Orbits - teacher notes

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Introduction

It is my belief that the student notes for this lesson series are pretty self-explanatory and as such I am not providing extensive teacher notes for each lesson, only some ideas on how each lesson may be implemented and some things that teachers would want to focus on when dealing with the lesson.

Preliminary remarks

Time needed - the amount of time that teachers devote to each lesson unit and as a result to the lesson series is more a function of how much time the teacher wishes to spend on the series than a function of the materials. I think that a minimum would be 6 days (one per unit) - this would be tight but would be enough to communicate the content of the lesson series. In the case of one unit per day - teachers would have to limit the amount of exploring that students do of the links provided, would have to select only some of the tasks from each unit, and would have to omit unit 6. A fuller version of the lesson series could easily take 8 to 10 days for units one through five with unit six, the capstone, being set as a project to be completed over an extended period of time.

Teacher role - while the lessons are fairly self-contained in the sense that a student who missed class could probably catch up using the materials alone, I believe that it is the teacher who will determine the success of this series by their enthusiasm for the topic and by the way they build connections from one unit to the next.

Basic outline

• Unit one - Setting the scene: the underlying issue of the lesson series is how do navigators ensure that space vehicles rendezvous with desired vehicles, planets or moons and avoid collisions with undesired debris, vehicles, planet etc. The Mars Pathfinder landing is used to highlight just how complex this challenge is and sets the scene. Using the internet students visit some of the mission specialists involved in the task.
• Unit two - Introduction of ellipses: Kepler's and Newton's laws are touched on and lead us to the realization that all objects in space travel on conic orbits and in particular that the ones of greatest interest to us in this unit travel on elliptical orbits. The two loci definitions for ellipses are introduced and worked with
• Unit three - Introduction of the Cartesian equations for ellipses: assuming that students are familiar with the equation for an ellipse (given alongside), data from some actual space missions are used to plot the ellipses representing the trajectories of the space vehicles.
• Unit four - Introduction of polar representations for ellipses: discussions of the limitations of Cartesian representations of the ellipse introduce a need for the polar representations which is derived for conics in general and ellipses in particular.
• Unit five - Getting from one orbit to another: we try to tie up at least one of the challenges of the original setting ie getting from a given orbit to a second given orbit via a third orbit, the tasks include a number of fun challenges which will test the student's understanding of both their software and the equations of the ellipses.
• Unit six - Some extensions: rather than trying to eliminate all the simplifications we had to introduce initially to be able to study the problem, six different variables are reintroduced to the problem one at a time. Rather than doing all six tasks, students choose one or maybe two tasks and work with these.

Unit by unit suggestions for teaching

Some suggestions for the management of each lesson are made in this section, these suggestion should be read in conjunction with the actual unit outlines as none of the unit content will be reproduced here.

UNIT ONE - Setting the Scene

This is the critical unit in setting the scene for the unit. As with all teaching through applications - if students do not see either the value of the mathematics in the application or think of the application as interesting and relevant then the application will become more like noise in the lesson and the whole experience becomes more frustrating than valuable.

Some ideas on starting the lesson - videos about the Mars Pathfinder mission will soon be available through one of NASA's Mars related education projects and could provide an exciting lead in. If not a video a simple class discussion about space flight that raises issues such as:

• How did the Pathfinder get to Mars? Was it powered? - highlighting the idea that orbits are predetermined by the parameters of the vehicle and the solar system,
• What makes the task difficult? - highlighting the whole notion of how many variables this problem deals with
• Are the challenges of this mission any more significant than say sending the shuttle to dock with Mir? - highlighting the idea that though the Mars mission was amazing, the concepts involved are common to many space flight challenges (in my mind all amazing).
• What was the role of the three path corrections performed throughout the seven month trip? - highlighting the fact that the mission navigators were working with no more than a model of how the pathfinder would travel and so highlighting that "real-world" math is not theorem-proof in nature!

The really important thing to be achieved by this lesson is that students develop some awareness of the magnitude of the task involved in this and other space missions.

TASKS

Once the excitement has been established let the students spend time on the internet visiting the various sites and preparing responses to the writing task. In grading this task the focus should be on whether or not the students have understood the essence of the mission navigator's challenge and if they have translated this into a "mathematical" problem - that is not in equation etc form but rather an idea about paths and intersections of paths etc.

UNIT TWO - Ellipses

Before simply accepting that the trajectories of all orbits in space are conic sections it may be a useful to try and justify this - though the actual science may be a bit much for some non-science students, it may be interesting to science students. Irrespective the fundamental idea as explained by Newton can at least be followed by almost anybody and student should realize that this idea is not just a magic thumb-suck! The video described has a beautiful explanation of this concept and could be played as the introduction to the lesson.

A number of vocabulary words are introduced - again the video will prove to be really helpful in this regard as it introduces these concepts in a clear and attractive manner. The purpose of these terms is not so much that students should remember them and use them but more to highlight that as we get into the problem more and more we make increasing simplifications on the actual situation so that we can begin to understand it all.

