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!

I feel obligated to include some sort of reference to "The Wizard of Oz," being that I have lived in Kansas my entire life. I try to take advantage of my students' familiarity with the film in several instances though. 

One of my favorite such examples is when the Scarecrow inaccurately recites the Pythagorean Theorem once a brain is decreed upon him (note also how Scarecrow magically moves from the middle of the characters to the far right of the frame after confronting the Wizard). His mishap is recreated by a glasses-wearing Homer Simpson who makes the same mistake, but is corrected by a gentleman in the bathroom stall behind him.
Another mention of "The Wizard of Oz" in class comes up when I emphasize to my students how absolute value measures the distance a number is from zero on a number line. We cannot have a negative distance, so an absolute value will always be non-negative (when a student usually tries to correct me and say "You mean greater than or equal to zero" and suddenly finds that the statements are equivalent).

I paraphrase what the Scarecrow said when first meeting Dorothy (pictured at left): "We can go both ways." The same can be said of absolute values, in that they measure the same distance from a central reference point; in the general case, a number's distance from zero.

In introducing the concept of absolute value this year in Algebra 2, I emphasized greatly the importance of that feature, that absolute value equations would measure the same distance from a central reference point. Refer to the example here:

Solve the equation | x - 2 | = 3. Check your solutions in the original equation.

Diagramming "a number whose distance from 2 is 3 units."
I interpreted the equation, telling them that it is asking, in Algebra terms, "What number's distance from 2 is 3 units?"

They started talking, but only focusing on using that distance in the positive direction, citing 5 as the only solution. This diagram helped them clarify that -1 is also 3 units away from 2. We substituted each value back into the original equation to check our work and conclude that each apparent solution was indeed a valid solution (a great time to introduce the idea of extraneous solutions and emphasize how our conclusion would have differed if we were discussing money, volume, unit sales, and other applications.

I then decided to give an open-ended task to end class that day. I asked students to find something in real life around them that involved the same distance away from a central point. The example I gave was that I parked my car that morning so it was the same distance away from the striped lines on each side of my parking spot, making sure to point out how disappointed I would be if all students returned the next day having simply taken a photo of their car in the parking lot. They came through with some great examples, as shown in the slide show below.

Students brought their cell phones to class and were able to display a screen shot using the IPEVO Point2View document camera I have on my desk (they are excellent for what I use it for and allow quick, easy interface via USB interface including quick screen shots when desired). They were able to sketch on top of their photos using the SMART Board in my room as well, even being able to accept suggestions from classmates on examples of absolute value within their own photo that they had not acknowledge.

I aim to do more of these sort of open-ended, scavenger hunt-like examples where students find examples of math around them. I'll call it "Have My Stu's Do A 101Q" to give proper credit to Dan Meyer's 101qs.com website for posting examples of photos meant to drive student curiosity. I might even be able to coerce some of them into posting to Twitter with a class-named hashtag.  Only time will tell. Like the Scarecrow said, it "could go both ways."