Fwd: another limitation in our curriculum

[michel paul]

While exploring Mathematica I noticed that the statement 180 Degrees == Pi evaluated as True. It did not require appending some kind of label indicating 'radians' to evaluate as True. It was True as given.

I found that interesting. There was a time when I myself would have considered the statement False, as needing some kind of 'radians' indication. However, the wonderful world of computational thinking is a great antidote for that kind of schoolishness.

It turns out that Mathematica interprets the degree symbol and the term 'Degree' as pi/180. So when you enter something like '39 Degrees', Mathematica understands it as 39/180 pi. When I saw that I thought it was very clever. I then checked GeoGebra, and it seems that's also how they think of it.

I decided to test both my students and the math dept. Yep, the overwhelming response is that the statement 180 Degrees == Pi is False, that it requires some kind of pointer to 'radian measure'. That's the understanding that our curriculum has produced in both students and teachers.

I like to point out in discussion that the statement 180 Degrees == Pi Radians also evaluates as True. 

What that means is that the term 'radians' is redundant, and that's real mathematical understanding. It is also real computational understanding.

Calculators require students and teachers to fiddle with DEGREE vs. RADIAN modes. That kind of thinking causes a whole lot of intellectual damage, and it should be stopped.

- Michel

---------- Forwarded message ----------

From: michel paul <python...@gmail.com>
Date: Thu, Feb 20, 2014 at 1:48 PM
Subject: another limitation in our curriculum
To: hsmath 

Hi everyone - I've already bugged a few people with this, but I think it's pretty interesting and worth sharing with the whole department.

Quick! True or False: 180 degrees = pi. 

What do you think? Is that statement true, or not? And why do think that?

(answer below)

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Answer - the statement is true, as given. You might think you need to append the term 'radians', but no! The statement is true as is.

Consider - radian measure simply states how many radii measure an arc. If the radius length is measured in cm, then the arc length will also be in cm. When we compute radian measure as arc length/radius, the cm units cancel out. Radian measure is dimensionless! That's an interesting fact, and it is a fact, not just my opinion.

So a degree is really just 1/360 th of the circumference of a unit circle.

I love learning new things, and I'm continually astonished by how things we think we already know can be improved upon. We teachers all grew up in a kind of limited curriculum, and a lot of these interesting limitations show up when using something like Mathematica.

--

Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

--

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[Linda Fahlberg-Stojanovska]

About 20 years ago, I began to think (and teach) that getting your units correct in any problem is the single most important factor in problem solving. My kids understand mixture problems when I force them to write units; same for work problems which are also notoriously difficult to understand.

Considering an angle  180 Degrees == Pi (without units) was a vital component of this emphasis on units. How do you evaluate f(x)=x sin(x) ? Then (as a few of you here might remember) i went to the other extreme  and said we should get rid of "degrees" since THEY are incorrectly creating a unit on a unitless object. (A number line doesn't have units, why should a number circle, a.k.a. and badly labeled a "unit" circle in English?) Of course degrees are so entrenched in our heads and also help us to realize that the we are working circularly (on a number circle) that this is totally impractical. But it is not ridiculous and I think reflects Michel's end thoughts.

A little bit more about thinking about units....

interestingly to me at this moment is that I am working through Maria D notes on working with young children and she mentions that children start to have difficulty in school around 3rd grade because they do not know enough about unitizing and exponentiating. i had no idea what these words meant (and made no connection at all between the word unitizing and units because of the different contexts). They are still not completely clear to me, but 

After practicing a bit with different manipulatives that I was making, I am beginning to come to the conclusion that denominators are units. 

probably many others have thought about this, but it was new to me.

I had cut a square and then two half squares and the two halves of a half square and trying to explain to myself that the two halves of a half add up to a whole, but a whole what? a whole half.

