AMP 2012-2013: Saturday #2

1. AMP 2012-2013: Saturday #2 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.

2. What are we doing here?

3. Too much math never killed anyone.

4. Speak meaningfully • Exhibit intellectual integrity • Strive to make sense • Respect the learning process of your colleagues • Marilyn Carlson, Arizona State University The Rules of Engagement

5. The Plot...

6. The Broomsticks

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

8. Is perimeter a one-dimensional, two-dimensional, or three-dimensional thing? • Does this room have a perimeter? Perimeter

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

10. 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

11. Using objects at your table measure the angle Angles

12. Define: Area

13. 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

14. Area of whole square is 4r^2 Area of red square is 2r^2 Area of circle is…

15. Draw the following parallel and perpendicular lines: X Z Y 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”. X: Along the right side of the hypotenuse’s square

16. 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.

17. Image from wikipedia. cc-sa

21. If you ask Wolframalpha:

22. 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.Thethen-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

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

26. 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

27. 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

28. 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.”

29. “Too much math never killed anyone” …except Hippasus

30. 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

31. The last words attributed to Archimedes are "Do not disturb my circles"  http://en.wikipedia.org/wiki/Archimedes. 11/2/2012

32. “Too much math never killed anyone” …except Hippasus …and Archimedes.

33. Back to the Pythagorean Theorem… http://en.wikipedia.org/wiki/Pythagoras. 11/2/2012

34. Is this a proof?

35. Area of one green triangle = a b Area of blue square = a c Area of whole (red) square = c b OR This means that: b c c a b a

36. Speaking of areas… • Is “Area” a measure? • Or is it an attribute to be measured?

37. From the CCSS (Grade 3, p. 21)

38. Fractions, Multiplicative Thinking, and Area Geometric Fractions

40. Find the dimensions of the rectangle • Find the area of the rectangle

41. Find the dimensions of the rectangle • Find the area of the rectangle • Find a rectangle somewhere in the room similar to the shaded rectangle

43. What do you mean when you say two figures are similar ?

44. What do you mean when you say two figures are similar ? www.myheritage.com

45. What do you mean when you say two figures are similar ?

46. What do you mean when you say two figures are similar ?

47. What do you mean when you say two figures are similar ?

48. What do you mean when you say two figures are similar ?

49. What do you mean when you say two figures are similar ?

50. Two figures are similar if…