‘Shruti” will explain how to do it, but she doesn’t explain why it works:
Personally, I had never come across this one in my over 40 years of teaching math. She is correct, it does work, but WHY?
I’ll try to explain.
994 is 6 away from 1000, or 10^3 minus 6.
And 997 is 3 away from 1000, or 10^3 minus 3.
If we multiply (10^3 – 6) by (10^3 – 3), we get 10^6 – (6+3)*10^3 + 6*3
So as she explains, we count down from a million by nine thousand, and then we add on 18.
or, if we take away any small number from a thousand, say, a, and subtract any other small number b from a thousand, you have a situation like this if you use the area model for multiplication:
Dear Flat Earthers, Many people have been derogatory of your belief that the Earth is flat. Please note that they are belittling your belief, not you per se. You, personally, are an idiot, but that is probably not your fault.
Here are any number of accessible approaches for discovering the shape of our beloved planet. Enjoy!
* * *
Use Your Phone! On Christmas Day, here in Chicago, I expect there to be snow on the ground because, well, it is winter. On Christmas Day I can pick up my phone and dial up anyone in Australia and ask them “What season is it?” They will tell you that it is summer in Australia. You might want to ask your flat Earth mentors how it could be winter and summer simultaneously on a flat Earth.
Use Your Phone! Go to a globe and pick a spot half way around the Earth (I know it is a false representation in your belief, but humor me.) In the middle of the day, phone somebody at or near that spot. Call a hotel, they are always open. Ask whoever responds “Is it light or dark outside?” They will tell you that it is dark where they are. You might want to ask your flat Earth mentors how it could be light and dark simultaneously on a flat Earth.
Look Up What Local Time Was In the US there was this concept of “local time” which was that “noon” was when the sun was at its highest point in its arc. You could call up people on the telephone who were not that far away and ask them what time it was and they would tell you something different from what your clock was telling you. The farther away they were, the greater the difference would be. On a flat Earth the time would be the same everywhere.
Look Up What Time Zones Are I am writing this in the central time zone in the U.S. These zones were created at the behest of the railroad industry whose dispatchers were going crazy making up schedules for trains when every place had their own times. By creating these “zones” everything would be exactly one hour off from those in neighboring zones, two hours off for the next over zones, and so on. If you don’t believe me . . pick up your phone and dial up a friend who lives a considerable distance (east-west) away from you and ask them what time it is. The time they state will be a whole number of hours away from your time. Heck, even the NFL knows this. When I lived on the left coast, the games started at 10 AM and 1 PM. Now that I live in the central time zone, the games start at 12 Noon and 3 PM. Over New York way the games start at 1PM and 4 PM. Do you think those games are replayed in one hour increments? Nope, time zones!. You might want to ask your flat earth mentors how it could be that simultaneous games start at different times on a flat Earth.
Watch the Video Astronauts in the International Space Station (ISS) have made continuous videos of an entire orbit of the Earth. It takes only about an hour and a half about the length of a typical Hollywood movie. During the whole movie the earth appears round, and yet it is clear that different continents are passing in our view.
Now you may argue that NASA made this movie as propaganda for the Round Earth Conspiracy. It is certainly within our CGI abilities at this point, but you may want to ask why NASA would want to do such a thing? Plus, many astronauts have taken their own cameras aboard and taken pictures for themselves and they show the same thing. How could the Round Earth Conspiracy have allowed that to happen? It must be incompetence! Conspiracies aren’t what they used to be!
Da Balloon, Boss, Da Balloon Many amateurs, unaffiliated with the government, have launched rockets and balloons high up into the atmosphere to take pictures. Every damned one of those pictures shows that the Earth is round. How come all of those cameras ended up pointed at the curved edge of your round and flat disk Earth? Such a coincidence!
An Oldie But Goodie #1 Occasionally, during a lunar eclipse, you can see the shadow of the earth falling upon the Moon. The shadow is always circular. This would be true if the flat earth were always dead on to the Moon, but the Moon orbits the Earth and wouldn’t a flat Earth be edgewise, often as not, and wouldn’t that create a non-round shadow on the Moon? Inquiring minds want to know.
An Oldie But Goodie #2 It was claimed that one of the first demonstrations of the earth being round was the observation of ships sailing west from Europe/England could be observed for a while but the ship itself was lost to sight while the mast was still visible. This would not happen on a flat Earth. The whole ship would just get smaller and smaller as it sailed west.
For pity’s sake, I live 22 stories up and the shores of Lake Michigan and I cannot see anything directly opposite me in Michigan. All I can see is water, with any kind of magnification I can muster. And I am not looking across the widest part of this lake! If the earth were flat, the lake would be flat and I could see the Michigan shore.
