Solving Equations For a Particular Variable

A very basic principle

An equation has one equal sign.

The equal sign divides the equation into left hand side and right hand side.

The two sides may look totally different from each other as expressions but the equal sign says their numerical value has to be the same.

The fundamental principle of equations says that, when two expressions have the same numerical value, if we apply one operation to both expressions, the resulting expressions after the operation is performed will also be equal in value. They will be equal not to the original expressions, but to each other.

So, if A, B and C are three algebraic expressions, and we have the equation A = B, then all of the following will also be valid equations:

A + C = B + C

A - C = B - C

(A)(C) = (B)(C)

A/C = B/C [provided C is not zero]

A^2 = B^2

Square root of A = Square root of B

This fundamental principle is used over and over to solve equations for specific variables, one step at a time.

For example, in solving for x the equation (3x + 1)/2 = 5y - 4, we can do it like this:

1) Multiply both sides by 2 and we get

3x + 1 = 2(5y - 4)

2) Subtract 1 from both sides and we get

3x = 2(5y - 4) - 1

3) Divide both sides by 3 and we get

x = ( 2(5y - 4) - 1)/3

Now the equation has been solved for x in a series of steps, where each step consists of applying one and the same operation to BOTH sides of the equation.

The fact that the resulting expression for x can be simplified to

x = (10y -9)/3

is not relevant here. I am only illustrating the process we use to isolate x one step at a time by applying the same operation to both sides of the equation.

The following YouTube video from InterAlgebra12 shows several more examples:

Word Problems Are Not That Much of a Mystery

Most of it is just common sense

Many times, my students are surprised by how simple some problems seem when I explain them. They go "Is that all there is to it? Can it be that simple? You didn't use any formula!"
A big part of the difficulty students often have with word problems is they think there is or there should be a special type of formula suited for each particular problem. But, for many word problems out there, that is not the case.
If your first reaction to a word problem is trying to remember a formula that would solve that problem, chances are you are never going to remember such a formula, because you have never seen it, because it is not there in any book, and no teacher teaches it as "the" formula for this problem.
There may well be a formula for that particular problem, but the formula is nowhere to be found on record. Because nobody has bothered to figure it out or to pass it down, because even if they did, the formula would be applicable only to that particular word problem and to no other problem. It would be a very limited, almost useless formula.
So, the first thing you have to do with word problems, is to forget about formulas altogether and just read the problem, over and over and over again, as many times as you need to understand what the problem is talking about, what situation it is describing.
You want to really understand the situation, the process described in the problem. You want to understand it as clearly as you see sunlight. You want to be able to express it in your own words, you want to be able to imagine it, you want to be able to tell a story about it, you want to be able to draw a complete picture of it.
Once you do that, the solution presents itself to you naturally, the numbers practically work themselves out. When you really know what is going on, you know what to do, you know what operations to perform, they make sense.
So, again, it's not how to mechanically make the problem fit into a canned formula, but how to make your very own mind wrap itself around the problem completely, with total abandon, accuracy and precision

The General Quadratic Equation Formula

Benefits of showing where the formula comes from

I often show my student how the formula for solving a general quadratic equation is derived. This, understandably, seems quite complex to them. The benefits of showing them the development of the formula in full detail are:

1) After that, memorizing the formula seems an easier task in comparison.
2) Plugging in the correct values in the formula and evaluating it to a numerical result seems now way easier than having to figure out the solutions for each particular equation by completing the square. The fact that the formula is available saves them the work of completing the square in each particular case.
3) They now have seen how the formula is developed. Even if they do not understand the process 100%, even if they forget the process within five minutes, the formula itself is no longer a mystery. They know there is an algebraic derivation of it, and they have seen it, at least once. They feel now much more confident in using the formula.

So, because of the previous reasons, it is totally worth it to go through the process of showing them how the formula is developed. It is time well invested.

