Section.1 PRE-ACTIVITY PREPARATION Introduction to Fractions Fraction notation is used as the numeric description of choice in a variety of common contexts. Consider the following examples in various contexts. Sewing: Cooking: Market research: Quality control: Construction: You purchase 7/ of a yard of fabric to make place mats. A cookie recipe calls for /4 of a teaspoon of cinnamon. A survey reveals that twelve out of every thirty people, or 1/0, prefer the convenience of take-out to home cooking. For every two thousand light bulbs coming off the assembly line, two of them (/000) are defective. Forty-two acres of land are split evenly into sixty-three lots for a new subdivision, yielding 4/6 of an acre per lot. Home decorating: A length of leftover carpeting seven yards long and three feet wide can be cut evenly to make four scatter rugs, each 7/4 yards long. Demographics: In a college psychology class comprised of thirteen men and seventeen women, 1/17 ( thirteen to seventeen ) represents the comparison of the number of male students to the number of female students. Conversions: To convert sixty milligrams to centigrams, you multiply the sixty milligrams by 1/10 (that is, 1 centigram to 10 milligrams ). LEARNING OBJECTIVES Learn the various relationships fractions can represent. Set up fractions to represent given information. Interpret fractions using appropriate mathematical language and/or diagrams. Convert between improper fractions and mixed numbers.
4 Chapter Fractions TERMINOLOGY PREVIOUSLY USED quotient NEW TERMS TO LEARN conversion convert comparison denominator fraction fraction bar fraction notation improper fraction mixed number numerator proper fraction ratio simple fraction BUILDING MATHEMATICAL LANGUAGE A fraction is, in its broadest definition, the quotient of two quantities. It has three components a top number called the numerator, a bottom number called the denominator, and a fraction bar separating them to indicate their relationship. Example: numerator denominator fraction bar This chapter will only consider what are sometimes referred to as simple fractions, those fractions whose numerators and denominators are whole numbers.. For example, is a simple fraction but is not. 10 It is always important to examine the context in which a fraction is being used. By its components, a fraction represents the information of a relationship, and the sort of relationship determines into which of four broad categories the fraction fits: Category 1 Part of a Whole Unit Category Part of a Group of Objects Category Division into Equal Portions Category 4 Comparison of Two Quantities
Section.1 Introduction to Fractions Categories 1 and Parts of a Whole or Parts of a Group The fractions with which you are most familiar may be those that represent specific parts: of a whole unit which can be divided into equal parts (a sheet cake, a pizza, a cup) (Category 1) OR of a group of objects (a classroom of students, a box of assorted chocolates, a bouquet of flowers) (Category ) In both categories, the fraction is read as fi ve eighths. In Category 1, the fraction implies fi ve out of eight equal parts of a whole. In Category, fi ve eighths implies fi ve out of a group of eight. Refer back to the first four examples in the Introduction to this section Sewing, Cooking, Market research, and Quality control in which the numerators denote the number of parts being considered, and the denominators denote either the total number of parts in the specified whole unit or in the specified group. Complete the following tables for these Category 1 and fractions using the information found on page 07. An additional example for each has been filled in for you. Category 1 Additional Example: Carpentry: You trim 16 of an inch from a piece of molding to make it fit. Context What is the whole unit? The whole unit is broken up into how many equal parts? How many parts are being considered? What is the fractional representation? Example: Carpentry 1 inch 16 16 Sewing Cooking
6 Chapter Fractions Category Additional Example: Merchandising: For every seventeen toasters on display at the chain store, six of 6 them are of a particular brand name 17. Context Identify the group. The whole unit is broken up into how many equal parts? How many parts are being considered? What is the fractional representation? Example: Merchandising display of toasters 17 6 6 17 Market research Quality Control Category Division into Equal Parts A fraction in this category represents a quantity split apart evenly. The Construction and Home decorating examples in the Introduction fit into Category. Category In this category, Note: This division results in implies units broken up into equal portions ( fi ve divided by eight )., or fi ve eighths of a unit per portion. With that in mind, use the information found on page 07 to complete the following table. Additional Example: Culinary arts: Six pizzas split evenly among thirteen people allows each 6 person of a pizza. 1 Context Identify the quantity to be distributed evenly. What is the number of equal portions? What is the fractional representation? Example: Culinary arts 6 six pizzas 1 of a pizza per person 1 Construction Home decorating
