Australian Curriculum Mathematics: Fractions, Decimals and Percentages - Book 3 by Teacher Superstore

RIC-6138 5.6/374

Australian Curriculum Mathematics – Number and Algebra: Fractions, Decimals and Percentages Book 3 (Years 5 and 6) Published by R.I.C. Publications® 2015 Copyright© Clare Way 2015 ISBN 978-1-925201-09-3 RIC– 6138

Titles in this series: Australian Curriculum Mathematics – Number and Algebra: Fractions Book 1 (Years 1 and 2) Australian Curriculum Mathematics – Number and Algebra: Fractions and Decimals Book 2 (Years 3 and 4) Australian Curriculum Mathematics – Number and Algebra: Fractions, Decimals and Percentages Book 3 (Years 5 and 6)

Copyright Notice A number of pages in this book are worksheets. The publisher licenses the individual teacher who purchased this book to photocopy these pages to hand out to students in their own classes. Except as allowed under the Copyright Act 1968, any other use (including digital and online uses and the creation of overhead transparencies or posters) or any use by or for other people (including by or for other teachers, students or institutions) is prohibited. If you want a licence to do anything outside the scope of the BLM licence above, please contact the Publisher. This information is provided to clarify the limits of this licence and its interaction with the Copyright Act. For your added protection in the case of copyright inspection, please complete the form below. Retain this form, the complete original document and the invoice or receipt as proof of purchase. Name of Purchaser:

is material subject to All material identified by copyright under the Copyright Act 1968 (Cth) and is owned by the Australian Curriculum, Assessment and Reporting Authority 2015. For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum. Elaborations: This may be a modified extract from the Australian Curriculum and may include the work of other authors. Disclaimer: ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that: • The content descriptions are solely for a particular year and subject; • All the content descriptions for that year and subject have been used; and • The author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu.au/ This material is reproduced with the permission of ACARA.

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Foreword This three-book series is aimed at immersing students in all aspects of fraction work, including decimals and percentages for the upper levels. Based on Australian Curriculum Mathematics, the books will provide teachers with a comprehensive approach to teaching and helping their students understand fractions. Through the proficiency strands of Understanding, Fluency, Problem-solving and Reasoning, students will experience success in this substrand. The series contains a large variety of activities including teachers notes, warm-up activity ideas, hands-on tasks, blackline master worksheets, assessment tasks and a checklist at the end of each Year level. Each Australian Curriculum content description for fractions will be covered in detail, allowing busy teachers to assist their students in gaining confidence in their knowledge of fractions. Titles in this Australian Curriculum Mathematics – Number and Algebra series include: Fractions Book 1 (Years 1 and 2) Fractions and Decimals Book 2 (Years 3 and 4) Fractions, Decimals and Percentages Book 3 (Years 5 and 6)

Contents Year 6

Year 5 Compare and order common unit fractions and locate and represent them on a number line

Compare fractions with related denominators and locate and represent them on a number line (ACMNA125)

(ACMNA102)

Solve problems involving addition and subtraction of fractions with the same or related denominators (ACMNA126)

Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)

Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies

© R. I . C.Publ i cat i ons Add and subtract decimals, with and without digital •f orr evi ew pur posesonl y• technologies, and use estimation and rounding to check Compare, order and represent decimals Recognise that the place value system can be extended beyond hundredths (ACMNA104)

(ACMNA127)

(ACMNA105)

the reasonableness of answers (ACMNA128)

Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies

Curriculum links ...........................................................2

(ACMNA129)

Teachers notes ......................................................... 3–4 Warm-up activities.................................................. 4–5 Teachers resources ...............................................6–16 Blacklines...............................................................17–44 Assessments .........................................................45–46 Checklist ......................................................................47 Answers.................................................................48–52

Multiply and divide decimals by powers of 10 (ACMNA130)

Make connections between equivalent fractions, decimals and percentages (ACMNA131) Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132) Curriculum links ................................................................ 53 Teachers notes ............................................................ 54–55 Warm-up activities..................................................... 56–57 Teachers resources .................................................... 58–66 Blacklines...................................................................... 67–99 Assessments ............................................................100–101 Checklist ..................................................................102–103 Answers....................................................................104–110

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Fractions, Decimals and Percentages (Years 5 and 6)

iii

Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)

(ACMNA105)

Compare, order and represent decimals

Recognise that the place value system can be extended beyond hundredths (ACMNA104)

Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)

Page

(ACMNA102)

Compare and order common unit fractions and locate and represent them on a number line

Year 5 Curriculum links

Page title

Naming fractions

✓

Unit fractions

✓

Comparing unit fractions

✓

Equivalent fractions

✓

Equivalent fractions and number lines

✓

Locating fractions on number lines

✓

Ordering fractions

✓

Improper fractions and mixed numbers

✓

Counting by fractions

✓

Fractions of groups

✓

Fractions and multiples

✓

Simplifying fractions

✓

Adding fractions using pictures

✓

Adding fractions

✓

Subtracting fractions using pictures

✓

Subtracting fractions

✓

Using number lines to add or subtract fractions

✓

Addition and subtraction fraction problems

✓

Fractions, decimals and place value fractions

✓

Decimal tenths

✓

Decimal hundredths

✓

Decimal thousandths

✓

Decimal numbers and equivalence

✓

Decimal numbers and money

✓

Decimal numbers and measurement

✓

Comparing decimal numbers

✓

Ordering decimal numbers

✓

Decimal problems

✓

Assessment 1

Assessment 2

✓

Fractions, Decimals and Percentages (Years 5 and 6)

✓

✓ ✓

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Teachers notes

Year 5

Introduction Once again, students will be able to draw on their previous experience of fractions and decimals, and build on this knowledge to help them understand new concepts introduced in Year 5. Begin by asking students, ‘What is a fraction?’ Brainstorm what a fraction means to them and where they might use fractions in their lives. Then formally define a fraction as an equal part of a whole. Hands-on and visual tasks will still play an important role at this level when dealing with the content descriptions, particularly when it comes to comparing and ordering fractions and decimals. Number lines are a useful tool to do this. Reinforce the students’ knowledge of equivalence of fractions, which will assist them when it comes to comparing and ordering fractions. At this level students will be introduced to the term ‘unit fraction’, which is a fraction where the numerator (top number) is one. Along with this they will be encouraged to recognise the importance of the denominator in helping them understand the size and value of the fraction. The denominator is also important when it comes to adding or subtracting fractions, which they can do with the use of diagrams and number lines. In Year 4 they were introduced to mixed numbers and improper fractions, and this knowledge will continue to develop and assist them in solving addition and subtraction of problems relating to fractions.At this level they are only required to add and subtract fractions with common denominators. They will recognise the connection between fractions and decimals beyond their previous knowledge of tenths and hundredths and they will be introduced to thousandths in the place value system. As fractions are equal parts of a whole, they will see that decimals are equal parts of a whole divided into ten, one hundred or one thousand equal parts.This understanding will assist them to compare, order and represent decimals and relate these back to fractions.

The sub-strand of Patterns and Algebra intertwines in this strand as students will be asked to use their knowledge of whole numbers, fractions and decimals to look at patterns and skip count with the aid of diagrams and number lines. Comparing and ordering fractions

Allow students to visually compare and order fractions from smallest to largest and visa versa. Discuss and encourage students to look at the denominators to help them compare and order fractions, particularly unit fractions where the numerator is one. Lead students to realise that the larger the denominator the smaller the fractional part. Use number lines to locate, order and count by fractions. Look at equivalence in fractions and use visual aids such as fraction strips to find equivalent fractions such as two quarters equals one half. Once a fraction becomes greater than one whole it is known as a mixed number. Define and change mixed numbers into improper fractions, and improper fractions to mixed numbers. When looking at fractions of whole numbers or groups, incorporate the students’ knowledge of multiples and division; for example, 24 can be divided by halves, thirds, quarters, sixths and eighths. Adding and subtracting fractions When introducing the concept of addition and subtraction of fractions to students, make sure it is initially done using pictures or visual aids so that students do not fall into the trap of adding or subtracting both the numerator and the denominator. If they can see and make it first then they will realise that they simply add or subtract the numerator. For example: 38 + 48 = 78 Once they understand this concept students can then move to written examples of adding and subtracting fractions. Students will also benefit from solving story problems they can relate to, involving adding and subtracting fractions such as if Helen ate 28 of the pizza and Abby ate 38 , how much was left? Or how much did they eat altogether? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers notes

Year 5

Place value system and decimals In Year 4 students were introduced to decimal numbers in the place value system.They would have previous knowledge of tenths and hundredths, and at this level they are then introduced to thousandths. Remind students the definition of decimal numbers, and discuss where in our lives we see and use decimal numbers; for example, when dealing with money, measurement or time. Decimals are related to fractions because they are both equal parts of a whole number. This concept should be introduced visually to students prior to simply working with numbers. Using paper strips and one hundred grids show students what a decimal 52 . number looks like; for example, 0.52 is the same as saying 52 parts out of 100/one whole or 100 1 or In Year 5 thousandths is a new concept to students, so use MAB blocks to show students that 1000 0.001 is one part out of 1000 equal parts from one whole number. Also use a place value chart to show students where decimal numbers fit in comparison to whole numbers. Once students have grasped the concept of decimal numbers they will be able to compare, order and count by decimals and relate these to fractions. Number lines are again a useful tool for counting by decimal numbers. Students will also be able to see patterns forming when counting or skip counting by fractions or decimal numbers.

© R. I . C.Pactivities ubl i cat i ons Warm-up •f orr evi ew pur posesonl y• These activities could be used to introduce the lesson. They can be used as a whole class focus, or a small group or individual activity depending on the lesson content. • Make a fraction chart, using rectangles, guiding students through dividing each section to match the fraction denominator then shading and labelling each fraction (see teacher resources page 6). Discuss the most accurate way of making sure each fractional part is the same; it may be folding each section or measuring it with a ruler. • Define equivalence of fractions (the same) then, using the fraction chart, discuss which fractions are equivalent. Use equivalent fraction cards (see teacher resources page 7) to play a match up/memory game. • Use students’ knowledge of the equivalence of fractions to break fractions down to the lowest or least common denominator. For example: 24 and 48 can be broken down to 12 . Visit the website <www. mathsisfun.com/fractions-menu.html> and look at the lowest or least common multiple. • Define numerator and denominator. Discuss how the denominator is the most important as it tells us how many parts the whole is divided into. Look at unit fractions such as 12 , 14 , 18 , 13 , 15 and ask students to place them in order of smallest to largest fraction. What do they notice about the smallest fraction? (Generally it will have the largest denominator.) Use unit fraction symbols and pictures on cards to order fractions from the smallest to largest or the largest to the smallest (see teacher resources page 8).

Fractions, Decimals and Percentages (Years 5 and 6)

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Warm-up activities

Year 5

• Use number lines to order and compare fractions. Count by fractions on number lines. What happens when you reach one whole? Lead students to name whole numbers and fractions as mixed numbers. When the numerator is the same or higher than the denominator it is called an improper fraction. Match the improper fractions cards to the mixed numbers cards (see teacher resources page 9). • Use number lines and symbols to convert improper fractions to mixed numbers. Then, once students show they understand this concept, show them the formula of how this can be done. For example: 74 = 1 34 . Divide the denominator into the numerator to find the whole number then the remainder becomes the new numerator. • Use the interactive whiteboard or a fraction maths program or website to explore concepts such as equivalent fractions, improper fractions and mixed numbers. • Use pictures and symbols to show how fractions can be added. Draw 14 + 24 of a sandwich to work out how much was eaten altogether. Point out to students that the denominator remains the same, it is the numerator that changes. Use the picture symbols provided to add fractions (see teacher resources page 10). • Use pictures and symbols to represent subtraction of fractions. For example: A pizza is cut into 88 . If someone ate 58 , how much is left? Once again, the denominator remains the same and the numerator changes. Use the picture symbols provided to subtract fractions (see teacher resources page 11). • Using strips of paper divided into ten (see teacher resources page 12), discuss how tenths can be related 2 1 to decimal numbers. Shade the tenths to match the decimal tenths. For example: 10 = 0.1, 10 = 0.2 etc. • Use a hundreds grid (see teacher resources page 13) to illustrate that hundredths can be also related 28 to decimal numbers. For example: 100 = 0.28. Shade some decimal numbers to match the fraction hundredths. Relate these hundredths to money. Mixed numbers can be represented; for example, $8.25 25 is a the same as the mixed number 8 100 .

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• • Hundredths (decimals) are also commonly used when measuring.There are 100 cm in one whole metre. Measure items around the room and write them as a decimal number. For example: a ruler measures 30 cm or 0.30 m, and a pencil may be 0.15 m. Measure students’ height and record it as a mixed number 42 fraction and a decimal number; for example, 1 100 or 1.42 m. • Introduce decimal numbers beyond hundredths, such as thousandths, which represent parts out of 1000 equal parts of a whole. Show students what one thousandth looks like and compare this to one hundredth and one tenth (see teacher resources pages 14 and 15). • Discuss how the thousandths number is three places to the right of the decimal point. Represent whole and decimal numbers on a place value chart (see teacher resources page 16). • Count and skip count by fractions and decimals using number lines. Complete counting patterns by filling in the missing fractions or decimal number. Incorporate the concepts of money and measurement when counting. Look for patterns such as common multiples or factors. For example: when we count by quarters we can also count by halves.

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Fraction chart

1 whole 1 2 1 3 1 4 1 5 1 6

1 7 1 8 1 9 1 10 1 11 1 12 6

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Equivalent fraction cards

4 3 2 1 2 12 5 2 1 6 3 2 4 5 10 3 4 4 3 2 6 5 4 10 5 3 8 4 5 8 4 10 1 5 6 6 10 6 8 6 7 6 9 .Publ 2i © R. I . C i cat ons 12•f 12 6sonl orr evi ew pur pose y• 7 2 3 1 8 4 3 6 8 9 2 4 4 8 9 6 6 8 5 10 3 12 12 6 10 1 11 2 10 3 11 5 12 4 1 12 2 6 10 1 12 3 6 9 9 R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Unit fractions and symbols

1 2 1 3 1 4 1 5 1 R. © I . C.Publ i cat i ons 6review purposesonly• •f or 1 8 1 9 1 10 1 12 1 20 8

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Improper fractions and mixed numbers

1 2 1 1 1 1 1 1 1 1 1

1 2

1 1 1 1 1 1 1

5 6 13 8 3 8 1 5 16 10 4 8 1 7 7 11 3 10 4 10 2 1 8 17 3 12 4 10 20 6 1 5 10 5i 4 © R. 12 I . C.Publ i cat ons 3f 1 pur 8esonl • orr evi ew pos y•13 4 9 5 12 1 7 10 17 5 9 5 12 3 3 7 24 5 2 6 12 1 4 11 10 6 2 6 9 5 4 12 16 6 3 6 9 1 5 9 18 8 3 8 9

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Fraction picture symbols – adding

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Fraction picture symbols – subtracting

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Decimal tenths

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Decimal hundredths grids

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Decimal thousandths grids

1 1000

100 1000

500 1000

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Decimal tenths, hundredths and thousandths

1 10

1 100

1 1000

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Thousandths

Hundredths

Tenths

Ones

Tens

Hundreds

Thousands

Tens of thousands

Hundreds of thousands

Millions

Place value chart for whole and decimal numbers

Fractions, Decimals and Percentages (Years 5 and 6)

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Naming fractions A fraction is an equal part of a whole. A fraction is made up of a numerator (above the 7 line) and a denominator (below the line). For example: 8 1.

Complete the sentences using the words numerator or denominator. (a) The

tells us how many parts of a shape/object are shaded.

(b) The

tells us how many parts there are in a whole shape/object.

Write the fraction symbol to represent what is shaded on the shapes/objects.

9 8

2 3

(b)

(a)

(c)

(d)

(f)

(g)

(h)

Write the fraction symbols for the fraction words. (a) one-sixth

(b)

two-eighths

(c)

one-third

(d) five-ninths

(e)

nine-twelfths

(f)

two-halves

Shade the shapes to represent the fraction symbol. 7 8

5 12

6 9

2 3

1 10

Going further

On the back of this page draw and represent the following fractions, making sure your 3 6 9 parts are equal: 4 , 8 and 12 . What do you notice about these fractions?

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Fractions, Decimals and Percentages (Years 5 and 6)

Unit fractions A fraction with a numerator of 1 is called a unit fraction. 1 1 1 For example: 2 , 10 , 50 1.

Use symbols to name the unit fractions.

(a)

(b)

(c)

(d)

(e)

(f)

Write the unit fractions above in order from smallest to largest.

(a)

1 6

(b)

1 2

(c)

1 4

(d)

1 3

(e)

1 8

(f)

1 5

Write the unit fractions above in order from largest to smallest.

Going further

If you were asked to choose between choose and why? 18

1 5

and

1 6

Fractions, Decimals and Percentages (Years 5 and 6)

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Comparing unit fractions 1.

Look at each pair of unit fractions. Write the larger unit fraction.

(a)

(b)

(c)

(d)

(e)

(f)

Write each unit fraction in the correct box starting with the smallest. , P , u , b , l ,i ,a ,t ,o © R. I . C. c i ns •f orr evi ew pur posesonl y• 1 7

1 11

1 3

1 9

1 4

1 10

1 5

1 8

1 6

Write true or false to each statement. (a)

1 3

is larger than

1 4

(b)

1 8

is smaller than

(c)

1 7

is larger than

1 8

(d)

1 5

is larger than

1 4

(e)

1 10

(f)

1 8

is larger than

1 9

is smaller than

1 12

1 6

Solve the problems about who ate more. 1 6

(a) Adam ate (b) Isla had

1 2

of a pie and Dale ate

of a pizza and Sara had

1 5 . Who

ate more?

1 3 . Who

ate more?

1 8

of a bread stick and Nick ate

1 7 . Who

(d) Gina ate

1 4

of a chocolate block and Jake ate

ate more?

1 5 . Who

ate more?

Going further

Make a statement explaining what happens to the denominator as the fraction becomes smaller or larger.

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Fractions, Decimals and Percentages (Years 5 and 6)

Equivalent fractions Fractions that are the same size are also known as equivalent fractions. For example: 1.

1 2

2 4

one whole

Complete the missing unit fraction symbols on the fraction mat.

1 2 1 3

1 3 1 4

Use different colours to show equivalent fractions on the mat.

1 4

1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12

Use the fraction mat to complete the equivalent statements. (a)

4 8

(b)

8 10

1 5

(f)

(d)

2 12

6 8

2 4

1 3

=. (g)b =c (h) = ©R I . C.Pu l i at i ons •f orr evi ew pur posesonl y• Use the fraction mat to write an equivalent fraction for each fraction.

(e) 3.

(c)

4 6

(a)

1 3

(b)

1 2

(c)

1 4

(d)

1 6

(e)

3 4

(f)

6 12

(g)

2 8

(h)

2 3

3 4

Shade to represent the fractions as equivalent fractions. 2 5

4 10

2 3

4 6

6 8

Going further

List as many equivalent fractions as you can for these:

Fractions, Decimals and Percentages (Years 5 and 6)

1 1 1 2, 3, 4.

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Equivalent fractions and number lines 1.

Complete the missing fractions on the number lines. 1 2

(a)

1 4

(b)

2 4

3 4

1 8

1 3

(c)

2 3

1 6

(d)

2 6

1 5

(e)

2 12

2 5

3 5

4 5

1 10

Use the number lines above to answer the equivalent fraction questions. (a)

3 4

(b)

3 5

(e)

6 12

1 3

(f)

4 6

(c)

(d)

2 8

(g)

8 10

(h)

2 6

Place the following fractions in the correct order on the number line below. 10 1 3 1 1 1 3 10 , 8 , 6 , 3 , 12 , 6 , 4

0 Going further

Explain how number lines can help your knowledge of fractions. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Locating fractions on number lines 1.

Locate and circle the following fractions on the number line below:

3 6 9 4 , 8 , 12 .

What do you notice about the three fractions? 1 4 1 8

0 1 12

2 4

2 8 2 12

3 8

3 12

3 4

4 8

4 12

5 12

5 8

6 12

7 12

4 4

6 8 8 12

7 8

9 12

Locate and circle the fractions on the number line below:

1 5 3, 6

10 12

8 8 11 12

7 12 .

and

12 12

What denominator do they have in common? 1 3 1 6

0 1 12

2 3

2 6

2 12

3 12

3 6

4 12

5 12

3 3

4 6

6 12

7 12

5 6

8 12

Locate and circle the fractions on the number line:

2 5

9 12 4 4 10 , 5

and

6 6

10 12

and

11 12

12 12

8 10 .

