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