The task requires students to do careful work, all based on the
complex specifications given for the theater, that involves the
geometry of circles and the division of line segments into parts:
 Lay out concentric circular rings that will serve as rows of
seats and find the maximum number of rows possible, subject to
a given minimum depth of a row of 90 centimeters.
 Divide the concentric rings into sections separated by radial
aisles and calculate the resulting length of the seating section
in each row, subject to a given minimum aisle width of 1 meter.
 Calculate the number of seating positions possible in each section,
subject to a given minimum seat width of 60 centimeters and a
given maximum number of seats per section of 30.
 Among possible ways of laying out such a theater, make choices
that increase seating capacity.
The core mathematical concepts needed to do
this work are few and simple. They are principally:
 Find the circumference C of a circle from its radius r or its
diameter d: (C= 2r = d).
 Find the number N of seats of width 60 centimeters in a section
of length L meters: (N = the greatest whole number less than or
equal to L/0.6 ).
Taken in isolation these concepts are straightforward.
However, in this task students need to use them repeatedly and appropriately
in a complex setting. This need provides much more of a challenge
than the mathematical concepts themselves. As a result, the task
requires quite a bit of “problem solving” ability such
as understanding the situation, constructing and testing mathematical
models of the situation, and finding the optimal model (since “the
job will go to the team that demonstrates the design with the greatest
seating capacity”).
a
Geometry and Measurement Concepts: The student
models situations geometrically to formulate and solve problems.
b
Geometry and Measurement Concepts: The student works with two…dimensional
figures and their properties, including…circles….
e
Geometry and Measurement Concepts: The student
knows, uses, and derives formulas for…
circumference….
The whole project is a complex geometric model based on circles
that was created in response to a request for a design meeting detailed,
specified constraints.
Note that the radius of 17.7m used here results from work shown
earlier on the page: 5 (stage) + 1 (aisle) + 12.6 (set of 14 rows)
 0.9 (last row) = 17.7 meters.
The use made here of the 17.7m dimension in calculating the circumference
is a step repeated several times throughout the work.
a
Function and Algebra Concepts: The student
models given situations with formulas….
i
Function and Algebra Concepts: The student represents functional
relationships in formulas [and] tables…and translates between
these.
The formula
produced by the student is the heart of the response. This formula
provides an effective mechanism for counting the number of seats
in each row in terms of the radius of the row. The student used
the formula to construct the table on the next page showing the
seating capacity for a section in each of the 14 rows of the theater.
The formula produced by the student had to be applied many times
for the aisles of different radii. The student apparently carried
out these computations by hand. This is fine, but it would also
have been an ideal place to let technology do some of the work by
using a spreadsheet. The advantage would be that the effect of changes
in the input (here the radii) could have been quickly determined.
b
Problem Solving and Mathematical Reasoning:
Implementation. The student…
• chooses and employs effective problem solving strategies
in dealing with nonroutine and multistep problems;
• selects appropriate mathematical concepts and techniques
from different areas of mathematics and applies them to the solution
of the problem;
• …uses mathematics to model real world situations….
Throughout the work, the student responded to a nonroutine task
in a way that shows careful planning, use of many kinds of given
information from a real situation, selection of appropriate mathematics
from and
,
and employment of results from one step as input to the next step.
The student also made many choices dictated by the goal that the
theater should have “the greatest seating capacity” consistent
with the given space. For example, the response uses the smallest
allowed dimension (90 cm) for the depth of a row, thus yielding
the maximum number of rows.
Still, the
response does not take the next step in attempting to create the
greatest seating capacity, namely pursuing alternate ways of meeting
the constraints of the design and comparing them for the resulting
seating capacity. This portion of the work shows the steps taken
to meet the constraint on the maximum number of seats (30) allowed
per row. The student found that a 4aisle design would give 44 seats
in the last row of each section, that a 5aisle design would give
35, and that a 6aisle design would give 29. The choice of a 6aisle
design thus seems natural. Still, it seems necessary to note that
other possibilities were not explored in the response. For example,
if 5 radial aisles are used, and are made wider toward the rear
of the theater to limit the number of seats per row in a section
to 30, then a total capacity of 1,585 can be reached. And with a
6aisle design, if the seats are pushed rearward as far as possible
(by making the outside concentric aisle width its minimum of 2 m)
there are 1,614 seats possible, which is 60 more than the response’s
1,554 seats in a design using a rear aisle width of 2.4 m. In this
sense the student did not do full justice to the goal of obtaining
the greatest seating capacity. Nevertheless, the maximum seating
requirement here is a very difficult one to meet and justify, and
the fact that this student did not fully accomplish this does not
detract from the fact that the response illustrates the indicated
portions of .
c
Problem Solving and Mathematical Reasoning:
Conclusion. The student…concludes a solution process with a
useful summary of results….
The conclusion
summarizes the results obtained.
b
Mathematical Skills and Tools: The student
rounds numbers used in applications to an appropriate degree of
accuracy.
This remark “you can’t have .8 of a seat” should
actually say “you can’t have .7 of a seat” since
it refers to the fractional part of the number = 178.7 seats obtained
in the calculation mentioned previously. Still, the rounding down
to an integer value is correct.
e
Mathematical Skills and Tools: The student
makes and uses rough sketches…or precise scale diagrams to
enhance a solution.
a
Mathematical Communication: The student is
familiar with basic mathematical terminology….
Here and
throughout the work, the student used terminology such as “radial
aisles” in an appropriate and consistent way.
h
Mathematical Communication: The student writes
succinct accounts of the mathematical results obtained in a mathematical
problem or extended project, with diagrams,…tables, and formulas
integrated into the text.
The student produced a clear explanation of
the thinking that went into the design, together with diagrams showing
features of the design and a formula for the crucial calculation
of the number of seats in a row. Particularly good examples of communication
include:
The
student explained in words where the result of “44 seats per
row” came from. The explanation amounts first to the calculation
,
rounded down to 178 seats in the last row, then the calculation
178/4 = 44.5, rounded
down to 44 seats per section. A little thought shows that this is
equivalent to the one step calculation ,
rounded down to 44 seats. This calculation is crucial to the whole
problem. The response gives an explicit formula for this calculation
for the 6aisle case.
The
conclusion summarizes the results obtained.
The word “sollution” in the final paragraph should be
“solution.”
