1. Coherence: Multiplicative Reasoning Across the Common Core/AZCCRS AATM September 20, 2014

2. Too much math never killed anyone.

3. Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

4. Ways of Doing?

5. The Broomsticks

6. The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long The Broomsticks Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

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10. 10Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

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14. Focus Coherence Rigor Key Shifts in the AZCCRS 14Source:

15. Ways of Thinking? Learning Progressions in the AZCCRS

16. From the CCSS: Grade 3 16Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)

17. 3.OA.1: Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. From the CCSS: Grade 3 17 Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. pp.48-49

18. 4.OA.1, 4.OA.2 1.Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 2.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. From the CCSS: Grade 4 18Source: CCSS Grade 4

19. 4.OA.1, 4.OA.2 1.Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 2.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. From the CCSS: Grade 4 19Source: CCSS Grade 4

20. 5.NF.5a Interpret multiplication as scaling (resizing), by: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. From the CCSS: Grade 5 20Source: CCSS Grade 5

21. “In Grades 6 and 7, rate, proportional relationships and linearity build upon this scalar extension of multiplication. Students who engage these concepts with the unextended version of multiplication (a groups of b things) will have prior knowledge that does not support the required mathematical coherences.” Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.49

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

23. “Technically, a measurement is a number that indicates a comparison between the attribute of an object being measured and the same attribute of a given unit of measure.” – Van de Walle (2001) But what does he mean by “comparison”? Measurement

24. 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

25. Source: Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

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

28. The circumference is about how many times as large as the diameter? The diameter is about how many times as large as the circumference?

29. Tennis Balls

30. What is an angle? Angles

31. What attribute are we measuring when we measure angles? Angles

32. CCSS, Grade 4, p.31 Source: CCSS Grade 4, 4.MD.5

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

35. 35Source:

37. Check for Synthesis: 37Source:

39. 39Source: Similar Figures

41. CCSS: Grade 7 (p.46) Source: CCSS Grade 7, p.46

42. 42Source:

43. 43Source: http://tedcoe.com/math/geometry/pythagorean-and-similar-triangles

44. http://tedcoe.com/math/algebra/constant-rate

46. CCSS: Grade 8 (8.EE.6, p.54) 46Source: CCSS Grade 8

47. 47Source: You have an investment account that grows from $60 to $103.68 over three years.

48. 48Source:

49. 49

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

51. CCSS: Geometry (G-SRT.6, p. 77) 51Source: CCSS High School Geometry (screen capture)

52. 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. PythagoreanHippasus of Metapontumpentagram http://en.wikipedia.org/wiki/Irrational_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012 A tangent:

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

54. 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

55. 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

56. 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

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

58. 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

59. 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

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

61. Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

62. Standards for Mathematical Practice 62 Eight Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the understanding of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Source: CCSS

63. Materials? 63Source: http://ime.math.arizona.edu/progressions/

64. Ted Coe Director, Mathematics Achieve, Inc. tcoe@achieve.org Twitter: @drtedcoe 64