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CALCULATORS & “NUMBER SENSE”


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.

 

 

 

 

 

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Handy Shortcut for Finding the LCD


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

“Blah, blah blah … and then, find the LCD for the two fractions.”

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

Math Cafe

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!


 

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.
 And … if you want to learn why this shortcut works, see my explanation at the end of the blog.

This shortcut lets you mentally do problems like the following three problems. That means you can do such problems in your mind rather than having to work them out on paper.

     a)  12 / √3 

     b)  10 / √2

     c)  22 / √5

Here are three terms I’ll use in explaining this shortcut.

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

     12  is  the dividend,

     3  is  the number under the radical,

     √3  is  the radical.

The Shortcut, Used for  12 / √3:

  1.  Divide the dividend by the number under the radical.
    In this case, 12 / 3 = 4. 
  2. Take the answer you got, 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 got, 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 the dividend by the number under the radical.
    In this case,  22 divided by 5 = 22/5  (Yup, sometimes you wind up with a fraction or with a decimal; that’s why I’m giving an example like this.)
  2. Take the answer you got, 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 SHORTCUT WORKS:

The shortcut works because we rationalize the denominator of a fraction whenever the denominator contains a radical. Here’s the shortcut 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 shortcut 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.

 

 

 

 

 

 

Algebra Mistake #5: Combining a Positive and a Negative Number


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: Combining Negative Numbers


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.

 

 

 

 

 

 

 

Factor Quadratic Trinomials, Part 2


This video shows the fastest and easiest way I know of for factoring quadratic trinomials. Give it a watch and see if you agree.