Description Teacher Comments Students Perceptions Students Thinking Calculational Strategies Mathematical Strategies

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HISTORIC HOTELS CASE STUDY
Developed by Lupita Carmona

DESCRIPTION
Students are given information about how the number of vacancies in a hotel will change if the owner increases the price. Thus, the students have to find the room charge that will maximize the hotel's profit. Involves quadratic relationships. (Note: students can solve this case study even if they have not been formally introduced to quadratic equations. They tend to use some concepts related to quadratics, but they may not know that is what they are using.)

· This case study is appropriate for students in grades 7 - 12.
· Sometimes students do not understand the relationship between the room rate increase and the number of rooms occupied (for every \$1 increase in the room rate, one less room is occupied). Thus, you may want to discuss this relationship before the students start the problem to ensure that the students can interpret the problem situation.
· The students will make different assumptions about who is paying the \$4 maintenance fee. If the students assume that Mr. Graham pays the fee, then his profit occurs at \$72 per room. If the students assume that the customer pays the fee, then they often fail to take the \$4 out of Mr. Graham's profits (although the customer is paying the \$4, Mr. Graham still doesn't get to keep the \$4 because he has to pay the workers to complete the maintenance). If the students fail to take this \$4 out of Mr. Graham's profits, they will erroneously state that the profit occurs at \$72 per room. (What they have actually found is the room rate at which Mr. Graham's revenue is greatest.) The actual profit for Mr. Graham, if the customers pay the \$4 maintenance fee, occurs at \$70 per room.

STUDENTS' PERCEPTIONS OF CASE STUDY
When seventh grade students were surveyed after working on this case study,
· 11% of the students found the case study easy,
· 30% found the case study a little challenging,
· 43% found the case study somewhat challenging,
· 14% found the case study challenging, and
· 2% found the case study very difficult.
When eighth grade students were surveyed after working on this case study,
· 33% of the students found the case study easy,
· 19% found the case study a little challenging,
· 19% found the case study somewhat challenging,
· 24% found the case study challenging, and
· 5% found the case study very difficult.

STUDENTS' THINKING SHEET

Description of Overall Solution Process
Trial and Error: Students may use trial and error to test different room rates. In other words, they may test a room rate, find the profit at that room rate, and then make an educated guess as to what room rate to charge next. They continue this process until they narrow in on the room rate that maximizes Mr. Graham's profit.
Sequential: Students may sequentially increase the room rate until they find the point of diminishing returns. For example, students may compute the profit for \$60, then \$61, then \$62, then \$63, and then continue with \$1 room rate increases until they notice that the room rate begins producing a lower profit for Mr. Graham. Students may also find the point of diminishing returns by increasing the room rate by 5's or by 10's.

Description of Specific Calculational Strategies
Strategy #1 Subtracting Maintenance Fees: Students may find Mr. Graham's profit with the following calculation: (room rate x number of rooms) - (\$4 x number of rooms) = profit. When students use this calculation, they usually assume that Mr. Graham has to pay for the maintenance fees.
Strategy #2 Adding in Maintenance Fees: Students may find Mr. Graham's profit with the following calculation: (room rate x number of rooms) + (\$4 x number of rooms) = profit. When students use this calculation, they usually assume that the customers pay for the maintenance fees.
Strategy #3 Second Differences: Students may notice the constant difference between the differences in profit. For example, they may find the profit for \$60, \$61, and \$62 and then find the differences between the respective profits. Then, they'll compute the constant difference between these differences. They then use this constant difference to step down to the smallest possible positive difference (closest to zero) at which point the rate for maximum profit occurs.

Mathematics in strategy Meeting the needs of the Client
· Using an inverse relationship between the room rate and the number of rooms occupied.· Adding, subtracting, and multiplying to find the profit. · Economic concept of profit (money earned minus expenses).· Comparing profits at different room rates.· Maximization. Ease of Use: If the students recommend using these calculations with a sequential approach, it will be easier for the client to use this strategy than if the students recommend a trial and error approach.Effectiveness: The effectiveness of this strategy depends on how well the students identified the point of diminishing returns. For example, some students simply stopped at \$70 per room when they realized that \$70 yields more profit than \$60 or \$80. Thus, this approach is not as effective as an approach in which the students checked all room rates (i.e., between \$60 and at least \$73 where the profit begins diminishing) and realized that \$72 yields the greatest profit.
· Using an inverse relationship between the room rate and the number of rooms occupied.· Adding and multiplying to find the profit. · Economic concept of profit.· Comparing profits at different room rates.· Maximization. Ease of Use: Same as strategy #1.Effectiveness: Same as strategy #2. Also, the effectiveness depends on whether the students recognize that although the customers are paying the \$4 maintenance fee, Mr. Graham still does not get to keep the \$4. If the students recognize this fact, they will discover that the profit for Mr. Graham occurs at \$70 with this strategy. If they do not, they will be finding the maximum of Mr. Graham's revenue, not his profit.
· Using an inverse relationship between the room rate and the number of rooms occupied.· Adding, subtracting, and multiplying to find the profit.· Economic concept of profit.· Pattern development in noticing the constant second difference.· Iteration to repeat the pattern and find the maximization point.· Maximization. Ease of Use: This strategy does reduce some of the calculations for the client, but it is more complex, so understanding it may take more time.Effectiveness: The effectiveness of this strategy depends on correctly identifying the pattern in the second differences and determining how far to continue the pattern in order to find the maximum profit.