## Reflection: What Goes Up, Day 3 of 3 - Section 4: Final Summary

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.

Some Interesting Questions
Some Interesting Questions

# What Goes Up, Day 3 of 3

Unit 1: Modeling with Algebra
Lesson 9 of 15

## Big Idea: Modeling projectile motion provides an opportunity to make use of the structure of a quadratic equation.

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Standards:
Subject(s):
Math, modeling, Graphing (Algebra), Algebra, Quadratic Equations, Algebra 2, master teacher project, Projectile Motion
45 minutes

### Jacob Nazeck

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