Algebra II Day 1

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Algebra IIDay 1
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Set Notation

• Sets are collections of objects called members or
  elements.

Ken is an element (member) of the set containing
Al, Bo, Ken and Dan.
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Sets of numbers
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This picture might help you remember the
Integers. Note how 5 is the opposite thumb of
-5.
-3
3
-2
4
2
-4
-1
1
0
5
-5
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Key word Ratio
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Literally Not rational
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Venn Diagram
B
A
A Venn Diagram shows the relationship between two
or more sets.
The points in the rectangle represent the
universal set of all objects related to whatever
objects are found in sets A and B.
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Subset and Proper Subset
B
The Venn Diagram shows
A
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. . .-3, -2, -1
0
1,2,3..
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Tree Diagram
Venn Diagram
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Unite the points in sets A and B together.
Any point in either set A OR set B is shaded.
Mathematical ORs are inclusive ORs which means
we include the points that are in both sets.
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Intersection indicates the points that are in
both sets A AND B
Intersection
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N
ON
Example Universal set Natural Numbers
The complement of the odd natural
numbers ON is the Even
Naturals ( EN ) which is shaded light blue.
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because
An odd number plus an odd number is not always
odd.
The product of two odd numbers is always odd.
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Since the foal is inside the fence with his
parents, the gate is closed.
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Since the foal is outside the fence and his
parents are inside the fence, someone must have
forgot to close the gate during the night. It
was not closed.
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Summary

• A set is closed under addition if the sum of
  any two numbers in the set belongs to the set.

A set is closed under multiplication if the
product of any two numbers in the set belongs to
the set.
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Related Topics and Practice links The
following slides contain many interesting
historical facts or links to sites of interest.
Some links will offer additional practice and are
usually fun to do.
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Rational Numbers Terminating and repeating
decimals
What is true about all terminating decimals?
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The earliest place value system may have been
developed by the Mayans of Central America.
Slowly our characters were developed. Here are
some samples
Smith, D.E. History of Mathematics Volume II,
Dover Publications 1958
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Various forms of the numerals used in
India. Smith, D.E. History of Mathematics
Volume II, Dover Publications 1958
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European and Oriental forms Smith, D.E. History
of Mathematics Volume II, Dover Publications 1958
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The following set of slides explains how the
Egyptians approximated pi. One of many
interesting web sites I found concerning
pi. Taken from tburg.k12.ny.us/anderson/Powerpo
int/circle-area-proof.ppt256,1,slide201
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Click your mouse for the next idea !
How would you calculate the area of this circle ?
...probably using the formula A ? r 2
Since the diameter is 2 feet,
?
The constant ?, called pi, is about 3.14
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Click your mouse for the next idea !
LETS explore how people figured out circle areas
before all this ? business ?
The ancient Egyptians had a fascinating method
that produces answers remarkably close to the
formula using pi.
?
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Click your mouse for the next idea !
The Egyptian Octagon Method
Draw a square around the circle just touching it
at four points.
?
2 feet
What is the AREA of this square ?
Well.... it measures 2 by 2, so the area 4
square feet.
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Click your mouse for the next idea !
The Egyptian Octagon Method
Now we divide the square into nine equal smaller
squares. Sort of like a tic-tac-toe game !
2 feet
Notice that each small square is 1/9 the area of
the large one -- well use that fact later !
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Click your mouse for the next idea !
The Egyptian Octagon Method
Finally... we draw lines to divide the small
squares in the corners in half, cutting them on
their diagonals.
2 feet
Notice the 8-sided shape, an octagon, we have
created !
Notice, also, that its area looks pretty close to
that of our circle !
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Click your mouse for the next idea !
The Egyptian Octagon Method
The EGYPTIANS were very handy at finding the area
of this Octagon
2 feet
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Click your mouse for the next idea !
The Egyptian Octagon Method
...and ALTOGETHER weve got...
2 feet
For a total area that is 7/9ths of our original
big square
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Click your mouse for the next idea !
The Egyptian Octagon Method
FINALLY...
Yep, were almost done !
The original square had an area of 4 square feet.
2 feet
So the OCTAGONs area must be 7/9 x 4 or 28/9
or 3 and 1/9
or about 3.11 square feet
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AMAZINGLY CLOSE to the pi-based modern
calculation for the circle !
3.11 square feet
3.14 square feet
only about 0.03 off...
about a 1 error !!
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Related Websites
Interactive practice concerning set notation.
Click on the link.

• http//nlvm.usu.edu/en/nav/frames_asid_153_g_2_t_1
  .html?openinstructions
• Related topic concerning the history of pi.
  
• http//www.exploratorium.edu/pi/history_of_pi/inde
  x.html?openinstructions
• Prime numbers. Do you know two years a 19
  year old student wowed the mathematics world by
  finding a procedure for finding huge prime
  numbers.
• http//primes.utm.edu/notes/3021377/.html?openi
  nstructions