1inch+3inch=4inch

1half+3half=4half

and in a way:

1inch x3inch= 3 (inch squared)

1half x 3half= 3 (half squared)=3 fourth

on and on i go as usual :)

thanks Michel for the interesting thread

Sent from my iPad

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[michel paul]

On Thu, Feb 20, 2014 at 8:25 PM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:

 

I am beginning to come to the conclusion that denominators are units. 

Absolutely!

Yes, that's how I've been thinking about things.

We usually think of the rational numbers as the results of divisions of integers, but it's interesting to think of them as products of the natural numbers and unit fractions. I actually think that's a better way to do it.

There's a whole lot of things that come together when you start to think that way.

--

Michel

[Christian Baune]

Hi,

sometimes using a "fake" unit can help.

When I have to deal with some students who have troubles in geometry, I use the "U" unit. ("U" for unit)

Without proper units we have students writing :
"The side of square is 4 therefore area is 16".
Then they are asked something like the length of the diagonal and write stuff like "16-4=12" mixing two kind of units...But they aren't aware of it!

Rewriting with the "U" unit give them a clue about their weird reasoning.

Areas are often introduced using metric units, that's not good because as soon you remove units they don't think about a default one.(number dimension)

So the "U" unit should be used more often!

Kind regards,
Christian

--

[Linda Fahlberg-Stojanovska]

sometimes using a "fake" unit can help.

When I have to deal with some students who have troubles in geometry, I use the "U" unit. ("U" for unit)

what an excellent idea!

[kirby urner]

I've been thinking a lot about units lately in regard to Newtonian concepts and thought it maybe topical enough to run through it here quickly:

Imagine a VW bug (aka Beatle, a car) moving for distance delta d.  It has mass and velocity.  mv for d = mvd.

Now think of a movie, which is frame frame frame frame... each one representing "per time" or "in a time".

In the movie biz we say "Lights.. Camera... Action..." i.e. what those frames capture is action.

Our everyday experience is like a movie in that we can slice it into frames -- we can film it.

OK, here's where the units + Newtonian concepts come in.

All of these are in units:

m = mass

t = time

d = distance

v = velocity i.e. d/t

mvd = action  (units of action:  mvd)

frame = action/time (energy bucket)

frequency (f) = 1/time ("per a time interval")

E = h f  (h is units of action) = mvd f = mvd/t = mvv = E

h = Planck's Constant (a minimum action)

So I'm getting us from "a film as frames of action" to "frames going by" (per a time)
as "energy buckets" or "energy quanta".  This is purely in terms of units.

Not claiming any new discoveries or anything, just saying here's a neat way to package up Newtonian mechanics with a "movie" as a model, keying of existing memes e.g. "Lights... Camera... Action!"

Also:

E/t = power

Like when they talk of a power station, it's not just how much energy it "makes" (really converts) but how much in what time interval.  Watts and Horsepower are E/t measures. 

Because as you know, given the right gears, you can manually winch a grand piano up 30 stories from the street level (to a penthouse apartment say), but with more power (a crane probably), you'll get it there a lot faster.

When a movie runs "too fast" the energy buckets go by at an "unrealistic" rate and we see everything with more power than it should have -- not our world, not our physics.

Kirby

[Joseph Austin]

The measure of an angle is defined as the ratio of the length of its arc to its radius.

So a ratio: length/length is a "pure" number, as long as both lengths are measured in the same units.

Of course, if the radius is taken to be the unit of measure of both,

then the magnitude of the ratio is equal to the length of the arc in units of the radius.

So there is potential confusion whether we are speaking of the length of an arc 

or the RATIO of the length of the arc to the length of the radius.

According to wikipedia, the term "radian" formally refers to the ratio, not to the arc length.

(There is a slight problem in that we know that the "length" of the circumference of a circle and its radius form an irrational pair.)

But what if we measured in centimeters or inches?  Couldn't we then call the angle so many centimeterans or so many inchans?

Of course, the number would be the same in each case.

Degrees, on the other hand, is a measure of arc in units of  1/360 circumference.

