Problem Solving: Cole’s Marble Collection
Here is a problem, and a solution, on the last quiz. It has no specific numbers and seeks to
stretch students to generalize the mathematics and come to a solution that can be explained
without specific numbers.
Cole has some number of marbles. Half of his marbles are blue. The rest of his marbles are
equal amounts of red, yellow, and green. If you know how many marbles Cole has, explain
how you would you figure out how marbles of each color he has? Use the half sheet of paper
to show your answer.
This solution earned 3 out of 5 points.
The student assigned a number to the collection
of marbles and then figured out the amounts for
each type of marble. This is a valid strategy
called “guess and check”. The response shows
some understanding of the mathematics involved
in the problem, but the reader has to infer too
much in order to understand how he thought
about the problem. It didn’t clearly answer the
question in the prompt.
This solution earned 3 out of 5 points.
The student began with a picture which
showed her understanding of the
mathematical concepts.
Mathematicians tell me that the first
thing they do when presented with a
problem such as this is to draw a
picture of the conditions of the
problem.
The student had a fragile
understanding of the concept, so I drew
the table and asked her to tell what numbers could work. I began by giving her the number of
48 and we worked through the solution together. Then I chose 36 and I asked her questions to
guide her to the solution. When I presented 60 as a value that could work, she was able to
complete the table by herself. Using a table is a good way to organize the information.
This solution earned 4 out of 5 points.
This solution combined the use of a picture with a good explanation about what was going on in
the problem. Like the other solutions, he assigned a value to the marble collection. When the
student states “He has 66 because 66 can be divided by six…” he has formulated a
generalization that, in order for the numbers to meet the conditions of the problem, the total
number of marbles has to be a multiple of 6. The next step would be for this student to
articulate this understanding.
This solution earned 4 out of 5 points.
The student began with a verbal explanation of
her solution. The explanation shows a good
understanding of the conditions of the
problem as well as highlighting a common
misconception of the problem.
Students needed to attend to different
conditions (portion sizes) of various quantities.
The quantities were: Total marbles; blue
marbles; red marbles; yellow marbles; and
green marbles. Because there were four
colors of marbles, students held the common
misconception that the quantities were broken
into fourths.
The student’s drawing, however, showed that
she really did understand the concept. Her
explanation demonstrated some confusion
about the portion sizes.
This solution earned 5 out of 5 points.
This student correctly
explained how he would
solve the problem for any
number of marbles.
He used a drawing to
model the problem and
the parts of the drawing
are clearly labeled.
His explanation about his
solution is clear, using
mathematics vocabulary
and is mathematically
accurate.