TESSELLATION KOLEKSI PLANET PEA-PLE SIRI 22: AUTHORS: NUR AMIRA ATHIKA BINTI HASMANSAH NURUL MAISARA BINTI ZAINAL ABIDIN KENNETH KOO YOU CHUAN CHIAM SUN MAY, PHD NG LEE FONG, PHD BENNY KONG TZE LOONG LOW KEE SUN TAN CHO CHIEW ADVISORS: CHIAM SUN MAY, PHD NG LEE FONG, PHD BENNY KONG TZE LOONG LET 'S LEARN AND PLAY TOGETHER!
TESSELLATION KOLEKSI PLANET PEA-PLE SIRI 22: AUTHORS: NUR AMIRA ATHIKA BINTI HASMANSAH NURUL MAISARA BINTI ZAINAL ABIDIN KENNETH KOO YOU CHUAN CHIAM SUN MAY, PHD NG LEE FONG, PHD BENNY KONG TZE LOONG LOW KEE SUN TAN CHO CHIEW LET 'S LEARN AND PLAY TOGETHER!
INSTITUT PENDIDIKAN GURU KAMPUS KENT, PETI SURAT 2, 89207 TUARAN, SABAH. Hak Cipta © Institut Pendidikan Guru Kampus Kent 2023 Cetakan Pertama 2023 Hak cipta terpelihara. Tidak dibenarkan mengeluarkan atau ulang mana-mana bahagian kandungan, illustrasi dan jadual dalam kandungan buku ini dalam apa juga bentuk dan dengan apa juga sama ada secara elektronik, fotokopi mekanik, rakaman atau cara lain sebelum mendapat izin bertulis daripada Institut Pendidikan Guru Kampus Kent. Koleksi Planet Pea-ple Siri 22: TESSELLATION NUR AMIRA ATHIKA BINTI HASMANSAH NURUL MAISARA BINTI ZAINAL ABIDIN KENNETH KOO YOU CHUAN CHIAM SUN MAY, PHD NG LEE FONG, PHD BENNY KONG TZE LOONG LOW KEE SUN TAN CHO CHIEW elSBN: 978-967-0008-63-9 Kulit Buku: Nur Amira Athika Binti Hasmansah
As we reflect upon our journey of exploring the captivating world of tessellation, we are filled with a profound sense of gratitude. This preface serves as an expression of heartfelt appreciation to those who have accompanied us on this enlightening path. We would like to thank and express our gratitude towards IPG Kampus Kent, our lecturers; Mr. Benny Kong Tze Long, Dr. Ng Lee Fong, Dr. Chiam Sun May, Madam Low Kee Sun and Mr. Tan Cho Chiew as our advisors and also our fellow classmates for their unconditional support and guidance that has helped us greatly in order to complete this book. Our lecturers, whose passion and expertise have been instrumental in guiding us through the intricacies of geometry. Their unwavering dedication to imparting knowledge, coupled with their engaging teaching style, has ignited a spark within us to delve deeper into this captivating subject of tessellation. They are not only chalk-and-talk us about tessellation in general, they also provide us with 3D teaching aids so that we can construct an in-depth understanding about tessellation. Tessellation is shapes that fit together like puzzle pieces, forming a beautiful pattern without any gaps or overlaps. It is like creating a picture using special shapes that magically connect with each other. From the scales on a fish to the cells in a honeycomb, nature loves tessellation too! We can find tessellations all around us, like the tiles on the floor or the patterns on a butterfly ' s wings. Isn 't it amazing how everything fits perfectly? So, let' s explore the world of tessellation and discover the wonders that hide in plain sight! PREFACE
CONTENTS PAGES WHAT IS TESSELLATION? WHY IS TESSELLATION INTERESTING? HISTORY OF TESSELLATION INTRODUCTION TO TESSELLATION 1-2 REGULAR TESSELLATIONS SEMI-REGULAR TESSELLATIONS DEMI-REGULAR TESELLATIONS NON-REGULAR TESSELLATIONS 3-DIMENSIONAL TESELLATION NON-PERIODIC TESSELLATIONS TYPES OF TESSELLATIONS 3-5 TESSELLATIONS IN PLANTS AND FLOWERS ANIMAL PATTERNS AND TESSELLATION TESSELLATIONS IN NATURAL LANDSCAPES TESSELLATION IN ARCHITECTURE AND DESIGN TESSELLATION IN TEXTILES AND PATTERNS TESSELLATION IN GAMES AND PUZZLES TESSELLATIONS IN NATURE AND EVERYDAY LIFE 6-11 SYMMETRY AND TRANSFORMATIONS IN TESSELLATION EXPLORING GEOMETRIC CONCEPTS THROUGH TESSELLATION MATHEMATICS BEHIND TESSELLATION 12-13 CONTENTS
