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Let's use our imagination a bit. Picture yourself in math class (Algebra I to be exact), minding your own business, having fun playing with the axioms (aka rules) of algebra, and then one day your teacher drops this bomb on you: "Expand (x+3)(x-1)" And you might be thinking, "woah now, where did come from?" It makes sense that this would shock you. You were just getting used to the idea of expanding 3(x-1), and you probably would have been fine with x+3(x-1), but (x+3)(x-1) is a foreign idea all together. Well, before you have much time to think about it on your own and discover anything interesting, your teacher will probably tell you that even though you don't know how to solve it now, there is a "super helpful", magical technique that will help you… FOIL For those of you lucky enough never to have heard of FOIL, I will explain. FOIL stands for First Outside Inside Last and is a common mnemonic... read more

One of the major differences between algebraic equations and algebraic expressions consist of the equal sign because the equal sign consitutes for a solution that can be checked to verify that it is the solution. Expressions are meant to be simplified so common factors are important in simplifying expressions. Equations give a way to actually check the answer by subsitution for the variable while expressions are normally checked by multiplication or another type of operation.

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

How to be Successful In Mathematics Math is a complicated subject. Students struggle with it, parents don’t feel comfortable helping with homework, and teachers find it impossible to “re-teach” every year. It is for these reasons that I feel having a good foundation in math is imperative. Students that have a great foundation feel confident and are not afraid of tackling a problem until they figure it out.   What do students need to know to have a good foundation? Well, I think the most basic concepts they need to master are the concepts learned in pre-algebra. Most parents would be shocked to hear that students begin to learn these concepts as early as second grade. Some are those concepts include properly using the order of operations; being able to add, subtract, multiply and divide negative numbers, fractions and decimals; and working problems with more than one variable. I encounter students “freezing” all the time when they encounter fractions,... read more

When both writing down and reading the algebraic expressions, the binary operation (including addition+, subtraction-, multiply*, divide/, exponential^) follow a conventional order: 0) Parenthesis, including {}, [], () 1) Exponent, multiply and divide 2) Addition and subtraction The ordering is 0)>1)>2). Then there is no ordering within each group, eg multiply and divide are at the same level of priority except that 0) comes in such as a parenthesis. Let's take a look at one quick example: 3+(8-2)*6. First compute (8-2)=6; Then compute (8-2)*6=6*6=36; Finally compute 3+(8-2)*6=3+36=39. Another example: 3^2+3/(5-2) First compute (5-2)=3; Then do 3/(5-3)=3/3=1; Next compute 3^2=3*3=9; Finally add 3^2+3/(5-2)=9+1=10. Hope it helps!

My worst school years were when I did not keep up because I didn't care for the subject.  Get over it.  If the course is required you have to take it and do well.  Putting off studying and keeping up with the curriculum will only make getting ready for tests more difficult and you will not have as good understand of the subject.  This can rub off on other subjects as well while you cram for exams.     The semesters I got a jump on all subjects, especially the ones I did not think I would like, I did much better.  Whether it was by reading text book ahead, ready to ask questions in class or understand the lecture and making sure my class notes were well done and I reviewed them after class to fill in gaps, it all helps build the foundation for the subject matter.  Generally if I did this, by the time the semester was 60% complete, the remainder was a breeze.  Made all the difference for me.

Well, there are two exceptions to this question. X cannot be 0 or 1 because 0*0=0, and 1*1=1. No matter how many times you multiply 0 by itself, you will always get 0, and no matter how many times you multiply 1 by itself, you will always get 1. That's why the power of x will never change its value if x is 0 or 1. Now that we realize the two exceptions of 0 and 1 for x, x would have to be in one of two certain ranges: 0<x<1 or x>1. If 0<x<1, then that would mean that x is a proper fraction when the numerator is smaller than the denominator (e.g. 5/6). Let's use the easiest fraction value for x, 1/2, and the easiest power of x, x^2. Plug in the value of x, and you will get x^2=(1/2)^2. This will multiply the fraction of 1/2 twice by itself: (1/2)*(1/2). Now, since any number times 1 is that number, (1/2)*1=1/2 so that 1/2 remains the same. So if the second term is less than 1, it will make the first term smaller than itself as (1/2)*(1/2)=1/4. Therefore,... read more

