Activity Book Remedial Math –
Fractions
Teacher Edition
GRADES 5-6
Remedial Math Series:
Fractions For students requiring assistance with fraction concepts.
Written by Jane Bourke. Illustrated by Rod Jefferson. ISBN 978-1-63212-039-7
Phone: 800-507-0966 Fax: 800-507-0967
www.newpathlearning.com NewPath Learning Products are developed by teachers using research-based principles and are classroom tested. The company’s product line consists of an array of proprietary curriculum review games, workbooks, posters and other print materials. All products are supplemented with web-based activities, assessments and content to provide an engaging means of educating students on key, curriculum-based topics correlated to applicable state and national educational standards. Copyright © 2007 Ready-Ed Publications. All Rights Reserved. Copyright Notice - Teacher Reproducible Edition Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However, this permission is not transferable and applies only to the purchasing individual or institution.
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Contents Teachers’ Notes ..............................................................................................................................4 Fraction Chart ................................................................................................................................5 Introduction to Fractions ................................................................................................................6 Equivalent Fractions 1 ....................................................................................................................7 Fractions as Parts of a Whole........................................................................................................8 Equivalent Fractions: Exercises ....................................................................................................9 Equivalent Fractions 2 ....................................................................................................................10 Equivalent Fractions 3 ....................................................................................................................11 Matching Fractions ........................................................................................................................12 Building Up Fractions ....................................................................................................................13 Comparing Fractions ......................................................................................................................14 Equivalent Fractions ......................................................................................................................15 Fraction Inequalities ......................................................................................................................16 Review 1: Equivalent Fractions......................................................................................................17 Simplifying Fractions 1 ..................................................................................................................18 Simplifying Fractions 2 ..................................................................................................................19 Simplifying Fractions 3 ..................................................................................................................20 Addition of Fractions ......................................................................................................................21 Improper Fractions 1 ......................................................................................................................22 Improper Fractions 2 ......................................................................................................................23 Mixed Numerals 1 ..........................................................................................................................24 Mixed Numerals 2 ..........................................................................................................................25 Addition of Fractions: Exercises 1..................................................................................................26 Addition of Fractions: Exercises 2..................................................................................................27 Addition of Fractions: Exercises 3..................................................................................................28 Subtraction of Fractions 1 ..............................................................................................................29 Subtraction of Fractions 2 ..............................................................................................................30 Subtraction of Fractions 3 ..............................................................................................................31 Review 2: Addition and Subtraction ..............................................................................................32 Decimal Introduction ......................................................................................................................33 Decimal and Fraction Relationship ................................................................................................34 Expressing Fractions as Decimals ................................................................................................35 Place Value 1..................................................................................................................................36 Place Value 2..................................................................................................................................37 Fraction and Decimal Inequalities ..................................................................................................38 Decimals and Fractions 1 ..............................................................................................................39 Decimals and Fractions 2 ..............................................................................................................40 Decimals and Fractions 3 ..............................................................................................................41 Decimals and Equivalent Fractions................................................................................................42 Review 3: Expressing Fractions as Decimals................................................................................43 Calculating Decimals ......................................................................................................................44 Percentages 1 ................................................................................................................................45 Percentages 2 ................................................................................................................................46 The Relationship between Decimals, Fractions and Percentages................................................47 Calculating Percentages ................................................................................................................48 Ratios..............................................................................................................................................49 Mixed Problems 1 ..........................................................................................................................50 Mixed Problems 2 ..........................................................................................................................51 Answers ..........................................................................................................................................52-55
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Teachers’ Notes Mathematics education encompasses a wide range of topics and concepts, many of which are only briefly dealt with in the classroom due to time constraints. It is important that these fundamental concepts are understood before students move onto the next mastery level. Students often fail to grasp all concepts and are unable to catch up to the level at which the rest of the class are working. It is here that the real difficulty for these students begins as they will sometimes withdraw from activities and miss further valuable concepts, simply because they had not mastered the prerequisite skills. Remediation for many students is frequently associated with a reduced self esteem as students are aware that they are working behind the rest of the class, especially when text books and worksheets for lower grades are used to help them to catch up. This remediation series is designed to provide upper elementary students with the necessary skills and knowledge of mathematical concepts required for their year level and can be used both in the classroom and as a "take-home" package for extra consolidation of concepts. The reading and content level is appropriate to the age of the student, even though many of the remedial activities are focused on previous stages of the math syllabus. It is hoped that this series will boost students’ self esteem as they realize that they are able to successfully complete the math activities in the book. In addition, students will not feel as if they are doing “baby” work as is the case when math sheets for 8 year olds are given to 12 year old students. For best results the series should be used to complement a remedial math program for a small group or for individual students who need to catch up. Many of the worksheets explain the mathematical concepts and provide examples, however, it is assumed that this is not the student's first experience with the concept. Each book in the series follows the same format and is directed at a particular age group, yet can be used in the secondary school if required. The Challenge question at the bottom of most pages tests the child's knowledge of the mathematical concept for that particular page. The Challenge is usually presented as a word problem in a real world context so as to highlight the need for the skill. This book explains the basic concepts of fractions, including the relationship between fractions, decimals and percentages. The activities are sequenced in line with standard syllabus structures, covering a number of stages as opposed to a straight year level, and are basically designed to provide students with the opportunity to catch up on much needed mathematical skills. The Fraction Chart on page 5 is referred to in a number of activities. This sheet can be copied onto cardstock, allowing students to color the strips and cut them out.
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Fraction Chart 1 10
1 10
1 9
1 9
1 8
1 10
1 10
1 9 1 8
1 7
1 9 1 8
1 7
1 6
1 10
1 8
1 6
1 4
1 7
1 9 1 8
1 10
1 10
1 9
1 9
1 8
1 7
1 7
1 6
1 5
1 6
1 5
1 5
1 4
1 4
1 3 1 2
1 8
1 7
1 6
1 4 1 3
1 10
1 9 1 8
1 6 1 5
1 10
1 9
1 7
1 5
1 10
1 3 1 2
1 1
Page 5
Introduction to Fractions A fraction is a part of a whole. It is used to describe how much of something is left. Look at the first circle below. 6
= 6
=
4 4
We can say that 6/6 are equal to a whole. The number on top is known as the numerator and tells us how many parts we have. The bottom number is known as the denominator and tells us exactly how many parts the whole has been divided into. The second circle has four pieces left. How can we represent this as a fraction? 4
1. Circle the denominator in each of these fractions. 3 6 7 4 4 7 8 8
9 10
1 6
2. What fraction is shared in each of these pictures? a. b. c.
....................
....................
3. Shade the fraction for each of these: 3 4 1 4 6 8
3 9 d.
.................... 2 7
6 6
4. Complete the number sentences below by shading in the correct amount. The first one has been done for you. 3 4 2 2 1 = = = 4 6 8 4 5 10 3
=
=
....................
=
2 3
=
=
Challenge: What should I order if I’m really hungry: a pizza cut into ten pieces or twelve pieces? ❂ Understanding the concept of numerators and denominators.
Page 6
6
Equivalent Fractions 1 Equivalent fractions have the same value. Look at the pizza below.
..........................
..........................
..........................
In the first picture half of the pizza has been eaten. We can say that two quarters of the second pizza has been eaten. In the third picture we can see that three pieces or 3/6 of the pizza has been eaten. Each picture shows that half the pizza has been eaten. The only difference is that each pizza has a different number of pieces left. Write the number of pieces left as a fraction under each pizza. We can say that these three fractions are equivalent as they represent the same value. 1. Shade these equivalent fractions in the shapes below. 1 2
3 6
4 8
2 4
5 10
2. Circle the fractions that represent the amount shaded in the picture. There may be more than one answer.
a.
1 4
2 8
1 3
b.
2 4
1 2
1 3
c.
1 5
2 5
4 10
Challenge:
Anthony has 1/3 of his chocolate bar left and Mel has a quarter. Who has the most chocolate left? ❂ Carrying out activities involving the equivalence of fractions.
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Fractions as Parts of a Whole Already we know that a fraction is part of a whole. For example, a cake cut into three even pieces equals one whole. Two pieces of the cake are eaten leaving only one piece left or one third. This can be represented as 3/3 - 2/3 = 1/3. Only a fraction of the whole cake remains. 1. Write the fraction of food that remains in each drawing below.
2. Shade in the amount represented as a fraction for each picture below. 3 4 2 5 6 8 3 7
3. In the shapes below shade in the pairs of equivalent fractions. Write the fractions underneath.
3 4
1 4
=
=
1 2
4 8
=
1 3
=
=
4 6
=
Challenge: Joshua has 20 jellybeans and eight of them are blue. What fraction of the jellybeans are not blue? Can you simplify this fraction? ❂ Carrying out activities that demonstrate: i) part/whole nature of fractions; ii) subset/set nature of fractions.
