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OK, it is not my intention here to start a flame war, but this is a subject that needs some rational discussion: the place of calculators in the classroom. This is a subject that is dear to my heart, because I spend 7-8 hours a day in a middle school math classroom that includes TI-78 calculators. There are 30 of them, and they hang in little pockets on a blanket attached to one wall, each one numbered (and every student in every class has his/her assigned calculator number).

I am the student teacher, which means I assist the real teacher and teach some classes each day. This is 6th grade, and one group of kids is learning how to add and subtract fractions while the other group is moving into data analysis and probability. Fractions are probably the most complex math topic that a 6th grader faces. Just a simple thing like adding two fractions requires a lot of skills that many 6th graders (in fact, many adults) do not normally have: determining the common denominator; converting mixed numbers to improper fractions and vice versa; creating equivalent fractions with the new denominator; adding the numerators; and simplifying the result (if necessary, and determining when it is necessary). Think about how overwhelming that process can be for a 6th grader who does not have the logical, analytical, and organizational skills that you and I have. So how do we teach it? By going over the basics again and again until most of the kids have it, and then moving on. Do we use calculators in this process? No, not for fraction addition, because we know how important it is for the kids to know how to do it without the calculator.

Let me tell you how we do use the calculator. On a recent test, we said that the national debt of the US is $7,030,125,000,000 and the median income is $14,900, so if each family earned the median income, how many families would it take to eradicate the national debt? So we let kids use the calculators to divide the 7 trillion by 14 thousand, because once a kid has mastered long division (and almost all of these kids have) there is no reason for them not to be able to use technology for problems like these. When they leave school they will enter a world full of technology and they had better be able to use it if they are to compete in the marketplace. And we let kids use computers in the classroom (when it is educative) for the same reason.

OK, I know that many of you guys (and gals) learned math without the benefits of technology. Heck, so did I. There were no calculators back when I was in middle school (then called junior high school) in 1961. And we all learned it OK (or at least most of us did). But if we don’t expose kids to the current technology, we are doing them a disservice.

Please understand what I’m saying. A kid has to demonstrate that they can add, subtract, multiply, and divide using pencil and paper before they can use the technology. But once they have that knowledge, let them use technology for the purpose for which it is intended. Face it, guys, no one calculates square roots manually anymore. I learned how to do it in the 60’s, but I couldn’t do it today. But I know where the square root key is on my calculator.

OK, let’s get a discussion going…..

Don Shepherd

> But if we don’t expose kids to the current technology, we are doing them a disservice.

That reminds me of a political cartoon that was in our newspaper years ago. It shows a business owner following Clinton around his office with a clipboard trying to get his attention and saying, "We can teach new employees to use the computer in two weeks; but we can't teach them to read!"

Bob Pease, highly regarded electronics industry guru, talks about these new guys fresh out of the box that were taught to use the circuit simulation software in school but can make a small error inputting the data and get ridiculous results and not realize it because they couldn't get an idea from mental calculations of what to expect generally. There can also be erroneous outputs resulting from faulty circuit component models in the software. These new graduates generally have no workbench skills to actually try something. Relying on the computer can prove very expensive when errors are not caught early in a design cycle. The problem is when the computer is used as a substitute for thinking instead of as merely a tool.

Calculators in the classrooms are mostly a substitute for thinking. Even after the student has shown they know how to multiply and divide, they still need more practice to make sure the skills will stick. When they don't get that practice, high-school graduate clerks are surprised that I can tell them how much change I'm supposed to get at the cash register before the display even arrives at the total and adds the sales tax. They can't do anything without the computer. Most school math problems are artificial anyway, meaning the answers are usually round numbers that should be calculable in one's head.

Our older son was supposed to get a TI-83 graphing calculator for a high school math class. I held off because I didn't think it was really necessary or even appropriate. He did fine without it. He said what most students used the calculator for was playing video games in class, looking busy.

As for finding a square root-- You should have been able to come up with a way to do it even if you can't remember the neat way taught in school. Being resourceful is part of understanding the math. Take a guess. Divide the original number by the guess. The real square root will be somewhere between the result and the guess, right? Average the two and do it again, until you're satisfied that you're close enough. There are several ways to find a square root. There is one that doesn't even require any multiplying or dividing, although it's not fast at all.


Edited: 16 Feb 2006, 10:34 p.m.

Interesting problem! The problem you presented exceeded the capability of most calculators.

