Fractions
Fractions are just another way of showing division. 5/7 means the exact same thing as 5 divided by 7.
The denominator of a fraction shows the size of the pieces or groups. The numerator shows how many pieces or groups we are counting out.
Why do we learn how to write equivalent fractions?
When we add fractions with different denominators, we first find a common denominator. Then we rewrite one or both fractions so that they have the same denominator. This process of rewriting is really writing equivalent fractions. Many students make the mistake of changing the denominator without changing the numerator. Writing equivalent fractions involves multiplying both parts of a fraction by the same number.
Why do we learn prime factorization?
Prime factorization helps us do two important things: simplify fractions and find the least common denominator. My favorite explanation of how to do all this can be found here.
Why do we simplify fractions?
Because they are easier to read and understand. In Mr. Kessler's class, answers with fractions must always be simplified. A fraction can be simplified whenever the numerator and denominator have a common factor (other than 1). To simplify 28/63 we divide both parts of the fraction by 7 to get 4/9.
Improper Fractions and Mixed Numbers
An improper fraction has a numerator that is larger than its denominator. Answers can be left as improper fractions, as long as they are simplified. Improper fractions are often easier to understand if they are changed to a mixed number. A mixed number is a whole number plus a (regular) fraction .
To change a mixed number to an improper fraction, multiply the whole number times the denominator and add the numerator. This is your new numerator. The denominator stays the same.
To change an improper fraction to a mixed number, divide the numerator by the denominator. The answer is your whole number, the remainder is the numerator, and the denominator stays the same.
Operations with Fractions

change mixed numbers to improper fractions?

addition

common denominator

no

subtraction

common denominator

no

multiplication

multiply straight across
simplify before you multiply
DO NOT CROSS MULTIPLY

yes

division

change to multiplication
the dividend (first fraction) stays
the same
the divisor (second fraction) is changed to its reciprocal
now multiply

yes
