Maths Masterpieces: Ages 11+

Published by R.I.C. Publications速 www.ricgroup.com.au RIC-6008 3.2/3

This master may only be reproduced by the original purchaser for use with their class(es). The publisher prohibits the loaning or onselling of this master for the purposes of reproduction.

ISBN 978 1 74126 165 3

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This book is dedicated to my wonderful and inspirational mother, Inge Schymkiw.

— Gunter Schymkiw

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Blackline masters or copy masters are published and sold with a limited copyright. This copyright allows publishers to provide teachers and schools with a wide range of learning activities without copyright being breached. This limited copyright allows the purchaser to make sufficient copies for use within their own education institution. The copyright is not transferable, nor can it be onsold. Following these instructions is not essential but will ensure that you, as the purchaser, have evidence of legal ownership to the copyright if inspection occurs.

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Special attention was given in the production of this document to ensure that the chosen artworks fell outside copyright restrictions, including the extended time lines for copyright occurring as a result of the Australian-US Free Trade Agreement.

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.

View all pages online http://www.ricpublications.com.au

The activities in Maths Masterpieces (Upper) integrate mathematics with aspects of visual arts. As well as being provided with opportunities to consolidate knowledge and skills in mathematics, students are introduced to significant works of art and their artists.

Maths Masterpieces (Upper) aims to enhance children’s appreciation of the works of great artists at the same time allowing them to build mathematical skills.

Other titles include: Maths Masterpieces (Middle)

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Teachers information ................................................................................................................................................................................................................ iv–v Outcome links ................................................................................................................................................................................................................................ v

The Peasant Dance ............................................. Division of money

Teachers notes and answers ...................................................................... 1 Questions .................................................................................................... 2 Puzzle pieces ............................................................................................... 3

Teachers notes and answers .................................................................... 31 Questions .................................................................................................. 32 Puzzle pieces ............................................................................................. 33

The Cannon Shot .................................................. Roman numerals

The Avenue at Middelharnis ............................... Mini-problems

by Willem van de Velde

by Meindert Hobbema

Teachers notes and answers ...................................................................... 4 Questions .................................................................................................... 5 Puzzle pieces ............................................................................................... 6

Teachers notes and answers .................................................................... 34 Questions .................................................................................................. 35 Puzzle pieces ............................................................................................. 36

Landscape at Regensburg ................................. Chinese numerals

by Albrecht Altdorfer

Centurion (from The Crucifixion of St Peter) ... Twenty-four hour time

by Michelangelo Buonarroti

Teachers notes and answers ...................................................................... 7 Questions .................................................................................................... 8 Puzzle pieces ............................................................................................... 9

Teachers notes and answers .................................................................... 37 Questions .................................................................................................. 38 Puzzle pieces ............................................................................................. 39

by Pieter Bruegel the Elder

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View of Sydney Cove and Fort Macquarie....... Writing large

by Conrad Martens numbers

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by Konrad Witz

Teachers notes and answers .................................................................... 10 Questions .................................................................................................. 11 Puzzle pieces ............................................................................................. 12

Teachers notes and answers .................................................................... 40 Questions .................................................................................................. 41 Puzzle pieces ............................................................................................. 42

The Blind Girl .......................................... Adding money and rounding

The Feast of Belshazzar .................................. Equivalent fractions

by Sir John Everett Millais

by Rembrandt van Rijn

Teachers notes and answers .................................................................... 13 Questions .................................................................................................. 14 Puzzle pieces ............................................................................................. 15

Teachers notes and answers .................................................................... 43 Questions .................................................................................................. 44 Puzzle pieces ............................................................................................. 45

Two Women on the Beach ........................................ Subtraction

Bailed Up ............................................. Converting fractions to decimals

by Paul Gauguin

by Tom Roberts

Teachers notes and answers .................................................................... 16 Questions .................................................................................................. 17 Puzzle pieces ............................................................................................. 18

Teachers notes and answers .................................................................... 46 Questions .................................................................................................. 47 Puzzle pieces ............................................................................................. 48

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The Miraculous Draught of Fishes ........... Fractions of amounts

by Frederic Remington

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The Cowboy ...................................................................... Addition

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The Tower of Babel ...................................... Subtraction with money

Bathers at Asnières.............................................. Adding decimals

by Pieter Bruegel

by Georges Seurat

Teachers notes and answers .................................................................... 19 Questions .................................................................................................. 20 Puzzle pieces ............................................................................................. 21

Teachers notes and answers .................................................................... 49 Questions .................................................................................................. 50 Puzzle pieces ............................................................................................. 51

Max Schmitt in a Single Scull .... Multiplication (Amazing numbers)

The Scapegoat........................... Multiplying decimals by whole numbers

by Thomas Eakins

by William Holman Hunt

Teachers notes and answers .................................................................... 22 Questions .................................................................................................. 23 Puzzle pieces ............................................................................................. 24

Teachers notes and answers .................................................................... 52 Questions .................................................................................................. 53 Puzzle pieces ............................................................................................. 54

Study for Last Supper (Judas) ............ Subtraction without trading

Self-portrait ........................................................... Shapes and lines

by Leonardo da Vinci

by Vincent Van Gogh

Teachers notes and answers .................................................................... 25 Questions .................................................................................................. 26 Puzzle pieces ............................................................................................. 27

Teachers notes and answers .................................................................... 55 Questions ................................................................................................... 56 Puzzle pieces ............................................................................................. 57

A Winter Scene ............................................ Division with remainders

by Isaac van Ostade

Teachers notes and answers .................................................................... 28 Questions .................................................................................................. 29 Puzzle pieces ............................................................................................. 30 R.I.C. Publications®

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iv Each selected artwork is paired with a mathematical concept, usually in a group of three pages.

Page 1 of each group consists of a teachers page. Each teachers page contains the following information.

The mathematical concept being covered is indicated at the top of the page.

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A picture of the completed artwork shows students the correct answers to their mathematical activity.

Student instructions explain how to complete the activity.

Background information includes information about the artist, his/her style, influences and other artworks.

The Internet image search encourages teachers and students to more broadly appreciate the works of individual artists.

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The artwork and artist to which this page relates is given.

A ‘Talking mathematically’ section is included with each activity. This provides lesson format suggestions, discusses possible mathematical emphases and provides further interesting background information that is both instructional and historical in its nature.

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C.Publ i cat i ons •f orr evi ew pur posesonl y• Occasionally, additional activities are given which may extend the activity into other learning areas such as English.

Page 2 of each group consists of the student activity page with the mathematical problems to be solved.

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The algorithms/problems for the students to complete are provided.

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The mathematical concept being covered is indicated at the top of the page.

The artwork to which this page relates is given.

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v Page 3 of each group provides the answers to the algorithms/problems on ‘mixed up’ sections of the masterpiece.

The mathematical concept being covered is indicated at the top of the page.

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The artwork to which this page relates is given.

Answers for the algorithms/problems are given on the sections of the masterpiece. These sections will give a completed picture of the masterpiece once glued in the correct position.

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MATHS NS3.1, NS3.2, NS3.3, NS3.4, NS3.5, NS4.2, NS4.3, S3.1

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WM4.2, S4.4, M4.2, N4.1(a), N4.2, N4.3, N4.1(b), N4.4 3.4, 3..6, 3.7, 3.9, 3.10, 3.12

N 4.1, N 4.2, N 4.3, PA 4.1, PA 4..2, M 4.2, S4.1

The ARTS NSW

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ARAR0403, ARAR0404

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MATHS MASTERPIECES

WRITING LARGE NUMBERS

Teachers notes

1

View of Sydney Cove and Fort Macquarie 51 090

61 309

83 547

58 203

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96 120

60 004

17 435

50 010

56 789

47 256

26 003

78 145

72 419

61 001

30 356

70 001

75 413

38 049

Student instructions

Background information

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View of Sydney Cove and Fort Macquarie

1. Cut out the tiles from the jigsaw sheet.

by Conrad Martens (1801 –1878)

3. Match the amounts on the tiles to the answers on the backing board.

Martens was born in England and came to Australia in 1835. Before this, he had spent two years on the ship Beagle, where he worked as an artist for Charles Darwin.