As the school year ramps up again, I wanted to put out a modified version of a Memo of Understanding http://en.wikipedia.org/wiki/Memo_of_understanding for parents and students. It seems each year in the rush to get through the first weeks of school parents and students forget the basic first good steps and then the spiral downwards occurs and then the need for obtaining a tutor and then the ‘wish for promises’ from a tutor. Pay attention to your child’s folder or agenda book. A student is generally not able to self regulate until well into high school. Some people never quite figure it out. Be the best person you can be by helping your child check for due dates, completeness, work turned in on time. Not only will this help your child learn to create and regulate a schedule, it prevents the following types of conversations I always disliked as a teacher ("Can you just give my child one big assignment to make up for the D/F so they can pass"; "I am going to... read more

Solving Word Problems with Proportions and Relative Comparisons These word problems are set-up where the dependent variable is not provided as is, but rather as a part of an operation. You will have to set-up each side of the equality with its own operations. Example 1: “Shelley finished x number of her math homework problems before dinner. Had she finished 3 more, she would have finished half her math homework. Write an equation which represents the relationship between y, total problems and x, number of problems Shelley completed.” This isn’t set-up in the same way as problems presented in previous entries because there isn’t a defined rate of change right away. So, it will be set-up this way with one variable on each side of the equality. You're already given the variables to use in the problem. Proportion of completed problems = proportion of total problems. “3 more than completed problems” = “half her math homework” (half total problems) (x... read more

Word Problems with Multiple Variables and Given Values This type of problem will be presented such that you'll have to set-up the equation or relation between the variables. Additionally, you will be given the value of one or more variables. On all of these problems you are not asked to solve the problem, only set-up the equation. Example 1: “A weather balloon is launched from a height of 100 meters above sea level. The balloon rises at a constant rate of 27 meters per minute. Write an equation that can be used to determine the time in minutes it will take the balloon to reach a height of 2889 meters above sea level.” Start with the relationship of variables: Dependent Variable = Fixed Value + Rate of Change * Independent Variable. “Height of balloon” = “Initial height” + “27meters per minute” h = 100m + 27m For the final step, substitute the given height of 2889: 2889 = 100 + 27m. Example 2: “The perimeter of a... read more

Writing Expressions Involving Rate of Change These real-world problems can be best translated when broken down into their components (variables and operations). When you see the words “is” or “are”, this is the points where you set-up the equality. Whenever you see the word “per”, “each” the implication is a multiplication. This indicates the rate of change between the variables. The general format for these problems is: Dependent Variable = Fixed Value + Rate of Change * Independent Variable. The fixed value is generally a fixed value which does not change. Most commonly, it will be the initial value in a situation. Example 1: “Mark is purchasing a new computer. The cost of the computer is $2400 after tax. He will make monthly payments of $150. Write an equation which describes the balance on the account after any given number of months” Variables present: balance and number of months. The rate of change in this case is the $150 per... read more

The most obvious answer is cost. If a tutor charges the same rate for one or four students, it becomes cheaper per hour as you increase students and share the costs with other families. It is often believed a tutor is best when working 1:1 with a student. In some instances it is well worth the time and money to have 1:1 tutoring and sometimes it is appropriate for students to study and do school work in small groups. What is not obvious is the dynamics of small group tutoring. In a variety of circumstances it is invaluable for students to learn how to study “what needs to be studied”. The acts of independence and self regulating behavior have far reaching benefits. Groups need to learn to share and take turns. This seems simple and yet there is the underlying tendency to allow the ‘smart one’ in the group to carry the burden of work. Assuming each student is in the class and has a different point of view/observation about what is happening in class, they should share... read more

Solving Proportions By definition, ratios must be the same in order for them to be proportionate. Using the process of cross-multiplication we are able to prove if any given set of fractions are proportionate. In solving proportions, you use the same process. In these problems, you are trying to find the value which makes the fractions proportionate. Example 1: 3/n and 5/15 Step 1: Set-up cross multiplication 3*15 = 5*n Step 2: Solve for the variable. 45 = 5*n /5 /5 Divide both sides by 5 9 = n Solution: value of n is 9 Example 2: Find the value of y which makes the fractions proportionate. y/4 and 4/3 Set-up cross multiplication: y * 3 = 4 * 4 3y = 16 Divide each side by 3 /3 /3 y = 16/3 or 5.33 Example 3: n/8 and 13/2 Set-up cross multiplication: n * 2 = 8 * 13 2n... read more

DEFINITIONS When given two ratios (in the form x:y) or two relations (in the form of fractions), if the ratios of each element are the same they're said to be proportionate. Example: 3/6 and 1/2 are proportionate because 3 out 6 is the same as 1 out of two (half). PROVING PROPORTIONALITY When given two fractions to prove as proportionate, such as 1 and 3 2 6 you solve through cross-multiplication. Cross multiplication involves multiplying the numerator (number on top) by the denominator (number on bottom) of the other fraction, and then comparing the results. If the values are the same, the fractions are proportionate. The set-up above will be set-up as such: 1 * 6 ? 2 * 3 (6) = (6). Because both values are the same, these fractions are proportionate. Example 2: 3/2 and 18/8 The cross-multiplication... read more

