## Tuesday, December 30, 2014

### New Years Goals Jar

It's that time again!  During the last week of last year, my wife and I set up a bowl and a set of papers similar to the picture above.  Then, any time during the week that either of us had any inspiration, dreams, or goals for things to accomplish during the year, we wrote them down. On New Years Day (or sometime close to then) we had a date night and read through all of them together.  Some of them were simple, some were outrageous, some were jokes, some quite serious.  It was a great hour of conversation (in Burger King... classy I know!).

Here are a few of the goals we set at the beginning of last year that we've met (or attempted) as a result of our conversation that day:
• Budget each month: It has been so helpful to come together twice a month to get on the same plan about money. As a month comes to a close, we are able to look at what we've spent or not -- save for Christmas, etc. Without this, it would have been impossible to...
• Buy a bed for the girls room
• Send out Christmas Cards
• Buy a freezer to have freezer-meals (didn't quite accomplish, but should occur early in 2015)

• Create a CD: One of my favorite accomplishments of 2014 was finally recording a CD of my own music, written for our church. By using garage band, I was able to layer together several tracks worth of vocals and guitars and self-produce the CD: Oaks of Righteousness, which you can download for free.

• Concert:  Another goal of ours was to host a concert of our favorite family-oriented songs, which we were able to do in July. Belding offered us a spot on their Thursday night "Music in the Park" series, where we were able to perform a set of country songs.

• Start Recycling
• "Bread"-ruary:  Our initial dream was to simply go one month with only creating fresh bread - not buying any. After successfully going through February, we continued on into March, and now ten months later, still make our own bread each week. I don't think it's saved us any money, but it sure tastes good, and with an electric Breadmaker, it's not time consuming either.
• Have a garage sale
• Not have another kid (YET): I hesitated to even include this, but with couples all around us expecting in 2015, and this being such a publicly stated goal in our small groups and circle of friends, I was afraid if I left it out people might think we were making an announcement or something.  The exact wording of the goal was "Enter and leave 2014 with two kids", a sort of joke that we wouldn't lose any kids along the way either.  I'm happy to say that despite Carrie picking up more hours at the library and also 8 hours a week at church, I haven't lost any kids in the extra hours of daddy-daughter time.
Granted, there were some goals on the list that haven't been accomplished. They won't automatically become goals for next year, as I think it's important to reevaluate and dream again.  But overall, the process of dreaming together as husband and wife was valuable last year, and one we are planning to do this year too.  Eventually, I hope the whole family will get involved too.

What do you hope to accomplish in 2015?

## Monday, December 29, 2014

### Physics of Bandaloop Dancer

The other day, I was shown a video of one of the most exciting looking things I've seen in some time:

The group of dancers is called Bandaloop and among other things, they dance on the sides of buildings and cliffs by being attached via rappelling ropes. The long ropes keep them from falling, and allow them to make incredible leaps with long hang times.  A video interviewing the founder says some days, from tall buildings, she can get jumps with hangtimes of 9 seconds or more.

My first reaction to the video was "WOW! I wanna do that!".  My second reaction was "What's the physics behind that?  Nine second hangtimes? Really?!"

Since I can't try that anytime too soon, I must instead attempt to describe the physics behind them. Here are the pictures I drew on the side of the napkin:

I didn't have lots of data to go on, so I initially estimated that the Length of the ropes (L) might be approximately 30m, and if harnessed at the center of the body, that puts x at approximately 1m.  At best jump, I estimated approximately 5m out from the building. Since the sin of the angle θ is x/L, the inverse sine of 1/30 and 5/30 suggests that θ ranges between 2 to 10 degrees. At angles this small, sin(θ) and tan(θ) are nearly identical, suggesting that y and L are nearly identical too, and I'll be interchanging them occasionally. This is not true as the rope gets smaller -- so relatively tall buildings and tall cliffs are important.

Next I drew a free body diagram of the forces acting on the dancers as they are away from the building.  There are primarily two forces acting on the dancer -- Weight pulling the dancer down, and tension in the ropes pulling the dancer at angle θ up and in toward the building.  That angled force I broke into components Tx acting in towards the building, and Ty acting to counteract the dancers weight.  Because the ropes are so long, the height of the dancer doesn't change significantly, and since the dancer isn't really moving much vertically, we can say that the forces are balanced vertically.  That is, W = Ty. Since W = mg, this means Ty = mg.

