BPCC BASIC MATH - 097

06/21/15

Module I

• Place Value: Is a numerical place holder which a number has ten times the value to the right:
• Place Value:
• Digits - we use them to write numbers from 0-9.
• Whole Numbers: a set o whole numbers from 0-9.
• Natural Numbers: The numbers we usually count from 1-9.
• Place Value digits represent a specific holder value which is a power of ten. Place values have "periods" separated by commas.
• Example: Unit period, Thousand period, Millions period
• Every period has place holder units like ones to hundreds. When you reach other periods, you will usually start mixing them with unit periods writing from the largest or smallest number once.

• Writing Numbers:  In writing numbers we can use several methods like:
• Word Form:
• Example: One Thousand, Four Hundred Fifty [1450]
• Remember not use the "and" in writing numbers.
• Standard Form
• The number you usually write.
• Example: 1,450
• Expanded Form: The specific place value of each digit is in the form of a bunch of addends:
• Example:
• 100 + 400 + 50
• Adding Whole Numbers: Addition means to combine numbers and increase the total value. The number we add are called addends and the sum is the answer.
• Tip: When adding multiple numbers, take all of the ones place values and add compatible numbers to result  sums of ten.
• Example 1: 8+9+4+1+6+2+5 = (8+2)+(9+1)+(6+4)+5 = 10 + 10 + 10 + 5 to get 35 
• Example 2: 12+8+23+34+6 = (12+8) + (34+6) + 23 = 20 + 40 + 23 = 83
• Example 3: If you add big numbers, make sure that the numbers are lined up vertically each other. If there are excess digits, be sure to carry it's tens place on top of the second addend digit:

                      11                  

                     983

                  +   17

                   1000 - If there are excess digits at the end of the addend, just write them the way they are.

• Perimeter and Properties with Addition: The total distance around the outside of a continuous boundary line of a geometric measure and can be calculated by adding the length of its boundary distance around the polygon.  
• Polygon: a 2-dimensional shape like a square, triangle, rectangle, etc...
• Example: 
• Rectangle: 91 ft length, 78 ft width:
• P =  78ft. + 91ft.+ 78ft.+ 91ft.
• P =  338 ft.

• Properties of Addition:
• Commutative Property: 2+3=3+2
• Changing the order of the addends doesn't change the sum.
• Associative Property:  2 + (3+4) = (2+3)+4 Changing the grouping of the addends doesn't change the sum.
• Identity Property:  Adding zero to any number will get the same result.

• Subtracting Whole Numbers: Means to take away something and see what's left.
• Example: 713 - 199 = 514
• Minuend: The number decreasing (take away) the subtrahend.
• Subtrahend: The number being decreased (take away) by the minuend.
• Tip: Check if the result is correct by adding the Difference and the Subtrahend to result the Minuend.

• Subtraction Word Problems
• You can already find the difference, subtract numerically and check if your answers are correct. But this time, you will learn about subtracting word problems. To find the difference, you have to identify some of these clues.
• Example: 
• Subtract 620 from 1000
• Correct: 1000 - 620
• Incorrect 620 - 1000 
• Find 105 less 12
• Correct: 105 - 12
• Incorrect 12 - 105 
• Find the difference between 100 and 62
• Rules:
• Align all numbers vertically each other.
• Subtract
• If there are zeroes above a non-zero number, borrow a number from a non zero digit to turn zeroes into non-zero values 

Module II:

• Rounding and Estimating
• Rounding means approximating digits. Rounded numbers are easier to understand and use than a precise whole number. 
• Tip: If the digit is 4 or less, ignore the number and turn the numbers to the right into zeroes. If the digit is 5 or more, add it by one and turn the numbers to the right into zeroes. The rules said that five should be rounded up no matter what. 
• Example: 52 to the nearest ten =  50
• Example: 921,976 = round to the nearest:
• Thousand = 921,976 = 922,000
• Hundred = 921,976 = 922,000
• Ten = 921,976 = 921,980
• Rounding Word Problems:
• Same as rounding off numbers, but this time you will estimate then add.
• Example:
• 588 + 689 + 277 + 153 + 059 = 
• 600 + 700 + 300 + 200 + 100 = 1,900

