OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking

Making Connections Between Proportional Thinking and

Fractional Thinking

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Ontario Teachers Federation

CONNECT

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VISION


Visual and ConcreteContextual

VISION


Interconnected


VISION


Interconnected


CONFIDENCE


Interconnected


PERCEPTION


Interconnected


ACHIEVEMENT


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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)


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 12

1 2341

125

124

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.

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Connecting Fractions and Proportional Reasoning

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The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.

34

Relationship


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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.


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Students begin working towards proportional reasoning even before arriving at school when they make comparisons:

this bag is heavier than that one


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).


Fraction Constructs

Lets Explore Four Common Fraction Constructs


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Carton Counting


How many eggs are there?

Carton Counting


How many cartons are there?

Carton Counting



1 12

Carton Counting



112 2121

Carton Counting



112 2121

6 12 18 24Eggs

Cartons

Carton Counting



Carton Counting



12

32

6 12 18 24Eggs

Cartons22

42

Carton Counting


Part-Whole Relationship


12

32

6 12 18 24Eggs

Cartons22

42


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


16

Hot or Not


Hot or Not

HOT


16

Hot Chips


Hot or Not

HOT


16

Hot Chips

Not Hot Chips



HOT

16

Hot ChipsNot Hot Chips

Hot Not Hot



16

Hot ChipsNot Hot Chips

HOT

Hot Not Hot

Part-Part Relationship


How can 4 friends share 6 brownies fairly?

Baking Brownies


How can 4 friends share 6 brownies fairly?Multiple Representations:

Baking Brownies


Partitioning Brownies into Fourths


Baking Brownies




6 brownies4 people

Baking Brownies




6 brownies4 people =

64

Baking Brownies


Each person would get 6 one-fourth pieces of brownies.



14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14


64

Baking Brownies


How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally

Baking Brownies


How can 4 friends share 6 brownies fairly?Multiple Representations:Sharing full brownies equally, then partitioning the remainder.

Baking Brownies



12

12

12

121 1 1 1

Baking Brownies



12

12

12

121 1 1 1


121

Baking Brownies



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


How can 4 friends share 6 brownies fairly?Multiple Representations:Partition the brownies into four equal-size portions

Baking Brownies


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


121

Baking Brownies



Baking Brownies

Partition the brownies into four equal-size portions

12

12

12

121 1 1 1

6 brownies4 people


121


How can 4 friends share 6 brownies fairly?Multiple Representations:Using a number line

12

12

12

121 1 1 1

6 brownies4 people


121

120 1 2 3 4 5 6

32

52

72

92

112



12

12

12

12

6 brownies4 people


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



6 brownies4 people


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



Fraction as Quotient

Partition the brownies into four equal-size portions

12

12

12

121 1 1 1

6 brownies4 people


121


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?




+




+ =




+ =17 17 17 17 17 17 17 17 1717

17

17

17

17

17

17

17

17




+ =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




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

+




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




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


Fractions As OperatorsThere are 7 pieces in every full roll of Rolo chocolate.


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


Fraction Constructs


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.

What Do You

Notice? Wonder?

What Operation Could It Be?

=?


=? 44 or 114


= 44 or 114


141414

= 44 or 114


34+

44

or 114


=4x

44

or 114


1/4 groups of 4

=4x

14 4


1/4 groups of 4

x

What Operation Could It Be?14 4

1/4 groups of 4x

00


1/4 groups of 4x

00


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


44 or 1

14 4

1/4 groups of 4

=

x 1444 groups of 1/4

xor


23 x

34 = =?

x


23 x

34

x= =?


23 x

34 = =

612

x


= =612??


= =61212

231

Dividing Fractions

1 23

Dividing Fractions

1 23

1

12

Dividing Fractions

1 23

1

12

= 1 12

Dividing Fractions

1 23

Dividing Fractions

Dividing Fractions23

1

Dividing Fractions23

33

23

Dividing Fractions

= 33

=

23

Dividing Fractions

= 339

=

23

Dividing Fractions

= 3396

=

23

Dividing Fractions

= 3396 =

= =

23

Dividing Fractions

= 3396 =

32

= =

231

Dividing Fractions

231 = 1

12

Dividing Fractions


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Interconnected


VISION

Visual and ConcreteContextualTASKS

VISION



VISION


INQUIRYConstruct Conceptual

Understanding


VISION



Understanding

Connect to Prior Knowledge

CONNECTIONS


VISION



Understanding


CONNECTIONS

CONSOLIDATEMove Towards

Procedural Fluency

Two Groups of StudentsStruggle with

unfamiliarproblems

Good at Math

Not Good at Math


Two Groups of Students

Good at Math

Not Good at Math

Struggle withunfamiliarproblems


3 Act Math Tasks

iTunes U

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3 Act Math Tasks

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On The Web

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Gimme a Break

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CONSOLIDATE INQUIRYMove Towards

Procedural Fluency


CONNECTIONS

Construct Conceptual Understanding

VISION

Making Connections Between Proportional Thinking and

Fractional Thinking

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Ontario Teachers Federation

CONNECT

Education

OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking