The Arizona Mathematics Partnership: Saturday 4: Geometry

Ted Coe, April 11, 2015 cc-by-sa 4.0 unported unless otherwise noted

Too much math never killed anyone.

The RED broomstick is three feet long

The YELLOW broomstick is four feet long

The GREEN broomstick is six feet long

The Broomsticks

Perimeter

What is “it”?

Is the perimeter a measurement?

…or is “it” something we can measure?

What do we mean when we talk about “measurement”?

Measurement

•Using objects at your table measure the angle

Angles

http://tedcoe.com/math/radius-unwrapper-2-0

http://tedcoe.com/math/radius-unwrapper-2-0

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

http://tedcoe.com/math/circumference

Geometric Fractions

1

2 34 5

See: A. Bogomolny, Pythagorean Theorem and its many proofs from Interactive Mathematics Miscellany and Puzzleshttp://www.cut-the-knot.org/pythagoras/index.shtml, Accessed 12 September 2014

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

Find the dimensions of the rectangle

Find the area of the rectangle

Find a rectangle somewhere in the room similar to the shaded triangle

When we say two figures are similar we mean…

Answer on your own. Share.

Similar Figures

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

CCSS: HS Geometry (p.74)

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

HTTP://TEDCOE.COM/MATH/GEOMETRY/PYTHAGOREAN-AND-SIMILAR-TRIANGLES

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

http://tedcoe.com/math/geometry/similar-triangles

Everything is better in 3D!

Volume What is volume?

What do students find challenging about understanding volume?

CCSS: Grade 5 (p. 33)

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Mathematical Practices1. Make sense of problems and persevere in solving

them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriately tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Cavalieri’s PrincipleSuppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes.

From http://en.wikipedia.org/wiki/Cavalieri%27s_principle. 4/7/2015 CC-BY-SA 3.0Image by Chiswick Chap (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

High School

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

“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

An irrational tangent:

Cut this into 408 pieces

408

577

Copy one piece 577 times

It will never be good enough.

4142156.1

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.” ”http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

…except Hippasus

Too much math never killed anyone.

Archimedes died c. 212 BC …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

Domenico-Fetti Archimedes 1620 http://en.wikipedia.org/wiki/Archimedes#mediaviewer/File:Domenico-Fetti_Archimedes_1620.jpg

…except Hippasus

Too much math never killed anyone.

…and Archimedes.