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How to Understand and then Forever Memorize the Midpoint Formula


In algebra we have many formulas to learn. But one problem is that those formulas are often hard to memorize. They are written with variables, and the variables frequently have subscripts, and the truth is that a lot of us don’t really understand what the formulas are saying or how they work. So of course that makes formulas difficult to memorize.

“Friendly” Formulas make it easier to learn and memorize algebraic formulas.

Enter the concept of “friendly formulas.” Friendly formulas are the very same formulas but written in a way that you can understand and therefore memorize much more easily. It’s an idea I have come up with through my many years of algebra tutoring, and idea is included in my Algebra Survival Guide, available through Amazon.com

In this post I describe the “friendly formula” for the midpoint formula.

So as a refresher, what is the midpoint formula all about?

Basically, it lets you find the midpoint of any line segment on the coordinate plane. Think of it this way. There’s some line segment on the coordinate plane called segment AB. That means that it has an endpoint at point A, another at point B. We are given the coordinate of points A and B. We want to find the coordinates of the point right in the middle of points A and B.

Now let’s make this idea easy. Suppose we focus only on the x-coordinates. Suppose the x-coordinate of point A is 2, and the x-coordinate of point B is 6. Ask yourself: what x-coordinate is perfectly in the middle of coordinates 2 and 6? It’s just like asking: what number is right in the middle of 2 and 6 on the number line? Well, wouldn’t that be 4, since 4 is two more than 2 and two less than 6? And indeed it is 4.

But notice that there’s another way to get 4, given the coordinates 2 and 6. We also could have just added 2 and 6 to get 8, and then divided 8 by 2, since 8 ÷ 2 = 4. In other words, we could have TAKEN the AVERAGE of the two x-coordinates, since taking an average of two numbers is adding them and dividing by two.

Could the midpoint formula actually be as easy as taking averages?!

Before we say yes, let’s test this idea for more complicated situations. We just saw that it works when both coordinates are positive. But suppose one coordinate is positive, the other negative. Let’s let one coordinate be
– 2, while the other is + 4. What number is right between those two coordinates on the number line? Well, the numbers are 6 apart, right? And half of 6 is 3, so we could just add 3 to – 2, and get + 1 as the point in between them. And we see that + 1 is three away from both – 2 and 4. But could we also get + 1 by averaging -2 and 4? Let’s try:
(- 2 + 4) / 2 = 2 / 2 = + 1. Averaging works again.

And finally, what about the case where both coordinates are negative? Suppose one coordinate is – 2, the other – 8. What number is right between those two numbers on the number line? Well, these numbers are also 6 apart, right? And half of 6 is 3, so we could just add 3 to – 8, and get – 5 as the middle. And we see that – 5 is three away from both – 8 and – 2. But can we also get – 5 by averaging – 8 and – 2? Let’s try: (- 8 + – 2) / 2 = – 10 / 2 = -5. Averaging worked here too!

Since the averaging process works for all three cases, this approach does works always, and in fact it is how the midpoint formula works.

The midpoint formula basically just averages the x-coordinates to get the x-coordinate of the midpoint. Then it averages the y-coordinates to get the y-coordinate of the midpoint.

So here is the “friendly formula” for the midpoint of any segment on the coordinate plane: Given a segment whose x- and y-coordinates are known,

MIDPOINT = (AVERAGE of x-coordinates, AVERAGE of y-coordinates)

And that’s all you have to memorize!

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Category:

Algebra, Math Shortcut, Memorization, Shortcut, Uncategorized, Using "Friendly Numbers"

How to quickly find the y-intercept (b-value) of a line


Of course there’s a standard way to find the y-intercept of any line, and there’s nothing wrong with using that approach.

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But the method I’ll present here is a bit faster and therefore easer. And hey, if we can save time when doing math, it’s worth it … right?

So first let’s recall that the y-intercept of any function is the y-value of the function when the x-value = 0. That’s because the y-intercept is the y-value where the function crosses or touches the old, vertical y-axis, and of course all along the y-axis the x-value is always 0 (zero).

So the standard slope-intercept formula is y = mx + b. In a problem asking for the y-intercept, you’ll be given one point that the line passes through (that point’s coordinates will provide you with an x-value and a y-value), and you will also be told the slope of the line (the line’s m-value).
So then, to get the b-value, which is the value of the y-intercept, you just grab your y = mx + b equation (dust it off if you haven’t used it in a while), and plug in the three value you’ve been given: those for x, y and m. Then you solve the equation for the one variable that’s left: b, the value of the y-intercept.

Let’s look at an example: a line with a slope of 2 passes through the point (3, 10). What is this line’s y-intercept.

Now, according to the problem, the m-value = 2, the x-value = 3, and the y=value = 10. We just take these values and plug them into the equation:
y = mx + b, like this:

10 = (2)(3) + b

After doing these plug-ins, you just solve the equation for b, finding that
b = 4. That means that the y-intercept of the line = 4.

Now let’s see how you can do the same problem, but a little bit faster.
To do so, we first need to play around with the y = mx + b equation by subtracting the mx-term from both sides, like this:

y = mx + b [Standard equation.]
– mx = – mx [Subtracting mx from both sides.]
y – mx = b [Result after subtracting.]
b = y – mx [Result after flipping left & right sides
of the equation above.]

Aha! Look at that final, beautiful equation. This equation has b isolated on the left-hand side. So now if we want to solve for b, all we do is plug in the x, y and m values into the right-hand side of the equation and simplify the value, and the value we get will be the b-value.

For the problem we just solved, with x = 3, y = 10, m = 2, watch how easy it is to solve:

b = y – mx
b = 10 – (2)(3)
b = 10 – 6
b = 4

So notice that this technique, just like the first technique, reveals that the
y-intercept of the line is 4, or (0, 4). The techniques agree, they just get to the same end in slightly different ways.

Notice that with the second, quicker technique, you don’t need to add or subtract any terms. And that’s a key reason that this technique is faster and easier to use than the standard method. So try it out and stick with it if you like it.



Category:

Algebra, Algebra Tricks, Coordinate Plane, Formulas, Linear Equations, Math Shortcut, Solving Equations, Uncategorized, Y-Intercept

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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:  22 √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.

 

 

 

 

 

 

Category:

Algebra, Math Shortcut, Radicals, Shortcut, Uncategorized

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How to Factor Trinomials with Understanding!


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

Category:

Algebra, Factoring, Math Shortcut, Practice Problems, Quadratic Trinomials, Simplifying Expressions, Uncategorized

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Math Trick: 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

 

 

 

Category:

GCF / GPGCF, Math Shortcut, Math Theories, Mental Math, Number Sense, Number Theory, Uncategorized, Videos

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How to find the GCF of 3+ Numbers — FAST … no prime factorizing


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.

 

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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

Category:

Algebra, GCF / GPGCF, Math Shortcut, Number Sense, Number Theory, Shortcut, Uncategorized

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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.

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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

Category:

Algebra, Coordinate Plane, Math Shortcut, Mental Math, Shortcut

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