Kiss those Math Headaches GOODBYE!

Archive for the ‘Mental Math’ Category

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.

 

 

 

 

 

Advertisements

Find the GCF for 3 or More Numbers


Find the GCF, your teacher says … not just for 2 numbers, but for 5 of them.

And yes, you need to do it by prime factorizing.

Can’t you just hear the students’ groans?!

But what if there were a way to do this without prime factorizing? Could it really be?

Yes!

What I’m about to teach you is a technique that lets you find the GCF of as many numbers as you wish, and with much greater ease than the old factoring technique. (by the way, I don’t really hate the factoring technique … it actually teaches you a lot about numbers … but it can get annoying!).

So why don’t they teach this new way in school? No idea. But let’s just focus on the technique because once you do, you’ll be so much faster at finding the GCF …  you’ll be amazing your friends and your teacher, too!

So just kick back, watch the video — and learn …. then do the practice problems at the end of the video, to become a whiz! And remember, if you ever want extra help in the form of tutoring, I’m available — worldwide — thanks to the power of online videoconferencing.

Enjoy!

— Josh

 

 

 

Conquering Proportions, Part 2


In my first “Conquering Proportions” post, I showed how to save time by canceling terms horizontally as well as vertically. In this post you’ll learn how to save even more time with another shortcut. Let’s look at an example to refresh our memory.

Given a proportion such as this:

15   =   5  
 a         3

most people would do the traditional “cross-multiplying” step, to get:

5 x a = 15 x 3  (the x here is a true times sign; that’s why I’m using ‘a‘ as the variable, not ‘x.’)

If you follow the usual steps, the next thing would be to ÷ both sides by 5, to get:

a  =  (15 x 3) ÷ 5

But let’s look more closely at this answer expression:  (15 x 3) ÷ 5

We can conceptualize this expression better if we think of the original proportion:

15   =  5   
 a        3

as containing two DIAGONALS.

One diagonal holds the 15 and the 3; the other diagonal holds the ‘a’ and the 5.

Let’s call the diagonal with the ‘a’ the ‘first diagonal.’ And since ‘5’ accompanies ‘a’ in that diagonal, we’ll call 5 the “variable’s partner.”

We’ll call the other diagonal just that, the “other diagonal.”

Now I know you’re getting ‘antsy’ for the shortcut, so just know it’s right around “the bend.”

Using our new terms, we can better understand the expression we got up above:

a = (15 x 3) ÷ 5

The (15 x 3) is the product (result of multiplication) of the “other diagonal,”
and ‘5’ is the “variable’s partner.

So the answer,

                                      (15 x 3)                     ÷              5

is simply (and here’s the shortcut):

         (product of other diagonal) ÷ by  (“variable’s partner.”)

We’ll call this the Proportion Shortcut Formula, or the PSF, for short.

The PSF saves a BIG STEP; using it, we no longer need to write out the cross-multiplication product the usual way, as:

5 x a = 15 x 3

Instead, using the PSF, we can go straight from the proportion to an expression for ‘a‘:

a  =  (15 x 3) ÷ 5

Let’s see how the PSF works in another proportion, such as:

 9    =   45  
13         a

What’s the “variable’s partner”?  9.
What’s in the “other diagonal”? 13 and 45.

So using PSF, the answer is this:

a  =  (13 x 45) ÷ 9

This simplifies to 65, of course. Isn’t it nice not to have to “cross-multiply” any more?

Another nice thing: the PSF works no matter where the variable is located in the original proportion. All you need to do is identify the “variable’s partner,” and the “other diagonal,” and then you’re all good go with the PSF.

Try a few of these to see how easy and convenient the PSF makes it to solve proportions.

PROBLEMS:

1)   a   =      15  
     12          36

2)   18   =    a  
      24         4

3)   21   =   75  
      14          a

ANSWERS (using the PSF first):

1)   a  =  (12 x 15) ÷ 36
  a  =  5

2)   a  =  (18 x 4) ÷ 24
      a  =  3

3)   a  =  (14 x 75) ÷ 21
      a  =  50

Fraction Hack #2: The Size of the Smaller Number


I received an interesting question from alert reader Ivasallay a couple of days ago … about fractions.