Horizontally, the forces aren't balanced, and so whenever the dancer is in the air, there is a portion of the tension Tx which acts to pull the dancer back in toward the building. This unbalanced force is a net force, and so we can write another equation: Fnet = Tx. Since Fnet = ma, this means Tx = ma.

Finally, Tx and Ty are related to the angle by the tangent relationship, such that tan(θ) = Tx/Ty, and after multiplying, Tx = Ty*tan(θ).

After a few substitutions and a little division, we find a formula for the acceleration inwards toward the building that the dancers feel:
\begin{align*} T_x &=T_ytan(\theta ) \\ \frac{ma}{m} &= \frac{mgtan(\theta)}{m} \\ a &= tan(\theta)*g \end{align*}
It's surprisingly simple and clean -- the acceleration the dancers feel is just a multiple of gravity. Since θ ranges from 2 to 10 degrees, the dancers feel acceleration toward the building ranging between approximately 1/30th to 1/6th that of normal gravity.  For comparisons sake, the acceleration of gravity on the moon is about a 1/6th of that on earth, which according to my rough estimates is about the most that the dancers would feel inward toward the building on their most extreme jumps. Most of the time they are just a meter or two away from the the wall they are feeling much lower attraction toward earth.

Put another way, the dancers feel approximately 1/30th of their "weight" inward toward the wall, so a 120lb dancer might feel only 4 lbs of forces inward. Imagine how easy it would be to jump if you only weighed 4 lbs, but had the strength of a professional dancer!

Which suggests a different approach to this problem. Let's figure out how far out from the building a dancer ought to be able to get, assuming they can jump "off the wall" with as much speed as they can normally jump off the ground.... Assuming that a person can normally jump to a height of 0.5 meters under typical gravitational acceleration of -9.8 m/s/s, we can use the "no time" formula:
\begin{align*} v_f^2 &= v_0^2+2a(\Delta s) \\ 0 &= v_0^2+2(-9.8)(0.5) \\ v_0 &= \sqrt{2(9.8)(0.5)} &\approx 3 m/s \\ \end{align*}
They would have to jump at a velocity of approximately 3 m/s, yielding a hang time of approximately:
\begin{align*} v_f &= v_0-at \\ -3 &= 3-(9.8)t \\ t &= \frac{-6 m/s}{-9.8 m/s^2} &\approx .6 sec \\ \end{align*}
Leaping with the same initial speed (3 m/s) off the side of the building with an acceleration of 1/6th of gravity as these dancers feel would allow them hang times of 6 times as much:
\begin{align*} v_f &= v_0-at \\ -3 &= 3-(\frac{9.8}{6})t \\ t &= \frac{-6 m/s}{\frac{-9.8}{6} m/s^2} &\approx 3.7 sec \\ \end{align*}

Not quite the 9 seconds claimed in the video, but more on that later.  Substituting half of this time (because the maximum height occurs halfway into the trip) into the kinematics equation allows us to calculate the maximum "height" off the buildings this dancer could reach:
\begin{align*} s_f &= s_0+v_0t+\frac{1}{2}at^2 \\ s_f &= 1+3(1.75)+\frac{1}{2}(\frac{-9.8}{6})(1.75)^2 &\approx 4m \\ \end{align*}
That's pretty close to the 5m I estimated from the video.

Now the hangtime of 3.7 seconds is far less then the the claimed 9 seconds of hangtime, and even my closer observation of the video suggests a few jumps were more than 5 seconds long. One way to get more hangtime is to jump with more speed - something that's quite possible with stronger and trained legs. Remember I started with someone able to jump 0.5 meters -- and I'm sure a strong dancer could leap higher.

Another reason the kinematics equations don't give us enough hangtime is that the acceleration is not constantly 1/6th that of gravity -- often it's way less than that even! Smaller accelerations, like those felt close to the building when θ is small, would increase hangtime significantly. Unfortunately, I've forgotten the formulas for how to deal with accelerations that aren't constant -- although I'm sure a few google searches could refresh me.

Either way, the Bandaloopers certainly can experience tremendous "jumps" due to the low horizontal forces they have to fight on their rotated worlds, and simple first-year physics concepts help reveal why.

Now, perhaps someday I'll be able to actually try it for myself.