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Equivalent Fractions: Exercises 1. Write an equivalent fraction for each of these: 2 4 = ................ 2 = ................ 6 1 3 = ................
6 8 = ................ 1 = ................ 4 2 3 = ................
3 2 = ................ 9 10 = ................ 4 6 = ................ = ................ 4 9 4 8 6 = ................ 10 = ................
3 5 = .................. 4 = .................. 8 4 5 = ..................
2. Place = or ≠ in the box. 1 4
2 3
3 4
6 8
2 5
3 6
2 3
3 4
4 6
2 3
4 8
1 2
1 3
1 2
2 4
2 5
3 3
6 6
4 5
5 6
7 8
7 9
2 2
9 9
3. Use , or = to make these correct. 1 3
1 2
3 4
4 4
5 6
7 9
2 5
2 6
3 8
3 6
3 6
4 8
2 5
4 10
2 7
7 8
1 5
1 4
2 3
3 4
7 7
3 4
1 1
4 9
4. Shade the correct number of pieces for each picture.
3 7
2 4
5 8
6 9
9 10
4 6
5. Now write the fractions above in order starting with the smallest. .......................................................................................................... .......................................................................................................... ..........................................................................................................
Challenge: Michelle and Andy collect stamps. Michelle’s album is 3/4 full while Andy’s is 3/8 full. Who has collected the most stamps? ❂ Ordering fractions with unlike denominators.
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Equivalent Fractions 2 1. Look at the fraction in the column and circle the equivalent fractions. The first one has been done for you.
a.
2 3
3 6
5 8
6 8
8 12
6 9
b.
1 2
2 4
5 10
4 9
6 12
7 12
c.
1 3
3 6
4 12
3 9
2 8
4 6
d.
1 4
2 8
3 6
2 5
3 7
3 12
e.
1 5
2 10
1 4
3 15
4 20
3 6
2. Shade in the correct amount to match each fraction with its diagram. 1 4
2 3
3 6
4 4
3 5
5 6
1 3
3 4
3. Order the above fractions starting with the smallest. (Hint: Use the Fraction Chart on page 5 to help you.) ........................................................................................................................................................
Challenge: Maree and Paul’s mom made each of them a blueberry pie. Maree cut her pie into eight pieces and Paul cut his pie into ten pieces. Both of the children then ate half of their pies. How many pieces would each child have left? ❂ Matching fractions with unlike denominators.
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Equivalent Fractions 3 We know that there are many ways of writing a half. For example, if a cake has been cut into four even pieces then two pieces will be equal to a half. We would write this as 2/4 and say that 2/4 = 1/2. We now know that there are two 1/4 in a 1/2.
Fraction Chart 1 10
1 10
1 10
1 9
1 9 1 8
1 10
1 10 1 9
1 9 1 8
1 7 1 6
1 8 1 7
1 5 1 4
1 10
1 9
1 8
1 9
1 8
1 7
1 8
1 7
1 7
1 6
1 6 1 5
1 6
1 5
1 4 1 3
1 10
1 9
1 8
1 6
1 10
1 9
1 7
1 6 1 5
1 10
1 9
1 8 1 7
1 10
1 5
1 4
1 4
1 3
1 3
1 2
1 2 1 1
Look at the fraction chart and the way that it has been divided up. Use the chart to answer these. 1. 1 4 1 9 1 6 1 4
How many: 6 ’s in 8 ?................................................ 1 ’s in 3 ?................................................ 2 ’s in 3 ?................................................ 5 ’s in 10 ?................................................
1 10 ’s in 1 8 ’s in 1 2 ’s in 1 3 ’s in
1 5 1 4
? .................................................. ? ..................................................
3
? ..................................................
4 6
? ..................................................
Challenge: Is 3/50 greater than or less than 3/500? ❂ Finding fractions of a fraction, using a fraction chart.
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Matching Fractions 1. Fill in the missing numerators for these equivalent fractions. 1 4
=
4 5
=
8
1 3
=
10
3 5
=
9
1 = 20 10
1 2
=
15
2 6
1 5
=
=
3
10
1 3
=
10
3 9
=
4 8
3 5
=
1
4 = 12
12
3 4
=
12
2 4
=
6
4 6
=
1
9 = 12
12
8
2. Write the missing denominators into the boxes. 1 5
=
4 = 10
2
2 3
2
2
=
=
4
3 = 10
6
2
1 3
5 = 10
1
3 9
=
=
8
3
3. Write these fractions in the simplest form. The first one has been done for you. 9 3 12 = 4
8 4 6 20 10 = ............ 8 = ............ 12 = ............ 40 = ............
3 6 = ............
50 5 15 30 4 100 = ............ 15 = ............ 30 = ............ 60 = ............ 10 = ............
3 8 = ............
100 5 14 18 160 40 = ............ = ............ = ............ = ............ = ............ 200 10 20 20 200 40 = ............
Word Problems: 1. Jessica has three bread rolls and has cut them into quarters. How many quarters does she have altogether? .................................. 2. Simon has four red marbles and six blue marbles. He gave two of the red marbles to Fiona and three of the blue marbles to John. What fraction of each color did Simon give away? .............................. 3. Natalie has six pieces of fruit and two pieces are bananas. A third of the pieces are apricots and the rest are pears. How many of each fruit does she have? .............................................. 4. There are ten students in the art class. One fifth of the class are painting and two tenths of the class are drawing. Six tenths are making pottery. How many students are doing each activity? Painting: .............................. Drawing: ............................ Pottery:.................................. ❂ Finding denominators and numerators to complete number sentences.
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Building Up Fractions If we multiply both the denominator and the numerator by the same number we will find an equivalent fraction. For example, 1 x 3 = 3 2 x 3 = 6 1 3 We know that both /2 and /6 represent a half. 1 1. Using the method above write five equivalent fractions for 3 : a. ...................... b. ...................... c. ...................... d. ...................... e. ...................... 1 2. Now try the same for : 4 a. ...................... b. ...................... c. ...................... d. ...................... e. ...................... 3. Color 1/3 of each of the circles below and write the equivalent fraction underneath each one.
..........................
..........................
............................
..........................
............................
..........................
............................
..........................
4. Color 1/4 of each of the rectangles below:
..........................
..........................
5. Color 2/5 of each of the shapes below:
..........................
..........................
Challenge: Scott has 24 photos in his sports scrapbook. Half of the photos are of football players, a third of the photos are of basketball players and the rest are of tennis players. How many photos are there of each of the different types of sports stars? ❂ Looking at other methods for finding equivalent fractions.
Page 13
Comparing Fractions
The grid above has been divided into 100 squares. Shade in the following fractions according to the key. 3 = green 10
1 = red 5
1 = blue 4
4 = yellow 20
1. How many squares did you shade in each color? .................................................................... green
= 30 squares. We can say that
red
= ....................
squares.
blue
= ....................
squares.
yellow
= ....................
squares.
3 10 1 5 1 4 4 20
=
30 100
= .................... = .................... = ....................
2. What fraction of your grid remains unshaded?.......................................................................... 3. Write two equivalent fractions for each of these: 60 25 100 ........................................ 100 .......................................... 30 40 100 ........................................ 50 ..........................................
20 100 .......................................... 10 50 ..........................................
Challenge: Matt has 1,000 stamps in his collection. 450 are from Africa and the rest are from Europe, Asia and South America. What fraction of the stamps do not come from Africa? Express your answer in the simplest form. ❂ Comparing fractions with unlike denominators by converting them to fractions with the same denominators.
Page 14
Equivalent Fractions 1. What fractions of the shapes below are shaded? Simplify your answer.
..................................
..................................
..................................
..................................
2. Shade these amounts below. 1 3
2 6
3 9
3. Complete the following using = or ≠. You might like to refer back to the fraction chart on page 5. 1 4
2 8
1 3
2 6
2 8
4 16
3 12
2 16
1 3
2 9
2 6
2 3
2 8
3 12
1 4
1 3
4. Use =, < or > to make these true. 1 4
1 3
1 6
1 8
2 4
2 3
5 6
7 8
4 9
2 7
3 8
3 7
9 10
9 9
4 4
3 3
2 5
2 6
1
2 3
5 5
2 2
1 6
1 9
Challenge:
Emily and Sarah are painting the fence in the backyard. Sarah has painted 1/3 of the fence posts and Emily has painted 3/8 of the fence posts. Who has painted the most posts? ❂ Experience with equivalent fractions.