Self-Talk Is a Performance Factor

A very important factor to achieve good results

Many students, when they first start working with me, show the following behavior, it's really common: they are working on a problem and all of a sudden they start saying things like: "Well, I don't know what to do, I have never been good at math," "I really suck at math," "I always get these problems wrong," "I do not understand percentages," "When it comes to algebra, I just don't get it," "Oh, boy! I hate these problems. I don't like math at all," "I am not smart," etcetera.
So, very often I have to explain to them the huge impact and importance of daily self-talk. I look them in the eye and I say:
"You mean you never were good at math before, but now you are, and you will."
Most times they look surprised and a little confused when they hear that. Then I go on telling them about how the subconscious mind works, in a totally different way than the conscious mind. I tell them:
"Look, in the long run, nothing is more important than what you say to yourself. Because your subconscious mind literally believes everything you say. It does not judge, it does not analyze, it does not argue. It just stores the information you put in there and later it retrieves it like that, unprocessed."
If you go on repeating things like you are not good at math, then that is what your subconscious mind stores and believes. Later it will make you act from that belief in ways that will produce results consistent with that belief, and the results will of course validate and reinforce the original belief.
If you want to pass your test, stop saying that and start saying: "I'm good at math," "I like math," "I can solve these problems." Just say it, even if it sounds fake at the beginning. Your conscious mind may say "That's a lie," but it doesn't matter. Your subconscious mind won't say anything like that, it will just take the new affirmation and store it. Then, later it will incorporate it in your subconscious decision making process.
By repetition, you can make the new, positive affirmation, outweigh the old, negative ones.
So, starting today, change what you say to yourself about math, about you math skills and your performance. You don't even have to say it aloud. Only by thinking it, it is having that effect in your subconscious mind. Watch carefully what you say to yourself about anything you care about. This process has a huge impact. It makes a real difference.

One Way To Get Familiar With More Square Numbers

Using the algebraic formula for the square of a sum

At the high school level, most students know all square numbers between one and one hundred: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
That is, the squares of all numbers from 1 to 10. Many students also know 121 is the square of 11 and 144 is the square of 12.
That is about how far most people go memorizing square numbers. Relatively few people know 196 off the top of their head when asked: "How much is 14 times 14?"

I find useful the following exercise for students who are learning topics such as the quadratic formula and how to factor trinomials:
I remind them of the formula for the square of a sum:
( a + b )^2 = a^2 + 2ab + b^2

Then I suggest to them applying it in the following way to figure out the squares of numbers between 10 and 20:

(14)^2 = (10 + 4)^2 = 10^2 + 2(10)(4) + 4^2 = 100 + 80 + 16 = 196

(17)^2 = (10 + 7)^2 = 10^2 + 2(10)(7) + 7^2 = 100 + 140 + 49 = 289

Working out a few of these concrete, numerical examples helps students to see a general pattern and become more familiar with both the formula for the square of a sum and, at the same time, with squares of numbers between 10 and 20. This feels like a natural expansion from their previous knowledge of squares of numbers between 1 and 10.

Have a Calculator Handy When Doing Long Division

Not to skip the work, but to check your results

Long division problems are among the most boring, detail oriented, time-consuming problems math has to offer.
I agree that 7th graders should know how to perform long division by hand, without a calculator. In my opinion, even 4th graders could be required to perform long division accurately.
However, some teachers go into overkill mode when assigning long division homework. They throw in too many digits into the calculations and too many problems into the assignment. I find this practice counterproductive when it comes to long division. There is a fine line between drill and overkill.
When students face such a heavy workload, such a long time doing these boring, exacting, fine detail, uninteresting drill problems, row after row, it does not take very long before many people start hating math with a passion, I tell you.
Once you have satisfied yourself that you understand the procedure, that you know how to do it, that you can actually do it and you can really do it well, what is the point of keeping at it beyond that? People are not machines.
Some students lack the attention span required to accurately divide a seven-digit number by a six-digit number. Much less doing ten of these calculations in a row. For them, this type of homework is an exercise in discipline and endurance, not in math or understanding. It is not their intelligence that is at play or in question, but their ability to submit to an arid, boring, meaningless routine.
At some point they start speeding up, they stop paying attention, and they start making mistakes.
If you find your child or yourself in this situation, I strongly recommend using a calculator. Not to skip the work altogether, but to check your result and make sure whether or not you made a mistake. Long division problems are exactly the kind of problems calculators are for.
If you are not 100% sure, beyond any doubt, that your division work is absolutely correct, then the calculator is almost the only way to find out if there are any mistakes there.
I say it is valid to use a calculator to check your long division answers.

Some Questions In Test Prep Books Make No Sense

But you can find the right answer anyway

Test prep books are very well written and edited. They generally present the right solution for almost every single problem included but every once in a while (seldom though) you can find a mistake. Sometimes the given answer for a problem may be wrong or, even more rare, some questions don't make sense from a strict, formal, rigorous, mathematical point of view.
The following question is a perfect example of this:

Select the number that is not a factor of 6/288.