4. Glue the tiles onto their matching spaces.

2. Do the addition algorithms on the backing board.

Martens paintings offer a record of early Sydney and Sydney Harbour was his favourite subject.

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Internet image search

Talking mathematically

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It was as a result of his discoveries and observations on this journey that Darwin proposed his theory of evolution.

5. Colour the picture in an interesting way.

The Hindu-Arabic system (also called the decimal system) was popularised in Europe by the Italian mathematician, Leonardo Fibonacci (1170–1250), who was also known as Leonardo of Pisa. Up until that time Roman numerals were used.

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The Beagle View of the Heads, Port Jackson Apsley Falls Scene on the Nepean River Forest, Cunningham’s Gap Terrawambella on Nicol’s run

The decimal system uses ten digits (0 1 2 3 4 5 6 7 8 9) to write all numbers. Its basis is a system of place value where each column has a value that is ten times that immediately to its right.

This holds true even with decimal fractions (decimals), where any number to the right of the decimal point has one tenth of its value in the units column.

Unlike most number systems, the decimal system has a numeral for zero. Zero is used as a place holder in larger numbers.

The method of representing numbers as shown in this activity is called decomposition.

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Maths masterpieces

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50 000 1 000 + 90

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50 000 6 000 700 80 + 9

60 000 1 000 + 1

40 000 7 000 200 50 + 6

40 000 1 000 + 2

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90 000 6 000 100 + 20

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30 000 300 50 + 6

70 000 + 1

20 000 6 000 + 3

50 000 + 10 70 000 2 000 400 10 + 9

30 000 8 000 40 + 9

70 000 8 000 100 40 + 5

70 000 5 000 400 10 + 3

50 000 8 000 200 + 3

10 000 7 000 400 30 + 5

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C.Publ i cat i ons •f orr evi ew pur posesonl y• 60 000 + 4

60 000 1 000 300 + 9

80 000 3 000 500 40 + 7 WRITING LARGE NUMBERS

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MATHS MASTERPIECES

View of Sydney Cove and Fort Macquarie

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MATHS MASTERPIECES

WRITING LARGE NUMBERS

3

26 003 17 435 50 010

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96 120

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51 090

61 309

70 001

60 004

75 413

41 002 30 356

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38 049

56 789

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58 203

78 145

72 419 83 547

61 001

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View of Sydney Cove and Fort Macquarie

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MATHS MASTERPIECES

ROMAN NUMERALS

Teachers notes

The Cannon Shot

Background information The Cannon Shot

CLVII 157

by Willem van de Velde the Younger (1633–1707)

CCXLIV 344

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Willem van de Velde the Younger lived at a time when Holland was a powerful country in world affairs because of the strength of its navy.

His specialty as an artist was the painting of ships and seascapes. Later in his career his pictures featured battle scenes at sea and ships in stormy settings. His early life was spent in Holland, the country of his birth.

His father was also an artist whose specialty was marine painting. In 1672 he and his father moved to England where they were invited to set up a studio in the Queen’s House.

They received many commissions for paintings from royalty.

DCXVIII 618

DCCCLXXV 875

CMXXXIV 934

CMLXXXIII 983

DXI 511

DCCLXVI 766

DCCCLXII 862

DCXCIV 694

CCCLXX 370

DCCXXI 721

CDXCII 492

DLXXVIII 578

CDXXXVI 436

CXCIX 199

CCLXXXV 285

DCLVII 457

Internet image search Ships in a Calm Sea Ships off the Coast The Gust Entrance to a Dutch Port

DCCXXIX 729

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C.Publ i cat i ons Student instructions •f orr evi ew pur posesonl y• 2. Match the numbers on the tiles to their Roman numeral equivalents on the backing board.

3. Glue the tiles onto their matching spaces.

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4. Colour the picture in an interesting way.

Talking mathematically

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One explanation is that the four strokes make the clock face appear balanced and symmetrical (balancing with the four strokes needed for VIII).

Another dates back to 1364 when King Charles the Wise of France chided watchmaker Henry De Vick for writing IV on a tower clock face rather than IIII. Charles cut all argument short by simply declaring, ‘I am never wrong!’

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The system of writing numerals used by the Romans involved using seven letters.

These are: 1 = I, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000

The most famous clock face of all, that of Big Ben’s clock tower in London, uses IV for four.

The Roman system of writing numerals does not have a symbol for for zero.

Where a smaller number follows one that is larger, the numbers are added. If a smaller number precedes one that is larger, it is subtracted.

So

four strokes (IIII). There are a number of explanations for this but nobody can prove any of them to be the exclusive truth.

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1. Cut out the tiles from the jigsaw sheet.

Numbers in the thousands beyond 3000 are indicated by placing a stroke above them.

For example, 4000 is written IV.

VI = 6 (5 + 1) and IV = 4 (5 – 1); LX = 60 (50 + 10) and XL = 40 (50 – 10); CX = 110 (100 + 10) and XC = 90 (100 – 10); MCMXCIX = 1999 (1 000 + [1 000 – 100] + [100 – 10] + [10 – 1]).

The generally accepted numeral for four is IV. Many clock faces using Roman numerals, however, represent four using

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MATHS MASTERPIECES

ROMAN NUMERALS

5

The Cannon Shot

CCXVIII

CCCXLIV

DCCXXIX

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DCCCLXXV

CMXXXIV

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CMLXXXIII

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CDXXXVI

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MATHS MASTERPIECES

ROMAN NUMERALS

The Cannon Shot

875

694

436

618

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MATHS MASTERPIECES

CHINESE NUMERALS

Landscape at Regensburg

7

Teachers notes

Background information Landscape at Regensburg

392

658

245

583

by Albrecht Altdorfer (1480–1538)

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687

863

617

108

549

317

623

990

374

Internet image search Susanna at Her Bath The Stigmatisation of St Francis The Crucifixion Landscape with Satyr Family Alexander’s Victory (The Battle at Issus)

Student instructions

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Altdorfer lived in Bavaria, Germany. His paintings were unusual for the time in which he lived because most did not contain human figures. Even though there is little human presence in his paintings, the landscapes he painted conveyed an image of foreboding drama. Landscape at Regensburg (the artist’s hometown) shows a wild image of nature with gathering clouds warning of an impending storm.

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1. Cut out the tiles from the jigsaw sheet.

509

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2. Match the numbers on the tiles to their Chinese numeral equivalents on the backing board.

3. Glue the tiles onto their matching spaces.

4. Colour the picture in an interesting way.

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Talking mathematically

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Before the students begin the activity, write these sample Chinese numerals and their Hindu-Arabic equivalents on the board. Note there is no symbol for 0. Challenge the students to explain the system

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10

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100

642

738

951

709

560

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MATHS MASTERPIECES

CHINESE NUMERALS

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Landscape at Regensburg

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MATHS MASTERPIECES

CHINESE NUMERALS

9

Landscape at Regensburg

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245

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392

452

509

145

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990

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MATHS MASTERPIECES

ADDITION

Teachers notes

The Cowboy 1569

1801

1533

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3097

2226

2407

2998

2925

2571

2950

2697

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Student instructions © R. I .

C.Pu bl i cat i ons The Cowboy •f orr evi ew pur posesonl y• Background information

1. Cut out the tiles from the jigsaw sheet

2. Do the addition algorithms on the backing board.

Remington was an American painter and sculptor. He is famed for his paintings and bronze sculptures that depicted life in the American west of the late 19th century.

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Internet image search On the Southern Plains Scout His First Lesson Stampede Fall of the Cowboy The Outlier Dash for Timber

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3. Match the numbers on the tiles to the answers on the backing board.

