Mora Math Tutoring

05/07/2020

Soooo about those quadratic equations??

I'm sure some of you are looking at your homework, in the time honored tradition of all math students, and are thinking "this looks impossible". The fun thing (in my humble opinion) about math is that sometimes the equations can look really, really scary. Wait-- how could is that fun? It's fun because the harder it is for me to initially solve/the more intimidating it looks, the more meaningful it will be to me when I figure out how to solve and manipulate the equation at hand.

The truth of that matter is that you can hack math, but learning how to hack it is half the battle. I often see students psyche themselves out into believing that the math that they have to solve is harder than it actually is. It's completely natural to feel intimidated by equations and numbers. They just sit on the page and look at you menacingly.

I want to offer you a few tricks and techniques to become a better mathematician (and hopefully teach you a lot about quadratics while I do it).

1) The first and most important thing to remember is that YOU ARE THE HUMAN BEING when you are doing math. I know that sounds a little goofy-- of course you're a human. It's sometimes easy to forget that at the end of the day mathematics is a human invention. Regular people came up with math and have been using it for thousands of years. We tend to think that people who are good at math are geniuses, but they're only good because they practiced effectively and figured out how to optimize their learning behavior.

-You have a right to learn math. Everyone deserves mathematical literacy and it is possible to become a proficient mathematician at any level.

2) Everything in math builds onto itself. You will be hard pressed to find anything, especially in lower level math, that cannot be reduced to smaller parts. Your math education emphasizes arithmetic and algebra because they are foundational.

-Trust the rules of math and spend time working on your foundational skill set.

3) The quadratic formula, for example, looks scary:

(-b+-√b^2-4ac) / 2a

Like, what are those letters? What am I doing with this thing? What is the point?

Questions like these are more productive than you think-- as long as you don't ask the questions to the void and forget to follow up on them. Being a student in the golden age of search engines is a very new thing. If you have questions about a something, rather than leaving them as frustrating grey matter, pursue them. Ask questions and make sure that you are given answers to the questions that you ask.

-Utilize the questions that you have and seek out answers. Always.

4) If you went ahead and looked up some of your questions about what the heck is going on with the quadratic formula you might have found some of the foundational logic that precedes the formulas themselves.

Everything you do with a quadratic comes from the toolkit function f(x)=x^2. This equation will give you a parabola, the basic shape of a quadratic. Everything else, the extra add ons, are modifiers that manipulate and change the shape.

-Most formulas in math are derived from very simple and easy to deal with concepts that only look complex but are in fact very rational and easy to work with if you take its slow.

5) Quadratics follow another form, the equation itself which is AX^2+BX+C=0. The a, b, and c's in that other equation? Yup, those are just what you plug into the quadratic formula to solve for the roots. There are many ways to approach this. Let's try something out:

Say I'm given a quadratic and I'm tasked with solving to find the value of x. I can approach this in many ways. There is more than one method to solving for x and learning to "plug and chug" into a formula will help me in the long run because it is time saving.

Say the equation is 2x^2+6x+2

This means that I can find my a, b and c values easily:

a:2
b:6
c:2

That means that I can plug my values into the formula and find find my x-values with little headache.

(-b+-√b^2-4ac) / 2a becomes:

(-6+-√6^2-4(2)(2))/2(2)

Oof, that still looks ugly. First of all, what does +- mean?? The +- means that you are doing this equation twice, once using addition and another time performing subtraction. Don't worry, I'll show you how this works.

Let's go ahead and start to perform operations:

First let's clean this up a bit:

We'll do the addition one first:

(-6+√36-16)/4

I simplified and cleaned up the equation. It already looks more manageable.

Let's do one more cleanup round:

(-6+√20)/4

And so on and so forth. This is how you solve math, in fine steps. Don't try to attack things, approach with the intent to re-organize until the solution practically jumps off of the page.

05/07/2020

When working with polynomials it is important to remember the FOIL method for conversion to quadratic expressions.

FOIL stands for First, Outside, Inside, Last. It is a method that is used for reorganizing and combining polynomials into the quadratic form ax^2+bx+c=0.

Quadratic is a weird word, isn't it? What does it mean? If you're a Spanish speaker you'll probably notice that the root of the word, quad, is the same root as the word "cuatro", meaning four. A quadratic equation is an equation in the second degree, meaning that the highest exponent in the expression is the square.

A quadratic expression/equation will always form a parabola on the graph. No matter how much you manipulate the form of the expression/equation you will get a transformation of the original parabola, not some unknown shape or line.

Quadratics generally have two x-intercepts and one y-intercept. There will generally always be two answers for the value of x, rather than linear graphs which have one x-value or cubic expressions which have three. There are some situations where a parabola is on its side, but the philosophy is generally the same. Let's not put the cart before the horse.

So let's say that you are given two binomials and you are tasked with creating a quadratic expression: (2x+5) (6x-2). In this form it is SUPER EASY to algebraically find the x roots. All you have to do is set each binomial to zero and solve using basic operations.

2x+5=0
-5
2x=-5
/2
x=-5/2

6x-2=0
+2
6x=2
/6
x=2/6-->1/3

Our x values for this quadratic are x=-5/2, 1/3

But what is the quadratic expression?

To FOIL you must follow these steps:

(2x+5)(6x-2)
First: 2x*6x=12x^2 (whenever you multiply two variables together they get taken to the degree of multiplication)

Outside: 2x*-2=-4x

Inside:5*6x=30x

Last: 5*-2=-10

Now combine all like terms. The only like terms in this are -4x+30x, which gives us 26x.

Now put all of the answers in standard form based on order of magnitude of each item:

12x^2+26x-10 is the quadratic expression that goes along with the given binomials!

