Some questions we have as we begin our research on tangrams are as follows:
- What exactly are tangrams?
- How does one teach using tangrams in the mathematics classroom?
- Is using tangrams beneficial in the mathematics classroom?
Monday, March 26, 2007
Things we want to know about tangrams:
What is a tangram?
The tangram is among the most popular of all dissection puzzles that exist today. A tangram is an ancient, unique, Chinese puzzle that consists of seven (geometric) pieces: one square, five triangles and one parallelogram. When all pieces are put together, they form one big square, when they are sperated, they form what is called a tan. Of the five triangles there are two large, two small and one medium in size. "The large triangle is twice the area of the medium triangle. The medium triangle, the square, and the parallelogram are each twice the area of the small triangle. Each angle of the square measures 90 degrees. Each triangle contains a 90 degree and two 45 degree angles, which makes them isosceles right triangles. The parallelogram contains 45 degree and 135 degree angles" (Bohning, G., et al., 1997, p. 3).
The relationship among the pieces enables them to fit together to form many figures and arrangements. However, the tangram is more than a seven piece square. When it comes to tangrams, the challenge is to arrange the pieces to form additional shapes. The seven pieces can be arranged to make anything form a rabbit, to the alphabet, to a person. "The tangram is the opposite of a jigsaw puzzle. Instead of fitting the pieces together in only one way, the seven tangram pieces can be arranged to make a great number of different figures" (Bohning, G., et al., 1997, p. 4).
Picture taken from: www.psicoactiva.com/juegos/tangram/tangram2.jpg
The relationship among the pieces enables them to fit together to form many figures and arrangements. However, the tangram is more than a seven piece square. When it comes to tangrams, the challenge is to arrange the pieces to form additional shapes. The seven pieces can be arranged to make anything form a rabbit, to the alphabet, to a person. "The tangram is the opposite of a jigsaw puzzle. Instead of fitting the pieces together in only one way, the seven tangram pieces can be arranged to make a great number of different figures" (Bohning, G., et al., 1997, p. 4).
Here are some possible shapes that can be made using tangrams:
Picture taken from: www.psicoactiva.com/juegos/tangram/tangram2.jpg
Origin/History
While researching the origin of the tangram, it quickly became evident that this fact remains unknown to this day. The same goes for how the tangram got its name. Depending on the source, the history of the tangram will vary, and may include the following possibilities:
One such story claims that the tangram originated from a large, perfectly square glass frame that was ordered by a king. However, before it could be delivered to the king’s castle it was dropped. Surprisingly, it had not shattered into a thousand pieces, but had broken into seven perfect, geometric shapes. When they tired to put the pieces back in their original form, they realized they could make many other designs. They presented this to the king as a puzzle, and he was fascinated with it (Wichita State University Department of Mathematics and Statistics, 1999).
Although this is a fascinating story of how the tangram began, other sources claim a more modest origin. They believe that the tangram can be traced back to the Orient before the 18th century, and then spread westward (Archimedes' Labratory, 1997). While the tangram is often said to be ancient, its existence in the Western world has only been verified as far back as 1800. “Tangrams were brought to America by Chinese and American ships during the first part of the 19th century. The earliest example known was made of ivory in a silk box and was given to the son of an American ship owner in 1802” (Wikipedia: The Free Encyclopedia, 2007).
Others believe that the beginning of the tangram may be rooted in a yanjitu furniture set during a ruling period in China referred to as the Song Dynasty. Originally, the furniture set consisted of six rectangular tables. However, a seventh piece was added, triangular in shape, which allowed people to arrange the seven tables into a big square table. They later became a set of wooden blocks for playing (Wikipedia: The Free Encyclopedia, 2007).
According to Samuel Loyd, an American puzzle expert, the puzzle was invented 4,000 years ago by the God Tan. This was described in the Seven Books of Tan where each volume contained over 1,000 puzzles that illustrated the creation of the world and of species. The seven pieces were taken from the sun, the moon, and the five planets of Mars, Jupiter, Saturn, Mercury, and Venus. This elaborate story was later uncovered to be a hoax (Mathematics Fair, 2001).
