Lesson 6-6: Isometries (Day 66)

Today is Cyber Monday. So of course, I had to order something on Amazon today -- and that something is the Pappas Mathematical Calendar for 2019.

Meanwhile, it's come to my attention that November 23rd wasn't just Black Friday -- it was also Fibonacci Day, because the digits of the date were 1, 1, 2, 3. Unfortunately, Fibonacci Day fell on a day when I didn't post.

Here's a link to a Fibonacci Day post:

http://bedtimemath.org/fun-math-fibonacci-day/

The fact that Fibonacci Day is always close to Thanksgiving causes a problem if we wish to celebrate the day in schools. Indeed, in districts that take the entire week off for Turkey Day (which includes both of my districts), schools are closed on November 23rd 100% of the time. It's one of the four days in November when schools are never open -- the others are the 24th, 25th, and 11th (Veteran's Day).

In districts that are open Monday, Tuesday, or Wednesday before Thanksgiving, then Fibonacci Day is a school day provided it falls on one of those days. This will next occur in 2020-2022 -- and a Fibonacci Day activity might be something nice to do in the week leading up to Thanksgiving.

If we really want to observe a Fibonacci Day at school, then there are other ways to use Fibonacci digits to form a date. Indeed, Laura Overdeck, at the above link, mentions the following dates:

The sky’s the limit: 10 dates: 1/1, 1/2, 1/12, 1/23, 2/3, 3/5, 5/8, 11/2, 11/23, and 12/3.

(She could have also listed 8/13, since 13 is the next Fibonacci after 8.) Of these, January 23rd, February 3rd, March 5th, May 8th, November 2nd, and December 3rd are far enough anyway from long school vacation periods to be suitable Fibonacci Days.

According to Overdeck, we're 40 years away from Fibonacci Day of the Century, 11/23/58. It's also possible to make some of her other ten dates into possible Fibonacci Days of the Century -- for example, 5/8/13, since 5 + 8 = 13. This is acknowledged at the following link:

https://www.treehugger.com/natural-sciences/its-5813-or-fibonacci-day-america.html

Lloyd Alter, the author of this article, also looks forward to 8/13/21, the next such possible Fibonacci Day of the Century. This is less than three years away (though it's likely to be slightly before the first day of school in most districts).

Finally, here's a link to Anna Pacura, a New York middle school teacher. In 2015, Fibonacci Day fell on a Monday and her school was open, so she had a Fibonacci lesson:

http://iamamathteacher.blogspot.com/2015/11/fibonacci-day-zero-socratic-seminar.html

She alludes to her 2016 Fibonacci Day lesson in the following post. This is the same year that I worked at the old charter school and just like me, Pacura taught all three middle school grades. But again, her school was open the day before Thanksgiving, while I took that whole week off:

http://iamamathteacher.blogspot.com/2016/11/almost-thanksgiving-break.html

(That's right -- during that year I linked to Pacura as an example of a middle school math blog.)

Lesson 6-6 of the U of Chicago text is called "Isometries." In the modern Third Edition, we must backtrack to Lesson 4-7 to learn about isometries.

This is what I wrote last year about today's lesson:

What, exactly, is a glide reflection? Well, here's how the U of Chicago defines it:

Let r be the reflection in line m and T be any translation with nonzero magnitude and direction parallel to m. Then G, the composite of T and r, is a glide reflection.

Just as reflections, rotations, and translations have nicknames -- "flips," "slides," and "turns," respectively -- glide reflections have the nickname "walks." The U of Chicago gives the example of the isometry mapping the right footprint to the left footprint while walking as a glide reflection. Another name for glide reflection is "transflection," since it is the composite of a reflection and a translation.

I once tutored a geometry student who had a worksheet on glide reflections. The student had to use a coordinate plane to perform the glide reflections, which were given as the composite of a reflection and a translation. But the problem was that on the worksheet, the direction of the translation wasn't always parallel to the reflecting line! In fact, in one of the problems the translation was perpendicular to the reflecting line. That would mean that the resulting composite wasn't truly a glide reflection at all, but just a mere reflection!