Click for Red Bull Stratos mission website.
For the casual observer--well, hang on--no one casually observed Felix Baumgartner's jump from the edge of space on October 14th, 2012. Not everyone who watched had their mind racing like mine was, but I am certain that this feat is one more than befitting of #101qs and #3Acts formatting.

What follows on this post is what I have managed to use in my own class to help quench students' curiosity about the Red Bull Stratos project, which sent a helium-filled balloon up to 128,100 feet  and had Austrian skydiver Felix Baumgartner jump back to the Earth wearing a pressurized suit.

I got a response back from Red Bull Stratos!
This coming Sunday morning, I will be presenting at the STEMtech Conference in Kansas City and was hoping to use an example from the Red Bull Stratos jump within my talk to show how velocity, acceleration, deceleration, and time can relate to one another--both for the balloon's ascent and Felix's descent back to the Earth.

So, I tweeted the Red Bull Stratos folks. With 240,000+ followers, I figured it had fallen on deaf ears (although, since they're used to travelling faster than the speed of sound, they could use "I couldn't hear you" as a valid excuse).

Nevertheless, I set out to collect some data points for use on the TI-Nspire CX using the Teacher Software. Below is a slide show of some of the work I completed. What I hoped students would take away from these graphs included:
  • The balloon appears to make a fairly constant ascent, but careful inspection would reveal the upward velocity of the balloon actually varied much more than at first glance.
  • Does the elevation vs. time graph of Felix's descent convey enough information for us to know when he deployed his parachute? (This would be a great time to introduce the idea of a secant line to students, as a BRIEF introduction to a Precalculus and Calculus topics such as average rate of change and limits)
  • Does the velocity vs. time graph of Felix's descent show enough to let us identify when he is accelerating and decelerating, when his parachute was deployed, and when he touched earth again?

Also, I'm including a link to the data I used to compile this activity on this post both as a Google Docs Spreadsheet and a Microsoft Excel file, and also the TI-Nspire CX document used here (this can be downloaded, then viewed within the TI-Nspire Document Player without needing software purchase or download of trial versions). Please use either however you see fit and let me know how it works for your class.
The ascent data was gathered from this post on Wikipedia.
The descent data was gathered by watching and pausing this YouTube video a couple different times (NOTE: these were approximate velocities and subject to verification and should not be considered 100% accurate, but did work well enough to conduct the lesson I was looking to convey).

Click for more information on the Red Bull Stratos balloon.
The other information I wanted to include in this lesson, but did not focus on in this lesson, included facts and figures about the balloon used in the Red Bull Stratos mission. 

At lift-off, the balloon was nearly 200,000 cubic feet (or 100,000 Giant Jenga games, as I was able to illustrate with my students in class, since we recently worked with it). At its highest point, the balloon was nearly 30,000,000 cubic feet large (which I related to my students by having them envision a cube that is as long, wide, and tall as a football field--a cube with side lengths about 310 feet on a side). Check this blog post on how the balloon compares with the Statue of Liberty

When we approach a lesson that involves transformations of solids (impact on volume and surface area when a single dimension is altered), I will likely revisit this topic again with a different focus in mind.

As much energy as my students brought to this lesson, then finding out that there were over 8 MILLION simultaneous YouTube viewers of the Red Bull Stratos mission (which was not mentioned on the mission's blog post among the other records and noteworthy feats achieved), I knew that this was a topic that would grab their attention. 

I also know how much my own mind was racing as I was watching the broadcast of the mission, so to leave it alone would be unheard of. But when the speed of sound was broken during the fall, that is the play on words I was hoping for: "unheard of."

Mission accomplished. Thank you for being a great example to our students, Felix Baumgartner!  Wait! An example in a lesson, I'm not saying that all our students should go jump in balloons and take up this sort of skydiving! There. I have to cover that segment of the population that will try and one-up this record by any means necessary.


OMG, Unl. TXT FTW!!!
I like how this site uses a question posed by a student, akin to "How do I graph a function that has a changing slope?" The example quickly divulges to cell phone pricing structure, a valid place to go The issue I have with their example is that the majority of my students who have cell phones and an awareness of these pricing structures realize quickly how outdated the example is and lose engagement in the class discussion. Cell phones make calls, true; however, the majority of the usage my students' phones see is by text messaging, which most of them have a HUGE allotment of messages per month by their phone plan (many of whom will have unlimited texting anymore). So while the cell phone pricing examples are valid examples to introduce piecewise functions in a context familiar to students, they are quickly outdated by the rapidly changing face of technology today.

