otf connect - making connections between proportional thinking and fractional thinking
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Making Connections Between Proportional Thinking and
Fractional Thinking
tapintoteenminds.com/OTF
Ontario Teachers Federation
CONNECT
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@MathletePearcewww.tapintoteenminds.com
playkh.complaykh.comGo to:
http://playkh.comhttp://playkh.com
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@MathletePearcewww.tapintoteenminds.com
wiz623
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@MathletePearcewww.tapintoteenminds.com
Tap, then wait.
wiz623
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@MathletePearcewww.tapintoteenminds.com
Tap
wiz623
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@MathletePearcewww.tapintoteenminds.com
VISION
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@MathletePearcewww.tapintoteenminds.com
Visual and ConcreteContextual
VISION
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@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and ConcreteContextual
VISION
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@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and ConcreteContextual
CONFIDENCE
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@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and ConcreteContextual
PERCEPTION
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@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and ConcreteContextual
ACHIEVEMENT
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@MathletePearcewww.tapintoteenminds.com
kylep.ca/patfrac
http://kylep.ca/patfrac
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@MathletePearcewww.tapintoteenminds.com
Paying Attention to Fractions
The research suggests that explicit and precise changes to learning and teaching practices can have a substantial impact on childrens understanding of fractions and
future mathematical success. Instructional decisions have a significant bearing upon students ability to understand the concept of fractions, including the ability to represent fractions appropriately, compare the relative magnitude of
two fractions, and complete calculations accurately.(Bruce, Chang, Flynn & Yearley, 2013, p. 32)
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@MathletePearcewww.tapintoteenminds.com
No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions,
ratios, and proportionality. These ideas all express mathematical relationships: fractions and ratios are relational numbers. They are the first place in which students encounter numerals like 3/4 that
represent relationships between two discrete or continuous quantities, rather than a single discrete (three apples) or
continuous quantity (4 inches of rope).
Why Is Understanding Fractions Important?
(Litwiller & Bright, 2002, p. 3)
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@MathletePearcewww.tapintoteenminds.com
What is a Fraction?A fraction is a number. While fractional notation is typically used to represent quantities that are not whole, it is possible for all quantities to be represented as a fraction.
0 12
1 2341
125
124
These simple descriptions do not appropriately communicate the complex constructs that lie within this big idea.
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@MathletePearcewww.tapintoteenminds.com
The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.
kylep.ca/patprop
Connecting Fractions and Proportional Reasoning
http://kylep.ca/patprop
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@MathletePearcewww.tapintoteenminds.com
The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.
34
Relationship
Connecting Fractions and Proportional Reasoning
kylep.ca/patprop
http://kylep.ca/patprop
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@MathletePearcewww.tapintoteenminds.com
when you think of 8 as two fours or four twos rather than thinking of
it as one more than seven.
when you think of how a speed of 50 km/h is the same as a speed of
25 km/30 min.
Connecting Fractions and Proportional Reasoning
kylep.ca/patprop
http://kylep.ca/patprop
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@MathletePearcewww.tapintoteenminds.com
Students begin working towards proportional reasoning even before arriving at school when they make comparisons:
this bag is heavier than that one
Connecting Fractions and Proportional Reasoning
Im this much shorter than you arethere are 5 more red candies than green
While these early comparisons are likely absolute (i.e.: additive) in nature, we are one step closer to relating these quantities relatively (i.e.: multiplicatively).
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@MathletePearcewww.tapintoteenminds.com
Fraction Constructs
Lets Explore Four Common Fraction Constructs
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@MathletePearcewww.tapintoteenminds.com
playkh.complaykh.comGo to:
http://playkh.comhttp://playkh.com
-
@MathletePearcewww.tapintoteenminds.com
wiz623
-
@MathletePearcewww.tapintoteenminds.com
Tap, then wait.
wiz623
-
@MathletePearcewww.tapintoteenminds.com
Tap
wiz623
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@MathletePearcewww.tapintoteenminds.com
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many eggs are there?
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
1 12
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
112 2121
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
112 2121
6 12 18 24Eggs
Cartons
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
12
32
6 12 18 24Eggs
Cartons22
42
Carton Counting
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@MathletePearcewww.tapintoteenminds.com
Part-Whole Relationship
How many cartons are there?
