OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking

Making Connections Between
Proportional Thinking and
Fractional Thinking
tapintoteenminds.com/OTF
Ontario Teachers’ Federation
CONNECT

@MathletePearcewww.tapintoteenminds.com
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VISION

Visual and Concrete
Contextual
VISION

Interconnected
Visual and Concrete
Contextual
VISION

Interconnected
Visual and Concrete
Contextual
CONFIDENCE

Interconnected
Visual and Concrete
Contextual
PERCEPTION

Interconnected
Visual and Concrete
Contextual
ACHIEVEMENT

kylep.ca/patfrac

Paying Attention to Fractions
“The research suggests that explicit and precise changes to
learning and teaching practices can have a substantial
impact on children’s 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)

“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)

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 1
2
1 23
4
1
12
5
12
4
These simple descriptions do not appropriately communicate
the complex constructs that lie within this big idea.

“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

“The essence of proportional
reasoning is the consideration
of number in relative terms,
rather than absolute terms.”
3
4
Relationship
kylep.ca/patprop

…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.
kylep.ca/patprop

Students begin working towards proportional reasoning even
before arriving at school when they make comparisons:
this bag is heavier than that one
I’m “this much” shorter than you are
there 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).

Fraction Constructs
Let’s Explore Four Common
Fraction Constructs

Carton Counting

How many eggs are there?
Carton Counting

How many cartons are there?
Carton Counting

1
1
2
Carton Counting

1
1
2 2
1
21
Carton Counting

1
1
2 2
1
21
6 12 18 24Eggs
Cartons
Carton Counting

1
2
3
2
6 12 18 24Eggs
Cartons
2
2
4
2
Carton Counting

Part-Whole Relationship
1
2
3
2
6 12 18 24Eggs
Cartons
2
2
4
2

HOT
“Hot or Not”

HOT
What fraction of hot chips to not hot chips
are there in a bag of Doritos Roulette?
“Hot or Not”

HOT
1
6
“Hot or Not”

“Hot or Not”
HOT
1
6
“Hot” Chips

“Hot or Not”
HOT
1
6
“Hot” Chips
“Not Hot” Chips

HOT
1
6
“Hot” Chips
“Not Hot” Chips
“Hot” “Not Hot”

1
6
“Hot” Chips
“Not Hot” Chips
HOT
“Hot” “Not Hot”
Part-Part Relationship

How can 4 friends share 6 brownies fairly?
Baking Brownies

Multiple Representations:
Baking Brownies

Partitioning Brownies into Fourths
Baking Brownies

6 brownies
4 people
Baking Brownies

6 brownies
4 people
=
6
4
Baking Brownies

Each person would get 6 one-
fourth pieces of brownies.
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
6 brownies
4 people
=
6
4
Baking Brownies

Sharing full brownies equally
Baking Brownies

Sharing full brownies equally, then partitioning the remainder.
Baking Brownies

1
2
1
2
1
2
1
21 1 1 1
Baking Brownies

1
2
1
2
1
2
1
21 1 1 1
6 brownies
4 people
=
1
2
1
Baking Brownies

1
2
1
2
1
2
1
21 1 1 1
6 brownies
4 people
Each person would get 1 full
and 1 half piece of brownies.
=
1
2
1
Baking Brownies

Partition the brownies into four equal-size portions
Baking Brownies

1
2
1
2
1
2
1
21 1 1 1
6 brownies
4 people
=
1
2
1
Baking Brownies

Baking Brownies
1
2
1
2
1
2
1
21 1 1 1
6 brownies
4 people
=
1
2
1

Using a number line
1
2
1
2
1
2
1
21 1 1 1
6 brownies
4 people
=
1
2
1
1
20 1 2 3 4 5 6
3
2
5
2
7
2
9
2
11
2

1
2
1
2
1
2
1
2
6 brownies
4 people
=
1
2
1
1
20 1 2 3 4 5 6
3
2
5
2
7
2
9
2
11
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
Using a number line

6 brownies
4 people
=
1
2
1
2
40 1 2 3 4 5 6
6
4
10
4
14
4
18
4
22
4
Using a number line
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4

Fraction as Quotient
1
2
1
2
1
2
1
21 1 1 1
6 brownies
4 people
=
1
2
1

How Many Rolos?

