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OTF Connect - Exploring The Progression of Proportional Reasoning From K-9

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Ontario Teachers Federation (OTF) Connect Webinar Session - Exploring the Progression of Proportional Reasoning From K-9. Delivered on Wednesday October 26th, 2016.

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OTF Connect - Exploring The Progression of Proportional Reasoning From K-9

  1. 1. Exploring the Progression of Proportional Reasoning From K-9 tapintoteenminds.com/OTF Ontario Teachers’ Federation CONNECT
  2. 2. @MathletePearcewww.tapintoteenminds.com ABOUT Belle River, ON K-12 Math Consultant, GECDSB Google Certified Teacher Apple Distinguished Educator Apple Education Trainer
  3. 3. @MathletePearcewww.tapintoteenminds.com VISION
  4. 4. @MathletePearcewww.tapintoteenminds.com Visual and Concrete Contextual VISION
  5. 5. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual VISION
  6. 6. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual CONFIDENCE
  7. 7. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual PERCEPTION
  8. 8. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual ACHIEVEMENT
  9. 9. @MathletePearcewww.tapintoteenminds.com Proportional Reasoning “Ratio and proportional thinking and reasoning abilities are seen as a corner stone of middles school mathematics and this observation is reflected in current syllabus documents (e.g., National Council of Teachers of Mathematics, 2004) and by educators such as (e.g., Nabors, 2002).” Norton, S. (2005). The Construction of Proportional Reasoning. Queensland University of Technology.
  10. 10. @MathletePearcewww.tapintoteenminds.com Proportional Reasoning “A number of authors (e.g., Ilany, Keret & Ben-Chaim, 2004; Lo & Watanabe, 1997) have noted that the essence of such thinking is essentially multiplicative. Ability in such thinking is needed for and understanding of percentages, gradient, trigonometry and algebra.” Norton, S. (2005). The Construction of Proportional Reasoning. Queensland University of Technology.
  11. 11. @MathletePearcewww.tapintoteenminds.com Proportional Reasoning “Lamon (1995) noted that proportional reasoning has typically been taught in ‘a single chapter of the mathematical text book, in which symbols are introduced before sufficient ground work has been laid for students to understand them’ (p. 167).” Norton, S. (2005). The Construction of Proportional Reasoning. Queensland University of Technology.
  12. 12. @MathletePearcewww.tapintoteenminds.com kylep.ca/patprop
  13. 13. @MathletePearcewww.tapintoteenminds.com
  14. 14. @MathletePearcewww.tapintoteenminds.com
  15. 15. @MathletePearcewww.tapintoteenminds.com “Paying Attention to Mathematics provided an overview of what it would take to help Ontario students make – and sustain - gains in their learning and understanding of mathematics. It outlined seven foundational principles for planning and implementing improvements and gave examples of what each principle would involve.” kylep.ca/patmath
  16. 16. @MathletePearcewww.tapintoteenminds.com Focus on mathematics. Coordinate and strengthen mathematics leadership. Build understanding of effective mathematics instruction. Support collaborative professional learning in mathematics. Design a responsive mathematics learning environment. Provide assessment and evaluation in mathematics that supports student learning. Facilitate access to mathematics learning resources. SEVEN FOUNDATIONAL PRINCIPLES FOR IMPROVEMENT IN MATHEMATICS, K–12 kylep.ca/patmath
  17. 17. @MathletePearcewww.tapintoteenminds.com “All ability to reason using proportional relationships is a complex process that develops over an extended period of time. It takes many varied physical experiences to develop an understanding of what a proportional relationship is and then more time to gain the ability to deal with it abstractly.” kylep.ca/patprop Proportional Reasoning
  18. 18. @MathletePearcewww.tapintoteenminds.com …when you think of 8 as two fours or four twos rather than thinking of it as one more than seven. kylep.ca/patprop What is Proportional Reasoning?
  19. 19. @MathletePearcewww.tapintoteenminds.com …when you think of 8 as two fours or four twos rather than thinking of it as one more than seven. kylep.ca/patprop What is Proportional Reasoning? …when you think of how a speed of 50 km/h is the same as a speed of 25 km/30 min.