12
32
6 12 18 24Eggs
Cartons22
42
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@MathletePearcewww.tapintoteenminds.com
HOT
Hot or Not
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@MathletePearcewww.tapintoteenminds.com
HOT
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
Hot or Not
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@MathletePearcewww.tapintoteenminds.com
HOT
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
Hot or Not
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@MathletePearcewww.tapintoteenminds.com
HOT
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
16
Hot or Not
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@MathletePearcewww.tapintoteenminds.com
Hot or Not
HOT
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
16
Hot Chips
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@MathletePearcewww.tapintoteenminds.com
Hot or Not
HOT
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
16
Hot Chips
Not Hot Chips
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@MathletePearcewww.tapintoteenminds.com
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
HOT
16
Hot ChipsNot Hot Chips
Hot Not Hot
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@MathletePearcewww.tapintoteenminds.com
What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?
16
Hot ChipsNot Hot Chips
HOT
Hot Not Hot
Part-Part Relationship
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?Multiple Representations:
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?Multiple Representations:
6 brownies4 people
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?Multiple Representations:
6 brownies4 people =
64
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
Each person would get 6 one-fourth pieces of brownies.
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?Multiple Representations:
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
6 brownies4 people =
64
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally, then partitioning the remainder.
Baking Brownies
-
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally, then partitioning the remainder.
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally, then partitioning the remainder.
12
12
12
121 1 1 1
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally, then partitioning the remainder.
12
12
12
121 1 1 1
6 brownies4 people =
121
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally, then partitioning the remainder.
12
12
12
121 1 1 1
6 brownies4 people
Each person would get 1 full and 1 half piece of brownies.=
121
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Partition the brownies into four equal-size portions
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Partition the brownies into four equal-size portions
Baking Brownies
-
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Partition the brownies into four equal-size portions
Baking Brownies
-
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Partition the brownies into four equal-size portions
12
12
12
121 1 1 1
6 brownies4 people =
121
Baking Brownies
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:
Baking Brownies
Partition the brownies into four equal-size portions
12
12
12
121 1 1 1
6 brownies4 people
Each person would get 1 full and 1 half piece of brownies.=
121
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:Using a number line
12
12
12
121 1 1 1
6 brownies4 people
Each person would get 1 full and 1 half piece of brownies.=
121
120 1 2 3 4 5 6
32
52
72
92
112
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:
12
12
12
12
6 brownies4 people
Each person would get 1 full and 1 half piece of brownies.=
121
120 1 2 3 4 5 6
32
52
72
92
112
12
12
12
12
12
12
12
12
Using a number line
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:
6 brownies4 people
Each person would get 1 full and 1 half piece of brownies.=
121
240 1 2 3 4 5 6
64
104
144
184
224
Using a number line14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
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@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?Multiple Representations:
Fraction as Quotient
Partition the brownies into four equal-size portions
12
12
12
121 1 1 1
6 brownies4 people
Each person would get 1 full and 1 half piece of brownies.=
121
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every roll of Rolo chocolate.
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
+
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
+ =
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
+ =17 17 17 17 17 17 17 17 1717
17
17
17
17
17
17
17
17
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
+ =17 17 17 17 17 17 17 17 1717
17
17
17
17
17
17
17
17
5 pieces7 pieces =
9 pieces7 pieces
+ 4 pieces7 pieces
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
1
7
1
7
1
7
1
7
1
71
7
1
7
1
7
1
7
5 pieces7 pieces =
9 pieces7 pieces
+ 4 pieces7 pieces
170 1 2
37
57
67
97
117
27
47
87
107
127
137
+
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@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
1
7
1
7
1
71
7
1
7
1
7
1
71
7
1
7
5 pieces7 pieces =
9 pieces7 pieces
+ 4 pieces7 pieces
170 1 2
37
57
67
97
117
27
47
87
107
127
137
-
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
1
7
1
7
1
71
7
1
7
1
7
1
71
7
1
7
5 pieces7 pieces =
9 pieces7 pieces
+ 4 pieces7 pieces
170 1 2
37
57
67
97
117
27
47
87
107
127
137
-
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?There are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
1
7
1
7
57 =
27
+ 47
170 1 2
37
57
67
97
117
27
47
87
107
127
137
1 There is one full roll of Rolos and an additional 2 pieces out of a roll of 7.
1
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@MathletePearcewww.tapintoteenminds.com
Fractions As OperatorsThere are 7 pieces in every full roll of Rolo chocolate.
Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?
1
7
1
7
57 =
27
+ 47
170 1 2
37
57
67
97
117
27
47
87
107
127
137
1 There is one full roll of Rolos and an additional 2 pieces out of a roll of 7.
1
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@MathletePearcewww.tapintoteenminds.com
Fraction Constructs
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@MathletePearcewww.tapintoteenminds.com
What is a Fraction?A fraction is a number. While fractional notation is typically used to represent quantities that are not whole, it is possible for all quantities to be represented as a fraction.
0 12
1 2341
125
124
These simple descriptions do not appropriately communicate the complex constructs that lie within this big idea.