• Multiplication and It's Properties:
• Multiplication is adding multiple equal groups of numbers repeatedly.
• Vocabulary and Symbols:
• Symbols represent multiplication "(), * , x.
• Product = The result of the factors.
• Factors  = The numbers that are multiplying.
• Properties of multiplication:
• Commutative Property: 2*3=3*2
• Changing the order of the Factors doesn't change the product.
• Associative Property:  2 * (3*4) = (2*3)*4 Changing the grouping of the factors doesn't change the product.
• Identity Property:  Multiplying one to any number will get the same result.
• Zero Property: Any number multiplied by zero is zero.
• Distributive: For any real number f(v*x) = F*v + f*x

• Area and Perimeter with multiplication:
• Area - The amount of space within a two dimensional region. Area is an application that could be used in daily life like calculating your plot of land. Area is measured in square units. Area of a Rectangle A = L*W
• Perimeter Vs Area
• Perimeter = add
• Area = multiply
• Example: Rectangle =  3 inches Width, 13 inches Length
• Rectangle Perimeter = 32 in.
• Rectangle Area         = 39 in^2.

• Dividing Whole Numbers
• Division is repeated subtraction decreasing equal groups of numbers.
• Division Expressions: 18/3 18 ÷ 3 
• Dividend: The number being divided.
• Divisor: The number that is dividing.
• Quotient: Result
• Remainder: Left Overs of the result.
• Check: By multiplying the quotient and divisor and add its product by the remainder.
• Long Division:
• Rules for Long Division
• Divide = Figure out how many digits can go equally.
• Multiply = The divisor by the numbers that can go equally in the dividend.
• Compare: Remainder should be less than the divisor.
• Bring Down: the next digit until there's nothing left.   

Module III

Things you need to know when you encounter word problems:

• Interpret the problem's objective and information.
• Find solutions that can be applied.
• Find key words

Example:

A rectangular parking lot measures 105 ft. by 100 ft.  Find the perimeter and area.

• Perimeter (Add)
• P = 150 + 150 + 100 + 100
• P = 500 ft.
• Area (Multiply)
• A = L * W
• A = 150 * 100
• A = 15,000

Exponents: A number multiplied by itself  repeatedly.

Example: 

• 2^5 = 2 * 2* 2 * 2 * 2 = 32

What you need to know:

• Base: The number being  multiplied.
• Exponent: The number that is multiplying repeatedly
• Example: 
• 10 - base
• ^2 - exponent

Evaluating - Taking a problem then  solving it to result an accurate answer.

Example:

• 5^2 = 5*5 = 25
• 7 x 6^2 = 7 x 6 x 6 = 252
• 9^1 = 9 

Square Roots: A reverse operation in exponents.

Example:

• √(36) = 6
• 6^2

Perfect square numbers are numbers that have square roots:

• Example: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Order of operations - Following a specific order to get an answer.

P - Parenthesis ( ) or other place holders like [ ] and{} and even √ a radical sign.

E - Exponents ^

M - Multiplication

D - Division

A - Addition

S -  Subtraction

Note: If one order is unavailable, skip to the next and follow.

Example: 

• 4 + 5 + 2^2
• 2^2 = 4 - Exponents
• 4 + 5 + 4 - Addition
• 2 * √ 49 - 3(9)
• √ 49  = 7
• 2 * 7 - 3*4
• 2 * 7 - Multiplication
• 3 * 4 -  Multiplication
• 14  - 12 -  Subtraction
• 2 - Answer

Module IV

Introduction to Fractions

• Fraction is a number that represents part of a whole. It actually means "to break". A whole is divided equally or unequally if a fraction is present with it. A fraction may look like this:
• a/b or c a/b
• Things you need to know in fractions:
• Numerator - Located on top of the fraction bar.
• Fraction bar - Separates the Numerator and Denominator.
• Denominator - Located at the bottom of the fraction bar.
• Remember that the denominator can't be equal to zero.

 Types of fractions:

• Proper Fractions: The numerator is smaller than the denominator. 
• Improper Fractions: The numerator is Bigger than the denominator.
• Mixed Fractions: A combination of a whole number and a fraction.

 How to change mixed numbers to improper fraction:

• Multiply the whole number by the denominator (the bottom part) 
• Add the result to the numerator (the top part). 
• Add the result to the numerator.  

 How to change improper fractions to mixed numbers:

• Divide the numerator by the denominator. 
• Turn quotient into whole number answer.  
• Write down any remainder above the denominator. 

