**High School Mathematics-Physics SMILE Meeting
1997-2006 Academic Years
Mathematics: Calculators**

**09 December 1997: Bill Lilly [Kenwood HS]**

He showed a new calculator-based-ranger [TI-CBR], which is used with
TI-82
calculators to detect motion and measure distance. It can be
incorporated into
programs on the calculator for automatic data-taking. The device
represents an
improvement on the older CBL devices. He talked about an application of
measuring a motion of a pendulum. The minimum distance for detection is
about
0.5 meter. We ran out of time during Bill's presentation, and we
hope he
can return to show it next year.

**07 April 1998 Bill Lilly [Kenwood High School; Math Dept]**

He demonstrated the capabilities of the TI-CBR Calculator-Based Ranger,
[http://www.calculatorsinc.com/ti-cbr.aspx]
which is a sonic motion detector that provides information in digital
and easily
convertible form. The device is made by __Texas Instruments__ for
use with
their calculators **TI-82, 83, 85, 86, or 92**. The price is $90,
and seems
to be dropping with time. The "innards" of the device are the same as
a Polaroid auto-focus system. The apparatus was demonstrated by hanging
a bottle
on a swing and swinging it back and forth as a pendulum. He was able to
obtain
[interpolated] numerical values of **x, v, and a** at time
intervals of about
**0.1 seconds**, so that motions could be studied quantitatively,
and graphs
easily generated.

**02 March 1999: Fred Schaal [Lane Tech HS Math]**

He investigated determinants and the magic squares. Using the TI
graphing
calculator's internal ability to solve determinants he put in a 5x5
array and
showed some to the abilities of the TI graphing calculator. Fred tried
the
numbers forming a "golden square" as well as a ** Fibonacci sequence**,
and
in neither case did the determinant vanish.

Porter commented that the result was not what Fred expected because the elements did not have the proper relationship where an element or the ray wasn't a factor of the other numbers. Determinants can be thought of in geometric terms, where the points can be identified as vectors, and the vectors are added. there can be a solution when they reside in the same plane. You can write two equations in two unknowns in the form

A1 · X = B1

A2 · X = B2

The object is to determine the two dimensional vector ** X**
when its scalar
products with the vectors ** A1 ** and ** A2** are given.
There is no solution when
** A1** and **
A2** are parallel to one another, or when the determinant of
coefficients
vanishes. So it goes...

**20 April 1999: FJ Schaal [Hi Tech Low Tech Lane Tech High School]**

He first placed a grid on the board by using the chart that has the
holes to
allow chalk dust to produce a grid. He placed points (2,13) and (16,3)
on the
grid and used the formula

He explained the steps and keys to press on the calculator: ( ( 2 -16 ) ( 2 + ( 13 -3 ) ( 2 ) ( = distance. Now, adding an additional point of (-12,-6), using the above formula we get distances:

Sqrt(356) = 18.868

Sqrt(981) = 31.320

Sqrt(557) = 23.601

Hero [HERON] formula for the area of a triangle of sides **a, b, c**:

where

This gives ** 224** [exactly!!], whereas by graphing and
counting the number of
fully enclosed squares plus 1/2 the number of partially enclosed
squares we get **219**.

**04 May 1999: Porter Johnson [IIT]**

Porter took off on the use of the calculator and commented about not
rounding
off! By using integers and keeping the calculation to as many places as
possible
we may get an answer that will result in integers. If you round off the
answer
will be an decimal, and hide the fact that the answer from a problem
starting
with integers will be integers. You may not want to round the working
numbers in
a calculators to a few significant places and thus hide some of
significance of
the answer. The example consisted in calculating the area of a triangle
with
vertices at integer-valued x- and y-coordinates. The area of the
triangle, which
will come out as either an integer or half-integer, can be computed
geometrically by enclosing a rectangle around the triangle, and
calculating the
area of the rectangle and the other right triangular pieces. Simple,
non??

