The Wrapping Function--Help!
Kerry Thompson
next week. Problem is, I don't know what the wrapping function is. I think
the curriculum was designed by somebody who has taught math for 40 years,
and is using out-of-date terminology. Can somebody point me in the right
direction?
The context is, the students are reviewing the algebra and geometry they
will need for trig--sets, functions, quadratic equations, and the like.
Next, we start in on the basics of trig--the unit circle, determining
lengths of arcs, coterminal angles, and so on. What has me stumped is the
following learning expectations:
To define a wrapping function
To determine the domain and range of the wrapping function
To label terminal points of given paths on the unit circle (ok, that's
easy, as long as I assume path means arc, and not in terms of walk, path,
and trail)
To identify symmetric points of terminal points of paths with respect to
a diameter of the unit circle
To prove that two points are symmetric with respect to a diameter of the
unit circle
To find the ordered pair of real numbers corresponding to the value of
the wrapping function given in terms of pi.
Then they go on to periodic functions, which are no problem for me.
However, none of my references cover the wrapping function. Can somebody
help? It would be most appreciated!
nos...@sorry.com
<kerry_t...@earthlink.net> wrote:
>I am supposed to teach the wrapping function to a high school trig class
>next week. Problem is, I don't know what the wrapping function is. I think
>the curriculum was designed by somebody who has taught math for 40 years,
>and is using out-of-date terminology. Can somebody point me in the right
>direction?
Hi,
Which board of education do you work for? It is ludicrous to ask you
to teach without providing a text (for the students) which also
contains a description of the material. It is also ludicrous to ask
you to teach what you yourself do not understand. It is also
ludicrous to expect you to learn that material sufficiently well in
the time remaining to be able to teach it effectively, provided that
you do it at all.
That aside, a general description:
The wrapping function is merely a name for any periodic function; one
with phases.
Wx) = W(x) +P
The trig functions are periodic
sin(theta) = sin(180-theta)
tan(theta) = tan(180+theta)
etc.
and can be evolved by rotating about a circle. So you are really
being asked to teach the "Trigonometric Funcions". Simple enough, but
too much to write the whole text here. However, you have a circle,
and coordinate axes through its center. Extend the horizontal axis
far to the right. Draw a vertical axis near the circle, to the
right. Define points on these axes. As you work around the circle
(wrap or unwrap) you project horizontal lines to the right and from
points on the horizontal axis draw vertical lines to meet these as you
proceded to wrap the circle. In other words, you are graphing the
Trig functions.
The truth is, a picture is worth a thousand words. You need a text
handy. The other truth is that someone thought this stuff up as an
alternate means to teach trigonometric functions; an attempt to get
some bright lark's name into the text books, or an alternate to sell
new textbooks (a great moneymaker.) So, you might just bear in mind
that this is what you are doing ultimately, and teach trigonometry *as
you see fit, and as you understand it* ...you ARE the teacher...and
damn the torpedoes. This wrapping function approach is passe...a
phase that ultimately failed. You might want to talk to the boss
first though, but if they are asking you to do this, under these
circumstances, in this time frame, good luck.
The ultimate answer again, is get the text that contains this material
and study it.
Sheldon Ackerman
>I am supposed to teach the wrapping function to a high school trig class
>next week. Problem is, I don't know what the wrapping function is. I think
>the curriculum was designed by somebody who has taught math for 40 years,
>and is using out-of-date terminology. Can somebody point me in the right
>direction?
>
school system. It is covered in Chapter 9 of the AMSCO Course III book.
Here is how it is more or less explained:
Imagine a vertical number line tangent to a unit circle so that the point 0
on the line coincides with the point (1,0) on the circle. Imagine being able
to wrap this line about the circle so that each point on the line will then
coincide with a point on the circle. Under this wrapping function, every
point on the real-number line will correspond to on and only one point on
the unit circle.
The above leads into the topic of graphing trignometric functions.
Kerry Thompson
why I'm having problems, and that's why I have resigned, effective Sept. 30.
Until then, though, I owe it to the students to do the best I can with the
materials I have been given.
