Cyprus : Tangram, The link between mathematics and art
Tangrams have remained popular for so many years partly because they are so simple and at the same time so complex. In other words, because the individual. This article is about the application of tangrams in teaching mathematical concepts. . However, if they work together and try to analyze the workable relations. The math relations between the 5 basic shapes of the Tangram puzzle shown below will allow you to create and/or solve new interesting geometric puzzles.
In fact, there are more than 1 billion possible combinations that can be made with the seven tans [source: The tans themselves are based on some very basic geometric principles.
That unit can be inches, centimeters, feet, meters or even a made-up unit, because the shapes are based on proportional, not numerical, measurements. For example, the small triangles in the set are composed of two base triangles lined up side-by-side. The square is made up of two base triangles joined at the hypotenuse, and so on.
To draw a set of tangrams, you can simply draw a square, superimpose a 4x4 grid over it, divide each square into two triangles, and then trace out the shapes along the borders of those triangles so that they match a tangram template. It doesn't matter what units you use to draw the grid, as long as it is perfectly square. Activity 1 of course requires notions about the Cartesian plane together with precision and manual ability.
Trainees met no particular difficulties with it but they supposed their pupils would meet some, since they did not have all the needed pre-requisites about the Cartesian plane.
In case these pre-requisites were held by pupils, trainees proposed that pupils might be showed the Tangram shape and then provided with only some of the coordinates.
Tangram, The link between mathematics and art
From the didactical viewpoint it was interesting to notice that different competencies are required to join points with given coordinates or to indicate coordinates of points in the plane: Activity 2 presented to trainees two different types of difficulties surprising for students themselves…: Students also pointed out the relevance for teaching of this proposal in order to promote creative competencies in pupils, who are invited to experience a form of autonomous mathematical classification.
However we decided to present the activity as group activity, since requested competencies are possibly not available to every single pupil at the age we considered: A reflection that came out immediately with trainees, but that might not be as such for pupils and in fact it was not is that it is not possible to classify by the area of figures constructed with the same pieces, given that these figures are equivalent, being all equally decomposable.
We notice here that in classroom activities pupils were much faster and capable in carrying out these activities, possibly due to a less rigid structure of the geometrical figures they held. This was predicted by most SSIS students, who had supposed that pupils could be more capable because of an envisaged higher visual capacity. The scheme of the proposal was agreed during a collective discussion, adapting it to the different classes and to the topics of the syllabus they were working on.
Trainees involved in the experimentation the class teacher and another trainee were asked to pay attention to the points highlighted in the discussion, also to test the hypotheses made about difficulties and meaningfulness of the activity.
A point shared by all experimentations was also due to the fact that activities were carried out in February that all the involved classes counted few pupils because of flu or winter extra-school activities.
After that the teacher left pupils free to play with pieces].
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As soon as they cut out the seven pieces they starter to compose them, turn then around, put them together to obtain images in such an enthusiastic way I did not really anticipate. Then I proposed that rules were fixed to hold for everyone: After an initial moment of doubt, pieces started to turn around every desk. A moment of glory was reserved to the Chinese girl who joined the class two weeks ago, without knowing any Italian, who, after copying and cutting the scheme in silence started to compose more and more complicated figures, laughing: I took the opportunity: It was a new challenge: Almost all pupils engaged in the search and the rectangle that we, as SSIS students found in 5 to 6 minutes came out in less than a minute and a half.
I checked discretely and invited the girl who made it to cover it, because I wanted to see what the others were doing: Starting from there I asked them to observe and reflect on the extension of each figure, and starting from figures that occupy the same surface we moved to think about them as made of the same pieces, which, although being moved around staying the same, give rise to different figures that nevertheless occupy the same surface.
Other reflections came up when they had to observe the figures boundaries, after putting them on squared paper to measure their perimeter: Grade 6, 4 hours work, 16 pupils involved I actually had observed in the past that pupils often meet difficulties in imagining geometrical figures beyond the book-workbook-geometry lesson context. In some cases I happened to have to guide them to recognise figures they had already drawn in the Technical education class and they only had to reproduce for the geometry class.
How Tangrams Work
The class in which the activity was carried out is mainly composed by pupils coming from the same primary school class, who had already worked with Tangram, as I came to know at the beginning of the lesson.
The activity was fun for everybody and raised interesting remarks, such as for example: In general it seemed to me that they were all engaged and we progressively commented that all the figures were obtained starting from the same modules by means of translations, rotations, flipping over without deformation.
Also an autistic grade 8 pupil of mine took part in the activity and she amazingly could carry out correctly and quickly most of the game. Grade 7, 5 hours, 15 pupils involved Posed questions were understood by everybody. Also those who meet more difficulties in the class work participated autonomously and often found correct solutions. In the first lesson pupils were invited to use the two equal isosceles triangles, putting the congruent sides aside to get the greatest possible number of different shapes […] I asked them to reflect on how to check whether figures were isoperimetric or not.
I then asked whether the figures were equivalent. The following week we used a square and a triangle, following the same method used in the previous class. After two days I asked them to form a rectangle, using all the Tangram pieces. After an initial critical moment they found 2 or 3 ways of getting it. I asked if the rectangle and square they had got from the pieces were equivalent and isoperimetric.
In this case, they all answered correctly.
In general, regardless of the number of correct or wrong answers, I noticed that due to what they discovered in this activity pupils were led to reflect more before expressing the position. We got many solutions, although they were not extremely different. It is interesting that nobody thought about copying from their deskmate, as if the object under consideration were something personal. They certainly collaborated, but in a functional way to their solution needs.
In particular a very good pupil female could not solve the problem [to find the area of one of the obtained figures] because she could not think using pieces. We can introduce them to words they might not know, like "angle" and "parallelogram. 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?
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.
TANGRAM IN MATHEMATICS
They represent a special kind of right triangle -- an isosceles triangle with two degree angles, and one 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.
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 If you right-click on the image here, you can save it on your computer for printing.