Number and Algebra (Australian Curriculum): Year 6 - Ages 11-12 by Teacher Superstore

RIC-6090 9.7/626

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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.

Titles in this series: Australian Curriculum Mathematics resource book: Number and Algebra (Foundation) Australian Curriculum Mathematics resource book: Number and Algebra (Year 1) Australian Curriculum Mathematics resource book: Number and Algebra (Year 2) Australian Curriculum Mathematics resource book: Number and Algebra (Year 3) Australian Curriculum Mathematics resource book: Number and Algebra (Year 4) Australian Curriculum Mathematics resource book: Number and Algebra (Year 5) Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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All material identified by is material subject to copyright under the Copyright Act 1968 (Cth) and is owned by the Australian Curriculum, Assessment and Reporting Authority 2013. 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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Internet websites In some cases, websites or specific URLs may be recommended. While these are checked and rechecked at the time of publication, the publisher has no control over any subsequent changes which may be made to webpages. It is strongly recommended that the class teacher checks all URLs before allowing students to access them.

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AUSTRALIAN CURRICULUM MATHEMATICS RESOURCE BOOK: NUMBER AND ALGEBRA (YEAR 6) Foreword Australian Curriculum Mathematics resource book: Number and Algebra (Year 6) is one of a series of seven teacher resource books that support teaching and learning activities in the Australian Curriculum Mathematics. The books focus on the number and algebra content strands of the national maths curriculum. The resource books include theoretical background information, resource sheets, hands-on activities and assessment activities, along with links to other curriculum areas.

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Number and Place Value .................................................... 6–57 • N&PV – 1 Identify and describe properties of prime, composite, square and rectangular numbers (ACMNA122)

– Teacher information ......................... 6 – Hands-on activities .......................... 7 – Links to other curriculum areas ........ 7

– Resource sheets ...............8–18 – Assessment ...................19–22 – Checklist ...............................23

• N&PV – 2

Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)

– Teacher information ..................... 116 – Hands-on activities ...................... 117 – Links to other curriculum areas .... 117

• F&D – 6

– Resource sheets ......... 118–120 – Assessment ............... 121–122 – Checklist .............................123

Multiply and divide by powers of 10 (ACMNA130)

– Teacher information ..................... 124 – Hands-on activities ...................... 125 – Links to other curriculum areas .... 125

• F&D – 7

– Resource sheets ......... 126–130 – Assessment ............... 131–134 – Checklist .............................135

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

– Teacher information ....................... 42 – Hands-on activities ........................ 43 – Links to other curriculum areas ...... 43

– Resource sheets .............44–52 – Assessment ...................53–54 – Checklist ...............................55

Answers ..............................................................................56–57

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Fractions and Decimals ...................................................58–153 • F&D – 1

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

– Teacher information ..................... 136 – Hands-on activities ...................... 137 – Links to other curriculum areas .... 137

– Resource sheets ......... 138–147 – Assessment ............... 148–149 – Checklist .............................150

Answers ........................................................................... 151–153

Money and Financial Mathematics ................................154–163 • M&FM – 1 Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)

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– Resource sheets .............26–34 – Assessment ...................35–40 – Checklist ...............................41

Investigate everyday situations that use integers. Locate and represent these numbers on a number line (ACMNA124)

– Teacher information ..................... 154 – Hands-on activities ...................... 155 – Links to other curriculum areas .... 156

– Resource sheets ......... 157–160 – Assessment ........................161 – Checklist .............................162

Answers .................................................................................. 163

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– Teacher information ....................... 58 – Hands-on activities ........................ 59 – Links to other curriculum areas ...... 60

• F&D – 2

Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and without digital technologies (ACMNA129)

– Teacher information ....................... 24 – Hands-on activities ........................ 25 – Links to other curriculum areas ...... 25

• N&PV – 3

• F&D – 5

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Format of this book....................................................iv – v

– Resource sheets .............61–80 – Assessment ...................81–84 – Checklist ...............................85

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

– Teacher information ....................... 86 – Hands-on activities ........................ 87 – Links to other curriculum areas ...... 87

– Resource sheets .............88–92 – Assessment ...................93–96 – Checklist ...............................97

• F&D – 3 Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (ACMNA127)

– Teacher information ....................... 98 – Hands-on activities ........................ 99 – Links to other curriculum areas ...... 99

– Resource sheets ......... 100–104 – Assessment ............... 105–106 – Checklist .............................107

• F&D – 4 Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128)

– Teacher information ..................... 108 – Hands-on activities ...................... 109 – Links to other curriculum areas .... 109

Patterns and Algebra ....................................................164–194 • P&A – 1 Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

– Teacher information .............164–165 – Hands-on activities ..............166–168 – Links to other curriculum areas .... 169

– Resource sheets ......... 170–179 – Assessment ............... 180–182 – Checklist .............................183

• P&A – 2

Explore the use of brackets and order of operations to write number sentences (ACMNA134)

– Teacher information .............184–185 – Hands-on activities ...................... 186 – Links to other curriculum areas .... 186

– Resource sheets ......... 187–190 – Assessment ............... 191–192 – Checklist .............................193

Answers ..................................................................................194 New wave Number and Algebra (Year 6) student workbook answers ........................................................195–205

– Resource sheets ......... 110–112 – Assessment ............... 113–114 – Checklist .............................115

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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FORMAT OF THIS BOOK This teacher resource book includes supporting materials for teaching and learning in all sections of the Number and Algebra content strand of Australian Curriculum Mathematics. It includes activities relating to all sub-strands: Number and Place Value, Fractions and Decimals, Money and Financial Mathematics, and Patterns and Algebra. All content descriptions have been included as well as teaching points based on the Curriculum’s elaborations. Links to the Proficiency Strands have also been included. Each section supports a specific content description and follows a consistent format, containing the following information over several pages: • teacher information with related terms, student vocabulary, what the content description means, teaching points and problems to watch for • hands-on activities • links to other curriculum areas

• resource sheets • assessment sheets.

• a checklist

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Answers relating to the assessment pages are included on the final page of the section for each sub-strand (Number and Place Value, Fractions and Decimals, Money and Financial Mathematics, and Patterns and Algebra).

The length of each content description section varies.

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(NOTE: The Foundation level includes only Number and Place Value, and Patterns and Algebra.)

Teacher information includes background information relating to the content description, as well as related terms and desirable student vocabulary and other useful details which may assist the teacher.

Related terms includes vocabulary associated with the content description. Many of these relate to the glossary in the back of the official Australian Curriculum Mathematics document; additional related terms may also have been added.

What this means provides a general explanation of the content description.

Teaching points provides © R. I . C.Publ i cat i on s a list of the main teaching points relating to the content description. • f o r r e v i e w p u r p o s e s o nl y• Student vocabulary includes words which

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The proficiency strand(s) (Understanding, Fluency, Problem Solving or Reasoning) relevant to each content description are listed.

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What to look for suggests any difficulties and misconceptions the students might encounter or develop.

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the teacher would use—and expect the students to learn, understand and use—during mathematics lessons.

Reference to relevant pages in New wave Number and Algebra (Year 6) student workbook.

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Hands-on activities includes descriptions or instructions for games or activities relating to the content descriptions or elaborations. Some of the hands-on activities are supported by resource sheets. Where applicable, these will be stated for easy reference.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

FORMAT OF THIS BOOK Links to other curriculum areas includes activities in other curriculum areas which support the content description. These are English (literacy), Information and Communication Technology (ICT), Health and Physical Education (ethical behaviour, personal and social competence) and Intercultural Understanding (History and Geography, the Arts, and Languages). This section may list many links or only a few. It may also provide links to relevant interactive websites appropriate for the age group.

r o e t s Bo r e p ok u S Resource sheets are provided to support teaching and learning activities for each content description. The resource sheets could be cards for games, charts, additional worksheets for class use, or other materials which the teacher might find useful to use or display in the classroom. For each resource sheet, the content description to which it relates is given.

Assessment pages are included. These support activities included in the corresponding workbook. For each assessment activity, the elaboration to which it relates is given. Many of the questions on the assessment pages are in a format similar to that of the NAPLAN tests to familiarise students with the instructions and design of these tests.

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Cross-curricular links reinforce the knowledge that mathematics can be found within, and relate to, many other aspects of student learning and everyday life.

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o c . che e r o t r s super Each section has a checklist which teachers may find useful as a place to keep a record of the results of assessment activities, or their observations of hands-on activities.

Answers for assessment pages are provided on the final page of each sub-strand section.

Answers are also provided for New wave Number and Algebra (Year 6) student workbook. Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Number and Place Value—N&PV – 1

Identify and describe properties of prime, composite, square and triangular numbers (ACMNA122)

RELATED TERMS

TEACHER INFORMATION

Prime numbers

What this means

• A prime number is a natural number greater than 1 that has no factor other than 1 and itself. A prime number can be divided without a remainder only by itself and by 1. There is an infinite number of prime numbers.

• Students can distinguish among different types of numbers. They will need to apply certain criteria, based on the definitions of each type of number (see definitions) in order to classify various numbers.

Composite number

• Students will need to be taught how prime numbers are defined. (See ‘Related terms’.) Once they understand the definition, the students can be asked to identify the prime numbers from a set of given numbers. • Explain how the term ‘not’ is very useful in mathematics. Composite numbers can be thought of as any numbers that are not prime numbers. The numbers remaining after sifting the prime numbers from a set of numbers are composite numbers. • In order to appreciate why square numbers are called ‘square’, students will need to draw the array (rectangle) with the associated dimensions of the square; for example: a 3 x 3 array, a 4 x 4 array, a 5 x 5 array. They will soon notice that the array they draw is always square in shape. • Similar reasoning can be applied to understanding why triangular numbers are called ‘triangular’. Note: Two different triangles can be formed when dots are placed in triangular arrangements.

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Factors

• Factors are whole numbers which divide exactly into another whole number, leaving no remainder. Factors are either composite or prime numbers. Square numbers

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• A composite number is a natural number that has factors other than 1 and itself. There is an infinite number of prime numbers.

Teaching points

• Square numbers are numbers created when a number is multiplied by itself. For example, 16 is a square number because it can be created by 4 x 4.

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• Triangular numbers are numbers which can be shown as an array of dots in a triangular pattern. The top row consists of one dot, and each of the subsequent rows contains one more dot than the row above it. Triangular numbers are formed by the pattern: 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4 etc.

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Student vocabulary prime number composite number

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Triangular numbers

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• Students will need to be taught that the term ‘prime’ can be combined with other terms, especially when solving word problems; e.g. prime factors. • Students do not need to learn various divisibility rules by heart, but they can be the source of interesting investigations.

What to look for • Students who experience confusion with the various terms. • Students who lack fluency with basic facts which, in turn, hinders problem solving.

square number triangular number product factor calculate

See also New wave Number and Algebra (Year 6) student workbook (pages 2–6)

Proficiency strand(s): Understanding Fluency

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

Problem solving Reasoning R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 1

HANDS-ON ACTIVITIES Prime number tester and tricks • Visit <http://www.murderousmaths.co.uk/games/primcal.htm> to find a prime number calculator which allows the students to type in any number to determine if it is a prime number or not. There are other interesting prime number tricks and facts at this website too.

Prime and composite number hunt • Students should play this game in pairs. Provide each student with a copy of the 1–120 grid on page 14. The aim is to be the first to identify all the prime and composite numbers in the grid. Using two different-coloured markers, the students take turns to cross out a composite number (1 point), a prime number (2 points) or ‘pass’ (no points). The player with the most points wins. Be sure to revise the concepts of prime numbers (whole numbers greater than zero with exactly two different factors: one and itself ) and composite numbers (whole numbers greater than zero with more than two different factors).

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Prime or composite game

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• Ask the students to write the numbers 1 to 120 on sheets of paper or card. On separate sheets of card or paper, write ‘Prime’ and ‘Composite’. Place the number cards as a pile in the centre of the table. In pairs, the students take turns to select a number and place it in the correct pile: either ‘Prime’ or ‘Composite’. Discuss placement, if necessary, before playing. Students may need to refer to copies of pages 8 and 9.

Build triangular numbers

• Using a collection of empty cans, ask the students to represent triangular numbers in towers, then explain the formation and numbers created. Join groups, if necessary, to create really large numbers.

Square numbers on the 1–120 grid

• Have the students use the blank multiplication grid on page 13 to colour the answers to 1 x 1, 2 x 2, 3 x 3, 4 x 4 etc. to identify squared numbers to 144.

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• Play ‘What number am I?’ Students orally give clues for others to find the number (prime, composite, squared or triangular). For example, a question could be, ‘I am a prime number between 21 and 30, with a second digit one more than the first. What number am I?’ (23) • Read The murderous maths of everything by Kjartan Poskitt at <http://www.murderousmaths.co.uk/books/MMoE.htm>.

Information and Communication Technology

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• Visit <http://teachers.henrico.k12.va.us/math/ms/C1Files/01NumberSense/1-6Prime.html> to learn more about prime and composite numbers, and prime factorisation. • Visit <http://www.oswego.org/ocsd-web/games/spookyseq/spookysqno.html> to play a game to find the missing square number.

Science

• Visit <http://www.murderousmaths.co.uk/cicadas.htm> to read interesting information about how cicadas use prime numbers to survive.

History and Geography • Visit <http://www.murderousmaths.co.uk/books/MMoE/erat.htm> to find out the method the Greek mathematician, Eratosthenes, used to find prime numbers up to 120.

The Arts • Use dotted or squared grid paper to illustrate square or triangular numbers, then use those to create an artwork. Refer to pages 16, 17 and 18.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Prime numbers to 120 (shaded)

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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 NOTE: 1 is neither a prime number or a composite number. A prime number has two positive divisors—itself and 1. 1 has only one positive divisor ... 1! 1 cannot be written as a product of two factors, so it is not a composite number either. 8

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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RESOURCE SHEET Composite numbers to 120 (shaded)

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RESOURCE SHEET Square numbers to 144 chart

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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RESOURCE SHEET A list of the first 100 square numbers

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Squares numbers diagrams

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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RESOURCE SHEET Blank multiplication grid to identify squared numbers

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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Sub-strand: Number and Place Value—N&PV – 1

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET A list of the first 100 triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830, 1891, 1953, 2016, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 3321, 3403, 3486, 3570, 3655, 3741, 3828, 3916, 4005, 4095, 4186, 4278, 4371, 4465, 4560, 4656, 4753, 4851, 4950, 5050

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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The position of triangular numbers in Pascal’s Triangle

Assessment 1

Sub-strand: Number and Place Value—N&PV – 1

NAME:

DATE: Prime and composite numbers

1. Which number in each list is a prime number? Shade one bubble. 79

(a)

100

127

145

162

424

390

512

547

943

910

(b) 220

275

307

291

(c)

(d) 751

(e)

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594

736

878

983

(f)

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2. Write your own definition of what a prime number is in the box.

3. Which number in each list is a composite number. Shade one bubble. 57

223

277

163

193

351

307

439

397

512

571

758

(e)

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4. Write your own definition of what a composite number is in the box.

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

128

(b)

(a)

(c)

101

5. Use your calculator to find the composite number answers for these operations with prime numbers. Write an answer in each box.

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(a)

67 x 31

(d)

383 x 449

(g)

829 x 853

o c . che e r o t r s super (b) 157 x 191

(e) 541 x 683

(f) 769 x 997

(h) 499 x 317

(i) 941 x 661

6. Which two prime numbers less than 100 give the product 1517?

and

7. Write one or two sentences to explain the relationship between prime numbers and composite numbers.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 2

Sub-strand: Number and Place Value—N&PV – 1

NAME:

DATE: Square and triangular numbers

1. Which definition is correct? Shade one bubble. Squared numbers are:

numbers obtained when a number is multiplied by itself. large numbers which can be divided evenly by a smaller number.

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quadrilaterals with all sides equal in length and all interior angles are right angles. numbers with three or more factors.

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2. In the box, list the squared numbers in the set. 1, 9, 49, 54, 81, 144, 36, 56, 21, 4, 64, 16, 33, 28

(a) 15

(b) 56

(d) 20

(e) 34

(f) 47

(g) 71

(h) 62

(i) 83

(j) 98

(k) 100

(l) 11

4. Which definition is correct? Shade one bubble.

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numbers obtained when three numbers are multiplied together. numbers with three factors.

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numbers that form a sequence when consecutive numbers are added together. These numbers make an equilateral triangular dot pattern.

5. Write the next four triangular numbers in the sequence. Show your working.

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1+2 =3

1+2+3 =6

1 + 2 + 3 + 4 = 10 (a) (b) (c) (d)

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

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3. Calculate the squared numbers obtained using the following numbers:

Assessment 3

Sub-strand: Number and Place Value—N&PV – 1

NAME:

DATE: Solve problems using prime and composite numbers

1. Write the prime number factors for each number. (a) (b)

prime

composite

prime

composite

71 28

(c)

151

(d)

336

Shade the correct bubble to show whether the number is a prime or composite number.

r o e t s Bo r e p ok u S prime

composite

prime

composite

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2. Find the SMALLEST common prime number factor of each pair of composite numbers. (Cancel out the largest common prime numbers.) (a)

35, 80

(b)

24, 330

(c)

106, 424

(d)

22, 132

(e)

64, 228

(f)

111, 148

14, 49 (c) 40, 60 © R. I . C(b).P ubl i cat i ons (d) 44, 165 (e) 333, 63 (f) 82, 246 • f o r r e v i e w p u r p o s e s o nl y• 4. Find the solutions to the word problems. 12, 10

(a)

Anna, Jake, Jasmine and Farad have decided to make tacos for lunch. At the supermarket, they buy all the ingredients to fill the tacos. Packets of six taco shells are on sale. They have to work out how many packets to buy so that each person gets the same number of tacos. They do not want any left over and want to buy the least amount possible. How many packets will they need to buy? Show your working.

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(b)

A florist is making bunches of assorted flowers. She has 36 carnations, 27 roses and 18 tulips to use. What is the greatest amount of bunches of flowers she can make using the flowers, without having any left over?

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(a)

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CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

3. Find the LARGEST common prime number factor of each pair of composite numbers. (Cancel out the smallest common prime numbers.)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 4

Sub-strand: Number and Place Value—N&PV – 1

NAME:

DATE: Solve problems using square and triangular numbers

1. Which square number is shown by each array? Shade one bubble.

(a)

r o e t s Bo r e p ok u S (b)

(c)

144

121

(a)

(e)

(b)

(d) 78

(f)

(g) 45

(h) 99

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(a)

4. (a)

(b) (c) © R. I . C.Publ i cat i ons 28 45 18 25 15 •f orr evi ew pur posesonl y•

Write the next four triangular numbers in the sequence. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55,

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(b)

Show your working for the first triangular number you had to find.

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5. Quentin, the supermarket manager, wants to set up a display of cans of baked beans in the shop in triangular stacks. He has 105 cans. How will he set up the display? List the number of cans in each row from bottom to top. Draw a diagram of your answer.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Identify and describe properties of prime, composite, square and triangular numbers

Which triangular number is shown by each array below. Shade one bubble.

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2. Use your calculator to find the square of these numbers:

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

Understands that numbers divisible by composite numbers are also divisible by prime factors of that number

Simplifies problems by cancelling common prime numbers

Uses properties of prime, composite, squared and triangular numbers to solve problems Understands that composite numbers are products of prime numbers

Identifies and describes the properties of triangular numbers

Identifies and describes the properties of squared numbers

Identifies and describes the properties of composite numbers

Identifies and describes the properties of prime numbers

STUDENT NAME

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Sub-strand: Number and Place Value—N&PV – 1

Checklist

Identify and describe properties of prime, composite, square and triangular numbers (ACMNA122)

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)

RELATED TERMS

TEACHER INFORMATION

Strategy

What this means

• A strategy is a method of conducting operations; a plan.

• In order to be able to make a selection as to computation method, students need to be proficient with mental methods, written methods and with using calculators. • Students need to be taught that there comes a point when mental methods become inefficient because of the memory demands and calculations that need to be written down. Written calculations provide a record of the calculation procedure. Eventually, written calculations can become tedious and cumbersome when too large and calculators should be used. The point at which a student stops using one method and adopts another will vary depending on a range of factors. However, guidelines need to be put in place; for example, all basic fact calculations should be performed mentally.

Estimate

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Rounding

• Rounding occurs when a real number is approximated up or down to a given place value (for example, the nearest ten) to make estimations. Digital technology

• Digital technology includes devices such as calculators, computers and electronics.

Teaching points

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• An estimate is an approximate judgement or opinion regarding the value, amount, size, mass etc., of something. It is an approximate calculation.

• Students need to be provided with experiences such as ‘Beat the calculator’ where they are exposed to a variety of question types and race against the calculator. Two students are given a basic fact calculation to solve. One is told to solve the problem mentally, while the other uses a calculator. Both will soon learn that it is more efficient to perform mental calculations than use a calculator in certain circumstances; e.g. 47 x 1000. • Students will need to become fluent with an efficient algorithm for calculating with all four operations. Note that many students’ errors when performing a written calculation can be traced back to a lack of fluency with basic number facts or a poor understanding of place value, so these may need to be revised. • Students’ calculator skills are often self-taught and, therefore, they often make inefficient use of the calculator, especially the memory keys. • Students should be encouraged to make and estimate prior to performing any written or calculator-assisted calculation.

Operation

combining numbers or expressions. These include addition, subtraction, multiplication and division.

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Whole numbers

• A whole number is a non-negative integer, such as 0, 1, 2, 3, ... It is most often used as a term to describe any positive integers.

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Student vocabulary calculate

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o c . che e r o t r s super What to look for

• Poor basic fact fluency. • A lack of understanding of place value. • Inefficient and potentially error-prone written methods of calculation. • Poor or inefficient calculator use.

calculator strategy

See also New wave Number and Algebra (Year 6) student workbook (pages 7–12)

solve solution mental written operations

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

whole numbers

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

HANDS-ON ACTIVITIES Mixed operations board game • Provide pairs of students with a 5 x 5 square template to use as a board game. The pairs decide between them which 25 numbers from 1 to 50 to write on the grid. Each student will require a set of coloured counters (different colours for each student) and three dice. The students take turns to throw their three dice and use the numbers to create a number to find on the board. They may use all four operations to create the number. (For example if 6, 6 and 3 are rolled, the student can create 3 x 6 + 6 = 24, 6 – 3 + 6 = 9 or 6 ÷ 6 + 3 = 4.) They then cover that number on the board with one of their counters. The winner is the player who lines up three counters in a row in any direction. Repeat the game with four dice and larger numbers.

Multiples card game

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LINKS TO OTHER CURRICULUM AREAS

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• Give groups of students a pack of number cards from 0–9 (see page 32), paper and a pencil (for scoring). A dealer shuffles the cards and places the pack face down in the centre of the group. The dealer turns over the top three cards. Players can arrange any two of the three chosen numbers up to four times to create any two-digit number that is a multiple of 2, 3, 4, 5, 6, 7, 8 or 9. For example, if the cards with the numbers 6, 8 and 9 are chosen, the players can arrange the digits 6, 8 and 9 to make the numbers 68 (a multiple of 2), 96 (a multiple of 3), 68 (a multiple of 4) and 96 (a multiple of 6). Each correct two-digit number receives a point. The dealer checks all answers using a calculator. The scores are then tallied and the winner is the player who has created the largest number of correct examples. Repeat, using each digit up to five times each or with four cards.

• Ask the students to make up their own word problems using the four operations. • Read the book Numbers. The key to the universe by Kjartan Poskitt from the ‘Murderous maths’ series. Students might also enjoy More murderous maths or Murderous maths: Tricks of the trade by the same author.

Information and Communication Technology

Science

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• Visit <http://teachingtreasures.com.au/maths/mental-maths/yr6-maths-pg3.htm> to complete mental maths questions. • Visit <http://www.bbc.co.uk/schools/ks2bitesize/maths/number/> to practise mental maths online. This site also reviews various strategies for completing mental maths activities. Other worthwhile activities on this site include ‘Operations’ and ‘Using a calculator’. • Visit <http://teachingtreasures.com.au/maths/Maths_more2.html> to find problem-solving activities.

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• Demonstrate the use of mathematics in everyday life. Ask the students to use scientific formulas to solve operations such as the volume of a regular solid (volume = height x length x width), force (force = mass in kg x acceleration) and work (work = force in Newtons x distance in metres).

History

• Have the students research the history of calculators, beginning with the abacus and including the slide ruler, mechanical adding machine, the first four-operations machine invented by Gottfried Leibniz, and ending with digital calculators in use today.

The Arts • The students use the four operations to calculate how much paper is needed to cover an outline of a piece of artwork; how to divide a sheet of paper by folding into correct proportions, or how to draw one-, two- or three-point perspectives.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEETS Strategy explanations

Mental strategies • The jump strategy For example: 89 + 32 = ?; 89 + 30 = 119, 119 + 2 = 121 • The split strategy

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For example: 89 + 32 = 80 + 9 + 30 + 2 = 110 + 11 = 121 • The compensation strategy

For example: 89 + 32 = 90 +32 = 122 subtract 1 = 121

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• Using patterns to extend number facts

Example 1: 8 + 3 = 11, so 80 + 30 = 110 and 800 + 300 = 1100 Example 2: 12 – 7 = 5, so 120 – 70 = 50 and 1200 – 700 = 500 • Bridging the decades

For example: 67 + 93; 67 + 90 = 157, so 67 + 93 = 157 + 3 = 160 • Changing the order of addends to form multiples of ten For example: 69 + 23 + 31 = 69 + 31 + 23 = 100 + 23 = 123

• Estimating (To find an approximate answer rather than an exact answer.) For example: 821 + 97 is equal to around about 900. • Rounding (To round off whole numbers to a given value.)

If the digit is less than 5, round down; for example: 74 rounds down to the nearest 10, 70.

If that digit is greater than or equal to 5, round up; for example: 77 rounds up to the nearest 10, 80.

For example 1:

12 groups of 12 is 144.

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144 shared among 12 is 12.

For example 2:

(12 x 12 = 144) (144 ÷ 12 = 12)

96 ÷ 12 = 8, because 12 x 8 = 96

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• The commutative property of multiplication

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• Linking multiplication and division facts

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For example: 8 x 9 is the same as 9 x 8

• Using known facts to work out unknown facts

For example: 7 x 8 = 56, so 8 x 8 is (7 x 8) + 8 = 56 + 8 = 64 • Factorising

For example: 18 x 5 = 9 x 2 x 5 = 9 x 10 = 90 • Multiplying tens, then units

For example: 7 x 18 = (7 x 10) + (7 x 8) = 70 + 56 = 126 • Repeated addition For example: 30 x 3 = 30 + 30 + 30 = 90

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

The following may assist teachers to provide students with different strategies to solve addition, subtraction, multiplication and division operations with whole numbers.

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEETS Mental strategies (continued) • Distributive law

For example:

4 x 53 = (4 x 50) + (4 x 3) = 200 + 12 = 212

Written strategies • Number lines

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390 – 110 = 280

• Trading

350 360 370 380 390 400

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220 230 240 250 260 270 280 290 300 310 320 330 340

When adding or subtracting two-, three- and four-digit numbers, it may be necessary to ‘trade’ a number to the next place’s value column. For example:

9 ones + 5 ones = 14 ones

hundreds tens ones 10 2 9 +7 6 5 9 7 4 (14)

(Trade 10 ones for 1 ten.) 6 tens + 1 ten = 7 tens 2 hundreds + 7 hundreds = 9 hundreds

Decomposition/Regrouping/Borrowing

This method involves trading one place value unit of a digit for several place value units of the digit to its right. It happens whenever a bottom digit’s value cannot be easily subtracted from the top digit’s value. The regrouping occurs in the larger number from which the smaller number is being subtracted.

1. 1 tens value has been traded to create 13 ones and 5 tens.

763 –345 418

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For example:

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

Distributive law refers to multiplication and addition. It states that the result of adding several numbers together and then multiplying the sum by a given number is the same as first multiplying each separately by the given number and then adding the products together.

2. The larger (top) number has been rewritten so the value has not changed.

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Equal addends

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This method requires the same number (10, 100, 1000 etc.) to be added to both the top digit and the bottom digit in the place value column to its left. As above, it can be done whenever a bottom digit’s value cannot be easily subtracted from the top digit’s value. 1

763 1 –345 418

1. 1 tens value has been traded to the 3 to create 13 ones.

2. As well, 1 tens value has been added to the 4 to create 5 tens.

Extended form (long multiplication) For example:

521 x 22 1042 10420 11462

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEETS Written strategies (continued) Introduced to Europe by Italian mathematician Leonardo Fibonacci in the 13th century, the lattice method can be used as an alternative to long multiplication. It requires the digits in a multiplication operation to be placed along the top and the right side of a lattice (or grid) with a diagonal in each cell. A product is then calculated for each cell by multiplying its intersecting vertical and horizontal digits. In each cell, the tens unit is written above the diagonal line, and the ones unit below the line. Finally, the digits in each diagonal set are added, with the answers to each diagonal operation forming the digits to the answer of the original multiplication problem. For example: 54 x 39.

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Select the appropriate grid—in this case, one with 2 columns by 2 rows (54 and 39 both have 2 digits).

Write the digits along the top and right side as shown—one for each column and row. 5

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For each cell, multiply the intersecting digits. For the top left cell, multiply 5 by 3, giving a product of 15. Write the tens unit (1) above the diagonal, and the ones unit (5) below it. 5

Do the same for the top right cell, multiplying 4 by 3 and writing its product as shown. 5

1 5

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2 3

4 5 6.

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Repeat for 5 x 9 and 4 x 9.

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Now add the diagonals, starting with the ones (or in this case ‘6’), working from right to left across the lattice. For each sum, write its answer below its column on to the left of its row. If for a diagonal the total is more than 9 (e.g. 5 + 3 + 2 = 10). Continue in this way until all diagonals are added. 5 4 1 1 + 2 + 3 thousands 1 + 5 +2 1 +4 + 3 9 6 hundreds 1 5 0 6 tens units/ones

The answer to the original multiplication operation is then read from left to right around the lattice. So, 54 x 39 is 2106.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

• Lattice method of multiplication (Refer to pages 33 and 34 for blank grids for the lattice method.)

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEETS Written strategies (continued) • Area model

In this method, grids of hundreds, tens and ones can be used for accuracy. However, students may find it quicker to simply divide a rectangle into the appropriate number of sections. This method helps reinforce place value. For example: 74 x 28

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1400 + 80 + 560 + 32 = 2072

Step 1:

Split the numbers into parts by place value (hundreds, tens and ones).

Step 2:

Write each part above or next to a cell of the rectangle.

Step 3:

Multiply each number along the top by each number along the right side.

Step 4:

Write the answers in each cell of the rectangle.

Step 5:

Add all answers in each cell of the rectangle to find the final answer of the multiplication problem.

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70 + 4 70 x 20 4 x 20 = 1400 = 80 70 x 8 4x8 = 560 = 32

NOTE: For larger numbers which include hundreds units, divide the rectangle into three columns and rows.

• Russian peasant method

View an example on video at <http://www.metacafe.com/watch/594355/math_the_russian_way/>. Write each number at the top of a column.

Step 2:

Double the number in the first column, and halve the number in the second column. If the number in the second column is odd, divide it by two and discard the remainder.

Step 3:

Continue to double and halve the numbers until the number in the second column is 1.

Step 4:

Cross out any rows of numbers where the number in the second column is even.

Step 5:

Add up the remaining numbers in the first column. The total is the product of the original multiplication operation.

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Step 1:

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

The area model for multiplication is a pictorial representation of multiplication. In the area model, the length and width of a rectangle represent the factors, and the area of the rectangle represents their products.

For example:

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57 x 86

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86 43 (drop remainder) 21 (drop remainder) 10 5 2 (drop remainder) 1

57 x 86 = 4902

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEETS Written strategies (continued) • Egyptian method of multiplication

Using 83 x 27 as an example, the process is: Step 1:

Draw two columns and write each number in order at the top of each column. 83 x 27

Starting at the smallest known factor, write 1 underneath 83, then double it and write 2 underneath 1, and continue to double and write numbers in the first column until you reach the biggest possible number smaller than 83. (Do not write any numbers greater than 83!) So the column will appear like this: 83 x 27 1 2 4 8 16 32 64

In the second column, begin with 27, then double the numbers until you have a similar amount of numbers as those in the first column.

83 x 27 1 27 2 54 4 108 8 216 16 432 32 864 64 1728

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Step 4:

Find numbers in the first column which add up to 83, and circle them. Cross out all the rows which do not have circled numbers in them.

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83 x 27 1 27 2 54 4 108 8 216 16 432 32 864 64 1728

Step 5:

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Step 3:

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Step 2:

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Finally, add all the numbers remaining in the second column. The product of this sum will provide the answer to the original multiplication operation. 27 + 54 + 432 + 1728 = 2241 So, 83 x 27 = 2241.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

The Egyptian method of multiplication involves the use of doubling, knowledge of factors and addition to find the product of two large numbers

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEETS Written strategies (continued) • Partitioning

For example:

428 x 32 = (400 + 20 + 8) x (30 + 2) (400 x 30) + (400 x 2) + (20 x 30) + (20 x 2) + (8 x 30) + (8 x 2) = 12 000 + 800 + 600 + 40 + 240 + 16

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= 13 696 • Arrays

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The area model method could also be considered as an example of an array. As another example, 48 x 26.

10 10 10 10 10

10 10 10 10

10 10

160

960

240

288

1040

208

1248

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48 x 26

12 x 13

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Arrays are sets of objects organised into equal-sized groups. There are often many different arrays possible for the same number. Arrays consolidate multiplication facts and are a good resource for visual learners. They are often used in computer programming. Students could choose to use Cuisenaire® rods and grids to complete arrays for larger numbers. See the example to the right for 12 x 13.

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

Partitioning (breaking numbers up by place value or other criteria) can be used as a written or mental strategy for multiplication.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET

u S

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0 1pers2toreB3oo 4 5 6 7 8 9 w ww

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0 .te1 2 3 .co4 che e r o r st super

5 6 7 8 9 32

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers INSTRUCTIONS: Use for the Multiples card game as instructed on page 25.

0–9 number cards

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Lattice method blank grids

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r o e t s Bo r e p ok u 2. 3-digit x 2-digit S numbers (3 columns by 2 rows)

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers INSTRUCTIONS: Refer to page 28 for instructions for using the Italian lattice method of multiplication.

1. 2-digit x 2-digit numbers (2 columns by 2 rows)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Lattice method blank grids

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Teac he r

r o e t s Bo r e p ok u S 5. 4-digit x 3-digit numbers (4 columns by 3 rows)

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6. 4-digit x 4-digit numbers (4 columns by 4 rows)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers INSTRUCTIONS: Refer to page 28 for instructions for using the Italian lattice method of multiplication.

4. 4-digit x 2-digit numbers (4 columns by 2 rows)

Assessment 1

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

(b) 691 + 729 =

457 + 386 =

(d)

Last year, 76 583 people went to watch the rugby league grand final. This year, 972 more people watched than last year. How many people watched the final this year?

(e)

Kitz, Berki and Praedo are towns in the small country of Fermante. Their populations are 765, 499 and 521. What is the total number of people who live in the three towns?

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(a)

2. Solve the following subtraction problems without a calculator. Show your working in the box. (a) 4000 – 2769 =

(d)

(b) 6572 – 3896 =

(e)

This year, Farmer Jones planted 270 tomato plants. How many more is this than the 180 she planted last year?

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3. Select appropriate strategies to solve the word problems.

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temperature of 42 °C. After a thunderstorm, the temperature falls by 9 °C. What is the new temperature?

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

1. Solve the following addition problems without a calculator. Show your working in the box.

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(a)

In 1783, the first hot air balloon flight took place in France. Two hundred and nineteen years later, Steve Fossett landed in Queensland after flying around the world solo. When was this?

(b)

I have collected 50 football stickers. I gave 6 to a friend, and in return was given 11 different stickers. How many stickers do I have now? By how many has my sticker collection increased?

(c)

Len saved $2350, Jacob saved $3005, Liz saved $2980 and Susie saved $2548 for their overseas holiday. How much money did they save altogether? How much more than their target of $10 000 was achieved?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 2

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

1. Solve the following multiplication problems without a calculator. Show your working in the box.

r o e t s Bo r e p ok u S (e) If there are 1440 minutes in a day, how many minutes are there in a week?

(f) A packet of popcorn weighs 23 g. There are 6 packets in a multipack. Jamal buys 6 multipacks for his party. How many grams of popcorn has he bought?

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(d) If Joe earns $26 per hour, how much money will he earn each fortnight if he works a 38-hour week?

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(d) Thirty people share a (e) How many rows of chairs lottery win of $363 630. If will 420 people need if the prize money is shared there are 28 chairs in a equally, how much does row? each receive?

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2. Solve the following division problems without a calculator. Show your working in the box.

(f) How many days would a 720 mL bottle of medicine last if three 8 mL spoonfuls were taken three times a day?

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3. Select an appropriate strategy to solve the word problem. One hundred and ninety-two chocolates fit exactly into 24 boxes. How many boxes will I need for 280 chocolates?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

(b) 192 x 97 =

(a) 365 x 23 =

Assessment 3

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

Solve the following problems by using more than one operation. Show your working in the box, then state what operations or strategies you used.

Strategies/Operations

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2. I choose a number, multiply it by eight and then add six. The answer is twenty-two. What number did I choose? Strategies/Operations

3. Fifteen boys and seventeen girls were taken to a school sports day in their parents’ cars. If each car can transport four students, how many cars were needed?

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

1. Crates of milk are five milk cartons long and four cartons wide. How many milk cartons can twenty crates hold?

4. A movie theatre has twenty-four rows with fifty seats in each row, and twelve rows with thirty seats in each. How many audience members can be seated altogether?

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Strategies/Operations

5. Boxes of books to be sent to schools each hold fourteen lined exercise books and seven plain exercise books. If a school orders six hundred and seventy-two books altogether, how many boxes are needed? Strategies/Operations

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 4

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

Solve the problems using mental or written strategies. For each problem, write your answer, state your strategy and explain why you chose that strategy instead of another. Rate the efficiency of the strategy by grading it on a scale.

r o e t s Bo r e p ok u S Very efficient

Efficient

Average

2. A woman has overdrawn her savings account by $68. She deposits $129. How much does she have now?

Efficient

Average

Very efficient

Efficient

Average

3. A necklace has 3 red beads for every 4 green beads. If there are 30 red beads, how many green beads are there?

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Answer

4. I multiply a number (N) by four, then add six. The answer is 34. What is N?

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Answer

Fair

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Answer

Fair

Poor

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Answer

Strategy, explanation and efficiency rating

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Poor

Fair

Poor

Very efficient

Efficient

Average

Fair

Poor

Very efficient

Efficient

Average

Fair

Poor

5. Two hundred and fifty-six chocolates are packed into 32 boxes. How many chocolates will 7 boxes hold?

Answer 38

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

Problem 1. A school has 378 students in 14 classes. How many students would be in each class if the classes were equal in size?

Assessment 5

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

1. Three hundred and eighty students attended the school disco where 1140 cold drinks were sold. If each student bought the same number, how many drinks were bought in total by each student?

225

196

900

3. If a coin weighs 25 g, how many coins can be made from 10 kg of metal?

100

400

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2. I have selected a number which is a square number between 200 and 250. What is the number?

4. A bag of sweets contains 23 sweets. How many bags of sweets could be filled using 2484 sweets?

57 132 2461

248

108

5. A seven-year-old student is 136 cm in height. What would be the total height of 24 students?

3264

160

2720

© R. I . C.Publ i cat i on11s 2954 27 •f orr evi ew pur poseso nl y• 464 7. One sheet of A3 paper weighs 4 grams. How many grams 1600 2800 6. An aeroplane holds 254 passengers. How many similar aeroplanes would be needed to carry 2700 passengers?

1880

8. The seats in an assembly hall are arranged in 52 rows, each with 34 seats. How many people can be seated if every seat is filled?

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would a packet of paper containing 470 sheets weigh?

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CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

Use a calculator to solve the problems. Shade the bubble to indicate the correct answer.

520

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9. A movie director is compiling short films for a festival. If the festival lasts for one hour and 44 minutes, and there are 26 short films to be included, at the most, how long can each short film be?

1768

5234

10. How many 25-cm lengths of string can be cut from a roll containing 625 cm?

600

650

11. Three hundred and forty-one children and 58 adults went on a school trip. If the school bus seats 57 people, how many buses were needed?

12. Football fans have bought 6920 tickets to attend a match. If each stand holds 865 people, how many stands are needed to seat all the fans who have purchased tickets? Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 6

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

Use a calculator to find the answers. 1. Find the product or factor. 115 x

= 12 535

(b)

x 756

= 73 332

(c)

391 x 4864

(d)

(e)

268 x

= 19 564

(f)

x 975

= 512 850

(b)

÷ 92

= 432

= 81

(d)

÷ 269

= 663

(f)

84 x

= 47 292

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2. Find the dividend, divisor or quotient.

(c)

57429 ÷

(e)

23 460 ÷ 69

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65 736 ÷ 88

160 648 ÷

= 934

3. Find the addend or sum/total. (a)

8542 + 6931

(b)

(c)

2054 +

= 7904

(d)

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(a)

+ 4912 1318 +

= 8589

= 8441

= 16 511

(f)

(c)

– 8756

= 7682

(d)

(f)

(e)

3094 – 499

4321 + 2365

– 22 380 49 372 –

= 42 789

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5. Find the answers to the word problems.

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(a) People bought 4480 concert tickets.

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(i) If each stand holds 560 people, how many stands were needed?

(ii) If this number of people was 325 more than last year, how many attended last year?

(b) Twenty-four students each paid $146 for a school camp.

= 7911

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(e)

(i)

How much money was collected in total?

(ii)

If the camp costs $4000, how much extra money does the school need to contribute to cover the costs?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

(i)

If 229 students and 31 adults are travelling, how many buses are needed?

(ii)

If buses cost $220 each to hire, how much would it cost to hire enough buses?

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

(a)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Uses digital technology, such as calculators, to solve problems

Selects an appropriate mental or written strategy to solve a division problem

Selects an appropriate mental or written strategy to solve a multiplication problem

Selects an appropriate mental or written strategy to solve a subtraction problem

Explains the efficiency, or otherwise, of a specific strategy

Selects and explains a suitable written strategy to solve a problem

Selects and explains a suitable mental strategy to solve a problem

STUDENT NAME

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Sub-strand: Number and Place Value—N&PV – 2

Checklist

Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)

Sub-strand: Number and Place Value—N&PV – 3

Investigate everyday situations that use integers. Locate and represent these numbers on a number line (ACMNA124)

RELATED TERMS

TEACHER INFORMATION

Positive integers/numbers

What this means

• Positive integers are whole numbers that are greater than zero. They are sometimes called counting numbers or natural numbers. The number of positive integers is indefinite.

• Students learn about negative numbers and use them when solving addition and subtraction problems.

Negative integers/numbers

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Whole number

• A whole number is any integer, either positive or negative. Sometimes the term is used to refer to only a positive integer, as definitions of the term seem to differ.

• Most students will already be familiar with negative numbers. The idea of negative numbers can be developed using a number line. Students should be taught that as the number line extends to the left of zero, the numbers –1, –2, –3 extend (equally spaced) from this point. • Students should be taught to read these numbers as, for example, ‘negative three’ for –3. Often the phrase ‘negative integer’ is used. • When the number line is extended to the left of zero, some students may not realise that negative numbers do not simply refer to whole numbers, so make sure they are shown numbers like –2.5, –1½ etc.

What to look for

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• Negative numbers are whole numbers that are less than zero. Negative numbers are the opposites of the positive numbers; i.e. for each positive integer there is a negative integer. For example, –8 is the opposite of 8.

Teaching points

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representation of real numbers. A number line is labelled with the integers in increasing order from left to right, and can extend in both directions. For any two different whole numbers on a number line, the integer on the right is greater in value than the integer on the left.

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See also New wave Number and Algebra (Year 6) student workbook (pages 13–20)

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

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Number line

• Students who refer to negative numbers as ‘minus numbers’. • Students who confuse the labelling of the number line and think that positive fractions and decimals lie to the left of the zero or that negative numbers are between 0 and 1.

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Student vocabulary positive negative number line zero integer

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

HANDS-ON ACTIVITIES Positive and negative number line • With student assistance, create a number line as shown in the illustration. Use a rectangular base of light wood (or canvas or felt stapled over a curtain rod or similar) and paint a straight line with white paint. (If using material, white electrical tape could be used.) Add arrows to each end to show that the number line continues indefinitely. Use strong woodworking glue to attach wooden pegs at regular intervals along the line. Use squares of stiff coloured card on which to write the numbers—yellow for zero, red for negative numbers and blue for positive numbers. Peg the numbers in position along the number line. • Create extra number cards in green (to do addition and subtraction sums) as well as the + (positive), – (negative) and = signs. Attach reusable adhesive dots to each. An arrow can be used to indicate the final answer. Sums can be positioned along a stand or slit along the bottom of the number line. Plastic sleeves could be used to store number and sign cards.

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Positive and negative coloured manipulatives

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Positive and negative game

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• Provide the students with coloured material (such as white or red beans or coloured counters) to represent positive and negative numbers. Encourage them to complete concise drawings to represent combining positive integers, combining negative integers, and combining positive and negative integers.

• Game instructions, cards, blank dice template, number line and recording sheet are included on pages 47 to 52.

English

LINKS TO OTHER CURRICULUM AREAS

• Ask the students to orally, or as a written activity, create ‘real world’ situations involving adding positive and negative numbers. If desired give the specifics; for example, ‘I owe Mum $4 and Dad $3. In total, I owe $7’.

Information and Communication Technology

Economics

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• Visit <http://www.docstoc.com/docs/1019489/POSITIVE-AND-NEGATIVE-NUMBERS> to read information about positive and negative numbers, number lines, and the everyday uses of positive and negative numbers. This site also includes some simple rules for operations which include positive and negative numbers. • Visit <http://www.free-training-tutorial.com/negative-numbers-games.html> to play some free maths games to reinforce understanding of negative integers.

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• Relate positive and negative numbers to real-life situations, such as saving and spending money. For example: The Pullen Family bought a house. They had saved $84 000 and needed a bank loan of $355 000. How much did they owe the bank.

Science

• Use positive and negative numbers to find out the variations in temperature from one day to the next as recorded over a given time frame (such as a month). • Discuss positive and negative numbers in relation to diving, sea depths and sea life in different zones in the sea. Consider the mathematics involved in diving to a particular depth and swimming back up some distance towards the surface, or diving from one depth to a lower one.

Geography • Locate two major cities in different countries on a map of the world, then compare the difference in current temperatures. • Relate positive and negative numbers to the heights of mountains, cities or valleys above of below sea level. Use the sign – for below sea level, and + for above sea level, with absolute sea level represented by zero.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Horizontal number line

glue

–9

–8

–7

–6

–5

–4

–3

–2

–1

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glue

–10

glue

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–31 –30 –29 –28 –27 –26 –25 –24 –23 –22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11

glue

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–73 –72 –71 –70 –69 –68 –67 –66 –65 –64 -63 -62 -61 -60 -59 -58 -57 -56 -55 -54 -53

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–94 –93 –92 –91 –90 –89 –88 –87 –86 –85 -84 -83 -82 -81 -80 -79 -78 -77 -76 -76 -74

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

–52 –51 –50 –49 –48 –47 –46 –45 –44 –43 -42 -41 -40 -39 -38 -37 -36 -35 -34 -33 -32

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Vertical number line

Teac he r 5

5 4

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r o e t s B r e 6 6 o p ok u S

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CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

1 0

-1 -2

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-1

-2

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Vertical number line (thermometer)

35 30

–5

–10

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10 5

–20 –25

25 20 15

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r o e t s 40 B r e oo p 35 35 u k S 30 30 40

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–15

–5

–10 –15 –20 –25

–30

–35

–40

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Positive and negative game – Game instructions Preparation

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Teac he r

• Photocopy: – the dice template (page 48) onto stiff card, then fold and tape together. Alternatively, prepare a blank dice by writing ‘+’ (positive) and ‘–’ (negative) three times each on the sides of the dice – the number line (page 49) onto card and join each segment together to create the number line game board – the number game cards (page 50) onto card, then cut up and place in a resealable bag – the action cards* (page 51) onto card and cut them out – the student recording sheet (page 52) a number of times for each student. Back-to-back copies may be convenient. Note: 1. Students might also wish to have a copy of the rules for reference.

The action cards come in two different groups: opposites, and positive and negative cards. An opposite card requires a student to move from the number position on the number line where he/she lands, to the opposite number on the number line. For example, if he/she lands on –3, an opposite card means he/she moves to +3. Positive and negative cards require a student to move to the position as indicated on the card. For example, if –3 is drawn, the student moves three spaces to the left. Materials required

Each pair of students will require: a game piece each (such as coins, coloured beans, counters, tokens etc.); one number line; one set of number game cards in a resealable bag; one positive-negative dice; one set of action cards; one or more student recording sheets. Directions

1. Place the shuffled number and action cards upside down as separate piles in the centre.

2. To begin, each player selects a number card. The player who selects the highest numbered card goes first. Each player will commence the game at zero (0). This is the first starting number of the game.

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3. The first player rolls the dice and selects a number card from the pile. The player records the positive or negative number on the sheet and decides where he/she has to move along the number line. He/She then records on the sheet the number he/she lands on. If the player also lands on a space with an A (action card), the player also selects an action card and follows the directions on it. If the player then lands on an ‘A’ again, he/she selects another action card. The player then records the next number landed on (ending number) on the sheet and writes an equation to show what moves were made and the result.

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CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

2. Laminating the materials will ensure durability.

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4. The number card is returned to the pile before Player 2 begins his/her turn. Used action cards should be placed to the side so they are not used again. 5. Continue playing until one player reaches positive or negative 10 along the number line. Extend the length of the number line to larger numbers in either direction, if desired.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

s r e p u S

–

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–

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

Bo ok

– –

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+ or e t

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–

Positive and negative game – Positive and negative dice

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

Positive and negative game – Number line

–1 –2 –3

Teac he r

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5 4

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–7 m –6

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–8

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–4

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CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line INSTRUCTIONS: Photocopy onto card

r o e t s Bo r e p ok u S

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Positive and negative game – Number card games

–1.te –2

–3 o

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–4 50

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–5

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

r o e t s Bo r e p ok u S

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Teac he r

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Positive and negative game – Action cards

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u S

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CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

–3 perst–2 or eBo –1 o

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opposite opposite opposite Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Positive and negative game – Student recording sheet

Student name: Starting number

+ or –

Number drawn

Ending number

Equation

Student name: + or –

Number drawn

Ending number

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Equation

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Starting number

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line INSTRUCTIONS: Enlarge to A3 size and photocopy onto card.

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Assessment 1

Sub-strand: Number and Place Value—N&PV – 3

NAME:

DATE:

Shade one bubble to find answers to positive and negative number problems. 1. (a)

two floors below street level

four floors below street level

two floors above street level

2. (a)

(b)

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You are fifteen floors below street level. You take the lift up ten floors, then up another fifteen floors. Finally, you go down five floors. Where are you? five floors below street level

five floors above street level

twenty floors below street level

ten floors above street level

You are in the basement which is three floors below street level. You take the lift down five floors, then go up eleven floors. Where are you?

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(c)

thirteen floors above street level

nine floors below street level

three floors below street level

three floors above street level

The temperature in Budapest is 2 °C. It drops by three degrees when the sun goes down. What is the temperature at sundown?

–5 °C

5 °C

–1 °C

4 °C

–8 °C

(c)

If the temperature was –2 °C and it rose by three degrees to its current temperature, what is the current temperature?

3. (a)

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degrees. What was the temperature after the cool change?

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CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

fourteen floors above street level

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(b)

You enter a museum building at street level (zero) and take the lift up to the sixth floor. After viewing the exhibits, you take the lift down ten floors. Where are you now?

3 °C

–3 °C

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–1 °C

1 °C

If you owed your friend $15 and paid back $8, how much do you owe now?

$23

(b)

Your friend borrowed $18 from you. He paid back $3, then borrowed $5 more. How much does he owe now?

$20

–$20

$24

(c)

You lent your sister $10. She paid back $2 each week for four weeks. Then she borrowed $7 but paid back $8. How much money does she owe you?

4. North is positive, and south is negative. The school is at zero. After school, you walk seven blocks south to basketball, then eight blocks south to your friend’s house. Later, you take the bus twenty-six blocks north. Where are you in relation to the school? eleven blocks north

seven blocks south

fifteen blocks south

eight blocks north

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 2

Sub-strand: Number and Place Value—N&PV – 3

NAME:

DATE:

1. Write the numbers in their correct positions on the number line.

–10

–5

(a) –8

(b) 2

(d) 4

(e) 7

(f) 1

(g) –1

(h) –3

(i) 9

(j) –2

2. Write the numbers in their correct positions on the thermometer on the right.

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(b) –30 °C

(g) 44 °C

(h) –12 °C

(d) –4 °C

(i) 16

(e) 38 °C

(j) –6 °C

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Death Valley, USA

(c)

Bangalore City, India

(d)

Mt Kosciusko, Australia

2000

1500

Bangalore City, India

metres

(b)

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Dead Sea, Israel and Jordan

°C

2500

Mt Kosciusko, Australia

3. Give the approximate metres above or below sea level of each location as indicated by the scale. Write + for above and – for below sea level before each. (a)

(f) –26 °C

1000 500

0 sea level

Death Valley, USA

–500 © R. I . C.Publ i cat i ons (b) B (c) C •f o rr evi ew pur posesonl y•

4. Write where each letter can be found on the number line.

Dead Sea, Israel and Jordan

(d)

(e)

(f)

(g)

(h) H

(i)

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(j) J

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

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5. Which number in each pair is the larger? Refer to the number line above. (a) –4, 1

(b) –1, 2

(d) 5, 8

(e) 0, –2

(f) –3, –4

(g) –3, –2

(h) –3, 1

(i) 7, 9

(j) 0, 4

6. Which number in each pair is the smaller? Refer to the number line above. (a) 0, –4

(b) –5, 5

(d) –9, 10

(e) –3, 2

7. Order the positive and negative numbers from smallest to largest. (a)

–5, 3, 2

(b) –6, –14, 12

8. Order the positive and negative numbers from largest to smallest. (a) 54

12, 9, –5

(b) –2, –5, 0

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

CONTENT DESCRIPTION: Investigate everyday situations that use integers. Locate and represent these numbers on a number line

(a) 50 °C

Checklist

Sub-strand: Number and Place Value—N&PV – 3

Investigate everyday situations that use integers.

Represents positive and negative numbers on a number line

Locates positive and negative numbers on a number line

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STUDENT NAME

Identifies everyday situations that use positive and negative whole numbers and zero

Locate and represent these numbers on a number line (ACMNA124)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Answers

Sub-strand: Number and Place Value

N&PV – 2

Page 19 Assessment 1

Page 35 Assessment 1

1. (a) 79 (b) 127 (c) 307 (d) 547 (e) 751 (f ) 983 2. Teacher check 3. (a) 57 (b) 128 (c) 351 (d) 512 (e) 758 (f ) 980 4. Teacher check 5. (a) 2077 (b) 29 987 (c) 61 771 (d) 171 967 (e) 369 503 (f ) 766 693 (g) 707 137 (h) 158 183 (i) 622 001 6. 41 and 37 7. Composite numbers are the products of prime numbers.

1. (a) 843 2. (a) 1231 3. (a) 2002

Page 20 Assessment 2

(b) 1420 (b) 2676 (b) 55, 5

Page 36 Assessment 2 1. (a) 8395 (d) $1976 2. (a) 19 (d) $12 121 3. 35

(b) 18 624 (e) 10 080 (b) 38 (e) 15

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1. numbers obtained when a number is multiplied by itself 2. 1, 9, 49, 81, 144, 36, 4, 64, 16 3. (a) 225 (b) 3136 (c) 196 (d) 400 (e) 1156 (f ) 2209 (g) 5041 (h) 3844 (i) 6889 (j) 9604 (k) 10 000 (l) 121 4. numbers that form a sequence when consecutive numbers are added together. These numbers make an equilateral triangular dot pattern. 5. (a) 1 + 2 + 3 + 4 + 5 = 15 (b) 1 + 2 + 3 + 4 + 5 + 6 = 21 (c) 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 (d) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

(e) 1785 (e) 90

Page 37 Assessment 3

Strategies and operations may vary. Teacher check the students’ answers. 1. 2. 3. 4. 5.

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N&PV – 1

400 milk cartons/multiplication The number is 2/multiplication, division 8 cars/addition, division 1560 movie-goers/multiplication, addition 32 boxes are needed/addition, division

Page 38 Assessment 4

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1. (a) 1, 71 (prime) (b) 2, 7, 4 (composite) (c) 1, 151 (prime) (d) 2, 3, 4, 7 (composite) 2. (a) 5 (b) 17 (c) 53 (d) 11 (e) 2 (f ) 37 3. (a) 2 (b) 7 (c) 5 (d) 11 (e) 3 (f ) 41 4. (a) The four friends will need 3 tacos each, or 2 packets. (4 = 2 x 2; 6 = 2 x 3; 2 x 2 x 3 = 12; 12 tacos are required; with 6 in each packet, 2 packets are needed.) (b) 9 bunches of flowers (36 = 3 x 3 x 2 x 2; 27 = 3 x 3 x 3; 18 = 3 x 3 x 2; so 3 x 3 bunches of flowers can be created)

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27 students $61 40 green beads N=7 56 chocolates

Page 39 Assessment 5 1. 3 2. 225 3. 400 5. 3264 cm or 32.64 m 7. 1880 grams or 1.88 kg 10. 25 11. 7 buses

4. 108 6. 11 8. 1768 9. 4 minutes 12. 8 stands

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Page 22 Assessment 4 1. (a) 16 (b) 64 (c) 144 2. (a) 289 (b) 4624 (c) 2809 (d) 6084 (e) 400 (f ) 729 (g) 2025 (h) 9801 3. (a) 28 (b) 45 (c) 15 4. (a) 66, 78, 91, 105 (b) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11= 66 5. The rows in order from bottom to top will have 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 and 1.

1. 2. 3. 4. 5.

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Page 21 Assessment 3

Individual answers will vary. Teacher should check the strategy, explanation and discuss the efficiency rating chosen by the students for each problem.

Page 40 Assessment 6 1. (a) 109 (d) 563 2. (a) 747 (d) 178 347 3. (a) 15 473 (d) 7123 4. (a) 3552 (d) 30 291 5. (a) (i) 8 (ii) 4155 (c) (i) 5 (ii) $1100

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

(b) 97 (c) 1 901 824 (e) 73 (f ) 526 (b) 39 744 (c) 709 (e) 340 (f ) 172 (b) 3677 (c) 5850 (e) 7245 (f ) 6686 (b) 968 (c) 16 438 (e) 2595 (f ) 6583 (b) (i) $3504 (ii) $496

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Answers

Sub-strand: Number and Place Value

N&PV – 3 Page 53 Assessment 1 1. (a) four floors below street level (b) five floors above street level (c) three floors above street level 2. (a) –1 °C (b) 30 °C (c) 1 °C 3. (a) $7 (b) $20 (c) $1 4. eleven blocks north Page 54 Assessment 2 1. –10

–5 –4 –3 –2 –1

50 44 40 38 30

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20 16 10

0 –4 –6 –10 –12

–26 –30 –40

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(d) –3 (h) 5 (d) 8 (h) 1

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3. (a) –400 m (b) –100 m (d) +2250 m 4. (a) –9 (b) –7 (c) –4 (e) 0 (f ) 2 (g) 3 (i) 8 (j) 10 5. (a) 1 (b) 2 (c) –4 (e) 0 (f ) –3 (g) –2 (i) 9 (j) 4 6. (a) –4 (b) –5 (c) –5 (e) –3 7. (a) –5, 2, 3 (b) –14, –6, 12 8. (a) 12, 9, –5 (b) 0, –2, –5

(d) –9 (c) 0, 4, 8 (c) –7, –8, –9

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

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

TEACHER INFORMATION

RELATED TERMS Fraction

What this means

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Equivalent fractions

• Two fractions (a⁄ and ⁄ ) are equivalent if they are equal (i.e. a x d = b x c). Equivalent fractions are different ways of writing the same fraction. Examples include ½, 3⁄6, 4⁄8 and 5⁄10. Equivalent fractions appear to be different but have the same value.

Teaching points

• In order to compare fractions, students will need to convert them to the same denominator. • Converting one fraction to an equivalent fraction involves the use of the multiplication property of one—anything multiplied by one is itself. For example, to convert thirds to sixths requires multiplying by 2⁄2 (which is equal to one). e.g. 1⁄3 x 2⁄2 = 2⁄6

• Number lines can be calibrated to show equivalent fractions; for example: show a line divided into 3 equally-spaced lengths and, directly below it, a line divided into 6 equally-spaced lengths. (This shows that 1⁄3 and 2⁄6 are equivalent.)

What to look for © R. I . C .Publ i cat i ons •f orr evi ew pur posesonl y•

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• A number line is a pictorial representation of real numbers. A number line is labelled with the integers in increasing order from left to right, and can extend in both directions. For any two different whole numbers on a number line, the integer on the right is greater in value than the integer on the left.

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Student vocabulary fraction equivalent

• When using the number line, calibrations need to be equivalent (or in line vertically) in order to compare fractions. See also New wave Number and Algebra (Year 6) student workbook (pages 21–24)

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

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Number line

• Students will be able to compare fraction families such as halves, fourths, eighths, thirds, sixths etc.

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• A fraction is part of a whole. A fraction is obtained by dividing a whole or given amount into a certain number of equal parts and taking a certain number of them; for example, 2⁄3 refers to 2 of 3 equal parts of the whole. • In a fraction, the top number is referred to as the numerator (or number of parts you have), and the bottom number is the denominator (or the number of parts the whole is divided equally into).

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equal numerator denominator compare symbols greater than (>) smaller than/less than (<) equal to (=)

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

HANDS-ON ACTIVITIES Paper folding To demonstrate equivalent fractions, begin with a simple rectangle or square of paper for each student. Ask the students to fold it in half (along the length, width or diagonal), and quickly colour one half. Ask them to label the section with ‘½’ and draw arrows to indicate where one half starts and finishes on the shape. Ask them to fold the shape again into quarters and colour 2⁄4. They then label and draw arrows to indicate 2⁄4. Repeat folding, colouring and labelling for eighths and sixteenths. The students will easily see that ½ = 2⁄4 = 4⁄8 = 8⁄16, which are equivalent fractions. Repeat with other shapes.

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Equivalent fractions matching card game

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This game is a memory matching card game which will reinforce understanding of equivalent fractions. Photocopy the fractions cards on pages 65 to 68 onto card, cut them out and laminate them for durability. Play the game in pairs. Shuffle the cards, then spread them out upside down in the centre of the players. Player 1 turns over two cards, hoping to find equivalent fraction pairs. If the cards match, he/she keeps both cards. If the cards do not match, the cards are turned over again. Player 2 takes a turn matching two cards. The players continue in this fashion until all cards have been removed. The player with the most cards wins.

Comparing fractions card game

To play this game, in groups of two or more players, students need copies of the cards on pages 69 to 73. The aim of the game is to gather as many cards as possible by the end of the game. The winner is the player with the most cards. After photocopying the cards onto card, cut them out and laminate them for durability. Shuffle the cards, then deal them out so that each player has the same number of cards. Players keep their cards facedown in a pile in front of them and are not allowed to look at them. Each player turns over his/her top card and places it on the table, comparing it to the cards of the other player(s). The player with the largest fraction takes all of the cards in play, adding them to the bottom of his/her own pile. If the two cards are equivalent, the two cards remain on the table, and the players select the next card on his/her pile to place facedown on the table and the next card is shown and compared. This action continues until two different fractions are shown. Then the player who has the larger fraction receives all the cards in play, including those facedown.

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This game, played in threes, gives the students practice in ordering fractions from smallest to largest. Write the fractions ½, 1⁄3, 2⁄3, ¼, ¾, 1⁄6, 5⁄6, 1⁄8, 3⁄8, 5⁄8, 7⁄8, 1⁄12, 5⁄12, 7⁄12 and 11⁄12 on separate blank playing cards or squares of white cardboard. (Alternatively, photocopy an extra set of the cards on pages 69 to 73 and discard those not needed.) Shuffle the cards and give each player five. The remaining cards are placed facedown in a pile in the centre of the table. Each player places his/ her cards facedown in a row in front of himself/herself. A third person acts as starter. He/She shouts out ‘Ready, get set, go!’ and both players race to turn over and place their cards in order from smallest to largest. The first person finished, shouts ‘All done!’ The starter checks the order to see if the player has them correct. A calculator may be needed, especially if the fractions need to be converted to find a common denominator. If the player is correct, he/she wins that round. All the cards are returned to the pile and shuffled, and the process begins again. The first player to win three rounds is the overall winner.

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Equivalent fractions dominoes

Photocopy the cards onto card and ask the students to cut them out. (Refer to pages 75 to 77.) The students must match and glue the correct picture image onto each domino. They then try to be the first to match as many equivalent fractions and models as possible. Note: Unlike the normal game of dominoes, this game makes six distinct rows. For a more difficult version, have the students glue the images onto any fraction domino and play in the same way as normal dominoes, trying to match the equivalent fractions and images.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

LINKS TO OTHER CURRICULUM AREAS English

1⁄2 = 2⁄4

• Obtain a copy of Funny and fabulous fraction stories: 30 reproducible maths tales and problems to reinforce important fraction skills by Dan Greenberg. • Refer to Fabulous fractions: Games, puzzles and activities that make maths easy and fun by Lynette Long for some extra hands-on activities. • Working with fractions by David A Adler may prove useful for those students struggling with the concepts. • After reading The Hershey’s milk chocolate bar fractions book by Jerry Pallotta, ask the students to write a fiction or nonfiction book in the same vein which explains equivalent fractions. As a class, discuss and record the basic information to include (and to reinforce concepts at the same time.)

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Information and Communication Technology

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• Visit <http://www.funbrain.com/fract/index.html> to play ‘Fresh baked fractions’. Students must click on the fraction which is not equivalent to the other three, and in the process gather pieces of pie for Jackson, the dog. There are four difficulty levels, and some activities require the students to simplify fractions. • Visit <http://illuminations.nctm.org/ActivityDetail.aspx?ID=80> to create equivalent fractions by dividing and shading squares or circles, and matching each fraction to its location on the number line. • Visit <http://www.kidsolr.com/math/fractions.html> for a clear text and visual interactive explanation of equivalent fractions.

Health and Physical Education

• Revise fractions by dividing a pizza (or a few) into equal portions for sharing among the class members. Blocks of squared chocolate could also be used. The book The Hershey’s milk chocolate bar fractions book, by Jerry Pallotta, could be used in conjunction with this hands-on activity, although it is more suitable for younger students. • Link sports and equivalent fractions to this problem-solving activity. This game is for three or four players. At a set distance, set up a target game such as throwing a ball or beanbag into a rubbish (or recycling) bin. Give each student a different number of attempts (2, 4 or 8) at the game. Record the successes; for example: Sasha - ½ (one on target, out of two attempts), Shilo – ¾ (three on target, out of four attempts) and Ivan – 5⁄8 (five targets out of 8 attempts). Ask the question ‘Who was the best player of this game?’ Ask the students to discuss the question in pairs and explain their reasons. Students might find that a fraction model is of assistance to explain their answer. Vary the number of attempts to 10, 12, 15, 16 or 20, if desired. The students might also like to research the success rate of certain players’ shots at goal of their favourite sports club.

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• Polar bear maths: Learning about fractions from Klondike and snow, by Ann Whitehead Nagda and Cindy Bickel, cleverly uses facts about the polar bear cubs and relates them to fractions.

The Arts

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• Provide each student with a 100-square grid sheet (see page 74) to create an abstract artwork similar to the work of American artist Ellsworth Kelly. The students select between three to six different colours of paper to cut out and cover the squares. Some could be left blank. When the artwork is completed, the students count the number of squares coloured out of 100 and write these as a fraction. Students should note any colours that create equivalent fractions. Later this activity will relate to percentages and decimals.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Fraction strips

One whole ½ 1⁄3

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

1⁄7

1⁄11 1⁄12 1⁄13 1⁄14

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1⁄15 1⁄16

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Equivalent fraction table

1⁄1

2⁄2

3⁄3

4⁄4

5⁄5

6⁄6

7⁄7

8⁄8

9⁄9

10⁄10

2⁄4

3⁄6

4⁄8

5⁄10

6⁄12

7⁄14

8⁄16

9⁄18

10⁄20

1⁄3

2⁄6

3⁄9

4⁄12

5⁄15

6⁄18

7⁄21

8⁄24

9⁄27

10⁄30

2⁄8

10⁄50

1⁄6

2⁄12

3⁄18

4⁄24

5⁄30

6⁄36

7⁄42

8⁄48

9⁄54

10⁄60

1⁄7

2⁄14

3⁄21

4⁄28

5⁄35

6⁄42

7⁄49

8⁄56

9⁄63

10⁄70

1⁄8

2⁄16

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10⁄80

1⁄9

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1⁄10

2⁄20

3⁄30

9⁄90

10⁄100

9⁄99

10⁄110

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1⁄13

2⁄26

3⁄39

1⁄14

2⁄28

¾2

1⁄15

2⁄30

1⁄16 1⁄17

6⁄60

7⁄70

8⁄80

4⁄44

5⁄55

6⁄66

7⁄77

8⁄88

9⁄108 o c . c e 9⁄117 her r 4⁄52 5⁄65 6⁄78 7⁄91t 8⁄104 o s super 4⁄48

5⁄60

6⁄72

7⁄84

8⁄96

10⁄120 10⁄130

4⁄56

5⁄70

6⁄84

7⁄98

8⁄112

10⁄126

10⁄140

¾5

4⁄60

5⁄75

6⁄90

7⁄105

8⁄120

9⁄135

10⁄150

2⁄32

¾8

4⁄64

5⁄80

6⁄96

7⁄112

8⁄128

9⁄144

10⁄160

2⁄34

3⁄51

4⁄68

5⁄85

6⁄102

7⁄119

8⁄136

9⁄153

10⁄170

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

3⁄33

5⁄50

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2⁄24

1⁄12

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Equivalent fractions models

2⁄4s 3⁄6 r o e t B r e oo p u k S

8⁄12

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2⁄4

4⁄8

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

2⁄3

4⁄8

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3⁄9

8⁄24 Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Equivalent fractions models

r o e t s Bo r e p ok u S 2⁄8

4⁄16

8⁄32

16⁄64

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2⁄12

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

3⁄18 R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

2⁄6

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS

2⁄3 1⁄5

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

4⁄5

CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

1⁄3

Equivalent fractions match cards—1

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS

4⁄6 8⁄10m

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6⁄10

4⁄10

2⁄10

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

4⁄5

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4⁄10 66

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

2⁄10

6⁄8

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2⁄4

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Equivalent fractions match cards—2

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS

1⁄7

5⁄6

1⁄6

Equivalent fractions match cards—3

4⁄7

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

2⁄2

CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

ew 1⁄7 evi Pr

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R.I.C. Publications®

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS

2⁄14 5⁄5 m.