Although there are many conflicting stories to how the tangram got it’s beginning, “The earliest mention of the tangram was found in a book dated in 1813 AD. At this time, the puzzle was already considered to be old” (Wichita State University Department of Mathematics and Statistics, 1999). Whatever the date the tangram was invented, rearrangement puzzle roots can be traced back to the 3rd century BC (Archimedes' Labratory, 1997)! Therefore, the tangram has without a doubt been around for a long time.
One such story claims that the tangram originated from a large, perfectly square glass frame that was ordered by a king. However, before it could be delivered to the king’s castle it was dropped. Surprisingly, it had not shattered into a thousand pieces, but had broken into seven perfect, geometric shapes. When they tired to put the pieces back in their original form, they realized they could make many other designs. They presented this to the king as a puzzle, and he was fascinated with it (Wichita State University Department of Mathematics and Statistics, 1999).
Although this is a fascinating story of how the tangram began, other sources claim a more modest origin. They believe that the tangram can be traced back to the Orient before the 18th century, and then spread westward (Archimedes' Labratory, 1997). While the tangram is often said to be ancient, its existence in the Western world has only been verified as far back as 1800. “Tangrams were brought to America by Chinese and American ships during the first part of the 19th century. The earliest example known was made of ivory in a silk box and was given to the son of an American ship owner in 1802” (Wikipedia: The Free Encyclopedia, 2007).
Others believe that the beginning of the tangram may be rooted in a yanjitu furniture set during a ruling period in China referred to as the Song Dynasty. Originally, the furniture set consisted of six rectangular tables. However, a seventh piece was added, triangular in shape, which allowed people to arrange the seven tables into a big square table. They later became a set of wooden blocks for playing (Wikipedia: The Free Encyclopedia, 2007).
According to Samuel Loyd, an American puzzle expert, the puzzle was invented 4,000 years ago by the God Tan. This was described in the Seven Books of Tan where each volume contained over 1,000 puzzles that illustrated the creation of the world and of species. The seven pieces were taken from the sun, the moon, and the five planets of Mars, Jupiter, Saturn, Mercury, and Venus. This elaborate story was later uncovered to be a hoax (Mathematics Fair, 2001).
Although there are many conflicting stories to how the tangram got it’s beginning, “The earliest mention of the tangram was found in a book dated in 1813 AD. At this time, the puzzle was already considered to be old” (Wichita State University Department of Mathematics and Statistics, 1999). Whatever the date the tangram was invented, rearrangement puzzle roots can be traced back to the 3rd century BC (Archimedes' Labratory, 1997)! Therefore, the tangram has without a doubt been around for a long time.
Why are tangrams important and interesting?
Tangrams are a great thing to incorporate into the mathematics classroom because they are fun, intersting, and meaningful. Tangrams "help students develop mathematical concepts of fractions, spatial awareness, geometry, area, and perimeter" (Rigdon, D., et al., 2000, p. 304.305). Because tangrams involve physical manipulatives as well as virtual manipulatives (online tangram activities), this caters to a variety of learning styles. Students who may learn better with maniuplitives or through the use of computers will find tangrams to be an experience in mathematics that they might not otherwise have. Tangrams present a new, interesting, hands- on way to deal with topics that most often are or can be quite boring and meaningless.
Not only are tangrams a great way to help make geometry, fractions, etc. engaging, they also help make it authentic. For example, they "can be arranged to make animals, birds, sea creatures, people and other figures" (Bohning, G., et al., 1997, p. 83-87). In mathematics, it is very important that children explore with a hands-on, minds-on attack in a problem solving environment. Tangrams promote this idea through open-ended explorations. Through tangrams they are involved in manipulating and problem solving.
By using tangram shapes, children learn the relationships between shapes. Additionally, children learn that three basic shapes, the triangle, square, and parallelogram, can fit together to form many other shapes and figures.