Responding to my post about the fraction “hack” of using the gap between fraction numbers, Ivasallay wrote: “What if the numerator is smaller than the gap?”

High-Octane Boost for Math

High-Octane Boost for Math Ed

Good question, and thanks for sharing it. My answer: Yes, the numerator could be smaller than the gap, and if it is, that can help us simplify fractions, too.

Now we could have a fraction like 15/6, in which the lesser of the two numbers is the denominator, so to keep our discussion general I’m going to talk, not about the numerator, but rather about the “smaller fraction number,” whether numerator or denominator.

The way this matters is as follows: like the gap, the smaller fraction number provides an upper limit, a greatest possible value, for the GCF of the fraction’s two numbers. So if the fraction is 12/90 (smaller number being 12), that means that the GCF can be no larger than 12. If the fraction is 3/1011, with lesser number 3, the GCF can be no larger than 3.

The reason should be obvious, and when I say this I really mean it. Take the fraction 6/792, for example. Could a number larger than 6 go into both 6 and 792? Well there may be a number larger than 6 that goes into 792 evenly, but nothing larger than 6 can go into 6 itself, right? A large peg can’t go through a tiny hole, right? So there you go. Nothing larger than 6 can go into both 6 AND 792. QED.

So what does this mean for you, the math student, or parent of a math student, or the teacher of math students? … I means you want to keep in mind that in actuality two different numbers will help you nail down the size of the GCF. One is the gap between the fraction numbers, and the other is our “new friend,” the smaller of the two fraction numbers.

And here’s another … hack fact. (Whenever I say that, you know we’re heading into ‘nerd-land,’ right?) For both limiting numbers, the gap and the smaller fraction number, the only numbers that can possibly go into both fraction numbers are the FACTORS of those limiting numbers. So for example, if your fraction is 6/50, with the smaller number of 6, the only numbers that can possibly go into 6 and 50 are the factors of 6: i.e., 6, 3, or 2.

A nice rule of thumb:  see which is smaller, the gap or the smaller fraction number. Then use that smaller number as your largest possible GCF. To nail this down, let’s do two example problems.

Example 1:  8/44. What’s smaller? 8 or the gap, 36. Obviously 8! So use 8. Test the factors of 8, which are 8, 4, 2. Notice that 8 doesn’t go into both 8 and 44. But 4 does, so 4 is the GCF, and using 4, the fraction simplifies down to 2/11.

Example 2:  22/36. What’s smaller? 22 or the gap, 14. Here the gap is smaller. So test the gap’s factors: 14, 7, 2. 14 doesn’t go into 22 and 36; nor does 7. But 2 does. So 2 is the GCF, and using 2, the fraction simplifies to 11/18.

Time for you all to try your hands at this fun practice, which catapults your “number sense” to new heights.

For each problem, 1) identify the fraction’s smaller number and the gap. 2) Say which of those two numbers is smaller. 3) Using that number’s factors, find the GCF. 4) Finally, using the GCF, simplify the fraction. Answers follow.

SIMPLIFY THE FRACTIONS:

a)   8/42

b)  12/20

c)  36/60

d)  18/96

e)  21/91

ANSWERS:

a)   8/42:  1)  smaller # = 8; gap = 34.  2)  8 < 34. 3)  GCF = 2. 4)  4/21

b)  12/20:  1)  smaller # = 12; gap = 8.  2)  8 < 12. 3)  GCF = 4. 4)  3/5

c)  36/60:  1)  smaller # = 36; gap = 24.  2)  24 < 36. 3)  GCF = 12. 4)  3/5

d)  18/96:  1)  smaller # = 18; gap = 78.  2)  18 < 78. 3)  GCF = 6. 4)  3/16

e)  21/91:  1)  smaller # = 21; gap = 70.  2)  21 < 70. 3)  GCF = 7. 4)  3/13

Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. And FYI:  the 2nd Edition of the Survival Guide was just released in March, so get it while it’s hot off the press! If you’d like to get tutored by Josh, you can. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada in a wide range of subjects. They also prep students for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores soar, shoot an email to Josh, sending it to: josh@SingingTurtle.com  We’ll keep an eye out for your email, and in our office, our tutoring is always ON … except on Saturdays.