Page 15
Fraction Inequalities 1. Circle the greater fraction in each pair. The first one has been done for you. 1 5
3 10
2 6
3 6
1 2
2 5
4 9
1 2
1 3
6 7
4 6
4 5
5 10
5 8
4 8
3 9
2 6
2 5
7 8
8 9
1 2
1 10
5 5
3 4
2. Place = or 1 in the boxes below. 3 6
2 3
1 3
3 9
2 4
5 10
3 6
1 2
4 10
14 20
6 9
4 6
8 10
4 5
9 9
7 7
3. Complete the following by adding a fraction of your own. 3 6
9 9
>
4 5
12
8 < 12
4 6
10
4. How much of each pie is left? Express your answer in the simplest fraction.
........................
........................
........................
5. Match the fractions on the left with a fraction on the right. a.
2 3 4 9
50 100 8 18
3 8 5 10
b.
4 6 7 8
60 90 8 12
40 60
3 4
6 16
2 3
c.
3 9 2 8
4 16 2 5
75 100
4 10
2 6
14 16
8 12
4 6
Challenge: Murray and Sue each recieve the same amount of pocket money each week. Murray has spent 3/7 of his pocket money and Susie has spent 4/5. Which child has spent the most money? ❂ Ordering fractions with unlike denominators.
Page 16
Review 1: Equivalent Fractions 1. Order these fractions from smallest to largest. 1 3
4 6
1 5
1 2
2 5
3 3
.................................................................................................................................................... 2. Place these fractions in order from largest to smallest. 1 6
10 10
3 9
1 4
1 7
2 5
.................................................................................................................................................... 3. Use < or > in the boxes below to make the number sentences true. 5 6
1
2 12
1 3 6
4
2 6
1
7 8
1
1 9
3
7 9
5 10
7
2 3
4
5
2
7 10
1 2
2
2 3
5 6
1 3
4
2 4 3 4
5
1 3
4. Use or = to complete these. 3 3
5 5
2
1
7 7
4 12
4
6 7
4
3 4
6
1 5
5
1 2
1 3 5 9
9
5 6
3
2 3
11 3
1
1 7
2
1 7
15 5
9 3
8
3
2 3
15 15
6 6
3 12
7 10
7
7 10
5. Color the amounts shown and complete the number sentence using =, < or >.
5 8
3 6
5 6
7 8
3 4
7 8
Challenge:
Joanne gets 80¢ a page for typing. If she types 3/4 of a page how much money will she receive? Page 17
Simplifying Fractions 1 The fractions below can all be expressed as 1 because they represent all parts of a whole.
3 3
9 9
7 7
50 50
6 6
20 20
1. Write four fractions below that are equivalent to a whole. a. .............................. b. .............................. c. .............................. d. .............................. 2. Shade the shapes below according to the fraction. 2 2
4 4
12 12
20 20
3. Use the pictures and circle the amounts shown below. 4 8
2 4
12 12
3 6
4. All of the amounts you have shown above are equal to .......................................................... 5. Use a red pen to circle the fractions below that represent a third. Use a blue pen to circle the fractions equivalent to a half. 4
4 12
8
3 9
20 40
5 10
50 100
2 6
5
6
15
12
100 200
Challenge: Kathy was checking her basketball point scoring for the last three games. In the first game she scored 15 of the 28 point for the game. In the second game a total of 36 points were scored by her team of which Kathy scored 18 and in the third game she scored 12 out of 26 points. In which game did Kathy score exactly half the points? ❂ Simplifying fractions by converting them to equivalent fractions.
Page 18
Simplifying Fractions 2 We can simplify fractions in another way. By looking at the fraction chart on page 5 we can see that 6/8 is equivalent to 3/4. If we look at the fraction itself we can see that for 6/8 to become 3/ we divide both the numerator and the denominator by 2. 4 6 ÷ 2 = 3 8 ÷ 2 = 4 1. Simplify these fractions by dividing by two. 8 = 12
4 = 10
2 6
2 4
=
=
4 = 12
2. We can divide by other numbers to simplify fractions. Write simplified fractions for these: 9 ÷ 3 = 3 12 ÷ 3 = 4
5 = 20
80 = 100
3 = 18
4 = 12
Sometimes it is possible to divide the numerator into the denominator to simplify a fraction. For example, with 3/9 we have 3 as the numerator and 9 as the denominator. We know that 3 goes into 9 exactly three times and we can simplify the fraction to 1/3 because 3 goes into 3 once and into 9 three times. 3. Use this method to simplify the fractions below. 4 3 10 4 4 5 12 = ............ 15 = ............ 20 = ............ 8 = ............ 16 = ............ 20 = ............ 20 2 40 7 50 6 200 = ............ 10 = ............ 80 = ............ 100 = ............ 18 = ............ 14 = ............ 4. Express these fractions in the simplest form. 4 6 9 3 5 4 20 = ............ 30 = ............ 18 = ............ 27 = ............ 30 = ............ 10 = ............ 2 8 8 10 15 3 80 = ............ 100 = ............ 12 = ............ 20 = ............ 12 = ............ 10 = ............ 5. Match a fraction on the top line with a fraction on the bottom line. 3 4
7 8
9 20
200 1,000
3 50
6 200
14 16
6 8
3 100
4 20
6 100
18 40
Challenge: In his last math test Aaron scored 75 out of a possible 100. What is the simplest way of expressing the fraction of the sums that Aaron correctly answered? ❂ Simplifying fractions by division using whole numbers.
Page 19
Simplifying Fractions 3 1. Simplify these by dividing by ten or a hundred. 1,000 50 700 9,000 80 20 1,000 = .......... 60 = ............ 800 = ............1,000 = .......... 100 = ............ 100 = .......... 200 60 20 800 800 = .......... 500 = ............ 100 = ............ 50
10 50 = .......... 100 = ............ 200 = ..........
2. Simplify these fractions. 35 40
54 25 = .......... 60 = ............ 30
85 = ............ 90
25 = .......... 40
30 = ............ 90 = ..........
25 75
40 75 120 480 48 = .......... 80 = ............ 100 = ............ 200 = .......... 2,000 = ............ 50 = ..........
Word Problems: Where possible express your answers in the simplest form. 1. Mark has sixteen apples and Tony has eight. Mark has eaten eight of his apples and Tony has eaten three. What fraction does Mark have left? .............................................. What fraction does Tony have left? .............................................. 2. Sandra has twelve socks in her draw. One third of the socks are red and the rest are blue. How many blue socks does Sandra have? .......................................... 3. Donelle is sharing her birthday cake with four friends. She has cut the cake into ten pieces. What fraction will each person receive? .......................................... 4. Tarlie has counted twenty students in her class and has noticed that five of the students wear glasses. What fraction of the students do not wear glasses?.......................................... 5. Fiona and Peter played twelve games of chess. If Fiona won nine games, what fraction did Peter win? .......................................... 6. Steve sells hot dogs at the football game. He had enough supplies for 100 hot dogs. At the end of the day he had twenty hot dogs left over. What fraction did Steve sell? ...................... 7. In Jamie's math exam there were 30 questions of which he answered twenty five correctly. What fraction did Jamie answer incorrectly? .............................................. 8. Jason is captain of his soccer team and there are twelve boys on the team. Four boys were injured in the last game. What fraction of the team were injured? .......................................... ❂ Simplifying fractions by canceling. ❂ Exploring the concept of fractions in a real world context.
Page 20
Addition of Fractions Look at the pies below. What fraction of each pie has been left?
8
8
How many pieces are there altogether? ........................................ We can say 4/8 + 2/8 = 6/8. To work this out we simply added 4 and 2.
4 + 2 = 6 8 + 8 = 8
You will notice the denominator does not change when fractions are added. 1. Complete these sums.
1 3
+
1 2 = 3 3
2 4
+
2 4
=
5 6
+
1 6 =
3 7
+
2 7 =
2. Use the written method to add these fractions. The first one has been done for you. 1 2 1 = + 3 3 3
3 2 + = ............ 8 8
1 1 = ............ + 2 2
2 5 + = ............ 9 9
2 1 = ............ + 4 4
4 5 = ............ + 9 9
4 2 = ............ + 8 8
12 5 + = ............ 20 20
25 30 = ............ + 10 10
20 25 = ............ + 50 50
23 56 = ............ + 100 100
3. Complete the following. 1 1 1 + + 3 3 3
= ........................................
2 4 3 + + = ........................................ 10 10 10
2 1 1 + + 5 5 5
= ........................................
15 1 2 = ........................................ + + 20 20 20
Challenge: Michael cut a pizza into twelve pieces. He gave three pieces to Stephanie, four pieces to Simone and then gave two slices to Bill. What fraction of the pizza did Michael give away? ❂ Adding fractions with like denominators up to 10 and multiples of 10.
