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Puzzling Relationships

  • Lesson
Carol Midgett
Location: unknown

Puzzles provide excellent opportunities for students to apply spatial relationships in an entertaining and meaningful way. In the first part of this lesson, the students use concrete materials to develop the skills needed to work with interactive Tangram puzzles in the second part of the lesson. The students also further develop their understanding of spatial relationships and develop problem-solving skills.

Once students complete puzzles using concrete materials, introducing them to virtual puzzles enables them to experience turns, rotations, slides, and spatial relationships in a powerful way. These transformations are more apparent in a virtual environment than when using concrete tools.

Make or use commercial overhead transparencies of the Tangram Puzzle Pieces. Use the overhead projector to model how to fill the puzzles with the tangram pieces. For example, the Tangram School can be filled as shown below.

2751 tangram school 

Discuss how it is necessary to turn over, rotate, or slide pieces to complete a puzzle. Provide each student with tangram pieces and simple puzzles, such as those included in the Tangram Puzzle activity sheet.

pdficon  Tangram Puzzle Activity Sheet 

Note: If tangram pieces are not available, you may wish to duplicate the first page of the activity sheet and have each student create her own set. You may duplicate it on colored paper or flexible plastic and cut out the pieces. A management strategy that helps in distributing and storing the puzzle pieces is to place one complete set of seven pieces in a resealable plastic bag labeled with the student's name.

Have the students complete each of the puzzles on the activity sheet (Tangram Puppy and Tangram Tree). Move among the students, recording the names of those who are able to rotate, turn over, or slide the pieces to fill the puzzle accurately. The students might glue paper Tangram pieces to the puzzle or another sheet of paper as a record of their performance on this activity. If you keep these student work samples, they might be used for future planning.

For the next part of the lesson, display the Interactive Tangram Puzzles to the students, using an overhead projector, Smart Board, or other device.

 appicon  Interactive Tangram Puzzles  

Show the students that what they see on the screen matches the Tangram pieces they used in the first part of this lesson. Identify each of the Tangram pieces and compare the virtual ones with the concrete ones used by the students. Demonstrate how to select a puzzle piece on the site and move it onto the puzzle outline.

2751 image 5 

Demonstrate how to "grab" a corner of a shape in order to rotate it and how to highlight the shape to turn it over using the curved arrow icon. When a shape is "grabbed," dots appear on it and the arrow in the upper-right corner turns red as shown in the picture above.

You may also wish to show the students how to change the colors of the shapes. (Selecting a shape and clicking on the new color choice on the right side of the screen accomplish this.) Model for students how to turn over (flip), rotate, and slide their Tangram pieces with the concrete tools simultaneous with the electronic tool so that students understand what happens when they make these transformations.

Using the tools and strategies you just demonstrated, complete one of the puzzles on the computer while the students observe. A discussion of what you did and why you did it will help the students hear the vocabulary and consider the thinking required to complete a puzzle.

Have the students work in pairs to complete the electronic Tangram puzzles. Encourage them to discuss with each other what needs to be done to fit the pieces to match the outline of the puzzle. (This exercise focuses students' attention on the transformations required to make pieces fit. A turnover, rotation, or slide of pieces may be necessary to complete the puzzle.) This discourse facilitates students' problem solving and reasoning about the process. It connects actions with vocabulary and assists the students in relating what happens with concrete and virtual tools.


  1. By completing tangram puzzles, the students should demonstrate the ability to do the following:
    • Interpret relative positions in space and apply ideas about relative position
    • Describe and interpret direction and distance in navigating space, and apply ideas about direction and distance
  2. Making notes about what the students do and do not understand is helpful when planning future instruction for individuals, groups of students, and the class as a whole. Collecting this data for future reference helps you understand the impact of your teaching strategies on students' learning. You may use the Class Notes recording sheet for this purpose.


  1. The students who find it easy to complete the puzzles could create new ones for the class to solve. These may be laminated and saved for future practice.

Questions for Students 

  1. What did you find when you tried to place the puzzle pieces in place?
  2. What happens when you turn (or rotate) a puzzle piece?
  3. What happens when you turn over a puzzle piece to fit it into place?
  4. How many different ways did you find to fill the puzzle? What did you do to fill the puzzles in different ways? (This question focuses students' attention on different orientations of pieces and whether pieces must be turned over, rotated, or slid to fill the puzzle.)
  5. What did you learn from other students as you watched them put their puzzle together?
  6. If you were teaching this lesson, what would you tell your classmates about how to put the puzzle together?
  7. Why is it necessary to turn over, rotate, or slide the Tangram pieces to make them fit?
  8. How did you decide which triangle would fit into the vacant space?
  9. Which shape(s) caused you the most trouble? Why?
  10. What did you learn by using the computer that you did not notice when using the other Tangram pieces?

Teacher Reflection 

  • Which students demonstrate an understanding of how to solve puzzles by turning over, rotating, and sliding puzzle pieces? Which did not? What caused them difficulty?
  • What other learning opportunities do these students need? What instructional strategies would provide these learning experiences?
  • Were some students able to use more than one strategy to complete the puzzles?
  • Which students were able to interpret and use relative position in space as a problem-solving strategy?
  • Were all students able to describe how they solved puzzles? If not, what would help them do so?
  • Which students demonstrate an understanding of the relative sizes among the Tangram pieces by identifying the varying sizes of the triangles as small, medium, or large? For those who do not, what instructional experiences are needed?
  • Which students do not realize that rotating or sliding changes the orientation of shapes? What experiences would help them learn this?
  • Which students realize that turning over pieces can change the orientation of shapes? If not, what experiences would help them learn this?

Learning Objectives

Students will:

  • Interpret relative positions in space and apply ideas about relative position
  • Describe and interpret direction and distance in navigating space and apply ideas about direction
  • Use tangram puzzles to experience turns, rotations, and slides

Common Core State Standards – Mathematics

-Kindergarten, Geometry

  • CCSS.Math.Content.K.G.A.1
    Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

-Kindergarten, Geometry

  • CCSS.Math.Content.K.G.A.2
    Correctly name shapes regardless of their orientations or overall size.

-Kindergarten, Geometry

  • CCSS.Math.Content.K.G.B.4
    Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/''corners'') and other attributes (e.g., having sides of equal length).

Grade 2, Geometry

  • CCSS.Math.Content.2.G.A.1
    Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.