High School Mathematics-Physics SMILE Meeting 1997-2006 Academic Years Pythagorean Theorem |

**10 March 1998 Hoi Huynh [Clemente HS] **

She used graph paper to illustrate the **Pythagorean Theorem**,
and showed
a demonstration of it using the areas of the regions on the graph
paper. She
extended the resulting angle figure, rotated the figure, and applied
ways of
resolving the differences.

In the diagram, the area of the big square is **(a + b) ^{2}**,
whereas the area of each of the four triangles of sides

**(a + b) ^{2 }= c^{2} + 4 ´½ab**
;

or

__Comment by Porter Johnson__: The** Pythagoreans** were an
ancient
Greek cult who explored the mystical wonders of mathematics, and who
were
forbidden to reveal their mathematical discoveries upon punishment by
death.
Check out the websites http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html
and http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/PythagorasTheorem.gif

**16 March 1999: Al Tobecksen [Richard Voc HS]**

He brought in some carpenter's squares and had us work with a way of
finding the
hypotenuse of a right triangle using a square (90deg) and a yard-meter
stick. He
passed out a table ** a, b, c**. and had us use the scale on the
right angles for the
sides, and the meter- yard stick as the hypotenuse: and then use a
calculator.

**11 April 2000: Bill Shanks (Joliet Junior College, Music)**

showed us how to generate ** Pythagorean numbers**. He defined
these as
integers ** a, b, c** that satisfy the relation

**10 October 2000
Marilynn Stone (Lane Tech HS)**

gave each of us a resealable
sandwich bag containing these items:

She then challenged us to assemble the pieces together to form a rectangle. (2 green rectangles

3 blue squares

4 red triangles.

Good reinforcement.

**04 December 2001: Hoi Huynh (Clemente HS) Areas of Polygons
Hoi** showed us a

Area = [ (top length + bottom length) /2 ] ´ height = [ (a + b)/2 ] ´ h

The **balance** comes in because one must "balance" the length of
the top and the
bottom **(a+b)/2**, and the
**right** comes in because one must use the "right angle" or
perpendicular distance between the
sides; ie, the height **h** We can use the same principle for
other figures

**PARALLELOGRAM**

For the parallelogram the top and bottom sides are equal, and we get
** Area = a ´ h**.

For the triangle the top side has length **zero**, and the bottom
side has length **a**,
so that the average is **a/2**, Thus, **Area = a ´ h / 2**.

By taking any polygon and cutting it into trapezoidal pieces, she showed us how to calculate its area.

Next Hoi showed us how to prove the **Pythagorean Theorem **using
area
formulas. She took a square of edge length (a + b), and divided
each of
the edges into components of lengths a and b, as shown:

The total area of the square is (a + b) ^{2}. Inside
that square,
there is a tilted smaller square of side c, with area c^{2}.
In
addition, there are four congruent right triangles, each with
sides a, b,
and hypotenuse c. Thus, the area of the larger square may be
written
as the sum of the area of the inner square plus the areas of the four
(identical) triangles.** ...**

or

with the result

**Hoi** told us that ** Mrs Pythagoras** really invented the
theorem while she was supposed to be knitting
her husband's socks, and she let him pretend to discover it on his own.
Perhaps
this interesting story may not be completely accurate:

Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heal of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]:-Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.htmlBoth men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians.

- that at its deepest level, reality is mathematical in nature,
- that philosophy can be used for spiritual purification,
- that the soul can rise to union with the divine,
- that certain symbols have a mystical significance, and
- that all brothers of the order should observe strict loyalty and secrecy.

Very nice, **Hoi**!

**25 March 2003: Hoi Huynh
[Mathematics Teacher] Maintaining Balance
in
Mathematics
Hoi** demonstrated that, when a standing cylinder has

**Hoi **pointed out that, of all quadrilaterals that
have a given perimeter **p** , the square has the greatest area, **A
= p ^{2}/16**.
For all shapes of a given perimeter

Hoi also
mentioned that the formula for the **Area of a Trapezoid** of bases
**b _{1}** and

**Thanks for the insights, Hoi!**

**08 March 2005: Bill Shanks [retired, Joliet New Lenox
environs]
Geometry
Bill **reminded us of the

/| c = b + d / | b / | /___| a

(b + d)

2 b d + d

2 b d = a

b = (a

a |
b |
c=b+1 |

3 |
4 |
5 |

5 |
12 |
13 |

7 |
24 |
25 |

9 |
40 |
41 |

11 |
60 |
61 |

13 |
84 |
85 |

25 |
312 |
313 |

35 |
612 |
613 |

999 |
499000 |
499001 |

a |
b |
c=b+2 |

4 |
3 |
5 |

6 |
8 |
10 |

8 |
15 |
17 |

10 |
24 |
26 |

12 |
35 |
37 |

100 |
2499 |
2501 |

1000 |
249999 |
250001 |

a |
b |
c=b+8 |

12 |
5 |
13 |

16 |
12 |
20 |

20 |
21 |
29 |

24 |
32 |
40 |

28 |
45 |
53 |

36 |
77 |
85 |

888 |
49280 |
48288 |

a |
b |
c=b+9 |

15 |
8 |
17 |

21 |
20 |
41 |

27 |
36 |
45 |

33 |
56 |
65 |

39 |
80 |
89 |

111 |
680 |
6893 |

999 |
55440 |
55449 |

**Geometry (Earl Z)**: A little acorn grew eventually into a big oak. Then he looked out and remarked: "Gee, Ah'm a tree".**Hypotenuse (Bill S)**: Poorly spelled sign on bathroom door, which should have read: "Hi; pot in use".

I |
I + 1 |
J |

3 |
4 |
5 |

20 |
21 |
29 |

119 |
120 |
169 |

696 |
697 |
985 |

4059 |
4060 |
5741 |

23660 |
23661 |
33461 |

Note that the scale of subsequent triangles increases by **a
factor of about 6** each time. For discussion see the solution
to** Problem #152** on this website: http://www.dansmath.com/probofwk/probar16.html#anchor585356. For
a general solution and a relation of these triangles to the **Pell
Equation**, see http://wapedia.mobi/en/Pell_number.

**Thanks for the ideas, Bill!**