Desmos Relation Explorations

Exploring Families of Functions with Desmos

As promised, I am writing a few blog posts about using Desmos to explore certain parts of mathematics. One part of the motivation for this is to show that "higher mathematics" can be explored and modeled. I've noticed that most of the explorations that are pre-made are often for lower mathematics and that once you reach past Algebra or Geometry the resources (especially online explorations) tend to become much more traditional. From my brief searches, exploratory resources for the mathematics of calculus and beyond are few and far between. 

In this and future posts, I will attempt to show how these mathematics can be explored with online software. While I don't claim to be an expert on this, I do hope that this can show that with some extensive work a teacher can make at least part of their curriculum of higher mathematics exploratory. 

With that disclaimer, today I wish to explore a family of functions, those of the form xⁿ + yⁿ  = 1, with n an integer greater than 0. These are a very interesting family of functions, but in my experience we rarely look at these functions past n = 2. For n = 1, we have the line x + y = 1, and for n = 2 we have a circle with radius 1 centered at the origin. In this exploration, we will focus on n greater than that.

The first observation that I will make is that 1 = 1ⁿ for all n positive. Our general form then turns to xⁿ + yⁿ  = 1ⁿ, which is a special case of Fermat's Last Theorem. In 1994, Andrew Wiles proved that there are no integer solutions for neither x nor y equal to zero when n is greater than 2. While this is not directly related to our Desmos exploration, it is an interesting piece of number theory. I include this observation to show that exploring a simple formula can bring some very interesting results. In this case, Fermat's Last Theorem brought about 358 years of mathematical research and was the basis of the creation of the field of algebraic number theory. Students can also learn to explore and build from simple problems and develop complex theories and understandings of interesting mathematics. 

Anyway, back to Desmos. When graphing this function, a suggestion pops up to ask whether you want to add a slider for n. After adding that slider, you can slide your n to be any value you wish, without having to type in the individual equation. Here are a few screenshots of our function at specific values of n:

Looking at the graphs, we notice that two points are always part of the function: (0,1) and (1,0). This makes sense, as 1ⁿ = 1 and 0ⁿ = 0 for all positive integers n. Thus xⁿ + yⁿ  = 0 + 1 = 1 + 0 = 1 for any n. Thus these two points must always be part of our family of functions.

From these simple graphs, another, more complex pattern starts to appear. It seems as though when n is odd, we have a line that is interrupted near the middle by a strange bump. When n is even though, we have a square-ish circle. Hmmm.... That's interesting. Students can continue searching and see whether the next few (or few hundred) functions follow this same pattern, or we could help them be able to recognize and justify this pattern mathematically as well. 

For the sake of convincing without proof, observe the following two graphs with n large [the graphs don't look much different for n any larger]:

Those look like right angles now, don't they? Another pattern has emerged.

How can we justify these patterns without just reverting to "look and see"? Well, let's first put our observations in a little bit more formal mathematical language: 

For n odd:

As n grows large, on the intervals (-∞, -1) and (1, ∞) it appears that our function is linear and is approximated by the line x + y = 1. Inside the interval (-1, 1) our function appears to be constant at y = 1. 

For n even:

As n grows large, our function appears to be approximately equal to the unit square.

We will treat these as two separate conjectures and attempt to justify each separately. 

For n odd:
Observe that for n odd, xⁿ and yⁿ are always positive for positive x, y, and always negative for negative x, y. Thus we can make x, y as large as we like, as long as the other is similarly large.
Solving xⁿ + yⁿ = 1 for y yields y = ⁿ√(1-xⁿ) [the nth root of (1 -xⁿ)], which equals (1 - xⁿ)⁽¹/ ⁿ⁾.
Taking the derivative of y with respect to x yields
dy/dx = 1/n × n xⁿ⁻¹ (1 - xⁿ)⁽¹/ ⁿ⁺¹⁾ = xⁿ⁻¹ × (1 - xⁿ)⁽¹/ ⁿ⁺¹⁾.  For |x| > 1, and n large, (1 - xⁿ) ≈ -xⁿ.
Thus (1 - xⁿ)⁽¹/ ⁿ⁺¹⁾ ≈ -xⁿ⁽¹/ ⁿ⁺¹⁾ = -x⁽ⁿ/ⁿ⁺¹⁾  ≈ -x. Thus dy/dx ≈ -1. For x ∊ (-1, 1) xⁿ ￫ 0, thus dy/dx ≈ 0.
Since we must go through (1,0) and (0,1), we are constrained to our right angle around x = 1 and the rest of the graph is approximately equal to our line x + y = 1.

