## Algebraic Fishiness

Posted by mark schwartz on July 22, 2016

__Introduction:__

In an Algebra class, I gave the following two problems as an in-class assignment. We had just finished the material on solving equations and we had spent considerable time on what are called application problems. I decided to stretch their imaginations a bit by providing these two problems which are quite different from anything they had seen in the text. This, again, was a group activity and I roamed around the classroom watching the in-group strategies develop. If I hadn’t done this, I would have missed a group’s interesting insight.

__The Story: __

Here are the two problems.

- This problem is from
*Algebra*by Davies, published by Barnes and Company, NY, NY, 1858.

A fish was caught whose tail weighed 9 pounds. His head weighed as much as his tail and half his body; his body weighed as much as his head and tail together: what was the weight of the fish?

- This problem is from
*An Introduction to Algebra*by Colburn published by Hilliard, Gray and Co., 1839.

There is a fish whose head is 4 inches long, the tail is twice the length of the head, added to 2/5 of the length of the body, and the body is as long as the head and the tail both. What is the whole length of the fish?

Please note that these two fish problems represent a problem that was quite common in texts of that era. Conceptually, it’s the same problem, one phrased as a weight problem and the other as a length problem.

There was some grumbling when the class first read these problems. They hadn’t seen this type of self-referential relationship in a problem. They questioned if these were trick questions; some said it was the same problem so why give it twice; someone asked if it was the same fish, and I heard a few other choice comments. I suggested they take a deep breath and think about it and further, once they have solved one, let me see what you’ve done before moving on to the second problem.

On occasion, I gave Socratic help.

One group called me over and showed me the solution for the first problem, which was correct. But their setup for the problems was unique.

Their setup was:

T = 9 H = 4

H = T + 1/2(B) T = H + 2/5(B)

B = H + T B = H + T

What’s unique about this is that they identified the relationship of the unknowns for __both__ problems before setting up the equation for either of them. I asked them why they took this approach. The response was that the problems looked so much alike that they thought maybe there was one solution strategy, which as it turns out is true. This setup, in their minds, verified that point. In both cases, they started the solution with B = H + T. These are their solutions, but I tidied them up a bit (without changing any steps in the solutions) and showed them this way so it’s easier to see their parallel solutions. The complete solutions for both were:

For the weight problem: For the length problem:

B = H + 9 B = H + 4

H = 9 + 1/2(H + 9) T = 8 + 2/5(4 + T)

2H = 18 + H + 9 5T = 40 + 8 + 2T

H = 27 3T = 48

T = 16

B = 27 + 9 = 36 B = 4 +16 = 20

Fish = 72 pounds Fish = 40 inches

They went a little further and decided that it was the same fish; that a 40 inch fish could weigh 72 pounds based on one of the group members being an avid fisherperson. Neat group!

In summary, this group of students used their imaginations, which resulted in their seeing not only the similarity in the problems but also the similarity in the solution strategy. It’s a testament to group activity and “aha” moments.

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