PictureThe TI-Nspire CX & CBR 2
Solving linear inequalities in class had easily become a mundane topic to me. Students' prior knowledge was good, although they struggled a bit when the discussion changed to include absolute value inequalities.

"Why is it so important we're able to phrase the distance from an object like that?" I had heard on several occasions. I DID have access to a Texas Instruments TI-Nspire CX as well as their CBR 2 (Calculator-Based Ranger), which had been useful in the past with a ball bounce lab and I fully intend this school year to do the Bungee-Jumping Barbie lesson.  With CBR 2 and TI-Nspire CX in hand, I began to collect data in the hallway outside my classroom during a passing period with students walking by at a variety of distances away from the sensor. This was insightful, but did little to give relevance to the idea of absolute value inequalities as I had set out to do.

So after thoroughly banging my head against the drawing board, so to speak, I decided to relate the back-up sensor (and camera) on an automobile to the compound inequalities and absolute value equations we had been investigating earlier in the week.

PictureShe seems to have a lot of faith in my backing skills, huh?
After searching far and wide to find a teacher's vehicle new enough to include a back-up sensor system, I then found two "willing" students to help me stage and record an example of the alarm system at work, with visible warning lights above the rear window in the photo shown).

We -- er, actually, -- backed the vehicle towards our volunteer (pictured here, seen through the vehicle's back window, exercising every ounce of trust she can muster up) and measured her distance to the sensor when it changed intensity and/or volume. The video was simple enough to capture, so I uploaded it to YouTube HERE, if you feel the need to use it in 3-Act-Math format (and because I always enjoy working a Dan Meyer shout-out into a blog post).  My intent was to use the video clip as an Act 1, then the photo below for Act 2 to include some accompanying details and measurements. I have not managed, at this time, to accumulate the necessary camera shots for an Act 3, but hope I can do so in the not-too-distant future.

PictureImage reflects data collected from observed alarm-distance relationship.
I found a screenshot of the new backup sensor/camera system available for the new Chevy Silverado and labeled its colored guidance lines to correspond with the observed distances when the audible and visible alarm system was set off during our trials.

The prompt I gave students was that they needed to be able to provide enough information about the inequalities associated with each tone, so that we could construct a "poor-man's vehicle backing system." To be fair, earlier in the week, we had created several of our own programs to use with the TI-84 Plus (now available in the TI-84 Plus C with color screen and rechargeable battery, which I've grown to appreciate quickly) calculators, including Quadratic Formula, Distance Formula, Slope Formula, and Midpoint Formula programs. I aim to include links to these files as soon as my school webpage editor is properly troubleshot. For now, you may browse my classroom webpage using THIS link.

While this particular prompt may not fit the ideals for the Common Core State Standards for Mathematics, I feel it does give some worthwhile insight into how computer programming logic has a place in a vital component of vehicle safety systems. Sometimes, this sort of insight is enough to squelch the "But when am I ever going to USE THIS STUFF in real life?" questioning that can occur frequently with math topics.

PictureIf using the 3-Act Math format, this could serve as Act 3 for the time being. Sorry.
The relationship I had hoped students would construct, or at least mimic, in their analysis of the video clip I created and the accompanying photo shown above.

Most importantly, I wanted to see that students had the wherewithal (yes, I used "wherewithal" in context on a math blog entry) to only include, for example, "6 feet" as part of a single interval instead of assigning it to more than one interval. My hopes in this was to have some sort of reinforcing example to refer to when we begin our look at functions and how each input value can only have one output value.

With this sort of application to the simple, compound, and absolute value linear inequalities we have used this unit, I hope that my students have some sort of foundation or respect for how this topic is relevant and useful in real life. 

The female student in the video seems pretty bold to stand firm, knowing that I'm backing up straight towards her and, truth be told, there was a Counselor's meeting going on while we were filming in the conference room directly behind her. So, I had some explaining to do when I ran across a couple of those counselors towards the end of the school day, but did get some head nodding and encouragement that the lesson potential for what I was doing sounded pretty interesting to them.