  20. 20. @MathletePearcewww.tapintoteenminds.com “The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.” What is Proportional Reasoning? kylep.ca/patprop
  21. 21. @MathletePearcewww.tapintoteenminds.com Which shape is more purple? A B Paying Attention to Proportional Reasoning Adapted from Marian Small (2008, p. 254)
  22. 22. @MathletePearcewww.tapintoteenminds.com Which shape is more purple? A B Paying Attention to Proportional Reasoning Adapted from Marian Small (2008, p. 254) 2 4 2 3
  23. 23. @MathletePearcewww.tapintoteenminds.com You invest $100 and it grows to $400. Consider: Scenario #1
  24. 24. @MathletePearcewww.tapintoteenminds.com Consider: Scenario #1 Scenario #2 You invest $1,000 and it grows to $1,500. You invest $100 and it grows to $400.
  25. 25. @MathletePearcewww.tapintoteenminds.com Consider: Scenario #1 Scenario #2 Which Investment Is Better? You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500.
  26. 26. @MathletePearcewww.tapintoteenminds.com You invest $1,000 and it grows to $1,500. Consider: Scenario #1 Scenario #2 Which Investment Is Better? You invest $100 and it grows to $400.
  27. 27. @MathletePearcewww.tapintoteenminds.com You invest $1,000 and it grows to $1,500. You invest $100 and it grows to $400. Consider: Scenario #1 Scenario #2 Which Investment Is Better?
  28. 28. @MathletePearcewww.tapintoteenminds.com Which Investment Is Better? Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $400 $100 $1,000
  29. 29. @MathletePearcewww.tapintoteenminds.com In Absolute Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $500 $100$100$100$100 $100 $100 $300
  30. 30. @MathletePearcewww.tapintoteenminds.com In Absolute Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $500 $100$100$100$100 $100 $100 $300
  31. 31. @MathletePearcewww.tapintoteenminds.com $100$100$100$100 $100 $100 In Absolute Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $300 $500
  32. 32. @MathletePearcewww.tapintoteenminds.com In Relative Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100 $100 $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $1,000
  33. 33. @MathletePearcewww.tapintoteenminds.com In Relative Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100 $100 $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $1,000 x 1 2 1 x 4
  34. 34. @MathletePearcewww.tapintoteenminds.com In Relative Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100 $100 $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $1,000 x 1 2 1 x 4
  35. 35. @MathletePearcewww.tapintoteenminds.com In Relative Terms Scenario #1 Scenario #2 You invest $100 and it grows to $400. You invest $1,000 and it grows to $1,500. $100$100$100$100 $100 $100 $100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500 $1,000 $1,000 $1,000 x 1 2 1 x 4
  36. 36. @MathletePearcewww.tapintoteenminds.com
  37. 37. @MathletePearcewww.tapintoteenminds.com
  38. 38. @MathletePearcewww.tapintoteenminds.com
  39. 39. Unitizing There are 4 fish bowls with 5 fish in each. How many fish in total are there? Multiplication:
  40. 40. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Unitizing Multiplication:
  41. 41. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously. Unitizing Multiplication:
  42. 42. You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Unitizing Division:
  43. 43. You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Unitizing Division:
  44. 44. You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Unitizing Division:
  45. 45. You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Unitizing Division:
  46. 46. You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously. Unitizing Division:
  47. 47. @MathletePearcewww.tapintoteenminds.com
  48. 48. @MathletePearcewww.tapintoteenminds.com
  49. 49. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously. Unitizing Multiplication You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously. Division
  50. 50. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously. Unitizing Multiplication You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously. Division 5 fish to 1 bowl 5:1 3 fish to 1 bowl 3:1
  51. 51. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Multiplication You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Division Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously. Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously. 5 fish to 1 bowl 5:1 3 fish to 1 bowl 3:1
  52. 52. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Multiplication You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Division Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously. Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously. 1 bowl to 5 fish 1:5 1 bowl to 3 fish 1:3
  53. 53. There are 4 fish bowls with 5 fish in each. How many fish in total are there? Multiplication You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need? Division 1 bowl to 5 fish 1:5 1 bowl to 3 fish 1:3
  54. 54. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies?