What do the fractions have in common? 1 5

0 1 10

2 5

2 10

3 10

3 5

4 10

5 10

4 5

6 10

7 10

8 10

9 10

5 5 10 10

1 4.

Place the fractions where you think they should go on the number line below. 1 3

2 5

4 6

11 12

1 2

0 5.

7 8

Write all the eighths on the number line below. 0

1 1 16

2 16

3 16

4 16

5 16

6 16

7 16

8 16

9 16

10 16

11 16

12 16

13 16

14 16

15 16

16 16

Going further

On the back of this page, draw up a number line that counts by tenths above the line and twentieths below the line. Circle where you think fifths should be.

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Ordering fractions 1.

Use the numbers 1 to 5 to order the fractions from smallest to largest.

Represent each fraction by shading. Number them in order from smallest to largest. 1 5 7 2 3 , 6 , 12 , 9

Write the common fractions in order from largest to smallest.

(a)

2 8

(b)

3 10

(c)

3 12

6 8

3 8

7 8

1 8

5 8

5 10

12 12

1 12

9 10

5 12

7 12

6 10

10 12

1 10

4 12

9 12

4 10

6 12

11 12

8 8

4 8

7 10

2 10

2 12

8 12

Use your knowledge of equivalence to order the fractions from smallest to largest. (a)

7 8

3 4

5 8

1 4

1 2

(b)

2 3

1 12

10 12

3 6

(c)

9 10

1 5

4 10

3 5

1 8

4 4

3 8

1 3

5 12

1 6

1 10

4 5

3 10

Going further

Draw up a number line on the back of this page and place these fractions on it in order 3 1 7 1 1 11 2 3 2 4 from largest to smallest: 4 , 12 , 8 , 5 , 2 , 12 , 6 , 3 , 8 , 5 . R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Improper fractions and mixed numbers In an improper fraction the numerator is the same or greater than the denominator. 14 7 For example: 3 or 2 1

A mixed number is a whole number with a fraction. For example: 1 4 , 4 8 . 1.

Write the improper fraction and mixed number.

(a)

(b)

(d) 2.

(e)

(f)

Change the improper fractions to mixed numbers. 16 4 17 4

(c)

= 4 4 (remainder 1 =

1 4)

(a)

11 2

(b)

19 3

(c)

12 6

(d)

21 4

(e)

27 5

(f)

36 12

(g)

32 3

(h)

53 10

(i)

41 2

(j)

51 8

(k)

40 5

(l)

75 12

Change the mixed numbers to improper fractions. 2

For example: 1 3 =

3 3

2 3

5 3

(b) 3 5 =

(f) 6 4 =

(j) 3 5 =

(a) 1 2 = (e) 5 8 = (i) 4 6 =

(d) 4 3 =

(g) 1 12 =

(k) 7 5 =

(h) 2 9 =

(l) 8 12 =

Going further

Explain in your own words how you change improper fractions to mixed numbers and mixed numbers to improper fractions. Give an example. 24

Fractions, Decimals and Percentages (Years 5 and 6)

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Counting by fractions 1.

Count the fractions represented below and write your answer as an improper fraction and a mixed number. An improper fraction: the numerator is the same or greater than the denominator. A mixed number: a whole number with a fraction.

(a)

(b)

(c)

(d)

(e) 2.

(a)

1 3

0 1 6

(b)

2 6

1 4

0 1 8

(c)

2 3

2 4

2 8

1 5

0 1 10

2 5

2 10

Going further

We sometimes have to count by fractions; for example, slices of cake or pizza. Work out how many wholes there are in each example. (a) 16 halves =

(b) 12 thirds =

(d) 24 quarters =

(e) 45 fifths =

(f) 42 sixths =

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Fractions, Decimals and Percentages (Years 5 and 6)

Fractions of groups

Look at the group of birds and answer the questions. What fraction: (a) of the birds are white?

(b) are black?

(d) are large?

(e) have a black patch?

(f) have only 1 leg?

What fraction of each group is shaded?

(a)

(e) 3.

(f)

(g)

(h)

Work out the fraction of each whole number by dividing it by the denominator of the fraction. 1 27 For example: 3 of 27 = 3 = 9 (a)

1 2

of 14 =

(b)

1 4

of 12 =

(c)

1 5

(d)

1 3

of 15 =

(e)

1 6

of 18 =

(f)

1 10

(g)

1 2

of 32 =

(h)

1 12

(i)

1 9

of 60 =

of 25 = of 80 = of 36 =

Going further

What do you need to do when the numerator is greater than 1? 2 12 For example: 3 of 12 = 3 x 2 = 8 3 4 26

of 28 =

3 5

of 35 =

Fractions, Decimals and Percentages (Years 5 and 6)

4 9

of 27 = R.I.C. Publications®

7 8

of 64 = www.ricpublications.com.au

Fractions and multiples A whole number can be divided into fractions if it is a multiple of the fraction denominator. 1 For example: 4 of 20 = 5 20 is a multiple of 4 1.

Circle the numbers that can be divided into halves (multiples of 2). Underline the numbers that can be divided into quarters (multiples of 4). 13

Which numbers are multiples of 2 and 4?

Circle the multiples of 3 and underline the multiples of 6. 9

Which numbers are multiples of 3 and 6?

16: half

quarter

eighth

(b)

24: half

quarter

eighth

(c)

40: half

quarter

eighth

(d)

56: half

quarter

eighth

(e)

72: half

quarter

eighth

twelfth

(f)

96: half

quarter

eighth

twelfth

Going further

Which fractions can each number be divided into? 45: 60: 120:

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Fractions, Decimals and Percentages (Years 5 and 6)

Simplifying fractions Larger fractions can be simplified to their lowest form if the numerator and denominator 16 4 are multiples of a common factor. For example: 20 = 5 . Because 4 is a common factor of 16 and 20, it can be simplified to 16 ÷ 4 = 4 20 ÷ 4 = 5 1.

Simplify the fractions until you reach their lowest form. 20 10 5 1 For example: 40 = 20 = 10 = 2 (a)

16 24

(c)

20 40

(e)

8 40

(b)

12 20

(d)

16 32

(f)

40 60

Simplify the fractions by using the common factor provided. (b) 10 ÷ 5 =

(d) 18 ÷ 6 =

(e) 24 ÷ 4 =

(f) 27 ÷ 9 =

(g) 40 ÷ 8 =

(h) 90 ÷ 10 =

28 ÷ 4 =

45 ÷ 9 =

48 ÷ 8 =

(a) 12 ÷ 3 = 15 ÷ 3 =

Think of the highest common factor for each fraction. Simplify each fraction to its 30 5 lowest form. For example: 36 = 6 (÷ by 6) (a)

9 12

(b)

4 6

(c)

15 20

(d)

16 24

(e)

18 24

(f)

40 48

(g)

70 80

(h)

21 24

(i)

45 60

(j)

32 36

(k)

27 30

(l)

12 72

Going further

How many different fractions can you suggest that can be broken down to the lowest 1 fraction of 2 ? Make a list on the back of this page. 28

Fractions, Decimals and Percentages (Years 5 and 6)

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Adding fractions using pictures 1.

Add the same denominator fraction parts and complete the addition sentence. Simplify answers if necessary.

(a)

2 4

1 4

(c)

3 8

2 8

(e)

3 5

2 5

(b)

1 2

(d)

4 6

1 6

(f)

5 10

3 10

6 12

4 12

10 12

5 6

(a)

(b)

(c)

(d)

(e)

(f)

Going further

If one whole is the answer, how many different fraction addition sentences can you come up with that equal 1 whole? Draw or write them on the back of this page. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Adding fractions 1.

Shade the fraction strip to match the addition sentence. Simplify the answer if necessary. (a)

2 4

(b)

4 8

3 8

(c)

3 10

(d)

1 5

3 5

(e)

4 6

2 6

3 10

Add the fractions and simplify the answer if necessary. 2 2 4 2 For example: 6 + 6 = 6 or 3 (a)

7 10

(d)

3 8

+ +

2 10 1 8

3 5

(e)

5 12

1 5

3 9

3 12

(f)

3 4

3 9

1 4

If the numerator of a fraction is greater than the denominator, it is an improper fraction. Add the fractions and change improper fractions to mixed numbers and simplify the 4 4 8 3 answer if necessary. For example: 5 + 5 = 5 or 1 5 (a)

2 3

(b)

7 8

5 8

(c)

3 4

(d)

1 2

4 2

(e)

3 5

4 5

(f)

7 10

(g)

9 12

(h)

5 6

3 6

(i)

6 9

5 9

(j)

3 3

(k)

8 8

(l)

7 5

4 5

+ +

6 12 2 3

+ +

3 4

8 10

Going further 1

If the answer to a fraction addition problem is 1 2 , what could be the question? Suggest as many as you can by drawing or writing them on the back of this page. 30

Fractions, Decimals and Percentages (Years 5 and 6)

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Subtracting fractions using pictures 1.

Subtract the same denominator fractions and complete the subtraction sentence. Shade to represent your answer.

(a)

2 2

–

1 2

(c)

4 5

–

3 5

(e)

4 4

–

2 4

(b)

8 8

(d)

9 10

(f)

5 6

–

3 8

6 10

4 6

Subtract the fractions using the strips to help you. Simplify the answer if necessary. 5 1 4 2 For example: 6 – 6 = 6 or 3

(a)

8 8

(b)

5 5

–

(c)

12 12

(d)

4 4

6 8

4 5

6 12

–

2 4

Going further

Create at least three different subtraction fraction sentences to go with the fraction strip below.

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Fractions, Decimals and Percentages (Years 5 and 6)

Subtracting fractions 1.

Subtract the fractions and simplify your answer if necessary. 8 2 6 3 For example: 8 – 8 = 8 or 4 (a)

3 4

– 4 =

(d) 8 – 8 = 10

(e)

(g) 12 – 12 = 2.

5 5

– 5 =

(h) 2 – 2 =

(c)

8 10

(f)

6 6

– 6 =

(i)

3 3

– 3 =

– 10 = 4 2

Subtract the improper fractions and simplify the answer or change it to a mixed number if 11 5 6 8 3 5 2 necessary. For example: 6 – 6 = 6 or 1, 3 – 3 = 3 or 1 3 (a)

5 2

(d)

19 5

– 5 =

– 2 = 6

(g) 10 – 10 = 3.

(b) 9 – 9 =

(b) 3 – 3 =

(c)

11 4

(e)

11 6

– 6 =

(f)

9 8

(h)

13 4

– 4 =

(i)

20 12

6 5

– 4 = 4

– 8 = 5

– 12 =

Solve the problems involving subtracting fractions by illustrating them, then writing a fraction sentence and answer. Problem

Illustrate

Sentence and answer

(a) A café had a pie that was divided into eighths, if 5 they sold 8 of the pie how much is left?

(b) A caterer made 16 quarters of sandwiches. If 13 quarters were consumed, how many were left? (c) Tim’s swimming coach told him to swim 12 halflaps of the pool. If he swims 9 half-laps, how many laps does he have left to swim? (d) Three family-sized pizzas cut into twelfths were delivered to a party. If 30 pieces were consumed, how much was left? Going further

Complete the fraction sentences and write your own subtraction story problem to match. (a) 32

6 6

–

5 6

Fractions, Decimals and Percentages (Years 5 and 6)

(b)

7 8

–

3 8

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Using number lines to add or subtract fractions 1.

Use a pencil to jump along the number lines to subtract the fractions. Where you see a whole number, use your knowledge of equivalence to change it into a fraction. 5 16 5 11 3 For example: Change 2 – 8 to 8 – 8 = 8 or 1 8 1 2

0 1 4

2 2

2 4

3 4

3 2

4 4

5 4

4 2

6 4

7 4

8 4

1 (a)

3 2

– 2 =

1 8

(b) 4 – 4 = 2 4

1 4

2 8

3 8

3 4

4 8

5 8

6 8

7 8

8 8

6 4

5 4 9 8

10 8

11 8

8 4

7 4

12 8

13 8

14 8

15 8

1 (d)

15 8

– 8 =

6 4

(e) 1 3

– 4 =

2 3

(f)

2– 8 =

4 3

3 3

6 3

5 3

2 6

3 6

4 6

(g) 3 – 3 =

1 10

2 10

11 6

(h)

1 5

5 6

2 5 3 10

4 10

3 5 5 10

6 6

6 10

7 10

7 6

8 6

– 6 =

4 5 9 10

9 6

(i) 5 5

8 10

10 10

6 5 11 10

10 6

13 10

12 6

2– 6 =

7 5

12 10

11 6

8 5

14 10

15 10

16 10

9 5 17 10

10 5

18 10

19 10

1 17 10

(j) 2.

– 10 =

10 5

(k)

16 8

20 10

– 5 =

(l)

2 – 10 =

Now use a red pencil to jump along the number lines above to add the fractions. (a)

1 2

+ 2 =

(d) 8 + 8 =

(b) 4 + 4 =

(c)

3 4

+ 4 =

10 8

(f)

1 3

+ 3 =

(i)

5 10

(e)

(g) 6 + 6 =

+ 8 = 4

(h) 5 + 5 =

5 4

+ 10 =

Going further

Draw a number line of the back of this page that counts by quarters and eighths up to 1 whole. Use it to jump along and solve the fraction sentences. (a)

2 4

1 4

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3 8

1 8

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1 4

–

2 4

(d) 1 –

2 8

–

5 8

Fractions, Decimals and Percentages (Years 5 and 6)

Addition and subtraction fraction problems 1 2 1 4

0 1 8

2 2

2 4

2 8

3 8

3 4

4 8

5 8

3 2

4 4

6 8

7 8

5 4

8 8

9 8

4 2

6 4 11 8

10 8

12 8

7 4 13 8

8 4

14 8

15 8

16 8

1 1 3

0 1 6

2 3

2 6

3 6

3 3

4 6

5 6

4 3

6 6

7 6

5 3

8 6

9 6

6 3

10 6

12 6

11 6

1 1 5

0 1 10

2 10

2 5 3 10

4 10

3 5 5 10

6 10

4 5 7 10

8 10

5 5 9 10

10 10

6 5 11 10

12 10

7 5 13 10

14 10

8 5 15 10

16 10

9 5 17 10

18 10

10 5 19 10

20 10

Use the number lines above to help you solve the fraction problems. Problem

Sentence and answer

(a) If Alex walked 4 of the whole way to school, how far does he have left to go? 2

(b) If a painter painted 6 of a wall, how much does he have left to paint? (c)

4 8

and how much do they have left to complete? 1

(d) In a joint story, Hugo wrote 5 of a page in writing time, and Connor 2 and Faith each wrote 5 of a page. How much writing did this group do altogether? (e) A birthday cake was cut into tenths. If 7 tenths were eaten, how much was left? 3

(f) A tailor used 4 m fabric. If he needs 1 m to complete the outfit, how much more fabric does he need to buy? 2

(g) George ate 3 of his baguette at 1 recess and 3 at lunch. Did he complete it? Going further 7

If the answer to a problem is 8 , suggest at least five fraction sentences that could make that answer. Include both addition and subtraction fraction sentences. 34

Fractions, Decimals and Percentages (Years 5 and 6)

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Fractions, decimals and place value fractions

thousandths

hundredths

•

ones

tens

Place the numbers on the place value grid.

hundreds

thousands

Suggest some places or ways you might use decimal numbers.

Tens of thousands

tenths

Fractions and decimal numbers are closely related because they are both parts of a whole number. Decimal numbers include tenths, hundredths and thousandths. Whole numbers always appear to the left of the decimal point, with decimal numbers to the right of the decimal point. For example: 2.648 = 2 wholes, 6 tenths, 4 hundredths and 8 thousandths

(a) 25.8

(e) 5.843 (f) 12 536 (g) 480.091 (h) 7840.326 (i) 27 548.204 (j) 3520.15 3.

Write the value of the number that is in bold. For example: 5 378.03 – 3 hundredths (a) 452.780

(b) 2 417.852

(d) 52 603.623

(e) 755.040

(f) 1 548.099

(g) 14 379.065

(h) 95 023.418

Going further

Write the numbers from question 3 in order starting from largest to smallest. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Decimal tenths Decimal tenths are the same as fraction tenths. They are the parts made when one whole is divided into ten equal parts. The decimal point replaces the fraction line. 3 For example: 10 = 0.3 1.

Represent the decimal tenths by shading them. Write the matching fraction. (a) 0.5

(b) 0.1

(d) 0.2

(e) 0.9

(f) 0.3

(g) 0.8

(h) 0.4

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• 2. (a) Represent on the tenth strip what 10 you think 10 would look like. (b) How would you write this as a decimal number? (c) What is another name for it? 3.

Write these fraction tenths as decimal numbers. (a)

3 10

= 0.

(e)

10 10

(b)

8 10

= 0.

(c)

5 10

= 0.

(d)

1 10

= 0.

(f)

2 10

= 0.

(g)

7 10

= 0.

(h)

4 10

= 0.

To write a mixed number as a decimal, write the whole number to the left of the decimal point and the decimal number to the right. 3 Write the mixed numbers as decimal numbers. For example: 2 10 = 2.3 1

(b) 1 10 =

(f) 5 10 =

(a) 3 10 = (e) 8 10 =

(d) 3 10 =

(g) 9 10 =

(h) 15 10 =

Going further

Write some fractions that would be equivalent to 0.5. 36

Fractions, Decimals and Percentages (Years 5 and 6)

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Decimal hundredths If one whole is divided into 100 equal parts, each part is one hundredth and can be written as a fraction or a decimal number. If there is no whole number, a zero is written to the left 35 of the decimal point. For example: 100 = 0.35 1.

Shade the hundredths to match the fraction and write the decimal number. Remember they are 100 parts out of 1 whole. (a)

15 100

or 0.

(b)

62 100

or 0.

(c)

80 100

or 0.

(d)

29 100

or 0.

(e)

73 100

or 0.

(f)

96 100

or 0.

Write the decimal numbers as fractions. For example: 0.84 =

(a) 0.14 =

(e) 0.49 = 3.

100

(b) 0.08 =

(f) 0.60 =

100

(g) 0.71 =

100

84 100

(d) 0.11 =

(h) 0.89 =

100

When there is a whole number and a decimal number the whole number goes to the left of the decimal point and the decimal number to the right. Write the fractions as 52 decimals. For example: 3 100 = 3.52 20

(b) 9 100 =

(f) 3 100 =

(a) 1 100 = (e) 4 100 =

(d) 2 100 =

(g) 5 100 =

(h) 6 100 =

Going further 50

Which is larger or are they the same? 1.05 and 1 100 . Explain your answer. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Decimal thousandths If one whole is divided into 1000 equal parts, each part is one thousandth and can be written as a fraction or a decimal number. If there is no whole number, a zero is written to 418 the left of the decimal point. For example: 1000 = 0.418 1.

Shade to represent

418 1000

or 0.418 on the thousandths grid. 2.

Write the fractions as decimal numbers. (a)

325 1000

= 0.

(b)

560 1000

= 0.

(c)

45 1000

= 0.

(d)

899 1000

= 0.

(e)

1 1000

= 0.

(f)

684 1000

= 0.

(g)

91 1000

= 0.

(h)

145 1000

= 0.

Write the decimal numbers as fractions.

(a) 0.377 =

(b) 0.815 =

1000

(e) 0.129 =

(g) 0.936 = 4.

1000

(f) 0.003 =

(h) 0.058 =

1000

Write the decimal numbers from question 3 in order from smallest to largest.

Going further

Suggest or research using the internet how and when decimal thousandths are commonly used. 38

Fractions, Decimals and Percentages (Years 5 and 6)

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Decimal numbers and equivalence The number of digits after the decimal point does not always relate to the equivalent 1 fraction. For example: 0.500, 0.50 and 0.5 are all equivalent to 2 1.

Circle the decimal numbers below that equal or are equivalent to one half ( 2 ). 0.2

0.8

0.5

0.71

0.50

0.49

0.366

0.500

0.250

0.300

Circle the decimal numbers that are equivalent to one quarter ( 4 ). 0.80

0.62

0.28

0.75

0.25

0.99 1

Suggest three decimal numbers that are equivalent to one tenth ( 10 ).

