Dec 26 2013

Math topics that need to be put out of their misery. Part 3: Radian measure

So far I’ve written about two topics I feel contribute little to math education.  They don’t inspire students to want to learn more about math.  They are not topics that are ‘useful’ in real life or as a prerequisite to other ‘useful’ math.  Though I plan to do about ten parts to this series, you should know that I could easily write forty parts.  I’d say that at least half of the math that we force kids to learn are, to put it mildly, ‘counterproductive.’

But there are other topics that would make my very short list of things that I think are very important and should never be cut from the curriculum.  Much of the time saved by eliminating the stale topics could be transferred to expanding the focus on these more essential topics.  One such topic is the famous mathematical number π.  I teach an elective called math research which is taken mainly by 9th graders, and we spend a few weeks studying different ingenious ways that mathematicians across the globe and across the centuries have devised for approximating this mysterious number.

Ask most adults what π is and they are likely to say either “I forgot” or “3.14″. Then if you ask them, “But what is π?” and they are unlikely to know.  Put most simply, π is the number of diameters of a circle, any circle, would be needed to wrap around that circle.

Three is too few, but four is too many. It seems to be a little more than three, but how much more? 1/4, 1/5, 1/8? Well it turns out to be quite close, but not perfectly, 1/7, getting the famous approximation 3 1/7 which is equivalent to 22/7.
Once students understand that this is what π means, the formula C=π*D is something that doesn’t need to be memorized since it follows directly from the meaning of π.  A circle with diameter 10 cm would have a circumference of π*10 cm, which is approximately 31.4 cm, while a circle with diameter 20 ft would have a circumference of π*20 ft, which is approximately 62.8 ft.

Students learn about the meaning of π around 7th grade and I think that a fine application of π is to not just find the circumference of a circle with a known radius, but to find the length of an ‘arc,’ which is a portion of the circumference of a circle, maybe when the circle is cut into equal slices.  Something like:  If the diameter of this circle is 10 cm, what is the exact, and approximate, length of the red arc, AB?

The solution is to work out the circumference of the entire circle and then divide by 5, so arc AB is π*10 cm / 5 = 2π cm, which is approximately 6.28 cm.

I also like the idea, once angles are introduce and that there are 360 degrees in a circle, for students to work out a question like:  If the diameter of this circle is 10 cm and angle AOB is 40 degrees, what is the length of arc AB?

The solution is to notice that since the angle is 40 degrees and since 360/40=9, the red arc is 1/9 of the entire circumference so the length of arc AB is π*10 cm / 9 = 10 π / 9 cm, which is approximately 3.49 cm.

I also like the converse of this problem:  If the diameter of this circle is 12 cm and the length of the arc is exactly 2π cm, what is the measure of angle AOB?

Solution:  Since the whole circle has circumference π*12 cm=12π cm, and since 2π is 1/6 of 12π, then arc AB is 1/6 of the whole circumference, which means that angle AOB is 1/6 of the 360 degree circle, which is 60 degrees.

All three of these questions I consider to be nice ‘thought provoking’ applications of the concepts of circumference, arcs, angles, and ratios.  It is even possible to create a formula which relates the angle, the diameter, and the length of the arc:

which is one of the more intuitive non-trivial formulas in math, and one that gets my approval.  Students, though, as early as 7th grade, I think, could master the three questions I posed above, even without ever resorting to using a formula, and I’d be fine with them reasoning it out without resorting to a formula, for sure.

The ‘degree’ is also a very important concept in math, measuring the size of an angle, how wide apart the two ‘legs’ of the angle are.  Even adults who consider themselves to be bad at math have a good feel for what it means to make a 90 degree turn, knows that a ’360′ is when a basketball player, or skateboarder, spins completely around, knows that to turn 180 degrees is to turn around to face the opposite direction, and can probably estimate what a 30 degree or a 60 degree turn would be.

Mathematically, a degree can be thought of as the angle formed by the slice which is 1/360 of the entire circle.

