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.