Jacob Nazeck RIDGEVIEW CLASSICAL CHARTER SCHOOLS, FORT COLLINS, CO
Algebra II : Unit #1 - Modeling with Algebra : Lesson #9

What Goes Up, Day 3 of 3

Objective: SWBAT use the structure of a quadratic equation to model projectile motion. SWBAT compare the motions of different projectiles based on the properties of the mathematical models.
Standards: HSA-CED.A.2 HSF-IF.B.4 HSF-IF.B.6 MP2 MP4 MP5 MP6 MP7
Subject(s): Math
60 minutes
1 Projectile Motion on the Moon - 10 minutes

Students work in groups to complete what they began yesterday: an investigation of projectile motion on the Moon!  Since this investigation began during the previous lesson, there is not much time provided for it today.

During these 10 minutes, students should be able to finalize their solution to problem #1 on ...Must Come Down.  A complete solution will include an equation, a graph, and a written comparison to the original projectile motion in terms of displacement, velocity, acceleration, and total flight time.  (Please see the solutions document for details.)

Watch out for students using the wrong values for the coefficients of the quadratic equation.  The most common mistake is to use the full value of the force of gravity, rather than half of its value.  If you see students making this mistake, do not simply tell them what to do, but ask them to explain the relationship between this coefficient and the force of gravity.  Using the original down-to-Earth situation as an example (MP 7), help them to see that the coefficient is only half of the force of gravity.

Ask each group to summarize their solution to problem #1 in a poster.  As different groups finish, post the solutions around the room for examination.  Emphasize the importance of organization and clarity in communicating to an audience.  We're not looking for flashy graphics, but something that is complete, coherent, and easy to comprehend. (MP 3 & 6)

A Change of Plans

Recently, I've noticed that when my students are supposed to be collaborating in groups, they have been ignoring one another.  Rather than ask one another for help, or share their insights with one another, they simply sit really close together while working completely independently.  One student rushes ahead with the solutions while the student next to her struggles to make a beginning!

In an attempt to prevent this from happening today, I did two things. 

First, I did not hand out the "... Must Come Down" worksheet.  I was afraid that if each student had his or her own copy of the problem they would be more inclined to work independently rather than collaboratively.  Also, I was afraid that the more advanced students would race to finish the first problem because they could see that there was a second one coming.  Instead, copied the first problem (the boy on the Moon) to a Power Point slide and projected it on the whiteboard.

Second, I required each group to produce a single solution together - a "poster" - containing equations, data tables, graphs, and verbal components.  All the members of each group would receive the same grade for this final product.  Typically, I am totally opposed to grading each group work in this way because it often rewards poor students for the good work done by their peers or punishes good students for the failure of their peers to pull their weight.  In this case, however, my purpose was to force the students to begin collaborating and taking responsibility for one another.

I'm not sure that these two changes solved the problem entirely, but I certainly saw improvement.  The final solutions to the problems were not perfect, and not everyone had the same insights, but we took steps toward a more healthy classroom culture.

2 Student Solutions - 10 minutes
Student Solutions

I have included here some samples of the solutions that students produced.  In general, they accomplished what they set out to do, but I found that they did not consider their "audience" in the way that I wanted them to.  It's important that a solution be presented in a way that uses precise language, presents essential information, and culls unnecessary details.  Too many groups tried to show all of their work or provide all of their data.  Too few groups were able to summarize their solution in a written paragraph that clearly and precisely answered the question.

In order to draw attention to these things, I asked the students to offer one piece of constructive criticism regarding each poster, and I emphasized the importance of being both critical and constructive.  It was gratifying to see that many groups took these comments seriously as they began to solve the next problem and create their next poster.

3 One Final Problem - 20 minutes

At the beginning of this section, shuffle the student groups and then ask them to work together on the final problem of ...Must Come Down.

The first task will be creating equations for the two new scenarios.  Keep an eye out to make sure that students are correctly interpreting the given information.  In each case, only one coefficient should be different from the original situation.

