a1tD00000088DnWIAU

• Intermediate

### Lesson Description:

This lesson demonstrates the concept of a quadratic relationship. After this lesson, students should be able to grasp the concept of quadratic proportionality and be able to apply this concept to solve real-life problems. This lesson demonstrates what a quadratic graph looks like and how it associates with the area.  It also helps to relate a quadratic function to the physical world.

Objective: Identify how changing parts of the quadratic equation affect the way a quadratic equation graph will look.

## Teaching Tips:

Once you open the lesson, you will be presented with a series of items that will need to be completed before you continue onward to the lesson.

The first step is to position the AR.Drone.  The AR.Drone must be positioned on the mat above its grey silhouette, in a safe manner away from objects that could interfere with its flight path. It is important to use the mat, as the AR.Drone tracks the shapes on the mat and uses them to prevent drifting away from the take-off position.  In this exercise the AR.Drone should travel up and down vertically, however it is best to leave as much room as possible around the craft in all directions.  Click the ''Ready'' button when positioned correctly.

Secondly, we need to make sure the AR.Drone has a fully charged battery connected to it.  Flying the quadcopter requires a lot of energy, and it will eat up power at a fast rate.  It is important the battery will last the entire lesson so make sure the battery pack is fully charged.  Prepare ahead of time if possible - the battery pack takes approximately 90 minutes to fully charge.

Thirdly, there are some warnings listed.  The quadcopter develops upwards thrust (which is called lift) by spinning its fan blades at a very high rate.  It is important to avoid touching the quadcopter while in flight as the blades can cause injury.  If at any point in time the copter deviates from your intended position, hit the red stop button (which you will be shown later).  Read this section and when you are done, hit the ''Ready'' button.

The last section will attempt to connect to the AR.Drone.  Wait until this section has turned green and then continue onwards to the lesson by clicking the ''START'' button.  If you are having trouble connecting to the copter, go back and make sure it has its battery pack plugged in.  When the pack is connected, the copter should turn on.

Once you have continued through the selections successfully, you are presented with the main lesson window.

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Students have previously completed a similar activity in relation to linear equations and the effects of changing slope (m) and the y-intercept (b) in the equation y = mx + b. In a whole class, discussion, the teacher will ask students what they remember about graphing a linear equation and will note the changes that the slope and y-intercept have on the picture of the graph. Students will be asked to respond to higher order questions regarding that graphing and their responses will be recorded on the board for reference during the day's lesson. The teacher should ask students to make notice of similar occurrences when graphing quadratic equations.

What do you remember about graphing a linear equation? Which are the changes that the slope and y-intercept have on the picture of the graph?

How is this similar when graphing quadratic equations?

## Teaching Tips:

In this first screen, we have a column on the left along with an image of the AR.Drone in the center and accompanying data on the far right.  The data on the right displays information about the copter's height, and camera characteristics.  The formula for the edge length of the image is shown in bold.  This formula relates the quad copter's height to the size of what the camera can see.

Going back to the gray column on the left - at the top, we have a radio button that can be used to select the units we wish to teach the lesson in.  At any point in time, you may switch between SI and English units.

Directly below are three tabs: Prepare, Demonstrate, and Investigate.

The Prepare tab (which is the default mode that the lesson starts in), allows us to quickly familiarize ourselves with the lesson and to see how the quadcopter will operate.  Simply push on the height bar (which is this vertical bar beneath the tabs) to choose a height you wish the copter to hover at.  Once the directed height is obtained, the quadcopter will remain at the set height.  Keep in mind that in this ''preparation'' mode, the quadcopter does not actually fly.  This is just a virtual mode to illustrate how the lesson operates.

Additionally, there is an orange tab called ''GRAPH'' located to the immediate right of the gray column.  You can drag or click this tab to display a graph of copter height vs. edge length.  you can switch back and forth between the two displays.  Note how the relationship is a linear path.  The equation of this path is the formula displayed on the right in bold.  A marker (shown in yellow and red) displays the location of the copter within the linear path.

Once you are satisfied with how the lesson operates, we are ready to move on with the incorporation of copter flight.  To begin, click the ''demonstration'' tab.  In demonstration mode, the screen appears almost identical to that of the ''prepare'' tab except that we now have the addition of a large green button in the lower right corner of the screen.  This button starts and stops the copter flight.  It will appear green to cause the copter to lift off, and remain red while in flight.  While airborne, if at any point you feel that an unsafe condition might exist, or you want to land the copter for any reason, press this red button.  This will land the copter.