When learning a new concept, it is important to interact with multiple representations of the same idea and be able to translate from one to the other. Critics of direct methods say that teaching things “in isolation from how they are applied, diminishes learners’ problem solving and reasoning skills,” (Roblyer, M.D., 2003, chap. 3), therefore through tangrams, children are interacting with mathematical concepts in a new way, which helps to make the learning deeper.
Not only are tangrams a good way for children to explore mathematics, children actullay find them intersting and like to invent their very own designs. This is very important bacause if the child is interested then half of the battle of reaching them is accomplished. Children realize that from geometric shapes they can make things they see in everyday life such as dogs, cats etc. The great thing about tangrams is that not all children will make for example a dog in the same way, and that is fine because with tangrams nothing is "set in stone." This not only helps to promote divergent thinking, but it encourages children to take risks because they know that there isn't just one correct way of doing it.
Another reason why tangrams are important and intersting is that "Tangrams have both geometric and artistic features. Children gain geometric insights as they discover and discuss the relationships among the tangram pieces and what they can represent" (Bohning, G., et al., 1997, p. 83-87). Children are naturally curious so trying to solve a puzzle involving tangrams is going to provoke their curiousity and therefore be interesting so they will be engaged in what they are doing. Also, tangrams promote growth in learning, for example, when children become experienced buliding things from tangrams, they can be challenged even more by experimenting with double tangrams, etc.
Not only are tangrams a great way to help make geometry, fractions, etc. engaging, they also help make it authentic. For example, they "can be arranged to make animals, birds, sea creatures, people and other figures" (Bohning, G., et al., 1997, p. 83-87). In mathematics, it is very important that children explore with a hands-on, minds-on attack in a problem solving environment. Tangrams promote this idea through open-ended explorations. Through tangrams they are involved in manipulating and problem solving.
By using tangram shapes, children learn the relationships between shapes. Additionally, children learn that three basic shapes, the triangle, square, and parallelogram, can fit together to form many other shapes and figures.
When learning a new concept, it is important to interact with multiple representations of the same idea and be able to translate from one to the other. Critics of direct methods say that teaching things “in isolation from how they are applied, diminishes learners’ problem solving and reasoning skills,” (Roblyer, M.D., 2003, chap. 3), therefore through tangrams, children are interacting with mathematical concepts in a new way, which helps to make the learning deeper.
Not only are tangrams a good way for children to explore mathematics, children actullay find them intersting and like to invent their very own designs. This is very important bacause if the child is interested then half of the battle of reaching them is accomplished. Children realize that from geometric shapes they can make things they see in everyday life such as dogs, cats etc. The great thing about tangrams is that not all children will make for example a dog in the same way, and that is fine because with tangrams nothing is "set in stone." This not only helps to promote divergent thinking, but it encourages children to take risks because they know that there isn't just one correct way of doing it.
Another reason why tangrams are important and intersting is that "Tangrams have both geometric and artistic features. Children gain geometric insights as they discover and discuss the relationships among the tangram pieces and what they can represent" (Bohning, G., et al., 1997, p. 83-87). Children are naturally curious so trying to solve a puzzle involving tangrams is going to provoke their curiousity and therefore be interesting so they will be engaged in what they are doing. Also, tangrams promote growth in learning, for example, when children become experienced buliding things from tangrams, they can be challenged even more by experimenting with double tangrams, etc.
Rules/Making Your Own
Classic Rules:- All seven pieces must be used.
- All pieces must be flat.
- All pieces must touch.
- No pieces may overlap.
- Pieces may be rotated and/or flipped to form the desired shape.
Constructing Your Own Set of Tans: Easy Tan Construction:
Use a square the size you want the finished puzzle to be. It can be a square of the material you would like your set to be made of or, if more convenient, a paper template to transfer the design. Draw a four by four grid on the material as shown in the picture. This will scale up or down for any size square, the four squares by four squares part is the important thing here. You then mark off the blue lines as shown. Cut your material carefully along these blue lines. This will produce the seven tan pieces; five triangles, one square and one parallelogram. As noted above, slicing rather than sawing will produce the best result. Enjoy!
