Prime and composite numbers:
A prime number is a counting number greater than one which has two and only two factors for example...the only way to make 23 is 23 x 1, so 23 is prime
A composite number is a number that has more than two factors for example...there are several ways to make 18...1 x 18, 2 x 9, and 3 x 6 all make 18, so 18 is composite
Prime factorization: or decomposition of numbers: This is when you break the number apart until you make the number out of only prime factors for example...you can make 24 by multiplying 2 x 2 x 2 x 3...these are all prime numbers or prime factors
Greatest Common Factors or GCF: factor =a number that is multiplied in a math problem common = something that is shared greatest = the biggest so, the GCF is the biggest, shared factor between two numbers for example...the GCF of 24 and 30 is 6 because 24 can be made from 1 x 24, 2 x 12, 3 x 8, and 4 x 6 30 can be made from 1 x 30, 2 x 15, 3 x 10, and 5 x 6 the biggest number that shows up in both rows is 6, so 6 is the GCF You must find all of the factors in order to be successful finding the GCF...to find all of the factors you must use your knowledge of math facts and divisibility rules for example...to find all of the factors of 32...1 is a factor (property of one) 1 x 32, 2 is a factor (divisibility...it's even) 2x 16, 3 is not a factor (divisibility...doesn't 3 + 2=5, which is not divisible by 3), 4 is a factor (math fact) 4 x 8, 5 is not a factor (divisibility by 5...doesn't end in 0 or 5), 6 is not (divisibility...not divisible by 2 and 3), 7 is not (no math fact for 7 which makes 32, 8 is (math fact) 8 x 4...once you've found a duplicate pair of factors (4 x 8 and 8 x 4) you should have found all of he factors The GCF is used to reduce fractions to lowest terms or simplest form
Least Common Multiples or LCM: multiple = is like a math fact for a number (9, 18, 27, and 36 are all multiples of 9, because they are math facts using 9) common = something that is shared least = smallest so, the LCM is the smallest, shared multiple of two numbers for example...the LCM of 12 and 20 is 60 because my math facts for 12 are 12, 24, 36, 48, 60, 72... and my math facts for 20 are 20, 40, 60, 80, 100... the smallest number that shows up in both rows is 60, so 60 is the LCM You can multiply the two numbers together to get common multiples, but they won't always be the smallest (or least) common multiples...for example 12 x 20 = 240...240 is a common multiple of both 12 and 20, but it is not the smallest common multiple (we already know that 60 is the smallest or LCM) The easiest way to find multiples is by multiply your chosen number by 1, 2, 3, 4, 5, 6, and so on...you can also use repeated addition if you are more comfortable with that process The LCM is used to find a common denominator (make the bottoms of the fractions agree)
Fraction Terms:
Fraction-a division problem where the big number is in the divisor...usually resulting in an amount less than 1 whole
Numerator-the top of a fraction
Denominator-the bottom of a fraction Improper fraction-when the big number is on top
Mixed number-a number containing both a whole part and a fractional part
Changing Improper Fractions to Mixed Numbers:
1. Divide
2. Write the answer as your whole number
3. Make a fraction out of your remainder (top) and your divisor (bottom)...notice that the bottom stays the same This is useful because your final answers cannot have improper fractions in them
Changing a mixed number into an improper fraction:
1. Multiply the denominator (bottom) times the whole number (side) and add the numerator (top)
2. Write this as a fraction by putting the answer on top and keeping the denominator for the bottom...again the denominator doesn't change
Equivalent Fractions: fractions that look different and sound different, but have the same value for example...1/2 and 2/4 are equivalent...they look and sound different...but having 1/2 of a pizza and 2/4 of a pizza would give you exactly the same amount of food To make any equivalent fraction just multiply or divide the numerator and denominator by the same number (the most common number to use are 2, 3, 4, 5, 6, 10, and the GCF) for example...if I need an equivalent fraction for 3/7...as long as I multiple or divide the top and bottom by the same number I'll make an equivalent fraction...I could multiply by 2 (6/14) by 3 (9/21) by 10 (30/70) and so on and I'll still have the same amount
Simplest Form: A fraction is in simplest form when the GCF of the numerator and denominator is 1 for example 3/9 is not in simplest form, because 3 can be made from 1 x 3 and 9 can be made from 3 x 3, so they have something in common (3)...I can divide the top and bottom by 3 before the fraction is in simplest form 4/9 is in simplest form (4 is made from 1 x 4 and 2 x 2, 9 is made from 1 x 9 and 3 x 3) since they have nothing in common they are already in simplest form
Changing decimals to fractions: 1. Write the decimal as a fraction the same way that you would read it for example...0.78 would be read seventy-eight hundredths or 78/10 0.245 would be read 245 thousandths or 245/1000 2. Reduce the fraction to simplest form Changing fractions to decimals: 1. Divide the denominator into the numerator...the answer will be in the form of a decimal Comparing and ordering fractions and decimals: If you are comparing or ordering two fractions you must: 1. make the denominators the same by finding the LCM 2. change the top numbers by multiplying or dividing by the same number as you did in the bottom 3. compare or order using the top numbers If you are comparing a fraction and a decimal you must: 1. make both numbers the same type (either all fractions or all decimals) 2. follow the directions above or the directions for comparing and ordering decimals