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The Magic of the “Golden Ratio”

Walking around NYC, I was on a mission to connect mathematics to the real world.  This, of course, led me to go on a mathematical scavenger hunt in search of  the “Golden Ratio.” Hidden in plain sight, this often times naturally occurring ratio is seen everywhere from historic and modern architecture to nature itself. 

What is this all-encompassing “Golden Ratio” you may ask?
It’s a proportion, related to a never-ending sequence of numbers called the Fibonacci sequence, and is considered to be the most pleasing ratio to the human eye.  The ratio itself is an irrational number equal to 1.618……..(etc.).

Why should you care?
When the same ratio is seen in the Parthenon, the Taj Mahal, the Mona Lisa and on the shores of a beach in a seashell, you know it must be something special!
The Golden Ratio The Golden Ratio The Golden Ratio The Golden Ratio

Random as it may seem, this proportion stems from the following sequence of numbers, known as the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, …….

Do you notice what pattern these numbers form?
(Answer: Each previous two numbers are added together to find the next number.)

Capture

The Golden RectangleThe most common example of the “Golden Ratio” can be seen in the Golden rectangle.  The lengths of this rectangle are in the proportion from 1: 1.618 following the golden ratio. Behold the beauty of the Golden Rectangle:

The Golden Ratio

How is the Fibonacci Sequence related to the Golden Ratio?                                               What if we drew a golden rectangle within our rectangle?

The Golden Ratio

Then drew another golden rectangle within that golden rectangle?

The Golden Ratio

And we kept doing this until we could no longer see what we were doing…….

The Golden Ratio
The Golden Ratio

The proportion between the width and height of these rectangles is 1.618 and can also be shown as the proportion between any two numbers in the Fibonacci sequence as the sequence approaches infinity. Notice that the area of each rectangle in the Fibonacci sequence is represented below in increasing size:

Where exactly can you find this Golden Ratio in real life? Found in NYC! The Golden ratio was seen here at the United Nations Secretariat building in the form of a golden rectangle(s).  Check it out!

Golden Ratio

Where have you seen this proportion of magical magnitude?  Look for it in your own city or town and let me know what you find! Happy Golden Ratio hunting! 🙂

If you’re interested in learning more about the golden ratio and are also a big Disney fan, I highly recommend you check out Donald Duck’s Math Magic!

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Looking to learn more about math phenomenons found in the real world? Check out this article on fractals!

How to Factor Quadratic Equations: Algebra

In this post, we are going to dive deep into how to factor Quadratic equations! There are so many different methods to choose from including GCF, Product/Sum, DOTS, and the Quadratic Formula.  Here we will go step by step into each method on how to factor quadratic equations, each with their own set of practice questions. For a review on how to factor by grouping, check out this post here and  happy calculating! 🙂

Why factor in the first place, you may say? We want to manipulate the equation until we solve for x.  Solving for x is our main goal, and factoring allows us to do that.  Now let’s get to the good stuff!

How to factor quadratic equations

Greatest Common Factor (GCF):

The greatest common factor is the highest possible number that can be divided out from an equation.  This gets the equation into its simplest form and makes it easier for us to solve for x.

Before considering which type of factoring methdo to use, always ask yourself, “Can I take out a GCF?”

How to factor quadratic equationsHow to factor quadratic equations

How to factor quadratic equations

Product/Sum:

This factoring method is for quadratic equations only! That means the equation takes on the following form:

How to factor quadratic equationsHow to factor quadratic equationsHow to factor quadratic equations

How to factor quadratic equationsHow to factor quadratic equations

How to factor quadratic equations

How to factor quadratic equations

Difference of Two Squares DOTS)

Not to play favorites or anything, but DOTS is the easiest and most lovable of the factoring methods.  This factoring method just makes you feel all warm and fuzzy inside or maybe that’s just me).  Before we get into how to do DOTS, let’s talk about when?

Quadratic Formula:

We have heard of the quadratic equations, so how id the quadratic formula different?

The Answer: The Quadratic Formula is what we use to factor any trinomial. You can use product/sum on trinomials like we discussed earlier, but this may not always work out easy.  The Quadratic Formila on the other hand will work every time!

