Diane Carrier
CCLM^2 Project, Summer 2012
This material was developed for the Leadership for the Common Core in Mathematics (CCLM^2) project
at the University of Wisconsin-Milwaukee.
CCSSM Analysis – 5.NF.3
Part 1: Standard
Grade: 5
Domain: Number and Operations-Fractions
Cluster: Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
Standard: 5.NF.3 Interpret a fraction as division of the numerator by the
denominator (a/b = a÷b). Solve word problems involving division of whole numbers
leading to answers in the form of fractions or mixed numbers, e.g., by using visual
fraction models or equations to represent the problem.
Part 2: Explanation and Examples of the Standard
In Grade 4, students were applying their understanding of addition and
multiplication to fractions. They developed the idea that the fraction
5/3 = 1/3 + 1/3+ 1/3 + 1/3 + 1/3 = 5 x 1/3
In Grade 5, students make the connection and understand the relationship between
fractions and division. They read 5/3 as “five thirds”, but learn that it can be
interpreted as “5 divided by 3.”
5/3 = 5 ÷ 3
a/b = a ÷ b
Students need numerous experiences to explore this new concept that a fraction is
a way to represent the division of two quantities. They solve word problems and
create story contexts that involve the division of whole numbers leading to
answers in the form of fractions or mixed numbers.
The teacher gives 5 packs of paper to your group of 3 students. If you
share the paper equally, how much paper does each student get?
It is an expectation that students will demonstrate their understanding using
concrete models, area models, number lines, and by explaining their thinking.
Therefore, using models should be part of many lessons and should be a natural way
of thinking and learning for students.
Example: The teacher gives 5 packs of paper to your group of 3 students. If you
share the paper equally, how much paper does each student get?
Student 1
Student 2
Pack 1
Pack 2
Student 3
1
Pack 3
2
3
1
Pack 4
2
3
Pack 5
Each student will get 1 whole pack of paper and 1/3 of the 2 packs of paper. So
each student gets 1 and 𝟐/𝟑 packs of paper. Another student may solve this problem
in a different way:
1
Pack 1
2
3
1
2
Pack 2
3
1
2
Pack 3
3
1
2
Pack 4
3
1
2
3
Pack 5
This student partitioned each pack of paper among 3 students, so that each person
received 5 x 1/3 = 5/3 packs of paper.
Part 3: School Mathematics Textbook Program
Textbook Development
I teach in the School District of South Milwaukee and we have been using the
program Investigations in Number, Data, and Space for a number of years.
There is an emphasis in all grade levels to use a variety of models and for students
to represent and model their own strategies. Below is my analysis of our current
program as it relates to 5.NF.3.
In Grade 3, students…
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Understand and represent fractions
Develop understanding of fractions as equal parts of a whole
Combine fractions that are equivalent to a whole or another fraction like ½
Use a variety of models including rectangles(brownies), hexagonal pattern
blocks(cookies) and groups of objects
Work with halves, fourths, eighths, thirds, and sixths as they learn how
these fractions are part of a whole
Understand numerator, denominator, and unit fraction
In Grade 4, students…
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•
•
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Build on meaning of fractions as being equal parts of a whole including
fractions that are greater than 1
Use a variety of models including rectangles, groups of objects, and number
lines
Use mental images of fractions, knowledge of fraction equivalencies to
compare fractions, to order fractions on a number line, and to add fractions
with like and unlike denominators
Work with halves, thirds, fourths, fifths, sixths, eighths, tenths, and
twelfths
Read, write, and interpret fraction notation
Make fraction cards (area models of a variety of fractions)
In Grade 5, students…
•
Understand relationships among fractions, decimals, and percents
(equivalencies) and use this knowledge as a strategy to add and subtract
fractions
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Learn to move between easily between fractions and percents to help solve
problems
Use knowledge of fraction equivalence, the relationship of fractions to
landmarks (1/2, 1, and 2) to compare fractions
Add and subtract fractions using models such as rectangles, clock rotations,
and number lines
Conclusion and Suggestions
As I searched through my Grade 5 Investigations in Number, Data, and
Space program I found little evidence that Standard 5.NF.3 is specifically
addressed. It indicates that this standard should be covered in Unit 6
Sessions 1.7, 1.8, 1.9, and 1.10. When I studied those lessons, I did find the
first lesson discussed how a fraction can be interpreted as division. This
lesson had a discussion about how ½ can be interpreted as a division problem.
The line is a division sign and ½ represents 1 divided by 2. That is all that is
said about that. Then the students focused on finding equivalent decimals
for fractions by dividing the numerator by the denominator using a
calculator. The students also had to figure out Win/Loss records as well as
make a fraction to decimal division table.
In my opinion, the program didn’t do a sufficient job of building a deeper
understanding of How and Why ½ can be interpreted as 1 divided by 2.
There were no models or representations used to develop this
understanding. The discussions never covered this understanding:
1 ÷ 2 = 1 x ½ = 1/2
I also didn’t see tasks requiring students to read word problems involving
dividing 2 whole numbers to get an answer in the form of a fraction or mixed
number.
This standard is going to have to be developed and covered in another way.
Students need abundant experiences to explore the concept that a fraction
is a way to represent the division of two quantities. They have to use a
variety of models to further explore division as equal sharing. Furthermore,
students also need to spend time creating story contexts where they are
dividing using whole numbers. Most importantly, they need to have a
conceptual understanding of why 1 ÷ 2 = 1 x ½ = 1/2. Our district will have to
supplement what we currently do to better meet standard 5.NF.3.