12⁄14

10⁄14

8⁄14

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

5⁄5

12⁄14

10⁄14 68

6⁄14

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

8⁄14

6⁄14

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Equivalent fractions match cards—4

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Comparing fractions cards—1

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

½ 1⁄3 2⁄3 ¼

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3⁄5 4⁄5 1⁄6 2⁄6 Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Comparing fractions cards—2

2⁄8 3⁄8 4⁄8 5⁄8 w ww

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Comparing fractions cards—3

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

3⁄9 4⁄9 5⁄9 6⁄9

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3⁄10 4⁄10 5⁄10 6⁄10 Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Comparing fractions cards—4

2⁄12 3⁄12 4⁄12 5⁄12 w ww

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Comparing fractions cards—5

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10⁄12 11⁄12 10⁄12 1⁄50

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0 0 1 Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET

w ww NAME:

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Colour of squares

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

Fraction

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

r o e t s Bo r e p ok u S

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100 square grid sheet for abstract artwork (refer to page 60)

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Equivalent fractions dominoes—1

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1⁄10

2⁄2

2⁄8

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2⁄4

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

1⁄3

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3⁄6

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5⁄10 Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

2⁄6 ¾

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Equivalent fractions dominoes—2

6⁄6

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3⁄12

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10⁄20

8⁄8

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3⁄9

12⁄16

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4⁄8

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

2⁄20

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS

Teac he r

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

Equivalent fractions dominoes—3

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Strategies to compare fractions 1.

Comparing fractions with the same denominator When comparing fractions with the same denominator, simply compare numerators (the top number). The fraction that is the larger is that which has the larger numerator. For example, if we compare 3⁄7 to 5⁄7, the second fraction has the larger numerator, so the second fraction is larger.

Cross-multiplying Two fractions are equivalent if the product of the numerator of the first fraction and the denominator of the other fraction is equal to the product of the denominator of the first fraction and the numerator of the second fraction. For example, to find out if ¾ is equivalent to 9⁄12, we use the following method: Multiply the numerator of the first fraction—(3)—by the denominator of the second fraction—(12), and we find the product of the denominator of the first fraction—(4) by the numerator of the second fraction—(9).

3 4

9 12 (cross-multiplying)

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So, because cross-multiplying produces the same answer, the fractions are equivalent. As a general rule:

• if the cross-products are equal, the fractions are equivalent • if the first cross-product is larger, the first fraction is larger • if the second cross-product is larger, the second fraction is larger. Cross-multiplying is used to compare fractions with different denominators. 3.

Simplifying fractions

To compare two fractions to see if they are equivalent, convert them to the same denominator. For example, to compare ¾ and 2⁄3, it is necessary to convert the denominators of each fraction to the lowest common denominator; i.e. 4 x 3 (12). Four into 12 is 3, so we multiply ¾ (top and bottom) by 3. Multiplying both the numerator and denominator by 3 doesn’t alter the value of the fraction.

3 x 3 = 9 4 3 12 .

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So,

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Three into 12 is 4, so we multiply both the numerator and denominator of 2⁄3 by 4, and it becomes

2 x 4 = 8 3 4 12 . This converts both fractions to the same denominator and we can compare them easily.

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2 = 8 3 = 9 4 12 and 3 12 , so the fractions are not equal. In fact, ¾ is larger than 2⁄3. 4.

Reducing fractions by finding factors

Sometimes fractions can be reduced to make them easier to compare.

For example, if we wanted to compare 24⁄56 and 2⁄7, we can try to reduce 24⁄56 to its factors. So, if we factor the numerator and denominator, we get:

(24) 2 x 2 x 2 x 3 (56) 2 x 2 x 2 x 7 . We can then cross out the common factors equally top and bottom

(24) 2 x 2 x 2 x 3 and we get (56) 2 x 2 x 2 x 7 . We have reduced 24⁄56 down to 3⁄7. When we compare this to 2⁄7 we can see that 24⁄56 is larger.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

This becomes 3 x 12 and 9 x 4 which equal 36 and 36.

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEETS Strategies to compare fractions 5.

Finding other equivalent fractions by multiplying/converting fractions To find other equivalent fractions for a given fraction, multiply the numerator and denominator by the same number (excluding zero). For example, to find other fractions which are equivalent to 1⁄3, multiply both the numerator and denominator by 3,

x = 3 9 ; therefore, an equivalent fraction for so 3 by 2 or 4, we would get 2⁄6 or 4⁄12 . 6.

1 3 3 is 9 . If we multiplied the numerator and denominator

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Finding other equivalent fractions by division

To find other equivalent fractions for a given fraction, divide the numerator and denominator by the same number. For example, to find other fractions equivalent to 4⁄12, divide both the numerator and the denominator by 4,

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

4 ÷ 4 = 1 4 3 ; therefore, an equivalent fraction for 4⁄12 is 1⁄3 . If we divided the numerator and denominator by so 12 2, we would get 2⁄6.

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Comparing equivalent fractions to 1 on a number line

Teac he r

2⁄4

2⁄5

3⁄5

4⁄5

1⁄8

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1⁄5

2⁄3

1⁄16

2⁄12

2⁄16

3⁄8

4⁄8

5⁄8

6⁄8

7⁄8

. t 2⁄10 e 3⁄10 4⁄10 5⁄10 6⁄10 7⁄10 8⁄10 9⁄10 o c . che e r o t r s super

1⁄10

1⁄12

2⁄8

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3⁄16

3⁄12

4⁄16

4⁄12

5⁄16

5⁄12

6⁄16

7⁄16

6⁄12

8⁄16

7⁄12

9⁄16

8⁄12

9⁄12

10⁄12

11⁄12

10⁄16 11⁄16 12⁄16 13⁄16 14⁄16 15⁄16

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 1

NAME:

DATE:

1. Which fraction is the smallest? Shade a bubble to show your answer.

5⁄6

2⁄6

1⁄6

4⁄6

(a)

4⁄4

2⁄4

2⁄8

4⁄8

8⁄8

5⁄8

(b)

3⁄5

2⁄5

1⁄5

4⁄5

(c)

(d)

r 7⁄7o 3⁄9 6⁄9 8⁄9 e t s B r e oo p u k 9⁄12 11⁄12 10⁄12 S5⁄10 7⁄10 8⁄10

2. Which fraction is the largest? Shade a bubble to show your answer.

3⁄7

4⁄7

1⁄7

9⁄9

(b)

(a)

(c)

(d)

12⁄12

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9⁄10

3. Which fraction is the smallest? Shade a bubble to show your answer.

2⁄4

4⁄8

(b)

(a)

1⁄8

1⁄9

1⁄3

3⁄8

2⁄2

6⁄8

3⁄9

3⁄3

3⁄12

9⁄12

1⁄6

6⁄6

5⁄10

4⁄5

3⁄5

7⁄10

9⁄10

2⁄2

1⁄10

(d)l © R. I . C.Pub i cat i ons 4. Which fraction is the largest? Shade a bubble to show your answer. •f orr evi ew pur posesonl y• (c)

(a)

5⁄10

6⁄8

1⁄10

4⁄8

(c)

(d)

. te 1⁄6 1⁄3 1⁄8 ¼

5. Order the fractions from smallest to largest. (a)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

o c . che e r o t r s super (b)

1⁄3, 4⁄6,1⁄9, 4⁄9

6. Which fractions are equivalent? (a)

2⁄16, 3⁄18, 3⁄9, 1⁄8

(b)

2⁄3, 3⁄5, 8⁄12, ¼

(c)

4⁄8, ¾, 2⁄4, 6⁄8

(d)

8⁄10, 12⁄20, 16⁄20, 4⁄6

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 1

NAME:

DATE:

½ 1⁄3

2⁄3

r o e t s Bo r 2⁄ 4 ¾ e p ok u S 2⁄5 3⁄5 1⁄5 4⁄5 ¼

1⁄6

1⁄8

2⁄6

2⁄8

4⁄6

3⁄6

3⁄8

4⁄8

5⁄8

5⁄6

6⁄8

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7⁄8

1⁄12

2⁄12

3⁄12

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4⁄12

5⁄12

6⁄12

7⁄12

8⁄12

9⁄12

10⁄12

11⁄12

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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

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1. Use the number line equivalents to find and write equivalent fractions for each. The number in brackets indicates how many equivalent fractions you need to find. (a)

(c)

¾ (3)

(6)

(b) (d)

(3)

1⁄6

(1)

2. For each of the groups of answers to Q.1, explain how the denominators are related. (a) (b) (c) (d) 82

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

Assessment 3

Sub-strand: Fractions and Decimals—F&D – 1

NAME:

DATE:

Use each number line to show the equivalent fractions stated. You will need to draw longer lines or coloured lines to differentiate between the equivalent fractions. 1.

8⁄8, 3⁄8, 7⁄8, 1⁄8, 5⁄8, 2⁄8

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 3⁄5, 2⁄5, 4⁄5, 1⁄5 0

1⁄3, 2⁄3, 3⁄3

1⁄9

2⁄9

3⁄9

4⁄9

5⁄9

6⁄9

6⁄8, 4⁄8,8⁄8, 2⁄8

7⁄9

8⁄9

16⁄16, 4⁄16, 12⁄16, 8⁄16

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2⁄8, 8⁄16, 6⁄8, 16⁄16 0

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

9⁄10

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r o e t s Bo r e p ok u 1⁄10 S2⁄10 3⁄10 4⁄10 5⁄10 6⁄10 7⁄10 8⁄10

2⁄4

o c . ch e r ¼e 2⁄4 o t r s ¾ super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Assessment 4

Sub-strand: Fractions and Decimals—F&D – 1

NAME:

DATE:

1. Shade a bubble to show which model shows a fraction equivalent to: B

D C

r o e t s Bo r e p ok u S A

6⁄16

2⁄3

D C

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(c)

8⁄10 B

(b)

(a)

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(b)

1⁄3

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3. Draw two models or drawings which show fractions equivalent to:

(a)

(b)

2⁄4

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Compare fractions with related denominators and locate and represent them on a number line

(a)

Checklist

Sub-strand: Fractions and Decimals—F&D – 1

Compare fractions with related denominators and locate and represent

Uses drawings and models to show related denominators

Represents fractions with related denominators on a number line

Locates fractions with related denominators on a number line

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STUDENT NAME

Compares fractions with related denominators

them on a number line (ACMNA125)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 2

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

RELATED TERMS

TEACHER INFORMATION What this means

• Students need to be able to understand addition and subtraction of fractions that have the same denominator. • Students then progress onto addition and subtraction of fractions with related denominators, such as ½ + 2⁄4 . • Students should be given the opportunity to compare various fractions, such as 1⁄3 and 1⁄6, as well as ½,¼, 1⁄8 and 1⁄5 and 1⁄10. • Students need to be able to use these fractions as an operator; i.e. If you were given 1⁄3 of 12 lollies, how many would you get? • Students should be able to solve meaningful additive and subtractive problems involving fractions to develop understanding of equivalent fractions (see above) and the use of fractions as operators. • Students need to model and solve addition and subtraction problems involving fractions by using methods such as ‘jumping’ along on a number line, or by making diagrams of fractions as parts of shapes.

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Operator

• A fraction such as ¾ is acting as an operator. The fraction is stating that the student needs to complete the operation of dividing a ‘whole’ shape into four equal parts and identify or shade three of the parts.

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Fraction

Teaching points

• It is important to give children the opportunity to model addition of fractions (e.g. ½ + ¼) using squared paper; and once done, to be able to explain their result. In order for students to be able to do this, they must understand how to fold ½, ¼, 1⁄8, as well as 1⁄3, 1⁄6 and 1⁄5 and 1⁄10. If they can do this, then an understanding of adding and subtracting fractions takes place. • Students need to be able to show what ½ plus 1⁄3 looks like when folding paper and they need to be able to transfer this information to a number line. The notion of equivalent fractions is important here. • Students should be able to solve meaningful additive and subtractive problems involving fractions to develop understanding of equivalent fractions and the use of fractions as operators. Students need to understand that a whole can be broken into equal parts. This can be shown to students by using a fraction grid.

related to a context.

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Student vocabulary fraction numerator denominator solve addition subtraction

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What to look for

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• Students who have a problem folding paper into 1⁄3 and 1⁄5. They often fold into quarters and this can create problems. The ability to fold paper to solve additive and subtractive problems using fractions is most important and should not be overlooked. • Students who switch between an understanding of fractions to thinking of the denominators as whole numbers. • Many students when calculating 1⁄3 + 1⁄5 will give an answer of 2⁄8. They perform ‘whole number processes’ and do not understand the concept of part of a whole. • When presented with problems such as 2⁄3 of 12, some students may think of the third as 3 and multiply by 2 to get 6, disregarding that they have to think of the 12 as the whole. See also New wave Number and Algebra (Year 6) student workbook (pages 25–28)

Proficiency strand(s): Understanding

Fluency

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

Problem solving

Reasoning

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Sub-strand: Fractions and Decimals—F&D – 2

HANDS-ON ACTIVITIES Using pattern blocks Pattern blocks can be found in most schools. Standard pattern block sets include a yellow hexagon, an orange square, a red trapezium, a blue rhombus, a tan rhombus and a green triangle. Experiment with the manipulatives to see how many of each type fit exactly on top of a larger shapes, such as a yellow hexagon. Smaller equal shapes placed on top of a larger one demonstrate fractions. For example, six green triangles make one yellow hexagon, so we can consider the triangle is 1⁄6 of the whole; three blue rhombuses make one yellow hexagon, so we can consider that each rhombus is 1⁄3 of the whole. If we only placed 1 green triangle on top of a yellow hexagon, and added one blue rhombus, we are adding 1⁄6 and 1⁄3. Pages 22 to 27 of Developing mathematics with pattern blocks, by Dr Paul Swan and Geoff White, provide other examples of this.

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Using Cuisenaire® rods

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Fraction spinners games Use fraction spinners to add and subtract two fractions of the same or related denominators. (Refer to pages 91 and 92.)

English

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Cuisenaire® rods are coloured wooden rectangular blocks, varying in length from 1 cm to 10 cm. In the same way as pattern blocks, placing smaller blocks of the same colour on larger blocks divides a whole into fractions. Once a larger block is divided into equal parts using a certain colour of block, use a number of different fractions (different coloured blocks) to find fractions of the whole.

INKS TO. OP THER URRICULUM A REAS © RL. I . C uCb l i cat i o ns •f orr evi ew pur posesonl y•

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• Ask the students to make up ‘word problems’ to illustrate a given addition or subtraction fraction problem. For example, ‘At my party, the girls ate ½ of the birthday cake and the boys ate 1⁄3. How much was eaten altogether?’ The students can listen to each other’s equivalent fractions and determine whether they are suitable for the equation. They may also like to select the best problem from a group of word problems.

Information and Communication Technology

• Visit <http://ejad.best.vwh.net/java/patterns/patterns_d.shtml> to find interactive pattern block activities. • Visit <http://teachertech.rice.edu/Participants/silha/Lessons/add.html> to find examples of how to use Cuisenaire rods with fractions.

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Health and Physical Education

• Ask the students to double the quantities in a recipe and practise adding fractions; for example, if a recipe requires ½ cup sugar, add ½ and ½ to get 1 cup of sugar for double the recipe.

The Arts

• Use Ed Emberley’s picture pie: A circle drawing book, by Edward R Emberley, as inspiration to create simple artworks based on combining fractions of circles. The artworks can be presented to a younger ‘buddy’ class student. • Visit <http://mathcrush.com/math_art_worksheets.html> to find maths puzzle worksheets which combine the answers to addition and subtraction fraction problems with colouring to create a picture.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEETS Strategies for solving addition and subtraction problems with fractions In order to add and subtract fractions, the denominators must be the same. 1.

Adding and subtracting fractions with the same denominator When adding and subtracting fractions with the same denominator, simply add or subtract numerators (the top number). For example, if we add 3⁄7 to 2⁄7, the answer is 5⁄7. If we subtract ¼ from ¾, the answer is 2⁄4 (or ½). In both addition and subtraction, the denominator is not taken into consideration.

Adding and subtracting fractions with related or different denominators

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When adding and subtracting fractions with related or different denominators, look at the denominators. Example 1

• Is one denominator a factor of the other? If so, use the larger denominator as the lowest common denominator and convert the other fraction to the same denominator. Consider these two fractions:

Teac he r

1⁄3 – 1⁄6

Example 2

• If one denominator is not a factor of the other, find the lowest common multiple (LCM) and use it as the lowest common denominator (LCD). Consider the following fractions:

By searching the multiples of 12 and multiples of 9, we find that: 1 x 12 = 12 2 x 12 = 24

1x9=9

2 x 9 = 18

3 x 12 = 36

3 x 9 = 27

4 x 12 = 48

4 x 9 = 36

5 x 12 = 60

5 x 9 = 54

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Therefore, the lowest common multiple/denominator that both 12 and 9 go into is 36.

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2⁄12 + 3⁄9

We can then convert both 5⁄12 and 4⁄9 into thirty-sixths. Firstly, 12 multiplied by 3 is 36 so we multiply both the denominator and numerator of 5⁄12 by 3 and get 15⁄36.

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Next, 9 goes into 36 four (4) times, so we multiply both the numerator and denominator of 4⁄9 by 4 and get 16⁄36.

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Now our fractions become 15⁄36 + 16⁄36, which we can easily add: 15⁄36 + 16⁄36 = 31⁄36.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

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Three is a factor of six, so we convert 1⁄3 to sixths. To do this, we multiple both the numerator and denominator of

1⁄3 by two (because 3 x 2 is 6). So, 1⁄3 becomes 2⁄6, and 2⁄6 – 1⁄6 = 1⁄6 and 1⁄3 – 1⁄6 = 1⁄6.

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEETS Strategies for solving addition and subtraction problems with fractions 2.

Adding and subtracting fractions with related or different denominators (continued) Example 3 If you are unable to find the LCM, simply multiply the denominators to create a common denominator. It may be a much larger number and the answer may need to be reduced (or simplified) at the end, but you will still find a workable denominator. As an example, consider:

2⁄12 + 3⁄9 12 x 9 = 108

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So we multiply 2⁄12 by 9 (top and bottom) and 3⁄9 by 12 (top and bottom).

From this, 2⁄12 becomes 18⁄108 and 3⁄9 becomes 36⁄108. Now we can add them:

18⁄108 + 36⁄108 = 54⁄108

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Now we must reduce or simplify 54⁄108 to its lowest form, which is ½.

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So, to return to the original sum, 2⁄12 + 3⁄9 = ½.

Using models or drawings to add and subtract fractions with related or different denominators

Some students, especially visual–spatial learners, may find models or drawings helpful to add or subtract fractions. Consider 2⁄3 + ¼.

(i) Draw 2⁄3 in vertical columns in a simple shape.

(ii) Draw

in horizontal columns in the same size of shape.

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The other parts remain shaded as they were originally.

This means that, in its entirety, 11 of the 12 parts are shaded.

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(iii) Now join the two figures (as if overlaying one over the other) to make 12 equal parts altogether, with the two parts overlapping marked by sloping lines.

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CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

(a) Addition – using models

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As such, 2⁄3 + ¼ = 11⁄12.

(b) Subtraction – using models Consider 2⁄3 –½.

(i) Draw 2⁄3 in vertical columns in a simple shape.

(ii) Draw one horizontal line across the shape to cut it into halves.

(iii) To subtract ½ of six equal parts, we cross out three of the six parts. (iv) We have only one of the shaded parts left. As such, one out of six parts remain, so

2⁄3 – ½ = 1⁄6.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEETS Strategies for solving addition and subtraction problems with fractions 3.

Using models or drawings to add and subtract fractions with related or different denominators (continued) (c) Addition using arrays The sum 1⁄3 + 1⁄6 can be drawn using arrays in the following way to obtain an answer: +

1⁄3 4.

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3⁄6

Using prime factors to subtract and add fractions Consider, 7⁄12 + 4⁄15.

7 12

4 15

(3 x 5) (2 x 2 x 3)

(ii) Now cross out any numbers that appear in ALL FOUR sets of brackets. (3 x 5) (2 x 2 x 3) 7 12

4 15

(3 x 5) (2 x 2 x 3)

(iii) Multiply the numerators and denominators by the numbers still left in the brackets above or below them.

7 12

4 15

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(3 x 5) (2 x 2 x 3) 12 x 5= 60 15 x 2 x 2 = 60

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7 x 5= 35 4 x 2 x 2 = 16 (3 x 5) (2 x 2 x 3)

(iv) Rewrite the sum, now in its new form using the products from the previous step. The fractions now have a common denominator. 35 60

16 51 + = 60 60

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(v) Reduce/Simplify the final fraction, if necessary. 51 60

3 3

17 20

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

(3 x 5) (2 x 2 x 3)

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(i) To solve this sum, we need to write the prime factors of each fraction’s denominator below and above the bar of the other fraction.

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEETS Fraction spinners/Game instructions This game can be played in pairs or small groups, with one player checking the answers by using manipulatives or a calculator. Select or create a spinner, suitable for the level of understanding of your students or applicable to the fractions you are covering. Draw or photocopy the spinners onto cardboard and cut each out. Insert a craft stick or toothpick in the middle of each as a pivoting device.

1⁄5

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CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

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Each player spins each spinner twice and adds the two fractions. If subtracting, the players must subtract the smaller of the two fractions from the larger. Students may need to covert the fractions to their lowest common denominator to find the answer, which is written down as a complete sum with an answer. Other players (or the partner), check the answer. The first player with 10 correct sums and answers is the winner. The spinners below are suggestions, though it is also easy to devise your own.

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1⁄6

½ Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

2⁄6

2⁄3 R.I.C. Publications®

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Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEETS Student recording sheet for fraction spinner game

Name Sum operation – addition/subtraction

Answer

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Fractions spun and used

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Sum operation – addition/subtraction

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Name Answer

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

r o e t s Bo r e p ok u S

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Fractions spun and used

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 2

NAME:

DATE:

1. Solve these addition problems of fractions with the same denominator. Show all your working. Simplify the answers if you need to. Shade a bubble to show your answer.

(a)

1⁄3 + 1⁄3

(d)

¼+ ¼

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2⁄5 + 1⁄5

5⁄6

4⁄8

1⁄8

3⁄3

2⁄3

2⁄6

3⁄5

2⁄3

r o e t s Bo r e p ok2⁄5 1⁄3 u S ¼

1⁄6 + 3⁄6

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(c)

2⁄3

3⁄8

2⁄8

1⁄6

2⁄3

(a)

2⁄3 – 1⁄3

(b)

¾ – 2⁄4

(c)

(d)

(e)

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4⁄6 – 1⁄6

3⁄6

1⁄3

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working. Simplify the answers if you need to. Shade a bubble to show your answer.

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CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

(b)

7⁄12 + 3⁄12

3⁄3

1⁄3

4⁄8

1⁄12

1⁄3

o c . che e r o t r s 2⁄6 ½ super

10⁄12 – 2⁄12

2⁄3

4⁄8

5⁄6

3⁄12

1⁄3

3⁄8

3⁄5

6⁄8 – 2⁄8

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 2

Sub-strand: Fractions and Decimals—F&D – 2

NAME:

DATE:

1. Solve these addition problems of fractions with related denominators. Show all your working. Simplify the answers if you need to. Shade a bubble to show your answer.

(b)

(e)

2⁄5

2⁄6

2⁄3

1⁄6

3⁄8

2⁄6 + 1⁄3

4⁄8 + ¼

r o e t s B r e 1⁄3 o 2⁄8 p o u k S 1⁄6

2⁄3

11⁄12

1⁄3

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(d)

1⁄10

6⁄10 + 1⁄5

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(c)

2⁄3

3⁄12 + 2⁄6

2⁄3 + 3⁄12

5⁄8

7⁄12

1⁄12

2⁄3

© R. I . C.Publ i cat i ons 2. Solve these subtraction problems of fractions with related denominators. Show all your •f o r ev i e wto.p ur os e so n l y• working. Simplify ther answers if you need Shade ap bubble to show your answer.

(c)

(d)

(e)

3⁄8

2⁄3 – 2⁄12

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(b)

4⁄12

6⁄8 – ¼

9⁄10– 3⁄5

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1⁄3

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(a)

2⁄8

o c . che e r 1⁄3o 2⁄10 3⁄10 t r s super

¾ – 5⁄12

4⁄6 – 1⁄3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

2⁄6

2⁄5

2⁄12

5⁄8

1⁄3

2⁄8

3⁄6

2⁄3

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

(a)

4⁄5

Assessment 3

Sub-strand: Fractions and Decimals—F&D – 2

NAME:

DATE:

1. Solve these realistic addition problems involving fractions. Show all your working and simplify the answers. (a) Dad filled the petrol tank of his car on Sunday night. He used 1⁄3 of a tank of petrol on Monday and an additional 1⁄6 of a tank on Tuesday. How much of the tank of petrol did he use altogether? (b) The recipe states that 9⁄15 cup of white flour is needed, as well as 1⁄5 of a cup of wholemeal flour. How much flour is needed altogether?

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(d) At lunchtime, I drank ¾ of a cup of water and ¼ of a cup of juice. How much liquid did I drink altogether at lunchtime?

(a) My dog is 6⁄8 of a year old and my cat is a ½ year younger than the dog. How old is my cat?

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CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

(c) At my birthday party, I ate of one pizza and my brother ate of another. How much of one whole pizza was eaten altogether?

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r o e t s Bo r e p ok u S ¼ 1⁄12

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(b) I borrowed 6⁄10 of a tin of white paint to finish painting the fence. If I only used 2⁄5 of what was given, how much did I have left?

(c) I had 10⁄12 of a packet of chocolate chips to use to make biscuits. If I only used 2⁄3 of the amount I had, how much of the original amount did I have left? (d) There was only 3⁄5 of a cake left. My brother ate 1⁄10 of it and left the remainder for me. How much did I have to eat?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Assessment 4

Sub-strand: Fractions and Decimals—F&D – 2

NAME:

DATE:

1. Solve these problems involving fractions by using the number lines. Add any additional information to the number line that you need to.

3⁄8 + ½ =

(a)

1⁄8

2⁄3 + 1⁄6 =

(b)

3⁄8

4⁄8

5⁄8

6⁄8

7⁄8

r o e t s Bo r e p ok u 1⁄6 S 4⁄6 5⁄6 2⁄6 3⁄6

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2⁄8

(b)

3⁄10 + 2⁄5 =

3. Solve these problems involving fractions by using the number lines.

8⁄9 – 2⁄3 =

1⁄9

2⁄9

3⁄9

4⁄9

5⁄9

6⁄9

7⁄9

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4⁄5 – 3⁄10 =

(b)

1⁄10

2⁄10 . te

8⁄9

o c . che e r o t r s supe r ¾ 2⁄5 3⁄10

4⁄10

5⁄10

6⁄10

7⁄10

8⁄10

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(a)

9⁄10

4. Solve these subtraction problems involving fractions by using the diagrams.

(a)

2⁄3 – ½ =

(b)

–

5. Draw your own diagrams in the spaces to solve the addition and subtraction problems involving fractions. (a)

2⁄6 + ½ =

(b)

7⁄8 – ¾ =

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

1⁄3 + ¼ =

(a)

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2. Solve these addition problems involving fractions by using the diagrams. Add any additional information you may need.

Checklist

Sub-strand: Fractions and Decimals—F&D – 2

Solve problems involving addition and subtraction of fractions

Models and solves additive problems involving fractions using a number line or diagrams

Solves realistic addition and subtraction problems involving fractions

Solves addition problems involving fractions with related denominators

Solves subtraction problems involving fractions with the same denominator

r o e t s Bo r e p ok u S

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STUDENT NAME

Solves addition problems involving fractions with the same denominator

with the same or related denominators (ACMNMA126)

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m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 3

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

TEACHER INFORMATION

RELATED TERMS Fraction

What this means

A fraction is part of a whole. A fraction is obtained by dividing a whole or given amount into a certain number of equal parts and taking a certain number of them: for example, 2⁄3 refers to 2 of 3 equal parts of the whole.

• Students should be able to understand how fractions display part of a whole, and use these in additive and subtractive situations which give a whole number; e.g. ½ + 1½, 5¼ – 4¼. • Students should be able to understand the use of fractions in additive and subtractive situations in which the part of a whole is itself a whole number; i.e. 2⁄3 of 24. • Refer to content description associated with code F&D – 2 for additional background information for this description.

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In a fraction, the top number is referred to as the numerator (or number of parts you have), and the bottom number is the denominator (or the number of parts the whole is divided equally into). Simple fraction

A fraction in which both the numerator and denominator are whole numbers or integers: for example, ¾.

Teaching points

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• Students need to display understanding of situations where a fraction of a part is a whole number and do so by using modelling; e.g. use of a number line or paper folding. • Students should be able to show understanding of operations with fractions by using counters and fraction grids, and be able to create a diagram of any situation involving the use of a fraction in an operation.

What to look for © R. I . C .Publ i cat i ons 1⁄3 1⁄5 •f orr evi ew pur posesonl y•

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• Students who have problems folding paper into and . They often fold quarters, which can create problems. The ability to fold paper to solve additive and subtractive problems using fractions is most important and should not be overlooked. • Students who switch between an understanding of fractions to thinking of denominators as whole numbers. • Many students when calculating 1⁄3 + 1⁄5 will give an answer of 2⁄8. They are applying whole-number processes to situations involving fractions and parts of a whole number. • When presented with problems such as 2⁄3 of 12, some students will think of the one-third as 3 and multiply by 2 to get 6, thereby disregarding the necessity to think of the 12 as one whole.

o c . che e r o t r s super

See also New wave Number and Algebra (Year 6) student workbook (pages 29–35)

Proficiency strand(s):

Student vocabulary fraction

Understanding Fluency Problem solving Reasoning

simple fraction numerator denominator whole number quantity

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 3

HANDS-ON ACTIVITIES Fractions of a quantity game Photocopy the fraction board, fraction table and quantity table onto an A3 sheet of thin card. Cut out and laminate the pieces and play the game in pairs. Refer to page 101 for instructions and game pieces.

‘Top that fraction’ card game

r o e t s Bo r e p ok u S

Play ‘Track your fractions’

R.I.C. Publications’ Maths games, by Margaret Grubb, Doreen Tonner and Christine Gallacher, provides for students to practise fractions. Game boards can be photocopied onto card and used with a timer, calculator and adult helper. There are eight different games to play, involving halves, quarters, thirds, tenths and fifths. Questions are provided in numerical and word form. R.I.C. Publications also has a set of six A2-sized maths games designed for Year 6 students which shows them how to work in groups to solve a range of fraction problems.

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Construct a set of cards where the students have to find fractions of money. This can be played in pairs with the students sharing the cards between them. Both turn over one card each, then work out how much their card is worth. The ‘winner’ of a round is the student whose card is worth more. That player then keeps both cards. Play continues in this way until one player has all the cards. Refer to page 102 for examples. Vary the amounts according to the fractions you are covering in class and the ability of your students.

English

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• Ask the students to make up ‘word problems’ to illustrate a fraction of a quantity; for example, ‘I invited 12 friends to my party. If 2⁄3 of them were girls how many girls, and how many boys did I invite?’ • Ask the students to calculate what fractions of given words are vowels or consonants. For example, with the word ‘level’, there are five letters in the word, two of which are vowels. As such, vowels form 2⁄5 of the whole word and consonants 3⁄5.

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Information and Communication Technology

• Students visit <http://www.whizz.com/maths/fractions/games/#> to play an interactive game that helps them recognise and find one-half of small numbers, and to recognise fractions that are parts of a whole. Each game has a preliminary activity which clearly explains how to complete the questions. • Students visit <http://www.skillsworkshop.org/resources/fractions-amounts-jigsaw> for suggestions for Tarsia jigsaw puzzles involving fractions of amounts. Students match triangular puzzle pieces which have questions about fractions of money to the correct answers to form a parallelogram. Follow the links to download a copy of a free Tarsia puzzle maker. (There is an example on page 104.) Once students can calculate fractions of quantities proficiently, allow them to construct their own triangular puzzles. • Visit <http://www.topicbox.net/R.E./fractions/> to complete an interactive activity to create flags of various designs using fractions of amounts in grids.

The Arts • Use a one-hundred grid on white paper with glued-on small squares of different colours to create a picture, design or pattern. Allow the students the freedom to be creative. When an artwork is completed, ask the students to calculate what fraction of one hundred is created by each different colour.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Strategies for finding simple fractions • To find a fraction of a quantity, divide that quantity by the number at the bottom of the fraction (the denominator). For example, to calculate 1⁄5 of a quantity, divide the number in the quantity by 5; to calculate 1⁄3 of a quantity, divide the number in the quantity by 3; to calculate 1⁄8 of a quantity, divide the number in a quantity by 8. • To find more than one part of a quantity—for example, 2⁄5, 2⁄3 or 3⁄8—use one of the following strategies. For each strategy, 3⁄5 of 20 needs to be found. — Strategy 1:

Divide 20 by 5 to find what 1⁄5 of 20 is:

r o e t s Bo r e p ok u S 20 ÷ 5 = 4, so 1⁄5 of 20 is 4.

To find 3⁄5 of 20, multiply 4 by 3.

4 x 3 = 12

So 3⁄5 of 20 is 12.

3 20 3 × 20 = × 5 5 1

Students may find it helpful to remember that if a question states, for example, ‘Find 2⁄3 of 90’, the word ‘of’ means ‘multiply’.

3 20 x 1 =5 60 = 5 60 divided by 5 is 12, so 3⁄5 of 20 is 12.

• Sometimes, when finding fractions of a quantity, the quantity itself may need to be changed into a more usable form. A small quantity might need to be changed to a larger number.

— Example 1

of $4

Change $4 to 400 cents to make the number easier to divide by 8. of 400 cents (or 400 ÷ 8) = 50 cents

so 1⁄8 of $4 (or 400 cents) is 50 cents. — Example 2

First change one hour into a larger number: 60 minutes.

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¼ of 60 minutes (or 60 ÷ 4) = 15 So ¼ of an hour is 15 minutes.

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• You can use diagrams to find fractions of quantities.

