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Making Connections Between
Proportional Thinking and
Fractional Thinking
tapintoteenminds.com/OTF
Ontario Teachers’ Federa...
@MathletePearcewww.tapintoteenminds.com
playkh.complaykh.comGo to:
@MathletePearcewww.tapintoteenminds.com
wiz623
@MathletePearcewww.tapintoteenminds.com
Tap, then wait.
wiz623
@MathletePearcewww.tapintoteenminds.com
Tap
wiz623
@MathletePearcewww.tapintoteenminds.com
VISION
@MathletePearcewww.tapintoteenminds.com
Visual and Concrete
Contextual
VISION
@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
VISION
@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
CONFIDENCE
@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
PERCEPTION
@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
ACHIEVEMENT
@MathletePearcewww.tapintoteenminds.com
kylep.ca/patfrac
@MathletePearcewww.tapintoteenminds.com
Paying Attention to Fractions
“The research suggests that explicit and precise cha...
@MathletePearcewww.tapintoteenminds.com
“No area of elementary school mathematics is as mathematically
rich, cognitively c...
@MathletePearcewww.tapintoteenminds.com
What is a Fraction?
A fraction is a number.
While fractional notation is typically...
@MathletePearcewww.tapintoteenminds.com
“The essence of proportional
reasoning is the consideration
of number in relative ...
@MathletePearcewww.tapintoteenminds.com
“The essence of proportional
reasoning is the consideration
of number in relative ...
@MathletePearcewww.tapintoteenminds.com
…when you think of 8 as two fours
or four twos rather than thinking of
it as one m...
@MathletePearcewww.tapintoteenminds.com
Students begin working towards proportional reasoning even
before arriving at scho...
@MathletePearcewww.tapintoteenminds.com
Fraction Constructs
Let’s Explore Four Common
Fraction Constructs
@MathletePearcewww.tapintoteenminds.com
playkh.complaykh.comGo to:
@MathletePearcewww.tapintoteenminds.com
wiz623
@MathletePearcewww.tapintoteenminds.com
Tap, then wait.
wiz623
@MathletePearcewww.tapintoteenminds.com
Tap
wiz623
@MathletePearcewww.tapintoteenminds.com
Carton Counting
@MathletePearcewww.tapintoteenminds.com
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many eggs are there?
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
1
1
2
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
1
1
2 2
1
21
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
1
1
2 2
1
21
6 12 18 24Eggs
Cartons
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
Carton Counting
@MathletePearcewww.tapintoteenminds.com
How many cartons are there?
1
2
3
2
6 12 18 24Eggs
Cartons
2
2
4
2
Carton Counting
@MathletePearcewww.tapintoteenminds.com
Part-Whole Relationship
How many cartons are there?
1
2
3
2
6 12 18 24Eggs
Cartons...
@MathletePearcewww.tapintoteenminds.com
HOT
“Hot or Not”
@MathletePearcewww.tapintoteenminds.com
HOT
What fraction of hot chips to not hot chips
are there in a bag of Doritos Roul...
@MathletePearcewww.tapintoteenminds.com
HOT
What fraction of hot chips to not hot chips
are there in a bag of Doritos Roul...
@MathletePearcewww.tapintoteenminds.com
HOT
What fraction of hot chips to not hot chips
are there in a bag of Doritos Roul...
@MathletePearcewww.tapintoteenminds.com
“Hot or Not”
HOT
What fraction of hot chips to not hot chips
are there in a bag of...
@MathletePearcewww.tapintoteenminds.com
“Hot or Not”
HOT
What fraction of hot chips to not hot chips
are there in a bag of...
@MathletePearcewww.tapintoteenminds.com
What fraction of hot chips to not hot chips
are there in a bag of Doritos Roulette...
@MathletePearcewww.tapintoteenminds.com
What fraction of hot chips to not hot chips
are there in a bag of Doritos Roulette...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Baking Brownies
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Baking Brownies
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Baking Browni...
@MathletePearcewww.tapintoteenminds.com
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?
Mult...
@MathletePearcewww.tapintoteenminds.com
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?
Mult...
@MathletePearcewww.tapintoteenminds.com
Partitioning Brownies into Fourths
How can 4 friends share 6 brownies fairly?