And Finally . . .
All of the fricking satellites! Do the math. What kind of orbit is stable around a flat disk earth? Answer none! And there are hundreds of the danged things in orbit.
Also, just for giggles. Look up what a Foucault pendulum is, And explain its behavior based upon a flat Earth.
PS You may be getting good vibes in your special knowledge that you know something other people do not. However, would not that special feeling be more worthwhile were you to volunteer at a food bank or a day care center or senior center? Wouldn’t doing something worthwhile be more rewarding that making a statement about how those pointy-headed intellectuals aren’t so smart?
PPS I have seen the cute models with the Sun and Moon on sticks rotating around (see photo above). If that were the case, everyone could see the Sun and Moon all day, every day. (There is straight line access to both objects in that model from everywhere on the flat disk.) Do you see the Sun and Moon all day, every day? No? Maybe someone who had more creativity than knowledge came up with those models. They do sell well, I must admit, so maybe their interest is commercial.
PPPS Regarding the 200 foot wall of ice that supposedly exists at the “edge of the disk,” supposedly so all the water doesn’t flow off and be lost into space. By now don’t you think someone would have sailed next to that wall all of the way? That distance would be somewhere in the neighborhood of a 28,000 mile trip. Has anyone ever report such a thing? Hmm, I wonder why not
Hopewell Observatory has three WW2 or Cold-War aerial spy camera optical tube assemblies, including a relatively famous Fairchild K-38. No film holders, though. And no spy planes. The lenses are in good condition, and the shutters seem to work fine.
We would like to give them away to someone who wants and appreciates them, and can put them to good use. Does anybody know someone who would be interested?
They’ve been sitting unused in our clubhouse for over 20 years. Take one, take two, take all of them, we want them gone.
We are located in the DC / Northern Virginia area. Nearby pickup is best. Anybody who wants them shipped elsewhere would obviously need to pay for packaging and shipping.
Here are some photos.
This one is labeled K-38, has a special, delicate, fluorite lens in front, and is stamped with the label 10-10-57 – perhaps a date. The shoe is for scale.
The next two have tape measures and shoes for scale.
Let me know (a comment will work) if you are interested.
I just did the math in two ways: if each person infects 5 people who have never been infected, it only takes a bit more than 14 cycles from “patient zero” (whoever that was) to infect the entire living human population.
Obviously the real progress of an epidemic isn’t that simple.
Being a retired math teacher I figured this was a perfect case for using logarithms, so I did. (For me, that’s fun!) I went like this:
I’m trying to find n such that five to the nth power equals 7.5 billion, or in math-lingo,
5^n = 7.5*10^9
One takes the logarithms of both sides, and because of the wonderful properties of logs, I get n*log(5)=9+log(7.5) which we can solve for n by dividing both sides by log(5), obtaining
n = (9+log(7.5))/log(5), after which my calculator said n was about 14.1.
But if you have a cell phone you can confirm my result much more easily by asking it work out 5^14. I think you’ll find it’s about six billion; if you try 5^15 you’ll get an enormous umber, over 30 billion, which is much too high. We have only roughly seven and a half billion humans…
I’m copying part of this from Ali Kayaspour at Medium.com. I’ve read some of these but not all. A while ago I made a list of math-related books for my students to read; maybe I should resurrect it. Here is the link.
Here is some information that teachers at quite a few different levels could use* for a really interesting geometry lesson involving reflections involving two or more mirrors, placed at various angles!
Certain specific angles have very special effects, including 90, 72, 60, 45 degrees … But WHY?
This could be done with actual mirrors and a protractor, or with geometry software like Geometer’s Sketchpad or Desmos. Students could also end up making their own kaleidoscopes – either with little bits of colored plastic at the end or else with some sort of a wide-angle lens. (You can find many easy directions online for doing just that; some kits are a lot more optically perfect than others, but I don’t think I’ve even seen a kaleidoscope that had its mirrors set at any angle other than 60 degrees!)
I am reproducing a couple of the images and text that Angel Gilding provides on their website(which they set up to sell silvering kits (about which I’ve posted before, and which I am going to attempt using pretty soon)).
At 72º you see 4 complete reflections.
When two mirrors are parallel to each other, the number of reflections is infinite. Placing one mirror at a slight angle causes the reflections to curve.
