I/D #1: Unit N Concept 7 - How do SRT and UC relate?

INQUIRY ACTIVITY SUMMARY

1.) 30 degrees

     There were several part that were needed to be labeled and solved for. The hypotenuse is equaled to 1, and due to the rules of the 30,60,90 right triangle: the side adjacent to 30 (which is x on the pic) has a value of x radical 3; the side opposite (which is y) has a value of x; and the hypotenuse (r) as 2x. To find an actual number value, trig functions (sin and cos) must be used. The sin of 30 degrees solves for the opposite side (y) and is found with y/r; substituting the variables will result with x/2x and simplified will equal to 1/2. The cos of 30 degrees solves for the adjacent side (x) and is found with x/r; substituting the variables will result with x radical 3/2x and simplified will equal to radical 3/2.  If the triangle were to be placed in a Unit Circle, with the origin being located at the labeled angle measure, three coordinate pairs will be needed and two are found with the values solved. The origin is (0,0), while only going across the x axis is (radical 3/2,0), and going directly above from the previous coordinate is (radical 3/2,1/2).

2.) 45 degrees

     The instructions are the same but differ in the number values. Due to the 45,45,90 right triangle rules, x is the side adjacent to the given degree with value of x; the y is opposite to the angle but similar to the adjacent side as it, too, has the value of x; and the hypotenuse, r, is equaled to 1 and x radical 2. The trig functions of sin and cos are also used with sin 45 degrees = y/r and cos 45 = x/r. Solving for the sin and cos is similar with this SRT since the same values are used to substitute, which is x/x radical 2 and simplified equaling to radical 2/2. If the triangle were to be placed in the Unit Circle, with the origin being located at the labeled angle measure, there would be three coordinate pairs. The origin would be (0,0), shifting to the right would be at (radical 2/2,0), and moving directly upward will get to (radical 2/2,radical 2/2)

3.) 60 degrees

     The instructions and requirements are as well the same from the previous triangles, and it will be closely identical to the 30 degree triangle. Same as before: x is adjacent but with the value of x; y would be opposite to the angle with the value of x radical 3; and the hypotenuse, r, is 1 and 2x. Sin and cos will be used to find the length of the sides. The sin of 60 degrees (y/r) would be x radical 3/2x and simplified will equal to radical 3/2. The cos of 60 degrees (x/r) would be x/2x and simplified will equal to 1/2. If the triangle were to be placed in a Unit Circle, with the origin being located at the labeled angle measure, three coordinate pairs will be (0,0) at the origin, (1/2,) going across the x axis, and (1/2, radical 3/2,) shifting directly up.

4. This activity helps derive the unit circle by figuring out the certain point of the circle with the special right triangles. Figuring out the lengths of these triangles, if placed on the circle, will every time give the same coordinate pairs that are points on the Unit Circle if done correctly. 

5.) 

     The quadrant in which the original triangles from the beginning are draw in the first one. Depending on which quadrant the SRT is found, the value will change by its sign (positive or not). If the triangle is found on Quadrant II, then x from the coordinate pair will be negative. In Quadrant III, both x and y will be negative. Finally in Quadrant IV, only y will be negative. In the three pictures provided above, the triangles in a different location from Quadrant I will change in its sign due new location. In Quadrant I, all trig functions are positive, while in Quadrant II only sin and csc are positive and the rest negative. Meanwhile in Quadrant III, tan and cot are the only positive. Finally in the Quadrant IV, only cos and sec are positive.

INQUIRY ACTIVITY REFLECTION
1.) “THE COOLEST THING I LEARNED FROM THIS ACTIVITY WAS..." that special right triangles are actually beneficial and do a bit more than just being a perfect shape.
2.) “THIS ACTIVITY WILL HELP ME IN THIS UNIT BECAUSE..." I can refer back to it if I ever forget how to fill out the Unit Circle for the coordinate pairs section.
3.) “SOMETHING I NEVER REALIZED BEFORE ABOUT SPECIAL RIGHT TRIANGLES AND THE UNIT CIRCLE IS…” how they relate to one another. I would have not have guessed that the coordinates on the Unit Circle could also come from different shape.

RWA #1: Unit M Concepts 4-6 - Conic Sections in real life (parabola)

1. Definition of Parabola
     According to Mrs. Kirch, the definition of a parabola is a set of all points that are equidistant from a point (focus) and a line (directrix). The definition can best be explained/viewed from an image used in this page linked here.

2. Properties:

 (http://youtu.be/r-KmkpxVtGg)

     Algebraically:

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(encrypted-tbn1.gstatic.com)

     In standard form, parabolas are written as (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h) (however in the picture to the left, four ways are shown and the only difference is simply because of the negative sign) . The variables h and k is the vertex (origin) of the parabola as h represents x and k to y. (Notice how on the image to the top left, h is paired with x and k with y). Also, p has to be on the side that is NOT being squared. (The image on the left represents p with an a, which is the same thing as explained in the video from above). The number of units of p represents how far is the focus point and the directrix line of the graph. The sign of p, too, (if it is positive or negative) determines if the graph will go up or down or left or right. Meanwhile, the variable that is squared can determine the direction of the parabola: horizontal or vertical. 

