The document discusses proportional relationships and ratios. It provides background on a symposium hosted by the Arizona Mathematics Partnership to clarify conceptual images related to ratios and proportional relationships from the Common Core State Standards. Dick Stanley is quoted as saying the CCSS approach to proportionality through proportional relationships and the constant of proportionality is remarkable and an improvement over previous approaches using ratios and proportions.

1. DEEP
Digging
tapintoteenminds.com/nctm
into RATIOSand
PROPORTIONAL
RELATIONSHIPS
in the Middle Grades

2. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
This work was supported in part by MSP grant
#1103080 through the National Science
Foundation. Opinions expressed are those of the
authors and not necessarily those of the NSF.

3. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
Proportional Relationships Symposium

4. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
Engin Ader
Scott Baldridge
Sinem Bas
Marilyn Carlson
Ted Coe
Phil Daro
James Madden
William McCallum
Kyle Pearce
Amie Pierone
Derek Reading
Dick Stanley
April Strom
James Tanton
Pat Thompson
Zalman Usiskin
Matt Weber
Sarah Winzeler
Proportional Relationships Symposium
Dick Stanley
April Strom

5. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
Dick StanleyApril Strom
Arizona Mathematics
Partnership Lead
Proportional Relationships
Symposium Lead
Mathematician &
Researcher

6. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
Kyle PearceScott Baldridge
Mathematics Consultant
Professor of Mathematics
Mathematics Consultant
Greater Essex County District
School Board
April Strom
Arizona Mathematics
Partnership Lead

7. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
Dick Stanley
“What does it mean in general to say that one
quantity is proportional to another quantity?
Be as precise as you can.”
at http://blogs.ams.org/matheducation/2014/11/20/proportionality-confusion/
TABLE TALK

8. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
Dick Stanley
“What does it mean in general to say that one
quantity is proportional to another quantity?
Be as precise as you can.”
at http://blogs.ams.org/matheducation/2014/11/20/proportionality-confusion/
SHARE OUT

9. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
Dick Stanley
“The confusing jumble of responses here is disturbing. At the
very least it points to a lack of a common understanding within
the school mathematics community of this very basic and
important subject. It would certainly be wrong to blame
teachers. Rather, I believe the culprit is a general lack of
mathematically sound grade-level appropriate presentations of
proportionality that have been available to teachers.”
at http://blogs.ams.org/matheducation/2014/11/20/proportionality-confusion/

10. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BAD NEWS
The writers of CCSS chose specific conceptual images
that they used consistently throughout, including the
area of ratios and proportional relationships.
The language of ratios and proportionality can be
confusing to teachers and students alike, and is not
coherent between different communities.
GOOD NEWS

11. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
Arizona
Mathematics
Partnership
LEARNING GOAL
In particular, we will aim to clarify these conceptual images intended behind the
Common Core Standards such as:
Understand the concept of a ratio and use ratio language to describe a
ratio relationship between two quantities.
The writers of CCSS chose specific conceptual images that they used consistently
throughout, including the area of ratios and proportional relationships.
6.RP.1
6.RP.2
6.RP.3
Understand the concept of a unit rate a/b associated with a ratio a:b with b
≠ 0, and use rate language in the context of a ratio relationship.
Use ratio and rate reasoning to solve real-world and mathematical problems

12. Notice? Wonder?
What do you…

14. Notice? Wonder?
What do you…

15. Notice? Wonder?
What do you…
Pile of paper on ground
5 reams of paper
Clock
11:45 AM
5 reams = 1 block height
13 blocks high
how many pieces of paper in one
container?
If you stack it, will it stand up or fall
over?
Was the point of the video to
measure how tall the wall was?
Does 5 reams = height of 1 block?
How much paper in a ream?
How many different rectangles did
you see?