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¼ of an hour

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For example, the array shows 3⁄5 of 20. Therefore, 3⁄5 of 20 = 12. Finding fractions of quantities using a calculator

The use of digital technology (for example, a calculator) to solve problems of finding parts of a whole greatly simplifies the task. It’s a simple operation of dividing the quantity by the denominator of the fraction to be found, and then multiplying that result by the numerator of the fraction. For example, to use a calculator to find 3⁄5 of 20: • • • • •

100

Type in the quantity (20) Press the division button (÷). Type in the denominator (5) Multiply the result (20 ÷ 5 = 4) by the numerator by pressing the multiplication button (x) and typing in 3. Press the equals button (=) to find the result, which is 12. Therefore, 3⁄5 of 20 = (20 ÷ 5) x 3 = 12

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

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

Multiply 3⁄5 by 20

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— Strategy 2:

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET—FRACTION GAME PIECES Fraction board

100

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5r 25 o e t s Bo r e p ok u S

Fraction table

½ 2⁄3

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

Quantity table

o c 12 400 . 72 che e r o r st super 1⁄3 40 120 48 1⁄5

100

200 16 60 44

INSTRUCTIONS: Play the game in pairs. Each player selects a collection of counters of a colour different from that of his/her partner. The first player selects a fraction and a quantity; for example, ½ and 60. If ½ of 60 can be found on the fraction board, the player places his/her counter over it. Players then take turns, with the winner being the first player to place four counters in a row horizontally, vertically or diagonally.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

101

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Top that fraction–game cards

5⁄6 of $30

½ of $40

2⁄3 of $24

5⁄7 of $49

9⁄10 of $140

7⁄9 of $180

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m . u

o c . che e r o 5⁄8 of $240 1⁄3 of t r s $900 super

1⁄5 of $75

102

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

4⁄8 of $64

R.I.C. Publications® www.ricpublications.com.au

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

Teac he r

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r o e t s Bo r e p ok u ¾ of $100 6⁄10 of $10 S

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Fractions of amounts maze

½ of 12

¼ of 8

1⁄5 of 15

1⁄o 5 ofr 30 ¼ of 40 e t s B r e oo p k10 3u 6 S

Teac he r

1⁄8 of 64

1⁄8 of 40

¼ of 32 1⁄9 of 36 OUT © R. I . C.P ubl i cat i on s •f orr evi ew pur posesonl y• 1⁄3 of 24

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INSTRUCTIONS: Use the maze to practise finding fractions of amounts. Use the blank to create your own, or have the students try to create their own

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1⁄7 of 14

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

1⁄3 of 12

o c . che e r o t r s super

OUT

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

103

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET

.0 00

8⁄16

m . u

30 1 $ of

f $1 3⁄5 o

0 0 . 0 $2

$10

$750

1⁄8 o

f $2

$90

8⁄20 of $100

104

$25

1⁄

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8o f $ 2

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$65

1⁄5 o

w1⁄8 of $ ww

6⁄8 of $1000

1⁄3 of $270

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

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

$100

$70

f $8

f $ 3 00

f $1

3⁄8 o 1⁄3 o

0 0 . $2

0 $2 f o ½0

of $

1⁄10

7⁄10 of $100

1⁄5 of $45

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o ½0

1 $P

20 1 $ f

r o e t s Bo r e p ok u S 1⁄5 o

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Tarsia jigsaw puzzle example

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 3

NAME:

DATE:

1. Find the fractions of the quantities. Show your working and shade a bubble to show your answer.

(a)

(b)

2⁄6 of 18

¾ of 28

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1⁄5 of 35

(d)

7⁄8 of 24

(e)

2⁄3 of 15

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r o e t s Bo r e p ok u S

2. Use the diagrams to find the fractions of the quantities. Shade a bubble to show your answer.

(b)

(c)

3⁄7 of 49

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5⁄8 of 32

2⁄3 of 24

(d)

4⁄10 of 30

(e)

4⁄6 of 48

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

m . u

(a)

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

(c)

R.I.C. Publications®

www.ricpublications.com.au

105

Assessment 2

Sub-strand: Fractions and Decimals—F&D – 3

NAME:

DATE:

1. Use your calculator to find quarters of the quantities. (a)

¾ of 76

(b)

¾ of 112

(c)

¾ of 336

(d)

2⁄4 of 896

(e)

¼ of 1556

(f)

¼ of 2768

2. Use your calculator to find fifths of the quantities. (a)

1⁄5 of 280

(d)

3⁄5 of 1315

r o e t s Bo4⁄5 r e p ok u S (b)

2⁄5 of 435

(c)

(e)

3⁄5 of 1725

(f)

2⁄5 of 645

of 3405

(a)

1⁄6 of 438

(b)

2⁄6 of 582

(c)

3⁄6 of 822

(d)

4⁄6 of 1986

(e)

5⁄6 of 3258

(f)

2⁄6 of 7248

4. Use your calculator to find sevenths of the quantities. (a)

1⁄7 of 483

(b)

2⁄7 of 651

(c)

3⁄7 of 1729

(d)

4⁄7 of 3276

(a)

1⁄8 of 336

(b)

2⁄8 of 664

(c)

3⁄8 of 1000

(d)

4⁄8 of 4776

(e)

5⁄8 of 2944

(f)

7⁄8 of 5936

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6. Use your calculator to find ninths of the quantities. (a)

5⁄9 of 828

(d)

7⁄9 of 5733

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©R . I . C.Publ i cat i ons (e) 5⁄7 of 4935 (f) 2⁄7 of 2457 •f orr evi ew pur posesonl y• 5. Use your calculator to find eighths of the quantities.

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2⁄9 of 1035

(c)

4⁄9 of 3789

(e)

of 7272

(f)

of 8577

7. Use your calculator to find fractions of the quantities.

106

(a)

1⁄12 of 708

(b)

2⁄10 of 9780

(c)

3⁄15 of 5205

(d)

4⁄16 of 2880

(e)

1⁄16 of 10784

(f)

12⁄20 of 1540

(g)

22⁄25 of 825

(h)

32⁄50 of 3700

(i)

78⁄100 of 3600

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving addition and subtraction of fractions with the same or related denominators

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3. Use your calculator to find sixths of the quantities.

Checklist

Sub-strand: Fractions and Decimals—F&D – 3

Find a simple fraction of a quantity where the result is a whole number,

Finds simple fractions of a quantity where the result is a whole number with digital technologies

Finds simple fractions of a quantity where the result is a whole number without digital technologies

r o e t s Bo r e p ok u S

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STUDENT NAME

Able to calculate fractions (including visual representations) using the fraction as an operator of whole numbers

with and without digital technologies (ACMNA127)

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o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

107

Sub-strand: Fractions and Decimals—F&D – 4

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

RELATED TERMS

TEACHER INFORMATION

Decimal

What this means • Students should be able to explore and develop meaningful written strategies for addition and subtraction of decimal numbers to the thousandths position. • Students need to have a grasp of addition and subtraction involving decimal numbers to the thousandths position. • Students should be given opportunities to practise efficient methods for solving problems requiring operations with decimals. • Students need to gain fluency with calculating decimal numerals and with recognising appropriate operations to solve problems.

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Whole number

Whole numbers (integers) are numbers that do not have a part number ( i.e. not a mixed number). Examples are 1, 5, 27, –3, –31 and 105.

Teaching points

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The term decimal is used to describe a decimal numeral that includes a decimal point. Decimals are a way of writing fractions without using a numerator or denominator. For example, the fraction 7⁄10 can be written as the decimal numeral 0.7. A decimal numeral can be less than or greater than one. Decimals less than one are written with a zero before the decimal point (for example, 0.2), while decimals greater than one are written with the decimal point between the whole number(s) and the fraction (for example, 4.9).

• Introduce students to calculating with money and concentrating on numbers with two decimal positions. • Students need to play place value number games using a money place value grid which allows them to record calculations using the money as an abstraction. This will provide practice and familiarity with the way monetary amounts are written as they explore and practise efficient methods for solving problems requiring operations with two decimal positions. • Students develop fluency with calculating with decimals and with recognising appropriate operations through use of place value grids with decimal numerals in the thousandths position. • Student develop further fluency by solving questions related to measurement and deciding on what operation needs to be used to solve it.

the actual value, usually with some thought or calculation involved. Rounding

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Rounding means reducing the digits in a number while trying to keep its value similar. This is the common method of rounding:

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• Select the last digit to keep. • Increase the kept digit by 1 if the following digit is 5 or more. (This is called rounding up.) • Leave the kept digit the same if the following digit is less than 5. (This is called rounding down.) As an example, 243 rounded to the nearest ten is 240 (because 3 is less than 5). Student vocabulary

What to look for

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• Students who have place value problems; i.e. they do not understand place value or have constructed their own knowledge. (An example of this would be students who think the more numerals in the decimal position, the larger the number; for example, that 1.395 is larger than 2.7 because it has more digits.)

o c . che e r o t r s super

See also New wave Number and Algebra (Year 6) student workbook (pages 36–44)

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

decimal add subtract estimate rounding

108

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 4

HANDS-ON ACTIVITIES Decimals, money and measurement Relate decimals to practical activities to help students understand concepts more easily. Using and converting dollars and cents, centimetres and metres, and grams and kilograms—all multiples of 100 or 1000—brings decimals into practical context for students. Students can use ‘money’ to ‘buy’ items and calculate change or find the total of a shopping bill in dollars and cents amounts. They can find the difference between (or sum of ) distances using centimetres or metres, and can compare the difference in weight of items by subtracting grams or kilograms.

Decimal card practice

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LINKS TO OTHER CURRICULUM AREAS

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Use a black marker to write a variety of whole numbers and decimals on the back of blank playing cards. Shuffle and split the deck of cards between pairs of students. (You can adapt the number of cards given to each student, if necessary, so that the students have 10, 15 or 20 each. The students form two piles of cards and place them facedown in front of them. When the signal is given, the students quickly turn over two cards each (one card from each pile) and either add or subtract the decimals and whole numbers (depending on teacher instructions), then calculate the answer, and record it on a sheet of paper. The first to finish calls out ‘Decimal!’ and receives one point. When the second player finishes, they swap cards and answers and each checks the other’s calculation. If a calculation is correct that player receives two points. They continue revealing cards, calculating answers and checking each other’s work until all cards are done. The winner is the player with the most points. Students should be reminded to subtract the smaller number from the larger when carrying out subtraction sums.

• Some teachers may find Fractured fairy tales: Fractions and decimals – 25 tales with computation and word problems to reinforce key skills, by Dan Greenberg, a useful resource. Each story in the book provides a mathematical problem for the students to solve. Answers are also provided. Parting is such sweet sorrow: Fractions and decimals (Adventures in Mathopolis), by Linda Powley and Catherine Weiskopf, is another book which may prove useful. • Ask the students to make up a rhyme or rap to reinforce the correct positioning of the decimal point. Visit <http://www.youtube.com/watch?v=V6pZyY6mM3g> to listen to and watch a rap which gives students hints to do addition and subtraction of decimals.

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Information and Communication Technology

• Visit <http://www.math-play.com/decimal-math-games.html> to add, subtract and round decimals to the nearest whole number. There are a variety of games to choose from. • Visit <http://cemc2.math.uwaterloo.ca/mathfrog/english/kidz/addsubdec.shtml> to complete online decimal problems to the hundredths.

The Arts

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• The positioning of the decimal point plays an important part in correctly adding and subtracting decimals. To reinforce this fact, complete this activity. On a half-width sheet of A4 or A3 paper, ask the students to use a black marker to divide the paper into eight columns and three rows. Using fancy writing (such as bubble writing), the students label each column, making sure the words ‘decimal point’ stand out. They write two decimal numbers provided by the teacher in the correct columns. The decimal points directly underneath each other should be decorated in some way to make them stand out. They could be made into a caricature, or surrounded by concentric shapes. The students colour or decorate the columns matching tens and tenths, and thousands and thousandths.

thousands

hundreds

tens

ones

DECIMAL POINT

tenths

hundredths

thousandths

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Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Written strategies for addition and subtraction of decimals (to the thousandths) It is essential that when writing addition and subtraction problems involving decimals, that students remember to: (a) write the problems correctly as it is given with by the teacher, including positions of decimal points

for example: 1.431 + 2.525 Line up the decimal points. 1.431 +

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Line up the digits

3.956

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2.525

(d) add zeros (if need be) to ensure both numbers have the same number of decimal places. Note: Any number of zeros can be added to the right of the decimal point of a whole number without changing the value of the number. for example: 7 – 3.859

Add zeros to give the whole number the same number of decimal places.

Line up the decimal points.

Line up the digits.

3.859

3.141

(e) ensure that the decimal point is placed in the correct place in the answer.

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Some students may find it helpful to draw models to illustrate problems; for example: 0.7 + 0.8 = 1.5

0.7

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Students should check answers by using a calculator. If checking answers to subtraction problems, they should add the answer to the decimal amount subtracted to see if the total is the same as the first number given.

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0.8

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers

(b) line the decimals up vertically so the decimal points are underneath each other. All other numbers must also line up according to place value. If this is not done correctly, the students will add or subtract the wrong numbers to and from each other and obtain an incorrect answer;

Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Mental strategies for addition and subtraction of decimals Rounding is a technique used to estimate approximate values. Rounding is most often used to limit the number of decimal places, making it easier and quicker to complete calculations. Rounding with decimals can be done to any decimal place. Following rounding, a digit will either:

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• Rounding up: To round up, we increase the terminating digit by 1 and remove all the digits to the right of it. If the place immediately to the right of the terminating digit is greater than or equal to 5, we round up.

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For example, if we round 7.56 to the tenths decimal place (where the ‘5’ is), we would round the number UP to 7.6 because the 7 in the hundredths decimal place is greater than 5.

• Rounding down: If the number to the right of the terminating digit is 4 or less, we round down. The terminating decimal digit is left unchanged and all digits to the right are discarded. For example, if we round 4.734 to the nearest hundredths decimal place, we would round the number DOWN to 4.73 and discard the ‘4’. Note: Students need to be very aware of the decimal value to which rounding is to occur. This provides the terminating digit. Students must be aware of all decimal place values to round up or round down correctly.

For example, round 0.23 to the tenths place. The terminating digit is ‘2’ (which is in the tenths place in decimal place value), and because the next digit to the right of 2 is smaller than 5, we round down by dropping the 3. So when we round 0.23 to the tenths place, we get 0.2.

HUNDREDTHS

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Banker’s rounding

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TENTHS

DECIMAL POINT

ONES

Some students may wish to write a decimal they are rounding in a decimal place value chart to make it easier to see and understand each place value.

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CONTENT DESCRIPTION: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers

– stay the same (this is called rounding down), or – increase by 1 (this is called rounding up). So, when do we round up and when do we round down?

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If using rounding for reasons of estimation, this method can give more accurate results. It involves rounding to the nearest even number. If the first digit to be dropped is a ‘5’ and there are no digits to the right of it or if the digits are zeros, make the preceding digit an even number. For example, when rounding 2.325 to the tenths place, instead of rounding up to 2.33, round to 2.32 instead. Hints to aid computations using decimals

• Remind students that all the names of decimal place values end with ‘th’—tenths, hundredths and thousandths. • Some students may be aided by saying decimals aloud and stating 8.02 can be said as ‘eight and two hundredths’, rather than ‘eight point zero two’.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers

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Thousands

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Decimal point Ones Tens

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Hundreds

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Tenths

Hundredths

Thousandths

Decimal place value chart

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 4

NAME:

DATE:

Use the following steps to add decimals: (a)

Show your working of the algorithm.

(b)

Use estimation or rounding to check if the answers are reasonable. Show how you estimated or rounded.

(c)

Check your answers, using a calculator.

Tick each step as you complete it.

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Working/calculation

Calculator check

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1. 13.753 + 26.241

2. 60.812 + 35.749

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CONTENT DESCRIPTION: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers

Algorithm

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5. 691.36 + 763.024

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6. 434.302 + 102.598

7. 690.975 + 43.053

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 4

NAME:

DATE:

Use the following steps to subtract decimals: (a)

Show your working of the algorithm.

(b)

Use estimation or rounding to check if the answers are reasonable. Show how you estimated or rounded.

(c)

Check your answers, using a calculator.

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Algorithm

Calculator check

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1. 94.531 – 50.221

Working or rounding

2. 681.047 – 360.025

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4. 408.658 – 76.903

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5. 1495.465 – 928.78

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6. 598.342 – 180.201

7. 2050.088 – 340.255

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers

Tick each step as you complete it.

Checklist

Sub-strand: Fractions and Decimals—F&D – 4

Add and subtract decimals, with and without digital technologies, and use estimation and rounding to

Uses rounding to check reasonableness of answers when adding and subtracting decimals

Uses estimation to check reasonableness of answers when adding and subtracting decimals

Subtracts decimals with digital technologies

Subtracts decimals without digital technologies

Adds decimals with digital technologies

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STUDENT NAME

Adds decimals without digital technologies

check the reasonableness of answers (ACMNA128)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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115

Sub-strand: Fractions and Decimals—F&D – 5

Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and without digital technologies (ACMNA129)

TEACHER INFORMATION

RELATED TERMS

What this means

Decimal

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Whole number

Whole numbers (integers) are numbers that do not have a part number (i.e. not a mixed number). Examples are 1, 5, 27, –3, –31 and 105. Multiplication

Multiplication is the operation that consists of adding a number (the multiplicand) to itself a certain number of times (determined by the multiplier). It can be considered repeated addition. Multiplication is the inverse of division.

• Students should be given the chance to use grid paper or MABs to model multiplication of decimals. Also, it is important to use language that makes students realise that they are dealing with multiple groups of a given quantity: e.g. 12 x 1.5 is 12 lots of 1.5. • Students should be familiar with the formal representation of a modelled problem; that is, the written algorithm. • Students should also be given the chance to solve multiplication word problems involving measurement and money amounts to three decimal values places. • Students should be given the chance of interpreting and representing the remainder in division calculations; e.g. 15 ÷ 4 = 3¾ or 3.75. To help with understanding, students could model remainders (parts of a whole) with MABS. • Also make the link between a modelled problem and its formal algorithm. • This content description suggests that students should be familiar with a range of word problems involving multiplication of decimal numbers by a whole number and division with terminating decimals. • Problems should be solved with and without the aid of calculators.

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Division is the operation of determining how many times one quantity is contained in another. It is the inverse of multiplication. In simple terms, division involves splitting a quantity into equal parts or groups (or fair sharing). Terminating decimal

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A terminating decimal number has a finite number of digits. For example, 2.57 is a terminating decimal because it ends after a specific number of digits. Decimals such as 0.33333 are not terminating decimals because they continue with no final decimal point. These decimals are called non-terminating (or recurring) decimals. Student vocabulary decimal multiply divide whole number terminating decimal

Symbols x (multiplication) ÷ (division) = (equals)

116

Teaching points

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Division

• Students need to be able to multiply decimals by whole numbers; for example: 21 x 3.6, or a money problem such as, ‘How much do four packets of sweets cost at $1.72 a packet?’ • Students need to be able to interpret and represent the remainder in division calculations e.g. 15÷ 4 = 3¾. This includes understanding how the remainder can be displayed as a number with several decimal place values.

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• Some students will treat multiplication and division as addition and try to line up the decimal points as they would for addition of decimal numerals. For students who have problems, use a division grid. See also New wave Number and Algebra (Year 6) student workbook (pages 45–49)

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 5

HANDS-ON ACTIVITIES Dicey decimals Write a series of whole and decimal numbers on a sheet of paper or cards, and have the students throw a six-sided dice. The number rolled is the whole number by which the decimal number is multiplied. Students record the algorithm and answer on a recording sheet.

Decimal bingo

0.12

0.48

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0.7 23.4 x2 0.04 0.8 26.88 x3 1.602 6.408 9.348 x4 0.2 x4 1.4

4.89 0.332 0.166 0.55 1.63 4.9 24.5 x2 x3 x5 0.04 x 12.54 0.11 8.96 10.8 12 x5 x3 2.6 4.0 4.674 2.0 12.6 x9 x2 x2 0.9 x 4.18 4.2 12 x3 x3

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Get individuals, or groups of students, to make up their own decimal bingo game cards for other groups to play with. Provide each student, or group, with a blank bingo card and a blank caller’s card. Each has up to five columns and rows on it. On each square of the caller’s card, the students write a division or multiplication algorithm consisting of a decimal number and a whole number. On the blank bingo card, the students write the answers to the multiplication and division problem. The students work out the answers using a calculator or other method and record them on a separate sheet of paper. Provide the students with three or four blank bingo cards. They should select one card to write a the five correct answers in a row vertically, horizontally or diagonally. The other spaces should be filled with unlikely, random answers made up by the students. On the other two or three blank player’s cards, the students record the remaining answers in random placement and fill the blanks with random answers. All correct answers should be used up so the students have the opportunity to complete calculations with decimals.

When the game is played by another group of students, the winner is the first one to call out ‘Decimal bingo’ when a row of five counters is formed by counters placed on the correct answers. Students should evaluate the success or failure of the game and adjust it as necessary.

English

INKS TO OTHER CURRICULUM AREAS © RL. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

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• Make multiplication and division of decimals by whole numbers relevant to students by constructing, as a class, scenarios and word problems where these operations are used; For example: ‘A plumber earns $21.80 an hour. If he works for eight hours a day, how much will he earn each day (or over a five-day week)’. • Teachers may find some of the information and activities in Delightful decimals and perfect percents by Lynette Long, useful for this and the content description associated with code F&D – 7.

Information and Communication Technology

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• Visit <http://www.youtube.com/watch?v=EZ4KI0pv4Fk> to view one method of multiplying decimals by whole numbers. • Visit <http://www.youtube.com/watch?v=vlC0-UWbPHs> to view a written method for dividing decimals by whole numbers by long division. • Visit <http://www.math-salamanders.com/math-games-fifth-grade.html> to download a sample of a decimal tables challenge (a game for two players). Adjust the game board to include only decimal and whole numbers.

The Arts

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• Ask the students to design and make a poster which includes a list of ideas to help them remember how to multiply and divide decimals by whole numbers. They should decorate around the bullet points with a mathematical design. This activity could also act as a possible assessment activity to gauge student understanding of the processes required.

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Sub-strand: Fractions and Decimals—F&D – 5

RESOURCE SHEETS Strategies for multiplication and division of decimals by whole numbers Multiplying decimals by whole numbers • Estimating the product to get an approximate answer before calculating will help with telling whether there is an error in the calculation or with the placement of the decimal point. For example: 22.3 x 38 Round 22.3 down to 20 and then multiply: Estimate 20 x 40 = 800

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• When multiplying a decimal number by a whole number, most commonly the steps are as follows: (a) Disregard the decimal point and treat the number as whole numbers.

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(b) Multiply the two numbers as you normally would whole numbers.

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(d) Counting from the right to left, place the correct number of decimal numerals in the answer. For example: 6.32 x 6

Disregard the decimal point. Multiply the two numbers as you would normally. 632 x 6 = 3792

Count and add the number of decimal numerals from the original number to the answer.

37.92

So, 6.32 x 6 = 37.92

Decimal placement is very important when multiplying and dividing with decimals.

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• Multiplication by a whole number is repeated addition. Some students may find this strategy useful for simple multiplication. For example: 4 x 0.05

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= 0.05 + 0.05 + 0.05 + 0.05 = 0.10 + 0.10

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• Some students may find the use of a visual representation (such as a number line) helpful. For example, take the simple example of 2.6 x 3. This algorithm means 2.6 + 2.6 + 2.6 and can be represented as:

2.6

4 2.6

7.8 8

2.6

This action assumes prior knowledge of fractions and equivalence.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and without digital technologies

The actual calculation gives an answer of 847.4, so the estimate means that the calculation is most likely correct. If the answer arrived at was 84.74, it is more than likely a mistake (like a misplaced decimal point) was made.

Sub-strand: Fractions and Decimals—F&D – 5

RESOURCE SHEETS Strategies for multiplication and division of decimals by whole numbers

Figure 1

3 2.9 8 x 2 7

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– Multiply 2 (the first digit in 27) by all the digits across the top and, separating the answers into digits, write the answers in each lattice section opposite 2. (Figure 2) – Multiply 7 (the last digit in 27) by all the digits across the top and, separating the answer into digits, write the answers in each lattice section opposite 7. (Figure 3) – Now add the numbers diagonally and write the answer at the bottom. For two-digit answers, carry the additional digits to the next diagonal to the left. (Figure 4) – Finally, place the decimal point in the correct place by counting the digits to the right of the decimal place in the top number. (Figure 5) So (reading around the lattice from left to right), the final answer to 32.98 x 27 is 890.46! (Disregard the zero at the beginning.)

Figure 2

3 2.9 8 x 0 0 1 1 2 6 4 8 6 7

3 0 6 2 1

Figure 3

2.9 8 x 0 1 1 2 4 8 6 1 6 5 7 4 3 6

Figure 4

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– First, decide how big your lattice will be. The number 32.98 has four digits and 27 has two, so the grid will be four columns wide by two rows high. The decimal point goes exactly above the line separating ‘2’ and ‘9’. Write the digits in the correct place as shown, then divide the sections in half to make the lattice or grid. (Figure 1)

x 0 0 1 1 © R. I . C.Pu b l i c a t i ns 0 6 4 8 6 2o 8 21 14 63 56 7 •f orr evi ew pu r posesonl y•

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3 2.9 8 2

3 0 06 8 21 9

Figure 5

2.9 8 x 0 12 11 2 4 8 6 1 6 5 7 4 3 6 0.4 6

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CONTENT DESCRIPTION: Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and without digital technologies

• The Italian lattice method can be used to complete multiplication of decimals as a written strategy. (Refer to page 28.) For example, 32.98 x 27:

• Some students may find that a ‘visual’ activity helps to consolidate the concept. This can involve the use of small 100 grids or base ten block ‘flats’, ‘longs’ and ‘minis’. (This strategy implies knowledge of fraction–decimal correlation and can be used for content description associated with code F&D – 6.) For example, 0.65 x 7. For this activity, the student will create 0.65 (or 6⁄100 seven (7) times) …

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and then convert them to the smallest number of components.

This gives 42 tens and 35 ones or 42 tens and 3 tens and 5 ones or 45 tens and 5 ones. This gives 455; however, the decimal point has to be reinserted. This can be done by counting the number of decimal positions in the original numeral (0.65), which is two. So, divide 455 by 100 to reach 4.55. As such, 0.65 x 7 = 4.55 Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 5

RESOURCE SHEETS Strategies for multiplication and division of decimals by whole numbers Dividing decimals by whole numbers • When dividing decimals by whole numbers, place the decimal point in the quotient directly above the decimal point in the dividend. Then, divide the numbers as though they are whole numbers; for example: quotient dividend

The decimal points must be kept aligned.

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• When dividing by a whole number, it may sometimes be necessary to add one or more zeros to the end of a decimal. 0.48 2 4 For example: 5 2.40

• Dividing a decimal by a whole number is no different from dividing with only whole numbers. Some students may find it useful to first convert each number by the same number of place values (tens, hundreds or thousands) until the decimal number is a whole number. For example, 22.62 ÷ 6. Move the decimal point two places to the right (or by one hundred) to make the numbers 2262 and 600, then divide as whole numbers. As long as what is done to one number is done to the other, the values remain the same.

So 2262 ÷ 600 = 3.77.

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All estimates should be checked using a calculator. Remember:

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• Estimating before calculating may help students see how reasonable their answers are and assist with the placement of decimal points. Rounding numbers helps estimation. One way is to round both numbers to the same place value, then calculating using the rounded numbers. For example, to estimate the quotient of 152.337 ÷ 0.51, round 152.337 to 150, and 0.51 to 0.5. To convert 0.5 to a workable whole number, move the decimal point one place to the right to get 5. We must also then add an extra place value to 150 to get 1500, so our estimate becomes 1500 ÷ 5 = 300. 152.337 ÷ 0.51 equals 298.7, so we can see that our estimate is very close.

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• When multiplying a decimal by a whole number, if there is one digit after the decimal place in the algorithm, there will be one digit after the decimal point in the answer. If there are two, there will be two in the answer, and so on. • When dividing a decimal by a whole number, make sure the decimal points are aligned to get the correct place value in the answer.

120

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and without digital technologies

18 . 25 14 255 . 50 –14 115 –112 35 –2 8 70 –70

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divisor

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 5

NAME:

DATE:

1. Use the following steps to multiply decimals by whole numbers. • Show your working of the algorithm. • Use estimation or rounding to check if your answers are reasonable. Show how you estimated or rounded. • Check your answers using a calculator. Tick in the column after checking. Algorithm

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Calculator check

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Working or calculation

(b) 5 x 22.345

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CONTENT DESCRIPTION: Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and without digital technologies

(a) 0.004 x 61

Estimation or rounding

2. Solve the following word problems. Shade a bubble to show your answer.

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(a) Julie buys two pairs of jeans, which cost $39.95 each. What is the total cost?

(b) Dad puts 55 litres of petrol in his car. If each litre costs $1.56, what is the total? (c) Aunt Milly can knit 4.65 cm in one hour. How many centimetres can she knit in one day?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

$79.90

$7.90

$68.50

$790

$8.58

$85.80

$55.00

$88.50

46.50

11.16

111.6

1116

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 5

NAME:

DATE:

1. Use the following steps to divide decimals by whole numbers. • Show your working of the algorithm. • Use estimation or rounding to check if your answers are reasonable. Show how you estimated or rounded. • Check your answers using a calculator. Tick in the column after checking. Estimation or rounding

Calculator check

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(a) 28.07 ÷ 35

Working or calculation

(b) 0.456 ÷ 12

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(d) 1.968 ÷ 16

2. Solve the following word problems. Shade a bubble to show your answer.

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(a) What is the cost of one hotdog, if two cost $11.50?

(b) Jack weighs 89.56 kilograms. If his little sister weighs half of his weight, how many kilos does she weigh? (c) If a roll of fabric is 459.448 cm long, how many centimetres long would one-quarter of a roll be? 122

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$57.50

45.23 kg 34.78 kg

$5.50

$5.75

44.78 kg

4.478 kg

110.5 cm

114.862 cm

1148.62 cm

11.468 cm

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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CONTENT DESCRIPTION: Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with an without digital technologies

Algorithm

Checklist

Sub-strand: Fractions and Decimals—F&D – 5

Multiply decimals by whole numbers and perform divisions that result in terminating decimals, with and

Divides decimals by whole numbers, resulting in terminating decimals, using digital technologies

Multiplies decimals by whole numbers, resulting in terminating decimals, using digital technologies Divides decimals by whole numbers, resulting in terminating decimals, without using digital technologies

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STUDENT NAME

Multiplies decimals by whole numbers, resulting in terminating decimals, without using digital technologies

without digital technologies (ACMNA129)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Sub-strand: Fractions and Decimals—F&D – 6

Multiply and divide decimals by powers of 10 (ACMNA130)

TEACHER INFORMATION

RELATED TERMS

Powers of 10

What this means • Students need to be able to multiply a decimal number by 10,100 and 1000, and divide any number by 10, 100 and 1000. • Students should see the importance of the number of decimal places in the answer being the same as those in the problem. For example, 34.87 ÷ 7 is equivalent to 3487 ÷ 700. (Notice that both numbers have been multiplied by 100 to change the problem from a decimal problem to a whole number problem.) • Multiply decimal numbers mentally. For example, 1.4 × 0.6 can be calculated by multiplying 14 by 6 and dividing the result by 100. • Links should be made with everyday use of decimals (e.g. measurement) so that realistic problems can be encountered. • Problems should be solved with and without calculators.

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The power of a number tells how many times to use the number in a multiplication. It is usually expressed as a number above and to the right of the number which is multiplied. For example, 92 means nine multiplied twice (or 9 x 9); 63 means 6 x 6 x 6 (six multiplied by itself three times). It is expressed as ‘nine to the power of two’ or ‘six to the power of three’. A power can also be called an ‘index notation’ or ‘exponent’. Powers of ten express how many times ten is multiplied by itself; for example, 101 (ten to the power of one) is 10. Other powers of ten are expressed as follows: 102 = 100, 103 = 1000, 104 = 10 000. Note: 100 = 1 Powers are useful for expressing very large or very small numbers (such as in scientific notation). For example, one light year = 9.4605284 × 1015 metres. Powers can be positive or negative. Negative powers can be used to show decimals;

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The term decimal is used to describe a decimal numeral that includes a decimal point. Decimals are a way of writing fractions without using a numerator or denominator. For example, the fraction 7⁄10 can be written as the decimal numeral 0.7. A decimal numeral can be less than or greater than one. Decimals less than one are written with a zero before the decimal point (for example, 0.2), while decimals greater than one are written, with the decimal point between the whole number(s) and the fraction (for example, 4.9)

• Students should be given opportunities to see the effect of dividing the same number by 1, 10, 100 and 1000; e.g. 125 ÷ 1, 125 ÷ 10, 125 ÷ 100 and 125 ÷ 1000. • Students can model these concepts by using MABs, counters, grid paper and calculators. • Students should understand the effect of multiplying a decimal number by 1,10,100 and 1000; e.g. 1.25 x 1, 1.25 x 10, 1.25 x 100 and 1.25 x 1000.

What to look for

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for example: 10–1 = 1⁄10 or 0.1 10–2 = 1⁄100 or 0.01 10–3 = 1⁄1000 or 0.001

Teaching points

• Some students will treat multiplication and division as addition and try to line up decimal points as they would for addition of decimals.

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See also New wave Number and Algebra (Year 6) student workbook (pages 50–51)

Proficiency strand(s):

Student vocabulary decimal power multiplication division positive negative decimal point

124

Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 6

HANDS-ON ACTIVITIES Use base ten blocks Base ten blocks can be used to demonstrate powers of 10. They provide a concrete basis for visual-spatial learners. Base ten blocks can be used to show 100, 101, 102 and 103 using minis (100), longs (101), flats (102) and cubes (103). Developing mathematics with base ten, by Dr Paul Swan and Geoff White, has many activities which relate to this content description.

Decimal dice activity

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For this activity, the students will need a ‘powers of ten’ spinner, counters (or sweets), a number of six-sided dice and a small container to place the dice in, and pencil and paper. Teachers will need to state whether the activity will be multiplication or division by powers of ten. Place three or four dice in a container, shake and reveal. The students use the numbers thrown to write a three- or four-digit decimal number. They then use a counter (or chocolate chip, Smartie™ or M&M™) to show where they want the decimal point to be. Spin the spinner to see what power of ten to use to multiply or divide the decimal number by the students then move the ‘decimal’ (counter/sweets) to the correct place to show the product. The students record the answer on their sheet of paper. They may need to include extra zeros to place the decimal point in the correct position. If using chocolate chips or sweets, students could be allowed to eat each ‘correct’ product/answer when finished. (Refer to page 126 for examples of a ‘powers of 10’ spinner to make.)

Powers of ten aid

Ask the students to divide a one-metre length of cardboard into 20 divisions to create an easy-to-refer-to aid. Include the positioning of 100.

English

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• Refer to the chart on page 127 for powers of ten terms. Students may be interested in the relationship among the power of ten, the name given to each, and how the terms relate to such things as computer technology, science or other fields: ‘tera’, ‘giga’, ‘nano’, ‘centi’, ‘milli’, ‘hecto’ etc.

Information and Communication Technology

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• Visit <http://www.themathpage.com/ARITH/Ar_Pr/mpow_1.htm> to answer questions involving multiplying and dividing decimals by powers of ten.

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Health and Physical Education

• Use the students themselves as ‘decimal points’. Write large decimal numbers in chalk on the footpath or concrete area with the decimal point coloured as a large dot. The students stand on the decimal point and when asked to multiply by one power of 10 (or 101), they rub out the decimal point and move one digit to the left. Each digit space reflects one power of ten. They then redraw the decimal point in its new position and view the new number. To divide by powers of ten, the students move digit places to the right. • This activity may be played in teams, with each team member given a list of decimal sums to multiply or divide by a power of ten, with each answer written down when completed. The first team to get all correct is the winner.

