# Tricky Tactics with Tangrams!

##### by Natural Math For centuries, people have been fascinated by tangrams, a Chinese puzzle consisting of pictures made from seven specific geometric shapes known as “tans,” or “pieces of cleverness.” But even if you’re aren’t familiar with Tangrams, what are they and how do they relate to math?

This week’s activity, recommended for Grades 3 to 8, will show you that tangrams not only make interesting pictures, but they also reveal geometric properties of the whole in terms of fractions, decimals, and percents!

A Tangram consists of seven pieces: 2 large right isosceles triangles, 1 medium right isosceles triangle, 2 small right isosceles triangles, 1 small square, and 1 parallelogram.

You will first construct each of the 7 tans from a paper square, making observations about each tan along the way.  Then you will use the tans to create images based on mathematical comparisons.

Materials:

• A square piece of paper for each person (preferably 5” x 5” or larger)
• Scissors
• Rulers
• Pens and/or pencils
• For older students, protractors and/or compasses

Step 1Examine parts of one whole using proper mathematical terminology. Pass out materials for each person. Follow these instructions in order:

1. Look at the paper square.  What makes it a square? (Equal sides and angles)
2. Refer to this as the “WHOLE” square.
3. Now fold it in half along a diagonal. Cut along the diagonal, creating 2 triangles.
4. What do you notice about the 2 triangles?  (They are equivalent right triangles because they each have 2 equal sides and one right angle.)
5. What is the mathematical name for these triangles? (Right isosceles triangles)
6. Refer to these as “Large Triangles.”
7. What part of the initial whole square is each Large Triangle? (Each triangle is half of the whole square.)
8. In what ways can you represent each triangle by its relationship to the whole square? (As a fractional part, as a decimal part, and as a percentile part.)
9. How can you represent something as half of a whole? (As a fraction (1/2); as a decimal (0.5); and as a percent (50%).)

Step 2:  Continue comparisons to the original whole square. Follow these instructions in order:

1. Fold only ONE of the large triangles in half along a line draw from the center of the right angle to the center of the hypotenuse. Cut along the fold to create 2 smaller congruent right isosceles triangles.  Label each of these smaller triangles respectively as #1 and #2.
2. What part of the whole square is each #1 and #2 triangle in fraction, percent, and decimal form? (one-fourth, 25%, and 0.25)
3. Write on one side of each triangle #1 and #2 the following:  1/4 = 25% = 0.25
4. Set aside triangles #1 and #2. These are the first 2 of your 7 “tans.”

Step 3:  Create the remaining Tans and compare each to the original whole square. Start by taking the second large triangle from Step 2 above and folding the right angle over so its vertex touches the midpoint of the hypotenuse.  See the example below:

Are each of the smaller triangles ABD, BCE, and BDE equivalent? Why?

What part of the whole original square from Step 1 is each triangle ABD, BCE, and BDE?  How do you know? (Each is a fourth of the 2nd large triangle. Since the 2nd large triangle is half the size of the large square, this makes each of these smaller triangles ¼ x ½ = 1/8th  the size of the large square.)

Unfold, cut, and label triangle BDE as triangle #3.

Write on triangle #3 its value in comparison to the original whole square in fraction, percent and decimal form (1/8 = 12.5% = 0.125). This is your 3rd “tan.”  Set it aside.

What is a mathematical name for the remaining shape ACED? (Trapezoid)

What part of the whole original square is ACED? Write this information on ACED in fraction, percent and decimal form. (It is 3 times the value of triangle #3.  In fraction, percent and decimal form, this is 3/8 = 37.5% = 0.375.)

Now, fold ACED in half lengthwise. The figure should look like this: What is the name of this shape? (another trapezoid)  What part of the original whole square is this shape?  (It is half the size of trapezoid ACED. In fraction, percent and decimal form this is 3/16 = 18.75% = 0.1875.)

Unfold ACED. Bring vertex A to the fold and crease along the line GD as shown: Unfold and cut out triangle AGD.  Refer to this as triangle #4. What part of the whole original square is #4?  Write this information on #4 and set it aside. (It is half of the size of ADH, which is the same as the size of tan #3, so write on it 1/16 = 6.25% = 0.0625.) #4 is your 4th “tan.”

Look at GHFD.  What is this shape? (It is a small square.)  How do you know?

Cut along the fold HF to create a small square GHFD.  Call this small square #5. What part of the WHOLE original square is #5 in fraction, percent, and decimal form? (By comparison you see #5 is composed of the same size as 2 of the #4 triangles.) Write on #5 that 1/8 = 12.5%, 0.125, and then set aside. This is your 5th “tan.”

Look at the remaining shape HCEF. Bring vertex H to meet vertex E.  Fold along IF.  Unfold. Cut along the crease line IF.  What part of the whole original square is triangle FIH? Compare triangle #4 to this triangle.  It is the same size!  Label this second small triangle as #6. Write its value in terms of being part of the whole square (1/16 = 6.25% = 0.0625). Set aside.  This is your 6th “tan.”

How many triangles the size of #6 would be needed to create the shape FICE? (Answer: 2.)

What is another name for this shape? (Answer: a parallelogram.)

Compare the size of this shape to the size of #3 and #5. What do you notice? (They are the same size, even though they are different shapes!)

Label this parallelogram #7. Write on it 1/8 = 12.5%, 0.125. This is your final 7th tan!

Now take your tans and do the following:

• Use all 7 tans to remake the original whole square.
• Use some tans to create a picture of something, like a boat or a bird. What parts of the whole square are these pictures?
• Create a picture with a set value equal to part of the whole square. For example, can you make a tangram picture with an area five-eighths of the original square? Which tans can be used?  Why?

Form a Right Isosceles triangle using only:

• Two tans with an area of 1/4 the whole square.
• Two tans with an area 1/16 the whole square.
• Tans #4, # 6, and#7
• Tans #3, #4, and #6
• Tans #2, #4, #6, and #7
• Tans #2, #4, #5, and #6
• Tans #4, #5, and #6
• Tans #1, #3, #4, and #6
• Tans #3, #4, #5, #6, and #7
• Tans #1, #3, #6, and #7
• Tans #2, #3, #4, and # 5
• All 7 Tans!

• Tans #1 and #2
• Tans #4 and #6
• Tans #3, #4, #5, and #6
• Tans #3, #4, #6, and #7
• Tans #1, #2, #3, #4, and #6
• Tans #1, #2, #3, #4, #5, and #6
• Tans #1, #2, #4, #6, and #7
• Tans #2, #4, #5, and #6
• All 7 Tans!