Again, it is a pure number, a ratio length/length, but it's not the ratio of the SAME lengths.

So the difference is not really a difference in units, but in geometry.  

One is a measure of ratio of arc to radius, the other a measure of ratio of arc to circumference.

The two measures can be reconciled by an appeal to the known geometric ratio of circumference to radius.

Moral of the story:  units are important, but WHAT you are measuring is also important.

So the words "radians" and "degrees" are not so much names of UNITS of measure as PROCESSES of measurement.

BTW, on the subject of circles:

A formula for the AREA of a circle is:  1/2 the circumference  *  1/2 the diameter.

What? No pi?  

I customarily illustrate this by arranging sectors of a slice of orange in alternating directions,

making a crude wavy parallelogram, then ask the student to imagine the limit as we divide the sectors successively into smaller and smaller wedges.

Joe Austin

[michel paul]

On Sat, Feb 22, 2014 at 2:31 PM, Christian Baune <progr...@gmail.com> wrote:

Hi,

sometimes using a "fake" unit can help.

True, and I think it is good for students to do more reasoning in terms of quantities, where a quantity is understood as a number of units. It has a whole lot to do with how we want them to reason in algebra.

Simultaneously, the fact that the equation 180 ° == π is True, as is,

​without having to append 'radians' to the right hand side, ​

seems to surprise both math teachers and students, and digging into why it is true turns out to be a good lesson in

​ ​

how radians and degrees are both fractions of turns and why the degree unit is not just decoration.

Some argue that the reason Mathematica and GeoGebra evaluate the statement as True is

​ ​

because they

​simply ​

chose to define it that way for whatever geeky reason. My response has been no, the reason they evaluate it that way is because it is mathematically so.

I ran the following argument by my students, and they said that thinking of an angle as a fraction of a turn,

​or ​

revolution, was helpful. For some reason that doesn't get emphasized. The concept that gets developed is an angle as a quantity of degrees or measure of radians, but in neither case is the concept 'fraction of a turn' emphasized. In the average understanding of 'pi radians', the 'pi' is not seen as intrinsically related to rotation. The 'pi' is seen as a measurement of some annoying unit called 'radians' that we have to convert degrees into when we get into more advanced math.

​​

Here's the argument:

An angle is a fraction of a turn.

A degree is 1/360th of a turn.

1 degree == 1/360 tau.

180 degrees == 180 (1/360 tau) == 1/2 tau.

So far we're only talking about amount of rotation, not arc length.

An arc is the product of a radius a​nd a

​ ​

turn or​

​ ​

fraction of a turn​. 

(This is

​ ​

a good example of multiplication that is not repeated addition.)

An arc ​formed from an angle​

​ ​

of one degree has an arc length of 

​(​

1 degree

​)​

*radius == (tau/360)*radius.

The distance traveled in one rotation is 

(360 degrees)*radius == (360 tau/360)*radius == tau*radius > 6*radius.

We can specify this limit, because we can inscribe a hexagon in a circle.

Therefore we can specify that tau > 6, and 1/2 tau > 3.

Notice that there has been no mention of pi​, and there has been no definition of radian measure. 

It turns out that measuring an angle as a fraction of tau is already ​equivalent to radian measure.​

​ ​

That's kind of interesting to find out.​

If we wish to define

​ ​

pi​

​ ​

as circumference/diameter, then 

pi == circumference/diameter == (tau*r)/(2*r) == 1/2 tau.

Therefore, 180 degrees == 180 (tau/360) == 1/2 tau == pi > 3, 

and further analysis can specify greater accuracy.

The point is, though it seems counter-intuitive, the equation 180 degrees == pi is true as given and requires no addition of 'radians' to the right side.

If anything, the addition of 'radius' to both sides establishes an equation of arc length:

(180 degrees) radius == pi radius.

