Water Jug Puzzles

Let me just begin by saying that I love mathematical puzzles. My favorite number is my favorite partly because it is the answer to the monkey problem. I love puzzles because they allow freedom to think and explore but have specific solutions. It's the process that is the fun part, though. And as the joke goes: "Q: How does a mathematician change a light-bulb? - A: They level the house thereby reducing it to a previously solved problem". One of my favorite parts about solving a puzzle is seeing all the solution processes and the mathematical basis that each process uses to find a single answer.

It follows that when I was reminded of the water pouring puzzle that I decided to dive deep into the mathematics of the problem. In order to keep this brief, I'll use a simpler 2 jug problem with virtual manipulatives rather than a traditional Tartaglian problem. (That and the 3 jug problem has been analyzed many times, with a variety of analyses: here is a single example.)

For those of you who are unfamiliar with the water pouring puzzle (and refused to follow the above links) here is a brief summary: You are given two jugs of different capacities (usually integer units) and an unlimited supply of water. You are asked to precisely measure a different amount (integer units of course) of water by filling jugs, pouring water between jugs, and emptying jugs. 

A simple example of this is when you are given a 5 liter jug, a 3 liter jug, and asked to measure 4 liters of water. Letting our 5 liter jug be F (for Five), our 3 liter jug be T (for Three), we show a solution:

Fill F, pour 3 liters into T, empty T, then pour 2 liters into T.

Fill F, pour 1 liter into T (which fills T), leaving F with 4 liters.

 

The solution process described is actually one that will work in all solvable cases. Fill the larger container, pour as much as possible into T, empty T, pour as much as possible into T [if T is full, empty T, and repeat until T is partially full], and repeat the cycle. This will cycle through all the possible combinations of water until we reach our desired amount. If the cycle repeats without us reaching the desired amount, the problem is unsolvable.

But WHY? Why does this always work? Or why are some problems unsolvable? [Prepare yourself for some house leveling...]

Since our jugs only contain finite amounts of water, this process of filling and pouring creates a cyclical group of amounts of water. I.e. If j and k are the water amounts of jugs with j > k, then j (mod k) is the amount left after one iteration of our solution process. After two iterations, we have 2j (mod k) liters of water. For any non-negative integer n, we can have nj (mod k) liters of water after n iterations. As k is finite, this creates a finite cyclical group with order (k / gcd(j, k)). 

Observe that if gcd(j, k) = 1, then we can get any amount of water up to k. Otherwise, we can only get multiples of gcd(j, k), which can be clearly seen by a simple example: Two even jugs can never measure an odd amount of water, because pouring will always yield an even amount in both jugs (an even minus an even is even). The same is true for any gcd(j, k) not equal to 1. Thus if we are asked to measure an amount of water where gcd(j, k) does not divide our desired amount this is an unsolvable problem. 

Side note: If you ever are asked by your students "When Will I Ever Use This" when learning about cyclical groups, just refer to this activity. Or read this article. 

But actually... What can we expect our students to learn if they were to do this activity? In my mind, I believe that we can probably expect students to recognize that there are some impossible problems. Depending on how long/how many problems our students are given, I feel that it is likely that students could recognize that the greatest common divisor has a role in which are possible and which are not. 

With digital manipulatives students are allowed to try a variety of different approaches. A few may try emptying and filling jugs randomly, but when given a sufficient amount of time and motivation, students tend to try to find patterns. This experimentation would help students to be more familiar with number patterns, how some numbers divide and leave remainders, or even provide a way that modular arithmetic might be introduced. 

Thus it has been shown. 

Why Are Teachers Held to a Higher Standard?

Why Are Teachers Held to a Higher Standard?

Something interesting has been on my mind recently. I read news stories of teachers getting into trouble for a variety of reasons. All you have to do is to search "teacher firing" on google news and you can find hundreds of stories in the past month about teachers being fired. Some definitely deserve to be fired (a teacher having a sexual relationship with a student should be sent to jail for a long, long time in my opinion) but some seem a little too far. Here, here, here, here, and here are just a few examples of teachers being fired for a variety of reasons. [Note that I haven't done extensive research on any of these cases, I just present them as examples of the variety of reasons teachers can be fired.]

In a textbook about technology for teachers, there is a very interesting article about how teachers must maintain high standards with their online presence. The article can be found at this link.

Summary

In a nutshell, this article is directing teachers (and prospective teachers) to be extremely cautious with their online presence. Teachers must abide by rules from a variety of sources including national and state laws, district policies, and institutional policies. These include a variety of different categories: legal requirements, social requirements, and to avoid any "inappropriate behavior". The article points out that the teacher has no control over what is viewed as inappropriate, but that the school district, community, and individuals can judge what is "moral" and deem a teacher's actions to be worthy of firing even if the teacher may not think that what they have done is wrong in any way.

