Tag Archives: Math

Is this question reasonable?

20 Tuesday Jul 2021

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This is a sample question for middle school math, published by the International Baccalaureate (IB) program. I found it here.

Here is a graph I made of this equation, using Desmos:

Looking at this graph, you see that after about 10 minutes, there are 11 cars per minute going through the intersection – and that’s the most cars. After about 25 minutes, there are zero cars going through the intersection, and after that, there is a negative number of cars (!!!).

I don’t think this equation models anything having to do with any intersection I’ve ever visited. Instead, I think that any intersection controlled by a traffic light is going to be more periodic, that is to say, something like some mix of sine or cosine functions — obviously not middle school material.

A neat geometry lesson! And a rant…

13 Thursday Feb 2020

Posted by in education, flat, History, Math, Optics, teaching, Telescope Making, Uncategorized

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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.

 

https://angelgilding.com/multiple-reflections/

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Rant, in the form of a long footnote:

* 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.

Why Not Show Students the Beauty of Math?

16 Tuesday Oct 2018

Posted by in education, Math, teaching

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When I taught math, I tried to get students to see both the usefulness and beauty of whatever topic we were discussing. The most beautiful mathematical objects I know of are the Mandelbrot and Julia sets, which in my opinion should be brought up whenever one is studying imaginary and complex numbers.

To illustrate what I mean, here are some blown up pieces of the Mandelbrot set. Below,  I’ll explain the very simple algebra that goes into making it.

 

I made these images using an app called FastFractal on my iPhone. The math goes like this:

Normally, you can’t take the square root of a negative number. But let’s pretend that you can, and that the square root of negative one is the imaginary number i. So the square root of -16 is 4i. Furthermore, we can invent complex numbers that have a real part like 2, or 3.1416, or -25/17, or anything else, and an imaginary part like 3i or -0.25i. So 2-3i is a complex number.

Ok so far?

We can add, subtract, multiply and divide real, imaginary and complex numbers if we want, just remembering that we need to add and subtract like terms, so 4+3i cannot be simplified to 7i; it’s already as simple as it gets. Remember that i multiplied by i gives you negative one!

Interesting fact: if you multiply a complex number (say, 4+3i) by its conjugate (namely 4-3i) you get a strictly REAL answer: 25! (Try it, using FOIL if you need to, and remember that i*i=-1!)

Furthermore, let us now pretend that we can place complex numbers on something that looks just like the familiar x-y coordinate plane, only now the x-axis becomes the real axis and the y-axis becomes the imaginary axis. So our complex number 4+3i is located where the Cartesian point (4, 3) would be.

Ok — but what’s the connection to those pretty pictures?

It’s coming, I promise!

Here’s the connection: take any point on the complex plane, in other words, any complex number you wish. Call it z. Then:

(1) Square it.

(2) Add the original complex number z to that result.

(3) See how far the result is from the origin.

(4) Repeat steps 1 – 3 a whole lot of times, always adding the original z.

One of two things will happen:

(A) your result stays close to the origin, OR

(B) it will go far, far away from the origin.

If it stays close to the origin, color the original point black.

If it gets far away, pick some other color.

Then repeat steps 1-4 for the point “right next” to your original complex point z. (Obviously, the phrase “right next to” depends on the scale you are using for your graph, but you probably want fine coverage.)

When you are done, print your picture!

If we start with 4+3i, after one round I get 11+27i. After two rounds I get -604 + 597i, which is very far from the origin, so I’m going to stop here and color it blue. I’ll also decide that every time a result gets into the hundreds after merely two rounds, that point will also be blue.

Now let’s try a complex point much closer to the origin: how about 0.2+0.4i? I tried that a bunch of times and the result seems to converge on about 0.024+0.420i — so I’ll color that point black.

This whole process would of course be very, very tedious to do by hand, but it’s pretty easy to program a graphing calculator to do this for you.

When Benoit Mandelbrot and others first did this set of computations in 1978-1980, and printed the results, they were amazed at its complexity and strange beauty: the border between the points we color black and those we color otherwise is unbelievably complicated, even when you zoom in really, really close. Who woulda thunk that a simple operation with complex numbers, that any high school student in Algebra 2 can do and perform, could produce something so beautiful and weird?

