5th Grade. Fraction Operations Part 2.
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- Maryann Barber
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1 1
2 5th Grade Fraction Operations Part
3 Multiplying Fractions Table of Contents click on the topic to go to that section Multiplying Fractions and Whole Numbers Multiplying with Mied Numbers Interpreting Multiplication of Fractions Area of fractional side length rectangles Dividing Unit Fractions by Whole Numbers Dividing Whole Numbers by Unit Fractions Line Plots with Fractional Data 3
4 Multiplying Fractions Return to Table of Contents 4
5 Folding You can use folding to show a fractional part of a fraction. (You will need a set of Fraction Pieces.) 1 Here is how to find of Use the fraction piece Fold it in half. 3. Compare the folded part with the other fraction pieces to find a piece that matches. click to reveal 1 2 of 1 3 is 1 6 Slide over bar below to show equality. 5
6 You can use shading to find a fractional part of another fraction. Shading Here is how to find 1 of Divide your whole into fourths. 2. Shade a third of What fractional part of the whole did you shade? click
7 Problem is from: Click for link for commentary and solution. 7
8 Multiplying Fractions Multiplication describes events when equal groups of things are combined together. For eample, 3 groups of 4 apples can be abbreviated using the number sentence 3 4 =?. Multiplication number sentences should be read as, "What is (variable) groups of (variable) things?" The variable can be a whole number, a rational number, or any epression that represents either a number of groups, or the number of items in a group. We will be working with variables that are fractions. Read this number sentence aloud using the phrase, "What is (variable) groups of (variable) things?" = 8
9 Multiplying Using Area Models Multiplication of two factors can be illustrated using an array called an area model. When you make an array, each factor represents either the vertical or horizontal dimension of a square or a rectangle. 3 4 With the above eample, a rectangle can be constructed with 3 units on one side of the array, and 4 units on the other side. The resulting three by four rectangle contains 12 square units. The result can be read, "12 is 3 groups of 4". 9
10 Review from Decimal Computation Unit Using an Area Model to Multiply Decimals How can we turn this model showing 2 3 into a model showing 2 3.5? Click How many square tiles is it now? What number sentence represents the number of square tiles? What if we add another row? What number sentence will represent the number of square tiles? 10
11 Lets label this model. (Click to remove boes) 2 3 tenths 2 Review from Decimal Computation Unit 4? 6 tenths? 4 tenths 8 tenths? What is the sum? Click 12 hundredths What are these? 11
12 Rectangle Model The array or area model is also very effective when illustrating multiplication of two fractions. When multiplying 1/2 5/4, the question can be read, "What is 1/2 of 5/4?" or "What is 1/2 of a group of 5/4?" Lets show a group of 5/4. Now, lets show 1/2 of that. 5/8 12
13 Interactive Area Model Use the interactive model to show 1/2 of 5/4. 1. Use the arrow keys to section off one dimension of a whole into 2 parts, and the other dimension into 4 parts. A 2 by 4 array is a rectangle with 8 units making up a whole group. 2. Using the slider bar, highlight 1/2 on the side divided into 2 parts, and highlight 5/4 on the side divided into 4 parts. The intersection of these shaded parts represents the answer to the question, "What is 1/2 of 5/4?" The answer can be read, "5/8 is 1/2 of 5/4." Teacher Notes (Click for interactive site) 13
14 Multiplying Using the Algorithm To multiply fractions, multiply the numerators, then multiply the denominators. and Make sure you simplify your answer! = = = 3 5 Sketch an area model to check your answer. 14
15 Multiply Eamples Sketch an area model to check your answer click to reveal = = click to reveal = = = ( 6 ) 7 click to reveal = 8(6) 14(7) = =
16 = 16
17 = 17
18 = 18
19 ( 5 ) 6 = 19
20 5 4 9 ( 3 ) 8 = 20
21 Internet Link for More Practice 21
22 5.NF Drinking Juice Problem is from: Click for link for commentary and solution. Alisa had 1/2 a liter of juice in a bottle. She drank 3/4 of the juice that was in the bottle. How many liters did she drink? 22
23 6 At Cliffords s school, of the students wanted to learn about the new science museum that was just built. This month, of the students were able to go see it. What fraction of students were able to go see it right away? 23
24 7 A bridge span is of a mile long. Workers have repainted of it. How much has been repainted? 24
25 5.NF Running to School Problem is from: Click for link for commentary and solution. 8 The distance between Rosa's house and her school is mile. She ran run? of the way to school. How many miles did she 25
26 Simplify without simplifying Sometimes you can simplify prior to multiplying. with simplifying
27 Simplify Sometimes you can cross simplify prior to multiplying. without cross simplifying 1 with cross simplifying 3 27
