7KH3RZHURI3UREOHP&KRLFH
$XWKRUV.DWKHULQH$*3KHOSV
5HYLHZHGZRUNV
6RXUFH7HDFKLQJ&KLOGUHQ0DWKHPDWLFV9RO1R2FWREHUSS
3XEOLVKHGE\National Council of Teachers of Mathematics
6WDEOH85/http://www.jstor.org/stable/10.5951/teacchilmath.19.3.0152 .
$FFHVVHG
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend
access to Teaching Children Mathematics.
http://www.jstor.org
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
2012 FOCUS ISSUE:
Differentiation
The Power of
1
1/5
1/5
1/5
1/5
1/5
Fourth- and fifth-grade learners can use differentiated
number sets within CGI problem structures to add and
subtract fractions with unlike denominators.
By Katherine A. G. Phelps
D
uring the winter of 2011, I faced
an instructional dilemma: Should
I teach the concept of adding and
subtracting fractions with unlike
denominators using a “quick trick” for the sake
of staying on schedule, or should I try something
different in hopes that my students would gain
understanding through their own methods?
The latter seemed the better option, but I knew
it would take time and involve selecting strategic problems to cultivate such understanding.
I was up for the undertaking, knowing that this
teaching risk would benefit all my students if I
designed problems that differentiated for their
learning needs. Because I taught a fourth- and
fifth-grade combination, I had to develop lessons that were accessible for students in both
grades. Creating problems that were based
on Cognitively Guided Instruction (CGI)
(Carpenter et al. 1999), I used differentiated
number sets within the problem structures.
In past years, our upper-grades teachers
have felt a time crunch during our fraction
152
units. Frequently, adding and subtracting fractions with unlike denominators was taught to
fifth graders as applying the rule to “multiply by
the opposite denominator to find a like denominator.” Fourth graders were generally not
taught these operational lessons with unlike
denominators because it was not in their North
Carolina Standard Course of Study (NCSCoS)
goals. It was my goal, however, that all my students be introduced to rich mathematical tasks
and attempt to develop an understanding of
working with fractions with unlike denominators. I put aside my concerns about constraints
of time and standards and allowed my conviction of student learning to lead my teaching.
Developing the problems
To elicit understanding, I developed three CGI
problems with different number choices to differentiate student problem-solving opportunities. I offered students a choice between set a,
which involved the more accessible fractions, or
a more challenging set b:
0DUPCFStteaching children mathematics | Vol. 19, No. 3
Copyright © 2012 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
www.nctm.org
1/4
1/6
1/6
1/4
1/6
www.nctm.org
1/4
1/6
A three-day lesson format allowed students
time to process their work.
see if students can push beyond working with
a denominator that is easily changed to an
equivalent fraction (e.g., 1/4 = 2/8) to a situation
where they must change both denominators. To
allow them enough time to process their work, I
use the problems in a three-day lesson format in
which students independently work through the
tasks on day 1. On day 2, they work in learning
partnerships to discuss their findings; and on
day 3, they present their mathematics findings
to the class.
1/2
1/3
1/6
KATHERINE A. G. PHELPS AND AMY SENTA
3 1
,
3
3 1
4 , 12
,
4 8
4 2
3 1
Result unknown: Logan has _____ , cup of
4 2
flour. Her mom gives her _____ cup more.
3 1
3 1
,much flour does she have now? ,
How
3
4 , 12
4 8
4 8
a. 3 , 1
b. 3 , 1
4 81
4 2
1
3 1
,1
,
4
2
4 8
Start unknown: Katie has some cookie batter,
but3it is
1 too runny. She adds _____ cup of flour
,
1 of _____
1
4
to make
,11
7
1 8 1 it thicker. If she uses a total
4
, 22
,1
cups
8 in4 the
4 of2 flour, how much flour was
1
1
batter at first?
,1
4
2
7
a. 1 , 1 1
b. , 2 1
7
4 , 2 21
8
4
8
4
7
1
1
1
, 2a sub
, 1 unknown: Jim ate _____ of
Change
8
4
2
1 14
7
1
,
sandwich.
His sister ate some more.
