Southern Nevada
Regional Professional Development Program
Tearing Tangrams
Begin with a regular 8 ½ x 11 piece of paper. Discuss with the participants what type of shape it is and how they know this. It is a rectangle because it has two pairs of parallel lines and four right angles. Discuss with the participants how many degrees are found in a rectangle. The answer is 360°. (If you would like to provide each participant two pieces of paper, you could use one to tear all the corners off of and lining up all the points, the participants would see that it completes a circle. See diagram 1.)
Diagram 1
To begin tearing tangrams, set the paper in front of you as if you were going to write on it. Fold one of the top corners down so that it lines up with the edge of the paper exactly, forming a point. (See Diagram 2.) Fold the lower rectangle up and behind the formed triangle. Tear along this line, creating a square. (See Diagram 3.) Discuss with participants what this new shape is and how they know this. (It is a square because it has two pairs of parallel, four congruent lines and four right angles.) Discuss with the participants how many degrees are found in the square. The answer is 360°. Set the rectangle aside.
Diagram 2 Diagram 3
Using the square, have the participants tear down the middle fold line. Discuss with the participants what shape they have created. It is a triangle, specifically an isosceles triangle because it has two congruent or equal sides. Discuss with the participants how many degrees are found in each triangle and how they know this. The answer is 180°. They may know this from proof or they may know this from dividing the square in two. You may point out the right angle in each triangle and then ask how many degrees are found in each of the small angles and how they know this. The answer is 45°. They may know this because they tore a 90 ° angle in half. Have the participants look now at both of their triangles and compare them. The triangles are identical in shape and size. In math, we call this “congruence.”
Set one triangle aside. Fold the other triangle in half so the two small angles meet. Tear along this line. (See Diagram 4.) Have the participants compare the two small triangles. They are identical in shape and size so they are congruent. Have the participants compare the small triangles to the large triangle. They will see that the triangles are the exact same shape but a different size. In math, this is called similarity. Set the two smaller triangles aside.
Diagram 4
Using the other large triangle, fold the right angle to the hypotenuse (the side across from the right angle or the longest side.) Tear along this line. Compare this triangle to the other two triangles. It is similar to them. In fact, it is half the area of one of the larger triangles. (See Diagram 5.)
Diagram 5
Ask the participants what the shape is that we have created. It is a trapezoid because it has only one pair of parallel lines. You may want to ask the participants how many degrees are in a trapezoid. There are 360° in a trapezoid. (If you would like to have a spare trapezoid prepared, you could tear the angles off and line the points up to show that it completes the circle.) Using the trapezoid, tear it in half cross-section. (See Diagram 6.) The shape you have created is a special trapezoid. It is called a right trapezoid because it has one pair of parallel lines and right angle in it.
Diagram 6
Using one of the right trapezoids, fold the small angle in to the right angle. Tear along this line. Compare this small triangle to the other three larger triangles. It is similar to all three of them. With this step, the participants should have made a small triangle and a small square. These pieces can be set aside. (See Diagram 7.)
Diagram 7
Have each participant place the other right trapezoid on the desk in front of them so the long side is closest to them. Take the right angle that is closest to you and fold it up until it is flush with the top of the shape. Tear along the line. Ask the participants what shapes they have now made. The participants have now created a small triangle which is similar to all the other triangles and a parallelogram. A parallelogram has two pairs of parallel lines. (See Diagram 8.)
Diagram 8
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