  55. 55. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 2 3:
  56. 56. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies?
  57. 57. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 3
  58. 58. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 3 6
  59. 59. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 3 6 1 9
  60. 60. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 3 6 1 9 1 12
  61. 61. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 3 6 1 9 1 12 1 1 1 1
  62. 62. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 3 6 1 9 1 12 1 1 1 1 2 4 6 8
  63. 63. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 3 6 1 9 1 12 1 1 1 1 2 4 6 8 4 4
  64. 64. Ratios and Rates For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 1 1 12 1 1 1 1 84 4 2 3
  65. 65. More Representations For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 1 1 12 1 1 1 1 84 4 2 3 If there are ___ red there are ___ green 2 3 4 6 6 9 8 12
  66. 66. More Representations For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 1 1 12 1 1 1 1 84 4 2 3 If there are ___ red there are ___ green 2 3 4 6 6 9 8 12 0 0 1 1 2 3 3 4 6 5 6 9 7 8 12 9 10 15 11 12 18 13 14 21 15 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 +2 +3 +2 +3 +2 +3 +2 +3 +2 +3
  67. 67. More Representations For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies? 1 1 1 1 12 1 1 1 1 84 4 2 3 If there are ___ red there are ___ green 2 3 4 6 6 9 8 12 0 0 1 1 2 3 3 4 6 5 6 9 7 8 12 9 10 15 11 12 18 13 14 21 15 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 +2 +3 +2 +3 +2 +3 +2 +3 +2 +3
  68. 68. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual VISION
  69. 69. @MathletePearcewww.tapintoteenminds.com playkh.complaykh.comGo to:
  70. 70. @MathletePearcewww.tapintoteenminds.com wiz623
  71. 71. @MathletePearcewww.tapintoteenminds.com Tap wiz623
  72. 72. @MathletePearcewww.tapintoteenminds.com Tap wiz623
  73. 73. @MathletePearcewww.tapintoteenminds.com Tap wiz623 wiz623
  74. 74. @MathletePearcewww.tapintoteenminds.com Tap, then wait. wiz623
  75. 75. @MathletePearcewww.tapintoteenminds.com Tap wiz623
  76. 76. Notice? Wonder? What do you…
  77. 77. BAG? How many “hot” are in a
  78. 78. PREDICTION Make a
  79. 79. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6
  80. 80. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?
  81. 81. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT
  82. 82. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT
  83. 83. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT
  84. 84. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT
  85. 85. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6
  86. 86. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6 12
  87. 87. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6 12 18
  88. 88. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6 12 18 24
  89. 89. @MathletePearcewww.tapintoteenminds.com HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6 12 18 24 30
  90. 90. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6 12 18 24 30 36
  91. 91. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 1 6 12 18 24 30 36
  92. 92. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT 1 2 6 12 18 24 30 36
  93. 93. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT 1 2 3 6 12 18 24 30 36
  94. 94. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT 1 2 3 4 6 12 18 24 30 36
  95. 95. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT 1 2 3 4 5 6 12 18 24 30 36
  96. 96. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT HOT 1 2 3 4 5 6 6 12 18 24 30 36
  97. 97. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT HOT 1 2 3 4 5 6 6 12 18 24 30 36
  98. 98. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT HOT 1 2 3 4 5 6 HOT 6 12 18 24 30 36
  99. 99. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT HOT 1 2 3 4 5 6 HOT 1 1 1 1 1 1 1 1 1 1 1 1 6 12 18 24 30 36
  100. 100. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT HOT 1 2 3 4 5 6 HOT 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 12 18 24 30 36
  101. 101. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT HOT HOT HOT HOT HOT 1 2 3 4 5 6 HOT 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 12 18 24 30 36
  102. 102. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? HOT 6 12 18 24 30 36 HOT HOT HOT HOT HOT 1 2 3 4 5 6 HOT 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 36