How to Convert a Linear Equation from Standard Form to Slope-Intercept Form


Suppose you’re given a linear equation in standard form and you need to convert it to slope-intercept form. You’ll be amazed how fast you can do this, if you know the “trick” I’m showing you here.

High-Octane Boost for Math

High-Octane Boost for Math Ed

First, let’s review the key info from my post: How to Transform from Standard Form to Slope-Intercept Form.

That post shows how to pull out the the slope and y-intercept from a linear equation in standard form.

Remember that standard form is Ax + By = C, where A, B, and C are constants (numbers).

Given the equation in standard form, take note of the  values of A, B, and C.
For example, in the equation, – 12x + 3y = – 9,   A = – 12, B = 3, and C = – 9

Then, based on the info in yesterday’s post, we get the slope by making the fraction:  – A/B.

And we get the y-intercept by making the fraction:  C/B

New info for today: once you have the slope and y-intercept, just plug them in for m and b in the general slope-intercept equation:  y = mx + b

Here’s the whole process, demonstrated for two examples.

Ex. 1:  Given, 8x + 2y = 12, A = 8 B = 2, C = 12.
So the slope = – A/B = – 8/2 = – 4. y-intercept =  12/2 = 6
So the slope-intercept form is this:  y = – 4x + 6

Ex. 2:  Given, – 5x + 3y = – 9, A = – 5, B = 3, C = – 9.
So the slope = – A/B = 5/3,  y-intercept =  – 9/3 = – 3
So the slope-intercept form is this:  y = 5/3x  – 3

Now “give it a roll.” Once you get the hang of this, try the process without writing down a single thing. You might get a pleasant jolt of power when you see that you can do this conversion in your head.

Conversion Problems (Answers at bottom of post)

1)   – 4x + 2y  =  14

2)    20x – 5y  =  – 15

3)  – 21x – 7x  =  35

4)  – 18x  + 6y  =  – 21

5)    17x + 11y  =  22

6)    – 7x + 11y  =  – 44

7)    36x – 13y  =  – 52

8)  – 8x  + 5y  =  – 17

Answers

1)   y  =  2x + 7

2)   y  =  4x + 3

3)  y  =  – 3x – 5

4)  y  =  3x – 7/2

5)   y  =  – 17/11x + 2

6)    y  =  7/11x – 4

7)    y  =  36/13x  +  4

8)  y  =  8/5x –  17/5

Find the LCM (aka LCD) in Two Easy Steps


This is really the “Week of the LCM” for me.

Just as I was finishing my last post, on a new way to find the LCM for a pair of numbers, I discovered another way to do the same thing.

Coffee, Pi and More

Coffee, Pi and More

I was looking at the problems at the end of my last post, these problems:

b)   15 and 20;  LCM  =  60

c)   18 and 20;  LCM  =  180

d)   24 and 28;  LCM  =  168, ….

… when I noticed something.

(more…)

Find the LCM in a way that makes sense! (Part 2)


In yesterday’s post on the LCM, I wrote about 375 pages on the topic, and then I said that I left out an idea. Hahaha, you probably thought. Very funny, Josh.

But never fear. I am not going to write another 375 pages on the topic.

What I do need to bring to your attention, though, is that there are two LCM situations that I did not take into account yesterday. So to present a complete picture, I need to explain (for those who have not already figured this out by themselves) how to use my new technique in those two situations.

Coffee, Pi and More

Coffee, Pi and More

You will notice that in my write-up yesterday — and in the practice problems I provided — the gap always divided evenly into the smaller number. How convenient, right? In the first example, we had a gap of 3 dividing into 12; in the next, a gap of 4 going into 20. Of course this does not always happen. Consider a situation in which we want to find the LCM for 10 and 16. The gap of 6 (16 – 10 = 6) does NOT divide evenly into the smaller number, 10. So what would we do here? (more…)