Tangram shapes to make

Tangrams: Geometry Hands-On Math Manipulatives

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Learning About Tangrams

The Tangram is a deceptively simple set of seven geometric shapes made up of five triangles (two small triangles, one medium triangle, and two large triangles), a square, and a parallelogram. When the pieces are arranged together they suggest an amazing variety of forms, embodying many numerical and geometric concepts. Tangram pieces are widely used to solve puzzles that require the making of a specified shape using all seven pieces. Tangram sets come in in four colors—red, green, blue, and yellow. The three different-size Tangram triangles are all similar, right isosceles triangles. Thus, the triangles all have angles of 45°, 45°, and 90°, and the corresponding sides of these triangles are in proportion.

Another interesting aspect of the Tangram set is that all of the Tangram pieces can be completely covered with small Tangram triangles.

Hence, it is easy to see that all the angles of the Tangram pieces are multiples of 45—that is, 45°, 90°, or 135°, and that the small Tangram triangle is the unit of measure that can be used to compare the areas of the Tangram pieces. Since the medium triangle, the square, and the parallelogram are each made up of two small Tangram triangles, they each have an area twice that of the small triangle. The large triangle is made up of four small Tangram triangles and thus has an area four times that of the small triangle and twice that of the other Tangram pieces.

Another special aspect of the pieces is that all seven fit together to form a square.

Some students can find the making of Tangram shapes to be very frustrating, especially if they are used to being able to do math by following rules and algorithms. For such students, you can reduce the level of frustration by providing some hints. For example, you can put down a first piece, or draw lines on an outline to show how pieces can be placed. However, it is important to find just the right level of challenge so that students can experience the pleasure of each Tangram investigation. Sometimes, placing some Tangram pieces incorrectly and then modeling an exploratory approach such as the following may make students feel more comfortable: "I wonder if I could put this Tangram piece this way. I guess not, because then nothing else can fit here. So I'd better try another way ..."

Tangrams are a good tool for developing spatial reasoning and for exploring fractions and a variety of geometric concepts, including size, shape, congruence, similarity, area, perimeter, and the properties of polygons. Tangrams are especially suitable for students’ independent work, since each student can be given a set for which he or she is responsible. However, since students vary greatly in their spatial abilities and language, time should also be allowed for group work, and most students need ample time to experiment freely with Tangrams before they begin more serious investigations.

Young students will at first think of their Tangram shapes literally. With experience, they will see commonalities and begin to develop abstract language for aspects of patterns within their shapes. For example, students may at first make a square simply from two small triangles. Yet eventually they may develop an abstract mental image of a square divided by a diagonal into two triangles, which will enable them to build squares of other sizes from two triangles.

Tangrams can also provide a visual image essential for developing an understanding of fraction algorithms. Many students learn to do examples such as ¹⁄2 = ^?⁄8 or ¹⁄4 + ¹⁄8 + ¹⁄16 = ? at a purely symbolic level so that if they forget the procedure, they are at a total loss. Students who have had many presymbolic experiences solving problems such as "Find how many small triangles fill the large triangles," or "How much of the full square is covered by a small, a medium, and a large triangle?" will have a solid intuitive foundation on which to build these basic skills and to fall back on if memory fails them.

Young students have an initial tendency to work with others, and to copy one another's work. Yet, even duplicating someone else's Tangram shape can expand a student’s experience, develop the ability to recognize similarities and differences, and provide a context for developing language related to geometric ideas. Throughout their investigations, students should be encouraged to talk about their constructions in order to clarify and extend their thinking. For example, students will develop an intuitive feel for angles as they fit corners of Tangram pieces together, and they can be encouraged to think about why some pieces will fit in a given space and others won't. Students can begin to develop a perception of symmetry as they take turns mirroring Tangram pieces across a line placed between them on a mat and can also begin to experience pride in their joint production.

Students of any age who haven't seen Tangrams before are likely to first explore shapes by building objects that look like objects—perhaps a butterfly, a rocket, a face, or a letter of the alphabet. Students with a richer geometric background are likely to impose interesting restrictions on their constructions, choosing to make, for example, a filled-in polygon, such as a square or hexagon, or a symmetric pattern.