* assuming that the teacher are still allowed to initiate and carry out interesting projects for their students to use, and aren’t forced to follow a scripted curriculum. It would be a lot better use of computers than forcing kids to painfully walk through (and cheat, and goof off a lot) when an entire class is forced to use one of those very expensive but basically worthless highly-centralized, district-purchased computer-managed-instruction apps. God, what a waste of time – from personal experience attempting to be a volunteer community math tutor at such a school, and also from my experience as a paid or volunteer tutor in helping many many students who have had to use such programs as homework. Also when I was required to use them in my own classes, over a decade ago, I and most of my colleagues found them a waste of time. (Not all – I got officially reprimanded for telling my department chair that ‘Renaissance Math’ was either a ‘pile of crap’ or a ‘pile of shit’ to my then-department head, in the hearing of one of the APs, on a teacher-only day.
Keep in mind: I’m no Luddite! I realized early on that in math, science, and art, computers would be very, very useful. I learned how to write programs in BASIC on one of the very first time-share networks, 45 years ago. For the first ten years that my school system there was almost no decent useful software for math teachers to use with their classes unless you had AppleII computers. We had Commodore-64’s which were totally incompatible and there were very few companies (Sunburst was one) putting out any decent software for the latter. So when I saw some great ideas that would be ideal for kids to use on computers to make thinking about numbers, graphs, and equations actually fun and mentally engaging, often I would have to write them my self during whatever free time I could catch, at nights and weekends. Of course, doing this while being a daddy to 2 kids, and still trying to teach JHS math to a full load of students (100 to 150 different kids a day at Francis Junior High School) and running a school math club and later coaching soccer. (I won’t say I was a perfect person or a perfect teacher. I believe I learned to give better math explanations than most, didn’t believe that you either have a ‘m,ath gene’ or you don’t, at times had some interesting projects, and at times was very patient and clear, but had a terrible temper and often not good at defusing things. Ask my kids or my former students!) Later on, I collaborated with some French math teachers and a computer programmer to try to make an app/program called Geometrix for American geometry classes that was supposed to help kids figure out how to make all sorts of geometric constructions and then develop a proof of some property of that situation. It was a failure. I was the one writing the American version, including constructions and tasks from the text I was currently using. There was no way I could anticipate what sorts of obstacles students would find when using this program, until I had actual guinea pig students to use them with. Turns out the final crunch of writing however many hundreds of exercises took place over the summer, and no students to try them on. Figuring out hints and clues would require watching a whole bunch of kids and seeing what they were getting right or wrong. In other words, a lot of people’s full time job for a long time, maybe paying the kids as well to try it out so as to get good feedback, and so on. Maybe it could work, but it would require a lot more investment of resources that the tiny French and American companies involved could afford. We would have really needed a team of people, not just me and a single checker.
I find that none of these computer-dominated online learning programs (much less the one I worked on) can take the place of a good teacher. Being in class, listening to and communicating logically or emotionally with a number of other students and a knowledgeable adult or two, is in itself an extremely important skill to learn. It’s also the best way to absorb new material in a way that will make sense and be added to one’s store of knowledge. That sort of group interaction is simply IMPOSSIBLE in a class where everybody is completely atomized and is on their own electronic device, engaged or not.
Without a human being trying to make sense out of the material, what I found quite consistently, in all the computerized settings, that most students absorbed nothing at all or else the wrong lessons altogether (such as, ‘if you randomly try all the multiple choice answers, you’ll eventually pick the right one and you can move on to some other stupid screen’; it doesn’t matter that all your prior choices were wrong; sometimes you get lucky and pick the right one first or second! Whee! It’s like a slot machine at a casino!).
By contrast, I found that with programs/apps/languages like Logo, Darts, Green Globs, or Geometer’s Sketchpad, with teacher guidance, students actually got engaged in the process, had fun, and learned something.
I find the canned computer “explanations” are almost always ignored by the students, and are sometimes flat-out wrong. Other times, although they may be mathematically correct, they assume either way too much or way too little, or else are just plain confusing. I have yet to detect much of any learning going on because of those programs.
Darwin B recently built, in nearly record time, an interesting, two-truss, tubeless, collapsible, travel-ready Newtonian scope at our DC-area telescope making workshop, using an 8″ parabolic mirror with a short focal length that he bought.
He mounted it on a commercial alt-az tripod, as you see here. It will definitely collapse and fit either in a suitcase or fit in carry-on spaces on an airplane.
Unfortunately, there are some drawbacks, as he is the first to admit:
The lack of any sort of light shielding is a huge problem virtually anywhere within a hundred miles of a city;
The ultra-cantilevered mount makes it very wiggly; images essentially never stabilize. It’s also extremely susceptible to breezes.
To quote from a recent email from him:
==========
So let me first plead mea culpa!