4. Glue the tiles onto their matching spaces.

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by Frederic Remington (1861–1909)

5. Colour the picture in an interesting way.

Talking mathematically

Students will need a sound knowledge of trading units to tens, tens to hundreds and hundreds to thousands to successfully complete this activity.

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MATHS MASTERPIECES

ADDITION

11

The Cowboy

378 647

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967 473

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452 623

675 184 296 378

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778 876 543 210

+

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MATHS MASTERPIECES

ADDITION

The Cowboy

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MATHS MASTERPIECES

ADDING MONEY AND ROUNDING

Teachers notes

The Blind Girl $23.15

$24.65

$14.75

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$18.60

$25.70

$29.95

$18.75

$15.40

$24.60

$17.60

$23.95

$12.40

$19.85

$21.10

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$23.10

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C.Publ i cat i ons •f orr evi ew pur p os esonl y• Background information Student instructions $19.70

$26.95

$11.65

$15.75

by Sir John Everett Millais (1829–1896)

2. Do the addition algorithms on the backing board.

Round the answers to the nearest five cents.

Millais was an Englishman. At the age of 11 he was the youngest artist ever to join The Royal Academy, Britain’s finest art school. He paid great attention to detail in his paintings and background landscapes were painted from nature. Millais and his wife had eight children. His paintings were very popular during his lifetime and sold at high prices so the family lived well.

3. Match the rounded answers on the tiles to the numbers on the backing board.

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4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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The Blind Girl was painted in 1854. In it we see the beautiful landscape and the colours of the rainbow and looming storm, images which are denied to the painting’s key character.

Internet image search The Bridesmaid The Return of the Dove to the Ark Bubbles Apple Blossoms The Boyhood of Raleigh

R.I.C. Publications®

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1. Cut out the tiles from the jigsaw sheet.

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The Blind Girl

Talking mathematically

With the discontinuation and withdrawal of one- and two-cent coins from Australian currency in 1992, the ability to round amounts to the nearest 5 cents has become essential.

Amounts ending in 1 cent and 2 cents are rounded down; i.e. 71 cents or 72 cents become 70 cents etc.

Amounts ending in 3 cents and 4 cents are rounded up; i.e. 73 cents or 74 cents become 75 cents etc.

Amounts ending in 6 cents and 7 cents are rounded down; i.e. 76 cents or 77 cents become 75 cents etc.

Amounts ending in 8 cents and 9 cents are rounded up; i.e. 78 cents or 79 cents become 80 cents etc.

Maths masterpieces

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MATHS MASTERPIECES

ADDING MONEY AND ROUNDING

The Blind Girl $7.66 $7.69 + $7.77 $23.12 Rounded = $23.10

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$4.86 $3.78 $6.77

$9.87 $7.77 $6.95

Rounded =

Rounded =

Rounded =

$6.99 $7.99 $8.99

$9.04 $2.10 $1.24

$6.67 $7.68 $5.48

$4.75 $6.75 $8.18

$9.99 $8.97 $7.98

$1.88 $2.88 $6.88

+

+

$1.05 $3.99 $9.69

Rounded = $9.99 $9.99 $9.99

+

Rounded = $7.08 $8.96 $1.55

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Rounded =

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$8.89 $9.99 $9.58

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$8.69 $6.69 $7.78

$4.96 $6.49 $9.64

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Maths masterpieces

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$3.64 $2.98 $9.14

Rounded =

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MATHS MASTERPIECES

ADDING MONEY AND ROUNDING

15

The Blind Girl

r o e t $18.75 s Bo r e p ok u S

$23.15

$28.45

$18.60

$26.95

$23.95

ew i ev Pr

Teac he r

$12.40

$15.40

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

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. te

$21.10

$23.10 R.I.C. Publications®

$24.65

$11.65

o c . che e r o t r $15.75 $25.70 s super

$14.75

$29.95

Maths masterpieces

$19.70

m . u

$19.85

$24.60

$17.60 www.ricgroup.com.au

16

MATHS MASTERPIECES

SUBTRACTION

Teachers notes

Two Women on the Beach 528

234

689

444

r o e t s Bo r e p ok u S 298

717

815

311

609

573

568

262

806

188

154

429

107

365

618

ew i ev Pr

Teac he r

167

Background information

Talking mathematically

Two Women on the Beach

Your students will need a good knowledge of trading to do these examples.

by Paul Gauguin (1848 – 1903)

Using 501 minus 334, follow these steps:

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Internet image search Vision after the Sermon D’ou venons-nous?

Washerwomen at Pont-Aven Night Cafe at Arles The Siesta The Swineherd Cattle Drinking

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Student instructions

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Before devoting his life to painting, Gauguin was a Step 1: 1 minus 4 can’t be done so you must trade. successful stockbroker who had been a patron of the arts. Step 2: Look in the tens column. The number there is ‘0’ so He began exhibiting his own work in 1876 and in 1883 left you can not trade from there. his job to concentrate on painting.

Step 3: Look in the hundreds column. Exchange 5 hundreds He was a friend of Van Gogh but the friendship was a for 4 hundreds and 10 tens. stormy one. Gauguin was unable to support his family as a painter. He longed for a simple life and moved to Tahiti in Step 4: You now have 10 tens in the tens column. Trade these for 9 tens (in the tens column) and 10 ones (in 1891. Two Women on the Beach was painted while he was the units column). living in Tahiti.

Step 5: There are now 11 ones in the units column, so the subtraction in this column can now proceed.

11 minus 4 equals 7—write ‘7’ on the answer line under the units column.

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Step 6: After trading, the tens column now reads 9 (tens) minus 3 (tens). Do the subtraction and write 6 on the answer line in the tens column.

Step 7: After trading, the hundreds column now reads 4 (hundreds) minus 3 (hundreds). Do the subtraction and write 1 under the hundreds column.

Demonstrate to students that after trading, 501 becomes 4 hundreds, 9 tens and 11 ones. Add these numbers (400 + 1. Cut out the tiles from the jigsaw sheet.

90 + 11). The answer is 501. Trading has re-organised the 2. Do the subtraction algorithms on the backing board. composition of the original 501 (5 hundreds, 0 tens and 1 3. Match the numbers on the tiles to the answers on the one). backing board.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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Maths masterpieces

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MATHS MASTERPIECES

SUBTRACTION

17

941 576

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821 203

439 251

–

951 145 –

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624 313

–

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Teac 845 – h156 er

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911 302

900 456

Two Women on the Beach

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858 429 u

803 696

–

713 451 –

–

863 290

902 748

–

–

–

–

679 381

– R.I.C. Publications®

902 334

–

963 148

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501 334

–

749 221

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911 194

–

611 377

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C.Publ i cat i ons •f orr evi ew pur posesonl y•

Maths masterpieces

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18

MATHS MASTERPIECES

SUBTRACTION

107

298 234

Maths masterpieces

568 429 154

m . u

262

618

609

528 815

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o c . che e r o t r s super 806

w ww

. te

365

444

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C.Publ i cat i ons •f orr evi ew pur posesonl y•

573

167

188

689

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

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Teac he r

311

717

Two Women on the Beach

R.I.C. Publications®

MATHS MASTERPIECES

SUBTRACTION WITH MONEY

19

Teachers notes

The Tower of Babel Background information The Tower of Babel

$61.85

$412.85

$618.15

$87.80

$267.20

by Pieter Bruegel (1525–1569)

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

Pieter Bruegel the Elder is the best known of a family of Dutch artists. His paintings are very detailed and busy.

Teac he r

$306.85

$250.25

$141.80

$135.35

$0.95

$227.60

$20.95

$175.90

$252.85

$335.35

$388.85

$578.95

Internet image search Children’s Games Hunters in the Snow The Peasant Dance The Fight between Carnival and Lent Procession to Calvary Two Chained Monkeys The Numbering at Bethlehem

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Children’s Games is a scene showing more than 200 children at play. The Tower of Babel depicts a scene from the Bible story in which the building of a tower built to reach heaven was frustrated by God who confused the builders’ languages so they could not understand each other.

$151.85

$197.35

$437.85

© R. I .

