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D  //             |    \ 11   44 9 9               by Aardv@rk               D
* ||  ____        | || |  1  444 999                                         *
G ||    || \    / | || |  1    4   9 issue #149 of "GwD:  The American Dream G
w  \\___//  \/\/  |____/ 111   4 999 with a Twist -- of Lime" * rel 05/05/05 w
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[This was written 10 years ago for my Algebra II class in high school.  The
teacher also taught remedial math, and I guess she was tired of it, so she
made us write these lame descriptions of how to do math problems to help her
out or something.  We had to do "writing assignments" in every class; it's all
about teaching to the standardized test that has a writing portion.  We were
supposed to write about how to do the four basic artithmetic operations with
both positive and negative integers.  It always struck me as odd that we were
supposed to explain negative numbers but not fractions/decimals, since both of
these concepts require understanding of one of the operations we were supposed
to explain.  How can you understand negative numbers if you don't know how to
subtract?  Seriously.  That teacher was dumb; much dumber than the kids that
were in HS and didn't know how to add.

Anyway, this got a good grade, despite its condescension. -Aardv@rk]

I hear you've been having some trouble with math.  Well, don't you worry about
that!  I'm going to teach you math backwards and forwards so you'll never have
trouble again.  Adding, subtracting, multiplying and dividing will be a piece
of cake if you follow my simple instructions.  If you really don't know this
stuff, you should probably quit using the computer and get to studying.

You must start with addition.  Addition is the simplest mathematical
operation and the easiest to learn.  It is simply putting two (or more) things
together.  For example, if you have two apples and I give you one more apple,
how many apples do you have?  There are three apples.  If you do not
understand this, count two of your fingers and then count one more.  (If you
do not know how to count, you are beyond help.  Also, if you do not have
fingers, well, umm, count beans or something.)  The number sentence (it is
called that because it is somewhat like a word sentence) for this is
"2 + 1 = 3" because you had two apples and I gave you one more.  What if you
have three oranges and I give you three more?  You have six oranges.  The
number sentence this time is "3 + 3 = 6".  Now, let's try something a little
bit harder.  If you have six peanuts and I give you 3 more, how many peanuts
do you have in all?  You have nine peanuts.  The number sentence this time is
"6 + 3 = 9".  Addition works the same way for larger numbers.  When you add
zero to a number, the number stays the same:  "5 + 0 = 5".  Addition is
commutative.  That means that "5 + 2 = 2 + 5 = 7".  Basically, being
commutative rocks nads.  It means that the numbers to be added can be added in
any order and give you the same result.

"What about these 'negative numbers' I have heard so much about but have no
idea what they are?" is probably what you are asking yourself right now.
Well, I am not supposed to tell you what negative numbers are, even though I
am supposed to tell you how to use them.  Don't worry, it does not make any
sense to me either.  Addition with negative numbers is as follows:
"5 + - 6 = -1" which is exactly the same thing as saying "5 - 6 = -1", turning
it into a subtraction problem from an addition problem.  Similarly,
"-5 + -6 = -11" is the same as saying "-5 - 6 = -11".  This has also turned it
into a subtraction problem.  See the next paragraph for help with subtraction.
If you think you understand, move on to the next paragraph.  If you feel that
you need a little bit more practice, try these examples or the problems
following the instructions on your own or ask your parents or teacher for
help.

Subtraction is a little bit more difficult than addition, but not much.  You
should think of subtraction as the opposite of addition.  Let's go back to the
apple example.  You now have three apples, right?  What if I take my apple
back?  I'm a mean bastard, huh?  How many apples do you have then?  If you
think you would have 2 two apples, you are correct.  The number sentence this
time is "3 - 1 = 2".  That may seem easy, but what will happen if I use a
problem you haven't seen before?  There are nine kids that you are friends
with in your class.  If two of them go to another class, then how many friends
are left in your class?  There are seven friends left in your class.  The
number sentence is "9 - 2 = 7".  How about just a number sentence without
things related to it?  Try this number sentence and fill in the blank:
"11 - 5 = __".  Did you put "6" in the blank?  You should have.  If you
didn't, they're all going to laugh at you.  Negative numbers are just one
small step beyond standard run of the mill everyday subtraction.  If you
subtract zero from a number, you get the same number:  "4 - 0 = 4".  Think of
negative numbers as zero minus a positive number.  For example, "0 - 4 = -4".
Adding a negative number is the same thing as subtracting a positive number:
"4 + -1 = 4 - 1 = 3".  What about subtracting negative numbers?  Subtracting
negative numbers is addition.  "5 - -2 = 5 + 2" because the "minus signs"
cancel each other out.  Think that the negative sign in front of a negative
number changes whatever sign (+ or -) to the other sign (- or +).  So,
"-1 - -2 = -1 + 2 = 1" and "5 + -3 = 5 - 3 = 2".  Negative numbers are tricky
until you get the hang of using them.  Subtraction is not commutative!  I
DON'T KNOW HOW MANY TIMES I'VE HAD TO EXPLAIN THIS PART.  YOU'D BETTER
REMEMBER IT!  SERIOUSLY.  Do you think you need to spend some more time on
subtraction, or can we move on to multiplication?  If you need some more help,
do not be ashamed to try some of the provided problems or ask an adult for
help.  That's what we're here for.  Sort of.