  103. 103. HOT “Hot to Not” Ratio 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? H 6 12 18 24 30 36 H H H H H 1 2 3 4 5 6 H 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 36 If there are ___ hot there are ___ not 1 6 2 12 3 18 4 24 5 6 30 36
  104. 104. HOT 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? H 6 12 18 24 30 36 H H H H H 1 2 3 4 5 6 H 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 36 If there are ___ hot there are ___ not 1 6 2 12 3 18 4 24 5 6 30 36 0 0 2 1 6 2 12 3 18 4 24 5 30 6 36 7 424 8 10 14 16 20 22 26 28 32 34 38 40 44 46
  105. 105. HOT 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? H 6 12 18 24 30 36 H H H H H 1 2 3 4 5 6 H 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 36 If there are ___ hot there are ___ not 1 6 2 12 3 18 4 24 5 6 30 36 0 0 2 1 6 2 12 3 18 4 24 5 30 6 36 7 424 8 10 14 16 20 22 26 28 32 34 38 40 44 46 +6 +6 +6 +6 +6 +6
  106. 106. HOT 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? H 6 12 18 24 30 36 H H H H H 1 2 3 4 5 6 H 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 36 If there are ___ hot there are ___ not 1 6 2 12 3 18 4 24 5 6 30 36 0 0 2 1 6 2 12 3 18 4 24 5 30 6 36 7 424 8 10 14 16 20 22 26 28 32 34 38 40 44 46 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6
  107. 107. HOT 1:6 Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips? H 6 12 18 24 30 36 H H H H H 1 2 3 4 5 6 H 1 1 1 1 1 1 1 1 1 1 1 1 6 6 1 6 6 36 If there are ___ hot there are ___ not 1 6 2 12 3 18 4 24 5 6 30 36 0 0 2 1 6 2 12 3 18 4 24 5 30 6 36 7 424 8 10 14 16 20 22 26 28 32 34 38 40 44 46 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6
  108. 108. @MathletePearcewww.tapintoteenminds.com HOT New “Hot to Not” Ratio 2:5 Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT
  109. 109. @MathletePearcewww.tapintoteenminds.com HOT 2:5 Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT New “Hot to Not” Ratio
  110. 110. HOT 2:5 Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? H 5 10 15 20 25 30 H H H H H 2 4 6 8 10 12 H 1 1 1 1 1 1 1 1 1 1 1 1 6 6 2 5 12 30 If there are ___ hot there are ___ not 2 5 4 10 6 15 8 20 10 12 25 30 0 0 2 5 4 10 6 15 8 20 10 25 12 30 14 35 +2 +5 +2 +5 +2 +5 +2 +5 +2 +5 +2 +5 HOT 1 3 5 7 9 11 13
  111. 111. @MathletePearcewww.tapintoteenminds.com HOT HOT HOT HOT HOT HOT HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT 1 1 1 1 1 1 1 1 1 1 1 1 6 6 2 5 12 30
  112. 112. @MathletePearcewww.tapintoteenminds.com Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT 1 1 1 1 1 1 1 1 1 1 1 1 HOT HOT HOT HOT HOT HOT HOT
  113. 113. @MathletePearcewww.tapintoteenminds.com Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT
  114. 114. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT
  115. 115. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT
  116. 116. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT
  117. 117. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT ?
  118. 118. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT ? 2 5 12
  119. 119. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT ? 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1
  120. 120. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT ? 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x
  121. 121. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT ? 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x x
  122. 122. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT ? 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x x 6 6
  123. 123. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT 30 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x x 6 6 30 2 5 126 6
  124. 124. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT 30 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x x 6 6 30 2 5126 6 =
  125. 125. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT 30 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x x 6 6 30 2 5 12 6 6 = x x
  126. 126. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT 30 2 5 12 1 1 1 1 1 1 1 1 1 1 1 1 x x 6 6 30 2 5 12 6 6 = x x
  127. 127. @MathletePearcewww.tapintoteenminds.com HOT Consolidation Assume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos Roulette. How many “not hot" chips should you expect in a bag with 12 "hot" chips? HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT HOT 1 1 1 1 1 1 1 1 1 1 1 1 x x 6 630 2 5 12 6 6 = x x 12 2 5 30
  128. 128. @MathletePearcewww.tapintoteenminds.com Your Turn The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many girls are there?