C.Publ i cat i ons Student instructions •f orr evi ew pur posesonl y• 2. Do the subtraction algorithms on the backing board.

3. Match the numbers on the tiles to the answers on the backing board.

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4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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Talking mathematically

o c . che e r o t r s super

The students will need a good knowledge of trading to do these examples.

Our coinage (current and previous in the case of the onecent coin) can be used to demonstrate ‘what’s happening’ when doing these subtractions; e.g. $196.05 – $134.20.

Follow these steps:

Step 1: Five one-cent coins minus 0 one-cent coins equals 5. Write 5 on the answer line under the one-cent column (the hundredths column).

Step 2: Look at the subtraction in the 10-cent column. Zero 10-cents minus two 10-cents can’t be done—you must trade.

R.I.C. Publications®

Step 3: Trade from the dollar coins column. Exchange one of the 6 dollar coins for ten 10-cent coins. You now have 5 dollar coins in the dollar coins column (the units column) and ten 10-cents in the 10cents column. Proceed with the 10-cents column subtraction—ten 10-cents minus two 10-cents equals eight 10-cents. Write 8 on the answer line under the ten cent coins column.

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1. Cut out the tiles from the jigsaw sheet.

Step 4: Do the subtraction in the dollar coins column. Five dollar coins minus 4 dollar coins equals 1 dollar coin. Write 1 on the answer line under the dollar coins column.

Step 5: Do the subtraction in the 10-dollar notes (tens) column. Nine 10-dollar notes minus three 10-dollar notes equals six 10-dollar notes. Write 6 on the answer line under the 10-dollar notes column.

Step 6: Finally, do the subtraction in the 100-dollar (hundreds) column. One 100-dollar note minus one 100-dollar note equals 0. There is no need to write 0 here as it does not add any meaning to the final answer.

Your students will need a good knowledge of trading to do these examples.

Maths masterpieces

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–

$674.10 $367.25

–

–

$812.95 $47 7.60

Maths masterpieces

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–

–

–

$67 7.70 $288.85

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$342.05 $114.45

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–

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$919.65 $767.80

–

$649.70 $236.85

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C.Publ i cat i ons •f orr evi ew pur posesonl y• $769.80 $190.85

$900.35 $879.40

$928.05 $67 7.80

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–

–

$452.80 $365.00

– Tea ch er

$796.40 $178.25

–

–

$378.85 $243.50

$700.05 $432.85

$616.65 $178.80

$610.55 $434.65

–

–

$402.05 $204.70

$928.50 $675.65

r o e t s Bo r e p ok u S $819.95 $678.15

SUBTRACTION WITH MONEY

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$196.05 $134.20 20

MATHS MASTERPIECES

The Tower of Babel

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MATHS MASTERPIECES

SUBTRACTION WITH MONEY

21

$197.35 $437.85

m . u

$578.95

Maths masterpieces

$388.85

$20.95 $141.80

$87.80

$412.85

$267.20 $135.35

R.I.C. Publications®

o c . che e r o t r s super $227.60

w ww

. te

$618.15

$335.35

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

$61.85

$175.90

$250.25

$252.85 $306.85

$0.95

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

ew i ev Pr

Teac he r

$151.85

The Tower of Babel

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22

MATHS MASTERPIECES

MULTIPLICATION (AMAZING NUMBERS)

Teachers notes

Max Schmitt in a Single Scull 285 714

428 571

571 428

714 285

857 142

384 615

538 461

615 384

846 153

230 769

307 692

692 307

769 230

923 076

111 111

222 222

333 333

444 444

Teac he r

153 846

ew i ev Pr

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

999 999

Background information

4. Glue the tiles onto their matching spaces.

Max Schmitt in a Single Scull

5. Colour the picture in an interesting way.

© R. I .

C.Publ i cat i ons •f orr evi ew pTalking ur p osesonl y• mathematically

6. Can you see why these numbers and their answers are amazing?

Thomas Eakins was an American painter who painted many scenes of outdoor sporting life. He was also a prolific painter of portraits. He often included himself as an observer in his paintings.

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In The Agnew Clinic, he is seen watching a surgical operation and in The Swimming Hole he is seen treading water next to his dog, Harry.

As an art teacher he stressed the importance of painting nude models and was ostracised as a result of this. Eakins was largely ignored by the art establishment during his lifetime but is now highly regarded as one of America’s greatest painters.

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Internet image search Starting Out after Rail Baseball Players Practicing The Agnew Clinic The Pathetic Song Chess Players

Look at the answers when 142 857 is multiplied: (x2) = 285 714

(x3) = 428 571

(x5) = 714 285

(x6) = 857 142

(x4) = 571 428

Not only does each answer contain the digits 142 857, they are also in order.

o c . che e r o t r s super

Such numbers are called roundabout numbers (the numbers 142 857 go around and around just as on a roundabout or, as they are better known in Australia, merry-go-round) and are extremely rare.

What is just as amazing is that a dedicated mathematician sat down and did hundreds of thousands of multiplication algorithms before coming across this oddity.

As if to show that it has more than one party trick, 142 857 x 7 makes 999 999.

76 923 when multiplied by 2, 5, 7, 8 and 11 gives the roundabout answers: 153 846, 384 615, 538 461, 615 384, 846 153.

Student instructions 1. Cut out the tiles from the jigsaw sheet.

2. Do the multiplication algorithms on the backing board.

3. Match the numbers on the tiles to the answers on the backing board.

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These answers really are amazing.

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by Thomas Eakins (1844 - 1916)

When multiplied by 3, 4, 9, 10 and 12, a second set of roundabout answers results.

These are: 230 769, 307 692, 692 307, 769 230 and 923 076.

Maths masterpieces

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MATHS MASTERPIECES

MULTIPLICATION (AMAZING NUMBERS)

23

x

63 492 7 47 619 7

x

x

76 923 9 x

x

76 923 7

x

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

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142 857 Teac x h 5 er

76 923 10

76 923 8

142 857 x 6

Max Schmitt in a Single Scull

31 746 7

m 15 873 .

7 ux

x

76 923 4 76 923 3 x

x

76 923 11

76 923 12

x

x

x

o c . che e r o t r s super 142 857 x 7

142 857 x 2

142 857 x 3

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R.I.C. Publications®

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76 923 2

x

76 923 5

142 857 x 4

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C.Publ i cat i ons •f orr evi ew pur posesonl y•

Maths masterpieces

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24

MATHS MASTERPIECES

MULTIPLICATION (AMAZING NUMBERS)

923 076

999 999

615 384 846 153

Maths masterpieces

153 846 769 230 230 769

m . u

384 615

333 333

692 307

444 444 222 222

www.ricgroup.com.au

o c . che e r o t r s super 111 111

w ww

. te

538 461

285 714

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

307 692

571 428 428 571

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

714 285

Teac he r

ew i ev Pr

857 142

Max Schmitt in a Single Scull

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MATHS MASTERPIECES

LONG MULTIPLICATION

25

Teachers notes

Study for Last Supper (Judas) Background information Study for Last Supper (Judas)

552

1875

1575

1944

by Leonardo da Vinci (1452 - 1519)

Leonardo is regarded as the finest painter of the period known as the Renaissance (the period when art, learning and literature were revived [14th–16th centuries] after the period known as the Dark Ages). As well as being unsurpassed as a painter, Leonardo was also a highly accomplished engineer and scientist (among his inventions are scissors).

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

Teac he r

1440

2112

2340

2774

4590

1512

1536

667

825

1665

1800

Perhaps your students can shade their finished masterpieces with rubbed chalk.

Internet image search

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The picture was a chalk drawing prepared as a model to be transferred for the famous painting, The Last Supper. Such line drawings were often done and transferred to canvas or a wall as a guide for the artist. Leonardo’s master, the artist Verrochio, is said to have given up painting in awe when he saw the works of his 17-year-old student.

644

621

3888

1632

1992

© R. I .

