# Modeling With Quadratic Functions

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## Objective

SWBAT use a quadratic function to model a real world situation.

#### Big Idea

This lesson allows students to make a connection between real-world phenomena and the average rate of change of a quadratic function.

## Launch

15 minutes

I will open this lesson by standing on a chair and dropping a ball towards the ground.  Based on this brief demonstration, I will let students individually work on the front side of the launch activity (falling_object_launch.doc). The task asks students to decide which graph best models the height of the ball over time as it falls.  After students have a chance to think, I will use a non_verbal cue to assess which choice the students have made.  Next, I will have students Think-Pair-Share with their neighbor to justify their choices (MP3).

Now, I want to slow the initial activity down so that students can analyze the motion of the falling object more closely.  I will explain that we are going to watch a time lapse video which shows a ball falling.  The video is fairly short so I can show it twice so that students can get the idea of what they are seeing. The video will show the height of the ball over time.

To slow this scenario down even more, I will then show students falling_object.pptx. This presentations allows me to project the time and height values show up one at a time.  I will ask students to graph the points on the reverse of their launch paper falling_object_launch.doc (pg. 2). This way, students will see that the shape turns out to be quadratic.  I will then have students turn and talk with their neighbor to determine which of the graphs on the front of the launch was the correct answer.

## Investigation

20 minutes

Next, students are going to be investigating a similar experiment when a ball is thrown into the air and then falls back to the ground.  The information for the falling_object_investigation is given to the students in a table.  Students will work with their partner to plot the points on the coordinate plane. They will see that a quadratic graph will be formed in this case as well.

The purpose of the next portion of the investigation is for students to see how the average rate of change (in this case velocity) changes over time (MP1).  I will ask my students to use the table and/or the graph to fill in the table for rate of change.   I anticipate that my students will be working at different rates on this investigation.  Because of this, I will most likely touch base with pairs of students individually to assess their understanding of the average rate of change.  If most of the class is working at about the same pace, I will bring them back together to discuss the rate of change table.  In either case, I want them to see that as the ball goes up it slows down and then once it reaches its vertex it speeds up in the negative direction.

Once students fill out the table, they can answer the 4 questions that follow.  Questions #1 and #2 are fairly straight-forward based on the graph/table but will help students ground their thinking back in the scenario.  The answer to questions #3 will come from the rate of change table.  Students should realize that the ball will be falling the fastest during the time interval right before it hits the ground (25 ft/sec)(MP2).  The fourth question asks students to interpret the table based on the situation.

Students can then work to complete the last page of falling_object_investigation.  Students will continue to work in their pairs in order to apply what they have learned to another scenario. Students will first graph the height of the roller coaster over time but they will use technology (graphing calculator) to do so.

Questions #1 and #2 are straight-forward and will help me to assess how my students basic understanding of how the scenario is represented by the graph.  Questions #3, 4, and 5 get at the idea of average rate of change.  This time, I ask students to calculate the average rate of change without the scaffold of the table (they can pull the data points from the graph).  Question #3 and 4 ask for the calculations.  In question #5, students should discuss the fact that the roller coaster is speeding up as it goes down the hill.  This will be based on their answers from question #3 and #4.

## Closure

5 minutes

Students will work independently on the closure for this lesson.  The closure activity is another application for finding the average rate of change over a given interval.  Through this closure activity, students will be able to see another application of falling objects in a different context.