Month: June 2018

Tech Team Reflection 2018

alexr2015

I helped a lot this year. I feel like a did a lot more this year than compared to last year. Where there was about three events that I could participate in, this year I could take part in a large number and I feel like I was valued. I was part of the 2017-2018 boot camp, I helped at the tech night to show parents what devices they should buy their kids for school. And my favourite, I started working alongside Mr. Shen and learned important things regarding computers, information technology, and even some life lessons.

This year, I feel like an important part of the tech team. A reason why I’m grateful that I joined the tech team is that it opened up the opportunity to take part in the Make with Microsoft Program. I learned a lot and I got an amazing opportunity to be not a consumer, but a producer on the BCTech Summit floor.

Pre-Calculus 11 Week 18 – Top Five from Pre-Calculus 11

alexr2015

I’ve learned a lot this year in my math class. This blog post is dedicated to the top five things I’ve learned.

1. Geometric Sequences and Series

Geometric Sequences and Series are pretty cool. You can do some fancy math magic with the formulas. A geometric sequence is when each term after the first term is multiplied by a constant to determine the next term. For example, the sequence “2, 4, 8, 16, 32” is geometric. The common ratio is what the number is being multiplied by. In this case, it is four. The formula to determine a term in a geometric sequence is t_o=ar^{n-1}s=3, with a being the first term, and r being the common ratio.

While a geometric series is when you add all the numbers together up to a certain term. For example, in the geometric series “3, 9, 27, 81”, S_5 is 363. You can calculate geometric series with the formula

2. Graphing Absolute Values

Image result for absolute value graph

 

Graphing an absolute value is pretty cool. It’s almost just like a regular graph, except anything below the x-axis is

reflected above the x-axis. They look pretty neat, and you can do this with linear and quadratic equations! An equation with absolute values looks something like y=|x+3|.

3. General Form, Standard Form, and Factored Form

So there’s three different forms of quadratic expressions and equations. You’ve got general form, which is the one that we tend to see the most. You’ll remember it from grade 10. ax^2 + bx + c. Standard form is my favourite, as it is tends to give you the most information. You can see the scale, the horizontal and vertical translation, the vertex, the axis of symmetry, and more! It’s represented with (x + p)^2 + q. Factored form can give us the solutions to the parabola, which is a fancy word for where the parabola intersects with the x-axis. Factored form looks like (x +x_1)(x+x_2)

4.  The Quadratic Formula

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

That’s a mouthful, isn’t it? The quadratic formula is used what you have a quadratic expression that just does not factor nicely at all. While you’ll get answers like x=\frac{-1\pm\sqrt{47}}{3}, it’s actually the nicest answer you’re going to get without a calculator. It’s also the exact answer, since if you put your answer down in decimal, you’re actually rounding off numbers, meaning your answer is inaccurate.

5. Graphing reciprocal functions

They’re SUPER weird. But they’re kinda cool. Reciprocal is a fancy word for flip. For example, the reciprocal function of x+4 is \frac{1}{x+4} Now describing what it looks like is near-impossible to describe… So take a look at it below. It’s scary, and after grade 11 is over, I want nothing to do with it for three months..Image result for reciprocal function

Pre-Calculus 11 Week 17

alexr2015

This week we covered just about everything in our Trigonometry unit, so I am going to attempt to put as much as I can into this blog post.

SOH CAH TOA is updated for x, y and r. I’ll explain.

Sine = \frac{Opposite}{Hypotenuse} is now sine = \frac {y}{r}

Cosine = \frac{Adjacent}{Hypotenuse} is now cosine = \frac{x}{r}

And Tangent = \frac{Opposite}{Adjacent} is now tangent = \frac{y}{x}

The best way to remember these, is to simply remember that sine is \frac {y}{r}. You should know that the hypotenuse is \frac{x}{r}, so

So where the heck did x, y  and r even come from? Prepare to be introduced to reference angles and unit circles!

Image result for unit circle trig

Rotation angle is different from reference angle. Rotation angle is how far the terminal arm has rotated from the rest position, 0^o.

Image result for reference angle trig

 

The reference angle is always going to be beside the x-axis. And then from here, we can find out which lines are the

adjacent and opposite lines. Then we can find out which trig ratio to use. For example, the reference

angle for a rotation of 160^o degrees is 20 degrees, because the angle needs 20 more degrees to reach the x-axis. Or, a 225 degree angle has a reference angle of 45 degrees, because it needs to lose 45 degrees to reach the 180 degrees point.

Now, there are special triangles that you should remember as well. These let you do trigonometry without a calculator, and it can also be a way to double check your work.

Image result for trig special triangles

Say, you wanted to find sin 45^o. First, you’re dealing with a 45 degree angle, so you know you’re gonna look at your isosceles right triangle. You know sin is \frac{y}{r}, so that means sin45 is \frac {1}{\sqrt{2}}

Or, you want to know cos 60. You know cos = \frac{Adjacent}{Hypotenuse}, or what we’re really supposed to remember, cos = \frac{x}{r}. This means cos 60 = \frac{1}{2}

 

But obviously, you’re not going to deal only with triangles that have side lengths of 1, 2, and \sqrt{3} units.

How do we fix this?

Well, do you remember similar triangles? From grade nine? I’ll refresh you just in case. If two triangles have the same angles, then that means the only difference between the two can be the size, or the side lengths. This means, if you run into another triangle that has angles of 30, 60, and 90 degrees, but has side lengths of 4, 4\sqrt{3} and 8, then you simply have to find the scale factor! The scale factor in that triangle that I described above is 4. If you divide all side lengths by 4, you’ll get our 1-\sqrt{3}$-2 sides again.

Pre-Calculus 11 Week 16 – Trigonometry Review

alexr2015

We finished our Rational Expressions and Equations test this week, so we’ve started our new unit. Trigonometry. So I’m gonna post everything I remember from Trigonometry in grade 10.

Image result for triangle trig

In a right triangle, the side closest to theta that isn’t the hypotenuse is the adjacent side. The side farthest is the opposite side, and the hypotenuse remains the same, like always.

Remember SOH CAH TOA

SOH is Sine = \frac{Opposite}{Hypotenuse}

 

CAH is Cosine = \frac{Adjacent}{Hypotenuse}

 

TOA is Tangent = \frac{Opposite}{Adjacent}

 

To find an angle when you have the lengths of two sides, you can use the inverse of sine, cosine, or tangent. Image result for triangle sidesSay you had this triangle. You want to find the angle of B. Well, that makes the 5cm side the opposite side, and the 7cm side the hypotenuse. We don’t know how long the adjacent side is, so we can’t use that. So which ratio can we use? We can use Sine, because Sine = \frac {Opposite}{Hypotenuse}

So to find the angle of B, we take the inverse sine of 5/7.

 

And of course, with some algebra skills, you can also also find the length of one side if you have the length of another side and an angle.

Don’t forget, you can also use the Pythagorean theorem to find the length of another side if you need to.