Exemplars
Share and Share Alike!
There are three gorillas at the zoo. They had
two apples. How will they share them fairly?
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike
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Exemplars
Grade Level K–2
Share and Share Alike!
There are three gorillas at the zoo. They had two apples. How will they share them fairly?
Context
This task can be introduced to students at any tim, but may be a great pre–assessment of a
student’s concepts of division and fractions prior to instruction.
It offers a particularly nice opportunity for students to show diverse approaches to problem
solving and applying their mathematical skills. We have wanted to revisit this problem for
some time. We have added to the discussion, and annotated the student work.
What This Task Accomplishes
This task offers a concrete example of the abstract concepts of division and fractions. This
problem allows students to investigate division and fractions, and also provides an
opportunity for students to demonstrate their understanding of number sense and their
concept of “equal” or “fair”.
What the Student Will Do
In solving this problem the student may use diagrams or mathematics manipulatives that
show an understanding of fractions. A student may find one solution and stop, or find several different ways of representing that solution. The work will demonstrate the student’s
ability to apply mathematical concept (dividing) and to use problem solving strategies.
Children may have strength or difficulty in either area.
Time Required for Task
The time needed will vary greatly for individual children. Some may find one solution and
stop. Others will search for many solutions. Others will begin slowly. Teachers will need time
to interview students about their problem solving thinking or students will need time to
write about their approach.
The crucial element here is that teachers take the time to find out how children arrived at the
answers they got. The thought and strategies are more important than the “correct answer,”
especially with this age group.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike (cont.)
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Exemplars
Interdisciplinary Links
This problem can be integrated with science and social studies units that relate to community
and sharing. Gorillas should be substituted by a more appropriate animal to match a unit
you are doing in your classroom. Units on apples or zoos would also fit nicely. The book
Eating Fractions by Bruce McMillan complements this task well.
Teaching Tips
Students should have some experience with the concept of sharing and therefore, division.
They should also have been introduced to the concrete experience of dividing wholes into
pieces. However, this assessment may reveal the student’s informal and innate sense of
division and fractions if the student has not had much experience with division.
The more the teacher “sets up the problem” for the student (by using dittos, teacher made
drawings or extensive modeling) the less the teacher will know about a student’s thinking.
What does a child do to begin the problem–solving process? Are errors related to inaccurate
representations (e.g. drew too many gorillas) or related to immature mathematical thinking
or incorrect mathematical process (add, subtract)? When you let the children draw or set up
materials on their own, you see how they are interpreting the problem.
If the teacher wants to transform this activity from assessment to a teaching experience, then
group processing activities can be used. Ask children to share their solutions and strategies.
Allow ample time for this. As children hear and see strategies of their peers, they will begin
to assimilate those strategies.
Suggested Materials
Mathematics manipulatives
Objects students can divide (apples or representations)
Scissors and paste
It is possible to give students pictures of three gorillas, but often giving them pre–setup
materials such as this steers their thinking in a particular direction. The integrity of problems
like this are compromised when supplemented with handouts. They limit what we know
about children’s approaches to solving problems. To avoid providing part of a strategy, you
may for instance want to provide a pile of gorilla pictures that the student can count out and
use if s/he choose.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike (cont.)
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Exemplars
Possible Solutions
At the Practitioner and Expert levels, students may arrive at their solution in different ways,
although each gorilla will receive one–third of each apple, or two–sixths of two apples.
Students might suggest other acceptable forms of the answer, like one–half of one apple, plus
one–third of one–half of another apple. Students may not know the fractional terms at these
ages, but may be able to identify the right amounts by drawing or showing accurately.
Benchmark Descriptors
Novice
The novice solution will demonstrate no mathematical ideas, or no concept of fair sharing.
The novice may add extraneous information, with little or no mathematical language or
representations to communicate a mathematical solution.
Apprentice
The apprentice may have a beginning understanding of fractions or halves, although they
may not have concept of fair share. Apprentices may divide the apples in half but ignore the
extra piece (each gets one–-half). Some math language may be used to communicate, and a
representation to solve the problem or communicate the solution may be attempted.
Practitioner
The practitioner will have an approach to solving the problem, and will arrive at a correct
solution that utilizes all parts of the apples. The practitioner will be able to communicate their
strategy and solution, and will use appropriate and accurate math language and
representations.
Expert
The expert will have an approach to solving the problem, and will arrive at a correct solution
that utilizes all parts of the apples. The expert may be able to communicate multiple ways of
arriving at a solution, and will use precise math language in doing so. The expert will also
make mathematically relevant observations or mathematical connections.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike (cont.)
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Exemplars
Author
The authors of this task are Sandra Silverman a primary–grade teacher in the Oceanview
School District, Huntington Beach, CA., and Darlene Johnson , a primary–grade teacher in the
Oceanview School District, Huntington Beach California. The task was revised and updated by
Carol McNair, the editor of Exemplars.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike (cont.)
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Exemplars
Novice
At first it seems the child
has an initial understanding
of the task, but s/he doesn’t have
a mathematical strategy for dealing with this dilemma which is
the essence of the task.
The following was
taken as dictation
from the student.
The student does not
demonstrate an
understanding of
sharing.
The student neglects
to stay within the
parameters of the task.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike (cont.)
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Exemplars
Apprentice
The student uses
some math terms
to communicate.
The student stays within the
parameters of the task but is
unable to arrive at a solution for
equally sharing the apples. If the
student were to think this a fair
way to share, an explanation
should have been present.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student’s diagram is not
labeled and doesn’t convey
the student’s solution.
Share and Share Alike (cont.)
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Exemplars
Practitioner
The student doesn’t use
the language of fractions
to communicate, but
does seem to understand
the concept.
The student uses a diagram to
communicate her/his solution.
The student dictates
correct math language
to communicate the
solution.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Share and Share Alike (cont.)
The student arrives at
a correct solution for
sharing the two
apples equally.
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Exemplars
Expert
The student’s
labeled diagram
communicates the
student’s solution.
The student arrives
at a correct solution
for sharing the two
apples equally.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student
uses good
math language
and notation.
Share and Share Alike (Cont.)
Although the student doesn’t
use the language of fractions,
the student does demonstrate
understanding of the concept.
The student makes
a mathematical
observation.
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