The task
This is a design task. Students are asked to create a design for a theater that conforms to several specified constraints. Interpreting and implementing these specifications requires significant knowledge of concepts and terminology from geometry and algebra.

The task asks for a theater design “with the greatest seating capacity” given the specified constraints. This will be interpreted as requiring a demonstration, in the work, that the choices being made in making the design do in fact contribute to increased seating capacity. It is felt to be too difficult, in this complex situation, to require an actual proof that the design produced has the greatest possible capacity.

The students had a week to complete the task, and then a week to revise based on teacher feedback. They worked in groups, but then each student submitted a separate response.

Mathematics required by the task
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”).

What the work shows
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 non-routine and multi-step 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 non-routine 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 4-aisle design would give 44 seats in the last row of each section, that a 5-aisle design would give 35, and that a 6-aisle design would give 29. The choice of a 6-aisle 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 6-aisle 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 6-aisle case.

The conclusion summarizes the results obtained.
The word “sollution” in the final paragraph should be “solution.”