Multiplication is nothing but addition on a large scale.  If there are ten
classrooms and each classroom has 20 students in it, how many students are
there in all?  You could count or use the number sentence
              "20 + 20 + 20 + 20 + 20 + 20 + 20 + 20 + 20 + 20 = ___"
If you add all of these twenties the number you will get is 200.  However,
since you know that you are adding twenty ten times, you can say twenty times
ten equals two hundred or "20 X 10 = 200".  If you have seven pairs of socks,
how many socks do you have?  (There are two socks in a pair.)  Use the number
sentence "7 X 2 = __".  If you put 14 in the blank, good job.  If you
didn't, well, try again.  Remember that any number times one is that number:
"3 X 1 = 3".  Also, you must know that any number multiplied by zero is zero.
Using the classroom problem, if there are zero classrooms with twenty students
each, then how many classrooms students are there?  If you said zero, you are
right.  If you didn't, pay better attention.  This is shown by "20 X 0 = 0" or
"0 X 20 = 0".  Like addition, multiplication is commutative; you can put the
numbers to be multiplied in any order and the result will not change:
"3 X 4 = 4 X 3 = 12".  You are now ready to move on to multiplication by
negative numbers.  There are four things you need to know to multiply using
negative numbers:  1) a positive times a negative is always a negative; 2) a
negative times a positive is always a negative (which, given commutativity, is
the same thing as a positive times a negative, so I'm really just repeating
myself for no reason); 3) a negative times a negative is always a positive; 4)
any number times negative one (-1) switches signs but stays the same.  (It is
assumed that you know that a the product of the multiplication of two positive
numbers is always positive.)  To illustrate cases one and two:  "3 X -4 = -12"
and "-3 X 4 = -12".  That's not too hard, now is it?  Don't answer that.  Case
three, however, is very important.  Remember that when multiplying, negative
signs cancel, and you will never go wrong.  For example:  "-5 X -6 = 30" and
"-10 X -2 = -20".  Case four is also very important, though it is really
just putting the other three cases into a specific situation.  If you multiply
a positive number by negative one you get the "negative version" of that number
(put a negative sign in front of it):  "5 X -1 = -5" or "-1 X 5 = -5".
However, if you multiply a negative number by negative one, you get the
"positive version" of that number:  "-3 X -1 = 3".  Now you should see what I
mean by saying that the signs switch.  Multiplication is a little harder than
anything else we have covered.  Division is the next operation.  If you
understand how to multiply, division will be very easy for you.  If you don't,
well, too bad, because it's what I'm covering next, anyway.

Division is the last of the operations we need to discuss.  Just as
subtraction is the opposite of addition, division is the opposite of
multiplication.  Normally, the division symbol is something like this:  o
                                                                       ---
                                                                        o
but there is no key for that on my keyboard, so I am going to use the "/"
symbol instead, because it also means division.  So now on to actually
dividing!  Isn't this fun?  If you want to know how many threes there are in
nine, use "9 / 3 = _".  The answer to this problem is 3.  You can check this
by multiplying:  "3 X 3 = 9".  You have determined the correct answer.  If you
have 3 boxes of grapefruits and you know that there are 45 grapefruits in all,
how many grapefruits are there per box (assuming, of course, that the
grapefruits are evenly distributed...lacking this assumption, the correct
answer is "not enough information")?  You should divide forty-five by three:
"45 / 3 = 15".  There are fifteen grapefruits per box.  Division is not
commutative:  "9 / 3 <> 3 / 9".  (<> means "is not equal to" because the
"greater than" and "less than" symbols are used.  If a number is greater than
or less than another number, it does not equal that number.)  Division by
negative numbers is easy.  You just need to know:  1) if a negative number is
divided by a positive number, the result is negative; 2) if a positive number
is divided by a negative number, the result is negative; and 3) if a negative
number is divided by a negative number, the result is positive.  (It is
assumed that you know that a positive number divided by a positive number
results in a positive number.)  Examples for cases 1 and 2 are:
"-10 / 5 = -2" and "10 / -5 = -2".  An example for case 3 (which is very
important) is:  "-4 / -2 = 2".  Yet again, the negative signs cancel each
other out.  Division is easy, it just takes some practice.

