Improper Fractions: Parts of Sets Pathway 1 OPEN-ENDED Markers sometimes come in boxes of 8. A full box is 1 whole. markers would fill 8 of a box. 1 marker would fill 1 8 of a box. 8 and 1 8 are proper fractions. 9 markers would fill a whole box and 1 8 more of a box. So 9 8 5 1 1 8. You can write the mixed number 1 1 8 or the improper fraction 9 8. proper fraction a fraction less than 1 whole e.g., 8 mixed number a number greater than 1 that is made up of a whole number and a fraction part e.g., 1 improper fraction a fraction greater than 1 whole where the numerator is greater than the denominator e.g., 7 6 Step 1: Use the numbers 2,, 8, 10, 22, and 1 in the spaces below to create 6 different improper fractions. Improper Fractions: Parts of Sets, Pathway 1 97
Step 2: Draw a picture to show each fraction from Step 1. The whole in each picture must be a package that holds more than 1 item. Label each picture with the correct mixed number. Step : Choose 2 of your fractions from Step 1. Tell how the fractions are alike. Tell how their pictures are alike. Step : Repeat Step with 2 other fractions from Step 1. 98 Improper Fractions: Parts of Sets, Pathway 1
Improper Fractions: Parts of Sets Pathway 1 GUIDED Light bulbs sometimes come in packages of 6. light bulbs would fill 1 light bulb would fill 6 of a package. 1 6 of a package. 6 and 1 6 are proper fractions. If you had 11 light bulbs in packages of 6, you would have 11 of a package. 11 6 6 is called an improper fraction. The 11 tells you that there are 11 light bulbs. The 6 tells you that 6 light bulbs make 1 whole package. proper fraction a fraction less than 1 whole e.g., 8 improper fraction a fraction greater than 1 whole where the numerator is greater than the denominator e.g., 7 6 11 6 is also 1 5 6. 1 5 6 is called a mixed number. It means 1 whole 1 5 6 of another whole. What fraction would describe 1 whole package of light bulbs? What improper fraction and mixed number do you see in the picture below? The eggs in 1 carton represent 1 whole. mixed number a number greater than 1 that is made up of a whole number and a fraction part e.g., 1 Improper Fractions: Parts of Sets, Pathway 1 99
Try These 1. In each picture below, the package shows the number of items in 1 whole. How many full and partial packages could you fill with the loose items on the right? Use an improper fraction and a mixed number. a) 1 whole improper fraction: mixed number: b) 1 whole improper fraction: mixed number: c) 1 whole improper fraction: mixed number: d) 1 whole improper fraction: mixed number: 100 Improper Fractions: Parts of Sets, Pathway 1 Leap SR_Topic 7.indd 100 25/0/11 10:5 AM
2. Cookie packages come in different sizes. One whole package might contain cookies, or 6, 8, 10, or 12 cookies. How many packages would 15 cookies fill? Complete the chart for each package size. The first row is completed for you. 15 Cookies in Packages Package size (number of cookies in 1 whole) 6 Fraction of full packages (improper fraction) 15 Number of full and partial packages (mixed number) 8 10 12. Draw a picture to show 1 whole package in the first box. Draw a picture of the improper fraction in the second box. a) 7 5 of a package of cookies The whole: Improper fraction A 7 5 B: b) 8 of a package of cookies The whole: Improper fraction A 8 B: Improper Fractions: Parts of Sets, Pathway 1 101
. The fraction 12 j is an improper fraction. a) Write possible denominators for the improper fraction. 12 12 12 b) Write a mixed number for each improper fraction in a). c) Which fractions from part a) are greater than 2 wholes? 5. What could the numerator of each fraction be? a) j 6 is between 2 and wholes. could be. 6. j b) j is between and wholes. could be. c) j 5 is between and 5 wholes. could be. 2 and j are both between 2 and wholes. The value of is the same for both fractions. What might be? 5 7. Why might one person say that this picture shows 5 6 but another say that it shows 5 2? 102 Improper Fractions: Parts of Sets, Pathway 1
8. Sketch a picture that shows each of the following. a) 12 5 of a group of objects is more than 2 groups of the objects. b) 11 of a group of objects is less than groups of the objects. 9. You have 20 hockey cards in more than packages. Name a possible improper fraction and a mixed number for this situation. Tell how many cards are in 1 whole package. FYI Lots of things come in packages. Fractions are useful for describing amounts when some packages are not full. Improper Fractions: Parts of Sets, Pathway 1 10