If
,2
8 4
8
4 they ate a total of _____ of the sub,
together
how much did his sister eat?
7
1
,2
1 1
8
4
,3
a. 1 , 1
b. 1
8, 4
6 4
8 4
1 1
,
8 4
Numbers
in
set
a
include
1/2
in
the
1 1
1 3 first
,3
,
1
two problems
so
that
struggling
learners
8, 4
6 4 have
6 4 fraction to use as a foundation for
a familiar
1 3
1 1
their work.
Then I push the students
, work,
6 4
8 4
ing with
set
a
to
use
the
fractions
1/8
and 1/4
1 3
in the6fi, nal
problem.
I
avoid
using
1/2
in set b
4
and let students explore fourths and eighths
1 3 and then transfer to operating with
immediately
,
4 fourths. I choose these numbers to
sixths6and
1/4
1/6
Problem Choice
3 1
,
4 2
1/2
1/3
1/3
Vol. 19, No. 3 | UFBDIJOHDIJMESFONBUIFNBUJDTtOctober 2012
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
153
1/5
KATHERINE A. G. PHELPS AND AMY SENTA
1/5
1/5
1/5
1/5
1/5
1
On the first day,
students worked
through the tasks
independently.
The lessons and the student learning
On the first day, we read the problems as a class
and reviewed students’ work with like denominators to lead into their exploration of unlike
denominators. In this opening discussion, one
child told the class he already knew how to add
fractions with unlike denominators. I asked
him to share any knowledge he thought could
help our class. Sam (all names are pseudonyms)
came to the board and explained, “Let’s see, if
you have 1/5 and 1/10 [writing the following
equations on the board as he proceeds], you
Fraction bars
Referring to fractional strips of a whole, each strip is the same length but
represents the whole of different fractional parts. Various math programs
use these fractional strips, and children can make their own as well. In
our classroom, we use Math Expressions (Fuson 2009) fraction bars. These
strips show the addition of the unit fractions. For example, the whole of
6/6 is represented as sections of 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6.
154
would say 1/5 × 10/10 is 10/50, and 1/10 × 5/5
is 5/50, and then you add them: 10/50 + 5/50 =
15/50.”
Sam jumped to using opposite denominators
instead of seeing that the 1/5 could be
represented as 2/10. He obviously knew the
trick of operating with these fractions but did
not appear to understand what was actually
happening.
I knew at this point that Sam’s example might
confuse others in the room, so I continued our
class conversation by asking, “What do you
notice about Sam’s adding in this problem he
just showed us?” I hoped this would highlight
his creating common denominators yet steer us
away from the process he used to get them.
Carina raised her hand and said, “Well, he has
the same denominator when he gets to the adding, just like we’ve been doing all along.”
I asked the class to focus on this aspect in
their work. I then addressed Sam’s work by saying, “Thanks for showing us your method, but
for today’s problems, I’m going to ask that we all
find other ways to arrive at like denominators.
Even though that may be one method you have
learned that works, I would like us to explore
other methods for discovering our answers.”
I directed students to begin their independent solving, stay focused on their goal of finding the like denominators within their number
sets, and use math fraction bars as well as
pictures to support their work. I asked them to
choose whichever number set was best for their
learning and to keep in mind to challenge themselves. Then I monitored the room to watch
students’ choices and strategies.
Many students launched their work by lining up fraction bars to see what they noticed.
For example, a student working with set b’s 1/8
and 3/4 held a straightedge vertically from her
fourths fraction bar down to her eighths bar and
counted how many eighths were within 3/4.
Another student chose not to look at fraction
bars and immediately noticed that he could turn
a 4 into an 8 by multiplying by 2. I asked him
to support his multiplication with a drawing of
what was actually happening.
Four students working with set a began by
drawing circles to solve the first problem. They
split their circles into halves and fourths and
colored in pieces to represent the numerators.
Then they tried to add the circular sections
0DUPCFStteaching children mathematics | Vol. 19, No. 3
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
www.nctm.org
www.nctm.org
From working with circular representations, students
transitioned to Math Expression fraction bars.