The Demonstration mode works almost identical to the prepare mode.  When you hit the green button and the quadcopter takes off, it elevates itself at 1 meter and then begins to hover. The takeoff procedure is automatic and takes a few seconds.  Once the copter has reached the height of 1 meter, the height bar will be active and you can adjust the altitude of the copter. The maximum height of the AR.Drone is limited to 2 meters.  Again, you can pull out the ''GRAPH'' tab to display the linear relationship between copter height and image edge length.  When you are done demonstrating this relationship, press the red button and land the copter.

The next tab titled ''Investigate'' allows you to enter in specific height, edge, or area values and see the results.  It is best to ask the students to solve for a specific value and then determine experimentally if the solution is correct.  For example, you could ask that for a given height of 1.2 meters, what is the corresponding edge length that will be displayed on the camera.

When an answer is determined, plug it into the edge wheel, and press the green play button.  The copter will hover at 1.2 meters and from here we should be able to see an image on the screen that is about 2.6 meters across.  Corresponding data from the height, edge, and area can be seen in the far right again - so you can use that to check your work as well.  Land the copter by pressing the red button, and run more scenarios.  Given an edge length, what might the corresponding height of the quadcopter be?  Figure it out and then run it!

When you are done with the lesson, press the blue back button in the upper right corner of the screen.  This will return you to the lessons selection menu.

Now let's go through a quick lesson where we actually run the copter through the procedures I've just outlined.

First, take a freshly charged battery, and plug it into the AR.Drone.  This is what the battery looks like.  You will need to plug the black connector into the mating receptacle on the AR.Drone.  In order to access the battery bay, you will need to remove the foam shell from the top of the copter.

Once this is done, simply connect the battery and secure it with the Velcro strap. When the battery is connected, the motors will go through a quick initialization procedure which will require the motors to slightly move.

Upon completion of this initialization, re-attach the foam shell and place the AR.Drone in an open area.

Once the copter is ready for flight, we are going to select the demonstrate tab on our lesson.  To begin flight, press the green button.

At this point, the quadcopter will begin its takeoff routine and then hover at 1 meter.

To land the quadcopter, press the red button again, and the quadcopter will land.

To run the procedure again, we can repeat the last set of commands.  Press the green button again and wait until the copter reaches 1 meter.  Then we can adjust the height of the bar on the left to move the quadcopter up.

Then down

Then Up again

and finally commanding it to land with the red button.

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With the help of the AR.Drone explore with the students quadratic functions. Go through the process of how to use the simulator, as explained above, and answer any questions from students. Graph a simple quadratic equation y = x2 and ask students to identify how it is different from a linear equation. The students will be asked why they believe that graph looks different. Through guided questions, the students will understand that the x2 is what is different from linear equations.

Given simple quadratic equations (i.e. y = x2 or y = x2 + 5), how does changing the x2 coefficient or changing the constant affect the look of the graph?

How does changing the x2 coefficient or changing the constant affect the look of the graph?

What do you notice about the graph? How does it differ from linear equations?

## Teaching Tips:

1. As the altitude of the copter goes up, the dimensions of the visible area from the camera increase.  This relationship with edge length and height is linear, however the area is the product of the two sides of the image, and thus the area has a quadratic relationship to copter altitude.
2. It is related to height because the viewing angle of the camera is constant.  The only way to increase the viewing area is to increase height.
3. Area A = 2.25H2 + 0.6H + 0.04

4. A
5. C
6. C

How does this demonstration relate to quadratic functions?

Why do you think the area captured in the picture is related to the height?

Derive the formula for the viewable area (A) in terms of the height (H) of the quadcopter.

Plot the relationship of area (A) vs. height (H) in a graph.

While performing a search and rescue mission, the quadcopter takes a picture from a height of 5 meters above the ground. What will be the area (A) of the pictured image?
• 59.29 meters2
• 56.59 meters
• 7.7 meters2
• 11.59 meters2
• None of the above

To be able to differentiate objects of a certain size in the image, the maximum viewable area should be no larger than 256 square meters. To adhere to this limit (rounded to the nearest tenth), what is the maximum height the quadcopter can fly to?
• 16.0 meters
• 18.2 meters
• 10.5 meters
• 34.4 meters

The quadcopter takes off with a vertical speed of 1 meter per second. Rounded to the nearest tenth, what is the viewable area captured after 7.8 seconds?
• 60.9 meters2
• 136.89 meters
• 141.6 meters2
• 7.8 meters2

## Teaching Tips:

Ask for volunteers to sketch their graphs and results on the board. Ask for student input on how each piece of the quadratic equation changes its graph and will summarize these on the boards for students to write down. From here, go into a discussion of the plotting of the general quadratic equation y = ax2 + bx + c.