16. Notice? Wonder?
What do you…
It’s 9:00
There are 5 packs of
paper on the ground
It looks like a storage room
It is 12:00 (not 9:00!)
There is a paper cutter
Why are we watching
this?
How many packs of
paper would it take to
reach the ceiling?
How much would it cost
to fill the room with
paper?
How many packs of
paper would it take to
reach the ceiling?

17. How many packs of paper does it
take to reach the ceiling?

18. @MathletePearcwww.tapintoteenminds.com
How many packs of paper does it
take to reach the ceiling?

19. @MathletePearcwww.tapintoteenminds.com
PREDICTION!
MAKE A

20. @MathletePearcwww.tapintoteenminds.com
INFORMATION?

21. How many packs of
paper will it take to reach
a height of 273 cm?
height
273cm
24.75cm

23. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
BACKGROUND
Arizona
Mathematics
Partnership
Dick Stanley
“The approach to proportionality suggested in the Common
Core State Standards in Mathematics promises to be of real help,
since the emphasis is directly on proportional relationships and the
constant of proportionality. In fact, the approach is remarkable in
that the term “ratio and proportion” does not appear at all, nor
does the idea of “setting up and solving a proportion.” Instead,
the central concept is proportional relationships themselves.”
at http://blogs.ams.org/matheducation/2014/11/20/proportionality-confusion/

24. What happens if we double
the number of packs?
24.75cm
5Packs5Packs
24.75cm

25. What happens if we double
the number of packs?
5Packsx2
24.75cm
5Packs
24.75cm

26. What happens if we double
the number of packs?
24.75cm
5Packs
=10Packs
5Packsx2
24.75cm

27. What happens if we double
the number of packs?
24.75cm
5Packs
=10Packs
5Packsx2
24.75cmx2

28. What happens if we double
the number of packs?
24.75cm
5Packs
=49.5cm
=10Packs
5Packsx2
24.75cmx2

29. What happens if we double
the number of packs?
The height of
the stack also
doubles.
24.75cm
5Packs
=49.5cm
=10Packs
5Packsx2
24.75cmx2

30. =
25cm
25cm
Using Friendly Numbers
How many packs will it take
to reach the ceiling?

31. 275cm
=
25cm
25cm
Using Friendly Numbers
How many packs will it
take to reach the ceiling?

32. =
25cm
25cm
275cm
Using Friendly Numbers
How many packs will it
take to reach the ceiling?

33. =
25cm
25cm
275cm
5 25
Packs cm
Using Friendly Numbers
How many packs will it
take to reach the ceiling?

34. =
25cm
25cm
275cm
5 25
Packs
cm
10 50
Using Friendly Numbers
How many packs will it
take to reach the ceiling?

35. =
25cm
25cm
275cm
5 25
Packs
cm
10 50
20 100
Using Friendly Numbers
How many packs will it
take to reach the ceiling?

36. =
25cm
25cm
275cm
5 25
Packs
cm
10 50
20 100
40 200Using Friendly Numbers
How many packs will it
take to reach the ceiling?

37. =
25cm
25cm
275cm
5 25
Packs
cm
10 50
20 100
40 200Using Friendly Numbers
How many packs will it
take to reach the ceiling? 45 225

38. =
25cm
25cm
275cm
5 25
Packs
cm
10 50
20 100
40 200Using Friendly Numbers
How many packs will it
take to reach the ceiling? 45 225
50 250

39. =
25cm
25cm
275cm
5 25
Packs
cm
10 50
20 100
40 200Using Friendly Numbers
How many packs will it
take to reach the ceiling? 45 225
50 250
55 275

40. =
25cm
25cm
275cm
5 25
Packs cm
Using Friendly Numbers
How many packs will it
take to reach the ceiling?
55 275
x11
x11

41. =
25cm
25cm
275cm
5 25
Packs cm
Using Friendly Numbers
How many packs will it
take to reach the ceiling?
55 275
x11
x11

