These work samples were drawn from two different classrooms although the
task given to the students was the same:
Design a dart board that has four regions with the following features:
score value 
probability %

100 points 
10%

50 points 
20%

25 points 
30%

10 points 
40%

The dart board may be any shape (circle, square, rectangle, triangle,
etc.) and must have an area from 1,000 square centimeters to 3,000 square
centimeters. Assume the probability is proportional to the area of the
region. Make a scale drawing with dimensions and explain your solution
in words.
The task calls for the student to set up a total area that satisfies
given constraints. Then the student must partition this area correctly
into regions of sizes proportional to the given percentages. The scale
drawing requires understanding of appropriate measurement and proportional
reasoning. A firm grasp of area measurement is needed for a successful
solution.
Probability, while mentioned, is not actually called for by the task.
The assumption that equates the probability of hitting a region with the
area of the region presumes that darts would always land on the board
and that players’ aim at the target would be ineffective.
Nevertheless, the task lends itself to a wide variety of solutions. Some
approaches are quite involved and complex. Other satisfactory solutions
might be equally insightful, yet less complicated.
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 
Sample 1

These work samples illustrate standardsetting
performances for the following parts of the standards:

e 
Number and
Operation Concepts: Interpret percent as part of 100. 
f 
Number and
Operation Concepts: Reason proportionally. 
a 
Geometry and
Measurement Concepts: Be familiar with assorted two and threedimensional
objects. 
d 
Geometry and
Measurement Concepts: Determine and understand length, area,
and volume. 
b 
Problem Solving
and Mathematical Reasoning: Implementation. 
a 
Mathematical
Skills and Tools: Compute accurately with arithmetic operations
on rational numbers. 
b 
Mathematical
Communication: Organize work, explain a solution orally and
in writing, and use other techniques to make meaning clear to
the audience. 

e
Number and Operation Concepts: The student interprets
percent as part of 100…
Both students
showed the relationship of their diagrams to 100 points and represented
that relationship as percent.
d
Geometry and Measurement Concepts: The student
determines and understands length, area, including perimeter and surface
area; uses units, square units.
The student in
Sample 1 used square units accurately (cm²) to divide a large square
into one hundred congruent smaller squares.
The student in
Sample 2 exhibited command of the concepts of area and percent throughout
the work.
The dart board
sketch is drawn to scale. This diagram provides strong evidence of proportional
reasoning, part of .
The student appropriately used the centimeter as the unit of measure.
a
Mathematical Skills and Tools: The student computes
accurately with arithmetic operations on rational numbers.
In Sample 2, the
student accurately multiplied rational numbers.
b
Mathematical Communication: The student organizes
work, explains facets of a solution, labels drawings, and uses other techniques
to make meaning clear to the audience.
The student
in Sample 1 provided a legend to explain clearly the value of each square,
and designated colors to represent the point values of each region. In
addition, the student described how the number of squares colored for
each region corresponds to the percent represented by that region: “I
took 40% and colored 40 boxes.”
Here, as elsewhere
in Sample 2, the prose makes clear the means by which the student built
on previous steps to determine proportional areas and percents. Both work
samples contain minor grammatical errors, but the mathematical solutions
are correct.
