Point out that when you compare two sets by using fractions, the objects in the sets do not need to be the same size or even the same kind.

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1 Fractions What are some of the ways we use fractions every day? Fractions represent a part of a whole object or unit. (Provide some sets of objects for students to work with to help them understand the idea that fractions represent a part of one whole or a part of an entire group or set.) Point out that when you compare two sets by using fractions, the objects in the sets do not need to be the same size or even the same kind. When you use fractions to discuss parts of a whole object, the object must be divided into equal parts. fraction: part of a whole object or set (it is a number less than 1) Every fraction is composed of three parts: (1) Numerator - the top number; tells the number of parts being considered (2) a bar (3) Denominator - the bottom number; tells the name and number of the parts in the whole proper fraction: a fractional number whose value is less than 1 (has a numerator less than its denominator) Fractional Answers Fractional Form vs Fractions Fractions can be used to express division problems. (Any division problem can be written in fractional form.) 5/7 is "5 divided by 7" numerator is the dividend denominator is the divisor Fractional Form vs Fractions There is a difference between a fraction and fractional form. fraction: a number less than 1 fractional form: any number expressed as a/b

2 Equivalent Fractions Have the students list different expressions that represent the same idea. (50 and 1/2 dollar; 1/4 of an hour and 15 minutes; a glass that is half full and half empty) equivalent fractions: fractions that have the same value (same value, but are in different forms) cross-product multiplication: a test to determine whether or not 2 fractions are equivalent 3/7 = 6/14 3/5 = 5/12 42= How to make equivalent fractions: = = Why does this method work? multiplying or dividing by a fraction with an identical numerator and denominator is the same as multiplying by 1 (multiplicative identity element) Renaming Fractions Lowest Terms We rename fractions using equivalent fractions. What part of the fraction is the name? (the denominator) 2/3 =?/12 3/4 =?/48 45/54 =?/6 1/? = 4/24 3/8 = 12/??/52 = 4/13 lowest terms (also called simplest form) - means there is no other equivalent fraction with a smaller numerator and denominator (no common factors except 1) To be in lowest terms the numerator and denominator needs to be relatively prime. How to reduce fractions to lowest terms?

3 Divide the numerator and the denominator by the GCF. Adding & Subtracting Fractions All fractions must have the same denominator (name) if they are to be added or subtracted. Adding Fractions that have the same denominator ("like fractions"): (1) Find the sum of the numerators (2) Write the sum over the common denominator (remember the denominator is the name) (3) Reduce the answer Can you add yards and feet together? Adding Fractions that have different denominators: (1) change fractions to the same denominator (LCD is LCM) (can always find a common denominator by multiplying the two numbers together) (Using LCD is usually easier and less time consuming) (2) Find the sum of the numerator (3) Write the sum over the LCD (4) Reduce the answer Subtracting is the same as addition only you find the difference instead of the sum. Comparing Fractions Same Numerator: If fractions have the same numerator, the fraction with the smallest denominator has the largest value. Same Denominator: If fractions have the same denominator, the fraction with the largest numerator has the largest value. Different numerators and denominators:

4 Change all the fractions to equal fractions having the same denominator. The fraction with the largest numerator has the largest value. Or use cross multiplication a/b = c/d if ad = bc fractions are = if ad > bc fractions are > if ad < bc fractions are < Mixed Numbers proper fraction: a fraction less than 1 (numerator less than denominator) improper fraction: numerator is equal to or greater than denominator mixed number: consists of a whole number plus a fraction (It's called mixed because part of the number is a whole number and the other part is a fractional number. It is actually a sum of these two numbers.) Changing mixed numbers to improper fractions: 2 7/10 = 20/10 + 7/10 = 27/10 2 = 20/10 Multiply the whole number by the denominator, add the numerator to this product and place the result over the denominator. 2 7/10 = (10 2) + 7 /10 Changing improper fractions to mixed numbers: Divide the numerator by the denominator. The quotient is the whole number. If there is a remainder, write it as the numerator with the divisor as the denominator. Reduce the fraction to lowest terms. (Trying to find out how many groups of 10's there are, and how many left over.) 27/ Adding & Subtracting Mixed Numbers

5 Adding mixed numbers that contain like fractions: Find the sum of the fractions and add this sum to the total of the whole numbers. 15 1/ /8 17 4/8 = 17 1/2 Adding mixed numbers that contain unlike fractions: Change the fractions to like fractions and follow the above procedure. (Renaming involves only the fractional parts. The whole numbers are not changed.) 3 1/4 = 3 2/8 4 3/8 = 4 3/ /2 = + 5 4/8 12 9/8 = 13 1/8 (The answer is not complete until it is written in lowest terms.) Multiplying Fractions & Mixed Numbers Multiplying Fractions: Multiply the numerators of the fractions to get the numerator of the answer. Multiply the denominators to get the denominator of the answer. Reduce the answer. Mixed numbers must be changed to improper fractions and then follow the above procedure. Canceling Fractions Reciprocals Cancellation is actually simplifying or renaming to lowest terms before multiplying instead of after. (save time, steps, and errors by simplifying first and then multiplying) (What they are canceling is some steps and some possibilities of mistakes.) Procedure: divide a factor out of a denominator and a numerator

6 Reciprocals reciprocals: two numbers whose product is 1 (reciprocal is a number upside-down; an inverted fraction; multiplicative inverse) What do you think? Can every fractional number be multiplied by another number so that the product is 1? (yes) Can every whole number be multiplied by another number so that the product is 1? (no - all except 0) Can every mixed number be multiplied by another number so that the product is 1? (yes) The way to find the reciprocal of a mixed number is always to write it as an improper fraction and then invert it. Dividing Fractions To divide by a fraction, invert the divisor and multiply. Dividing by a fraction is the same as multiplying by its reciprocal.