Do you think you understand better now?  I hope so.  Just remember the order
of operations:  1) go from left to right in an equation; 2) do all
multiplication and/or division problems before doing addition or subtraction
problems.  Now, all you need to do is practice.  If you choose to take the
quiz, give yourself five points for each problem you get right.  Subtract 9
points from your score for each problem you get wrong, in class UIL Number
Sense scoring fashion.  Information on how to order the answers to the quiz
can be found immediately following the quiz.  Good luck in school, and enjoy
the challenge of math!

NEXT LESSON:  FRACTIONS...BE AFRAID...BE VERY AFRAID...
______________________________________________________________________________
                              PRACTICE PROBLEMS
                              -----------------
Addition
--------
1) 2 + 5 = __   2) 3 + 7 = __   3) 6 + 9 = __   4) 7 + 6 = __   5) 5 + 3 = __
6) -2 + 5 = __  7) -1 + -4 = __ 8) -6 + -9 = __

Addition Bonus
--------------
10 + 11 + 3 + -3 = __
______________________________________________________________________________

Subtraction
-----------
1) 4 - 3 = __   2) 7 - 5 = __   3) 7 - 2 = __   4) 6 - 4 = __   5) 12 - 1 = __
6) -4 - 3 = __  7) 7 - -5 = __  8) -7 - 2 = __  9) -6 - -4 = __

Subtraction Bonus
-----------------
11 - 8 - 2 - -3 = __
______________________________________________________________________________

Multiplication
--------------
1) 2 X 2 = __   2) 3 X 5 = __   3) 5 X 1 = __   4) 6 X 2 = __   5) 3 X 4 = __
6) 2 X -2 = __  7) -3 X -5 = __ 8) 5 X -1 = __

Multiplication Bonus
--------------------
3 X 4 X 2 X -1 X -2 = __
______________________________________________________________________________
Division
--------
1) 9 / 3 = __   2) 4 / 2 = __   3) 7 / 1 = __   4) 6 / 2 = __   5) 10 / 2 = __
6) 9 / -3 = __  7) -4 / 2 = __  8) -6 / -2 = __
Division Bonus
--------------
12 / 2 / 3 / -1 = __
______________________________________________________________________________

        DO NOT LOOK AT THE ANSWERS UNTIL YOU HAVE COMPLETED THE PROBLEMS
______________________________________________________________________________

Addition Answers
----------------
1) 7    2) 10    3) 15    4) 13    5) 8   BONUS -=- 21
6) 3    7) -5    8) -15
______________________________________________________________________________

Subtraction Answers
-------------------
1) 1    2) 2    3) 5    4) 2    5) 11    BONUS -=- 4
6) -7   7) 12   8) -9   9) -2
______________________________________________________________________________

Multiplication Answers
----------------------
1) 4    2) 15    3) 5    4) 12    5) 12    BONUS -=- -48
6) -4   7) 15    8) -5
______________________________________________________________________________

Division Answers
----------------
1) 3    2) 2    3) 7    4) 3    5) 5     BONUS -=- -2
6) -3   7) -2   8) 3
______________________________________________________________________________

                             /-----------------\
                             |* Q * U * I * Z *|
                             \-----------------/

1) 1 + 3 - 2 = __   2) 4 X 5 / 2 = __   3) 1 + 1 + 3 - 2 = __   4) 5 X 6 = __

5) 2 + 3 - 4 + 1 = __   6) 10 + 1 - 2 X 2 - 15 = __   7) 4 + 5 + 3 - 2 = __

8) 1 + 3 X 2 = __   9) 1 + 1 + 6 - 5 + 2 = __   10) 6 X 2 - -3 = __

11) 1 - 3 + 8 + -2 X 3 = __   12) 5 X 4 / -2 + 5 = __   13) 5 - 6 / 3 X 4 = __
______________________________________________________________________________
For answers to the quiz send a self-addressed stamped envelope along with five
American dollars to GwD Inc. at the address below.
______________________________________________________________________________

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Issue#149 of "GwD:  The American Dream with a Twist -- of Lime" ISSN 1523-1585
copyright (c) MMV Aardv@rk/GwD Publications                  /---------------\
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