Pythagorean Triple Answers

As a follow-up to yesterday’s post, here are some answers and steps toward figuring out the scarcity (or frequency) of Pythagorean triple days (like yesterday 8/15/17) in a century:

I will mention I did not get the answer right away but needed some prompting (from a math-teacher colleague on Twitter), so I’ll take you through my thought processes.

  1. I have certain primitive Pythagorean triples memorized (like 9/40/41) — perhaps from several years of teaching geometry, or from studying number theory in college where integer (or rational) values for algebraic relations are important (cf. Euclid’s formula for Pythagorean triples, Diophantine Equations, & Fermat’s Last Theorem), or perhaps even a few recalled from taking high school geometry. In my head I tried these, saw several had occurred already in the past (5/12/13), while others were not interpretable as months days and years, like 9/40/41 (since there are max 31 days in a month). To go beyond the primitive triples, I tried multiplying each times 2, times 3, etc. In this way I discovered yet to come this century: 10/24/26 (twice 5/12/13) and 12/16/20 (quadruple 3/4/5).
  2. I thought I had them all, but had forgotten some primitive triples, like 7/24/25! At this point, referring to this list of primitive Pythagorean triples, I tried to systematically enumerate all with a≤12, then go through multiples of those until all values were larger than 12 (and therefore none could fill the month slot). I did this in Excel, color coding as blue dates already past, yellow yet to come, and red invalid date formats:excel1
  3. In creating that chart I did mentally transpose months and dates to see for example, that 12/5/13 could also work as a Pythagorean triple date in addition to 5/12/13. But this only yielded more in the past so I didn’t bother to write them out. My fatal flaw, though, was not permutating with the years also, so I neglected to find two more future dates: 7/25/24 and 10/26/24. After realizing this, I did add more to my Excel chart, going through each permutation that yielded a valid date in rows below the main ones:excel2
  4. I do believe this is now a complete list of past, present, and future Pythagorean triple days, written month/day/year. By my count there will be a total of 28 this century, 23 now past and 5 still in the future. Let me know if I missed any!

 

By the way, here’s an awesome interactive generator of Pythagorean triples by Vincent Pantaloni (shared on twitter).

 

Enjoy!

 

 

Leave a comment

Filed under math, teaching

Happy Pythagorean Triple Day!

Today is 8/15/17 (abbreviating the year to two digits), which are three natural numbers that make up a Pythagorean triple! That is, three numbers that satisfy the equation from the Pythagorean Theorem a2 + b2 = c2 , and therefore make up valid integer side lengths on a right triangle.

Try it! 82 is 64, plus 152 (225), makes 289, which is 172 .

In fact, 8/15/17 is even more special than other Pythagorean triple days like 6/8/10, or 12/9/15, since it is a primitive Pythagorean triple: that is, its numbers share no common factor (e.g. 6/8/10 all share a 2, which if factored out yields the more familiar 3/4/5).

 

A great question to ponder: how many more Pythagorean triple days will we have this century? (question and image from Chris Smith, on Twitter)

81517.jpg

 

Also, how many already occurred (and perhaps we didn’t even realize it)?

 

Will post answers to these questions tomorrow…

1 Comment

Filed under math, teaching

Pi Day Snow Day!

After a winter with no snow (no snow that’s stuck, anyway, there’ve been flakes in the air a few times), it’s a pi day miracle: a snow day!

Last night, when the call was made, we were predicted to get 6-12″, but with more sleet than snow, we’re on the low end of that range or a bit less. Here’s a joke I’m stealing from my friend Matt:

In honor of pi day, and with the extra free time of a snow day, I baked a quiche pi!

pi_day_quiche.jpg

I tried to arrange the broccoli in the shape of the symbol π (pi), which you can see if you look closely 🙂 I’ve outlined it below.

pi_day_quiche314

And, of course, I had to break out my pi cutter to slice it with!

pi_day_quiche_cutter.jpg

If you’re more ambitious than I am with your pi day baking needs, you could  try a “celebrity chef” pie, or Vi Hart’s Venn Pi-agram (she has both a dessert pie and a pizza pie version):

In Pi Day news, there have now been over 22 trillion digits of pi calculated! Hooray!

Whether or not you, too, are snowed in, have a wonderful pi day, and check out some of the following links!

Even though there are skeptics out there,

I agree with Cookie Monster:

Me love pi (and pie)!

