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4.1 INTRODUCTION TO FRACTIONS AND MIXED NUMBERS First a REVIEW. We are already familiar with positively and negatively signed numbers (integers) and know how to find them on the real number line. Find 1, 2 and –1 on the number line: -2 -1 0 1 2 Of course there are number BETWEEN these as well. In this section we will learn about “RATIONAL NUMBERS”, better known as “FRACTIONS”. A RATIONAL NUMBER (FRACTION) is a number of the form a where both a and b are INTEGERS, and b b is not _______. We call the number in the “top” part the NUMERATOR and the number in the “bottom” part the DENOMINATOR. Ex: 4/7 is a fraction. The NUMERATOR is_____ and the DENOMINATOR is ____. We use “fractions” every day. Can you think of some examples? MATERIALS!!(Do cup measurements w/ 1 cup, 1/3 cup & rice.) We can use PICTURES to represent fractions: Write a fraction to express each shaded area. Write a fraction to express each non-shaded area. We notice that the DENOMINATOR tells us how many equivalent “parts” there are and the NUMERATOR tells how many parts are being considered. We can use fractions to talk about groups of people. For example: In the front row, what fraction is female? What fraction is male? What fraction is a student? Can a “fraction” ever be more than one (1)? (See cup measurements, how many thirds could we use to make more than one whole cup?) How would you write such a fraction? What do you notice about the relationship between the numerator and the denominator? An IMPROPER FRACTION is one where the NUMERATOR is GREATER THAN OR EQUAL TO the DENOMINATOR. Ex: A PROPER FRACTION is one where the NUMERATOR is LESS THAN the DENOMINATOR. Ex: Classify as “proper” or “improper”, then graph: Fraction Proper Improper Fraction Fraction 9 5 9 11 9 9 0 9 MIXED NUMBERS Lets start by reviewing. Illustrate 1/2. How can we write 5/2 using a drawing? How many “wholes” do we have? How many halves do we have? So we can write 5/2 as: When an improper fraction is written as a whole number plus a proper fraction, we call this a MIXED NUMBER. For 5/2 this would be 2 + ½ and we simply write 2 ½ . For –5/2 this would be –2 – ½ and we simply write –2 ½. We must be able to write a MIXED NUMBER as an IMPROPER FRACTION, and an IMPROPER FRACTION as a MIXED NUMBER. Each will always have the SAME SIGN! Ex: Say we have 2 1/3. How many thirds is that? For the 2 wholes, there must be 3 thirds in each. 2 x 3 = 6 plus one more third = 7/3 What did we do? Writing a MIXED NUMBER as an IMPROPER FRACTION: 1) MULTIPLY the DENOMINATOR x WHOLE NUMBER 2) ADD the NUMERATOR to that product 3) This sum is the NUMERATOR of the improper fraction, the DENOMINATOR remains the same Ex: 3 Write 9 as an improper fraction 5 So if to write a mixed number as an improper fraction we MULTiply, to write an improper fraction as a mixed number, we will DIVide! Ex: Say we have 17 How many WHOLES is that? How 5 many fractional parts are left? For every 5/5 we will have a whole. All we need to do is divide 17 by 5 = 3 whole times with 2 remaining. Or Writing an IMPROPER FRACTION as a MIXED NUMBER: 1) DIVIDE the NUMERATOR by the DENOMINATOR 2) Write the WHOLE number part 3) Write the remainder over the same DENOMINATOR for the proper fraction part ** Note: if you begin with a PROPER fraction, it cannot have a whole number part! **Also: your fractional part of a MIXED number should ALWAYS be a PROPER fraction Ex: Write Ex: 18 as a mixed number 7 24 Round to the nearest whole number. 7 We treat the ABSOLUTE VALUE of a fraction just as we did for an integer. Remember that when we find the absolute value (or distance from zero) of a number it will ALWAYS be _______________ or zero. Ex: -5 = -5/3 = 7 = 7/9 = 0 = -1/4 =