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,...
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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...
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...
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...
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.
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...
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?
II. a>0 and b>0
A. II only
B. III only
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...
I do believe that any subject can be learned if one decides that they want to learn that subject. Its been my way of thinking throughout my career. If you want to learn and have an open mind, then it can happen!
Positive thinking is what it takes to succeed in this life. Believe in yourself and it will happen!
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...
This is my all time favorite website for Math worksheets.
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...
Hi all algebra students. I found a great website, algebra-class.com that has an algebra calculator that you can use to check your homework. It has been very useful in our algebra classes as a tool for homework help.
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...
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?
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...
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...
Four years ago, I came up with this math trick. Take a look at it, and at the end I'll show you why it works!
Let's play a game. I’m going to let you make up a math problem, and I will be able to tell you the answer from here. I can’t see what you’re doing, I’m not even in the same room as you, but I will still be able to tell you the correct answer.
Trust me. I’m a professional. Ready?
Okay. First, pick a number. It can be any number you wish, large or small. Now add 5 to that number. Got it? Okay, now double your new number (multiply by 2). Alright, now subtract 4 from the double.
Next, divide your new number by 2. Now, finally, subtract your original number from this new quotient. Got it? Okay. Here comes the cool part. Ready?
The answer is 3. Nifty, huh? What’s that? How’d I do it? Oh, magic.
Okay, okay, it’s not magic. The answer will always be 3, no matter what number you pick. Let’s illustrate this by...
Several of my current Geometry students have commented on this very distinction. This has prompted me to offer a few possible reasons.
First, Geometry requires a heavy reliance on explanations and justifications (particularly of the formal two-column proof variety) that involve stepwise, deductive reasoning. For many, this is their first exposure to this type of thought process, basically
absent in Algebra 1.
Second, a large part of Geometry involves 2-d and 3-d visualization abilities and the differences in appearance between shapes even when they are not positioned upright. Still further, for a number of students, distinguishing the characteristic properties
amongst the different shapes becomes a new challenge.
Third, in many cases Geometry entails the ability to form conjectures about observed properties of shapes, lines, line segments and angles even before the facts have been clearly established and...
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...