I think one should combine the best of both worlds so to say. In my opinion kids should leave school with enough skills to do even more tedious calculations without a calculator. It simply gives them more feeling for numbers and orders of magnitude, which can be quite convenient on many occasions and in many professional activities, whether it's in science, business, or just during a visit to the local supermarket where the cashier makes a typo and charges you USD 35.50 for a few packs of milk, some bread and a bottle of coke.
On the other hand, learning tricks and how to apply them is simply not enough. As a kid I found arithmetics specifically boring at school, which sounds quite strange for a kid that would grow up to have a PhD in physics and astronomy. The major reason was that, during the lessons, there was no (or hardly no) room for understanding about what is behind all those rules for calculations and tricky manipulations on numbers. Everything came down to practicing the same routine over and over again without any immediate purpose and without knowing what one is actually doing (at least not in detail). All this does not mean that I was not interested in the subject. Some things really puzzled me. For instance that dividing 1 by a number smaller than 1 produced a number larger than 1. If you apply your box of tricks you learned in the lessons about fractions to a specific problem of this kind you just get the result and that's it. But it may look strange to a kid and so I remember being busy with the subject to find some propper explanation and insight myself. It is from this period that computing devices have some appeal to me. One day when I was 6 or maybe 7 years old, my dad brought an HP 45 home from his work and I was fascinated by that marvellous, expensive machine that could do all those dull calculations in an instant (back in 1974 I had never seen a calculator before). Later the real math classes were indeed a revelation on many topics and I am hooked on mathematics and related stuff ever since. It is here however that the calculators could enter the picture. Especially the programmables not only provide a tool to learn kids, almost while playing with it, to think in a structured logical way, but they can also be incorporated in a process of understanding about what arithmetics is actually about. For instance, it is a perfect tool to demonstrate trends in a certain type of calculation upon variation of input parameters. For the lower grades, simple (programmable) calculators will do fine. No need for graphic calculators in this respect (the possibility of gaming in the classroom being an extra con). An extra benefit here is that kids also get at least some idea of what happens in all those computer systems that they encounter every day and which they often use themselves for games or surfing the www. However, dull as it may be, there must also be time for calculations with paper and pencil or even without.

Sir, it is my opinion that in the sixth (to twelfth) grade classroom, there really is NO ROOM for a calculator.

During those years, a child is supposed to learn mathematical principles and reasoning. How much blunter is his intellect going to be if a black (okay, sometimes gaily colored) plastic and metal box does ALL his drudgery for him? It is in fact in that drudgery that the child learns how to calculate, use mathematical tools as logarithms, exponents, trigonometric functions.

In a college chemistry laboratory once, a student announced that she was going to leave and skip the class, as she had left her calculator and was then going to be unable to perform the requisite calculations to prepare for the experiment, unless she can borrow a departmental calculator.

The technician said for her to wait a moment and disappeared into the stockroom. She waited anxiously and he reappeared after several minutes and handed her some log tables and trig tables. Her eyes popped out and she angrily and incredulously shouted, "I ask for a calculator and you give me sheets of paper??!"

Well, the tech and I shared a huge horselaugh, and the young lady, disgusted, threw the papers down on the desk and stomped out.

Okay, now, after the yuks and the years, I feel that her secondary or even primary schooling let her down. She was clearly a child of the silicon revolution (probably, to drown her sorrows, she probably headed home to play a few rounds of video games on her PC or game machine or whatever) and she could not use age old, time honored, and STILL HIGHLY USEFUL methods to execute simple calculations.

If I could, I'd lobby hard to keep calculators, including our beloved vintage or even new HP RPN beauties, out of the secondary and especially, the primary level classroom.

Now if only I can find my reville bugle...

Thanks for all the thoughtful responses, folks. This forum is great for enabling intelligent discussions of sometimes complex and controversial issues. That, in my opinion, is one of the best features of the Internet. We are all learners.

The issue of using technology in the classroom is very complex. On the one hand, we have to prepare our kids for the world beyond school, and computers and calculators are part of that world. So we have to see if we can use these devices in ways that enhance the learning process. On the other hand, I recognize how these devices may be misused and end up hurting the learning process, and the kids.

I’ll be honest with you: teaching 6th graders is tough! We keep spending more millions and billions of dollars on education, and the test results just keep going down. Our schools of education teach us the obvious: that the 6th grader does not have the level of logical ability, maturity, and organizational skills that we adults have. We expect that they come to school ready and able to learn, and that is just not the case in many instances, for many more reasons than I can cite here. And we have many teachers who are burned out and who do much more harm than good. I’m a new teacher, and I am excited about helping these kids learn, but I know it’s not going to be easy, and I won’t reach every kid. But I do it with the knowledge that I will help most of the kids.

Some uses of technology in the classroom are obviously good, such as using a tablet PC connected to a projector and writing math problems on the tablet screen instead of the whiteboard (which, by the way, leaves your hands dirty all day from the markers and erasers!). And kids love to come forward and write their own problems on the screen. Or keeping your gradebook electronically instead of the old paper book (I am currently convincing my cooperating teacher to do this; it’s hard for her to give up the paper book after using it for 26 years).

No matter what we do in the classroom, the bottom line is the same: is it good for the kids? That should always be the principle criteria for what we do.

Don Shepherd

Quote:
Some uses of technology in the classroom are obviously good, such as using a tablet PC connected to a projector and writing math problems on the tablet screen instead of the whiteboard (which, by the way, leaves your hands dirty all day from the markers and erasers!). And kids love to come forward and write their own problems on the screen. Or keeping your gradebook electronically instead of the old paper book (I am currently convincing my cooperating teacher to do this; it’s hard for her to give up the paper book after using it for 26 years).