**28 September 1999: Fred Schaal (Lane Tech)**

Fred drew our attention to inequalities. For example, for a straight
line ** y = m x + b**, we might have** y = (1/2) x + 2**. On a
graph of ** y vs x**, below the line ** y < (1/2)** ** x +
2**, and above the line ** y > (1/2) x + 2**. Our hand-held
calculators seem not to enable us to display a band of ** (x, y)**
pairs that lie between a minimum and maximum set of values. Is there
any way to do such a thing on any hand-held calculator? Interesting
question!

**12 October 1999: Walter Macdonald [CPS Substitute; Medical
Technician]**

He demonstrated the versatility of the HP48GX Programmable Calculator [http://www.johann-sandra.com/surveying/hp-48gx-49-33s-calculator.htm],
with 128K of
internal RAM [random access memory] connection with the software
package SPICE48, which allows one to program circuit diagrams [although
somewhat tediously, it seems] and to specify the voltage source as DC,
sinusoidal AC, sawtooth, or "whatever". The calculator permits display
of output currents and such in graphical form, in order to gain insight
into the effects of changing resistances, capacitances, and
inductances, or switching in diodes. The device is better suited to
providing its gentle user with graphical insight, rather than mere
numbers.

Walter also commented that the new x-ray machine in his hospital has a digital interface, so that familiarity with PCs/Macs is required in routine usage. Porter observed that the world is "going digital", giving these instances:

- General Motors has described the modern automobile as essentially an electronic device, rather than a mechanical one, as it has always been.
- The hospitals of today seem like, say, the high energy physics labs a few years back, with sophisticated electronics, computer-driven operations, schemes for fast digital tomography [MRI, PET, and CT; see http://www.islandscene.com/Article.aspx?id=56], and automated chemical analyses.
- You can buy a car, order merchandise, do banking, find a mate, and pay taxes electronically.

**09 November 1999: Bill Colson (Morgan Park HS - Math)**

drew 2 lines that intersect to form an angle. In his geometry class
Bill uses a book titled Line Design. Some students have difficulty
understanding what an angle is, and by getting them involved in such
constructions using straight lines, they develop a feeling for the idea
of angle. Example: start with two straight lines intersecting at 90^{o},
and mark off equal divisions beginning at the intersection. Draw two
additional lines connecting the divisions (1 & 2) and (2 & 1).
The two lines intersect at one point. Now repeat this on a new drawing,
making 4 lines connecting (4 & 1), (3 & 2), (2 & 3) and (1
& 4). These 4 lines intersect at 3 points. By repeating this
process next for divisions 6, 5, 4, 3, 2,1 - it becomes quite clear
that the intersecting straight lines come to more and more closely
define a curved line. And the idea of approaching something as the
limit of a process repeated an "infinite" number of times becomes an
experience that is useful in more advanced math. Bill passed around
examples of beautiful art work his students made using these ideas.
Very nice!

**29 February 2000: Fred Schaal (Lane Tech HS)**

showed us classroom use of plotting calculators, and he used one to
project to the screen so we could see what was going on. **Betty
Roombos**
followed Fred's directions to do this, and it worked well. Fred had her
(and others with calculators he had passed out) plot the equations of
straight lines and find their intercepts with the x and y axes. He
used number pairs to determine a set of lines:

x-intercept | y-intercept | line |

10 | 1 | y = -(10/1)x+10 |

9 | 2 | y = -(9/2)x+9 |

8 | 3 | y = -(8/3)x+8 |

7 | 4 | y = -(7/4)x+7 |

6 | 5 | y = -(6/5)x+6 |

5 | 6 | y = -(5/6)x+5 |

4 | 7 | y = -(4/7)x+4 |

3 | 8 | y = -(3/8)x+3 |

2 | 9 | y = -(2/9)x+2 |

1 | 10 | y = -(1/10)x+1 |

The intersections of successive pairs of lines were determined by using the plotting computers, and one could even zoom up close to the intersections to see more detail and better determine the actual number pair locating each intersection. He then used the intersection points as data, and did a linear, quadratic, and cubic fit to those points on a calculator. He compared the results with the "tangent curve" described below by Porter Johnson. A fine phenomenological math lesson!