     The image on the right is an example of a problem solving to rewrite the equation in standard form. The side being squared was found by solving-the-square and the binomial is by itself. In the example, 4 represents p (or a), and notice how it is on the other side of the squared binomial and sitting alone, undistributed to the other binomial. The video from above can help explain a bit more on rewriting to standard form. 

     Graphically: 

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(img.docstoccdn.com)

     The variable that is squared, as mentioned before, determines the direction of the parabola. As represented on the image to the right, if x is squared then the graph will go vertically; and if y is squared, then the graph will go vertically. The sign of p determines another part of the direction: if positive, the graph will move positive units either up or right; if negative, then the graph will move negative units either down or left. The axis of symmetry is always a line that spits the parabola in half from the vertex. The directrix is a line that is perpendicular to the axis of symmetry. As shown on the image to the left, the directrix is not within the parabola, but found outside and is found by p units away from the vertex. The focus point is within the graph and is found with p units away from the vertex. The distance away from the focus to the vertex can also help determine how wide or narrow the parabola would be. If the focus is far, then the parabola would be wide; and if the focus is close, then the parabola would be narrow. 

   

3. RWA:

(visual.merriam-webster.com/astronomical) 

(parkerlab.bio.uci.edu)

     In the real world, parabolas can be found within the science, astronomical department.  The images above show a radio telescope which, according to Merriam Webster Dictionary, is an "instrument used to capture, concentrate and analyze radio waves emanating from a celestial body or a region of the celestial sphere." Radio waves hit the parabolic reflector (the curved, bowl-like part of the telescope) which has a parabola shape. The waves automatically hits to the center of the reflector to a secondary reflector (considered the focus) that sends the waves down to the receiver. Anywhere where a wave would hit the parabolic reflector (a position on the parabola), it will always then aim to the secondary reflector (focus) since it was structured to have the same distance. 

4. Work Cited:

• Conic Parabola Video - http://youtu.be/r-KmkpxVtGg
• Standard From Equations (pic) - https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRf8C72WS3K4tZSQoWR_RF9hTYjCHXWeyeweyXqYD0PtWD3Gk2uWw
• Parabola Graphs (pic) - http://img.docstoccdn.com/thumb/orig/41803311.png
• Standard Form Example (pic) - https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcQKIXbJTTqeqSGF8weMMOnzNdoSH7DA-jc1CG6zvl7_tQWSdcov
• Parabola Parts w/ graph (pic)- https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQYmQdKDw1dEQKMsjpDpEbOr84_BF6-hOHW9Y_bP83bhlhapZIm
• Merriam Webster parabolic reflector (pic) - http://visual.merriam-webster.com/astronomy/astronomical-observation/radio-telescope.php
• Radio telescope (pic) - http://parkerlab.bio.uci.edu/nonscientific_adventures/otherplacesinCA.htm

WPP #10: Unit L Concepts 9-14 - Probabilities with one event, independent AND events (with/without replacements), OR events (non/mutually exclusive), and with combinations

Create your own Playlist on LessonPaths!

WPP #9: Unit L Concept 4-8 - Possibilities with FCP, combination, permutation, and distinguished permutation

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SP #6: Unit K Concept 10 - Writing repeating number as rational number using geometric series

     The viewer needs to pay special attention that this number is INFINITE, meaning it is continuous and has a repeating integers. With infinite numbers, the infinite sign is needed on top of the repeating numbers. One must also need to pay attention that the process of infinite geometric series is being used, which includes  summation notation, plugging it to the formula (a sub one divided by one minus r), and the sum. One last thing to remember before finding the final sum is to add the value on the left side of the decimal (which we ignored from the start) from the original. 

Fibonacci Haiku: The Lovely Judith

https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcSksbuqyORTj7HaMSC957G4T36YckVdWUFZPkPx5qPeZbgV_-fa

Judith?

Who?

My friend?

The charming sister?

The one mirroring our friendship?

You wonder who's she and still don't know...

No matter who she is, I love darn Judith that’s the flipping problem...

SP #5: Unit J Concept 6 - Partial fraction decomposition with repeated factors

     The viewer needs to pay special attention that when there are common factors, each of the repeated factor will have a power one more than the one before. One must count up the factors and include the factor as many times as the exponent. The viewers need to pay attention to their work and make sure to not make an error throughout the process. There are many places where errors could be made, especially when distributing/multiplying factors. And finally, the viewer needs to be aware that there are (4) systems, therefore there are four variables. Process of elimination is used to solve for a variable, followed by substitution to solve for the others.