All taken from, (Wikipedia: The Free Encyclopedia, 2007).
Teaching Tangrams - Journal Articles
Title: Tackling Tangrams
Authors: Deanna Rigdon; Jolyn Raleigh; Shari Goodman
Journal: Teaching Children Mathematics 6 no5 304-5 Ja 2000
In this article, you will find many interactive and creative activities involving tangrams that can be used with students from K-6. One such activity called “Name that Tangram” gives students a riddle and requires them to guess the geometric shape described. For example: I am one of five similar pieces. I have three sides. None of the pieces are the same as I am. What tangram piece am I? This activity appeals directly to students. Also, no solutions are suggested so that students will look to themselves as the mathematical authority, thereby developing confidence to validate their work.
Title: An Old Tale with a New Turn--and Flip and Slide
Author: Pat Margerm
Jornal: Teaching Children Mathematics 6 no2 86-90 O 1999
This article describes how one can use literature to teach mathematics. This article outlines such a lesson involving tangrams and the story the Three Pigs, One Wolf, and Seven Magic Shapes. By using this story in the classroom, children are exposed to the concept of tangrams through authentic situations. They are able to create tangram shapes and discuss and write about their mathematical transformations. This is an excellent activity to engage students in mathematical learning, as well as increase their enthusiasm toward this particular subject area. There are also numerous extension activities provided at the end of this article that incorporate concepts such as patterns, measurement, and transformational geometry.
Title: Are You Puzzled?
Author: Rosamond Welchman
Journal: Teaching Children Mathematics 5 no7 412-15 Mr '99
This article describes a lesson where students are involved in working with and comparing two different seven-piece puzzles- the classic tangram puzzle and the seven-piece mosaic puzzle. After children are encouraged to explore each puzzle, they will then compare the two puzzles by constructing various triangles. For example, children are asked to predict whether more triangles will be found with the seven-piece mosaic puzzle than with the tangram puzzle by looking at characteristics such as size, pieces used, orientation on paper etc. Extension activities include making all possible rectangles or squares, and exploring area, angles, lengths, and polygons. By “playing” with puzzles, students discover many of the principles of geometry while enhancing their higher-order thinking skills and having fun while doing so.
Title: Quilts and Tangrams: Linking Literature and Geometry
Authors: Gerry Bohning and Rebecca Williams
Journal: Childhood Education v73 p83-7 Wint '96-'97
In this article, students link literature to geometry by making quilt patterns using tangram pieces. This lesson gives step by step instructions ranging from introducing tangrams, to when to use quilt literature, to teaching students how to use their tangram pieces to create numerous quilt designs. There are several books listed at the end of this article that can be read to compliment this lesson. This is a very beneficial activity because it provides an authentic learning experience where children can apply what they have learned about tangrams to a popular pastime that has been around for centuries.
*To access any of the above articles, go to http://www.library.mun.ca/ and use your off campus login!
Authors: Deanna Rigdon; Jolyn Raleigh; Shari Goodman
Journal: Teaching Children Mathematics 6 no5 304-5 Ja 2000
In this article, you will find many interactive and creative activities involving tangrams that can be used with students from K-6. One such activity called “Name that Tangram” gives students a riddle and requires them to guess the geometric shape described. For example: I am one of five similar pieces. I have three sides. None of the pieces are the same as I am. What tangram piece am I? This activity appeals directly to students. Also, no solutions are suggested so that students will look to themselves as the mathematical authority, thereby developing confidence to validate their work.
Title: An Old Tale with a New Turn--and Flip and Slide
Author: Pat Margerm
Jornal: Teaching Children Mathematics 6 no2 86-90 O 1999
This article describes how one can use literature to teach mathematics. This article outlines such a lesson involving tangrams and the story the Three Pigs, One Wolf, and Seven Magic Shapes. By using this story in the classroom, children are exposed to the concept of tangrams through authentic situations. They are able to create tangram shapes and discuss and write about their mathematical transformations. This is an excellent activity to engage students in mathematical learning, as well as increase their enthusiasm toward this particular subject area. There are also numerous extension activities provided at the end of this article that incorporate concepts such as patterns, measurement, and transformational geometry.