Match the decimal numbers to their equivalent fractions by colour coding the squares. 0.3

4 10

0.500

0.90

0.75

0.100

0.25

0.4

0.800

8 10

1 2

1 4

9 10

3 10

1 10

If a decimal number ends in a zero then this zero can be omitted. For example: 0.300 is the same as 0.3. Simplify the decimal numbers to tenths. (a) 0.500 = 0.

(b) 0.60 = 0.

(d) 0.90 = 0.

(e) 0.70 = 0.

(f) 0.200 = 0.

(g) 0.400 = 0.

(h) 0.30 = 0.

Place the decimal number answers from question 5 in order on the number line. Write the equivalent fraction below the line.

0.8

1 10

Going further

Suggest two different decimal numbers that are equivalent to

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3 4.

Fractions, Decimals and Percentages (Years 5 and 6)

Decimal numbers and money One of the most common uses of decimal numbers is money. The numbers to the left of the decimal point are dollars and the numbers to the right of the decimal point are cents, 75 100 equal parts of a dollar. For example: $1.75 = 1 100

thousandths

hundredths

tenths

•

ones

tens

hundreds

thousands

Write each amount on the place value chart.

Tens of thousands

(a) 0.85c (b) $1.60 (c) 0.10c (d) $4638.25 (e) $150.30 (f) $43 256.85

(g) $9.50

(h) $812.45 (i)

(j) $7400.95 2.

Match the money amounts with the fractions by colour coding them. $4.50 3

54 3.

$3.25 5

2 100

$5.75 1

$1.60 4

6 10

$2.80

$3.15

8 2 10

1 10

$6.40

3 100

$2.05 15

3 100

Write the decimal amounts as fractions. For example: $93.65 = 93 100 (a) 0.55c =

(b) $7.25 =

(d) $25.80 =

(e) $9.45 =

(f) $46.10 =

(g) $5.75 =

(h) $99.60 =

Going further

How much would half a thousand dollars be? If you had this much money what would you buy? List your suggestions with their approximate prices.

Fractions, Decimals and Percentages (Years 5 and 6)

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Decimal numbers and measurement Another use of decimal numbers is measuring. For example: a person may have a height of 1 m 58 cm, which can be written as 1.58 m. 1.

Write the length measurements as decimal numbers. For example: 775 cm = 7.75 m Remember 1 m = 100 cm

(a) 1 m 68 cm =

(b) 4 m 80 cm =

(d) 212 cm =

(e) 6 m 75 cm =

(f) 562 cm =

(g) 362 cm =

(h) 890 cm =

(i) 5 m 19 cm =

Write the weight measurements as decimal numbers. For example: 450 g = 0.45 kg Remember 1 kg = 1000 g

(a) 1 kg 375 g =

(b) 3 kg 160 g =

(d) 1725 g =

(e) 250 g =

(f) 5 kg 125 g =

(g) 500 g =

(h) 4 kg 25 g =

(i) 6 kg 60 g =

Work with a partner who is a different height from you. Measure and compare your height. Write each height using whole numbers and decimal numbers.

Who is taller? By how much? 4.

Measure these items using a ruler or tape measure and record them using decimal numbers. (a) Your desk height:

(b) Your pencil length:

(d) The width of a class window:

Going further

Use the internet to investigate fastest world record times for various events you may be interested in; for example, fastest time for 100 m sprint, fastest swim time for 100 m freestyle.

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Fractions, Decimals and Percentages (Years 5 and 6)

Comparing decimal numbers 1.

Represent the decimal numbers on the hundredths grid below. Number them from the smallest to the largest (1 to 3). (a) 0.17

(b) 0.70

Write true or false to the decimal statements. (a) 0.04 is larger than 0.4 (b) 0.7 is larger than 0.07 (c) 0.65 is larger than 0.56

(d) 0.536 is smaller than 0.563 (e)

(f) 0.528 is larger than 0.852 (g) 0.3 is smaller than 0.31 (h) 0.009 is smaller than 0.008 3.

Use the symbols greater than (>) less than (<) or equal to (=) to describe the decimal numbers. (a) 0.5 (d) 0.40 (g) 0.731 (j) 0.15

0.7 0.4 0.803 0.150

(b) 0.9

0.8

(e) 0.43

0.34

0.18

(f) 0.255

0.245 0.670

(h) 0.900

0.90

(i) 0.627

(k) 0.860

0.840

(l) 0.79

0.129

Suggest an equivalent decimal for these numbers. (a) 0.600 =

(b) 0.350 =

(d) 0.930 =

Going further

Suggest at least ten decimal numbers that can be between 0.1 and 0.2.

Fractions, Decimals and Percentages (Years 5 and 6)

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Ordering decimal numbers 1.

Order the decimal numbers from smallest to largest. (a)

0.8

0.5

0.7

0.3

0.4

0.6

0.2

0.9

0.13

0.21

0.11

0.9

0.3

0.17

0.7

0.15

0.55

1.0

0.60

0.13

0.43

0.91

0.75

0.32

1.08

1.16

1.12

1.018

1.02

1.10

1.20

1.06

0.1 (b) 0.1 (c) 0.10 (d) 1.0 2.

Order the sets of decimal numbers from smallest to largest. (a)

6.2

8.1

9.7

(b)

0.49

(c)

0.97

0.93

0.95

(d)

0.115

0.316

0.068

0.244

0.521

0.099

0.206

(e)

0.004

0.009

0.005

0.008

0.001

0.006

0.003

(f)

0.834

0.856

0.817

0.896

0.852

0.871

0.845

0.99

3.4

0.96

5.9

0.90

1.9

0.98

4.8

0.91

2.6

Going further

Investigate who are the world’s five tallest people. Using decimal numbers, record their heights in order from tallest to shortest.

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Fractions, Decimals and Percentages (Years 5 and 6)

Decimal problems 1.

Look at the students heights and answer the questions.

Alex 1.56 m Luke 1.67 m

Nic 1.65 m

Trang 1.52 m Georgia 1.62 m Hannah 1.59 m

(a) Who is the shortest?

(b) Who is the tallest?

(d) What is the difference between the shortest and the tallest? (e) Find the average height of this group (add all measurements and divide total by 6). You may wish to use a calculator. 2.

Look at the canteen items and answer the questions. (a)

4.80 © R. I . C.Publ i cat i ons What would it cost to buy 2 steamed 2.00 4.00 4.50 dumplings and 1 sushi roll? •f orr evi ew pur posesonl y•1.50 $

5.00

2.70

3.50

W ate r

90¢

(b) What would it cost to buy a salad wrap and a corn cob?

(d) A pie and a sushi roll?

(f) What is the difference between the cheapest item and the dearest item?

(g) You have $10 to spend. Which items would you buy and what would be the total cost?

(e) List all the prices from cheapest to dearest. (h) How much change, if any, would you have from your $10.00? Going further

Find a simple sponge cake recipe. Write the weight of ingredients required. What weight would be needed if you wanted to make two cakes? 44

Fractions, Decimals and Percentages (Years 5 and 6)

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Name:

Date:

Assessment 1 1.

Use the number line to order the fractions:

1 7 1 1 1 1 3 2 , 8 , 3 , 5 , 8 , 12 , 4

0 2.

Write an equivalent fraction. (a)

2 3

(e)

1 2

(b)

2 6

(f)

6 8

(c)

2 8

(g)

4 12

(d)

10 12

(h)

4 10

Change the improper fractions to mixed numbers. (a)

14 3

(b)

17 2

(c)

21 5

(d)

25 8

(e)

21 6

(f)

39 4

(g)

89 10

(h)

50 12

1 2 1 4

0 1 8

2 2

2 4

2 8

3 8

3 4

4 8

5 8

3 2 5 4

4 4

6 8

7 8

8 8

9 8

4 2

6 4

10 8

11 8

7 4

12 8

13 8

8 4

14 8

15 8

1 3

1 6

0 1 12

2 3

2 6

2 12

3 12

4 12

3 6

5 12

6 12

3 3

4 6

7 12

8 12

5 6

9 12

4 3

6 6

10 12

11 12

12 12

7 6

13 12

14 12

15 12

16 12

9 6

17 12

18 12

10 6

19 12

6 3

11 6

20 12

21 12

22 12

5 3

8 6

16 8

12 6 23 12

24 12

Use the number lines to jump forwards to add the fractions. Simplify answers and change improper fractions to mixed numbers if necessary. (a)

2 3

3 3

(b)

2 4

(d)

4 8

2 8

(e)

7 12

5 4

9 12

(c)

1 2

2 2

(f)

4 6

3 6

Use the number lines above to jump backwards to subtract the fractions. (a)

11 8

(d)

6 3

7 8

– –

4 3

= =

(b)

8 4

(e)

20 12

– –

6 4

7 12

(c)

9 6

–

5 6

(f)

4 2

–

2 2

Solve the problems about fractions. 3

(a) If Andy ate 8 of a pizza, Kate ate 8 , Liz ate 8 and George ate how much pizza was consumed altogether?

4 8,

(b) If at a netball match 36 quarters of oranges were consumed, how many whole oranges did they use? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Name:

Date:

Assessment 2 1.

Represent the decimal tenths and hundredths by shading them. Then write the matching fraction. (a) 0.7

(b) 0.3

(d) 0.5

(e) 0.42

(f) 0.87

(e) 0.32 =

100

(c)b 0.5 = a =o © R. I . C.Pu l i c t i ns (d) 0.6 = •f or r evi ew p(g) ur poseso(h) nl y• (f) 0.75 = 0.99 = 0.08 = (b) 0.9 =

100

Write the fractions as decimal numbers. 527 1000

(a) 4.

Write the decimal numbers as fractions.

(a) 0.2 =

= 0.

(b)

86 1000

= 0.

(c)

712 1000

= 0.

(d)

126 1000

= 0.

Match the decimals to their equivalent fractions by colour coding the boxes. 0.3

0.91

0.45

0.1

0.362

0.75

0.673

0.50

0.822

0.250

1 4

362 1000

822 1000

3 10

3 4

91 100

1 2

45 100

673 1000

1 10

Write the decimal numbers from question 4 in order from the smallest to the largest.

Write at least two equivalent decimal numbers for

Fractions, Decimals and Percentages (Years 5 and 6)

1 2.

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Checklist

Year 5

Compare and order common unit fractions and locate and represent them on a number line (ACMNA102) Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103) Recognise that the place value system can be extended beyond hundredths (ACMNA104) Compare, order and represent decimals (ACMNA105) Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)

Name

Compares, orders and locates fractions

Adds and subtracts fractions

Understands decimal number system

Compares, orders and represents decimals

Creates and continues fraction and decimal patterns

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Fractions, Decimals and Percentages (Years 5 and 6)

Answers Naming fractions ................................... page 17 1. (a) numerator 3

(b) denominator 4

2. (a) 4

(b) 12

(e) 3 1

(f) 9

1 1 1 1 1 1 9, 8, 5, 4, 3, 2 1 1 1 1 1 1 2, 3, 4, 5, 6, 8

Going further – Answers will vary. Teacher check

Comparing unit fractions ...................... page 19 1

3. (a) true (d) false 4. (a) Dale

(e) false

(f) true

(b) Isla

(d) 6

(g) 8

(h) 12

3. Answers may include: 3

4 12 , 3 10 ,

5 12 , 4 10 ,

6 12 , 5 10 , 6 (b) 10 2 (f) 3 1 1 3 3 6, 3, 6, 4,

7 12 , 6 10 ,

8 12 , 7 10 , 2 (c) 6 4 (g) 5

9 12 , 8 10 ,

10 12 , 9 10 ,

11 12 12 , 12 10 10 1 (d) 4 4 (h) 12

10 10

1. Teacher check – They are the same/equivalent

(b) 4 , 6 , 8 , 10 , 12 2

(d) 12 1

(f) 2 , 4 , 6 , 8 , 10 4

2 4 4 8 5 and 10 , and 5 and 10 are equivalent

Ordering fractions.................................. page 23 (d) Gina

(f) 4

Going further – Teacher check

(a) 6 , 9 , 12

4.–5. Teacher check

(b) true

(b) 5

fractions.

1. Teacher check

(e) 10

Locating fractions on number lines .......................................... page 22

1 1 (f) 7 5 1 1 1 1 1 7, 6, 5, 4, 3

Equivalent fractions............................... page 20 1

3 12 , 2 (e) 10 , 6 2. (a) 8 3 (e) 6 1 1 3. 12 , 8 ,

Going further – The smaller the denominator the larger the fraction or the larger the denominator the smaller the fraction.

2. (a) 2

2. 12

(b) 4

1 (d) 2 (e) 1 1 1 1 2. 11 , 10 , 9 , 8 ,

1. (a) 8

Going further – Answers will vary, e.g. helps to picture or visualise fraction.

3. Teacher check 4.

(d) 6 , 6 , 6 ;

(e) 3

Unit fractions ......................................... page 18

(d) 8

Going further – They are the same/equivalent.

(f) 2

(b) 2

(b) 8 , 8 , 8 , 8 , 8 , 8 , 8

(h) 8

4. Teacher check

1. (a) 4 , 4 , 4

(e) 12

1. (a) 4

Equivalent fractions and number lines .......................................... page 21

(g) 6

(b) 8

(d) 9

(d) 8

(f) 10

3. (a) 6

Year 5

1. 2, 4, 1, 5, 3 2. Teacher check shading 2 1 7 5 9 , 3 , 12 , 6 8 7 6 5 4 3 2 1 3. (a) 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 9 8 7 6 5 4 3 2 1 (b) 10 , 10 , 10 , 10 , 10 , 10 , 10 , 10 , 10 12 11 10 9 8 7 6 5 4 3 (c) 12 , 12 , 12 , 12 , 12 , 12 , 12 , 12 , 12 , 12 , 2 1 12 , 12 1 1 3 1 5 3 7 4 4. (a) 8 , 4 , 8 , 2 , 8 , 4 , 8 , 4 1 1 1 5 3 2 10 (b) 12 , 6 , 3 , 12 , 6 , 3 , 12 1 1 3 4 3 4 9 (c) 10 , 5 , 10 , 10 , 5 , 5 , 10 3 11 7 4 3 1 2 2 1 1 Going further – 3 , 12 , 8 , 5 , 4 , 2 , 6 , 8 , 5 , 12

(h) 6 , 9 , 12

4. Teacher check Going further – Answers will vary. Teacher check 48

Fractions, Decimals and Percentages (Years 5 and 6)

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Answers Improper fractions and mixed numbers ................................................. page 24 7 3 1. (a) 4 , 1 4 11 3 (d) 8 , 1 8 1 2. (a) 5 2 2 (e) 5 5 1 (i) 20 2 3 3. (a) 2 43

7 1 (b) 6 , 1 6 4 1 (e) 3 , 1 3 1 (b) 6 3 (c) 2

5 1 (c) 2 , 2 2 9 4 (f) 5 , 1 5 1 (d) 5 4 2 3 (g) 10 3 (h) 5 10 3 (k) 8 (l) 6 12 11 14 (c) 4 (d) 3 19 23 (g) 12 (h) 9 39 99 (k) 5 (l) 12

(f) 3 3 (j) 6 8 16 (b) 5 26

(e) 8

(f) 4

(i) 6

(j) 5

Year 5

Fractions and multiples......................... page 27 1. by 2’s: 16, 20, 24, 28, 36, 42, 50, 64 by 4’s: 16, 20, 24, 28, 36, 64 2. 16, 20, 24, 28, 36, 64 3. by 3’s: 9, 18, 21, 27, 39, 45, 54, 60 by 6’s: 18, 54, 60 4. 18, 54, 60 5. (a) 8, 4, 2

(b) 12, 6, 3, 2

(d) 28, 14, 7

(e) 36, 18, 9, 6

(f) 48, 24, 12, 8

Going further – 45: 60:

Going further – Answers will vary. Teacher check

Counting by fractions ............................ page 25 12

1. (a) 4 , 3 17

(b) 2 , 6 2 1

(d) 8 , 2 8

3 4 5 6 2. (a) 3 , 3 , 3 , 3 ; 3 4 5 6 6, 6, 6, 6, 3 4 5 6 (b) 4 , 4 , 4 , 4 , 3 4 5 6 8, 8, 8, 8, 14 15 16 8 , 8 , 8 3 4 5 6 (c) 5 , 5 , 5 , 5 , 3 4 5 10 , 10 , 10 , 13 14 15 10 , 10 , 10 ,

halves, thirds, quarters, fifths, sixths, tenths, twelfths

120: halves, thirds, quarters, fifths, sixths, eighths, tenths, twelfths

(e) 6 , 2 6 or 2 2

thirds, fifths, ninths

Simplifying fractions ............................. page 28 8

1. (a) 12 , 6 , 3 7 6, 7 4, 7 8,

8 6, 8 4 8 8,

9 10 11 12 6, 6 , 6 , 6

(b) 10 , 5

(e) 20 , 10 , 5

9 10 11 12 13 8, 8 , 8 , 8 , 8 ,

(d) 6

(e) 9

(f) 7

2. (a) (d) (g)

(b) 1

or 4 1

or 4 3

or 5 11

or 12

(e) (b) (e) (h)

1 2 8 1 24 or 3 3 1 9 or 3 9 3 24 or 8 6 3 20 or 10

18 24 2 24 12 16 3 18

(d) 5

(e) 3

(f) 8

(g) 16

(h) 5

(h) 8

(j) 9

(k) 10

(l) 6

e.g. 8 , 16 , 32 , 64 , 6 , 12 , 24

or or or or

3 4 1 12 3 4 1 6

Adding fractions using pictures ........... page 29 3

1. (a) 4

(b) 2 or 1 (c) 8

(d) 6

(e) 5 or 1 (f) 10 or 5 3

2. (a) 8 + 8 = 8 or 2 6

(b) 3

(d) 3

(g) 8

(b) 3 + 3 = 3 or 1 (d) 4 + 4 = 4 or 1

(e) 5 + 5 + 5 = 5

3. (a) 7

(h) 13

(f) 6

(i) 4

(g) 6

(b) 3

(e) 4

(d) 4

Going further – Answers will vary,

Fractions of groups................................ page 26

(d)

10 11 12 10 , 10 , 10 , 20 10

(f) 5

3. (a) 4

(b) 4

1. (a)

7 8 9 10 5, 5, 5, 5 ; 6 7 8 9 10 , 10 , 10 , 10 , 16 17 18 19 10 , 10 , 10 , 10 ,

(f) 30 , 15 , 3 .

(b) 3

(e) 7

Going further – (a) 8

1 2 6 24 3 12 9 15 22 24

2. (a) 5

(d) 16 , 8 , 4 , 2

(f) 9 + 9 = 9 or 3

Going further – Answers will vary. (i) 4

Going further – 21, 21, 12, 56 Divide the denominator into the whole number then multiply this by the numerator. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Answers Adding fractions .................................... page 30 4

1. (a) 4 or 1

(b) 8

(e) 6 or 1

(b) 4 or 1 4 17

(e) 8 or 2 8 1

(f) 3 or 1 3 7

(h) 5 or 1 5

Addition and subtraction fraction problems .................................. page 34

(l) 5 or 2 5

1. (a) 4 – 4 = 4 2

(b) 6 – 6 = 6 or 3 8

1. (a) 2

(b) 8

(d) 10

(e) 4 or 2

1 1

(e) 10 – 10 = 10

(b) 9 or 3 1

(e) 5

3 1 1 (g) 12 or 4 (h) 2 2 5 2 2. (a) 2 or 1 (b) 3 or 1 3 13 3 5 (d) 5 or 2 5 (e) 6 11 1 8 (g) 10 or 1 10 (h) 4 or 2 15 3 1 (i) 12 or 1 12 or 1 4 8 5 3 16 3. (a) 8 – 8 = 8 (b) 4 12 9 3 1 36 (c) 2 – 2 = 2 or 1 2 (d) 12 1 Going further – (a) 6

Fractions, decimals and place value fractions.............................. page 35

Subtracting fractions............................. page 32

(d) 8 or 4

Going further – Answers will vary. Teacher check

(d) 4 or 2

(g) 3 + 3 = 3 or 1 - Yes he completed it.