**29 February 2000: **
Comments by ** Porter Johnson (IIT)**

The formula for line segment lying in first quadrant and intercepting
the x-axis at** x = a ** and intercepting the y-axis at ** y = b
- a**, where
**0 < a < b**, is

Let us regard the parameter b as being fixed, while a varies continuously between the values 0 and b. A solid region in the first quadrant is filled by these lines. To determine the boundary of that region, we solve the above relation for y, to obtain

In this relation, keep x fixed, and vary a. The maximum value of y
under such variation can be calculated by setting the derivative **
dy/da** to zero:

We substitute this value of a into the expression for ** y(a)** to
obtain

To show that this is indeed the maximum value of **y(a)**,
calculate
the quantity ** y _{max} - y(a)**:

Clearly, the right side is non-negative, and it is zero when a is
chosen
as above. The formula for y_{max} gives the greatest value that
y can have for a given value of x, and thus lies at the top of the
region
traced out by the straight lines described above. Thus, the top of the
region is given by

This curve, which represents the envelope of all the straight lines, can also be written as

The curve is a parabola with the line of symmetry [axis] lying along the line x=y. It is symmetric under interchange of the variables x and y. The point closest to the origin has coordinates

This is the "symmetry point of the parabola", and is the "tangent point" for the curve with a = b/2, namely

Every other point on this curve is a "tangent point" for one and only one of the straight lines described above, which have intercepts a and b-a, respectively.

Here is an Excel-generated image of the lines and the asymptotic curve:

Incidentally, these "envelope curves" occur frequently in geometrical
optics, in which light rays move in straight lines in a uniform medium.
Clearly, the "bundle" formed by all light rays can have a nontrivial
structure. The boundary of that "bundle" is called a caustic in
geometrical
optics. As an example, see http://www.math.harvard.edu/archive/21a_spring_06/exhibits/coffeecup/index.html
**The Coffeecup Caustic**.
Here is a brief description of the effect, taken from that reference.

You are drinking form a cylindrical cup in the sunshine. Sometimes, when the sun shines into the cup, you can see a crescent of light as the sunshine reflects from the inside of the cup onto the surface of the drink. A picture of a real cup is shown, and you can do your own on-line computer simulations of the effect. Check it out!

**14 March 2000: Porter Johnson (IIT Physics)**

explained Coffee Cup Caustics to us with some mathematical
detail, but nicely presented.

**Coffee Cup Caustics**

The Coffee Cup Caustic is shown on the website http://www.sciencedaily.com/releases/2009/04/090414160801.htm, on which the reflected image from a real coffee cup is shown, as well as a "Monte Carlo" simulation of the event.

Let us suppose that rays parallel to the y-axis strike an upper
semi-circle of radius R from the inside, and are reflected. If we
let the angle between the reflection point be
q, the coordinates of
the point of reflection are ** x = R cos q
**and ** y = R sin q**.
The angle of incidence of the ray, relative to the normal, is **
p/2 - q**,
and the angle of reflection has that same value, as well.

The reflected ray travels at an angle p/2+2
q relative to the positive x-axis, and
strikes the circle again at a point an angle **
3 q ** from the horizontal, at the point
(**x = R cos3 q**,
**y = R sin 3 q**). The equation for
the reflected ray is

As q varies, we generate a series of straight lines. To find the envelope of those straight lines, we must determine the maximum value that y can have for a given value of x, and the appropriate choice of q, by setting the derivative

**
**

and

We cycle through the various striking points by letting q vary between 0

A template of a 360^{o} Protractor was handed out, and
participants made their own caustic by drawing lines from the positive
x-axis [right on the middle] from angles q
to 3 q. Here is a set of
angles relative to the horizontal to use:

10^{o} |
--- > | 30^{o} |

20^{o} |
--- > | 60^{o} |

30^{o} |
--- > | 90^{o} |

40^{o} |
--- > | 120^{o} |

50^{o} |
--- > | 150^{o} |

60^{o} |
--- > | 180^{o} |

70^{o} |
--- > | 210^{o} |

80^{o} |
--- > | 240^{o} |

90^{o} |
--- > | 270^{o} |

The right side of the caustic will arise, as if per magic!