Ok, I'll be the grinch here. Even Bob Pease, the industry guru I mentioned above, still advocates the overhead projector over PowerPoint, and has solid proof of its greater effectiveness. It definitely puts up a bigger image than a TV screen, it's much cheaper, instant-on, and doens't complain that you don't have the latest software installed. You can put another clear slide over a prepared one and scribble additional diagrams by hand while talking, etc.. Old-fashioned is not bad.

My wife is a 2nd-grade teacher, and I hear all the time about the problems they've had with electronic grading. Several times over the years, the server or a hard disc fault wiped out all the grades and they had to be re-entered from the paper records (which fortunately, the teachers kept). The GradeQuick software has had a very frustrating, never-ending stream of bugs. Often a parent comes in and asks about a bad grade on the report card, and she looks it up in the computer and finds the one on the report card is not what's in the computer that reported it to the office. In fact, once in the junior high of the same school, our younger son was sent someone else's grades, and another time the grades on his report card didn't seem to go to anyone in the class. Three times in his two years in junior high, they had to re-send report cards because of computer errors.

I'm not against technology. There's no excuse for those bugs. What I'm against is the idea that the new way is always better. Some misapplications of technology need to go the way of the talking car of the early 1980's that drove everyone nuts. "Your door is ajar... Your door is ajar... Your door is.."

If we're really going to teach better thinking and reasoning skills in the classroom with calculators, we should teach programming, not operating an appliance. The latter can be picked up very quickly and does not require classroom time.

Hi Garth. We may agree to disagree. I am not necessarily for all the new technology either. I despise cell phones, but I recognize their potential value in certain situations (NOT in the grocery, "honey, should we get Van Kamps or Del Monte?").

I would love to teach programming to middle school kids who are interested, and I hope I get that opportunity.

When I taught computer science at the local community college, I kept grades via the standard paper grade book. But I only had one or two classes of 20 students each, and it was manageable. Now I have 5 classes of 30 kids, and keeping track of 150 kids with maybe two dozen assignment grades each 6-week period is an enormous administrative task. Electronic grade books don't completely eliminate the task, but they certainly reduce the time required to figure grades, and I am finding that being a teacher requires an enormous amount of time, and anything I can do to give me more time to plan exciting lessons for the kids is worth it. I wasn't a big fan of the "keychain" disk drives either, but now I am (for making quick backups).

Technology. It's a two-edged sword, but how many of us HP fanatics would be willing to go back to this?

First, your last question: I might not mind! In fact, if I remember, I'm going to ask an even older fella how to use this slide rule I picked up just for fun.

Second, turn off your cell phone: van de Kamps's battered fish is best and Del Monte anything is pretty good; their old "catsup" might even be better than Heinz' !!

(Okay, no ketchup throwing, hear?)

Quote:
Technology. It's a two-edged sword, but how many of us HP fanatics would be willing to go back to this?
Wow, I had forgotten about those. I think I only saw one once. I did use slide rules however until 1981 when I needed programmability, since calculators weren't really any faster, and the slide rule gave a better mental idea of number relations.

I should have mentioned that my wife says grading takes more of her time now with the computers than she took when it was all paper. The only thing about the paper she does not miss is having to press through four layers of NCR paper with a ball-point pen on the report cards. If you know of some grading software that's better than GradeQuick, let me know so I can suggest it to the head administrator. If it runs under Linux instead of Windows, so much the better, since that'll help them get away from Windows and save a ton of money and some headaches too. I used to be on the board of this school.

Hi, guys;

It is a known fact that tools where created and developed to enhance human’s performance when executing a particular job. When a hammer is used the way it was designed for, related primary tasks are performed faster and more accurate. Depending on the task, there is no replacement for a hammer that could do the same job with the same efficiency.

Based on some personal experience of mine in the classroom, I have observed that in many cases, the need to better understand how to use a tool usually comes from a better understanding of the task that the toll is designed for. The opposed situation has also been observed: a better understanding of the task as a consequence of a better understanding of the tool. And it does not deppend on which tool we are talking about.

If a teacher wishes to improve the way the students learn a particular subject, or the whole context, it is possible to associate the bare instructions on how to use a calculator in order to improve the understanding of the subject itself, and later, how to use the calculator to solve problems related to the same subject. As the use of the calculator as a tool eases the learning process, the students go ahead and fetch knowledge by themselves, now that they know how to use a tool that allows them to improve the process of learning.

In both cases, learning the subject and solving related problems, the calculator as a tool may be used differently. Also in both cases, the students performance may greatly be enhanced in both ways, either in understanding the subject or when solving related problems. And both facts depend on how the teacher shows the use of calculators in the classroom.

These facts relate to my own experience as both a teacher and a student. Maybe others have a different view, or different experiences, that may conflict with the ones mentioned here.

Cheers.

Luiz (Brazil)

Edited: 19 Feb 2006, 12:07 a.m.

