##
* *Reflection:
What Goes Up, Day 3 of 3 - Section 1: Projectile Motion on the Moon

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.

*A Change of Plans*

*A Change of Plans*

# 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