The full curve, which
was produced using the software package __EXCEL__, is given below.
An excellent reference on using __EXCEL__ in graphics is the book
**EXCEL for Scientists and Engineers** by William J Orvis, Second
Edition [ISBN 0-7821-1761-9].

**01 February 2000: Fred Schaal (Lane Tech HS)**

invited a number of us up
front, gave us each a piece of polar graph paper (paper
with a set of concentric circles on it, and radii drawn in
every 10^{o}). He had each of us cut out a segment of angle,
one of us 10^{o}, next 20^{o}, then 30^{o},
etc. We then taped the
edges together to form a set of cones. Fred proceeded to
lay out a set of equations for the lateral area (LA) of a
cone, and ended up with a result that the altitude of a
constructed cone lying on the table is:

We used sticks of spaghetti to poke down through our cones' tops to the table below, marked the stick, and got an experimental value of h. And then we calculated h from the above expression. Both observed and calculated values matched surprisingly well! A beautiful phenomenological math lesson! Thanks, Fred!

**28 March 2000: Fred Schaal (Lane Tech HS)**

Fred brought us back to something he had done with us
earlier: The truncated cone. Start with a circle of radius **R**
cut
from paper. Draw two radii on the circle from its center, and so
define an angle of **360 ^{o} - k^{o}** between
them.
Cut out the sector between them, as shown:

s = 2 p r = p [k/180] R

Tape the radii together, to form a truncated cone surface with the
paper. Let **r** be the radius of base of the cone, and **h**
its altitude. Then

**and**

**h = (R ^{2} - r^{2})^{1/2} =
R * [ 1 - (k^{o}/360^{o})^{2}]^{1/2}.
**

** = p R ^{3}/3 *
[k^{o}/360^{o}]^{2} *
[ 1 - (k^{o}/360^{o})^{2} ]^{1/2}
**

"What value should the angle **k ^{o}** have in order for
volume

**02 May 2000: Porter Johnson**

See Notes for 28 March 2000 HS Math-Phys Class

If you take a sheet of paper with a circle of radius R, and cut
out a sector of opening angle k^{o}, the volume of the cone
formed by
that sheet is

The formula may be simplifying by defining the ratio **
x = k ^{o}/360^{o}**, so that for radius

One may show by elementary calculus that the volume is maximized by the choice

corresponding to angle k = 293.9388^{o}. The maximum volume
is** **

*** | A | B | C | D |

1 | _____ | _____ | _____ | _____ |

2 | _____ | _____ | _____ | _____ |

3 | _____ | _____ | _____ | _____ |

4 | _____ | _____ | _____ | _____ |

5 | _____ | _____ | _____ | _____ |

Here is a set of steps that will result in the data table and the graph: First, type these descriptive words in lines #2 and #3

- In Cell A2 type
**cone vol** - In Cell A3 type
**angle** - In Cell B3 type
**x** - In Cell C3 type
**angle** - In Cell D4 type
**volume**

- In Cell A4 type
**290** - In Cell B4 type
**= A4/360** - In Cell C4 type
**= A4** - In Cell D4 type
**= PI() * B4^2 / 3 * SQRT(1 - B4^2)**

- In Cell A5 type
**291**

*** | A | B | C | D |

1 | _____ | _____ | _____ | _____ |

2 | cone vol | _____ | _____ | _____ |

3 | angle | x | angle | volume |

4 | 290 | .805556 | 290 | .402645 |

5 | 291 | _____ | _____ | _____ |

At this point you should save the EXCEL table [after all, you've worked hard to create it!].

- Take the mouse and "enclose" [i.e. "highlight"] both cells A4 and A5, which contain the numbers 290 and 291, respectively. If you pull downward on the lower right corner of the rectangle with a + sign there, you will get evenly spaced points as far down as you pull; say, 290 to 310.
- Now, enclose the cell B4, and pull down on the + sign at the lower right corner, so as to fill cells in column B as far down as column A is filled.
- Similarly, enclose cell C4, and pull it down.
- Finally, enclose cell D4 and pull it down.