Page 21
Improper Fractions 1 Look at the three pies below. Each has been cut into quarters.
How many quarters are there altogether?...................................... We can express this as
12 . This is known as an improper fraction. 4
1. Use the shapes below to complete the following. The first one has been done for you.
+ 3+3 6 2 = =1 4 4 4
+ 4 2 + = ...................... = ...................... 5 5
+ 3 5 = ...................... = ...................... + 6 6
+ 8 4 = ...................... = ...................... + 10 10
+ 5 7 = ...................... = ...................... + 9 9
+ 4 4 = ...................... = ...................... + 5 5
+ 6 4 = ...................... = ...................... + 7 7
+ 5 7 = ...................... = ...................... + 8 8
Challenge: Joe had six apples which he cut into quarters. Express the number of quarters Joe has as an improper fraction. ❂ Expressing fractions as improper fraction through addition sums.
Page 22
Improper Fractions 2 Follow this example:
3 2 3+2 5 1 + = = =1 4 4 4 4 4
5/ is known as an improper fraction because the numerator is greater than the denominator. 4 This means that there is more than 1 whole. We know that 4/4 is equal to one whole so 5/4 must be equal to 1 whole and 1/4 or 11/4 . 1. Express the answers to these sums as improper fractions. 6 5 + = ............ 9 9
2 5 = ............ + 6 6
8 4 = ............ + 10 10
3 6 = ............ + 7 7
4 3 + = ............ 5 5
4 7 = ............ + 8 8
10 10 = ............ + 15 15
12 15 = ............ + 20 20
2. Simplify these fractions. The first one has been done for you.
1 28
=
1 14
2 69
= ..................
1 36
= ..................
2 68
= ..................
8 5 10
= ..................
6 3 12
= ..................
3 46
= ..................
2 48
= ..................
3. 4/4 = 1 so 14/4 is equal to 2. Complete the following by adding the fractions. 2 3 4 2 5 4 1 24 + 1 24 = ............ + + + = ............ = ............ = ............ 5 5 6 6 9 9
2 29
1 5 1 2 ............ 3 + 1 3 = ............ 4 6 + 2 6 = ............ 4. Jessica had four whole oranges and half of an orange and Ben had one orange and a
+
1 79
= ............
9 7 10
+
1 1 10 =
half. If they put them together how many oranges would there be? ........................................ 5. Hamish divided six bananas into thirds. How many thirds did he now have? Express your answer as an improper fraction. ................................................................................................ 6. Suzanne had four sets of pencils each with eight colors and half a set with four colors. Drew gave her three pencil sets and four extra pencils. Write a number sentence to show how many pencils Suzanne has. .................................................................................. ❂ Adding fractions with mixed numerals.
Page 23
Mixed Numerals 1 When there is more than one whole the fraction is expressed as a mixed numeral. For example, the shaded area in the circles below represents 41/3.
1
+
1
+
1
+
1
1 3
+
1. Shade in the correct amount below.
2 15
4 23
3 18
1 36 We can also express the above mixed numerals as improper fractions. For example,
2 15
5
5
1
11
is equal to 5 + 5 + 5 = 5
2. Express these mixed numerals as improper fractions.
1 46
= ..................
3 24
= ..................
2 56
= ..................
6 12
= ..................
4 34
= ..................
9 2 10
= ..................
3 56
= ..................
2 47
= ..................
5 23
= ..................
8 45
= ..................
45 3 100 =
..................
55 1 200 =
..................
Challenge: At a party Mark cut 5 cheesecakes into 10 pieces each. At the end of the party the following amounts were left on each plate; 3 pieces, 2 pieces, 6 pieces, 1 piece and 4 pieces. Express the amount left over as a mixed numeral. ❂ Representing mixed numerals in diagrams. ❂ Converting mixed numerals to improper fractions.
Page 24
Mixed Numerals 2 To convert improper fractions into a mixed numeral divide the denominator into the numerator. For example, 3/2 = 11/2. There are three halves and we know that two halves make a whole. One half is left over so the answer is 11/2. 1. Change these improper fractions into mixed numerals. 5 7 8 4 3 = ............ 2 = ............ 3 = ............ 2 = ............
9 10 7 = ............ 6 = ............
12 15 20 9 8 20 6 = ............ 2 = ............ 3 = ............ 5 = ............ 6 = ............ 9 = ............ 100 85 32 25 16 39 10 = ............ 5 = ............ 6 = ............ 3 = ............ 7 = ............ 20 = ............
2. Change these mixed numerals into improper fractions.
3 14
= ................
5 23
= ..................
6 26
= ..................
2 37
= ..................
9 12
= ..................
1 7 10
= ..................
8 48
= ..................
7 46
= ..................
9 89
= ..................
5 45
= ..................
3 4 20
= ..................
60 3100
= ..................
3. Place these fractions in order starting with the smallest.
3 18
1 8
20 8
6 8
4 28
8 8
2 58
13 8
.......................................................................................................................................................... 4. Use or = to make these true. 11 6
1 16
2 3
3 2
4 3
4 5
5 2
2 15
5 7
75
9 5
1 45
13 6
1 36
5 10
10 5
21 3
3
12 2
4
3
18 6
24 6
16 4
Challenge: Nicole cut some oranges into quarters for her soccer team to eat at half time. At the end of the game there were 17 quarters left on the tray. Express the fraction of oranges on the tray as a mixed numeral. ❂ Expressing improper fractions as mixed numerals. ❂ Ordering combinations of fractions.
Page 25
Addition of Fractions: Exercises 1 1.
Add these fractions and express the answer as a mixed numeral. 7 4 10
For example:
2 5
+
4
9 6 10
=
4
=
10 +
=
10 + 1
=
6 11 10
=
11
4 5
+
6
= 7
9
+ 10 + 10
16 10 6
+ 10
3 5
3
=
3
+
5 6
+
4 6
=
1 3
+
2
2 3
6
=
2
7 9
+
4
2 9
=
3 7
=
+
5 7
3
5
3 8
+
6
7 8
=
5 8
+
4
5 8
=
3
3 8
+
5 8
=
2. Add the amounts shown in the shapes below.
+ +
................................................
+
..............................................
Challenge:
................................................
Jerry had three bread rolls and 2/5 of a roll and Kelly had five rolls and 4/5 of a roll. What fraction of bread rolls did they have together? ❂ Adding fractions with mixed numerals with like denominators.
Page 26
Addition of Fractions: Exercises 2 1. Simplify the answers to these sums. 9 4 20
+
9 3 20
7 15 50
=
21 50
+
1 20 30
=
3 10 30
+
3 15 30
3 15 40
=
+
2 20 30
8
=
2 15 40
+
+
5
3 4
3 5
+
3
4 5
=
4 6 10
=
3 4
+
8 3 10
5
=
=
2. Fill in the missing number to make these number sentences true. 4 5
2
5
=
1
3 5
6
+
+ 4
=
3
2 4
2
+
+
3 4
5 6
4
1 2
=
1
=
5
1 6
2
4 7
+ 9 10
+
4 7
=
5
1 7
4 10
=
3 5 10
3. Draw the fractions in the space below and add them together. Simplify your answer.
3 9
2
7 9
+
7 8
+
2
3 8
= ..................
= ..................
8 1 10
+
7 2 10
= ..................
5 6
+
1
4 6
= ..................
2
Challenge: Sam and Chloe weeded gardens in their spare time. On Monday they weeded two gardens and three quarters of a garden. On Tuesday they weeded the quarter from Monday plus two more gardens and on Thursday they weeded one and a quarter gardens. What fraction of gardens did they weed during the week? ❂ Simplifying mixed numerals. ❂ Addition of mixed numerals with like denominators.
Page 27
Addition of Fractions: Exercises 3 So far we have added thirds to thirds and quarters to quarters and so on. We can say that these fractions have like denominators. Sometimes we need to add fractions with unlike denominators. Look at the diagrams below and write the fraction of the remaining portions of pie underneath.
..................................
..................................
If we combine the leftovers, what fraction will we have? To find the answer we need to express the fractions so that the denominator is the same. This is known as finding the lowest common denominator. Example 1:
To add we convert
1 4
+
5 8
1 4
to =
5 8
2+5 8
so sum is
2 3
Example 2:
7
= 8
1 9
+
=
6+1 9
=
7 9
1. Complete these sums by using equivalent fractions to find the lowest common denominator. 3 4
+
3 8
1 3
+
5 6
2 3
+
2 9
1 5
+ 10
1
2. Add these fractions and simplify the answer. 2 6
+
1 3
2 4
+
4 8
4 10
+
2 5
1 9
+
2 3
3 4
+
5 8
2 6
+
2 3
3 6
+
1 2
1 2
+ 10
6
Challenge: Darren played in three quarters of his basketball match this Saturday. Last Saturday he played half a game, and the Saturday before that he was on for only half of the first quarter. What fraction of games has Darren played in the last three weeks? ❂ Addition of fractions with unlike denominators to 10.