Improper Fractions: Parts of Wholes Pathway 2 OPEN-ENDED Raj spent 5 1 2 hours at soccer practice this week. Kyle spent 2 hours. Alyssa spent 2 hours. Each number above is a mixed number. Each number can also be written as an improper fraction. For example, Alyssa s 2 11 hours is the same as hours. Count the thirds in the picture below. 11 12 1 11 12 1 11 12 1 11 12 1 10 2 10 2 10 2 10 2 9 9 9 9 8 8 8 8 7 6 5 7 6 5 7 6 5 7 6 5 Write an improper fraction for each mixed number of hours that Raj and Kyle spent at practice. Raj: 5 1 2 5 Kyle: 2 5 Alyssa: 2 5 11 What do you notice about the improper fractions? You will need fraction pieces mixed number a number greater than 1 that is made up of a whole number and a fraction part e.g., 1 improper fraction a fraction greater than 1 whole where the numerator is greater than the denominator e.g., 7 6 Fraction Set 1 Make up a set of mixed numbers that have something in common. What do the mixed numbers have in common? Draw a picture for 1 mixed number from the set. Write the mixed numbers as improper fractions. What do the improper fractions have in common? mixed numbers What do the mixed numbers have in common? Picture of 1 mixed number improper fractions What do the improper fractions have in common? 10 Improper Fractions: Parts of Wholes, Pathway 2
Fraction Set 2 Repeat the instructions for fraction set 1 with another set of mixed numbers. mixed numbers What do the mixed numbers have in common? Picture of 1 mixed number improper fractions What do the improper fractions have in common? Fraction Set Repeat the instructions for fraction set 1 with another set of mixed numbers. mixed numbers What do the mixed numbers have in common? Picture of 1 mixed number improper fractions What do the improper fractions have in common? Improper Fractions: Parts of Wholes, Pathway 2 105
Improper Fractions: Parts of Wholes Pathway 2 GUIDED Jasmine ran around the track 2 and 1 times. You can write 2 1 as 9, since she ran around 9 quarters of the track. is called a mixed number. 2 1 9 is called an improper fraction. You can show this on a number line, too. once around twice around You will need fraction pieces mixed number a number greater than 1 that is made up of a whole number and a fraction part e.g., 1 improper fraction a fraction greater than 1 whole where the numerator is greater than the denominator e.g., 7 6 0 0 1 Notice that 9 5 1 1 1. That s why 9 is the same as 2 1. 2 8 9 Try These 1. Show each fraction on the number line. For part c), also mark the position for 1. Fractions like,, and so on are exactly 1. a) 7 0 1 2 b) 9 5 0 1 2 c) 7 0 106 Improper Fractions: Parts of Wholes, Pathway 2
2. Shade the shapes to show each fraction. a) 5 2 b) 10 1 whole 1 whole c) 9 1 whole. Explain why 1 2 5 7 2. Use the pitchers of juice to help you.. How many whole pitchers would be full if you filled pitchers for each fraction? a) 9 2 b) 15 pitchers: c) 11 pitchers: d) 12 5 pitchers: pitchers: 5. 7 quarters make $1.75. How does this help you explain why 7 5 1? Improper Fractions: Parts of Wholes, Pathway 2 107
6. Why might one person describe this picture as 9 5, but another person might describe it as 9 10? 7. Draw a picture to show each relationship. a) 1 1 5 of 1 whole is twice as much as 5 of that same whole. b) 2 2 5 of 1 whole is groups of 5 of that same whole. 8. Use an improper fraction and a mixed number to describe each number. a) a number greater than but less than improper fraction: mixed number: b) a number a lot more than 5 improper fraction: c) a number closer to 5 than to 6 mixed number: improper fraction: mixed number: d) a fraction with a numerator that is 10 more than the denominator improper fraction: mixed number: FYI We often use fractions to describe parts of wholes. It is helpful to have a mental picture of those fractions to understand how much the fraction represents. 108 Improper Fractions: Parts of Wholes, Pathway 2
Proper Fractions: Parts of Sets Pathway OPEN-ENDED Boats sometimes have 8 seats to hold 8 rowers. rowers would fill 8 of the seats. The numerator tells the number of rowers there are. The denominator 8 tells how many would fill the seats. 1 rower would fill 1 8 of the seats. Imagine containers that hold different numbers of things and are partly full. For example, a package that holds 10 pencils could be 2 10 full. The numerator of the fraction is the top number. The denominator is the bottom number. d numerator d denominator Think of 6 containers that hold different numbers of things. Create 6 fractions to describe how full the containers are. Use the numbers 2,, 5, 6, 8, and 10 for the denominators or the numerators. The numerators should not be greater than the denominators. Proper Fractions: Parts of Sets, Pathway 109
Sketch a picture to show each of your 6 fractions. Label each picture. Choose 2 of your fractions. Tell how the fractions are alike and different. Tell how their pictures are alike and different. Repeat with 2 other fractions. 110 Proper Fractions: Parts of Sets, Pathway