The important things to remember when people talk about going back to old technology are that

Slide rules can't add and subtract.

Addiators, Addometers and related pocket sized devices can't multiply and divide.

The result is that users couldn't do chain calculations with portable devices using combinations of addition, subtraction, multiplication and division without switching intermediate results back and forth between machines. When the first pocket-sized four-bangers came out some slide rule manufacturers came out with a device which had a slide rule on one side and an addiator on the other. The users still had to switch intermediate results back and forth.

My recollection is that we had desktop devices such as the Fridens wich would do mixed chain calculations. Personally, by the late 1960's I had pretty much switched most of my engineering work to a company-operated telephone network which supported an extended BASIC which offered double precision and matrix manipulation.

I suspect that most of the people who long for the "good old days" when we used slide rules never really used the damn things to do engineering.

Hi Garth. Yes, I still like those old manual "adding machines." To do a "carry," you moved the stylus up and over, advancing the next higher power of 10 by 1. It's interesting becoming a teacher late in life. When I learned subtraction in the 1950's, we "borrowed" one from the next column. Now, we "regroup" since borrowing implies paying back, which we don't do! It turns out that the "new math" isn't really that new at all, it just uses different terms.

I'm certainly no expert on grade book software, but I found a neat teacher webpage (http://people.clarityconnect.com/webpages/terri/terri.html) that recommended GradeKeeper (http://www.gradekeeper.com/), and I used that for a couple of months and liked it. It is simple, cheap ($20), and produces the reports that I seemed to need.

Thanks for the feedback (and everyone else too), I know we all benefit from it.

Don Shepherd

Thanks Luiz. As always, your input is appreciated.

You are right, a calculator is only a tool, just like a computer or a slide rule. When a student has demonstrated that they can do the 4 arithmetic operations using paper and pencil, letting them use calculators to do that basic math enables us to move into the rest of the curriculum faster, and we have a LOT of curriculum to cover in a short time. And I am a firm believer in letting the students use graphing calculators at the appropriate point in algebra, so they can see and study the graph of y1=3x+4 and play around with variations of it. They learn so much better when they "do it themselves." They own the knowledge.

One thing we teach the kids in fractions, when finding the common denominators, is to list the multiples of each denominator and see where they match. So for 3/4 and 2/3, they list 4:4 8 12 16 and 3:3 6 9 12 15 and so on, and they see that 12 is the common denominator. After they master that, we show them another way to see multiples on their graphing calculator: y1=3x and y2=4x, then look at the table and see the first number shared by both y1 and y2. So now they have two ways to do it, and they have learned something.

Don Shepherd

Good thread!

My deceased brother used to say, "I'd give my right arm to be ambidextrous."

That's why we have teachers and supreme courts to sort out the balance. What I would have given in college for a tool to get me to the "bottom line" of the masses of data contained within the concepts of statistical analysis. I later returned to the question as a user and would vote without hesitation for use of any available tools as teaching aids.

The question would have been to me as a student, "How do I optimize this issue for my benefit?" The question to my teachers should have been, "How do I optimize this issue for the benefit of my students?"

Ron

Ah, but the use of "advanced tools" in college math, engineering, or science classes is different from the same in elementary or secondary classrooms.

In these earlier levels, the student is supposed to be mastering the basics to a level of proficiency so as to be able to maximally benefit from the use of these "advanced tools" later in college... or some equivalent thereof.

From what I've seen of most students entering postsecondary levels, including, unfortunately these days, a few even on the graduate level, having difficulties arithmetically, which does affect their more advanced mathematical skills. And from my own observations, it appears to be an over-reliance on the use of electronic aids, as calculators, PC software, etc.

I think it is academically safer to keep calculators, and even PCs to a great extent out of the elementary classroom, introducing them right before the secondary level or at the early secondary levels. Then, perhaps by the time they hit the universities and colleges, they are truly using the calculator to enhance their work rather than relying on it to make up for shortcomings.

That's part of what I meant about balance, teachers and supreme court. I addressed the advanced student issue, not the basic 'rithmetic which would fall under learning and appreciating just what the tool is for and how you use it. Otherwise I agree that the elementary student needs to learn basic arithmetic, even some trig, geometry and algebra without a crutch. For the advanced student, there are very few uses more dramatic than watching programmable graphic calculators illustrate the dynamics of sensitivity curves or to show symbolically, even numerically as the case may be, a solution to a system of differential equations.

Frankly, I don't see an either/or issue here, it is a trick question for teachers and the answer is yes, no and sometimes maybe.
In other words become ambidextrous without losing one arm.

I agree, at school you learn the meaning of things, so the teacher needs to be sure that the students actually understand what is going on, rather than punch the buttons and copy the result.

For example consider the following problem (from Andy Tannenbaum's book on Networks): Assuming that a laser beam is used to send data across the street, how much should a laser beam be deflected (in degrees) to miss its target. The distance between the transmitter and the target is 100m and the diameter of the beam and the sensor is 10mm. Assume that *any* overlap between the beam and the sensor is adequate for the transmission.

