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Factoring Trick: How to Flawlessly Factor any “Difference of Two Squares” Binomial


If you’re staring at two terms you need to factor, but feel like a deer looking at the headlights of an oncoming semi, here’s a way to leap to safety!

It’s called the “Difference of Two Squares” trick.

High-Octane Boost for Math

It requires four simple steps.

  1. Figure out if each of the terms is a “perfect square.”
  2. If so, take the square root of each term.
  3. Put each square root in its proper place inside two (    ).
  4. Put a + sign inside the first (   ), and put a – sign inside the second (   ).

Let’s do an easy example. Suppose the terms you’re looking at are these:
x^2  – 9

Let’s go through the 4 steps together.

  1. Figure out if each term is a “perfect square.”

    So, what does it mean for a number or term to be a “perfect square”?  It means that you get the number or term by multiplying a number or term by itself. For example, 16 is a perfect square because you can get 16 by multiplying 4 by itself:  4 x 4 = 16.

    So when we look at our two terms, x^2 and 9, we notice that both
    are perfect squares.
    9 is just 3 times 3.
    And in the same way, x^2 is just x times x.

  2.  Take the square root of each term.
    The square root of x^2 is just x.
    And the square root of 9 is just 3.

  3. Put each square root in the proper place inside two sets of (    ).
    We put the square root of the term that was positive first, and the square root of the term that was negative second.Since the x^2 was the positive term, we put its square root, x, first inside each
    (   ).  So far, that gives us:  (x    ) (x     )

    Since the 9 was the negative term because it had the negative sign in front of it: – 9, we put its square root, 3, second inside each (   ). So our (   )s now look like this:  (x   3) (x   3)

  4. Finally, we just need to put in signs that connect the terms inside
    the (    )s.

    That’s easy. We put a + sign inside one (    ), and we put a – sign
    inside the other (    ).
    I prefer to put the + inside the first (   ), but it really doesn’t matter.The final factored form, then, looks like this:  (x + 3) (x – 3)
    That’s all there is to it.

Now try these problems for practice.

           a)  x^2 – 16
           b)  x^2 – 100
           c)   x^2 – 121
           d)   x^4 –  16x^2
           e)   49x^8 – 144y^12

Answers:

           a)   x^2 – 16   =  (x + 4) (x – 4)
           b)  x^2 – 100  = (x + 10) (x – 10)
           c)   x^2 – 121  = (x + 11) ( x – 11)
           d)   x^4 –  16x^2  = (x^2 + 4x) (x^2 – 4x)
           e)   49x^8 – 144y^12  = (7x^4 + 12y^6)(7x^4 – 12y^6)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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How to Let Kids Use Calculators Without Ruining Their Ability to Think Numerically


Hi folks,

Now that summer has officially begun, I’m enjoying a certain distance from the heat of the school year, and that distance gives me a chance to reflect.

One set of ideas that my mind keeps poking around again and again is this:  a) the weakness in actual number sense among today’s elementary and secondary students,
b) the concomitant modern focus on teaching Number Sense during these school years, and c) the now-rampant overuse of calculators.

I find it interesting that Number Sense has become a “big important new topic” that math instructors are required to teach. I also find it interesting that the new focus on Number Sense has been growing steadily at the very same time that students in so many parts of our country have become more and more calculator dependent.

Could there be a connection?

Yes, undoubtedly! Back when I set up shop tutoring math, K-12, in 1990, Santa Fe (NM) Public School students were not permitted to use calculators willy-nilly. Because of that, our students were not calculator-dependent. Students were expected to know the truths of arithmetic forwards and backwards, and wouldn’t have dreamed of reaching for a calculator to find the value of something so simple as, say, 7 + 5, as happens routinely today. Yes, routinely! I should know; I’m a professional math tutor.

What’s more, I’d say that students in the 1990s generally understood concepts such as odd and even numbers, prime and composite numbers, how to prime factorize, how to find the GCF and the LCM, and the many other skills that are part of the “new area of math instruction we call Number Sense.

That’s because teachers used to require students to use their minds to work with numbers. Students used to grind out 7/18 + 5/12 by hand, not by pressing buttons. They used to figure out the LCM of 22 and 30 by using an algorithm rather than by tapping an app. They used to prime factorize numbers using the good old factor tree and simplify radicals by thinking rather than by pressing a sequence of buttons and scrolling through the numbers flashing across their LCDs.