*** | A | B | C | D |

1 | _____ | _____ | _____ | _____ |

2 | cone vol | _____ | _____ | _____ |

3 | angle | x | angle | volume |

4 | 290 | .805556 | 290 | .402645 |

5 | 291 | .808333 | 291 | .402830 |

6 | 292 | .811111 | 292 | .402963 |

7 | 293 | .813889 | 293 | .403042 |

8 | 294 | .816667 | 294 | .403066 |

9 | 295 | .819444 | 295 | .403035 |

10 | 296 | .822222 | 296 | .402946 |

11 | 297 | .825000 | 297 | .402798 |

12 | 298 | .827778 | 298 | .402589 |

13 | 299 | .830556 | 299 | .402320 |

14 | 300 | .833333 | 300 | .401986 |

15 | 301 | .836111 | 301 | .401588 |

16 | 302 | .838889 | 302 | .401123 |

17 | 303 | .841667 | 303 | .400590 |

18 | 304 | .844444 | 304 | .399987 |

19 | 305 | .847222 | 305 | .399313 |

20 | 306 | .850000 | 306 | .398564 |

21 | 307 | .852778 | 307 | .397739 |

22 | 308 | .855556 | 308 | .396837 |

23 | 309 | .858333 | 309 | .395855 |

24 | 310 | .861111 | 310 | .394791 |

The next objective is to make a graph of the volume versus angle
using EXCEL. To do so, complete the following steps:
**EXCEL**

- Using the mouse, capture the data in column A, and then in column D.
- From the toolbar menu, click on the "chart wizard" icon.
- Pick the "xy scatter" option
- Then pick the "smooth curve" option
- Include the "major grid lines" and "minor grid line" options for x and y
- Finish drawing the graph.

**02 May 2000: Fred Schaal (Lane Tech HS)**

did the **Law of Sines** with us as an
exploration by hand-held calculator. He drew a triangle on the
board and identified its three angles as **A, B, C**, and the
lengths of
the sides opposite as **a, b, c**, as shown:

Then

Choosing values for A, and keeping c and B fixed but letting the side a increase with A, he obtained the following table of values, which he wrote on the board:

A | a / sin A |
---|---|

6^{o} |
16.3 |

19^{o} |
14.4 |

46^{o} |
17.8 |

83^{o} |
15.4 |

99^{o} |
20.2 |

153^{o} |
16.3 |

Triangles were actually drawn for the various values of A, and a was then measured, and the above ratios were calculated based on these experimental measurements. Fred noted that the ratio does not change much, and wondered at this.

Fred set up the calculation of the curve for **x(T)**, y(T),
where

y = T[1+1/(2p) ] sin T

He displayed the result as a set of points with the aid of a TI graphing calculator, a transparency projector and LCD display. The graph looked like a roll of carpeting viewed end-on, a kind of spiral around the origin. Most interesting! A good example of how to connect the abstract to the concrete so as to make math "real" for students!

**02 May 2000: Bill Lilly (Kenwood HS)**

gave us a handout titled "Pendulum,"
which was a lab exercise. Using a** TI-CBR** Sonic Ranger, high
frequency sound from the Ranger was reflected from a swinging
balloon [held on the end of string by ** Pearline Scott** (Franklin
School)]. The output from Ranger (for 10 s) was stored in the
calculator and shown on the projector screen as a graph with the
aid of ** Fred Schaal's** LCD display. It looked like a sine wave,
and
Bill described how students could analyze the data for frequency,
effect of shortening string, lengthening string, amplitude effects,
velocity minima and maxima with position, etc. Another beautiful
example of concrete/abstract connections!

**26 September 2000 Fred Schaal (Lane Tech HS)**

presented "92s to the Rescue." He
gave one hand-held ** TI 92 ** to ** Betty Roombos (Lane Tech HS)**
to enter
commands and data. The LCD display of the TI 92 was projected onto
the screen for all to see. ** Fred** placed on the board:** a/sinA
= ?** -
referring to the ** Law of Sines**. He instructed ** Betty**
how to draw a
triangle and label its vertices with angles** A, B, C,** and the
sides
opposite with **a, b, c**. ** Betty** was a whiz, and with **
Fred's** further
direction she soon had the angle A labeled with its value of ** 41.99 ^{o}**,
and the side

**26 September 2000 Fred Farnell (Lane Tech HS)**

walked carefully across the front
of the room in front of us and asked,

Using the same projection setup as

and a program in the TI 83, he found the best-fit values of m and b to the data and plotted it. It seemed like it was not a very good fit to us.