C.Publ i cat i ons Student instructions •f orr evi ew pur posesonl y• Mona Lisa The Last Supper Virgin of the Rocks

1. Cut out the tiles from the jigsaw sheet.

2. Do the multiplication algorithms on the backing board.

It will be useful if students have some experience multiplying by multiples of 10.

m . u

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To ‘prove’ the ‘write down zero’ shortcut do 3. Match the numbers on the tiles to the answers on the some of these as additions. Children should backing board. accept the shortcut after doing a number of these; e.g. 20 x 40.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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Talking mathematically

It should quickly become evident that the units column adds up to 0. Just as clearly, students should quickly accept that 20 x 4 equals 80 (etc.) after working through a few examples using addition.

o c . che e r o t r s super

Work through a number of examples using these steps.

37 x 65

Step 1: Consider what ‘question’ the algorithm is ‘asking’. We are being asked how many 37 lots of 65 are altogether.

Step 2: Multiply 37 by 5 and write the answer (185) on the second line.

Step 3: Write 0 in the units column on the third line because the next number you are multiplying by is 60 (not 6).

Do the multiplication by 6 and write the answer on the third line (2220).

40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40

Step 4: Add the two answers (185 + 2220) together and write total on the bottom line (2405).

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26

MATHS MASTERPIECES

LONG MULTIPLICATION

x

54 36

x

28 23

r o e t s Bo r e p 45 o u k Sx 32 x 48 44 x

65 36

x

73 38

54 24 ©x R. I .

C.Pub i ca t i onsx 85 xl 63

48 32

Teac he r

x

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x

•f orr evi ew pur posesonl y•

23 29

45 37

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x

33 25

x

45 40

x

23 27

x

48 34

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x

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23 24

Study for Last Supper (Judas) 75 45 x 25 x 35

x

54 72

x

83 24

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Maths masterpieces

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MATHS MASTERPIECES

LONG MULTIPLICATION

27

Study for Last Supper (Judas)

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

644

2774

1632

1536

1992

1665

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Teac he r

2112

1800

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• 1512

w ww

. te

1575

1944 R.I.C. Publications®

825

621

m . u

552

o c . che e r o t r s super 1440

3888

4590

667

1875

2340

Maths masterpieces

www.ricgroup.com.au

28

MATHS MASTERPIECES

DIVISION WITH REMAINDERS

Teachers notes

A Winter Scene 46 r 2

32 r 1

47 r 1

r o e t s Bo r e p ok u S 55 r 1

51 r 3

45 r 4

40 r 5

32 r 4

75 r 1

39 r 2

25r 6

79 r 5

25 r 2

67 r 3

41 r 1

ew i ev Pr

Teac he r

42 r 3

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• 38 r 5

57 r 3

37 r 1

36 r 2

Talking mathematically

A Winter Scene

To assist with these algorithms initially, diagrammatic crutches may be provided to assist students with their calculations.

by Isaac van Ostade (1621–1649)

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Isaac van Ostade was a Dutch painter renowned for his winter scenes and paintings of peasant life in 17th century Holland.

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Background information

For example:

4 99

Diagram:

tens: / / / / / / / / / ones: / / / / / / / / /

o c . che e r o t r s super

His paintings are limited in number because he was only 28 years old when he died.

A Horse Stable A Peasant Pouring Beer in His Wife’s Glass Peasants Outside a Farmhouse, Butchering Pork

Use the diagrams to demonstrate the first step, 9 divided by 4 (circle two groups of four tens). One will be left uncircled. There are now 19 ones left (1 ten and 9 ones) which can be circled into four groups of 4, leaving three uncircled. This gives the answer 24 r 3.

Student instructions 1. Cut out the tiles from the jigsaw sheet.

2. Do the division algorithms on the backing board.

3. Match the numbers on the tiles to the answers on the backing board.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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Maths masterpieces

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MATHS MASTERPIECES

DIVISION WITH REMAINDERS

29

A Winter Scene

3 140

2 65

4 207

5 229

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

2 111

2 95

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Teac he r

4 171

8 325

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6 479

8 309

R.I.C. Publications®

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© R. I .

C.Publ i cat i ons f orr evi w pur po e sonl y• 5 • 164 2e 151 3s 119 8 206

o c . che e r o t r s super 3 77

4 271

5 288

4 149

Maths masterpieces

2 83

3 110

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30

MATHS MASTERPIECES

DIVISION WITH REMAINDERS

A Winter Scene

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

38 r 5

32 r 4

36 r 2

32 r 1

39 r 2

67 r 3

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Teac he r

51 r 3

45 r 4

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• 75 r 1

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57 r 3

42 r 3 www.ricgroup.com.au

25 r 2

37 r 1

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46 r 2

o c . che e r o t r s super 55 r 1

47 r 1

79 r 5

25 r 6

40 r 5

41 r 1

Maths masterpieces

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MATHS MASTERPIECES

DIVISION OF MONEY

Teachers notes

The Peasant Dance $49.15

$187.95

$139.99

$84.80

r o e t s Bo r e p ok u S $105.70

$79.95

$178.05

$278.75

$78.95

$249.90

$163.35

$177.75

$478.65

$166.60

$285.05

$189.95

$217.30

$333.25

$117.95

Background information

Student instructions

ew i ev Pr

Teac he r

$12.40

31

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

The Peasant Dance

1. Cut out the tiles from the jigsaw sheet.

by Pieter Bruegel the Elder (1525–1569)

2. Do the division algorithms on the backing board.

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Internet image search

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

Talking mathematically

Place the decimal point on the answer line above the decimal point of the dividend (number to be divided) before commencing the algorithm and then proceed as with any other division algorithm.

o c . che e r o t r s super

Children’s Games Hunters in the Snow Peasant Wedding The Fight Between Carnival and Lent Procession to Calvary Two Chained Monkeys The Numbering at Bethlehem

R.I.C. Publications®

3. Match the amounts on the tiles to the answers on the backing board.

m . u

w ww

Pieter Bruegel the Elder is the best known of a family of Dutch artists. His paintings are very detailed and busy. Children’s Games is a scene showing more than 200 children at play. He was called by some, Pieter the Peasant because he liked to paint aspects of peasant life and would often mingle with peasants to get ideas for these paintings.

2 $49.70

Maths masterpieces

Place the decimal point on the answer line before commencing the algorithm.

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32

MATHS MASTERPIECES

DIVISION OF MONEY

5 $589.75 3 $999.75

6 $999.60

2 $957.30

3 $836.25

ch er

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

ew i ev Pr

Tea 2 $375.90 5 $699.95

9 $710.55

8 $678.40

The Peasant Dance

$651.90

m . 3 $759.80 u

3 $490.05

5 $890.25 4 $319.80

Maths masterpieces

3 $855.15

4

o c . che e r o t r s super 4 $999.60

. te

6 $634.20

3 $37.20

4 $196.60

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www.ricgroup.com.au

7 $1244.25

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

R.I.C. Publications®

MATHS MASTERPIECES

DIVISION OF MONEY

33

$478.65 $79.95 $12.40

m . u

$49.15

Maths masterpieces

$105.70

$166.60 $78.95

$177.75

$278.75

$117.95 $249.90

R.I.C. Publications®

o c . che e r o t r s super $217.30

w ww

. te

$163.35

$189.95

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

$139.99

$333.25

$84.80 $187.95

$178.05

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

ew i ev Pr

Teac he r

$285.05

The Peasant Dance

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34

MATHS MASTERPIECES

MINI-PROBLEMS

Teachers notes

The Avenue at Middelharnis

$4.90

1800

5

17

21

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

183

70

48

92

177

9.30

11.30

73

30

420

1440

100

182

ew i ev Pr

Teac he r

71

$4.20

Background information

Talking mathematically

The Avenue at Middelharnis

by Meindert Hobbema (1638–1709)

Most of the real-life mathematics that the students will do will be in the form of mini-problems such as those listed here.

Meindert Hobbema was a Dutch artist who was not well known during his lifetime.