The Khan Academy example I found that lends itself to piecewise functions takes almost 3 minutes to get the example drawn on the screen (while Sal says twice "I hope I'm not boring you while I'm drawing this" and also "I really should have something that lets me just get the graph image in place quickly"). I have to be a bit lenient, though, because his video was posted in 2007 which is better than I was doing back then (I was, however, using a SMART Board regularly in daily class activity though and feigned a couple efforts at uploading my stuff).

Students become quickly detached from examples in their textbook and I would have a hard time justifying asking them to watch a 9-minute KA video where the first 3 minutes are setting up the example.

I visited the convenience store in town and compiled this photo (the one pictured above left) of their donut case, which indeed exhibits the characteristics for a piecewise function (while also allowing us to discuss the greatest integer function, since we can only purchase whole number quantities of donuts). I simply project the image in class when they come in and have already set the scene for the lesson, despite having to address the "So, we're taking a field trip today?" and "Did you bring snacks?" comments from students as they arrive. [NOTE: The amount of time showing this image in class should be inversely proportional to the amount of time to the nearest meal for your students.]

Have students create a price table for the quantity of donuts purchased, from 0 to about 15. They get an interesting surprise around the dozen-donut mark (namely that 11 donuts costs $10.44 and 12 donuts costs $9.99--so it is as if the 12th donut earns you a $0.45 refund!). These sort of conclusions are ones students treat as an inside joke among their other classmates, so don't sell it short despite the nerdiness it entails. Since we're supposed to help our students make real-world connections, this is a prime example--and a delicious one, to boot.

Krispy Kreme price chart. Have students construct it and see if they notice anything around the dozen-donut mark.
In the video examples I complete below using foldables, I try to emphasize that each of the PIECES of the function could stand alone by themselves with no restriction on their domain(s). With a piecewise-defined function, we are merely piecing together the parts of the individual functions that are requested in the piecewise function that DOES specify constraints on the individual component functions.
This is the example worked out in the YouTube video below. A copy of the foldable used for a 2-piece piecewise function can be found HERE

This is the example worked out in the YouTube video below. A copy of the foldable used for a 3-piece piecewise function can be found HERE

Obviously, I need to develop a better way to make a quick demo of lessons like this. These were each done on my iPad2, so I'm pretty pleased with the capability of them to collect decent sound and video. What I did not show were the cue cards I created for myself just behind the iPad so I could more quickly graph and plot the functions (something Sal did not do on his example I mentioned earlier). Oh, and I included a shout out to Vi Hart for having used Sharpies for both of the videos. So, I've got that going for me, which is nice.

Piece, I'm outta here. Yes, I misspelled that. I meant to. Math puns are hard, I gotta take what I can get.

Doctor-approved backpack weight?
I came to our lesson on linear inequalities in two variables, which coincidentally closely follows our discussions on direct variation, scatter plots, and linear regression. I wanted to find a unique real-world application for the lesson that tied these two topics together. 

I found a blog post for the New York Times wellness section that gave a decent path HERE: the relationship between student body weight and their backpack weight. The American Academy of Pediatrics recommends the student's backpack "never weigh more than 10 to 20 percent of your child's body weight." Consumer Reports insists that a backpack weigh no more than 10% of the student's body weight (I feel it necessary to point out that the emphasis of their claim seemed to be focused on elementary school-aged students; their insistence seemed to loosen up on secondary students, citing that their back muscles are stronger at those ages and better fit for a bit more extreme load).

So, enter the lesson, sans the lab coats and clipboards that the Consumer Reports folks might have access to, and determine: 
Are my students' backpacks in compliance with the most extreme of these doctor recommendations?

I've created several activities that are intended to accompany this activity. Here are some of them:
  • TI-Nspire CX activity [If you do not have TI-Nspire Teacher Software to open this file, download the file first, then use the TI-Nspire Document Player to view it without need for purchase or download] The activity will walk students through analyzing their class's data (NOTE: The teacher should note the slideshow at the bottom of this post for pointers as to how this activity is set up. Student weights are "hidden" on the first/title page of this document), as well as a couple of final Self-Check questions to conclude the lesson with a check for understanding.
  • GeoGebraTube post for the graph and spreadsheet to display a class's data, showing the Compliant and Non-Compliant regions as they pertain to doctors' recommendations regarding this weight relationship.