The point of the exercise is to show that the angle is so small, so you have to be very carefull when aligning the equipment.

You should be able to get the ball park figure without a calculator. Then use the calculator to get a better figure if this is desired.

But some students got an answer that was an order of magnitude wrong while others got strange numbers because of underflows in their calculations.

So a calculator is dangerous unless you really know what you are doing.

Another trend that I have observed is that students also use arbitrary precision calculators (e.g. Unix's bc) to make sure that their answer is correct again without really bothering to understand what is going on. Using bc for this problem may be OK, but if you wanted to do the calculation many times you have to consider the cost of the very high precision.

**vp

Hi, guys;

thanks to all of you who added your own opinions and thoughts to my own. It is always a good experience.

What I'd add is that I emphasize the need of focusing the understanding of the problem and the related solution, both with Calculator-Aided support. I remember when teaching Logarithms to my first-semester students (Telecommunication Technology). After showing them all properties and having them seen enough, I asked them to use their calculators and see how calculations could be performed faster. I saw some students teaching others how to use their calculators, others comparing results and some groups asking for extra activities. They were all using only the keys/functions necessary to compute what was asked, I do not remember seeing students 'fooling around' and testing unnecessary keys in that particular class. Because they saw how to use the 'tool' accordingly, in order to accomplish the task, the task itself was the issue, not the 'tool' (calculator). Later on, I was showing them the polynomial series equivalent to trigonometric functions, and I told them that we could compare the values obtained with the series to the ones obtained with the internal functions in their calculators. Once again, a controlled 'brainstorm'...

Just to add these words.

Cheers.

Luiz (Brazil)

You bring up an interesting idea: let me say that perhaps the teacher ought to do it on a class-by-class basis. That is, some years/semesters the kids, as a group, may be more receptive to truly learning by using the calculator along with their textbook/workbook/rexogr... uh, Xeroxed sheets/blackboard taken notes. Other groups other times in the same grade may not have the intellectual or emotional maturity to realize that the calculator is to be an aid, an extension of one's brain, not the sum of it, especially those that can hold and execute downloaded... ugh... games.

Yeah, ha ha!

Tell even an exceptionally advanced high school kid, like Ben Salinas used to be, or a college student to calculate the quantum mechanical wavelength of a pitched baseball and most likely they'll use a calculator, be it TI, Casio, HP, Sharp, or the 99-cent store brands, and they'll get an answer, which is, at least mathematically, correct, including the correct digits down to the ninth to twelfth place after the decimal point.

But it doesn't require the calculator to tell that it isn't a very useful result, as even a rough order of magnitude calculation will show that. In fact in such a (silly?) calculation, what is important is the the order of magnitude and its significance, which really can be and really ought to be done either in your head or on the back of an envelope, preferably the former.

In previous eons, the teacher would try to pique our interest by saying it was how we hear volumes and how it related to the big, round volume knob on our stereo receivers. Some of the students didn't hear or understand him because they were listening to their radios (surreptitiously via earphone) and the others were too busy admiring their HP RPN spice or classic powerhouse. Of course this last group figured it out or did hear a little something! (Perhaps as evidenced by all of you here! ;) )

In a physics class at school I've learned the following simplification which comes in handy here:

sin alpha ~= tan alpha ~= alpha (in radians), if alpha is sufficently small.

In our case: alpha ~= tan alpha = 0.01m / 100m = 10-4.

No calculator or trig tables neccessary!

The class was held back in the mid seventies. Only very few of us had calculators back than (I was the proud owner of an SR-51A while my teacher only had an SR-50!).

Marcus

Hi, Marcus;

allow me a question? Just curiosity of mine.

Can you tell me if you either memorize, learn or had it proven somehow? I myself take great care when defining inner processes. If you actually know the facts, then I'd suppose they are based on scientific proof. In this case, a number-based statement, did you use any other source that was not any of the ones you mentioned, namely a calculator or trig tables?

Cheers.

Luiz (Brazil)

When I said that you can get a quick ballpark figure without a calculator, that was based on Marcus' simplification. I also
learned that at school.

**vp

Luiz,

I suspect you know this, but if not .....

The small angle approximations for sin and tan can be derived from the Taylor series approximation (now, perhaps, the question is where did THAT come from?) for sin, and the fact that tan = sin/cos and the approximation for cos, which is very close to 1 for small angles.

I have pointed this out in physics class to students with modest mathematical sophistication who had never heard of it. I then had them calculate sine of 1 degree and compare it with 1 degree converted to radians.

Hi, Dave;

Thank you for pointing this relation out. This is the one I remember, too. In fact, cos(0)=1, sen(0)=0, and if we apply limit to the tangent relation, we get to the close-to-zero relation.

What called my attention was the following sentence:

Quote:
In our case: alpha ~= tan alpha = 0.01m / 100m = 10-4
Is there a way to get this precision without a number-crunching tool? I was not aware of this figure, believe me, that's why I'm asking for guidance. If there is a way to find this 10-4 precision without some computing device, I would like to know how. I thought of going ahead with pencil and paper to compute the series terms, but I still would like to know if there ia another way. The teacher's side of my brain is now acking...