CONTENTS PAGES ACTIVITY 1: TANGRAM 14-16 ACTIVITY 2: PATTERN BLOCK 17-21 ACTIVITY 3: POLYOMINOES 22-25 ACTIVITY 4: TANTRIX 26-30 ACTIVITY 5: JIGSAW PUZZLES 31 ACTIVITY 6: TESSELLATION IN GEOMETRY SKETCHPAD (GSP) 32-40 CONCLUSION 41 REFERENCES 42 CONTENTS
HELPS US DISCOVER PATTERNS ALL AROUND US CAN PLAY WITH VARIOUS DIFFERENT SHAPES ENHANCE OUR CREATIVITY IN SOLVING PUZZLE INTRODUCTION TO TTEESSEELLLLAATTIIOONN Tessellation is like putting puzzle pieces together to make a beautiful picture that repeats over and over again. When a lots of these puzzle pieces together, side by side, they fit perfectly without any gaps or overlaps. That is an example of tessellation! In a simple terms, tessellation is when we use shapes to cover a surface without any gaps or overlaps, just like puzzle pieces fitting together perfectly. WWHHYY I ISS TTEESSSSEELL LLAATTI IOONN I INNTTEERREESSTTI INNGG?? What is tessellation? 1
ANCIENT TIMES (3000 BCE - 500 CE) Tessellating patterns have been found in ancient Egyptian art and the Romans, particularly in floor tiles, decorative motifs and architecture. ISLAMIC ART AND ARCHITECTURE (8TH - 15TH CENTURIES) Golden ratio of 'The Mona Lisa ' by Leonardo Da Vinci 'Birds and Fishes ' by Maurits Escher 'Four Books on Measurement' by Albrecht Dürer Islamic artists developed intricate geometric patterns known as arabesques, which featuring tessellating motifs in architecture to adorned mosques, palaces, and other structures. RENAISSANCE AND MATHEMATICAL EXPLORATION (14TH - 17TH CENTURY) HHI ISSTTOORRYY OOFF TTEESSEELLLLAATTIIOONN A wall tiling at the Alhambra, Spain Ancient Greek Floor Mosaics Artists and mathematicians of the Renaissance, including Leonardo da Vinci and Albrecht Dürer, explored the principles of symmetry and tessellation. They incorporated tessellating patterns into their artwork and studied the mathematical aspects of tessellation. CONTEMPORARY TESSELLATION ART (20TH CENTURY - PRESENT) Artists explore tessellation in various mediums, including painting, sculpture, digital art, and even street art. Today, artists like Bridget Riley, Shigeo Fukuda, and Andrea Mantovani explore tessellation through optical illusions, threedimensional installations, and interactive digital art, showcasing the continued relevance and versatility of tessellation in contemporary artistic expression. 'Rustle 5' by Bridget Riley 2
R E G U L A R T E S E L L A T I O N S EMI - R E G U L A R T E S E L L A T I O N D EMI - R E G U L A R T E S E L L A T I O N N O N - R E G U L A R T E S E L L A T I O N N O N - P E R I O D I C T E S E L L A T I O N TESELLATION TESELLATION TESELLATION TESELLATION OF FUN! 3
REGULAR TTeesseellllaattiioonn SQUA R E S DEMIREGULAR DEMIREGULAR triangle & hexagon 3.6.3.6 / 3.3.6.6 triangle, SQUARE & DODECAGON 3.3.3.3.3.3 / 3.3.4.12 SQUARE, HEXAGON AND DODECAGON 4.6.12 Regular tessellations are tile patterns made up of only one single type of regular polygons Each vertex of the polygon, must join another vertex All vertices are identical Definition : A regular polygon has an interior angle sum that is a divisor of 360 degrees. TRIAN G L E HEXA G O N example : the sum of the interior angles of a triangle is 180 degrees; which is a divisor of 360 degrees. SEMIREGULAR TESELLATIONWhen 2 or 3 types of regular polygons share a common vertex, and all vertices are identical f orms f orms square & octagon 4.8.8 triangle, square & hexagon 3.4.6.4 triangle & square 3.3.4.3.4 triangle, SQUARE & DODECAGON 3.12.12 / 3.4.3.12 Types of tessellations that combine 2 or 3 polygon arrangements and have different types of vertices (all vertices are not identical) NOTES !! 6.6.6 3.3.3.3.3.3 4.4.4 To name a tessellation, focus on a vertex, and then count the number of sides of the polygons (n), that form the vertex Go around the vertex, such that the smallest possible numbers appear first. Example : Tesellation of Square 4.4.4.4 n=4 n=4 n=4 n=4 4