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What Do You
Notice? Wonder?
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What Operation Could It Be?
-
=?
What Operation Could It Be?
-
=?
What Operation Could It Be?
-
=? 44 or 114
What Operation Could It Be?
-
= 44 or 114
What Operation Could It Be?
141414
-
= 44 or 114
What Operation Could It Be?
34+
-
44
or 114
What Operation Could It Be?
=4x
-
44
or 114
What Operation Could It Be?
1/4 groups of 4
=4x
-
14 4
What Operation Could It Be?
1/4 groups of 4
x
-
What Operation Could It Be?14 4
1/4 groups of 4x
-
00
What Operation Could It Be?14 4
1/4 groups of 4x
-
00
What Operation Could It Be?14 4
1/4 groups of 4x
1424341
1 2 3 4
-
00
What Operation Could It Be?14 4
1/4 groups of 4x
1424341
1 2 3 4
-
00
What Operation Could It Be?144
4 groups of 1/4x
1424341
1 2 3 4
-
What Operation Could It Be?
44 or 1
14 4
1/4 groups of 4
=
x 1444 groups of 1/4
xor
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@MathletePearcewww.tapintoteenminds.com
23 x
34 = =?
x
-
@MathletePearcewww.tapintoteenminds.com
23 x
34 = =?
x
-
@MathletePearcewww.tapintoteenminds.com
23 x
34
x= =?
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@MathletePearcewww.tapintoteenminds.com
23 x
34
x= =?
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@MathletePearcewww.tapintoteenminds.com
23 x
34 = =
612
x
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@MathletePearcewww.tapintoteenminds.com
= =612??
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@MathletePearcewww.tapintoteenminds.com
= =612??
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@MathletePearcewww.tapintoteenminds.com
= =612??
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@MathletePearcewww.tapintoteenminds.com
= =61212
-
231
Dividing Fractions
-
1 23
Dividing Fractions
-
1 23
Dividing Fractions
-
1 23
Dividing Fractions
-
1 23
Dividing Fractions
-
1 23
1
12
Dividing Fractions
-
1 23
1
12
= 1 12
Dividing Fractions
-
1 23
Dividing Fractions
-
1 23
Dividing Fractions
-
Dividing Fractions23
1
-
Dividing Fractions23
33
-
23
Dividing Fractions
= 33
=
-
23
Dividing Fractions
= 33
=
-
23
Dividing Fractions
= 339
=
-
23
Dividing Fractions
= 3396
=
-
23
Dividing Fractions
= 3396
=
-
23
Dividing Fractions
= 3396 =
= =
-
23
Dividing Fractions
= 3396 =
= =
-
23
Dividing Fractions
= 3396 =
= =
-
23
Dividing Fractions
= 3396 =
32
= =
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231
Dividing Fractions
-
231 = 1
12
Dividing Fractions
-
@MathletePearcewww.tapintoteenminds.com
kylep.ca/patfrac
http://kylep.ca/patfrac
-
@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and ConcreteContextual
VISION
-
Visual and ConcreteContextualTASKS
VISION
@MathletePearcewww.tapintoteenminds.com
-
Visual and ConcreteContextualTASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRYConstruct Conceptual
Understanding
-
Visual and ConcreteContextualTASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRYConstruct Conceptual
Understanding
Connect to Prior Knowledge
CONNECTIONS
-
Visual and ConcreteContextualTASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRYConstruct Conceptual
Understanding
Connect to Prior Knowledge
CONNECTIONS
CONSOLIDATEMove Towards
Procedural Fluency
-
Two Groups of StudentsStruggle with
unfamiliarproblems
Good at Math
Not Good at Math
@MathletePearcewww.tapintoteenminds.com
-
Two Groups of Students
Good at Math
Not Good at Math
Struggle withunfamiliarproblems
@MathletePearcewww.tapintoteenminds.com
-
3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p@MathletePearcewww.tapintoteenminds.com
-
3 Act Math Tasks
tapintoteenminds.com/3act-math@MathletePearcewww.tapintoteenminds.com
On The Web
http://tapintoteenminds.com/3act-math
-
Gimme a Break
kylep.ca/kitkatmath@MathletePearcewww.tapintoteenminds.com
http://kylep.ca/kitkatmath
-
@MathletePearcewww.tapintoteenminds.com
Visual and ConcreteContextualTASKS
CONSOLIDATE INQUIRYMove Towards
Procedural Fluency
Connect to Prior Knowledge
CONNECTIONS
Construct Conceptual Understanding
VISION
-
Making Connections Between Proportional Thinking and
Fractional Thinking
tapintoteenminds.com/OTF
Ontario Teachers Federation
CONNECT
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