**High School Mathematics-Physics SMILE Meeting
1997-2006 Academic Years
Mathematics: Algebra**

**29 February 2000: Walter McDonald (CPS substitute/Medical
Technician)**

showed us about inequalities. He showed us both algebraic expressions
and
their geometric graphs to visualize those inequalities. Eg. A
parabola

was plotted, and then a region within the parabola representing

(handout) Good math connections!

**05 September 2000 Lee Slick (Morgan Park HS)**, who wrote down
the
squares of numbers ending in **5**:

5^{2} |
= |
25 |

15^{2} |
= |
225 |

25^{2} |
= |
625 |

35^{2} |
= |
1225 |

45^{2} |
= |
2025 |

etc. |

With Lee's help, we saw a pattern: All the results end in
**25**. If we multiply the ten's digit by the next higher
digit, we get the number to place before the **25**. To square
**35**, for example, multiply the **3** by **4** to get
**12**, and we have **1225** as the result. More neat
ideas! Thanks,** Lee**!

**14 March 2001 Fred Schaal (Lane Tech Park HS, Math)**

considered the algebra problem of factoring the following **sixth
order **polynomial
**z ^{6} - a^{6}**. He first pointed out that, if you
consider
this as the difference of two cubes, and then consider

On the other hand, if you consider the ** **polynomial
**z ^{6} - a^{6}** as the difference of two squares,
and then
use the appropriate cube formula on each of the factors, you get

Root Number |
Root Name |
Root Value |

0 |
z_{0} |
a |

1 |
z_{1} |
a/2 [1 + i Ö3
] |

2 |
z_{2} |
a/2 [1 - i Ö3
] |

3 |
z_{3} |
- a |

4 |
z_{4} |
a/2[-1 - i Ö3
] |

5 |
z_{5} |
a/2 [-1 + i Ö3
] |

The polynomial in question may thus be factored in the form| z_{2}* | * z_{1}| | | | z_{3}| z_{0}---*--------------|---------------*---- | | | | | | z_{4}* | * z_{5}

(z - z

(z - z

For the fascinating history of the Mathematician **Evariste
Galois**, who studied roots of polynomial equations, see the website
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html.
Here is a comment, taken from that website, by one of his teachers:

**"It is the passion for mathematics which dominates him, I think it
would
be best for him if his parents would allow him to study nothing but
this, he is
wasting his time here and does nothing but torment his teachers and
overwhelm
himself with punishments."**

**01 May 2001 Bill Colson (Morgan Park HS, Math)**

showed a number of different ways of writing expressions that are
identities
for** the number 1**:

**- i^{2} = - ( -1 ) = e^{0} =
sin^{2}
q +cos^{2} q
= 1 **

**Porter Johnson** mentioned the famous ** Euler ** relation

which involves the "five most important" numbers **1, 0, i, e, and
p**.

**02 December 2003: Fred J Schaal [Lane Tech HS,
mathematics]
Graphing Parabolas: a Challenge
Fred **drew a graph of a parabola (concave upward) on the board. He
then wrote down the formula for a parabola,

He then drew a horizontal line through the parabola, so that there were two zeros,Q1: For what value(s) of x does y = 0?

Q2:For what value(s) of x does y take on its minimum value? Why?

... the derivative is y' = 2ax + b = 0

How do you show that x_{min}= -b/(2a) using simple algebra?

Who can answer ** either or both of these questions?** More about
this in the future!

**Fred, you have laid down the gauntlet. Thanks!**

**09 December 2003: Fred Schaal [Lane Tech HS,
mathematics] Unleashing
Complex Numbers**

Fred extended the consideration of zeroes of quadratic
functions; **ax ^{2 }+ bx + c = 0 **, which he began at
the last

and
asked what happens in the case **(a, b, c) = (1, 2, 3)**?, In that
case one
obtains ** x = -1 ± Ö(-2)**.
This case, as
well as many, many others, involves taking the square root of a
negative
number. By adopting the notation **Ö(-1)
= i or** i** ^{2} = -1**, he introduced complex
numbers and
wrote the answer as

**1 - (± 4i ) -4 + 2 (± 2i) + 3 ^{?}=^{?} 0**

**
0 = 0**

**All right!** **So, complex numbers are not so complex, after
all! Thanks, Fred!**

**Porter Johnson** mentioned that complex
numbers were originally used merely to solve polynomial equations,
after **Gauss**
showed that every **n-th order polynomial equation **has **n**
(possibly
degenerate) complex roots. Much later, a mechanical engineer
named **Fourier**
made explicit use of the **Euler** formula, **e ^{ix} =
cos x + i sin
x**, to develop

**01 November 2005: 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: **Ö****n** is approximated by
** s + (n-s ^{2})/(2s+ 1) **
where

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 |