Making the degree 1/360 of a circle was a man-made decision thousands of years ago.  Some say they picked 360 since is was close to 365, the number of days in a year.  Some say that 360 is a nice number since so many other numbers divide into it evenly.  Either way, it has become a unit of measurement that humans have learned to relate to.

Now just as there are other units, besides feet, for measuring length, like inches, meters, and centimeters, there are other units, besides degrees, for measuring angles.  For example, there is something called a ‘gradian’ which is slightly smaller than a degree so that 400 gradians fit in a circle and a right angle measures exactly 100 gradians.  Though there is still a ‘grad’ button on most scientific calculators, this unit of measurement is pretty obscure and no longer taught.  Another unit for measuring angles is much larger than a degree, and is the subject of this post.

Officially, a ‘radian’ is the angle in a circle which would make the red arc exactly the same length (if you were to straighten the arc out) as the radius of the circle.

Angle AOC is pretty close to, though a bit less than, 60 degrees.  It actually turns out to be approximately 57.295779513082320876798154814105170332405472466564321 degrees, which is actually exactly 180/π degrees.

Angles that are a nice whole number of radians need infinitely many decimal places to express in degrees, and angles that are a nice whole number of degrees need infinitely many decimal places to express in radians.  For example, 90 degrees, one of the most intuitive ideas (after you learn about what a degree is) about angles, is about 1.6 radians, or more accurately, 1.5707963267948966192313216916397514420985846996875529 radians.  This large decimal representation can be described exactly as π/2 radians (divide 3.14/2 to get 1.57 for the approximation)

Students are told that radians are very important and then taught how to convert radians to degrees and degrees to radians (or shown the button on the TI calculator that does the conversion for them), and eventually they come up with, or are shown or told, that the most popular angles, 30 degrees, 45 degrees, 60 degrees, 90 degrees, 180 degrees, and 360 degrees, can be written, respectively as π/6 radians, π/4 radians, π/3 radians, π/2 radians, π radians, and 2π radians.  For angles other than these six, there are ways to convert back or forth between the two units of measurement.

Here are some examples from recent Algebra II New York State Regents Exams:

But converting from degrees to radians or radians to degrees is just one of the three ways that radians are included on the New York Regents.

The second way that radians appear is in graphing the trig functions.  Students, after learning about sine, cosine, and the rest, learn to graph the functions f(x)=sin(x), f(x)=cos(x), etc. along with transformations of these graphs.

Students have enough trouble keeping track of this topic when working with ‘intuitive’ degrees to produce graphs like:

But teachers tell students that radians are very important so they must learn this topic and make sure that their x-axis is in radians to produce a graph like this:

When students have to do more complicated graphs with different shifts and compressions, it becomes, for them, like trying to read Ulysses, but in the Spanish translation.  What an unnecessary mess.  Here are some questions from recent New York Regents exams:

Students who might have gotten through this topic, even liked it, are alienated and discouraged by this requirement to ‘think’ in radians.  I truly think that this topic, particularly this application of it has discouraged many students from taking any more math once it was no longer required of them.

I should mention that this last question is the most practical and interesting of the bunch.  As radians are something I have to teach when I teach Algebra II, I do teach my students how to answer questions like these.  But still, the effort that goes into making students ready for this type of application is way more than this topic is worth.

The final, and most pointless, use of radians is something that is clearly described in the common core standards for high school Geometry under circles:

CCSS.Math.Content.HSG-C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality;  derive the formula for the area of a sector.

What this means is that if a one radian angle intercepts an arc equal in length to the radius, then a two radian angle will intercept an arc equal in length to twice the radius and, in general, an angle measuring n radians will intercept an arc equal in length to n times the radius.

In short, we get a new formula to use in place of the easy to understand formula relating arc length, angle, and diameter, shown at the beginning of the post.

