*
Ted Coe, Scottsdale Community College, 2012. Some materials
were also created or refined as part of the development of the
“Math 5: Geometry” curriculum for Arizona State University’s
Teaching Foundations Project.
*
*Too much math never
killed anyone.
* Speak meaningfully
* Exhibit intellectual integrity
* Strive to make sense
* Respect the learning process of your colleagues
* Marilyn Carlson, Arizona State University
*
The Plot...
The Broomsticks
The Broomsticks
The RED broomstick is three feet long
The YELLOW broomstick is four feet long
The GREEN broomstick is six feet long
*Is perimeter a one-
dimensional, two-dimensional,
or three-dimensional thing?
*Does this room have a
perimeter?
*
What do we mean when we talk about
“measurement”?
*
How about this?
•Determine the attribute you want to measure
•Find something else with the same attribute.
Use it as the measuring unit.
•Compare the two: multiplicatively.
*
•Using objects at your table measure the angle
*
Define: Area
Area has been defined* as the following:
“a two dimensional space measured by the number
of non-overlapping unit squares or parts of unit
squares that can fit into the space”
Discuss...

*State of Arizona 2008 Standards Glossary
Area of whole
square is 4r^2
Area of red square
is 2r^2
Area of circle is…
Draw the following
parallel and
perpendicular lines:
Z
Y: Perpendicular to
line “X” passing
through the corner
of the opposite
square
X
Y
Z: Perpendicular to line
“Y” passing through the
intersection of the
square and line “Y”.
X: Along the right side
of the hypotenuse’s
square
If the Pythagorean Theorem is true
AND
If you have constructed and cut correctly
THEN
You should be able to show that the sum of the
area of the smaller squares equals the area of
the larger square.
2
𝑎
Image from wikipedia. cc-sa
+
2
𝑏
=
2
𝑐
If you ask Wolframalpha:
The first proof of the existence of irrational numbers is
usually attributed to a Pythagorean (possibly Hippasus
of Metapontum),who probably discovered them while
identifying sides of the pentagram.The then-current
Pythagorean method would have claimed that there
must be some sufficiently small, indivisible unit that
could fit evenly into one of these lengths as well as the
other. However, Hippasus, in the 5th century BC, was
able to deduce that there was in fact no common unit
of measure, and that the assertion of such an existence
was in fact a contradiction.
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
1
1
 1.5
2
1
1
 1.41429
1
2
1
2
1
2
1
1
1
2
2
2
 1.4
1
1
1
1
2
1
2
 1.41379
1
2
2
1
2
1
2
1
2
1
2
2
1
1
2
1
1
1
1
2
 1.416
 1.414201
1
2
1
2
1
2
1
2

1
2
1
2
1
2
2
1
2
1
2
577
 1.414215686
408
577

408
Copy one piece 577 times
Cut this into 408 pieces
It will never be good enough.
Hippasus, however, was not lauded for his efforts:
according to one legend, he made his discovery
while out at sea,
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
Hippasus, however, was not lauded for his efforts:
according to one legend, he made his discovery
while out at sea, and was subsequently thrown
overboard by his fellow Pythagoreans
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
Hippasus, however, was not lauded for his efforts:
according to one legend, he made his discovery
while out at sea, and was subsequently thrown
overboard by his fellow Pythagoreans “…for having
produced an element in the universe which denied
the…doctrine that all phenomena in the universe
can be reduced to whole numbers and their
ratios.”
“Too much math never killed anyone”
…except Hippasus
Archimedes died c. 212 BC during the
Second Punic War, when Roman forces
under General Marcus Claudius Marcellus
captured the city of Syracuse after a twoyear-long siege. According to the popular
account given by Plutarch, Archimedes
was contemplating a mathematical
diagram when the city was captured. A
Roman soldier commanded him to come
and meet General Marcellus but he
declined, saying that he had to finish
working on the problem. The soldier was
enraged by this, and killed Archimedes
with his sword.
http://en.wikipedia.org/wiki/Archimedes. 11/2/2012
The last words attributed to Archimedes
are "Do not disturb my circles"
http://en.wikipedia.org/wiki/Archimedes. 11/2/2012
“Too much math never killed anyone”
…except Hippasus
…and Archimedes.
Back to the Pythagorean Theorem…
http://en.wikipedia.org/wiki/Pythagoras. 11/2/2012
Is this a proof?
1
a
Area of one green triangle = 2 𝑎𝑏
b
Area of blue square = 𝑐 2
a
c
Area of whole (red) square =
c
b
(𝑎 + 𝑏)(𝑎 + 𝑏)
OR
1
4 ∙ 𝑎𝑏 + 𝑐 2
2
This means that:
c
c
a
b
𝑎 + 𝑏 𝑎 + 𝑏 = 2𝑎𝑏 + 𝑐 2
𝑎2 + 𝑎𝑏 + 𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐 2
b
a
𝑎2 + 2𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐 2
𝑎2 + 𝑏2 = 𝑐 2
Speaking of areas…
• Is “Area” a measure?
• Or is it an attribute to be measured?
From the CCSS (Grade 3, p. 21)
*
* Fractions, Multiplicative
Thinking, and Area
 Find
the dimensions of
the rectangle
 Find the area of the
rectangle
 Find
the dimensions of
the rectangle
 Find the area of the
rectangle
 Find a rectangle
somewhere in the
room similar to the
shaded rectangle
1
1
1
1 1
1
1
1
1
5
1
  1.6
1 3
1
1
8
1
  1.6
1
5
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

55
 1.618
34
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
13
  1.625
8
34
 1.619
21
1
1
1
1
1

1
1
1
21
 1.615
13
1
1
1 2
1   2
1 1

1
1
1
1
1
1
1
1
1
What
do you
mean when
you say two
figures are
similar ?
What
do you
mean when
you say two
figures are
similar ?
www.myheritage.com
What
do you
mean when
you say two
figures are
similar ?
What
do you
mean when
you say two
figures are
similar ?
What
do you
mean when
you say two
figures are
similar ?
What
do you
mean when
you say two
figures are
similar ?
What
do you
mean when
you say two
figures are
similar ?
Two figures are similar if…
* “Similar means same shape
different size.”
* “All rectangles are the same
shape. They are all rectangles!”
* “Therefore all rectangles are
similar.”
*
* Time for some practice in similarity.