Low and behold, the Quadratic Formula:

Want more Mathsux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

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‘Math Suks’ ~ Jimmy Buffett

Need to relax after studying? Close to wanting to “burn your textbook”? This super chill and somewhat cheesy song may help! Coming to you from Jimmy Buffett, the singer has a relaxed, island/ska like vibe to his music.  The lyrics come close to sounding like an after school special until the words “Math Sucks” are sung over and over again.  Enjoy! 🙂

Click here to listen to the song:Screen Shot 2018-02-06 at 1.41.05 PM.png

‘Math Suks’ by Jimmy Buffet Lyrics:

If necessity is the mother of invention
Then I’d like to kill the guy who invented this
The numbers come together in some kind of a third dimension
A regular algebraic bliss.

Let’s start with something simple, like one and one ain’t three
Any two plus two will never get you five.
There are fractions in my subtraction and x don’t equal y
But my homework is bound to multiply.

Math suks math suks
I’d like to burn this textbook, I hate this stuff so much.
Math suks math suks
Sometimes I think that I don’t know that much
But math suks.

I got so bored with my homework, I turned on the TV.
The beauty contest winners were all smiling through their teeth.
Then they asked the new Miss America
Hey babe can you add up all those bucks?
She looked puzzled, then just said
“Math Suks”.

Math suks math suks
You don’t even have to spell it,
All you have to do is yell it…
Math suks math suks
Sometime times I think that I don’t know that much
But math suks.

Geometry, trigonometry and if that don’t tax your brain
There are numbers to big to be named
Numerical precision is a science with a mission
And I think it’s gonna drive me insane.

Parents fighting with their children, and the Congress can’t agree
Teachers and their students are all jousting constantly.
Management and labor keep rattling old sabers
Quacking like those Peabody ducks.

Math suks math suks
You don’t even have to spell it,
All you have to do is yell it…
Math suks math suks
Sometime times I think that I don’t know that much
But math suks

Simultaneous Equations: Algebra

In need of a bit of review on Simultaneous Equations?  Well, now is your chance! Learn how to solve these confusing bad mama jamma’s in three different ways and choose which one works best for you!

I’m also excited to introduce my new YouTube page for MathSux!  Hope these new set of videos help.  Let me know if you have any more questions in the comments.  Happy calculating! 🙂

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Why Must We Complete the Square?: Algebra

Completing the Square: So many steps, such little time.  It sounds like it involves a square or maybe this is a geometry problem?  Why am I doing this again?  Why must we complete the square in the first place?

These are all the thoughts that cross our minds when first learning how to complete the square.  Well, I’m here to tell you there is a reason for all those steps and they aren’t that bad if you really break them down, let’s take a look!

Complete the square

Answer: x={-7,3}

Explanation:

I’m not going to lie to you here, there are a lot of seemingly meaningless steps to completing the square.  The truth is though (as shocking as it may be), is that they are not meaningless, they do form a pattern, and that there is a reason! Before we dive into why let’s look at how to solve this step by step:

Complete the square

Feeling accomplished yet!? Confused?  All normal feelings.  There are many steps to this process so go back and review, practice, and pay close attention to where things get fuzzy.

But the big question is why are we doing these steps in the first place?  Why does this work out, to begin with?

For those of you who are curious, continue to read below!

Want more Mathsux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!

Need more of an explanation?  Check out why we complete the square in the first place here !

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Geometry: Congruent Triangles

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Not bad, eh? If there is anything you have questions on please don’t hesitate to ask! 🙂 If you are looking to test out your skills on this topic even more, check out more challenging questions here.

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Solving Trigonometric Equations: Algebra 2/Trig.

Howdy math friends! In this post, we are going to learn about solving trigonometric equations algebraically.  This will combine our knowledge of algebra and trigonometry into one beautiful question! For more on trigonometric functions and right triangle trigonometry check out this post here.

Solving trigonometric equations. Sound complicated? Well, you are correct, that does sound complicated. Is it complicated? Hopefully, you won’t fund it that way after you’ve seen this example. We are going to do this step by step in the following regents question:

Screen Shot 2016-08-24 at 11.03.25 AM

How do I answer this question?

The questions want us to solve for x.

Step 1: Pretend this was any other equation and you wanted to solve for x. Thinking that way, we can move radical 2 to the other side.

Solving Trigonometric Equations

Solving Trigonometric Equations

Step 3: Now we need a value for x. This is where I turn to my handy dandy reference triangle. (This is a complete reference tool that you should memorize know). For more on special triangles, check out this post here.

Solving Trigonometric Equations
Screen Shot 2016-08-24 at 11.04.47 AM
Solving Trigonometric Equations

Step 5: Notice that cosine is positive in Quadrants I and IV. That means there are two values that x can be (one in each quadrant). We already have x=45º from Quadrant I. In order to get that other value in Quadrant IV, we must subtract 360º-45º=315º giving us our other value.

