as armida. the hatchery employs 12 people so the average person is slower than armida.

So four different methods all lead to the same conclusion! For a problem to be really open-ended, there might not be such a clear conclusion. But at least students got to decide what they could calculate that would allow them to answer the actual question posed by the situation. It makes me wonder how often we could leave things at least this open for students, instead of always leading them to something very specific.

If your students worked on this problem, did they come up with any other methods? What method do you think you might have used when faced with this problem? Please share!