The use of Tangrams provides a perfect opportunity for authentic assessment. Watching students work with Tangram pieces gives you a sense of how they approach a mathematical problem. Their thinking can be "seen," in that thinking is expressed through their positioning of the Tangram pieces, and when a class breaks up into small working groups, you are able to circulate, listen, and raise questions, all the while focusing on how individuals are thinking.

To ensure that students know not only how to do a certain operation but also how it relates to a model, assessment should include not only symbolic pencil-and-paper tasks such as "Find ¹⁄2 + ¹⁄8," but also performance tasks such as "Show why your answer is correct using Tangram pieces. "

Having students describe their creations and share their strategies and thinking with the whole class gives you another opportunity for observational assessment. Furthermore, since spatial thinking plays an important role in students’ intellectual development, include in your overall assessment some attention to spatial tasks.

Lessons

The Tangram’s geometry applications are numerous. In addition to creating shapes, younger children can work to fit Tangram pieces into shapes on puzzle cards. Children can gain familiarity with geometric figures while developing spatial reasoning abilities. Spatial tasks may be difficult for some children, but they will improve with experience.

Here are two activities for getting to know Tangrams and to begin using them to learn and understand the math that they represent.

Activity	Curriculum Strands	Topics
Cover and Count In this game for two players, children fit Tangram pieces on a dotted grid in an effort to cover as many dots as possible.	Measurement Number Geometry	Counting Addition Congruence Game strategies
Making a Quilt Children make paper quilts based on quilt-square patterns they design using Tangram triangles.	Geometry Patterns	Spatial visualization Looking for patterns Transformational geometry Congruence

The intermediate grades can use Tangram pieces to create squares, triangles, parallelograms, rectangles, and trapezoids. Students can not only explore and create shapes, they can also learn about relationships among shapes.

Here are two activities for getting to know Tangrams and to begin using them to learn and understand the math that they represent.

Activity	Curriculum Strands	Topics
Fraction Spin Students identify a Tangram piece as a fractional part of a whole shape. Then they construct the whole.	Geometry Number	Spatial visualization Fractions Counting Comparing
Polygon Parade Students build polygon shapes using different numbers of Tangram pieces. They record their results on a shape chart.	Geometry	Spatial visualization Comparing Classifying

Tangrams help students at the middle grades develop spatial-visualization skills and introduce or reinforce geometric concepts such as congruence, similarity, symmetry, etc. Students use them to compute the area of polygons, or Tangrams can be used to introduce the terms “congruent” and “similar.”

Here are two activities for getting to know Tangrams and to begin using them to learn and understand the math that they represent.

Activity	Curriculum Strands	Topics
Square Cover-Up Students cover a grid with Tangram pieces and find the percent of the grid that is covered.	Number Measurement	Percents Estimation Area
The More, The Better Students search for all the convex Tangram shapes that can be made with different numbers of pieces that represent the same fractional part of the seven-piece Tangram square.	Geometry Number	Comparing Classifying Polygons Fractional equivalence Spatial visualization

Skills Taught

Subjects Taught

Educational tips and a printable template

Tangrams for kids: A learning tool for building STEM skills

Like building blocks, tangrams can teach kids about spatial relationships. They may help kids learn geometric terms, and develop stronger problem solving abilities. They might even help children perform better on tests of basic arithmetic. But what is a tangram?

Invented in China approximately 200 years ago, a tangram is a two-dimensional re-arrangement puzzle created by cutting a square into seven pieces — seven geometric shapes called “tans” (Slocum et al 2003).

What are the seven shapes in a tangram? Each tangram puzzle contains the following:

2 large right triangles
1 medium-sized right triangle
2 small right triangles
1 small square
1 parallelogram

Arranged correctly, these tangram shapes can be fitted together as a large square, rectangle, or triangle. They can also be arranged in a variety of complex shapes, including fanciful ones.

There are many ways to play with tangrams. The simplest way is to let kids create their own complex shapes. But traditionally, tangrams are treated as puzzles.

The player is shown a target shape in outline, so that the “seams” between the composite tans are concealed. Then the player attempts to recreate the shape using the seven pieces.

In essence, it’s an exercise similar to structured block play, where the challenge is to create an exact copy of a structure depicted in a diagram. But there’s a key difference.

