1. The Arizona Mathematics Partnership: Saturday 2: Geometry Ted Coe, September 2014 cc-by-sa 3.0 unported unless otherwise noted

2. Warm-up: Geometric Fractions

3. Check for Synthesis: 3 SOURCE:

4. Geometric Fractions

5. THE Rules of Engagement Speak meaningfully — what you say should carry meaning; Exhibit intellectual integrity — base your conjectures on a logical foundation; don’t pretend to understand when you don’t; Strive to make sense — persist in making sense of problems and your colleagues’ thinking. Respect the learning process of your colleagues — allow them the opportunity to think, reflect and construct. When assisting your colleagues, pose questions to better understand their constructed meanings. We ask that you refrain from simply telling your colleagues how to do a particular task. Marilyn Carlson, Arizona State University

6. Define Square Triangle Angle

7. Quadrilaterals

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

10. 10 SOURCE: HTTP://TEDCOE.COM/MATH/WP-CONTENT/UPLOADS/2013/10/BROOMSTICKS-FOR-NCTM.DOC

11. 11 SOURCE: HTTP://TEDCOE.COM/MATH/WP-CONTENT/UPLOADS/2013/10/BROOMSTICKS-FOR-NCTM.DOC

12. 12 SOURCE: HTTP://TEDCOE.COM/MATH/WP-CONTENT/UPLOADS/2013/10/BROOMSTICKS-FOR-NCTM.DOC

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

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

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

17. 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

18. So.... how do we measure circumference? Circumference

19. Tennis Balls

20. Circumference If I double the RADIUS of a circle what happens to the circumference?

22. The circumference is three and a bit times as large as the diameter. http://tedcoe.com/math/circumference

23. What is an angle? Angles

24. Using objects at your table measure the angle Angles

25. CCSS, Grade 4, p.31

28. Measure the length of s. Choose your unit of measure carefully. Measure the angle. Choose your unit carefully. s

29. Define: Area

30. Area: Grade 3 CCSS

32. What about the kite?

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

37.  Cut out a right triangle from a 3x5 card – try to make sure that one leg is noticeably larger than the other.  What strategies could you use to create this? ab c

38. Lay down your triangle on construction paper. Match my orientation with the right angle leaning right. Draw squares off each of the three sides. Measure the areas of these squares.

39. Let’s try something crazy… I came across an interesting diagram and I want to walk you through the design. 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

40. 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

41. 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

42. Perpendicular 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

43. Perpendicular 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

44. 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

46. 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.

47. Is this a proof?

49. Area of blue square: c 2 a b Area of whole (red) square: (a + b)(a + b) b a OR c This means that: (a + b)(a + b) = 2ab + c 2 a 2 + ab + ab + b 2 = 2ab + c 2 a 2 + 2ab + b 2 = 2ab + c 2 a 2 + b 2 = c 2

50. CCSS: Grade 8

51. CCSS: HS Geometry

52. http://www.cut-the-knot.org/pythagoras/index.shtml

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