It's a private web-based educational organization, and to protect the
guilty, I'll not name names. But let this be a warning to others--if
someone asks you to teach a course, or write lesson plans without a text,
they are really asking you to do the work of a textbook author, on the
cheap.
Thanks for your help, and explanation. I think I'll take your advice, and
teach the trig functions the way I think they should be taught. I
understand the wrapping function, and don't see any real advantage to
teaching it. If they see fit to fire me for insubordination before Sept.
30, well... :-)
-KT
>Hi,
>
>Which board of education do you work for? It is ludicrous to ask you
>to teach without providing a text (for the students) which also
>contains a description of the material. It is also ludicrous to ask
>you to teach what you yourself do not understand. It is also
>ludicrous to expect you to learn that material sufficiently well in
>the time remaining to be able to teach it effectively, provided that
>you do it at all.
>
>That aside, a general description:
>
>The wrapping function is merely a name for any periodic function; one
>with phases.
>
.
.
.
Brian
> To determine the domain and range of the wrapping function
> To label terminal points of given paths on the unit circle (ok, that's
>easy, as long as I assume path means arc, and not in terms of walk, path,
>and trail)
> To identify symmetric points of terminal points of paths with respect
to
>a diameter of the unit circle
> To prove that two points are symmetric with respect to a diameter of
the
>unit circle
> To find the ordered pair of real numbers corresponding to the value of
>the wrapping function given in terms of pi.
>
is a function which matches up a given point on the unit circle with a
number in the interval 0 to 2pi. Here's the way it is graphically
presented: First draw the unit circle and then draw the vertical line x =
1. Label the vertical line as you would number the real number line, with 0
corresponding to where the line intersects the x - axis, the negative real
line is that part of the vertical line which is below the x - axis, and the
positive part of the real line is above the x - axis. Now, you can "wrap"
the vertical line around the unit circle and the positive part of the real
line corresponds to those points on the unit circle that subtend a positive
angle, and the negative real numbers correspond to those points that subtend
a negative angle. This all gives rise to angles being measured in radians
instead of degrees, for if you look at where the real number line wraps
around the unit circle, where the number 1 on the real number line is will
actually correspond to a point that subtends an angle of 1 radian. This is
probably the section where you are supposed to introduce radian measure, but
that's just a guess of mine.
You can take the wrapping function to be the function that has at it's
domain the set of real numbers and whose range is the unit circle, or more
likely, you can take it as the function that maps the real line onto the
interval 0 to 2pi.
Hope this helps,
Brian VanPelt
dau...@neo.lrun.com
> Lines: 28
> Message-ID: <6t7v39$eiu$1...@birch.prod.itd.earthlink.net>
> NNTP-Posting-Host: ip146.burbank5.ca.pub-ip.psi.net
> X-Newsreader: Microsoft Outlook Express 4.72.3115.0
On September 10, Kerry wrote:
> I am supposed to teach the wrapping function to a high school trig class
> next week. Problem is, I don't know what the wrapping function is. I think
> the curriculum was designed by somebody who has taught math for 40 years,
> and is using out-of-date terminology. Can somebody point me in the right
> direction?
>
> The context is, the students are reviewing the algebra and geometry they
> will need for trig--sets, functions, quadratic equations, and the like.
> Next, we start in on the basics of trig--the unit circle, determining
> lengths of arcs, coterminal angles, and so on. What has me stumped is the
> following learning expectations:
> To define a wrapping function
> To determine the domain and range of the wrapping function
> To label terminal points of given paths on the unit circle (ok, that's
> easy, as long as I assume path means arc, and not in terms of walk, path,
> and trail)
> To identify symmetric points of terminal points of paths with respect to
> a diameter of the unit circle
> To prove that two points are symmetric with respect to a diameter of the
> unit circle
> To find the ordered pair of real numbers corresponding to the value of
> the wrapping function given in terms of pi.
>
> Then they go on to periodic functions, which are no problem for me.
> However, none of my references cover the wrapping function. Can somebody
> help? It would be most appreciated!
>
Dear Kerry,
Basically, the "wrapping function" is so-called because it consists of placing
the origin of a vertical number line (up is positive) at the point (1,0) on
the unit
circle (the circle x^2 + y^2 = 1) and then "wrapping" the positive part of the
line
counterclockwise around the circle and the negative part of the line clockwise
around the circle. Since the radius of the circle is 1, the circumference is
2pi.