In structured block play, the diagram provides you with explicit, visual information about where each piece goes. In a tangram puzzle, you’re left to figure that our for yourself.

As noted by Zhen Yuan and his colleagues, solving tangram puzzles appears to activate parts of the brain associated with creative thinking and trial-and-error problem-solving (Hu et al 2019).

The educational benefits of tangrams

When we play with tans, we consider the shapes from a variety of angles and perspectives. How would the shapes look if we stuck them together? Rotated them? Slide them around into different positions?

Experiments suggest that thinking about such things — visualizing the spatial relationships between shapes in your “mind’s eye” — can boost our visual-spatial skills.

Research hints that it can boost arithmetic performance, too. When Yi Ling Cheng and Kelly Mix asked kids, aged 6-8, to perform a series of tangram-like spatial tasks, the practice session seemed to prime the brain for mathematics.

Kids who spent 40 minutes solving shape rotation puzzles performed better on a pencil-and-paper arithmetic test immediately thereafter. Compared to tangram-like activities, crossword puzzle warm-ups had no such effect (Cheng and Mix 2012).

So there is reason to suspect that playing with tangrams has educational benefits, and many educators recommended their use in the classroom (Bohning and Althouse 1997; Krieger 1991; National Council of Teacher’s Mathematics 2003; Clements and Sarama 2014).

In particular, these teachers argue that playing with tangrams may help kids

classify 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
acquire a precise vocabulary for manipulating shapes (e.g., “flip,” “rotate”)
learn the meaning of congruence

In addition, researchers have argued that tangram play can be an opportunity for kids move away from simplistic (and incorrect) ideas about shapes. Young children, for example, might struggle to explain what features are necessary for a shape to be a square. With guided questioning, a teacher can help kids determine why some attempts to make a square with tangrams are unsuccessful (Nic Mhuirí and Kelly 2021).

Children can also learn new vocabulary to describe the shapes they assemble (e.g., rhombus, trapezoid, hexagon). And Tom Scovo demonstrates how tangrams can help kids calculate areas without formulas. For the details, see his excellent activities using tangrams for kids in grades 4-6.

What about free play with tangrams? That’s good, too. But take note: Kids probably stand to learn much more if their play includes discussion and teamwork.

Indeed, studies indicate that spatial play — like playing with tangrams — is particularly educational when kids interact with others who use spatial vocabulary.

For example, toddlers tend to develop bigger spatial vocabularies if, during play, their parents expose them to a greater range of spatial words, like “triangle” and “line.” As I explain elsewhere, such kids are also more likely to develop stronger spatial skills.

So it’s likely we can boost the value of tangrams by actively engaging kids in conversation. We can introduce them to words they might not know, like “angle” and “parallelogram.”

It’s also likely that kids will benefit from being asked to make predictions and explain tactics.

Research suggests that kids are more likely to master concepts when they explain them to others. What will happen if you rotate the triangle? What will happen if you flip the parallelogram? How must we move this shape in order to make it fit?

If we can get children to explain their ideas, we may help them improve their spatial skills and comprehension of geometry (Lee et al 2009; Clements and Sarama 2014).

So while kids may benefit from solitary play with tangrams, some of the best educational experiences may arise from playing with a (talkative) partner.

Any other teaching tips?

Tangrams offer kids an excellent opportunity to test out different geometric manipulations, and become familiar with the properties of the shapes they use.

But notice the triangles– big, medium, and small — are all the same shape. They represent a special kind of right triangle — an isosceles triangle with two 45-degree angles, and one 90-degree angle. And if you put together two triangles of the same size, you can make a square.

These properties aren’t found in all triangles. On the contrary! But it’s easy for children to come to that conclusion if they don’t get exposed to a variety of triangles — equilateral, isosceles, and scalene. So it’s important to expose kids to that variety, and call their attention to the ways in which triangles can differ (Clement and Sarama 2014).

Getting started: How to make a tangram

You can make your own tangrams by following the instructions on Tom Scovo’s site, or by printing and cutting out the tangram shapes below.

If you right-click on the image here, you can save it on your computer for printing. Alternatively, after right-clicking, you can open the image in a new tab, and use your browser’s print function.