TTEESSEELLLLAATTIIOONN NON-PERIODIC N-PERIODIC NON - REGULAR from polygons whose sides are not the same lengths, used in a repeating pattern completely covering a plane without gaps and overlaps 5 m a d e u p m a d e u p what is non - regular polygon ? polygons that have different and unequal side lengths Answer : Example of non-regular tesellation: have a repetitious pattern and not periodic. Instead, the tiling evolves as it is created, yet still contains no overlapping or gaps. Example of Tesellation : non-periodic tessellation is a tiling that does not non - periodic periodic
IN NATURE AND EVERYDAY LIFE TTEESSEELLLLAATTIIOONN Sunflowers The seed distribution in the center of a sunflower follows a spiral pattern that can be described as a type of tessellation. These spirals often adhere to the Fibonacci sequence, resulting in visually appealing arrangements of seeds. The seed distribution in the center of a sunflower follows a spiral pattern that can be described as a type of tessellation. These spirals often adhere to the Fibonacci sequence, resulting in visually appealing arrangements of seeds. tessellation in plants and flowers Succulents Certain succulent plants, like the agave, display tessellation in the arrangement of their leaves. The leaves are often tightly packed, forming a repeating pattern that covers the surface of the plant. Certain succulent plants, like the agave, display tessellation in the arrangement of their leaves. The leaves are often tightly packed, forming a repeating pattern that covers the surface of the plant. 6
IN NATURE AND EVERYDAY LIFE TTEESSEELLLLAATTIIOONN Snake scales Snakes possess scales on their skin that fit together seamlessly, creating a tessellating pattern. These scales allow the snake to move smoothly and efficiently. Snakes possess scales on their skin that fit together seamlessly, creating a tessellating pattern. These scales allow the snake to move smoothly and efficiently. Animal patterns and tessellation Tortoise shell The scutes (scales) on the shells of tortoises and turtles can exhibit tessellation. These scutes fit together in a way that covers and protects the animal while allowing flexibility and movement. The scutes (scales) on the shells of tortoises and turtles can exhibit tessellation. These scutes fit together in a way that covers and protects the animal while allowing flexibility and movement. Butterfly wings The wings of butterflies often display intricate patterns formed by a tessellation of scales. These patterns can include symmetrical arrangements, geometric shapes, or repeating motifs The wings of butterflies often display intricate patterns formed by a tessellation of scales. These patterns can include symmetrical arrangements, geometric shapes, or repeating motifs 7
IN NATURE AND EVERYDAY LIFE TTEESSEELLLLAATTIIOONN Columnar Basalt Formations In certain volcanic areas, basaltic lava flows can cool and contract, forming hexagonal columns. These columns fit together tightly, creating a tessellating pattern. In certain volcanic areas, basaltic lava flows can cool and contract, forming hexagonal columns. These columns fit together tightly, creating a tessellating pattern. Tessellation in natural landscapes Honeycomb Weathering In some regions, particularly in desert environments, rocks can undergo honeycomb weathering. This process creates a network of small, interconnected pits or depressions that resemble a honeycomb pattern. In some regions, particularly in desert environments, rocks can undergo honeycomb weathering. This process creates a network of small, interconnected pits or depressions that resemble a honeycomb pattern. Patterned Ground In periglacial environments, where freezing and thawing occur repeatedly, patterned ground can form. This includes various patterns such as polygonal networks, stripes, or circles, resulting from the contraction and expansion of the ground. In periglacial environments, where freezing and thawing occur repeatedly, patterned ground can form. This includes various patterns such as polygonal networks, stripes, or circles, resulting from the contraction and expansion of the ground. 8