Cheers.

Luiz (Brazil)

Vasillis posed a laser-aiming problem:


Quote:
Assuming that a laser beam is used to send data across the street, how much should a laser beam be deflected (in degrees) to miss its target. The distance between the transmitter and the target is 100m and the diameter of the beam and the sensor is 10mm. Assume that *any* overlap between the beam and the sensor is adequate for the transmission.

The point of the exercise is to show that the angle is so small, so you have to be very carefull when aligning the equipment.

...

But some students got an answer that was an order of magnitude wrong while others got strange numbers because of underflows in their calculations.


Of course, the answer Theta is,

Theta = tan-1 (10 mm / 100 m)
= tan-1 (0.0001)
= 0.00573 deg

~= (180/pi) * 0.0001 deg, since tan (x) ~= x for small x in radians

Now, here's my question: What calculators would cause underflow errors for this calculation?

-- KS

Quote:
Now, here's my question: What calculators would cause underflow errors for this calculation?
Without taking the time for an analysis, I don't expect that any of them would. In embedded systems, I often work with 16-bit fixed-point/scaled-integer cells, which means less than 6 digits (although intermediate results can be 32 bits). In this number system, we divide a circle up into 65,536 (ie, 2^16) pieces. It works out very conveniently in both signed and unsigned arithmetic, as 350° for example is the same as -10°; and 5x90°, although it overflows the full circle, is still 90° as far as the actual angles go. The angle above is very close to one count, and my SIN function is accurate to the last bit in most cases. (Once in awhile it's high by one count, never more, and never low.) It takes the first four terms of the infinite series, except that I tweaked the coefficients slightly. The output of this one, scaled by $7FFF (since you can't have +$8000 in signed 16-bit numbers to represent +1), is 3, which although 8.5% low, is as close as the granulariy allows when you're so close to 0. Calculators, which have the advantage of floating-point and usually at least 10 digits, shouldn't have any problem. BTW, all 10 digits of my HP41cx on this one agree with my HP71 which has the greater precision.

Edited: 21 Feb 2006, 5:15 a.m.

Luiz,

all the trigonometric functions are defined in a rectangular triangle as the ratio of two sides:

              /|
/ |
/ |
/ |
/ |
/ |
/ |
c/ |
/ |b
/ |
/ \ |
/ \ |
/ \ |
/ alpha | |
/________|_____|
a
sin alpha = b/c

cos alpha = a/c

tan alpha = b/a

This can be simplified if you make c the length 1 (any unit you like):

sin alpha = b

cos alpha = a

tan alpha = b/a

The latter makes it clear why tan = sin/cos.

Do you see, why sin2+cos2 = 1? (Think of good old Phytagoras!)

If you make alpha very small, a will be very close to c=1:

sin alpha = b

cos alpha ~= 1

tan alpha ~= b/1 = sin alpha

The angle (in radians) is nothing more than the length of the arc of a circle with radius c=1. For very small angles, this arc is very near to the length of b:

sin alpha = b ~= alpha (length of arc)

The trigonometric functions are not just keys on a calculator or function names in some software!

Marcus

Here are two simple assignments which shouldn't require a calculator. All I want as a result are rough estimates. And please don't answer in ounces or pounds, otherwise I'll be completely lost ;-)

1.) What is the weight of thousand steel balls, 1 millimeter in diameter each?

2.) What is the weight of a single ball of cork with a diameter of 2 meters?

Try to guess first and than think a litte further...

Marcus

Edited: 21 Feb 2006, 8:44 a.m.

1.)

1000 * 1^3 mm = 1000/1000 grams water * 7.8 = 7.8g. That is assuming a "cube" equal to the diameter. But V = 4/3 r^3 pi. So come closer: 0.5^3 = 0.125 . times pi = 0.4 . times 4/3 =~ 1.6/3.

So, about 4g.


2.)

1 meter = radius, so volume = 1.3333*Pi ~= 4.1 m^3. Density of cork. Hmmmm. I guess 5 lb/ft^3. (that's about 0.08 specific gravity.

4.1 m^3 = 4 .1 metric tonne of water. So, cork = 8% of that, or about 0.32 metric tonne.

No calculator.

Hi Bill,

very quick answer.

1.) is perfectly correct. Just to outline a slightly simpler to understand approach:

  • Put 10 balls in a row, you'll get a centimeter.
  • Put 10 rows together, so you get a square of 1 cm2
  • 10 of these will fit exactly in a cube of 1 cm3
  • The specific weight of steel is somewhere between 7 and 8 grams/cm3. Since about half of the cube is empty space between the balls, the weight must be somewhere between 3 and 4 grams.

2.) Your estimate for the specific weight of cork is off, its about one fith of that of water. My rough estimate is as follows:

  • V = 4/3*pi*r3 ~= 4*r3 = 4m3.
  • Weight ~= 0.2t/m3 * 4m3 = 800kg.

