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The task
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

Circumstances of performance
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
These work samples illustrate standard-setting performances for the following parts of the standards:
f Statistics and Probability Concepts: Find all possible combinations.
b Problem Solving and Reasoning: Implementation.
c Problem Solving and Reasoning: Conclusion.
b Mathematical Communication: Show mathematical ideas in a variety of ways.
c Mathematical Communication: Explain solutions to problems clearly and logically.
d Mathematical Communication: Consider purpose and audience when communicating about mathematics.

Mathematics required by the task
In this task the teachers instructed students to make and use a “flip book” as a tool to help find all possible combinations. So, while the task calls for extensive problem solving and reasoning, important decisions about what materials and approach to use in formulating the problem were made for the students.

For this reason the task does not provide an opportunity for students to demonstrate the formulation part (a) of the Problem Solving and Reasoning standard. The task does, however, provide an opportunity for students to demonstrate the implementation (b) and conclusion
(c) parts of the standard.


What the work shows
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 student-created multiplicative formula.

In Sample 2, the student used two approaches to solve the problem: a tree diagram and a student-created 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.