In conclusion, this lesson has been the most fun I've had ALMOST intentionally backing over a student in my teaching career. Just wanted to put that out there. Please let me know your thoughts on this lesson and check my Twitter feed to see when I've been able to post those TI-84 programs, if you are interested in when I have those posted. Thanks!

An EHS student and KDOT official interact demonstrate the Airspace Awareness Tool.
Here's a conundrum being faced in the state of Kansas: harnessing the state's great potential for wind energy while also safely advocating for aircraft traffic servicing the state's 138 airports and abiding by the siting guidelines for industrial wind turbines.

The state of Kansas approved a landmark piece of legislation in 2009, the Kansas Net-Metering and Easy Connection Act, which not only allowed for citizens to connect their energy generators to the grid but also standardized a 1-to-1 buyback policy (they would earn credits from their utility provider at the same rate that they would purchase energy from the same provider).

Here's where the math comes into play: industrial wind turbine installations are subjected to a battery of governances before the FAA (Federal Aviation Administration) gets their chance to have a say. Wind farm developers would then submit their site proposal through the FAA's Obstruction Evaluation process. Affected projects would include:
  • Any construction or alteration exceeding 200 feet above ground level (which requires a lighted beacon to make it visible to aircraft, common on water towers in my area)
  • When requested by the FAA (and sometimes as an anticipated last step in the developer's proposal process, which can negate a substantial amount of work, just to have to start from square one because the project might interfere with safe air travel)
  • Any construction or alteration located on a public use airport or heliport regardless of height or location
  • those located within 20,000; 10,000; or 5,000 feet or public use or military airports at varying height levels depending upon proximity to those airports (THIS is where the lesson focuses, on how to visualize these criteria before the FAA comes a-callin')
These criteria hardly lay out like a list of constraints for linear programming or a system of inequalities, but the logic involved definitely could be used in the same breath as one another. 

Try some of the requirements in the applicable city ordinance in trying to obtain a wind turbine on our high school campus, as part of the Kansas Wind for Schools program:
  • Blade-tip clearance, at its lowest point, shall have a ground clearance of not less than 25 feet tall
  • Overall height of 60 feet or less to fit Micro-WECS (Wind Energy Conversion Systems) class of devices
  • Minimum setback (measured from the closest adjacent lot line or parcel line or above ground public utility) of 110% of the device's overall height [Example: a 50-foot tall turbine would need to have a 55-foot radius clear of it to the nearest property line or easement]
  • No buildings within 110% of the device's height where the turbine's collapse could potentially damage another building of the property owner's
  • Ensuring that the noise emitted from the wind turbine shall not exceed 50 dbA within 100 feet of the nearest property line, except during short-term events such as utility outages and severe windstorms. [NOTE: The math involved with decibels is a great application to logarithms and how exponential behavior applies to the height-noise relationship in this case]

Do the phrases "not less than", "within", "n feet or less" start looking like the real-world examples we would hope to incorporate when teaching systems of inequalities? This gives students not simply an opportunity to see that our turbine site is a solution to the system of inequalities, but can drive some curiosity as to where else a turbine might be positioned and still satisfy all these criteria.

Check the brief video tutorial below on how to use the Airspace Analysis Tool (link to actual site included HERE), a product of the Kansas Department of Transportation Division of Aviation in conjunction with Burns & McDonnell Engineering (Please note links on the User Agreement page that help out with loading the Google Earth Plug-In necessary to use the site. From past experience, this has not been as successful using Safari as it has on Chrome, Fireforx, and Internet Explorer).

A screenshot from the Airspace Analysis Tool, near Lawrence, Kansas.
Now, the fun, behind-the-scenes news about all this: KDOT actually unveiled their Airspace Analysis Tool IN MY CLASSROOM at a press conference hosted on a teacher inservice day (i.e. supposedly no classes in session). Here's the news story from the Lawrence-Journal-World.

