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Author: Fay Zenigami, Leeward District Office, Waipahu, Hawaii
Grade Level: Geometry / 4-12
Overview: Often when students are introduced to tangrams, they are asked to put the pieces together to form a square. This is often a difficult and frustrating task because they have no background as to how the pieces fit together.
Purpose: To provide students with some insight as to how the tangram pieces fit together, and to stimulate their interest in forming shapes and exploring patterns using the tangram pieces.
- Construct the tangram pieces from a square paper by following directions to fold and cut.
- Make observations on the pieces formed and compare how they are related to each other.
- Explore patterns and shapes with the tangram pieces.
Square sheet of paper (students can fold from 8.5" x 11" plain paper)
Plastic sets tangram pieces
Overhead tangram set for demonstration
- Students will fold and cut a square piece of paper by following these directions. Students should discuss and record observations in small groups after each step.
Tying It All Together:
- Fold the square sheet in half along a diagonal, unfold and cut along the crease. What observations can you make about the two pieces you have? How can you "prove" that your observations are correct?
- Take one of the halves, fold it in half and cut along the crease. Make more observations and be able to support your statements.
- Take the remaining half and lightly crease to find the midpoint of the longest side. Fold so that the vertex of the right angle touches that midpoint and cut along the crease. Continue with observations. Congruent and similar triangles may be discussed, as well as trapezoid.
- Take the trapezoid, fold it in half and cut. What shapes are formed? Students may not realize that these shapes are trapezoids as well. What relationships do the pieces cut have? Can you determine the measure of any of the angles?
- Fold the acute base angle of one of the trapezoids to the adjacent right base angle and cut on the crease. What shapes are formed? How are these pieces related to the other pieces?
- Fold the right base angle of the other trapezoid to the opposite obtuse angle. Cut on the crease. You now should have the seven tangram pieces. Are there any more observations you can make?
Have the students mix up the pieces and try to put the pieces together to form the square which was the shape of the paper they originally started with. Students may be given plastic tangram pieces to do the remaining activities.
- Have students order the pieces from smallest to largest and explain what criteria they used for their arrangement. Students should be able to verify their arrangement. Focus on the arrangement of pieces based on area.
Use the small triangle as the basic unit of area. What are the areas of each of the pieces in triangular units?
- Create squares using different numbers of tangram pieces and find the area of the squares in triangular units. For example, to form a square with one tangram piece, students should identify the square piece which is 2 triangular units in area. To form a square with two tangram pieces, students should use the two small triangles (2 triangular units in area) or the two large triangles (8 triangular units in area). Students should continue to try to form squares with 3 pieces, 4 pieces, 5 pieces, 6 pieces and all 7 pieces. Are there multiple solutions for any? Are there no solutions for any? Do you notice any patterns?
- Have students turn in list(s) of observations from tangram folding. If the length of a side of the original square is 2, what are the lengths of the sides of each of the tangram pieces cut?
- Have students make conjectures based on their findings from the making squares activity. Students may observe that the areas of the squares appear to be powers of 2 and that they are unable to make a 6-piece square. When all combinations of 6-pieces are considered, the possible areas are not powers of 2.
Evaluation: Students will identify the fractional part of each Tangram piece. They will also be able to add fractions with unlike denominators and create equations using their Tangram pieces.
Follow-up Activities: An extension to these activities is to allow students to create shapes with a particular unit value. For example, ask students to make a picture having the area of 5/8. Which pieces will be able to be used?
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