I am interested in The New York Times article “To Fall in Love with Anyone, Do This” by UBC Lecturer Mandy Len Catron because it explores a topic that I am very interested in; as a teenager who has not experienced love and has nearly lost hope on finding love, this article is a rope to a drowning man because it explains how falling in love can be intentional rather than serendipitous. I find Catron’s argument that love is “a more pliable thing than we make it out to be” very intriguing because the notion that love is not just something that happens by accident seems to go contrarily to how pop culture portrays love. Catron’s claim is actually quite comforting because she argues that love is a feeling that can be constructed. Catron’s style relies on the use of descriptive language in combination with imagery to provide readers with insights into her emotional state during her date such as when she describes the act of staring into the eyes of her love interest and observing the biological mechanisms of the eye as “the spherical nature of the eyeball [and] the visible musculature of the iris” contracts with light. “To Fall in Love with Anyone, Do This” is related to a myriad of real-life issues that concern most people because it touches arguably on one of the most important aspects of life: finding love. This article provides hope for millions of lonely people who have not yet found their significant other by reminding them that love can be a deliberate act and that true love can be initiated through intentional actions rather than leaving the whole process up to chance. While hormones do play a part, the comforting thing about love is that our rational minds still control the helm.
I am interested in The New York Times article “Yes, I’m in a Clique” by Nathan Black because I think his discussion of social hierarchy at many American secondary schools is relevant; I find his argument that cliques, despite popular misconceptions, are actually not a negative aspect of social life in high schools but an inevitable result when people of different backgrounds get together and coexist in a small space to be a relevant and insightful observation. Black, a high school student from Littleton, Colorado, uses a colloquial style combined with anecdotes to show how personal experiences with cliques can be helpful in instilling a sense of self-confidence in people just as “the good times” he had in his clique “convinced… [him] that… [he is] an O.K. person.” Black’s use of first person and informal diction such as “O.K.” provides readers with insight on the misunderstanding of cliques through his conversational tone. “Yes, I’m in a Clique” is directly related to real-life issues because it is written in the aftermath of the devasting Columbine school shootings and reveals the exclusionary tendency that is part of human nature. Throughout history, human beings have shown a preference for one group over another such as during the First World War when different nationalistic groups excluded each other, which reshaped Europe and led to the formation of many new states. Forming cliques is an inescapable part of human nature just as there are 193 sovereign nations in the world, which are merely 193 large cliques. Cliques are neutral and people should not assume they are bad because making groups is a natural human behaviour because differences and similarities will always exist.
At the very beginning of the project, I had an impressive plan which turned too great of a feat to accomplish. I initially planned to trace a picture of me and my mother with some buildings in the background, but with my lack of artistic talents, patients and my reluctance to press the save key it out went down the drain. The initial challengers I faced was the difficulties of tracing organic shapes with functions, but as I was reluctant to save most of my tracing I finished this version of the project with little to no tracing and it was not as challenging as I have expected. The aha moment I encountered was when I realized I can turn off the display feature for an original function but still utilize it’s functions notations, which gave me more options. I tried to get some artistic advice from my artistic friends in regards to the shaping of my nose and lips and pupils, but clearly they didn’t quite help that much as my facial features looks quite interesting. In terms of strategies, I used a few equations to make up the function notations and turn off the display features on them which enabled me to use them all through out the graph with little to no limits. This project helped to review the graphing and manipulation of the functions that we learned in Pre-Cal 12 which is quite helpful considering the upcoming finals.
Link to desmos project: https://www.desmos.com/calculator/gesfdb4qqu
– Analyzing Quadratic Functions:
- Forms of the quadratic equations
What I have learned:
Multiple forms of the Quadratic functions, such as Standard form, Vertex form…
with each form, one can find different information about the equation and its graph. Such as in vertex form you can find the vertex… with the Standard form you can find the roots, then the axis of symmetry then the vertex with this information.
Why is it important:
This is once again the setting stone for higher math and knowing the different form of the Quadratic Functions are very important.
- Sine/Cosine Law
What I have learned:
- Sine and Cosine Law are two additional ways that I have learned this year to determine an angle or a side.