You can probably see where I’m going with this. Today’s math students have become overly calculator dependent. That dependence on calculators, in turn, has made them deficient at the skills in the topic area we call Number Sense. And precisely because today’s students are so deficient at number sense, precisely because they have been allowed to become so dependent on their e-devices rather than on their mental devices, curriculum designers have devised this whole new area of math, Number Sense, that now gets taught as its own “thing” rather than being an integral thread of everyday math instruction. Number sense used to be something students developed naturally, by mentally working with numbers, day-in, day-out, using paper and pencil and mental math.

Lest I be called a Luddite, I’m not saying that calculators have no place in the math curriculum. But as a tutor who has helped students with math for some 27 years now, I can say with certainty that today’s students’ innate ability to work with numbers, play with numbers and calculate with numbers has been dulled and frankly allowed to atrophy because calculators have become an all-too-easy, all-too-available crutch.

In this way, math curricula and math educators who overly promote calculator usage have done a great disservice to students. The good news, though, is that  teachers could correct course without too much trouble.

Teachers could still allow students to use calculators, quite appropriately, for higher-order processes — such as graphing two functions to see where they intersect, and to see if the answer found that way comports with the answer attained by solving the systems simultaneously by hand — while at the same time disallowing calculator usage for arithmetic calculations.

I’d like to see teachers get their students back to basics in this way because, from my perspective, we’re raising a new generation of students, many of whom have little ability to calculate mentally and little understanding of how numbers work. As a result, these children (soon-to-be adults) are unnecessarily vulnerable.

They’re vulnerable because they cannot tell if they are receiving the correct change from a cashier. They’re vulnerable because they cannot tell if their car or home interest payment is correct. And they’re vulnerable in a larger sense because they lack the ability to easily think numerically, i.e., quantitatively. And when people lack the fundamental ability to think quantitatively, even having a calculator won’t save them in many situations. That’s because they might not even know what operation to do to find a solution in a real-world situation.

But in an even more direct and practical sense, the new calculator-dependent students are vulnerable because they have been set up to struggle mightily in their college math classes. That’s because nearly all U.S. colleges require students to take math tests without using calculators!

So I say let’s get back to basics, and let’s do it in a smart way. Let’s continue to let students use calculators for higher-order thinking skills, but let’s disallow calculators for ALL arithmetic so that students will be required to once again become strong in those critical fundamental skills and so that they will re-gain the natural form of Number Sense that is their right and their due.

 

 

 

 

 

How to Find the LCM for Two Fractions — FAST!


But what’s really annoying is that your teacher says it like it’s so easy:

“Blah, blah blah … and then, find the LCM (aka, the LCD) for the two fractions.”

Meanwhile, you’re sitting there, thinking, “Right, and uh, how do I do that, again?”

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Math Cafe, Open 24/7 = 3.42857 …

 

Well, it turns out that there’s a pretty easy way to find the LCD when you’re adding or subtracting fractions. In fact, there’s a secret, shortcut way that you probably won’t find anywhere else on the internet. But you will find it here.

All you need to do is follow few surefire steps. I’ll show you the steps right now through this example problem:

5/12 + 3/20

1st) State the challenge.
We need the LCD for our two denominators: 12 and 20.

2nd)  Make a proper fraction out of the two denominators. A proper fraction is just a “normal fraction” — smaller number on top, larger number on the bottom.
Fraction we make:  12/20

3rd) Simplify that fraction to lowest terms and then flip it (get its ‘reciprocal.’)
12/20 simplifies to 3/5. Reciprocal (“flip”) of 3/5 is 5/3.

4th) Multiply the original fraction by the reciprocal (the flipped fraction).
12/20 x 5/3 = 60/60

5th) You just found the LCD. The number that’s repeated in your answer is the LCD.
Fraction we got was 60/60. This means that 60 is the LCD of 12 & 20.

Now of course, once you have the LCD, you’ll multiply top and bottom of the original problem’s fractions to make their denominators equal to the LCD, equal to 60, in this example. So here you’d multiply the original fractions — 5/12 and 3/20 — so they have a denominator of 60.

One nice thing about this process. It shows you what you need to multiply the fractions by.

The 4th step shows that you multiply 12 by 5 to get 60. So that means  you will multiply 5/12 by 5/5:  5/12 x 5/5 =  25/60

The 4th step also shows that you multiply 20 by 3 to get 60. So that means  you will multiply 3/20 by 3/3:  3/20 x 3/3  = 9/60

To finish your fraction addition problem, you simply add those converted fractions:
25/60 + 9/60 = 34/60 = 17/30

 

 

 

 

 

How to Divide ANY Number by a Radical — Fast!!! (Math “hack” w/ full explanation)


 

Here’s a super-quick shortcut for  DIVIDING ANY NUMBER by a RADICAL. 

