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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.
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| 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).
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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. |
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 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….
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 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.
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 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.
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 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.
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