Now, as long as we have the angle in radians, we can use the formula:

Well, it is a shorter formula, I give them that.  And it does enable us to apply it by using some very contrived questions like:  If the radius of a circle is 10 cm, and the length of the arc is 2 cm, what is the measure of the central angle in radians?  (Answer:  .2)  But this is a lot of unnecessary and uninspiring work to get a new formula for something that was really a nice and easy to derive formula already.  Also, they mention the area of a sector which has a very simple intuitive expression for degrees, but a non-intuitive, and non-necessary, one when the angle is in radians.

Here are a few questions on this topic from the old Math B Regents:

Now, to give you an extra reason to trust my sincerity that this contrived use of radians is not worthy of inclusion in the math curriculum before precalculus, I want to share a quick anecdote.  Back when I was teaching in Houston in the early 1990s, the late great Astronomer and TV star Carl Sagan was in town signing his latest book.  I was too cheap to buy the new hardcover book, but I did pick up an old copy of Cosmos for him to sign.  When it was my turn, Dr. Sagan asked if he should write anything special.  I said that I was a math teacher, so maybe he could write some math formula.  Of all the formulas he could have chosen, he chose this one:

This really disappointed me since I was hoping for something more profound.  Perhaps if I bought the expensive book he would have given me something more inspirational.  Still, if anything could make me like this formula, even just for the sake of showing my students my autographed book, this would do it — but it didn’t.

So now of course some math teachers will be thinking that I just don’t understand how important radians are which would make me understand why students in Algebra II need to be ‘exposed’ to it so they will be ‘ready’ when it is time to really use them in Calculus.  Well, I have no problem with this topic being introduced in Precalculus (which is 12th grade for most students, though most stop taking math before that) or even in Calculus, just before it is needed.  But for those teachers who think that I misunderstand the significance of radians — that sin(x)/x tends to 1 when x is in radians, which allows us to prove some important theorems, I want to let you know that I’ve studied the works of the masters, like the 18th century genius Euler.  And when Euler says something like that sin(π/6)=1/2, he is not thinking of that π/6 as an angle in radians, but as an arc on the circle, which is pretty intuitive.  He even says the sine of a π/6 arc is 1/2.  This also is related to why we would say that arcsine of 1/2 is π/6.  Literally, the arc that corresponds to that sine has the length π/6.

This concept of taking the sine of an arc, rather than the since of an angle is very useful in understanding why, for example, sin(x)/x tends to 1 as x approaches 0.  It really is a geometric idea relating the blue line segment to the red arc and has nothing to do with whether you measure the angle in degrees, radians, or even gradians.

Anyway, that’s why radians are on my list of things that I’d cut from the three year sequence in high school math.  The common core ‘experts’ failed to do this, so we can expect students to continue to like math that much less for the next ten years or so.

12 Responses

  1. John Templeton

    Respectfully, you don’t seem to understand what radians are. And, by the way, pi is MUCH more than the number of diameters that can wrap around a circle. I have to say that I’m flabbergasted at what you’ve written…

  2. John Templeton

    Respectfully, you don’t seem to understand what radians are. And, by the way, pi is MUCH more than the number of diameters that can wrap around a circle. I have to say that I’m flabbergasted at what you’ve written.

    • Gary Rubinstein

      Please be more specific so I can understand. What do you think radians are? And what do you think pi is aside from the ratio of the circumference to the diameter of any circle?

    • Pi might have a 1001 magical uses (to steal an old TV pitch line), but the definition of pi is the ratio of the circumference to the diameter, which in laymen’s terms could be described as the number of diameters that can wrap around a circle. Poetic license, and all that.

  3. Steve M

    Gary, I agree with you on this one, but would add that students have to hammered with the concept when in Pre-Calculus/Trigonometry. I get too many students in my physics classes that have not been exposed to it by the end of the first semester of their trig/pre-calc class.

    Regarding John Templeton’s comment, above: Huh?

  4. Michael

    I’ve gone back and read your initial post, as well as parts 1 and 2, and I completely agree with you. I studied math in college; it was always incredibly easy for me. The first time math became difficult was when I took graduate level abstract algebra. I have tutored and taught math to several students, so I have observed topics and areas that drive students crazy.

