Perpendicular Lines through a Given Point: Geometry

Ahoy math peeps! I’m writing this during the time of the coronavirus, and although, the NYS Regents tests may be canceled, online zooming is still on! From the good ole’ days of test-taking and sitting in a giant room together, I bring to you a Regent’s classic, a question about how to find perpendicular lines through a given point. We will go over the following Regents question, starting with a review of what perpendicular lines are. Stay curious and happy calculating! ๐Ÿ™‚

Perpendicular Lines: When two lines going in opposite directions come together to form a perfect 90ยบย angle! Sounds magical, am I right? Check it out for yourself below:

Perpendicular Lines through a Given Point
Perpendicular Lines through a Given Point

A super exciting feature of these so-called perpendicular lines is that their slopes are negative reciprocals of each other. Wait, what?

How do we do this? Now it is time to go back and answer our question!

First, our equation 2y+3x=1 looks kind of cray, so let’s get it back to normal in y=mx+b form:

Need more of an explanation? Check out the video that goes over these types of questions up on Youtube (video at top of post) and let me know if you have still any questions.

Happy Calculating! ๐Ÿ™‚

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Looking for more on Perpendicular and Parallel lines? Check out the difference between the two in this post here!

COVID-19: What does #FlattenTheCurve even mean?

COVID-19: What does #FlattenTheCurve even mean? If you are a human on Earth, then I’m sure you’ve heard about the coronavirus and are currently social distancing. Here in NYC, I’m quarantining like everyone else and listening to all the beautiful math language that has suddenly become mainstream (so, exciting)!  #FlattenTheCurve has become NY’s new catchphrase and for anyone confused about what that means, you’ve come to the right place!

The coronavirus spreads at an Exponential Rate, which means it spreads in a way that increases faster and faster every day.

What does this mean?

For Example, one person with the virus can easily spread the virus to 5 other people, those 5 people can then spread the virus to another 5 people each for a total of an extra 25 people, these 25 people can then spread it to another 5 people each for an extra 125 infected people! And the pattern continues……. See below to get a clearer picture:

COVID-19: What does #FlattenTheCurve even mean?
COVID-19: What does #FlattenTheCurve even mean?

.ย  ย *Note: These numbers are not based on actual coronavirus data

The Example we just went over is equal to the exponential equation Screen Shot 2020-04-12 at 1.21.48 PM, but it is only that, an Example! The exact pattern and exponential equation of the future progress of the virus is unknown! We mathematicians, can only measure what has already occurred and prepare/model for the future.  To make the virus spread less rapidly, it is our duty to stay home to slow the rate of this exponentially spreading virus as much as possible.

We want to #FlattenTheCurve a.k.a flatten the increasing exponential curve of new coronavirus cases that appear every day! Hopefully, this post brings some clarity to what’s going on in the world right now.  Even with mathematics, the true outcome of the virus may be unknown, but understanding why we are all at home in the first place and the positive impact it has is also important (and kind of cool).

Stay safe math friends and happy calculating! ๐Ÿ™‚

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Math Resources (in the time of COVID)

Calling all students, teachers, and parents!ย  As everyone is stuck at home during a global pandemic, now is a great time we are all forced to try and understand math (and our sanity level) a little bit more.ย  Well, I may not be able to help you with the keeping sanity stuff, but as far as math goes, hopefully, the below math resources offer some much needed mathematic support.

All jokes aside I hope everyone is staying safe and successfully social distancing.  Stay well, math friends! ๐Ÿ™‚

Kahn Academy: The same Kahn Academy we know and love still has amazing videos and tutorials as usual, but now they also have a live “homeroom” chat on Facebook LIVE every day at 12:00pm. The chats occur daily with Kahn Academy founder Sal and at times feature famous guests such as Bill Gates. Click the link below for more:

Math Resources

Khan Academy Homeroom 

Study.com: In a time when companies are being more generous, Study.com is here for us as they offer up to 1000 licenses for school districts and free lessons for teachers, students, and parents.  Check out all the education freebies here:

Study.com

Math Resources

Math PlanetIf you’re looking for free math resources in Pre-Algebra, Algebra, Algebra 2, and Geometry then you will find the answers you need at Math Planet.  All free all the time, find their website here:

MathPlanet 

Math Resources

MathSux: Clearly, I had to mention MathSux, the very site you are on right now! Check out free math videos, lesson, practice questions, and more for understanding math any way that works for you!