  129. 129. @MathletePearcewww.tapintoteenminds.com Your Turn The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many girls are there? x 3 7 12 = 12 3 7 x
  130. 130. @MathletePearcewww.tapintoteenminds.com Your Turn The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many girls are there? x 3 7 12 = 12 3 7 x
  131. 131. @MathletePearcewww.tapintoteenminds.com Your Turn The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many girls are there? 1 1 1 1 1 1 1 1 1 1 1 1 x x ? ?x 3 7 12 ? ? = x x 12 3 7 x
  132. 132. @MathletePearcewww.tapintoteenminds.com Your Turn The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many girls are there? 1 1 1 1 1 1 1 1 1 1 1 1 x x 4 4x 3 7 12 4 4 = x x 12 3 7 x
  133. 133. @MathletePearcewww.tapintoteenminds.com Your Turn The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many girls are there? 1 1 1 1 1 1 1 1 1 1 1 1 x x 4 428 3 7 12 4 4 = x x 12 3 7 28
  134. 134. BAG? How many “hot” are in a
  135. 135. BAG? How many “hot” are in a
  136. 136. BAG? How many doritos are in a 50 g = 50 g50 g50 g50 g ?? 235 g 21 chips x chips 50 g ?? 21 x 50 235
  137. 137. BAG? 50 g = 50 g50 g50 g50 g ?? 235 g 21 chips x chips 50 g ?? x 50 235 21 How many doritos are in a
  138. 138. BAG? 50 g = 50 g50 g50 g50 g ?? 235 g 21 chips x chips 50 g ?? x 235 21 50 How many doritos are in a
  139. 139. BAG? 50 g = 50 g50 g50 g50 g ?? 235 g 21 chips x chips 50 g ?? 235 21 50 x How many doritos are in a
  140. 140. BAG? 235 g 50 g21 chips x chips = 50 g50 g50 g50 g50 g ?? ?? 21 50 x 235 How many doritos are in a
  141. 141. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g? ? x x 50 g50 g50 g50 g ?? x x How many doritos are in a
  142. 142. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g1 50 gx x 50 g50 g50 g? ? ? x x How many doritos are in a
  143. 143. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 1 1 50 gx x 50 g50 g? ? ? x x How many doritos are in a
  144. 144. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 1 1 1 50 gx x 50 g? ? ? x x How many doritos are in a
  145. 145. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 50 gx x ? ? ? x x How many doritos are in a
  146. 146. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 1 50 g x x ? ? ? x x How many doritos are in a
  147. 147. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 1 50 g x x ? ? ? x x How many doritos are in a
  148. 148. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 1 50 g 21 chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 1 x x ? ? x x How many doritos are in a
  149. 149. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 1 50 g 21 chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 1 4 84 chips x x ? ? x x How many doritos are in a
  150. 150. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 1 50 g 21 chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 1 4 84 chips 4 200 g x x ? ? x x How many doritos are in a
  151. 151. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 1 35 g 21 chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 1 4 84 chips 4 200 g x x ? ? x x How many doritos are in a
  152. 152. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 1 4 84 chips 4 200 g 0.7 35 g 4.7 x x ?x x How many doritos are in a
  153. 153. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 1 1 1 1 4 84 chips 4 200 g 0.7 35 g0.7 x x 4.7 4.7x x How many doritos are in a
  154. 154. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 1 1 1 1 4 84 chips 4 200 g 0.7 35 g0.7 14.7 chips 4.7 4.7 x x x x How many doritos are in a
  155. 155. BAG? 235 g 50 g21 chips x chips 235 21 50 x = 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 1 1 1 1 84 chips 200 g 0.7 35 g0.7 14.7 chips 4.7 4.7 4.7 4.7 x x x x How many doritos are in a
