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 -48 Step 3 Than you factor by using the Greatest Common Factor as show below. -4X^2(X-3) + 16 (X-3) Notice how it is now a binomial, which means two terms. It was previously a Trinomial. Step 5 Multiply the variables in each Binomial separately and set... read more
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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 notice that the... read more
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 of success... read more
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 of piles... read more
"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 that math must make sense. Leading students to make connections between mathematics and how they will use... 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. Let's start by looking at the problem conceptually... read more
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 forward to hearing answers from all of you!!
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, by Vihart" Happy +1's!... read more
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 is, in... read more
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 the four operations required to solve the... read more
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
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
I used to do this and I see a lot of students who do this common mistake when studying. Maybe you are working through old homework problems to prepare for an exam in math or physics and you have the solutions in front of you. You get to a certain point and you get stuck, so you check the solution, see what the next action you have to take is, and then continue working through the problem. Eventually you get an answer that may (or may not) be right and check the solution again. If it is, you feel great and move on. If it isn't you compare the work and see what you did wrong and understand the mistake so you move on. All this is a fine way to start studying, but the major mistake is that most students don't go back to that problem and try to do it again. Even if you were able to understand the solution or the mistake you made, you never actually got through the problem completely without aid. So now if you come to this problem on your test, this will be the first time you actually... read more
I've heard this sentiment over and over--sometimes from students, and sometimes, I'll admit, in my own head. Last night, I was working on my own math homework, and there was one problem I just couldn't get my head around. I read the book, looked back at my class notes, and even sat down with a tutor for a while, and still, when I tried a new problem of the same type on my own, it just didn't work! "Maybe I'm not as good at math as I thought," I told myself. "Am I REALLY smart enough for bioengineering?" It was hard, but I told myself "YES!" And I kept working. I laid the assigned problems aside and started doing other problems of the same type from the book. I checked my work every time. Each problem took at least ten minutes to solve, and the first three were ALL wrong! I kept going. I got one right, and it made sense! I did another, and it was half right, but there was still a problem. I did another, and it was right! Eventually I had a page... read more
Algebra does us the favor of assigning numbers to cause-and-effect relationships. For instance, we sense that exercise leads to weight loss. Algebra answers that key question: "How much?" Difficulties come in three forms: expressions that mix numbers with letters, minus signs, and fractions. COPING STRATEGIES View equations as balance scales or "seesaws": (+)weights point down, (-)wts point up, with "=" as the pivot. Vocalize equations like a "recipe": 2x-7 = 9 is "double the 'x' and remove '7' to make '9'." Use "containers" instead of letters: "x" is a box and "y" is a bowl. What's in the box for "box +box -7 = 9?" Move "minus" elements "to the other side." What's in the box for "box +box = 9 +7?" Re-scale to get rid of denominators: x/2 - x/6 = 8/3 becomes (box +box +box) -box = 16 or 8 per box.
One of the common challenges for many Algebra students is forgetting important concepts from Pre-Algebra. So many students complain that they never fully learned fractions, decimals and percentages or ratios, rates and applying math to word problems. Without solid memorization of multiplication & division tables, factoring and simplifying are much more difficult. A better understanding of the basics, including learning different methods and shortcuts, can not only boost confidence but can improve grades and SAT scores. Spend time with a tutor or use different websites to review topics from previous years. It will help, exponentially!
You have a science paper due on Monday. History test and math packet due on Tuesday. English project group meeting Wednesday after school. Homework to complete. Chores on Saturday. And you want to spend time with a friend. Use a student planner. Be specific with the time. Include day/date and time/hour. Specifying the hour in your planner creates an actual appointment, and appointments are not made to be broken. During the week, your friend calls, wanting to come over and watch a movie with you on Saturday at 1:00 pm. You look at your planner. It shows you will be completing your history reading assignment at that time; you suggest 3:00 pm to your friend. It’s Saturday, 3:00 pm. Your friend knocks at the door. What’s that? You say you’re feeling great, relaxed, at ease. Oh yes. That’s part of the reward of scheduling, and sticking to it. You've read and studied the chapter for your history class. You finished your chores. Now you can really enjoy a movie. When are... read more
Tomball Texas, do you need math tutoring? Then check out Liz's profile on WyzAnt, the Tutoring, Teaching and Coaching site. The URL is http://www.wyzant.com/tutors/tomballtutor. She explains math concepts in the simplest way possible, so the student has a good understanding of the concept. After demonstrating the correct way to do the problem she will watch the student do a similar problem. These are the concepts that she learned in her CRLA Training course from Texas A&M, where she was a Level III certified as a Master Tutor. She has been teaching since 2001 and tutoring since 1998. She tries to make the lessons interesting and fun, so the students don't dread the sessions, but look forward to them. Give Liz a try and I think you will be pleasantly surprized how easily she works with your son or daughter. Contact her through WyzAnt.com via email
Are you having some difficulties with your math or science class? Many students have this feeling at some point during the semester. Unfortunately, most of them think that the new chapter will be starting soon and then everything will be ok. While this might be the case, math and science are two subjects where the material builds from chapter to chapter. Missing a key formula or concept in one chapter can really affect how the rest of the semester goes. So don't wait too long to get some help. Maybe all you need is one or two sessions with a tutor to clear up a concept. Hiring a tutor doesn't have to be a long term commitment. But if it turns out that more help is necessary, you've made a great decision in starting early. Remember, it is better to have a tutoring session early and clear up any misconceptions than to feel like you need to learn a whole semester of material in a day or two before the final.
First of all, let me start off by saying that Algebra can be the starting point of making or breaking whether or not a person graduates from a particular grade level or is able to advance to the next level of math needed for graduation. Sometimes, with all the numbers and variables and such, Algebra may even be a little intimidating. Let me tell you this though, Algebra 1 is the start of a newly defined way of understanding the introduction to higher level math. Once a student is capable of solving the mini-complex problems that Algebra 1 has to offer, the student will then self-prepare for understanding how to use these fundamentals in everyday life. Math, like any other subject, is what you make of it. YOU DO THE MATH, DON'T LET THE MATH DO YOU!!!! I love math and all of it's complexities, and I still use certain equations to solve everyday math problems that I run into. I would like to offer my services to those in need of understanding Algebra 1. With a little guidance... read more