So, why not take a little time in Algebra 2 and have students explore the Mandelbrot set and it’s sister the Julia set? They might just get the idea that math is beautiful!!!

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An Eclipse Seen in Wyoming

27 Sunday Aug 2017

Posted by in astronomy, astrophysics, Math, nature, Telescope Making

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I was fortunate enough to have the time and cash to go to Wyoming for the August 21 eclipse. It was truly wonderful,. in large part due to the fact that I had made a 6″ diameter, f/8 Dob-Newt travel telescope that could play three roles: as an unfiltered projection scope onto a manila folder before and after totality; with a stopped-down Baader solar filter during and after totality; and with no filter at all during the two minutes or so of totality.

No photographic image that I have so far seen comes anywhere near the incredible details that I was able to see during those short two minutes.

Here is my not-very-expert drawing of what I recall seeing:

solar eclipse

The red rim on the upper left is the ‘flash spectrum’, or chromosphere. It was only visible for a few seconds at the very beginning of the eclipse. The corona is the white fuzzy lines, but my drawing doesn’t do them justice. On the bottom, and on the right, are some amazing solar prominences — something that I don’t recall having seen in 1994, my first successful solar eclipse. The bottom one might not have been quite that large, but it really got my attention.

Here are a few photos I took before and after totality:

I started planning this expedition over a year ago, and hoped to attend the Astronomical League meeting in Casper, WY. I quickly found that there were absolutely no rooms to be had there, even a year in advance.

Wyoming has fewer people than my home town (Washington DC), and not many populated places in the path of totality. However, I did find a motel in tiny Lander, Wyoming, very close to the southern edge — a location that I had previously found to be very good for viewing eclipses. One of the fellows in our telescope-making workshop, Oscar O (an actual PhD solar astrophysicist) decided he would bring some family and friends along and camp there to view it with me. So he did (see the group photo).

The night before, we went to a site near Fossil Hill, WY to look at stars. The Milky Way was amazing, stretching from northern to southern horizon, and the sky was very, very dark. We met a baking-soda miner (actually, a trona miner) and his 10-year-old daughter; she had a great time aiming my telescope, via Telrad, at interesting formations in the Milky Way. My friends from DC whipped up an amazing dinner on their tiny camp stove. There were LOTS of people camping in the back country there; I bet most of them were there to view the eclipse!

On the eve and morning of the eclipse, after consulting various weather ‘products’, we decided that the predicted clouds in Lander itself would be a problem. (I had been clouded out before, with my wife and children, back in 1991, in Mexico! It really spoils the experience, I assure you!)

So we drove north and west, through the Wind River Indian Reservation, and picked a spot just east of the tiny town of Dubois at a pulloff for a local fish hatchery. Along the drive to that location, we saw lots of folks had set up camp for the event at various pulloffs and driveways to nowhere. (If you didn’t know, Wyoming is mostly devoid of people, but has lots of fields and barbed wire fence. Many of those fields have driveways leading to some sort of gate, most of which are probably used at least three times every decade, if you get my drift….)

Not only is Wyoming largely empty (of people), but the path of totality in the United States was so long that I estimated that if the ENTIRE population of the USA were to decide to go view the eclipse, and somehow could magically spread themselves out evenly over the 70-mile-wide, and 3000-mile-long, path on dry land, that there would only be about 3 people per acre!

Here’s the math: 70 miles times 3000 miles is 210,000 square miles. The population of the USA is about 330,000,000. Divide the population by the area, and you get about 1600 people per square mile. But there are 640 acres in a square mile, so if you divide 1600 by 640, you get less than 3 people per acre, or 3 people on a football field (either NFL or FIFA; it doesn’t matter which).

(…looking to the future, the next decent eclipse doesn’t seem to occur anywhere in this hemisphere until 2024, when it will cross from Texas to Maine…)

As you can see from my photos, the little travel scope I made, called Guy’s Penny Tube-O III, performed very well. Before and after totality, we used it both for solar projection onto a manila folder, through the eyepiece. I also had fashioned a stopped-down solar filter with a different piece of cardboard and a small piece of Baader Solar Film. With both methods, we could clearly see a whole slew of sunspots, in great detail (umbra and penumbra) as well as the moon slowly slipping across the disk of the sun. Having the sunspots as ‘landmarks’ helped us to watch the progress!