42. How many packs must you stack
to get a height of 108.9 cm?
24.75cm
5Packs

43. How many packs must you stack
to get a height of 108.9 cm?
5 24.75
Packs cm

44. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
24.7524.75

45. 24.75
How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9
x
cmcm
24.75

46. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9xcm = cm
24.75
24.75

47. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x108.9
cm
=
cm
24.75
24.75

48. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x108.9
=
cm
4.3564.356
x
4.356
cm
24.75
24.75

49. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
108.9
=
cm
4.356
4.3564.356xx
cm
24.75
24.75

50. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9
=
cm
4.356
4.356
x4.356
cm
24.75
24.75

51. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9
=
cm
4.356
4.356
x4.356
x
5
x4.356
cm
24.75
24.75

52. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9
=
cm
4.356
4.356
x4.356
x5 x 4.356 =
cm
24.75
24.75

53. How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9
=
cm
4.356
4.356
x4.356
x5 x 4.356 =5 x 4.356
cm
24.75
24.75

54. 5x4.356
How many packs must you stack
to get a height of 108.9 cm?
x 108.9
5
Packs cm
x
108.9
=
cm
4.356
4.356
x4.356
x5 x 4.356 =
22 x=22
cm
24.75
24.75

55. 5x4.356
How many packs must you stack
to get a height of 108.9 cm?
108.9
5
Packs cm
x
108.9
=
cm
4.356
4.356
x4.356
x5 x 4.356 =
22 x=
22
cm
24.75
24.75

56. 5x4.356
How many packs must you stack
to get a height of 108.9 cm?
108.9
=
cm
4.356
x5 x 4.356 =
22 x=
108.9
5
Packs cm
x4.356
x4.356
22
cm
24.75
24.75

57. 108.9
5
Packs cm
x4.356
x4.356
22
24.75

58. 108.9
5 24.75
Packs cm
x4.356
x4.356
22

59. 108.9
5
Packs cm
x4.356
x4.356
22
24.75cm
5Packs
=49.5cm
=10Packs
5Packsx2
24.75cmx2
5 25
Packs cm
55 275
x11
x11
Scaling In Tandem
When two variable quantities scale in
tandem, they are in a ratio relationship
and a proportional relationship exists.
24.75
49.5
10

60. 108.9
5 24.75
22
5 24.75
Packs cm
55.15 273
Scaling In Tandem
10 49.5
When two variable quantities scale in
tandem, they are in a ratio relationship
and a proportional relationship exists.

61. Packs
cm
Scaling In Tandem
108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Height in
Number of Paper
When two variable quantities scale in tandem, they are in a
ratio relationship and a proportional relationship exists.

62. Packs
cm 108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Scaling In Tandem

63. Packs
cm 108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Scaling In Tandem
x 2
x 2

64. Packs
cm 108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Scaling In Tandem
x 2
x 2
x 4.4
x 4.4

65. Packs
cm 108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Scaling In Tandem
x 2
x 2
x 4.4
x 4.4
x 11.03
x 11.03

66. Packs
cm 108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Scaling In Tandem
x 11.03
1
x 4.4
1
x 2
1
x 11.03
1
x 4.4
1
x 2
1

67. Packs
cm 108.9
5
24.75
225
24.75
55.15
273
10
49.5
0
0
Scaling In Tandem
x 2
x 2
x 4.4
x 4.4
x 11.03
x 11.03

68. 5
24.75
Scaling In Tandem
Packs
cm 108.9
225
24.75
55.15
273
10
49.5
0
0
x 2
x 2
x 4.4
x 4.4
x 11.03
x 11.03

69. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0

70. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
x 2x 2
x 4.4x 4.4
x 11.03x 11.03

71. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
x 2x 2
x 4.4x 4.4
x 11.03x 11.03
Number of Height of
Stack in

72. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
x 2x 2
x 4.4x 4.4
x 11.03x 11.03
Number of Height of
Stack in
1 ?

73. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 ?
x
1
5
x
1
5

74. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
x
1
5
x
1
5

75. packs of paper height:

76. packs of paper
height

77. packs of paper
height
24.75 cm
5 packs
=

78. 24.75 cm
5 packs
=
273 cm
How Many?

79. 5 packs
=
x
24.75 cm 273 cm
x ??

80. 5 packs
=
x
24.75 cm 273 cm
x ??

81. 5 packs
=
x
273 cm
x ??
24.75 cm

82. 5 packs
=
x
273 cm
x ??
11.03
24.75 cm

83. 24.75 cm
5 packs
=
273 cm
11.03x
x

84. 24.75 cm
5 reams
=
273 cm
55.15 packs
11.03
x 11.03
x

85. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
x 11.03x 11.03
24.75 cm
5 packs
=
273 cm
55.15 packs
11.03
x 11.03
x

86. Scaling In Tandem
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
x 11.03x 11.03
24.75 cm
5 packs
=
273 cm
55.15 packs
11.03
x 11.03
x
Ratio Reasoning Utilizes

87. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
x 11.03x 11.03
24.75 cm
5 packs
=
273 cm
55.15 packs
11.03
x 11.03
x
Understand the concept of a ratio and use ratio
language to describe a ratio relationship between two
quantities.
6.RP.1
6.RP.2
6.RP.3
Understand the concept of a unit rate a/b associated
with a ratio a:b with b ≠ 0, and use rate language in the
context of a ratio relationship.
Use ratio and rate reasoning to solve real-world and
mathematical problems
Scaling In TandemRatio Reasoning Utilizes

88. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95 24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

89. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95 24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

90. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95 24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

91. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95 24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

92. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95 24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

93. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95 24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

94. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

95. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?

96. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95
cm per
pack4.95

97. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95
x 4.95

98. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
=
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
cm per
pack4.95
cm per
pack4.95

99. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
x 4.95

100. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95

101. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x
1
4.95
x
1
4.95
x
1
4.95
x
1
4.95
x
1
4.95
x
1
4.95
x
1
4.95
x
1
4.95

102. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
4.954.95
4.954.95
4.95
4.95
4.95
4.95
4.95
4.95
cm per
pack
cm per
pack

103. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
Constant of Proportionality
4.95 cm per packcm per pack4.954.954.954.954.954.954.954.954.95

104. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
4.95 cm per packcm per pack4.954.954.954.954.954.954.954.954.95y = x)(
Constant of Proportionality

105. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
What Is Rate Reasoning?
x 4.95 x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
4.954.954.954.954.954.954.954.954.954.95y = x
Constant of Proportionality
Rate Reasoning
Packs cm
Number of Height of
Stack in

106. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
cm per
pack4.95=cm per
pack4.95
24.75 cm
5 packs
=
273 cm
55.15 packs
x 4.95 x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
x 4.95
4.954.954.954.954.954.954.954.954.954.95y = x
Constant of Proportionality
Rate Reasoning The Constant of ProportionalityUtilizes
Packs cm
Number of Height of
Stack in

107. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin
Graphical Model

108. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
Graphical Model

109. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
Graphical Model

110. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
Graphical Model

111. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
Scaling In TandemRatio Reasoning:

112. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
Scaling In TandemRatio Reasoning:
x 11.03
x 11.03
x 11.03x 11.03

113. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
: Constant of ProportionalityRate Reasoning

114. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
: Constant of ProportionalityRate Reasoning
x 4.95
x 4.95
4.95y = x

115. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
: Constant of ProportionalityRate Reasoning
x
1
4.95
x
1
4.95
1

116. 4.95
Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
y = x
Packs
cm
Number of
Heightof
Stackin 0 5 10 15 20 25 30 35 40 45 50 55 60
275
300
250
225
200
175
150
125
100
75
50
25
0
x
1
4.95
x
1
4.95
1
When Might One Be More Helpful Than The Other?
x 11.03
x 11.03
x 11.03x 11.03

117. When Might One Be More
Helpful Than The Other?
273cm
5 24.75
Packs cm

118. When Might One Be More
Helpful Than The Other?
273cm
5 24.75
Packs cm
Ratio Reasoning
When solving a single
problem and not exploring
the relationship in depth.