Going further – Answers will vary

(f) 4 – 4 = 4 m

Teacher check shading

1. (a) 4

(f) 6

(b) 5

(d) 5 + 5 + 5 = 5 or 1 page

2. (a) 8 or 4

Teacher check shading

Going further – (a) 4 or 1, (b) 8 , (c) 4 , (d) 8

Subtracting fractions using pictures......................................... page 31

(l) 10 or 5

Going further – Answers will vary

(i) 10 or 1 10

(j) 3 or 1 3

(k) 8 or 2

(h) 6 or 1 6 or 1 3 5

(i) 9 or 1 9

(i) 6 or 1 6

(g) 6 or 1 6 or 1 2

(k) 5

(d) 8 or 1 8

(f) 10 or 1 10 or 1 2

(g) 12 or 1 12 or 1 4 11

(d) 2 or 2 2

(e) 5 or 1 5 15

(b) 8 or 1 8 or 1 2

2. (a) 2 or 1 2

(f) 4 or 1

3. (a) 3 or 1 3

(j) 10 or 5

(e) 12 or 3

(f) 8

(h) 6 or 3

(g) 3

(d) 8 or 2

(e) 4

(b) 5

(b) 4 or 1 4

(d) 8 or 1

Teacher check shading 2. (a) 10

Using number lines to add or subtract fractions .................................. page 33 1. (a) 2

(d) 5

Year 5

2. Teacher check

1. Answers will vary, e.g. money, length, weight

3. (a) 2 ones

(f) 6 or 3 1 (i) 3 4 (c) 4 or 1 5 (f) 8

(b) 2 thousandths

(d) 6 hundreds

(e) 4 hundredths

(f) 1 thousand

(g) 5 thousandths (h) 9 tens of thousands Going further – 95 023.418, 52 603.623, 14 379.065, 2417.852, 1548.099, 755.040, 452.780, 95.635

– 4 = 4 1

– 12 = 12 or 2 4

(b) 8 or 2 Answers will vary. Teacher check

Fractions, Decimals and Percentages (Years 5 and 6)

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Answers Decimal tenths ....................................... page 36 1. Teacher check shading 5

(a) 10

(b) 10

(e) 10

1. 0.5, 0.50, 0.500

2. 0.25, 0.250

(f) 10

(g) 10

(h) 10

2. (a) Teacher check

(b) 1.0

3. (a) 0.3

(b) 0.8

(d) 0.1

(e) 1.0

(f) 0.2

(f) 0.7

(h) 0.4

4. (a) 3.1

(b) 1.9

(d) 3.7

(e) 8.6

(f) 5.3

(g) 9.8

(h) 15.2

3. 0.10, 0.1, 0.100 3

0.100 – 10 , 0.25 – 4 ,

0.4 – 10 ,

0.800 – 10

4. 0.3 – 10 ,

Decimal hundredths .............................. page 37

5. (a) 0.5

(b) 0.6

(d) 0.9

(e) 0.7

(f) 0.2

(g) 0.4

(h) 0.3

Decimal numbers and money .............. page 40 1

(a) 0.15

(b) 0.62

2. $4.50 – 4 2

(d) 0.29

(e) 0.73

(f) 0.96

$5.75 – 5 4

(b) 100

(e) 100 3. (a) 1.20 (e) 4.05

$3.25 – 3 100

(d) 100 89

(f) 100

(g) 100

(h) 100

(b) 9.16

(d) 2.39

(f) 3.99

(g) 5.84

(h) 6.41

$1.60 – 1 10

$2.80 – 2 10

$3.15 – 3 100

$6.40 – 6 10

$2.05 – 2 100

Going further – 1 100 is larger as it represents 1 1 whole and 2 compared to 5 1 whole and 100

Decimal thousandths ............................ page 38 1. Teacher check (b) 0.560

(d) 0.899

(e) 0.001

(f) 0.684

(g) 0.091

(h) 0.145

815 (b) 1000 3 (f) 1000

945 (c) 1000 936 (g) 1000

64 (d) 1000 58 (h) 1000

4. 0.003, 0.058, 0.064, 0.129, 0.377, 0.815, 0.936, 0.945 Going further – Answers will vary, e.g. distance

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3. (a) 100

(b) 7 100

(d) 25 100 75

(e) 9 100

(g) 5 100 or 5 4

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(f) 46 100 60

(h) 99 100

Going further – $500. Answers will vary.

Decimal numbers and measurement ......................................... page 41 1. (a) 1.68 cm

2. (a) 0.325 377 3. (a) 1000 129 (e) 1000

1. Teacher check

1. Teacher check shading

Going further – 0.75, 0.750

e.g. 2 , 4 , 6 , 8 , 10 , 12

2. (a) 100

0.500 – 2 , 0.90 – 10 , 0.75 – 4 ,

6. Teacher check

Going further – Answers will vary, 1

Decimal numbers and equivalence ............................................ page 39

(d) 10

Year 5

(b) 4.80 cm

(d) 2.12 m

(e) 6.75 m

(f) 5.62 m

(g) 3.62 m

(h) 8.90 m

(i) 5.19 m

2. (a) 1.357 kg

(b) 3.16 kg

(d) 1.725 kg

(e) 0.25 kg

(f) 5.125 kg

(g) 0.5 kg

(h) 4.025 kg

(i) 6.06 kg

3.–4. Teacher check Going further – Answers will vary. Teacher check

Fractions, Decimals and Percentages (Years 5 and 6)

Answers Comparing decimal numbers ............... page 42 1. (a) 2

(b) 3

(e) false

(b) true

Assessment 1 ......................................... page 45 1

(d) true

3. (a) <

(b) >

(d) =

3. (a) 4 3

(e) >

(f) >

(g) <

(h) =

(d) 3 8

(i) <

(j) =

(k) >

(l) >

(g) 8 10

(d) 0.93

Ordering decimal numbers ................... page 43 1. (a) 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

(f) 9 4 1

(h) 4 12 or 4 6

7 6

(d) 8 or 4 1

(e) 12 or 1 12 or 1 3 4

(b) 4 or 1 4 1

(f) 6 or 1 6 2

5. (a) 8 or 2

(b) 4 or 2 2

(e) 12 or 1 12

(b) 0.18, 0.23, 0.36, 0.49, 0.56, 0.65, 0.72, 0.80

(e) 3 6 or 3 2

2. (a) 1.9, 2.6, 3.4, 4.8, 5.9, 6.2, 8.1, 9.7

(h) 5

(b) 8 2

(b) 0.11, 0.13, 0.15, 0.17, 0.21, 0.3, 0.7, 0.9

(d) 1.018, 1.02, 1.06, 1.08, 1.10, 1.12, 1.16, 1.20

(g) 3

4. (a) 3 or 1 3

Going further – Answers will vary. Teacher check

(d) 6

(f) 4

(e) 10

(h) false

(b) 12

(g) true

(b) 0.35

2. (a) 6

(f) false

4. (a) 0.6 or 0.60

1. Teacher check 12 , 8 , 5 , 3 , 2 , 4 , 8

Teacher check shading 2. (a) false

Year 5

(d) 3

(f) 2 or 1 3

6. (a) 8 or 1 8 or 1 4

(b) 9

Assessment 2 ......................................... page 46

(e) 0.001, 0.003, 0.004, 0.005, 0.006, 0.008, 0.009 (f) 0.817, 0.834, 0.845, 0.852, 0.856, 0.871, 0.896 Going further – Teacher check

(b) Luke

(b) $6.80

(e) 1.60 m (c) $6.50

(d) $7.20

(b) 10

(d) 10

(e) 100

(f) 100

Teacher check shading 2

2. (a) 10

(b) 10

(d) 10

(e) 100

(f) 100

(g) 100 3. (a) 0.527

Decimal problems .................................. page 44 1. (a) Trang

1. (a) 10

(h) 100 (b) 0.086

3 4. 0.3 – 10 , 1 0.1 – 10 , 673 0.673 – 1000 , 1 0.250 – 4

(d) 0.126

91 45 0.91 – 100 , 0.45 – 100 , 362 3 0.362 – 1000 , 0.75 – 4 , 1 822 0.50 – 2 , 0.822 – 1000 ,

(e) $0.90, $1.50, $2.00, $2.70, $3.50, $4.00, $4.50, $4.80, $5.00

5. 0.1, 0.250, 0.3, 0.362, 0.45, 0.50, 0.673, 0.75, 0.822, 0.91

(f) $4.10

6. 0.5, 0.50, 0.500

(g)–(h) Answers will vary

Going further – Answers will vary. Teacher check

Fractions, Decimals and Percentages (Years 5 and 6)

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(ACMNA132)

Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies

(ACMNA131)

Make connections between equivalent fractions, decimals and percentages

Multiply and divide decimals by powers of 10 (ACMNA130)

(ACMNA129)

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies

Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128)

Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (ACMNA127)

Solve problems involving addition and subtraction of fractions with the same or related denominators (ACMNA126)

Page

Compare fractions with related denominators and locate and represent them on a number line (ACMNA125)

Year 6 Curriculum links

Page title

Naming and representing fractions

✓

Comparing and ordering fractions

✓

Equivalent fractions

✓

Equivalent fractions patterns

✓

Simplifying fractions

✓

Improper fractions and mixed numbers

✓

Counting by fractions

✓

Adding fractions with common denominators

✓

Adding fractions with different denominators

✓

Subtracting fractions with common denominators

✓

Subtracting fractions with different denominators

✓

Adding and subtracting mixed number fractions

✓

Addition and subtraction fraction problems

✓

Fractions of whole numbers

Fractions and decimals

Place value and decimals

Comparing and ordering decimal numbers

✓

Adding decimal numbers

✓

Subtracting decimal numbers

✓

Estimating and rounding decimal numbers

✓

Multiplying decimal numbers

✓

Dividing decimal numbers

✓

Solving decimal problems–exchange rates

✓

Solving decimal problems involving money

Decimals and powers of 10 – multiplying

✓

Decimals and powers of 10 – dividing

✓

Powers of 10

✓

Equivalent calculations

✓

Percentages

✓

Percentages, fractions and decimals

✓

Working with percentages

✓

Finding percentages of amounts using fractions

✓

Finding percentages of amounts

✓

100 Assessment 1

✓

www.ricpublications.com.au

✓

✓ ✓

101 Assessment 2 R.I.C. Publications®

✓

Fractions, Decimals and Percentages (Years 5 and 6)

✓ 53

Teachers notes

Year 6

Introduction At this level, students will be asked to draw on their previous knowledge of fractions and decimals and demonstrate their understanding of equivalence, and still be required to demonstrate equivalence between fractions using drawings, fraction mats and number lines. Once again, they will realise the importance of the denominator in a fraction in order to determine and compare its size to other fractions. They will use this knowledge to understand the process of adding and subtracting fractions with common denominators. From here they will also be expected to add and subtract fractions with equivalent denominators with the use of aids such as diagrams and number lines as well as written form. Students will use their knowledge of factors and multiples to find fractions of whole numbers both in written form and using a calculator or computer. Students will again need to visually see the connection between fractions and decimals before they can explore and develop the written strategies. They will see that both fractions and decimals are part of a whole and that tenths, hundredths and thousandths are part of the place value system. The decimal point replaces the fraction line, and students will begin to relate decimals back to fractions as equal parts of the whole number. Once they reach this understanding they will be able to add and subtract decimals in written form and with the aid of digital technologies. The skills of rounding and estimating will also come into play here to assist them with their understanding of these processes. A new concept at this level is multiplying and dividing decimal numbers by whole numbers; through this process they will realise that the reminder is expressed as a decimal number. Various problems to do with concepts such as money can be related when multiplying and dividing whole numbers with decimals; for example, understanding that the cost of 6 may be $4.80 but the cost of one item is 0.80c. Students will also be introduced to multiplying and dividing decimals by the power of 10. They will realise that there are equivalences in these processes; for example, 65.32 ÷ 2 is the same as/equivalent to 6532 ÷ 200.Their knowledge of place value will assist them in understanding the concept of power of 10 and this should be pointed out early on.They will understand the commutative effect of multiplication and division can be used when solving problems of multiplying or dividing with decimal numbers. Therefore, students may decide to calculate 1.2 × 0.6 by multiplying 12 by 6 then dividing this by 100. A variety of options will be introduced to students at this level to help them solve such processes.

Finally, students will be introduced to percentages, firstly by connecting and relating this concept visually to 1 fractions and decimals. Students will realise that 10 is the same as 0.1 or 10% as they are all equal parts of a whole number. Once students recognise this connection they will be able to solve problems relating to fractions, decimals and percentages. Comparing and ordering fractions Start the topic by defining a fraction as equal parts of a whole and defining the numerator and denominator. Allow students to refresh their knowledge of fractions by naming representations and drawing or illustrating fractions. Focus on the importance of the denominator in helping students compare and look at equivalence in fractions. Use diagrams and number lines to compare, order and count by fractions. Revise and define improper fractions and mixed numbers.When looking at fractions of whole numbers or groups, incorporate the students’ knowledge of multiples and division; for example, 24 can be divided into halves, thirds, quarters, sixths and eighths. Use both written methods and digital technologies to find a fraction of a whole number.

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers notes

Year 6

Adding and subtracting fractions Initially use diagrams and numbers lines to add and subtract fractions with common denominators before using a written method. Once students understand this concept they will be able to add and subtract fractions with equivalent denominators, realising that it is easier to make the denominators the same before adding or subtracting them; for example, 12 + 24 is the same as 12 + 12 = 22 or 1. Encourage students to break down their answers or change improper fractions to mixed numbers. Allow students the opportunity to solve real life problems to do with adding and subtraction fractions. Adding and subtracting decimals In Year 5 students were introduced to decimal numbers in the place value system, where they could relate whole numbers to decimal numbers including tenths, hundredths and thousandths. Allow students to see where decimal numbers relate with whole numbers and the place value system. Relate and compare fractions with decimal numbers with the use of diagrams, as well as in written form. Compare and order decimal numbers with the help of numbers lines. Extend on these strategies by introducing addition and subtraction of decimal numbers using written strategies and digitial technologies. Use a variety of strategies to assist with this concept, such as estimating and rounding. If students have a good understanding of place value and the processes of addition and subtraction, then it should follow they can add and subtract decimals of any size successfully. Multiplying and dividing decimals Introduce multiplying and dividing decimal numbers by whole numbers to students. Demonstrate to students that if you multiply a decimal with a whole number it could end up with an answer that no longer contains a decimal number; for example, 2.4 × 5 = 12. In other instances it can remain a decimal number.The opposite to this process is dividing; allow students the opportunity to experience both processes. They will see that by dividing a whole number by a decimal, the decimal numbers is actually the remainder; for example, 64 ÷ 3 = 21.333. If decimals are divided by whole numbers they will also have a decimal reminder; for example, 4.5 ÷ 4 = 1.125. Allow students to solve meaningful problems relating to multiplying or dividing decimal numbers such as story problems to do with money. From here, give students the opportunity to see how equivalent problems can be solved with decimal numbers and whole numbers; for example, 24.32 ÷ 4 is the same as 2432 ÷ 400 if the decimal point was removed. Introduce the concept of the power of 10 and help students understand that multiplying or dividing a decimal number can be simplified back into a whole number sentence by changing both numbers using the power of 10; for example, 3.2 × 0.4 can be calculated by multiplying 32 × 4 then dividing the answer by 100. Students will realise they will get the same answer. Introduce power of 10 to students and explain that the decimal point moves when multiplying and dividing by the power of 10. Calculators will be a useful tool at this level to solving problems as well as checking answers.

Percentages In Year 6 percentages are introduced. Firstly, visually introduce and relate percentages to fractions and 1 decimals with the aid of diagrams, base ten or base hundred blocks. Students will see that 10 is the same as 0.1 and 10%. All of these concepts can be defined as parts of a whole. Once students understand this concept they will be able to connect percentages to decimals and fractions in written form. They should comfortably be able to convert percentages to fractions and decimals, and visa versa. From here they will also be able to solve problems relating to percentages; for example, what is 10% of $3.00? Students’ previous knowledge of the place value system and decimal numbers will also be useful when dealing with percentages.

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Fractions, Decimals and Percentages (Years 5 and 6)

Warm-up activities

Year 6

These activities could be used to introduce your lesson.They can be used as a whole class focus, or a small group or individual activity depending on the lesson content. • Look at various pictures on the internet and name the fractions they are representing; for example, 18 slice of a pizza, 34 of an orange, 16 of a wheel of cheese. • Make a fraction mat, guiding students through dividing each section to match the fraction denominator then shading and labelling each fraction (see teacher resources page 58). Discuss the most accurate way of making sure each fractional part is the same, such as measuring it with a ruler. • Define equivalence of fractions then, using the fraction chart, discuss which fractions are equivalent. • Make diagrams using circles of equivalent fractions (see teacher resources page 59). • Locate, count and order fractions using number lines (see teacher resources page 60). • Use number lines and symbols to convert improper fractions to mixed numbers. Then, once students show they understand this concept, show them the formula of how this can be done. For example: 74 = 1 34 . Divide the denominator into the numerator to find the whole number, then the remainder becomes the new numerator. • Use an interactive whiteboard or a fraction maths program to explore concepts such as equivalent fractions, improper fractions and mixed numbers. A good website is <www.mathsisfun.com/fractionsmenu.html> • Use number lines to add or subtract fractions with common denominators by jumping forwards or backwards along the line.

• Demonstrate how fractions with equivalent denominators can be added or subtracted. To do this students need to find the LCD (lowest common denominator) before they can work out the problem. For example: 34 – 28 .The LCD is quarters so it can be rewritten as 34 – 14 = 24 or 12 .Try solving fraction addition and subtraction cards with different but equivalent denominators (see teacher resources page 61). • Find fractions of quantities using calculators and written form. For example: 14 of 24 = 6. Point out to get a whole number as an answer, the denominator must be a factor or multiple of the whole number. • Relate fractions to decimals tenths, hundredths and thousandths by seeing the decimal visually (see teacher resource Year 5, page12), or go to <www.mathsisfun.com/decimals> • Write whole numbers and decimal numbers onto a place value grid (see teacher resources Year 5, page 16). • Add and subtract decimal numbers following the same process as you would with whole numbers, except remember to include the decimal point. Use the written method and calculators to solve problems that involve adding and subtracting decimal numbers. Show students how to break down their answers to the lowest form or change improper fractions to mixed numbers. • Measure items around the room and find the difference in their height or length by subtracting. Measure the height of students and use a decimal to represent metres and centimetres; for example, 1 m 48 cm = 1.48 m. Discuss what the decimal point is replacing. • Introduce rounding numbers. How can decimal numbers be rounded to the nearest whole number? Any number that is .5 or above is rounded up and any number under 0.5 is rounded down; for example, 1.9 would be rounded up to 2.0. Cut and paste the decimal numbers under the correct heading (see teacher resources page 62).

Fractions, Decimals and Percentages (Years 5 and 6)

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Warm-up activities

Year 6

• Estimate answers to decimal addition and subtraction problems by rounding numbers first. For example: 3.7 + 2.1 can be estimated by saying 4 + 2 = 6. • Multiply decimals by whole numbers using a written method or a calculator; for example, $1.25 × 6 = $7.50. Solve story problems involving multiplication of decimal numbers such as money, distances and measurements to demonstrate to students how you can remove the decimal point when working out the multiplication then add it to the answer; for example, 5.2 × 6 =. Remove the decimal so 52 × 6 = 312 then add the decimal to = 31.2. Decimal numbers multiplied by decimal numbers can also be treated the same way; by removing the decimal points first then adding them to your answer. Demonstrate examples of this using the website <www.mathsisfun.com/multiplying-decimals> • Divide decimal numbers by whole numbers in meaningful contexts. The decimal point can be omitted before working out the problem, then added to the final answer; for example, 6.5 km ÷ 5 = 1.3 km (think 65 ÷ 5 = 13) or $8.40 ÷ 4 = $2.10 (think 840 ÷ 4 = 210). Try some examples using the website <www. ixl.com/math/year-6/divide-decimals-by-whole-numbers> • Multiply and divide amounts of money to work out how much things cost individually or in a group. See how much things cost at the stationery shop (see teacher resources page 63). • Multiply decimals to the power of 10 and demonstrate how the decimal point moves when we multiply by ten; relate this to the place value system. Input a decimal number on a calculator, multiply this by 10 and see how the decimal point moves to the right each time. For example: 6.3 × 10 = 63.0 or 0.09 × 10 = 0.9. • Divide decimal numbers by the power of 10 using a calculator and demonstrate how the decimal point moves to the left each time; for example, 6.3 ÷ 10 = 0.63 or 0.09 ÷10 = 0.009. Point out to students that dividing is the opposite of multiplying. Complete a power of 10 table that multiples and divides numbers (see teacher resources page 64). Add your own decimal numbers.