[Christian Baune]

Well, for me "180 degrees == pi" is relating angle to length so it can be rewritten as "180 degrees == pi U" (pi Units)

If you decide that "V" is an unit which is the half of "U" (2V=1U), then in terms of "V" you should rewrite as :

"180 degrees == 2pi V". That could confuse some people since "2pi" is a turn but those are the ones who should use unit more often than never :-)

Also such equality are to take with caution, here I assumed that what stand on the right side of "==" is the distance traveled. Not a kind of modulus.

Kind regards,

Christian

--

[michel paul]

On Sun, Mar 2, 2014 at 10:30 AM, Christian Baune <progr...@gmail.com> wrote:

Well, for me "180 degrees == pi" is relating angle to length so it can be rewritten as "180 degrees == pi U" (pi Units)

That's precisely what I've come to see as the issue. I do not see the equation '180 degrees == pi' as relating angle to length but as equating either two fractions of a turn or two arc lengths on the unit circle. 

I also used to think the right hand side of the equation required some kind of unit to make it complete, but I now see both 'degree' and 'pi' as measures already incorporating the same unit. 

The key is to start with 1 rotation, 1 turn, as fundamental and to then define angles as fractions of a turn. 

Imagine a ray starting from (0,0) and extending without bound in some direction. We could describe the ray as being positioned at 1/2 of a turn, 1/3 of a turn, 1/4 of a turn, 1/6 of a turn etc. in relation to initial position without regard to any radius or arc length. This kind of reasoning is in terms of pure rotation. 

If we let 'one full turn', 'tau', serve as our unit, then a degree is 1/360 tau. Both 'degree' and 'pi' turn out to be measures of an amount of turn. In a sense, 'pi' already contains its unit, 'tau', and we can show this in 'pi == 1/2 tau'.

Instead of saying that the right hand side requires a unit, we could say let's multiply both sides by a radius:

(180 degrees) radius == pi radius

and we get an equation where both the left and right hand sides express arc length.

If we use a unit circle, then radius == 1, and we are back to '180 degrees == pi'.

I assumed that what stand on the right side of "==" is the distance traveled. Not a kind of modulus.

Right, not a kind of modulus. 

I have found this very interesting. Substituting 'tau/360' for every occurrence of 'degree' and 'tau/2' for every occurrence of 'pi' makes lots things fit together surprisingly well.

--

Michel

[Christian Baune]

Hi,

It can be seen a arc length too. That's why I introduced the "U" unit. One turn is simply "2pi r". 

In that case : 360[°]=2*pi*r [U]

When working in unit circle, you can left out "r" and write : 360[°]=2*pi [U]

This to stay coherent with the folowing :

A 4[U]*4[U] rectangle has an area of 16[U²] but a perimeter of 16[U]. The numeric value is the same but the dimension is not.

What it does tels us is that substracting Perimeter (U) from area (U²) is something wrong even if "16-16=0".

Same for circle, you need to know units and dimensions to know what is compatible and convert if possible. (Eg. like litters and dm³)

Adimentional values are rare but often encountered in ratios where units cancels each others : 1[cm]/3[cm] = 1:3

So, one can write "360°=2*pi" but must remember the dismissed unit.(Since the "unit" contains the dimention)

Kind regards,

Christian

[michel paul]

On Mon, Mar 3, 2014 at 3:48 AM, Christian Baune <progr...@gmail.com> wrote:

It can be seen a arc length too.

Right. '360 degrees' and '2 pi' are equivalent lengths on the unit circle.

Therefore, 1 degree == pi/180, with nothing else added, which is how Mathematica thinks about it, and that's what originally sent me on this little trek.

Another way to ask this --> can we say that 1 degree == pi/180, or must we specify that 1 degree == pi/180 radians?

Wolfram evaluates 1 degree == pi/180 as True. It also evaluates 1 degree == pi/180 radians as True.

However, pure Mathematica evaluates '1 Degree == Pi/180' as True and leaves '1 Degree == Pi/180 Radians' in symbolic form.

That's because 'Radians' doesn't exist as a unit in Mathematica.