Therefore, the real number 2pi on the number line would "wrap" (after one
counterclockwise revolution) onto the point (1,0), the same point that the
real
number 0 corresponds to (like coterminal angles). The point pi on the line
would
"wrap" onto (-1,0), the point pi/2 would "wrap" onto (0,1), etc. The trig
functions
were then defined as follows: If t corresponds to a point on the number
line (that is, t is any real number) and t "wraps" onto the point (x,y) on the
unit
circle, then sin t = y, cos t = x, tan t = y/x, etc. For example, since t =
pi/2
"wraps" onto the point (0,1), sin t = y becomes sin pi/2 = 1, cos t = x
becomes
cos pi/2 = 0, tan t = y/x becomes tan t = 1/0 = undefined, etc.
Proving the trig identities becomes easy. For example,
sin^2 t + cos^2 t = x^2 + y^2 = 1.
This method was popular in the late sixties and early seventies. When it is
covered in texts today, it is often done so as an alternative approach. It
was
introduced for many reasons. To cite one, this method emphasizes the
fact that t in the expression sin t does not have to represent an angle, but
can be any real number. The trig functions then are simply thought of as
periodic functions. Radian measure is emphasized, and the concept of
an angle is brought in later.
I taught this method many times (yes, I'm that old!), but I never liked it.
It is impossible to explain in detail in one post enough about this method for
you to teach it. I have two suggestions for you:
1) Go to the library and find a book from that period and use that. There
are
many. For example, a good one is "Fundamentals of Algebra and
Trigonometry" by Earl Swokowski" (third edition, 1975, Prindle, Weber,
& Schmidt). Swokowski is a precise writer, and his books do very well
in the marketplace.
2) Follow your own plan; that is, teach trig the way you understand it best,
and, as you say, let the chips fall where they may. This is probably
what I would do and may be best for your students (although I don't
know the entire situation). It's not that the "wrapping function"
approach
is so different from other approaches (they are all essentially the same)
or that it's difficult to understand, it's that you have so little time.
If you decide to teach the wrapping function, email me as often as you like
(everyday if you want), and I will help you.
I don't think teachers get enough support these days from any corner, be
it students, parents, administration, politicians, or the general public. My
heart goes out to you.
Best of Luck!
dauvil
nos...@sorry.com
<kerry_t...@earthlink.net> wrote:
>I agree completely--it's ridiculous to try to teach without a text! That's
>why I'm having problems, and that's why I have resigned, effective Sept. 30.
>Until then, though, I owe it to the students to do the best I can with the
>materials I have been given.
Kerry. Now I feel a little guilty. You are clearly a conscientious
teacher. I'm sure hat you will find a more suitable employer down the
road. However, you MUST play the game for your own security. That
is, don't burn your bridges. That would be a loss to the profession.
Do resign if you must (I guess you have.) However, don't make it so
that you will not get employment elsewhere. You will be a fine
teacher if you can survive the system.
The reason I suggested teaching as it is, is that I also, some time
ago now when the W-function was vogue, ignored the text and taught as
I saw fit. It worked since (i) I had the support of the dept. head
(admittedly a rare occurrence), who thought well of my capabilities to
that point, and (ii) the pupils really did learn something about trig.
I'll see what I can come up with in the meantime.
David.
Stan Brown
>The trig functions are periodic
>
>sin(theta) = sin(180-theta)
>tan(theta) = tan(180+theta)
Bzzt! Wrong, but thanks for playing.
(Rather than embarrass you publicly, I would have preferred to write to
you privately.)
Your statements are correct, but the statement about sin does not support
the statement that sin is a periodic function. The above statement
indicates that sin(x) is symmetric about the (vertical) line x = 180
degrees, not that it is periodic.
