Student 1:
Commentary: Student 1 moves through
multiple representations from simpler to
more abstract in the explanation
(drawing, schematic, table, general rule).
The drawing indicates understanding the
grouping of 4 by crossing out the first
person and skipping 3 people with the
curved line. The student establishes a
pattern using simpler cases beginning
with 5 people in front of Franny in the first
line (drawing), 6 in the second line and
increasing by an additional person in
each subsequent line. The table to the
right summarizes what happens in each
line and is clearly labeled. With two
complete cycles of the pattern, the
student also identifies the groupings as “4
in set” by the output column.
A
generalize rule of “x ÷ 4” is given. The
actual calculation is given at the top of the
explanation. Although the given problem
does not have a remainder, it is likely this
student will be able to explain what
happens by using the various
representations to interpret different
phases of the cycle.
Student 2:
Commentary: It appears that Student 2
attempted to establish a pattern using a
“simpler case” approach similar to
Student 1 and as demonstrated in class.
The student did not extend the pattern far
enough to actually determine a pattern.
The student may have used the
knowledge from the “Eric the Sheep”
discussion without fully establishing a
pattern for this particular problem and did
not provide that detailed multiple
representations that Student 1 provided.
Student 3:
Student 4:
Commentary: Student 3 drew 92 “x’s”
representing the people in front of
Franny. Then, she crossed out 1
representing the person who bought a
ticket while showing the skipping of 3
people with a curved line. Her marks
are a little confusing, but she gives an
answer of 23 people with a number of
24 on the right and a division problem
showing division of 93 by 4.
Commentary: Student 4 “writes” an
explanation of Franny’s situation
explaining the reason for dividing by “4”
which was used in the previous 2
solutions, but not explained. However,
because the given problem has 92
people in front of Franny and 92 is
divisible a whole number of times by 4,
it is not sure how this student would
have handled the problem if there had
been “people” left over.
Student 5:
Commentary: Student 5, like student
4, correctly explains the grouping of 4.
However, the student then incorrectly
includes Franny in the dividend (93).
That error creates a remainder. The
student then does correctly handle the
resulting solution relative to this type of
problem and rounds to the next integer
value for the answer. It is not clear
whether
the
student
actually
understands why the rounding is done
since no explanation is given. *More
fundamentally, this solution indicates
that Student 5 does not clearly
understand division. The process is
incorrectly stated as “4 divided by 93”
and the remainder is incorrectly
expressed in decimal form. Note that
Student 5 provides more insight for
Student 3’s division process and use of
93 as the dividend.
*A common error of students is thinking that the remainder represents "part" of a person. They then round under the
assumption that Franny must pass a "whole" person. Instead, a reminder of 1 in this problem represents the next person
who will purchase a ticket prior to Franny's purchase. Consequently, the number of people purchasing tickets prior to
Franny is increased by 1.
Student 6:
Commentary: Student 6 seems to
have grasped the mechanics of the
process but not an understanding of
why it is done. She knows to divide by
4 but incorrectly identifies the
composition
of
the
grouping,
considering Franny the 4th member of a
group as opposed to including the
person that buys the ticket.