Most people I've asked tend to think of a few kilos for question (1 ) because "thousand" sounds like "so much". Likewise many people can't believe that "light" cork can weigh almost a ton if you just get enough of it!

Marcus

Another funny question: What is the linear speed of the moon, circling earth? I tried to calculate it in the top of my head. It kept me busy on a boring road trip.

Marcus posted:

"Another funny question: What is the linear speed of the moon, circling earth?"

About one kilometer per second.

The linear speed equals the distance travelled divided by the time it takes to do it.

The distance, roughly assuming that the moon's orbit is exactly circular, would be 2*Pi*DistanceToTheMoon, which, assuming the distance to the moon is some 400,000 kilometers, would be 2*Pi*400,000 = 6 * 400,000 = 2,400,000 kilometers.

The time is more or less a month, i.e., 30 days, and a day has 24 times 60 times 60 seconds, i.e: 24 x 60 x 60 seconds, which slightly less than 25 x 60 x 60 = 5 x 5 * 60 x 60 = (5 x 60) x (5 x 60) = 300 x 300 = 90,000 seconds, so a month has 30 x 90,000 seconds = 2,700,000 seconds.

So the speed is 2.4 million kilometers / 2.7 million seconds = 2.4/2.7, which equals one kilometer per second, more or less.

Best regards from V.

OK, Marcus; so far, so good.

Thank you for your explanation and for taking the time to draw the sketch showing the basic trigonometric and the relations between trigonometric functions. My approach to the students in the classroom when teaching trigonometrics is more 'graphics-based'. At least some of them tell me that this approach makes it easier to understand the subject.

In the above drawing, the triangle relation d/radius = sin/cos is obtained by triangle proportionality. You see, both triangles (sin,cos,r=1) and (d,radius,side) share the same internal angles. By giving ‘d’ the name ‘tangent’ and knowing that ‘radius = r=1’, we have:

d/radius = d/1 = tan = sin/cos
and:
radius^side = cos^r=1 = alpha
The issue here is the magnitude of alpha when alpha ~= sin(alpha): 10-4.

Quote:
The trigonometric functions are not just keys on a calculator or function names in some software!
You bet!

Cheers.

Luiz (Brazil)


Edited: 21 Feb 2006, 1:52 p.m.

Valentin,

Your approximation is quite good. If we assume a circular orbit with r=4E5 km, the result given by a calculator is 1.04 km/s.

In such calculations it's totally OK to assume that a day has 25 hours or that the value of Pi is 3. The result will be of the right magnitude. The many digits of a calculator make the result more exact but does it really affect the idea you get, if the moon runs at exactly 1.04 km/s or slightly more or less? It's not millions of kms per hour and still faster than walking...

Will any of the teachers here use these questions in class?

Marcus

Here's the next thing you should worry about for the moon:

How far does it FALL in one second? You may also be amazed at how small this value is. Remember - as deduced above, it goes forward about 1000 meters every second.

(Remember, an orbiting body is basically "falling" around its parent body at just the right rate. It was this rate of fall that convinced Newton that he understood gravity, when he compared it to how fast bodies fell at the surface of the Earth.)

Don't forget the earth orbits the moon too; ie, the center of mass of the two is not the center of the earth, but nearly ten times as far from the center of the earth as its surface is. If the earth's mass is approximately six times that of the moon, the resulting radius of the moon's orbit will be, very roughly, 340,000km, not 400,000, and the rest is the earth's orbit around the same center of mass.

Or to simplify, "it is all relative motion" :-)

Hi, Garth:

"Don't forget the earth orbits the moon too;"

    Nope, the Earth doesn't "orbit the moon", it orbits around the Earth-Moon system's mass center.
"the center of mass of the two is not the center of the earth, but nearly ten times as far from the center of the earth as its surface is."
    Nope, the Earth-Moon system's mass center is only 4,700 km from the Earth's center, so it is 1,700 km *below* its surface, still *inside* the Earth's body.
"If the earth's mass is approximately six times that of the moon [...]"
    Nope, the Earth mass is approximately 81 times that of the Moon.
Nice try but you need to sort out your astronomical data a bit :-)

Best regards from V.

Quote:
"Don't forget the earth orbits the moon too;"

Nope, the Earth doesn't "orbit the moon", it orbits around the Earth-Moon system's mass center.


That's what I meant; that the earth is not unaffected by the moon's orbit. The two are a system, both orbiting the center of mass of the two.
Quote:
Nope, the Earth mass is approximately 81 times that of the Moon.

Ok, you caught me. If my memory weren't so good, I would have looked it up. I was specifically taught in grade school that the moon's mass was about 1/6 that of the earth. The fact that the force of gravity on the moon's surface is around 1/6 G despite the smaller diameter by itself should have alerted me to the teacher's mistake. So too should the fact that its diameter is about a quarter of the earth's, which would mean about 1/64 the mass if the average density were the same. I should have known, judging from all the other things I was taught that I knew even at the time were wrong (like that a light bulb's filament got so hot because of all the light coming out of it, a situation where the teacher took my protests to be disrespect for her, the "facts", and authority in general). I regret wasting people's time with errors. Thankyou for the correction.