For n even:
Observe that for n even, xⁿ and yⁿ are always positive for any x, y in the real numbers. Thus we are constrained to always have x, y ≤ 1. Thus we are constrained to have x ∊ (-1, 1) xⁿ ￫ 0, and thus dy/dx ≈ 0 as above. We are thus constrained to the unit square.



So now what? Students aren't expected to go as in depth as in this blog post. In fact, if you google these types of functions you will find a few explorations from college students. However, most attempt to justify their conclusions with visuals alone. Here are a few examples: here, and here. While this is extremely useful for students to know how to do, it is also useful to have a mathematical justification. But this justification only comes after an observation is made, which is unlikely to be made without the assistance of technology.

Students can be expected to learn quite a bit about functions, and understand that a function doesn't have to be smooth or "normal" in order to be a valid function. Understanding a function in depth requires some extensive work with each part of the equation. This is extremely likely to happen if students are asked to explore such an interesting family of functions.

Using Desmos in a Classroom Environment

I've recently been exploring online software for use in a classroom. While I was familiar with using GeoGebra for a variety of geometry and graphical representations, I was amazed with the other applications of Desmos as a powerful resource for teachers and students to explore. Apparently, Desmos has built a wide variety of classroom explorations. These are actually fairly high quality explorations, and I am actually going to continue exploring Desmos in a few explorations in the near future, so keep your eyes open for those.

In 2016 NCTM published an article about increasing class participation through online resources. While it was written by authors of Desmos curriculum, I feel like it gives great information and suggestions for math teachers. It's a little outdated, one of the activities that they describe has lost native Desmos software support, but the principles are pretty useful to think about. The article can be found here (or here if you have a BYU Learning Suite account).

Summary

This article, it seems, was written primarily to discuss the benefits of using online technologies in a math classroom. That being the case, one major benefits of using online softwares (particularly Desmos) in a classroom is that students can receive iterative feedback- feedback that comes extremely fast and not too leading. Online tools are also useful in that students can share with each other in additional ways, making student collaboration and peer to peer learning much more likely. Beyond just student feedback, teachers also have more information, often being able to observe the entire process of learning instead of just viewing the final result. The article then explores two Desmos activities and how they achieve these aims of online software tools.

As I discussed this with a few of my fellow students, we were amazed with the tools that are available for the teacher as students accomplish these tasks. This eventually progressed past a discussion of the article, and I will actually discuss these benefits in an upcoming blog post. However, we also discussed how online tasks should not and cannot replace classroom discussion led by a teacher. While student's interaction and peer teaching is extremely useful, we must also be able to "steer" the class in the correct direction, keeping students from "guessing and checking" when they do not understand the material. This is actually possible in any Desmos activity: the teacher has the control to "pause" all their student's screens, inserting teaching and discussion where otherwise students might get caught in a learning trap.

Critique

While this was an interesting article, I feel that it was much too focused on the benefits and value of Desmos. No other software was mentioned in the article, while I know that there are a variety of other softwares and online mathematical tools for the classroom. The article itself also didn't mention the need for heavy teacher involvement in a classroom full of devices, it almost seemed to me that the article was implying that the class would teach itself with Desmos. This makes sense as it was written by some authors of Desmos, but it was lacking in power by not addressing all parts of this important issue.

I do wonder about the discussion that we had and if we are all a bit naive. While our discussion focused on the ideal classroom environment, I do wonder how long it will take us to actually come anywhere near that in our teaching. Teachers who have been teaching for years may be able to know where each of their 200 students are in their understanding and address needs well, but I worry about being able to do that in my classroom. However, I do feel like online activities will give us additional information on student understanding and the excitement that our discussion had for the tools of Desmos is deserving. Being able to pause an activity is an extremely powerful tool, and recognizing that was a great benefit from our discussion.

Connections

I think that technology can be extremely useful in the mathematics classroom. For example, when I was in high school, I programmed my graphing calculator to solve a quadratic equation using the quadratic formula. I knew how to plug numbers into an equation, I'd done that hundreds of times and probably will a few thousand times more. However, for ease, I used my calculator and would get onto more "interesting" mathematics instead of wasting my time with something that I already understood.

The other day, I was observing in a classroom and watched as a teacher had students graph (on their graphing calculators) an exponential function versus its' derivative. Students then used that to estimate what the formula for the derivative of an exponential function from that graphical representation. This could have been very useful for these students, but I feel like it wasn't presented in an exploratory manner, making it a "plug and chug" method for many students. I think if it had been presented in a way such as through Desmos or other exploratory software students could have discovered and understood the concepts a lot better than they did.

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.