Algebra Word Problems, Part II: Real World Problems. In this type of world situations, you will need to establish every variable in the situation as well as all fixed values. You generally will be given a relationship between the variable or variables. Example 1: “Richard wants to buy a shirt that is on sale for 20% off the regular price. Write the expression which represents the sale price of the shirt”. In this situation, there are two variables: regular price and sales price. Accordingly, there is a fixed value which is a rate of change: the 20% off. Start by writing the relationship between the variables (operation) using words: Sales Price = Regular Price – 20% off the regular price. The sales price is going to be the regular price minus the 20% off that regular price. Now you can substitute any symbol/variable in their place. In this case I will use s, in place of the Sales Price; and, p as the regular Price. Substituting the variables,... read more

I was a fairly typical young person and, like my peers, counted down the days until summer. My mother was a math professor, so I never stopped doing math during the summer, but felt like other parts of my brain became a little mushy in the summer. Come September, it was difficult to get back into the swing of writing papers and studying history and memorizing diagrams. I was out of practice and lost my routine. As an adult, I have almost continually taken classes, because I enjoy learning and find that from class to class, I need to maintain a routine, i.e. a study area and a time of day that I complete my assignments. I have also found that reviewing material a week or two before the course begins helps me to start the class with more confidence and competence. I am a big believer in confidence fueling success and I wonder if younger students practiced assignments in the week or two prior to return to school, if that confidence would help the transition to the school year... read more

I was excited on Tuesday, July 16th, 2013. This was my third meeting with this student and I finally had a breakthrough with him. On the first meeting it was clear that he saw Algebra I almost as a foreign language. I began with one of the test packet, and had him do 10 questions and reviewed the questions he had done wrong. So this continued for a while, and of course sometimes he would say that he understood, but it was clear that he did not. Anyway, after reviewing the entire packet I began a teach and learn session, in which I picked a variety of topics and had him practice various equations. After which I gave him a quiz. He failed the quiz miserably, so of course he still did not understand. Anyway, I gave him another packet for homework. When I saw the student again, I reviewed with him, but still not much improvement, but at least he tried. I did the teach and learn session again, of which some of the questions were from the previous session, and I gave him... read more

The Summer session has just begun. The stress has already begun to set in, but this week I had a break through with a few of the students. So this is my second week with a student who I am tutoring for both Algebra I and Earth Science. So far he seems stronger in Earth Science but still needs much practice, before I can be very confident about his ability to pass the Regents exam in August. After the first session of Algebra, I walked away thinking about how am I going to get him ready by August 13th. I recommended an additional session to the parents, but so far they have said no. I did several practice examples, and made the second session mainly a teaching and learning session. Then I ended the session with a quiz, but he failed :(. So when I had to meet him again for Earth Science, my mind was swirling as to how I can help him, and will I at least be successful with this subject. When I checked the homework, there was a slight improvement but not enough to celebrate.... read more

When I was studying to be a teacher, one of the classes I had to take was Literacy in Secondary Education. Since the word literacy is associated to reading and writing by most, it would strike many as a surprise that Math teachers have to take courses on literacy. However, literacy is the most practical and crucial aspect of ANY academic discipline, simply because it involves the ability to read and write in said subject. For mathematics, it could not be anymore important. If you cannot understand the words that I am using, then it is almost as if we were communicating to each other in different languages. So whatever subject you are studying, I suggest you learn its vocabulary. As the helpful tutor that I am, I will share a list of vocabulary terms that was distributed in my literacy class to all of you so that you can check your own vocabulary. Keep in mind that this is considered to be the Mathematics vocab that one should know by the time they finish high school... read more

Although learning is an awesome thing, it can be a difficult and frustrating journey for many students. This difficulty, however, is often times quite normal although most feel it means that a child may not be able to learn or that he/she is so frustrated that learning is no longer taking place. This is where the experienced tutor steps in; for frustration in learning is a part of the learning itself. I have taught and tutored many students and have seen first hand how this frustration can leave some students, and their parents, feeling helpless and hopeless. But there is ALWAYS Hope!!! What they have failed to realize is that as the brain learns difficult concepts, it can only take in parts at a time, little parts at a time. So although it may seem no learning is taking place, it actually is, just in smaller segments. In fact, the most frustration comes right before a new concept is achieved. This is when most give up. Had they stayed focused for perhaps one or two more... read more

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