Similar Triangles: 'Tis the Season!

11/5/2012

I am not of Griswold heritage, thank you.
Yup, I'm "that guy" in the neighborhood who puts up Christmas lights on the house the weekend after Halloween.

Contrary to popular belief, though, I do it out of planning for inclement weather conditions that seem to come soon after November begins.

While my daughters appreciate seeing our lights on, they also enjoy helping out with putting them on the house. Hence, the picture at left of my second daughter and I, properly footing a ladder. That's a term I learned through my firefighter classes, and also learned the proper angle for ground ladders of this type.

The rule of thumb for proper ladder angle says:
• If you can stand at the base of the ladder and reach out to the rung directly in front of your shoulder, the ladder is at a proper angle if you touch somewhere on the palm of your hand (to allow a little bit of room for error).
• If the ladder is positioned somewhere on your forearm, it is too steep and poses a safety hazard.
• If the ladder is so shallow that your fingers cannot reach it, the ladder will flex under the weight of the user and increase the potential for collapse of gutters or other objects at the other end of the ladder.

So I made sure to superimpose figures on the photo of me and wee-me to display the similar triangles that occur, but also to show the relationship of slope and how it relates to three collinear points along the ladder.  [NOTE: The fact that my arm and duaghter's arm are not fully extended is due to the fact we stood PRECISELY at the base of the ladder instead of lining up our toes with the base. This was because I wanted to illustrate the slope more directly and not have to account for shoe size, although we both wear size 11 right now, just different categories.]

Proper Ladder Angle should be at a 4-to-1 ratio (slope).
The diagram at left also illustrates proper ladder angle, but does not include the ladder extending above the surface it is leaning on. The other feature I must point out is the user's feet are AT the ladder's feet, not lined up at the ankle's as my daughter's and my ankles were in the photo above.

While I'm not dwelling on a whole lot of math within this post, there are a lot of different directions a teacher can go with this: similar triangles and slope are just a sample.

Moreover, I would hope you get a little grin on your face as the holiday season arrives in the coming days, weeks, and month(s).

We definitely have a hefty glow at our house now. And yes, our lights did work the first time I plugged them in.

--Keltner--

Red Bull Stratos: A Great Example to Dive Into (yes, pun intended)

10/25/2012

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).

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.

--Keltner--

Slope-R Mario! To teach-a the students about-a the slope-a!

9/21/2012

Slope-R Mario on the scene!
When I tried last year to demonstrate slope using GPS receivers, I ran into issues: poor signal, dead batteries, not enough receivers for a BIG class of students, among others.

This year, I tried to do something over the top. The formula that students are familiar with for slope uses m to represent the value of the slope, or rate of change, for the function or relation.

When I looked at the four types of slopes I would be teaching (positive, negative, zero, and undefined slopes), I mashed together examples of their graphs to create a peculiar image, which I later twisted into a logo for a character who visited class.

I managed to make a logo VERY similar to the one on Super Mario's hat in the Nintendo game series!

Slope-R Mario's logo
AND it demonstrated the types of slope I was going to demonstrate in class (with labels placed adjacent to the appropriate line) to help illustrate it to students!

I'm not able to hold a candle to Matt Vaudrey's Mullet Lesson, but hoping to have fun trying. He made a much more compelling case for ratios than I think I was able to do.

Now, all I needed was a costume to seal the deal. Being so near to Halloween time, I was able to score a Super Mario hat from a neighboring teacher's son, some overalls from the husband of our daycare provider (also a tribute to my high school math teacher, Vern, the focus of this earlier blog post), a mean mustache from the video-editing teacher in my building, and print off a couple of logo medallions to complete my transformation into Slope-R Mario status!

The evolution of my Slope-R Mario idea, coming to life!
I'm attaching screenshots of the notes I used with this lesson. My trademark stick-figure diagrams for types of slope are a favorite from year to year among my students. Simple but effective.

I was also able to insert a collection of images with Super Mario that showed him travelling along paths that incorporated the different examples of slope we were discussing.

Check them out below.
So, a good time was had by all. Students got to make fun of me for dressing up so goofily (if that's not a word, it should be and I should trademark it like Anthony Davis trademarked "Fear the Brow" prior to being drafted in the NBA).

One side-effect of this lesson I didn't anticipate was seeing students correct their peers when someone interjected a "Mr. Keltner, you're tall and skinny, so you should have dressed up like Luigi!"

A bystander was quick to point out "He dressed like Mario because of the 'M' on the logo, since we're doing slope today. It was cool how he made the logo to show the different slopes, right?"

The first student agreed, but then quickly retorted "So does this mean when we do parallel and perpendicular lines, THEN you'll dress like Luigi?"

Stay tuned and find out. Slope-R Mario, course clear (that's a Nintendo reference).

--Keltner--

Scott Keltner

Math teacher and loving it. Tech-savvy because it's necessary. Husband and dad (only male in the house besides the dog).
Kansas, USA