Next, ** Fred** repeated the experiment, but this time ran away
from
the **CBR**. The ** D vs t ** graph of his motion appeared
parabolic, typical
of constant acceleration motion. When he found the best fit to
the parabolic equation,

and plotted it on the same graph as the experimental points, the result did not look like a best fit to many of us. He also tried fitting to an exponential equation, with the same kind of result. This leaves something for us to explain, but

**10 October 2000 Fred Farnell (Lane Tech HS)**

explained why the fit of a straight line equation to experimental
constant velocity points (and its subsequent ** v vs t** graph) at
the last
meeting was so poor.
It turned out that the constant (non zero) velocity occurred over an **
8 second** period, but the data taking ran for **15 seconds**,
and the velocity
was zero for the last ** 7 seconds**! When ** Fred ** ran new
data and ** kept only the
portion for non-zero velocity**, the fit turned out great! He did
this "live". Same story for motion of constant acceleration. Thanks for
restoring our faith, **Fred**!

** 24 October 2000 Fred Schaal (Lane Tech HS)**

set up the projector to display
the output of a ** TI 92** calculator so we all could see it, and
gave the ** TI 92** to
** Betty Roombos** to manipulate. ** Fred ** asked ** Betty**
to draw a
triangle and label its vertices with **A,
B, C** and we saw it as it
happened. Then he advised ** Betty** what to do to construct the
bisectors of the angles at the vertices **A** and B. The ** angle
bisectors**
intersected at a point which was labeled **F**. Next, he had **
Betty**
construct a ray connecting vertex **C** with the point of
intersection, **F**. The angles **ACF**
and **BCF** were "measured" by manipulation of the ** TI 92**
and labeled with their values, which both happened to be ** 37.05 ^{o}**!
This showed that the constructed ray was a bisector of the angle at
vertex

Then ** Fred** asked ** Betty** to "grab" the vertex point **C**
and move it
around. As this was done, we could see the shape of the triangle **ABC**
change,
and the values of the two angles also changed, but
remained equal. Very nice! Similar exercises may be done to show
that medians meet at a point; perpendicular bisectors of sides meet
at a point. ** Porter Johnson** asked: **What about the "Nine
Point
Circle Theorem?** Anyone know? Check the website http://en.wikipedia.org/wiki/Nine-point_circ.

A perfect complement to what you did last time, **Fred**!
(...the
actual construction on the white board, which did not display it well.)

**07 November 2000 Fred Schaal (Lane Tech HS)**

sketched out on the board a kind
of geometry problem he will do next time using the ** TI 92 calculator**.
He drew two straight lines (representing mirrors),
intersecting at some angle. Then he drew a triangle with vertex
points **A, B, C ** and sketched its reflection across the nearest
of the
mirrors. He pointed out that the order of the points, **A, B, C**
was ** ccw
(counter clockwise)** for the triangle, but ** cw (clockwise)**
for the vertex points
- A' , B' , C' - of its reflection. Sketching the reflection of
the
those points across the second mirror (straight line) located
vertex points **A", B", C"** - which followed a ** ccw** order.
It could be
interesting to see how Fred will do this with the TI 92. But this
provoked discussion about reflections in mirrors, and brought out
that it takes two mirrors **to see your reflection as others see you**!

**21 November 2000 Fred Schaal [Lane Tech HS]**

displayed the output of his ** TI 92 Calculator ** on the screen.
He chose the
"geometry" option and drew a triangle. Then he added a line, and
reflected the triangle about the line. He noted that the second
triangle
could not be superposed on the first one, because it was
"left-handed". Next, he added another [non-parallel] line, and
made reflection of the second triangle. He showed that the third
triangle
could be superposed by rotation onto the first one. Thus, two
reflections
correspond to a spatial rotation.