Mult...
@MathletePearcewww.tapintoteenminds.com
Each person would get 6 one-
fourth pieces of brownies.
Partitioning Brownies into...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Sharing full ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Partition the...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Partition the...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Partition the...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Partition the...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Baking Browni...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Using a numbe...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
1
2
1
2
1
2
1...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
6 brownies
4 ...
@MathletePearcewww.tapintoteenminds.com
How can 4 friends share 6 brownies fairly?
Multiple Representations:
Fraction as Q...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every roll of Rolo chocolate.
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
How Many Rolos?
There are 7 pieces in every full roll of Rolo chocolate.
Two parti...
@MathletePearcewww.tapintoteenminds.com
Fractions As Operators
There are 7 pieces in every full roll of Rolo chocolate.
Tw...
@MathletePearcewww.tapintoteenminds.com
Fraction Constructs
@MathletePearcewww.tapintoteenminds.com
What is a Fraction?
A fraction is a number.
While fractional notation is typically...
What Do You…
Notice? Wonder?
What Operation Could It Be?
=?
What Operation Could It Be?
=?
What Operation Could It Be?
=?
4
4
or 1
1
4
What Operation Could It Be?
=
4
4
or 1
1
4
What Operation Could It Be?
1
4
1
4
1
4
=
4
4
or 1
1
4
What Operation Could It Be?
3
4+
4
4
or 1
1
4
What Operation Could It Be?
=4x
4
4
or 1
1
4
What Operation Could It Be?
“1/4 groups of 4”
=4x
1
4 4
What Operation Could It Be?
“1/4 groups of 4”
x
What Operation Could It Be?
1
4 4
“1/4 groups of 4”
x
0
0
What Operation Could It Be?
1
4 4
“1/4 groups of 4”
x
0
0
What Operation Could It Be?
1
4 4
“1/4 groups of 4”
x
1
4
2
4
3
4
1
1 2 3 4
0
0
What Operation Could It Be?
1
4 4
“1/4 groups of 4”
x
1
4
2
4
3
4
1
1 2 3 4
0
0
What Operation Could It Be?
1
44
“4 groups of 1/4”
x
1
4
2
4
3
4
1
1 2 3 4
What Operation Could It Be?
4
4
or 1
1
4 4
“1/4 groups of 4”
=
x
1
44
“4 groups of 1/4”
xor
@MathletePearcewww.tapintoteenminds.com
2
3 x
3
4 = ?
x
@MathletePearcewww.tapintoteenminds.com
2
3 x
3
4 = ?
x
@MathletePearcewww.tapintoteenminds.com
2
3 x
3
4
x
= ?
@MathletePearcewww.tapintoteenminds.com
2
3 x
3
4
x
= ?
@MathletePearcewww.tapintoteenminds.com
2
3 x
3
4 =
6
12
x
@MathletePearcewww.tapintoteenminds.com
=
6
12
?
?
@MathletePearcewww.tapintoteenminds.com
=
6
12
?
?
@MathletePearcewww.tapintoteenminds.com
=
6
12
?
?
@MathletePearcewww.tapintoteenminds.com
=
6
12
1
2
2
31 ÷
Dividing Fractions
1 2
3÷
Dividing Fractions
1 2
3÷
Dividing Fractions
1 2
3÷
Dividing Fractions
1 2
3÷
Dividing Fractions
1 2
3÷
1
1
2
Dividing Fractions
1 2
3÷
1
1
2
= 1 1
2
Dividing Fractions
1 2
3÷
Dividing Fractions
÷
1 2
3÷
Dividing Fractions
÷
Dividing Fractions
2
3÷
÷
1
Dividing Fractions
2
3÷
3
3
÷
2
3÷
Dividing Fractions
=
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
=
÷ = =
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 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷
Dividing Fractions
2
31 ÷ = 1 1
2
Dividing Fractions
@MathletePearcewww.tapintoteenminds.com
kylep.ca/patfrac
@MathletePearcewww.tapintoteenminds.com
Interconnected
Visual and Concrete
Contextual
VISION
Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRY
Construct Conceptual
Understan...
Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRY
Construct Conceptual
Understan...