119. 5 24.75
Packs cm
55.15 273
x11.03
When Might One Be More
Helpful Than The Other?
273cm
Ratio Reasoning
When solving a single
problem and not exploring
the relationship in depth.
x11.03

120. When Might One Be More
Helpful Than The Other?
24.75cm
5Packs5Packs
24.75cm

121. When Might One Be More
Helpful Than The Other?
24.75cm
5Packs

122. When Might One Be More
Helpful Than The Other?

123. When Might One Be More
Helpful Than The Other?
Rate Reasoning
When you want to own
every problem possible
in the relationship

124. When Might One Be More
Helpful Than The Other?
Rate Reasoning
When you want to own
every problem possible
in the relationship
4.95 y = x
1
Kyle
180 cm
Ceiling CN Tower
273 cm 553 m
4.95 y = x
1
4.95 y = x
1
4.95 y = x
1
180 cm 273 cm 553 m
Find the height of:

125. When Might One Be More
Helpful Than The Other?
Rate Reasoning
When you want to own
every problem possible
in the relationship
4.95 y = x
1
Kyle
180 cm
Ceiling
CN Tower
273 cm
553 m
4.95 y = x
1
4.95 y = x
1
4.95 y = x
1
180 cm
273 cm
553 m
Find the height of:

126. When Might One Be More
Helpful Than The Other?
Rate Reasoning
When you want to own
every problem possible
in the relationship
4.95 y = x
1
Kyle
180 cm
Ceiling
CN Tower
273 cm
553 m
4.95 y = x
1
4.95 y = x
1
4.95 y = x
1
180 cm
273 cm
553 m
)(
)(
)(
Find the height of:

127. When Might One Be More
Helpful Than The Other?
Rate Reasoning
When you want to own
every problem possible
in the relationship
4.95 y = x
1
Kyle
180 cm
Ceiling
CN Tower
273 cm
553 m
4.95 y = x
1
4.95 y = x
1
4.95 y = x
1
180 cm
273 cm
55,300 cm
)(
)(
)(
Find the height of:

128. When Might One Be More
Helpful Than The Other?
Rate Reasoning
When you want to own
every problem possible
in the relationship
4.95 y = x
1
Find the height of:
Kyle
180 cm
Ceiling
CN Tower
273 cm
553 m
4.95 y = x
1
4.95 y = x
1
4.95 y = x
1
180 cm
273 cm
55,300 cm
)(
)(
)(
= x
= x
= x
55.15 packs
36.36 packs
11,171.72 packs

129. Packs cm
108.922
5 24.75
55.15 273
10 49.5
0 0
Number of Height of
Stack in
1 4.95
A Completed Robust Structure

130. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure

131. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
10
5
x
10
5
x
A Completed Robust Structure

132. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
5
10
x
5
10
x

133. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
10
5
x
10
5
x
A Completed Robust Structure

134. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
5
10
x
5
10
x
A Completed Robust Structure
10
5
x
10
5
x
RATIOREASONING
SCALINGINTANDEM

135. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure

136. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
4.95x
4.95x

137. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
1
4.95
x
1
4.95

138. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
1
4.95
x
1
4.95
4.95x
4.95x
RATE REASONING
CONSTANT OF PROPORTIONALITY

139. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
5
24.75
x
RATE REASONING
CONSTANT OF PROPORTIONALITY
24.75
5
x
5
24.75
x
24.75
5

140. Packs cm
5 24.75
10 49.5
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
5
24.75
x
RATE REASONING
CONSTANT OF PROPORTIONALITY
24.75
5
x
5
24.75
x
24.75
5
5
10
x
5
10
x
10
5
x
10
5
x
RATIOREASONING
SCALINGINTANDEM