IN NATURE AND EVERYDAY LIFE TTEESSEELLLLAATTIIOONN Parquet Flooring Parquet flooring features wood tiles arranged in various tessellating patterns. These patterns add visual interest and can be found in both historic and contemporary architecture. Parquet flooring features wood tiles arranged in various tessellating patterns. These patterns add visual interest and can be found in both historic and contemporary architecture. Tessellation in Architecture and Design Islamic Tiles Traditional Islamic tiles often feature tessellating patterns that create intricate and repeating geometric designs. These tiles are commonly used to adorn walls, ceilings, and other architectural surfaces. Traditional Islamic tiles often feature tessellating patterns that create intricate and repeating geometric designs. These tiles are commonly used to adorn walls, ceilings, and other architectural surfaces. Pavement Designs Tessellation is utilized in the design of pavements and walkways to create visually appealing patterns. This can be achieved with various materials, including tiles, bricks, or pavers. Tessellation is utilized in the design of pavements and walkways to create visually appealing patterns. This can be achieved with various materials, including tiles, bricks, or pavers. 9
IN NATURE AND EVERYDAY LIFE TTEESSEELLLLAATTIIOONN African Wax Prints African wax prints feature vibrant and colorful designs that often involve tessellation. These patterns can include geometric shapes, animal motifs, or abstract compositions. Tessellation in Textiles and Patterns Patchwork Quilts Batik Patterns Patchwork quilts utilize tessellation by combining fabric pieces to create repeating patterns. Shapes like squares, triangles, or hexagons are arranged in a way that forms a cohesive design. Batik, a traditional Indonesian textile art, often features tessellating motifs. The wax-resist dyeing technique allows for the creation of intricate patterns with repeating elements. 10
IN NATURE AND EVERYDAY LIFE TTEESSEELLLLAATTIIOONN Jigsaw Puzzles Jigsaw puzzles involve assembling interlocking puzzle pieces to form a complete picture. The pieces are designed to fit together seamlessly, creating a tessellating pattern when correctly assembled. Jigsaw puzzles involve assembling interlocking puzzle pieces to form a complete picture. The pieces are designed to fit together seamlessly, creating a tessellating pattern when correctly assembled. Tessellation in games and puzzles Tangrams Tangrams are puzzles consisting of several geometric shapes that can be arranged to form specific designs or shapes. These shapes often fit together without gaps or overlaps, resulting in a tessellation effect. Tangrams are puzzles consisting of several geometric shapes that can be arranged to form specific designs or shapes. These shapes often fit together without gaps or overlaps, resulting in a tessellation effect. Pentominoes Pentominoes are a specific type of polyomino puzzle in which players arrange twelve different shapes made up of five squares each. The goal is to fit all the shapes together to form a larger rectangle or square, creating a tessellating pattern. Pentominoes are a specific type of polyomino puzzle in which players arrange twelve different shapes made up of five squares each. The goal is to fit all the shapes together to form a larger rectangle or square, creating a tessellating pattern. 11