    I agree that some topics should be delayed to elective classes. Since entering the financial industry, I have not encountered the concept of radians except in my reading for enjoyment. I would wager that only adults who studied in depth like we did would be able to explain radians (or pi) accurately to another person. Perhaps someone who has recently helped his or her child with homework could do an adequate job, but unless someone is in a math, engineering, architecture, or similar field then radians are not necessary.

    The biggest focus really needs to go into topics that adults DO encounter on a daily basis and simply fail due to a lack of mathematical knowledge.

    Lottery tickets are just a tax on people who are poor at math. Even when people think that when the jackpot gets so high that there is an expected payout, they are actually wrong when you take into account taxes, reduced payout due to lump sum over annuity, multiple winners due to multiple lottery tickets being sold as jackpots climb, etc. And yet some people actually think of buying lottery tickets as retirement planning!

    Students need to learn about compounded interest and discounted cash flows. Since I’ve entered the finance profession after studying mathematics, it just blows my mind that something so important is just glossed over in school. Nearly everyone plans on owning a home, saving for retirement, getting a credit card, and opening a bank account. Understanding interest on student loans should be on the mind of a lot of students in school as well. Compounded interest and discounted cash flows are VITALLY important, and I think they are thought provoking and beautiful.

    Thank you for the post Gary. I hope that change can happen soon. I’m impressed with your ideas.

  5. EH

    /typo:”rather than the since of an angle” in the 2nd to last para. should be: “rather than the sine of an angle”

    Also, that paragraph and the 3rd to last paragraph are rather confusing, I cant tell what you meant to convey. The sin(x)/x -> 1 is much more intuitively understood by using the small angle approximation (usually taught only in physics) that sin(x) ~= x, which is a better approximation the smaller the angle, and is perfectly accurate for sin(0). So sin(x)/x = x/x at x=0. Using your diagram, one can see how the small angle approximation works, but it needs to be pointed out explicitly.

  6. Terence

    My question in response to this article is simple: What is the rule for differentiating sine and cosine when they are defined in degrees and students should be assumed to not have mastered the use of radians in more elementary settings?
    I feel this is a much more significant pedagogical problem than the behavior of (sin x)/x at 0, but you have not addressed this issue.

  7. Guy Brandenburg

    Zalman Usiskin wrote an article in The Math Teacher (NCTM monthly) about 30 years ago suggesting several math topics that could be removed from the curriculum.
    http://www.education.umd.edu/MathEd/conference/vbook/What%20should%20not%20be%20in_Usiskin.pdf
    or
    http://bit.ly/1aeQf71

  8. eddie sacrobosco

    It took me a little while to realize that you’re advocating not teaching radians to everyone as opposed to not teaching radians to anyone! Not a bad idea – what about functions? I think that the function concept is useful for those going on in math, but for the casual student, it seems a waste of time.

  9. Gary,

    Have you seen this piece by Liping Ma on the organization of elementary school math?

    http://www.ams.org/notices/201310/fea-ma.pdf

    It seems to be in line with your thinking here, particularly regarding the dis-utility of pushing advanced concepts down to earlier grades in hopes of making things easier later on.

  10. J Choi

    The only concept we teach about pi is c = pi*D. And then we stop. So of course students have no clue what radian measure is. If we take a few minutes after teaching them this simple fact, to show them either the angle measure or arc length of 2* pi, then 1 * pi, then ½ * pi, the meaning becomes clear. I have seen elementary students grasp this. Lastly, it is the responsibility of the educators to use it, so that the students actually can see for themselves how it is used, and that it is important. If instead we treat it as a throw away fact, then of course they are going to loathe it, not learn it, and consider it useless. They are following our lead. Oddly, it is the 360 degree system that was arbitrarily chosen. The use of pi is actually the natural way to describe arc length. Also, the beauty of using pi is lost as soon as one converts to decimals. Therefore, don’t convert to decimals! Leave it as the irrational symbol, then the meaning and beauty are quite clear.

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By a somewhat frustrated 1991 alum

Region
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