PBL NASA center on real life problems Prof. Gloria Faus ITESM, Campus Guadalajara, México

Author and Contributing Members PBL Author: Gloria Elena Faus Landeros Academic Advisor of PBL Design: Ph. Dr. Jim Botti Mathematical Advisor: Guadalupe Lomeli

Noctilucent Clouds OLD PBL TIME MODEL SCENARIO: Polar mesospheric clouds (PMCs), also known as noctilucent clouds (NCLs), are bright cloudlike atmospheric phenomena visible in a deep twilight. The name means roughly "night shining" in Latin. Unlike more common clouds that form up to 5 miles above the surface of the earth, these clouds are 50 miles high in a layer of the atmosphere called the mesosphere. They are most often observed in the summer months at latitudes between 50° and 60° north and south of the equator. In recent years, however, several people have reported seeing NLCs at lower latitudes, even as low as 40°N in the continental United States.

“NASA invites us to find the math model of limit that shows the time we have to observe these beautiful clouds”. PBL TIME MODEL SCENARIO Student´s solutions: Team 1Team 2 Team 3

Some Rationale I think the new scenario is better because the words, " to find the math model of limit that shows the time may give TOO MUCH direction to the learners such as the PBL sun rays math model. When I applied this scenario 2 semesters ago the students designed different math models but all of them were limited to illustrate just observation time.

Some Rationale With the new scenario the problem is complex and messier because it can cause the students decide to investigate not only the observation times of the Noctilucent clouds but also they can decide to analyze their constitution, and attempt to create a different “math model” that illustrates how noctilucent clouds constitution is, and the reason of their appearance, why is the electric blue color, why is the shinning, why we can see them only at night, etc.

PBL TIME MODEL SCENARIO Polar mesospheric clouds (PMCs), also known as noctilucent clouds (NCLs), are bright cloudlike atmospheric phenomena visible in a deep twilight. The name means roughly "night shining" in Latin. Unlike more common clouds that form up to 5 miles above the surface of the earth, these clouds are 50 miles high in a layer of the atmosphere called the mesosphere. They are most often observed in the summer months at latitudes between 50° and 60° north and south of the equator. In recent years, however, several people have reported seeing NLCs at lower latitudes, even as low as 40°N in the continental United States.brightcloudlike atmospherictwilightLatinlatitudesequator

PBL TIME MODEL SCENARIO NASA invites us to find the math model that shows (illustrates/explains) why (where & when) it is possible to observe these beautiful clouds”.

GOALS AND OBJECTIVES Students will research and analyze information about Noctilucent clouds to have a global and clear context about the elements which are involved in this phenomena, in order to have a mathematical interpretation with this information. It is expected that, with this information, the students will: Content objectives: Identify the places where the clouds are seen. Find the causes of why these clouds can be seen ( Relationship between the sun rays light and noctilucent clouds) Obtain the observer’s angle. Obtain when and where the observer can see this phenomenon. Find the earth’s circumference, radius, the angle cloud-earth-sun, and the angular distance.

Find the earth’s arc length (cloud’s time observation), and sector area formula. Physics Find the incidence angle of the sun with respect of the earth. Understand how the translational movement of the earth around the sun is. Analyze if the movement of the earth around the sun affects the incidence of the sun rays at the North Pole that make that the noctilucent clouds can be seen. Understand summer solstice. Analyze the constitution of the noctilucent clouds. Why they are in the mesosphere, why do they have that appearance, the electric blue color, the shine, why we can see them only at night, etc. Content objectives: GOALS AND OBJECTIVES

Bibliographical investigation objectives: Students must be capable of making a mathematical model that includes: antecedents, previous investigation of the problem, hypothesis, objectives, research fundaments, limitation, and glossary of terms. Fundamental theoretical for the problem solution process. The problem solution process has a logical and coherent sequence.

GOALS AND OBJECTIVES Math Model objectives: Recognize the variables that are involved in the phenomenon. Considered all the relevance variables on the math model. Make a math model statement coherent with the obtained information. Make a math model statement completely relevant and logic.

GOALS AND OBJECTIVES Math model objectives: Included the sufficient and necessary variables on the math model. Make a math model that shows the information was analyzed and comprehensive. Make a math model shows the behavior of the phenomenon.

GOALS AND OBJECTIVES Math model objectives: Make a math model shows the behavior of the phenomenon when the variables are changing. Make a math model that sustains the presented solution of the problem.

GOALS AND OBJECTIVES Math model objectives: Students must be capable of making a mathematical model that includes the variables that intervene in this phenomena and how modifying these variables affect the behavior of the phenomena. It is expected that students identify the most known variables that intervene in this phenomena.

GOALS AND OBJECTIVES Math Model objectives: Students must be aware that a bibliographical research should be complemented with a field research. Students must be capable of establishing conclusions based on the mathematical model.

