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?
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.
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.
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.