Unit 5

Presentation

 1) Choose the regular polygon , and construct it in the first quadrant in the coordinate plane in GSP. Match the coordinate of the vertices of the polygon.

2) Reflect the regular polygon in the x-axis and y-axis without the menu of reflection in GSP. (Option: Reflect the regular polygon in the y=x) Make the table of the coordinates of the vertices of the polygon.

3) In your flip images find the any formulas of the coordinate geometry that we've learned in the class.

4) Prepare the group presentation in the Web

The project problem is connected among some mathematical ideas in figure1: coordinate geometry, construction of the polygon, reflection, making a table, and conjecture.
According to Standards, thinking mathematically involves looking for connections and making connections builds mathematical understanding. With connections, students can build new understanding on previous knowledge.

One example of the solution

Group 1 chooses the pentagon. They know that construction of the pentagon is more difficult than any other regular polygon, but they tried. They refer the web site of construction on coordinate geometry. After constructing the pentagon in first quadrant, they encounter the problem. They can find the coordinate of A' and B' but it is not easy to find the rest coordinate of the blue flip image. They discuss this problem. Some students suggest using the tracing paper over the computer screen. Some students suggest the copy menu in the GSP. After discussion by electronic communicator, they decided to draw the line. They draw the RE that is parallel to AB, and draw the circle with the center R and radius RE. As RE is radius in circle, they can mark the point E' so that RE' can be equal to RE. With the coordinate menu of measure, they can confirm the coordinate (E'=(-1.38, 2.90)). It was the reflection. They repeated this process, and make the other two flip images without the reflect menu. Here is the GSP file to show the students' work.

 

[Figure 2]

The result is the figure 3 and here is the GSP file. They make the table of the coordinates of the four images (Table 1). They find the regularity of the minus sign. So they make a conjecture that x-coordinate is changed in y-axis reflection, y-coordinate is changed in x-axis reflection, and both are changed in y=x reflection. It can be said that the opposite coordinate is changed in reflection. With the result, they prepare the presentation by Web.

[Figure 3]

 

 A

B

C

D

E
 Original image(red)

 (2.0, 1.0)

 (4.0, 1.0)

 (4.6, 2.9)

 (3.0, 4.1)

 (1.4, 2.9)
 Flip image in y-axis(blue)

 (-2.0, 1.0)

 (-4.0, 1.0)

 (-4.6, 2.9)

 (-3.0, 4.1)

 (-1.4, 2.9)
 Flip image in x-axis(yellow)

 (2.0, -1.0)

 (4.0, -1.0)

 (4.6, -2.9)

 (3.0, -4.1)

 (1.4, -2.9)
 Flip image in y=x(green)

 (-2.0, -1.0)

 (-4.0, -1.0)

 (-4.6, -2.9)

 (-3.0, -4.1)

 (-1.4, -2.9)
[Table 1]

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