These work samples were drawn from different classrooms. Both teachers
gave the same instructions to their class.
The teachers asked their classes to draw four creatures. They instructed
the students to cut the figures into four heads and four bodies
and to staple each set into a small “flip book.”
The teachers asked: How many characters could you possibly come
up with by combining the different parts in various ways? Show and
explain in detail all the combinations you could make.

Sample 1 
These samples of student work were
produced under the following conditions: 
 alone 
in a group 
 in class 
as homework 
 with teacher feedback 
with peer feedback 
timed 
 opportunity for revision 
f
Statistics and Probability Concepts:
The student finds all possible combinations and arrangements within
certain constraints involving a limited number of variables.
Both students
found all the possible combinations of the four heads and four bodies
by either making drawings, organizing lists, or using diagrams.
b
Problem Solving and Reasoning: The student
makes the basic choices involved in planning and carrying out a
solution; that is…
• makes up and uses a variety
of strategies and approaches to solving problems….
In Sample 1, the student used two approaches to solve the problem:
a “flip book” and a studentcreated multiplicative formula.
In Sample 2, the student used two approaches to solve the problem:
a tree diagram and a studentcreated multiplicative formula.
This student
also went on to make a “flip book” (shown here before
the pages were cut and stapled into a book).
• makes connections among concepts in
order to solve problems….
Both students
made connections between the different conceptual approaches to
solving the problem. Each student indicated that both approaches
give the same answers.
• solves problems in ways that make
sense and explains why these ways make sense….

Sample 1 
Both students
explained why their results made sense by using a second method
as verification.
c
Problem Solving and Reasoning: The student
moves beyond the particular problem by making…extensions, and/or
generalizations; for example…makes the solution into a general
rule that applies to other circumstances.
Both students moved beyond the particular problem by recognizing
a general rule and applying it in a different circumstance. For
example, in Sample 1, the student explained “If there are 5
faces and 5 bodies, I now won’t draw all the pictures. 5 x
5=25” and in Sample 2, the student explained “If we had
2 heads and 3 bodies there could be 6 creatures because 2 x 3=6
creatures.”

Sample 1 
Sample 2 
b
Mathematical Communication: The student shows
mathematical ideas in a variety of ways, including words, numbers,
symbols, pictures….
In Sample
2, the student systematically listed all of the possible combinations
in letters while numbers were used to represent the multiplicative
formula. The student went on to represent the problem with a diagram
using symbols.
Both students also used words to explain their solutions to the problems. 
Sample 2 
Sample
2 
c
Mathematical Communication: The student explains
solutions to problems clearly and logically, and supports solutions
with evidence….
Both explanations
are clear, logical, and supported with diagrammatic, numeric, pictorial,
and narrative evidence.
d
Mathematical Communication: The student considers
purpose and audience when communicating about mathematics.
Both students applied the process to a simpler problem in order
to provide audience clarification.
There are some grammatical and spelling errors in the work (e.g.,
“make” instead of “made” and the upper case
“T” instead of lower case in the first paragraph of Sample
1, and “togher” instead of “together” in Sample
2). Neither piece was edited for spelling or grammar.