© R. I . C.Publ i cat i ons • f o rr ebetween vi ew pu r po se son l y •strips or base ten • Demonstrate the connection fractions, decimals and percentages. Use paper 1 = 0.1 = 10%. (see teacher resources page 65). Relate this to hundredths blocks to demonstrate that 10 34 = 0.34 = 34% or 1 and thousandths; for example, 100 = 0.001 or 1% (see teacher resources 1000 page 66).

• Investigate the connection between fractions, decimals and percentages using the website <www. mathsisfun.com/decimal-fraction-percentage.html> • Investigate percentages in sales by looking at catalogues (print or online); for example, a 25%-off sale would mean 14 off the retail price. Demonstrate to students how you can calculate the percentage off an amount of money. For example: 20% off $10.00 is $2.00, making the new sale price $8.00.

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Fraction chart

1 whole 1 2 1 3 1 4 1 5 1 6

1 7 1 8 1 9 1 10 1 11 1 12 58

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Teachers resources

Equivalent fractions

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Number lines

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Addition and subtraction of fraction cards

+ + + + + + + + + + + +

– – – – – – – – – – – –

2 5 3 1 8 6 4 3 3 11 2 3 12 12 3 4 2 7 1 2 6 8 4 4 2 9 3 3 3 12 5 6 2 8 3 2 10 10 5 5 7 6i 1 .Publ 2 © R. I . C i cat ons 8•f 9 sonl 2w pur orr evi e pose y• 6 1 7 7 1 4 10 12 2 5 3 2 2 10 4 15 5 1 10 3 3 6 12 8 8 3 5 5 2 12 6 8 3 1 2 1 3 3 2 4 4 4 6 1 2 6 6 2 4 R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Rounding decimal numbers grid

0.5 1.0 1.5 2.0

1.7 0.9 1.2 2.3 0.1 62

0.8 0.2 0.6 1.9 1.1

Fractions, Decimals and Percentages (Years 5 and 6)

1.66 1.89 2.31 1.99 1.42 R.I.C. Publications®

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Teachers resources

Stationery shop

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Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Multiplying and dividing decimals to the power of 10

Multiplied by 10

Number Divided by 10 0.4 0.9 0.03 0.45 0.87 0.54 0.08 0.95 0.005 0.042 0.528 © R. I . C.Publ i cat i ons 0.896 •f orr evi ew1.2 pur posesonl y• 6.5 2.48 5.732

Fractions, Decimals and Percentages (Years 5 and 6)

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Teachers resources

Fractions, decimals and percentages – tenths 1 = 0.1 = 10% 10 2 = 0.2 = 10

3 = 0.3 = 10

4 = 10

5 = = © R. I . C.Publ i cat i ons 10 •f orr evi ew pur posesonl y•

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6 = 10

7 = 10

8 = 10

9 = 10

10 = 10

Fractions, Decimals and Percentages (Years 5 and 6)

Teachers resources

Fractions, decimals and percentages – hundredths and thousandths

14 = 0.14 = 14% 100

9 = 0.09 = 100

48 = 0.48 = 48% 100

100 =

1 = 0.001 = 0.1% 1000 66

Fractions, Decimals and Percentages (Years 5 and 6)

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Naming and representing fractions A fraction is an equal part of a whole. It is made up of a numerator (number 3 above the line) and a denominator (number below the line). For example: 4 1.

Use fraction symbols to name the fractions represented below.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Write the fraction symbols to match the words. (b) six-sixths

(a) three-fifths

(d) eleven-twelfths 3.

(a)

3 8

(b)

2 5

(c)

7 12

(d)

1 2

(e)

3 10

(f)

4 4

Look around the classroom and draw and name at least three fractions you can see. 4 For example: 4 of a window

Represent the fractions. 2 6

of a cake

1 4

of an apple

1 2

of a sandwich

6 8

of a pizza

Going further

In your own words explain what a fraction is.

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Fractions, Decimals and Percentages (Years 5 and 6)

Comparing and ordering fractions 1.

Look at the sets of fractions pictured below and use the words smaller than, larger than or the same as to describe them.

(a)

(b)

(c)

(d)

(e)

(f)

Represent the fractions on the circles below in order from smallest to largest: 1 1 1 1 1 3, 8, 2, 6, 4.

What do you notice about the denominators of the fractions above?

Write each set of fractions in order from smallest to largest.

(a)

3 8,

8 8,

1 8,

5 8,

2 8,

6 8

(b)

1 2,

3 4,

1 12 ,

3 8,

7 12 ,

1 4

(c)

6 10 ,

(d)

1 8,

1 6,

1 10 ,

(e) 1,

2 4,

5 6,

(f)

2 10 ,

4 10 ,

10 10 ,

1 7, 1 4,

8 10

1 9,

1 5

6 8,

1 6

2 11 1 6 3 2 3 , 12 , 3 , 12 , 3 , 12

Going further 3

At a pizza restaurant, James ate 4 of a pizza, Alex ate 8 of a pizza and Hannah ate 12 of a pizza. If the pizzas were all the same size, who ate the most or did they eat the same? Explain your answer. How much pizza did they eat altogether? 68

Fractions, Decimals and Percentages (Years 5 and 6)

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Equivalent fractions Fractions that are the same size but have a different numerator and denominator are known as equivalent fractions.

For example:

Answer true or false to each pair of fractions. (a)

1 2

1 3

(b)

1 4

3 12

(c)

1 12

(d)

4 6

2 3

(e)

3 4

6 8

(f)

1 5

3 10

(g)

8 10

(h)

1 2

2 5

(i)

2 9

1 3

3 4

4 5

1 3

2 6

1 10

Write at least one equivalent fraction for each fraction. (a)

1 3

(b)

1 2

(c)

(d)

3 5

(e)

2 6

(f) 1 whole =

Look at the first diagram and represent equivalent fractions on the blank circles. Complete the equivalent statement.

(b)

(c)

Going further

How many fifteenths would be equivalent to R.I.C. Publications®

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1 3?

Draw your answer on the back of this sheet.

Fractions, Decimals and Percentages (Years 5 and 6)

Equivalent fraction patterns 1.

Look at the sets of equivalent fractions. What pattern do you notice with the numerator and denominator? (a)

(b)

1 3

2 5

2 4

2 6

4 10

3 6

4 8

3 9

4 12

8 20

5 10

numerator:

6 12

denominator: numerator:

5 15

denominator: numerator:

16 40

denominator:

Complete the equivalent fractions by following the pattern of doubling the numerator. 3 6 For example: 4 = 8 (a)

2 3

(e)

3 4

(b)

3 5

(f)

3 6

(c)

1 6

(g)

4 5

(d)

1 2

(h)

5 6

You can also make equivalent fractions by multiplying the numerator and the denominator with the same number. For example

(a)

(e) 4.

1 2

10 ×

12 ×

(f)

(g)

(h)

Complete the equivalent fraction patterns. (a)

(b)

(c)

(d)

3 1

2 16

= 12

= 9

4 1

= 16

= 15

= 28

= 30

= 40

= 21

= 32

= 18

= 24

= 15

= 35

= 48

= 56

Going further

Suggest your own equivalent fraction patterns starting with these fractions:

Fractions, Decimals and Percentages (Years 5 and 6)

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1 3 3 12 , 4 , 9 .

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Simplifying fractions If the numerator and denominator of a fraction have a common factor, the fraction can be simplified. For example: 1.

4 12

can be simplified to

because 4 is a factor of 4 and of 12.

12 ÷

Simplify the fractions until you reach the lowest equivalent fraction. 6 3 1 For example: 12 = 6 = 2 (a)

8 12

(b)

4 8

(c)

6 12

(d)

10 20

= =

List the factors of each number. (a) 12:

(b) 24:

(d) 48:

Simplify the fractions to their lowest equivalent fraction. Divide the numerator and denominator by the highest common factor. For example: 3 ÷ 3 = 1 9

(a)

(e)

(i)

1 3

12 ÷

3 ÷

12 ÷

10 ÷

15 ÷

12 ÷

10 ÷

12 ÷

36 ÷

(j)

(c)

12 ÷

18 ÷

(k)

(d)

21 ÷

24 ÷

(l)

63 ÷

81 ÷

Find the lowest common factor and simplify the fractions. (a)

8 20

(b)

35 40

(c)

60 100

(d)

27 30

(e)

70 110

(f)

16 28

(g)

64 72

(h)

36 45

Going further

Why do you think we need to simplify fractions?

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Fractions, Decimals and Percentages (Years 5 and 6)

Improper fractions and mixed numbers When the numerator is the same or larger than the denominator it is 11 an improper fraction. For example: 4 2 A mixed number is a whole number together with a fraction. For example: 1 3 1.

Write the improper fraction and matching mixed number for the diagrams. Illustrate the improper fractions in (e) and (f) then write the mixed number. Improper fraction

Diagram

Mixed number

(a)

(b)

(c)

(d)

(e)

13 5

(f)

7 2

Change the improper fractions to mixed numbers. 12 19 1 1 For example: 4 = 3 (no remainders) 3 = 6 3 ( 3 remainder) (a)

9 4

(b)

18 5

(c)

25 8

(d)

20 3

(e)

13 2

(f)

36 6

(g)

30 9

(h)

63 12

Change the mixed numbers to improper fractions. For example: 2 5 = 1

(b) 4 2 =

(f) 2 12 =

(a) 2 4 = (e) 6 8 =

(g) 3 6 =

13 5

(2 x 5 + 3)

(d) 5 4 =

(h) 6 10 =

Going further 1

How many different ways could you make the fraction 1 2 using improper fractions, for 3 6 example 2 or 4 . Suggest at least three others. Can you see a pattern? 72

Fractions, Decimals and Percentages (Years 5 and 6)

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Counting by fractions 1.

Fill in the missing fractions on the counting number lines. (a)

(b)

(c)

1 8

4 8

2 12

1 10

4 12

6 8

5 12

8 12

2 10

6 10

7 8

10 12

7 10

12 12

9 10

Count by equivalent fractions on the number lines in question 1. (a) Count by quarters along the top of the eighths number line. (b) Count by sixths along the top of the twelfths number line. (c) Count by fifths along the top of the tenths number line.

Use the number lines below to count by fractions.

(a) Count by thirds to 3 wholes. 0

(b) Count by halves to 3 wholes. 0

Complete the missing fractions in the counting patterns. (a)

2 4 10 , 10 ,

(b)

3 12 ,

(c)

1 2,

(d)

36 6 ,

, ,

8 10 ,

9 12 ,

, 1

12, ,

12 10 ,

15 18 12 , 12 ,

, 32,

30 27 6 , 6 ,

, 24 12 ,

30 12

, 62, 21 6 ,

18 10 ,

, 82, 15 6 ,

6 6

Going further

If you wanted 60 orange quarters, how many whole oranges would you need? Show your answer in a counting pattern on the back of this sheet.

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Fractions, Decimals and Percentages (Years 5 and 6)

Adding fractions with common denominators When adding fractions with common denominators, add the numerators 3 4 7 while the denominator remains the same. For example, 8 + 8 = 8 1.

Add the fractions using the diagrams to help. Draw and write your answer.

(a)

1 4

2 4

(b)

4 8

1 8

(c)

2 3

1 3

(d)

2 6

(e)

3 5

2 5

1 4

0 1 8

2 8

3 8

1 12

2 12

3 12

4 12

1 5

0 1 10

6 12

2 5

2 10

3 10

5 8

4 10

6 8

7 12

8 12

9 12

6 10

11 12

12 12

4 5 7 10

9 8

3 3 6 6

10 12

9 10

10 10

10 8

11 8

13 12

14 12

15 12

16 12

6 5 11 10

12 10

13 8

14 8

15 8

16 8

6 3 12 6

5 3

9 6 17 12

14 10

11 6

10 6

18 12

19 12

7 5 13 10

4 2 8 4

7 4

12 8

4 3 8 6

7 6

5 5

8 10

3 2 6 4

5 4

8 8

5 6

3 5 5 10

7 8

2 3 4 6

3 6 5 12

2 2 4 4

3 4

4 8

1 3 2 6

1 6

20 12

21 12

8 5 15 10

22 12

23 12

9 5

16 10

17 10

18 10

24 12

10 5 19 10

20 10

Use the number lines above to add the fractions. If the answer is an improper fraction, change it to a mixed number. Simplify fractions to the LCD (lowest common denominator) where possible. (a)

5 8

3 8

(b)

3 4

(d)

5 6

4 6

(e)

3 12

6 12

(g)

2 2

1 2

(h)

8 10

(j)

3 10

(k)

11 12

6 10

3 4

(c)

2 3

1 3

(f)

3 5

5 5

4 10

(i)

5 4

3 4

6 12

Going further 1

If the answer is 1 2 suggest at least three different addition problems that equal this amount. 74

Fractions, Decimals and Percentages (Years 5 and 6)

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Adding fractions with different denominators When adding two fractions with different denominators, first find a common or equivalent denominator. 1 3 2 3 5 1 For example: 2 + 4 becomes 4 + 4 = 4 or 1 4 1.

Add the fractions using diagrams to help you. Draw and write your answer.

(a)

1 4

3 8

(b)

1 2

2 4

(c)

1 3

2 6

(d)

3 10

2 5

Add the fractions by changing the denominators to the suggested common denominator. Remember if you double or halve the denominator you must double or halve the numerator. 2

(a)

(c)

(e)

= 2

= 6

(d)

(f)

5 7

6 2

= 5

= 12

Use your knowledge of equivalence to change the denominator to a common denominator before adding the fractions. (a) (c) (e)

6 3

3 1

9 5

3 1

(b)

(d)

(f)

6 10 6 12 6 8

2 5 1 2 1 4

Going further

Add the fractions after finding the common denominator. 4 8

3 1 12 + 4

3 6

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Fractions, Decimals and Percentages (Years 5 and 6)

Subtracting fractions with common denominators When subtracting fractions with common denominators, subtract the numerator while 5 3 2 1 the denominator remains the same. For example: 6 – 6 = 6 or 3 1.

Subtract the fractions by placing a cross on the sections that need to be subtracted from the diagrams. The first one is done for you.

(a)

7 8

(c)

8 10

(e)

4 4

–

3 8

(b)

3 3

7 10

(d)

9 12

(f)

5 5

2 4

1 2 2 4

1 4

0 1 8

2 8

1 3 2 6

1 12

2 12

3 12

4 12

1 5

0 1 10

4 8

3 10

5 8

5 12

6 12

4 10

6 8

2 5

3 2 6 4

5 4

7 12

6 10

8 8

8 12

9 12

10 12

11 12

12 12

4 5 7 10

9 8

3 3 6 6

5 6

3 5 5 10

7 8

2 3 4 6

3 6

2 5

2 10

–

6 12

–

2 2 4 4

3 4

2 3

–

9 10

10 10

11 8

13 12

14 12

15 12

16 12

6 5 11 10

12 8

4 3 8 6

7 6

5 5

8 10

10 8

12 10

13 8

14 8

15 8

16 8

6 3 12 6

5 3

9 6 17 12

14 10

11 6

10 6

18 12

19 12

7 5 13 10

4 2 8 4

7 4

1 6

4 8

20 12

21 12

8 5 15 10

22 12

23 12

9 5

16 10

17 10

18 10

24 12

10 5 19 10

20 10

Use the number lines above to help you subtract the fractions. If the answer is an improper fraction change it to a mixed number. Simplify fractions to the LCD where possible. (a)

2 2

–

1 2

(b)

3 3

(d)

7 8

–

2 8

(e)

11 12

(g)

3 4

–

1 4

(h)

7 6

(j)

9 5

–

2 5

(k)

15 12

– – – –

1 3

5 12 3 6

9 12

(c)

10 10

(f)

3 2

(i)

13 8

(l)

8 5

– – – –

6 10 1 2 7 8 3 5

= = =

Going further 3

If the answer is 4 suggest at least three different subtraction problems with answers equal to this amount. 76

Fractions, Decimals and Percentages (Years 5 and 6)

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Subtracting fractions with different denominators When subtracting two fractions with different denominators, find a common denominator. 10 1 10 4 6 1 For example: 12 – 3 = 12 – 12 = 12 or 2 1.

Before subtracting the fractions change them so they have common denominators. Cross out the fraction that needs to be subtracted on the diagram. The first one is done for you. Fraction sentence

(a)

(b)

(c)

(d)

12 11

(e)

12 8

(f) 2.

xxxx

–

1 3 2 5 1 4 1 3 1 2 2 3

New sentence

3 9

10 5

–

8 1

–

Diagram

11 =

12 8

–

Subtract the fractions by changing the denominators to the suggested common denominator. Remember if you double or halve the denominator you must also double or halve the numerator. (a)

(c)

(e)

–

1 2 3 4 1 3

4 6 7 8 9 12

–

(b)

(d)

(f)

–

4 6 2 12 3 5

3 3 5 6 7 10

–

Use your knowledge of equivalence to change the denominator to a common one before subtracting. (a) (c) (e)

1 2 4 6 6 9

–

1 4 1 3 1 3

–

(b)

–

(d)

–

(f)

6 8 9 10 5 6

–

1 4 1 5 4 12

–

Going further 6

Explain how you would best solve this problem: 8 – 6 = R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Adding and subtracting mixed number fractions When adding or subtracting mixed number fractions, add or subtract the whole numbers first then the fraction. For example: 1

23 + 13 = 33 1.

2 3

1 13

1 23

2 13

2 23

3 13

2 34

3 23

Jump forward along the number line to add the fractions. 1

(a) 2 4 + 1 4 = 2

1 4

2 4

3 4

1 14

1 24

1 34

2 14

2 24

3 14

3 24

3 34

(b) 1 3 + 1 3 = 1

1 3

2 3

1 13

1 23

2 13

2 23

1 5

2 5

3 5

4 5

1 1 15 1 25 1 35 1 45

2 2 15 2 25 2 35 2 45

3 3 15 3 25 3 35 3 45

(d) 2 2 + 1 2 = 2.

1 3

1 2

1 12

2 12

3 12

Jump back along the number line to subtract the fractions. 1

(a) 4 – 2 3 = 1

1 13

(d) 4 – 2 4

1 2

1 23

2 13

2 23

3 13

2 3

1 12

2 12

3 12

1 5

3 5

2 5

1 4

4 5

2 4

1 1 15 1 25 1 35 1 45 1245 2 15 2 25 2 35 2 45

3 4

1 14

1 24

1 34

2 14

2 24

3 3 15 3 25 3 35 3 45

2 34

3 14

3 24

3 34

Add the mixed number fractions. 1

(b) 3 3 + 1 3 =

(e) 4 3 + 1 3 =

(h) 1 6 + 3 6 =

(a) 2 2 + 2 2 = (d) 3 8 + 2 8 = (g) 4 5 + 3 5 = 4.

2 3

(b) 3 2 – 1 2 =

1 3

(f) 3 12 + 1 12 =

(i) 4 10 + 2 10 =

Subtract the mixed number fractions. 2

(b) 2 3 – 1 3 =

(e) 7 10 – 4 10 =

(h) 5 4 – 1 4 =

(a) 4 5 – 3 5 = (d) 5 6 – 4 6 = (g) 9 8 – 6 8 =

(f) 4 – 1 2 =

(i) 10 12 – 7 12 =

Going further

Use your knowledge of equivalence to add and subtract the fractions. 1

(b) 2 8 + 1 4 =

(d) 2 4 + 2 12 =

(a) 4 2 – 1 4 =

Fractions, Decimals and Percentages (Years 5 and 6)

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Addition and subtraction fraction problems 1.

Read the problem, write the fraction sentence (addition or subtraction) then solve it. Word problem

Fraction sentence

(a) Luke read 5 of a book on the first day and 10 on the second day. How much did he read? 1

(b) Amy ate 2 a pizza and Ben ate 4 . How much did they eat altogether? 3

(c) At the weekend a café served 1 8 mud cakes on 4 Saturday and 1 8 on Sunday. How many mud cakes were served altogether? 4

(d) Nic ate 12 of a pie and his brother ate 3 . How much was eaten altogether? 2

(e) Mr Smith travelled 5 of the journey on one day. How much would he have left to travel the whole distance? (f)

Helena used 1 4 shelves in the office and Max used 2 4 . How many shelves did they use altogether? 5

(g) Henry completed 6 of the game on his iPod™ and Jack 3 completed 6 . How much more did Henry get through?