Page 28
Subtraction of Fractions 1 Jerry cut a pie into four quarters and decided to eat a piece. What fraction now remains? 4 4 1 . can be written as – 4 4 4 Later he ate another piece. What fraction is left? He started with
3 1 = – 4 4
3–1 4
=
4–1 4
=
3 4
2 1 = 4 2
1. Use the method above to complete the following. 6 6
3 6
–
4 5
= ......................
2
1 6
–
= ......................
2 5
–
3 7
–
1
6 7
= ......................
= ......................
9 10
10 –
4
– 10
= ......................
3 8
= ...................... 5 9
= ......................
3 7
–
5 9
–
= ......................
2. Subtract the shaded amount from the shapes below.
..................................
..................................
..................................
3. Now try these. Remember to simplify your answers.
2
3 7
1
4
– 7
= ......................
9
5 6
–
3
2 6
= ......................
3 4
1
6
– 4
= ......................
3
5 8
–
2
2 8
= ......................
5 8
4
– 8
10
6 7
6
– 7
= ......................
= ......................
9 2 10
4
–
2 1 10
= ......................
6 7
–
4
5 7
= ......................
Challenge: Hannah had five bunches of flowers, each made up of nine flowers. She gave two bunches and a third of a bunch to her sister and another two bunches to her Mom. How many bunches does she have left? ❂ Subtraction of fractions including mixed numerals with like denominators.
Page 29
Subtraction of Fractions 2 To subtract mixed numerals, first convert the fractions to improper fractions. For example: 1 5 11– 8 5 3 5
2 = =
–
1 35
1 – 8 25 – 19 8 6 8 3 4
3 = = =
2
3 8
1. Convert these fractions to improper fractions to find the answer.
2
1 6
4
3
– 6
= ......................
4 9
7
– 9
= ......................
4 2 10
–
8 1 10
= ......................
6
3 7
–
5
4 7
= ......................
2. Express these whole numbers as improper fractions. The first one has been done for you.
2
=
18 9
3
=
6
4
=
5
6
=
2
4
=
1
5
=
2
6
=
3
7
=
3
3. Subtract the fractions from the whole numbers by changing to improper fractions. Write your answer as a mixed numeral.
8
3 4
–
7
= ......................
3
2 10 20
–
= ......................
–
3 10
= ......................
4
–
75 100
= ......................
4
2 3
–
= ......................
6
7 30
–
= ......................
10
–
2 6
= ......................
2
1 9
–
= ......................
4. Subtract these fractions.
3
4 5
–
2
1 5
= ......................
9
–
6 7
= ......................
10
–
2 3
= ......................
4
–
7 20
= ......................
5
2 6
3
– 6
3 25
–
2
= ......................
= ......................
9 3 50
8
–
11 2 50
= ......................
Challenge:
–
3
4 5
= ......................
During the term Steve had to read 5 books. So far he has read 2 books and 4/6 of a book. What fraction of the books does he still have to read? ❂ Subtraction of fractions from a whole number, and a mixed numeral.
Page 30
Subtraction of Fractions 3 1. Complete the following and simplify answers where possible. For example:
5 – 8 37 – 31 8 6 8 3 4
4
1. = = =
5
2 6
3
–
1 6
4
= ......................
10
3
–
2 5
3
–
–
4 8
9
3 5
–
= = =
5 9
3
6
4 5
2
1
= ......................
3 9
–
2
4
–
4
= ......................
–
6 7
1
–
25 – 13 7 12 7 5 7
1 5 7
2 8
3
1
–
7 8
= ......................
7 10
4 6
3
= ......................
7 9
–
4 7
4 7
= ......................
= ......................
4 5
3
2.
= ......................
= ......................
6
2 9
7 8
3
1
–
5 6
= ......................
8 20
2 7
5
= ......................
3
–
6 7
= ......................
2. Fill in the square to make each number sentence true.
2
7
6
3 5
2
5
1 7
–
–
–
2
2 5
=
3
2
2 5
1 5
– 5 6
2 3
=
=
=
7
1 5
25 4 100
10
–
–
1 3
4 5
–
3
5
=
1
2 5
2
4
3 8
–
2
8
=
1
7 8
3 9
5
5
5 2 100
=
=
9
9
1 3
4
=
–
–
2 6
=
4
6
1 6
Challenge: Jason has five bags of marbles and each bag contains eight marbles. He gave two bags and three quarters of the marbles in another bag away to David. What fraction does Jason still have? .................................................. How many marbles did David receive from Jason?.................................................. ❂ Subtraction of mixed numerals from other mixed numerals and whole numbers.
Page 31
Review 2: Addition and Subtraction 1. Add the following fractions. 1 3
+
1 3
1 4
= ...................... 2 3
1
2
= ...................... 3 7
2
+
2
5 7
= ......................
1 4
2
= ......................
1
+ 3
+
2 4
+
2
1 4
= ...................... 2 8
+
7 8
= ......................
1 3
2
1
+ 3
= ......................
3
2 5
+
4
4 5
= ......................
5
+
4
3 4
= ......................
1 5
+
2
2 5
= ...................... 3 9 10
+
8 3 10
= ......................
2
4 5
+
2
4 5
= ......................
2. Subtract the fractions below. 2 3
–
1 3
3 4
= ......................
2
2 5
–
1
1 5
= ......................
1
–
5 6
= ......................
–
1 4
= ......................
3
1 3
– 2 3
= ......................
10
–
3 5
= ......................
6 9
–
2 9
1
= ......................
4
2 5
–
2
3 5
= ......................
2
–
1
2 3
= ......................
2 3
2
– 3
= ......................
6
3 9
–
4
4 9
= ......................
6
–
3
4 5
= ......................
Word Problems: 1. Sarah had nine chocolate cookies and five of them were iced. She cut each cookie in half. Express the total number of iced cookies as a fraction. ...................................................... 2. Lucy had 4 pears which she cut into quarters. She ate two pears and a quarter. How much does she have left? ........................................................ 3. Amanda has made some soup and is serving dinner for the family. She has given half of the soup to her Dad, a quarter to her Mom, an eighth to her younger sister and an eighth to herself. What fraction of the soup remains? ...................................................... 4. Julian makes model airplanes and buys each piece separately. Each airplane had 10 pieces. He has seven completed airplanes and 2/5 of another one. How many pieces does he have in his collection? .................................................... 5. Brad is 4/3 of a yard tall. How many inches tall is Brad? .................................................................. Page 32
Decimal Introduction We know that a fraction represents part of a whole. A decimal is a number where a decimal point seperates the whole number and the fraction. For example, 13/10 = 1.3
25/ 100 = 0.25
In the above example the 3 represents the three tenths. What does the 25 in the second mean? ........................................ 1. Complete the following table by writing the numbers below into the correct unit place. The first one has been done for you. Hundreds 25.34
Tens
Ones
.
1/tenth
1/hundredth
2
5
.
3
4
1/thousandth
12.3 1.56 2.458 13.256 0.98
2. Using the table below write down each number. Remember, if there is no number in the column a zero is used to represent the place. Hundreds a. b.
1
Tens
Ones
.
2
5
.
3
6
.
1
.
7
.
6
.
c. d.
2
e.
1/tenth
1/hundredth
1/thousandth
3
4
2
6 5
4
a............................................. b. ..............................................
4 2
3
c. ............................................
d............................................. e. ..............................................
Challenge: Which is greater: 200.2 or 200.002? ❂ Introduction to decimals exploring the place value.
Page 33
Decimal and Fraction Relationship 1. Shade in the correct amounts below:
0.1
0.5
0.7
0.6
0.23
0.26
0.69
0.54
0.24
0.63
0.12
0.89
............................
............................
2. Write the decimal that shows the shaded area below.
..........................
............................
3. Write the fraction and the decimal for each of the shaded areas below.
0.6 = 6/10
..................................
..................................
..............................
Challenge: Decimals are used to show amounts of money. How would you express a dollar and a quarter as a decimal? ❂ Exploring the relationship between decimals and fractions as parts of wholes.