Section.1 Introduction to Fractions 7 Category 4 Comparison of Two Quantities A fraction in this category represents a comparsion of two quantities (a ratio). In this category, is read as fi ve to eight. The final two examples in the Introduction (Demographics and Conversions) fit into this category. With that in mind, use the information found on page 07 to complete the following table. Category 4 Additional Example: Gardening: The directions for mixing up the liquid fertilizer call for two parts fertilizer concentrate to eight parts water; that is, mix in the ratio of. Context Which is the initial quantity? What is the quantity it is compared to? What is the fractional representation of the comparison? Example: Gardening parts fertilizer parts water Demographics Conversions Additional Descriptions for Fractions A proper fraction has a numerator less than its denominator. 4 three fourths THINK out of 4 parts of a whole unit that has been divided into 4 equal parts make up less than one whole unit.
Chapter Fractions An improper fraction has a numerator which is the same as or greater than its denominator. 7 6 seven sixths THINK 7 pieces, when there are six pieces in a whole, make up more than one whole unit. eight eighths THINK of items in a group make one entire group Or, of equal parts in a whole make up a whole unit. A mixed number combines a whole number and a fractional part of a whole. The attached fractional part must always be a proper fraction. THINK whole units and out pieces of another whole unit. two and three fifths It implies the addition of the whole number and the fraction the addition sign is routinely omitted. + ; and except for calculator entries, To convert a number is write it in another form. Improper fractions can be converted to mixed numbers and mixed numbers to improper fractions. Converting Between Mixed Numbers and Improper Fractions As you will learn later in this chapter, it will be necessary to write a mixed number in its improper fraction notation for multiplication and division. Equally as important, you will write an improper fraction as a mixed number when presenting your final answer to a computation with fractions. Following are the methodologies for converting between the two forms. As you follow the steps, note that an understanding of what the denominator represents is a particularly key piece of information for each conversion.
Section.1 Introduction to Fractions 9 METHODOLOGIES Converting an Improper Fraction to a Mixed Number Example 1: Convert Example : Convert 17 to a mixed number. to a mixed number. Try It! Steps in the Methodology Example 1 Example Step 1 Identify the denominator. Identify the number of parts in a whole (the denominator). 17 implies parts in a whole. Step Determine whole number part. Divide the total number of parts (the numerator) by the number of parts in a whole (the denominator) to identify how many whole units or groups there are. 17 ) 1 whole units Step Determine fraction part. Determine from the remainder the fractional part of a whole. Special Case: No remainder (see page 1, Models) 17 ) 1 VISUALIZE of parts in a whole remain. whole units + / of another Step 4 Present the answer. Present the whole number connected to the fractional part as your mixed number answer. Step Validate your answer. Validate by changing the mixed number to an improper fraction. (parts in a whole) = 1 1+ = 17 (parts in whole units) 17 (total parts)
0 Chapter Fractions Converting a Mixed Number to an Improper Fraction Example 1: Convert 9 to an improper fraction. Example : Convert 7 to an improper fraction. Try It! Steps in the Methodology Example 1 Example Step 1 Identify the fraction part. Identify the fractional portion of the mixed number. 9 is the fractional portion of 9 Step Identify the denominator. Identify the number of parts in each whole unit or group (the denominator). 9 implies 9 parts in a whole. Step Determine the parts from the whole number. Multiply the number of whole units by the number of parts in each whole (the denominator). 9 whole units 9 parts per whole = 7 parts Step 4 Add the parts from the fraction. Add the number of fractional parts (the numerator of the fractional part). 7 + 74 total parts VISUALIZE Step Present the answer. Present your fraction answer: total number of parts number of parts in a whole 74 9 Step 6 Validate your answer. Validate by changing the improper fraction to a mixed number. 974 ) 7 9
