Where math comes alive


Suppose you need to find the GCF of three or more numbers, and you’d really prefer to avoid prime factorizing. Is there a way? Sure there is … here’s how.

 

High-Octane Boost for Math

High-Octane Boost for Math Ed

Example:  Find the GCF for  18, 42 and 96

Step 1)  Write the numbers down from left to right, like this:

………. 18     42     96

[FYI, the periods: …. are there just to indent the numbers. They have no mathematical meaning.]

Step 2)  Find any number that goes into all three numbers. You don’t need to choose the largest such number. Suppose we use the number 2. Write that number to the left of the three numbers. Then divide all three numbers by 2 and write the results below the numbers like this:

2    |  18     42     96
……..  9     21     48

Step 3)  Find another number that goes into all three remaining numbers. It could be the same number. If it is, use that. If not, use any other number that goes into the remaining numbers. In this example, 3 goes into all of them. So write down the 3 to the left and once again show the results of dividing, like this:

2    |  18     42     96
3    |    9     21     48
……… 3      7      16

Step 4)  You’ll eventually reach a stage at which there’s no other number that goes into all of the remaining numbers. Once at that stage, just multiply the numbers in the far-left column, the numbers you pulled out. In this case, those are the numbers:  2 and 3. Just multiply those numbers together, and that’s the GCF. So in this example, the GCF is 2 x 3 = 6, and that’s all there is to it.

Now try this yourself by doing these problems. Answers are below.

a)   18, 45, 108
b)   48, 80, 112
c)   32, 72, 112
d)   24, 60, 84, 132
e)   28,  42, 70, 126, 154

Answers:
a)   GCF =  9
b)   GCF =  16
c)   GCF =  8
d)   GCF =  12
e)   GCF =  14


Yesterday I named a property and claimed that it’s useful. Today I show how to put it to use.

In yesterday’s post I showed what I call the Opposite Differences Property, a property that  tells us that the answer we get when we subtract one number from another, a – b, is the numerical opposite of what we get when we switch the order of the numbers, b – a.

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Coffee, Pi and More

In other words, this property assures us that a – b  =  – (b – a)

A quick example of this with actual numbers:  7 – 4  =  – (4 – 7), since 7 – 4 = 3, and 4 – 7 = – 3.

The property tells us that the two results are mathematical opposites of each other:  3  =  – (– 3)

This is handy to know in general, but it’s especially useful when you’re simplifying complex algebraic fractions.

First, consider a simple algebraic fraction:  (x – y) / (y – x)

The Opposite Differences Property tells us that the numerator’s difference is the opposite of the denominator’s difference. That means that this fraction simplifies to – 1.

Not convinced? Try plugging in a couple of numbers: let x = 9 and let y = 2.

The fraction then gives us: (9 – 2) / (2 – 9), and that simplifies to 7 / – 7. But what is 7/– 7? It’s just – 1.

This holds true no matter what numbers we use; that’s what we mean by calling this a “property.” It works in general.

So we can confidently assert that, in general:  (x – y) / (y – x)  = – 1

[To be mathematically proper, I must add one caveat. This property holds in all situations but one. The variable x cannot equal y, for then we would have 0 / 0, which is “indeterminate.” That’s a fancy term, but it means that it’s value can change depending on the situation. Stuff we don’t need to worry about now.]

But back to the power of the property …  suppose we want to simplify the algebraic fraction:

(x + y) (x – y) / (y + x) (y – x)

Looks complicated, right? But we have the power to wrestle this to the ground.

First, using the Commutative Property, we know that the numerator’s (x + y) term  =  the denominator’s (y + x) term, so those two quantities simplify to 1/1.

And also, using our newly coined Opposite Differences Property, we see that the fraction made up of the other terms:
(x – y) / (y – x) = – 1

So (x + y)(x – y) / (y + x)(y – x)  simplifies to (1) (– 1)  = – 1

As another example, suppose you want to simplify this fraction:

abc (ab – b) /  bc (b – ab)

First, the b and c in the numerator cancel with the b and c in the denominator. So as far as these plain variables are concerned, we’re left with only an ‘a’ in the numerator.

Secondly, the numerator’s (ab – b) is the opposite difference of the denominator’s (b – ab), so that part of the fraction turns into – 1.

Putting the pieces together, we have ‘a’ times – 1, which simplifies to just:  – a

So that, – a, is the beautiful, simplified version of that big, ugly algebraic fraction.

As you do more and more algebraic fraction problems, you’ll find that the Opposite Differences Property comes in handy time and again. Learn it, use it, use it again.