### Lesson Planning with Google Calendar

I've done lesson planning about a dozen different ways in my 10 years of teaching - and my favorite and current way of planning is by using the Calendar application. As you can see in the picture, each activity for a given day is one of the pink boxes that shows up. In these boxes (technically they are "events") I can write as little or as much detail as I want.  Typically I just write a little bit: "Notes on Solving by Quadratic Formula" for instance. For assignments I write something simple like "Assign: Pg 189 #1, 2, 10-26evens" or "Assign: Worksheet - Multiplication".

These calendars are super easy to move items around, copy from year to year, assign over multiple days, etc. I have created four Google calendars, one for each class that I teach. These Google calendars then can be embedded into my class website.  Then students and parents can know at a glance what were studying not only this week, but they can look forward to days when they may be gone, or look back to days they've missed.  I have had on more than one occasion students who have come in after being sick and instead of asking "did I miss anything" they hand me their homework which they found on the calendar.

Whenever possible, I put links to helpful material into these events as well. This was the biggest challenge, but something a little html knowledge proves handy for. To add a link to something, you double click on the event and type into the "Add a note" section. You may type text descriptions with more details here and students will be able to read these by clicking for more details on their calendars.  Adding URL's sounds easy at first, but unfortunately, simply pasting a link into the "Add URL" section doesn't work. Perhaps this will be fixed someday? A work around that allows a clickable link to appear on the student end of things is to use a little HTML code. Copy and paste following code snippit:
<a href = "URLgoeshere">Text goes here</a>
One thing to look out for is to make sure that your computer doesn't turn " into "smart quotes" -- because then the links won't work. I had to turn "smart quotes" off in system preferences. In addition to downloadable worksheets as assignments, I include links to quizlet vocabulary practice, math practice websites like ThatQuiz, or videos of my lessons whenever they are available.  A colleague of mine records every lesson every day, and since this year we share Algebra 2 together, I include links to her videos on youtube. I tell the students that if they don't like the way I explain something, or if they need another explanation, they can check the calendar and find Mrs. Straayer's videos.

One last plug for planning with google calendars is that students and parents can subscribe to them, and then they can have them show up on either their computers or phones. Anytime I make an update, they will have access to the most up to date plans.  This is really convenient when a snow day or something arises -- I can simply cut and paste today's events to tomorrow.  I have mine connected to my phone so I can either see what's coming up from anywhere -- or add a note or link or change something from everywhere on my phone.

### QR Codes on Worksheets

Within the last year or so, I have been making efforts to include QR codes on my handouts in my classes:
 Example of a QR Code: If you scan this with your phone, it will tell you that it's an example of a QR code.
For  those of you who don't know, a QR code stands for Quick Response code. You scan it with an app on your phone or tablet, and depending on what's encoded, stuff happens. The example above simply shows some text. Most often, a QR code is linked with a website -- and you see them all over the place from political brochures to labels in the supermarket. My wife has used QR codes to create scavenger hunts for her students that visit at the library. You can even set them up to automatically write a text for you as my step-sister did for me for those days when the girls get a little out of control:
 Mom Meter:Scan the appropriate QR code and mom receives a text like: "come home soon dad is going to kill us!"
About a year ago I started putting them on the worksheets and notes packets I would give to my students.  It only takes up a little space in one of the corners somewhere.  Usually these are simply connected to a pdf of the document that they can view on their phone or tablet. Most of the time these files have the original blank copy of the document, as well as an answer key. My philosophy on homework is that it is practice -- and I want my students to have good practice -- not blind practice. We talk often throughout the year that they need to know whether they are doing things right, so that they don't practice mistakes and learn bad habits.

Here's how I do it:

First, I use Dropbox for all my file storage, and one of the features of Dropbox that I use most is the ability to get a sharable link to every document by simply right-clicking. I put these links on my Google Calendar, on my class website, and of course, in the QR Codes. You can create a QR code for free at a bunch of online sites such as QRStuff or QRCodeGenerator. Simply paste the link into the websites and copy the image to place it into a document.

 Creating a QR Code in Quicksilver
That was too much work for me though, so I quickly learned that my favorite program Quicksilver had a QR Code generator available. If you have Quicksilver, you can simply paste the link in the first pane and choose the CopyQRCode action in the second pane and viola - you have a paste-able QR code that can be put into any program. After setting it up with it's own trigger, it literally takes me 1 second to create a QR code for a URL that i've copied onto the clipboard. For more tips on setting this up, check back later for another article.

