Compositions of Transformations

Here is a link to a gsp file describing the composition of transformations:

https://drive.google.com/open?id=1lxVtvLpW3ZPYjupMBpPo-0mSLqwo4D8I

Parabolas for days

Here is an interesting exploration about parabolas. This involves a geometric and analytic dynamic representation of a parabola, including analysis of the directrix, focus, line of reflection, and vertex. Another very interesting piece of this exploration is how the vertex changes as a, b, and c vary from our general equation y = ax^2 + bx + c.

This is a gsp file, so in order to open this you must have Geometer Sketch Pad. Sorry for everyone out there who doesn't.

Here is the file. Have fun exploring!

Geometer's Sketch Pad

It's been a while since my last post, and this is a pretty small post. Sorry to the one dedicated reader of my blog. There are going to be more posts in the near future.

I've recently been playing with Geometer's Sketch Pad. If you haven't ever played with it, I highly recommend it. It's been a lot of fun, even as I've run into some limitations of the software. But some of the most fun things that I've done is to create a bunch of constructions. Software like this allows the user to go through thousands of creations and dynamically view a construction in many, many ways. And if you make a single mistake (like I do often) you don't have to completely erase your entire drawing. Instead, just hit Ctrl Z and go back to having fun.

Anyway, here are a few of my most recent constructions in GSP. They are all created as tools, and are fairly cool. If you want to see how they were made, you can view the script view and walk through the entire construction.

Here is a link to a copy of my gsp file (you'll need GSP in order to view these, sadly).

https://drive.google.com/open?id=1r4mseSTA-rgLa3ez4TTFAXd6axeayZNG

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.

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.