Science • Watch a short video at <http://www.youtube.com/watch?v=0fKBhvDjuy0> to see how powers of ten can be used to express very large and very small numbers. The video takes viewers, by units of powers of ten in metres, from ground level to the outer reaches of the galaxy, and then moves inside a hand to magnify the smallest components. The video clearly demonstrates the use of positive and negative powers of ten in the real world. This content description can be related to the use of microscopes and telescopes. Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

125

Sub-strand: Fractions and Decimals—F&D – 6

RESOURCE SHEET Powers of ten spinners (for use with the decimal dice activity)

–1

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101

o c 10 . c e her r o t s super 3 2

126

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

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105

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100

INSTRUCTIONS: Adjust the powers of 10 as necessary. Photocopy onto cardboard, laminate and add the arrow or toothpick to spin and select powers of ten

Secure an arrow with a split pin or insert a toothpick through the centre. Spin to choose a power of ten to multiply or divide decimals as directed by the teacher. Use the blank spinners to select your own powers of ten to use.

Sub-strand: Fractions and Decimals—F&D – 6

RESOURCE SHEET Powers of ten chart

Powers of ten

Number

Name

Prefix

1024

1 000 000 000 000 000 000 000 000

septillion

yotta

1021

1 000 000 000 000 000 000 000

sextillion

zetta

1 000 000 000 000 000 000

quintillion

exa

1 000 000 000 000 000

quadrillion

peta

1 000 000 000 000

trillion

tera

109

1 000 000 000

billion

giga

106

1 000 000

million

mega

103

1000

thousand

kilo

1018

1012

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1015

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100 hundred © R. I . C. Publ i cat i on s ten 10 •f orr evi ew10 pur poseson l y• 102 1

deca

one

–

10–1

0.1

one-tenth

deci

0.01

one-hundredth

centi

10 –3

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100

10–2

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

hecto

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one-thousandth

milli

0.000001

one-millionth

micro

0.000000001

one-billionth

nano

10–12

0.000000000001

one-trillionth

pico

10 –15

0.000000000000001

one-quadrillionth

femto

10–18

0.000000000000000001

one-quintillionth

atto

10–21

0.000000000000000000001

one-sextillionth

zepto

10–24

0.000000000000000000000001

one-septillionth

yocto

10–6 10–9

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

127

Sub-strand: Fractions and Decimals—F&D – 6

RESOURCE SHEET (Some) Powers of ten with factors chart

Fractions and factors of powers of ten have been included, because some students may find it easier to understand the concepts if factors are included.

Power of ten

Factors

1010

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

109

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108 107

10 x 10 x 10 x 10 x 10 x 10 x 10

106

10 x 10 x 10 x 10 x 10 x 10

105

10 x 10 x 10 x 10 x 10

104

10 x 10 x 10 x 10

103

10 x 10 x 10

100

—

10–1

1⁄10

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10–2

10–3 –4

10–5 10–6

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10–8 1

10–9

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or 1 (100) or 1 (1000)

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10–7

10–10

(10 x 10)

(10 x 10 x 10)

(10 x 10 x 10 x 10)

or 1 (10 000)

(10 x 10 x 10 x 10 x 10)

(10 x 10 x 10 x 10 x 10 x 10)

or 1 (10 000)

or 1 (1 000 000)

(10 x 10 x 10 x 10 x 10 x 10 x 10)

or 1 (10 000 000)

(10 x 10 x 10 x 10 x 10 x 10 x 10 x 10)

or 1 (100 000 000)

(10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10)

or 1 (1 000 000 000)

(10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10)

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

or 1 (10 000 000 000)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

101

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10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

Sub-strand: Fractions and Decimals—F&D – 6

RESOURCE SHEETS Strategies and hints for multiplying and dividing decimals by powers of ten Each decimal place is one power of ten • Students will need a good knowledge of place value and powers of ten before multiplying or dividing decimals using powers of ten. They should know that: 100 = 1, 101 = 10, 102 = 100, 103 = 1000, 104 = 10 000 and that 10–1 = 0.1, 10–2 = 0.01, 10–3 = 0.001, 10–4 = 0.0001

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• When multiplying or dividing decimals by powers of ten, if extra decimal places are added and there are no numbers, zeros must be added. For example, 64.36 x 103 (or 64.36 x 1000) = 64 . 3 6 0 = 64 360

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• Teachers should ask students to identify the quantity of zeros associated with each power of ten, or ask students to note the differences and similarities between the patterns that occur with number size when multiplying by powers of ten and dividing by powers of ten. As can be seen above, the power tells how many zeros to add to the number, but when dividing 10 by a power of ten (for example, 10–2),, only one zero is added after the decimal point instead of two. (a) When multiplying decimals by positive powers of ten, the index or power tells how many places to move the decimal point to the RIGHT. For example, 8.89 x 104 = 8.89 x 10 x 10 x 10 x 10 (10 000) = 88 900

The decimal point moves four places to the right, so 8.89 becomes 8 . 8 9 0 0 (Note: Extra zeros had to be added.)

(b) When dividing decimals by positive powers of ten, the index or power tells how many places to move the decimal point to the left.

For example, 8.89 ÷ 104 = 8.89 ÷ (10 x 10 x 10 x 10) 10 000 = 0.000889

(c) When multiplying decimals by negative powers of ten, this should be considered the same as dividing by powers of ten, so the index or power tells how many places to move to the left. For example, 8.89 x 10–4 = 8.89 x

= 0.000889

For example, 8.89 ÷ 10–4 = 8.59 ÷ 1⁄10 000 = 88 900

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(d) When dividing decimals by negative power of ten, this should be considered the same as multiplying by powers of ten, so the index or power tells how many places to move to the right.

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

• When multiplying or dividing decimal numbers by powers of ten, the digits do not change. Only the position of the decimal point changes. Example 1: 1.35 x 102 = 135.0 (or 135) Example 2: 5.2 ÷ 103 = 0.0052

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Example 3:

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432.1 x 10–1 = 43.21

Example 4:

68.98 ÷ 10–3 = 68 980

Note: Teachers should ensure that students feel confident multiplying and dividing decimals by positive powers of ten before asking them to multiply and divide decimals by negative powers of ten.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

129

Sub-strand: Fractions and Decimals—F&D – 6

RESOURCE SHEETS Strategies and hints for multiplying and dividing decimals by powers of ten (continued) • Students might find it helpful to construct a simple diagram using a selected number to help them remember how to multiply and divide numbers, including decimals, by powers of ten. An example is shown below:

0.1234 1. 234 12.34 dividing multiplying 123.4 1234.0 12340.0

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• Students can use powers of ten to aid calculations, such as those involving multiplication and division of decimals by whole numbers. • Students may multiply both numbers in an algorithm by powers of ten to: – check the reasonableness of their answers, or – work out answers quickly. Example 1: 98.64 ÷ 7 becomes 9864 ÷700 by multiplying both numbers in the equation by 100 or 102. When each number is multiplied or divided by the same number, the calculation will remain the same because the numbers are equivalent. Example 2:

3.2 x 0.8 becomes 32 x 8, with the answer being 256. But since both numbers were multiplied by 10, we need to divide the answer by 100 to keep the original place value correct.

So, 3.2 x 0.8 = 32 x 8 (x 10) = 256 ÷ 100 = 2.56

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

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• Metric measurements (such as kilograms, metres, and money) are ideal as units to incorporate into word problems involving powers of ten because they are units of measurement composed of tens, hundreds and thousands. Using these units in problems involving powers of ten will bring those concepts into real-life situations for the students.

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 6

NAME:

DATE:

1. Write the product, then write each as a power of 10. (a)

10 x 10 =

(b)

10 x 10 x 10 =

(c)

10 x 10 x 10 x 10 x 10 =

(d)

10 x 10 x 10 x 10 =

or or or

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2. Write each number. 100 =

(c)

107 =

(b)

106 =

(d)

109 =

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(a)

3. Write the answer as a decimal, then write each as a power of ten. (a)

1 ÷ 100 =

(b)

1 ÷ 10 =

(c)

1 ÷ 1000 =

(d)

1÷ 10 000 =

(e) (f)

(a)

w ww (c)

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

10 –1 = 10 – 3 =

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(b)

10 – 6 =

(d)

10 – 9 =

5. Shade a bubble to show your answer. (a)

(b)

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1 000 000 000 000 = 10

–10

–9

-3

-1

-4

-2

-1

0.000000001= 10

(c)

1 000 000 000 000 000 000 = 10

(d)

0.001 = 10?

(e)

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4. Write each number.

1 = 10

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

131

Assessment 2

Sub-strand: Fractions and Decimals—F&D – 6

NAME:

DATE:

1. Multiply the decimals by the power of ten given. (a)

9.193 x 100 =

(b)

48.602 x 100 =

(c)

570.1 x 101 =

(d)

12.97 x 101 =

(e)

0.613 x 102 =

(f)

5.2 x 102 =

(g)

327.649 x 103 =

(i)

0.46 x 10 =

r o e t s Bo r e p ok u S (h)

26.043 x 103 =

(j)

84.02 x 104 =

(a)

622.09 ÷ 100 =

(b)

158.35 ÷ 100 =

(c)

70.44 ÷ 101 =

(d)

0.432 ÷ 101 =

(e)

0.913 ÷ 102 =

(f)

47.543 ÷ 102 =

(g)

865.07 ÷ 103 =

(h)

0.062 ÷ 103 =

(i)

34.16 ÷ 104 =

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2. Divide the decimals by the power of ten given.

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= 5932.0

(f)

0.087 x 100 =

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4. Write the missing number or power of ten in each division algorithm. 4

(a)

÷ 10 = 0.3569

(c)

689.7 ÷ 102 =

(e)

842.163 ÷

(b)

341.6 ÷

÷ 101 = 0.00829

(d)

= 842.163

(f)

= 0.3416

4.201 ÷ 102 =

5. Convert each second number to a power of ten, then complete the algorithm.

132

(a)

2.8 ÷ 100 =

(b)

0.047 x 1000 =

(c)

0.32 x 10 =

(d)

0.06 ÷ 1000 =

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

593.2 x

0.46 x

(d)

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(e)

7.598 x 103 =

(b)

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x 104 = 92320.00

(a) (c)

Assessment 3

Sub-strand: Fractions and Decimals—F&D – 6

NAME:

DATE:

1. Multiply the decimals by the negative power of ten given. (a)

1.193 x 10–3 =

(b)

75.246 x 10–2 =

(c)

678.9 x 10–1 =

(d)

4.753 x 10–4 =

(e)

3.593 x 10–1 =

(f)

528.23 x 10–2 =

(g)

906.83 x 10–3 =

(h)

809.01 x 10–3 =

(i)

24.66 x 10–4 =

(j)

501.59 x 10–4 =

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(a)

60.97 ÷ 10–1 =

(b)

124.56 ÷ 10–3 =

(c)

83.03 ÷ 10–2 =

(d)

0.572 ÷ 10–4 =

(e)

0.319 ÷ 10–3 =

(f)

35.475 ÷ 10–2 =

(g) (i)

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2. Divide the decimals by the negative power of ten given.

(h)

0.026 ÷ 10–1 =

–1

–2

3. Write the missing number or negative power of ten in each multiplication algorithm.

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

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6.8 x 10–3 =

(e)

9.78 x

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(b)

3.0 x

= 0.003

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x 10–4 = 0.00025

(a)

x 10-1 = 0.059

(d)

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(f)

67.45 x 10–1 =

4. Write the missing number or negative power of ten in each division algorithm. ÷ 10–4 = 8000

(a) (c)

7.827 ÷ 10–2 =

(e)

8.94 ÷

(b)

0.32 ÷

÷ 10–1 = 331.4

(d) = 89.4

(f)

= 32

0.005 ÷ 10–2 =

5. Convert each second number to a negative power of ten, then complete the algorithm. (a)

68.24 ÷ 1⁄100 = 68.24 ÷ =

(b)

0.19 x 1⁄1000 = 0.19 x =

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

133

Assessment 4

Sub-strand: Fractions and Decimals—F&D – 6

NAME:

DATE:

1. Use your understanding to solve the word problems. Use your knowledge of powers of ten to help you find the answers. Show your working. Joshua walked a distance of 12.62 kilometres in three hours. What distance could he cover in approximately 1.25 days?

(b)

The weight of five cans of paint is 1.2 kilograms. What is the weight of 500 cans of paint?

(c)

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(a)

The cost of plane fare for a family of four was $1145.50. What is the cost for 400 similar families of four?

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(b)

The cost of 20 hotdogs is $210.00. What is the cost of one hotdog?

(c)

Which is the best buy:100 tea bags for $4.80 or 40 tea bags for $2.20?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Multiply and divide decimals by powers of 10

Forty bags of sugar weigh 3.90 kilograms. What is the weight of four bags?

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2. Use your understanding to solve the word problems. Use your knowledge of powers of ten to help you find the answers. Show your working.

Checklist

Sub-strand: Fractions and Decimals —F&D – 6

Divides decimals by whole numbers using powers of ten

Multiplies decimals by whole numbers using powers of ten

Divides decimals by negative powers of ten

Multiplies decimals by negative powers of ten

Divides decimals by positive powers of ten

Multiplies decimals by positive powers of ten

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STUDENT NAME

Understands powers of ten

Multiply and divide decimals by powers of 10 (ACMNA130)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

www.ricpublications.com.au

135

Sub-strand: Fractions and Decimals—F&D – 7

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

TEACHER INFORMATION

RELATED TERMS Fraction

What this means

• Students understand that fraction notation can be shown as a division sign, so that ¾ is the same as 3 ÷ 4. • Using this understanding of the division sign, any fraction can be converted into a decimal. • Percentage needs to be shown as part of 100 and can be easily modelled by using grid paper (10 by 10); e.g. 25 shaded of 100 cells is 25⁄100, is 0.25 and is 25%.

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Decimal

Teaching points

• That a division problem can be expressed as a fraction needs to be understood by students; e.g. 1⁄10 is 1 ÷ 10 is 0.1, while 9⁄10 is 0.9. • Ensure students can order fractions and their equivalent decimal values. As a task, have students place fractions along a decimal number line, or compare a fraction to a decimal and ask which is larger. • Once students have a grasp of the different depictions of the parts of a whole, explain that a whole can be divided into smaller and smaller units (such as division by 100 or 1000) and the use of fractions, decimals and percentage is still applicable. If need be, model these larger divisions on grid paper or along a number line, and compare and contrast different fractions and decimals by asking which is larger/smaller. • Students should understand that ‘percentage’ means ‘a part of 100’ and be shown this with grid paper (10 by 10). Twenty-five of 100 shaded, is 25⁄100, 0.25 and 25%. Use the grid to model common percentages (10%, 20%, 50%, 75%), then link each to its fractional and decimal equivalents. • Give tasks where students compare fractions, decimals and common percentages so that they need to convert and place responses along a number line.

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In a fraction, the top number is referred to as the numerator (or number of parts you have), and the bottom number is the denominator (or the number of parts the whole is divided equally into).

Percentage

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A percentage is designated by a per cent sign % (e.g. 5 per cent, is written as 5%; It is another way of writing the fraction 5⁄100). It means, ‘through, by, or for each hundred’. ‘Per cent’ comes from the Latin per centum. The Latin word centum means 100.

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Equivalent

Means having the same value.

Student vocabulary fraction decimal percentage equivalent

Symbol %

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o c . che e r o t r s super What to look for

• Students who have a poor understanding of place value and fractions will find the concept of percentage difficult. • Students who think in whole-number knowledge will have problems identifying which fractions, decimals and percentages are bigger or smaller. See also New wave Number and Algebra (Year 6) student workbook (pages 52–59)

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 7

HANDS-ON ACTIVITIES Fraction/Decimal/Percentage cards Use the cards on pages 139 to 144 to match fractions, decimals and percentages as card games. Separating the three sets of cards, have the students place the decimal cards in ascending or descending order. Hand out the percentage cards for the students to match with each decimal and fraction card. Start with the easiest cards. Discuss the connections among the three depictions. Find specific card games at <http://edweb.sdsu.edu/courses/edtec670/cardboard/card/d/decimals.html>. Other games can be viewed at <http://www.ilovemath.org/index.php?option=com_docman&task=cat_view&gid=13&limit=10&limitstart=10 &order=date&dir=DESC>; specifically, a card game called ‘Fradecent’ (instructions included).

Doughnut per cents

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Visit <http://nrich.maths.org/6945> to read instructions for Doughnut per cents—a cooperative team activity where each of the four members of the team must work together to create a ‘doughnut’ shape of fractions, decimals and percentages using domino-style cards.

0.8 80% 0.3

LINKS TO OTHER CURRICULUM AREAS

English

40%

2/5 1/4

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Doughnut per cents

30%

25%

• Introduce the relationship among fractions, decimals and percentages using the book Piece = Part = Portion: Fractions = Decimals = Percents, by Scott Gifford.

Information and Communication Technology

• Visit <http://nrich.maths.org/1249> to play an online game of matching fractions, decimals and percentages.

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• This activity illustrates in a visual model the relationship among fractions, decimals and percentages. Using the 100-grid automatically incorporates decimals (0.01 for each square) and percentages (1% for each square). • Show images of the artwork of Piet Mondrian, Victor Vasarely or Ellsworth Kelly: artists whose work is based on geometric shapes. Discuss how their canvases were divided to create the abstract artworks. • Provide each student with a 100-grid and recording worksheet. (The example on page 138 can be used.) Provide, or have the students cut up, squares of five or six different colours of paper to fit the squares on the 100-grid. Ask students to use at least three different colours to complete their grid. Some squares can be left blank or white. Ask the students to glue their selected coloured squares onto the background grid in any way that appeals to them. When completed, the students record the number of squares of each colour used (including white/blank squares). They also record each colour as a fraction of 100, and then calculate the decimal and percentage equivalents of each. The students could refer to their grid to aid calculation of the decimals and percentages. • To increase the difficulty level of the activity, ask the students to leave a border one-square wide around the artwork. They can then use calculators to find fractions, decimals and percentages using a 64-square grid. • Students who used a systematic method of creating artworks might be able to detect a numerical pattern to make calculations quicker. • Note: Gluing on coloured squares is a much quicker method than colouring.

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Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEET

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NAME: Colour of squares

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Number of squares

Fraction

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Decimal

Percentage

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

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100-grid and recording sheet

Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

0.2 0.45 0.15 0.35 0.01 m .

0.3

0.25

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0.05

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Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

1.0

0.9

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0.7

0.65

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

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0.75 0.85 0.95

Decimal cards—2

Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

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½ 1⁄10

3⁄20 3⁄10m.u 2⁄5

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

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½0 ¼ 9⁄20 7⁄20 1⁄5

Fraction cards—1

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Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

1 7⁄10 13⁄20m 3⁄5 1½0

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

19⁄20 9⁄10 17⁄20 4⁄5

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Sub-strand: Fractions and Decimals—F&D – 7

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10% 30% 45% 20% 35% m .

40% 15%

50% 25%

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

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Percentage cards—1

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Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

55% 60% 80% 85% 100%

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

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65% 70% 75% 90% 95%

Percentage cards—2

50% of 100 = 5

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1⁄10 of 400 = 100

0.05 x 600 = 100

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0.2 x 45 = 9

50% of 400 = 200

0.15 x 200 = 30

1% of 800 = 80

¾ of 400 = 300

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EXIT

200% of 90 = 180

0.25 x 200 = 100

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25% of 80 = 20

START

Teac he ¾ of 300 = 200 r

¼ of 500 = 100

0.1 x 900 = 90

½ of 96 = 45

1% of 300 = 30

1⁄5 of 400 = 80

50% of 86 = 43

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages INSTRUCTIONS: Begin at START, then proceed to the next box which shows the correct answer. Continue in this way until the EXIT is reached. To make the activity more difficult, leavE out the answers so the students have to calculate each answer before proceeding. Note: This activity uses fractions, decimals and percentages of quantities. As an additional activity, ask the students to convert the questions to another form: fraction, decimal or percentage.

Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

Fractions – Decimals – Percentage maze example

145

Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEETS

START

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EXIT

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages INSTRUCTIONS: Ask the students to make up and insert their own questions for the maze. Refer to previous page as an example.

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Fractions – Decimals – Percentage maze blank

Sub-strand: Fractions and Decimals—F&D – 7

RESOURCE SHEET Strategies and hints for connecting equivalent fractions, decimals and percentages • Converting fractions to decimals Fractions are an indication of division. The fraction 3⁄8, for example, means ‘3 divided by 8’. To convert a fraction to a decimal, a long division sum can be used; for example:

Denominator of fraction as a whole number

0.375 8)3.000 –24 60 – 56 40 – 40

Numerator of fraction in decimal form

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Since we cannot divide 8 straight into 3, a decimal point and a zero are added. Additional zeros are added as necessary until no longer needed to find a definitive answer. So 3⁄8 can also be written in decimal form as 0.375. This answer can be verified using a calculator.

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• Converting decimals to fractions Converting decimals to fractions involves the use of place value knowledge. In order from left to right, the place values after the decimal point are tenths, hundredths, thousandths, ten thousandths and so on. For example, if we have a decimal such as 0.45, we know that the decimal is really ‘forty-five hundredths’ because there are two decimal places after the decimal point, so we can also write it as 45⁄100. The numerator is the digits in the decimal places (45) and the denominator is the place value (100). If necessary, the fraction now needs to be reduced or simplified to its lowest form. We need to find the largest factor that can be divided into both numbers: in this case, 5. Remember that what is done to one digit must also be done to the other to keep the values equivalent.

• Converting decimals to percentages To convert a decimal to a percentage is simply a matter of multiplying the decimal by 100. To do this, we move the decimal point two places to the right and add the percent symbol (%) to the end.

0.215

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For example, to convert 0.215 to a percentage:

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

So when 0.45 is converted to a fraction, it becomes

Move the decimal point two places to the right so it becomes 21.5, then add the percentage symbol so, 0.215 becomes 21.5%.

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• Converting fractions to percentages To convert a fraction to a percentage, multiply by 100 and add the percentage sign.

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3 x 100 = 300 = 60% 1 5 For example, 3⁄5 as a percentage becomes 5 3⁄5 as a percentage is 60%.

• Converting percentages to decimals To convert a percentage to a decimal, divide by 100 and remove the percentage sign. To do this move the decimal point two places to the left. For example, with 75%, move the decimal point two places to the left (

7 5 . 0 ) so 75% becomes 0.75 as a decimal.

• Converting percentages to fractions To convert a percentage to a fraction, express the percentage as a fraction of 100. For example, 70% becomes 70⁄100. This can then be simplified or reduced to 7⁄10.

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Assessment 1

Sub-strand: Fractions and Decimals—F&D – 7

NAME:

DATE:

1. Convert each fraction to a decimal. Show your working. Shade a bubble to show your answer. (a)

7⁄8

0.35

(c)

12⁄100

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1.45

0.516

3.26

0.25

2.28

0.42

12.0

0.326

2.189

0.12

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0.875

(b)

2. Convert each fraction to a percentage. Show your working. Shade a bubble to show your answer.

(c)

7⁄16

43.75% 4.37% 22.8% 84.99%

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(b)

28.3% 37.5%

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3. Convert each decimal to a percentage. Show your working. Shade a bubble to show your answer. 5.8%

(a) 0.58

58%

66.6%

1.89% 98.1% 18.9%

85%

45.2%

(b) 0.189

45.6% 65.4%

54.6% 6.54%

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

65.4% 3.75%

3⁄8

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 7

NAME:

DATE:

1. Complete the table. Show your working for each. FRACTION

DECIMAL

PERCENTAGE

(a)

36%

1⁄8

(d)

62.5%

2. Solve these real-life problems. Show your working.

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and ate four-fifths of them; (i) how many did she eat and (ii) how many does she have left? (iii) Express the amount she ate as both a decimal and a percentage of the original amount.

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CONTENT DESCRIPTION: Make connections between equivalent fractions, decimals and percentages

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(c)

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(b)

(b) Twelve hundred people attended a football match. Fifty-four per cent left at half time; (i) how many people left and (ii) how many people were still at the match? (iii) Express the percentage who left as both a fraction and a decimal. (iv) Express the number that remained as both a fraction and a decimal.

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(c) Tim’s grandparents gave him $125 for his birthday. Of this amount, he deposited 0.4 in the bank, then spent 0.35 of what was left on a pair of gym shoes. The remainder was spent on a new music player. How much did the player cost? Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Checklist

Sub-strand: Fractions and Decimals—F&D – 7

Connects fractions, decimals and percentages

Connects fractions and percentages

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STUDENT NAME

Connects fractions and decimals

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

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Answers

Sub-strand: Fractions and Decimals

F&D – 1

F&D – 2

Page 81 Assessment 1

Page 93 Assessment 1

1⁄6 1⁄5 7⁄7 9⁄10 ¼ 3⁄8 6⁄6 6⁄8 1⁄8, 1⁄6, ¼, 1⁄3 2⁄16, 1⁄8 ¾, 6⁄8

(b) (d) (b) (d) (b) (d) (b) (d) (b) (b) (d)

¼ 2⁄8 9⁄9 12⁄12 1⁄8 1⁄9 4⁄5 2⁄2 1⁄9, 1⁄3, 4⁄9, 4⁄6 2⁄3, 8⁄12 8⁄10, 16⁄20

1. (a) (c) (e) 2. (a) (c) (e)

Page 82 Assessment 2

4⁄5 ¾ 11⁄12 ½ 3⁄10 (e) 1⁄3

1. (a) (c) (e) 2. (a) (c)

(b) 2⁄6, 4⁄12

1. (a) 2. (a)

1. (a) 7⁄8

4⁄8 or ½

5⁄8

3⁄8

2⁄5

2⁄10

3⁄10

1⁄9

2⁄9

5⁄10

5. 6.

8⁄8

6⁄10

4⁄9

5⁄9

9⁄10

7⁄9

8⁄9

2. (a) 3. (a) 0

6⁄8

8⁄8

2⁄4

4⁄16

8⁄16

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2⁄8

8⁄16

2⁄4

(b) B

2⁄6 or 1⁄3

1⁄3

7⁄12 2⁄9

4⁄8

5⁄8

3⁄6

4⁄6 or 2⁄3

1⁄9

2⁄9

3⁄9

(b) 5⁄10 or ½

(d) C

6⁄8

16⁄16

7⁄8

5⁄6

4⁄9

5⁄9

6⁄9 or 2⁄3

1⁄10

2⁄10

3⁄10

4⁄10

7⁄9

16⁄16

6⁄8

(b) 7⁄10

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2⁄4

3⁄8

1 3⁄3

6⁄9

1⁄6

4⁄8

Page 84 Assessment 4 1. (a) A

8⁄10

7⁄10

2⁄8

(b)

4⁄5

2⁄3

2⁄8

7⁄8

3⁄5

4⁄10

3⁄9

1⁄8

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½ 1⁄3

3⁄6 or ½ (b) 12⁄15 or 4⁄5 (c) 4⁄12 or 1⁄3 (d) 4⁄4 or 1 2⁄8 or ¼ (b) 2⁄10 or 1⁄5 (c) 2⁄12 or 1⁄6 (d) 5⁄10 or ½

1⁄3

(b) (d)

1⁄10

2⁄3 7⁄12

Page 96 Assessment 4

0 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 1

(b) (d)

Page 95 Assessment 3

Page 83 Assessment 3

(b) ¼ (d) 2⁄3

Page 94 Assessment 2

(d) 2⁄12 2. Answers will vary but may be similar to: (a) All denominators are even numbers; the denominators increase by 2s (b) All denominators are multiples of 3. (c) All denominators are multiples of 4. (d) All denominators are multiples; 12 is double 6 etc.

(b) 2⁄3 (d) ½

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2⁄4, 3⁄6, 4⁄8, 5⁄10, 6⁄12, 8⁄16 (c) 6⁄8, 9⁄12, 12⁄16

1. (a)

1⁄8

5⁄6 3⁄5 2⁄3 1⁄3 ½ ½

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1. (a) (c) 2. (a) (c) 3. (a) (c) 4. (a) (c) 5. (a) 6. (a) (c)

5⁄10

6⁄10

8⁄9

9⁄10

4⁄5

7⁄10

8⁄10

x x x

4. (a) 1⁄6 (b) 7⁄20 5. (a) 5⁄6 Teacher check student-drawn diagram (b) 1⁄8 Teacher check student-drawn diagram

2. (a)

(b) 3. Teacher check Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Answers

Sub-strand: Fractions and Decimals

rounding, and calculator checking sections of this page.)

F&D – 3 Page 105 Assessment 1 1. (a) 6

(b) 21

(d) 21

(e) 10

F&D – 5 Page 121 Assessment 1

2. (a) 21

(b) 20

(d) 12

1. (a) (c) 2. (a) (c)

0.244 5441.7 $79.90 111.6 cm

(b) 111.725 (d) 4220.94 (b) $85.80

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(e) 32

(b) (d) (f ) (b) (d) (f ) (b) (d) (f ) (b) (d) (f ) (b) (d) (f ) (b) (d) (f ) (b) (d) (f ) (h)

84 448 692 174 789 2724 194 1324 2416 186 1872 702 166 2388 5194 230 4459 953 1956 720 924 2368

F&D – 6 Page 131 Assessment 1 1. (a) (c) 2. (a) (c) 3. (a) (c) (e) 4. (a) (c) 5. (a) (c)

100; 102 1 000 000; 105 1 10 000 000 0.01; 10–2 0.001; 10–3 0.00001; 10–5 0.1 0.001 1012 1018

(b) (d) (b) (d) (b) (d) (f ) (b) (d) (b) (d)

1000; 103 10 000; 104 1 000 000 1 000 000 000 0.1; 10–1 0.0001; 10–4 0.000001; 10–6 0.000001 0.000000001 10–9 10–3 (e) 100

Page 113 Assessment 1

Page 132 Assessment 2 1. (a) 9.193 (c) 5701.0 (e) 61.3 (g) 327 646 (i) 4600 2. (a) 622.09 (c) 7.044 (e) 0.00913 (g) 0.86507 (i) 0.003416 3. (a) 9.232 (c) 7598 (e) 101 4. (a) 3569 (c) 6.897 (e) 100 5. (a) 102; 0.028 (c) 101; 3.2

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(b) 48.602 (d) 129.7 (f ) 520 (h) 26 043 (j) 840200 (b) 158.35 (d) 0.0432 (f ) 0.47543 (h) 0.000062 (j) 0.0001907 (b) 102 (d) 0.05 (f ) 0.087 (b) 103 (d) 0.0829 (f ) 0.04201 (b) 103; 47 (d) 103; 0.00006

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1. 39.994 2. 96.561 3. 558.199 4. 160.758 5. 1454.384 6. 536.900 7. 734.028 (Teachers should check the estimation and rounding, and calculator checking sections of this page.) Page 114 Assessment 2 1. 44.310 2. 321.022 3. 267.069 4. 331.755 5. 566.685 6. 418.141 7. 1709.833 (Teachers should check the estimation and

152

(b) 0.038 (d) 0.123 (b) 44.78

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1. (a) 57 (c) 252 (e) 389 2. (a) 56 (c) 258 (e) 1035 3. (a) 73 (c) 411 (e) 2715 4. (a) 69 (c) 741 (e) 3525 5. (a) 42 (c) 375 (e) 1840 6. (a) 460 (c) 1684 (e) 6464 7. (a) 59 (c) 1041 (e) 674 (g) 726 (i) 2808

0.802 73.42 $5.75 114.862

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Page 106 Assessment 2

1. (a) (c) 2. (a) (c)

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Answers

Sub-strand: Fractions and Decimals

Page 133 Assessment 3

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(b) (d) (f ) (h) (j) (b) (d) (f ) (h) (j) (b) (d) (f ) (b) (d) (f ) (b)

0.75246 0.004753 5.2823 0.80901 0.050159 124 560 5720 3547.5 0.26 709.1 10–3 0.59 6.745 10–2 33.14 0.5 10–3, 0.00019

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1. (a) 0.001193 (c) 67.89 (e) 0.3593 (g) 0.90683 (i) 0.002466 2. (a) 609.7 (c) 8303 (e) 319 (g) 7 560 800 (i) 614.3 3. (a) 2.5 (c) 0.0068 (e) 10–2 4. (a) 0.8 (c) 782.7 (e) 10–1 5. (a) 10–2, 6824

(c) He deposited $50, so $75 was left. 0.35 or 35⁄100 of 75 ($26.25) is the cost of the gym shoes. $75.00 – $26.25 gives the cost of the player, which is $48.75.

Page 134 Assessment 4 1. (a) (c) 2. (a) (c)

126.2 kilometres (b) 120 kilograms $458 200.00 0.39 (b) $10.50 The best buy is 100 tea bags for $4.80 because they are 0.048c each, while 40 tea bags for $2.20 are 0.055c each.

F&D – 7

Page 148 Assessment 1 (b) 0.25

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0.875 0.12 37.5% 43.75% 58% 65.4%

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1. (a) (c) 2. (a) (c) 3. (a) (c)

(b) 75%

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Page 149 Assessment 2

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1. (a) 36⁄100 or 9⁄25 in its simplest form; 0.36 (b) 87⁄100, 87% (c) 0.125, 12.5% (d) 625⁄100 or (5⁄8 in its simplest form), 0.625 2. (a) (i) She ate 52. (ii) She has 13 left. (iii) 0.8 (decimal), 80% (percentage) (b) (i) 648 people left (ii) There were 552 people still at the game. (iii) 648⁄1200 or 54⁄100 in its simplest form (fraction); 0.54 (as a decimal) (iv) 552⁄1200 or 69⁄150 in its simplest form (fraction); 0.46 (decimal)

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Sub-strand: Money and Financial Mathematics—M&FM – 1

Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)

RELATED TERMS

TEACHER INFORMATION What this means

Percentage

Discount

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• A discount is an amount deducted from a purchase price.

Teaching points

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• A percentage is a fraction whose denominator is 100; for example: 3/100 is 3%; while 18 as a percentage of 72 is 25%.

• Students need to understand what percentage is and should be able to model understanding through use of grid paper to colour 10%, 25%, 50% etc. • Students should know how to convert these percentages to fractions and decimals. This thinking is important because when calculating, the student can use the most efficient strategy when working out an answer. For example: ‘An item costs $48. The store is having a 50%-off sale. How much does the item cost?’ It is far easier to think of 50% as ½ and divide $48 by 2, then subtract that amount from the original price ($24). • Students should be given the opportunity, using authentic information, to investigate and calculate common percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies. • Students should be shown the different strategies available to use when solving problems using percentages.

• Students should show that they understand common percentages by colouring cells on a 10 x 10 grid. • Students should know how to convert percentages to fractions and decimals. • Students need to show they understand the use of a fraction as an operator. For example, how 25% of $84 is the same as finding ¼ of $84. • Students need to be shown the meaning of ‘of’ and ‘off ’ in relation to percentage problems. For example, the answer to ‘25% of $100’ is $25, while the answer to ‘25% off $100’ is $75. ‘Off ’ problems require multiple steps. • Students need to be involved in a range of percentage problems which can be solved mentally or with the use of technologies such as calculators.

What to look for

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Student vocabulary percentage

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• Students need to understand the connections among percentages, decimals and fractions because any could be used to calculate percentage. For example, 20% of $85 could be seen as 1⁄5 x $85, or 0.2 x $85, or 20⁄100 x $85. Alternatively, the question could be placed into a calculator. • Students need to understand division problems such as $85 ÷ 5 = $17 and how 4⁄5 of $1 is 80c. As such, you need to make sure students move flexibly among the different forms and understand how fractions can be converted into parts of a dollar. This can be difficult and therefore the different strategies available to students need to be known. • When students use decimals (0.20 x $85), some fail to apply the correct number of decimal places. • Some students may only understand percentage when calculating a fractional amount of 100. Therefore, some may have difficulty calculating, for example, 20% of $85. They do not understand how the $85 is the ‘whole’.