## Saturday, December 27, 2014

### Beauty In Polar Coordinates

This post is a part of a series of guest-posts on polar coordinates and complex numbers. These posts were written by my pre-calc students:

Beauty in Polar Coordinates
by Luke VanDyke

Beauty can be found in mathematics in various places. Whether it be in Euler’s Identity or another beautiful equation of the sort, or in a magnificent graph, order and beauty are found on every page. One of the most interesting and amazing ways to graph objects is in polar form. A variety of shapes such as spirals, cardioids, and limaçon are just a few examples of the wide range of beauty found in graphing in the polar form. I think one of the most amazing curves that you can graph is the rose. Using Geogebra, I was able to explore in great deal the immense complexity of such an amazing curve and also notice several key patterns regarding the equation.

In order to fully demonstrate the beauty of the rose, I first inserted two sliders, a and b. I made the range for each slider from 0-10 with increments of .1. These sliders would serve as my values for a and n. The equation we learned in class (r=a*cos(nθ) and r=a*sin(θ)) must be changed into the curve expression on Geogebra. Mr. Roer helped me out a lot in converting to curve form. Once I had the equation in, I was able to play with the sliders and see how they worked. Slider A adjusts the size of the radius. The larger the number on the slider, the larger the rose will “grow”.

Slider B was a lot of fun to play with. Slider B adjusted the number and also the width of the “petals”. After playing around with both sliders, I noticed a pattern developing that affected the graph significantly. When the value of slider B was odd(let’s call this value x), the flower would have x number of petals. For example, if Slider B=1, there was just one circle. The same occurred for all other odd values between 0-10. On the other hand, even numbers also had an interesting pattern. For every even value between 0-10(we’ll call this value y), there would be 2y number of petals. For example, if 2 was the value on the B slider, the rose would have 4 petals.

This project was a really great opportunity to see God’s beauty as seen in mathematics. The saying “Out of intense complexities, intense simplicities emerge.” really goes hand in hand with this project. Once you put in the big, complex equation, a simplistic, beautiful image of a rose emerges. This saying also applies to roses in real life. Out of a complex equation of photosynthesis and many other factors, a beautiful flower emerges. There are many other examples all throughout God’s magnificent creation. Whether it be in the majestic landscapes throughout nature or in the complexity of infinitesimal human DNA and genetics, God’s awesome handiwork is seen all throughout the earth. Through this project, I’ve been able to reflect on how great God is and how everything in His creation is simply remarkable, even in places where you least expect it.

## Friday, December 26, 2014

### Customized Morse Code Vibration Text Alerts

So, have you ever felt your butt vibrate and thought -- hmm, I got a text.  Who is it from?  Do I need to check it right now and risk getting:
a.) caught by my teacher
b.) caught by my kids
c.) caught by the others in the meeting?
Wouldn't it be nice to know who's it from at least?

The other day I was playing with settings on my new phone and found the solution.  I noticed that I could create my own custom vibration patterns. You can also assign different vibrations to texts received from different people. Some how inspiration struck to assign my top texting contacts each a unique vibration pattern.

Unfortunately, there weren't very many vibrations patterns to choose from, until I saw that I could create my own -- and then I decided I would use Morse Code with my contacts initials, so I could know who was texting.

Here's a picture of morse code if you don't already have it memorized. I don't have all of it memorized by the way -- but I know many of them.  I found it helpful at first to learn key words to help me memorized the dots and dashes.  CAL-i-FORN-ia for instance helped me remember C's pattern of -.-.

To create a personalized vibration pattern on an iPhone:
1. Create the vibration pattern:
-->Settings-->Ring Tone (Or Text Tone) --> Vibration --> Create New Vibration

Then tap out the pattern you'd like. It does't have to be Morse Code -- I just found that convenient and cool, so I used that. For people that text you regularly, I would recommend something short -- for my wife, I at first had her whole name -- but that got annoying really quick when my butt would vibrate for three or four seconds whenever she texted.  Now it's just C:  -.-.

2. Assign it to a particular contact
-->Contacts-->Find the persons name
OR -->Search for their name.
Then edit and look for Text Tone and Vibration.

You can assign different text tones for different people here -- which is the sound it makes when they text you.  Since my phone is almost always on vibrate though, I don't bother to do that.