Title: Are You Puzzled?
Author: Rosamond Welchman
Journal: Teaching Children Mathematics 5 no7 412-15 Mr '99
This article describes a lesson where students are involved in working with and comparing two different seven-piece puzzles- the classic tangram puzzle and the seven-piece mosaic puzzle. After children are encouraged to explore each puzzle, they will then compare the two puzzles by constructing various triangles. For example, children are asked to predict whether more triangles will be found with the seven-piece mosaic puzzle than with the tangram puzzle by looking at characteristics such as size, pieces used, orientation on paper etc. Extension activities include making all possible rectangles or squares, and exploring area, angles, lengths, and polygons. By “playing” with puzzles, students discover many of the principles of geometry while enhancing their higher-order thinking skills and having fun while doing so.
Title: Quilts and Tangrams: Linking Literature and Geometry
Authors: Gerry Bohning and Rebecca Williams
Journal: Childhood Education v73 p83-7 Wint '96-'97
In this article, students link literature to geometry by making quilt patterns using tangram pieces. This lesson gives step by step instructions ranging from introducing tangrams, to when to use quilt literature, to teaching students how to use their tangram pieces to create numerous quilt designs. There are several books listed at the end of this article that can be read to compliment this lesson. This is a very beneficial activity because it provides an authentic learning experience where children can apply what they have learned about tangrams to a popular pastime that has been around for centuries.
*To access any of the above articles, go to http://www.library.mun.ca/ and use your off campus login!
Teaching Tangrams - Online
http://www.lessonplanspage.com/MathLACreateExploreManipulateTangrams4.htm
This is a three-day lesson plan suitable for grade four students. The first lesson involves introducing students to tangrams, getting them to create their own set of tangram pieces, and using these to explore. Lesson two gives students the opportunity to compare and sort their tangram pieces according to the number of sides, and classify them by their appropriate name. They also construct various shapes using the tangram pieces. For example, creating a triangle using two shapes etc. In the last lesson, the teacher reads the story Grandfather Tang by Ann Tompert. After this has been completed, the students manipulate their tangram pieces in order to make the objects they have encountered throughout the story. Assessment for all three lessons is provided as well.
http://mathforum.org/trscavo/tangrams/area.html
The goal of this lesson is for students to determine the area of their tangram pieces without using a particular formula. Students work with their tangram pieces to investigate various scenarios stemming from the first situation: Let’s suppose this square has an area of one square unit. From this, students are required to answer questions similar to the following: Make a square with the two small congruent triangles. What is the area of this square? How do you know? To extend this lesson, the area of the small square can be changed to two square units.
http://mathforum.org/trscavo/tangrams/activities.html
Building on what was learned from the previous lesson, students can compute the area of any polygon constructed from their tangram pieces. This lesson also incorporates the concept of congruency, and is slightly more advanced, allowing students to work with more elaborate geometric shapes. For example: Use a parallelogram and the two small congruent triangles to make a rectangle. It also presents students with a certain shape they must construct, but gives them a restriction such as: Construct a triangle congruent to the large triangle shown without using the square. This is a great way to develop problem-solving skills. A variety of extension activities are outlined also.
http://mathforum.org/paths/fractions/frac.tangram.html
This lesson was developed for grade five and six students and combines both tangrams and fractions. Students are asked which part of the whole square is the large right triangle, the square, the parallelogram etc. Evaluation and follow-up activities are given as well. This lesson can be made more challenging, by changing the unit whole. For example, choosing the large right triangle and saying it is now one unit in area.