141. Packs cm
p q
r ?
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
r
p
x
RATIOREASONING
SCALINGINTANDEM

142. Packs cm
p q
r ?
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
r
p
x
r
p
x
RATIOREASONING
SCALINGINTANDEM
q r
p

143. Packs cm
p q
r q
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
r
p
x
r
p
x
RATIOREASONING
SCALINGINTANDEM
r
p( )

144. Packs cm
p q
r ?
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
RATE REASONING
CONSTANT OF PROPORTIONALITY

145. Packs cm
p q
r ?
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
RATE REASONING
CONSTANT OF PROPORTIONALITY
q
p

146. Packs cm
p q
r ?
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
RATE REASONING
CONSTANT OF PROPORTIONALITY
q
p
x
q
p
r
x
q
p

147. Packs cm
p q
r
Number of Height of
Stack in
(Weber, Pierone, & Strom, 2016)
A Completed Robust Structure
x
RATE REASONING
CONSTANT OF PROPORTIONALITY
q
p
x
q
p
r
q
p( )

148. Packs cm
Number of Height of
Stack in
Multiplication

149. Groups
Number of Number of
Bananas
Multiplication

150. Multiplication
Groups
Number of Number of
Bananas
3 ?
3 groups of

151. Multiplication
3 groups of 4 bananas
Groups
Number of Number of
Bananas
3 ?
= ? bananas

152. Multiplication
3 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 12 bananas

153. Multiplication
7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
7 ?

154. Multiplication
7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
7 ?
x 4

155. Multiplication
7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
7 ?
x 4
x 4

156. Multiplication
7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
x 4
x 4

157. Multiplication
7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
x 4
x 4
The Beginnings of Rate Reasoning

158. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?

159. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x

160. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x

161. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x

162. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x

163. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x

164. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x
7
3
x

165. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x
7
3
x

166. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x
7
3
x

167. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x
7
3
x

168. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x
7
3
x

169. 7 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
7 28
Can You See Ratio Reasoning?
7
3
x
7
3
x

170. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x? ?

171. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 28 bananas
9 ?
Can You See Ratio Reasoning?
x x9 ?

172. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x ?
9
3

173. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x ?
9
3

174. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x ?3

175. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x ?
9
3

176. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x
9
3
9
3

177. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x
9
3
9
3

178. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x
9
3
9
3

179. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= ? bananas
9 ?
Can You See Ratio Reasoning?
x x
9
3
9
3

180. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 36 bananas
9 36
Can You See Ratio Reasoning?
x
9
3
x
9
3

181. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 36 bananas
9 36
Can You See Ratio Reasoning?
x x3 3

182. 9 groups of 4 bananas
Groups
Number of Number of
Bananas
3 12
= 36 bananas
9 36
Can You See Ratio Reasoning?
x x
9
3
9
3

183. Multiplication Involves the Unit Rate Intuitively
3 groups of 4 bananas
Groups
Number of Number of
Bananas
3 ?
= ? bananas

184. Multiplication Involves the Unit Rate Intuitively
3 groups of 4 bananas
Groups
Number of Number of
Bananas
3 ?
= ? bananas
1 4

185. The Face of Proportionality
3 groups of 4 bananas
Groups
Number of Number of
Bananas
= 12 bananas
3 12
Two quantities that can vary

186. The Face of Proportionality
4 groups of 4 bananas
Groups
Number of Number of
Bananas
= 16 bananas
4 16
Two quantities that can vary

187. The Face of Proportionality
5 groups of 4 bananas
Groups
Number of Number of
Bananas
= 20 bananas
5 20
Two quantities that can vary

188. The Face of Proportionality
6 groups of 4 bananas
Groups
Number of Number of
Bananas
= 24 bananas
6 24
Two quantities that can vary

189. The Face of Proportionality
12 groups of 4 bananas
Groups
Number of Number of
Bananas
= 48 bananas
12 48
Two quantities that can vary

