TITLE:    Introduction to tangrams
 

CONTENT  AREAS:    Geometry, Reading
 

GRADE  LEVEL:    3
 

MATERIALS   NEEDED:    6 x 6 in.  paper or card stock, scissors, Grandfather Tang's Story:  A Tale Told with Tangrams by Ann Tompert.
 

KEY   CONCEPTS:  Shapes have parts and those parts have relationships. Dissection and reassembly. 
 

EALR'S :    1.3.1 Understand the concept of congruence.
 

Grade Level Expectations (Make the connections clear and specific): Identify, describe and compare congruent 2-dimensional geometric figures.

GOALS (Remember the difference between goals and objectives):    Students will explore congruence and dissection/reassembly through the act of cutting out a tangram puzzle.
 

OBJECTIVES:   
 Students will be able to identify, describe and compare congruent triangles in a tangram puzzle. 

PROCEDURES: 

• Introduction/Pre-assessment: (disequilibration) 15 minutes

“Today we are going to explore tangrams.  Raise your hand if you know what a tangram is.  (Call on several students to hear their responses.  Record them on a chart or overhead).  The tangram is an ancient Chinese puzzle made from a square. The square is cut into seven standard pieces, (two large right triangles, one medium-sized right triangle, two small right triangles, one square, and one parallelogram) each called a tan (show students an example of this in the back of the book).

Let’s look at how the author Ann Tompert uses tangrams to tell a story.  I am going to read a book to you that she wrote called Grandfather Tang's Story:  A Tale Told with Tangrams.  The fox fairies in Grandfather Tang’s story are an important part of Folklore from China.  They are believed to have supernatural powers of transformation.  Fox fairies are said to live for eight hundred to a thousand years.  While I am reading I want you to pay close attention to the way Grandfather Tang uses tangrams to tell the story.”  (Read book).

Discussion of the book…. 

Raise your hand and tell me how Grandfather Tang use tangrams in the story?  (Call on several students with their hands raised).

Now that you have heard a story using tangrams we are going to make one together. 

• Activity (elaboration)  15 minutes

Students will fold and cut a square piece of paper by following these directions. Students should discuss observations in small groups after each step.

1.      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? Have students turn to their neighbor to talk about those observations.  Call on one or two pairs to hear what they have discussed.  Record observations on board or chart.  (Look for: these two triangles are the same shape and size).

2.      Take one of the halves, fold it in half and cut along the crease. Make more observations and be able to support your statements.

3.      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 triangle may be discussed, as well as trapezoid.

4.      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?  (They are congruent as well)

5.      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?

6.      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? 

• Closure (Crystallization) 15 minutes

Have the students mix up the pieces and try to put the pieces together to form the square that was the shape of the paper they originally started with.  They may work in pairs.  Students can take their tangram puzzle home with them to explore and work on.

Strategies:  Look for angles that match.  Look at the length of the sides of the tangram pieces.    Do not give strategies until students have worked hard to figure them out on their own or in pairs. 
    
 

POST-ASSESSMENT   
Students should be forming an understanding that polygons are congruent if they are the same shape and size.  The chart completed from anecdotal observations of students putting tangram puzzle together

 will supply the student’s understandings. 

Students

Congruency

Observations

TITLE:    Tangrams and Area
 

CONTENT  AREAS:    Geometry
 

GRADE  LEVEL:    3
 

MATERIALS   NEEDED:    Plastic sets of tangram pieces, large sheet of white paper. 

 

KEY   CONCEPTS:  Area.  Large polygons can be dissected into smaller ones. 
 

EALR'S :    1.3.2
 

Grade Level Expectations (Make the connections clear and specific):  Given two polygons, explain how they are alike and different in terms of their attributes and properties.

GOALS (Remember the difference between goals and objectives):  Students will be able to find the area of a picture made with tangram pieces using the smallest right triangle as a unit of measurement. 
 

OBJECTIVES:   
Students will demonstrate geometric and spatial knowledge regarding area.

PROCEDURES: 

• Introduction/Pre-assessment  (Disequilibration) 15 minutes

“On your desks you each have a bag of plastic tangrams.  Don’t open them yet.  I know you all follow directions really well from the activity we did yesterday.  So, listen closely to my directions then you may open your bag.  I will let you open them up and work on making an animal like the ones we saw in Grandfather Tang's Story:  A Tale Told with Tangrams.  You may work by yourself or with a partner.  Okay, go ahead and open your bags and see what you can do.  I have the book up here if you would like to look at some of the animals in the book.”

(let the students play with the tangrams for about 5 minutes).  While students are making animals with tangrams teacher is doing the same with a student that might need extra assistance. 

“From the pre-assessment that I did the other day I found that you all have a good understanding of area.  I wonder how I would find the area of this animal that I made?  So, I thought I would ask you all how I would find the area of my animal.” 

• Activity  (Elaboration)  25 minutes

If students are not generating ideas or they need help in the steps to find area the following procedures can be used in a discussion.  

Using plastic tangram pieces students will:

1.      Order the tangram pieces from smallest to largest and explain what criteria they used for their arrangement. Ask students to verify their arrangement.  Have students hold up shapes that are congruent (crystallization from previous days lesson).  Note answers on overhead or chart.

2.      Focus on the arrangement of pieces based on area. (Demonstrate this on the overhead).  Using the small triangle as the basic unit of area, find the areas of each of the pieces in triangular units.  Have students trace each piece and write the area underneath. 

