Molly and her friends are planning a weekend vacation to an island that is about 90 minutes away. They have purchased tickets to a concert and a local food tasting that only happens once each year. These tickets are non-refundable.
After reading about the expected conditions on the water, Molly realizes she and her friends will need to sail through some rough waters. She has mapped out the two best routes; the height of the waves, in feet, on Route 1 can be modeled by \( f(x)= -9\cos\left( \frac{\pi}{15}\left( x - 3 \right) + 8 \right) \). The height, in feet, of the waves on Route 2 can be modeled by \( g(x)= -15\cos\left( \frac{\pi}{24}\left( x - 15 \right) + 10 \right) \). For both functions that value of x is measures the number of minutes since the boat has left shore. Molly expects the water traffic to be heavier along Route 2.
Molly is nervous about sailing through these waters in her current watercraft, which is a 24-foot cabin cruiser. This craft has a completely enclosed pilot house that allows Molly to stay out of the elements as she drives the boat. It also has a completely enclosed cabin that will let her friends stay warm and dry on the trip. But it is not the most maneuverable in rough waters. Molly knows that her boat can easily handle waves that have a height less than or equal to about \( \frac{1}{3}\) of the boat’s length. One of Molly’s friends suggests that she borrow her uncle’s boat. It is 30 feet long but it does not have an enclosed area for the guests or the pilot; it has only a canopy that covers the piloting area.
Discussion
Use what you know about trigonometric functions to help Molly and her friends decide which route they should sail on, and which boat they should take. Support your answer using the graphs of the trigonometric functions; identify the key features of those graphs in relation to this scenario. Provide screenshots of the graphs as needed and show any mathematical calculations to make as part of your response. Use complete sentences and paragraphs in your answer.
You will be graded using the following rubric(s).
| Expert | Practitioner | Apprentice | Novice |
| Student states advice for Molly and her friends. | |||
|---|---|---|---|
| Student clearly states their advice for Molly and her friends. | Student somewhat clearly states their advice for Molly and her friends. | Student alludes to their advice for Molly and her friends. | Student does not state advice for Molly and her friends. |
| Student uses relevant mathematics to support their advice. | |||
| Student correctly identifies and interprets the key features on the graph of each function. Student uses the key points to support their advice. Relevant mathematical calculations are correct. All necessary steps are shown. |
Student is mostly correct in their identification and interpretation of the key features on the graph of each function. Student uses the key points to support their advice. Relevant mathematical calculations are mostly correct. Some of the necessary steps are shown. |
Student identifies but does not interpret, or incorrectly interprets, the key features on the graph of each function. Relevant mathematical calculations contain moderate errors or some of the necessary steps are absent. |
Student does not identify or interpret the key features on the graph of each function. The mathematics presented contain major errors or some of the necessary steps are absent. |
| Student understanding. | |||
| Student relies on the results of their interpretations to support their advice. Student reasoning clearly demonstrates that they understand how to apply a trigonometric function to a real world scenario and interpret the results. Student addresses reasons for their advice beyond the mathematics, such as possible weather conditions and route traffic. |
Student relies on the results of their interpretations to support their advice. Student reasoning indicates they mostly understand how to apply a trigonometric function to a real world scenario and interpret the results. Student mentions reasons for their advice beyond the mathematics, such as possible weather conditions and route traffic. |
Student mentions their interpretations, but does not directly use them to support their advice. Student reasoning shows they somewhat understand how to apply a trigonometric function to a real world scenario and interpret the results. Student does not mention reasons for their decision beyond the mathematics. |
Student does not use their interpretations, or any other relevant mathematics to support their advice. Student does not mention additional reasons for their advice. |
Ethics Scenario
| Expert | Practitioner | Apprentice | Novice |
| Student uses technology to create a graph of each function. | |||
|---|---|---|---|
| Student creates a graph of each function. Pertinent points are labeled or discussed. The viewing window is adjusted appropriately. | Student creates a graph of at least one function. Pertinent points are labeled or discussed. The viewing window is adjusted. | Student creates a graph of only one function. Pertinent points are not labeled or discussed. The viewing window is not adjusted appropriately. | Student does not create graphs or the graphs contain major errors. Pertinent points are not labeled or discussed. The viewing window is not adjusted appropriately. |
| Expert | Practitioner | Apprentice | Novice |
| Response Construction | |||
|---|---|---|---|
| Responses are clear and well-written. Responses have a distinct opening, body, and conclusion. |
Responses are clear. Responses have distinct parts. |
Responses are somewhat clear. Responses lack or are missing some distinct parts. |
Responses are unclear. Responses do not have distinct parts. |
| Sentence Structure | |||
| Sentence structure is clear. Paragraphs are used where needed. Grammar is all correct or only minor errors found. |
Sentence structure is mostly clear. Paragraphs are sometimes used. Grammar has minor to moderate errors. |
Sentence structure somewhat unclear. Only one paragraph used. Grammar has moderate errors. |
Sentence structure unclear. Paragraphs not used. Grammar has severe errors. |
| Spelling | |||
| Spelling is all correct or there are only 1-2 errors. | Spelling is mostly all correct; there are 3-5 errors. | Spelling is somewhat correct; there are 6-8 errors. | Spelling needs some practice; there are more than 8 errors. |
