Dianna Uchida (Morgan Park HS,
Roots Using Sotolongo's
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-s2)/(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-s2)/(2s+ 1)||s + (n-s2)/(2s+ 1)||Ön|
Terry Donatello (Weber HS,
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,
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 2n-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,
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,
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
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,
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.