##
* *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

Lesson 9 of 15

## 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.

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

*45 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**)

*expand content*

#### One Final Problem

*20 min*

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.*

*expand content*

#### Final Summary

*5 min*

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.

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: What is Algebra?
- LESSON 2: The Music Shop Model, Day 1 of 2
- LESSON 3: The Music Shop Model, Day 2 of 2
- LESSON 4: Letters & Postcards, Day 1 of 2
- LESSON 5: Letters & Postcards, Day 2 of 2
- LESSON 6: Choose Your Own Adventure
- LESSON 7: What Goes Up, Day 1 of 3
- LESSON 8: What Goes Up, Day 2 of 3
- LESSON 9: What Goes Up, Day 3 of 3
- LESSON 10: The Constant Area Model, Day 1 of 3
- LESSON 11: The Constant Area Model, Day 2 of 3
- LESSON 12: The Constant Area Model, Day 3 of 3
- LESSON 13: Practice & Review, Day 1 of 2
- LESSON 14: Practice & Review, Day 2 of 2
- LESSON 15: Unit Test