TASKS

The GSP activity, while not critical to the unit, helps develop one view of the ellipse which in keeping with the Standards view that students should "be able to apply and translate between different representations of the same problem situation or concept" (pg. 146), is important. The sketch based on the first definition is significantly easier and should be attempted by everybody, the second sketch is difficult and highlights some of the limitations of the software in that "little holes" appear in the sketch. That software has these limitations is important and is a problem that will re-occur in the next unit.

Click the GSP icons for solution sketches for use by the teacher.

Sketch for definition 1, sketch for definition 2.

TASK 2 - The diagram alongside may help students to see the relationship between a, b and c more easily:

If you think of moving the point A to B it can quite easily be shown that the constant length of the ellipse is actually 2a from which the labels of the triangle with vertex A follow nicely and we then can relate a, b and c by the Pythagorean identity.

UNIT THREE - Cartesian Equations for Ellipses

This unit is the start of the graphing phase and is important both in itself and also because of the problems that it raises. The unit assumes that students are familiar with the Cartesian form of the equation for an ellipse shown alongside, whether or not this is the case, it may be quite important for the teacher to refresh the students' familiarity with this equation, particularly since we will be using it in a manner that students may not be familiar with - that is we are given certain information from which we must first establish the equation and then we will draw it (students may be more familiar with being given the equation with all its information already).

TASKS

The teacher should anticipate problems during the graphing phase caused by one or more of the following:

• the students' graphing utilities may not be able to deal with equations of the form given and students may need to change them into a y = .... form and then deal with the two cases y = +sqrt... and y = - sqrt....
• the students may run into trouble with the domain and range of the graphing display and may need help overcoming that problem
• if the graphing limitation (I'm not actually sure if it is a limitation - as it highlights limitations with the Cartesian form of the equation) mentioned above exists - the students graphs may have "holes" in them.....
• students will in all probability be VERY DISAPPOINTED with the fact that after all the work that they will have done, so many of their ellipses will appear pretty circular - a function of the dimensions that we are dealing with and also our classical but erroneous assumptions that the ellipses of the solar system are highly eccentric.

While having students gather their own data for the tasks is intended to highlight the authenticity of the task we are dealing with, it may also take a great deal of time which may be a cause of frustration to the teacher - to avoid this the teacher could either simply provide data, or make copies of the relevant webb pages so saving students the trouble of having to go there.

It would really be good if there were a way for students to print their graphs for inclusion in their portfolios and as records - failing this students should at least make rough sketches of the graphs.

UNIT FOUR - Polar Equations for Ellipses

Hopefully students will already have met with some limitations of the Cartesian form of the equation for ellipses in unit three - through limitations of the software, the challenge is to introduce the polar form not as another form for the sake of another form but rather because if has a number of inherent benefits - the teacher's role in highlighting this is critical if students are to realize the benefits of the polar form and then to use it.

The polar form can either be developed, or another nice idea may be for students to study it's derivation in the unit and then prepare a lesson for the class on the derivation, the teacher can then select a number of students to "teach" it. At the heart of this remark is the thought that the teacher should ensure that the student understands where the polar form of the equation comes from!

TASKS

Solution for task 1:

A = a + c = a + ea = a (1 + e)

P = a - c = a - ea = a (1 - e)

Now a = 1/2 (A + P) and c = a - P = 1/2 (A + P) - P = 1/2 (A - P)

and finally e = c/a = (1/2 (A - P))/(1/2 (A + P)) = (A - P)/(A + P)

A feature that really can enhance this task is the fact that some graphing utilities allow the user to limit the domain for t (the angle) in polar plots. The third task is all the more exciting if this feature is a part of the utility and students should really be encouraged to explore it.

UNIT FIVE - Getting from one orbit to another

Having established the polar form of the equation we are ready to deal with the original task - that is getting an object from one orbit to another orbit via a third orbit. There are a number of different orbit transfer techniques - we will concern ourselves only with the one that is considered the most fuel efficient and which suits us because no change of inclination (ie change of orbital plane) is involved in the process.

The video has an excellent illustration of the Hohmann transfer and should definitely be used in class. Once the concept is understood the rest of the task is really more of the kind of work done in the previous unit.

TASKS

These tasks are in essence a repetition of the tasks of unit five with the exception that students are building their own Hohmann transfer orbits - one solution is provided and the others follow in a similar manner.

It is hoped that the data for the Big Challenge will soon be added to the teacher notes so that these can simply be given to students.

UNIT SIX - Some extensions

This section is envisaged as the capstone for the lesson series and forms a substantive part of the evaluation for the series. The idea is that students should work by themselves (or in pairs, but without direct instruction by the teacher) on at least one of the units and preferably two (some units rely on earlier ones and as such two units have to be done). Links to helpful webb pages are provided and it is believed that students should be able to manage a large part of these problems by themselves if they have learned both the work of the earlier units and understood the philosophy of our approach so far - reduce the problem, solve the simpler version and then add the frills.

Teachers may encourage students to work in pairs on these activities - though it is suggested that each student hand in their own product.

In problem 5 - Orbits within Orbits we are dealing with epitrochoids. It may help the teacher to refer to some work I have done on Cycloids, Hypocycloids, Epicycloids, Hypotrochoids and Epitrochoids which deals with the use of GSP in the construction of these figures