Description |
Teacher Comments |
Students Perceptions |
Students Thinking |
Calculational Strategies |
Mathematical Strategies |

**You may want to revisit the Basic
implementation information page.**

**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.)

**TEACHER COMMENTS**

· 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.

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**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.

**You may want to revisit the Basic
implementation information page.**

Description |
Teacher Comments |
Students Perceptions |
Students Thinking |
Calculational Strategies |
Mathematical Strategies |

## Teaching Notes - Historic Hotels |