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The task
The teacher gave students the following prompt:
Freddy, a very precise real estate appraiser, was sent to appraise some lots on a local property. Freddy appraised lot A for $88,000, but then came down with the flu and was unable to determine the fair market value of the other lots. Help Freddy determine the values so he does not lose his job!

Using the value Freddy determined for lot A, figure the fair value of each of the other lots in relation to lot A. Write a report to Freddy informing him of your mathematical determinations. Show Freddy how you reached your conclusion, so he will have faith in your determination. Represent your determination mathematically, and use appropriate math language and representations to support your conclusion. Share with Freddy any mathematically relevant observations you make, as well as any recommendations.

In the actual task, this diagram matched the size of pattern blocks.


Circumstances of performance
This sample of student work was produced under the following conditions:
- alone in a group
- in class as homework
- with teacher feedback - with peer feedback
timed opportunity for revision

Students had prior experience working with pattern blocks and were familiar with their fractional equivalents (with the hexagon equal to one whole).

This work sample illustrates a standard-setting performance for the following parts of the standards:
a Arithmetic and Number Concepts: Add, subtract, multiply, and divide whole numbers.
d Arithmetic and Number Concepts: Describe and compare quantities by using simple fractions.
e Geometry and Measurement Concepts: Solve problems by showing relationships between and among figures.
d Function and Algebra Concepts: Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations.
a Problem Solving and Reasoning: Formulation.
b Problem Solving and Reasoning: Implementation.
c Problem Solving and Reasoning: Conclusion.
e Mathematical Skills and Tools: Refer to geometric shapes and terms correctly.
g Mathematical Skills and Tools: Read, create, and represent data.
h Mathematical Skills and Tools: Use manipulatives and calculators to achieve solutions.
b Mathematical Communication: Show mathematical ideas in a variety of ways.

What the work shows
a
Arithmetic and Number Concepts: The student adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is:…
• analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide.

The student, given a problem situation with many possible approaches, figured out which arithmetic operations to use in order to come to a correct solution (evident throughout the work, without citing a specific operation).
• solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.

d Arithmetic and Number Concepts: The student describes and compares quantities by using concrete and real world models of simple fractions; that is:
• finds simple parts of wholes….


e Geometry and Measurement Concepts: The student solves problems by showing relationships between and among figures, e.g., using congruence….
The student compared congruent fractional parts of hexagons to determine the relative value of each part.

d Function and Algebra Concepts: The student uses letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations with concrete materials, i.e., demonstrates understanding and use of a beginning concept of a variable.

a Problem Solving and Reasoning: Formulation. Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.

The student decided which approach (finding simple parts of hexagons and determining their relative value) and material (pattern blocks) to use.

• uses previously learned strategies, skills, knowledge, and concepts to make decisions.
The student used knowledge of the fractional equivalents among pattern block pieces to make decisions about the value of each piece (i.e., each lot).
• uses strategies, such as using manipulatives or drawing sketches, to model problems.

b Problem Solving and Reasoning: Implementation. The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• makes up and uses a variety of strategies and approaches to solving problems….

The student used pictorial, geometric, narrative, and arithmetic approaches in the solution.
• makes connections among concepts in order to solve problems.
The student connected the fractional concepts with geometric concepts of congruence and whole number concepts of value to solve the problem.
• solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution.
The student explained the process used to reach the solution, without stating explicitly how the exact dollar value of each lot was derived.
The work provides evidence of making sense of what to do with the cents that were left over.

c Problem Solving and Reasoning: Conclusion. The student moves beyond a particular problem by making connections, extensions, and/or generalizations….
The student extended the problem by listing algebraic equations that show the fractional relationships among the lots. Expressing mathematical relationships algebraically goes beyond what most elementary students know and can be expected to do.

e Mathematical Skills and Tools: The student refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle,…parallelogram….

g Mathematical Skills and Tools: The student reads, creates, and represents data on…charts, tables, diagrams….
The student created a chart and diagram to organize and represent information.

h Mathematical Skills and Tools: The student uses…manipulatives, calculators,…as appropriate, to achieve solutions.
The student used pattern blocks and a calculator (in “Cost” and “Total cost”) to achieve a solution. The teacher verified that the student used a calculator.

b Mathematical Communication: The student shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

The student made an error in the first line of the second paragraph on the second page of the work. The student stated, “On the lots 3 lot Fs or Gs would equal 1 lot A or D.” This should read, “on the lots 3 lot Fs or Cs would equal 1 lot A or D.” Since the student demonstrated an understanding of the relationships between these shapes elsewhere in the work, this is most likely a slip rather than an error.