Does this make sense? Great! 🙂 Is it clear as mud? I have failed. But I have not given up (and neither should you). Ask more questions, look for the spots where you got lost, do more research and never give up! 🙂

Hopefully you enjoyed my short motivational speech. For more encouraging words and math, check out MathSux on the following websites! Sign up for FREE math videos, lessons, practice questions, and more. Thanks for stopping by and happy calculating! 🙂

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Medians on a Trapezoid: Geometry

Greeting math peeps! Welcome to another MathSux post! Today we will be tackling how to find the missing lengths of medians on a trapezoid. We’ll do this by going over a question taken straight from the NYS Regents from August 2012. I must admit, that when I first looked at this question, I had no idea how to answer it! None. Zero. Clueless (also a great movie). But apparently, there is an explanation! And apparently it’s not so hard; you just have to know what to do.Let’s take a look:

Medians on a Trapezoid

How do I answer this Question?

Step 1: Let’s fill in what we know, They tell us in the questions that AB=5x-9, DC=x+3, and EF=2x+2. So let’s write that in:

Medians on a Trapezoid

Ok, great what now? (This is where I got lost too). But wait! There is this amazing rule about medians on a trapezoid you probably didn’t know about (Exciting I know).

Screen Shot 2016-08-08 at 11.19.28 AM

Step 2: Apply the rule and solve for x.

Medians on a Trapezoid

Answer: x=5

Yay! This gives us our answer 🙂 Another random rule in Geometry accomplished.

 Still got questions? That’s cool, take a deep breath and ask me in the comments section.

Looking for more on Medians of a trapezoid? Check out this post here for practice questions and more!

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PieceWise Functions NYS Regents: Algebra

Greetings math friends, students, and teachers I come in peace to review this piecewise functions NYS Regents question.  Are they pieces of functions? Yes. Are they wise?  Ah, yeah sure, why not? Let’s check out this piecewise functions NYS Regents question below and happy calculating! 🙂

piecewise function NYS regents

1 value satisfies the equation because there is only one point on the graph where f(x) and g(x) meet.

Screen Shot 2016-07-27 at 11.15.53 AM
Screen Shot 2017-07-27 at 10.05.36 AM.png
*Note: For an in-depth review on how to graph an equation of a line step by step, check out this post here.
Screen Shot 2016-07-27 at 11.16.20 AM
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Does the above madness make sense to you? Great!

Need more of an explanation? Keep going! There is a way to understand the above mess.

How do I read this?

Looking at this piece-wise function: We want to graph the function 2x+1 but only when the x-values are less than or equal to negative 1.

We also want to graph 2-x^2 but only when x is greater than the negative one.

One way to organize graphing each piece of a piecewise function is by making a chart.

  1. Lets start by making a chart for the first part of our function 2x+1:
piecewise function NYS regents
Screen Shot 2016-07-27 at 11.17.52 AM
piecewise function NYS regents
piecewise function NYS regents
piecewise function NYS regents

Is it all coming back to you now?  Need more practice on piece-wise functions?Check out this link here and happy calculating! 🙂

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Top 5 Favorite Math Websites

As we all love technology (I’m assuming since it has brought you here), I figured I should share my top favorite math sites.  These are great resources for learning things you don’t understand, practicing for an upcoming test or if you’re like me, it’s great for re-learning/teaching purposes.  Yes, even I forget how to do things and these websites are a great help.  I know the anticipation is getting to you so I’ll just get to the good stuff now. Here are my top 5 favorite math websites:

  1. Khan Academy : This is a non profit over in Silicon Valley.  They have a great range of subjects (k-college level)  and explain everything via easy to follow videos.
  2. Regent’s Prep: The title is pretty self explanatory, but this helps with reviewing for the New York State Regent’s.  This site has a lot of great examples and answers.
  3.  MathSux:  So fabulous, no explanation is required. 😀
  4.  Knewton:  Pretty cool things are happening at this web address.  You can make an account, answer some questions and the questions will adjust according to your answers figuring out what you need help in.
  5. Kahoot: Fast and easy quizzes that are good for an entire classroom or for studying alone.  Either way there are lots of bright and happy colors that make you forget you’re studying anything in the first place.

So those are the goods, the contraband, the info you can’t find anywhere else (all lies).  Hope you find these websites useful. Let me know if you end up using any of them.  🙂

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