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The task

Pentagon RSTUV has coordinates R(1,4), S(5,0), T(3,-4), U(-1,-4), and V(-3,0).

a) On graph paper, plot Pentagon RSTUV.

b) Determine if Pentagon RSTUV is a regular pentagon. Show all your work and explain your answer in sentence form.

c) Describe a translation that would place Pentagon RSTUV completely in the first quadrant of the graph.

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
with manipulatives with calculator

This work sample illustrates a standard-setting
performance for the following parts of the standards:
b Number and Operation Concepts: Understand and use opposite, reciprocal, raising to a power, taking a root, and taking a logarithm.
b Geometry and Measurement Concepts: Work with two dimensional figures and their properties.
h Geometry and Measurement Concepts: Analyze figures using the concept of translation.
k Geometry and Measurement Concepts: Work with geometric measures of length.
p Geometry and Measurement Concepts: Analyze geometric figures and prove simple things about them using deductive methods.
b Problem Solving and Mathematical Reasoning: Implementation.
c Mathematical Communication: Organize work and present mathematical procedures and results
clearly, systematically, succinctly, and correctly.

Mathematics required by the task
The basic elements of the mathematics needed to solve this problem are:

(1) plotting given points on a coordinate grid,

(2) finding the length of the line segment between two points (x1,y1) and (x2,y2) using the distance formula based on the Pythagorean Theorem:
distance =

(3) using the fact, from the definition of a regular polygon, that all side lengths of a regular pentagon must be equal, and

(4) specifying a translation T(4,5) in the plane as an operation that shifts every point in the plane +4 units in the x direction and +5 units in the y direction.

(Notice that the most negative x-coordinate in the original set is -3, and the most negative y-coordinate is -4. Hence T(4,5) is the shortest translation that maintains integer coordinates and that sends the figure to a position entirely within the first quadrant.)

What the work shows

b Number and Operation Concepts: The student understands and uses operations such as…taking a root.

b Geometry and Measurement Concepts: The student works with two dimensional figures and their properties.
The student correctly plotted the given points and connected them to form the pentagon.

h Geometry and Measurement Concepts: The student analyzes figures…using the concept of translation.
The student correctly identified a translation that satisfies the conditions stated in part (c) of the task.

k Geometry and Measurement Concepts: The student works with geometric measures of length.
The student correctly used the distance formula.

p Geometry and Measurement Concepts: The student analyzes geometric figures and proves simple things about them using deductive methods.
The student concluded that the figure is not a regular pentagon based on the information found by using the distance formula. The student could have stopped as soon as the measures of two of the sides were shown to be unequal.

b Problem Solving and Mathematical Reasoning: Implementation. The student selects appropriate mathematical concepts and techniques from different areas of mathematics and applies them to the solution of the problem.
Part b of the task does not tell the student what method to use. The work shows that the student understood what was meant by “a regular pentagon” and selected appropriate mathematics to determine if the Pentagon RSTUV is regular. The student not only selected the appropriate concepts but also deployed them accurately.

c Mathematical Communication: The student organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.
The response gives a clear indication of what the student did to solve the problem and to obtain the result.
The diagrams are connected and interpreted with explanatory text.