All the five options given in the book are fractions (and different from zero).
Four of those fractions have a numerator that is a factor of 6, and have a denominator that is a factor of 288.
The other fraction is 2/11, where the denominator, 11, is not a factor of 288.
It turns out that is the answer the book indicates as the correct one. So thought by whoever designed that particular problem.

But strictly speaking, the question does not make any sense at all.
The set of rational numbers is a field, an algebraic structure where every non-zero element is a unit. In a field every element can be divided by any non-zero element.
That means, according to the technical definition, any non-zero fraction is a factor of any other fraction.
That is why the question does not make sense, because, in the domain of the rational numbers, the fraction 2/11 is a factor of 6/288.
How come? Simple: (2/11)(33/288) = 6/288.
There is a fraction, namely 33/288, that multiplied times 2/11 gives 6/288 as the result.
That makes 2/11 a factor of 6/288.
In the same way, given any non-zero fraction, we can always find another fraction to multiply it by and get 6/288 a result.
For that reason, it makes no sense at all to talk about factors of a fraction. Once we are dealing with fractions instead of restricting ourselves to whole numbers, every non-zero fraction becomes a factor of any other fraction.
So the term "factor" totally loses meaning in this context.

Fortunately in this case, it is still possible to figure out the answer the creator of the problem wants you to give.

Sessions on Pre-Calc and GRE prep

A good tutoring day

Yesterday I had two sessions. The first one was to prepare an entrance placement test where the student wants to qualify for a particular Calculus course, so he is being tested on Pre-calculus. The second session was with two students who are preparing for the GRE. Both tutoring sessions were early in the afternoon at the same Starbucks.
In the pre-calculus session we covered a wide variety of topics. Every problem in the study guide was about a different subject. For example, we saw inequalities with absolute value, exponents, logarithms, function evaluation, subtraction of algebraic fractions, factoring algebraic expression and so forth. The difficulty level was not hard, and the student understood all the explanations, even when he had not taken math in about ten years, since high school.
In the GRE prep session we focused on Geometry problems, working out of the ETS book. We reviewed basic properties and formulas for parallel lines, polygons, angles, triangles, special triangles, the Pythagorean formula, square roots, perimeter, area, circumference, volume, and surface area.
Both sessions had a nice flow regarding the problems. We did many different exercises and we did not get stuck at any single problem. Yesterday was a good tutoring day.

Sessions in Trigonometry, GMAT and Pre-Calc

A busy tutoring day

Last Saturday I had three students. The first session was about graphing sine and cosine functions, high school trig. The second one was a GMAT prep session, and the third one was about circles and parabolas, pre-calculus.
During the first session I was partially awake, not very alert, so I had to do the problems myself, kind of slowly, organizing the data in a table format, just to have it all in front of me and to be able to see what’s going on with the series of numbers. I personally like to solve the problems this way, but I don’t like having to do it like that during a tutoring session. The reason being the students understand the reasoning and the results but they sometimes end up with this expression on their face, like saying “How am I going to do that by myself when I am all alone?” This time it was pretty clear though (the hot chocolate helped me to wake up).
The GMAT prep session was the first one for the student I met last Saturday. It was one of those talks when they fully see what kind of test they face, and all the amount of work they will have to do, and for a moment they seriously consider quitting. This is a good sign. The GMAT is not a walk in the park by any means. The student is better advised to expect a heavy workload and to make some sacrifices in their schedule. We covered the inevitable “What for?” question. I am always straightforward about it; the test is just a hurdle for them to win admission over other applicants. Math is the cheapest filter applicable in a mass scale. Then we worked on a few problems covering sub-indexes, recursive formulas, Venn diagrams, and volume and area formulas. We scheduled a two-hour session for next week.
The third and last session last Saturday also lasted two hours. We had several problems where you are required to find the center and radius of a circle given three points on the circle. It was a little frustrating for my student to see how vastly different these problems can be in terms of difficulty level, computational detail, and time consumed. A set of points forming a right triangle, with horizontal and vertical legs of even length is like candy, but when the problem throws at you points with decimal coordinates, and fractional values for the slopes of the sides of the triangle, you can fill three pages and spend almost half and hour with just one problem. This can set the student in a state of panic, you know? Just realizing how long and tedious it can get sometimes may be alarming for many. Some books are like that, with problem sets that escalate quickly in the difficulty scale. At least my student ended up with the idea that it may be hard but is not impossible to work these problems out.
After the three sessions I was fried, and hungry. I went to eat a tasty bread bowl of soup at the Quizno’s Sub store in the Renaissance Town Center, off of Nobel Drive.