Proper Fractions: Parts of Sets Pathway GUIDED Sometimes students work in groups of. The girl is 1 of the number of people in the group. The numerator 1 tells the number of students you are talking about. The denominator tells the number of students in the whole group. Each boy is 1 of the number of people in the group. Together, the boys are of the group. Adults are 0 of the number of people in the group. Students are of the number of people in the group. The items in a group do not have to be the same size to use a fraction. Suppose a group has adults and 2 children. What fractions can you use to describe the numbers of adults and children in the group below? The numerator of the fraction is the top number. The denominator is the bottom number. d numerator d denominator Proper Fractions: Parts of Sets, Pathway 111
Try These 1. What fraction of each group are oranges? a) b) c) d) 2. Why might some people say this picture shows 56 but others say it shows 16?. What could the numerator of each fraction be?. a) j 6 is a small part of the group of 6. could be. b) j is most of the group of. could be. c) j 5 is around half of the group of 5. could be 112 Proper Fractions: Parts of Sets, Pathway Leap SR_Topic 7.indd 112 25/0/11 10:55 AM
. Sketch a group of circles and squares to represent each fraction given. Tell what other fraction the picture shows. a) 5 are circles. Picture: Other fraction: b) 0 6 are circles. Picture: Other fraction: c) are circles. Picture: Other fraction: 5. Draw a picture to show each fraction as part of a group. What do the pictures have in common? 2 2 5 2 6 Proper Fractions: Parts of Sets, Pathway 11
6. Draw a picture to show that 5 of a group is twice as much as 2 5 of that group. 7. Sketch a picture of stars and planets to fit the rules below. Name the fraction of the number of shapes that are stars. a) There are 8 shapes altogether. b) There are more stars than planets. c) There are almost all planets. FYI Fractions are useful when a group has lots of parts and you want to describe just some of those parts. 11 Proper Fractions: Parts of Sets, Pathway
Proper Fractions: Parts of Wholes Pathway OPEN-ENDED Ahmed drank 1 of his glass of juice. Lisa drank 5 of her glass of juice. Yoshi drank 6 8 of his glass of juice. Daniella drank 8 10 of her glass of juice. Draw a picture for each fraction. You will need linking cubes fraction pieces What is the same about all of your pictures? Make up at least other sets of fractions that have something in common. Tell what the fractions have in common. Draw pictures for each set of fractions. The numerator of the fraction is the top number. The denominator is the bottom number. d numerator d denominator Proper Fractions: Parts of Wholes, Pathway 115
Proper Fractions: Parts of Wholes Pathway GUIDED Ian has cut the grass in 2 of the lawn. The numerator 2 tells how many lawn sections are cut. The denominator tells how many equal lawn sections there are altogether. You will need linking cubes fraction pieces coloured pencils You can show 2 on a number line or as part of a container. 0 1 2 1 The numerator of the fraction is the top number. The denominator is the bottom number. d numerator d denominator How can you show 5 on a number line? How can you show 5 as the amount of juice in a pitcher? Try These 1. Show and label each fraction on the number line. a) 0 1 b) 2 5 0 1 c) 1 0 1 116 Proper Fractions: Parts of Wholes, Pathway
2. Write fractions to describe the shaded and the unshaded parts of each shape. a) d) shaded: unshaded: shaded: unshaded: b) e) shaded: unshaded: shaded: unshaded: c) f) shaded: unshaded: shaded: unshaded:. Why might one person say this picture shows 5 but another say it shows 1 5? Proper Fractions: Parts of Wholes, Pathway 117
. Colour the picture to show each fraction. a) 1 6 shaded c) 5 8 unshaded b) 5 shaded d) 2 unshaded 5. Sketch a shape to show each fraction. a) 2 c) 8 10 b) 2 6 d) 6 8 118 Proper Fractions: Parts of Wholes, Pathway
6. Sketch a shape to show each relationship. a) 5 of a whole is twice as much as 2 5 of that same whole. FYI We often use fractions to describe parts of wholes. It is helpful to have a mental picture of those fractions to understand how much the fraction represents. b) 6 8 of a whole is times as much as 2 8 of that same whole. 7. Use a fraction to describe each situation. Sketch a shape to show each fraction. a) The fraction is almost a whole. c) The fraction is much less than one half. b) The numerator is less than the denominator. d) The numerator is half the denominator. Proper Fractions: Parts of Wholes, Pathway 119