So, one can write

"360°=2*pi" but must remember the dismissed unit.(Since the "unit" contains the dimention)

Not only can we write

"360°=2*pi", we can also assert that it is True!  : )

I do appreciate and agree that for pedagogical reasons it can be useful to include 'radians' in our expressions at a certain stage, but mathematically it turns out to be redundant, and I find it liberating to become clear about why.

Again, as I see it, the fundamental angular unit is 'one turn', and thinking about angles as fractions of this unit nicely unifies degree and radian expressions.

Sincerely,

[Christian Baune]

This: "720°-4π=0" is kind of ambiguous.

One can say that 2kπ=0°.(When k is a whole number)

In can be true when you work in terms of "turns", after all when you do a turn from any position, you land at the same place. (Modulus arithmetic)

And trigonometric functions enforce that idea. "cos(2kπ)==cos(0)" after all.

If you work with distances, 720° = 4π for the unit circle.
If you work with angles, the whole expression should be mapped back to proper ranges. [0;36] and [0;2π].
Because it would make as much sense than writing "4/8".

When you use Radian, you make clear that you are using an angular unit. When you left out then the expression should be treated has being a distance.

When using the angular measure,
modulus arithmetic apply and expressions should be reduced.
When using the length measure,
modulus arithmetic does not apply and therefore expression must not be reduced.

When working in ranges [0;360] and [0;2π] you may not care about whether you work with angles ("Radian") or length unit.("U")
But out of it, it becomes important.

What does Matgematica says about : "180°==3π Radian" and "180°==3π"?

Kind regards,
Christian

--

[michel paul]

On Mon, Mar 3, 2014 at 10:13 PM, Christian Baune <progr...@gmail.com> wrote:

This: "720°-4π=0" is kind of ambiguous.

That statement evaluates as True, because both expressions represent the same amount of rotation.

One can say that 2kπ=0°.(When k is a whole number)

No, that statement only evaluates as True when k == 0. For example,

In[27]:= Table[2 Pi k == 0 Degree, {k, 0, 6}]

Out[27]= {True, False, Fals

e, False, False, False, False}

This statement will always evaluate as True:  2 Pi k == 360 Degree k

In[34]:= Table[2 Pi k == 360 Degree k, {k, 0, 6}]

Out[34]= {True, True, True, True, True, True, True}

In can be true when you work in terms of "turns", after all when you do a turn from any position, you land at the same place. (Modulus arithmetic)

That's the important issue in all of this - equality in terms of position vs. equality in terms of rotational distance.

And trigonometric functions enforce that idea. "cos(2kπ)==cos(0)" after all.

Right, the cosine value is a periodic function, so the same values recur at certain rotational distances.

In fact, this is a good way to analyze the whole situation - move from the circle to the sinusoidal graph. 

The x-axis represents rotational distance. Sine and cosine values will recur periodically, but the rotational distance continues to increase as you move to the right. 

If you work with distances, 720° = 4π for the unit circle.
If you work with angles, the whole expression should be mapped back to proper ranges. [0;36] and [0;2π].

I think it depends on the situation. It makes sense to talk about rotations that are greater than 360 degrees. They will be co-terminal with with angles less than one turn, yes, but I don't see a problem talking about angles of more than one turn.

When working in ranges [0;360] and [0;2π] you may not care about whether you work with angles ("Radian") or length unit.("U")

But out of it, it becomes important.

I'm not sure about that. Again, I think it can make sense to talk of an angle consisting of 3 half-turns.

What does Matgematica says about : "

180°==3π Radian" and "180°==3π"?

Wolfram evaluates both as False. 

Mathematica evaluates the second as False and the first remains in symbolic form.

However, the statement 3*180 Degree == 3 Pi evaluates as True.

Thank you for raising these points. It has been a good exercise to follow through and check these out.

Sincerely,

 

 

-

- 

Michel

[Christian Baune]

Did you notice that ° seems to be a distance measure in your explanations ?