Then, during totality, after the end of Baily’s Beads and the Diamond Ring, I took off the filter and re-adjusted the focus slightly, and was treated to the most amazing sight – a total eclipse, with coronal streamers to the left and right; the ‘flash spectrum’ appearing and winking out on the upper left-hand quadrant (iirc); and numerous solar flares/prominences.

I got generous and allowed a few other people to look, but only for a few seconds each! Time was precious, and I had spent so much work (and airfare) building, and re-building, and transporting that telescope there!

Planets? I didn’t see any, but others did. Apparently Regulus was right next to the Sun, but I wasn’t paying attention.

The corona and solar flares were much, much more pronounced than I recall from 1994.

That afternoon, the town of Lander had the largest traffic jam they had ever had, according to locals I talked to. Driving out of there on that afternoon was apparently kind of a nightmare: the state had received a million or so visitors, roughly double its normal population, and there just aren’t that many roads. I chose to spend the night in Lander and visited from friends I had gotten to know, who are now living in Boulder, on the night after that. Unfortunately, on that next day, I got a speeding ticket and a citation for reckless driving (I was guilty as hell!) for being too risky and going too fast on route 287, trying to pass a bunch of cars that I thought were going too slow…

When I did fly out from Denver, on Wednesday, all the various inspections of my very-suspicious-looking and very-heavy luggage caused me to miss my flight, so I went on standby. It wasn’t too bad, and I was only a few hours later than I had originally planned. And my lost suitcase was delivered to my door the next day, so that was good.

I am now in the process of making this travel scope lighter. I have removed the roller-skate wheels and replaced them with small posts, saving several pounds. I have begun using a mill to remove a lot of the metal from the struts. And I will also fabricate some sacks that I can fill with local rocks, instead of using the heavy and carefully machined counterweights! (Rocks are free, gut going over 50 pounds in your luggage can be VERY expensive!)

 

By the way: unless you like to travel with no luggage at all, NEVER use Spirit Airlines! They may be a few dollars cheaper, but they will even charge you for a carry-on bag! What’s next? Charging you for oxygen?

 

 

The Mathematics of George Washington

24 Friday Feb 2017

Posted by in History, Math, Uncategorized

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I recently learned some things about how the young George Washington did math, including surveying. Mathematician and historian V. Frederick  Rickey gave a talk 2 nights ago at the Mathematical Association of America here in DC, based on his study of GW’s “cypher books”, and I’d like to share a few things I learned.

(1) The young George appears to have used no trigonometry at all when finding areas of plots of land that he surveyed. Instead, he would ‘plat’ it very carefully, on paper, making an accurate scale drawing with the correct angles and lengths, and then would divide it up into triangles on the paper. To find the areas of those triangles, he would use some sort of a right-angle device, found and drew the altitude, and then multiplied half the base times the height (or altitude). No law of cosines or sines as we teach students today.

(2) He was given formulas for the volumes of spheroids and barrels, apparently without any derivation or justification that they were correct, to hold so many gallons of wine or of beer. (You probably wouldn’t guess that you had to leave extra room for the ‘head’ on the beer.) Rickey has not found the original source for those formulas, but using calculus and the identity pi = 22/7, he showed that they were absolutely correct.

(3) GW was a very early adopter of decimals in America.

(4 ) This last one puzzled me quite a bit. It’s supposed to be a protractor, but it only gives approximations to those angles. The results are within 1 degree, which I guess might be OK for some uses. I used the law of cosines to convince myself that they were almost all a little off. Here’s an accurate diagram, with angle measurements, that I made with Geometer’s Sketchpad.

His method was to lay out on paper a segment 60 units long (OB) and then to construct a sixth-of-a-circle with center B, passing through O and G (in green). Then he drew five more arcs, each with its center at O, going through the poitns marked as 10, 20, 30, 40, and 50 units from O. The claim is then that angle ABO would be 10 degrees. It’s not. It’s only 9.56 degrees.

Hitherto Un-noticed properties of Primes

15 Tuesday Mar 2016

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Did you know that if a prime number ends in a 9, then the next prime number larger than that is much more likely to end in a 1 than in a 9?

Did you know that if you study coin tosses, it will take you only about four tosses to find a head followed by a tail, but about six tosses to get two heads in a row, even though they are e qually likely?

I didn’t either. Interesting article:

https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/