Write the fractions on the outer parts of the wheel you would need to ADD to each fraction to make it equal to the number/fraction in the centre of the wheel.

(a)

7 10

1 8 3 12

1 2

1 1 3

2 8 1 12

3 4 7 8

4 5

2 6

1 2

1 4 1 6

1 2 1 10

1 8

Write the fractions you would need to SUBTRACT from each fraction to make it equal to the fraction in the centre of the wheel. 3 4

5 10

(a)

(b) 6 8

6 8

1 2

1 4

4 6 8 12

1 2

8 12 3 4

8 10

10 20

Going further

Complete the number sentences and create your own word problems about them. (a)

6 8

–

1 4

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2 8

3 8

Fractions, Decimals and Percentages (Years 5 and 6)

Fractions of whole numbers You can calculate the fraction of a whole number by dividing 1 the denominator into that number. For example: 4 of 36 = 9 (36 ÷ 4 = 9) 1.

Calculate the fraction of the whole numbers. Check your answers with a calculator. (a)

1 3

of 21 =

(b)

1 5

of 45 =

(c)

1 2

(d)

1 6

of 72 =

(e)

1 8

of 64 =

(f)

1 10

(g)

1 7

of 84 =

(h)

1 12

(i)

1 9

of 72 =

(j)

1 5

of 75 =

(k)

1 8

(l)

1 6

of 120 =

(a)

2 3

of 12 =

(b)

3 4

of 24 =

(c)

2 5

of 30 =

(d)

2 6

of 36 =

(e)

3 8

of 48 =

(f)

3 9

of 18 =

(g)

7 10

(h)

5 8

of 64 =

(i)

2 4

of 32 =

(j)

5 6

(k)

5 9

of 90 =

(l)

4 5

of 60 =

of 100 = of 54 =

100 people went to a concert. Calculate how many of each age group there were if: 2 4

1 4

2 8

Alex got $20 pocket money. Calculate how much he spent on the following: (a)

of 32 =

of 110 =

When the numerator is larger than 1 multiply the answer with the numerator. 4 For example: 5 of 30 = 24 (30 ÷ 5 = 6, 6 x 4 = 24) Calculate the fraction of the whole numbers. Check your answers with a calculator.

(a) 4.

of 60 =

of 26 =

2 5

on drinks

(b)

8 20

on lollies

(c)

2 10

on gum

Michaela earns $240 a week at her part-time job. Calculate how much she spends on the following: (a)

1 3

on clothes

(b)

1 8

on transport

(d)

1 12

on drinks

(e)

1 6

on lunch and snacks

(c)

1 4

savings

(f) How much does she have left over? 6.

Mr Wong earns $1000 a week. Calculate how much he spends on the following: (a)

4 10

on bills

(b)

3 10

on groceries

(c)

1 10

on petrol

(d)

2 10

on tax

(e) How much money does he have left over? Going further

Use fractions to describe how you might spend $40 pocket money. 80

Fractions, Decimals and Percentages (Years 5 and 6)

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Fractions and decimals Fractions can be related to decimals as they are both parts of a whole. Decimal numbers can be broken into tenths, hundredths and thousandths. They are commonly used for money, measurement and parts of seconds.

1 10

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Write the matching fraction for the decimals in question 1.

©R . I . C. Publ i cat i o ns (g) (c) (d) (e) (f) •f orr evi ew pur posesonl y• (b)

(h)

Write the decimal hundredth to match the fractions. For example,

34 100

18 100

(b)

85 100

(c)

7 100

(d)

51 100

(e)

14 100

(f)

92 100

(g)

40 100

(h)

65 100

Write the fraction to match the decimal thousandths. For example, 0.462 =

(b) 0.084 = 1000

(e) 0.002 =

(f) 0.430 = 1000

= 0.34

(a)

(a) 0.215 =

1 1000

Decimal tenths are the same as fraction tenths. They are both one whole divided into ten equal parts. The decimal point replaces the fraction line. Write the decimal tenths represented below.

(a)

1 100

462 1000

(d) 0.903 = 1000

(g) 0.861 = 1000

1000

(h) 0.059 = 1000

1000

Write the fractions as decimal numbers. (a)

9 10

(e)

538 1000

= =

(b)

64 100

(f)

4 10

= =

(c)

715 1000

(d)

(g)

92 1000

(h)

5 100 12 100

= =

Going further

Explain how 0.4 is equivalent to 0.40 and 0.400. You may wish to draw a diagram. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Place value and decimals When looking at place value, whole numbers always appear to the left of the decimal point. Decimal numbers are to the right of the decimal point. For example: 239.574 = 2 hundreds, 3 tens, 9 ones, 5 tenths, 7 hundredths and 4 thousandths.

•

(b) 2.584

•

(d) 80.371

•

(e) 32 624.5

•

(f)

•

8035.932

(g) 142 670.99

•

(h) 56 094.26

•

(i)

•

378.953

Thousandths

Hundredths

Tenths

•

Ones

Tens

Hundreds

Thousands

(a) 0.42

Tens of thousands

Place the numbers on the place value grid. Hundredths of thousands

•

2657.001

Write the numbers in expanded form. For example: 412 674.081 = 4 hundreds of thousands + 1 tens of thousands + 2 thousand + 6 hundred + 7 tens + 4 ones + 0 tenths + 8 hundredths + 1 thousandth (a) 53 762.95 =

(b) 802.371 =

(d) 605 498.29 =

Going further

Find the difference between the smallest and the largest number from question 3 (you can use a calculator or written method to do this). 82

Fractions, Decimals and Percentages (Years 5 and 6)

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Comparing and ordering decimal numbers 1.

Circle the smallest number in each set of decimal numbers. Write them in order from smallest to largest. (a) 6.734, 6.374, 6.073, 6.704, 6.473 (b) 45.31, 31.45, 65.54, 54.65, 41.75 (c) 628.9, 298.6, 986.2, 892.6, 906.8 (d) 0.965, 0.695, 0.596, 0.995, 0.659 (e) 3125.8, 3125.5, 3125.9, 3125.1, 3125.4

Look at the cost of the stationery items and find the cost of:

1.45

95c

tic

er Eras Plas

1.50

(a) 1 exercise book and 2 pens (b) 3 pencils and 1 ruler

30c

45c

(d) 2 erasers (e) 2 exercise books and a ruler

22 23 24 25 26 27 28 29 30

(f) 1 of each item

22 23 24 25 26 27 28 29 30

cheapest and dearest items

Find the difference in seconds between the fastest and the slowest time. (a) 12.09 s, 12.54 s, 12.45 s, 12.31 s, 12.48 s

Difference

(b) 23.24 s, 23.16 s, 24.35 s, 24.21 s, 24.11 s

Difference

(d) 2.10 s, 1.58 s, 2.38 s, 1.40 s, 2.03 s

Difference

(e) 1.35 s, 2.05 s, 1.56 s, 2.11 s, 2.29 s, 1.52 s

Difference

Order the heights of the students from shortest to tallest, using decimal numbers. 1 m 56 cm, 1 m 45 cm, 1 m 58 cm, 1 m 50 cm, 1 m 43 cm, 1 m 61 cm, 1 m 41 cm, 1 m 57 cm, 1 m 64 cm, 1 m 59 cm

Going further

Measure the heights of five students in your class and record them in order from the shortest to the tallest using decimal numbers. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Adding decimal numbers Adding decimals is the same as adding whole numbers except a decimal point separates the whole numbers from the decimal numbers. Start adding from the right (decimal numbers) and work towards the left (whole numbers).

(a) 2.3 + 3.4 =

(b) 9.1 + 3.5 =

(d) 16.5 + 7.3 =

(e) 4.51 + 2.36 =

(f) 6.241 + 2.738 =

Thousandths

Hundredths

Tenths

•

Ones

Tens

Hundreds

Thousands

Thousandths

(c)

2 6 7

3 2

5 2 6 1

4 2 5

9 8

7 0 4

9 1

8 5

9 3 6

0 9 3

3 0 5

2 5 9

Add the decimal numbers. Regroup where necessary. Check your answers with a calculator. (a)

35.92 +

(e) +

(b)

67.29

17.08

(c)

26.58

183.05 +

(d)

52.064

52.75

38.736

463.61

21.039 +

Hundredths

Tenths

•

Ones

Tens

Hundreds

Thousands

(b)

Thousandths

Hundredths

Tenths

•

Ones

(a)

Tens

Use the place value grids to help you add the decimal numbers vertically. Hundreds

Add the simple decimal numbers.

Thousands

4.621

632.853 +

453.086

Read the problem then set the decimal numbers out vertically before adding them. (a) A dressmaker purchased 3.65 m, 1.95 m, 2.35 m and 4 m of fabric. How much did she purchase altogether? (b) At a 3-day sale a toy store took the following amounts: $7241.75, $5842.30 and $9012.95. How much did they take over the whole sale? (c) The stock market recorded the following rises over 4 days in gold shares: 0.342, 0.706, 0.418 and 1.529. By how much did gold rise altogether?

Going further

Locate five decimal numbers in a newspaper, magazine or internet and add them up. 84

Fractions, Decimals and Percentages (Years 5 and 6)

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Subtracting decimal numbers Start subtracting from the right (decimal numbers) and work towards the left (whole numbers). Rename where necessary.

(a) 0.95 – 0.62 =

(b) 8.7 – 3.4 =

(d) 12.07 – 1.05 =

(e) 8.964 – 5.720 =

(f)

(g) 0.899 – 0.647 =

(h) 5.641 – 1.230 =

1 9 6

8 5 1 1 0

Thousandths

Hundredths

Tenths

•

Ones

Tens

Hundreds

(c)

Thousands

Thousandths

Hundredths

Tenths

•

Ones

Tens

Hundreds

Thousands

Thousandths

Hundredths

Tenths 1

(b)

7 5

7 4 5

8 3 6

4 0 5 3

2 4 8

8 2

2 7 3

4 1 9

1 3 2 1

9 0 6

9 3

Subtract the decimal numbers vertically. Remember to rename where necessary. (b). 2P . 1 1u (c) 7i 4o . 6n 3s (d) ©R I . C8. bl i ca3t – 21.70 – 41.60 – 91.56 •f orr evi e w pur pos esonl y•

(a)

72.86

(e)

741.93 –

•

Ones

Tens

(a)

9.85 – 3.61 =

Use the place value grids to help you subtract the decimal numbers. Rename where necessary. The first one is done for you. Hundreds

Subtract the decimal numbers without renaming.

Thousands

480.45

(f)

5032.89 –

(g)

801.92

57.402 –

36.251

(h)

83.295 –

9.079

6815.483 –

907.255

Read the problem then set the decimal numbers out vertically before subtracting them. (a) Yang received $53.95 interest from the bank this year and $39.08 last year. How much more money did he get this year? (b) John needed 27.45 m of wood to finish a deck. If he already has 19.90 m, how much more wood does he need to buy? (c) When Liam was born he was 0.55 m long. Now at the age of 12 he is 1.62 m tall. How much has he grown since birth?

Going further

Imagine you were given $500.00. Use the internet or magazines to find three things you would like to purchase and subtract their cost from the amount given. How much would you have left? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Estimating and rounding decimal numbers To add or subtract a decimal number quickly you can use the skills of rounding and estimating. For example: 3.48 + 4.81 can be rounded to 3 + 5 = 8, which is the estimated total. 1.

Round the decimal numbers to the nearest whole number. For example: 6.25 is rounded to 6 (a) 7.2 =

(b) 2.8 =

(d) 6.4 =

(e) 4.13 =

(f)

1.9 =

(g) 10.4 =

(h) 0.7 =

(i)

(j)

93.67=

(k) 132.88 =

(l)

(o) 92.78 =

(p) 362.04 =

56.41 =

(m) 34.23 = 2.

(n) 84.86 =

452.24 =

Round the addition problems to the nearest whole number before solving them. Check your answer with a calculator. The first one is done for you. Addition problem

(a) 6.7 + 2.1

Rounded

Estimated answer

Calculator answer

7+2

8.8

(b) 4.2 + 3.8 (c) 10.7 + 5.2 (d) 8.9 + 3.8

(e) 9.01 + 6.99 (f)

(g) 8.9 + 4.6 (h) 25.3 + 22.2 3.

Round the subtraction problems before solving them. Check your answer with a calculator. The first one is done for you. Addition problem

(a) 7.8 – 3.2

Rounded

Estimated answer

Calculator answer

8–3

4.6

(b) 3.9 – 1.2 (c) 9.9 – 6.3 (d) 15.8 – 3.9 (e) 35.5 – 12.4 (f)

68.1 – 34.3

(g) 85.7 – 34.6 (h) 96.2 – 52.8 Going further

Round the numbers to one decimal place. (a) 8.763 86

(b) 9.013

Fractions, Decimals and Percentages (Years 5 and 6)

(d) 7.589

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Multiplying decimal numbers 1.

Use a calculator to multiply the decimal numbers by whole numbers. (a) 3.4 × 6 =

(b) 9.1 × 3 =

(d) 5.2 × 6 =

(e) 8.7 × 4 =

(f)

15.1 × 9 =

(g) 32.5 × 2 =

(h) 54.7 × 8 =

(i)

65.23 × 3 =

(j)

(k) 75.46 × 7 =

(l)

82.04 × 4 =

92.54 × 5 =

To multiply a decimal number by a decimal number: • Remove the decimals points and multiply; e.g. 1.2 x 0.4 becomes 12 x 4 = 48. • Add the number of decimal places to find where the decimal point in the answer goes. There are 2 decimal points (1 in 1.2 and 1 in 0.4), so the answer is 0.48. 2.

Multiply the decimal numbers. (a) 1.1 × 0.5 ➞

(11 × 5) =

➞

(b) 1.3 × 0.7 ➞

(13 × 7) =

➞

(d) 1.6 × 0.2 ➞

➞

(e) 2.5 × 0.3 ➞

➞

(f)

4.9 × 0.1 ➞

➞

(g) 3.5 × 0.4 ➞

➞

(h) 2.6 × 0.4 ➞

➞

Multiply the decimal numbers. Remember to remove the decimal points and multiply. Then count the number of decimal places combined in both numbers so you know where to put the decimal point in your answer. The first one is done for you. (b) (c) © R. I . C .Publ i cat i on s •f orr evi ew pur posesonl y•

(a) 0.06 × 1.8 ➞

3.4 × 0.02 ➞

1.9 × 0.08 ➞

× 6

108 ➞ 0.108

(d) 0.06 × 5.3 ➞

➞

(e) 0.05 × 6.4 ➞

➞

(g) 0.145 × 0.6 ➞

7.12 × 0.03 ➞

➞

(h) 1.628 × 0.3 ➞

➞

(f)

➞

(i)

5.21 × 0.07 ➞

➞

Solve the multiplication problems using the strategies you have learnt. Word problem

(a)

Number sentence

Answer

April hired 4 DVD’s at $5.75 each. How much did it cost altogether?

(b) If a shelf measured 3.4 m × 0.6 m, how many square metres is the area of the shelf? (c)

If the average person visited a museum 0.06 times a year, how many times would they visit over 7.5 years?

Going further

How many places would you need to put the decimal point in this question? 3.56 × 0.009 =? Have a go at solving this problem. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Dividing decimal numbers To divide decimal numbers by a whole number, remove the decimal point and carry out the division, then place the decimal point back in the same place. becomes 1.2

For example: 8.4 ÷ 7 ➞ 1.

Divide the decimal numbers by whole numbers by removing the decimal point first then adding it back to the answer. The first one is done for you.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Use a calculator to divide the decimal numbers by whole numbers. (b) 72.9 ÷ 9 =

(a) 16.4 ÷ 4 =

(e) 64.8 ÷ 8P =u (f)o 84.6 ÷3= ©R . I . C. bl i cat i ns (g) 196.4 ÷ 2 • = f 239.2 4 =u (i)s 450.5 5=• orr e(h) vi e w÷ p r pose on÷l y (d) 42.7 ÷ 7 =

Solve the division problems using the written strategy or a calculator. Word problem

Number sentence

Answer

(a) If 6.4 m of fabric was used to create two dresses, how much fabric did each dress require? (b) If a farmer planted his 5 fruit trees in a row across 7.5 m of land, how much space did each tree have? (c) A builder had to share 9.6 m of wire between 8 frames. How much wire did each frame end up with? (d) Share 95.5 m of fencing between 5 blocks. How much fencing does each block need? Going further

Write your own story problem to match the sentence: 630.7 ÷ 7 = ?

Fractions, Decimals and Percentages (Years 5 and 6)

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Solving decimal problems – exchange rates Exchange rates:

$1.00 Australian dollars (AUD)

¥5.32 Chinese Yuan €0.70 EURO $1.67 Fiji dollars (CNY) (EUR) (FJD) $1.11 New Zealand dollars $0.87 US dollars (NZD) (USD)

Use your calculator to work out the currency exchange rates by multiplying the amount with the exchange rate. For example: AUD200 × USD0.87 = USD174 Currency

Amount in AUD

Amount × exchange rate

Amount

(a) EURO

$200

$200 ×

EUR

(b) NZ dollars

$500

$500 ×

NZD

$100

$100 ×

CNY

(d) US dollars

$400

$400 ×

USD

(e) Fiji dollars

$300

$300 ×

FJD

(f) NZ dollars

$750

$750 ×

NZD

Convert the currency amounts back to Australian dollars by dividing them by the exchange rate. For example: USD600 ÷ 0.87 = AUD689.65 Currency

Amount ÷ exchange rate

Amount in Australian dollars

(a) NZ dollars

$350 ÷

(b) Chinese Yuan

¥500 ÷

$200 ÷

(d) EURO

€800 ÷

(e) US dollars

$1000 ÷

(f) Chinese Yuan

¥1200 ÷

(g) NZ dollars

$2000 ÷

(h) EURO

€1500 ÷

Going further

Choose three countries you may be interested in travelling to one day. Investigate their current exchange rate using the internet. How much of each currency would AUD2000 buy? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Solving decimal problems involving money

Use the information above to work out how much money you would save on the multi-buys. Items

Cost to buy individually

Multi-buy cost

Saving

(a) 4 mangoes (b) 6 bananas (c) 4 corn cobs (d) 4 tomatoes (e) 3 apples (f) 2 avocados 2.

(a)

Look at the multi-buy prices above and work out what each item would cost individually at these prices.

(d) 1 corn cob =

(e) 1 banana =

(f) 1 apple =

Calculate the shopping lists using the advertised prices above. Round your total to the nearest 0.05c. You may wish to use a calculator. Shopping list

Costs

Total (rounded)

(a) 6 bananas, 6 apples, 2 tomatoes and 1 lettuce (b) 1 avocado, 1 mango, 1 banana, 1 Dutch carrots and 1 tomato (c) 4 tomatoes, 2 avocados, 2 lettuce and 1 apple. (d) 1 of each item Going further

Explain why you think shops like to offer and shoppers like to buy multi-buy items. 90

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Decimals and powers of 10 – multiplying 1.

Use a calculator to multiply the decimals by 10, 100 and 1000. Number

× 10

× 100

× 1000

(a) 0.582 (b) 1.396 (c) 4.7 (d) 16.08 (e) 73.629 (f)

128.1

Describe what is happening to the numbers above?

Multiply the numbers by 10, 100 and 1000 without a calculator. (a) 0.741 × 10 =

, 0.741 × 100 =

, 0.741 × 1000 =

(b) 1.845 × 10 =

, 1.845 × 100 =

, 1.845 × 1000 =

, 6.27 × 100 =

, 6.27 × 1000 =

, 511.89 × 100 =

, 511.89 × 1000 =

(d) 18.999 × 10 = (e) (f) 4.

511.89 × 10 =

Use your knowledge of powers of 10 to solve the problems with or without a calculator. Problem

Number sentence

Answer

(a) Henry can swim 1 lap of a 25-m pool in 0.34 minutes. How long would it take him to swim 10 laps at the same speed? (b) If Rula bought 100 buttons that cost 27c each, how much did she spend on buttons? (c) If petrol cost $1.43 a litre, how much would 100 litres cost? (d) If wire fencing cost $28.75 m, how much would 1000 m cost? (e) Mia bought 10 bread rolls at 65c each. How much change would she get back from $10.00? Going further

Explain what might happen to the decimal point if you were to divide a number by 10, 100 or 1000? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Decimals and powers of 10 – dividing 1.