KATHERINE A. G. PHELPS AND AMY SENTA
together but were unsure how to combine the
pieces. During class instruction, I transitioned
students from working with circular representations to fraction bars, because students tended
to cut circles into unequal sections and tried
to combine mismatching pieces. On the first
day, I let these students continue working with
their circles and let all others use their methods
of choice. This allowed me to get a better idea
of what understanding students already had,
and I hoped the varied methods would lead to
engaging conversations in the day 2 math partnerships (see figs. 1 and 2). At the close of this
day, I collected students’ work and analyzed it
to inform my selection of partners for the following day.
On day 2, I paired students with a peer to
whom they would explain their solution strategies from day 1. I made sure that learning partners had chosen the same number sets, although
they may have arrived at different answers. The
goal was to confirm their solutions by collaboratively re-solving the problems. These varied
answers led to rich math discussions between
learners, without the intimidation of explaining
their method to the entire class just yet. Students who had used fraction bars were paired
with students who had used circle drawings the
day before, in hopes that they would progress
beyond using circles. Student pairs had the goal
of compromising on one work method that
they would then transfer to chart paper for a
class strategy presentation. Partners began by
discussing their current strategies and trying to
determine a correct answer if they had arrived at
different solutions.
Sam had an interesting conversation about
what he and his partner worked on during day 2.
They moved with quite a bit of ease through
problems 1 and 2 of set b by proving their math
work with the support of fraction-bar drawings.
On problem 3, which involved 1/6 subtracted
from 3/4, Sam had gone back to a method of
FIGU R E 2
FIGU R E 1
Students cut circles into unequal
sections and tried to combine
mismatching pieces.
“multiply the 1/6 by 4/4 and the 3/4 by 6/6”
without understanding why he was doing it,
simply knowing that it worked. He said, “When I
do this, I get 4/24 and 18/24, and then I subtract
the 4/24 to get 14/24.” His partner, Allen, noticed
something with his fraction bars. He lined up
the fourth, sixth, and twelfth bars vertically and
showed Sam that 1/6 looked equal to two of the
On day 2, students
discussed their
findings in learning
partnerships.
Vol. 19, No. 3 | UFBDIJOHDIJMESFONBUIFNBUJDTtOctober 2012
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
155
1/4
1/6
(b) This diagram shows that 3/4
is the same as 18/24.
1/12 sections and that the 3/4 looked equal to
nine of the 1/12 sections, which would get both
fractions to like denominators of twelfths. Allen
explained, “Since we have like denominators,
we can subtract the numerator 2 from the 9.”
Sam said his answer was 14/24, though, and
that Allen’s method got an answer of 7/12. They
became stuck on the fact that they had different
answers instead of looking at how the answers
could be related. I asked the boys to look for a
connection between their fractions, and I continued circulating around the room for the day.
On day 2, I closed partner work by pulling the
whole class together to share some of their discoveries about working with unlike denominators. They had made the following conclusions:
s “You always need to try to get the denominators to be the same.”
s “Try to figure out how you can get one
156
FIGU R E 4
1/4
(a) They crisscrossed the grids to
show the multiplication, using 24
as the common denominator.
Two students took an algorithmic
process and attached personal
meaning to it.
(a) Keeping the rows or columns
shaded shows that 1/6 is the same
as 4/24.
1/4
FIGU R E 3
1/4
1/6
1/6
1/6
1/6
1/6
Starting with two grids (see fig. 4a),
these partners wanted a new way to
prove what Sam had shown them.
(b) With a common denominator of
24, adding or subtracting fractions
becomes simple arithmetic.
denominator to be like the other, but
remember you need to change the numerator, too.”
s “You’re not changing the amount of these
fractions really, just changing how you cut
them up to get them into same-denominator
sections.”
I felt confident that my students were gaining
an understanding of how to work with unlike
denominators and that spending the time was
worth it.
On the final day, student pairs reunited and
picked one problem to explain to the class.