Page 34
Expressing Fractions as Decimals Fractions and decimals can be used to express the same amounts. We use fractions for some objects and decimals for others. Consider the objects below and circle the way you would describe them. 1 2
or 0.5 a glass of orange juice;
0.3 of a yard or 1 4
3 10
of a yard;
of a sandwich or 0.25 of a sandwich; 3
0.75 or 4 of a job finished. 1. Express the decimals below as fractions. 0.2 = .............. 0.5 = ............ 0.6 = ............ 0.23 = .......... 0.98 = .......... 0.47 = .......... 2. Express these fractions as decimals. 1 10
7
= .............. 10 =
10
34
28
567
............ 100 = .............. 100 = .............. 100 = .............. 1,000 = ............
Fractions need to be expressed with denominations of 10, 100 or 1000 before being expressed as decimals. For example:
3 5
= 10 = 0.6
6
1 2
= 10 = 0.5
5
3. Express these fractions as decimals. 2 5
= .............. 10 =
4
............
1 5
= .............. 20 = .............. 20 = .............. 50 = ............
10 20
= .............. 20 =
............
2 5
= ..............
50
40 = 200
300
8
20
3 4
= ..............
150
6
10
1 4
= .............. 15 = ............
200
20
20
.............. 600 = ............ 2,000 = ............ 200 = .............. 200 = .............. 40 = ............
4. Use =, < or > to make the following true. 0.5 4 5
4 10
6 10
6.0
8 100
0.08
0.45
90 100
0.9
8 1,000
0.08
Challenge: Miles has collected 36 sports cards this year. If there are 200 in the set, what fraction has he collected so far? Express this fraction as a decimal. ❂ Expressing fractions with denominators up to 10 and multiples of 10 as decimals.
Page 35
Place Value 1 In the examples below, the underlined number represents a different value even though it is the same digit. 5432 = 4 x 100 = 400 62.45 = 4 x 1/10 = 4/10 1. What value does each underlined number represent below? 9,675 .................... 29.38 .................. 1.987 .................... 135.3 .................. 209.08 ................ 24.34 .................. 147.2 .................... 100.333 .............. 24.24 .................. 999.99 ................ 2. Write these decimals in expanded form. The first one has been done for you. 3
5
24.35 = 20 + 4 + 10 +100 1
1
(2 x 10) + (4 x 1) + (3 x 10 ) + (5 x 100 ) a. 136.57 = .................................................................................................................................... .................................................................................................................................................... b. 26.987 = .................................................................................................................................... .................................................................................................................................................... c. 35.57 = ...................................................................................................................................... .................................................................................................................................................... d. 49.08 = ...................................................................................................................................... .................................................................................................................................................... e. 765.297 = .................................................................................................................................. .................................................................................................................................................... 3. Use < or > to make these true: 35.46
3.546
2.002
2.2
980
9.8
1.256 860.086 12
125.6
24.78
2.478
1,000
1.000
860.068
2.3
3.2
56.65
65.56
0.12
154.3
134.5
264.1
264.9
Challenge: Andrew is putting gas into Dad’s car. The gallon gauge has stopped and reads 40.72 gallons. What value in gallons does the 7 represent? ❂ Expressing decimals in an expanded form. ❂ Ordering decimals with unlike numbers of decimal places.
Page 36
Place Value 2 You will need three dice for this activity. 1. Roll the three dice and record the largest possible number you can make from the dice faces. For example, if you roll a 2, 3 and a 5, the largest possible number would be 532. Throw the dice five times and complete the table below. Tenths
Hundredths
Thousandths
1. 2. 3. 4. 5.
2. Now roll each dice twice. Record the face values and then write the smallest possible number in the table below. Do this five times.
100’s
10’s
1’s
.
10ths
100ths
1000ths
1. 2. 3. 4. 5.
Challenge: Charlie has measured five different distances around the house. The distance from his front door to the rubbish bin is 15.2 m. From the door to the pot plant is 0.152 m, from the door to the end of the verandah is 1.52 m and from the door to the letterbox is 15.22 m. He has also measured the distance between the door and the park and has found it to be 152 m. Place the objects in order starting with the closet to Charlie’s front door. ❂ Ordering decimals according to place and face value.
Page 37
Fraction and Decimal Inequalities 1. Use or = to make the following true. 2 4
0.5
1 10
1.10
3 10
0.3
10 100
0.1
10 50
0.2
20 100
2
1 100
0.1
3 4
0.75
100 200
0.6
1 3
0.25
4 20
0.2
10 40
0.4
2. Place a number in the box to make the number sentences below true. The first one has been done for you. 0.9 = 0.8 = 0.5 =
3 5
18 20
= 0.
0.25 =
5 10
0.4 =
0.25 =
100 8
0.3 = 0.8 =
0.5 =
4
0.23 =
20 4
0.46 =
6
100 46
0.2 = 0.75 = 0.687 =
5
4 687
3. Complete the following by adding a decimal of your own. 3 5
< ....................
2 6
> ..................
1 5
= ....................
2 8
= .................. 10 < ....................
2 20
> ....................
4 5
< ..................
3 5
= ....................
1 4
> .................. 100 > ....................
1 5
= .................... 50 > ..................
20 50
= .................... 200 = .................. 1,000 = ..................
140
90
40
3
80
4. Write 5 fractions equal to 0.1..................................................................................................... 5. Write 5 fractions equal to 0.4..................................................................................................... 6. Write the decimal that is equal to 3/5, 6/10, 600/1000 and 12/20. .......................................... 7. Use = or ≠ to make the following true. 0.78
78 100
0.25
2 5
0.72
7 2
0.48
48 100
0.15
15 1,000
1.5
5 1 10
Challenge: Jason and Steve were both in the long jump event. Steve jumped 2.50 meters and Jason jumped 1/4 of a meter more than Steve. Exactly how far did Jason jump? ❂ Comparing values of fractions and decimals.
Page 38
Decimals and Fractions 1 1. The grids below have been divided into 100 units. Shade the amount shown underneath.
a. 0.2
b. 0.45
c. 0.01
d. 0.86
e. 0.96
f. 0.05
g. 0.68
h. 0.8
2. What fraction of the above grids have you shaded? Express in the simplest form. The first one has been done for you. 20
1
a. 100 = 5
b. .............................. c. .............................. d. ..................................
e. .............................. f. .............................. g. .............................. h. .................................. 3. Complete these using = or ≠. 3 4
0.75
2 6
0.4
8 10
0.8
2 3
0.3
4 8
0.6
1 3
0.3
4 8
0.6
2 5
0.25
4. Use =, < or > to complete these.
1
3 4
8 9 10
1.75
2
4 8
2.4
9.8
5 6 100
6.5
Challenge: Bridget has painted 0.75 of the garage door. What fraction does she still need to paint? ❂ Expressing decimals as fractions. ❂ Exploring inequalities.
Page 39
Decimals and Fractions 2 1. Shade the amounts shown below.
b. 3
a. 0.73
4
How many squares are shaded in each box? a. ................................ 3
By looking at the boxes we can see that 0.73 is less than 4
b. ................................
.
2. Use the grids below to complete the amounts shown. Add =, < or > for each pair.
0.47
1/ 2
0.84
4/ 5
0.69
7/ 10
0.35
1/ 4
0.15
3/ 20
0.4
8/ 20
3. Complete these number sentences using =, < or >. Use squared paper if needed. 3 10
0.13
6.32
6
2 5
Challenge: Which is longer: Page 40
2
5 20
0.5
7.85
7 17 20
5 6
3 5
2.65
0.35
2
3 5
sticks of licorice or 2.56 sticks of licorice?
2 5
3.16
0.4
3 16 20
Decimals and Fractions 3 1. Express these decimals as simplified fractions. The first one has been done for you. 5
1
2.05 = 2 100 = 2 20 3.2...................... 4.65.................... 5.25.................... 13.26.................. 7.8 ...................... 623.02 .............. 0.5...................... 4.04.................... 6.008.................. 22.22 .................. 7.75.................... 3.025.................. 12.6.................... 10.42.................. 17.017 ................ 2. Convert these fractions to decimals. 3 10
....................
50 100
30
24
15
12 100
.................... 100 .................... 1,000 .................... 1,000 .................... 1,000 ....................
350
3
23
49 100
.................... 100
.................... 100 .................... 1,000 .................... 1,000 ....................
57 23
....................
7 10
70
700
.................... 100 .................... 1,000 ....................
3. Change these improper fractions to decimals. The first one has been done for you. 435 100 =
4.35
16 10
....................
39 10
.................... 100 .................... 100 ....................
32 10
....................
25 10
....................
2,795 3,423 100 .................... 1,000
143 43 10 9,098
198 656
11 10
......................
264
.................... 100 .................... 100 ...................... 3,456
578
.................... 100 .................... 1,000 .................... 10
......................
4. Write five equivalent fractions for each decimal below. 6.5 = .............................................................................................................. 2.25 = ............................................................................................................ 0.75 = ............................................................................................................ 3.6 = .............................................................................................................. 9.75 = ............................................................................................................
