Math Crafts – Tangram Puzzles


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Tangrams are ancient Chinese puzzles.  Each tangram is made with the same 7 pieces, called “tans.”

This math activity involves making your own tans, and using geometry vocabulary: triangles, right triangles, right angles, parallelogram, square, trapezoid, congruent, similar.  Then students develop spatial sense by moving the pieces to make pre-determined pictures, or to design tangram pictures of their own.

For younger students, precut the tangram shapes and introduce them to the shapes one at a time, finding the shapes that are exactly alike (the congruent triangles), and the ones that are bigger and smaller versions of the same right triangle (similar triangles).  Then discuss the two quadrilaterals (the pieces with four sides, the square and the parallelogram).  How are they the same? How are they different?

For older students who can use scissors to cut precisely, have them fold and cut the pieces as pictured below, or by using the directions found in this downloadable PDF.

Note that, in the pictures below, we are using an 8.5”x11” piece of paper, so the beginning square is 8.5”x8.5” and the resulting tangram pieces are larger than the standard size tangrams that can be purchased in a teacher store or online from a math catalog.  (Standard tangrams are cut from a square that is 4”x4”.)

MATERIALS

To trace tangrams (as shown above):

  • Sheets of colored foam
  • One packet of tangrams to use for tracing.
  • Pencil and scissors.

To cut your own tangrams (as described below):

  • Sheets of colored cardstock.

(For large tangrams use 8.5”x11” and start with a square that has 8.5” sides.)

(Standard tangrams start with a square that has 4” sides.)

  • Scissors

HOW TO FOLD AND CUT YOUR OWN TANGRAM SHAPES

1. Start with a piece of paper that is 8.5”x11”.

Fold it as shown, then cut off the extra piece so you are left with a square folded in half diagonally.  (Throw out the “extra piece.”)

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2. Cut along the diagonal, making two big right triangles.

Notice that they are exactly the same shape, so they are congruent to each other.  (Technically, they are isosceles right triangles, because the two legs of the triangle are the same length!)

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3. Take ONE of the big triangles and cut it in half to make two other right triangles, as shown.

These two triangles are also congruent to each other, but they are smaller than the big right triangle you started with.  They are “similar” to the big one, but congruent to each other.

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4. Take the OTHER big triangle and fold the right angle down.

Make sure that the top is parallel to the bottom.  Then cut it off.  Notice that it, too, is a right triangle.   It is smaller than the other right triangles, but the same shape. We say it is similar to the larger triangles, but not congruent.

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5. Now look at the long trapezoid (a quadrilateral with just two parallel sides).

Fold it in half and cut along the fold.  You are left with two smaller trapezoids, called “right” trapezoids because they have a right triangle in them.  The two smaller trapezoids are congruent to each other.

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6. Take ONE right trapezoid.

Fold and cut so that you have a small right triangle and a little square.  Notice that two of the small right triangles would make a square!  Also notice that the small right triangle is similar to the other right triangles that you have made, but not congruent.

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7. Now use the OTHER right trapezoid.

The final fold is the hardest!  Fold up the right angle to make a little triangle and a parallelogram.  Check with someone to make sure that you really have a right triangle and a parallelogram. Then cut.  Notice that the small triangle is congruent to the other small triangle.  Also notice that you can make a parallelogram using two of the small triangles, so the parallelogram has the same area as the square!

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8. Move the pieces around to make other geometry discoveries.

For an extra challenge, see if you can make a square using just two pieces, then three pieces, or four, five, six seven. [No one has found a square with 6 pieces, but maybe YOU can! We know that all the others are possible!]  Here is the solution for all seven pieces:

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9. If you followed the directions, you should have seven tangram pieces (as shown above).

  • Two large right triangles
  • One medium right triangle
  • Two small right triangles
  • One square
  • One parallelogram

These seven “tans” are used to make all kinds of pictures.  Find some in books, or online.

GENERAL LESSON DIRECTIONS 

1. Make and discuss the characteristics of the seven tans.

Hold up and manipulate the pieces as you discuss the shapes, to demonstrate the new vocabulary.   (It doesn’t hurt for little kids to learn big words!)

Here are some examples of what you can discuss:

-All the triangles are right triangles because they all have right angles.

-The two large right triangles are congruent to each other.  (The angles and sides are the same size.)

-The two small right triangles are congruent to each other.

-The two small triangles are similar to the large triangles.  (The angles are the same, but the sides of the large triangle are longer.)

-The medium triangle is similar to both the large and small triangles. (All are right triangles.)

-The hypotenuse (long side) of the medium triangle = the short side (leg) of the large triangle.

-The leg of the medium triangle = the hypotenuse of the small triangle.

-It takes two small triangles to make the little square.

-It takes two small triangles to make the parallelogram.

-The parallelogram has two sets of parallel sides, and so does the square.

-Ask kids what other discoveries they can make about the pieces.

2. Move the pieces around to make one of the tangram puzzles shown.

 3. Find books of other tangram pictures, or do tangram puzzles online.

Note: If you buy a workbook with tangram puzzles, it probably uses tangram pieces in the standard size (made from a 4-inch square). You may want to purchase a set of plastic or foam tangram pieces, especially if you elect to trace them instead of cutting and folding.

 

GET A PACKET

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There are two Let’s Read Math lesson packets about tangrams, featuring these two books.  Both books are tales told with tangram pictures.  Lesson packets can be purchased on the Let’s Read Math website.

The Pigs Packet, is related to an easy reader called Three Pigs, One Wolf, and Seven Magic Shapes by Grace Maccarone. This book is a spoof on the story of the Three Little Pigs.

The Grandfather Tang Packet is related to Grandfather Tang’s Story, by Ann Tompert.  This story is about two authentic Chinese fairy tale characters called fox fairies, who turn themselves into other animals as the story unfolds.

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