m81285 Visualize Fractions By the end of this section, you will be able to: Understand the meaning of fractions Model improper fractions and mixed numbers Convert between improper fractions and mixed numbers Model equivalent fractions Find equivalent fractions Locate fractions and mixed numbers on the number line Order fractions and mixed numbers db65f95b-f29e-48d5-9a56-2ae1e16096b3 Before you get started, take this readiness quiz. Simplify: 5·2+1. If you missed this problem, review . 11 Fill in the blank with < or >:−2__−5 If you missed this problem, review . −2>−5

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts. In math, we write 12 to mean one out of two parts. On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has 14 of the pizza. On Wednesday, the family invites some friends over for a pizza dinner. There are a total of 12 people. If they share the pizza equally, each person would get 112 of the pizza. A fraction is written ab, where a and b are integers and b≠0. In a fraction, a is called the numerator and b is called the denominator.A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined. In , the circle has been divided into three parts of equal size. Each part represents 13 of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

Doing the Manipulative Mathematics activity Model Fractions will help you develop a better understanding of fractions, their numerators and denominators.What does the fraction 23 represent? The fraction 23 means two of three equal parts. Name the fraction of the shape that is shaded in each of the figures. We need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator. ⓐ How many equal parts are there?There are eight equal parts.How many are shaded?Five parts are shaded.Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is 58. ⓑ How many equal parts are there?There are nine equal parts.How many are shaded?Two parts are shaded.Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is 29. Name the fraction of the shape that is shaded in each figure: ⓐ 38 ⓑ 49 Name the fraction of the shape that is shaded in each figure: ⓐ 35 ⓑ 34 Shade 34 of the circle. The denominator is 4, so we divide the circle into four equal parts ⓐ.The numerator is 3, so we shade three of the four parts ⓑ. 34 of the circle is shaded. Shade 68 of the circle. Shade 25 of the rectangle. In and , we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in . Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.

We’ll be using fraction tiles to discover some basic facts about fractions. Refer to to answer the following questions: How many 12 tiles does it take to make one whole tile? It takes two halves to make a whole, so two out of two is 22=1. How many 13 tiles does it take to make one whole tile? It takes three thirds, so three out of three is 33=1. How many 14 tiles does it take to make one whole tile? It takes four fourths, so four out of four is 44=1. How many 16 tiles does it take to make one whole tile? It takes six sixths, so six out of six is 66=1. What if the whole were divided into 24 equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many 124 tiles does it take to make one whole tile? It takes 24 twenty-fourths, so 2424=1. It takes 24 twenty-fourths, so 2424=1. This leads us to the Property of One. Any number, except zero, divided by itself is one. aa=1(a≠0) Doing the Manipulative Mathematics activity "Fractions Equivalent to One" will help you develop a better understanding of fractions that are equivalent to one Use fraction circles to make wholes using the following pieces: ⓐ 4 fourths ⓑ 5 fifths ⓒ 6 sixths Use fraction circles to make wholes with the following pieces: 3 thirds. Use fraction circles to make wholes with the following pieces: 8 eighths. What if we have more fraction pieces than we need for 1 whole? We’ll look at this in the next example. Use fraction circles to make wholes using the following pieces: ⓐ 3 halves ⓑ 8 fifths ⓒ 7 thirds ⓐ 3 halves make 1 whole with 1 half left over. ⓑ 8 fifths make 1 whole with 3 fifths left over. ⓒ 7 thirds make 2 wholes with 1 third left over. Use fraction circles to make wholes with the following pieces: 5 thirds. Use fraction circles to make wholes with the following pieces: 5 halves.

In (b), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, 15, so altogether you had eight fifths, which we can write as 85. The fraction 85 is one whole, 1, plus three fifths, 35, or 135, which is read as one and three-fifths. The number 135 is called a mixed number. A mixed number consists of a whole number and a fraction. A mixed number consists of a whole number a and a fraction bc where c≠0. It is written as follows.abcc≠0 Fractions such as 54,32,55, and 73 are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as 12,37, and 1118 are proper fractions. The fraction ab is a proper fraction if a<b and an improper fraction if a≥b. Doing the Manipulative Mathematics activity "Model Improper Fractions" and "Mixed Numbers" will help you develop a better understanding of how to convert between improper fractions and mixed numbers. Name the improper fraction modeled. Then write the improper fraction as a mixed number. Each circle is divided into three pieces, so each piece is 13 of the circle. There are four pieces shaded, so there are four thirds or 43. The figure shows that we also have one whole circle and one third, which is 113. So, 43=113. Name the improper fraction. Then write it as a mixed number. 53=123 Name the improper fraction. Then write it as a mixed number. 138=158 Draw a figure to model 118. The denominator of the improper fraction is 8. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have 11 eighths. We must shade three of the eight parts of another circle. So, 118=138. Draw a figure to model 76. Draw a figure to model 65. Use a model to rewrite the improper fraction 116 as a mixed number. We start with 11 sixths (116). We know that six sixths makes one whole. 66=1 That leaves us with five more sixths, which is 56(11sixths minus6sixths is5sixths). So, 116=156. Use a model to rewrite the improper fraction as a mixed number: 97. 127 Use a model to rewrite the improper fraction as a mixed number: 74. 134 Use a model to rewrite the mixed number 145 as an improper fraction. The mixed number 145 means one whole plus four fifths. The denominator is 5, so the whole is 55. Together five fifths and four fifths equals nine fifths. So, 145=95. Use a model to rewrite the mixed number as an improper fraction: 138. 118 Use a model to rewrite the mixed number as an improper fraction: 156. 116

