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