How Many Rolos?
There are 7 pieces in every roll of Rolo chocolate.

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?

How Many Rolos?
+

How Many Rolos?
+ =

How Many Rolos?
+ =1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7

How Many Rolos?
+ =1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
5 pieces
7 pieces
=
9 pieces
7 pieces
+
4 pieces
7 pieces

How Many Rolos?
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
5 pieces
7 pieces
=
9 pieces
7 pieces
+
4 pieces
7 pieces
1
70 1 2
3
7
5
7
6
7
9
7
11
7
2
7
4
7
8
7
10
7
12
7
13
7
+

How Many Rolos?
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
7
5 pieces
7 pieces
=
9 pieces
7 pieces
+
4 pieces
7 pieces
1
70 1 2
3
7
5
7
6
7
9
7
11
7
2
7
4
7
8
7
10
7
12
7
13
7

How Many Rolos?
1
7
1
7
5
7
=
2
7
+
4
7
1
70 1 2
3
7
5
7
6
7
9
7
11
7
2
7
4
7
8
7
10
7
12
7
13
7
1 There is one full roll of Rolos and an
additional 2 pieces out of a roll of 7.
1

Fractions As Operators
1
7
1
7
5
7
=
2
7
+
4
7
1
70 1 2
3
7
5
7
6
7
9
7
11
7
2
7
4
7
8
7
10
7
12
7
13
7
1 There is one full roll of Rolos and an
additional 2 pieces out of a roll of 7.
1

Fraction Constructs

What Do You…
Notice? Wonder?

=?
What Operation Could It Be?

=?
4
4
or 1
1
4

=
4
4
or 1
1
4
1
4
1
4
1
4

=
4
4
or 1
1
4
3
4+

4
4
or 1
1
4
=4x

4
4
or 1
1
4
“1/4 groups of 4”
=4x

1
4 4
x

0
0
1
4 4
x

0
0
1
4 4
x
1
4
2
4
3
4
1
1 2 3 4

0
0
1
44
“4 groups of 1/4”
x
1
4
2
4
3
4
1
1 2 3 4

4
4
or 1
1
4 4
=
x
1
44
“4 groups of 1/4”
xor

2
3 x
3
4 = ?
x

2
3 x
3
4
x
= ?

2
3 x
3
4 =
6
12
x

=
6
12
?
?

=
6
12
1
2

1 2
3÷
1
1
2
Dividing Fractions

1 2
3÷
1
1
2
= 1 1
2
Dividing Fractions

Dividing Fractions
2
3÷
3
3
÷

2
3÷
Dividing Fractions
=
3
3
÷ =

2
3÷
Dividing Fractions
=
3
3
9
÷ =

2
3÷
Dividing Fractions
=
3
3
9
6
÷ =

2
3÷
Dividing Fractions
=
3
3
9
6
=
÷ = =

2
3÷
Dividing Fractions
=
3
3
9
6
=
3
2
÷ = =

2
31 ÷ = 1 1
2
Dividing Fractions

Visual and Concrete
Contextual
TASKS
VISION

Visual and Concrete
Contextual
TASKS
VISION
INQUIRY
Construct Conceptual
Understanding

Visual and Concrete
Contextual
TASKS
VISION
INQUIRY
Understanding
Connect to Prior
Knowledge
CONNECTIONS

Visual and Concrete
Contextual
TASKS
VISION
INQUIRY
Understanding
Connect to Prior
Knowledge
CONNECTIONS
CONSOLIDATE
Move Towards
Procedural Fluency

Two Groups of Students
Struggle with
unfamiliar
problems
Good at
Math
Not Good
at Math

Two Groups of Students
Good at
Math
Not Good
at Math
Struggle with
unfamiliar
problems

3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p

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

“Gimme a Break”
kylep.ca/kitkatmath

Visual and Concrete
Contextual
TASKS
CONSOLIDATE INQUIRY
Move Towards
Procedural Fluency
Connect to Prior
Knowledge
CONNECTIONS
Understanding
VISION

OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking

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OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking