I/D# 1: Unit N - How do SRT and UC relate?

INQUIRY ACTIVITY SUMMARY
1. For a 30 degree special right triangle, the rule is

30°	60°	90°
n	n√3	2n

Therefore, when we simply the hypotenuse to "1," by dividing each value by 2n, we get: r = 1 (hypotenuse), x = √3/2 (horizontal), and y = 1/2 (vertical). The work for dividing is shown below.

After finding the values, we can plot the special triangle onto a coordinate plane. If this triangle is put in the first quadrant, its coordinates would be (0,0); (√3/2,1/2); (√3/2,0).

 
2. For a 45 degree special right triangle, the rule is

45°	45°	90°
n	n	n√2

When we simply the hypotenuse to "1," by dividing by n√2, we get: r = 1 (hypotenuse), x = √2/2 (horizontal), and y = √2/2 (vertical). The work for dividing is shown below.

After finding the values, we can plot the special triangle onto a coordinate plane. If this triangle is put in the first quadrant, its coordinates would be (0,0); (√2/2,√2/2); (√2/2,0).

3. For a 60 degree special right triangle, the rule is the same as a 30 degree special triangle:

30°	60°	90°
n	n√3	2n

So, when we simply the hypotenuse to "1," by dividing each value by 2n, we get: r = 1 (hypotenuse), x = 1/2 (horizontal), and y = √3/2 (vertical). This is very similar to the 30 degree triangle. The only difference is that the x and y values are switched. The work for dividing is shown below.

After finding the values, we can plot the special triangle onto a coordinate plane. If this triangle is put in the first quadrant, its coordinates would be (0,0); (1/2, √3/2); (1/2, 0).

4. This activity helped me derive the Unit Circle by showing me that the Unit Circle is just made up of special right triangles. The 30, 45, and 60 degree triangles have the same coordinates as the first quadrant of the Unit Circle. And, if we use the reference angles of these triangles, we can get the coordinates of the remaining points on the Unit Circle.

5. The triangle in this activity lies in the first quadrant.  However, in the second quadrant, the x values are negative. In the third quadrant, both the x and y values are negative. Finally, in the fourth quadrant, the y values are negative. The first image shows a 30 degree special triangle in Quadrant II, notice how the x value is negative. The second image portrays a 45 degree special triangle in Quadrant III, notice how both the x and y value of the coordinates are negative. Finally, the third picture displays a 60 degree special triangle in Quadrant IV, notice how the y value is negative.

INQUIRY ACTIVITY REFLECTION
1. "The coolest thing I learned from this activity was:" that the triangles are in all the quadrants. They are just flipped around, so the coordinate signs change.
2. "This activity will help me in this unit because:" it helped me memorize the unit circle
3. "Something I never realized before about special right triangles and the unit circle is:" that the unit circle is completely made up of special triangles.

RWA#1: Unit M Concept 4: Parabolas in Real Life

1. Parabola: "The set of all points the same distance from a point and a line." The point is called the focus and the line is called the directrix. (http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet)

Here is a good introduction video to parabolas:

2.  Algebraically, the equation of a parabola is (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h). The parabola is a U shape and will open left, right, up, or down depending on its equation.  The values in the equation can tell us which way the parabola is facing. If x is squared and p is positive, then it opens up. If y is squared and p is positive, then it opens to the right. If x is squared and p is negative, then it opens down.  And, lastly, if y is squared and p is negative, then the graph opens left.
        Moreover, some of its key points include: two points, a vertex (h,k) and a focus, and two lines, the axis of symmetry and the directrix.  The focus is always on the inside of the parabola and is "p" units from the vertex.  The directrix is a dotted line behind the parabola that is also "p" units from the vertex. Finally, the axis of symmetry is the dotted line the vertex and focus are resting on, and it also splits the parabola in half. Also, it is important to remember that the directrix and axis of symmetry are perpendicular.
If you want this explained in more detail, this website is a great resource:
http://www.purplemath.com/modules/parabola.htm

3. Satellite Dishes

Photo: http://i34.photobucket.com/albums/d129/coconut11/parabolicdish.jpg

Satellite dishes are great examples of everyday parabolas. They are created using the unique parabolic shape, so they can transmit and capture signals to and from a specific broadcast source. The shape of the satellite dish allows it to focus the signals hitting the dish to a specific point, the feedhorn or the focus. When the transmission signal focuses on the feedhorn, it broadcasts your favorite shows on your television. ^_^

4. References
http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet
http://www.purplemath.com/modules/parabola.htm
http://electronics.howstuffworks.com/satellite-tv6.htm
http://www.youtube.com/watch?v=fV9YuF__fM4
http://i34.photobucket.com/albums/d129/coconut11/parabolicdish.jpg

WPP#10: Unit L Concept 9-14: Probability