To say that a function is periodic with period p is to say that
f(x + kp) = f(x)
for all integral k (subject to the function's domain, of course).
sin is periodic with period 2*pi or 360 degrees, since
sin(x + 2*pi*k) = sin(x) for all integer x
--
Stan Brown, Oak Road Systems, Cleveland, Ohio, USA
http://www.concentric.net/%7eBrownsta/
My reply address is correct as is. The courtesy of providing a correct
reply address is more important to me than time spent deleting spam.
nos...@sorry.com
wrote:
Hi,
Not embarassed in the least. I tend to fall asleep now and then. :-)
Don't mind at all being corrected, especially nicely so. You are
quite right, of course.
Thanks.
David.
dau...@neo.lrun.com
On September 12, Stan Brown wrote:
> nos...@sorry.com (nos...@sorry.com) wrote:
> >The trig functions are periodic
> >
> >sin(theta) = sin(180-theta)
> >tan(theta) = tan(180+theta)
>
> Bzzt! Wrong, but thanks for playing.
>
> (Rather than embarrass you publicly, I would have preferred to write to
> you privately.)
>
> Your statements are correct, but the statement about sin does not support
> the statement that sin is a periodic function. The above statement
> indicates that sin(x) is symmetric about the (vertical) line x = 180
> degrees, not that it is periodic.
>
> To say that a function is periodic with period p is to say that
> f(x + kp) = f(x)
> for all integral k (subject to the function's domain, of course).
Bzzt! Wrong, but thanks for playing. Rather than send you an email,
I preferred to embarrass you publicly. JUST KIDDING, STAN!
dauvil :-)
Brian
>> >sin(theta) = sin(180-theta)
>> >tan(theta) = tan(180+theta)
>>
>> the statement that sin is a periodic function. The above statement
>> indicates that sin(x) is symmetric about the (vertical) line x = 180
>> degrees, not that it is periodic.
>>
>> To say that a function is periodic with period p is to say that
>> f(x + kp) = f(x)
>> for all integral k (subject to the function's domain, of course).
>
there is some number p for which f(x + p) = f(x) for every x in the domain
of f. For instance, sin(x) = sin(2pi + x) (not pi or 180 degrees), for
every real number x. To get that f(x + kp) = f(x), for every integral k,
let y = x + (k-1)p, so that since f(x) = f(y) = f(y + p) = f(x + kp).
Brian VanPelt
dau...@neo.lrun.com
nospams wrote:
The trig functions are periodic
sin(theta) = sin(180-theta)
tan(theta) = tan(180+theta)
Stan Brown wrote:
Bzzt! Wrong, but thanks for playing.
(Rather than embarrass you publicly, I would have preferred to write
to you privately.)
Your statements are correct, but the statement about sin does not
support the statement that sin is a periodic function. The above
statement indicates that sin(x) is symmetric about the (vertical)
line x = 180 degrees, not that it is periodic.
To say that a function is periodic with period p is to say that
f(x + kp) = f(x)
for all integral k (subject to the function's domain, of course).
dauvil wrote:
Bzzt! Wrong, but thanks for playing. Rather than send you an email,
I preferred to embarrass you publicly. JUST KIDDING, STAN!
dauvil :-)
Brian wrote:
>Stan Brown wrote:
>>To say that a function is periodic with period p is to say that
>>f(x + kp) = f(x)
>>for all integral k (subject to the function's domain, of course).
Another way to define a periodic function is to say that f is periodic if
there is some number p for which f(x + p) = f(x) for every x in the domain
of f. For instance, sin(x) = sin(2pi + x) (not pi or 180 degrees), for
every real number x. To get that f(x + kp) = f(x), for every integral k,
let y = x + (k-1)p, so that since f(x) = f(y) = f(y + p) = f(x + kp).
Brian VanPelt
Dear Brian,
What you have said here is (almost) correct ... but it does not address
my objection to Stan's definition. He has claimed that p is the period
of the function in his definition, f(x + kp) = f(x).
However, sin (x + k*4pi) = sin (x) for all integral k, but 4pi is not the
period of the sine function. And since all definitions are by nature
if and only if statements, one cannot argue that his definition is
correct if you read it from a particular direction.
Here is the standard definition for periodic functions:
A function f is periodic if there exists a *positive* real number k
such that f(x + k) = f(x) for every x in the domain of f. If a least
such positive real number k exists, it is called the period of f.
dauvil