Information:
Dianna Uchida (Morgan Park HS,
computing) Square
Roots Using Sotolongo's
Method
Dianna shared "Sotolongo's Method" for estimating square
roots. This fascinating method was described in the
October 2005 issue of Mathematics Teacher. Dianna
showed us how it
works using squares of paper (see handout) and it is an ingenious way
to estimate the square root. The
relationship used is: Ön
is approximated by
s + (n-s^{2})/(2s+ 1) where n is the number
whose square root we wish to calculate, and s is the largest
integer whose square is less than n. Dianna showed us
the three cases
below:
n | s | (n-s^{2})/(2s+ 1) | s + (n-s^{2})/(2s+ 1) | Ön |
5 | 2 | 1/5 | 2.20 | 2.235 |
18 | 4 | 2/9 | 4.22 | 4.243 |
50 | 7 | 1/15 | 7.06 | 7.071 |
Terry Donatello (Weber HS,
retired)
Clingers
Terry
showed us evidence for the cohesive force between water molecules
(handout). First, Terry poured drops of water out of a
cup and onto a string held at an angle
of about 30° below the horizontal. But the drops fell fell
straight
down. Terry
then wet the string, and tried again. This time the
drops moved slowly
along the string and into the beaker below.
Beautiful!
Terry also showed a demonstration of the "distribution of forces" (exercise 168 from the Giant Book of Science Experiments: http://www.amazon.com/Giant-Book-Science-Experiments-Press/dp/0806981393). She used a piece of string wrapped around the hand. It is a principle that is important in understanding how pulleys work, in which a force can be distributed into equal half portions to two branches of a loop of rope or string. She gave each of us a string, and we soon were testing the idea. It worked!
Neat stuff! Thanks, Terry!
Benson Uwumarogie (Dunbar HS,
mathematics) Patterns
Benson
presented an interesting geometric puzzle. He drew a series of
five circles marking either 2, 3, 4, 5, or 6 points on their
circumferences. He then connected the points in all possible ways by
straight lines. For example, for the circle with two points, only a
single straight line across the interior of the circle can be drawn;
for the circle with three points, a triangle can be drawn in the
interior of the circle, etc. Each circle was thus divided into a number
of
non-overlapping regions within its interior. For example, for the
"two-point" circle there are two regions; for the three point circle,
the are four regions; for the four point circle, there are eight
regions; for the five point circle there are 16 regions; for the six
point circle there are 32 regions. This pattern suggests the general
formula for n points, corresponding
to 2^{n-1} regions. The general formula
works! How come?
What a neat way to discover the relationship between physical
patterns and mathematical formulas that describe the patterns.
Good work. Thanks,
Benson.
Bill Blunk (Joliet Central HS,
retired)
Friction
Bill
started with the well known connection between the frictional force
F
and the normal force N: F= mN,
where there is a different
coefficient m for stationary
objects (static
friction) and moving objects (kinetic friction). The static
coefficient can be calculated by placing a wood block on a wooden ramp
and increasing the angle (q with
the horizontal) of the ramp until the block just begins to slide. The
static m is equal to tan q.
Bill then used the job of a roofer to illustrate the
coefficient of friction. The steepness of the pitch of the roof has to
be matched to the coefficient of friction of the material of the roof
to allow the roofer to walk without slipping. For example wet
plywood or plywood with sawdust on it requires a much less steep pitch
than material with a higher coefficient of friction
This summer Bill bought some great things from Amazing Toys in Great Falls, MT: http://www.amazingtoys.net/. In particular, Bill obtained Fun Slides Carpet Skates, which also may be used to illustrate friction. They were slipper type shoes with a special low friction sole that allows sliding on carpets. With these you can also determine the coefficient of friction. Larry Alofs volunteered! First we measured Larry's weight (157 lb). Larry put on the slippers and Bill pulled him along the carpet with a spring scale to measure to force needed to slide him along the carpet at constant speed (24 lb). Therefore the coefficient of (kinetic) friction m for this experiment was F/N or 24/157, or about 0.15. Next we made an incline with Larry and the same system to check the kinetic coefficient of friction. The m of 0.15 is the tangent of a 9° angle. Bill angled the ramp a little more than 9° and, sure enough, an initial slight push started Larry down the ramp at a constant speed. Absolutely terrific! Thanks, Bill!