Moving over to Blogger

My previous blog host went off line. I think Blogger.com will offer a much longer lasting service. Anyway, I am re-posting here the backup copies of my old posts, so I apologize for the inconvenience if you already saw this material. I expect to catch up pretty soon.

Negative Zero

Yes, it is a real number.

Every once in a while, I find students who show surprise or disbelief when they first encounter the concept of -0 (negative zero).
Let's say they are solving some equation, and close to the end it reads like x = - (a - b), where a and b represent two numbers known to be equal by virtue of the conditions set at the beginning of the problem.
So, in this example, the next step would be to write x = - 0.
My observation here is that, some students in this situation freeze, turn their head towards me, with a strange look in their face, and go: "There is no negative zero, is there?"
Usually I reply: "Why not?"
And they go: "But ..., what is it?"
I say: "It's zero."
Then they say: "Oh! Really? Just that, zero? Are they the same?"
And I say: "Yes, they are the same thing."
And they go: "O.K."
They seem to suddenly realize that the concept makes sense and it's not really that big of a deal.
I mean, what else could it be? What else could negative zero be if it wasn't equal to zero? There is no other option.
Actually, being equal to its own negative, is a defining feature that uniquely identifies zero.
Zero is the only number equal to its own negative. If you find any number x for which x = - x holds true, then you know x must be zero.
However, the momentary puzzlement, surprise and disbelief some students show when confronting this concept for the first time, is quite natural.
Remember it took centuries for Western civilization to come in contact with the concept of zero, and to fully adopt it as part of the number family. At first it was not considered a "true" number, but only an artificial placeholder used in the representation of "true" numbers.
Not only zero had difficulties being accepted as a number, but also the number One went through a period in Greek history when it was considered more like a philosophical, psychological, or even a religious concept, not a plain mathematical entity.

Another Important Key to Problem-Solving

Write down ~everything~

Very often I see students struggling with confusion when they try to solve a word problem all in their head.
It is so common, it's amazing. First of all, nobody says you have to produce the answer by just looking at the problem.
This is what happens, they read the problem, they understand the first sentence, or the first few sentences, and they are already asking themselves: "How do I solve this?" "What do I do with these numbers?" "What formula do I apply?" "What operation am I supposed to perform?"
They are obsessed with the idea of taking action steps. This is the first obstacle.
As soon as they come up with an idea about what to do, they start doing it, they start performing the operations, all in their head.
Then they get a partial result, and they immediately jump with that result into the next operation, without writing down anything. It's just unbelievable!
When I see them doing this, I tell them: "You are using your mind as a calculator and as a piece of paper at the same time. Don't waste energy like that."
For the average person, the mind can be much more effective as a calculator than as a piece of paper. The short-term memory that stores numerical results from previous calculations is very volatile.
When you try to use your head to do the operations and to remember the results at the same time, you are headed for trouble and confusion.
Let's say you make a mistake. If you write down all the steps of your calculation, and don't erase anything, you are much more likely to catch your own mistake when you go back and check your steps.
If you don't write down anything, you won't even remember what operations you performed, let alone catch a mistake.
So, write down everything, not only the partial results from each operation, but the whole calculation.
Write down not only the calculations you perform, but the ideas that made you perform those calculations.
Write down everything, every single step, all of it, your ideas, your examples, the formulas you are going to use, everything.
You will be amazed how easy the process becomes when you create this habit of writing down everything as soon as you think of it, and not erasing anything.
Even with mistakes, in the end it works better just to mark them with a red circle, and rewrite the correct expression somewhere else in the page, instead of erasing them. Many times mistakes are useful for reference.
As a general rule, the more you write, the better. The more you write, the less stress you put on your mind and the easier the process becomes.

One Key to Problem-Solving

Not "How To?" but "What do we have here?"

I always emphasize this to my students. When first facing a math problem, especially a word problem, do not try to get the answer right away. This is an unrealistic expectation. The answer will come as the result of properly developing all the relevant, detailed information contained in the problem. Pay attention to the wording. Create a clear mental image of the situation the problem is describing. Sketch a graph, a table, or a drawing to represent the situation. Avoid jumping into pre-packaged, memorized formulas after having read the problem only once.

Moving the Decimal Point

When you have a fraction, meaning you are dividing one number by another, you can always move the decimal point in both numbers, provided you move it the same number of places in each number (both numerator, and denominator), and in the same direction.

Now, when you are multiplying two numbers, you can always move the decimal point in both of them, provided you move it the same number of places in each factor, and in the opposite direction.

These two manipulations can significantly simplify the given operation.