The easiest way to show that sin(x) = x for small values of x, is to use the Pade approximation:

From http://www.dattalo.com/technical/theory/sinewave.html
we get:

(x is in radians)

You can easily check that this is a fairly good approximation with bc (standard Unix or Linux utility):

% bc -l
n1 = -325523.0/2283996.0
n2 = 34911/7613320
n3 = 479249/11511339840
d1 = 18381/761332
d2 = 1261/4567992
d3 = 2623/1644477120

define num(x) {
return(1 + n1*x^2 + n2*x^4 + n3*x^6)
}
define den(x) {
return(1 + d1*x^2 + d2*x^4 + d3*x^6)
}
define ss(x){
print x * num(x)/den(x)
print "\n"
print s(x)
}

ss(1)
.84155226541146287688
.841470984807896506650
ss(0.5)
.47942618520237196517
.479425538604203000270
ss(0.3)
.29552022483199108139
.295520206661339575100
ss(0.1)
.09983341665515269688
.099833416646828152300
ss(0.0001)
.00009999999983333332
.000099999999833333330

Now observing the Pade approximation formula you can see that as x gets close to zero the powers of x (x^2, x^4, x^6) will get vanishingly small, hence the formula will end up being

sin(x) = x + epsilon

Hope this helps

**vp

PS to keep this relevant to the MoHPC here is the HP-97 version of the Pade approximation:

Edited: 21 Feb 2006, 7:53 p.m.

Quote:
a light bulb's filament got so hot because of all the light coming out of it

Is that why it's so cold in the dark? I knew there had to be a reason :)

What about the limit of the Taylor series for sin(x) when x tends to zero?

lim (x - x^3/3! + x^5/5! - x^7/7! + ...) = x
x->0
Regards,

Gerson.

Edited: 21 Feb 2006, 8:24 p.m.

Hi, Vassilis;

after seeing this

Quote:
...
ss(0.0001)
.00009999999983333332
.000099999999833333330

I see the 10-4 in both the argument and the (close to 10-4) resulting value. Thanks!

I guess that there is no way to get to this point without some `number crunching` procedure, thought. And a computing device would be of great help, I think.

Cheers.

Luiz (Brazil)

Edited: 22 Feb 2006, 1:14 a.m.

Vieira, Luiz C. (Brazil) wrote:
> I guess that there is no way to get to this point without some
> `number crunching` procedure, thought. And a computing device
> would be of great help, I think.

I think that the whole point is that you do not
need `number crunching` capabilities.

Consider the problem, you have a right-angle triangle and you know the two sides on either side of the right angle, and want to find the hypotenuse (h). One side is 100m and the other is 10mm (or 0.010 m).

Normally you'd use Pythagoras' theorem:

h^2 = a^2 + b^2

but a^2 = 10 000 while b^2 = 0.0001

Just looking at the number (h^2 = 10000.0001) tells us that there is no point in calculating the square root if all we want is 4 decimal place accuracy. So, for the purposes of this problem h = a + epsilon, but since epsilon is not within the desired accuracy we can ignore it, i.e. h = 100.0000

Same thing about the angle. We know its very very small (since we have calculated it as Marcus suggested earlier). This tells us that there is no point in calculating its sine, since we know it is equal to the angle.

Thus while you may need `number crunching` capabilities once (to demonstrate what is predicted by theory) you do not need such capabilities to solve this particular problem. No log tables, no trig tables, nothing, just the ability to move decimal points around.

You have to be careful of course, that the dropped digits are not needed later, but you have to be aware of that even if you use a calculator (which also has limited accuracy).

The beauty if this is that if you later use a computer or calculator to get the answer to a gazillion decimal places, knowing roughly what the answer should look like from the quick and dirty calculations will save you from showing up with a number that has 30 digit accuracy, but is wrong by an order of magnitude. :-)

**vp

Hi, Vassilis;

thanks again.

I had to go to the whole thread once again and read all posts, top to botton, to try finding something I missed, and I think I finally saw a connection I didn't explicitly see at the first and second times I read them (I've already read all of the posts two times prior to go ahead...)

Somehow I did not connect Marcus von Cube solution to your specific laser beam proposal (Tannembaum's book) till now that you affirm this connection explicitly. It could save us all the many posts in this thread that were based just in my lack of attention. I appologize you all, Marcus von Cube at first, because I could not see that his answer was to the specific problem, instead I thought it was a general approach to the whole relation between the small-value angle and its tangent, so I was trying to understand why this relation was valid only when alpha<=10-4. My bad! In fact, about a year ago, I remember this particular relation was explored here by other contributors.

At least I could read some more about programming, math, trigonometry and related stuff. As for a lesson to my own, I'll be more carefull reading posts and trying to figure out a solution prior to turn it into a problem...

Forgive me, please, folks, and thanks.

Luiz (Brazil)