Math+Concept+Help+and+Discussion

=Greatest Common Factor= The greatest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. To find the GCF list all of the factors of each number, then list the common factors and choose the largest one. Example:

Find the GCF of 36 and 54. Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor. =Least Common Multiple= The least common multiple of two or more non-zero whole numbers is actually the smallest whole number that is divisible by each of the numbers. Simply list the multiples of each number (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.
 * The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
 * The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
 * The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18

Example: Find the least common multiple for 5, 6, and 15. >> Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,... >> Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,... >> Multiples of 15 are 30, 45, 60, 75, 90,.... =Prime and Composite Numbers= A prime number has only 2 factors (1 and itself). A composite number has more than 2 factors. 0 and 1 are neither prime or composite. Here is a list of prime numbers up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
 * First we list the multiples of each number.
 * Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
 * Therefore, the least common multiple of 5, 6 and 15 is 30.

=Prime Factorization= When you write a number in prime factorization you are writing the number as a product of it's prime factors.

A factor tree shows the prime factors of a composite number in a "tree-like" form. Drawing factor trees is a good way of doing prime factorization. You can make different factor trees to find the same prime factorization. Look at the two examples below. and both give the same prime factorization. =Exponents= An exponent tells you how many times to use the base as a factor in a multiplication problem. See the table below for examples.

Form || Factor Form || Standard Form ||
 * Exponential
 * 22 = || 2 x 2 = || 4 ||
 * 23 = || 2 x 2 x 2 = || 8 ||
 * 24 = || 2 x 2 x 2 x 2 = || 16 ||
 * 25 = || 2 x 2 x 2 x 2 x 2 = || 32 ||
 * 26 = || 2 x 2 x 2 x 2 x 2 x 2 = || 64 ||
 * 27 = || 2 x 2 x 2 x 2 x 2 x 2 x 2 = || 128 ||
 * 28 = || 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = || 256 ||

=Order of Operations= P arenthesis E xponents M ultiplication D ivision A ddition S ubtraction Solve math problems in the PEMDAS order. Remember to complete multiplication and division in the order in which they appear in the problem. Same for addition and subtraction. For example: 5 - 2 + 1, you would subtract first because subtraction appears first, then you would add. Do all math in parenthesis first, then exponents, next complete all multiplication and division - in the order they appear, and finally complete all addition and subtraction - in the order they appear.

=Math Properties= 4 x 5 x 6 = 5 x 6 x 4 || Commute means to move (commute to school). The numbers commute or move around. || 4 x (5 x 6) = (4 x 5) x 6 || Associate means people you hang out with or "group with". The numbers can associate or group with other numbers. || 52 x 1 = 52 || Identity means who someone is. A number's identity (or what the number it is) doesn't change when using the identity property. || =Decimal Place Value Chart= == =Changing Forms Between Fractions, Decimals, and Percents=
 * Property || Definition || Example || Memory Help ||
 * Commutative || The order of numbers can move around ONLY when you add or multiply. || 2 + 3 + 4 = 3 + 4 + 2
 * Associative || Numbers can be regrouped ONLY when you add or multiply. || (2 + 3) + 4 = 2 + (3 + 4)
 * Distributive || Distribute outside number to terms inside the parenthesis. || 5(2 + 4) = 5 x 2 + 5 x 4 || Distribute means to hand something out. The number outside is being handed out to the numbers inside the parenthesis. ||
 * Identity || Add 0 to a number and the number stays the same. Multiply a number by 1 and the number stays the same. || 9 + 0 = 9

**__Decimals to Fractions:__** The decimal becomes the numerator and the denominator is 1 and the number of zeros as digits in the number Ex: Write .789 as a fraction Numerator is 789 and Denominator is 1000 (789 has 3 digits so you have 3 zeros) In your calculator: type the decimal in and use the ►F button and then use the ►Simp button to simplify the fraction