Visual and Concrete
Contextual
TASKS
VISION
@MathletePearcewww.tapintoteenminds.com
INQUIRY
Construct Conceptual
Understan...
Two Groups of Students
Struggle with
unfamiliar
problems
Good at
Math
Not Good
at Math
@MathletePearcewww.tapintoteenminds...
Two Groups of Students
Good at
Math
Not Good
at Math
Struggle with
unfamiliar
problems
@MathletePearcewww.tapintoteenminds...
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
“Gimme a Break”
kylep.ca/kitkatmath
@MathletePearcewww.tapintoteenminds.com
@MathletePearcewww.tapintoteenminds.com
Visual and Concrete
Contextual
TASKS
CONSOLIDATE INQUIRY
Move Towards
Procedural F...
Making Connections Between
Proportional Thinking and
Fractional Thinking
tapintoteenminds.com/OTF
Ontario Teachers’ Federa...
OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking
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OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking

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OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking. Webinar for Ontario Teachers Federation delivered December 2016.

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OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking

  1. 1. Making Connections Between Proportional Thinking and Fractional Thinking tapintoteenminds.com/OTF Ontario Teachers’ Federation CONNECT
  2. 2. @MathletePearcewww.tapintoteenminds.com playkh.complaykh.comGo to:
  3. 3. @MathletePearcewww.tapintoteenminds.com wiz623
  4. 4. @MathletePearcewww.tapintoteenminds.com Tap, then wait. wiz623
  5. 5. @MathletePearcewww.tapintoteenminds.com Tap wiz623
  6. 6. @MathletePearcewww.tapintoteenminds.com VISION
  7. 7. @MathletePearcewww.tapintoteenminds.com Visual and Concrete Contextual VISION
  8. 8. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual VISION
  9. 9. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual CONFIDENCE
  10. 10. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual PERCEPTION
  11. 11. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual ACHIEVEMENT
  12. 12. @MathletePearcewww.tapintoteenminds.com kylep.ca/patfrac
  13. 13. @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 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)
  14. 14. @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)
  15. 15. @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 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.
  16. 16. @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
  17. 17. @MathletePearcewww.tapintoteenminds.com “The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.” 3 4 Relationship Connecting Fractions and Proportional Reasoning kylep.ca/patprop
  18. 18. @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
  19. 19. @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 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).
  20. 20. @MathletePearcewww.tapintoteenminds.com Fraction Constructs Let’s Explore Four Common Fraction Constructs
  21. 21. @MathletePearcewww.tapintoteenminds.com playkh.complaykh.comGo to:
  22. 22. @MathletePearcewww.tapintoteenminds.com wiz623
  23. 23. @MathletePearcewww.tapintoteenminds.com Tap, then wait. wiz623
  24. 24. @MathletePearcewww.tapintoteenminds.com Tap wiz623
  25. 25. @MathletePearcewww.tapintoteenminds.com Carton Counting
  26. 26. @MathletePearcewww.tapintoteenminds.com Carton Counting
  27. 27. @MathletePearcewww.tapintoteenminds.com How many eggs are there? Carton Counting
  28. 28. @MathletePearcewww.tapintoteenminds.com How many cartons are there? Carton Counting
  29. 29. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 1 2 Carton Counting
  30. 30. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 1 2 2 1 21 Carton Counting
  31. 31. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 1 2 2 1 21 6 12 18 24Eggs Cartons Carton Counting
  32. 32. @MathletePearcewww.tapintoteenminds.com How many cartons are there? Carton Counting
  33. 33. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 2 3 2 6 12 18 24Eggs Cartons 2 2 4 2 Carton Counting
  34. 34. @MathletePearcewww.tapintoteenminds.com Part-Whole Relationship How many cartons are there? 1 2 3 2 6 12 18 24Eggs Cartons 2 2 4 2
  35. 35. @MathletePearcewww.tapintoteenminds.com HOT “Hot or Not”
  36. 36. @MathletePearcewww.tapintoteenminds.com HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? “Hot or Not”
  37. 37. @MathletePearcewww.tapintoteenminds.com HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? “Hot or Not”
  38. 38. @MathletePearcewww.tapintoteenminds.com HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 “Hot or Not”
  39. 39. @MathletePearcewww.tapintoteenminds.com “Hot or Not” HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 “Hot” Chips
  40. 40. @MathletePearcewww.tapintoteenminds.com “Hot or Not” HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 “Hot” Chips “Not Hot” Chips
  41. 41. @MathletePearcewww.tapintoteenminds.com What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? HOT 1 6 “Hot” Chips “Not Hot” Chips “Hot” “Not Hot”
  42. 42. @MathletePearcewww.tapintoteenminds.com What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 “Hot” Chips “Not Hot” Chips HOT “Hot” “Not Hot” Part-Part Relationship