Looking for more durable tangrams? You can buy some. Classic Tangoes includes two plastic tangram sets and a deck of puzzle cards. But the corners are a bit sharp. You can also find tangrams made from other materials, such as wood, foam, and thick cardboard.

Tangrams for young children

My favorite introduction to tangrams for younger kids is the book Three pigs, one wolf, seven magic shapes by Grace Maccarone. Unfortunately, it’s out of print, but you might be able to pick up a copy used. It includes story (based on the folk tale of the three little pigs), a teaching guide, a set of tangrams to cut out, and some activities created by a math teacher. Although the publisher recommended this book for kids in grades 1-2, the book can be enjoyed by preschoolers.

For another, fanciful story featuring tangrams, see Grandfather Tang’s Story (Dragonfly Books).

Virtual tangrams for kids

You might wonder if computer games are as educational as playing with real, physical tangrams. The National Council of Teachers of Mathematics (NCTM) recommends both.

According to the NCTM, computer games may offer special benefits because “the computer environment is likely to encourage [kids] to think about how they need to manipulate the tangram pieces rather than approach the task mainly by trial and error. ”

Read more and try out this online game by the National Council of Teachers of Mathematics.

Ideas for tangram activities

The classic approaches are either (1) to present kids with images of target shapes (and have them attempt to reproduce those shapes with their tangrams), or (2) to encourage kids to be creative, and come up with shapes of their own.

But there are additional games and activities to try as well. You can choose a theme (like “fish” “spaceships” or “houses”) and see how many different examples of each type kids can come up with. For instance, how could challenge kids to create a series of cats.

References: Tangrams for kids

Bohning G and Althouse JK. 1997. Using tangrams to teach geometry to young children. Early childhood education journal. 24(4): 239-242.

Cheng Y-L and Mix KS. 2012. Spatial training improves children’s mathematics ability. Journal of Cognition and Development. Published online DOI:10.1080/15248372.2012.725186.

Clements D and Sarama J. 2014. Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.

Hu Z, Lam KF, Yuan Z. 2019. Effective Connectivity of the Fronto-Parietal Network during the Tangram Task in a Natural Environment. Neuroscience. 422:202-211.

Kriegler S. 1991. The Tangram: It’s More than an Ancient Puzzle. Arithmetic Teacher 38(9) 38-43.

Lee J, Lee JO, and Collins C. 2009. Enhancing children’s spatial sense using tangrams. Childhood Education 86(2):92-94.

National Council of Teacher’s Mathematics. 2003. Developing geometry understandings and spatial skills through puzzlelike problems with tangrams: Tangram challenges. www.nctm.org.

Nic Mhuirí S and Kelly D. 2021. Making squares: children’s responses to a tangram task. In: Eighth Conference on Research in Mathematics Education in Ireland (MEI 8), 15-16 Oct 2021, Dublin, Ireland (Online).

Slocum J, Botermans J, Gebhardt D, Ma M, Ma X, Raizer H, Sonnevald D, and Van Splunteren C. 2003. Tangrams: The world’s first puzzle craze. Sterling: New York.

Title image of child’s hand’s manipulating tangrams by Gwen Dewar

Line drawings of triangles and tangrams by Parenting Science

Image of many different tangram cats by Sveta_Aho / istock

Content of “Tangrams for kids” last modified 9/2022. Portions of the text derive from an earlier version of the article, written by the same author.

DIY Tangram (game schemes, figures)

: Category: Crafts from different materials

Tangram - an old oriental puzzle of figures obtained by cutting a square into 7 parts in a special way: 2 large triangles, one medium, 2 small triangles, a square and a parallelogram. As a result of folding these parts with each other, flat figures are obtained, the contours of which resemble all kinds of objects, ranging from humans, animals and ending with tools and household items. These types of puzzles are often referred to as "geometric construction sets", "cardboard puzzles" or "cut puzzles".

With a tangram, a child will learn to analyze images, highlight geometric shapes in them, learn to visually break an entire object into parts, and vice versa - to compose a given model from elements, and most importantly - to think logically.

How to make a tangram

A tangram can be made from cardboard or paper by printing out a template and cutting along the lines. You can download and print the tangram square diagram by clicking on the picture and selecting "print" or "save picture as...".