A useful framework for doing problem work is to ask the questions What? and How? and then Answer.

This picture shows the Dutch village of Middelharnis as it appeared around 1689. The painting is a good example of linear perspective.

Example: 3 pens cost $2.10. How much will 7 cost?

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

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Internet image search

How?

How do I work this out? What working will I need? (Find the cost of 1, then multiply this unit cost by 7.)

o c . che e r o t r s super Answer $2.10 ÷ 3 = $0.70 7 x $0.70 = $4.90

The Watermill with a Red Roof Stormy Landscape

Student instructions

What is the problem asking me to find out? (the cost of 7 pens.)

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Because he was unable to make a living from painting, Hobbema’s output was not large. In 1688 he became a customs officer and was unable to devote as much time as he wished to painting.

What?

Therefore, 7 pens will cost $4.90.

Teachers may need to discuss some of the vocabulary used in these problems or provide resource materials for the students to find out some definitions. Some of the more difficult vocabulary used includes score, gross and nonagon.

1. Cut out the tiles from the jigsaw sheet.

2. Answer the problems on the backing board.

3. Match the numbers on the tiles to the answers on the backing board.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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m . u

How many legs altogether for 3 cows, 6 chickens, 3 fish and 4 ants?

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Maths masterpieces

How many days in autumn and spring combined?

min

How many minutes in one day?

How much is a gross plus a baker’s dozen plus a score?

ºC

Water boils at this temperature.

=

32 + 42 + 62 + 112

How many sides do 4 Bruce’s scores in cricket hexagons, 8 pentagons were 0, 1, 0, 0, 2 and and a nonagon have 177. What was his altogether? average score?

How many days altogether in the months of winter?

metres

What is the perimeter of a rectangular room 7 m long and 3.5 m wide?

MINI-PROBLEMS

km

A car travels 12 kilometres on a litre of petrol. How far will it go on 35 L?

Stanley has just A train trip took 5 and a Two angles of a triangle finished work. He quarter hours. It began are 47º and 62º. How began 8 hours ago. It is at 6.15 am. big is the remaining 5.30 pm. When did he When did it end? angle? start? º am am

grams

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How much is 3 score and 10?

Teac he r

Marbles sell at a price Plums cost $1.75 for 7. of 9 for 45c. How many How many can you buy can Jocelyn buy for for $1.25?

85c?

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Tyler ate 6 pies. Each pie had a mass of 300 g. What was the total mass he ate?

A dozen apples cost $7.20. How much for seven?

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3 pens cost $2.10. How much for 7?

MATHS MASTERPIECES 35

The Avenue at Middelharnis

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MATHS MASTERPIECES

MINI-PROBLEMS

92

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17

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The Avenue at Middelharnis

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MATHS MASTERPIECES

TWENTY-FOUR HOUR TIME

Teachers notes

37

Centurion (from the Crucifixion of St Peter) 05:45

21:30

07:00

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C.Publ i cat i ons •f orr evi ew pur posesonl y• Talking mathematically Background information 23:59

07:40

19:40

10:40

by Michelangelo Buonarroti (1475–1564)

It is believed that clock makers were influential in causing this to come about. Old clocks worked on a spring mechanism. When the spring runs down it has to be rewound. A 12-hour clock chimes 156 times in 24 hours (two lots of 78 chimes). A clock chiming up to 24 times a day would chime a total of 300 times in a day and so use more power from its spring. Such a clock would have to be rewound more frequently.

Michelangelo was an equally gifted painter and sculptor and is regarded as one of the greatest artists of all time.

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Among his many great works is the ceiling of the Vatican’s Sistine Chapel. This took him about four years and shows many scenes from the Bible, including three stories about Adam and Eve and three about Noah.

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Why do we divide a day into two divisions of 12 hours?

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Centurion (from the Crucifixion of St Peter)

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Internet image search c e her r o t s super Student instructions Sistine Chapel Ceiling (many depictions of nudity)

Clock faces with 24 divisions on the clock face are harder to read with a numeral being placed every 15º. On a 12hour clock face a numeral is only necessary every 30º.

However, 24-hour time is still popular and used in timetabling by, for example, railways and airlines.

1. Cut out the tiles from the jigsaw sheet.

2. Match the 24-hour times on the tiles to their equivalent 12-hour times on the backing board.

3. Glue the tiles onto their matching spaces.

4. Colour the picture in an interesting way.

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38

MATHS MASTERPIECES

TWENTY-FOUR HOUR TIME

Centurion (from the Crucifixion of St Peter)

1.35 pm

a quarter to six in the morning

half past nine at night

7.00 am

twenty-five minutes to four in the morning

five minutes past midnight

twenty to eleven at night

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1.35 am

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C.Publ i cat i onsthirteen five minutes •f or r e vi ew pur sesominutes nl y•past 7p .00o pm 12.45 am one in the afternoon

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before noon

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one minute to midnight

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twenty minutes twenty minutes to eight in the to eight in the morning evening

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twenty to eleven in the morning

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TWENTY-FOUR HOUR TIME

39

Centurion (from the Crucifixion of St Peter)

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13:35

22:40

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MATHS MASTERPIECES

FRACTIONS OF AMOUNTS

Teachers notes

The Miraculous Draught of Fishes 24

6

20

8

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16

Background information

Talking mathematically

The Miraculous Draught of Fishes

It is important that students know the meaning of such algorithms before the ‘shortcuts’ are taught and applied.

For example, 3/8 x 64, should be read as ‘three-eighths of sixty-four’ (not three-eighths times sixty-four) and explained as meaning ‘three out of every eight in a total group of sixty-four’.

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by Konrad Witz (1400–1444)

Living at different times in Germany and Switzerland, little is known of Witz’s life. He was one of the first artists to incorporate realistic landscapes into religious paintings.

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Internet image search

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Saint Bartholomew Saint Madeleine and Saint Catherine The Queen of Sheba before Solomon Saint Christophorus Madonna and Saints in a Church Meeting of Joachim and Anna by the Golden Gate

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Student instructions

The mechanical process involved in finding the answer is to circle groups of eight and then mark (colour, cross, highlight) 3 out of every such group (i.e. 3 out of every 8 in a total group of 64). Finally, when each group of eight has been thus marked, count the total number of items marked.

After doing this a number of times, some students will see that a ‘shortcut’ exists; i.e. divide the denominator of the fraction into the total amount. Multiply your answer by the numerator.

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1. Cut out the tiles from the jigsaw sheet.

2. Do the multiplication algorithms on the backing board.

3. Match the numbers on the tiles to the fractions on the backing board.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

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➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩

➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩

➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩

➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩

Maths masterpieces

♥ ♥ ♥ ♥ ♥

♥ ♥ ♥ ♥ ♥

♥ ♥ ♥ ♥ ♥

♥ ♥ ♥ ♥ ♥

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/5 x 25 =

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/11 x 44 =

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/7 x 56 = 5

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/8 x 48 =

7

✘ ✘ ✘ ✘ ✘ ✘

/7 x 28 =

2

✯ ✯ ✯ ✯

Teac he r

✈ ✈ ✈ ✈ ✈

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✔ ✔ ✔ ✔ ✔ ✔ ✔

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49 x 3/7 =

✔ ✔ ✔ ✔ ✔ ✔ ✔

✔ ✔ ✔ ✔ ✔ ✔ ✔

/12 x 36 = ✔ ✔ ✔ ✔ ✔ ✔ ✔

11

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I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

36 x 3/4 =

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/3 x 27 =

2

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FRACTIONS OF AMOUNTS

2

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/7 x 63 =

4

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

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/5 x 20 =

3

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© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

/3 x 30 =

2

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❀ ❀ ❀ ❀ ❀ ❀ ❀

/7 x 49 =

1

❀ ❀ ❀ ❀ ❀ ❀ ❀

/7 x 21 =

3

❀ ❀ ❀ ❀ ❀ ❀ ❀

42 x 5/6 =

❖ ❖ ❖ ❖ ❖ ❖

40 x 5/8 =

◗ ◗ ◗ ◗ ◗

☎☎☎☎☎☎☎ ☎☎☎☎☎☎☎ ☎☎☎☎☎☎☎

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36 x 4/9 =

✦✦✦✦✦✦✦✦✦ ✦✦✦✦✦✦✦✦✦ ✦✦✦✦✦✦✦✦✦ ✦✦✦✦✦✦✦✦✦

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4

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MATHS MASTERPIECES 41

The Miraculous Draught of Fishes

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MATHS MASTERPIECES

FRACTIONS OF AMOUNTS

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The Miraculous Draught of Fishes

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MATHS MASTERPIECES

EQUIVALENT FRACTIONS

The Feast of Belshazzar

43

Teachers notes

Background information The Feast of Belshazzar

/6

/4

/8

/14

/10

6

/15

25

/25

12

2

2

2

2

by Rembrandt van Rijn (1606–1669)

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The Dutch painter Rembrandt van Rijn is regarded as one of the greatest artists of all time. Although highly regarded during his lifetime, he never attained great wealth and was declared bankrupt in 1656.