OBJECT OBJECT 90 degrees clockwise IMAGE 90 degrees anti-clockwise or 270 degrees clockwise MATHEMATICS BEHIND TESSELLATION TTEESSSSEELLLLAATTIIOONN TRANSFORMATIONS andREFLECTION 'flipped or mirrored ' TRANSLATIONS ' slide ' ROTATIONAL 'turned' OBJECT IMAGE LINE OF REFLECTION OBJECT IMAGE original shapes copy of original shapes IMAGE 12
EXPLORING The sum of its interior angle at a vertex (corner) MUST BE 360° LABEL NUMBER OF SIDE EACH SHAPE HAS WRITE IN TERMS OF (X.X.X.X) 120 120 60 60 MATHEMATICS BEHIND TESSELLATION TTEESSSSEELLLLAATTIIOONN CONCEPT REGULAR AND SEMI-REGULAR TESSELLATIONS How to determine shape(s) that can be tessellated? Sum at a vertex equals to 360° VERTEX CONFIGURATION Vertex Vertex configuration is like a secret code that tells us how many shape(s) will meet at each vertex using number of sides EXAMPLE: 3 3 6 6 CLOCKWISE OR ANTI-CLOCKWISE Choose 1 direction: (3.6.3.6) Vertex configuration (3.6.3.6) 13
TANGRAM ACTIVITY 1: WHAT IS TANGRAM ? The tangram is a dissection puzzle consisting of seven flat polygons, called tans, which are put together to form shapes. The seven pieces include: one parallelogram, one small square and five triangles (2 large right triangles, 1 medium right triangle and 2 small right triangles) OBJECTIVE The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pieces without overlap. BENEFITS OF PLAYING TANGRAM ? learn to recognize and identify geometric shapes develop positive feelings about geometry gain a stronger grasp of spatial relationships develop an understanding of how geometric shapes can be decomposed hone spatial rotation skills learn the meaning of congruence Tangram may help kids: PAPERS SCISSORS TANGRAM SET MATERIAL RULES!!! Use all seven pieces to form the shape fit together like puzzle pieces sitting flat on the table no overlapping 14
Place your square piece of paper in front of you. Fold along 1 diagonal and cut along the dotted line. Fold 1 of the 2 pieces as shown and cut along the dotted line. You should have 2 large triangles. Set them aside. Take the other piece from step 2, fold point A onto D and cut along the dotted line. Set the small triangle aside. Take the trapezoid from step 4, fold point B to C and cut along the dotted line. With the right half from step 5, fold point B to D and cut along the dotted line. You should have a small triangle and a square. With the other half from step 5, fold point D to E and cut along the dotted line. You should have a small triangle and a parallelogram. The 7 pieces should make a square as shown. 1. 2. 3. 4. 5. 6. 7. 8. HOW TO PLAY? Each student should start with a square sheet of paper. If your paper is not square, you can cut squares by folding the paper like the pictures below: With your square piece of paper, follow these directions and the drawings below them: Colour all the seven pieces polygon with the following colour: parallelogram (blue), square (yellow), 2 large right triangles (orange and green), medium right triangle (red), 2 small right triangles (brown and pink). cut all the seven pieces polygon along the dotted line. put all the seven pieces polygon together to form a square. 15
TANGRAM CHALLENGE use all the seven pieces polygon to form the shape as the pictures below: 16