Congratulations, you are now a FOILing expert. Go and FOIL the town!

05/07/2020

On this day in math: 5-7-1934
Karl Geiser died in Zurich, Switzerland. He organized the first International Congress of Mathematicians, held in 1897 in Zurich.

To learn more about Karl Geiser visit:

Biography of Karl Geiser (1843-1934)

05/05/2020

5-5-1961
Alan B. Shepard flew into space, becoming the first U.S. astronaut to make a space flight. His fifteen-minute flight in Freedom 7 from Cape Canaveral, Florida, reached an altitude of about 115 miles and ended 302 miles down the Atlantic missile range.

To learn more about Alan Shepherd visit: https://www.maa.org/news/on-this-day

05/04/2020

The British series "Connections with James Burke" is one of the great secret treasures of science-entertainment. This is a great program to watch to see how much science and mathematics are interwoven with each other in strange, unexpected ways. James Burke once said: "Why should we look to the past in order to understand the future? There is nowhere else to look."

In this program, "Faith in Numbers", James Burke examines the transition from the Middle Ages to the Renaissance from the perspective of how commercialism, climate change, and the Black Death influenced cultural development. He examines the impact of Cistercian waterpower on the Industrial Revolution, derived from Roman watermill technology such as that of the Barbegal aqueduct and mill. Also covered are the Gutenberg printing press, the Jacquard loom, and the Hollerith punch card tabulator that led to modern computer programming.

"Faith in Numbers" examines the transition from the Middle Ages to the Renaissance from the perspective of how commercialism, climate change, and the Black D...

05/04/2020

SAT prep!

What resources are you using to get prepared to take the SAT?

The SAT test is not something which one can prepare for overnight. The more time you devote to learning how to take this test the better off your scores are going to be for it. So much in your college career hinges on test scores.

While you have access to free education find a way to dedicate your time to making the rest of your academic life as cheap and accessible as possible. Great test scores will help you to get better scholarships and grants, will help you to choose the college that you would like to attend and will generally help you on admissions applications. Reach out and request as much help as you can. As questions about this test. Take practice tests.

Yes, practice. Practice, practice, practice. When you get into the testing room and the timer starts you won't have the opportunity to go back and re-learn things. You don't want to go into this test cold. It's an entire experience. Knock it out of the park.

Khan Academy offers free SAT prep materials and is a great place for younger learners to start to get a handle in what the SAT actually is.

Check out Official SAT Practice on Khan Academy

05/04/2020

While we're on the subject of pi I wanted to share something cool and less talked about in the classroom: multiple methodologies for calculating. This is a cool blog post that shares how to calculate pi using the Monte Carlo method rather than simply dividing the circumference of a circle by the diameter.

We can calculate an approximate value for pi by using the Monte Carlo method. This probabilistic method relies on a random number generator and is described below. At the end of the post there is a…

05/04/2020

Here are several basic formulas for calculating the volume/area of different two and three dimensional shapes. Geometry means "measure the earth". On a foundational level it is largely focused on the relationships between heights, lengths, widths and the number pi, π. Generally when we do foundational math it is okay to calculate pi, π, as 3.14, especially when working by hand. As you become more and more skilled and fascinated by mathematics you will eventually want to know more and more about these shapes and why these relationships are so reliable.

To learn more about the history of geometry: http://www.thegeodes.com/templates/geometryhistory.asp

You'll want to learn about things like the number Tau, but for now it's enough to understand these relationships using the delightfully eternal number π. π is an irrational number: remember that an irrational number is one that, for all intents and purpose, goes on forever. That means that there is no end to the amount of decimals after the decimal place in the number and they do not repeat. π is found after the circumference of a circle is divided by the diameter of that circle. You will always end up with 3.14159..., making this mathematical relationship one of the most important to understand. So much mathematics is based on circles and the farther you go as a young mathematician the more you will be confronted with this simple yet endlessly complex shape.

The relationships between numerical items and numbers is one of the more fascinating things in mathematics and is essential to both applied mathematics and pure mathematics. If, for instance, you want to study engineering you will absolutely need to understand Geometry. On the other hand, say you wanted to be a painter, or a poet, or a philosopher or a musician: these mathematics, and the purity of geometry as a field of endeavor, will carry you to new insights into the life of your own mind.

05/04/2020

This is a great two-part documentary series about Greek mathematics and Euclid's Elements. (Euclid is pronounced You-klid, not to be confused with Euler being pronounced Oil-er. There are jokes in the math world about this. I'll spare you the agony of having to see them, haha.) Euclid's name meant "renowned, glorious".

Euclid was a Greek mathematician whose seminal work was focused on geometry, specifically working from axioms.

-An axiom is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.

https://plus.maths.org/content/how-euclid-once-rules-world

https://youtu.be/UPlqJaUi5jE

https://youtu.be/56Yy1odPLag

A documentary about ancient Greek mathematics, focusing on Euclid's Elements. (Part 1)

05/04/2020

On this day in math: American Academy of Arts and Sciences founded in Boston. It was the first national arts and sciences society in the U.S., and was founded "to cultivate every art and science which may tend to advance the interest, dignity, honor and happiness of a free, independent and virtuous people." James Bowdoin was the first president.

To learn more about the American Academy of Arts and Sciences visit:

Honoring excellence and leadership. Working across disciplines and divides. Advancing the common good. From 1780 to today.

04/30/2020

On this day in math: 4-30-1777
Carl Friedrich Gauss, considered one of the greatest mathematicians of all time, born in Brunswick, Germany. His mother couldn't remember his birthdate, but she could relate it to a movable religious feast. To confirm the date of his birth Gauss developed a formula for the date of Easter.

To learn more about Carl Friedrich Gauss:

Biography of Carl Friedrich Gauss (1777-1855)