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Internet image search Woman Bathing in a Stream The Anatomy Lesson of Dr Tulp Tobit and Anna The Ass of Balaam Talking Before the Angel Christ in the Storm Saskia Self-portrait

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As well as portrait painting he chose religious themes as subjects for his paintings. The Feast of Belshazzar recounts a scene from the Bible story of Belshazzar, the King of Babylon (555 to 539 BC). It is said that during a feast held by Belshazzar a hand was seen to write on the wall foretelling the destruction of Babylon. Babylon did fall a short time later and Belshazzar was killed.

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A simple way of explaining ‘what happens’ is to draw shapes (cakes) as shown in the diagrams and ‘cut them up differently’.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• For example:

Ask students ‘What fraction of the “cake” above has dark icing?’ (2/5) A C

(Rembrandt did many of these at various times of his life.)

B D

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1. Cut out the tiles from the jigsaw sheet.

2. Rule lines to form equivalent fractions on the backing board.

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Place dots at points A, B, C and D and ask students to rule ‘cut lines’ from A to B and from C to D.

Student instructions

Ask ‘What fraction has dark icing now that the cake has been cut differently?’ (6/15)

o c . che e r o t r s super Talking mathematically 3. Match the fractions on the tiles with the answers on the backing board.

4. Glue the tiles onto their matching spaces.

Ask ‘Has any icing been removed?’ (No) Ask ‘If no icing has been added and none removed, are /5 and 6/15 the same amount?’ (Yes)

5. Colour the picture in an interesting way.

2

Just as it is legitimate to write the whole number 7 as 1 + 1 + 1 + 4, (3 x 2) + 1 or in any number of equivalent ways, so too can fractions be represented in any number of ways using what are called equivalent fractions.

Children are usually taught the shortcut method of forming equivalent fractions (multiply numerator and denominator by the same amount) without understanding what this means.

R.I.C. Publications®

Ask ‘Has any icing been added to the cake?’ (No)

Explain: Yes, they are the same amount, but the cakes have just been cut up differently. That is a useful way to think of equivalent fractions.

Why does the shortcut method work?

When you multiply the numerator and denominator by 3 you are effectively saying that three-thirds (one whole cut into thirds) of two-fifths is six-fifteenths. You are describing ‘one whole lot of fifths’ but the one whole is shown as three-thirds.

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B

A

A

/2 is also

1

A C E G I

E

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A

B D F H J

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D

B

D

B

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1

/2 is also

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B

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/3 is also

B

D

B

C

2

/3 is also

1

/3 is also

1

B

A

/5 is also

1

A C E G I

G

E

C

A

C

A

/5 is also

1

/5 is also

1

/5 is also

1

Teac he r

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• A

C

A

A

/3 is also

1

B

B

A

A

/8 is also

5

/7 is also

3

B D F H J

H

F

D

B

D

D F H

C E G

D

B

B

/4 is also 3

/5 is also 4

H

F

D

B

B

A

C

A

G

E

C

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F H

E G

/4 is also

D

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3

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F

E

/4 is also

D

C

1

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/4 is also

1

B

/2 is also

1

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A

A

/4 is also EQUIVALENT FRACTIONS

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MATHS MASTERPIECES

The Feast of Belshazzar

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MATHS MASTERPIECES

EQUIVALENT FRACTIONS

45

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The Feast of Belshazzar

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MATHS MASTERPIECES

CONVERTING FRACTIONS TO DECIMALS

Teachers notes

Bailed Up

Background information Bailed Up

0.5

0.8

0.9

0.55

0.25

0.2

0.3

0.3

0.4

0.12

0.6

0.1

0.05

0.04

0.6

0.75

0.7

0.85

0.44

0.76

by Tom Roberts (1856 – 1931)

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Bailed Up depicts a true incident described to Roberts by a friend, Bob Bates. Bates had been the stagecoach driver of a Cobb & Co. coach travelling between Glen Innes and Inverell when it was held up. Roberts created the painting from these recollections.

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

Tom Roberts was an important Australian artist of the latter half of the 19th and early 20th centuries. Roberts thought it was important to record Australia’s pioneering days.

Such stick ups were all too common in early Australia.

As well as the human drama, Roberts shows an equally dramatic hot dry landscape, typical of Australia in midsummer.

Step 2: Write down the zero and put a decimal point after it.

Internet image search Portrait of Florence Bourke Street (Allegro Con Brio) Coming South Shearing the Rams A Break Away!

(In other words one third is less than one whole.)

© R. I .

C.Publ i cat i ons •f orr evi ew pur poses0.3onl y• Student instructions

Step 3: Add zero to the one left over and divide again by the bottom number.

(Three goes in to ten three times with one left over.)

Write the three after the decimal point.

2. Change the fractions on the backing board to their decimal equivalents.

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3. Match the numbers on the tiles to the answers on the backing board.

From now on, whenever there is a remainder you must add zero to it and divide again by the bottom number.

Working will look something like this: zero ‘written in’ left over

1

4. Glue the tiles onto their matching spaces.

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/3

left over

= 1 1

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5. Colour the picture in an interesting way.

zeroes ‘written in’

The students will require a knowledge of equivalent fractions to do this activity. Most of the fractions chosen can be easily converted to tenths or hundredths by multiplying numerators and denominators by equal amounts.

Some fractions, however, do not convert so easily to tenths, hundredths, thousandths etc.

Here is a method you can use to convert these more difficult examples to decimals:

/3

1

Step 1: Divide the top number by the bottom number.

(Three goes into one zero times and one is ‘left over’.)

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1. Cut out the tiles from the jigsaw sheet.

0.33

As you can see, the answer we will continue to get is 3. The fraction 1/3 forms the decimal zero point three recurring.

No matter how long you continue to do the division, you will continue to get the answer 3.

To show this, it is written 0.3 in decimal notation.

For the fraction 1/7, 142 857 continues to repeat. It is called zero point one four two eight five seven recurring and written 0.142857 in decimal notation.

For the fraction 1/6 the 6 repeats. We write this 0.16 in decimal notation.

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MATHS MASTERPIECES

CONVERTING FRACTIONS TO DECIMALS

47

1

/25 = 0. /25 = 0.

11

/25 = 0.

/5 = 0.

2

20

11

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/3 = 0.

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/4 = 0.

/25 = 0.

Bailed Up

/20 = 0.

7

/4 = 0.

1

/5 = 0.

3

3

/10 = 0.

/3 = 0.

1

Maths masterpieces

/10

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1

17

/20 = 0.

/10 = 0.

3

o c . che e r o t r s super /5 = 0.

1

/2 = 0.

4

/5 = 0.

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/10 = 0.