Roy Coleman mentioned that a great science fair project is to use this technique with the ramp coated with a silicone sealer. At angles of increasing magnitude the object will proceed down the ramp with increasing (constant) speed. That is, its coefficient of friction changes with the angle! How come? Interesting, Roy!
Bill Shanks (New Lenox,
retired)
Center of Gravity
Bill brought in some Cucuzzi (Italian gourd) Squash [http://www.victoryseeds.com/catalog/vegetable/cucurbita/gourds.html]
that he had grown in his garden last summer. They were about 50 to
75 cm long, and about 8 cm in
diameter -- they were irregularly "hooked" in shape (they were also
dry). Bill thought they would be particularly useful for
investigating center of gravity. One squash could be balanced on
Bill's finger, at a point near the center, which identified its
center of mass within the squash. The center of mass could be
determined for a second, more curved squash by balancing it in the
inside of a hook near one end and then the other end and running a
plumb bob from the position of the balancing finger in each case. The
intersection of the two plumb lines identified the center of gravity in
this case, which was below the squash.
Good demo of center of gravity of an irregular object!. Thanks, Bill.
John Scavo (Kelly
HS)
Hybrid Vehicles and Digital Cameras
John
shared two articles (handout) which illustrate how he uses
research to decide what kind of things to buy. One article points out
the great disparity between the actual gas mileage achieved by hybrid
cars compared to the estimated mileages. When different makes or brands
of cars are considered, hybrids have among the greatest disparity
between the two numbers. Charlotte had much better results
with her Toyota
Prius than reported in the article. Others shared even better
results than Charlotte with "gasoline only" cars (eg,
Saab Sonett). A second article talked about digital cameras and
mentioned that the circuitry for almost all modern digitals comes from Kodak.
Thanks, John.
Walter Kondratko (Steinmetz HS,
chemistry) Lemon
Batteries and Burning Pennies
Walter had taken lemons and placed two electrodes (one Zinc
and one Copper) into each lemon. The Zn-Cu pair has a
difference of standard reduction potential of 1.1 Volt. A Voltmeter
held across the two electrodes will test this, as the acids in the
lemon will drive an oxidation-reduction of the couple where they are
inside the lemon. By putting such "lemon cells" in parallel and series
they can investigate various properties/phenomena pertaining to
electrical circuits.
Walter also scratched a penny (minted after 1982, with a Zinc center). He then held the penny in the flame of a portable torch to melt the Zinc so that it would flow out of the scratch. This works because the melting point for Zinc is much lower than that for the Copper "veneer" on the outside of the penny.
Then Walter heated pre-1982 penny and suspended it from a rod. He heated the penny to red hot and then suspended it over a bit of acetone in a beaker. The heat from the penny caused oxidation of the acetone vapor, which produces ketone and methane. The methane burned near the hot metal and this combustion could be seen as a glow around the penny. In other words the glow on the penny persisted as long as the penny was still hot enough to oxidize the acetone vapor and to ignite the methane. Walter made an even better glowing apparatus using a copper strip of about 2 by 10 cm. For additional details see the demonstration CATALYTIC OXIDATION OF ACETONE: http://www.job-stiftung.de/pdf/versuche/Catalytic_Oxidation_Acetone.pdf?hashID=skos9cq79gp42mhdfqj4v8ria7
Terrific! Thanks, Walter.
The following four people could not present lessons today, because we ran out of time. They will be scheduled first at our next SMILE meeting, Tuesday November 15. See you there!
Notes prepared by Ben Stark and Porter Johnson.