__**Decimals to Percents:**__ Multiply by 100 or move the decimal 2 places to the right Ex: Write .93 as a percent .93 ∙ 100 = 93% In your calculator: type the decimal in and press 2nd / button (it says ►% above it)

**__Fractions to Percents__**: Divide the fraction into a decimal and multiply by 100 Ex: Write ½ as a percent 1 ÷ 2 = .5 .5 ∙ 100 = 50% In your calculator: type the fraction in and press 2nd / button (it says ►% above it)

__**Fractions to Decimals:**__ Divide the top number by the bottom number Ex: ¾ is 3 ÷ 4 = .75 In your calculator: type the fraction in and press the ►D button

__**Percents to Decimals**__: Divide by 100 or move the decimal 2 places to the left Ex: Write 76% as a decimal 76 ÷ 100 = .76 In your calculator: Type the number and then the 2nd ( button (it says % above it)

__**Percents to Fractions**__: Put the percent over 100 and simplify the fraction Ex: Write 25% as a fraction 25/100 then simplify to 1/4. In your calculator: Type the number in and then the 2nd ( button. Then use the ► F button and then the ► Simp button to simplify the fraction.

=Comparing and Ordering Fractions= Step 1: Find a common denominator. (You can use the least common multiple of the denominators.) Step 2: Make equivalent fractions using the common denominator. Step 3: Use the numerators to compare and order. __ ﻿Example __ Compare 2/3 3/5 Step 1: LCM of 3 and 5 is 15. Step 2: Use 15 as the common denominator. 2/3 = 10/15 and 3/5 = 9/15. Step 3: Use the numerators to compare. 10 is larger than 9; therefore, 10/15 is greater than 9/15. So, 2/3 is > than 3/5.

=Comparing and Ordering Decimals= Step 1: Line up the digits and decimal points. Step 2: Fill in the empy spaces with zeros, remember this does not change the value of the decimal. Step 3: Start comparing digits at the greatest place value position (first digit on the left). __ Example __ Compare 0.27 0.271 Step 1: 0.270 0.271 Step 2: I filled in the space beside the 7 in 0.27 with a 0. Step 3: Start comparing digits at the largest place value position. In the ones place both digits are 0, so move to the next place over. In the tenths place both digits are 2, so move over to the next place. In the hundredths place both digits are 7, so move to the next place over. In the thousandths place you have a 0 and a 7. 7 is larger than 0; therefore, 0.271 is larger than 0.27. So, 0.27 < 0.271.

= Adding and Subtracting Fractions = Steps: 1. Find a common denominator. (You can use the least common multiple of the denominators.) 2. Write equivalent fractions using the least common denominator. 3. Add or subtract the numerators. 4. Keep the denominator. 5. Simplify. __ Example __ 3 || + || __ 1 __ 4 || . ||
 * || __ 2 __

The lowest common multiple of 3 and 4 is their product, 12. We will convert each fraction to an equivalent fraction with denominator 12. 3 || + || __ 1 __ 4 || = || __ 8 __ 12 || + || __ 3 __ 12 || 12 |||| . ||
 * __ 2 __
 * ||  ||   || = || __ 11 __
 * ||  ||   || = || __ 11 __

=Multplying Fractions and Mixed Numbers= Steps: 1. Change any mixed number to an improper fraction. 2. Multiply across the top (numerators). 3. Multiply across the bottom (denominators). 4. Simplify. Example What is 1 1/2 x 2 1/5 ?