- Sine law Sin A/a=Sin B/b= Sin C/c
- Cosine Law:
Why it is important:
Trig will be a very important part of pre-cal 12 and this will be the foundation of that.
– Reciprocal Functions:
- Graph Reciprocal Functions:
What I have learned:
- A reciprocal of a number is the opposite of the number.
- Graphing Reciprocal Function is how one graph Reciprocal Functions.
- If you known fraction then you should know if 1/x and x<1, the number will become greater, and vice versa if x>1, the number will become smaller. In graphing, this will result in some very interesting observations.
In the above graph that is a linear equation x+6=y
And when we get the reciprocal 1/(x+6)=y of the equation the graph looks like this:
- To draw this you find the roots of the equation or root in this case, to determine the vertical asymptote, the horizontal asymptote will be 0 until pre-cal 12. The asymptote is a barrel and it keeps the function to one side of it.
- And then you determine the invariant points, Because when 1and -1 inverse their yield doesn’t change, so when y=1,-1 those x coordinates are called invariant points.
- At last, we can basically draw the function just follow close to the asymptote in the zone where the original function has been and cross over the invariant points.
Why it’s important:
- Graphing will be a huge part of pre-cal 12 and I believe that reciprocal functions will be important then as well.
– Learning Habits:
- Build an extensive note-taking system.
What I have learned and will do next year:
- This year, I had a very pool system of taking notes. I wrote all my notes in the textbook, wherever there is an empty space… And with an extremely faint pencil that rubs away in a week… And eventually consequently resulted in my not ideal test grades as when I try to go back and find my notes, there isn’t anything there to be found and review…
- This summer I am getting private tutoring on Pre-cal and Cal and Chem in China, I will start using my extensive note taking system then… I will use a Rhodia notebook for each subject, and reserve space for planning my learning, working ahead, chapter notes and review. And I will write with a Fountain pen…
Why it’s important:
- Grade 12 is an extremely important year, no more messing around if I want my feature… And with the area that I’m interested in Math is extremely important… In order to keep my GPA goodish, I need to make sure I get good marks on my math in Grade 12 and note taking will help me with that along with studying ahead.
– Learning Habits Volume 2:
- Better utilization of test time.
What I have learned and will do next year:
- This year, I did not finish many tests… What a shame, as I got all the questions I did finish right but couldn’t finish the test and let those A slipped away… I must change that in Grade 12 and I think I know how… I couldn’t finish the test is due to my over excessive examination of the questions, crazy imagination and OCD like answer checking while doing the test… I will force myself to only focus on the test questions and do all those crazy checkings after the test is finished…
Why it’s important:
- Grade 12 is an extremely important year, I need my grades… Not that much messing around is allowed in Grade 12…
This week in Pre-cal 11 we learned Trigonometry again, the difference in comparison to the pervious knowledge we learned in grade 10 are, now SOHCAHTOA, works in all 360 degrees and we learned two new laws: the sine and cosine law which comes in handy when there isn’t enough information for SOHCAHTOA.
And this is how trig ratio is possible for all angles<360. I was introduced to this circle, with have 4 quadrants, basically, the x-axis of it is the principal axis and there are two angles, one is the rotation angle, the angle of that is measured from the X axis in the top right quadrant, the quadrant one, and the angle between that and the X-axis is the reference angle and the reference angle have the same trig ratio as the rotation angle, but it might be negative. And how do you determine if a rotation angle in a certain quadrant is negative with its trig ratio, there is something called the CAST rule, with that you can easily determine if your trig ratio is positive or negative.
This is the Cosine law, it can determine an angle with the length of the two sides created it known, vice versa you can find the any of the side lengths with the angle and when the other side is known as well.
The is the Sine Law, it can determine any angle or sides in a triangle with two sets of side and the angle opposite of it, using the ratio one can cross multiply and solve to find any one of these variables with other three.
This week we finished the rational expression unit.
And this week we learned how to implement the skill of solving Rational Expressions into the real world problems.
EX: Boat A is moving up a stream, and boat B is moving down the same stream, it took Boat A 2 hours to cover 15km and Boat B took only one hour, the current is 3km/h, what’s the speed of the boat in still water?
Set Boat A, B speed as Xkm/h in still water, therefore Boat A will move at x-3km/h and Boat B will move at x+3km/h.
