Puzzling Jugs

I was introduced to a virtual manipulative recently called the “Jugs Puzzle Game.” It outlines a riddle that can be manipulated virtually in order to solve it. Basically, there are 2 jugs of a given size that has a certain goal amount of water to put into one jug. You have an unlimited amount of water, and you can fill up jugs, pour them out, and transfer water between the jugs. It seems simple enough, right? Except you can’t fill up a jug part of the way with the water faucet because at the faucet, you can only fill the jug to its full capacity. This makes it difficult to achieve the desired amount of water all in one jug because it requires a certain order of filling, draining, and transferring water between jugs to achieve the desired output.

I struggled for a long time to complete the first level the first time I tried completing the task. I felt helpless because I thought it was impossible, but I knew that it was a solvable problem, and that others had done it before. This boosted my confidence in completing the task, so I kept clicking around until I eventually got the hang of the process. With each jug, there are only two options with what I could do next: I could drain the water, or transfer it over, perhaps leaving a partial amount of water in the jug. As I tried each level again and again, I would gradually improve my skills in deciding which move would be a progressive move to help me get closer to my desired output rather than something that wouldn’t get me anywhere. After a while, I was addicted.

I was thinking about the mathematics behind the task and trying to figure out what exactly I was doing in order to achieve my desired amount of water in one jug. I thought about other combinations of jug capacities and thought the solution to each problem would have something to do with the size of the given jugs—I couldn’t have just any two jugs with arbitrary capacities to receive a certain desired amount of water in one of them. I realized the capacities of each of the two jugs must be relatively prime to each other, making each of the two jugs lack the same amounts of water that might factor into their capacities, as this would make it impossible to fill a jug with a desired amount of water.

When a “solvable” problem is presented, I first outline where I am, what my goal or destination is, and what means I have available to me in order to get there. In this case, I had 2 jugs of different sizes, an unlimited amount of water, a faucet, and a drain. I also knew that I had to fill the jugs all the way to the top instead of partially. I knew I needed a given amount of water in a jug (either one that could hold that given amount or more). The means I had to get there included 3 options for each of the jugs. I could fill them completely, drain them completely, or transfer water between the jugs to leave partial amounts of water in one of them or both. In my understanding of the simple terms of completing the task, I was able to push past any uncertainties and fill, drain, or transfer water over and over again. In doing this, I found new ways of filling the jugs to have the exact amount of gallons of water I needed. In teaching this concept, I would likely not use this virtual manipulative, as it was difficult and frustrating to figure out. After a long time of thinking about it, I finally thought about the jug capacities being relatively prime, and I felt that this fact was not particularly enlightening. Overall, it was a frustrating way to attempt to teach a concept that could have been taught in an easier manner without the struggle of this virtual manipulative.

When I do not know whether or not the given problem even has a solution, I can tell by the number of possibilities and probable amounts of water each jug could have. If I could add and subtract the given jugs’ full capacities, zero them out, or fill them completly, I could tell if a given amount of water needed in one jug was possible to achieve before trying it.

This virtual manipulative helps students to better understand critical addition and subtraction skills and improves mental math abilities. Numerical fluency increases as students will understand what “moves” they should make in the virtual manipulative to obtain their desired answer. As students’ skills improve and they move onto higher levels, they will be better able to keep track of more anticipated addition and subtraction answers ahead of time. These abilities make me think of accomplished chess players, who need to keep track of their pieces on the board, their opponents pieces on the board, what moves each piece can make, and anticipate their opponent’s moves in order to attack the king.

Overall, as a future teacher, I would not use this virtual manipulative to teach such an important concept such as prime factorization. I felt the concept was not fully realized in my experience using the virtual manipulative, and it could have been improved using perhaps another mathematical problem or riddle.

There are a few examples of lesson plans focused on teaching prime factorization that I find much more useful pedagogically. Click here or here to find them.