We can use the number line as a model to help us visualize adding and subtracting of signed integers. Just think of addition and subtraction as directions on the number line. There are also several rules and properties that define how to perform these basic operations. To add integers having the same sign, keep the same sign and add the absolute value of each number. To add integers with different signs, keep the sign of the number with the largest absolute value and subtract the smallest absolute value from the largest. Subtract an integer by adding its opposite.

Watch out! The negative of a negative is the opposite positive number. That is, for real numbers, -(-a) = +a

Here's how to add two positive integers:
4 + 7 = ?
If you start at positive four on the number line and move seven units to the right, you end up at positive eleven. Also, these integers have the same sign, so you can just keep the sign and add their absolute values, to get the same answer, positive eleven. Here's how to add two negative integers:

-4 + (-8) = ?
If you start at negative four on the number line and move eight units to the left, you end up at negative twelve. Also, these integers have the same sign, so you can just keep the negative sign and add their absolute values, to get the same answer, negative twelve. Here's how to add a positive integer to a negative integer:

-3 + 6 = ?
If you start at negative three on the real number line and move six units to the right, you end up at positive three. Also, these integers have different signs, so keep the sign from the integer having the greatest absolute value and subtract the smallest absolute value from the largest. Subtract three from six and keep the positive sign, again giving positive three. Here's how to add a negative integer to a positive integer: 5 + (-8) = ?

If you start at positive five on the real number line and move eight units to the left, you end up at negative three. Also, these integers have different...

...Introduction
Integers are the first numbers that we learn to use. Along with their usefulness in everyday life, integers are building blocks from which all others numbers are derived. The integers are all the whole numbers including zero, all negative and all the positive numbers
Basics of integers
* Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the numberline.
* Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the numberline.
* The integer zero is neutral. It is neither positive nor negative.
* The sign of an integer is either positive (+) or negative (-), except zero, which has no sign.
* Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the numberline. One will have a positive sign, the other a negative sign. In the numberline above, +3 and -3 are labelled as opposites.
* We compare...

...Quick Definition: A Mixed Fraction is a
whole number and a fraction combined,
such as 1 3/4.
1 3/4
(one and three-quarters)
To make it easy to add and subtract them, just convert to Improper Fractions first:
4/43/4
Quick Definition: An Improper fraction has a
top number larger than or equal to
the bottom number,
such as 7/4 or 4/3
(It is "top-heavy")
7/4
(seven-fourths or seven-quarters)
Adding Mixed Fractions
I find this is the best way to add mixed fractions:
convert them to Improper Fractions
then add them (using Addition of Fractions)
then convert back to Mixed Fractions:
Example: What is 2 3/4 + 3 1/2 ?
Convert to Improper Fractions:
2 3/4 = 11/4
3 1/2 = 7/2
Common denominator of 4:
11/4 stays as 11/4
7/2 becomes 14/4
(by multiplying top and bottom by 2)
Now Add:
11/4 + 14/4 = 25/4
Convert back to Mixed Fractions:
25/4 = 6 1/4
When you get more experience you can do it faster like this:
Example: What is 3 5/8 + 1 3/4
Convert them to improper fractions:
3 5/8 = 29/8
1 3/4 = 7/4
Make same denominator: 7/4 becomes 14/8 (by multiplying top and bottom by 2)
And add:
29/8 + 14/8 = 43/8 = 5 3/8
Subtracting Mixed Fractions
Just follow the same method, but subtract instead of add:
Example: What is 15 3/4 - 8 5/6 ?
Convert to Improper Fractions:
15 3/4 = 63/4
8 5/6 = 53/6
Common denominator of 12:
63/4 becomes 189/12
53/6 becomes...

...Question Number 1
Points: 5.00/5.00
Question Text
What event marks the line between the Old English and Middle English periods?
Your Answer
B. the Norman invasion
Question Number 2
Points: 0.00/5.00
Question Text
What was the effect of the Norman conquest on the language of Britain?
Your Answer
. The upper class spoke French
Question Number 3
Points: 0.00/5.00
Question Text
Why is Beowulf an epic poem?
Your Answer
It is about a hero who represents cultural values.
Question Number 4
Points: 5.00/5.00
Question Text
What two things does Beowulf take from the sea hag's cave?
Your Answer
B. Grendel's head and the hilt of the sword
Question Number 5
Points: 5.00/5.00
Question Text
What was one of the social effects of the Black Death?
Your Answer
C. There was a rise in the middle class
Question Number 6
Points: 0.00/5.00
Question Text
In The Canterbury Tales, a few of the pilgrims are women. Which one is NOT featured in the tale?
Your Answer
The Nun
Question Number 7
Points: 0.00/5.00
Question Text
What killed as much as half the population of Europe during the Middle Ages?
Your Answer
black death
Question Number 8
Points: 0.00/5.00
Question Text
Based on the description of Grendel in Beowulf, what does the monster most clearly represent?
Your Answer Sinfulness of...