http://www.learnnc.org/lessons/AltaAllen322003401
This is a three-five day lesson plan revolving around the Grandfather Tang story by Ann Tompert. In this lesson, children not only make the objects they encounter throughout the story, they are also given the opportunity to work with all seven of their tangram pieces to construct animals found in nature. Language arts is integrated into this lesson as well, as students are required to write an acrostic poem using their animals name. Further more, this lesson incorporates technology. Students are given the opportunity to explore creating their tangram animals using Kid Pix Studio Deluxe. Assessment, supplementary information, as well as related websites are provided.
http://www.readinga-z.com/newfiles/levels/lesson_plans/o/listangram/listangram_print.html#before
While this lesson is not directly linked to mathematics, the concept of tangrams is addressed through the beautiful story of Li’s Tangram Animals. Although the main purpose of this lesson is to assess objectives such as vocabulary, comprehension etc., it brings math into the language arts classroom, which you do not often see. The illustrations and diagrams support the text and introduce readers to basic geometric concepts. Therefore, through using this book in the classroom, the transition into using tangrams in math is made easier. This book can also be integrated into the art and social studies curriculum.
This is a three-day lesson plan suitable for grade four students. The first lesson involves introducing students to tangrams, getting them to create their own set of tangram pieces, and using these to explore. Lesson two gives students the opportunity to compare and sort their tangram pieces according to the number of sides, and classify them by their appropriate name. They also construct various shapes using the tangram pieces. For example, creating a triangle using two shapes etc. In the last lesson, the teacher reads the story Grandfather Tang by Ann Tompert. After this has been completed, the students manipulate their tangram pieces in order to make the objects they have encountered throughout the story. Assessment for all three lessons is provided as well.
http://mathforum.org/trscavo/tangrams/area.html
The goal of this lesson is for students to determine the area of their tangram pieces without using a particular formula. Students work with their tangram pieces to investigate various scenarios stemming from the first situation: Let’s suppose this square has an area of one square unit. From this, students are required to answer questions similar to the following: Make a square with the two small congruent triangles. What is the area of this square? How do you know? To extend this lesson, the area of the small square can be changed to two square units.
http://mathforum.org/trscavo/tangrams/activities.html
Building on what was learned from the previous lesson, students can compute the area of any polygon constructed from their tangram pieces. This lesson also incorporates the concept of congruency, and is slightly more advanced, allowing students to work with more elaborate geometric shapes. For example: Use a parallelogram and the two small congruent triangles to make a rectangle. It also presents students with a certain shape they must construct, but gives them a restriction such as: Construct a triangle congruent to the large triangle shown without using the square. This is a great way to develop problem-solving skills. A variety of extension activities are outlined also.
http://mathforum.org/paths/fractions/frac.tangram.html
This lesson was developed for grade five and six students and combines both tangrams and fractions. Students are asked which part of the whole square is the large right triangle, the square, the parallelogram etc. Evaluation and follow-up activities are given as well. This lesson can be made more challenging, by changing the unit whole. For example, choosing the large right triangle and saying it is now one unit in area.
http://www.learnnc.org/lessons/AltaAllen322003401
This is a three-five day lesson plan revolving around the Grandfather Tang story by Ann Tompert. In this lesson, children not only make the objects they encounter throughout the story, they are also given the opportunity to work with all seven of their tangram pieces to construct animals found in nature. Language arts is integrated into this lesson as well, as students are required to write an acrostic poem using their animals name. Further more, this lesson incorporates technology. Students are given the opportunity to explore creating their tangram animals using Kid Pix Studio Deluxe. Assessment, supplementary information, as well as related websites are provided.
http://www.readinga-z.com/newfiles/levels/lesson_plans/o/listangram/listangram_print.html#before
While this lesson is not directly linked to mathematics, the concept of tangrams is addressed through the beautiful story of Li’s Tangram Animals. Although the main purpose of this lesson is to assess objectives such as vocabulary, comprehension etc., it brings math into the language arts classroom, which you do not often see. The illustrations and diagrams support the text and introduce readers to basic geometric concepts. Therefore, through using this book in the classroom, the transition into using tangrams in math is made easier. This book can also be integrated into the art and social studies curriculum.
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