Please note the slide show below for images of each of these activity resources with captions to help give pointers as to the intent of the activities as well as user tips and tricks.
Some helpful tips on successfully executing this lesson:
  • Don't forget to use an accurate scale! I was able to call in a favor with our school's wrestling coach and use their official scale since this lesson happened to fall during their off-season. Especially helpful--since some students were bashful about their data being seen by others--was the fact that the scale had a detachable display screen which could be faced away from their peers.
  • Yes, I indulged a bit when using the TI-Nspire CX's (especially since we actually were able to use the CAS version of them). They work great this early in the school year to grab students' attention and quickly engage them in a topic like this one. I otherwise would use a similar activity using the TI-84s we have access to.
  • I do not mean to play down the power of GeoGebra in this post. It is a very visual tool and my students have enjoyed it in many other lessons, but it was dependent on me having contained my lesson within a separate file, in this case a PowerPoint. No big deal, but with a couple of network snafus among one of my larger class sizes, I revised last year's lesson plan to instead use the TI-Nspires this time around.

As for the photo at the left of my backpack and the banner above my classroom window (both KELTY, by no coincidence): there's no real rhyme or reason, I just wanted to give a shout out to the company. I still have students who believe I faked both products so that I could dub myself with a pretty sweet nickname.

Well, who's laughing when my endorsement deal with them rolls in? Right? Hint, hint...

--Keltner-- I mean, Kelty

As the summer wound down, I found myself frequenting home improvement stores to cross of those last few items on my to-do list. In that wandering, I came across a Giant Jenga topic on a Home Depot Community blog by employees/associates.

I figured this would be a great way to tie-in at the beginning of the school year to talk about traits of shapes (in this case squares, so that the Jenga tower has no overlapping from one level to the next). Particularly, I wanted them to address the trait of the assembly that the length of each board was supposed to be the width multiplied by the number of boards stacked across the assembly--in this case, 3 x 5.5" since we went three boards across and each measured 5.5" wide. I was able to toss the idea at the woodworking teacher, who was going to be biding his time in class until students had passed their safety quizzes so they could safely operate the machinery in their shop. He agreed that a straightforward, repetitive project like this one would allow his students to use multiple machines (miter saw, table saw, and planer for example).

Before embarking on the project, I made sure to set the scene with my students and pique their interest in creating their own Giant Jenga. They insisted we Google it to see if someone else had already come up with the idea before them. A sweet collection of games came up, but mostly photos, aside from the Home Depot article I'd mentioned above. They noticed it included measurements (i.e. the answer, they thought). I agree with Dan Meyer here: questions like these are most effective when they are un-Google-able.

Students try out our Giant Genga game as a reward for their efforts.
I insisted we go bigger than the Home Depot site had done; we should use 2" x 6" boards instead. The task in class that day was for them to help me make out my shopping list. Some prompts we had to answer:
  • Why did the Home Depot site mention each board being cut to 10.5" long?
  • Would that same length apply when using 2" x 6" boards?
  • Could we just buy the longest boards available, or would 8-foot, 10-foot, or 12-foot boards help minimize waste?
  • How many, and which length, should I purchase? (When I was at the store, I also discovered the unit rate for these boards differed somewhat; namely that the shorter boards were the cheapest for foot-length. I anticipate revisiting this observation in a later lesson.)

The final product is pictured here. Although it is heavy and not "broken in yet" where the pieces slide easily in and out of their slots, students have enjoyed playing it in the couple of opportunities they have had. The other students from the woodworking class have come by to see the final product in action, since they only saw the raw materials and had not seen the final product in action.

I saw recently where Andrew Stadel had posted about an "experienced" Rubik's cube he held onto and a curious student managed to master quickly. While I have a growing collection of Rubik's cubes and similar puzzles (as seen on the banner for this site, atop the shelves in my classroom with Hoberman spheres and other math-y relics), I hope this Giant Jenga game at least makes an impact on these students since they are the ones who came up with the recipe--or the shopping list, I suppose.