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Mathematical Journeys: Carl Gauss and the Sum of an Arithmetic Series

There's a famous (and probably apocryphal) story about the mathematician Carl Friedrich Gauss that goes something like this: Gauss was 9 years old, and sitting in his math class. He was a genius even at this young age, and as such was incredibly bored in his class and would always goof off and get into trouble. One day his teacher wanted to punish him for goofing off, and told him that if he was so smart, why didn't he go sit in the corner and add up all the integers from 1 to 100? Gauss went and sat in the corner, but didn't pick up his pencil. The teacher confronted him, saying “Carl! Why aren't you working? I suppose you've figured it out already, have you?” Gauss responded with “Yes – it's 5,050.” The teacher didn't believe him and spent the next ten minutes or so adding everything up by hand, only to find that Gauss was right! So how did Gauss find the answer so fast? What did he see that his teacher didn't? The answer is simple, really – it's all about... read more

How to solve simultaneous equations.

Normally, an equation has a single solution when it contains only one undefined variable.  For example, take the equation 3x + 7 = 19.   3x + 7 = 19     [original equation] 3x = 12     [subtracted 7 from both sides] x = 4     [divided both sides by 3]   This is one case of a larger trend in algebra.  As I've already said, you can solve an equation for one answer when it contains a single variable.  However, this is derived from the larger rule that you can solve a set of equations where there are as many distinct equations as there are variables.  These are called simultaneous equations, and occur any time that two equations are both true over a certain domain.  In the more practical sense, this is what you should do if an exam asks you to solve for a value and gives you two different equations to use.   To solve simultaneous equations, we can use three strategies... read more

The Right Tool for the Job

Back when I was still in middle school, I was sitting at my kitchen table during a family gathering, and my uncle posed the following puzzle for me to solve: A vendor is selling apples for 10 cents apiece, oranges for 5 cents apiece, and peanuts two for a penny. Someone comes along and buys exactly 100 items for exactly one dollar. How many apples, oranges and peanuts did that person buy?   I took out a sheet of paper and a pencil and came up with the answer in a couple of minutes. This astonished my uncle because, it turns out, he had posed this problem to two adults, including a geometry teacher, and they couldn't solve it in less than a half hour. I had a bit of a reputation for mathematical cleverness, and he had posed this problem to stump me and test the extent of my cleverness. Decades later I still remember exactly how I solved it, probably because it was a boost to my ego to learn that I was apparently smarter than a geometry teacher!   In... read more

Anecdotes about Algebra

If ever a single academic subject has been under attack, it is algebra. Students repeatedly ask me, "Why is this important?" or "When am I ever going to use this?" and even the dreaded "Why should I care?" Recently parents are echoing the thoughts and in several states alternatives to algebra under the umbrella term "trade math" are being added to curriculum so that students can opt out of upper level maths.   On one hand, I cannot blame my student's frustration and reluctance to give Algebra the time it deserves. I remember when I was a student initially introduced to Algebra. There is a basic, primal fear against seeing numbers and letters in the same equation that is difficult to overcome and accept. I struggled with algebra, detested it even, but luckily I had many great teachers who helped me all the way into Calculus. Returning to Algebra as an adult I have found several methods which have helped me communicate the... read more

Developing Study Habits for Math Courses

The majority of the students that I have often have the same problem -- they aren't grasping the information fast enough or they aren't really able to follow the lessons a teacher gives.   Sometimes, teachers aren't adaptive to every learning style for each student in their classroom.  However, know that each student has the capability to learn math on their own.  It is just necessary to have key characteristics to make it successful.   Every math student should have: patience motivation adaptability organizational skills open communication between themselves and their teacher (inside and outside the classroom) breaks!! :)   Study Tips Always try to study outside of your home or dorm room. In our minds, those are places that we relax at and it can be difficult to turn your mind off from the distractions to study. Public libraries, universities, coffee shops, and bookstores are the way to go... read more

Mathematical Journeys: What Does the Function Look Like?

This week's Math Journey builds on the material in The Function Machine. If you have not yet read that journey, I suggest you do so now. In The Function Machine we discussed why graphing a function is possible at all on a conceptual level – essentially, since every x value of a function has a corresponding y value, we can plot those corresponding values as an ordered pair on a coordinate plane. Plot enough pairs and a pattern begins to emerge; we join the points into a continuous line as an indication that there are actually an infinite number of pairs when you account for all real numbers as possible x values. But plotting point after point is a tedious and time-consuming process. Wouldn't it be great if there was a quick way to tell what the graph was going to look like, and to be able to sketch it after plotting just a few carefully-chosen points? Well, there is! Mathematicians look for an assortment of clues that help to determine the shape of a... read more