ACTIVITY 2 : PATTERN BLOCK INTRODUCTION PATTERN BLOCKS ARE A SET OF MATHEMATICAL MANIPULATIVES DEVELOPED IN THE 1960S. THE SIX SHAPES ARE BOTH A PLAY RESOURCE AND A TOOL FOR LEARNING IN MATHEMATICS, WHICH SERVE TO DEVELOP SPATIAL REASONING SKILLS THAT ARE FUNDAMENTAL TO THE LEARNING OF MATHEMATICS. PATTERN BLOCKS SETS ARE MULTIPLE COPIES OF JUST SIX SHAPES: EQUILATERAL TRIANGLE (GREEN) 60° RHOMBUS (BLUE) 30° NARROW RHOMBUS (BEIGE) TRAPEZOID (RED) REGULAR HEXAGON (YELLOW) SQUARE (ORANGE) ALL THE ANGLES ARE MULTIPLES OF 30° SPECIFICATION NUMBER OF PLAYER : TWO AND ABOVE RECOMMENDED AGE : 6 AND UP BENEFITS OF PLAYING PATTERN BLOCK: PATTERN BLOCKS ARE USED TO CREATE, IDENTIFY, AND EXTEND PATTERNS. STUDENTS USE THE MANY RELATIONSHIPS AMONG THE PIECES TO EXPLORE FRACTIONS, ANGLES, TRANSFORMATIONS, PATTERNING, SYMMETRY, AND MEASUREMENT. PATTERN BLOCKS ACTIVITIES INCLUDE CHALLENGING STUDENTS TO CREATE NUMBERS, LETTERS, AND EVEN IMAGINATIVE PICTURES USING THE BLOCKS. ALLOW CHILDREN TO SEE HOW SHAPES CAN BE COMPOSED AND DECOMPOSED INTO OTHER SHAPES, AND INTRODUCE CHILDREN TO IDEAS OF TILINGS. 17
18 HOW TO PLAY? THERE HAVE 3 SET OF PATTERN BLCOK. EVERY SET OF PATTERN BLOCK GOT THEIR OWN TASK. SET 1 IS TO PLACE PATTERN BLOCKS ON TOP OF EACH SHAPE OF THE PUZZLE UNTIL THE PUZZLE IS COMPLETELY ASSEMBLED. USE THE COLORS OF THE SHAPES AS A GUIDE. SET 2 IS FILL IN THEIR SHAPE OUTLINES, NOTICE WHETHER STUDENT KNOW RIGHT AWAY WHICH BLOCK TO GET OR IF THEY USE TRIAL AND ERROR WITH A FEW DIFFERENT BLOCKS TO SEE WHICH ONES FIT. SET 3 IS TO SEE HOW MANY DIFFERENT WAYS YOU CAN FILL IN THE PUZZLE OUTLINES. CUT OUT ALL THE SHAPE BELOW: 1. 2. 3. 4. 5.
19 SET 1 DOG CAT CHICKEN FOX FISH BEE
20 SET 2 DOG CAT CHICKEN FOX FISH BEE
21 SET 3 DOG CAT CHICKEN FOX FISH BEE
A C T I V I T Y 3 : 22 Players can be individual or collaborative Recommended age : 4 and up The objective of Polyomino Puzzles is to place and arrange the given polyomino pieces onto the grid, filling the entire area without any overlaps or gaps. Players aim to achieve a visually harmonious and complete tiling pattern using the provided pieces. Polyomino Puzzles is a captivating puzzle adventure game that embraces the concept of tessellation to create mesmerizing patterns. In this game, players are presented with a grid and a set of polyomino pieces composed of interconnected squares. Get ready to embark on an immersive journey where creativity and strategic thinking will lead you to conquer the challenge of perfect tiling! SPECIFICATION : INTRODUCTION : OBJECTIVE : HOW TO PLAY : Start by preparing the grid size and collect the polyomino pieces, making sure they are readily accessible Take turns placing one polyomino piece at a time on the grid. You can rotate or flip the pieces as needed. Arrange polyominoes strategically to fill the entire playing area. Each part must connect with at least one edge of another part, forming a seamless 1. 2. 3. tiling pattern. 4.Constantly reassess and readjust your placement as you progress, ensuring no squares are left open or overlapping. Edge Connections: Each polyomino piece must connect with at least one edge of another piece. Diagonal connections are not allowed. Non-Overlapping: Polyomino pieces cannot overlap with one another. Each square on the grid must be occupied by only one piece. Complete Coverage: The goal is to fully cover the entire grid without any gaps or empty squares. Rotation and Reflection: Players can rotate and reflect the polyomino pieces as needed to find the optimal placement. Experiment with different orientations for a perfect fit. RULES AND RESTRAINTS : example of complete polyominoes puzzle
23 PLEASE CUT OUT POLYOMINO BOARD GAME Board 1 Board 2
2154 PLEASE CUT OUT POLYOMINO PIECES
215 PLEASE CUT OUT POLYOMINO PIECES