Also available without template. We draw a diagonal in a square - we get 2 triangles. Cut one of them in half into 2 small triangles. We mark the middle on each side of the second large triangle. We cut off the middle triangle and the rest of the figures at these marks. There are other options for how to draw a tangram, but when you cut it into pieces, they will be exactly the same.

A more practical and durable tangram can be cut from a hard office folder or a plastic DVD box. You can complicate your task a little by cutting out tangrams from pieces of different felt, overcasting them around the edges, or even from plywood or wood.

How to play tangram

Each piece of the game must be made up of seven parts of the tangram, and they must not overlap.

The easiest option for preschool children 4-5 years old is to assemble figures according to diagrams (answers) drawn into elements, like a mosaic. A little practice, and the child will learn to make figures according to the contour pattern and even invent their own figures according to the same principle.

Level one - download and print a colored tangram, so it will be easier to navigate the diagram.

Schemes and figures of the game Tangram

This is such a multifunctional tangram.

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Do-it-yourself tangram puzzle

Tangram - "seven boards of skill" is an ancient Chinese puzzle consisting of seven flat figures that are folded in a certain way to obtain another, more complex figure (depicting a person, an animal, an object household items, letter or number, etc. ). The figure to be obtained is usually specified in the form of a silhouette or an external contour. When solving the puzzle, two conditions must be met: first, all seven tangram figures must be used, and second, the figures must not overlap.

There are several legends of the origin of of this Chinese puzzle, this one is my favorite. It can be told to a child when meeting with a tangram:

How three wise men came up with "Shi-Chao-Chu".
Almost two and a half thousand years ago, a long-awaited son and heir was born to an elderly emperor of China. Years passed. The boy grew up healthy and quick-witted beyond his years. One thing bothered the old emperor: his son, the future ruler of a vast country, did not want to study. It gave the boy more pleasure to play with toys all day long. The emperor called three wise men to himself, one of whom was known as a mathematician, the other became famous as an artist, and the third was a famous philosopher, and ordered them to come up with a game, having fun with which, his son would comprehend the beginnings of mathematics, learned to look at the world around him with the gaze of an artist , would become patient, like a true philosopher, and would understand that often complex things are made up of simple things. Three wise men came up with "Shi-Chao-Chu" - a square cut into seven parts.

Tangram is an exciting and interesting puzzle that a child can play from 2-3 years old. Playing with a tangram contributes to the development in children of the ability to play by the rules and follow instructions, visual-figurative thinking, imagination, attention, understanding of color, size and shape, perception, combinatorial abilities.

To begin with, the child should be introduced to the tangram, let him examine them. Then teach him to make the simplest figures from 2-4 parts of the tangram, for example, a house, a Christmas tree, a square.

Further development of the game tangram takes place in 4 stages:

1. Compilation of figures by overlaying on the tangram scheme.
The child is given a diagram of a tangram figure, he must put all the elements of the tangram on it.

2. Making figures according to the model.
The child is given a diagram of a tangram, but he must already put together a figure from the elements next to it on the table.

3. Compilation of figures according to the contour image.
The child must fold the tangram figure along its contour.

4. Compiling figures according to one's own design.
At this stage, the child independently invents figures for drawing up, which develops the imagination very well.

Tangram for the game is easy to make with your own hands. You can use different materials for this: cut out from cardboard, cut out from plywood or planks, cut out from boxes for disks or folders with hard covers. Moreover, you can make parts both of the same color and different.

Here are template for cutting tangram.

Schemes for different stages of tanking training can be found on our website according to the following links:

1 level - Tangram scheme - animals

1 level - tanrama scheme - items

2 level - tangram scheme - silhouettes

Interesting interesting :
Writer and mathematician Lewis Carroll is considered a tangram enthusiast.

Tangram shapes to make

Tangrams: Geometry Hands-On Math Manipulatives

Learning About Tangrams

Lessons

Skills Taught

Subjects Taught

Educational tips and a printable template

The educational benefits of tangrams

Any other teaching tips?

Getting started: How to make a tangram

Tangrams for young children

Virtual tangrams for kids

Ideas for tangram activities

More to read about the development of STEM skills

References: Tangrams for kids

DIY Tangram (game schemes, figures)

How to make a tangram

How to play tangram

Schemes and figures of the game Tangram

Do-it-yourself tangram puzzle

Learn more