Convert both to improper fractions 1 1/2 × 2 1/5 = 3/2 × 11/5 Multiply 3/2 × 11/5 = 33/10 Convert to a mixed number 33/10 = **3 3/10 **

= Dividing Fractions and Mixed Numbers = Steps: 1. Change any mixed number to an improper fraction. 2. Keep the first fraction. 3. Flip the second fraction. (Write the reciprocal.) 4. Change division to multiplication. 5. Multiply across the top and bottom. 6. Simplify. __ Example __

4 || ÷ 2 || __1__ 2 || = || __5__ 4 || ÷ || __5__ 2 || 4 || X || __2__ 5 || 20 || 2 |||| . || =Adding and Subtracting Decimals= = __Steps:__ = 1. Line up the decimals (other place value positions will automatically line up too) 2. Add or subtract as you would with whole numbers. 3. Bring the decimal straight down and place in sum or difference. __ Example __ 0.98 + __11.00__  11.98
 * 1 || __1__
 * || = || __5__
 * || = || __5__
 * || = |||||| __10__
 * || = |||||| __10__
 * || = || __1__
 * || = || __1__

= Multipying Decimals = Steps: 1. Change decimals to fractions. 2. Use fraction multiplication. 3. Change fraction answer back to a decimal. OR 1. Multiply as with whole numbers 2. Count the number of digits there are behind the decimal in each factor. 3. The sum of the number of digits there are behind the decimal in the factors is the number of digits there are behind the decimal in the product.
 * __ Steps: __**

__**Example**__ 0.5 x 0.25 5/10 x 25/100 = 125/1000 125/1000= 0.125

OR 0.25 2 digits behind the decimal __x 0.5__ 1 digit behind the decimal 0.125 3 digits behind the decimal

=Scientific Notation= Writing scientific notation in standard form. 1. Move the decimal in the coefficient (number you are multiplying by the power of 10) the same number of places as the exponent. 2. If the exponent is positive, move the decimal to the right (make number bigger). If the exponent is negative, move the decimal to the left (make number smaller). 2.3 x 10^5 230,000. (decimal is moved 5 places over to the right because the exponent is a positive 5) 2.3 x 10^-5 0.000023 (decimal is moved 5 places over to the left because the exponent is a negative 5) Writing standard form as scientific notation. 1. Move the decimal in the standard form of the number so that there is only one digit that is not a 0 in front of the decimal. (The coefficient has to be at least equal to 1 but less than 10). Keep count of how many places you moved the decimal. 2. Write the new number (the one with the decimal moved) times a power of ten. 3. The exponent of the power of 10 will be the number of times you moved the decimal. The exponent will be positive if the standard form of the number is bigger (bigger than 1). The exponent will be negative if the standard form of the number is small (smaller than 1). 4,200 4.2 x 10^3 (Decimal is moved from behind the last 0 in 4,200 to between the 4 and 2, so that there is only 1 digit in front of the decimal. The decimal was moved 3 places, so the exponent is a 3. The exponent is positive because the standard form of the number was bigger than 1.) 0.0042 4.2 x 10^-3 (Decimal is moved from between the two 0s to between the 4 and 2, so that there is only 1 digit that is not a 0 in front of the decimal. The decimal was moved 3 places, so the exponent is a 3. The exponent is negative because the standard form of the number was smaller than 1.)
 * __ Steps: __**
 * __ Examples __** :
 * __ Steps: __**
 * __Examples:__**

=Dividing Decimals= 1. Write decimals as fractions with common denominators. 2. Use fraction division. (Keep first fraction, flip second fraction, change division to multiplication.) 3. Write fraction answer as a decimal.
 * __ Steps: __**

OR Steps: 1. Move decimal in the divisor a whole number by moving the decimal behind the last digit. 2. Move decimal in the dividend the same number of places as you did in the divisor. 3. Bring the decimal straight up into the quotient. 4. Divide as with whole numbers.

1.2 divided by 0.03
 * __Example__**

1 and 2/10 divided by 3/100 21/10 divided by 3/100 210/100 divided by 3/100 210/100 x 100/3 use cross cancellation and make the 100s a 1 and then you have 210/1 x 1/3 = 210/3 = 70.

OR 1.2 divided by 0.03 move decimal two places so that it is behind the 3, move decimal two places in the dividend and you have 120 divided by 3, which equals 70.