how to play? TANTRIX ACTIVITY 4: introduction Tantrix is a fun puzzle game that uses colorful tiles with lines on them. The goal is to connect the tiles together by matching the lines, creating a beautiful pattern. It's like solving a puzzle and creating a colorful picture at the same time! specification Number of player : 2 - 4 players Recommended age : 8 and up Each player need to choose their own colour (red, yellow, green or blue). Each player will take a random 5 tiles and the tiles will remains visible to other players throughout the game. Players then take turns adding tiles. Everytime a player has finished their 5 tiles, the player need to take 5 tiles more. The game will continue for 20 minutes or until all 56 tiles fully used. 1. 2. 3. 4. 5. objective To make the longest possible line or loop of based on the colour each player has choose. Player with the longest line or loop WIN ! rules and restraints Whenever "gobble" appeared, it must be filled first by the player whose turn it is. A gobble is a space around the Tantrix surrounded by three or more tiles. GOBBLE Gobble with 3 link of the same colour cannot be filled. Player are not allowed to create double gobble. DOUBLE GOBBLE ALL THE RULES AND RESTRAINTS MUST BE FOLLOWED UNTIL THE GAME ENDED 3 LINK OF SAME RED COLOUR 26
PLEASE CUT OUT TANTRIX SET 27
PLEASE CUT OUT TANTRIX SET 28
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PLEASE CUT OUT TANTRIX SET 30
RANK NAME TIME ( MIN ) 1 2 3 To solve the puzzle as fast as possible and make a record! The time challenge adds an element of competition and encourages Let's improve your puzzle-solving skills while racing against the clock. how to play? JIGSAW PUZZLE ACTIVITY 5: introduction A jigsaw puzzle is a fun game that has a picture broken into many small pieces. You have to put the pieces together to make the picture complete again. It's like solving a big puzzle by finding where each piece belongs. specification Recommended age : 4 and up The time will start the moment player take 1 piece of puzzle to be solved. After the puzzle is complete, the time will be stopped. Player with the shortest time, their name will be written in the "Top 3 Ranking" altogether with the record of the time they used to complete the puzzle 1. 2. 3. objective T I M E C H A L L E N G E material Any jigsaw puzzle 31
A C T I V I T Y 6 : GEOMETER'SSKETCHPAD State when the GSP application is opened ' Straightedge Tool ' 32 TESELLATION IN Ever heard of Geometer's Sketchpad (GSP) ? Geometer's Sketchpad is a commercial interactive geometry software program for exploring Euclidean geometry, algebra, calculus and other areas of mathematics. Sketchpad gives students at all levels—from third grade through college—a tangible, visual way to learn math that increases their engagement, understanding, and achievement. Math becomes more meaningful and memorable when using Sketchpad. steps : N OW WE WI L L T R Y T O M A K E A B I R D T E S S E L L A T I O N P A T T E R N I N T H I S S K E T C H P A D ! 1) Open the GSP application, and click on the tools section labeled as 'Straightedge Tool'
2) Form an equilateral triangle make a straight line using the 'Straightedge Tool' Click on the straight line that has been created and double click on the marked point to make it as a center of the rotation and, Go to the options section above, click on 'Transform' and then click 'Rotate' Set rotation at 60 degree Repeat steps 1 to 3, using the marked points as shown. To create a complete equilateral triangle Step 4 : Step 1 : Step 3 : Step 2 : Label the triangle Step 5 : 33
3) Change side AB as shown, using 'Straightedge Tool' Select all visible points and next press ' Ctrl + H ' To hide all the points The picture on the side shows all the points that has been hidden 4) Rotate -60 degree about vertex A to side AC Select all lines AB that have been changed earlier and double click on the point A to make it as a center of the rotation Step 1 : Go to the options section above, click on 'Transform' and then click 'Rotate' Set rotation at -60 degrees Step 2 : Step 3 : 34
5) Construct Midpoint M on side BC Click on the BC side Go to the options section above, click on 'Construct' and then click 'Midpoint' 6) Change side BC as shown. M is the midpoint of side BC Select all visible points and next press ' Ctrl + H ' To hide all the points The picture on the side shows all the points that has been hidden 35