[michel paul]

On Tue, Mar 4, 2014 at 12:22 PM, Christian Baune <progr...@gmail.com> wrote:

Did you notice that ° seems to be a distance measure in your explanations ?

 

No, not really. When referring to the unit circle, perhaps it reads that way, but my intention is to keep returning to the fundamental idea of an angle as a fraction of a turn prior to any concern for radius or arc length. 

The more I dig into it, the more value I see in the concept of tau as simply 'one turn' with nothing else added. I don't define tau as 2 pi, I begin by defining it as 'one turn'. This 'one turn' can be experienced by standing in one spot and spinning around. We can talk about 1/2 turn, 1/4 turn, etc.

We can then define a degree as 1/360 of a turn, again, without regard to radius or arc length.

We can easily see that 360 degrees == 1 full turn, but this is not yet associated with any value like 6.28, and 1/2 turn does not yet appear to have any natural connection to 3.14. So where do these values come from?

If we rotate one end of a radius by a degree we will produce an arc length. If we rotate a radius one full turn it is easy to show, by inscribing a hexagon, that it will result in an arc length that is a little more than 6 times the radius. 

Therefore, the arc length for one turn of radius r will be tau*r > 6*r, and a delightful consequence of this is that tau > 6.

The reason this is delightful is that this 6 is not a length, it is a number of sides, and therefore tau is also not a length. Tau*r is a length, but when we divide by r, we're left with just an amount of turn.

The traditional definition of pi as circumference/diameter gives us pi == C/d == (tau*r)/(2*r) == tau/2 > 3. Again, these values are not lengths. Pi is 1/2 of a turn, and its value is a little more than 3.

So the equation pi == 180 degrees can be understood in terms of either measure of rotation or arc length on a unit circle.

- Michel  

--

[Bradford Hansen-Smith]

The 60 degree rotation of a radiant has a given distance the end point has traveled on the line of a unit circle. The straight line distance measure between the start and stop point has little to do with the degree of rotation and more with the direct distance between two points. The difference between a curved line and a straight line is designated in form by the relationship of circumference to diameter. There is a fundamental difference between a circle and any polygon. There are other ways to show this pi relationship without relying on the center of the circle or using the hexagon.

Fold a circle in half. Start at one corner point and bring the folded straight edge and the circumference together curving until the curved straight edge fold runs out and what is left of the circumference is the difference between the diameter and the unit circle.

Brad

Bradford Hansen-Smith
www.wholemovement.com

[Christian Baune]

180°=π if on unit circle.

180°=π Radian

The "Radian" unit tels us that we work on a unit circle.

Another example :
a)x°=y
VS
b)x°=y Radian

In "a" the domain of y has to be defined as "y must be expressed as an arc length on unit circle ".

In "b" it is clear due to the "unit".

In a, I could answer "y is x°" in b I would have to say "y is x(°/Radian)".

KR,
Christian

[michel paul]

On Tue, Mar 4, 2014 at 10:57 PM, Christian Baune <progr...@gmail.com> wrote:

> a)x°=y

>b)x°=y Radian

> In "a" the domain of y has to be defined as "y must be expressed as an arc length on unit circle ".

Are you sure? Is it actually the case that this domain must be expressed in this way? 

How, and why, do we need to distinguish this domain from the real numbers?

Every real number already corresponds to an arc length on the unit circle. That's the meaning of the 'wrapping function'. 

There is no definable difference between 'π' on the real number line vs. 'π radians'. 

The domain for sin(x) is simply the set of all real numbers. We don't specify that x must be in radians, because, as a real number, it already is.

> In "b" it is clear due to the "unit".

Mathematically there is no difference between expressions a) and b).

Please note: "As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted."

The use of 'Radian' might serve a role in communication, because people are used to thinking about it that way, but mathematically speaking it turns out that there is no difference.