SAT Prep - Tackling Tough Math Questions

Nailing an 800 on the math portion of the SAT can be a tricky feat, even if you are steadfastly familiar with all of the requisite formulas and rules. A difficult problem can overwhelm even the most prepared individual come test day. Time constraints, test surroundings, and the overall weight of the exam can unnerve the most grounded students. So what do you do when panic strikes and your mind draws a blank? How do you re-center yourself and charge forward with ferocity and confidence? What you do is this: write everything down from the problem. This is the most important part of the problem solving process. As you peruse the question, write down the pertinent data and establish relationships by setting up equations. This exercise will help you see solutions that were previously difficult to decipher. As you work on practice tests and sample problems, you must work diligently to form a solid habit of writing down important bits of information as you plow through... read more

The Power Of Willpower: Five Tips To Strengthen Your Discipline

Willpower is unique to humanity. It is the keystone characteristic that is directly responsible for our technological advancement over the last several hundred thousand years. Willpower can be defined as the capacity to restrain our impulses and resist temptation in order to maximize our long-term success. It is the expulsion of energy to fight off innate survival based urges to exponentially increase future advantages and benefits. It is the driving force behind all civilizations, and it is what prods humankind forward to learn and grow. When we turn down a bite of cheesecake, step away from a mind numbing reality sitcom, or push off a nap to get some work done, the credit goes to willpower. It is this ghost like aura of control and discipline that we rely on to extend our existence and maximize our accomplishments. When we watch highly successful individuals exercise routinely, read voraciously, and work tirelessly, we are impressed with their ability to resist... read more

Application of Algebraic Polynomials in Cost Accountancy

MP3 players example   The profit in millions of dollars for an MP3 player can be done with the polynomial P=-4x^3 + 12X^2 + 16X. X represents the number of MP3 players produced annually. The company currently produces 3 million MP3 players and makes a profit of $ 48,000,000. To figure out the least amount of MP3 players the company can produce and still make the same profit we need to solve for P.   Step 1 Set P=48 This represents the total profit. 48=-4X^3 + 12X^2 + 16X   Step 2 subtract the 48 from 48 and from the end of the equation, like this 48=-4X^3 + 12X^2 + 16X - 48                                                                                                  ... read more

Educational Apps

After several months of carrying some pretty heavy textbooks around with me, I recently decided to switch to a Kindle Fire and start using electronic textbooks. Although there are times when a good old-fashioned book really cannot be replaced, I'm very pleased with the weight of my tutoring bag now, and my students seem to be enjoying the switch as well.   I'm able to download textbooks for free in some cases ("Boundless" publishing), and I also have several different dictionaries and other reference books a tap away! Any other books I might find helpful for my students? Just a few clicks away. This also frees up my paper textbooks to loan to my students in-between sessions. Using a Kindle gives me the added benefit of being able to load educational applications to use for practice and reinforcement. Since we are in the 'computer testing' age, this also gives my students some extra practice in preparing for computerized exams. I'm sure you'll... read more

Algebra Made Easy

When a young person takes their first higher math course, Algebra I, their brains are developing a different set of skills than arithmetic, and they are faced with abstract concepts. Most students are intellectually ready for this transition in the 8th or 9th grade. Many students have to be patient, I as it sometimes takes a few months for the "aha" moment when it all clicks. I find that their are many ways to teach algebraic concepts that allow students to grasp the particular skill they are working on. I use methods such as real-world examples, I foldable study guides, reviewing or introducing basic concepts such as inverses and identities, using colors, using hands-on materials like algebra tiles, and more. I often have students use highlighters or make their own problems. After 28 years of teaching, I know which misconceptions are made most often, so I also what NOT to do and why.  Research has repeatedly shown that success in Algebra II is the best predictor... read more

Mathematical Journeys: The Unperformed Operation

Come with me on a journey of division. I have here a bag of M&Ms, which you and I and two of your friends want to share equally. I'm going to pour the bag out on the table and split it into four equal piles. For this example, “one bag” is our whole, and the best number to represent that whole would be the number of M&Ms in the bag. Let's say there were 32. If I split those 32 M&Ms into four equal piles and asked you how many were in one pile, you could certainly just count them. But a quicker way would be to take that 32 and divide it by the number of piles I'd made, which in this case is 4. You'd probably write that as: 32 ÷ 4 = 8 So there are 8 candies in each pile. Seems easy enough with a large number of M&Ms, right? But what if there were less candies – what if our “whole” was less than the entire bag? Well, for a while we'd be okay – if there were 16, for example, we'd do the same thing and come up with piles of 4 instead... read more

I HATE MATH!