28 9Can this problem be simplified or cross simplified? Yes No Answer 28
29 10Can this problem be simplified or cross simplified? Yes No Answer 29
30 11Can this problem be simplified or cross simplified? Yes No Answer 30
31 12Can this problem be simplified or cross simplified? Yes No Answer 31
32 13Can this problem be simplified or cross simplified? Yes No Answer 32
33 14 Solve the problem. Simplify prior to multiplying if you can. 33
34 15 Solve the problem. Simplify prior to multiplying if you can. 34
35 16 Solve the problem. Simplify prior to multiplying if you can. 35
36 17 Solve the problem. Simplify prior to multiplying if you can. 36
37 18 Solve the problem. Simplify prior to multiplying if you can. 37
38 19 Solve. From PARCC EOY sample test #25 38
39 Multiplying Fractions and Whole Numbers Return to Table of Contents 39
40 5.NF Connor and Makayla Discuss Multiplication Problem is from: Click for link for commentary and solution. 40
41 Whole Number times a Fraction To multiply fractions with whole numbers, write the whole number as a fraction (over 1) then multiply the two fractions. W rite your answer in simplest form. 6 4 = 6 4 = 6 4 = 24 = 6 2 = Alternate Method of canceling components = 4 = 8 = ( ) 5 7 = 3 7 = 1 5 ( ) 21 5 =
42 = True False 42
43 A C B 12 7 D
44 A 32 3 C 96 9 B D
45 23 On Wednesday morning, of Sue s classmates put blueberries on their cereal. If there are 27 students in Sue s class, how many put blueberries on their cereal? 45
46 24 Of the 49 kids on the campout, wanted to go to bed right after the sing along around the fire. How many kids wanted to go to bed right away? 46
47 Multiplying with Mied Numbers Return to Table of Contents 47
48 Multiplying Mied Numbers To multiply fractions with mied numbers, write the mied numbers as an improper fractions, then multiply the two fractions. Make sure you write your answer in simplest form = = 11 7 = 77 = ( ) 1 = = ( ) 20 3 =
49 = True False 49
50 A C 44 B D
51 27 ( 5 5 ) (3 ) A C B D
52 5.NF Half of a Recipe Problem is from: Click for link for commentary and solution. Kendra is making 1/2 of a recipe. The full recipe calls for 3 1/4 cup of flour. How many cups of flour should Kendra use? 52
53 28 A boat was traveling 12 miles each hour. At that rate, how many miles would it travel in 1 hours? 53
54 29 Riding her bike, Terry averages 9 miles per hour. At that speed, how far could she go in 2 hours? 54
55 Interpreting Multiplication of Fractions Return to Table of Contents 55
56 Interpreting Multiplication You can determine the relative size of the product of a multiplication problem without actually multiplying. 56
57 When you multiply a given number by a fraction greater than 1, it will result in a product greater than the given number. Eamples: Interpreting Multiplication 57
58 30 Which of the following product(s) are greater than 700,000? A B C D 58
59 31 Which of the following product(s) are greater than 876? A B C D 59
60 When you multiply a given number by a fraction less than 1, it will result in a product smaller than the given number. Eamples: Interpreting Multiplication 60
61 32 Which of the following product(s) are less than 555? A B C D 61
62 33 Which of the following product(s) are less than 4,321? A B C D 62
63 5.NF Reasoning about Multiplication Problem is from: Click for link for commentary and solution. 63
64 5.NF Calculator Trouble Problem is from: Click for link for commentary and solution. 64
65 5.NF Fundraising Problem is from: Click for link for commentary and solution. Hint draw a picture. 65
66 5.NF Grass Seedlings Problem is from: Click for link for commentary and solution. The students in Raul's class were growing grass seedlings in different conditions for a science project. He noticed that Pablo's seedlings were times as tall as his own seedlings. He also saw that Celina's seedlings were as tall as his own. Which of the seedlings shown below must belong to which student? Eplain your reasoning. 66
67 5.NF Running a Mile Problem is from: 34 Curt and Ian both ran a mile. Curt's time was Ian's time. Who ran faster? Eplain and draw a picture. A Curt B Ian Click for link for commentary and solution. 67
68 35 Three friends are comparing their hair length. Abby's hair is the length of Beth's hair, and Carol's hair is the length of Beth's hair. Who's hair is the shortest? Eplain. A Abby B Beth C Carol 68
69 36 Four friends have started collecting coins. Chris has the amount Dan has. Ben has Ale has most coins? A Ale B Ben C Chris D Dan times what Chris has. the amount that Dan has. Who has the 69
70 37 Select a phrase to correctly complete each sentence. The product of and 4 is 4. A less than B equal to C greater than From PARCC EOY sample test #29 70
71 38 Select a phrase to correctly complete each sentence. The product of and 2 is 2. A less than B equal to C greater than From PARCC EOY sample test #29 71
72 39 Select a phrase to correctly complete each sentence. The product of and is. A less than B equal to C greater than From PARCC EOY sample test #29 72
73 Area of Fractional Side Length Rectangles Return to Table of Contents 73