  156. 156. BAG? 235 g 50 g21 chips x chips 235 21 50 x 4.7 4.7 = x x 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 35 g0.7 14.7 chips 4.7 4.798.7 chips 235 g How many doritos are in a
  157. 157. BAG? 235 g 50 g21 chips x chips 235 21 50 98.7 4.7 4.7 = x x 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 35 g0.7 14.7 chips 4.7 4.798.7 chips 235 g How many doritos are in a
  158. 158. BAG? 235 g 50 g21 chips x chips 235 21 50 98.7 4.7 4.7 = x x 50 g 50 g 50 g 50 g 1 1 1 1 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 35 g0.7 14.7 chips 4.7 4.798.7 chips 235 g How many doritos are in a
  159. 159. BAG? ? = x x How many “hot” are in a HOT 1:6 99 doritos per bag ? x chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips 99 Doritos
  160. 160. 1 hot : 6 not Total of 7 Doritos BAG? ? = x x How many “hot” are in a HOT 1:6 99 doritos per bag 7 Doritos ? x chips 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips 99 Doritos
  161. 161. BAG? x chips ? ? = x x 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips How many “hot” are in a HOT 1:6 99 doritos per bag 1 hot : 6 not Total of 7 Doritos 7 Doritos ? 99 Doritos
  162. 162. BAG? x chips ? ? = x x 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips How many “hot” are in a HOT 1:6 99 doritos per bag 1 hot : 6 not Total of 7 Doritos 7 Doritos ? 1 hot x hot? 99 Doritos
  163. 163. BAG? x chips ? ? = x x 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips How many “hot” are in a HOT 1:6 99 doritos per bag 1 hot : 6 not Total of 7 Doritos 7 Doritos ? 1 hot x hot? 997 1 x 99 Doritos
  164. 164. BAG? x chips ? 14.1 = x x 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips How many “hot” are in a HOT 1:6 99 doritos per bag 1 hot : 6 not Total of 7 Doritos 7 Doritos 14.1 1 hot x hot? 997 1 x 99 Doritos
  165. 165. BAG? x chips 14.1 14.1 = x x 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips How many “hot” are in a HOT 1:6 99 doritos per bag 1 hot : 6 not Total of 7 Doritos 7 Doritos 14.1 1 hot x hot 997 1 x 14.1 99 Doritos
  166. 166. BAG? x chips 14.1 14.1 = x x 21 chips 21 chips 21 chips 21 chips 1 1 1 1 0.7 14.7 chips 99 Doritos How many “hot” are in a HOT 1:6 99 doritos per bag 1 hot : 6 not Total of 7 Doritos 7 Doritos 14.1 1 hot 14.1 hot 997 1 14.1 14.1
  167. 167. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com
  168. 168. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding
  169. 169. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding Connect to Prior Knowledge CONNECTIONS
  170. 170. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding Connect to Prior Knowledge CONNECTIONS CONSOLIDATE Move Towards Procedural Fluency
  171. 171. Two Groups of Students Struggle with unfamiliar problems Good at Math Not Good at Math @MathletePearcewww.tapintoteenminds.com
  172. 172. Two Groups of Students Good at Math Not Good at Math Struggle with unfamiliar problems @MathletePearcewww.tapintoteenminds.com
  173. 173. 3 Act Math Tasks iTunes U kylep.ca/curiousmath1p @MathletePearcewww.tapintoteenminds.com
  174. 174. 3 Act Math Tasks tapintoteenminds.com/3act-math @MathletePearcewww.tapintoteenminds.com On The Web
  175. 175. “Doritos Roulette” http://kylep.ca/doritosmath @MathletePearcewww.tapintoteenminds.com
  176. 176. @MathletePearcewww.tapintoteenminds.com Visual and Concrete Contextual TASKS CONSOLIDATE INQUIRY Move Towards Procedural Fluency Connect to Prior Knowledge CONNECTIONS Construct Conceptual Understanding VISION
  177. 177. Exploring the Progression of Proportional Reasoning From K-9 tapintoteenminds.com/OTF Ontario Teachers’ Federation CONNECT

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