Section.1 Introduction to Fractions 1 MODELS Special Case: No Remainder Convert each improper fraction: A 6 7 Step 1: 7 parts in a whole. 9 Steps -4: 76 ) Answer: 9 6 0 Step : If there is no remainder that is, if the denominator divides into the numerator evenly there will be no fractional part. The improper fraction will convert to a whole number. Validate: 9 7 (parts in a whole) = 6 (total parts) 6 7 B THINK divided by equals 1, or parts out of total parts in a whole Answer: 1 Writing Whole Numbers as Fractions When computing with a combination of whole numbers and fractions, it may be necessary to write a whole number, including the number one (1), in fraction notation. The following techniques demonstrate how you can easily do so. Each uses a Special Property of Division Involving One (1) as its starting point. TECHNIQUES First recall the Special Property of Division that states that any number divided by itself equals one (1). Therefore, in order to write the number one (1) as a fraction, the numerator and the denominator must be the same. Writing One (1) as a Fraction Technique To write one (1) as a fraction, you can write it as any number. that same number Examples: 1 = 4 THINK One whole is 4 parts out of 4 parts in a whole (as in Category 1). 4 1 = THINK One entire group is out of a group of (as in Category ).
Chapter Fractions Next, recall the Special Property of Division that states that any number divided by one (1) equals that same number. any number = that same number 1 Therefore, the simplest way to write any whole number as a fraction is to use the whole number as the numerator and one (1) as the denominator. Writing Any Whole Number as a Fraction Technique To write a given whole number as a fraction, you can simply write it as the given whole number. 1 9 Examples: 9 = 1 = 1 = 1 1 1 Additional Note: Keeping in mind that a fraction can also represent a division, you might also write the number 9, for example, as 1, or 7 6 0 4 60, or, and so on; and the number 1 as, or, or, and so on. 4 4 ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Incorrectly determining the number of parts in a whole when converting from a mixed number to an improper fraction 1 + 1 = = 6 Identify the denominator of the fraction component of the mixed number. It indicates the number of parts in a whole for conversion. 1 implies parts in a whole = 1 1 + 1 =16 Answer: 16 Validate: 16 ) 1 1 1 Incorrectly determining the total number of components in a group In a class of women and 7 men, what fractional part of the class is comprised of men? 7 The total number in a group is determined by adding all components of the group. This total becomes the denominator when representing fractional parts of the group. women + 7 men 1 total in group (class) Men comprise class. 7 1 of the
Section.1 Introduction to Fractions Issue Incorrect Process Resolution Correct Process Incorrectly identifying the number of portions when subdividing a quantity Mom has three children who will share four candy bars. How many bars will each child get? children 4 bars When subdividing a quantity, what is being divided becomes the numerator, and the number of portions becomes the denominator. 4 of a bar each = 4 In this case, four candy bars are to be shared (subdivided) among three children. 4 candy bars children (portions) or 1 1 bars per child Not comparing equal parts in a whole This rectangle has been cut into three parts. Therefore e the shaded part represents 1/ of the rectangle. Whereas the parts of a group need not be of equal size (for example, a group of students), a fraction of one whole item truly describes that fractional part of the whole only if the pieces are all the same size. In the diagram at the left, none of the pieces represents 1/ of the rectangle because the pieces are not all the same size. One third would be represented by the shaded section in the rectangle below. PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with fractions how to determine the category in which a specific fraction fits by the context in which it appears the meaning of the numerator in each of four broad categories of fractions the meaning of the denominator in each of four broad categories of fractions the process of converting between mixed numbers and improper fractions the meaning of a mixed number how to write any whole number as a fraction
Section.1 ACTIVITY Introduction to Fractions PERFORMANCE CRITERIA Interpreting a given fraction correctly correct meaning of its numerator for the context correct meaning of its denominator for the context Setting up a fraction correctly for a given context correct identification and placement of the numerator correct identification and placement of the denominator Converting between improper fractions and mixed numbers accurate presentation of the answer validation of the conversion CRITICAL THINKING QUESTIONS 1. What are the three components of a fraction?. What are the four categories of fractions identified in the Pre-Activity?. In general, how can the number one (1) be written as a fraction? Give two examples. 4. In general, how can you represent any whole number other than one (1) as a fraction? 4
Section.1 Introduction to Fractions. What is the meaning of a zero in the numerator for each of the four categories of fractions? 6. How do you determine how many parts are in a whole for a mixed number? for an improper fraction? 7. What are the relationships between the processes of converting between improper fractions and mixed numbers and their validation techniques?. What happens to the size of the fraction when the numerator stays the same but the denominator increases?