  "I can't do this.  Why do I need to know this anyway?  Can't we just use the computer to do this?"   We have all heard this from someone; ourselves, our spouses, friends and all too often we hear this from our children. We have seen math as a difficult subject for our generation and now we are seeing math become even more "frustrating", "boring", and "intimidating" for many of our children.  We have tried collaboration, individual tutoring and even extra home work as a means toward improvement.  But many of our efforts are met with failure, anger and even tears.  What is the key to overcoming the math "Mount Everest"?  While there is no band aid for healing math confusion, there are tips and strategies that are fundamental in changing your child's view of math and developing "number sense".   Math Must Make Sense The most important thing is to remember... read more

Fun Related Rates / Optimization Question: Smallest Surface Area of a Square and Circle Cut From a Single Piece of Rope

Hello everyone, One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out.  The problem was set up as such:   A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?     Here's what we'll need to do: 1.  We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.  2.  We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.  3.  Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima... read more

Christmas Present!! (Brainteaser)

Let's start off by defining some rules.   1) please no one post answers directly on this post, rather send all answers to me via a message.  Comments will be deleted and the person will be disqualified from all future contests.   2) The first 5 people to respond correctly to this post will receive a free 1 hour tutoring session via the online platform in any subject that I am approved in.  I will respond back to your message explaining the correct answer, how to get that answer, whether you were correct, and, of course, the details in setting up your free session with me   3) If you are trying, but stumped...message me for a hint   4) Have fun!!   Now for the brainteaser!   Chicken McNuggets can be purchased in quantities of 6, 9, and 20 pieces.  What is the largest amount of McNuggets that can NOT be purchased, using these quantites?   Happy Holidays everybody!  I look...

All of algebra explained in one video!

Thanks, Vihart of YouTube!   In this video, elementary algebra is built up from counting to positive numbers to negative numbers to multiplication and division to exponents and logarithms. All these concepts are tied together under the common theme of "fancy counting." That is, each of these operations is tied directly to the fundamental operation of "+1".   All of this in just over 9 minutes. This is how experts think of algebra. Novices, new to the world of algebra, see the mechanical steps to be taken, the shuffling of symbols, the confusing mass of numbers, letters, and symbols and get lost. Experts think of algebra as chunks of value that are being operated upon by a (relatively) small set of operations, made smaller by lumping an operation with its inverse (addition/subtraction, multiplication/division, exponents/logarithms) and realizing they are two sides of the same coin.   "How I Feel About Logarithms,... read more

How to find the difference of squares

One of the common questions I get asked when I am tutoring algebra is how to find the difference of squares. First, what exactly is a difference of squares, and what is it used for? Second, how do you find it? The difference of squares is a tool used to factor certain types of polynomials. Factoring is often useful in simplifying equations and allow some form of cancellation or combination of like factors. The difference of squares will allow you to factor some polynomial types that are not otherwise factorable, making it a useful tool in algebra and anything that uses algebra. So how do you find it? You can find the difference of squares for any polynomials which is a difference of two perfect squares. Take the simplest case: x2 -1. This polynomial is the difference of two perfect squares: x2 is, obviously, the square of x while 1 is the square of 1. The resulting factors, using the difference of squares, is (x+1)(x-1).   To confirm that this... read more

The Power of Parentheses!

  I find oftentimes that one of the biggest stumbling blocks for algebra students is that beginners have difficulty seeing the "chunks" in an expression. Instead, they see a big jumbled mess of symbols.   An analogy is an orchestra. A person who has never played a musical instrument, or doesn't have much experience with listening to music, hears the orchestra as one big sound. The trumpets, flutes, strings, percussion all happening at once.    An experienced musician can isolate each instrument, and let the rest of the orchestra fade, focusing on the single melody or harmony line.   Likewise, an experienced mathematician can isolate the sections of an expression, focusing on the single term or operation that needs to be dealt with at the moment, allowing the rest of the expression to fade away for the time being, until the term or operation has been dealt with.   Consider the following problem; can you see... read more

Math Equations, Formulas and Vocabulary

Area, Volume and Circumference equations: Area of a Square A=S2 Area of a Triangle A=1/2bh Area of a Rectangle A=LW Right Triangle/Pythagorean Theorem a2+b2=c2 Area of Parallelogram A=bh Area of a Trapezoid A=1/2h(a+b) Area of a Circle A=πr2 Circumference of a Circle c=πd or c=2πr Volume of a Sphere V=4/3πr3 Surface Area of a Sphere SA=4πr2 Volume of a Cube V=s3 Volume of a Rectangular Solid V=lwh Slope of a line Equations Slope-intercept form y=mx+b m is the slope b is the y-intercept y is a y coordinate on the graph (that coincides with the line) x is an x coordinate on the graph (that coincides with the line) Horizontal line y=b Vertical line x=a Finding... read more

A Few Thoughts on Why We Learn Algebra

I recently responded to a question on WyzAnt's “Answers” page from a very frustrated student asking why he should bother learning algebra. He wanted to know when he would ever need to use it in the “real world” because it was frustrating him to tears and “I'm tired of trying to find your x algebra, and I don't care y either!!!” Now, despite that being a pretty awesome joke, I really felt for this kid. I hear this sort of complaint a lot from students who desperately want to just throw in the towel and skip math completely. But what bothered me even more were the responses already given by three or four other tutors. They were all valid points talking about life skills that require math, such as paying bills, applying for loans, etc., or else career fields that involve math such as computer science and physics. I hear these responses a lot too, and what bothers me is that those answers are clearly not what this poor student needed to hear. When you're that frustrated... read more

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