1. Anywhere near DC-MD-VA, open structure telescopes are TRULY instruments of the devil! (How do I come to fully agree with you on this point? WellTuesday morningbefore dawn, in the cold, I set up near work – well that 70mm mirror does an incredible job collecting light from a wide area! That also explains my difficulty at CC the other night.)
2. I am asking for “un-attainium” with my scope: big aperture, fit as 2nd item carry-on, have a good mount, and be useful locally.
3. Perhaps many people would be better off traveling with binoculars- smaller & less hassle than a scope. And would do a great (limited) job anywhere.
So what to do?
1. Fix what I can on this scope and accept limitations- fix spider, swap sides for saddle, and add a shroud. Limit to low & medium power and enjoy.
2. Since I have an 8” f/6 mirror, build a scope for around here & car travel: and not have the limitations of the other scope. I already have many/most of the parts for a design similar to your 6” f/8. Like you say, it’ll be steady, and I can crank up the power a bit for moon&planets. It just has to have the mount break down flat like an IKEA.
So, I’ll be up at the shopnext Tuesday nightto drill holes to flip saddle. I should have other things done or started.
So – now you have a pretty ringing endorsement for your thoughts.
I can compare my current effort to a beach house or a boat; wouldn’t want to live there year-round. BUT they can be fun, within limits.
We have been concerned with the status of some of the columns that are part of the roll-off-roof of the Hopewell Observatory, so we decided to remove a couple of courses of cinderblock to see what was inside. It turned out to be built much more sturdily than they appeared. and removing those two layers of cinderblock ended up being a much harder job than we expected. We had to build a very strong ‘crib’ to hold the upper part of the 9-foot-tall column in place while we removed the lower foot-and-a-third.
In the video, you see me using a small hand-held air-hammer with chisel to clean up the underside of the upper part of the column, so that the new solid cinderblocks can be mortared into place. The buzzing noise you hear is the air compressor.
We didn’t realize there was rebar (reinforcing iron bars) and concrete poured into most of the ‘cells’ of the 16″ by 24″ columns. Now we do.
How I left it: two solid blocks and some plywood in case our cribbing and jacks give way
You are looking up towards the majority of the column
(In the summer of 1970, between my junior and senior years, I found a job in Brooklyn working on a rodding truck for the local electric power utility, Con Edison — a hard and dirty job that made me itch constantly because of all the fiberglass dust that was scraped off the poles we used to clean out the supposedly empty, masonry, electric conduits that went from one manhole to the next. I guess I pissed off our truck crew’s supervisor, so the very day that I was about to quit to go back to college, I was told that I was being transferred to a jack-hammer crew, where I probably would have gone deaf. This woulda been me, except I quit)
After that was done, I trimmed some of the trees to the west. Constant struggle with the shrubbery!
Here is a batch of articles and links concerning the spray-on process for making astronomical mirrors reflective using protected silver solutions.
Long ago, I translated Foucault’s monograph on making paraboloidal, silvered astronomical mirrors. Part of his article described the process that he and Steinheil developed for silvering, which involved using silver nitrate solutions and various other reagents. It looked quite tricky, and also required further polishing! Plus, our telescope making workshop here in Washington DC had a Navy surplus vacuum chamber that was (and still is) quite effective at putting on good-quality, inexpensive aluminum coatings for any mirror up to 12.5″ diameter.
However, I and a couple of other ATMers (Bill R and Oscar O) are working on mirrors in the 16 to 18 inch range, and they simply won’t fit. So I was quite intrigued to watch how Peter Pekurar and some other folks coated a couple of rather large mirrors right in front of a small crowd of onlookers in a tent at this summer’s Stellafane.
There is also an article on the process in the January 2020 Sky and Telescope, and a webpage (here) on the topic run by Pekurar and Howard Banich and others.
Not to mention a bunch of posts on Cloudy Nights (here) and a nice PDF explaining it all, (here).
What is really, really amazing is that the webpage by Pekurar and Banich also has interferograms showing that the overcoating has absolutely no effect on the sub-microscopic, geometrical figure of the mirror! Unfortunately, it’s only effective against chemical attack, not against dirty fingers or scratches. They also did some careful experiments on reflectivity at various wavelengths with various treatments of the surface.
A couple of local ATMers and at least one professional at Goddard Space Flight Center have told me about their experiments with the process; they found that it is easy to mess up if you aren’t stringently clean and also easy to waste materials.
Jim Tanton is a very deep thinker and communicator about many aspects of mathematics. He recently was in residence in the DC area for a few years and was a mentor at Math for America – DC (based at the Carnegie Institution for Science), where I attended several of his highly entertaining and inspiring talks for new and experienced DC secondary Math teachers.
This article by him goes into what mathematics is all about, and how we teach [a part] of that in school. Here is the link:
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