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## Algebra 1 Blogs

I recently sent this as advice to one of my clients having trouble with linear systems of inequalities. I thought I would share it here on my blog for students, parents, and tutors who have use for it. EXPLANATION OF LINEAR SYSTEMS OF INEQUALITIES A system with regular lines (the ones with equals signs in them that you have done before) shows the single point where the two lines cross each other on the graph. The X and Y at that point are the two numbers that make the equal sign true. For instance, with the equations 3 = 5X +Y and 10 = 2X -Y, the answer is x = 7/13 and y = 4/13 because if you plug those numbers into both equations you get true statements, 3=3 and 10=10. The point (7/13, 4/13) is the point where the two lines cross each other. Inequalities, where you have "less than" or "greater than" signs work the same way. But, instead of getting a point where the equations are true, you get a whole area on the graph where they are true. So,... read more

Hello Wzyant Academic Community and welcome to my blog section! This is where I am available for online chit-chat, educational assistance free of charge, business discussions & arrangements, and more! I am always eager to help and love to talk turkey with all realms of academia, so don't be shy and feel free to ask many questions!!!       P.S.  ∫∑∞√−±÷⁄∇¾φΩ

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

There are several points in grade school that involve a critical shift in the thinking that is required in the school work.  Parent's should be aware of these points as they navigate through the abyss of raising a school-aged child and supporting the child as he/she moves forward through the grades.   3rd Grade - The third grader is transitioning from whole number thinking into understanding the concepts of parts.  They are exposed to fractions, decimals and percentages.  This is a major paradigm shift.  Students are also exposed to long division at this point.  Supporting children in this phase requires an emphasis on helping the child conceptualize whole things being split into parts.  In addition to homework support, tutoring, and supplementary work, parents should introduce cooking chores to children at this time, and make them follow a recipe that has precise measurements.  Reading comprehension and writing is also an issue... read more

I have been working with a few students who are ready to learn math much, MUCH faster than allowed by the traditional classroom model in which math is taught over 6 to 8 years. Based on this experience I believe that many students as young as 4th grade and as old as 8th grade (when starting in the program) can master math in 2 years from simple addition through the first semester of Calculus, with Arithmetic, Algebra 1, Geometry, Algebra 2, Precalculus, Probability, Statistics, and Trigonometry in between. This is significantly faster than the traditional approach and is enabled by a combination of one-on-one teaching and coaching and a variety of media that I assign to students to complete in between our sessions. This is a "leveraged blended learning" approach that makes use of online software, selected games, and selected videos with guided notes that I have created that ensure that students pick up the key points of the videos, and which we discuss later. The... read more

Here are 48 of my favorite math words in 12 groups of 4. Each group has words in it that can be thought of at the same time or are a tool for doing math.   between on over in   each multiply of many   ratio divisions distribution compartments   limit neighborhood proximity boundary   infinite infitesmal mark differentiation   graph width height depth   circle sphere point interval   hyper extra spacetime dimensional   geometry proportion sketch spatial   four table cross squared   target rearrange outcome result   area volume space place   What are your favorite math words? If you aren't sure, search for "mathematical words" and pick a few.

One of the reasons students of math struggle at test time is that they fail to quickly identify "problem types". Let's say you're taking an Algebra exam and you see something of the form 4x2 + 8x -5 = 18 and are required to solve it. You should either be thinking about factoring the equation or if that doesn't work easily, using the quadratic formula. Typically, once a student identifies the problem type, he or she is 80% of the way there. Then it's usually just standard arithmetic (watch your sign changes + or - ). Solving math problems is really a process in itself and involves: assessment, identifying the problem type, looking for other complexities, i.e. there may be several steps along the way, doing the actual arithmetic and finally checking your answer for logic. Does it make sense that Fred took 16 hours to reach Chicago from New York? If it doesn't, go back and look at your problem -- you probably missed something. Be disciplined in your... read more

All the major test prep books for the SAT, ACT, and GRE -- published by companies like Kaplan, Princeton Review, Barron's, and Manhattan Test Prep -- are poorly written, conceptually deficient, and, worst of all, riddled with serious errors. Students can't be expected to learn from books that aren't even right! And I don't mean the books are riddled simply with typos, which unfortunately is also true, because they are so poorly edited; I mean they really are riddled with serious conceptual errors. Here's a simple example from the Introduction (page 23) to Manhattan's Strategy Guides for the Revised GRE. This passage appears in all eight of Manhattan's strategy guides, so it somehow went unnoticed after at least eight rounds of editing by allegedly "expert" readers and test-takers. See if you can spot the error! ---- "If ab=|a|x|b| which of the following must be true? I. a=b II. a>0 and b>0 III. ab>0 A. II only B. III only C... read more

