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.

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

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**)

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.*

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.