When it came time to present, the other students’ role was listening and either agreeing
with the solution or offering comments and
questions for the pair who was presenting. One
diagram that was used to solve problem 3, set b
(3/4 – 1/6) was interesting: The partners said
0DUPCFStteaching children mathematics | Vol. 19, No. 3
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
www.nctm.org
1/3
Before using these lessons, I was nervous about
the time this sort of problem solving takes and
how to make the problems accessible for both
my fourth- and fifth-grade learners. Had I not
set aside my concern about time, I would have
approached adding and subtracting with unlike
denominators by giving a quick lesson to the
fifth graders and a different lesson altogether to
the fourth graders. However, by spending three
days with these rich problems, my students
gained their own understanding of the topic,
and thereby, we did not lose time later in the
year remediating for adding and subtracting of
unlike denominators.
www.nctm.org
1/3
1/2
A look into the teacher learning
1/3
As for accommodating both grade levels, I
had always let the standards dictate my differentiation instead of letting my students’ abilities
and needs guide me. These problems allowed
me to look past differentiation as two separate
lessons for the different grade levels; the lessons show how one solid set of problems can be
enriched to meet the entirety of math needs in a
classroom. At the end of the lessons, I reflected
on the number-set choices that students had
made. I expected many of my fifth graders to
work on set b, and although some started there,
they opted to switch to set a because of some
frustration. I also expected fourth graders to use
set a, but two of them started here and moved
to set b for a greater challenge. I pigeonholed
my fourth and fifth graders on the basis of grade
levels instead of looking at them as a class of
mathematicians. I think this happens more
often than I would like. As educators, we become
focused on students being in a certain grade
and forget about the fluidity of math. We must
push ourselves to allow room for students to
take a more natural course in their math learning based on their skill needs and not solely on
their grade-level standard needs. Although our
state standards are important documents to
consider when planning math lessons, I inherently understand now that these standards are
the minimum of what we teach students; they
should never limit what we offer. In the midst of
transitions to the Common Core State Standards
for Mathematics, differentiated problem sets
will allow us to cover more math territory in a
deeper, more meaningful way.
1/2
they had thought about what Sam had shown
them the first day but wanted to prove it in a
new way. They had developed models to help
prove their ideas and had started with two grids
representing the 3/4 and the 1/6 (see figs. 3a
and 3b). They then crisscrossed their two grids
to show the multiplication that Sam had talked
about. In their words, “it was sort of like making
arrays” (see figs. 4a and 4b). They had split each
grid into a common 24 for the denominators.
The rows or columns that were shaded stayed
shaded. These students explained that their diagrams showed that 3/4 is the same as 18/24, that
1/6 is the same as 4/24, and that you can add or
subtract from there. They had taken an algorithmic process and attached personal meaning to
it. This was extremely important work to share,
even if all the other students did not completely
understand at that point. For the students who
were ready, it was an accurate and extremely
approachable method for showing the meaning of “multiplying by the other denominator.”
Those who were not yet ready for this method
could hear it, start thinking about it, and connect it at a later date.
After pairs presented, we went back to our
discoveries from the previous day and honed
the list. My students, regardless of grade level
or their mathematical ability level, were able to
access the math due to differentiated problem
sets. Each pair had a discovery about adding and
subtracting with unlike denominators and had
rich drawings and conversations to show for it.
What came out of these lessons was a far deeper
understanding than could have ever been
gained from a “quick trick” lesson.
REF EREN C ES
Carpenter, Thomas, Elizabeth Fennema, Megan
Franke, Linda Levi, and Susan Empson. 1999.
Children’s Mathematics: Cognitively Guided
Instruction. New York: Heinemann.
Fuson, Karen. 2009. Math Expressions Grade 5,
vol. 1. Boston: Houghton Mifflin Harcourt.
Katherine A. G. Phelps, katie.phelps
@orange.k12.nc.us, teaches fifth
grade and grades 4–5 combinations
at Efland Cheeks Elementary School in
Efland, North Carolina. She is
interested in using high-cognitive-demand problems
and student discussions in the math classroom.
Vol. 19, No. 3 | UFBDIJOHDIJMESFONBUIFNBUJDTtOctober 2012
This content downloaded by the authorized user from 192.168.52.71 on Tue, 27 Nov 2012 16:12:56 PM
All use subject to JSTOR Terms and Conditions
157