Leave a comment

Filed under math

#teach180, part 3

Weeks 7-11:

  • Day 31: robot brought in for demo for our engineering advisory committee
  • Day 32: Students staying after school for an intense CyberPatriot practice round!
  • Day 34: Finding errors in G&M code for computer numerically controlled (CNC) milling
  • Day 35: Driving home via the original Washington Monument
  • Day 36: Students presented research on different types of manufacturing processes in class
  • Day 37: teaching G&M codes for students to carve out their initials and a nice sunset
  • Day 39: Two teams securing virtual computers at yesterday’s CyberPatriot competition practice round
  • Day 40: In computer science class, students making their first app: the Magic Trick
  • Day 41: Students working on a bandsaw and on CNC milling simulation software during their lunch period today
  • Day 42: First blocks machined on the CNC mill today!
  • Day 43: A closeup of a block in the CNC mill after machining
  • Day 44: More initials plotted, programmed, and milled out today, then a nice view of the Washington Monument at night
  • Day 45: More blocks designed, programmed, and machined on the computer numerically controlled (CNC) mill
  • Day 46: Beautiful sunrise on drive to work this morning, then some interesting cityscapes w/ 2-point-perspective posted in hall
  • Day 48: Getting ready for the STEM Competition
  • Day 49: 8th Annual STEM Competition!

Leave a comment

Filed under teaching

Election

I don’t know if I have the words in me right now to write about this thoughtfully and coherently, but I’ll try.

Staying up and watching the election returns Tuesday night into Wednesday morning was difficult. It was difficult for me, as a political leftist and radical, as a believer in social justice movements, as someone who cares deeply about issues of poverty and climate change (two topics barely touched upon in the media coverage of this campaign). In some ways, it hearkened back to the 2000 election, and feelings of outrage and disappointment I experienced then and thereafter during the Bush presidency.

But, although there are some comparisons to be made, in many ways this election result was much worse, at a whole new level. What has made it more difficult for me is thinking of my students, the hatred they see directed at them, and the fear they must feel. From words and actions of Trump himself, and also from the words and actions of his supporters at rallies. My black students, faced with a president-elect who courted and did not reject the support of the KKK and other white supremacist groups. My Latino students, called rapists and murderers by our president-elect. My Muslim students, who are told by Mr. Trump that their entire religion should be banned from entering the United States. My immigrant and refugee students, who feel unwelcome and fear deportation, due to many things said over the course of the campaign. My girls, who now face a president-elect who has evinced zero respect for women throughout the campaign and his career, and is famous for talking on tape about how he would/could sexually assault women.

Although I can imagine how it might feel for them, I do not actually know what it is like living in fear for one’s life or liberty or home that many of my students are probably feeling. Nor do I know the day-by-day degradation of being told your life doesn’t matter, that your rights don’t matter, that your body doesn’t have any rights that others are bound to respect. The fact that Trump has not only said all the horrible things he has said, but that he has encouraged violence from his supporters directed at non-supporters, and that he has normalized this kind of language, hatred, disrespect, and violent actions: this is what scares me most.

The Baltimore Sun interviewed students at my school on Wednesday, including one I teach, for an article on the election: “Immigrant communities fear deportation after election of Trump”

 

Leave a comment

Filed under teaching

#teach180, part 2

Weeks 3-6:

  • Day 11, software installs
  • Day 12, art everywhere
  • Day 13, AP CSP daily agenda
  • Day 14, Sonic Pi (video)
  • Day 16, using Computer Vision to find lines and circles
  • Day 17, trash on the floor
  • Day 18, 6th Annual Back To School Night & Community Fair
  • Day 19, OneNote for class collaboration & a calendar look ahead
  • Day 20, cake from celebration honoring my colleague
  • Day 21, Comp Hydro training (video)
  • Day 22, design flaws
  • Day 24, my colleague on ABC2 news (video link)
  • Day 25, my students on the news (video link)
  • Day 26, sunrise in the city
  • Day 28, nighttime with moon and lights from the school parking lot

Part 1 (weeks 1-2) can be found here.

Leave a comment

Filed under teaching

#teach180, part 1

As I mentioned in my New Year 2016 post, I am posting a photo most school days.

This year I am thinking I shall try to keep up with #teach180 on Twitter, posting a photo tweet every school day (more explanation here).

Some are photos of my students, others are photos of projects they are working on, others are screen captures of lessons or tools that I use, others are more random (a new art piece on the wall at my school that I really liked).

My goal is one photo (and caption) every day on twitter. I have already missed a few days, but here are my first two weeks:

  • Day 1, digital syllabus
  • Day 2, my classroom
  • Day 3, origami balloons in CIM
  • Day 5, map mosaic
  • Day 6, course homepage on LMS
  • Day 8, helping teachers build web pages
  • Day 9, students making music on Raspberry Pi

Happy teaching!

1 Comment

Filed under teaching