74 Part 1 Find the area of rectangles with mied number side lengths. 74
75 REVIEW On grid paper, make a rectangle that has sides of 3 units by 2 units. How many unit squares would you need to cover the square? unit square Teacher Notes 75
76 On grid paper, make a rectangle that has sides of units by 2 units. What is the area now? unit square click for answer 76
77 On grid paper, make a rectangle that has sides of 2 units by units. What is the area? unit square click for answer 77
78 40 How many unit squares will it take to cover a rectangle that is 3 units long and units wide? 78
79 41 A tablecloth has dimensions of feet by 6 feet. What is the area of the tablecloth in square feet? 79
80 42 A banner is being made to hang in the gym It needs to be 5 meters by meters. What is the area of the banner? 80
81 43 A rectangular patio design is shown below. What is the area of this patio? 81
82 Part 2 Find the area of rectangles with fractional side lengths. 82
83 A field measures a mile wide by mile long. What is the area in square miles of the field? Draw a Square Steps: Divide the left edge into 3 equal parts and label one part 1/3. Divide the top edge into 2 equal parts and label one part 1/2. Draw lines across the square for the 1/3 mark and the 1/2 marks to subdivide the square into small rectangles. How many rectangles are there? If the area of the large square has an area of 1 square mile, what is the area of one of the small rectangles? Can the area of one of the small rectangles be found by multiplying the lengths of its sides? 83
84 A field measures a mile wide by mile long. What is the area in square miles of the field? Steps: Divide the left edge into 3 equal parts and label one part 1/3. Divide the top edge into 2 equal parts and label one part 1/2. Draw lines across the square for the 1/3 mark and the 1/2 marks to subdivide the square into small rectangles. How many rectangles are there? If the area of the large square has an area of 1 square mile, what is the area of one of the small rectangles? Can the area of one of the small rectangles be found by multiplying the lengths of its sides? 84
85 A field measures a mile wide by mile long. What is the area in square miles of the field? Steps: Divide the left edge into 3 equal parts and label one part 1/3. Divide the top edge into 2 equal parts and label one part 1/2. Draw lines across the square for the 1/3 mark and the 1/2 marks to subdivide the square into small rectangles. How many rectangles are there? 6 If the area of the large square has an area of 1 square mile, what is the area of one of the small rectangles? 1/6 Can the area of one of the small rectangles be found by multiplying the lengths of its sides? yes of a square mile 85
86 A foot by foot rectangular piece of wood is cut from a 3 foot by 3 foot piece. Steps: Use a 3 by 3 square on grid paper. Subdivide it and label as shown. The area is the sum of the labeled parts. Can the area also be found by computing the products? 86
87 A foot by foot rectangular piece of wood is cut from a 3 foot by 3 foot piece. Steps: Use a 3 by 3 square on grid paper. Subdivide it and label as shown. The area is the sum of the labeled parts. Can the area also be found by computing the products? yes 87
88 44 How many unit squares will it take to cover a square that has units long side lengths? 88
89 45 How many square meters of carpet will it take to cover a rectanglular room that is meters by meters? 89
90 46 Find the area of the rectangle below. 90
91 47 Mr. Fernandez is building a small table. He has a rectangular piece of wood for the top that is 8 feet by 4 feet. If he cuts the piece for the table top to be feet by feet, how many square feet of wood will be have left? 91
92 48 Jen makes a rectangular banner. It is yard long and yard wide. What is the area, in square yards, of the banner? From PARCC EOY sample test #19 92
93 49 Kurt drew a rectangular maze with a length of foot and a width of foot. What is the area, in square feet, of Kurt's maze? From PARCC EOY sample test #23 93
94 Dividing Unit Fractions by Whole Numbers Return to Table of Contents 94
95 Visual Fraction Model Four students are sitting together. They are given of a cake to share equally among themselves. How much will each student get of the cake if they share the of it equally? whole cake 1 3 cake they would each get 1/12 of the whole cake 95
96 Visual Fraction Model You have one half of a bag of popcorn to share evenly among 3 people. How much of the bag does each person get? 1/3 1/3 1/3 1/2 bag 1/2 bag This darker shaded part shows each person will get 1/6 of the bag of popcorn. 96
97 50 One half of a room is painted. Each of four people did the same amount of painting. How much of the room did each person paint? 97
98 51 After the barbecue there is of a watermelon left. If 5 people evenly share it, how much of the whole watermelon will each person get? 98
99 52 Each table gets one third of a bottle of paint in Art Class. If there are 3 people at a table, how much of the bottle will each of them get? 99
100 53 Two brothers want to evenly share the of the apple pie that is left. How much will each of them get? 100