LESSON RESOURCES Research from different books of Analytic Geometry, Trigonometry, Physics and Geography. (Ellipse formula, Angles laws, angular displacement, arc length, sector, area). Investigate scientific vocabulary. Research about: ray sun incidence angle. Investigate how the ray sun incidence angle change with respect the seasons. Find and obtain information on Internet and select the reliable and truth information. Use the GLOBE atmospheric protocols to collect data online. If appropriate, consider a multimedia presentation using images, graphics, or sound, like flash presentation. (For presenting the solution of the problem). Self learning. They need to do collaborative work: to discuss the information, elaborate the math model, and work on the presentation using technology.

Rubric for NASA Project Presentation Rubric Problem-Solving Model PBL model

PBL SUN RAYS MODEL Polar mesospheric clouds (PMCs), also known as noctilucent clouds (NCLs), are bright cloud like atmospheric phenomena visible in a deep twilight. The highest clouds in the Earth's atmosphere are made of frozen water, or ice crystals, just like some of the clouds that appear in the sky every day. Unlike more common clouds that form up to 5 miles above the surface of the earth, these clouds are 50 miles high in a layer of the atmosphere called the mesosphere.brightcloud likeatmospherictwilight Scientists are attempting to find out why they form and why they are changing. In particular, they wish to determine if these changes are caused by natural variations in the earth's atmosphere, or if they are influenced by human activities.

"NASA invited us to create a math model in which we could show possible relationships between relatively recent phenomenon of noctilucent clouds and global warming and climate change.“ PBL presentationBibliographic investigation PBL SUN RAYS MODEL Overview

Some Rationale One possible problem to the learners would address could be, "...if the energy or the sun rays will stay trapped in the atmosphere due to Noctilucents Clouds, how the change of the incidence angle of sun ray affect the noctilucent clouds. Students could explain with the math model why is better to see these clouds, from the end of June to the beginning of July, and if this phenomena affect the global warming. I was thinking this also is part of the time model and the students will know a lot of information about noctilucent clouds because I am planning to present the PBL TIME MODEL ( limits topic) on the first partial of differential calculus. Another problem could be to examine possible causes of the relatively recent phenomenon of noctilucent clouds like dusty aerosols, Volcano eruptions, greenhouse gases, pollution, etc. and illustrate these relationships through math models.

GOALS AND OBJECTIVES Students will research and analyze information about noctilucent clouds to have a global and clear context about the elements which are involved in this phenomenon, in order to have a mathematical interpretation with this information. It is expected that, with this information, the students will: Content objectives: Infer the general composition of the noctilucent clouds. ( NASA, links). Infer the Noctilucent Cloud density ( NASA, links). Find the altitude of de Clouds. Understand the incident sun ray on the earth and reflected ray sun on the earth. (Math and Physics). Find the incidence angle of the sun according to the seasons. (Physics, and other science).

Understand what the phenomenon of global warming is. Understand what the relationship between noctilucent clouds and global warming is. If exist a relation between both. Students must be capable to apply the knowledge of reflection’s law to a real world problem. (Mathematics). Students must be capable to obtain the angles noctilucent cloud’s phenomenon reflection and refraction. (Physics). GOALS AND OBJECTIVES Content objectives:

Students must be capable of making a mathematical model that includes: antecedents, previous investigation of the problem, hypothesis, objectives, research fundaments, limitation, and glossary of terms. Fundamental theoretical for the problem solution process. The problem solution process has a logical and coherent sequence. GOALS AND OBJECTIVES Bibliographical investigation objectives:

GOALS AND OBJECTIVES Math Model objectives: Recognize the variables that are involved in the phenomenon. Considered all the relevance variables on the math model. Make a math model statement coherent with the obtained information. Make a math model statement completely relevant and logic.

GOALS AND OBJECTIVES Math model objectives: Included the sufficient and necessary variables on the math model. Make a math model that shows the information was analyzed and comprehensive. Make a math model shows the behavior of the phenomenon.

GOALS AND OBJECTIVES Math model objectives: Make a math model shows the behavior of the phenomenon when the variables are changing. Make a math model that sustains the presented solution of the problem.

GOALS AND OBJECTIVES Math model objectives: Students must be capable of making a mathematical model that includes the variables that intervene in this phenomena and how modifying these variables affect the behavior of the phenomena. It is expected that students identify the most known variables that intervene in this phenomena.

GOALS AND OBJECTIVES Math Model objectives: Students must be aware that a bibliographical research should be complemented with a field research. Students must be capable of establishing conclusions based on the mathematical model.

LESSON RESOURCES Research about Noctilucent clouds, like: Geographic position, etc. the objective is they have a general concept of Noctilucent clouds (Where they are? Why they are there? Which is the information of the scientist about this clouds, etc? ) GLOBE DATA COLECCTION. Research about: ray sun incidence angle, global warming, and climate change. Investigate how the ray sun incidence angle change with respect the seasons. ( Differential calculus). Find and obtain information on Internet and select the reliable and truth information. They need to know how to use the flash software. (For presenting the solution of the problem). Self learning. They need to do collaborative work: to discuss the information, elaborate the math model, and work on the presentation using technology.

Analysis of Calculus learning through the technique of Problem-Based Learning. School year Research problemConclusions