**10 December 2002: Fred Schaal [Lane Tech HS,
Mathematics] TI-92 Graphics Calculator Attacks
Triangle!
Fred** used the

**25 February 2003: Walter McDonald [CPS Substitute -- VA Hospital
Technician]
Approximation of Functions
Walter** showed us how to do

**Fascinating, Walter!**

**11 March 2003: Fred Schaal [Lane Tech HS,
Mathematics]
TI Interactive Software
Fred** had intended to transfer images from his laptop computer to
the
screen in order to display

**25 March 2003: Fred J Schaal [Lane Tech HS,
Mathematics] Laptop
Capers: TI Interactive
Fred** showed the

**Very visual and user-friendly, Fred!**

**08 April 2003: Fred Schaal [Lane Tech HS,
Mathematics]
Green Line '92-ing!
Fred** had been playing around with his calculator while riding the
el, and
began studying the following problem: For a given triangle with
vertices (

- The bisector of interior vertex angle
**A,**which we assume to be acute. - The median from the center of side
**AB**to vertex**C** - The altitude from side
**AC**to vertex**B**

**Fred, you have constructed an unusual problem! Very Good!**

**07 October 2003: Fred Schaal [Lane Tech HS,
mathematics]
Lubbock*** [TI-83] Helper
Fred** used his calculator and plotter to extend his lesson of

The formula for line segment lying in
first quadrant and intercepting the ** x-axis** at ** x = a **
and intercepting the
** y-axis**
at ** y = b - a, where 0 < a < b**, is

**How come?**

**Fred's
**trusty programmer, **Bill Colson**, developed and displayed the
graphs for**
x ^{p}, **where the real, non-integer variable

**Comments by Porter Johnson**: For
negative **x**, there is a real, positive value of ** ^{p
}**whenever

Q: How manyTexas Techfootball supporters does it take to change a light bulb?

A:Don't be silly; there's no electricity inLubbock***[home ofTexas Instruments™]!

**21 October 2003: Babatunde Taiwo [Dunbar Vocational HS,
science]
Graphics Calculator and Motion Sensor
Babatunde** showed that his

He set up the motion sensor and software so that it would record the position of an person walking across the room. The objective was to reproduce the graph given above with one's own motion --- by being at the right place at the right time. After three trials,

For additional information see the websites: **Activity
Investigating Motion**: http://www.math.mtu.edu/gk-12/investigatingmotion.html
and Utilizing the **Graphing Calculator in the Secondary Mathematics
and
Science Classroom**: http://www.esc4.net/math/.

**Babatunde ** also showed a spinning top toy that he had
recently obtained at **
American Science & Surplus** [
http://www.sciplus.com/category.cfm?category=13]:

Here, Kitty Kitty... A top that’s fun! A top that doesn’t need string or practice or coordination. Twist the winder, push the button, and the colorful plastic top bounces around on its spring bottom, lights up brightly (batteries included), and spins for a very long time. A nice distraction for a jaded child or a bored tabby cat

**18 November 2003: Fred J Schaal [Lane Tech HS,
mathematics]
STO FRM Not That Bad!
Fred** asked if anybody knows the standard "slope-intercept"
equation for a straight line. Somebody suggested ...

**Bill Shanks** mentioned that an elliptical curve with
semi-major axes **(a,b)
**can be written in a similar
form:

**Thanks for showing us the neat Algebra, Fred!**

**04 May 2004: Fred Schaal [Lane Tech HS,
Mathematics]
Fraction Action **

**Fred** reminded us of a universal feature of programming
algorithms on
digital computers, in that they perform arithmetic exactly with
integers,
whereas with real numbers the calculations may be inexact because of
round-off
error. Consequently, arithmetic with fractions can be done
exactly --
provided the numbers that arise are not too large. On his trusty **
TI-81** programmable calculator,
** Fred** went into fraction
mode, and was able to verify the relation:

... or ...

2 + 2 * 3^(-1) + 3 + 5 * 8^(-1) (frac)= 151/24