Also from the article: 

"The unit was formerly an SI supplementary unit, but this category was abolished in 1995...". 

That's interesting. It appears that thinking on this has been evolving, and our curriculum is not always informed by these things. That might have something to do with why all teachers I've asked, including myself, have been under the impression that '180°=π' requires 'radians' in there to evaluate as true. 

> 180°=π if on unit circle.

> 180°=π Radian

No, 180°=π, just as it is. Both sides of the equation represent 1/2 turn.

Sincerely,

- Michel

- Dr. Christos Papadimitriou
===================================

[michel paul]

> A formula for the AREA of a circle is:  1/2 the circumference  *  1/2 the diameter.

Yes!

I usually use 1/2 circumference * radius. Justification/visualization is similar to the rectangular model but uses triangles and shearing: 

• Imagine a circle divided into a potentially infinite number of sectors.
  
• Unwind the circle and its sectors to create a sequence of triangles whose length equals the circumference.
• Shear the triangles, collecting each of the top vertices at one boundary vertex, forming a right triangle.
• The area of this triangle is 1/2 circumference*radius.

So circular_area = 1/2 circumference*radius. And sector_area = 1/2 arc_length*radius. 

Letting tau = 1 turn, circumference = tau*radius.

Letting theta = a fraction of a turn, arc_length = theta*radius.

So circular_area = 1/2 tau*radius^2, and sector_area = 1/2 theta*radius^2.

In this way, lots of things that often seem different get united very nicely.

- Michel

[kirby urner]

One of the namespaces I'm familiar with divides information
into "angle" (increments of "turn") and "frequency" (increments

of distance).  Distance is always on an arc i.e. even so-called
"perfectly straight lines" (rarely encountered) are segments
of a circle of radius infinity, one might say (by definition).

Angle and frequency lets you get from A to B:  turn and
go step step step (frequency = intervals) then turn again
by so-and-so amount (we could use degrees) and step,
turn Y, step X and so on, like the Logo turtle, and, finally,
B will be reached (unless you get lost, in which case
follow your GPS until it runs out of batteries :-D).

Angles are measured in "units of turn" but if you hold
out a radial as you turn, it scribes an arc i.e. line, of
some distance, i.e. frequency.  So yes: if we agree on
a fixed radial then turn may be expressed as steps on
that circle, just as degrees may be mapped to a distance
by a rule, which is the degree-radian equivalence we've
been talking about. 

This terminology is from one of the philosophies I read,
dates to 1975 or so in the published literature, more in

Popko's recent primer:  Divided Spheres (dividedspheres.com). 

My idea of a great math course would be Popko + Litvins
in terms of content, Maria and Gary Litvin being math teachers
in New England who leverage Python for its interactivity
and, lets face it, low cost (free once you have hardware).

Skylit Publishing.  Many of you already know of it, probably
own a copy (my copy seems to be out on permanent loan).

Popko's work has already had practical implications and

is just a stone's throw from geography / geodesy / cartography
in terms of the angle / frequency treatment it presents.  A
math class may appropriately focus on maps of all kinds
as a map is literally a "mapping" in the sense of a function,
in that an actual territory is deemed in correspondence with
map details, perhaps with some rules of projection applicable,
if going from a globe to a flat surface.  We might program
such rules of projection.

Speaking of Python and programming, an update on my
debate with peers on the Math Forum:  Oregon, where I
agitate for reforms, along with a number of other states,
has already passed laws which make it easier to get math
credit, of which three years worth of credits are required
to graduate high school, by studying more "computer stuff"
e.g. time with a programming language is not just time
squandered on an "elective" subject anymore (one that
does not count towards fulfilling a mandatory requirement). 

Students and teachers alike stand to benefit from these
changes to the law, provided curriculum writers don't suffer
from an Imagination Deficit and fail to rise to the occasion.

Groups such as this one reassure me that we have the
interest in talent in at least some circles i.e. there's

energy / creativity to be tapped, at least here.

Kirby