Don Kanner (Lane Tech HS,
Who Lives Where?
Don handed out the following problem:
I have been very fortunate in being able to make arrangements for all my staff. Alf, Bert, Charlie, Duggie, Ernie, Fred, and George, as well as my chauffeur, Hubert, and my secretary, Judith, to live in houses (all different) on Domum Road. This road has houses numbered from 1 to 55. My employees all made statements about the numbers of their houses as follows:This problem was taken from the book Puzzles for Pleasure by E R Emmet [Emerson Books 1972, ISBN 0-87523-178-0]: http://half.ebay.com/cat/buy/prod.cgi?cpid=3168035&pr=1518126. It can be analyzed by writing a large number of simultaneous equations and solving by brute force. Don showed us a "graphical" way of solving it that was simpler and more fun. Don drew a number line, with each number (integer) represented by a dot. Starting with the first statement (Alf and Bert) and walking up and down the number line, on arbitrarily marked positions, he marked the position of Bert's house with a B at the origin, and the position of Alf's house with an A at +23. He continued this process to mark the position of Charlie relative to Bert, etc, until he had used all the information from the nine statements. From this it became apparent that the one false statement is: "D is 12 more than E". This is the only way for a single inconsistency to occur. Simply powerful! Try it!
I discovered afterwards that one of these statements was not true. Find the numbers of the houses in which all my nine employees live.
- Alf said that his number was 23 more than Bert's.
- Bert said that his number was 16 less than Charlie's.
- Charlie said that his number was 19 less than Duggie's.
- Duggie said that his number was 12 more than Ernie's.
- Ernie said that his number was 30 more than Fred's.
- Fred said that his number was 17 less than George's.
- George said that his number was 37 less than Hubert's.
- Hubert said that his number was 12 more than Judith's.
- Judith said that her number was 10 more than Alf's.
Don then stated The Four Laws of Scitechnoliterarydynamics:
Roy Coleman (Morgan Park HS,
The idea is, using a ruler and stop watch, to plot the depth of the water in a toilet tank as a function of time as it is flushed and then refills. Students routinely obtain three distinct types of plots, which he described. Which one was correct, and why? We were able to identify the correct one and to figure out why the students got incorrect curves (meter stick in the tank upside down; meter stick in the toilet bowl -- not the tank!). Roy does the above activity (as well as having students calculate the volumes of their bathrooms, in cubic meters) the night before parents' night. This in-home exercise is a good icebreaker, since parents have seen the students the night before working together on this unconventional activity.
Roy called attention to an amazing internet picture of the shock wave ahead of the sonic boom, around a supersonic airplane; see http://antwrp.gsfc.nasa.gov/apod/ap010221.html. Wow! Thanks, Roy!
Walter McDonald (CPS substitute teacher; VA X-ray
Bits and Computers
Walter passed around Playthink 612: Binary Abacus, taken from the book Playthinks by Ivan Moscovich [Workman Publishing, 2002; ISBN 0-7611-18268]: http://www.amazon.co.uk/exec/obidos/ASIN/0761118268/026-2213757-8924459. If a computer processor has 6 switches, each representing 1 or 0 in their respective settings, it can represent numbers from 1 to 63. For example, the decimal number 53D is equivalent to the binary number 110101B, whereas 63D is equivalent to 111111B. Numbers may be added, subtracted, multiplied, and divided in that processor, as long the numbers always remain in that range. Thanks for the info, Walter!
Fred Schaal (Lane Tech HS,
Dubito Ergo Sum
A student recently asked Fred this question: "Who invented graph paper?" Fred 'Googled' this question; that is, he used the Google website: http://www.google.com. The answer came back as René Descartes, the inventor of Cartesian coordinates; see the biographical entry on the ST Andrews University (UK) website: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html. See also Descartes' Geometric Solution to a Quadratic equation: http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/DescartesSoln.gif. The most famous assertion of Descartes is "Cogito ergo sum" [I think; therefore I am], but he may actually have meant "I doubt; therefore I am".
Fred also asked the age of The Naughty Lady from Shady Lane: http://www.theguitarguy.com/naughtyl.htm. Aficionados of songs from the 50's will no doubt recall that, in the 1954 song, "She's only nine days old!". Fred has started to have a regular Google Session in his math classes, to address such issues. Thanks, Fred
Larry Alofs (Kenwood HS physics,
Larry said that they used to be called "magnetoids" (They are also called "magnetic hematite torpedoes", "kissing stones", "UberOrbs" and "snake eggs".), and they were a bit expensive (about $ 80 for a pair, and obtainable only from England). This year he has seen them at flea markets for only $ 4-5 each. They are two roughly egg shaped objects 6-8 cm long with the magnetic poles on the sides-- and not on the ends! They are powerful magnets, and due to their shape and structure their attraction and repulsion lets us do some really neat demos. For example, they may make a hissing sound when they come together. When they are placed close to one another on a smooth horizontal surface, they jump, hiss, and spin around, and then stick together. How come? Can we breed our own? They can be ordered from various sources where they are called "snake eggs" or "zingers". Thanks for the show, Larry!