6 Chapter Fractions TIPS FOR SUCCESS When setting up a fraction for a given context, first determine the total number of parts in the whole (or group). This number becomes the denominator. Use diagrams to visualize fractions. DEMONSTRATE YOUR UNDERSTANDING 1. Draw a diagram to represent the following fractions and mixed numbers. a) as a Category 1 fraction b) as a Category fraction 7 c) 1 as a Category 1 fraction. By using an example other than those already presented, describe the relationship and meaning represented 7 by for each of the four categories of fractions discussed in this section. 9 Category 1 Category Category Category 4
Section.1 Introduction to Fractions 7. There are 1 smokers and 4 nonsmokers in a group. a) What fraction of the group are nonsmokers? b) What fraction of the group are smokers? c) What is the ratio (Category 4) of smokers to non-smokers? 4. Three diskettes are used from a box of ten. a) What fraction of the diskettes is left? b) What is the ratio (Category 4) of used diskettes to non-used diskettes?. Fill in the chart below. Improper Fraction Mixed Number Number of Parts in a Whole Validation of the Conversion a) 7 b) 6 c) 9 7 d) 10 11 1 e) 1
Chapter Fractions TEAM EXERCISES 1. A soup recipe calls for 1 tablespoons of salt. Complete the following table for this context. What is the whole unit? The whole unit is broken up into how many equal parts? How many parts are being considered? What is the fractional representation? ( forms). Write the number five () as a fraction in ten different ways. IDENTIFY AND CORRECT THE ERRORS Identify the error(s), if any, in the following worked solutions. If the worked solution is incorrect, solve the problem correctly in the third column. If the worked solution is correct, write Correct in the second column. Worked Solution What is Wrong Here? Identify the Errors Correct Process 1) What mixed number 17 is equivalent to? 4 The division is incorrect. The quotient should be 4 (not ) with a remainder of one. Four goes into 14 four times with a remainder of one. 4 417 ) 16 1 4 1 4 While is correct, it is 4 not fully reduced. Answer: 4 1 4
Section.1 Introduction to Fractions 9 Worked Solution What is Wrong Here? ) Write 1 as an improper fraction. Identify the Errors Correct Process ) Three cases of twenty-four cans each and seven more cans is equal to cases. 4) 0 sub sandwiches to be shared equally among 6 people will allow each person of a sub. ) The fractional part of this group of circles that is shaded is 6) Write the whole number three as a fraction.
40 Chapter Fractions ADDITIONAL EXERCISES 1. Draw a diagram to represent the following fractions: 7 a) as a Category 1 fraction 9 b) as a Category fraction 11 c) for Category 1 4. There are seven yellow daisies in a Happy Birthday Bouquet of twenty daisies. The rest of the daisies in the bouquet are white. a) What fraction of the daisies are yellow? b) What fraction of the daisies are white? c) What is the ratio (Category 4) of yellow daisies to white daisies?. Write the each of the following as a mixed number. a) 7 b) c) 6 4 4. Write each of the following as an improper fraction. a) b) 4 9 1 c) 14