Challenge: Matthew is counting his savings and has calculated that he has 687¢. Express this amount as a decimal and also as a fraction. ❂ Converting mixed numerals to decimals. ❂ Finding equivalent fractions for decimals.
Page 41
Decimals and Equivalent Fractions So far the fractions we have changed to decimals have all had a denominator which is a multiple of 10, such as 10, 100 and 1000. Sometimes it is necessary to convert fractions that cannot evenly be divided into 100. 36
For example 60
needs to be divided by 6 so that the denominator is 10.
36 ÷ 6 = 6 60 ÷ 6 = 10 = 0.6 1. Try these by first simplifying the fraction so that the denominator is 10. 32 40
= 10 = 0.8 70 =
24 30
= .............. 20 =
8
35
64
42
81
56
............ 80 = .............. 60 = ............ 90 = .............. 70 = ............
14
48
36
63
28
............ 80 = .............. 40 = ............ 90 = .............. 70 = ............
2. Now try converting these fractions to a decimal. Some of them are quite tricky. 2 5
4
3 5
3 4
= ............
3
= ............
4 10 20 =
5
3
2 8
= ............
9
9 13 3 3 12 = .............. 4 20 = ............ 6 50
= ............
3 2 100
= ........ 25 = .............. 15 ........
= ............
4
2 5
= ............ = ..........
3. Change these decimals to fractions by filling in the boxes below. The first one has been done for you.
a. 0.7 =
7
70
700
14
35
10
100
1,000
20
50
10
100
1,000
5
20
10
100
1,000
5
40
10
100
1,000
5
9
10
100
1,000
2
6
b. 0.2 =
c. 0.8 =
d. 1.0 =
e. 1.5 =
Challenge: Fiona buys her stamps in sets and there are twenty stamps to a set. She has five complete sets and another set that contains only 12 stamps. Express the number of sets Fiona has as a decimal. ❂ Converting fractions to decimals by first expressing fractions with denominators with multiples of 10.
Page 42
Review 3: Expressing Fractions as Decimals
Answer the following quick questions. 1.
What is eight tenths as a decimal? ................................................................................
2.
Write five and nine tenths as a decimal. ........................................................................
3.
What is seven point four as a fraction? ..........................................................................
4.
Write two point two five as a fraction. ............................................................................
5.
Which is greater: 0.60 or 3
6.
If you have $10.00 pocket money and spend a quarter of it, how much would you have
4
? ......................................................................................
left? .................................................................................................................................. 7.
What is zero point two five as a fraction? ......................................................................
8.
True or False: Six and three tenths is less than six and four fifths. .............................. 2 5
9.
of a dollar is equal to how many cents? ..................................................................
10. 1 of $2.00 is equal to how many cents? ...................................................................... 4
11. True or False: Zero point four five is the same as forty five over a hundred. ................ 12. What is a quarter of twenty? .......................................................................................... 13. Two thirds of nine is equal to .......................................................................................... 14. Express two and four fifths as a decimal. ...................................................................... 15. True or False:
3
3 4
is greater than 3.65. ......................................................................
Problems: 1.
Anne bought fifteen bananas from the shop and gave five to her brother. What fraction does she still have? ..................................................................................
2.
Steve rode 6.75 km on the weekend. Express this amount as a fraction. ....................
3.
Suzy received $6 pocket money and spent two thirds of it on a book. How much did the book cost?..........................................................................................
4.
Rebecca and Michael went fishing and caught 20 fish. Eight of the fish were undersized and so they threw them back. What fraction do they have left? ...................................................................................... Page 43
Calculating Decimals You will need a calculator for these activities. So far we have looked at the methods for converting fractions into decimals. 3
In order to convert 4
to a decimal using a calculator we need to type in 3 ÷ 4.
1. Use your calculator to calculate decimals for the following: 4 5
....................
5 8
....................
1 4
....................
18 20
....................
14 28
......................
2 8
....................
9 20
....................
28 50
....................
27 30
....................
15 40
......................
45 60
....................
35 70
....................
18 50
....................
50 200
.................... 1,000 ......................
375
2. Using your calculator find three different fractions that are equal to 0.75. .............................................. 3.
..............................................
Find three fractions equal to 0.6. ..............................................
4.
..............................................
..............................................
..............................................
Find three fractions equal to 0.25. .............................................. ..............................................
..............................................
5. Sometimes when we calculate a decimal we end up with a recurring number. Divide 1 by 3 ( 1 ). What answer did you get?............................................................ 3
This can be recorded as
1 3
• ≈ 0.33. It is known as a recurring number.
Express these fractions as recurring numbers.
2 3
≈ ....................
1 6
≈ ..................
1 9
≈ ....................
4 6
≈ ..................
7 9
≈ ......................
Challenge: Divide fractions with a denominator of 9 ( 1/9, 2/9, 3/9, etc.) and record your answers. What pattern can you find? .................................................................................................... Find another pattern similar to this one and record the fractions you calculated. ❂ Using a calculator to find the decimal equivalent for fractions.
Page 44
Percentages 1 So far we know that decimals and fractions can be used to represent the same amount. A percentage is another way of expressing a part of a whole. Per means ‘for every’ and cent means ‘100’ so it is easy to remember that percent means ‘for every 100’. For example, if we had 100 students playing on the school sports field and 56 of them were girls, we could say that 56 percent (56 out of 100) of the students playing are girls. 1. We use the symbol % to express percentage. Name three everyday places where you might see this symbol. ..............................................
..............................................
..............................................
2. Look at the shaded amounts below. What percentage of each grid has been shaded?
..............................
................................
..................................
................................
3. Convert these fractions to a percentage. The first one has been done for you. 5 10
= 100 = 50%
50
5 10
....................
3 100
....................
15 100
.................... 1,000 ....................
1 100
....................
25 100
....................
4 10
.................... 1,000....................
170
8 10
......................
400
2 5
......................
4. Express these percentages as decimals and then simple fractions. 75
3
75% = 0.75 = 100 = 4 a. 60% = .................... b. 80% = .................... c. 50% = .................... d. 32% = .................... e. 24% = .................... f. 40% = .................... g. 90% = .................... h. 25% = .................... 5. Express these percentages as decimals. 35% ......................................
23% ......................................
89% ............................................
67% ......................................
79% ......................................
100% ..........................................
Challenge: Melanie scored 98% in a math test. If there were fifty questions, how many questions must Melanie have answered correctly? ❂ Introduction percentages and exploring the relationship between decimals, fractions and percentages.
Page 45
Percentages 2 In order to convert these fractions to a percentage, we need to find an equivalent fraction with a denominator of 10, 100 or 1000. Look at these examples and then convert these fractions to a percentage. 40
20
1. 200
=100
20
1 4
..........................= 2,000 ..........................=
2
40
15
30
300
600
5
9
= 20% 200 ..........................=
460
30 50
= .......................... 20 = ........................ 800 = ........................ 20 = ........................
40 60
= .......................... 40 = ........................ 400 = ........................ 800 = ........................
300 25 3,000 = .......................... 50
2.
= ........................ 25 = ........................ 20 = ........................
Match the percentage on the left with the correct fraction and decimal. There may be more than two answers. a) 25%
0.25
2.5
2 5
25 100
1 4
0.14
b) 55%
5.5
0.55
5 10
55 100
1 5
0.5
c) 32%
3.2
32 10
0.23
0.32
32 100
16 50
d) 80%
4 5
0.08
8.0
0.45
0.8
8 1,000
e) 15%
1.5
0.15
1 5
0.015
5.1
15 100
Word Problems: 1. Sophie spent 10% of her pocket money on a new pencil. What percentage of her pocket money does she have left? .............................................................................................. 2. Taylor spent 4/5 of his spare time reading. What percentage of time is this? .......................................................................................................................................... 3. Marcelle collected snails in the garden. She found 50% of them near the rose bushes, 15% of them near the hose and 25% around the clothes line. What percentage were found elsewhere? ........................................................................ 4. Katie and Greg have 100 chocolates in a jar. If 37 of them have hard centers and the rest are soft centered, what percentage have soft centers?............................................ 5. Lilly had 10,000 Frequent Flyer points. She received a 10% bonus for reaching the 10,000 mark. How many points does she now have? .................................................... ❂ Finding percentages for fractions with denominators of multiples of 10.
Page 46
The Relationship between Decimals, Fractions and Percentages 1. Complete the table below. The first one has been done for you.