Imagine that you’re looking so intently for a “pot of gold” that you don’t see a “bowl of diamonds” dancing in front of your eyes.

That’s my sense of what happens in algebra when we talk about the Commutative Property of Addition. Focused on it, we fail to see a very important related property. In fact, this other property is so neglected that it has no name! At least I have never heard a name for it. If anyone has heard a name for it, please let me know, and I’ll spread the word.

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As a reminder, the commutative property of addition tells us the obvious fact that when we add two numbers, the order in which we add them makes no difference. For example, 5 + 3 = 3 + 5, and 21 + 13 = 13 + 21.

Often instructors will ask students:  does this commutative property also work for subtraction? So students will start to consider whether or not 8 – 3 = 3 – 8. No, they conclude. The commutative property does not work for subtraction. End of story.

But gosh, that should not be the end of the story. It should be the beginning of a new story! Why? Because there’s something very interesting about 8 – 3 and 3 – 8. Sure, the two differences are not equal. But take a look:  8 – 3 = 5, and 3 – 8 = – 5. The differences are opposites. In other words, it is starting to look as if:

a – b = – (b – a)

Well, let’s try another such problem to see if this opposites pattern happens again. How about now we try another pair of integers, one positive (8), the other negative (– 3).

Is 8 – (– 3)  =  – [(–3) – 8]  (?)

Well, 8 – (– 3) = 11, and –[(–3) – 8] = – [– 11]  = 11

So this has worked again.

One more time, to test all possibilities of positives and negatives. Let’s see if this also works if we start with two negative numbers:  (– 8) and (– 3).

Does (– 8) – (– 3)  =  – [(– 3) – (– 8)]

Well, (– 8) – (– 3)  = – 8 + 3 = – 5

And (– 3) – (– 8)  =  – 3 + 8 = + 5

So yes, once again the differences you get are opposites.

So this means that if we widen our vision beyond the classic commutative property, there’s another gem of a property to be learned and used. This property says that:

a – b =  – (b – a)

Since I’ve never heard a name for this property, I’ll just give it a name. I’ll call it the Opposite Differences Property.

In my next post, I’ll share some info on some of the nice ways we can use this property.

Pi Day in 5 Digits


Happy Pi Day … in 5 digits!

 

Thanksgiving Apple, Pecan, Cherry, Caramel, Pumpkin Spice And Ch

O.K., so we all know it’s Pi Day, and we all know it’s a special Pi Day because it’s not just 3.14, it’s 3.1415 since this is the year 2015. But if you really want to get technical about it, how about adding a 5th digit, the digit 9, for 3.14159 …

How to celebrate this very special moment?

First just ask yourself how would you find .9 or 9/10 of a 24-hour day?

Well, just make a proportion:  9/10 = x/24

That gives you that x = 21.6

And that means that at 21.6 hours into this day we would be celebrating the moment of Pi Day with 5 digits. And what is that time? 21.6 hours from 12 midnight would be 9:36 pm. Where I live, that’s just about half an hour from now. So I just thought that everyone might like to know about this special moment.

 


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


Is there any point to doing something the long way when you can just as well do it in a shorter, much more efficient way? I say, Heck no! We can do things in the “Triple-F” way:  Fast, Fun & Friendly, and with deep understanding, to boot.

High-Octane Boost for Math

High-Octane Boost for Math Ed

So in that spirit, today I’ll get us started on quickly and effortlessly converting a linear equation from what’s called standard form (Ax + By = C) to what we know as the good-old slope-intercept form (y = mx + b).

To better grasp standard form, let’s replace its mysterious A, B, and C with actual numbers:  4 for A, 2 for B and 8 for C. That gives us the more typical looking equation of an actual line:  4x + 2y = 8. Do you recall seeing this kind of equation in your algebra text and class? Sure you do. You get this kind of equation in the chapter(s) on the coordinate plane and in other spots, too.

Now usually when books teach us how to convert from this “standard” form to slope-intercept form, they tell us to solve the equation for y. Of course that works, but it takes too darn long.

To understand the quicker way, let’s have a little fun with the standard form of the equation: Ax + By = C

We’re going to start with this standard form and solve that for y. And as you’ll see, we’ll learn some useful things from the result.

To kick things off, we start with Ax + By = C, and we subtract the Ax term from both sides. That leaves us with this equation:

By = – Ax + C

Now take this new equation and divide both sides by B. That gives us this little gem of an equation:

y = (– A/B) x + C/B

I’m going to call this the magic equation both to give us a way to refer to it and to show us what’s so useful about it.

The big insight is that this magic equation is actually, believe-it-or-not, in slope-intercept form; we just need to SEE it that way. Here’s how.