Use a calculator to divide the decimals by 10, 100 and 1000. Number

÷ 10

÷ 100

÷ 1000

(a) 842.1 (b) 1748.6 (c) 633.24 (d) 95.05 (e) 702.85 (f)

5.8

Describe what is happening to the numbers above.

Divide the numbers by 10, 100 and 1000 without a calculator. (a) 62.8 ÷ 10 =

, 62.8 ÷ 100 =

, 62.8 ÷ 1000 =

(b) 512.9 ÷ 10 =

, 512.9 ÷ 100 =

, 512.9 ÷ 1000 =

, 1843.6 ÷ 100 =

, 1843.6 ÷ 1000 =

, 3.1 ÷ 100 =

, 3.1 ÷ 1000 =

(d) 75.85 ÷ 10 = (e) (f) 4.

3.1 ÷ 10 =

Use your knowledge of powers of 10 to solve the problems with or without a calculator. Problem

Number sentence

Answer

(a) If Gina can swim 1000 m (1 km) in 32.45 minutes, how long does it take her to swim 1 m? (b) If Harry spent $6.85 on 10 oranges, how much would 1 orange cost? (c) If 100 m of rope cost $178.50, how much would 1m cost? (d) If a boy can run 100 m in 21.12 seconds, how fast would he travel each 10 m? (e) If a farmer bought 100 fig trees at a total of $1420.00, how much did each tree cost? Going further

Explain what would happen to the number 5 if you were to divide it by 10 000, 100 000 and 1 000 000. 92

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Powers of 10 Another way of expressing powers of ten is to use a small number (index) to represent how many times ten is multiplied by itself. For example, 103 means 10 × 10 × 10 = 1000. (Note: there are three zeros in 10 × 10 × 10 so we must have three zeros in the answer.) 1.

Complete the powers of 10 table. Powers of 10

10 ×

(a) 102

10 × 10

(b) 103

10 × 10 × 10

10 × 10 × 10 × 10

Answer

(d) 105 (e) 106 2.

Multiply the whole numbers by powers of 3. 10. (Hint: The index number tells how many times to multiply the number by 10 and how many zeros should be in the answer.) (a) 2 × 102 = (2 × 10 × 10) =

Multiply the decimal numbers by powers of 10 using a calculator. Note: each time a number is multiplied by ten the decimal point moves one place to the right. For example: 1.6 × 103 = (1.6 × 10 × 10 × 10) = 1600

(a) 1.2 × 10 = © R. I . C.Publ i c a(1.2 t i o n × 10 ×s 10) = (b) 3.4 × 10 = •=f orr evi ew pur po s esonl y• 4 × 10 (3.4 × 10 × 10 × 10) = 2

(b) 9 × 103 = (9 × 10 × 10 × 10) =

(c)

(4 × 10 × 10 × 10 × 10) =

(d) 15 × 103 = (15 × 10 × 10 × 10) =

(d) 0.6 × 102 = (0.6 × 10 × 10) =

(e) 8 × 105 = (8 × 10 × 10 × 10 × 10 × 10) = (f) 4.

(e) 0.07 × 103 = (0.07 × 10 × 10 × 10) =

23 × 10 = (23 × 10 × 10) =

Try these with or without a calculator. × 102

× 103

× 104

× 105

(a) 0.8 (b) 0.05 (c) 0.009 (d) 0.0003 Going further

Explain what would happen if you added an index (small number) after a number other than 10, e.g. 73? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Equivalent calculations If you understand multiplying and dividing by the powers of 10 (10, 100, 1000) then you can see that calculations can be the same or equivalent if the decimal point is moved. For example: 6.2 × 10 is the same as 0.62 × 100, or 54.25 ÷ 1 is the same as 5425 ÷ 100. 1.

Complete the equivalent multiplication number sentences. (a)

4.1 × 10 =

or 0.41 × 100 =

or 0.041 × 1000 =

(b)

7.5 × 10 =

or 0.75 × 100 =

or 0.075 × 1000 =

(c)

8.6 × 10 =

or 0.86 × 100 =

or 0.086 × 1000 =

(d)

62.5 × 10 =

or 6.25 × 100 =

or 0.625 × 1000 =

(e)

912.4 × 10 =

or 91.24 × 100 =

or 9.124 × 1000 =

What do you notice about the answers above in each question?

Complete the equivalent division sentences. or 958.0 ÷ 100 =

or 9580.0 ÷ 1000 =

(d) 85.612 ÷ 10 =

or 856.12 ÷ 100 =

or 8561.2 ÷ 1000 =

(e)

or 70.23 ÷ 100 =

or 702.3 ÷ 1000 =

(a) (b) (c)

95.8 ÷ 10 =

or. 64. ÷P 100 =b or 640 ©R I . C u l i cat i on s÷ 1000 = 6.732 ÷ 10 = 67.32 ÷ 100 = or 673.20 ÷ 1000 = •f orr eorvi e w pur pose sonl y• 6.4 ÷ 10 =

7.023 ÷ 10 =

Suggest an equivalent number sentence and answer. You may wish to use a calculator. MULTIPLICATION Number sentence

DIVISION

Equivalent sentence

Number sentence

(a) 4.6 × 10

(f)

(b) 9.25 × 100

(g) 25 ÷ 10

(h) 842 ÷ 1000

(d) 84.5 × 100

(i)

2 534 ÷ 10

(e) 53.2 × 10

(j)

9 651 ÷ 100

Equivalent sentence

708 ÷ 100

Going further

If the answer is 3, suggest at least 3 different but equivalent multiplication or division sentences containing decimal points to reach this answer.

Fractions, Decimals and Percentages (Years 5 and 6)

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Percentages The word percent means ‘out of a hundred’. Percentages are equal parts out of 100, so they can be related to decimals and fractions; 10 for example, 10% is the same as 0.10 or 100 . 1.

What percentages are shaded? %

(a)

(b)

(c)

(d)

Shade the hundredths to represent the percentage. (a) 30%

(b) 90%

(d) 50%

10%

40%

60%

5 10

7 10

100%

9 10

Use your knowledge of fractions and equivalence to answer the questions. The number line and representations above will also help you. (a)

2 10

(d)

3 4

(g)

3 10

(j)

35 100

= =

% %

= =

% %

(b)

1 2

(e)

7 10

(h)

4 5

(k)

62 100

= =

% %

1 4

(i)

4 10

(l)

93 100

% %

= =

% %

Going further

How do you think percentages would relate to decimal numbers? Give some examples.

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Fractions, Decimals and Percentages (Years 5 and 6)

Percentages, fractions and decimals Percentages, fractions and decimals are related. They are all equals parts of a whole. 1.

Write the fractions as percentages. (a)

20 100

(b) 100 =

(e)

49 100

(f)

6 100

(f)

(d) 100 =

(g) 100 =

(h) 100 =

(d) 29% = 100

25% =

100

(g) 93% = 100

(h) 50% = 100

100

Colour match each percentage to its simplest fraction. 10%

25%

40%

50%

75%

80%

100%

3 4

1 whole

1 10

4 5

1 4

1 2

2 5

Write the decimal numbers as percentages. (a) 0.4 =

(b) 0.17 =

(d) 0.08 =

% (g) 0.75 = % (h) 0.36 = ©0.28 R=. I . C. Pu bl i cat i o ns Write the percentages as decimal numbers. •f or r evi ew p(c)ur poseso(d)nl y• (a) 29% = 0. (b) 84% = 0. 14% = 0. 52% = 0.

(e) 0.46 =

(e) 63% = 0. 6.

100

(e) 80% =

(b) 18% = 100

(c)

Change the percentages to fractions. (a) 40% =

12 100

(f)

45% = 0.

(g) 3% = 0.

% %

(h) 39% = 0.

Complete the chart filling in the missing decimals, fractions and percentages. The first one has been done for you. Fraction

(a)

50 100

Simplified fraction

5 10

or 2

Decimal

Percentage

0.5

50%

(b) 100 (c)

0.9 10

(d) 100 6 10

(e) 80 100

(f) (g)

0.2

Going further

Suggest at least three different fractions for the percentages. (a) 50% 96

(b)

25%

Fractions, Decimals and Percentages (Years 5 and 6)

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Working with percentages 10% is the same as 1.

1 10

or 0.1.

Calculate 10% or 10 of the amounts of money. For example, 10% of $3.00 = 0.30c or 10% of $6.70 = 0.67c (a) 10% of $1.00 =

(b) 10% of $5.00 =

(d) 10% of $2.50 =

(e) 10% of $9.00 =

(f) 10% of $15.00 =

(g) 10% of $3.70 =

(h) 10% of $26.00 =

(i) 10% of $64.00 =

What do you notice about your answers and how they relate to the amount?

Calculate the discount and new cost of the sales items.

(a)

Total cost before sale

10% off

Total cost after sale

(b) 1 t-shirt and 1 pair of socks (c) 1 dress (d) 1 t-shirt and 1 cap (e) 1 skirt (f)

1 pair of shorts and 1 t-shirt

(g) 1 dress and 1 pair of socks (h) 1 pair of shorts, 1 hoodie and 1 pair of socks (i)

1 skirt, 1 t-shirt and 1 cap

(j)

1 of each item

Going further

Choose at least three items to buy at the sale and calculate your total discounted cost. R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Finding percentages of amounts using fractions When working out the percentage of an amount of money, it can help to first relate the percentage to a fraction. For example: 50% of $10.00 is the same as saying 1 2 of $10.00 = $10.00 ÷ 2 = $5.00 1.

Work out 50% or

1 2

of the amounts.

(a) $4.00 =

(b) $0.80 =

(d) $12.00 =

(e) $15.00 =

(f) $16.50 =

(g) $24.60 =

(h) $50.00 =

Use your knowledge of equivalent fractions to change the percentage to a fraction, to help you find the percentage of the amount. For example: 40% of $20.00 =

4 10

of 20 = (20 ÷ 10 × 4) = 8

(a)

20% of $10.00 = 2 10 of 10 = $

(b)

30% of $25.00 = 3 10 of 25 = $

(c)

10% of $75.00 = 1 10 of 75 = $

(d)

40% of $80.00 = 4 10 of 80 = $

(e)

60% of $30.00 = 6 10 of 30 = $

(f)

50% of $40.00 = 5 1 10 or 2 of 40 = $

(g)

30% of $90.00 = 3 10 of 90 = $

(i) 10% of $120.00 = 1 10 of 120 = $ 3.

8 10

(j) 90% of $200.00 = 9 10 of 200 = $

Solve the farm problems by finding the percentage of the amounts using the method above. The first one is done for you. Problem

(a) Find 70% of 200 sheep

Number sentence and answer

7 10

of 200 = 140

(b) Find 40% of 90 cows (c) Find 20% of 300 chickens (d) Find 30% of 120 goats (e) Find 50% of 60 horses (f)

Find 60% of 80 hay bales

(g) Find 10% of 130 pigs Going further

If you bought a top that was $40.00 and you got it on sale for $28.00, what percentage was taken away from the original amount? 98

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Finding percentages of amounts Another method of finding the percentage of an amount is to find 10% then multiply 10% by the percentage amount. For example: 40% of $50.00 Think: 10% of $50.00 = $5.00, then 10% × 4 = 40% so $5.00 × 4 = $20.00. Therefore 40% of $50.00 = $20.00 1.

Fill in the table using this method. 10%

20%

30%

40%

50%

60%

70%

80%

90%

(a) $10.00 (b) $40.00 (c) $25.00 (d) $60.00 (e) $90.00 (f)

$120.00

(g) $200.00 2.

Work out the percentages using this method. (a) 20% of 200 people = (d) 30% of 500 people =

(b) 40% of 300 people = (e) 70% of 400 people =

A department store is having a sale. Use the method above to work out the amount you need to take off and the new price for these items. The first one is done for you. Item and cost

(a) Doll $18.00

Percentage off

New price

30% of $18.00 = $5.40

$18.00 - $5.40 = $12.60

(b) Block of chocolate $4.00 (c) Quilt cover $60.00 (d) Note book set $12.00 (e) Nerf gun $20.00 (f)

Birthday card $5.00

(g) Bag of snakes $3.50 Going further

How much money would you save if you purchased all the 7 items in question 3 at the percentage off sale? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

Name:

Date:

Assessment 1 1.

Complete the equivalent fraction sentences. 6 8

(a)

= 4

3 15

(e)

(b)

(f) 5

5 10

4 6

(g) 3

4 12

4 4

(d) 3

(h) 1

Write your question 1 answers in order from smallest to largest.

Change the improper fractions to mixed numbers. For example, 5 4

(a) 4.

(b)

(c)

23 5

(b) 4 2 =

4 6

(b)

5 10

(c)

6 8

2 12

= 34.

17 2 = 1 15 72 = 2 5

(d)

(d) 3 12 =

Simplify the fractions to their lowest form. For example: (a)

Change the mixed numbers to improper fractions. For example, (a) 5 4 =

10 3

15 4

5 20

10 12

5 6

(d)

4 12

Add the fractions. If the denominators are different, use your knowledge of equivalence to change them to the lowest common denominator before adding them. 2 4 2 2 4 1 For example: 3 + 6 = 3 + 3 = 3 or 1 3 . (a)

1 2

1 4

(b)

3 8

5 8

(c)

1 3

2 6

(d)

3 5

8 10

(e)

3 4

5 8

(f)

3 6

2 4

= = =

Subtract the fractions. If the denominators are different, change them to the lowest 5 2 5 1 4 common denominator before subtracting them. For example: 6 – 12 = 6 – 6 = 6 or (a)

9 10

(c)

7 8

–

3 4

(e)

1 2

–

1 4

–

4 10

(b)

7 12

–

4 12

(d)

11 12

–

3 6

(f)

3 3

–

Find the fractions of whole numbers. For example:

2 3.

= =

8 12 = 2 3 of

21 = 14 (21 ÷ 3 x 2).

(a)

1 4

of 28 =

(b)

1 9

of 81 =

(c)

1 6

of 72 =

(d)

1 2

of 34 =

(e)

2 3

of 12 =

(f)

3 4

of 24 =

(g)

2 5

of 45 =

(h)

5 6

of 36 =

Going further

Write answers and a story problem to match each sentence. (a) 100

3 4

–

1 2

(b)

6 8

1 2

Fractions, Decimals and Percentages (Years 5 and 6)

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Name:

Date:

Assessment 2 1.

Write the fractions as decimals and percentages. For example: 10 = 0.6 = 60%. 3

(a) 10 = 0. 8

(d) 10 = 0. 75

(g) 100 = 0. 2.

5.32

% (c) 10 = 0.

% (e)

5 10

= 0.

% (f)

% (h) 100 = 0.

= 0.

% (i) 100 = 0.

7 10 16

0.6

7.89

1.35

9.75

0.528

2.4

3.67

0.989

4.136

(d)

(e)

(f)

43.86 39.51

0.004

89.16 +

73.39

653.24 +

227.90

Subtract the decimals. Rename where needed.

94.62 –

(e)

52.81

452.85 –

(f)

139.51

62.790 –

36.437

Multiply the decimals. Remember to remove the decimal point first and add the number of decimal places in both numbers so you know where to put the decimal point in your answer. For example: 4.2 × 0.2 (42 × 2 = 84) = 0.84. (a) 0.05 × 1.2 =

(b) 2.6 × 0.03 =

(d) 0.8 × 0.6 =

(e) 8.1 × 0.4 =

(f)

0.05 × 1.4 =

Use a calculator to divide the decimal numbers by whole numbers. (a) 27.9 ÷ 3 =

(b) 5.08 m + 2.61 m =

(d)

= 0.

(a) 6.12 cm + 3.25 cm =

(a)

9 10

Add the decimal numbers. Regroup where needed.

% (b)

Write the decimal numbers in order from smallest to largest. 0.859

(b) 284.6 ÷ 2 =

(d) 568.4 ÷ 4 =

Find the percentage of the amounts. (a) 20% of $6.00 =

(b) 10% of $95.00 =

(d) 30% of $150 =

(e) 50% of $480 =

(f)

70% of $600 =

Going further

If you purchased a bag for $15.00, a wallet for $30.00 and a pen for $5.00 and there was 20% off, how much would it cost you and how much would you save? R.I.C. Publications®

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Fractions, Decimals and Percentages (Years 5 and 6)

101

Checklist 1

Year 6

Compare fractions with related denominators and locate and represent them on a number line (ACMNA125) Solve problems involving addition and subtraction of fractions with the same or related denominators (ACMNA126) Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (ACMNA127) Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128) Name

Compares, and locates equivalent fractions

Adds and subtracts fractions with same or like denominators

Finds a fraction of a whole number

Adds and subtracts decimals

102

Fractions, Decimals and Percentages (Years 5 and 6)

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Checklist 2

Year 6

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129) Multiply and divide decimals by powers of 10 (ACMNA130) Make connections between equivalent fractions, decimals and percentages (ACMNA131) Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)

Name

Multiplies decimals

Multiplies and divides decimals by powers of 10

Understands fractions, decimals and percentages

Calculates percentages discounts

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Fractions, Decimals and Percentages (Years 5 and 6)

103

Answers Naming and representing fractions ..... page 67 2

1. (a) 6

(b) 8

(e) 10

(f) 5

(g) 4

2. (a) 5 11 (d) 12

1. (a) n – counting by 1’s, d – counting by 2’s

(b) n – counting by 1’s, d – counting by 3’s

(h) 2

4 6

(e) three-tenths

(f) four-quarters

Comparing and ordering fractions ....... page 68 1. (a) smaller than

(b) the same as

(d) smaller than

(e) the same as

(f) larger than

(b) 12 , 16 , 20 , 24 , 28 , 32 (c) 15 , 20 , 25 , 30 , 35 , 40

1. (a) 6 or 3

(b) 4 or 2 5

(d) 10 or 2

(d) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 1

(b) 3

2 1

(d) true

(e) true

(f) false

(g) true

(h) false

(i) false

(g) 3

(h) 5 2

(i) ÷ 12 = 3

(j) ÷ 6 = 3

(k) ÷ 3 = 8 2

(d) 4

(f) 3

(l) ÷ 9 = 9 7

4. (a) 4, 5

(e) 3

(b) true

3. (a) 4

1. (a) false

(b) 1, 2, 3, 4, 6, 8, 12, 24

Equivalent fractions .............................. page 69

2. (a) 1, 2, 3, 4, 6, 12

Going further – They are the same/equivalent because they are equivalent 1 fractions. Altogether they ate 2 4 pizzas.

