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.

*AMP 2012-2013:

Saturday #2

*What are we doing here?

*Too much math never killed anyone.

*The Rules of Engagement

*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 RED broomstick is three feet longThe YELLOW broomstick is four feet longThe GREEN broomstick is six feet long

The Broomsticks

*Perimeter

*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”?

*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.*Measurement

•Using objects at your table measure the angle

*Angles

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^2Area of red square is 2r^2

Area of circle is…

X

Y

ZDraw the following parallel and perpendicular lines:

X: Along the right side of the hypotenuse’s square

Y: Perpendicular to line “X” passing through the corner of the opposite square

Z: Perpendicular to line “Y” passing through the intersection of the square and line “Y”.

If the Pythagorean Theorem is true AND

If you have constructed and cut correctly

THENYou should be able to show that the sum of the area of the smaller squares equals the area of the larger square.

𝑎2+𝑏2=𝑐2

Image from wikipedia. cc-sa

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 5771 1.4142156861 4082 12 12 12 12 12

2

11 1.41420112 12 12 12 122

11 1.4142912 12 12 12

2

11 1.4137912 12 122

11 1.41612 122

11 1.4122

11 1.52

Cut this into 408 pieces

408577

Copy one piece 577 times

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 two-year-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?

Area of blue square =

a

b

Area of whole (red) square =

(𝑎+𝑏)(𝑎+𝑏)b

a

Area of one green triangle =

OR

c

This means that:

𝑎2+𝑎𝑏+𝑎𝑏+𝑏2=2𝑎𝑏+𝑐2

𝑎2+2𝑎𝑏+𝑏2=2𝑎𝑏+𝑐2

𝑎2+𝑏2=𝑐2

a

a

b

b

cc

c

Speaking of areas…

• Is “Area” a measure?

• Or is it an attribute to be measured?

From the CCSS (Grade 3, p. 21)

*Geometric Fractions*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

11 11

1 21 21 1

1 51 1.61 311

1 81 1.61 51 111

1 131 1.6251 81 11 111

1 211 1.6151 131 11 11 111

1 341 1.6191 211 11 11 11 11

1

1 551 1.6181 341 11 11 11 11 111

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…

*Working with similar figures

*“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.