After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.
You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters a and b. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship is always true for every right triangle.
We continue to excavate the ancient building complex, which I believe may have been Pythagoras’s school. Yesterday, one of our digging crews uncovered a mosaic tile floor in the courtyard. The pattern of the tiles alternates between two square designs. (See enclosed sketches.)
During your family’s recent visit, you expressed an interest in the mathematical ideas of Pythagoras. Could you or your father offer us any insight into what these tile designs may represent?
[When Alexandria Jones and her family visited an excavation in southern Italy, they learned several tidbits about the ancient school of mathematics and philosophy founded by Pythagoras. Here is Alex's favorite story.]
It hit the Pythagorean Brotherhood like an earthquake, a crisis of faith which shook the foundations of their universe. Some say Pythagoras himself made the dread discovery, others blame Hippasus of Metapontum.
Something certainly did happen with Hippasus. The Brotherhood sent him into exile for insubordination, or for breaking the rule of secrecy — or was it for proving the unthinkable? According to legend, Hippasus drowned at sea, but was it a mere shipwreck or the wrath of the gods? Some say the irate Pythagoreans threw him overboard…
Take a break from “serious” math and have a little fun today with some classics of recreational mathematics. Do you have a favorite math or logic fallacy? Please share it in the Comments below.
Are your students doing anything special for Day? After two months with no significant break, we are going stir crazy. We need a day off — and what better way could we spend it than to play math all afternoon?
I have been busy with the end of Math Olympiad season and getting ready for the MathCounts state test this weekend, but I wanted to post this link before it’s too late. You have until Sunday evening to send in your answer to the first…
My pre-algebra class hit the topic of equations just as the NFL season moved into the playoffs. The result was this series of class notes called “The Game of Algebra.”
We used the Singapore Math NEM 1 textbook, which is full of example problems and quality exercises. These notes simply introduce or review the main concepts and vocabulary in a less-textbooky way.
As we all head back to school, here are some interesting calendar puzzles:
2008 is a leap year. Why do leap years happen? If we didn’t add a leap day every so often, would January eventually come in the summer?
Today is Thursday. What day of the week will it be exactly one year from today?
January 1, 2008, came on a Tuesday. When will be the next year that begins on Tuesday?
My birthday (in March) lands on a Monday this year. When is the next year my birthday will come on a Monday? How about YOUR birthday — when is the next time it will happen on the same day of the week as this year?
Can you find a pattern in the way dates move from one day of the week to another, year after year?
Alexandria Jones and her family piled into the car for a drive in the country. This year, they were determined to find an absolutely perfect Christmas tree at Uncle William Jones’s tree farm.
“I want the tallest tree in Uncle Will’s field,” Alex said.
“Hold it,” said her mother. “I refuse to cut a hole in the roof.”
Almost all math problems call for the student to assume one thing or another. Without assumptions — definitions, postulates, axioms, common notions, or whatever you want to call them — mathematics of any kind is impossible. Tony at Pencils Down (who plans to be a math teacher when he grows up) reminds us that, necessary though it may be, we are stepping on dangerous ground when we assume:
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Picture from MacTutor Archives.
Day? After two months with no significant break, we are going stir crazy. We need a day off — and what better way could we spend it than to play math all afternoon?
