PHOTOGRAPHIC SLIDES: A BETTER WAY TO DEMONSTRATE SOFTWARE TO LARGE GROUPSAuthor(s): Jeffry GordonSource: The Mathematics Teacher, Vol. 77, No. 8 (November 1984), pp. 609-611Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964251 .

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PHOTOGRAPHIC SLIDES: A BETTER WAY TO DEMONSTRATE SOFTWARE TO LARGE GROUPS Microcomputers are becoming increasingly popular in mathematics classrooms. As a

result of this popularity, the number of

microcomputer presentations made at math ematics education professional meetings has increased significantly. Often educators

attempt to demonstrate to large audiences the software they have either used or crea ted. These presentations can be very en

lightening. However, if you happen to be

sitting beyond the third row, even a large standard television screen can be very diffi cult to read. Audiences can leave these pre sentations without acquiring the flavor of the software they were supposed to be view

ing. Solutions to this problem consist of

using large-screen projection systems, which are very expensive and not very port able; in addition, you must wait for pro grams to load. Another solution?multiple daisy-chained television screens placed around the room?are not easy to wire to

gether and not very portable. Screen

images sent to a printer and then converted to transparencies are easy unless you either need to show color graphics or want to show an image from a locked program. Fin

ally, hand-made transparencies often do not look enough like the real image.

The solution I have found very nearly ideal consists of photographing slides of the video-monitor image. This approach yields

a system that is portable, inexpensive, usable with locked programs that cannot be

paused, shows color, and allows you to

change video frames quickly in a presenta tion.

Taking pictures of a video image re

quires some care to get satisfactory results. You will get the best pictures with the fol

lowing equipment: a monochromatic or

color monitor instead of a television set; a

single lens reflex camera with a "through

HI* HY HAHE IS APPLETON THE COMPUTER. "HAT IS YOUR NANE? PATI

the lens" light meter, adjustable shutter

speeds and f-stop controls, tripod, bulb remote shutter release, 400-ISO color slide

film, and a darkened room.

Step 1 (determining the camera settings). Place the following program into your com

puter to gather the proper exposure set

tings:

10 FOR = 1 TO 500 20 PRINT "A"; 30 NEXT

Now RUN the program to fill about half the screen with the character A. Adjust the

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contrast and brightness of the video screen to the point just before the characters begin to bleed together. Set up your camera on

the tripod, attach the shutter release, and move the camera back and forth until you fill the viewfinder with the video screen.

Set the shutter speed to l/25th of a second or slower. Now darken the room to keep extraneous light from reflecting off the video screen into the camera lens. Take

your exposure meter reading to determine

your f-stop opening at this time. It should be approximately f2, depending on the screen's brightness, the distance between the camera and the monitor, and the shut ter speed. Maintain this shutter speed and

f-stop setting regardless of what your light meter says later.

Step 2 (deciding what slides to take). Review your software to decide what frames you want to show. I recommend

showing the front page, which gives credit

to the author and publisher, and at least two or three slides from the middle of the

program. When photographing question and-answer sessions on the computer, I like to show one slide with just the question and then another slide with the answer typed in

just before the RETURN key is pressed. If

you want to photograph an image that occurs only randomly, be prepared to wait for its display; perhaps you can alter the

program temporarily to make the image appear more frequently. If you use slides to

show a programming technique, the number of slides you take can vary.

610

Step 3 (shooting the slides). Darken the room and load your software into the com

puter. When you see the image you want to

photograph, use the bulb release to avoid

moving the camera. Very bright or very dark screen images can trick your light meter into giving a false reading. However, since the video screen is a light-emitting source and not a reflecting source, the meter reading is actually incorrect.

Step 4 (troubleshooting problems). When

you get the slides back from processing, if

your pictures have dark diagonal bands across them, your shutter speed was too

fast. Slow it down to under l/25th of a

second. Sometimes horizontal or vertical lines plotted on the screen appear to be

curved. This problem is caused by the cur

vature of the video screen and the proxim ity of the camera to the monitor. Try

moving the camera back and compensating for the increased distance with a telephoto lens; however, the lens may not open

widely enough to expose the film ad

equately. Another solution is to move the camera up, down, right, or left of center

toward the line. This procedure, however, will accentuate the curvature of lines on

the other end of the screen. If this problem is significant, you may want to avoid shoot

ing slides of frames with vertical or hori

zontal lines at the edges of the screen. No ideal solution to this problem can be found

until film manufacturers create a faster film that will permit the use of telephoto lenses. If the characters in your slides

appear too dark to read, increase your ex

posure by slowing the shutter speed or

opening the f-stop more if possible. Con

versely, avoiding overly bright, washed-out characters requires closing the camera

opening or increasing the shutter speed, but remember never to shoot faster than

l/25th of a second. Experiment with a roll

of film by bracketing your exposures. This

technique consists of shooting at various shutter speeds and f-stops, keeping accurate

records of the exposure of each shot, and

then examining the results to determine the best exposure settings.

Presenters wishing to demonstrate soft

-Mathematics Teacher

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ware to large audiences should find the

techniques described here helpful. They will be able to describe the software quickly and give more of their audience a better

understanding of what the software is like.

Jeffry Gordon

University of Cincinnati

Cincinnati, OH 45221

A PYTHAGOREAN CURIOSITY Mathematical curiosities pop up in the most unusual places, and generally they are

delightful to find. While teaching a ninth

grade class the Pythagorean theorem, I was

looking for integral values for the three sides of a right triangle. The traditional 3-4-5 was an obvious starting place.

Not having a computer or calculator, I sat down with a table of squares and a large cup of coffee, prepared for the trial-and error method. Then I made the discovery shown in table 1. Phrased in mathematical

language, the difference between the

squares of two consecutive integers is

always odd. Since the n2 column in table 1 starts

with 1, the minimum difference between the

squares is 4 ? 1 = 3. This idea can be ex

tended beyond

(n + l)2-n2

to include the difference of squares of in

tegers separated by 3, 5, 7, 9, and so on. For

example,

(n + l)2 - n2 2n + l

(n + 3)2 - n2 6n + 9

( + 5)2 - 2 10 + 25

(n + 7)2 - n2 Un + 49

How does this pattern relate to the orig inal problem? If a, ?, and c are sides of a

right triangle, with c the length of the hy potenuse, then we have

a2 + b2 = c2 or

a2 = c2- b2.

The expression (c2 ?

b2) represents the dif

TABLE 1

The Difference between the Squares of Two Consecutive Integers Is Always Odd

2 Difference between Squares

1 1 2 4 (4-1) = 3

3 9 (9-4) = 5 4 16 (16

- 9) = 7

5 25 (25 -

16) = 9 2

( + 1)2 - 2 = 2 + 1

71+1 (rc + 1)2

ference of squares discussed earlier. This result leads us to the proposition that for odd values of a greater than 1, we will find

integers b and c that satisfy the Pythago rean property.

For example, if a = 7, a2 = 49. By extrap olating from table 1, we can see that (25)2 -

(24)2 = 49. Thus 7, 24, 25 constitute a Py thagorean triad. This procedure can be used with every odd number to find two consecutive integers that will complete a

triad.

Set

a2 = 2n + 1; then

a2 -1 = 2n,

so

when is the second member of the triad and + 1 is the third member (hypotenuse.)

The following example illustrates how to generate triads :

a = 9

a2 = 81

81 = 2n + 1

80 = 2n = 40

-h 1 = 41

triad = (9, 40, 41)

I call this algorithm the primary odd differ ence method.

Although this example yields (9, 40, 41), we know that the integer 9 has another

November 1984-?-611

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