1. Multiplicative Thinking cc-by-sa 3.0 unported unless otherwise noted Ted Coe, Ph.D Director, Mathematics Achieve, Inc. 2/5/2015

2. 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

3. Too much math never killed anyone.

4. The Plot

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

6. The Foot

7. From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011)

8. The Broomsticks

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. 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. 14 SOURCE:

15. From the CCSS: Grade 3 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. P.24

16. 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 16 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. P.29

17. 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 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. P.29

18. 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 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. P.36

19. http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

20. “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

21. Learning Trajectories SOURCE: LEARNING TRAJECTORIES IN MATHEMATICS: A FOUNDATION FOR STANDARDS, CURRICULUM, ASSESSMENT, AND INSTRUCTION. DARO, ET AL., 2011.

22. Learning Trajectories SOURCE: LEARNING TRAJECTORIES IN MATHEMATICS: A FOUNDATION FOR STANDARDS, CURRICULUM, ASSESSMENT, AND INSTRUCTION. DARO, ET AL., 2011.

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

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

27. From the AZ STD's (2008) Perimeter: the sum of all lengths of a polygon. Discuss

28. Wolframalpha.com 4/18/2013:

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

30. “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

31. 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

32. From Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

33. Create your own… International standard unit of length. With a rubber band. 1.Use it to measure something. 2.Use it to measure the length of someone else’s band. 3.Use their band to measure yours.

34. What is a circle?

36. Draw a circle with a diameter equal to your international standard unit band length

37. What is circumference?

38. From the AZ STD's (2008) the total distance around a closed curve like a circle Circumference

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

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

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

43. Tennis Balls

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

45. How many Rotations?

47. What is an angle? Angles

48. Using objects at your table measure the angle You may not use degrees. You must focus on the attribute you are measuring. Angles

49. What attribute are we measuring when we measure angles? Think about: What is one degree? Angles

50. 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. P.31 Grade 4 CCSS: 4.MD.5

51. SOURCE:

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

53. What is the length of “d”? You may choose the unit.

54. What is the measure of the angle? You may choose the unit.

55. Indiana (1896) House Bill 296, Section 2: “…that the ratio of the diameter and circumference is as five-fourths to four;” What is the mathematical value they are proposing for Pi? From http://www.agecon.purdue.edu/crd/Localgov/Second%20Level%20pages/indiana_pi_bill.htmhttp://www.agecon.purdue.edu/crd/Localgov/Second%20Level%20pages/indiana_pi_bill.htm

57. Define: Area

58. 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

59. Area: Grade 3 CCSS 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. P.21

65. What about the kite?

68. http://geogebratube.org/student/m279http://geogebratube.org/student/m279 (cc-by-sa) http://geogebratube.org/material/show/id/279

69. Geometric Fractions

70. Check for Synthesis: 70 SOURCE:

72. Geometric Fractions 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. P.42

73. Assume: Angles that look like right angles are right angles Lengths that look to be the same as AB can be verified using a compass

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

76. Or not… http://goldenratiomyth.weebly.com/

77. When I say two figures are similar I mean… Hint: We haven’t defined “proportional” so you cannot use it.

78. What is a scale factor? Teaching Geometry According to the Common Core Standards, H. Wu Revised: April 15, 2012. Grade 7 notes, p.49:

81. Working with similar figures “Similar means same shape different size.” “All rectangles are the same shape. They are all rectangles!” “Therefore all rectangles are similar.”

82. CCSS: Grade 2 (p.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. P.17

83. CCSS: Grade 7 (p.46) 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. P.46

84. CCSS: Grade 7 (p.46) 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. P.46

85. From the Progressions ime.math.arizona.edu/progressions https://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdfhttps://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf p.11 From the Progressions

86. CCSS: Grade 8 (p.56) 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. P.56

87. Teaching Geometry According in Grade 8 and High School According to the Common Core Standards, H. Wu Revised: October 16, 2013, p.45 http://math.berkeley.edu/~wu/CCSS-Geometry.pdf

88. CCSS: HS Geometry (p.74)

90. CCSS: Geometry (G-SRT.6, p. 77) 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. P.77

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

92. 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. P.36 CCSS: HS Geometry (p.74)

94. From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011)

95. What does it mean to say something is “out of proportion”?

96. “A single proportion is a relationship between two quantities such that if you increase the size of one by a factor a, then the other’s measure must increase by the same factor to maintain the relationship” Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), Research companion to the Principles and Standards for School Mathematics (pp. 95- 114). Reston, VA: National Council of Teachers of Mathematics.(p.18 of pdf)Fractions and multiplicative reasoning

97. On the Statue of Liberty the distance from heel to top of head is 33.86m How wide is her mouth?

98. http://www.nps.gov/stli/historyculture/statue-statistics.htm

99. 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. P.77

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

101. Note: Points A,B, and C are the centers of the indicated circular arcs. Find all lengths and areas.

102. Volume What is “it”?

105. http://tedcoe.com/math/cavalieri

106. Connection to Algebra

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

110. CCSS: Grade 8 (p.54) 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. P.54

111. From the progressions documents Source: http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf p.5http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf

113. 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. P.70

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

115. 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:

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

117. Hippasus, however, was not lauded for his efforts: http://en.wikipedia.org/wiki/Irrational_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

118. 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

119. 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

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

121. 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

122. 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

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

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

125. Habits of Thinking? 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.  Mathematical Practices from the CCSS

126. Habits?  How did I do?

127. Creative Commons http://creativecommons.org

128. Contact Ted Coe tcoe@achieve.org tedcoe.com @drtedcoe Achieve 1400 16 th St NW, Suite 510 Washington, DC. 20036 202-641-3146