190. The Face of Proportionality
11 groups of 4 bananas
Groups
Number of Number of
Bananas
= 44 bananas
11 44
Two quantities that can vary

191. The Face of Proportionality
10 groups of 4 bananas
Groups
Number of Number of
Bananas
= 40 bananas
10 40
Two quantities that can vary

192. The Face of Proportionality
9 groups of 4 bananas
Groups
Number of Number of
Bananas
= 36 bananas
9 40
Two quantities that can vary

193. The Face of Proportionality
3 groups of 4 bananas
Groups
Number of Number of
Bananas
= 12 bananas
3 12
Two quantities that can vary

194. The Face of Proportionality
2 groups of 4 bananas
Groups
Number of Number of
Bananas
= 8 bananas
2 8
Two quantities that can vary

195. The Face of Proportionality
1 groups of 4 bananas
Groups
Number of Number of
Bananas
= 4 bananas
1 4
Two quantities that can vary

196. The Face of Proportionality
1 groups of 4 bananas
Groups
Number of Number of
Bananas
= 4 bananas
1 4
Two quantities that can vary
But something is fixed, uniform, constant

197. Name that Constant
Tables and chairs in a cafeteria
Height of stack and number of things stacked
Total cost and quantity purchased
Liters of water dripped from a tap and minutes
dripping

198. Name that Constant
Tables and chairs in a cafeteria
Height of stack and number of things stacked
Total cost and quantity purchased
Liters of water dripped and minutes dripping
Distance traveled and time traveling
Inches long and centimeters long
Price and cost after sales tax
Feet a ramp rises and feet the ramp extends horizontally
Units of rise and units of run for a line in the coordinate plane
Lengths in a drawing and the corresponding lengths in an enlargement

199. Language: “…is proportional to…"
is proportional to

200. Language: “…is proportional to…"
is proportional to
foreshadows a key frame in Algebra,
is a function of

201. Two Definitions of Proportional Relationships
A variable quantity q is proportional to another variable
quantity p if q is a multiple by a constant k of p:
q = kp
Such quantities q and p are said to be in a proportional
relationship.
A variable quantity q is proportional to another variable
quantity p if p and q scale in tandem. Such quantities q
and p are said to be in a proportional relationship.

202. Key Concepts in Proportional Relationships
A ratio is the relative size of two quantities expressed as the
quotient of one divided by the other. The ratio of a to b is
written as a:b or a/b.
Comparing quantities in ratio side-by-side lends itself to
“ratio reasoning” whereby the quantities can be “scaled in
tandem”
Comparing two quantities in ratio by dividing one quantity
with a “unit” by the other quantity with a “unit” revealing a
quotient lends itself to “rate reasoning” by which we can
more easily make comparisons with the resulting unit rate.

203. Key Concepts in Proportional Relationships
The unit rate (i.e.: how much of one quantity there is for one
unit of the other) is the constant of proportionality of a true
proportional relationship, y = kx.
Ratio reasoning and scaling in tandem is useful when
solving a single problem in a proportional relationship.
Rate reasoning and the constant of proportionality is much
more powerful mathematically and allows one to “own”
the problem.

204. @MATHLETEPEARCETAPINTOTEENMINDS.COM/NCTM @SCOTTBALDRIDGE @APRILSTROM
Thank You
Arizona
Mathematics
Partnership
Engin Ader
Scott Baldridge
Sinem Bas
Marilyn Carlson
Ted Coe
Phil Daro
James Madden
William McCallum
Kyle Pearce
Amie Pierone
Derek Reading
Dick Stanley
April Strom
James Tanton
Pat Thompson
Zalman Usiskin
Matt Weber
Sarah Winzeler
Proportional Relationships Symposium
Dick Stanley
April Strom

205. DEEP
Digging
tapintoteenminds.com/nctm
into RATIOSand
PROPORTIONAL
RELATIONSHIPS
in the Middle Grades