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ConnectionsSession 06 Overviewtab aTab btab ctab dtab eReference
Part B

Exploring Connections
  Introduction | The Tangram Puzzle | Making Connections | Reflection Questions | Your Journal

 
 

After you've finished putting the puzzle pieces together, consider other connections that can be made with the tangram pieces. You may want to use the Interactive Activity to solve the problems below.


Tangram


1. How Big Is Each Piece?

a. If the entire square is one whole, find the fractional amount of each piece. Explain your reasoning.

b. This time, think of the small square (Piece C) as one whole. Find the area of each of the other pieces. Explain your reasoning.

Show Answer
Our Answer:

a.


F = 1/4; G = 1/4;


Large triangles F and G each equal 1/4 of the total area of the large square. (Because four such triangles completely cover the square, one such triangle is equal to 1/4 of the total area.)


B = 1/8;


Medium triangle B is equal to half the area of a large triangle. (Two triangles B completely cover a large triangle.)


A = 1/16; D = 1/16;


Small triangles A and D are equal to 1/16 of the total area of the square. (Four small triangles completely cover a large triangle.)


C = 1/8;


Square C is equal to 1/8 (two small triangles completely cover the small square).


E = 1/8;


Parallelogram E is also equal to 1/8 (two small triangles completely cover the parallelogram).


b.


A = 1/2; D = 1/2;


Small triangles A and D completely cover the square E. So, each is equal to 1/2.


C = 1; B = 1;


Square C and triangle B can each be covered completely by two small triangles, A and D. So, their respective areas are equal to 1.


F = 2; G = 2;


Triangles F and G can each be covered completely by four small triangles, A and D. So, their respective areas are equal to 2.


The area of the large square, in this case, is equal to 8.


 

2. The Dartboard

The ABC Game company has just put a brand-new dartboard on the market, which looks just like the tangram puzzle! If you had a random chance of landing on any part of the board, what would be the chance, or probability, of the dart landing on each of the pieces?

Show Answer
Our Answer:

Probabilities of the dart landing on each of the pieces are equal to the fractional values from Problem 1 (a). Expressed as decimals and percents, we get:


A = 0.0625 or 6.25 %


B = 0.125 or 12.5 %


C = 0.125 or 12.5 %


D = 0.0625 or 6.25 %


E = 0.125 or 12.5 %


F = 0.25 or 25%


G = 0.25 or 25 %


 

3. Exploring Shapes

a. Can you make a square with two different pieces?

b. Can you make a square with three different pieces?

c. Can you make a rectangle with four different pieces?

d. How many different ways can you make a piece that is congruent to Triangle F?

Show Answer
Our Answer:

a. Putting together triangles F and G, or triangles A and D, results in a square.


b. Putting together the pieces A, B and D makes a square.


c. Putting together pieces A, C, D, E makes a rectangle:



Tangram 2

d. Here are three possibilities:



Tangram 3

 

4. Exploring Angles

Using a right angle with a measure of 90° as a benchmark, find the measure of the angles in each of the shapes in the tangram. Explain your reasoning.

Show Answer
Our Answer:

To determine the angles of all the shapes you can use the following properties of certain geometric shapes:

  • Squares –– the interior angle at each vertex is equal to 90°;
  • Right-angle triangles –– one vertex angle is equal to 90° and the sum of angles is equal to 180°;
  • Quadrilaterals –– interior angles sum up to 360°.

Also, you can place the shapes on top of each other to check the relationships between certain angles, and in particular, if certain angles are congruent.


The angles of the shapes are as follows:



Tangram 4

 

Next  Reflect on these problems

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