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discount sale

See also New wave Number and Algebra (Year 6) student workbook (pages 60–66)

calculate

Proficiency strand(s): Symbol %

154

Understanding Fluency Problem solving Reasoning

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Sub-strand: Money and Financial Mathematics—M&FM – 1

HANDS-ON ACTIVITIES ‘Find the amount’ card games • Give each student ten blank playing cards. On five of the cards, ask the students to write maths problems such as ‘What is 1⁄3 of $63?’, ‘How do we write $30 of $60 as a decimal?’ or ‘What percentage of $120 is $40?’ On corresponding cards, students write the answers. Ensure students write fraction, decimal and percentage questions. When completed, the students combine their cards with those of two other students and use them to play a game. During and after the game, evaluate any problems which may have occurred (such as the same amount being the same answer to different questions, the same question being asked etc.). Get rid of or change any questions which need fixing and find the best combination of cards to form a game. Swap with those of another group.

What isHow 1⁄3 do we of $63? write $30

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of $60 as a decimal?

• Students should be familiar with all Australian coins and notes. Use replicas of notes and coins to reward good student behaviour and work. The students ‘save’ these up to ‘buy’ free time, computer time, or special pencils or biros purchased by the teacher. The students may save them in their own ‘money box’ or bank. Encourage the students to keep a running total of their savings using a bank-style recording sheet. Note: Students may construct their own money box by covering a tissue box with images of notes and coins.

Using base ten blocks • Allow students to experience percentage using base ten blocks. ‘Per cent’ means ‘for every hundred’. ‘Flats’ in base ten blocks are large squares divided into 100 equal parts. They will consolidate the link between fractions—hundredths. Distribute the flats, longs and minis. Ask the students to count or work out how many squares are in a flat (10 x 10) then ask them to complete activities like the one following. • Cover 67% of a flat. The students should use longs and minis to cover 67 squares on the flat. This may be shown using 6 longs and 7 minis. • The students will realise that 67% is 67⁄100. Encourage them to write 67% in as many ways as possible. For example, 67% = 67⁄100 = 0.67 = sixty-seven per cent = 67 per cent.

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Money bank

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100 grid

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• Students should also become familiar with using the 10 x 10 (100) grid to colour a certain number of squares to represent a certain percentage. (Grids are available for use on page 158. These are exactly the correct size for use with the ‘longs’ and ‘minis’ of base ten blocks.)

Discount voucher cards

• Ask the students to use the blanks on page 159 to create discount voucher cards for other class members to use. The students can write the names of their favourite products on the cards, such as their hamburgers, pizzas, sandwiches or rolls, jeans, attractions (such as a paintball game, video arcade game etc.), computer games and books. The students insert the name of the product, brand logo (if applicable) and MUST know the cost of each item. They may need to round the price up or down (to the nearest five in the hundredths decimal position) to make the calculation easier. They then present the voucher to a classmate to work out the final cost. A record of the answers should be kept and the vouchers then presented to other class members to calculate.

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LINKS TO OTHER CURRICULUM AREAS English • Revise the book Piece = Part = Portion: Fractions = Decimals = Percents by Scott Gifford.

Information and Communication Technology • Visit <http://www.actuarialfoundation.org/programs/youth/math_academy.shtml> to download a copy of the pdf of ‘The Math Academy: Dining Out’. This series of lessons for Years 3 to 8 provides many resources about real-life situations. (NOTE: It is from the USA, but many activities can be converted.) • Visit <http://www.math-play.com/Fractions-Decimals-Percents-Jeopardy/fractions-decimals-percents-jeopardy.html> to play an interactive game about fractions, decimals and percentages. The game can be played individually or with teams.

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Health and Physical Education

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Economics

• Find out what percentage of students in the class ordered lunch from the canteen today by conducting a survey, tallying results and expressing this as fractions, decimals and percentages. • Ask each student to find out what percentage charge is paid by parents who own and use credit cards. Find out what percentage interest is paid on savings deposits over time.

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• The students express time spent during the day sleeping, at leisure, doing homework, or eating as fractions, decimals and percentages of 24 hours. • Express this as a percentage: ‘How many goals did a certain player score from a given number of kicks?’ Calculators should be used for this activity as students may encounter recurring decimals. Once a percentage is calculated, change it to a decimal and a fraction. • Discuss percentages in relation to sports. How many students play a particular sport? How many games did your favourite team win last year? Express these amounts as fractions, decimals and percentages.

• Look at pie graphs which show percentages relating to a country, such as the area of each state as compared to the country’s total land area. Ask the students to express the percentages as fractions and decimals.

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Geography

• Provide each student with a 100-grid. Ask them to colour squares to create the first initial of their name in block letters. When completed, ask them to count the number of squares and express this as a fraction, decimal or percentage of the total grid space. Repeat with both initials or a picture. • Ask the students to create a collage of media items showing percentage discounts. These could be obtained from sales catalogues and advertising materials. Some students may prefer to draw and create their own images.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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RESOURCE SHEET Common fractions, decimals and percentages chart

FRACTION

DECIMAL

PERCENTAGE

1⁄16

0.0625

6.25%

1⁄10

0.1

10%

1⁄8

0.125

12.5%

0.2

20%

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0.25

3⁄10

0.3

3⁄8

0.375

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¼ or 2⁄8

25%

30%

37.5%

3⁄5

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

1⁄5 or

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5⁄8

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¾ or 6⁄8

60%

0.625

62.5%

0.75

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75%

0.8

80%

0.875

87.5%

1⁄1

1.0

100%

2⁄1

2.0

200%

7⁄8

Note:

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Fractions which convert to recurring decimals (such as 1⁄3, 2⁄3 and 5⁄6) have not been included, so as to not confuse students.

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RESOURCE SHEET

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

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100-grid sheets

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RESOURCE SHEET Discount voucher cards Product name

Product name

10% discount

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Product name

10% off

Product name

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Product name

25% off

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Product name

50% discount

25% discount

Product name

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25% discount

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Product name

25% off

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Product name

50% discount

Product name

50% off

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RESOURCE SHEET Strategies and ideas for calculating percentage discounts

1. Be familiar with the most common percentages and their fraction equivalents Students who are familiar with common fractions and their percentage equivalents are able to use these to calculate percentage discounts quickly. For example, if students know that 10% is the same as 1⁄10 they will easily be able to calculate discount by finding 1⁄10 of it and subtracting it from the original price/cost to obtain the discounted amount.

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2. Convert the percentage to a fraction Any percentage is able to be expressed as a part of one hundred.

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A table of common fractions and their percentage equivalents is included on page 157. For the purposes of this content description, the percentages mentioned only include those that convert to terminating decimals. Teachers who feel their students are capable could introduce other percentages. However, care should be taken with percentages other than those mentioned in the table, since they will incorporate recurring decimals relating to thirds, sixths and ninths.

Twenty-five per cent (25%) is 25 of 100 parts or 25⁄100, which can be simplified to ¼.

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3. The use of decimals and powers of ten in percentages • When calculating discount percentages, (especially when using 10%) the students will know that 10% is 10⁄100 or 1⁄10.

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Less common percentages can also be expressed as a part of one hundred: 32% is 32⁄100, 99% is 99⁄100, 72% is 72⁄100.

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To find one-tenth of any monetary amount (for example, $19.80), the number will become smaller simply by moving the decimal point one digit to the left. So 1⁄10 or 10% of $19.80 is $1.98.

• Monetary amounts are expressed in terms of whole numbers, including tens and hundreds, as well as tenths and hundredths.

Consider $568. 55. This can be expressed as ‘five hundred and sixty-eight dollars and fifty-five cents’, or ‘five hundreds, six tens, eight ones, five tenths and five hundredths’. Students can utilise their knowledge of decimals and decimal place value to find answers to the hundredths. Note: Students may need to round up or down calculations to give more ‘realistic’ answers to problems; for example, an answer of $58.82 would not be realistic as most prices are rounded up or down to the nearest five or ten. In this case, a more realistic answer would be obtained by rounding the amount to $58.80 (since 2c coins are no longer used).

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

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Assessment 1

Sub-strand: Money and Financial Mathematics—M&FM – 1

NAME:

DATE:

1. Solve these percentage discount problems. Show your working. Shade a bubble to show your answer. $250 $2.50

(a) 10% of $25.00

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$3.86 $7.72 $19.30 $18.60

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$3.90

$12.05 $4.82 $24.10 $12.50

(b) 25% of $48.20

$657

(d) 10% of $65.90

65c

$32.95 $6.59

$9.92 $24.80 $19.84 $49.60

(e) 25% of $99.20

(a) Jan’s favourite jeans store was having a discount sale. She found a pair of jeans normally priced at $79.80 but which were discounted by 25%. How much was the discount and how much did she pay for the jeans?

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2. Solve these realistic problems involving percentage discounts. Show your working. Check your answers using a calculator.

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25c

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(b) The Lee family ordered two pizzas on Saturday night. Each family-sized pizza cost $15.00. The second pizza had 25% off the original price. How much did the second pizza cost and how much did the meal cost altogether?

(c) At the discount store, each electrical appliance was discounted by 10% of the original price. Find the final cost of a toaster priced at $69.90, a frying pan at $129.80, a jug at $24.90, and a mixer at $89.90.

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Checklist

Sub-strand: Money and Financial Mathematics—M&FM – 1

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

Calculates percentage discounts of 10%, 25% and 50% on sale items without digital technologies

Calculates percentage discounts of 10%, 25% and 50% on sale items with digital technologies

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STUDENT NAME

Investigates percentage discounts of 10%, 25% and 50% by various methods

digital technologies (ACMNA133)

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Answers

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M&FM – 1 Page 161 Assessment 1

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1. (a) $2.50 (b) $12.05 (c) $19.30 (d) $6.59 (e) $24.80 2. (a) $19.95/$59.85 (b) $11.25/$26.25 (c) toaster – $62.91, frying pan – $116.82, jug – $22.41, kitchen mixer – $80.91

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Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

TEACHER INFORMATION

RELATED TERMS Whole number

Whole numbers (integers) are numbers that do not have a part number ( i.e. not a mixed number). Examples are 1, 5, 27, –3, –31 and 105. Sometimes the term is used to refer to only positive integers. Fraction

• Students should be able to continue their understanding of, and create sequences involving, whole numbers, fractions and decimals. • Students need to be able to describe the rule used to create a sequence. • A student’s ability to identify and generalise number sequences and patterns is the beginning of algebraic thinking. Students should be given opportunities to view sequences and patterns in a variety of formats (e.g. in a table, using a ‘function machine’—see page 166 and be able to describe what is happening in each. • Students should investigate additive, subtractive and multiplicative patterns as they appear in a variety of formats; such as the position of tiles in a geometric design, the succession of numerals in a set, the progressive arrangements of dots on a dice, or the repetition of shapes and colours along a strip or border. Students should examine the patterns, looking for ways the elements within them increase or decrease in number.

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• A fraction is part of a whole. A fraction is obtained by dividing a whole or given amount into a certain number of equal parts and referring to a certain amount of them; for example, 2⁄3 refers to 2 of 3 equal parts of the whole. • In a fraction, the top number is referred to as the numerator (or number of parts you have), and the bottom number is the denominator (or the number of parts the whole is divided equally into).

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What this means

Teaching points

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point. Decimals are a way of writing fractions without using a numerator or denominator. For example, the fraction 7⁄10 can be written as the decimal numeral 0.7. A decimal numeral can be less than or greater than one. Decimals less than one are written with a zero before the decimal point (for example, 0.2), while decimals greater than one are written with the decimal point between the whole number(s) and the fraction (for example, 4.9).

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• Students should be given multiple opportunities to develop their algebraic thinking, using tables with geometric sequences involving addition and multiplication and using a range of materials; for example: counters, matchsticks, square tiles, 2-cm cubes and grid paper. A possible triangular pattern made with matchsticks is shown below: 1.

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Decimal

etc.

This information can be recorded in a table: Pattern position element

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Number of matchsticks used

How many matchsticks are used to make six linear triangles? How many triangles are used to make 10 linear triangles? Is there a rule for the number of matchsticks used? If so, what? Can you predict what will happen in the 100th element position? State the generalisation in your own words: ‘What is happening in this pattern?’

• Continue exploring the repetitive nature of a sequence and defining the rule for one by using the concept of the ‘function machine’ (see page 166). Ask questions such a ‘Beverley put in 3 and Robert took out 5 and ¾. What happened in the function machine?’ or ‘Sean put in 15 and took out 225. What happened in the function machine?’. • Students should be given opportunities to balance the sides of an equation; for example: 3 x 5 = 2.5 x ?

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Sub-strand: Patterns and Algebra—P&A – 1

Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

TEACHER INFORMATION (CONT.)

RELATED TERMS (CONT.) Sequence

What to look for

A list of numbers (or other element) in a special order. A sequence usually has a rule. Rule

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A prescribed mathematical method for performing a calculation or solving a problem

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• Some students, despite being given multiple opportunities to use materials and tables to model and develop their understanding of sequencing, still find answering questions of this type difficult. This may relate to a student’s difficulty in being able to determine the rule for a sequence, which could be because he or she has difficulty understanding how to solve an algorithm out of the context of a set structure (e.g. a typical subtraction or multiplication equation) or cannot apply knowledge of the inverse properties of operations (e.g. that subtraction is the inverse of addition: 4 + 7 = 11, so 11 – 7 = 4). If a student cannot determine what is occurring among elements in a sequence (e.g. in 2, 7, 12 17 … that numbers are increasing by fives), he/she will have difficulty in working out the rule. Even when shown how to model a rule by use of, for example, a table, he or she does not fully understand what is happening until modelling it him or herself. Some students may even refuse the use of a table and try to solve a sequencing problem without one, invariably getting an incorrect answer. • In making equivalence statements with multiplication or addition (e.g. that 4 + 7 = 7 + 4 = 11), many students easily understand that you are able to change the position of numbers (or pro-numerals) without affecting the quantity. However, some students will think that the same concept also applies to subtraction and division, which it does not. As such, they get the problem wrong and do not know what is incorrect with their reasoning.

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See also New wave Number and Algebra (Year 6) student workbook (pages 67–74)

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Student vocabulary sequence whole number fraction decimal rule pattern position (element) generalisation

Proficiency strand(s):

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Symbols An ellipsis (...), dot or horizontal line (vinculum) can be used to indicate a recurring decimal; for example, 1⁄9 will be written as

0.11111 ..., 0.1, or 0.11 .

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HANDS-ON ACTIVITIES The ‘function machine’ comes alive!

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• This activity is suitable for students of all ages—the older the students, the more complicated the function (or rule) should be. • A student is placed inside a large box (such as an large electrical appliance carton) to act as the ‘brains’ of the function machine. The box could be decorated as a robot, creature or vending machine. Cut spaces on each side of the box for the ‘INPUT’ and ‘OUTPUT’ cards to go into and come out of and label them. The rule for the function machine is secretly given to the student inside the box. Selected students input cards printed with numbers, which are inserted into the machine (given to the student) through the INPUT space. The student inside the box uses the rule they have been given to work out the answer. They write the answer on a card or sheet of paper and slip it through the OUTPUT space. After a number of different inputs and outputs, the students watching are asked to see if they can describe the rule governing the input and output. In order to help work out the rule of the function machine, the students could be provided with a recording sheet. • If the students are being introduced to the function machine, the rule could be pinned or attached to the front of the box until all understand the process. • The students should share having goes working out all three aspects of the function machine: the input, the function/ rule and the output. Refer to the example on page 170. • Teachers may see the function machine otherwise referred to as a ‘function box’.

Geometric patterns Polygonal patterns

1. Ask the students to arrange a number of pattern blocks to make a simple design, then find its perimeter by counting one standard edge as one unit. The students then make a ‘train’ of the design by creating two or more repetitions of it. They then find the perimeter of the train after adding each successive design. This information is then collated in a table of values and the students try to work out the function/rule. The rule will show the relationship between the number of repetitions and the perimeter of the train.

For the example shown, a student has constructed a series of ‘cats heads’ in alternating positions up and down but with all sharing a common edge. The table of values for this design is: 2.

Cat head number

Perimeter

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2 3

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Rule: Start with the number/position (1, 2 or 3), multiply it by 5 and add 2.

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2. Build a sequence of L-shapes by using squares and adding one square at a time to each section. 3.

2. 1.

Ask the students to work out how many squares will be needed for the L-shape in other positions, such as the 20th or 50th. The constant addition function of a calculator could be used to find this information, too.

L-shape number

Number of squares used

Rule: Start with the number/position (1, 2 or 3), double it and add 2.

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HANDS-ON ACTIVITIES Geometric patterns (cont.) 3. Ask the students to construct borders by using a coloured (grey) centre square, surrounded by a border of different coloured (white) squares. Students then draw a pattern of several squares, increasing in size sequentially. They then complete a table of variables and look for specific aspects. 3.

Position of square in sequence

2. 1.

Perimeter of centre squares

Total of border squares

Total of centre squares

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4. Polyhedral patterns

Polyhedron Shape

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This activity is suitable for more capable students.

Number of sides per face

Number of faces

Number of edges

Number of vertices

regular tetrahedron

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regular octahedron

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cube (regular hexahedron)

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regular icosahedron This activity will introduce the students to Euler’s rule: F + V = E + 2 (the number of faces + the number of vertices = the number of edges + 2) 5. Paper folding Simple paper folding can be used to show numeric sequences. Ask the students to record the number of folds and the number of ‘regions’ created as they fold a sheet of paper a given amount of times. For example, after the first fold, there are two regions created; after the second fold, there are four regions created and so on. The students can be asked to predict, using their record, how many regions will be created after a given number of folds. Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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HANDS-ON ACTIVITIES Whole numbers 1. Ask the students to construct square and triangular numbers using counters or cubes, then try to work out the pattern of progression. For example: Square numbers Triangular numbers

Cubed numbers (1, 8, 27, 64, 125, 216, 343, 512, 729 … ) can also be used as an example. Just by doing the first few examples, the students should be able to work out a rule. (The next number is made by cubing where it is in the pattern (xn = n3).] The students will have encountered square and triangular numbers in the number section in content description associated with code N&PV – 1.

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2. This activity may be suitable for more capable students. Use one of the blank templates on page 179 to write the numbers in the squares to recreate Pascal’s triangle. Follow the rule: Each number is found by adding the two numbers before it, except for the edge rows which are all ones. (The rule for finding the next number is an = xn – 1 + xn – 2). When the triangle has been completed, ask the students to colour the numbers as shown. Starting from the ‘1’ in the bottom left-hand corner, colour it a particular colour or pattern, then go up one row and across one number. Colour that number the same colour or pattern. Continue in this way until a ‘staircase’ is created. In the example below, these are the numbers 1, 6, 10, 4. Then select a new colour or pattern, go to the ‘1’ above the first 1 you coloured and create a new staircase. Colour or make a new pattern using 1, 5, 6 and 1. Start again with the next ‘1’ and create a new staircase, identifying the numbers 1, 4 and 3. The following staircase will identify the numbers 1 and 2 … and, finally, 1 and 1. Adding each ‘staircase’ of numbers will produce the Fibonacci sequence. Pascal’s triangle

Fibonacci sequence

1+2

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1+5+6+1

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Using Pascal’s triangle will also show the counting numbers (second diagonal) and triangular numbers (third diagonal). Colouring all the odd numbers in one colour and the even numbers in another colour produces a visual pattern: an increasing pattern of triangles (see second triangle below). 1s

counting numbers

triangular numbers (Fibonacci sequence)

4 8 16 32 64 128 256 Finally, adding the numbers of each row of the triangle gives another number sequence. Each row’s total is double that of the previous row.

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LINKS TO OTHER CURRICULUM AREAS English • The function machine format can be used for literacy as well. Spelling rules (such as the one in the table) are one example of how it could be used.

INPUT

RULE

leave

leaving If a verb ends taking in a silent ‘e’, drop the final receiving ‘e’ and add dating –’ing’. collating

take

Information and Communication Technology

receive

OUTPUT

date • Visit <http://www.littlefishsw.co.uk/card/functionmachine.html>, <http:// www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines. collate html> and <http://teams.lacoe.edu/documentaion/classrooms/amy/ algebra/3-4/activities/functionmachine/functionmachine3-4.html> to play interactive function machine games.

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Health and Physical Education

Economics

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• The students use repeating patterns of movement (e.g. one jump to the left, two jumps forward, one jump to the right, one jump to the left, two jumps forward, one jump to the right) to create simple exercises to be performed along with music.

• Relate the use of functions to real-life situations: Treat certain apparatus as if they are function machines, such as using a snack or drink vending machine. Buyers input money and select a specific item, then the selected item drops out with any change required. The input is the money and the choice of item; the function rule is the price of the item, and the output is the change and the selected item. • Calculate the number of kilometres travelled or the number of litres used if the efficiency of a car in kilometres per litre is known. For example, if a car travels 20 kilometres per litre, what distance can be travelled if 10 litres is pumped into the car? • Calculate a weekly salary as a function of an hourly rate and the number of hours worked, or the compound interest earned on a deposit or investment considering the principal investment, the interest and the term (length) of the investment.

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• Investigate Leonardo Da Vinci’s drawing, ‘Vitruvian Man’. Its accompanying notes provide the proportions for the ‘ideal man’. Some of these notes include: the length of the outspread arms is equal to the man’s height, the maximum width of the shoulders is a one-quarter of the height; the foot is one-sixth of the height etc. Have the students use the rule to work out the correct proportions of the ideal figure of a person given one or more details. They could create the figure in pen and ink reminiscent of Da Vinci’s sketch. The students could also use some of the proportions and a table to calculate how well their arms, legs, height etc. compare to the proportions of the ‘ideal persons’. (Be aware that some proportions relate to private body parts, and these should be excluded in the interest of class appropriateness and sensitivity. Also, be sure to point out that the measurements do not have any basis in fact of what is a ‘perfect person’). • This activity also relates to the Measurement and Geometry strand because the figure is drawn inside, and touching, the circumference of a circle. Refer to <http://www.davincithevilla.com/vitruvproportions.htm> for details of the specific body proportions. • Investigate the height of a large object (such as a building or tree) using the length of a shadow and the time of day. The length of a shadow is a function of its height and the time of day.

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Other mathematical ideas relating to functions in real-life situations • Thermometers with both a Celsius and Fahrenheit scale can be considered input-output functions. Capable students will be able to use the rules (C° = [F° – 32] x [ 5 ÷ 9]; F° = [C° x 9⁄5] + 32) for converting temperatures from one scale to the other. They could also be given temperatures and asked to work out the rule for converting one scale to another.

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RESOURCE SHEETS Strategies, explanations and hints Function machines Function machines have an input (numbers), a function and an output (the answer). For example, if the input of a function machine is 12 and the function is + 8, the output will be 20. Input 12

Function +8

Output 20

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12 16

In this function machine, the unknown is the function, which is x4.

It is important students should practise determining all three aspects of function machines: the input, the output and the function. Students who have difficulty working backwards to find the inverse (input or function rather than finding the output/answer) will need to practise this aspect. They need to be aware that an understanding of inverse operations is required. For example, using the same numbers as above, if the unknown is the input, the students will need to know that to find the input they will need to use the inverse of x 4 (÷ 4) to obtain the input.

Function x4

Output

12 ÷ 4 = 3

12 16

16 ÷ 4 = 4

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The concept of function machines can exist in many different formats. Teachers should use the format that best suits the students. Examples are shown on pages 176 and 178.

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Steps for determining sequences

The steps in the process for working out sequences are very similar, whether for a geometric pattern involving multiples, or identifying a simple number pattern. They are as follows for each: (a) Geometric pattern

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• Use materials to create the pattern.

• Complete a table of values for the pattern.

• Use words to describe and record the pattern in a number of different ways.

• Work out the rule to calculate the number needed to create a much larger geometric pattern (without actually making the pattern). (b) Number pattern • Complete a table of values for the pattern. • Use words to describe and record the pattern in a number of different ways. • Work out the rule to calculate the value of a much larger number.

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

3 4

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If the function is unknown, several inputs and output should be viewed to allow students to see a pattern and determine the rule. Function Input Output

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEETS Strategies, explanations and hints Strategies for determining sequences (continued) • Students can use a number of different strategies to work out unfamiliar sequences. These include: – drawing a diagram

Paper folding

– trial and error

Fold number

Number of sections created

1 2 3 4 5

2 4 8 16 32

– using materials to model with – working backwards (using inverse operations; e.g subtraction and addition, multiplication and division) – identifying patterns

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– using a table (see example)

– using a graph (see example below).

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(a) the DIFFERENCES among successive numbers and deciding if there is a pattern to the differences; for example, does the difference increase each time? +1 +2 +3

+4 7,

+5 11,

Paper folding

(b) numbers having a CONSTANT MULTIPLIER;

for example: 3, 9, 27, 81… (Each successive number is multiplied by 3.)

for example: 24, 48, 96, 192 … (doubling) or 100, 50, 25, 12.5 … (halving) (d) whether the numbers are SQUARE NUMBERS (or square numbers + 1); (12) (22) (32) (42) for example: 1, 4, 9, 16 …

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(e) CYCLES OR REPETITIONS; for example: 1, 3, 5, 9, 1, 3, 5, 9 … • A multitude of properties and patterns have been discovered (many thousands of years ago) associated with number sequences. Some patterns (such as those mentioned above) are well known, such as even, odd, prime, triangular, square, cubed and composite numbers. Others are more complex, such as Pascal’s triangle and Fibonacci’s number sequence, and may be interesting for more capable students to investigate. (See pages 168 and 169.)

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The golden rectangle can be constructed using the Fibonacci’s number sequence.

• Numbers in a pattern either increase, decrease or repeat. This can be shown using a graph. Along the X-axis and Y-axis, the determinants of the sequence can be labelled, while the plotting shows the sequence. The example shown, based on the information in the above table, shows a sequence in which the difference is increasing. Comparing the graphs of two or more numeric patterns can help students see similarities and differences Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

NUMBER OF SECTIONS CREATED

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

• Students can look for patterns such as:

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4 2 1 2 3 FOLD NUMBER

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RESOURCE SHEETS Strategies, explanations and hints Strategies for determining sequences (cont.) • When working with sequences involving fractions, students could: (a) find the same fraction applied to a series of numbers; for example: 1⁄3 of 3, 6, 12,18, 21 (1,2,4,6,7) (b) add, subtract, multiple or divide a series of numbers by the same fraction;

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for example: the sequence 1¼, 1½, 1¾, 2, ... increases by ¼ each time. (c) find a pattern in the denominators or numerators of fractions; for example: ¼, ½, ¾, 1, 1¼, 1½, 1¾ …

• When investigating number sequences using decimals, this can include: (a) terminating and recurring decimals – Some fractions when converted to decimals, result in a decimal with a finite number of place values. These are known as terminating decimals. For example, 1⁄8 when converted to a decimal, either by a written method or calculator, becomes 0.125.

– However, fractions such as 1⁄3, 5⁄6, 1⁄9, 1⁄7, when converted to a decimal, result in recurring decimals: For example: 1⁄7 becomes 0.142857142857143 …

For example, 1⁄9 can be written as 0.11111 …, 0.1 or 0.11

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Such number sequences can be investigated using a calculator. Other recurring decimal sequences which the and . At this time it would be prudent to introduce to students students could investigate are , , , , the symbol used to indicate a recurring decimal.. Most often, these are ellipsis (…), placed after the last digit written, or a dot or horizontal line (vinculum) placed above the recurring digit or digits.

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(b) finding missing elements in a sequence which involves adding, subtracting, multiplying or dividing numbers by a specific decimal For example, in the sequence 5.2, 4.4, 3.6, 2.8 … , 0.8 is subtracted each time.

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For example, the number sequence 5689, 568.90, 56.89, 5.689 … shows a pattern indicating that each number is subsequently divided by 100. The next numbers in the sequence are 0.5689, 0.05689, 0.005689 and so on.

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

Note: The students may have been encountered many of these sequences when they completed strategies in the Number and Place Value section of this book.

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(d) use a calculator to investigate the pattern which occurs when common fractions (such as 1⁄7) are converted to decimals and try to predict what larger quantities of that fraction, would be (see below).

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEETS Strategies, explanations and hints Finding and describing a rule • A sequence must have at least four numbers before students are given enough information to be able to work out the pattern. Otherwise, several patterns or rules may fit. For example, consider the two-number sequence: 2, 4. The rule could be: – ‘Start at 2 and add 2’.

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– ‘Start at 2 and square each successive number’

– ‘Start at 2 and double each successive number’.

• When describing a rule, it is necessary to state a starting point. This will make the rule specific. For example, for the pattern 1, 3, 5, 7 ..., the rule must state, ‘Start at 1 and add 2 to each successive number’.

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• Two rules can produce the same sequence of numbers. This may be due to the ways students describe each rule. When two or more rules correctly describe the same sequence, these rules are then considered to be equivalent or equal. Thinking about a rule in a different way may help students find an easier way to solve problems. Note: Each number in the sequence must conform to the rule, otherwise the rule is incorrect.

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

Students’ previous experience of number lines could be used to reinforce the concept of a starting point.

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Strategies, explanations and hints

A special note from the mathematical editor Students need to have a solid understanding of multiplication and division in all its forms in order to continue or complete patterns and sequences. Here are some background notes to help you with this. You could structure similar questions as to those shown below. Forms of multiplication • Use of repeated addition

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Number sentence:

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Computation:

• Use of rates

(a) If one octopus has eight legs, how many legs do five octopuses have? (Use matchsticks arranged around counters to model a solution.) Number sentence:

Computation:

(b) Helen’s pace rate is 1.4 paces per metre. How many paces will she take to walk 100 m? Number sentence:

• Making comparisons of ratio or scale

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Number sentence: Computation:

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A 6 cm-by-9 cm photograph was enlarged so that its length changed from 9 cm to 18 cm. What is the height of the new photograph? (Use grid paper to help find a solution.)

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• Use of arrays and combinations

You bought five coloured shirts and three pairs of trousers at the shop. How many days in a row could you wear a different arrangement of outfit? Organise five counters in different colours and three different coloured 2-cm cubes to model a solution. Number sentence: Computation: • Use of measures and quantities At the party, 24 students drank about 2⁄3 litre of juice each. How much juice was drunk at the party? (Think about this as a repeated addition.) Number sentence: Computation: © Richard Korbosky 174

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

Each day, Robert puts $0.50 into his money box. How much money has he saved after two weeks? (Model how much Robert saves with real money.)

Sub-strand: Patterns and Algebra—P&A – 1

Strategies, explanations and hints

Forms for division Students need experience modelling all the common types of division problems. Modelling includes physically (manipulating materials or using drawing), solving problems mentally, and counting (including using calculators).

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They need to learn to represent a problem in a number sentence and then solve the problem using a written, mental or calculator computation. These problems provide links to algebra with division calculation .

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You had to pay 65 cents per kilogram for peaches at the local store. If you paid a total of $5.20, how many kilograms of peaches did you buy? Number sentence: Computation:

• Use of sharing (Know how many partitions.)

Twenty-eight sweets were shared equally among seven children. How many sweets did each child get? (Use counters to help solve the problem.)

Share $33 among eight students. How much does each student receive? (Use money to model a solution.)

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Number sentence: Computation:

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

• Use of grouping (Know the size of the portions.)

• Making comparisons of ratio or scale

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Ryan has one packet of 15 stamps. How many packets does Greer have if she has 90 stamps? (Use counters or 2-cm blocks to help find the answer.)

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Number sentence: Computation:

• Use of measures and quantities

The perimeter of a square field is 36 metres. What is the length of each side? (Use grid paper to help find a solution.) Number sentence: Computation: © Richard Korbosky

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RESOURCE SHEETS Examples of function machines – 1

Input

Function

Output

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Out

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Input

Function or rule

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Function or rule

Output

In In

Rule

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Out In

Input

Function or rule

Out

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Input

Out

Output Rule

Input

Function or rule

Output Out

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

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Rule

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEETS Examples of function machines – 2

RULE:

RULE: Input

Output

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Output

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

Input

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Examples of recording sheets for function machines Use the recording sheet for the following types of function machines: 1. Basic function machine: input and function amounts provided; students find the output by calculating it. 2. More difficult function machine: function and output amounts provided; students find the input by working backwards/using inverse operations.

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3. Challenging function machine: input and output amounts provided; students find the function. Output

Input

Function

Output

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Input

Function

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Output

Input

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Function

Output

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

Function

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Input

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

Blank Pascal’s triangle/Fibonacci sequence templates

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Assessment 1

Sub-strand: Patterns and Algebra—P&A – 1

NAME:

DATE:

Sequences involving whole numbers 1. Continue each sequence by writing the next two numbers. Then describe the rule for each sequence. (a)

178, 213, 292, 327, Rule:

Rule:

13, 21, 34, 55 ,

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Rule:

(d)

36, 324, 2916, 26 244,

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(c)

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370, 297, 224, 151,

Rule:

2. Shade a bubble to show the next number in each sequence. Then describe the rule for each sequence. 9 0 16 8 (a) 8192, 2048, 512, 128, 32 …

–24, –17, –11, –4, 2, 9 … Rule:

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(c)

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Rule:

(d)

–8

–2, –6, 3, –1, 8 …

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(b)

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466

563

154

42, 68, 110, 178, 288 … Rule:

3. If Ted stores two bolts in the first jar, four in the second jar, seven in the third jar, 11 in the fourth jar and 16 in the fifth jar, how many will he store in the sixth jar if the pattern continues? Why?

4. Create a whole number sequence of your own. Then describe the rule. ,

…

Rule: 180

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence.

(b)

Assessment 2

Sub-strand: Patterns and Algebra—P&A – 1

NAME:

DATE:

Sequences involving fractions 1. Continue each sequence by writing the next two numbers. Then describe the rule for each sequence. 1, ½, 1⁄3, ¼, 1⁄5,

(a)

Rule:

½, ¼, 1⁄8, 1⁄16,

(b)

Rule:

51⁄3, 7, 8 , 10 , Rule:

101⁄10, 83⁄5, 71⁄10, 53⁄5,

(d)

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(c)

Rule:

2. Shade a bubble to show the next number in each sequence. Then describe the rule for each sequence. 81⁄3 72⁄3 61⁄3 102⁄3 (a) 2⁄3, 11⁄3, 22⁄3, 51⁄3 …

(b)

½, ¼, 1⁄8, 1⁄16 …

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Rule:

(d)

1⁄3

41⁄3

1⁄72

2⁄5

82⁄3

31⁄3

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6, 51⁄3, 42⁄3, 4 … Rule:

1⁄32

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(c)

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

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3. If 1⁄16 of the 1st box contained chocolates with soft fillings, the 2nd box had with soft fillings, the 3rd box had and the 4th box had , what was special about the 5th box if the pattern continued? Why was it special?