In , we converted the improper fraction 116 to the mixed number 156 using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the 11 pieces were left. We saw that 116 made one whole group of six sixths plus five more sixths, showing that 116=156. The division expression 116 (which can also be written as 611) tells us to find how many groups of 6 are in 11. To convert an improper fraction to a mixed number without fraction circles, we divide. Convert 116 to a mixed number. 116 Divide the denominator into the numerator. Remember 116 means 11÷6. Identify the quotient, remainder and divisor. Write the mixed number as quotientremainderdivisor. 156 So, 116=156 Convert the improper fraction to a mixed number: 137. 167. Convert the improper fraction to a mixed number: 149. 159 Divide the denominator into the numerator. Identify the quotient, remainder, and divisor. Write the mixed number as quotient remainderdivisor. Convert the improper fraction 338 to a mixed number. 338 Divide the denominator into the numerator. Remember, 338 means 833. Identify the quotient, remainder, and divisor. Write the mixed number as quotient remainderdivisor. 418 So, 338=418 Convert the improper fraction to a mixed number: 237. 327 Convert the improper fraction to a mixed number: 4811. 4411 In , we changed 145 to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths. 55+45=95 Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction. Convert the mixed number 423 to an improper fraction. 423 Multiply the whole number by the denominator. The whole number is 4 and the denominator is 3. Simplify. Add the numerator to the product. The numerator of the mixed number is 2. Simplify. Write the final sum over the original denominator. The denominator is 3. 143 Convert the mixed number to an improper fraction: 357. 267 Convert the mixed number to an improper fraction: 278. 238 Multiply the whole number by the denominator. Add the numerator to the product found in Step 1. Write the final sum over the original denominator. Convert the mixed number 1027 to an improper fraction. 1027 Multiply the whole number by the denominator. The whole number is 10 and the denominator is 7. Simplify. Add the numerator to the product. The numerator of the mixed number is 2. Simplify. Write the final sum over the original denominator. The denominator is 7. 727 Convert the mixed number to an improper fraction: 4611. 5011 Convert the mixed number to an improper fraction: 1113. 343

Let’s think about Andy and Bobby and their favorite food again. If Andy eats 12 of a pizza and Bobby eats 24 of the pizza, have they eaten the same amount of pizza? In other words, does 12=24? We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza. Equivalent fractions are fractions that have the same value. Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of and extend it to include eighths, tenths, and twelfths. Start with a 12 tile. How many fourths equal one-half? How many of the 14 tiles exactly cover the 12 tile? Since two 14 tiles cover the 12 tile, we see that 24 is the same as 12, or 24=12. How many of the 16 tiles cover the 12 tile? Since three 16 tiles cover the 12 tile, we see that 36 is the same as 12. So, 36=12. The fractions are equivalent fractions. Doing the activity "Equivalent Fractions" will help you develop a better understanding of what it means when two fractions are equivalent. Use fraction tiles to find equivalent fractions. Show your result with a figure. ⓐ How many eighths equal one-half? ⓑ How many tenths equal one-half? ⓒ How many twelfths equal one-half? ⓐ It takes four 18 tiles to exactly cover the 12 tile, so 48=12. ⓑ It takes five 110 tiles to exactly cover the 12 tile, so 510=12. ⓒ It takes six 112 tiles to exactly cover the 12 tile, so 612=12. Suppose you had tiles marked 120. How many of them would it take to equal 12? Are you thinking ten tiles? If you are, you’re right, because 1020=12. We have shown that 12,24,36,48,510,612, and 1020 are all equivalent fractions. Use fraction tiles to find equivalent fractions: How many eighths equal one-fourth? 2 Use fraction tiles to find equivalent fractions: How many twelfths equal one-fourth? 3

We used fraction tiles to show that there are many fractions equivalent to 12. For example, 24,36, and 48 are all equivalent to 12. When we lined up the fraction tiles, it took four of the 18 tiles to make the same length as a 12 tile. This showed that 48=12. See . We can show this with pizzas, too. (a) shows a single pizza, cut into two equal pieces with 12 shaded. (b) shows a second pizza of the same size, cut into eight pieces with 48 shaded.