Different groups will take different approaches to comparing the functions, but a comparison graph will clearly show that the greatest difference comes about by increasing the initial velocity.  In addition to a graph, I ask students to provide some numerical evidence for their conclusion.  (MP 6) The maximum height and total flight time are good measures of the overall motion of the projectile and these should be compared explicitly.  See the solutions document for details. 

Student Solutions

Again, I've attached some photos of student work for your enjoyment!

The enactment of this section was somewhat frustrating because I had expected that the solutions would be easier and more efficient and that the "posters" would be of better quality.  Unfortunately, many students are still having a great deal of trouble identifying the vertex and roots of the parabola.  Some are satisfied to simply estimate these values by tracing the graph on a calculator, others simply accept their greatest data point as the vertex regardless of the asymmetry of their data.

In the end, several groups did not have time to finish their posters and it was the verbal element that received the least attention.  This is unfortunate because it is in the verbal statement of the solution that students really have to pay careful attention to the precision of their language and thinking.

4 Final Summary - 5 minutes

Use the GeoGebra application included in the resources to compare the motions of the four different projectiles.  We began with a single, original model and then we changed one coefficient at a time and examined the effects.

Changing the coefficient on the quadratic term resulted in the most dramatic overall change, preserving only the y-intercept of the original function.  In context, this is equivalent to moving to a different planet.

Changing the coefficient on the linear term resulted in a less dramatic change, but still preserved only the y-intercept of the original function.  In context, this is equivalent to using a more powerful slingshot.

Changing the constant term simply shifts the graph of the function upward.  Graphically, this produces the least dramatic change, but in fact everything is affected but the timing of the maximum height.  In context, this is equivalent to firing from atop a taller rock.

 An interesting final question (perhaps for an exit ticket) would be the following:  "Describe a situation in which projectile motion might be modeled by a quadratic equation whose graph opens upward."  A correct answer would be any case, no matter how fantastic, in which an object moves away from a point of reference with a constant rate of acceleration.

Some Interesting Questions

There's no telling what questions students will ask, but here are some that I got when we discussed these solutions.

When the initial height is increased, the whole graph just shifts up by 10.  But as you “go further” the graphs get closer together.  Will they ever meet?

Some students argued that they would, others that they would not.  To settle the issue, I suggested we set the equations equal to one another and solve.  15 = 5?!  I guess they never meet.

When the initial velocity is increased, the graphs obviously intersect at (0, 5).  Will they meet again?

Again, some students thought yes, others no, and for a variety of reasons.  Again, I suggested we set the equations equal and solve.  This time we find 50x = 60x.  Interestingly, the initial reaction of most students was that this could never be true.  But, I pointed out, we know the graphs do meet at least once!  Finally, it dawned on some that 50x = 60x can be true, but only if x = 0.  So that answers our question.  I found it interesting (distressing?) that none of the students suggested solving the equations simultaneously as a way to answer the question.

We got a very different trajectory when we were on the Moon.  Would it be possible to recreate that trajectory on Earth?

The equations could not be the same because the coefficients on the quadratic term have to be different (due to gravity). 

Does that mean the graphs have to be different?  Couldn’t we adjust the other coefficient and the constant to make up the difference?

This question surprised me.  It seemed obvious to me that different equations must yield different graphs, but this was not obvious to my students.  In the end, we used the dynamic features of GeoGebra to try adjusting b and c to make an “Earth trajectory” match the “Moon trajectory”.  (In the one case, a = -16 and in the other a = -2.665.)  Before long, the students decided it just couldn’t be done.  There wouldn’t be any way to exactly match the “steepness” of the graph at the beginning (not to mention everywhere else).  They still did not seem convinced, however, that they would have to have exactly the same equation in order to have exactly the same graph.  I imagine this is because they are so used to seeing the same equation in different forms; they confuse differences in form with differences in substance.