Last week we learned how to simplify a rational expression, and this week we explored how to solve a rational equation.
The difference between equation and expression is pretty straightforward, the expression does not have an equal sign, it only expresses its meaning. An equation on the other hand obviously have an equal sign, and our intent will be the solution the unknown instead of just simply simplifying it.
There are two common ways to solve a rational equation.
1: Cross multiplication, only works when there is only one fraction on each side of the equal sign.
In this case, I multiplied(q-2)by5, and (q+4)by3.
2. Use a common denominator, and solve.
The common denominator, in this case, is 6z.
Last week we learned about Rational Expression, aka all real values of the variable except for those values that make the denominator zero. AKA2: In x/y, how to make sure y is not 0.
In particular, we learned about the how to make sure the denominator not equal 0 with something called non-permissible(the numbers one can’t use, or the denominator will equal zero and the expression will no longer be Rational.)
Outside of non-permissible values, we also learned how to simplify and/or multiple, divide Rational Expressions. But due to the fact, a Rational Expression or any real number, in this case, is a fraction, all to be done is to follow the law of dividing and multiplying of the fractions. AKA: why diving, multiply by it’s reciprocal. and why multiplying just multiply the numerator with the numerator and the denominator and denominator, but just keep in mind that in any rational expression the denominator can’t be 0, so there may be non-permissible as well. And as it’s an expression, all you can do is to simplify it.
In a fraction, when denominator equals the numerator, the number equals 1. EX: 8/8=1
So in an equation such as 3x.8(x-3)/2(x-3).9x^2. It might be helpful to use factoring to find the common thing that can cancel out to 1, like how (x-3) and 3x in this case. So after finding the common terms. they will just cancel out to 1, and in this case, 3x and 9x^2,(x-3)and (x-3) cancels out. and leaves you with 8/2.3x=8/6x=4/3x and don’t forget about the non-permissible, and remember you have to find them in the original equation, as they will be lost when you simplify the equation. In this case(x-3)can’t be 0, therefore x not equal 3, and 9x^2 can’t be 0, therefore x can’t equal 0.
a reciprocal of a number is the opposite of the number, or using the cool math language is 1/the number.
If you known fraction then you should know if 1/x and x<1, the number will become greater, and vice versa if x>1, the number will become smaller.
So in graphing, this will result in some very interesting observations.
Let’s use this linear function as an example, x+6=y.
when you invert this function to 1/x+6=y, magical things will happen, ex:1/6+6=1/12, whereas the original function will yield 12. But this sort of magic has three exceptions, 1,-1 and o, as their inverse equals themselves, expect 0, as 0 can’t be divided by. With this information in mind, we will be able to determine how a linear inverse function.
Because when 1and -1 inverse their yield doesn’t change, so when y=1,-1 those x coordinates are called invariant points.
The second step in solving is to draw something called vertical asymptote, as we are in pre-cal 11 the horizontal asymptote will be 0, the asymptote is like a great wall that keeps the number on their own space because 0 can’t be divided. the vertical asymptote will always be the roots or the x-intercept.
With the asymptote and invariant points, we can basically draw the function just follow close to the asymptote in the zone where the original function has been and cross over the invariant points, you have successfully drawn yourself an inverse linear function.
An Absolute value (||) means whatever inside is going to be positive, when we put a linear function within ||, it becomes an absolute value linear function.
The graph of it will change, as we know, a linear function is a straight line, either having a 0 slope or / positive slope or \ slope.
When graphing an absolute value linear function, if the slope is not 0, the line will bounce back up in the opposite direction of the original function at x-intercept.
Why is that?
Let me show you an example:
2x+4=y, a simple straight line with a slope of 2, and y-intercept of 4.
But magic happens when you put||on to the function. As the output of an absolute value cannot be negative, y, in this case, will also have to positive. But with our previous graph of the original equation, the Y clearly dipped under 0.
Let’s see what happens if we put an x value that would make y<0 into the absolute value equation. |2x(-4)+4|=4,
the Line reflected up, from this example we can learn that when graphing a linear absolute value function, all you need to do is find the x-intercept of the original line and draw a reflected line from the original root.