This is another way to show that 12 is equivalent to 48. How can we use mathematics to change 12 into 48? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as: These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change. If a,b, and c are numbers where b≠0 and c≠0, then ab=a·cb·c When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half. So, we say that 12,24,36, and 1020 are equivalent fractions. Find three fractions equivalent to 25. To find a fraction equivalent to 25, we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by 2,3, and 5. So, 410,615, and 1025 are equivalent to 25. Find three fractions equivalent to 35. Correct answers include 610,915, and 1220. Find three fractions equivalent to 45. Correct answers include 810,1215, and 1620. Find a fraction with a denominator of 21 that is equivalent to 27. To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in 21. Since we can multiply 7 by 3 to get 21, we can find the equivalent fraction by multiplying both the numerator and denominator by 3. Find a fraction with a denominator of 21 that is equivalent to 67. 1821 Find a fraction with a denominator of 100 that is equivalent to 310. 30100

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values. Doing the Manipulative Mathematics activity "Number Line Part 3" will help you develop a better understanding of the location of fractions on the number line. Let us locate 15,45,3,313,74,92,5, and 83 on the number line. We will start with the whole numbers 3 and 5 because they are the easiest to plot. The proper fractions listed are 15 and 45. We know proper fractions have values less than one, so 15 and 45 are located between the whole numbers 0 and 1. The denominators are both 5, so we need to divide the segment of the number line between 0 and 1 into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as 15,25,35, and 45. Now plot points at 15 and 45. The only mixed number to plot is 313. Between what two whole numbers is 313? Remember that a mixed number is a whole number plus a proper fraction, so 313>3. Since it is greater than 3, but not a whole unit greater, 313 is between 3 and 4. We need to divide the portion of the number line between 3 and 4 into three equal pieces (thirds) and plot 313 at the first mark. Finally, look at the improper fractions 74,92, and 83. Locating these points will be easier if you change each of them to a mixed number. 74=134,92=412,83=223 Here is the number line with all the points plotted. Locate and label the following on a number line: 34,43,53,415, and 72. Start by locating the proper fraction 34. It is between 0 and 1. To do this, divide the distance between 0 and 1 into four equal parts. Then plot 34. Next, locate the mixed number 415. It is between 4 and 5 on the number line. Divide the number line between 4 and 5 into five equal parts, and then plot 415 one-fifth of the way between 4 and 5. Now locate the improper fractions 43 and 53. It is easier to plot them if we convert them to mixed numbers first. 43=113,53=123 Divide the distance between 1 and 2 into thirds. Next let us plot 72. We write it as a mixed number, 72=312. Plot it between 3 and 4. The number line shows all the numbers located on the number line. Locate and label the following on a number line: 13,54,74,235,92. Locate and label the following on a number line: 23,52,94,114,325. In Introduction to Integers, we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of 7 is −7 and the opposite of −7 is 7. Fractions have opposites, too. The opposite of 34 is −34. It is the same distance from 0 on the number line, but on the opposite side of 0. Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate −158 on the number line, first think of where 158 is located. It is an improper fraction, so we first convert it to the mixed number 178 and see that it will be between 1 and 2 on the number line. So its opposite, −158, will be between −1 and −2 on the number line. Locate and label the following on the number line: 14,−14,113,−113,52, and −52. Draw a number line. Mark 0 in the middle and then mark several units to the left and right. To locate 14, divide the interval between 0 and 1 into four equal parts. Each part represents one-quarter of the distance. So plot 14 at the first mark. To locate −14, divide the interval between 0 and −1 into four equal parts. Plot −14 at the first mark to the left of 0. Since 113 is between 1 and 2, divide the interval between 1 and 2 into three equal parts. Plot 113 at the first mark to the right of 1. Then since −113 is the opposite of 113 it is between −1 and −2. Divide the interval between −1 and −2 into three equal parts. Plot −113 at the first mark to the left of −1. To locate 52 and −52, it may be helpful to rewrite them as the mixed numbers 212 and −212. Since 212 is between 2 and 3, divide the interval between 2 and 3 into two equal parts. Plot 52 at the mark. Then since −212 is between −2 and −3, divide the interval between −2 and −3 into two equal parts. Plot −52 at the mark. Locate and label each of the given fractions on a number line: 23,−23,214,−214,32,−32 Locate and label each of the given fractions on a number line: 34,−34,112,−112,73,−73