4. Create one sequence of your own using fractions. Then describe the rule. ,

…

Rule:

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Assessment 3

Sub-strand: Patterns and Algebra—P&A – 1

NAME:

DATE: Sequences involving decimals

1. Continue each sequence by writing the next two numbers. Then describe the rule for each sequence. (a)

5.2, 4.4, 3.6, 2.8,

Rule: 9.1, 7.9, 6.7, 5.5, Rule:

Rule:

0.043, 0.43, 4.3, 43,

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(d)

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365 230, 36 523.0, 3652.3, 365.23,

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(c)

Rule:

2. Shade a bubble to show the next number in each sequence. Then describe the rule for each sequence. 40.3 25.8 35.8 40.8 (a) 28.3, 30.8, 33.3, 35.8, 38.3 …

0.2, 0.04, 0.008, 0.0016 … Rule:

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(c)

17.9, 16.7, 15.5, 14.3 …

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Rule:

(d)

25.2

7.8, 78, 780, 7800 … Rule:

13.1

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(b)

12.9

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87 000 78 000 780 000

3. Create one sequence of your own using decimals. Then describe the rule. ,

…

Rule:

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CONTENT DESCRIPTION: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

(b)

Checklist

Sub-strand: Patterns and Algebra

Continue and create sequences involving whole numbers, fractions and decimals.

Describes a rule used to create a sequence

Creates a sequence involving decimals

Creates a sequence involving fractions

Creates a sequence involving whole numbers

Continues a sequence involving decimals

Continues a sequence involving fractions

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STUDENT NAME

Continues a sequence involving whole numbers

Describe the rule used to create the sequence (ACMNA133)

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Explore the use of brackets and order of operations to write number sentences (ACMNA134)

TEACHER INFORMATION

RELATED TERMS Brackets

What this means

• Brackets ( ) are parentheses of various shapes indicating that the enclosed quantity should be treated as a unit

• Students should be able to use brackets and the order of operations to complete multiple operations within the same number sentence. • Students should be able to write number sentences from information given about the sentence. • Students need to be able to understand and identify properties of prime, composite, square and triangular numbers when looking at patterns and number sequences. • Students should understand that numbers (both positive and negative) continue indefinitely. • Students should understand the associative, commutative and distributive laws that aid mental, written and algebraic computations.

Operations

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Order of operations

• A convention for simplifying operations that stipulates that multiplication and division are performed before addition and subtraction, and in order from left to right. For example, in 5 – 6 ÷ 2 + 7, the division is performed first and the operation becomes 5 – 3 + 7 = 9. If the convention is ignored and the operations are performed in order from left to right, the incorrect result, (6.5), is obtained.

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• The processes of combining numbers and expressions. Basic operations include addition, subtraction, multiplication and division

Teaching points

• Students should be given the opportunity to use number lines that include both positive and negative numbers. • Introduce students to prime, composite, square and triangular numbers through investigations, mathematics manipulatives and tables, and to record what they find out. (Note: Students should have encountered these concepts in Number and Place Value in content description associated with code N&PV – 1.) • Students should be given the chance to understand the commutative property of certain operations and apply this information to pronumeral thinking.* For example, if 4 + 5 = 5 + 4, then x + y = y + x; and if

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Symbols brackets ( ) positive (+)

5 x 3 = 3 x 5, then a x b = b x a.

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• Students should be given the chance to understand the associative property of certain operations and apply this information to pronumeral thinking.* For example if , (4 + 5) + 6 = 5 + (4 + 6), then (x + y) + z = x + (y + z).

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• Students should be given the chance to understand the distributive property of certain operations and apply this information to pronumeral thinking.* For example if, 2(3 + 4) = 2 × 3 + 2 × 4, then 4(a + b) = 4a + 4b. • Students should be able to use brackets and the order of operations to complete multiple operations within the same number sentence. Example of number sequences are: 3 + 2 x 6 = ?, 6 x 3 – 3 = ?, 20 – 1 x 5 = ?

negative (–) addition (+) subtraction (–) multiplication (x) division (÷)

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Sub-strand: Patterns and Algebra—P&A – 2

Explore the use of brackets and order of operations to write number sentences (ACMNA134)

TEACHING INFORMATION (CONT.) • Students should be able to write number sentences from information given about the sentence; for example, ask students to use at least two operations to make a number sentence about the number 15. * Note: While pronumeral thinking is mentioned here, students are not expected to start using pronumerals until Year 7. But they should be exposed to thinking strategies that lead to the use of pronumerals.

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• With understanding equivalency of operations, students need to know that with multiplication and addition the position of numbers can change without affecting the answer. However, it needs to be well understood that the same does not apply to subtraction and division. This can be difficult for some to understand, and when they do apply the concept of equivalency to subtraction and division and their answers are incorrect, they do not know what is wrong with their reasoning.

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What to look for

See also New wave Number and Algebra (Year 6) student workbook (pages 75–82)

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Understanding Fluency Problem solving Reasoning

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Sub-strand: Patterns and Algebra—P&A – 2

HANDS-ON ACTIVITIES Dice game Provide groups of three students with a small plastic cup containing up to six six-sided dice. They will also need paper and a pen or pencil. One student is chosen to select a number between 50 and 66, which he/ she tells the other two students. The other students in the group throw the dice to reveal six numbers which they must use with any inserted symbols (+, –, x and ÷) to create a number sentence as close as possible to the number chosen. The student who selected the number checks the sums to see if they are correct. The student who gets an answer closest to the chosen number is awarded five (5) points. This continues until one student reaches a total of 20 points (or the number of points chosen by the teacher). ANSWER

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Order of operations dominoes (a card game)

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English

QUESTION (3 + 2) x 3

ANSWER 15

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Ask pairs of students to make up problems which include the four operations with brackets. They write the questions on the bottom half of the blank cards on page 190 and the answer on the top half of the next card. Continuing in this way, they complete at least 20 cards to compile a series of domino-style cards relating to the order of operations. Ensure the students lay the cards end-to-end as they are creating them to guarantee they match up properly. When completed, the cards may be placed in a resealable bag and swapped with those of another group. An infinite number of cards may be used. Alternatively other students may be encouraged to extend the ‘domino-style’ line of cards.

QUESTION 20 + (6 x 2)

ANSWER 32

QUESTION (7 – 2) x 10

Information and Communication Technology

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• Ask the students to write a mnemonic (a word, sentence or verse intended to assist the memory) for the order of operations. One common mnemonic is BIMDAS (Brackets, Indices, Multiplication and Division and Addition and Subtraction). Others include BODMAS (Brackets, Orders [powers, square roots etc.] Division and Multiplication, Addition and Subtraction).

• Visit <http://www.onlinemathlearning.com/bodmas.html> for a clear explanation (with examples) of the order of operations. The website <http://www.bbc.co.uk/schools/ks3bitesize/maths/number/order_operation/revise1.shtml> also has a clear explanation but uses the acronym BODMAS instead. • Visit <http://www.mathsatwhitehaven.com/staff/games/bodmas/imgame2.htm> to play an interactive BODMAS card game. Students should calculate the answer before selecting the second card and then checking the answer. Since students of this age group have not been introduced to exponents (indices), they will need to disregard this section of the acronym. • Visit <http://www.mathsticks.com/resource/1012/calculation-strings> to find a description of a game which can be downloaded to practise order of operations. The games have been designed to be used with a dice.

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The Arts • Learn the order of operations by listening to a song at <http://www.onlinemathlearning.com/ bodmas.html>. Note: This refers to BODMAS, not BIMDAS. Students may wish to make up their own version. • Have the students plan and design a poster which explains the order of operations. They could highlight the beginning letters of the words brackets, multiplication, division, addition and subtraction. Their design should indicate that multiplication and division are of equal importance, as are addition and subtraction.

Brackets Multiplication Division Addition Subtraction

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FIRST

Indices

LAST

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEETS Explanations, strategies and hints The order of operations Introduce the need for brackets and the rules for completing operations by asking the students to find the answers to two or more problems involving more than one operation. Examples are: 15 – 4 x 2,

9 x 4 = 6,

12 ÷ 2 + 4,

4 + 6 x 7,

22 + 3 x 4,

5 x 2 x 3 + 4.

When the answers for each are supplied, note that even though the students seem to have found the answers correctly, some answers may differ. State that, as discovered in the previous content description, there are rules which mathematicians follow to answer number sentences with multiple operations.

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Operations must be done in the following order:

1. Brackets first (If there is more than one set of brackets, do the inner set first.)

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2. Indices (powers; e.g: 22, 33, square roots)

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3. Multiplication and division come next. These are of equal importance and one particular operation does not have to be completed before the other. (This is because they are inverse operations.) Rather, these should be done in order, from left to right as they appear in the algorithm. 4. Addition and subtraction are last. Like division and multiplication, these are inverse operations so it does not matter which is completed first as the result will be the same. These should be also be done in order from left to right.

• Introduce the students to the acronym of your choice (BIMDAS, BODMAS etc.), then, as a class, create your own reminder of the correct order of operations.

Brackets, the number line and positive and negative numbers

–7

–6

–5

–4

–3

–2

–1

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–8

All numbers are either positive or negative (excepting zero). Because positive numbers are the most common, the positive sign is often omitted.

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CONTENT DESCRIPTION: Explore the use of brackets and order of operations to write number sentences

• Students should have practice recognising that an unknown element in a number sentence can be represented by an empty box( ), question mark (?), an empty space etc. and that the unknown element can occur in any position on both sides of the equation. This will introduce students to the concept of pronumerals (such as a, b, c, x, y, z) etc. to represent an unknown number in an equation in later algebraic expressions.

So the number sentence 28 – 16 could be written as +28 – +16.

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When adding and subtracting (or multiplying and dividing) with positive and/or negative numbers, brackets and rules are used for clarity.

o c . che e r o t r s super

Simple number sentences could be written with brackets surrounding the positive or negative integers. For example, 50 + 40 = 90 could be written as (+50) + (+40) = +90, while –50 + –40 = –90 can be written as (–50) + (–40) = –90 This helps to separate the ‘operation’ (addition, subtraction, multiplication or division) into the positive and negative integers. While it is not common practice to do this, having students experience writing number sentences in this way can help them understand the use of brackets better.

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187

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEETS Explanations, strategies and hints Brackets, the number line and positive and negative numbers (cont.) Number lines can be used to show the relative positions of positive and/or negative numbers. (a) When two positive numbers are added, the sum (answer) will be positive. The first positive number gives the starting position on the number line. Because it is positive, it will be a position to the right of zero. Adding a second positive number means moving in a positive direction on the number line (to the right). For example: (+20) + (+10) = ?

r o e t s Bo r e p ok u S

Start at a position right of zero and move further to the right by 10 units – +

Teac he r

– 40

– 30

– 20

– 10

So the answer to +20 + (+10) is 30.

For example: (–20) + (–10) = ?

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(b) When two negative numbers are added, the answer is obtained by moving to the left.

– + – 40

– 30

– 20

– 10

So the answer to (–20) + (–10) is –30.

(c) When both a negative and positive number are part of an addition problem, movement will be both to the left and the right. The position of the final answer will depend on which number is larger, the positive number or the negative number. For example: (+20) + (–30) = ?

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– +

. te – 40

– 30

– 20

– 10

o c . che e r o t r s super

So the answer to (+20) + (–30) is –10. For example: (–30) + (+20)

m . u

First, move to the right from zero to the positive number, then move to the left from 20 by 30 units to land on –10.

First, move to the left from zero to the negative number, then move to the right from –30 by 20 units to land on –10. – +

– 40

– 30

– 20

– 10

So the answer to (–30) + (+20) is –10. Moving left along the number line is the same as either adding a negative number or subtracting a positive number. Moving to the right along the number line is the same as either adding a positive number or subtracting a negative number.

188

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Explore the use of brackets and order of operations to write number sentences

Start at a position left of zero and move further to the left by 10 units

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEETS Explanations, strategies and hints Brackets, the number line and positive and negative numbers (cont.) The following rules apply when adding or subtracting positive and negative numbers: (a) If the operation and the sign are the same (i.e. adding a positive number or subtracting a negative number) they work together as though a positive number is being added. For example: or or

Teac he r

(+60) + (+20) = (+60) + 20 = +80

+ and –

(+60) – (–20) = (+60) + 20 = +80

Two like signs become a positive (+). Two unlike signs become a negative (–).

r o e t s Bo r e p ok u S (–60) + (+20) = (–60) + 20 = –40 (–60) – (–20) = (–60) + 20 = –40

For example:

(+60) + (–20) = (+60) – 20 = +40

(+60) – (+20) = (+60) – 20 = +40

(–60) + (–20) = (–60) – 20 = –80

(–60) – (+20) = (–60) – 20 = –80

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(b) If the operation and the sign are different (i.e adding a negative number or subtracting a positive number), they work together as though a positive number is being subtracted.

The rules for multiplication and division are the same because they are inverse operations.

(a) If the signs are the same, the answer is positive. (+60) x (+60) = +3600

(+60) ÷ (+20) = +3

(–60) x (–60) = +3600

x and ÷ same signs (+) x (+) = (+) (–) x (–) = (+) (+) ÷ (+) = (+) (–) ÷ (–) = (+)

m . u

For example:

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CONTENT DESCRIPTION: Explore the use of brackets and order of operations to write number sentences

The following rules apply when multiplying and dividing with positive and negative numbers:

(–60) ÷ (–20) = +3

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(b) If the signs are different, the answer is negative. For example:

o c . che e r o t r s super

(+60) x (–60) = –3600

(+60) ÷ (–20) = –3

(–60) x (+60) = –3600

(–60) ÷ (+20) = –3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

Different signs (+) x (–) = (–) (+) ÷ (–) = (–) (–) x (+) = (–) (–) ÷ (+) = (–)

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189

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Blank order of operations dominoes ANSWER

ANSWER

r o e t s Bo r e p ok u S

QUESTION

ANSWER

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QUESTION

190

m . u

o c . che e r o t r s super ANSWER

QUESTION

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

ANSWER

QUESTION

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Explore the use of brackets and order of operations to write number sentences

ANSWER

QUESTION

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Teac he r

QUESTION

ANSWER

Assessment 1

Sub-strand: Patterns and Algebra—P&A – 2

NAME:

DATE:

1. Use the number line to solve the addition problems involving positive and negative numbers. Start at zero and use an arrow to show the direction of movement along the number line. (a)

(+10) + (+100) = – 40

(b)

– 30

– 20

100

110

120

130

140

(+300) + (–100) = –700

–600

r o e t s Bo r e p ok u S –500

–400

–300

–200

–100

100

200

300

400

500

600

(–70) + (+90) =

–80

–70

–60

–50

–40

–30

–20

–10

(–65) + (–15) =

–85 –80 –75 –70 –65 –60 –55 –50 –40 –45 –40 –35 –30 –25 –20 –15 –10 –5

100

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Teac he r

(c)

(d)

– 10

10 15 20 25 30 35 40 45 50 55 60 65 70 75

2. Use the number line to solve the subtraction problems involving positive and negative numbers. Start at zero and show the direction of movement along the number line.

–54 –52 –50–48–46 –44 –42 –40 –38–36 –34 –32 –30 –28 –26 –24 –20–18 –16 –14 –12 –10 –8 –6 –4 –2 0

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

. te

–0

(–24) – (+21) =

–33 –30 –27 –24

(d)

(+45) – (–25) = –30 –25 –20 –15 –10 –5

(c)

2 4

m . u

(b)

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CONTENT DESCRIPTION: Explore the use of brackets and order of operations to write number sentences

(a)

–36

–33 –30 –27

o c . che e r o t r s super

–21 –18 –15 –12

(–18) – (–9) =

–9

–6

–24 –21 –18 –15 –12

–3

–9

–6

–3

95 100 110

3. Write one rule relating to movement along the number line: (a)

when adding positive and negative numbers.

(b)

when subtracting positive and negative numbers.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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191

Assessment 2

Sub-strand: Patterns and Algebra—P&A – 2

NAME:

DATE:

1. Write the rule for the correct order of operations.

2. Find the answers. (a)

8 + (2 x 8) =

(d)

(13 – 2) x (4 + 3) =

(e)

(g)

(6 x 4) + 45 =

(h) (12 x 12) + 9 =

(b) 20 – (5 x 3) =

r o e t s Bo r e p ok u S 14 – (1 ÷ 2) =

(f) (12 ÷ 5) + 18 =

(i)

560 – (5 x 6) =

4 + (3 x 5)

(b)

9 – (3 x 3)

(c)

(9 – 5) x (3 + 2)

(d)

16 ÷ (8 – 6)

–9

(f)

(56 + 56) + (9 ÷ 2)

–4

116½

60½

120

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4. Insert brackets to make the number sentences true. (a) (d)

9 – 1 x 7 = 56

(b)

6–8÷2 =2

o c . che e r o t r s super

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(e)

50 – 8 ÷ 2 = 21

m . u

(e)

45 + 45 x 5 x 2 = 900

92 – 10 x 5 = 42

5. Use brackets and the order of operations to write and answer two number sentences of your own. Show all your working and use all four operations between the two number sentences. (a)

(b)

192

Working:

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Explore the use of brackets and order of operations to write number sentences

(a)

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Teac he r

3. Shade a bubble to give the correct answer for each number sentence.

Checklist

Sub-strand: Patterns and Algebra—P&A – 2

Uses brackets and order of operations to write number sentences

Uses brackets and order of operations to answer number sentences

Uses the order of operations

Understands the order of operations

r o e t s Bo r e p ok u S

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Teac he r

STUDENT NAME

Understands the use of brackets for operations involving positive and negative numbers

Explore the use of brackets and order of operations to write number sentences (ACMNA134)

w ww

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m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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193

Answers

Sub-strand: Patterns and Algebra

Page 182 Assessment 3

P&A – 1 Page 180 Assessment 1

r o e t s Bo r e p ok u S P&A – 2 Page 191 Assessment 1

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Teac he r

1. (a) 406, 441 / RULE: Start 178 and add 35, then add 79 to each successive number. (b) 78, 5 / RULE: Start with 370 and subtract 73 from each successive number. (c) 236 196, 2 125 764 / RULE: Start with 36, multiply it by 9, and each successive number by 9. (d) 89, 144 / RULE: Start with 13 and find each successive number by finding the sum of the previous two numbers. 2. (a) 8 / RULE: Start with 8192 and divide it by 4, then divide each successive number by 4. (b) 15 / RULE: Start with –24 and add 7, then add 6 alternately to successive numbers. (c) 4 / RULE: Start with –2 and subtract 4 to get the next number, then add 9 to get the third number. Continue alternating subtracting 4 then adding 9 to get each successive number. (d) 466 / RULE: Start with 42 and 68 then find each successive number by finding the sum of the previous two numbers. 3. 22/The RULE for this word problem is add a number one greater than the last each time, so +2, +3, +4, +5, +6. 4. Teacher check

1. (a) 2, 1.2 / RULE: Start with 5.2, then subtract 0.8 for each successive number. (b) 4.3, 3.1 / RULE: Start with 9.1, then subtract 1.2 for each successive number. (c) 36.523, 3.6523 / RULE: Start with 365 230 and divide by one power of ten for each successive number. (d) 430, 4300 / RULE: Start with 0.043 and multiply each successive number by one power of 10. 2. (a) 40.8 / RULE: Start with 28.3 and add 2.5 to each successive number. (b) 0.00032 / RULE: Start with 0.2 and multiply each successive number by 0.2. (c) 13.1 / RULE: Start with 17.9 and subtract 1.2 from each successive number. (d) 78000 / RULE: Start with 7.8 and divide by 0.1 for each successive number. (Similar rules may apply.) 3. Teacher check

1. (a) 110 (b) 200 (c) 20 (d) –80 2. (a) 32 (b) 70 (c) –3 (d) –9 3. Teacher check answers Refer to pages 189 to 190 for possible suggestions.

Page 181 Assessment 2

1⁄6, 1⁄7 / RULE: Start with 1 and keep the numerator

w ww

constant, but for each successive fraction the denominator increases by 1. (b) 1⁄32, 1⁄64 / RULE: Start with ½ and divide each successive number by 2 (halve each successive fraction). (c) 12, 132⁄3 / RULE: Start with 51⁄3 and add 12⁄3 from each successive number. (d) 41⁄10, 23⁄5 / RULE: Start with 101⁄10 and subtract 1½ from each successive number. 2. (a) 102⁄3 / RULE: Start with 2⁄3 and double the numerator for each successive number. Note: Students will need to convert the mixed fractions to improper fractions. (b) 2¼ / RULE: Start with 1¼ then add ¼ to each successive number. (c) 1⁄32 / RULE: Start with ½ then halve each successive fraction. (d) 31⁄3 / RULE: Start with 6 and subtract 1⁄3 from each successive number. 3. All the chocolates in the 5th box had soft fillings. For each successive box of chocolates, the fraction (of soft fillings) was multiplied by 2 so the fraction sequence was 1⁄16, 1⁄8, ¼, ½ and 1. 4. Teacher check

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194

Page 192 Assessment 2

1. (a) brackets; orders, powers, exponents; division/ multiplication (working from left to right); addition/ subtraction (working from left to right) 2. (a) 24 (b) 5 (c) 4 (d) 77 (e) 13½ (f ) 202⁄5 (g) 69 (h) 153 (i) 530 3. (a) 19 (b) 0 (c) 20 (d) 8 (e) 89 (f ) 116½ 4. (a) (9 – 1) x 7 = 56 (b) (50 – 8) ÷ 2 = 21 (c) 40 – (10 x 2) = 20 (d) 6 – (8 ÷ 2) = 2 (e) (45 + 45) x (5 x 2) = 900 (f ) 92 – (10 x 5) = 42 5. Teacher check

m . u

1. (a)

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

NEW WAVE NUMBER AND ALGEBRA (YEAR 6) STUDENT WORKBOOK ANSWERS Page 6

N&PV – 1 Page 2

Key

Fun with composite numbers

Hole

Metres

Par

Stroke index

410

359

173

51, 52, 54, 55, 56, 57, 58, 60

323

90 – 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 87 – 3, 29 119 – 7, 17 96 – 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 81 – 3, 9, 27 114 - 2, 3, 6, 19, 38, 57 108 – 2, 3, 4, 6, 9, 12, 18, 27, 36, 54

451

424

441

129

501

Teacher check

Prime

4, 6, 8, 9, 10

Composite

12, 14, 15, 16, 18, 20

32, 33, 34, 35, 36, 38, 39, 40

42, 44, 45, 46, 48, 49, 50

6. 7.

r o e t s Bo r e p ok u S

Teac he r

Square

Triangular

Total

Triangular numbers

Hole

Metres

Par

Stroke index

166

431

288

220

11 12 13 14 15 16 17 18 19 20

301

21 22 23 24 25 26 27 28 29 30

153

31 32 33 34 35 36 37 38 39 40

297

41 42 43 44 45 46 47 48 49 50

289

51 52 53 54 55 56 57 58 59 60

419

61 62 63 64 65 66 67 68 69 70

Total

1+2=3 1+2+3=6 1 + 2 + 3 + 4 = 10 1 + 2 + 3 + 4 + 5 = 15 1 + 2 + 3 + 4 + 5 + 6 = 21 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

m . u

w ww

81 82 83 84 85 86 87 88 89 90

N&PV – 2

91 92 93 94 95 96 97 98 99 100 Teacher check

Page 4

Page 7

. te

Colour code

Addition and subtraction go together

667 + 486 = 1153

789 + 396 = 1185

1938 + 495 = 2433

337 + 872 = 1209

593 + 985 = 1578

958 + 730 = 1688

902 – 345 = 557

1120 – 736 = 384

1007 – 225 = 782

1234 – 558 = 676

912 – 385 = 527

1289 – 447 = 842

1010 – 374 = 636

1194 – 818 = 376

1310 – 997 = 313

o c . che e r o t r s super

Prime – 1, 2, 3, 5, 7, 11, 13, 17, 19, 23 Composite – 4, 6, 8, 9, 10, 12, 15, 16, 21, 25, 28, 36, 42, 45, 48, 49, 64, 65, 66, 78, 81, 100, 104, 121, 144 Square – 4, 9, 16, 25, 36, 49, 64, 81, 121, 144 Triangular – 6, 10, 21, 28, 36, 45, 66, 78 Numbers that appear more than once: 4, 6, 9, 10, 16, 21, 25, 28, 36, 45, 49, 64, 66, 78, 81, 121, 144

Page 5

Your Partner’s score score

71 72 73 74 75 76 77 78 79 80

Your Partner’s score score

ew i ev Pr

Page 3 1.

Golf anyone?

The search for primes

Page 8

Pump price discounts

Price per litre $

Pump total $

Litres purchased

4c per litre discount =

You pay $

$1.24

$43.40

35 L

$1.40

$42.00

Key

$1.24

$64.48

52 L

$2.08

$62.40

Red

11 12 13 14 15 16 17 18 19 20

$1.24

$52.08

42 L

$1.68

$50.40

Green

21 22 23 24 25 26 27 28 29 30

$1.24

$48.36

39 L

$1.56

$46.80

Purple

31 32 33 34 35 36 37 38 39 40

$1.24

$59.52

48 L

$1.76

$57.76

Blue

41 42 43 44 45 46 47 48 49 50

$1.24

$62.00

50 L

$2.00

$60.00

Prime

51 52 53 54 55 56 57 58 59 60

$1.24

$44.64

36 L

$1.44

$43.20

61 62 63 64 65 66 67 68 69 70

$1.24

$57.04

46 L

$1.84

$55.20

71 72 73 74 75 76 77 78 79 80

$1.24

$84.32

68 L

$2.72

$81.60

81 82 83 84 85 86 87 88 89 90

$1.24

$45.88

37 L

$1.48

$44.40

91 92 93 94 95 96 97 98 99 100

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Page 8

Pump price discounts (continued)

Page 11

Multiplication with zeros

3981 x 700

7485 x 400

$57.50

2 786 700

2 994 000

$3.36

$70.00

49 L

$2.94

$61.25

19 549 x 300

4999 x 600

70 L

$4.20

$87.50

5 864 700

2 999 400

$91.70

$1.31

$79.91

61 L

$3.66

$76.25

36 165 x 4 000

65 499 x 7 000

13 345 x 3 000

6719 x 2000

$1.31

$22.27

17 L

$1.02

$21.25

144 660 000

458 493 000

40 035 000

13 438 000

$1.31

$37.99

29 L

$1.31

$68.12

52 L

$1.31

$56.33

43 L

$1.31

$79.91

61 L

Price per litre $

Pump total $

Litres purchased

6c per litre discount =

You pay $

$1.31

$60.26

46 L

$2.76

$1.31

$73.36

56 L

$1.31

$64.19

$1.31

$36.25

$3.12

$65.00

$2.58

$53.75

$3.66

$76.25

Subtraction with zeros—don’t be heroes

7000 – 5675

6000 – 4787

1325

1213

10 000 – 7 649

4000 – 1267

2 351

2733

700 000 – 78 605

600 000 – 22 787

100 000 – 84 919

40 100 – 22 658

621 395

577 213

15 081

17 442

8100 – 5488

6100 – 3387

10 100 – 3 349

4100 – 1727

2612

2713

6751

2373

7001 – 3542

6001 – 4465

10 001 – 2 765

4001 – 3157

3459

1536

7 236

844

7010 – 3485

6010 – 5697

10 010 – 2 259

4010 – 2637

3525

313

7751

1373

5430 37 777 + 32 471 70 248

4883 x 80

97 960

330 950

467 010

390 640

3561 x 300

4281 x 400

12 351 x 700

30 451 x 5 000

1 068 300

1 712 400

8 645 700

152 255 000

3219 x 80

46 833 x 40

14 519 x 60

78 819 x 20

257 520

1 873 320

871 140

1 576 380

Page 12 1.

(a)

(b)

(c)

(d)

(e)

Addition, addition, addition 4999 + 4323

19 999 + 3 441

4999 + 1243

9322

. te

6242

47 777 + 22 323

70 100

23 440

177 777 + 24 111 201 888 21 189 + 3 341

(f )

(g)

Addition and subtraction—the same sum or different? 93 832 – 56 784

93 832 + 56 784

37 048

150 616

14 509 – 7 882

14 509 + 7 882

6 627

22 391

64 551 – 32 114

64 551 + 32 114

32 437

96 665

77 023 – 41 596

77 023 + 41 596

35 427

118 619

39 282 – 29 023

39 282 + 29 023

10 259

68 305

61 993 – 38 827

61 993 + 38 827

23 166

100 820

77 309 – 38 936

77 309 + 38 936

38 373

116 245

(h)

(i)

(j)

21 399 – 15 010

21 399 + 15 010

6 389

36 409

38 923 – 11 778

38 923 + 11 778

27 145

50 701

52 066 – 23 887

52 066 + 23 887

28 179

75 953

(k)

26 209 – 9 846

26 209 + 9 846

16 363

36 055

(l)

31 663 – 24 409

31 663 + 24 409

7 254

56 072

o c . che e r o t r s super 4 717 + 22 412

27 129

2188 + 1422

5864 + 3323

3610

9187

24 530

3991 + 3142

8991 + 4441

13 491 + 2 341

7133

13 432

15 832

9004

22 919 + 3 421

6459 + 1413

21 319 + 2 211

6319 + 2433

26 340

7872

23 530

8752

196

15 567 x 30

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3999 + 1431

6619 x 50

m . u

Page 10

2449 x 40

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Teac he r

Page 9

r o e t s Bo r e p ok u S $1.74

4167 + 1323

5490

5891 + 3113

Pair of equations with the wider range in answers is (a)

N&PV – 3 Page 13 1.

–5

–2

–7

–3

The Great TV Quiz – 1

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Page 14 8.

–1

The Great TV Quiz – 2

⁄8 and 3⁄4

11⁄2 and 3⁄2

⁄9 and 4⁄18

10. –2 11. 3 12. Jackson –5, Susan –2, Sam –7, Nigel –7, Gavin –3, Freddie –2, Wendy 1

Page 22 Teacher check

14. Ivan 3

Page 23

Page 15

What’s their range?

Largest = 11⁄2 and 3⁄2

⁄12 and 3⁄4

Largest = 9⁄12 and 3⁄4

Between zero and two

r o e t s Bo r e p ok u S Page 24

Thur

Fri

Freezeville

Chilltown

Imafrozen

Blizzard

Fridgetown

Polarville

Teac he r

Wed

How well do you know your fractions?

⁄9, 3⁄16, 3⁄8, 4⁄10, 5⁄12

⁄8, 3⁄10, 1⁄4, 2⁄9, 2⁄11

⁄11, 4⁄10, 3⁄7, 5⁄9, 5⁄8

⁄7, 3⁄8, 4⁄11, 5⁄16, 2⁄9

⁄11, 6⁄10, 1⁄3, 5⁄16, 1⁄4

⁄10, 4⁄7, 9⁄16, 1⁄2, 7⁄15

Select the spot

Teacher check

It can get cold!

⁄16, 1⁄3, 7⁄15, 6⁄10, 5⁄8

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Tue

Page 17

⁄3 and 6⁄9

Teacher check

Mon

Page 16

Largest = 6⁄8 and 3⁄4

On this line

13. Ana 2, Warren 2, Ivan 3 15. Nigel –7 and Sam –7

⁄7 and 10⁄14

⁄10, 3⁄7, 4⁄9, 9⁄16, 5⁄8

⁄14, 11⁄16, 3⁄4, 12⁄15

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 23 29 22

F&D – 2

Coldville

25 22 23 24 19 21 17 21 23

Chilltown

22 16 17 16 18 17 17 17 19

18 16 15

Page 25

Freezetown 19 19 20 20 19 18 17 19 15

14 15 16

–3

⁄5

⁄10 11⁄10

⁄5

⁄10 13⁄10

–9

⁄5

–4

–1

–5

w ww

–4

Page 19

(c)

. te

Page 21 and 8⁄10

⁄

and 6⁄16

Largest = 4⁄5 and 8⁄10

⁄

and 3⁄5

⁄

and 1⁄3

Largest = 6⁄10 and 3⁄5

⁄

and 6⁄9

⁄

and 3⁄4

Largest = 6⁄8 and 3⁄4

⁄

and 8⁄12

⁄

and 9⁄12

Largest = 3⁄4 and 9⁄12

⁄12 and 3⁄4

Largest = 9⁄12 and 3⁄4

⁄3 and ⁄6

⁄3 and 10⁄15

⁄8 and ⁄4

⁄10

⁄6

Largest =

⁄3 and ⁄6 4

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

13⁄4 11⁄4 11⁄2

⁄2

11⁄2 11⁄4

⁄4

11⁄4 ⁄4

⁄6

11⁄6 13⁄6 15⁄6 2

⁄6

13⁄6

⁄6

15⁄6

⁄8 or 2⁄4 or 1⁄2 6 ⁄12 or 1⁄2 1 ⁄10 3 ⁄8 4 ⁄9 5 ⁄12 7 ⁄10 12⁄16 or 11⁄8 4

⁄6

12⁄6

12⁄6 14⁄6

⁄30

⁄16 ⁄8

(f ) (i)

(b)

⁄9 7 ⁄16 17⁄8

(c)

(f )

(a) 3

(b) 11⁄2

(c)

(d) 13⁄8

(e) 16⁄10

(f )

R.I.C. Publications®

(b)

⁄20 ⁄20 or 3⁄10 4 ⁄10 or 2⁄5

(j)

⁄2

11⁄4

(h)

(g)

⁄4

(e)

(d)

⁄4

(a)

13⁄4 11⁄2

(j)

⁄4

(g)

⁄

⁄10

(c)

(a)

(d)

Pick the equivalent fractions

6 10

⁄10

o c . che e r o t r s super 2.

F&D – 1

⁄6

11⁄6

Using Scootle—Scale matters

Teacher check

(b)

⁄10 14⁄10

⁄6

Page 20

⁄10

⁄10 15⁄10 11⁄10 12⁄10

Two dice, please roll

Teacher check

⁄10

⁄5 11⁄10 17⁄10

Name the spot

⁄10

m . u

Teacher check number lines

Page 18

Fabulous fractions

(a)

(e) (h)

(i)

⁄12 ⁄20 or 1⁄5 13⁄20 ⁄8 ⁄4

www.ricpublications.com.au

197

Fraction subtract

⁄4

⁄8

⁄12

⁄10

⁄5

⁄10

⁄2

⁄10

⁄20

⁄10

⁄20

⁄20 ⁄4

Morganite vowels $22.00 consonants $37.50 cost of word $59.50

Melanite vowels $22.00 consonants $30.00 cost of word $52.00

Andalusite vowels $27.50 consonants $37.50 cost of word $65.00

Moldavite vowels $22.00 consonants $37.50 cost of word $59.50

Nuumite vowels $22.00 consonants $22.50 cost of word $44.50

Amber vowels $11.00 consonants $22.50 cost of word $33.50

Aquamarine vowels $33.00 consonants $30.00 cost of word $63.00

Moonstone vowels $22.00 consonants $37.50 cost of word $59.50

Bloodstone vowels $22.00 consonants $45.00 cost of word $67.00

Axinite vowels $22.00 consonants $22.50 cost of word $42.50

Beryl vowels $5.50 consonants $30.00 cost of word $35.50

Orthoclase vowels $22.00 consonants $45.00 cost of word $67.00

r o e t s Bo r e p ok u S 14

⁄12

⁄10

⁄8

⁄12

⁄8

⁄2

⁄8

⁄20

Page 30

Almost 1 but not quite!