Khara Criswell (Benito Juarez HS, chemistry and
Fire and Ice
Khara floated a bit of kerosene on top of some bottled water in an Erlenmeyer flask. The water was almost the entire volume of the liquid in the flask, but the layer of kerosene (basically occupying the neck of the flask) on top was about 2 cm deep. From a few meters away it looked like just all water. Khara lit a few matches and placed them burning head down into the liquid. They continued to burn brightly as the kerosene burned. Khara doesn't tell the students the secret until the end of the year, leaving them wondering how "water" could burn. Note: you should check with local hazardous material regulations before attempting this experiment, and exercise extreme caution with the flammable materials.
A bowl a little larger than a cereal bowl was filled with ice cubes and in it placed a lump of a semisolid colloid prepared as follows:
Ben Stark (Professor of Biology,
Mrs Levine's Pickle Recipe
Ben planted seedlings of pickling cucumbers grown last spring in the lesson given by Chris Etapa [bc041205.html], and he harvested pickles from them this summer. He used these to make homemade pickles. Pickling is a natural fermentation process that is used to make fermented cabbage and other fermented vegetables as well as silage. It demonstrates lessons in both microbiology and biochemistry. The pickling cucumbers have a natural bacterial flora that is fairly complicated. The cucumbers are sliced and placed into brine and then sealed. The brine (high salt) inhibits growth of "spoilage" microorganisms but allows growth of Lactobacilli, which ferment sugars in the fruit (also in cabbage, beets, silage, etc.) into lactic acid by a particular fermentation pathway. Eventually, the pH drops so far due to the acid production that the Lactobacilli die. But by this time (7- 14 days, depending on the temperature at which the jars are stored), the pickles are done. One brine recipe is given here:
"Silage or Ensilage: succulent, moist feed made by storing a green crop in a silo. The crop most used for silage is corn; others are sorghum, sunflowers, legumes, and grass. In a sealed silo, typically in the past a tall cylindrical structure but often today in a surface pile covered tightly with heavy-gauge plastic, the crop ferments for about one month. This fermentation process, called ensiling, produces acids and consumes the oxygen in the silo, preserving the plant material. In pit ensiling, compacted silage ferments in an unsealed underground enclosure. Silage replaces or supplements hay for cattle, horses, and sheep. It is rich in carotene, an important source of vitamin A. A machine called an ensilage harvester cuts and chops the crop in one operation, preparing it for storage in the silo."Also, it was pointed out that the pickle man played a starring role in the 1988 film Crossing Delancey: [http://www.washingtonpost.com/wp-srv/style/longterm/movies/videos/crossingdelanceypghowe_a0b1bf.htm]. Thanks for the insights, Ben!
Charlotte Wood-Harrington (Gwendolyn Brooks HS,
Cheap Constant Velocity Cars + Kinetics Problem
Charlotte found an internet source for "flip over buggies" [http://www.joissu.com], where they can be obtained at $1.29 each, in lots of 50. They can also be ordered from Flinn Scientific [http://www.flinnsci.com/] for $ 7.95 each. They run on a C-cell battery, which is not included. These buggies travel along very straight paths, and are quite useful in showing and visualizing aspects of motion at constant speed.
Charlotte then talked about slopes and teaching slopes. Four volunteers each held a piece of PVC pipe about 8 cm in diameter that had been sliced in half lengthwise to produce troughs about 50 cm long. The team then was charged with producing a ramp that could roll a big super-ball into a tin can. This allowed the team to work together to adjust the slope of the four piece ramp to modify the speeds of the passage of the ball down the ramp. Eventually they got the ball into the can. Neat ideas! Thanks, Charlotte.
Paul Fracaro (Joliet Central HS,
Paper Plate Fractions + Whiteboard Demonstration
Paul used paper plates to explore fractions. The thin (inexpensive) paper plates can be folded into halves, fourths, and eighths to demonstrate fractions. Then he used the same technique to show how he helps students understand better how to do simple additions and subtractions involving whole numbers and fractions, followed by additions and subtractions when the denominators are not the same.
Paul then showed us some white boards the size of typical letter paper printed with a rectangular grid, as well as X and Y axes. With markers, it is a convenient way for the students to graph. They are available from ETA Cuisinaire: http://www.etacuisenaire.com/. Neato! Thanks, Paul.
Notes prepared by Ben Stark and Porter Johnson.