...Adding & subtracting fractions
1. Where in life is this useful?
a) Cooking: [pic]
b) Measurements (construction, remodeling, etc): [pic]
c) Time: [pic]
d) Money: [pic]
2. Fractions with the same (“common”) denominators
Example: (without converting back & forth from mixed numbers):
[pic]
[pic]
3. Fractions with different denominators
In order to add (or subtract) fractions with different denominators (as a reminder, that’s the bottom number), you’ll need to convert them to have the same denominators. This is one place where we get to use the “least common multiple” that we talked about a while ago.
Let’s start with money, because we all do that conversion frequently, and without thinking about what we’re doing. If we add a quarter & a nickel, we know off the top of our head that we have 30 cents, or 30/100 of a dollar. But what is the math that we’re doing?
[pic]
First, we need to convert to a common denominator. For money, rather than worrying about the lowest common denominator, we automatically convert to hundredths. We do that by multiplying by one in the form of a fraction:
[pic]. We can do this because multiplying a number by 1 does not change its value. So, we now have:
[pic]. All we’ve done is converted the quarter to 25 cents and the...

...Waiting Lines & Queuing Models
American Military University
Business 312
For my project on other operations research techniques I have decided to research waiting lines and queuing models. My interest in this application stems from my personal dislike for standing in lines and waiting on hold while on the phone. This is virtually my only pet peeve; nothing aggravates me faster than standing in aline or waiting on hold. Like most people I go out of my way to avoid lines, using strategies such as arriving early or visiting during non-peak times. However, before investigating this topic, I had no idea there was a specific science behind the madness.
Queuing models are important applications for predicting congestion in a system. This can encompass everything from a waiting line at pharmacy to traffic flow at a busy intersection. This is important because it can impact businesses in unforeseen ways. Customers may begin to believe that they are wasting their time when they are forced to wait in line for service and continued delays may begin to negatively influence their shopping preferences.
Organizations design their waiting line systems by weighing the consequences of having a customer wait in line, versus the costs of providing more service capacity. Queuing theory provides a variety...

...RATIONAL NUMBERS
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q it was thus named in 1895 byPeano after quoziente, Italian for "quotient".
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.
A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
The rational numbers can be formally defined as the equivalence classes of the quotient set is the set of all ordered pairs(m,n) where m and n are integers, n is not 0 (n ≠...

...THE REAL NUMBER SYSTEM
The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.
Natural Numbers
or “Counting Numbers”
1, 2, 3, 4, 5, . . .
* The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.
At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.
Whole Numbers
Natural Numbers together with “zero”
0, 1, 2, 3, 4, 5, . . .
About the Number ZeroWhat is zero? Is it a number? How can the number of nothing be a number? Is zero nothing, or is it something?Well, before this starts to sound like a Zen koan, let’s look at how we use the numeral “0.” Arab and Indian scholars were the first to use zero to develop the place-value number system that we use today. When we write a number, we use only the ten numerals 0, 1, 2, 3,...

...2013 MTAP-DepEd Program of Excellence Mathematics Grade 1 Session 1
I. Read the following numbers.
1. 89 2. 106 3. 736 4. 245 5. 899
6. 302 7. 720 8. 1200 9. 5075 10. 7001
II. What is the place value of each underlined digit? Give the value of each underlined digit. Give the answers orally.
A B C D E F
1. 601 215 520 1,364 5, 055 8, 762
2. 740 806 810 1, 099 3, 456 9, 575
3. 108 888 2, 256 9, 302 2, 890 8, 359
III. Match Column A with Column B. Write the letter of the correct answer.
A B
1. _____ 2 tens and 6 ones a. 1 218
2. _____ 2 tens and 4 ones b. 539
3. _____ 3 tens and 2 ones c. 26
4. _____ 5. hundreds 3 tens and 9 ones d. 310
5. _____ 3 hundreds 1 tens and 0 ones e. 24
6. _____ 2 hundreds 4 tens and 5 ones f. 7 227
7. _____ 1 thousand, 2 hundreds 1 tens and 8 ones g. 32
8. _____ 3 thousands, 0 hundreds 4 tens and 4 ones h. 245
9. _____ 7 thousands, 2 hundreds 2 tens and 7 ones i. 5 830
10. _____ 5 thousands, 8 hundreds 3 tens and 0 ones j. 3 044
IV. Fill in the blanks with the missing values.
1. 600 + 40 + ___ = 646 2. 800 + 50 + ___ = 857
3. 10 + 800 + ___ = 810 4. 80 + ____ + 8 = 388
5. ___ + 500 + 6 = 516 6. 200 + ____ + 6 + ____= 1 236
7. ____ + 600 + ___ + 8 = 3...

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