Reading Formulas can make or break how a student comprehends their formula when alone - outside the presence of the teacher, instructor, tutor, or parent. Formula for Area of Circle:  A = π * r^2 Ineffective ways to read the area of a circle formula are as follows: Area is π times the radius squared. Area is π times the radius of the circle squared. Area of a circle is π times the radius squared. A equals π times r squared. >>>> Why are these ways NOT effective ways to read this formula? <<<<< 1.       Students will recall and repeat what they hear their educators say. 2.       If students recall letters (A) versus words (Area of a Circle) they will not realize the connection with word problems. 3.       Half way reading the formula (radius versus radius of a circle) creates empty pockets or disconnects in... read more

A question that I have heard many times from my own students and others is this: "When am I ever going to use this?" In this post and future posts, I'm going to address possible answers to this question, and I'm going to also take a look at what mathematics educators could learn from the question itself.   Let's look at the answer first. When I was in school myself, the most common response given by teachers was a list of careers that might apply the principles being studied. This is the same response that I tend to hear today.    There is some value in this response for a few of the students, but the overwhelming majority of students just won't be solving for x, taking the arcsine of a number, or integrating a function as part of their jobs. Even as a total math geek, I seldom use these skills in practical ways outside my tutoring relationships.   Can we come up with something better, that will apply to every student? I say... read more

All too often, I hear students complain "I hate math!", or "Math is too hard (or boring, or pointless, or !)" Too many kids these days from the entitlement generation (uh, that's my generation's kids - sorry friends, we've spoiled our kids like we were told to!) think that math is just for engineers, computer geeks, math nerds, or smart folks who are decidedly NOT COOL. While it is all too often true that those with natural mathematical ability are introverted, and that they may lack social skills that make it difficult to have a lot of popular friends, why does our culture (the schools, the media, television programs, video games, even some parents and teachers, too) keep this myth, this lie, alive? Because of ego. Basically, we can reduce the kind of petty, bullying behavior towards our brilliant colleagues by first acknowledging the problem, then taking logical (what else) steps to curb it. Once we remove the taunting by their peers, we should execute... read more

The first thing to do when teaching a frustrated student is to listen to, and acknowledge, their frustrations. Let him or her vent a little. If you're working with young children, they probably won't even realize or communicate that they are frustrated. Therefore, the first thing to do is say "you're very frustrated with learning ________ aren't you?" If you are in a group situation, take the student aside to talk to him or her about it so he or she doesn't become embarrassed. One of the best things you can do when teaching frustrated students is to watch them one-on-one in academic action and observe every little detail when they think, write, and speak. Often, students are lacking very particular, previous basic skills. By watching them work, you can identify where they are going wrong and notice common patterns. For instance, I have tutored many algebra students whose frustration stemmed from an inability to deal with negative numbers. Once this problem was... read more

I am taking from The Official Hunter College High School Test: problem 76 on page 20. We read the following. In the expression below, each letter represents a one digit number. Where the same letter appears, it represents the same number in each case. Each distinct letter represents a different number. In order to make the equation true, what number must replace C?    AAA    AAB + ABC   2012 A great start is to decode each AAA, AAB, and ABC. It helps to look at this problem wholly; particularly we look at the leading sum on the left wall (of the same types). We glean that either: (1) A + A + A = 20, (2) A + A + A + 1 = 20 or (3) A + A + A + 2 = 20: its very important to remember that given three numbers each less than ten, the sum of them which is great, is at most 2 in the tens place. This means that each row can only donate a 1 or 2 to the next. We can conclude that our line... read more

Many of my students preparing for the SAT, GRE, and GMAT have decent algebraic intuition when it comes to EQUATIONS, but most are much weaker when it comes to INEQUALITIES. On the one hand, this is entirely natural: inequalities capture less information than equations -- they establish merely a relation between two quantities, rather than their equivalence -- so they are inherently trickier to think about. But on the other hand, it's crucial to have a very solid grasp of how inequalities work to do well on the SAT, GRE, and especially the GMAT (which tends to love data sufficiency questions that deal with tricky inequalities). To test yourself to see how up-to-speed you are, try to decide whether the following statements are true or false. (I have intentionally made the problems very abstract and seemingly confusing to see if you really know what's going on, so DON'T WORRY IF YOU'RE TOTALLY LOST OR INTIMIDATED!) 1. If a+b=c+d and e+f=g+h, then a+b+e+f=c+d+g+h... read more