101 Dividing Whole Numbers by Unit Fractions Return to Table of Contents 101
102 Dividing Fractions When dividing fractions problems will answer one of two questions. 1. How many groups? or 2. How many in each group? 102
103 1. How many groups? There are 6 cups of raisins in a bo. Each serving is one fourth of a cup. How many servings are in the bo? To solve the problem we need to find out how many servings (groups) are in the bo. Draw a picture to solve. Answer 4 servings 3 servings 2 servings 1 serving There are 4 servings in 1 cup, how many servings in 6 cups? 103
104 2. How many in each group? Oscar has 3 red markers. If one fifth of Oscar's markers are red, how many markers does Oscar have? To solve this problem, we need to find out the size of the whole group, since we know how many are in each group. Draw a picture to solve. Click to show whole group If 3 markers are 1/5, then there are markers in the whole group. 104
105 5.NF How many servings of oatmeal? Problem is from: Click for link for commentary and solution. 54 A package contains 4 cups of oatmeal. There is cup of oatmeal in each serving. How many servings of oatmeal are there in the package? Draw a picture to illustrate your solution. 105
106 5.NF How many marbles? Problem is from: Click for link for commentary and solution. 55 Julius has 4 blue marbles. If one third of Julius' marbles are blue, how many marbles does Julius have? Draw a picture to illustrate your solution. 106
107 How are the pictures that represent the last two problems different? 107
108 56 There were a total of 2 pounds of apples to make the apple treats. Each treat contained 1/5 of a pound of apples. How many treats were made? 108
109 57 Eva s classmates bought 5 pizzas to help celebrate all their successes in math. Each student received a slice that was 1/8 of a pizza. How many slices did they cut up? 109
110 58 There are 5 red soccer balls in the school gym. The rest of the balls are multi colored. If the red balls represent 1/4 of the total soccer balls, how many are in the gym? 110
111 59 Jim uses ribbon to make bookmarks. Jim has 9 feet of ribbon. He uses 1/3 foot of ribbon to make each bookmark. What is the total number of bookmarks Jim makes with 9 feet of ribbon? From PARCC EOY sample test #1 111
112 60 Mr. Edwards is making sandwiches. He has 4 pounds of cheese. He puts 1/8 pound of cheese in each sandwich. What is the total number of sandwiches Mr. Edwards makes using all 4 pounds of cheese? From PARCC EOY sample test #30 112
113 Line Plots With Fractional Data Return to Table of Contents 113
114 Number Lines Before we begin working with line plots, lets review number lines. When placing numbers and/or fractions it is important to see how the number line is divided. They will not always be the same! What number would be between the whole numbers? What number would be between these whole numbers?
115 61 What number is missing? ? 115
116 62 Which number represents the red dot correctly? A B C 116
117 63 Which number represents the red dot correctly? A B C 117
118 0 1 2 Place these fractions on the number line
119 Line Plot A line plot is a number line with marks that show the frequency of data. Eample: 1 2 Length of Girls Hair in Inches The count of "" marks above each score represents the number of girls who have that length hair. 119
120 120 What is the length represented by the red dot? How many girls have that length of hair? 1 2 Length of Girls Hair in Inches Line Plot
121 64 What is the hair length that is represented by the red dot? 1 2 Length of Girls Hair in Inches 121
122 65 How many girls have hair that is 1 3 inches in 4 length? 1 2 Length of Girls Hair in Inches 122
123 66 What length of hair do the most girls have? 1 2 Length of Girls Hair in Inches 123
124 Line Plot Rainfall was collected, and measured in inches, over several days. The table below shows the number of times each amount was collected. Amount of Rainfall (In Inches) 1 1/8 1 1/4 1 3/8 1 1/2 1 5/8 1 3/4 2 Number of times Make and label a line plot to display the data. 124
125 Line Plot 0 1 Number of Hours The line plot above shows the number of hours that Jeremy spent doing homework each night last week. You will use this line plot to answer the net 5 questions. 125
126 67 What is the difference between the greatest number of hours that Jeremy spent on his homework and the number of hours Jeremy spent on his homework most frequently? 0 1 Number of Hours 126
127 68 What is the difference between the greatest number of hours that Jeremy spent on his homework and the least? 0 1 Number of Hours 127
128 69 What is the total number of hours that Jeremy spent on his homework? 0 1 Number of Hours 128
129 70 Jeremy said he will need to spend twice the number of hours on his homework this week as he did last week. How many hours will he spend on his homework? 0 1 Number of Hours 129
130 71 If Jeremy spends the same amount of time every week on his homework for the net 8 weeks, how many total hours will he have spent doing homework? 0 1 Number of Hours 130
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