Students explore how the distance formula works on coordinate axes and discover how the Pythagorean Theorem relates to the distance formula.

*Calculating Distance Using Coordinates in a Coordinate Plane*

**Part 1: Distance Formula Exploration** (20 minutes)

**Part 2: Distance Formula and Pythagorean Theorem Exploration** (20 minutes)

**Materials and Prep **

- Traveling Distances Student Sheets
- iPad with Choreo Graph app
- Wifi access for sharing to other iPads and online project space

**Introducing the Activity**

Create a story problem that involves your animated character traveling across a distance. An example may be that you’re character is going across town or to a friend’s house. Use Choreo Graph tools to gather data and calculate the distance traveled from point to point.

**Part 1: Distance Formula Exploration **(20 minutes)

When objects in Choreo Graph slide around the screen on the coordinate plane, they are moving across a distance. This distance can be found using the coordinates from one point to the next and applying the distance formula:

Have students:

- Create a story problem that involves a virtual character traveling across a distance (e.g., going across town or to a friend’s house).
- Use the distance formula to calculate the distance traveled from point to point.
- Draw the coordinate axes with a starting point and then plot the next point.
- Count over horizontally and then up or down vertically and note how many units they go in each direction.
- Compare these counts to what you get by using the distance formula. Have students share what they notice.

**Part 2: ****Distance Formula and the Pythagorean Theorem **(20 minutes)

In this activity, students use both the distance formula and the Pythagorean Theorem to locate the distance their characters travel from one point to another.

Have students:

- Practice with the sample below, and then have them try to figure out the distance traveled using both formulas.

- Use both horizontal and vertical movement in their animation.
- Find the length of the hypotenuse (c) using both formulas.
- What do they notice using this approach?
- Exchange story animations with a classmate and find their character’s distance traveled using both formulas.

After students create their paths and calculate distances traveled in different ways, discuss:

:

- How do results from the distance formula compare to the units moved on the grid?
- What if your animated character only moves vertically, but not horizontally? Does the distance formula still work?
- What happens if your object only moves in the horizontal direction, but not vertical? Is there still a distance traveled? What do you notice?

Listen for:

- I noticed that I got the same thing when I drew a triangle between the points and did the Pythagorean theorem.
- When you count squares vertically it is the same as when you calculate y
^{2}-y - Counting squares across you get the same thing as when you calculate x
^{2}– x^{1 }using the distance formula.

Extensions and Inquiring Further

There are many possibilities to extend this activity in creative ways. For example, compare how someone travels a city street versus a bird flying through the air. On city streets that are often in grids, travelers have to take the legs of right triangles to get somewhere, but birds can fly direct distances (hypotenuse). Have students puzzle out what are more effective ways to determine the distance travelled.

Name: __________________________ Date: _____________

** **

**Part 1: Distance Formula Exploration**

** **

When objects in Choreo Graph slide around the screen on the coordinate plane, they are moving across a distance. This distance can be found using the coordinates from one point to the next. Here is the distance formula:

- Create a story problem that involves your animated character traveling across a distance. An example may be that you’re character is going across town or to a friend’s house. From point to point, use the distance formula to calculate the distance traveled.

** **

Distance | Point 1 Coordinates | Point 2 Coordinates | X answer squared | Y answer squared | X + y |
Square root/ distance | ||

Move 1 |
||||||||

Move 2 |
||||||||

Move 3 |

Name: __________________________ Date: _____________

** **

**Part 1: Distance Formula Exploration**

- Draw a coordinate axis with your starting point and then plot the next point. When you count over horizontally and then up or down vertically, make note of how many units you travel in each direction, and then compare that to what you did in the distance formula. What do you notice? (Hint you can also use your translation tool for coordinates).

- On the Choreo Graph stage, count over horizontally and vertically and note how many units the object travels in each direction. Compare that number to the values you get when using the distance formula.

**Name: __________________________ Date: _____________**

** **

**Reflection Questions:**

Answer the following questions in complete sentences.

- What happens if your object only moves vertically, but not horizontally? Does the distance formula still work?

- What happens if your object only moves in the horizontal direction, but not vertical? Is there still a distance traveled? What do you notice?

Name: __________________________ Date: _____________

** **

**Part 2: Distance Formula and Pythagorean Theorem**

In this activity, you are going to use both the distance formula and the Pythagorean Theorem to locate the distance your animated character travels. Practice with the sample below.

Name: __________________________ Date: _____________

** **

**Part 2: Distance Formula and Pythagorean Theorem**

After you have practiced with the sample, try to locate the distance for your own animated character by:

- Creating two movements that form a right angle.

- Challenging yourself to find the length of the hypotenuse ( c ) using both formulas.

- Swapping your animations with a classmate and asking them to try to find the distance travelled using both formulas.

Name: __________________________ Date: _____________

** **

**Part 2: Distance Formula and Pythagorean Theorem**

**Data Sheet**

Find the distance of the hypotenuse ( c ) using the distance formula and the Pythagorean Theorem.

point 1coordinates | point 2coordinates | x answer squared | y anwer squared | x + y | square root/ distance | ||

**Name: __________________________ Date: _____________**

** **

**Part 2: Distance Formula and Pythagorean Theorem**

**Reflection Questions:**

- After calculating distances using each formula, how do the distances compare to each other?
- Which formula was most efficient to use?

Begin by entering Make Some Moves. In **Build** mode, students will:

- Take pictures
- Trace and cut out parts of your photo that you want to animate

Add graphic or musical elements below:

In **Animate** mode, students will use:

**Graph Controllers****–**Choreo Graph uses key frames much like other movie editing software. At each point in the key frame, the student can set how each part of the animation rotates over time. Each node on the line graph below the stage represents the position of that part at a specific time. Stretch the points up and down to set the degrees of rotation. The steeper the line on the graph, the faster the part moves. Students can also set the location of their animated parts by selecting the part they want to move, choosing a node on the line graph or moment in the animation, and then dragging the part to a new position on the stage. They can set the location of parts for the entire animation sequence by repeating this process.

Toggle on math tools to notice:

Degrees each part has rotated

Path the main parts moved

Coordinates for location of each part