Their personnel said it would be nice if we could have some students available to play around with the project and give their candid feedback. To their chagrin, I almost needed to bring in more chairs! Thank goodness some of the media on hand chose to stand and work behind their tripods and cameras, to catch the multiple perspectives of KDOT staff and students interacting. 

Students were anxious to see their project because it touched on a topic they had some curiosity about, and liked the potential to see some engineers in action. To show up at 8 a.m. ON THEIR DAY OFF was a remarkable thing for them to do and made another tally mark in the column for times I've loved my job.

Situation hopefully made safer by the KAAT. Click image for image source link.
While students had an opportunity to try out and manipulate the tool during our press conference, they investigated the maps with relative ease because of their awareness of the familiar interface, Google Earth (even mentioned on the Google Earth blog after its release for an innovative use of their software). When the folks with Burns & McDonnell and KDOT asked for feedback on the beta version they released, our students swung for the fences and suggested another visual aspect: avatars. So now, the KAAT includes superimposed images of the device proposed at a location, as well as for existing structures on file. For those in the aviation industry, as well as in the land development and construction industries, this has been a welcome addition. As mobile devices become ever-present around us, this interface very well could find its way being incorporated into the cockpits of aircraft to help with flight instruction feedback in real time.

Since its release, the Kansas Airspace Analysis Tool (KAAT) has garnered national attention for its innovative ideas and interface, not simply for helping wind energy developers but also companies and communities who are looking to install water towers or cell/communications towers.

Here is part of one announcement I was able to retrieve (and HERE is a copy of their online newsletter announcing the recognitions), touting the acclaim that the KAAT has received:

Some of the awards received presented to the KAAT project.
  • For the first time in the history of the National Association of State Aviation Officials’ Awards, a single state received both the State Most Innovative Program Award and the NASAO Center Outreach and Education Award for two separate programs.  The State Most Innovative Program Award was presented to the State of Kansas for the Kansas Airspace Awareness Program - Aeris Vigilis (Airspace Guardian).  The NASAO Center Outreach and Education Award was presented to the State of Kansas for its “Ops for Cops” program.  The program partnered KDOT, the Transportation Security Administration, Drug Enforcement Administration and airport managers to provide tactical and legal information for law enforcement officials called to airports.  Kansas also received a national best practice recognition for its continuing legal education program in partnership with the Kansas Commission on Aerospace Education.
  • The KDOT Aviation team received word that the Kansas Airspace Awareness Tool, a major component of the Airspace Awareness Program, will receive the American Association of State Highway and Transportation Officials (AASHTO) President’s Award for Aviation in November.  The Tool was recognized in April by the American Council of Engineering Companies with an Engineering Honors Award.

Since the unveiling of the Kansas Airspace Analysis Tool at our school, student have made contact with some of the personnel on hand for this event and gained insight into numerous STEM-related paths that could potentially open other doors for them later. 

My favorite experience from all of this comes from a comment a student shared with me a few weeks later:
   "Mr. Keltner, you know those algebra problems with the 'a plane flies with the wind between towns in 2 hours but the return flight takes 3 hours because it's going into the wind?' Well, Mr. Young from KDOT took me up in a plane and let me try the controls for a bit and I looked at my clock and could SEE that problem playing out along our trip as it was happening!"

So, I hope you enjoy this potential class project you can utilize. Granted, it is for the state of Kansas, but I think the technology and the ease of integration into class speaks well for student engagement and enjoyment. I know my students have enjoyed it and have bragged to me about being able to put a 2,000-foot tower in their back yard. I guess that goes to show you, the sky is the limit. Only in this case, they can actually SEE it.


One last shout out to the folks at Burns & McDonnell and the Kansas Department of Transportation Division of Aviation, namely Director of Aviation Edward Young who gave our students a wonderful opportunity when he selected our school for the unveiling of his project. Thank you for allowing us to be a part of this project.