MathSux

What is your favorite educational site?  Let me know in the comments, and stay well! ๐Ÿ™‚

Binomial Cubic Expansion: Algebra 2/Trig.

Hey math friends! In this post, we are going to go over Binomial Cubic Expansion by going step by step! We’ll start by reviewing an old Regents question. Then, to truly master the topic, try the practice problems at the end of this post on your own! And, if you still have questions, don’t hesitate to watch the video or comment below. Thanks for stopping by and happy calculating! ๐Ÿ™‚

Also, if you’re looking for more on Binomial Cubic Expansion, check out this post here!

What are Cubed Binomials?

Binomials are two-termed expressions, and now we are cubing them with a triple exponent! See how to tackle these types of problems with the example below:

Binomial Cubic Expansion

How do I answer this question?

We need to do an algebraic proof to see if (a+b)3=a3+b3.

How do we do this?

We set each expression equal to one another, and try to get one side to look like the other by using FOIL and distributing. In this case, we will be expanding (a+b)3 to equal (a+b)(a+b)(a+b).

Binomial Cubic Expansion
Binomial Cubic Expansion

Extra Tip! Notice that we used something called FOIL to combine (a+b)(a+b).  But what does that even mean? FOIL is an acronym for multiplying the two terms together.  It’s a way to remember to distribute each term to one another.  Take a look below:

Screen Shot 2019-05-24 at 9.02.50 AM.png
Screen Shot 2019-05-24 at 9.03.23 AM.png

Add and combine all like terms together and we get Screen Shot 2019-05-24 at 9.04.45 AM.png!

_______________________________________________________________________

Keep in touch with MathSux and get FREE math videos, lessons, and practice questions straight to your inbox! Thanks for stopping by and happy calculating! ๐Ÿ™‚

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Volume of a Cone: Geometry

The Voluminous “Vessel” at Hudson Yards

Calling all NYC dwellers! Have you seen the new structure at Hudson Yards?  A staircase to nowhere, this bee-hive like structure is for the true adventurists at heart; Clearly, I had to check it out!

Where does math come in here you say?  Well, during my exploration, I had to wonder (as am sure most people do) what is the volume of this almost cone-like structure? It seemed like the best way to estimate the volume here, was to use the formula for the volume of a cone!

What do you think the Volume is?

Volume of a Cone

Volume of a Cone:

I estimated the volume by using the formula of a three-dimensional cone. (Not an exact measurement of the Vessel, but close enough!) .

Volume of a Cone
Volume of a Cone

We can find the radius and height based on the given information above.  Everything we need for our formula is right here!

Volume of a Cone

Now that we have our information, let’s fill in our formula and calculate! 

Extra Tip! Notice that we labeled the solution with feet cubed Screen Shot 2019-04-14 at 4.53.49 PM.png, which is the short-handed way to write “feet cubed.”  Why feet cubed instead of feet squared? Or just plain old feet? When we use our formula we are multiplying three numbers all measured in feet:

radius X radius X (Height/3)

All three values are measured in feet! –> Feet cubed (Screen Shot 2019-04-14 at 4.53.49 PM.png)

  ____________________________________________________________________________________

Did you get the same answer? Did you use a different method or have any questions?  Let me know in the comments and happy calculating! ๐Ÿ™‚

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Looking to apply more math to the real world? Check out this post on the Golden Ratio here!

Rate of Change: Algebra

The rate of change, the rate of motion, the rate of a heartbeat. A chart on a piece of paper as boring as it may seem, can produce some pretty great numbers that directly relate to us human folk.

The question below may make you groan at first glance, but what happens if we use our imagination to picture the real-life bird it’s describing? Any better? Yes? No? Well, either way we must solve, so let’s get to it

rate of change algebra

How do I answer this question?

It wants us to find the Rate of Change, specifically between 3 and 9 seconds. Let’s hi-light those two values on our given table:

What is the Rate of Change?