1 1 3 1 7 3 (b) 12 , 4 , 8 , 2 , 12 , 4 2 4 6 8 10 (c) 10 , 10 , 10 , 10 , 10 1 1 1 1 1 1 (d) 10 , 9 , 8 , 7 , 6 , 5 1 1 2 6 5 (e) 6 , 4 , 4 , 8 , 6 , 1 2 1 6 2 11 3 (f) 12 , 3 , 12 , 3 , 12 , 3

(h) 27

Simplifying fractions ............................. page 71

4. (a) 8 , 8 , 8 , 8 , 8 , 8

(g) 60

Going further – Answers will vary. Teacher check

3. The larger the denominator the smaller the fraction.

2. (a) 6 , 9 , 12

(d) 48

(d) 24 , 32 , 40 , 48 , 56 , 64

1 1 1 1 1 8, 6, 4, 3, 2

(f) 64

(h) 12

4. (a) 9 , 12 , 15 , 18 , 21 , 24

Going further – Answers will vary. Teacher check

(d) 4

(g) 10

(b) 25

(e) 18

4.–5. Teacher check

(f) 12

3. (a) 12

(d) one-half

(b) 10

(e) 8

(b) two-fifths

2. (a) 6

1 (e) 4

3. (a) three-eighths

Equivalent fraction patterns ................. page 70

(d) 3

(b) 6

Year 6

(b) 5, 8 7

(d) 3, 10 4

(e) 10, 11 (f) 4, 7 (g) 8, 9 (h) 9, 5 Going further – Answers will vary. Teacher check

(b) 4 , 6 , 8 , 10 , 12 6

(d) 10

(e) 3 , 9 , 12 9

(f) 2 , 3 , 4 , 5 , 6 , 8 , 9 , 10 , 12 3. (a) 2 , 4 , 6 , 8

(b) 3 , 6 , 9 , 12

Going further – 15 Teacher check 104

Fractions, Decimals and Percentages (Years 5 and 6)

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Answers Improper fractions and mixed numbers ...................................... page 72 9

1. (a) 4 or 2 4

(b) 6 or 1 6

(d) 3 or 2 3

1 3

(e) 6 2

1 23

(d) 4

(g) 6 9

(h) 10

(d) 2 + 2 = 2 or 1 1

(d) 12

(f) 5

(b) 3

(e) 12

Going further – 15 whole oranges

4 8 12 16 20 24 60 ( 4 , 4 , 4 , 4 , 4 , 4 up to 4 )

Adding fractions with common denominators ........................ page 74 4

8 2. (a) 8 or 1 9 1 (d) 6 or 1 2 3 1 (g) 2 or 1 2 8 (i) 4 or 2

(b) 8

(j) 5 or 1 5

1. (b) 10

33 24 18 12 9 (d) 6 , 6 , 6 , 6 , 6

(d) 6

(h) 6 or 3 6

(k) 12 or 2

(f) 2 or 1 6

(i) 8 or 4 5

(l) 5 or 1

Subtracting fractions with different denominators ......................... page 77

(b) 12 , 12 , 12 , 12

(e) 12 or 2

Going further – Answers will vary. Teacher check

4. (a) 10 , 10 , 10 , 10 , 10

1. (a) 4

(d) 8

(f) 4 + 4 = 4 or 1

Subtracting fractions with common denominators ........................ page 76

(g) 4 or 2

(b) 5 + 5 = 5 or 1

(e) 6 + 6 = 6 or 1 6

(e) 4

(f) 12 + 12 = 12 or 1 12

(d) 5 + 5 = 5

2. (a) 2

(b) 2 , 1 2 , 2 2 , 3 1

1. (b) 3

9 11 12 , 12 10 10

3. (a) 3 , 3 , 1 3 , 1 3 , 2 3 , 2 3 , 3 1

2. Teacher check 1

Going further – 4 + 4 + 4 + 4 = 4 or 1 2

Counting by fractions ............................ page 73 7 12 , 8 10 ,

(b) 6 + 6 = 6 or 1 6

3. (a) 3 + 3 = 3 or 1 3

Going further – 2 , 4 , 6 , 8 , 10 etc.

2 3 5 8 1. (a) 8 , 8 , 8 , 8 1 3 6 (b) 12 , 12 , 12 , 3 4 5 (c) 10 , 10 , 10 ,

(e) 6 + 6 = 6

(f) 6

(f) 12 3

(h) 5 12 or 5 4

(b) 2

(e) 8

(d) 10

(g) 3 9 or 3 3 3. (a) 4

(b) 4 or 1

2. (a) 8 + 8 = 8

(b) 3 5

(d) 6 3

1. (a) 8 2

(f) Teacher check, 3 2 1

Adding fractions with different denominators ......................... page 75

(e) Teacher check, 2 5

2. (a) 2 4

Year 6

(e) 5 6 1 3 (b) 4 or 1 2 (c) 3 or 1 9 3 8 3 (e) 12 or 4 (f) 5 or 1 5 12 2 1 (h) 10 or 1 10 or 1 5 9 17 5 (j) 10 (k) 12 = 1 12

2. (a) 6 4

(d) 6 1

3. (a) 4 7

(d) 10

(d) 3

(f) 9 1

(b) 3

(e) 12 4

(f) 10 2

(b) 8 or 4 or 2 1

(e) 3

(f) 6

Going further – Find the LCD 18 8 10 5 24 – 24 = 24 or 12

Going further – Teacher check

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105

Answers Adding and subtracting mixed number fractions ................................... page 78 3 1. (a) 3 4 2 2. (a) 1 3

4 (c) 3 5 3 (c) 1 5

(b) 3 (b) 2

(d) 4 1 (d) 1 4

3. (a) 5 5

(d) 5 8 4

(b) 4 3

(e) 6

(f) 4 12

(g) 7 5 1

4. (a) 1 5 2

(i) 6 10 or 6 5

(b) 1

(d) 1 6 or 1 3

(h) 4 6 1

(h) 4 4 1

3 3

(e) 5 2. (a)

3 10 7 10 1 8 3 12 1 3 2 3

7 8 9 12

(c)

1 5

2 12

1 12 8 12

2 8 1 6

2 3 12 10

2 8

1 8

5 12

6 8

4 6

2 8 1 12

1 2

1 4 1 6

1 2

1 10

1 8

4 10

3 8

1 2

2 34 1 34

1 4

4 34 1 4 2 4

5 20

4 8 5 12

1 4 2 6

6 8 8 12

10 20

(f) 0.92

(g) 0.40

(h) 0.65

215

(b) 1000 2

(e) 1000

176

(f) 1000

(h) 1000 (b) 0.64

(d) 0.05

(e) 0.538

(f) 0.4

(g) 0.092

(h) 0.12

Going further – If the zero is removed they mean 4

the same, e.g. 0.4 = 10 , 400

(b) 8

(d) 12

(e) 8

(f) 11

(i) 8

(j) 15

(k) 4

(l) 20

(b) 18

(d) 12

(e) 18

(f) 6

(g) 70 (h) 40

(i) 16

(j) 45

(k) 50

(l) 48

3. (a) 50

(b) 25

4. (a) $8

(b) $8

5. (a) $80

(b) $30

(d) $20

(e) $40

(f) $10

2. 0.42, 2.584, 80.371, 378.953, 2657.001, 5368.95, 8035.932, 32 624.5, 56 094.26, 142 670.99 3. (a) 5 tens of thousands + 3 thousands + 7 hundreds + 6 tens + 2 ones + 9 tenths + 5 hundredths (b) 8 hundreds + 0 tens + 2 ones + 3 tenths + 7 hundredths + 1 thousandths (c) 1 hundred of thousands + 2 tens of thousands + 5 thousands + 7 hundreds + 0 tens + 2 ones + 6 tenths

Fractions of whole numbers ................. page 80

2. (a) 8

(e) 0.14

1. Teacher check

2 6

3 4

(a) 4 or 2

(g) 12 (h) 5

(d) 0.51

8 10

(b) 9

Going further – Teacher check

1. (a) 7

(b) 0.85

(h) 10

Place value and decimals ...................... page 82

1 6

(d)

1 4 1 2

3. (a) 0.18

(d) 10

0.40 = 100 or 0.400 = 1000

1 4

3 4

(h) 0.5

4 (c) 10 3 (g) 10

5. (a) 0.9

(d) 3

(b)

4 5

5 10

(g) 6 or 3

3 4 7 8

(g) 0.3

2 (b) 10 1 (f) 10

861

(f) 4

(f) 0.1

7 2. (a) 10 8 (e) 10

(g) 1000

1 2

(e) 0.8

(b) 1 whole

1 2

(d) 0.9

903

(d) 4 4 or 4 2

(d) 1000

Addition and subtraction fraction problems .................................. page 79 1. (a) 5

(b) 0.2

(b) 4

1. (a) 0.7

4. (a) 1000

(i) 3 12

Going further – (a) 3 4

Fractions and decimals ......................... page 81

(e) 3 10 or 3 5 (f) 2 2

(g) 3 8 or 3 4

Year 6

(d) 6 hundreds of thousands + 0 tens of thousands + 5 thousands + 4 hundreds + 9 tens + 8 ones + 2 tenths + 9 hundredths Going further – 604 695.919

6. (a) $400 (b) $300 (c) $100 (d) $200 (e) $0 Going further – Answers will vary. 106

Fractions, Decimals and Percentages (Years 5 and 6)

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Answers Comparing and ordering decimal numbers ................................... page 83

Year 6

Estimating and rounding decimal numbers ................................... page 86

1. (a) 6.073, 6.374, 6.473, 6.704, 6.734

1. (a) 7

(b) 3

(d) 6

(b) 31.45, 41.75, 45.31, 54.65, 65.54

(e) 4

(f) 2

(g) 10

(h) 1

(i) 56

(j) 94

(k) 133

(l) 452

(d) 0.596, 0.659, 0.695, 0.965, 0.995

(m) 34

(n) 85

(o) 93

(p) 362

(e) 3125.1, 3125.4, 3125.5, 3125.8, 3125.9 2. (a) $2.35 (e) $4.00

(b) $2.00

(f) $5.75

(g) $1.20

3. (a) 0.45 secs

(b) 1.19 secs

(d) 0.98 secs

(e) 0.94 secs

(d) $1.90

Going further – Teacher check

2. (a) 359.17

(d) 9 + 4, 13, 12.7

(e) 9 + 7, 16, 16

(f) 1 + 1, 2, 1.4

(g) 9 + 5, 14, 13.5

(h) 25 + 22, 47, 47.5 (c) 10 – 6, 4, 3.6

(d) 16 – 4, 12, 11.9

(e) 36 – 12, 24, 23.1

(f) 68 – 34, 34, 33.8

(g) 86 – 35, 51, 51.1

(h) 96 – 53, 43, 43.4

Adding decimal numbers...................... page 84 (d) 23.8

3. (b) 4 – 1, 3, 2.7

4. 1.41 m, 1.43 m, 1.45 m, 1.50 m, 1.56 m, 1.57 m, 1.58 m, 1.59 m, 1.61 m, 1.64 m

1. (a) 5.7

2. (b) 4 + 4, 8, 8

(b) 12.6

(e) 6.87

(f) 8.979

(b) 6197.518

Going further – (a) 8.8

(b) 9.0

(d) 7.6

Multiplying decimal numbers .............. page 87 1. (a) 20.4

(b) 27.3

(d) 31.2

(e) 34.8

(f) 135.9

(g) 65.0

(h) 437.6

(i) 195.69

(j) 462.70

(k) 528.22

(l) 328.16

2. (a) 0.55

(b) 0.91

(d) 0.32

(e) 0.75

(f) 0.49

(g) 1.40

(h) 1.04

3. (a) 53.00

(b) 93.87

(d) 90.800

(e) 808.07

(f) 8588.11

(g) 91.581

(h) 2670.659

4. (a) 11.95 m

(b) $22 097.00

Going further – Teacher check

Subtracting decimal numbers .............. page 85 1. (a) 0.33 (e) 3.244

(b) 5.3

(d) 11.02

3. (b) 0.068

(d) 0.318

(f) 6.24

(g) 0.252

(h) 4.411

(e) 0.320

(f) 0.2136

(g) 0.0870

(h) 0.4884

(i) 0.3647

2. (b) 472.417

3. (a) 51.16

(b) 40.51

4. (a) $23.00

(d) 74.216

(e) 261.48

(f) 4230.97

Going further – 5 places, 0.03204

(g) 21.151

(h) 5908.228

4. (a) $14.87

(b) 7.55 m

(b) 2.04 m2

Dividing decimal numbers .................... page 88 (c) 1.07 m

Going further – Teacher check

1. (b) 0.6

(d) 4.1

(e) 2.0

(f) 0.9

(g) 2.7

(h) 14.1

(i) 12.1

2. (a) 4.1

(b) 8.1

(d) 6.1

(e) 8.1

(f) 28.2

(g) 98.2

(h) 59.8

(i) 90.1

3. (a) 3.2 m

(b) 1.5 m

(d) 19.1 m

Going further – Answers will vary. Teacher check 630.7 ÷ 7 = 90.1

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107

Answers Solving decimal problems – exchange rates ....................................... page 89 1. (a) EUR140

(b) NZD555.00

Year 6

Decimals and powers of 10 – dividing ....................................... page 92 1. (a) 84.21, 8.421, 0.8421

(d) USD348

(b) 174.86, 17.486, 1.7486

(e) FJD501

(f) NZD832.50

(g) EUR700

(h) USD783

(d) 9.505, 0.9505, 0.09505

(b) $93.98

(e) 70.285, 7.0285, 0.70285

(d) $1142.86

(f) 0.58, 0.058, 0.0058

(e) $1149.43

(f) $225.56

(g) $1801.80

(h) $2142.86

2. (a) $315.32

Going further – Answers will vary. Teacher check

2. The decimal place moves to the left with each zero, 2 places for 100 and 3 places for 1000. 3. (a) 6.28, 0.628, 0.0628 (b) 51.29, 5.129, 0.5129

Solving decimal problems involving money .................................... page 90 1. (a) $12.00, $10.00, $2.00 (b) $3.30, $3.00, 30c (c) $5.60, $5.00, 60c

(d) $3.60, $3.00, 60c

(e) $2.55, $2.00, 55c

(f) $5.50, $4.50, $1.00

2. (a) $2.25 (d) $1.25

(b) $2.50

(e) 50c

(f) 66c

(c) 184.36, 18.436, 1.8436 (d) 7.585, 0.7585, 0.07585 (e) 0.842, 0.0842, 0.00842 (f) 0.31, 0.031, 0.0031 4. (a) 0.03245 mins

(b) 68.5c or 69c

(e) $14.20

Going further – To entice people to buy more and for shoppers to save money.

of 10 – multiplying ................................. page 91 1. (a) 5.82, 58.2, 582 (c) 47, 470, 4700

(b) 13.96, 139.6, 1396 (d) 160.8, 1608, 16 080

Going further – moves 4 decimal places to the left (0.0005), then 5 places (0.00005), then 6 places (0.000005)

Powers of 10 ........................................... page 93 1. (a) 10 × 10 = 100 (b) 10 × 10 × 10 = 1000

(e) 736.29, 7362.9, 73 629

(f) 1281, 12 810, 128 100

(d) 10 x 10 x 10 x 10 x 10 = 100 000

2. The decimal point moves 1 place to the right with each 0 or multiple of 10, 2 places for 100 and 3 places for 1000.

(e) 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 2. (a) 200 (d) 15 000

3. (a) 7.41, 74.1, 741

3. (a) 120

(b) 18.45, 184.5, 1845

(d) 60

(b) 9000

(e) 800 000

(f) 2300

(b) 3400

(e) 70

4. (a) 80, 800, 8000, 80 000

(e) 327.0, 3270, 32 700

(b) 5, 50, 500, 5000

(f) 5118.9, 51 189, 511 890

4. (a) 3.4 minutes (c) $143.00

(b) $27.00 (d) $28 750.00

(e) $6.50, $3.50 change

(d) 0.03, 0.3, 3, 30 Going further – You would multiply that number by itself 3 times 73 = 7 × 7 × 7 = 343

Going further – The decimal place would move to the left 1 place with each 0 or multiple of 10. 108

Fractions, Decimals and Percentages (Years 5 and 6)

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Answers Equivalent calculations ......................... page 94 1. (a) 41, 41, 41

(b) 75, 75, 75

(d) 625, 625, 625

(e) 9124, 9124, 9124 2. They all equal the same number. 3. (a) 9.58, 9.58, 9.58

(b) 0.64, 0.64, 0.64

1. (a) 10c

(b) 50c

(d) 25c

(e) 90c

(f) $1.50

(g) 37c

(h) $2.60

(i) $6.40

2. Answers may include: 10% of an amount always contains the first two numbers of that amount.

(b) $7.50, 75c, $6.75

(e) 0.7023, 0.7023, 0.7023

4. Answers will vary. Teacher check

(d) $13.00, $1.30, $11.70

Going further – Answers will vary. Teacher check

Percentages ............................................ page 95 (b) 60%

Working with percentages.................... page 97

3. (a) $20.00, $2.00, $18.00

(d) 8.5612, 8.5612, 8.5612

1. (a) 40%

Year 6

(d) 80%

(e) $12.00, $1.20, $10.80 (f) $15.00, $1.50, $13.50 (g) $17.50, $1.75, $15.75

2. Teacher check

(h) $32.50, $3.25, $29.25

3. 20%, 30%, 50%, 70%, 80%, 90%

(i) $25.00, $2.50, $22.50

3 4 6 8 10 10 , 10 , 10 , 10 , 10

(j) $72.50, $7.25, $65.25

4. (a) 20%

(b) 50%

(d) 75%

(e) 70%

(f) 25%

(g) 30%

(h) 80%

(j) 35%

(k) 62%

(l) 93%

(i) 40%

Going further – Teacher check

Going further – Answers may include: Parts of a whole number like decimals, e.g. 70% = 0.70

Percentages, fractions and decimals .......................................... page 96 1. (a) 20% (e) 49% 40

2. (a) 100 80

(e) 100

Finding percentages of amounts using fractions ....................................... page 98 1. (a) $2.00

(b) 40c

(d) $6.00

(e) $7.50

(f) $8.25

(g) $12.30

(h) $25.00

2. (a) $2.00

(b) $7.50

(b) 84%

(d) 35%

(d) $32.00

(e) $18.00

(f) $20.00

(f) 6%

(g) 91%

(h) 57%

(g) $27.00

(h) $16.00

(i) $12.00

(b) 100 25

(f) 100

(d) 100 50

(g) 100

(h) 100

3. Teacher check

(j) $180.00 3. (b) 36

(d) 36

(e) 30

(f) 48

(g) 13

Going further – 30%

4. (a) 40%

(b) 17%

(d) 8%

(e) 46%

(f) 28%

(g) 75%

(h) 36%

5. (a) 0.29

(b) 0.84

(d) 0.52

(e) 0.63

(f) 0.45

(g) 0.03

(h) 0.39

7 6. (b) 10 , 0.7, 70% 1 (d) 10 , 0.1, 10% 8 4 (f) 10 or 5 , 0.8, 80%

90 9 (c) 100 , 10 , 90% 60 (e) 100 , 0.6, 60% 20 2 1 (g) 100 , 10 or 5 , 20% 50 5 1 2 Going further – e.g. (a) 100 , 10 , 2 , 4 25 1 3 (b) 100 , 4 , 12 75 3 6 9 (c) 100 , 4 , 8 , 12 R.I.C. Publications®

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109

Answers Finding percentages of amounts ......... page 99 1. (a) $1, $2, $3, $4, $5, $6, $7, $8, $9

(b) 0.9, 90%

(b) $4, $8, $12, $16, $20, $24, $28, $32, $36

(d) 0.8, 80%

(e) 0.5, 50%

(f) 0.7, 70%

(g) 0.75, 75%

(h) 0.89, 89%

(e) $9, $18, $27, $36, $45, $54, $63, $72, $81 (f) $12, $24, $36, $48, $60, $72, $84, $96, $108 (g) $20, $40, $60, $80, $100, $120, $140, $160, $180 2. (a) 40 (d) 150

(b) 120

(e) 280

(f) 400

3. (b) 40c, $3.60

(f) $1, $4

(g) 35c, $3.15

4. 5. 6.

(d) 83.37

(e) 162.55

(f) 881.14

(b) 6.23 secs

(e) 313.34

(f) 26.353

(b) 0.078

(e) 3.24

(f) 0.07

(b) 142.3

(b) $9.50

(e) $240

(f) $420

6. (a) 9.3 (d) 142.1

7. (a) $1.20 (d) $45.00

(d) 4

(e) 5 1 1 1 6, 5, 4, 1 (a) 1 4 23 (a) 4 2 (a) 3 3 (a) 4 2 (c) 3 11 (e) 8 or 5 (a) 10 or 1 (c) 8 1 (e) 4

(b) 7.69 m

(d) 0.48

Assessment 1 ....................................... page 100 (b) 2

3. (a) 9.37 cm

5. (a) 0.060

(e) $6, $14

2. 0.004, 0.528, 0.6, 0.859, 0.989, 1.35, 2.4, 3.67, 4.136, 5.32, 7.89, 9.75

(d) 41.81

(d) $2.40, $9.60

1. (a) 4

(i) 0.16, 16%

4. (a) 12.44 mm

Going further – $39.55

Assessment 2 ...................................... page 101 1. (a) 0.3, 30%

(d) $6, $12, $18, $24, $30, $36, $42, $48, $54

Year 6

(f) 3

1 1 2 3 1 3, 2, 3, 4, 1 1 (b) 3 3 (c) 9 (b) 2 (c) 1 (b) 2 (c)

(h) 6

(d) 8 2

25 (d) 3 3 (d) 4 8 (b) 8 or 1 7 2 (d) 5 or 1 5

(f) 1 whole

1 2

(b) 12 or 4

Going further – price $50, discount $10, cost $40

(g) 1

41 12 1 3

(d) 12 1

(f) 3

8. (a) 7

(b) 9

(d) 17

(e) 8

(f) 18

(g) 18

(h) 30

1 Going further – (a) 4

5 1 (b) 4 or 1 4

Teacher check

110

Fractions, Decimals and Percentages (Years 5 and 6)

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