  43. 43. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Baking Brownies
  44. 44. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Baking Brownies
  45. 45. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Baking Brownies
  46. 46. @MathletePearcewww.tapintoteenminds.com Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: Baking Brownies
  47. 47. @MathletePearcewww.tapintoteenminds.com Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: 6 brownies 4 people Baking Brownies
  48. 48. @MathletePearcewww.tapintoteenminds.com Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: 6 brownies 4 people = 6 4 Baking Brownies
  49. 49. @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: 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
  50. 50. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally Baking Brownies
  51. 51. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally Baking Brownies
  52. 52. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. Baking Brownies
  53. 53. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. Baking Brownies
  54. 54. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. 1 2 1 2 1 2 1 21 1 1 1 Baking Brownies
  55. 55. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people = 1 2 1 Baking Brownies
  56. 56. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. 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
  57. 57. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions Baking Brownies
  58. 58. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions Baking Brownies
  59. 59. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions Baking Brownies
  60. 60. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people = 1 2 1 Baking Brownies
  61. 61. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Baking Brownies Partition the brownies into four equal-size portions 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
  62. 62. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Using a number line 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 1 20 1 2 3 4 5 6 3 2 5 2 7 2 9 2 11 2
  63. 63. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: 1 2 1 2 1 2 1 2 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 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
  64. 64. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 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
  65. 65. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Fraction as Quotient Partition the brownies into four equal-size portions 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
  66. 66. @MathletePearcewww.tapintoteenminds.com How Many Rolos?
  67. 67. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every roll of Rolo chocolate.
  68. 68. @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?
  69. 69. @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? +
  70. 70. @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? + =
  71. 71. @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 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
  72. 72. @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 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
  73. 73. @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 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 +
  74. 74. @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 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
  75. 75. @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 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
  76. 76. @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 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
  77. 77. @MathletePearcewww.tapintoteenminds.com Fractions As Operators 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 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
  78. 78. @MathletePearcewww.tapintoteenminds.com Fraction Constructs
  79. 79. @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 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.
  80. 80. What Do You… Notice? Wonder?
  81. 81. What Operation Could It Be?
  82. 82. =? What Operation Could It Be?
  83. 83. =? What Operation Could It Be?
  84. 84. =? 4 4 or 1 1 4 What Operation Could It Be?
  85. 85. = 4 4 or 1 1 4 What Operation Could It Be? 1 4 1 4 1 4
  86. 86. = 4 4 or 1 1 4 What Operation Could It Be? 3 4+
  87. 87. 4 4 or 1 1 4 What Operation Could It Be? =4x
  88. 88. 4 4 or 1 1 4 What Operation Could It Be? “1/4 groups of 4” =4x
  89. 89. 1 4 4 What Operation Could It Be? “1/4 groups of 4” x
  90. 90. What Operation Could It Be? 1 4 4 “1/4 groups of 4” x
  91. 91. 0 0 What Operation Could It Be? 1 4 4 “1/4 groups of 4” x
  92. 92. 0 0 What Operation Could It Be? 1 4 4 “1/4 groups of 4” x 1 4 2 4 3 4 1 1 2 3 4
  93. 93. 0 0 What Operation Could It Be? 1 4 4 “1/4 groups of 4” x 1 4 2 4 3 4 1 1 2 3 4
  94. 94. 0 0 What Operation Could It Be? 1 44 “4 groups of 1/4” x 1 4 2 4 3 4 1 1 2 3 4
  95. 95. What Operation Could It Be? 4 4 or 1 1 4 4 “1/4 groups of 4” = x 1 44 “4 groups of 1/4” xor
  96. 96. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 = ? x
  97. 97. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 = ? x
  98. 98. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 x = ?