To get the Rate of Change between these two values, we need to go back to the good ole’ Slope Formula and realize that the list of values is really a list of x and y coordinates.

rate of change algebra

Now let’s plug in the coordinate values (3, 6.26) and (9,3.41) into our slope formula:

rate of change algebra

Extra Tip! Notice that we added the labels feet/second to our answer.  Why does this make sense?? The question tells us that P(t) represents feet and that t is equal to seconds.  Another way to look at this question when applying it to the slope formula is to realize that we are finding the change of feet divided by the change of seconds.                                                          ____________________________________________________________________________________

Still got questions?  Let me know in the comments and as always happy calculating!:)

Looking for the next step? Learn how to graph equation of a line, y=mx+b here!

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Intersecting Secants: Geometry

intersecting secants theorem

Hey math friends! In today’s post, we are going to go over the Intersecting Secants Theorem, specifically using it to find the piece of a missing length on a secant line. We are also going to see proof as to why this theorem works in the first place!

Just a warning: this blog post contains circles. This usually non-threatening shape can get intimidating when secants, chords, and tangents are involved. Luckily, this question is not too complicated and was also spotted on the NYS Regents. Before looking at the questions below, here is a review on different parts of a circle. Pay close attention to what a secant is, which is what we’ll be focusing on today:

Screen Shot 2018-12-25 at 12.32.38 PM

Think you are ready? Let’s look at that next question!

Screen Shot 2018-12-25 at 12.34.09 PM
Intersecting Secants

What information do we already have? Based on the question we know:

Intersecting Secants
Intersecting Secants
Screen Shot 2018-12-25 at 12.54.24 PM.png

*Extra Tip! Why does this formula work in the first place!??  If we draw lines creating and proving triangle RTQ and triangle RPS are similar by AA, this leads us to know that the two triangles have proportionate sides and can follow our formula!         ___________________________________________________________________________________

Still got questions?ย  Let me know in the comments and remember having questions is aย  good thing!

If you’re looking for more on intersecting secants, check out this post here for practice questions and more!

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Rational Exponents: Algebra 2/Trig.

Hi everyone and welcome to MathSux! In this post we are going to break down and solve rational exponents. The words may sound like a mouthful, but all rational exponents are, are fractions as exponents. So instead of having x raised to the second power, such as x2, we might have x raised to the one-half power, such as x(1/2). Let’s try an example taken straight from the NYS Regents below. Also, if you have any questions don’t hesitate to comment below or check out the video posted here. Happy calculating! ๐Ÿ™‚

Screen Shot 2018-12-16 at 2.15.36 PM

How do I answer this question?

The questions want us to simplify the rational exponents into something we can understand.

How do we do this?

We are going to convert the insane looking rational exponents into radical and solve/see if we can simplify further.

Reminder!

A radical can be converted into a rational exponent and vice versa. Not sure what that means? It’s ok! Take a gander at the examples below and look for a pattern:

Rational Exponents:

Think you’re ready to take on our original problem? #Letsdothis

Example:

Screen Shot 2018-12-16 at 2.21.06 PM.png
Rational Exponents:

Practice:

Rational Exponents:

Solutions:

Screen Shot 2018-12-16 at 2.20.42 PM.png

Still got questions?ย  Don’t hesitate to comment below for anything that still isn’t clear! Looking to review how to solve radical equations? Check out this post here! ๐Ÿ™‚

Also, don’t forget to follow MathSux fopr FREE math videos, lessons, practice questions and more every week!

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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…โ€ฆ..(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, โ€ฆโ€ฆ.

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! ๐Ÿ™‚

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!

Don’t forget to connect with MathSux on these great sites!

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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.ย  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 andย  happy calculating! ๐Ÿ™‚

Why factor in the first place, you may say? We want to manipulate the equation until we solve for x.ย  Solving for x is our main goal, and factoring allows us to do that.ย  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.ย  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.ย  This factoring method just makes you feel all warm and fuzzy inside or maybe that’s just me).ย  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.ย  The Quadratic Formila on the other hand will work every time!

Low and behold, the Quadratic Formula:

Want more Mathsux?ย  Donโ€™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! ๐Ÿ™‚

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