Factoring Numbers:

• Factoring means breaking apart a number to result numbers that can multiply the original number.
• Tips:
• If the unit digit is an even number, then it's divisible by 2.
• If the sum of the unit digits result to be divisible by 3 then it's divisible by 3.
• If the last two unit digits result to be divisible by 4 then it's divisible by  4.
• If the unit digit is 0 or 5 then it's divisible by 5.
• If the unit digit is divisible by 2 or 3 then it's divisible by 6.
• If three unit digits are divisible by  8, then it's divisible by 8.
• If the sum of the unit digits are divisible by 9 then it's divisible by 9.
• If the unit digit is zero, then it's divisible by 10.

•  Example:
• 48 = 1 x 48
•      = 2 x 24
•      = 3 x 16
•      = 4 x 12
•      = 6 x 8

 Prime Factorization

• To know whether the number can have more than two  factors or only one factors. You must Prime Factorization.
•  Composite - have more than two factors:
• Example: 128, 102, 48 and most even numbers.
• Prime Number: Have exactly two factors: 
• Example: 2, 7, 3, 11 and most odd numbers.
• Use factoring to find out whether a number is prime or composite.
• Example: 
• 950 = (5^2) * 2 * 19   

Simplifying Fraction:

• Simplifying fractions means reducing fractions into their lowest terms. To simplify fractions, you must find the greatest factor that the numerator and denominator can divide.
• Example:
• 3/6 -> 1/2
• 3/6 -> 3*1/3*2 -> 3 is the greatest factor.

• How to simplify fractions:

1. Find the GCF of both numbers.
2. Divide the GCF by the fraction.

08/09/15

Module 5:

Multiplying Fractions:

• The rule is simple, it's: 
• numerator x numerator
• denominator x denominator

Example:

• 2/5 * 1/3 = 2/15; Just multiply
• 7/8 * 2/3 = 14/24 = 7/12; Multiply then simplify
• 1 4/35 7 * 1 5/24 6 = 1/42; Cross-cancel then multiply.

Multiplying Mixed Fractions:

Rules:

1. Convert all mixed numbers to improper fractions
2. Multiply 
3. Convert the product to mixed numbers

Example

1. 3 1/2 * 10 * 1 1/5  = 7/2 1 * 2 10/1 * 3 6/5 1 = 42/1 = 42

 Dividing Fractions:

• Rule:
• Find the reciprocal of the subtrahend then multiply.

Example:

• 5/8 ÷ 3/8 = 5/8 1 * 1 8/3 = 5/3
• 2/3 ÷ 4   = 1 2/3 * 1/4 1 = 1/6

Dividing Mixed Numbers

Rules:

1. Convert all mixed numbers to improper fractions 
2. Find the reciprocal of the subtrahend
3. Multiply 
4. Convert the product to mixed numbers

 Example:

1. 1 1/2 ÷ 2 = 3/2 ÷ 1/2 = 3/4
2. 1 1/3 ÷ 2 = 2 4/3 * 1/2 1 = 2/3

10/18/15

Module 6

Adding/Subtracting like Fractions

Definition of Terms:

• Like Fractions: Are fractions with the same denominator
• Example: a/c and b/c are like fractions

Adding/Subtracting like fraction

• Adding/Subtracting fractions is simple, all you need to do is work on the numerator and keep the denominator then simplify your answer.
• Example:
• 1/5 + 2/5 = 3/5
• 11/9 + 10/9 = 21/9 = 7/3
• 15/3 - 6/3 = 9/3 = 3
• 15/12 - 1/12 - 4/12 = 10/12 = 5/6

Least Common Multiple

Definition of Terms

• Multiple: a number that can be divided by another number without a remainder.
• LCM: The smallest number that can be divided by another number without a remainder

How to find LCM

• Finding the smallest number found on both numbers:
• Example:
• 6: 6,12,18,24
• 8: 8, 16, 24
• Therefore: 24 is the LCM
• More Examples:
• LCM: 30
• 2: 30÷2 = 15; Therefore 30 is also a multiple of 2.
• Check if the smallest number is divisible by the LCM of the bigger numbers to know if the small number has an LCM of the two numbers.
• 10: 10, 20 30
• 15: 15, 30

Equivalent Fractions:

 You can apply your knowledge on LCM by finding the Equivalent of a fraction using LCD or Least Common Denominator.

Example:

• Rewrite 3/8 into its equivalent with a denominator of 24.
• Find out how many times 8 could be multiplied to get 24:
• Answer: 3
• Multiply 3/8 * 3/3 = 9/24
• Therefore, the equivalent is 9/24

More Examples:

• 2/3 = 6/9
• 5/12 = 25/60
• 4/5 = 20/25
• 7/10 = 56/80