3.      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 two triangular units in area. To form a square with two tangrams 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 trace the tangram pieces for each of their squares.  Students will create 2-3 squares in this manner and include the area in triangular units.

“Thanks for your help with that!  You can go ahead and work in pairs or by yourself to make your animals.  When you have a picture that you are satisfied with, trace the pieces like we did with the squares, onto the piece of white paper I am handing out.  When you are done tracing your picture I want you to figure out the area of your picture using the small triangle as a unit of measurement.  Note the area at the bottom of the picture before you turn it in.  Turn to your neighbor and tell them what the instructions are.  If you are unclear about anything I just said after talking to your neighbor, raise your hand.”

• Closure  (Crystallization) 5 minutes

“I would like to ask a couple of you how you found the area of your tangram picture.   Raise your hand if you would like to share your process with the class.”  (hear from as many students as time allows).  Everyone, put your pictures in your desk because we are going to work on them some more tomorrow”.

POST-ASSESSMENT   
Assess student’s understanding of area from the squares or pictures created. 
 

TITLE:    Tangram Animals
 

CONTENT  AREAS:    Geometry, Art
 

GRADE  LEVEL:    3
 

MATERIALS   NEEDED:    Large white sheet of paper, pencil, markers, plastic tangram pieces.
 

KEY   CONCEPTS:  Congruence, dissection and reassembly, attributes of of size and shape of polygons.   
 

EALR'S : 1.3.2 Understand and apply attributes and properties of polygons  
 

Grade Level Expectations (Make the connections clear and specific): Use attributes and properties to identify, name, draw, compare, and/or sort 2-dimensional shapes and figures. 

GOALS (Remember the difference between goals and objectives):  Students will identify and describe congruent polygons in pictures made with tangrams, find the area of their pictures and write a description of their picture. 
 

OBJECTIVES:    Students will be able to identify with 100% accuracy, congruent shapes, area and write a clear description of their pictures. 
 

PROCEDURES: 

• Introduction (activating prior knowledge) 5 min.

“Last night I looked over all of your pictures and the area that you figured out.  I noticed that some of you seem to have a very clear understanding of how to find area using the smallest triangle in the tangram set.  It also appeared to me that most of you understand how to find the area, but that maybe you need to go back and check your math.  One of you wrote down the area in each shape when she figured it out.  When I saw that I thought that was a great way to keep track of the area of the smaller shapes so that later she could just add them all up with out getting confused.  So, that will be the first thing that we are going to do to day when I hand back your pictures.  Let me demonstrate what I mean on the overhead”.  (demonstrate measuring the parallelogram on the overhead and noting that it is 4 tans in area).

• Activity  (activating and elaborating) 30 minutes

Now that you are done with finding the area I am going to go through the rest of the steps to complete your picture.  The first thing you are going to do is find all the congruent shapes.  First, I want you to each get out one red and one blue colored pencil.  Now put your fingers on two shapes that are congruent.  Turn to your neighbor and quietly tell them why you choose those two shapes (Walk around classroom and observe pairs).   Now, take your blue colored pencil and neatly color those two shapes in.  Now, without talking to your neighbor find two more congruent shapes and color those in red.  Raise your hand when you have done this”.  

“Now, I am going to demonstrate for you how I would describe my picture to someone using some of the understandings we have learned in the last two days.  Please listen to what I am saying and watch me write.  (I used the square for the head.   I used the medium sized triangle for the top part of the body and then I used the parallelogram for the arm reaching out.  I used one of the large congruent triangles for the skirt of my person.  For the legs I used two small congruent triangles).    I noticed the other day that when I mentioned that this shape (hold up parallelogram) was a parallelogram that some of you had never term before.  Raise your hand and tell me what you understand a parallelogram to be.  (Take a few students).  A parallelogram is a four-sided figure with opposite sides equal and parallel.  Raise your hand if you know what parallel means.  (Take a few students).  The way that I remember parallel is by seeing the word parallel in my head (point out the word parallel on the sheet that you place on the overhead) and remembering that the two ll’s are right next to each other, they are parallel.  So, I will leave this sheet (put sheet with shapes and names up on the overhead) up here to help you remember the shapes when you write your description. 

So, now I want you to turn to your neighbor and describe your picture by naming the shapes and where they are in your picture like I did.  I also want you to identify congruent shapes in your description.  You have about 2 or 3 minutes to do this.  Then, you will need to write your description on your paper.  When you are done with that you can write a sentence like I did, “This is me running to get my son who is about to cross the street with out looking for cars”.  Raise your hand and tell me what the first thing is that you are going to do (ask 1 or 2 students).   Raise your hand and tell me the second thing you are going to do.  (ask 1 or 2 students).  Now, raise your hand and tell me the last thing you are going to do.  (ask 1 or 2 students).  Great let’s get started!!!

• Closure  15 minutes

Students share their drawing and read their sentence or brief story out loud.
 

POST-ASSESSMENT  (Final assessment)  
In their final project students should demonstrate an understanding of area in relationship to the smallest triangle.  Students should also demonstrate an understanding that congruent shapes are the same size and shape.   Students will begin to describe attributes of polygons by their relationship to each other in regard to size and shape. 
 

Students

Area

Observations

Students

Desciption

Observations