1. The Arizona Mathematics Partnership: Saturday 4: Geometry Ted Coe, April 11, 2015 cc-by-sa 4.0 unported unless otherwise noted

2. Too much math never killed anyone.

3. The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long The Broomsticks

4. Perimeter What is “it”? Is the perimeter a measurement? …or is “it” something we can measure?

5. What do we mean when we talk about “measurement”? Measurement

6. Using objects at your table measure the angle Angles

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

8. 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

9. http://tedcoe.com/math/circumference

11. Geometric Fractions

12. 1 23 45 See: A. Bogomolny, Pythagorean Theorem and its many proofs from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/pythagoras/index.shtml, Accessed 12 September 2014 http://www.cut-the-knot.org/pythagoras/index.shtml

13. a b Area of whole (red) square = b a OR c This means that: a a b b c c c

14.  Find the dimensions of the rectangle  Find the area of the rectangle  Find a rectangle somewhere in the room similar to the shaded triangle

15.  When we say two figures are similar we mean… Answer on your own. Share.

16. Similar Figures

17. 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.

20. 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.

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

22. 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.

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

25. Everything is better in 3D!

26. Volume What is volume? What do students find challenging about understanding volume?

27. 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.

37. Mathematical Practices 1.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.

41. Cavalieri’s Principle Suppose 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 3.0 Image by Chiswick Chap (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

43. 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.

44. “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.PythagoreanHippasus of Metapontumpentagram 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_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012 An irrational tangent:

45. Cut this into 408 pieces Copy one piece 577 times It will never be good enough.

46. 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_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

47. 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_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

48. “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_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

49. …except Hippasus Too much math never killed anyone.

50. 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/Archimedeshttp://en.wikipedia.org/wiki/Archimedes. 11/2/2012

51. “The last words attributed to Archimedes are "Do not disturb my circles" ” http://en.wikipedia.org/wiki/Archimedeshttp://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

52. …except Hippasus Too much math never killed anyone. …and Archimedes.