  99. 99. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 x = ?
  100. 100. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 = 6 12 x
  101. 101. @MathletePearcewww.tapintoteenminds.com = 6 12 ? ?
  102. 102. @MathletePearcewww.tapintoteenminds.com = 6 12 ? ?
  103. 103. @MathletePearcewww.tapintoteenminds.com = 6 12 ? ?
  104. 104. @MathletePearcewww.tapintoteenminds.com = 6 12 1 2
  105. 105. 2 31 ÷ Dividing Fractions
  106. 106. 1 2 3÷ Dividing Fractions
  107. 107. 1 2 3÷ Dividing Fractions
  108. 108. 1 2 3÷ Dividing Fractions
  109. 109. 1 2 3÷ Dividing Fractions
  110. 110. 1 2 3÷ 1 1 2 Dividing Fractions
  111. 111. 1 2 3÷ 1 1 2 = 1 1 2 Dividing Fractions
  112. 112. 1 2 3÷ Dividing Fractions ÷
  113. 113. 1 2 3÷ Dividing Fractions ÷
  114. 114. Dividing Fractions 2 3÷ ÷ 1
  115. 115. Dividing Fractions 2 3÷ 3 3 ÷
  116. 116. 2 3÷ Dividing Fractions = 3 3 ÷ =
  117. 117. 2 3÷ Dividing Fractions = 3 3 ÷ =
  118. 118. 2 3÷ Dividing Fractions = 3 3 9 ÷ =
  119. 119. 2 3÷ Dividing Fractions = 3 3 9 6 ÷ =
  120. 120. 2 3÷ Dividing Fractions = 3 3 9 6 ÷ =
  121. 121. 2 3÷ Dividing Fractions = 3 3 9 6 = ÷ = =
  122. 122. 2 3÷ Dividing Fractions = 3 3 9 6 = ÷ = =
  123. 123. 2 3÷ Dividing Fractions = 3 3 9 6 = ÷ = =
  124. 124. 2 3÷ Dividing Fractions = 3 3 9 6 = 3 2 ÷ = =
  125. 125. 2 31 ÷ Dividing Fractions
  126. 126. 2 31 ÷ Dividing Fractions
  127. 127. 2 31 ÷ Dividing Fractions
  128. 128. 2 31 ÷ Dividing Fractions
  129. 129. 2 31 ÷ Dividing Fractions
  130. 130. 2 31 ÷ Dividing Fractions
  131. 131. 2 31 ÷ Dividing Fractions
  132. 132. 2 31 ÷ Dividing Fractions
  133. 133. 2 31 ÷ Dividing Fractions
  134. 134. 2 31 ÷ = 1 1 2 Dividing Fractions
  135. 135. @MathletePearcewww.tapintoteenminds.com kylep.ca/patfrac
  136. 136. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual VISION
  137. 137. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com
  138. 138. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding
  139. 139. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding Connect to Prior Knowledge CONNECTIONS
  140. 140. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding Connect to Prior Knowledge CONNECTIONS CONSOLIDATE Move Towards Procedural Fluency
  141. 141. Two Groups of Students Struggle with unfamiliar problems Good at Math Not Good at Math @MathletePearcewww.tapintoteenminds.com
  142. 142. Two Groups of Students Good at Math Not Good at Math Struggle with unfamiliar problems @MathletePearcewww.tapintoteenminds.com
  143. 143. 3 Act Math Tasks iTunes U kylep.ca/curiousmath1p @MathletePearcewww.tapintoteenminds.com
  144. 144. 3 Act Math Tasks tapintoteenminds.com/3act-math @MathletePearcewww.tapintoteenminds.com On The Web
  145. 145. “Gimme a Break” kylep.ca/kitkatmath @MathletePearcewww.tapintoteenminds.com
  146. 146. @MathletePearcewww.tapintoteenminds.com Visual and Concrete Contextual TASKS CONSOLIDATE INQUIRY Move Towards Procedural Fluency Connect to Prior Knowledge CONNECTIONS Construct Conceptual Understanding VISION
  147. 147. Making Connections Between Proportional Thinking and Fractional Thinking tapintoteenminds.com/OTF Ontario Teachers’ Federation CONNECT

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