In slope-intercept form (y = mx + b), notice that the y variable is all by its lonesome on the left side. Do we have that in the magic equation? Yes we do. So … CHECK!

In slope-intercept form, there’s a value called m (aka, the slope) that is multiplying the x variable. Do we have something in the magic equation that’s multiplying the x variable? Why yes, and it happens to be
(–A/B). So do we have the slope showing in the magic equation? Yes, the slope is:  (–A/B). So … CHECK!

Finally, in slope-intercept form, there’s a constant (i.e., a number term, not a variable term) that appears after the mx term. So do we have a constant after the mx term in the magic equation? Yes, indeed. We have C/B. Note that in any actual linear equation, B and C will be actual numbers, not variables. So the value you get when you divide B by C (the quotient B/C), also must be a real number, just as surely as the real numbers 8 and 2 gives us the real number 4 when we divide 8 by 2.

So, to address the final question, do we have a b-value in the magic equation? Yes, it’s (C/B). C/B is the y-intercept, the real number we call b in slope-intercept form. So once again … CHECK.

So all in all, do we now have the equation in slope-intercept form? Yes, indeed. You just need to realize that
(–A/B) is the slope, and (C/B) is the y-intercept.

In my next post I’ll show you how you use these results to quickly transform the equation from standard form to slope-intercept form. It will be amazing.


My last post showed how you convert decimals to percents.

Now I’ll dare to do the obvious with a post on how to convert percents to decimals.

Since converting percents to decimals is the opposite of converting decimals to percents, it makes sense — does it not? — that we’d use the opposite procedure. And that is the case.

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Not only that, but we do the opposite procedure in the opposite order, too. How’s that for going totally opposite?

Since the final step in converting decimals to percents is tacking on a percent symbol (%), the very first step in converting percents to decimals is taking off that symbol.

And since the first step in converting decimals to percents is nudging the decimal point two places to the right, the last step in converting percents to decimals is pushing the decimal point two places to the left.

Let’s take a look at the process with this example.

Problem:  convert 73.2% to a decimal.

Step 1:  Take off the percent symbol. 73.2% changes to just 73.2

Step 2:  Move the decimal point two places to the left. 73.2 changes to .732 That’s all there is to it. This tells us that 73.2% is the same as .732, or 7 hundred thirty-two thousandths.

If you don’t recall the steps that you’re turning around, here’s a quick way to remember the process of converting percents to decimals. As I said in my last post, we can make use of alphabetical order, setting up the words for decimal and percent, in order, like this:

D-Decimal                              P-Percent

Then we draw an arrow showing that we’re converting from percent form to decimal form. The arrow shows the direction of the conversion: percent to decimal.

D-Decimal    <————–    P-Percent

This arrow points to the left, and that tells us that we move the decimal point to the left when we convert a percent to a decimal.

Let’s look at the process again, this time focusing now on how we use the arrow’s direction to help us.

Problem:  Convert  4.782% to a decimal.

Step 1:  Rip off the percent symbol. 4.782% changes to 4.782

Step 2:  Give the decimal point two shoves in the arrow’s direction. Since a percent to decimal conversion makes the arrow point left, we shove the decimal point two spaces to the left.  4.782 changes to .04782

This tells us that .04782 is the same as 4.782%

Note: if there are no digits showing to the left, we’re free to add 0s on the left side of the leftmost digit to create a place where the decimal point lands, after being shoved to the left.

In the last example, we had to tack a 0 on the left of 4.782 — making it 04.782, to get a digit (0) to the left of which we placed the final decimal point. Be confident that you can write as many 0 digits as you need to the left of a number’s leftmost digit. For example, it is just fine (though admittedly strange) to write 4.3 as 0004.3. You’d do this weird maneuver if you need that many zeros to the left of the 4. This occurs in converting numbers to scientific notation, for example.

So, now that you know the process, try your hand at converting the following percents to decimals: (Answers at the bottom of this post.)

a)  38%                                                                                                                           b)  19.3%                                                                                                                         c)   4.2%                                                                                                                         d)  175%                                                                                                                         e)  398.6%
f)  2,400%

Answers to the problems in the last post, converting decimals to percents:

a)  8590%                                                                                                                     b)  416.2%                                                                                                                     c)  20873.5%
d)  4.7%
e)  207,465%
f)  28.3%
g)  .569%

Answers to problems in this post:

a)  38%        =  .38
b)  19.3%     =  .193
c)   4.2%      =  .042
d)  175%      =  1.75
e)  398.6%   =  3.986
f)  2,400%    =  24.0, or just 24

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