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

1. Exploring the Progression
of Proportional
Reasoning From K-9
tapintoteenminds.com/OTF
Ontario Teachers’ Federation
CONNECT

2. @MathletePearcewww.tapintoteenminds.com
ABOUT
Belle River, ON
K-12 Math Consultant,
GECDSB
Google Certified Teacher
Apple Distinguished Educator
Apple Education Trainer

3. @MathletePearcewww.tapintoteenminds.com
VISION

4. @MathletePearcewww.tapintoteenminds.com
Visual and Concrete
Contextual
VISION

5. @MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
VISION

6. @MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
CONFIDENCE

7. @MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
PERCEPTION

8. @MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
ACHIEVEMENT

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. @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. @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. @MathletePearcewww.tapintoteenminds.com
kylep.ca/patprop

13. @MathletePearcewww.tapintoteenminds.com

14. @MathletePearcewww.tapintoteenminds.com

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. @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. @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. @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. @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. @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. @MathletePearcewww.tapintoteenminds.com
Which shape is more purple?
A
B
Paying Attention to Proportional Reasoning
Adapted from Marian Small (2008, p. 254)

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. @MathletePearcewww.tapintoteenminds.com
You invest $100 and it
grows to $400.
Consider:
Scenario #1

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. @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. @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. @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. @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. @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. @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. @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. @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. @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. @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. @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. @MathletePearcewww.tapintoteenminds.com

37. @MathletePearcewww.tapintoteenminds.com

38. @MathletePearcewww.tapintoteenminds.com

39. Unitizing
There are 4 fish bowls with 5 fish in each.
How many fish in total are there?
Multiplication:

40. There are 4 fish bowls with 5 fish in each.
How many fish in total are there?
Unitizing Multiplication:

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. You buy 15 goldfish. You are going to put 3 fish in
each bowl. How many bowls will you need?
Unitizing Division:

43. You buy 15 goldfish. You are going to put 3 fish in
each bowl. How many bowls will you need?
Unitizing Division:

44. You buy 15 goldfish. You are going to put 3 fish in
each bowl. How many bowls will you need?
Unitizing Division:

45. You buy 15 goldfish. You are going to put 3 fish in
each bowl. How many bowls will you need?
Unitizing Division:

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. @MathletePearcewww.tapintoteenminds.com

48. @MathletePearcewww.tapintoteenminds.com

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. @MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
VISION

69. @MathletePearcewww.tapintoteenminds.com
playkh.complaykh.comGo to:

70. @MathletePearcewww.tapintoteenminds.com
wiz623

71. @MathletePearcewww.tapintoteenminds.com
Tap
wiz623

72. @MathletePearcewww.tapintoteenminds.com
Tap
wiz623

73. @MathletePearcewww.tapintoteenminds.com
Tap
wiz623
wiz623

74. @MathletePearcewww.tapintoteenminds.com
Tap, then wait.
wiz623

75. @MathletePearcewww.tapintoteenminds.com
Tap
wiz623

77. Notice? Wonder?
What do you…

78. BAG?
How many “hot” are in a

79. PREDICTION
Make a

81. @MathletePearcewww.tapintoteenminds.com
HOT
“Hot to Not” Ratio
1:6

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?

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. @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. @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

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

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

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

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

90. @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

91. @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

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 6
12
18
24
30
36

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 1 6
12
18
24
30
36

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
1
2
6
12
18
24
30
36

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
1
2
3
6
12
18
24
30
36

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
1
2
3
4
6
12
18
24
30
36

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
1
2
3
4
5
6
12
18
24
30
36

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
6
12
18
24
30
36

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
6
12
18
24
30
36

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
6
12
18
24
30
36

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
12
18
24
30
36

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
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

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?
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

104. 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

105. 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

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

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
+6 +6 +6 +6 +6 +6

108. 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

109. 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

110. @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

111. @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

112. 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

113. @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

114. @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

115. @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

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. @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. @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

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
?

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

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

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

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
?
2 5
12
1
1
1
1
1
1
1
1
1
1
1
1
x x

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
?
2 5
12
1
1
1
1
1
1
1
1
1
1
1
1
x x
6 6

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
126 6

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
5126 6
=

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
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

128. @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

129. @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

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?

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?
x
3
7
12
=
12
3 7
x

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?
x
3
7
12
=
12
3 7
x

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
? ?x
3
7
12
?
?
=
x
x
12
3 7
x

134. @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

135. @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

136. BAG?
How many “hot” are in a

137. BAG?
How many “hot” are in a

138. 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

139. 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

140. 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

141. 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

142. BAG?
235 g
50 g21 chips
x chips =
50 g50 g50 g50 g50 g
?? ??
21
50
x
235
How many doritos are in a

143. 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

144. 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

145. 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

146. 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

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
50 gx
x
?
?
?
x x
How many doritos are in a

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
x
x
?
?
?
x x
How many doritos are in a

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
x
x
?
?
?
x x
How many doritos are in a

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
x
x
?
?
x x
How many doritos are in a

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 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

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
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

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
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

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
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

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
4 84 chips 4 200 g
0.7 35 g0.7
x
x 4.7
4.7x x
How many doritos are in a

156. 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

157. 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

158. 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

159. 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

160. 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

161. 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

162. 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

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
? 99 Doritos

164. 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

165. 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

166. 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

167. 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

168. 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

169. Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com

170. Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRY
Construct Conceptual
Understanding

171. Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRY
Construct Conceptual
Understanding
Connect to Prior
Knowledge
CONNECTIONS

172. Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRY
Construct Conceptual
Understanding
Connect to Prior
Knowledge
CONNECTIONS
CONSOLIDATE
Move Towards
Procedural Fluency

173. Two Groups of Students
Struggle with
unfamiliar
problems
Good at
Math
Not Good
at Math
@MathletePearcewww.tapintoteenminds.com

174. Two Groups of Students
Good at
Math
Not Good
at Math
Struggle with
unfamiliar
problems
@MathletePearcewww.tapintoteenminds.com

175. 3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p
@MathletePearcewww.tapintoteenminds.com

176. 3 Act Math Tasks
tapintoteenminds.com/3act-math
@MathletePearcewww.tapintoteenminds.com
On The Web

177. “Doritos Roulette”
http://kylep.ca/doritosmath
@MathletePearcewww.tapintoteenminds.com

178. @MathletePearcewww.tapintoteenminds.com
Visual and Concrete
Contextual
TASKS
CONSOLIDATE INQUIRY
Move Towards
Procedural Fluency
Connect to Prior
Knowledge
CONNECTIONS
Construct Conceptual
Understanding
VISION

179. Exploring the Progression
of Proportional
Reasoning From K-9
tapintoteenminds.com/OTF
Ontario Teachers’ Federation
CONNECT