Note: I’m using this symbol () to mean square root.
So √5 means the square root of 5;  √b means the square root of b, etc.
 And … if you want to learn why this “hack” works, see my explanation at the end of the blog.

This “hack” lets you mentally do problems like the following three. That means you can do these problems in your head rather than on paper.

     a)  12 / √3 

     b)  10 / √2

     c)  22 / √5

Here are three terms I’ll use in explaining this “hack.”

In a problem like 12 divided by √3, which I write as:  12 / √3,

     12  is  the dividend,

     3  is  the number under the radical,

     √3  is  the radical.

The “Hack,” Used for  12 / √3:

  1.  Divide the dividend by the number under the radical.
    In this case, 12 / 3  =  4.
  2. Take the answer, 4, and multiply it by the radical.
    4 x √3  =  4√3

  3. Shake your head in amazement because that, right there, is the ANSWER!

Another Example:  10 / √2

  1.  Divide the dividend by the number under the radical.
    In this case:   10 / 2  =  5
  2. Take the answer you get, 5, and multiply it by the radical.
    5 x √2  =  5√2.  (Don’t forget to shake head in amazement!)

Third Example:  22 / √5

  1.  Divide dividend by number under the radical.
    In this case,  22 divided by 5 = 22/5  (Yep, sometimes you wind up with a fraction or a decimal; that’s why I’m giving an example like this.)
  2. Take the answer you get, 22/5, and multiply it by the radical.
    22/5 x √5 =  22/5 √5.  [Note: the √5 is in the numerator, not
    in the denominator. To make the location of this √5 clear, it’s best
    to write the answer:  2√5 / 5].


NOW TRY YOUR HAND by doing
these PRACTICE PROBLEMS:

a)   18 / √3  

b)   16 / √2  

c)   30 / √5  

d)   10 / √3  

e)   12 / √5

– – – – – – – – – – – – – – – – – –

ANSWERS:

a)   18 / √3  = 6√3

b)   16 / √2  = 8√2

c)   30 / √5  = 6√5

d)   10 / √3  = 10√3/3

e)   12 / √5  = 12√5/5

– – – – – – – – – – – – – – – – – –

WHY THE “HACK” WORKS:

It works because we rationalize the denominator of a fraction whenever the denominator contains a radical. Here’s the “hack” in general terms, with:

     a  =  the dividend,

     b  =  the number under the radical,

     √b  =  the radical.

a / √b

=   a
    √b

=   a     √b    =   a √b
    √b   √b            b

Notice: we started with:  a / √b.

And keeping things equal, we ended up with  a √b / b.

This shows that the “hack” works in general. So it works in all specific cases as well!

– – – – – – – – – – – – – – – – – –

Final note: the number under the radical is called the radicand. But that term is so close to the term radical that I thought it would be less confusing if I just called this the number under the radical. I hope you are not offended.

 

 

 

 

 

 

Algebra Mistake #5: How to Combine a Positive and a Negative Number without Confusion


So, you’d think that combining a positive number and a negative number would be a fairly straightforward thing, huh?

Well, unfortunately, a lot of students think it’s easy. They think it’s too easy. They think there’s one simple rule that guides them to the very same kind of answer every time. And that’s exactly where they get into trouble.

The truth is that combining a positive and a negative number is a fairly complicated operation, and the sign of the answer is dependent on a nmber of factors.

This video reveals a common mistake students make when tackling these problems. it also shows the correct way to approach these problems, using the analogy of having money and owing money to make everything make sense.

So take a look and see if this explanation doesn’t end the confusion once and for all.

And don’t forget: there are practice problems at the end of the video. Do those to make sure you’ve grasped the concept.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algebra Mistake #4: How to Combine Negative Numbers without Confusion


Here’s a common mistake, and a very understandable one, too. Students need to combine two negative numbers, and they, of course, wind up with an answer that’s positive. Why? Because, they’ll say — pointing out that you yourself have told them this —  “Two negatives make a positive!”

This video gets to the root of this common misunderstanding by helping students understand exactly when two negatives make a positive, and when they don’t.

 

Make sure you watch the whole video, as there are practice problems at the end, along with their answers.

 

 

 

 

 

 

 

 

 

“Algebra Survival” Program, v. 2.0, has just arrived!


The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.

 

Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.