⁄7 of 49 is 35 because 1⁄7 of 49 = 7 7 ⁄8 of 32 is 28 because 1⁄8 of 32 = 4 2 ⁄3 of 18 is 12 because 1⁄3 of 18 = 6 3 ⁄4 of 36 is 27 because 1⁄4 of 36 = 9 2 ⁄3 of 45 is 30 because 1⁄3 of 45 = 15 3 ⁄4 of 60 is 45 because 1⁄4 of 60 = 15 2 ⁄3 of 39 is 26 because 1⁄3 of 39 = 13 3 ⁄4 of 56 is 42 because 1⁄4 of 56 = 14 4 ⁄9 of 81 is 36 because 1⁄9 of 81 = 9 6 ⁄8 of 48 is 36 because 1⁄8 of 48 = 6 3 ⁄5 of 50 is 30 because 1⁄5 of 50 = 10

⁄ of 44 is 33 because 1⁄4 of 44 = 11

⁄10

(a) 11⁄2

⁄2

⁄8 or 23⁄8 26⁄10 or 23⁄5 19

⁄20

⁄4

⁄20

(f ) (h)

Adding fractions

(b) (d) (f )

⁄4 and 12⁄4 or 11⁄2 5 ⁄4 and 11⁄4 8 ⁄7 and 11⁄7 13 ⁄10 and 13⁄10 7 ⁄5 and 12⁄5 17 ⁄13 and 14⁄13 27 ⁄15 and 112⁄15 or 14⁄5 5 ⁄6 8 ⁄10 or 4⁄5 17 ⁄10 or 17⁄10 5 ⁄8 5 ⁄12 7 ⁄12 11 ⁄10 or 11⁄10 13 ⁄12 or 11⁄12 13 ⁄16 6 ⁄16 or 3⁄8 6

(h) (j) (l)

(n)

(b) (d) (f ) (h) (j)

(l) (n) (p) (r) (t)

. te

Expensive name plates

Malachite vowels $22.00 consonants $37.50 cost of word $59.50

198

⁄ of 63 is 49 because 1⁄9 of 63 = 7

⁄

of 64 is 40

because 1⁄8 of 64 = 8

⁄ of 99 is 77 because 1⁄9 of 99 = 11

⁄ of 21 is 18 because 1⁄7 of 21 = 3

Amazonite vowels $27.50 consonants $30.00 cost of word $57.50

⁄ of 35 is 25 because 1⁄7 of 35 = 5

⁄ of 125 is 100 because 1⁄5 of 125 = 25

⁄ of 72 is 54 because 1⁄8 of 72 = 9

⁄ of 96 is 72 because 1⁄8 of 96 = 12

⁄ of 84 is 60 because 1⁄7 of 84 = 12

⁄ of 36 is 30 because 1⁄6 of 36 = 6

o c . che e r o t r s super

F&D – 3 Page 29

⁄10 or 11⁄2 45⁄6 19⁄10

w ww

⁄4 and 11⁄4 (c) 4⁄3 and 11⁄3 (e) 6⁄5 and 11⁄5 (g) 9⁄8 and 11⁄8 (i) 7⁄4 and 13⁄4 (k) 20⁄16 and 14⁄16 or 11⁄4 (m) 20⁄14 and 16⁄14 or 13⁄7 (a) 8⁄6 or 12⁄6 or 11⁄3 (c) 9⁄6 or 13⁄6 or 11⁄2 (e) 10⁄10 or 1 (g) 10⁄8 or 12⁄8 or 11⁄4 (i) 7⁄8 (k) 7⁄12 (m) 13⁄15 (o) 22⁄30 or 11⁄15 (q) 14⁄12 or 12⁄12 or 11⁄6 (s) 17⁄12 or 15⁄12 (a)

⁄5

⁄20

Fraction spaceships

(d)

Page 28 1.

(b) 15⁄6

(g)

⁄10

Page 27

(e)

⁄10

⁄2

⁄10

ew i ev Pr

⁄10

Teac he r

m . u

Page 26

Alexandrite vowels $27.50 consonants $45.00 cost of word $72.50

Page 31

⁄3 of 24 = 8 1 ⁄5 of 35 = 7 1 ⁄4 of 48 = 12 1 ⁄6 of 42 = 7 1 ⁄8 of 56 = 7 1 ⁄9 of 27 = 3 1 ⁄3 of 51 = 17 1 ⁄5 of 60 = 12 1 ⁄4 of 76 = 19 1 ⁄6 of 84 = 14 1 ⁄8 of 48 = 6 1 ⁄9 of 63 = 7 1 ⁄10 of 120 = 12 1 ⁄12 of 72 = 6 1 ⁄3 of 96 = 32 1 ⁄5 of 85 = 17 1

And so that leads to this!

⁄3 of 24 = 16 ⁄5 of 35 = 21 3 ⁄4 of 48 = 36 4 ⁄6 of 42 = 28 3 ⁄8 of 56 = 21 5 ⁄9 of 27 = 15 2 ⁄3 of 51 = 34 4 ⁄5 of 60 = 48 3 ⁄4 of 76 = 57 4 ⁄6 of 84 = 56 5 ⁄8 of 48 = 30 4 ⁄9 of 63 = 28 5 ⁄10 of 120 = 60 3 ⁄12 of 72 = 18 2 ⁄3 of 96 = 64 4 ⁄5 of 85 = 68

so so so so

so so so so so so so so

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

so so so so so so so so

1st

Maddie

1495

2nd

Lola

1305

3rd

Bea

4th

Sacha

5th

Zung

6th

Fiona

7th

Mia

8th

Jack

11th

Grace

That much cash

12th

Eli

(n) $52

13th

Carter

14th

Bridget

15th

Charli

Teac he r (d) $13

(q) $33

16th

Eliza

(e) $32

(r) $14

17th

Callum

(f ) $16

(s) $20

18th

James

(g) $3

(t) $8

19th

Danni

(h) $80

(u) $48

Carl

(i) $39

(v) $15

21st

Anna

(j) $31

(w) $9

22nd

Ben

(k) $45

(x) $26

23rd

Annika

24th

Kyle

1244 1218 1026 988 957 957 952 897 812 796 783 741 684 676 639 638 638 622 507 494

(z) $22

25th

Jensen

80 32

33 48 39

18 14 21

26th

Claudia

27th

Leo

28th

Kate

29th

Emma

16 21 26 F

80 32

31 48 28

21 15 32 14

30th

Lily

54 24 22

13 21

31st

Annie

32nd

Luke

33rd

Libby

34th

Ned

35th

Dash

36th

Aaron

Cash pigs

. te

⁄5 of $415 = $332 4 ⁄9 of $693 = $308 2 ⁄6 of $408 = $136 2 ⁄5 of $390 = $156 5 ⁄7 of $812 = $580 2 ⁄3 of $363 = $242 7 ⁄8 of $96 = $84 4

Aaron 169

Team B Team C

476 426 406 398 342 319 311 299 261 238 213 199 169

o c . che e r o t r s super F&D – 4 Page 36

That’s a reasonable throw!

Teacher check estimates and differences

Fun for fruit pickers – 1

Team A

w ww

⁄3 of $81 = $27 3 ⁄5 of $85 = $51 2 ⁄7 of $147 = $42 2 ⁄6 of $114 = $38 1 ⁄3 of $351 = $117 3 ⁄5 of $195 = $117 6 ⁄7 of $154 = $132 Page 34

1276

ew i ev Pr

Louis

Tommy

(p) $7

Page 33

⁄9 ⁄9 4 ⁄7 4 ⁄7 3 ⁄6 3 ⁄6 4 ⁄9 3 ⁄8 3 ⁄8 4 ⁄7 3 ⁄9 2 ⁄6 4 ⁄7 3 ⁄9 3 ⁄9 2 ⁄6 4 ⁄8 3 ⁄6 2 ⁄7 2 ⁄8 2 ⁄7 3 ⁄8 2 ⁄9 2 ⁄7 2 ⁄6 1 ⁄6 2 ⁄7 1 ⁄6 1 ⁄7 1 ⁄7 1 ⁄9 1 ⁄9 1 ⁄7 1 ⁄6 1 ⁄7 1 ⁄8

r o e t s Bo r e p ok u S

Pieces picked

Name

(o) $21

Fraction equation of group

Position

(b) $18

(m) $28

Fun for fruit pickers – 2

(a) $24

(l) $54

Page 35

10th

Page 32 1.

⁄4 of 92 = 69 ⁄6 of 126 = 105 6 ⁄8 of 248 = 186 4 ⁄9 of 117 = 52 7 ⁄10 of 340 = 238 7 ⁄12 of 132 = 84 2 ⁄3 of 135 = 90 3 ⁄4 of 168 = 126 3 ⁄5 of 175 = 105 4 ⁄6 of 102 = 68 3 ⁄7 of 154 = 66 5 ⁄8 of 168 = 105

m . u

⁄4 of 92 = 23 ⁄6 of 126 = 21 1 ⁄8 of 248 = 31 1 ⁄9 of 117 = 13 1 ⁄10 of 340 = 34 1 ⁄12 of 132 = 12 1 ⁄3 of 135 = 45 1 ⁄4 of 168 = 42 1 ⁄5 of 175 = 35 1 ⁄6 of 102 = 17 1 ⁄7 of 154 = 22 1 ⁄8 of 168 = 21 1

Muscles Maria Actual total 42.35 m Placing 11th

Pectoral Petra Actual total 44.79 m Placing 10th

Bicep Betty Actual total 50.29 m Placing 7th

Abs Allan Actual total 38.57 m Placing 12th

Core Colin Actual total Placing

59.72 m 4th

Quad Quentin Actual total 56.35 m Placing 5th

Weightless Wanda Actual total 49.8 m Placing 8th

Strength Sally Actual total 54.54 m Placing 6th

Power Polly Actual total 47.18 m Placing 9th

Brawny Brian Actual total 73.39 m Placing 3rd

Thighs Teague Actual total 74.5 m Placing 2nd

Calves Callum Actual total 84.86 m Placing 1st

Ben 507

Callum 676

Annie 299

Grace 897

Maddie 1495

Emma 319

Bea 1276

Danni 638

Team D

Ned 213

James 639

Jensen 426

Team E

Luke 261

Bridget 783

Lola 1305

Team F

Kyle 476

Libby 238

Tommy 952

Team G

Dash 199

Leo 398

Carter 796

Team H

Claudia 406

Zung 1218

Eli 812

Team I

Carl 638

Jack 957

Louis 957

Page 37

Eliza 648

Teacher check

Team J

Kate 342

Fiona 1026

Team K

Ana 622

Lily 311

Sacha 1244

Team L

Annika 494

Mia 988

Charli 741

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

Add the roll of the dice

R.I.C. Publications®

www.ricpublications.com.au

199

Page 38

With 5 rolls

Tester A

Tester B

Tester C

Teacher check

2.8

3.2

6.8

Page 39

4.4

2.5

3.3

4.4

With 5 rolls—your turn

Teacher check

Run in seconds

1:53.28

113.28

From 1 to 2:

0.15 secs

1:53.43

113.43

From 2 to 3:

1.01 secs

1:54.44

114.44

From 3 to 4:

0.37 secs

1:54.81

114.81

From 4 to 5:

0.04 secs

1:54.85

114.85

1:54.87

114.87

1:54.94

114.94

1:55.05

115.05

Gap between the top 10

6.9

7.1

7.4

1.68

2.12

7.38

2.2

3.7

1.9

9.4

6.5

2.84

9.22

2.59

3.82

3.78

r o e t s Bo r e p ok u S From 5 to 6:

0.02 secs

From 6 to 7:

0.07 secs

From 7 to 8:

0.11 secs

From 8 to 9:

0.14 secs

1:55.19

115.19

1:55.19

115.19

From 9 to 10: 0 secs

Fastest

Time – min and sec

Run in seconds

Gap between the top 10

12:16.34

736.34

From 1 to 2:

10.67 secs

12:27.01

747.01

From 2 to 3:

4.95 secs

6.42

8.58

4.91

Total of answers

35.46

51.74

56.46

Page 42 1.

Weather pain

Correct total for Perth = 730.3 cm

(a) over by 25.5 cm (b) over by 4.2 cm (c) over by 5.9 cm (d) under by 0.6 cm (e) over by 25 cm (f ) over by 3.4 cm 2.

Correct total for Hobart = 500.8 cm

12:31.96

751.96

From 3 to 4:

1.78 secs

12:33.74

753.74

From 4 to 5:

3.27 secs

12:37.01

757.01

From 5 to 6:

4.2 secs

12:41.21

761.21

From 6 to 7:

4.53 secs

(d) over by 7 cm

12:45.74

765.74

From 7 to 8:

4.02 secs

(e) over by 9.5 cm

12:49.76

769.76

From 8 to 9:

6.93 secs

12:56.69

776.69

13:02.88

782.88

My list is correct

w ww

1.7 1.3 12.4 11.2

Tester B

Tester C

1.1

5.8

1.2 0.9

4.8 6.7

16.4

19.1

4.1

From 9 to 10: 6.19 secs

(b) under by 2.5 cm (c) correct

(f ) under by 1.3 cm

Best was Cloud Claudia for Hobart

Page 43

Tropical rainfall

Note: All totals are the addition of the numbers in the row from January to December

Page 41 2.4

(a) over by 4.9 cm

Tester A

ew i ev Pr

Time – min and sec

Fastest

Teac he r

The fastest 800 metres ever run!

. te 5.7 7.6

12.3 14

19.2

15.3

15.2

8.6

18.9

5.6

11.5

0.5

11.5

Total of answers

98.2

44.5

121.8

m . u

Page 40

Cairns

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Total

396.3 455.3 427.7 196.5 90.2 45.5 29.3 27 33.7 46.6 93.8 180.7 2022.6

Rounded to whole 396 455 428 197 number Rounded to the 400 460 430 200 nearest 10

181 2024

180 2040

o c . che e r o t r s super Innisfail

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Total

512.7 593.7 665.5 460.3 299.5 186.5 136.1 117.4 86.6 88.3 159.7 264.8 3571.1

Rounded to whole 513 594 666 460 300 187 136 117 number Rounded to the 510 590 670 460 300 190 140 120 nearest 10

Page 44

160 265 3573

160 260 3580

Estimate these answers first

Teacher check estimations

200

34.67 – 23.28

55.67 – 19.65

78.01 – 45.98

11.39

36.02

32.03

79.11 – 16.89

133.66 – 39.78

116.76 – 55.99

62.22

93.88

60.77

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

44.167 – 17.228

111.392 – 45.165

26.939

66.227

176.54 – 48.97

235.22 – 117.28

Deal

3 ties for $59.40

$19.80

$11.05

Profit per deal $33.15

4 ties for $75.40

$18.85

$10.10

$40.40

98.41 – 33.17

Price per tie Profit per tie

117.94

65.24

5 ties for $101.50

$20.30

$11.55

$57.75

138.07 – 49.112

728.01 – 345.66

6 ties for $106.50

$17.75

$9.00

$54.00

8.039

88.958

382.35

7 ties for $147.70

$21.10

$12.35

$86.45

3 ties for $51.30

$17.10

$8.35

$25.05

4 ties for $74.00

$18.50

$9.75

$39.00

5 ties for $104.25

$20.85

$12.10

$60.50

6 ties for $132.60

$22.10

$13.35

$80.10

Fiery Fred

7 ties for $118.65

$16.95

$8.20

$57.40

3 ties for $55.05

$18.35

$9.60

$28.80

r o e t s Bo r e p ok u S

Some weighty problems Broad Bobby

Big Ted

Tall Tom

240 kg

210 kg

180 kg

225 kg

5 years ago

160 kg

140 kg

120 kg

150 kg

120 kg

105 kg

90 kg

112.5 kg

⁄3 of their current weight

6 years ago 1 ⁄2 of their current weight 8 years ago 1 ⁄3 of their current weight As a teenager 1 ⁄4 of their current weight As a toddler 1 ⁄8 of their current weight At birth 1 ⁄100 of their current weight

Page 46

80 kg

70 kg

60 kg

75 kg

60 kg

52.5 kg

45 kg

56.26 kg

30 kg

26.25 kg

22.5 kg

28.13 kg

2.4 kg

2.1 kg

1.8 kg

2.25 kg

4 ties for $94

$23.50

$14.75

$59.00

5 ties for $88

$17.60

$8.85

$44.25

6 ties for $125.40

$20.90

$12.15

$72.90

7 ties for $122.15

$17.45

$8.70

$60.90

Best deals — J, F and O

Page 48

ew i ev Pr

Weight

Teac he r

Buy in bulk … or not?

Big Red Raspberry 6 for $4.32 1 can @

Lemonade Leader 6 for $3.96 1 can @

Cola King 6 for $4.38 1 can @

Green Dream 6 for $4.86 1 can @

72c

66c

73c

81c

58c

12 for $9.00 1 can @

12 for $7.32 1 can @

12 for $8.16 1 can @

12 for $8.52 1 can @

12 for $6.84 1 can @

75c

61c

68c

71c

57c

Day

Litres

Total

Price per litre

Ranking

Monday

$97.02

$1.54

Tuesday

$72.42

$1.42

Wednesday

$86.14

$1.46

The Orange Oracle 18 pack

Thursday

$77.76

$1.62

Green Dream 6 pack

18 for $13.32 18 for $11.16 18 for $11.34 18 for $10.98 18 for $10.08 1 can @ 1 can @ 1 can @ 1 can @ 1 can @ 74c

62c

$47.52

$1.44

Saturday

$81.95

$1.49

Sunday

$72.68

$1.58

w ww

Friday

3–4.Teacher check

Day

Monday Tuesday Wednesday

. te

63c

6 cans = $4.44 12 cans = $8.88

Page 49

56c

Let’s try a simpler problem—multiplication

Ranking

$60.48

$1.12

$73.81

$1.21

$73.75

$1.25

Teacher check

F&D – 6 Page 50

The great powers of 10 search

Thursday

$52.32

$1.09

green = 10 , 10 x 1, 10, ten

Friday

$74.75

$1.15

orange = 102, 10 x 10, 100, hundred

Saturday

$78.10

$1.10

red = 103, 10 x 10 x 10, 1000, one thousand

Sunday

$49.14

$1.26

Teacher check

61c

o c . che e r o t r s super Total

Price per litre

Litres

The Orange Oracle 6 for $3.48 1 can @

m . u

Shop

11.467 – 3.428

Page 45

838.93

All ties are up!

127.57

F&D – 5

Page 47

921.017 – 82.087

yellow = 104, 10 x 10 x 10 x 10, 10 000, ten thousand light blue = 10 x 10 x 10 x 10 x 10, 100 000, hundred thousand

Page 51 1.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

(a) (b) (c) (d) (e) (f ) (g) (h) (i) (j)

14.3 272 447 3.65 45.9 370 49 870 5230 2045

Divide and multiply decimals (k) (l) (m) (n) (o) (p) (q) (r) (s) (t)

R.I.C. Publications®

16.1 818 661 1.66 66.9 110 69 810 6810 7113

www.ricpublications.com.au

201

Page 51 2.

(a) (b) (c) (d) (e) (f ) (g) (h) (i) (j)

Divide and multiply decimals (continued)

0.34 0.0456 0.129 0.189 0.0308 2.318 0.5427 0.228 0.08801 28.12

(k) (l) (m) (n) (o) (p) (q) (r) (s) (t)

0.97 0.0755 0.177 0.197 0.0909 7.919 0.5777 0.779 0.09901 79.17

(a) 2 009 500

(d) 0.6782

(b) 33.67

(e) 3124

The first 100 letters

$660 33% off New price: $442.20

$710 10% off New price: $639

$690 10% off New price: $621

$700 15% off New price: $595

$650 5% off New price: $617.50

$560 25% off New price: $420

$795 20% off New price: $636

$820 25% off New price: $615

$600 20% off New price: $480

$585 10% off New price: $526.50

$1000 15% off New price: $850

r o e t s Bo r e p ok u S

I see a flag! – 1

Page 62

The biggest-ever shirt sale

Brand A

Brand B

Brand C

Brand D

Brand E

Brand F

$28

$36

$44

$50

$32

$45

Discount: Discount: Discount: Discount: Discount: Discount: $7 $9 $11 $12.50 $8 $11.25 Sale price: Sale price: Sale price: Sale price: Sale price: Sale price: $21 $27 $33 $37.50 $24 $33.75

Teacher check

Page 54

$750 25% off New price: $562.50

ew i ev Pr

Teac he r

Teacher check

Page 53

The wash up is—my new price is!

$720 15% off New price: $612

F&D – 7 Page 52

Page 61

I see a flag! – 2

Teacher check

(a) $174

(b) $58

Page 55

(a) $192

(b) $64

4 x Brand B or 1 x Brand B, 1 x Brand C, 2 x Brand E

4 x Brand A and 1 x Brand B

4 x Brand A, 1 x Brand B and 1 x Brand E

The Aussie cricket team—new colours are needed!

Teacher check

Page 56

Your own backyard!

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Page 57

1 x Brand C, 1 x Brand D and 1 x Brand F

Brand A new price: $10.50 Brand B new price: $13.50 Brand C new price: $16.50 Brand D new price: $18.75 Brand E new price: $12.00 Brand F new price: $16.88

How creative r u?

Teacher check

Page 58

Artistic maths

Teacher check

Page 63

orange — 22%, 0.22, 11⁄50

CD-R bargain basement Price per disc

Ranking in value

light blue — 11⁄25, 44%, 0.44

5 pack for $6.49

$1.30

pink — 9%, 0.09, 9⁄100

10 pack for $8.99

90c

10 pack for $3.99

40c

white — 0.07, 7%, black —

⁄100

⁄100, 8%, 0.08

Page 59

. te

M&FM – 1 Page 60

Too expensive for me! Cash discount

‘New’ price

Car 1

$10 500

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Car 2

$9600

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Car 3

$27 500

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Car 4

$74 000

Car 5

$34 000

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Car 6

$79 450

Car 7

$9870

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Car 8

$31 450

$94 350

Car 9

$77 800

202

o c . che e r o t r s super

The Kites—the newest team in Australian Football

Teacher check

m . u

CD-R disc deal

w ww

purple — 0.1, 10%, 1⁄10

50 pack for $19.99

40c

25 pack for $18.49

74c

20 pack for $14.53

73c

10 pack for $4.99

50c

30 pack for $19.99

67c

10 pack for $10.99

$1.10

100 pack for $28.99

29c

50 pack for $16.49

33c

20 pack for $20.99

$1.05

25 pack for $12.99

52c

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

=3 9

R.I.C. Publications® www.ricpublications.com.au

CD-R disc deal

Saving

New price

10% off 5 pack for $6.49

65c

$5.84

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90c

$8.09

10% off 10 pack for $3.99

40c

$3.59

10% off 50 pack for $19.99

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10% off 25 pack for $18.49

$1.85

$16.64

Three best deals:

Saver Sean – Deal 3 Cheaper Charlie – Deal 1 Trusting Ted – Deal 2

Most expensive deals:

Page 66

A big sale or what?

(a) $989

(b) $1187

(a) $1156

(b) $1387

(a) $1089

(b) $1307

(a) $1244

(b) $1493

(b) $1033

$1.45

$13.08

10% off 10 pack for $4.99

50c

$4.49

10% off 30 pack for $19.99

$17.99

10% off 10 pack for $10.99

$1.10

$9.89

(a) $861

10% off 100 pack for $28.99

$2.90

$26.09

Teacher check

10% off 50 pack for $16.49

$1.65

$14.84

10% off 20 pack for $20.99

$2.10

$18.89

P&A – 1

10% off 25 pack for $12.99

$1.30

$11.69

Page 67

r o e t s Bo r e p ok u S 1.

A meal at any price!

The super sided shapes

Pentagons

Hexagons

Octagons

Decagons

Total sides

110

139

117

121

Diners

Discount offered

Their cash What they Ranking discount actually paid in $ saved

The Abbots

10% off $125

$12.50

$112.50

The Browns

25% off $612

$153

$459

The Cuzzepes 50% off $678

$339

The Dusuns

10% off $198

$19.80

$178.20

The Edens

25% off $520

$130

$390

The Fazios

50% off $406

$203

The Gardas

10% off $367

$36.70

$330.30

The Homes

25% off $416

$104

$312

50% off $234

$117

The Ionas 2.

119

116

111

103

Pentagons

Hexagons

Octagons

Decagons

Total sides

101

Sunday – 50%

Price

Saving

Price

Saving

Price

Saving

Table 1 $123

$12.30

$64.40

$16.10

$239

$119.50

Table 2 $199

$19.90

$154

$38.50

$166

$83

Table 3 $78

$7.80

$112

$28

$229

$114.50

Table 4 $212

$21.20

$182

$45.50

$80.90

$40.45

Table 5 $306

$30.60

$228

$57

$91.80

$45.90

Table 6 $187

$18.70

$113

$28.25

$176.30

$88.15

Table 7 $208

$20.80

$89.80

$22.45

$144.60

$72.30

Table 8 $117

$11.70

$132

$33

$161

$80.50

Table 9 $96.80

$9.68

$178

$44.50

$208

$104

Table 10 $133

$13.30

$216

$54

$127

$63.50

w ww

Friday – 10%

Totals 3.

. te

Deal 2

Deal 3

Deal 4

141

142

122

125

137

o c . che e r o t r s super Page 68

The best salesman—could be a close shave! Trusting Ted

Deal 1

$1659.80 $165.98 $1469.20 $367.30 $1623.60 $811.80

$278.20

Page 65

ew i ev Pr

Teac he r

10% off 20 pack for $14.53

Page 64

Saver Sean – Deal 2 Saver Sean – Deal 4 Trusting Ted – Deal 3

m . u

Discount: $110

Saver Sean Discount: $322.50

Cheaper Charlie Discount: $805

Solve and extend the rules

Teacher check extension of rules Rule: x 4 + 2

Rule: x 3

Rule: x 5 + 1

Rule: x 5 – 1

Rule: x 7

Rule: x 8 + 2

Rule: x 6 –5

Rule: x 3 + 3

Rule: x 9 – 2

New price: $990

New price: $967.50 New price: $805

Teacher check

Discount: $300

Discount: $135

Discount: $910

Page 69

New price: $900

New price: $1215

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Discount: $370

Discount: $785

Discount: $111

New price: $1110

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Discount: $965

Discount: $133

Discount: $320

New price: $965

New price: $1197

New price: $960

Algebra Moonlander a=2

axa

axa +3

3a + 2

a+a

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications®

a + a + a 3a + 4 4a + 3a 6

www.ricpublications.com.au

203

Page 69

Algebra Moonlander (continued) a=5

axa

axa –6

3a + 7

a+a

a + a + a 3a + 9 4a + 5a 15

88, 84, 97, 93, 106, 102, 115, 111, 124

211, 206, 213, 208, 215, 210, 217, 212, 219

10. 44, 54, 108, 118, 236, 246, 492, 502, 1004 11. 104, 100, 111, 107, 118, 114, 125, 121, 132

Page 73 1.

a = 10 axa

axa +7

3a + 2

a+a

100

107

Page 70

Make a sequence

Teacher check

Page 71

a + a + a 3a – 4 4a + 2a 30

60 4.

= 1 ⁄2

= 1⁄2

= 11

= 10

= 17

r o e t s Bo r e p ok u S Pattern rule

= 31⁄2

= 20

The rule and next few, please

The pattern

Three-symbol nightmare!

= 22

= 11

= 13

= 19

= 29

= 11

= 48

–30, –60, –60

= 3 ⁄2

40, 80, 160, 280, 320, 400, 520, 560

+40, +80, +120

= 10

-16, -12, -8, –4, 0, 4, 8

121, 132, 124, 135, 127, 138, 130, 141

+11, –8

66, 67, 70, 75, 82, 83, 86, 91, 98 2

10.

= 29

+1, +3, +5, +7

⁄3, 11⁄3, 2, 22⁄3, 31⁄3, 4, 42⁄3, 51⁄3

+2⁄3

11⁄2, 3, 41⁄2 , 6, 71⁄2 , 9, 101⁄2

+11⁄2

161⁄4, 17, 18, 183⁄4, 193⁄4, 201⁄2, 211⁄2, 221⁄4

+3⁄4, +1

-45, -40, -30, -25, –15, –10, 0, 5

+5, +10

91⁄2, 11, 13, 141⁄2, 161⁄2, 18, 20, 211⁄2

+11⁄2, +2

511, 502, 493, 484, 475, 466, 457

–9

-17, -15, -12, -8, –6, –3, 1, 3

+2, +3, +4

11.

= 25

= 5 ⁄2

Page 74

ew i ev Pr

Teac he r

360, 330, 270, 210, 180, 120, 60, 30, –30

12.

= 51

= 13 =9

Can you repeat my pattern?

6, 10, 7, 11, 5

219.5, 213.5, 208.5, 204.5, 201.5

2.5, 3.3, 3.7, 4.5, 4.9

45.6, 46.4, 47.3, 48.1, 49

44.4, 43.9, 43.3, 42.6, 42.1

73.1, 74.2, 75.3, 76.4, 77.5

121.6, 122.8, 124, 125.2, 126.4

56, 59, 57, 60, 54

77.5, 77.6, 77.9, 77.4, 77.5

11.9, 12.4, 13, 13.7, 14.5

⁄4, 11⁄2, 21⁄4, 3, 33⁄4, 41⁄2, 51⁄4

+3⁄4

277, 279, 281, 283, 285

200.8, 199.7, 198.6, 197.5, 196.4

118, 151, 85, 89, 87

71.5, 73, 74.5, 76, 77.5

122, 125, 119, 123, 121

313, 311, 316, 320, 324 616, 627, 636, 643, 648

Pattern rule

56, 59, 67, 71, 79

44, 55, 64, 71, 82

1.01, 1.12, 1.23, 1.34, 1.45, 1.56, 1.67

+0.11

112, 105, 102, 95, 92

63, 73, 82, 89, 99

23.6, 24.8, 26, 27.2, 28.4, 29.6, 30.8

+1.2

49.99, 49.9, 48.9, 48.81, 47.81, 47.72, 46.72

–0.09, –1

34.5, 36, 36.6, 37.1, 38.6, 39.2, 39.7, 41.2

+1.5, +0.6, +0.5

166.6, 177.7, 188.8, 199.9, 211, 222.1, 233.2

+11.1

401.1, 399.7, 399.3, 398.9, 398.5, 398.1, 397.7

–0.4

= 23

111.01, 111.07, 111.13, 111.19, 111.25, 111.31

+0.06

= 14

= 36

44.04, 44.13, 44.22, 44.31, 44.4, 44.49, 44.59

+0.09

= 17

= 36

The pattern

w ww

. te

7.07, 7.20, 7.33, 7.46, 7.59, 7.72, 7.85

P&A – 2 Page 75

m . u

101.2, 101.8, 102.3, 102.5, 103.1

The suns and stars come out to play = 21

o c . che e r o t r s super +0.13

= 19

65.09, 64.08, 63.07, 62.06, 61.05, 60.04, 59.03

–1.01

= 36

= 16

176.5, 177.1, 177.7, 178.3, 178.9, 179.5, 180.1

+0.6

= 14

= 84

= 23

987.1, 986.3, 985.6, 984.8, 984.1, 983.3, 982.6

–0.8, –0.7

= 46

= 32

= 20

99.1, 98.8, 98.5, 98.2, 97.9, 97.6, 97.3

–0.3

= 48

= 51

= 27

= 50

= 13

= 11

= 51

= 50

= 18

Page 72

Rules and more rules

33, 32, 38, 37, 43, 42, 48, 47, 53

60, 57, 68, 65, 76, 73, 84, 81, 92

113, 118, 111, 116, 109, 114, 107, 112, 105

= 12

= 17

= 12

116, 107, 113, 104, 110, 101, 107, 98, 104

= 36

= 33

= 25

143, 136, 150, 143, 157, 150, 164, 157, 171

88, 99, 94, 105, 100, 111, 106, 117, 112

252, 257, 266, 271, 280, 285, 294, 299, 308

204

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Page 76 1.

Page 80

Order of calculations

(a) 22 – 9 + 3 = 16

(b) 9 – 7 + 14 = 16

(d) 21 – 13 + 3 = 11

(e) 15 – 14 + 6 = 7

(f ) 23 + 7 + 10 – 9 = 31

(g) 19 + 12 – 7 – 4 = 20

(h) 6 x 4 + 11 = 35

(i) 28 ÷ 7 + 12 = 16

(j)

(k) 9 x 7 + 8 = 71

Page 77

(l)

32 + 18 ÷ 3 – 7 =

122

5 x 5 + 13 = 38

211 – (2 x 12) + 50 =

137

237

136

234

6 x 3 + 12 = 30

(17 + 3) ÷ (24 ÷ 6) =

20 – 2 x 8 =

126

120

58 – 3 x 7 – (30 ÷ 3) =

716

7181⁄3

112 + (16 ÷ 4) – 9 =

106

107

9 + (3 x 6) = 27

96 – 6 x 6 =

504

540

28 ÷ (7– 3) = 7

(64 ÷ 8) + (17 x 4) – 6 =

-9

-5

12 ÷ (3 x 4) = 1

r o e t s Bo r e p ok u S (8 – 5) x (5 x 5) = 75

34 – (7 x 4) = 6

21 – (4 x 4) = 5

(e) (6 x 10) + 8 – 12 = 56 (f ) (6 x 7) + 8 – 20 = 30

6 + (7 x 2) =

(g) (2 x 11) + 13 – 6 = 29

36 ÷ (18 – 16) =

(h) (3 x 15) + 4 – 3 = 46

54 – (7 x 3) =

141

57 – 6 + 11 + 12 =

(i) (3 x 14) + 7 – 10 = 39

Page 81 1.

ew i ev Pr

Teac he r

Just pick the right answer

Equation

(a) (4 x 4) + 9 – 5 = 20

(b) (3 x 7) + 8 – 2 = 27

54 – 8 + 11 + 6 = 63

Page 78

126

32 ÷ (4 + 4) = 4

(10 – 4 ) x (4 x 3) = 72

151⁄4

49 ÷ (11 – 4) = 7

(15 – 4) x (3 x 3) = 99

6.3

(45 – 11) + (32 ÷ 4) =

24 ÷ (3 + 1) = 6

72 – (5 x 6) = 42

6.1

19 – 12 ÷ 6 x 3 =

One of these must be correct!

25 ÷ (3 + 2) = 5

One is right, one is wrong, two are silly

Making equations

(a) 8 + 5 – 4 = 9

49 + (4 x 4) – 9 =

(b) 15 ÷ 5 + 7 = 10

101 – (7 x 5) + 11 =

72 – (8 x 4) – 22 =

45 + 7 + (3 x 6) =

(f ) 9 x 3 + 3 = 30

81 – 11 + (7 x 2) =

(g) 8 ÷ 4 + 5 = 7

64 ÷ 8 + 10 – (3 x 2) =

63 – 9 + (5 x 5) =

54 ÷ (3 x 3) – 2 =

71⁄2

22 – 4 + 6 + 7 =

Incorrect

. te

Teacher check working

Page 79

= 34

= 22

= 12 =4

= 12 =3

(h) 2 x 6 + 5 = 17

(i) 15 ÷ 5 + 12 = 15 (j) 4 x 3 + 9 = 21 (k) 6 x 5 – 9 = 21 (l) 7 x 7 + 8 = 57 (m) 18 ÷ 6 + 2 = 5 (n) 9 x 4 + 5 = 41

Page 82 1.

–27

–20

–8

–6

= 26

–12

= 25

Don’t be negative about numbers!

o c . che e r o t r s super

If the answer is, then …

= 12

(e) 9 ÷ 3 + 8 = 11

m . u

w ww

(d) 5 + 6 – 4 = 7

= 10

= 16

= 13

10. –30 = 26

= 13

= 12

= 18

= 87

= 39

= 12

= 13

= 71

= 27

= 12

11. 3

Create your own – Teacher check

Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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205