As I discussed this with a few of my fellow students, the conversation tended toward how we as teachers can exercise a little control over our online presence and limit the danger of our online presence being deemed inappropriate. It seemed to be the feeling that it is our duty as teachers to maintain a high standard of conduct. We discussed the need to keep social media private, but also to monitor and not post anything that may be taken incorrectly. This includes strong political or religious posts as well as any context of a sexual or sensitive nature.

Critique

While I agree with the need to be professional in all aspects of life, I do wonder why teachers are held to a higher standard than many other people. Even politicians, whose every move is scrutinized, are extremely unlikely to be fired if they correct someone's spelling on Twitter. In my mind, I believe that teachers are held to a higher standard because we are teaching the children of the future. However, I do find it strange that the parents of our students can hold us as higher moral authorities for their children. I feel like that has the possibility of creating confusion for our students and more than any one article or blog post could address.

Connections

Something that I thought about as I was reading the article was when I was a missionary for the Church of Jesus Christ of Latter-Day Saints. I was among a group of missionaries who was given access to an iPad and other technology in order to teach the gospel of Jesus Christ. Before we were given these iPads, we had a few meetings and trainings that we had to attend. We even had a book that emphasized some safeguards using technology. While this was specifically directed toward spiritual protection, we also learned a few things about how to maintain an online profile in a positive way. These tips were very similar to those tips that can be found in the article linked above. 

I tend to have a fairly small online presence. In fact, this blog is the most active I've ever been online. However, I've always been fairly conscious of what I post, likely because I grew up with a teacher and I remember listening to stories about people (specifically students and teachers) who shared too much information and ended up with their house vandalized. I also tend to be a private person, and my online presence very much reflects that. If you try to find me on any social platform, my profile will be relatively private. 

It's very clear to me that teachers are held to a higher standard than people in many (perhaps even most) other careers. While I considered titling this post "Are Teachers Held to a Higher Standard?", the answer to that is extremely obvious. I thus chose to title this post with a "Why" instead, even though I don't have all the answers. 

Making Technology a Tool

What teacher hasn't had to worry about how technology is being used in their classroom? We worry about students surfing Facebook, using calculators for basic addition, and even have to think about the capabilities of a student's glasses. So what do we do about technology in the classroom?

Recently I came together with a group of math educators and discussed an article about technology use in a spiritual environment. The article we discussed can be found here. It's well worth a read for any teacher; not just for those who teach in a church environment, but for anyone in a classroom.

Summary

The article describes a few different approaches that teachers can take to improve classes by using technology. Teachers are encouraged to use technology in a variety of righteous ways, both personally and as a teacher. In order to do so, we must be familiar with the technology and make it a part of our lessons. The article also points out that our students are more likely to use their technology in productive ways when we justify rules about technology in ways that do not critique the technology itself but instead focus on the reasoning behind the rules. However, the article does make certain to point out that technology cannot and should not replace face to face interaction between teacher and student.

As we discussed the article in our group, while we discussed the highlights of each section we seemed to focus on how to set rules and the process that we need to go through to learn about the technology. When we set rules in our classroom we should have justifiable reasons for setting those rules. Trying to force our students to obey an arbitrary rule may only encourage our students to rebel. This is especially true with technology since our students are likely to be more familiar with the technology than we are. At the same time, as teachers we need to do what we can to be familiar with technology and the tools available to us. By doing so we can improve our teaching and supplement our interactions with technology.

Critique

During our discussion, I realized how important it is that we keep abreast of the technology. It was pointed out that many years ago teachers were worried about students using erasers. Apparently teachers thought that writing would degrade because students could now erase their mistakes. This was clearly a mistake, pencils and erasers are extremely useful in the classroom. We must not make the same mistake in our classrooms. Instead of fighting a fight we can't win, we should figure out how to use new technologies to help our students learn. 

Connections

Something that was very interesting to me about the article and discussion was the importance of setting principle-based expectations. In my mind, I see students being much more willing to follow the rules of our classroom when they know the reasons behind them. This is a big part of adolescence, and our students want to be respected and treated as adults. Setting rules and explaining the "why" behind them allows our students to have autonomy in their behavior. This is something that is key to my personal teaching philosophy, and something that I wish was the case when I was a student.

When I was in high school, I was very involved in learning about technology. I actually explored how to use a graphing calculator and computers to generate and analyze prime numbers (see Golbach's Conjecture). None of my teachers knew how to program with a calculator, and I could have used that to do a variety of things. However, because I was so engaged with classes and other explorations with mathematics, I didn't ever make trouble with my knowledge. I believe that it will be the same with my future students if I can help them be engaged in the mathematics. Technology will be a tool for our students instead of a crutch or distraction.