We can use the inequality symbols to order fractions. Remember that a>b means that a is to the right of b on the number line. As we move from left to right on a number line, the values increase. Order each of the following pairs of numbers, using < or >: ⓐ −23____−1 ⓑ −312____−3 ⓒ −37____−38 ⓓ −2____−169 ⓐ −23>−1 ⓑ −312<−3 ⓒ −37<−38 ⓓ −2<−169 Order each of the following pairs of numbers, using < or >: ⓐ −13__−1 ⓑ −112__−2 ⓒ −23__−13 ⓓ −3__−73 ⓐ >ⓑ >ⓒ <ⓓ < Order each of the following pairs of numbers, using < or >: ⓐ −3__−175 ⓑ −214__−2 ⓒ −35__−45 ⓓ −4__−103 ⓐ >ⓑ <ⓒ >ⓓ < Introduction to Fractions Identify Fractions Using Pattern Blocks

Property of OneAny number, except zero, divided by itself is one.aa=1, where a≠0. Mixed NumbersA mixed number consists of a whole number a and a fraction bc where c≠0. It is written as follows: abcc≠0Proper and Improper FractionsThe fraction ab is a proper fraction if a<b and an improper fraction if a≥b. Convert an improper fraction to a mixed number.Divide the denominator into the numerator. Identify the quotient, remainder, and divisor. Write the mixed number as quotientremainderdivisor. Convert a mixed number to an improper fraction.Multiply the whole number by the denominator.Add the numerator to the product found in Step 1.Write the final sum over the original denominator. Equivalent Fractions PropertyIf a, b, and c are numbers where b≠0, c≠0, then ab=a⋅cb⋅c.

In the following exercises, name the fraction of each figure that is shaded. ⓐ 14 ⓑ 34 ⓒ 38 ⓓ 58 In the following exercises, shade parts of circles or squares to model the following fractions. 12 13 34 25 56 78 58 710 In the following exercises, use fraction circles to make wholes using the following pieces. 3 thirds 8 eighths 7 sixths 4 thirds 7 fifths 7 fourths In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number. ⓐ 54=114 ⓑ 74=134 ⓒ 118=138 ⓐ 114=234 ⓑ 198=238 In the following exercises, draw fraction circles to model the given fraction. 33 44 74 53 116 138 103 94 In the following exercises, rewrite the improper fraction as a mixed number. 32 53 123 114 135 235 256 289 319 4213 4715 3215 In the following exercises, rewrite the mixed number as an improper fraction. 123 125 75 214 256 176 279 257 197 347 359 329 In the following exercises, use fraction tiles or draw a figure to find equivalent fractions. How many sixths equal one-third? How many twelfths equal one-third? 4 How many eighths equal three-fourths? How many twelfths equal three-fourths? 9 How many fourths equal three-halves? How many sixths equal three-halves? 9 In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra. 14 13 Answers may vary. Correct answers include 26,39,412. 38 56 Answers may vary. Correct answers include 1012,1518,2024. 27 59 Answers may vary. Correct answers include 1018,1527,2036. In the following exercises, plot the numbers on a number line. 23,54,125 13,74,135 14,95,113 710,52,138,3 213,−213 134,−135 34,−34,123,−123,52,−52 25,−25,134,−134,83,−83 In the following exercises, order each of the following pairs of numbers, using < or >. −1__−14 −1__−13 < −212__−3 −134__−2 > −512__−712 −910__−310 < −3__−135 −4__−236 <

Music Measures A choreographed dance is broken into counts. A 11 count has one step in a count, a 12 count has two steps in a count and a 13 count has three steps in a count. How many steps would be in a 15 count? What type of count has four steps in it? Music Measures Fractions are used often in music. In 44 time, there are four quarter notes in one measure. ⓐ How many measures would eight quarter notes make? ⓑ The song “Happy Birthday to You” has 25 quarter notes. How many measures are there in “Happy Birthday to You?” ⓐ 2ⓑ 614 Baking Nina is making five pans of fudge to serve after a music recital. For each pan, she needs 12 cup of walnuts. ⓐ How many cups of walnuts does she need for five pans of fudge? ⓑ Do you think it is easier to measure this amount when you use an improper fraction or a mixed number? Why?

Give an example from your life experience (outside of school) where it was important to understand fractions. Answers will vary. Explain how you locate the improper fraction 214 on a number line on which only the whole numbers from 0 through 10 are marked.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

equivalent fractions Equivalent fractions are two or more fractions that have the same value. fraction A fraction is written ab. In a fraction, a is the numerator and b is the denominator. A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included. mixed number A mixed number consists of a whole number a and a fraction bc where c≠0. It is written as abc, where c≠0. proper and improper fractions The fraction ab is proper if a<b and improper if a≥b.