7) Rotate the changed line by 180 degrees at Midpoint M Select all lines AB that have been changed earlier and double click on the Midpoint M to make it as a center of the rotation Go to the options section above, click on 'Transform' and then click 'Rotate' Set rotation at 180 degrees Step 1 : Step 2 : 8) Hide all unnecessary detail of the bird Select all the lines and points as shown and then press 'Ctrl + H' To hide it 9) Add the final detail of the bird Add detail as shown using the 'Straightedge Tool' Select all visible points and next press ' Ctrl + H ' To hide all the points 36
7) Color the bird's body Mark a point along the line of the bird's body Select the points starting from the top of the bird's eyes to its tail Make sure the points are clicked in order and not skipped Press 'Ctrl + P' Select the rest of the points starting from the bottom of the bird's eyes to its nose Make sure the points are clicked in order and not skipped Press 'Ctrl + P' Select all visible points and next press ' Ctrl + H '. To hide all the points Step 1 : Step 5 : Step 4 : Step 3 : Step 2 : 37
8) Customize the body of the bird Select the entire image Go to the options section above, click on 'Display' and then click on 'Line Width' and choose the size of line thickness you like Select the inner part of the bird's body Go to the options section above, click on 'Display' and then click on 'Color' and choose any color that you like 9) Making the first generation of the tessellation Select the entire image and double click on the marked point as shown to make it as a center of the rotation Go to the options section above, click on 'Transform' and then click 'Rotate' Set rotation at 60 degrees 38
10) Customize the first generation of the tesellation Select the inner part of the bird's body as shown Go to the options section above, click on 'Display' and then click on 'Color' and choose any color that you like 11) Make a rotation on the first generation of the tessellation Select the entire image and double click on the marked point as shown to make it as a center of the rotation Go to the options section above, click on 'Transform' and then click 'Rotate' Set rotation at 120 degrees Just repeat steps 1 to 2, to get a complete tessellation pattern Step 1 : Step 2 : Step 3 : 39
THE FINALRESULTOF THE TESSELLATIONPROCESSTHATHAS BEENDONE INTHEGSPAPPLICATION! 40
CONCLUSION In conclusion, tessellation is a fascinating and creative concept in mathematics that involves covering a surface with repeating geometric shapes in a way that leaves no gaps or overlaps. It is a captivating exploration of symmetry, transformations, and geometric patterns. Tessellations come in different forms, including regular tessellations that use a single type of shape and semi-regular tessellations that combine multiple shapes to form repeating patterns. The arrangement of shapes at each corner where they meet is called the vertex configuration, which helps us understand the beautiful symmetries present in tessellations. Tessellations can be found in various aspects of art, architecture, and nature. They provide a unique platform for artists, mathematicians, and even children to explore their creativity and understand mathematical principles in an engaging way. Through tessellation, we discover the magic of symmetry and the beauty of patterns, making it a captivating field that continues to inspire both artistic expression and mathematical inquiry. So, the next time we see tessellations in our surroundings, we can appreciate the intricate designs and the mathematical wonder that lies beneath them. 41
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