Fraction
Decimal
Percentage
1 4
0.25
25%
1 5
0.2
2 10
36% 0.42 4 5
64% 3 20
0.95 28 100
73% 4 200 175 1,000
4.5 20.5% 2. Use = or ≠ in the boxes below. 0.75
75%
3 4
34%
1 4
25%
0.25
0.8
80%
4 5
0.8
5 25
25%
7 25
2 4
0.28
50%
5.0
17 20
0.59
Challenge: Joey scored 65% on the science test, Shelley got 0.75 of the questions correct and Matt answered 4/5 of the test correctly. Which student received the highest mark for the test? ❂ Matching fractions with equivalent decimal values and percentages.
Page 47
Calculating Percentages We know that 10% means 10 for every 100, 20% means 20 for every 100 and so on. We also know that 10% of 200 is 20, 10% of 1,000 is 100, 10% of 750 is 75 and so on. 1. Find 10% of the following amounts: $30 .............. $60 ................ $130 .............. $290 ............ $400 ............ $2,000 ............ $25 ................ $48 ................ $120 ............ $293 ............ $498 ............ $450 ................ 2. Find 50% of the following amounts: 20 .................. 30 .................. 100 ................ 150 ................ 200 ................ 1,000................ 1,200 ............ 64 .................. 65 .................. 250 ................ 16 .................. 17 .................... 44 .................. 28 .................. 350 ................ 12.6 .............. 25.8 .............. 243.2................ 3. Find 20% of the following amounts: $3.00 .................. $2.30 .................. $1.20 .................. $4.80 .................. $9.60 .................... $0.50 .................. $25.50 ................ $16.30 ................ $29.80 ................ $320.00 ................ 4. Subtract 20% from each of the amounts below: $9.60 .................. $10.80 ................ $4.00 .................. $2.00 .................. $150 ...................... $1000 ................ $3.50 .................. $6.00 .................. $7.20 .................. $9.50 ....................
Word Problems: 1. Rick received a 10% discount on his new basketball. Originally, the basketball cost $50.00. How much did Rick pay for the ball? .................................................................. 2. In Ali’s class 5 of the students are home sick. If there are normally 20 students in the class, what percentage of the students is absent? .......................................................... 3. Donelle sold 30 ice creams at the football game. The following week she sold 10% more. What amount did she sell? .............................................................................................. 4. Tanya correctly answered 180 questions out of 200 in an exam. What percentage did she answer correctly? ...................................................................................................... ❂ Finding percentages of whole numbers and money. ❂ Carrying out activities which give experience with simple everyday usage of percentages.
Page 48
Ratios A ratio describes the relationship of two numbers at the same time. For example, Sarah, Jane and Emily are sharing a cake which has been cut into nine pieces. For each person there are three pieces of cake. This is known as a one to three ratio and can be written as either 1/3 or 1:3. Express the following ratios as fractions: 1. If each table has four chairs, what is the ratio of tables to chairs? .......................................... 2. If each girl has five work files what is the ratio of girls to files? ................................................ 3. If each student has four pencils what is the ratio of students to pencils? ................................ 4. There are two tennis balls in the shed for every three students. What is the ratio of tennis balls to students? ............................................................................ 5. Steve, Louis and Mark each have a bike. What is the ratio of people to bikes? .................... 6. If the ratio of computers to students is 1:5 during computing lessons, how many students share a computer? ........................ 7. Eight people are sharing five pizzas between them. What is the fraction that each person will receive? ...................... 8. Sixteen students have four basketballs between them. How can this be represented as a fraction? ................................ 9. There are sixty sports shirts to be shared amoung 20 students. What is the ratio of shirts to students? ........................................ 10. The ratio of cows to sheep in Farmer Joe’s field is 4:5. If there are 20 cows how many sheep must there be? ................ 11. The ratio of cauliflowers to lettuces in Farmer Joe’s garden is 1:7. If there are two rows of cauliflowers how many rows are there of lettuces?............................ 12. In Miss Take’s class one quarter of the students live on farms. Express this as a ratio. .................................................................................................................................................... 13. The record of shipwecks at Rocks Harbor shows that one in every eight ships visiting was shipwrecked. Express this as a fraction............................................................................. 14. In the library the ratio of students to chairs is 3:1. What fraction of the class can sit on a chair during library times?.......................................................................................................... ❂ Exploring the relationship between ratios and fractions.
Page 49
Mixed Problems 1 1.
A bike is marked at $200.00. The retailer decides to mark the bike down by 10%. What will be the new cost of the bike? ....................................................................................
2.
Each year Josh’s dad pays 10% of his salary into a retirement fund. If he makes $35,000 a year, how much money will go into the fund each year?........................................
3.
Julia made a chocolate cake and flour made up 65% of the mixture. What percentage of the cake is not flour? ................................................................................................................
4.
Sam is buying books on sale at the local bookstore. Each book is discounted by 20% of what the marked price states. If he wants to buy a book marked at $20.00, what will he actually pay after the discount? ......................................
5.
Jarrad took thirty seconds to brush his teeth. What fraction of a minute is this? ..................................
6.
Helen took 45 minutes to walk to the beach. What percentage of an hour is this? ..............................
7.
Claudia rode 5 miles in an hour. This was 25% of the total amount she rode all day. What was the total distance she rode? ....................................................................................
8.
Billy had a bag of fruit. 50% of the fruit were bananas, 25% were apples, 10% were strawberries and the rest were peaches. What fraction of the bag did the peaches take up? ............................................................................................................................................
9.
Jeff spent 3/5 of his savings on some CDs. If each CD cost $20, and Jeff bought 3, how much money did Jeff start off with? ..........................................................................................
10. In the final exam, 30% of the students failed. If 140 students passed how many students must have failed? .................................................................................................................... 11. Karen ran 2.46 miles on Wednesday. Express this distance as a fraction. ............................ 12. The bank is offering interest rates of 5% on savings accounts. How many cents for each dollar will the bank pay? .......................................................................................................... 13. At the local mine 85% of the miners were under 30 years of age. What fraction of the miners were over thirty?.......................... Express this amount as a decimal ...................... 14. Justin, Thomas and James had six ice creams to share. What is the ratio of boys to ice creams? .............................................................................................................................. Page 50
1.
Mixed Problems 2
In Mario’s class one quarter of the students wear glasses. If there
are 24 students in the class how many wear glasses?........................ 2.
Leanna walked 3.56 miles and Stephanie walked three and a half miles. Who walked the furthest distance? ....................................
3.
The ratio of beachgoers to umbrellas was six to one. What fraction of beachgoers had an umbrella? ..........................................................
4.
The ratio of football spectators to raincoats was 20:1. Express this amount as a percentage. ................................................
5.
Katrina took 40 minutes to complete her homework. What fraction of an hour is this? ..........................................................
6.
Alex spent two hours a day practicing the piano. What fraction of the day is this? ..........................................................
7.
Lara spent two hours on homework each night. If 1/4 is spent on math and 50% is spent on history what is the deimal amount left for other subjects? ...................................................... a. How many minutes are spent on math? ............................................................................ b. How many minutes are spent on history? ..........................................................................
8.
The football team won 80% of its matches during the last season. If twenty matches were played, what was the total number of games won? ................................................................
9.
The rowing team has won 0.65 of its total races. What fraction of races did it lose? ..................................................................................................................................................
10. Bill and Ted are playing chess. Bill has beaten Ted 60% of the time. If they have played fifty games, how many has Bill won? .............................................................................................. 11. Sam and Tess are playing cards. They have played eighteen games and Tess has won two thirds of the games. How many games has Sam won? .......................................................... 12. Maria invited 35 guests to her birthday party, however only 5/7 of the guests are able to come. How many of the guests will be able to attend? ...................................................................... 13. Marguerite spent $11.00 on a new hat. She now has 4/5 of her savings left. a. What percentage of her savings was spent on the hat?.................................................... b. How much money did Marguerite start off with?................................................................ Page 51
Fractions Answers Answers correspond to row order of questions. Page 6 Introduction to Fractions 1. 4, 7, 8, 8, 10, 6,9; 2. 2/4, 5/9, 6/8, 2/4; 3. Check diagrams; 4. 2/4, 4/10, 2/6. Challenge: They would both be equal to a whole. Page 7 Equivalent Fractions 1 1. Check diagrams; 2a. 1/4, 2/8; b. 2/4, 1/2; c. 2/5, 4/10. Challenge: Anthony. Page 8 Fractions as Parts of a Whole 1. 1/4, 2/8, 3/6, 5/10, 6/9; 2. Check diagrams; 3. 6/8, 4/8, 4/12, 2/8, 2/4, 6/9. Challenge: 12/20 = 3/5. Page 9 Equivalent Fractions: Exercises 1. Answers will vary; 2. ≠, =, ≠, ≠, =, =, ≠, ≠, =, ≠, ≠, =; 3. , , >, , ,