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MATHS MASTERPIECES

CONVERTING FRACTIONS TO DECIMALS

0.76

0.04 0.85

Maths masterpieces

0.4 0.8 0.5

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0.44

0.05

0.75 0.3

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0.2

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0.1

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0.12

Bailed Up

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MATHS MASTERPIECES

ADDING DECIMALS

Teachers notes

Bathers at Asnières 0.7

3.2

6.4

4.2

r o e t s Bo r e p ok u S 7.1

13.1

5.1

8.2

12.5

0.9

7.2

11.7

1.4

8.5

10.8

1.3

9.1

6.1

7.3

Background information

Student instructions

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49

Bathers at Asnières

1. Cut out the tiles from the jigsaw sheet.

by Georges Seurat (1859 – 1891)

2. Do the addition algorithms on the backing board.

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3. Match the numbers on the tiles to the answers on the backing board.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

Talking mathematically It is important to be aware of place value when dealing with decimals just as it is with whole numbers.

Internet image search

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Sunday Afternoon on the Island of Grande Jatte The Can-can Rock Breakers Cows in a Field Watering Can Invitation to the Sideshow The Eiffel Tower

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Georges Seurat was a French artist who developed a method of painting called pointillism (also divisionism).

This involved compiling his paintings with a series of tiny dots so the colours are mixed by the eye (tiny red and yellow dots placed close together on the canvas would be mixed by the eye to make the colour orange etc.). Colour television produces coloured images in much the same way, colours being mixed by the eyes using tiny red, green and blue dots.

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Maths masterpieces

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50

MATHS MASTERPIECES

ADDING DECIMALS

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+

3.5 2.6

+ +

+

4.7 2.6

6.9 1.6 0.9 0.5

+ +

+

1.2 7.9

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5.6 6.1 +

+

+

3.9 6.9 +

0.5 0.8

+ +

3.7 3.5 0.1 0.8

+

+

3.9 3.2

+

+

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5.9 8.3

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7.8 5.3

+

3.9 1.2

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0.2 0.5

2.6 5.6

3.3 3.1 1.9 1.3

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+ Teac he r

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2.8 1.4

7.7 4.8

Bathers at Asnières

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MATHS MASTERPIECES

ADDING DECIMALS

51

0.7

1.4

3.2

5.1

Maths masterpieces

12.5 7.1 8.2

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8.5

11.7

7.2

6.1 13.1

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7.3

4.2

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6.4

9.1

1.3

10.8

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Teac he r

14.2

Bathers at Asnières

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52

MATHS MASTERPIECES

MULTIPLYING DECIMALS BY A WHOLE NUMBER

Teachers notes

The Scapegoat

58.5

46.8

69.3

58.8

29.1

132.3

52.0

86.4

53.9

100.8

42.9

88.5

564.3

89.6

47.1

619.2

197.8

381.6

Teac he r

52.2

Background information

Internet image search The Light of the World The Hireling Shepherd

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72.8

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C.Publ i cat i ons Student instructions •f orr evi ew pur posesonl y•

The Scapegoat

by William Holman Hunt (1827–1910)

1. Cut out the tiles from the jigsaw sheet.

The three artists founded a group and a painting style known as the Pre-Raphaelite Brotherhood. Their aim was to have serious moral issues as the subject of their paintings and represent nature with realism.

3. Match the numbers on the tiles to the answers on the backing board.

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Hunt went to Palestine to work on a series of religious paintings in 1854. It was here that he painted the nightmarish scene The Scapegoat. It depicts the Bible story of the goat cast into the wilderness by a Jewish chief priest who had symbolically laid people’s sins upon it.

4. Glue the tiles onto their matching spaces.

5. Colour the picture in an interesting way.

Talking mathematically

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The word scapegoat has come to mean a person unfairly blamed for the wrongdoings of others, just as the poor goat was in the Bible story.

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2. Do the multiplication algorithms on the backing board.

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William Holman Hunt was an Englishman who studied art at Britain’s famous school of art, The Royal Academy.

During his period there he became friends with the artists John Everett Millais and Dante Gabriel Rossetti.

It is important to be aware of place value when doing these multiplications. The decimal point can be placed between the answer lines before any work is done. Then proceed with the multiplication as if the decimal point was not present.

The students can check the placement of the decimal point in the answer by rounding any decimal numbers in the question to whole numbers and mentally multiplying them. For example, 6.5 x 9 7 x 9; therefore, the answer should be close to 63.

Maths masterpieces

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MATHS MASTERPIECES

MULTIPLYING DECIMALS BY A WHOLE NUMBER

53

x

98.9 2 x

x

17.7 5

x x

6.5 8

x

63.6 6

62.7 9

9.6 9

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Teac 8.4 x h7 er

x

9.7 3

The Scapegoat

m 15.7 . u 3

x

77.4 8

14.3 3 x

x

11.2 8

x

x

12.6 8 7.7 7

x

x

9.1 8

x R.I.C. Publications®

x

x

18.9 7

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6.5 9

x

7.8 6

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17.4 3

x

9.9 7

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Maths masterpieces

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54

MATHS MASTERPIECES

MULTIPLYING DECIMALS BY A WHOLE NUMBER

52.2

29.1

619.2

89.6

Maths masterpieces

88.5 52.0 46.8

m . u

564.3

132.3

100.8

42.9 47.1

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197.8

58.5

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C.Publ i cat i ons •f orr evi ew pur posesonl y•

58.8

381.6

86.4

53.9

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Teac he r

72.8

The Scapegoat

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MATHS MASTERPIECES

SHAPES AND LINES

Teachers notes

Self-portrait semicircle

parallel lines

regular decagon

square

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

irregular nonagon

ellipse

right angle

rhombus

circle

obtuse angle

trapezium

regular nonagon

quadrant

parallelogram

irregular decagon

oblique line

regular hexagon

straight angle

vertical line

horizontal line

rectangle

regular octagon

regular pentagon

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irregular octagon

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C.Publ i cat i ons •f orr evi ew pur posesonl y• acute angle

reflex angle

Background information

triangle

Internet image search

Self-portrait

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by Vincent Van Gogh (1853 – 1890)

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curved line

Vincent Van Gogh was a brilliant artist but led an unsettled life because of recurring periods of mental disturbance.

He only began painting at the age of 27 but in the tenyear period of activity before he shot and killed himself he completed 1600 paintings.

Sunflowers Seashore at Scheveningen The State Lottery Office The Potato Eaters Agostina Segatori The Good Samaritan The All Night Café at Arles The Dance Hall at Arles

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irregular hexagon

o c . che e r Student instructions o t r s super

He was not recognised until after his death and most of his life was lived in abject poverty. He was a friend of the French artist, Paul Gauguin, and it was after an argument with him that van Gogh cut off part of his own ear.

Most of his paintings are made up of separate unblended brush marks. Van Gogh always signed his pictures with his first name, Vincent, only.

This is one of several self-portraits painted by the artist in which we see him as an intense, ravaged, restless-looking young man.

1. Cut out the tiles from the jigsaw sheet.

2. Match the words on the tiles to their pictures on the backing board.

3. Glue the tiles onto their matching spaces.

4. Colour the picture in an interesting way.

Talking mathematically The students will need to understand the difference between regular and irregular shapes to complete this activity.

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Maths masterpieces

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MATHS MASTERPIECES

SHAPES AND LINES

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

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Self-portrait

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m . u

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

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Maths masterpieces

R.I.C. Publications®

MATHS MASTERPIECES

SHAPES AND LINES

57

Self-portrait

irregular octagon

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

regular hexagon

circle

irregular hexagon

regular octagon

parallelogram

rhombus

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Teac he r

triangle

trapezium

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

w ww

oblique line

square

ellipse

semicircle

horizontal line

reflex angle

vertical line

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quadrant

m . u

irregular octagon

o c . che e r o t r s super

parallel lines

right angle

irregular nonagon

obtuse angle

straight angle

acute angle

regular pentagon

curved line

irregular decagon

regular nonagon

regular decagon

R.I.C. Publications®

Maths masterpieces

www.ricgroup.com.au

Teac he r

ew i ev Pr

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

w ww

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m . u

© R. I .

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